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THE SCIENTIFIC PAPERS OF
JAMES CLERK MAXWELL
Edited by W. D. NIVEN, M.A., F.R.S. Two Volumes Bound As One
DOVER PUBLICATIONS, INC., NEW YORK

All rights reserved under Pan American and International Copyright Conventions.
Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto,
Ontario.
Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, London W. C. 2.
This Dover edition, first published in 1965, is an unabridged and unaltered republication of the work first pubhshed by Cambridge University Press in 1890. This edition is published by special arrangement with Cambridge University Press.
The work was originally pubhshed in two separate volumes, but is now published in two volumes bound as one.
Library of Congress Catalog Card Number: A53 -9813
Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street
New York, N. Y. 10014

THE SCIENTIFIC PAPERS OF
JAMES CLERK MAXWELL
Edited by W. D. NIVEN, M.A., F.R.S, Volume One

TO HIS GRACE
THE DUKE OF DEVONSHIRE K.G.
CHANCELLOR OF THE UNIVERSITY OF CAMBRIDGE FOUNDER OF THE CAVENDISH LABORATORY
THIS MEMORIAL EDITION
OF
THE SCIENTIFIC PAPERS
OF
THE FIRST CAVENDISH PROFESSOR OF EXPERIMENTAL PHYSICS
IS
BY HIS GRACE'S PERMISSION
RESPECTFULLY AND GRATEFULLY DEDICATED

SHORTLY after the death of Professor James Clerk Maxwell a Committee was
formed, consisting of graduate members of the University of Cambridge and
of other friends and admirers, for the purpose of securing a fitting memorial of

him.

The Committee had in view two objects : to obtain a likeness of Professor Clerk Maxwell, which should be placed in some public building of the University ; and to collect and publish his scattered scientific writings, copies of which, so far as the funds at the disposal of the Committee would allow, should be presented to learned Societies and Libraries at home and abroad.

It was decided that the likeness should take the form of a marble bust. This was executed by Sir J. E. Boehm, R.A., and is now placed in the apparatus room of the Cavendish Laboratory.

In carrying out the second part of their programme the Committee obtained the cordial assistance of the Syndics of the University Press, who willingly consented to publish the present work. At the request of the Syndics,
Mr W. D. Niven, M.A., Fellow and Assistant Tutor of Trinity College and now Director of Studies at the Royal Naval College, Greenwich, undertook the

duties of Editor.

The Committee and the Syndics desire to take this opportunity of

acknowledging their obligation to Messrs Adam and Charles Black, Publishers

of the ninth Edition of the EiicyclopcEdia Biitannica, to Messrs Taylor and Francis, Publishers of the London, Edinburgh, and Dublin Philosophical Maga-

zine and Journal of Science, to Messrs Macmillan and Co., Publishers of Nature and of the Cambridge and Dublin Mathematical Joui-nal, to Messrs Metcalfe and Co., Publishers of the Quarterly Journal of Pure and Applied

Mathematics, and to the Lords of the Conmiittee of Council on Education,

Proprietors of the Handbooks of the South Kensington Museum, for their

courteous consent to allow the articles which Clerk Maxwell had contributed to

these

publications

to

be

included

in

the

present

work ;

to

Mr

Norman

Lockyer

for the assistance which he rendered in the selection of the articles re-printed

from Nature; and their further obligation to Messrs Macmillan and Co. for

permission to use in this work the steel engravings of Faraday, Clerk Maxwell,

and Helmholtz from the Nature Series of Portraits.

Numerous and important Papers, contributed by Clerk Maxwell to the Transactions or Proceedings of the Royal Societies of London and of Edinburgh, of the Cambridge Philosophical Society, of the Royal Scottish Society of Arts, and of the London Mathematical Society; Lectures delivered by Clerk Maxwell at the Royal Institution of Great Britain pubHshed in its Proceedings; as well as Communications and Addresses to the British Association published in its Reports, are also included in the present work with the sanction of the above mentioned learned bodies.
The Essay which gained the Adams Prize for the year 1856 in the
University of Cambridge, the introductory Lecture on the Study of Experimental Physics delivered in the Cavendish Laboratory, and the Rede Lecture delivered before the University in 1878, complete this collection of Clerk Maxwell's scientific
writings.
The diagrams in this work have been re-produced by a photographic process from the original diagrams in Clerk Maxwell's Papers by the Cambridge Scientific Instrument Company.
It only remams to add that the footnotes inserted by the Editor are enclosed between square brackets.
Cambridge, Augv^t, 1890.

PEEFACE.
CLERK MAXWELL'S biography has been written by Professors Lewis Campbell and
Wm. Garnett with so much skill and appreciation of their subject that nothing further
remains to be told. It would therefore be presumption on the part of the editor of his papers to attempt any lengthened narrative of a biographical character. At the same time a memorial edition of an author's collected writings would hardly be complete without some account however slight of his life and works. Accordingly the principal events of Clerk Maxwell's career will be recounted in the following brief sketch, and the reader who wishes to obtain further and more detailed information or to study his character in
its social relations may consult the interesting work to which reference has been made.
James Clerk Maxwell was descended from the Clerks of Penicuick in Midlothian, a well-known Scottish family whose history can be traced back to the IGth century. The first baronet served in the parliament of Scotland. His eldest son, a man of learning, was a Baron of the Exchequer in Scotland. In later times John Clerk of Eldin a member of the family claimed the credit of having invented a new method of breaking the enemy's line in naval warfare, an invention said to have been adopted by Lord Rodney in the battle which he gained over the French in 1782. Another John Clerk, son of the naval tactitian, was a lawyer of much acumen and became a Lord of the
Court of Session. He was distinguished among his Edinburgh contemporaries by his ready
and sarcastic wit.
The father of the subject of this memoir was John, brother to Sir George Clerk of
Penicuick. He adopted the surname of Maxwell on succeeding to an estate in Kirkcud-
brightshire which came into the Clerk family through marriage with a Miss Maxwell. It cannot be said that he was possessed of the energy and activity of mind which lead
to distinction. He was in truth a somewhat easy-going but shrewd and intelligent
man, whose most notable characteristics were his perfect sincerity and extreme benevolence.
He took an enlightened interest in mechanical and scientific pursuits and was of an essentially practical turn of mind. On leaving the University he had devoted himself
to law and was called to the Scottish Bar. It does not appear however that he met
mth any great success in that profession. At all events, a quiet life in the country

X

PREFACE.

presented so many attractions to his wife as well as to himself that he was easily induced to relinquish his prospects at the bar. He had been married to Frances, daughter of Robert Cay of N. Charlton, Northumberland, a lady of strong good sense and resolute
character.

The country house which was their home after they left Edinburgh was designed by John Clerk Maxwell himself and was built on his estate. The house, which was named Glenlair, was surrounded by fine scenery, of which the water of Urr with its rocky and wooded banks formed the principal charm.

James was bom at Edinburgh on the 13th of June, 1831, but it was at Glenlair
that the greater part of his childhood was passed. In that pleasant spot under healthful influences of all kinds the child developed into a hardy and ccirageous boy. Not precociously clever at books he was yet not without some signs of future intellectual strength, being remarkable for a spirit of inquiry into the caupjs and connections of the phenomena around him. It was remembered afterwards when he had become distinguished, that the questions he put as a child shewed an amount of thoughtfulness which for his years was very unusual.

At the age of ten, James, who had lost his mother, was placed under the charge of

A relatives in Edinburgh that he might attend the Edinburgh Academy.

charming account

of his school days is given in the narrative of Professor Campbell who was Maxwell's

schoolfellow and in after life an intimate friend and constant correspondent. The child is

father to the man, and those who were privileged to know the man Maxwell will easily
— recognise Mr Campbell's picture of the boy on his first appearance at school, the home-

made garments more serviceable than fashionable, the rustic speech and curiously quaint

but often humorous manner of conveying his meaning, his bewilderment on first undergoing

the routine of schoolwork, and his Spartan conduct under various trials at the hands of

his schoolfellows. They will further feel how accurate is the sketch of the boy become

accustomed to his surroundings and rapidly assuming the place at school to which his

mental powers entitled him, while his superfluous energy finds vent privately in carrying

out mechanical contrivances and geometrical constructions, in reading and even trying his

hand at composing ballads, and in sending to his father letters richly embellished with

grotesquely elaborate borders and drawings.

An event of his school-days, worth recording, was his invention of a mechanical method of drawing certain classes of Ovals. An account of this method was printed in the
Proceedings of the Royal Society of Edinburgh and forms the first of his writings collected in the present work. The subject was introduced to the notice of the Society by the celebrated Professor James Forbes, who from the first took the greatest possible interest in Maxwell's progress. Professor Tait, another schoolfellow, mentions that at the time when the paper on the Ovals was written. Maxwell had received no instruction in Mathematics beyond a little Euclid and Algebra.

PREFACE.

aa

In 1847 Maxwell entered the University of Edinburgh where he remained for three
sessions. He attended the lectures of Kelland in Mathematics, Forbes in Natural Philosophy, Gregory in Chemistry, Sir W. Hamilton in Mental Philosophy, Wilson (Christopher North) in Moral Philosophy. The lectures of Sir W. Hamilton made a strong impression upon
him, in stimulating the love of speculation to which his mind was prone, but, as might have been expected, it was the Professor of Natural Philosophy who obtained the chief share of his devotion. The enthusiasm which so distinguished a man as Forbes naturally inspired in young and ardent disciples, evoked a feeling of personal attachment, and the Professor, on his part, took special interest in his pupil and gave to him the altogether unusual privilege of working with his fine apparatus.
What was the nature of this experimental work we may conjecture from a perusal of
his paper on Elastic Solids, written at that time, in which he describes some experiments made with the view of verifying the deductions of his theory in its application to Optics. Maxwell would seem to have been led to the study of this subject by the following cir-
cumstance. He was taken by his uncle John Cay to see William Nicol, the inventor of
the polarising prism which bears his name, and was shewn by Nicol the colours of unannealed glass in the polariscope. This incited Maxwell to study the laws of polarised light and to construct a rough polariscope in which the polariser and analyser were simple glass reflectors. By means of this instrument he was able to obtain the colour bands of unannealed glass. These he copied on paper in water colours and sent to Nicol. It is gratifpng to find that this spirited attempt at experimenting on the part of a mere boy was duly appreciated by Nicol, who at once encouraged and delighted him by a present of a couple of
his prisms.
The paper alluded to, viz. that entitled "On the Equilibrium of Elastic Solids," was
read to the Royal Society of Edinburgh in 1850. It forms the third paper which Maxwell addressed to that Society. The first in 1846 on Ovals has been abready mentioned. The second, under the title "The Theory of Rolling Curves," was presented by Kelland in 1849.
It is obvious that a youth of nineteen years who had been capable of these efforts must have been gifted with rare originality and with great power of sustained exertion. But his singular self-concentration led him into habits of solitude and seclusion, the tendency of which was to confirm his peculiarities of speech and of manner. He was shy and reserved with strangers, and his utterances were often obscure both in substance and in his manner of expressing himself, so many remote and unexpected allusions perpetually obtruding themselves. Though really most sociable and even fond of society he was essentially reticent and reserved. Mr Campbell thinks it is to be regretted that Maxwell did not begin his Cambridge career eai'lier for the sake of the social intercourse which he would have found it difficult to avoid there. It is a question, however, whether in losing the opportunity of using Professor Forbes' apparatus he would not thereby have lost what was perhaps the most valuable part of his early scientific training.

XU

PREFACE.

It was originally intended that Maxwell should follow his father's profession of advocate, but this intention was abandoned as soon as it became obvious that his tastes lay in a direction so decidedly scientific. It was at length determined to send him to Cambridge and accordingly in October, 1850, he commenced residence in Peterhouse, where however he
resided during the Michaelmas Term only. On December 14 of the same year he migrated
to Trinity College.
It may readily be supposed that his preparatory training for the Cambridge course was far removed from the ordinary type. There had indeed for some time been practically
no restraint upon his plan of study and his mind had been allowed to follow its natural bent towards science, though not to an extent so absorbing as to withdraw him from
— other pursuits. Though he was not a sportsman, indeed sport so called was always repugnant
— to him he was yet exceedingly fond of a country life. He was a good horseman and a
good swimmer. Whence however he derived his chief enjoyment may be gathered from the account which Mr Campbell gives of the zest with which he quoted on one occasion the lines of Bums which describe the poet finding inspiration while wandering along the banks of a stream in the free indulgence of his fancies. Maxwell was not only a lover of poetry but himself a poet, as the fine pieces gathered together by Mr Campbell abundantly testify. He saw however that his true calling was Science and never regarded these poetical efforts as other than mere pastime. Devotion to science, already stimulated by successful endeavour, a tendency to ponder over philosophical problems and an attachment to English
— literature, particularly to English poetry, these tastes, implanted in a mind of singular
strength and purity, may be said to have been the endowments with which young Maxwell began his Cambridge career. Besides this, his scientific reading, as we may gather from his papers to the Royal Society of Edinburgh referred to above, was already extensive and varied. He brought with him, says Professor Tait, a mass of knowledge which was really immense for so young a man but in a state of disorder appalling to his methodical
private tutor.
Maxwell's undergraduate career was not marked by any specially notable feature. His private speculations had in some measure to be laid aside in favour of more systematic
study. Yet his mind was steadily ripening for the work of his later years. Among those with whom he was brought into daily contact by his position, as a Scholar of Trinity College, were some of the brightest and most cultivated young men in the University. In the genial fellowship of the Scholars' table Maxwell's kindly humour found ready play, while
in the more select coterie of the Apostle Club, formed for mutual cultivation, he found a field for the exercise of his love of speculation in essays on subjects beyond the lines of the ordinary University course. The composition of these essays doubtless laid the foundation of that literary finish which is one of the characteristics of Maxwell's scientific writings. His biographers have preserved several extracts on a variety of subjects chiefly of a speculative character. They are remarkable mainly for the weight of thought contained in them but occasionally also for smart epigrams and for a vein of dry and sarcastic humour.

PREFACE.
These glimpses into Maxwell's character may prepare us to believe that, with all his shyness, he was not without confidence in his own powers, as also appears from the account which was given by the late Master of Trinity College, Dr Thompson, who was Tutor when Maxwell personally applied to him for permission to migrate to that College. He appeared to be a shy and diffident youth, but presently surprised Dr Thompson by producing a bundle of papers, doubtless copies of those we have already mentioned, remarking " Perhaps these may shew you that I am not unfit to enter at your College."
He became a pupil of the celebrated William Hopkins of Peterhouse, under whom his course of study became more systematic. One striking characteristic was remarked by his
contemporaries. Whenever the subject admitted of it he had recourse to diagrams, though his fellow students might solve the question more easily by a train of analysis. Many illustrations of this manner of proceeding might be taken from his writings, but in truth it was only one phase of his mental attitude towards scientific questions, which led him to proceed from one distinct idea to another instead of trusting to symbols and
equations.
Maxwell's published contributions to Mathematical Science during his undergraduate career
were few and of no great importance. He found time however to carry his investigations into regions outside the prescribed Cambridge course. At the lectures of Professor Stokes*
he was regular in his attendance. Indeed it appears from the paper on Elastic Solids, mentioned above, that he was acquainted with some of the writings of Stokes before he entered Cambridge. Before 1850, Stokes had published some of his most important contributions to Hydromechanics and Optics ; and Sir W. Thomson, who was nine years' Maxwell's senior in University standing, had, among other remarkable investigations, called special attention to the mathematical analogy between Heat-conduction and Statical Electricity. There is no doubt that these authors as well as Faraday, of whose experimental researches he had made a careful study, exercised a powerful directive influence on his mind.
In January, 1854, Maxwell's undergraduate career closed. He was second wrangler, but shared with Dr Routh, who was senior wrangler, the honours of the First Smith's Prize.
In due course he was elected Fellow of Trinity and placed on the staff of College Lecturers.
No sooner was he released from the restraints imposed by the Trinity Fellowship
Examination than he plunged headlong into original work. There were several questions he was anxious to deal with, and first of all he completed an investigation on the Transformation of Surfaces by Bending, a purely geometrical problem. This memoir he presentel to the Cambridge Philosophical Society in the following March. At this period he also set about an enquiry into the quantitative measurement of mixtures of colours and the causes of colour-blindness. During his undergraduateship he had, as we have seen, found time for the study of Electricity. This had already borne fruit and now resulted in the first of his important memoirs on that subject,—the memoir on Faraday's Lines of Force.
• Now Sir George Gabriel Stokes, Bart., M.P. for the University.

—

Xiv

PREFACE.

The number and importance of his papers, published in 1855—6, bear witness to his assiduity during this period. With these labours, and in the preparation of his College lectures, on which he entered with much enthusiasm, his mind was fully occupied and the work was congenial. He had formed a number of valued friendships, and he had a variety of interests, scientific and literary, attaching him to the University. Nevertheless, when the chair of Natural Philosophy in Marischal College, Aberdeen, fell vacant, Maxwell became a candidate. This step was probably taken in deference to his father's wishes, as the long summer vacation of the Scottish College would enable him to reside with his father at Glenlair for half the year continuously. He obtained the professorship, but unhappily the kind intentions which prompted him to apply for it were frustrated by the death of his father, which took
place in April, 1856.

It is doubtful whether the change from the Trinity lectureship to the Aberdeen professorship was altogether prudent. The advantages were the possession of a laboratory and the long uninterrupted summer vacation. But the labour of drilling classes composed chiefly of comparatively young and untrained lads, in the elements of mechanics and physics, was
not the work for which Maxwell was specially fitted. On the other hand, in a large college like Trinity there could not fail to have been among its undergraduate members, some of the
most promising young mathematicians of the University, capable of appreciating his original
genius and immense knowledge, by instructing whom he would himself have derived ad-
vantage.

In 1856 Maxwell entered upon his duties as Professor of Natural Philosophy at Marischal College, and two years afterwards he married Katharine Mary Dewar, daughter of the
Principal of the College. He in consequence ceased to be a Fellow of Tiinity College,
but was afterwards elected an honorary Fellow, at the same time as Professor Cayley.
— During the yeai*s 1856 60 he was still actively employed upon the subject of colour
sensation, to which he contributed a new method of measurement in the ingenious instrument known as the colour-box. The most serious demands upon his powers and upon his time were made by his investigations on the Stability of Saturn's Rings. This was the subject chosen by the Examiners for the Adams Prize Essay to be adjudged in 1857, and
was advertised in the following terms:

"The Problem may be treated on the supposition that the system of Rings is

exactly or very approximately concentric with Saturn and symmetrically disposed about

the plane of his equator and different hypotheses may be made respecting the physical

constitution of the Rings. It may be supposed (1) that they are rigid; (2) that they

are

fluid

and

in

part

aeriform ;

(3)

that

they consist of masses

of matter

not

materially

coherent. The question will be considered to be answered by ascertaining on these

hypotheses severally whether the conditions of mechanical stability are satisfied by the

mutual attractions and motions of the Planet and the Rings."

PREFACE.

XV

"It is desirable that an attempt should also be made to determine on which of the above hypotheses the appearances both of the bright rings and the recently discovered dark ring may be most satisfactorily explained; and to indicate any causes to which a change of form such as is supposed from a comparison of modem with the earlier observations to have taken place, may be attributed."
It is sufficient to mention here that Maxwell bestowed an immense amount of labour in working out the theory as proposed, and that he arrived at the conclusion that "the only system of rings which can exist is one composed of an indefinite number of unconnected particles revolving round the planet with different velocities according to their respective distances. These particles may be arranged in a series of narrow rings, or they may move about through each other irregularly. In the first case the destruction of the system will be very slow, in the second case it will be more rapid, but there may be a tendency towards an aiTangement in narrow rings which may retard the process."
Part of the work, dealing with the oscillatory waves set up in a ring of satellites, was illustrated by an ingenious mechanical contrivance which was greatly admired when exhibited before the Royal Society of Edinburgh.
This essay, besides securing the prize, obtained for its author great credit among scientific men. It was characterized by Sir George Airy as one of the most remarkable applications of Mathematics to Physics that he had ever seen.
The suggestion has been made that it was the irregular motions of the particles which compose the Rings of Saturn resulting on the whole in apparent regularity and uniformity, which led Maxwell to the investigation of the Kinetic Theory of Gases, his first contribution to which was read to the British Association in 1859. This is not unlikely, but it must also be borne in mind that Bernoulli's Theory had recently been revived by Herapath, Joule and Clausius whose writings may have drawn Maxwell's attention to the
subject.
In 1860 King's College and Marischal College were joined together as one institution, now known as the University of Aberdeen. The new chair of Natural Philosophy thus created was filled up by the appointment of David Thomson, formerly Professor at King's College and Maxwell's senior. Professor Thomson, though not comparable to Maxwell as a
physicist, was nevertheless a remarkable man. He was distinguished by singular force of
character and great administrative faculty and he had been prominent in bringing about
the fusion of the Colleges. He was also an admirable lecturer and teacher and had done much to raise the standard of scientific education in the north of Scotland. Thus the choice made by the Commissioners, though almost inevitable, had the effect of making it appear
that Maxwell failed as a teacher. There seems however to be no evidence to support such
an inference. On the contrary, if we may judge from the number of voluntary students
attending his classes in his last College session, he would seem to have been as popular as a professor as he was personally estimable.

XVI

PREFACE.

This is also borne out by the fact that he was soon afterwards elected Professor of

Natural Philosophy and Astronomy in King's College, London. The new appointment had the advantage of bringing him much more into contact with men in his own department

of science, especially with Faraday, with whose electrical work his own was so intimately
— connected. In 1862 63 he took a prominent part in the experiments organised by a

Committee of the British Association for the determination of electrical resistance in

absolute measure and for placing electrical measurements on a satisfactory basis. In the

experiments which were conducted in the laboratory of King's College upon a plan due

to Sir W. Thomson, two long series of measurements were taken in successive years. In

the

first

year,

the

working members

were

Maxwell,

Balfour Stewart and

Fleeming Jenkin ;

in

the second, Charles Hockin took the place of Balfour Stewart. The work of this Committee

was communicated in the form of reports to the British Association and was afterwards

republished in one volume by Fleeming Jenkin.

Maxwell was a professor in King's College from 1860 to 1865, and this period of his life is distinguished by the production of his most important papers. The second memoir on Colours made its appearance in 1860. In the same year his first papers on the Kinetic Theory of Gases were published. In 1861 came his papers on Physical Lines of Force
— and in 1864 his greatest memoii' on Electricity, a Dynamical Theory of the Electro-
magnetic Field. He must have been occupied with the Dynamical Theory of Gases in 1865,
as two important papers appeared in the following year, first the Bakerian lecture on the Viscosity of Gases, and next the memoir on the Dynamical Theory of Gases.
The mental strain involved in the production of so much valuable work, combined with the duties of his professorship which required his attention during nine months of the year, seems to have influenced him in a resolution which in 1865 he at length adopted of resigning his chair and retiring to his country seat. Shortly after this he had a severe illness. On his recovery he continued his work on the Dynamical Theory of Gases, to which reference has just been made. For the next few years he led a quiet and secluded life at Glenlair, varied by annual visits to London, attendances at the British Association meetings and by a tour in Italy in 1867. He was also Moderator or Examiner in the Mathematical Tripos at Cambridge on several occasions, ofiBces which entailed a few weeks' residence at the University in winter. His chief employment during those years was the prepai-ation of his now celebrated treatise on Electricity and Magnetism which, however, was not published till 1873. He also wrote a treatise on Heat which was
published in 1871.

In 1871 Maxwell was, with some reluctance, induced to quit his retreat in the country and to enter upon a new career. The University of Cambridge had recently resolved to found a professorship of physical science, especially for the cultivation and teaching of the subjects of Heat, Electricity and Magnetism. In furtherance of this object her Chancellor, the Duke of Devonshire, had most generously undertaken to build a laboratory and furnish it with the necessary apparatus. Maxwell was invited to fill the

PREFACE.

XVU

new chair thus formed and to superintend the erection of the laboratory. 1871, he delivered his inaugural lecture.

In October,

The Cavendish Laboratory, so called after its founder, the present venerable chief of the family which produced the great physicist of the same name, was not completed for practical work until 1874. In June of that year it was formally presented to the University by the Chancellor. The building itself and the fittings of the several rooms were admirably contrived mainly by Maxwell himself, but the stock of apparatus was smaller than accorded with the generous intentions of the Chancellor. This defect must be attributed to the anxiety of the Professor to procure only instruments by the best makers and with such improvements as he could himself suggest. Such a defect therefore required time for its removal and afterwards in great measure disappeared, apparatus being constantly added to the stock as occasion demanded.

One of the chief tasks which Maxwell undertook was that of superintending and directing the energies of such young Bachelors of Arts as became his pupils after having acquired good positions in the University examinations. Several pupils, who have since acquired distinction, carried out valuable experiments under the guidance of the Professor. It must be admitted, however, that the numbers were at first small, but perhaps this was only to be expected from the traditions of so many years. The Professor was singularly kind and helpful to these pupils. He would hold long conversations with them, opening up to them the stores of his mind, giving them hints as to what they might try and what avoid, and was always ready with some ingenious remedy for the experimental troubles which beset them. These conversations, always delightful and instructive, were, according to the account of one of his pupils, a liberal education in themselves, and were repaid in the minds of the pupils by a grateful affection rarely accorded to any teacher.

Besides discharging the duties of his chair, Maxwell took an active part in conducting the general business of the University and more particularly in regulating the courses of study in Mathematics and Physics.

For some years previous to 1866 when Maxwell returned to Cambridge as Moderator in the Mathematical Tripos, the studies in the University had lost touch with the great scientific movements going on outside her walls. It was said that some of the subjects most in vogue had but little interest for the present generation, and loud complaints began to be heard that while such branches of knowledge as Heat, Electricity and Magnetism, were left out of the Tripos examination, the candidates were wasting their time and energy upon mathematical trifles barren of scientific interest and of practical results. Into the movement for reform Maxwell entered warmly. By his questions in 1866 and subsequent years he infused new life into the examination ; he took an active part in drafting the new scheme introduced in 1873 ; but most of all by his writings he exerted a powerful influence on the younger members of the University, and was largely instrumental in bringing about the change which has been now effected.

XVIU

PREFACE.

In the first few years at Cambridge Maxwell was busy in giving the final touches to his great work on Electricity and Magnetism and in passing it through the press. This work was published in 1873, and it seems to have occupied the most of his attention for the two previous years, as the few papers published by him during that period relate chiefly to subjects forming part of the contents. After this publication his contributions to scientific journals became more numerous, those on the Dynamical Theory of Gases being
perhaps the most important. He also wrote a great many short articles and reviews which made their appearance in Nature and the Encyclopcedia Britannica. Some of these
essays are charming expositions of scientific subjects, some are general criticisms of the works of contemporary writers and others are brief and appreciative biographies of fellow workers in the same fields of research.
An undertaking in which he was long engaged and which, though it proved exceedingly interesting, entailed much labour, was the editing of the "Electrical Researches" of the Hon.
Henry Cavendish. This work, published in 1879, has had the eflfect of increasing the reputation of Cavendish, disclosing as it does the unsuspected advances which that acute physicist had made in the Theory of Electricity, especially in the measurement of electrical quantities. The work is enriched by a variety of valuable notes in which Cavendish's views and results are examined by the light of modern theory and methods. Especially valuable are the methods applied to the determination of the electrical capacities of conductors and condensers, a subject in which Cavendish himself shewed considerable skill both of a mathematical and experimental character.
The importance of the task undertaken by Maxwell in connection with Cavendish's papers will be understood from the following extract from his introduction to them.
"It is somewhat difficult to account for the fact that though Cavendish had prepared a complete description of his experiments on the charges of bodies, and had even taken the trouble to write out a fair copy, and though all this seems to have been done before 1774 and he continued to make experiments in Electricity till 1781 and lived on till 1810, he kept his manuscript by him and never published it."
"Cavendish cared more for investigation than for publication. He would under-
take the most laborious researches in order to clear up a difficulty which no one but himself could appreciate or was even aware of, and we cannot doubt that the result of his enquiries, when successful, gave him a certain degree of satisfaction. But it did not excite in him that desire to communicate the discovery to others which in the case of ordinary men of science, generally ensures the publication of
their results. How completely these researches of Cavendish remained unknown to other men of science is shewn by the external history of electricity."
It will probably be thought a matter of some difficulty to place oneself in the position of a physicist of a century ago and to ascertain the exact bearing of his experiments. But Maxwell entered upon this undertaking with the utmost enthusiasm and

PREFACE.

XIX

succeeded in completely identifying himself with Cavendish's methods. He shewed that
Cavendish had really anticipated several of the discoveries in electrical science which have been made since his time. Cavendish was the first to form the conception of and to measure Electrostatic Capacity and Specific Inductive Capacity; he also anticipated Ohm's law.
The Cavendish papers were no sooner disposed of than Maxwell set about preparing a new edition of his work on Electricity and Magnetism; but unhappily in the summer term of 1879 his health gave way. Hopes were however entertained that when he returned to the bracing air of his country home he would soon recover. But he lingered through
the summer months with no signs of improvement and his spirits gradually sank He was
finally informed by his old fellow-student, Professor Sanders, that he could not live more than a few weeks. As a last resort he was brought back to Cambridge in October that he might be under the charge of his favourite physician, Dr Paget*. Nothing however could be done for • his malady, and, after a painful illness, he died on the 5th of November, 1879, in his 49th year.
Maxwell was thus cut oflf in the prime of his powers, and at a time when the departments of science, which he had contributed so much to develop, were being every day extended by fresh discoveries. His death was deplored as an irreparable loss to science and to the University, in which his amiable disposition was as universally esteemed as his genius was admired.
It is not intended in this preface to enter at length into a discussion of the relation which Maxwell's work bears historically to that of his predecessors, or to attempt to estimate the effect which it has had on the scientific thought of the present day. In some of his
papers he has given more than usually copious references to the works of those by whom
he had been influenced; and in his later papers, especially those of a more popular nature which appeared in the Encyclopoedia Britannica, he has given full historical outlines of some of the most prominent fields in which he laboured. Nor does it appear to the present editor that the time has yet arrived when the quickening influence of Maxwell's mind on
modem scientific thought can be duly estimated. He therefore proposes to himself the duty
of recalling briefly, according to subjects, the most important speculations in which Maxwell
engaged.
His works have been arranged as far as possible in chronological order but they fall naturally under a few leading heads; and perhaps we shall not be far wrong if we place first in importance his work in Electricity.
His first paper on this subject bearing the title "On Faraday's Lines of Force" was read before the Cambridge Philosophical Society on Dec. 11th, 1855. He had been previously attracted by Faraday's method of expressing electrical laws, and he here set before himself
the task of shewing that the ideas which had guided Faraday's researches were not inconsistent with the mathematical formulae in which Poisson and others had cast the laws of
Now Sir George Edward Paget, K.C.B.

PREFACE.
Electricity. His object, he says, is to find a physical analogy which shall help the mind to grasp the results of previous investigations "without being committed to any theory founded on the physical science from which that conception is borrowed, so that it is neither
draw aside from the subject in the pursuit of analytical subtleties nor carried beyond the
truth by a favorite hypothesis."
The laws of electricity are therefore compared with the properties of an incompressible fluid the motion of which is retarded by a force proportional to the velocity, and the fluid
is supposed to possess no inertia. He shews the analogy which the lines of flow of such
a fluid would have with the lines of force, and deduces not merely the laws of Statical Electricity in a single medium but also a method of representing what takes place when the
action passes from one dielectric into another.
In the latter part of the paper he proceeds to consider the phenomena of Electromagnetism and shews how the laws discovered by Ampere lead to conclusions identical with those of Faraday. In this paper three expressions are introduced which he identifies with the components of Faraday's electrotonic state, though the author admits that he has not been able to frame a physical theory which would give a clear mental picture of the various connections expressed by the equations.
Altogether this paper is most important for the light which it throws on the principles which guided Maxwell at the outset of his electrical work. The idea of the electrotonic state had afready taken a firm hold of his mind though as yet he had formed no physical
explanation of it. In the paper "On Physical Lines of Force" printed in the Philosophical Magazine, Vol. xxi. he resumes his speculations. He explains that in his former paper he had found the geometrical significance of the Electrotonic state but that he now proposes "to examine magnetic phenomena from a mechanical point of view." Accordingly he propounds
his remarkable speculation as to the magnetic field being occupied by molecular vortices, the axes of which coincide with the lines of force. The cells within which these vortices rotate are supposed to be separated by layers of particles which serve the double purpose of transmitting motion from one cell to another and by their own motions constituting an electric current. This theory, the parent of several working models which have been devised to represent the motions of the dielectric, is remarkable for the detail vnth which it is worked out and made to explain the various laws not only of magnetic and electromagnetic action, but also the various forms of electrostatic action. As Maxwell subsequently gave a
more general theory of the Electromagnetic Field, it may be inferred that he did not desire
it to be supposed that he adhered to the views set forth in this paper in every particular; but there is no doubt that in some of its main features, especially the existence of rotation round the lines of magnetic force, it expressed his permanent convictions. In his treatise on "Electricity and Magnetism," Vol. ii. p. 416, (2nd edition 427) after quoting from Sir W. Thomson on the explanation of the magnetic rotation of the plane of the polarisation
of light, he goes on to say of the present paper,

—

PREFACE.

XXI

"A theory of molecular vortices which T worked out at considerable length was
published in the Phil. Mag. for March, April and May, 1861, Jan. and Feb. 1862."
- " I think we have good evidence for the opinion that some phenomenon of rotation is going on in the magnetic field, that this rotation is performed by a great number of very small portions of matter, each rotating on its own axis, that axis being parallel to the direction of the magnetic force, and that the rotations of these various vortices are made to depend on one another by means of some mechanism between them."
"The attempt which I then made to imagine a working model of this mechanism must be taken for no more than it really is, a demonstration that mechanism may
be imagined capable of producing a connection mechanically equivalent to the actual
connection of the parts of the Electromagnetic Field."
This paper is also important as containing the first hint of the Electromagnetic Theory of Light which was to be more fully developed afterwards in his third great memoir
" On the Dynamical Theory of the Electromagnetic Field." This memoir, which was presented
to the Royal Society on the 27th October, 1864, contains Maxwell's mature thoughts on a subject which had so long occupied his mind. It was afterwards reproduced in his Treatise with trifling modifications in the treatment of its parts, but without substantial changes in its main features. In this paper Maxwell reverses the mode of treating electrical phenomena adopted by previous mathematical writers; for while they had sought to build up the laws of the subject by starting from the principles discovered by Ampere, and deducing the induction of currents from the conservation of energy, Maxwell adopts the method of first arriving at the laws of induction and then deducing the mechanical
attractions and repulsions.
After recalling the general phenomena of the mutual action of cuiTents and magnets
m and the induction produced in a circuit by any variation of the strength of the field
which it lies, the propagation of light through a luminiferous medium, the properties of dielectrics and other phenomena which point to a medium capable of transmittmg force
and motio^i, he proceeds.
"Thus then we are led to the conception of a complicated mechanism capable of a vast variety of motions but at the same time so connected that the motion of one part depends, according to definite relations, on the motion of other parts, these teotions being communicated by forces arising from the relative displacement of their connected parts, in virtue of their elasticity. Such a mechanism must be subject
to the laws of Dynamics."
On applying dynamical principles to such a connected system he attains certain general propositions which, on being compared with the laws of induced currents, enable him to identify certain features of the mechanism with properties of currents. The induction of
currehts and their electromagnetic attraction are thus explained and connected.

XXll

PREFACE.

In a subsequent part of the memoir he proceeds to establish from these premises the general equations of the Field and obtains the usual formulae for the mechanical force on currents, magnets and bodies possessing an electrostatic charge.

He also returns to and elaborates more fully the electromagnetic Theory of Light.
His equations shew that dielectrics can transmit only transverse vibrations, the speed of propagation of which in air as deduced from electrical data comes out practically identical with the known velocity of light. For other dielectrics the index of refraction is equal

to the square root of the product of the specific inductive capacity by the coefficient of

magnetic induction, which last factor is for most bodies practically unity. Various comparisons have been made with the view of testing this deduction. In the case of paraffin wax and

some of the hydrocarbons, theory and experiment agree, but this is not the case with

glass and some other substances. Maxwell has also applied his theory to media which

are not perfect insulators, and finds an expression for the loss of light in passing through
a stratum of given thickness. He remarks in confirmation of his result that most good

conductors are opaque while insulators are transparent, but he also adds that electrolytes

which transmit a current freely are often transparent, while a piece of gold leaf whose
resistance was determined by Mr Hockin allowed far too great an amount of light to pass. He observes however that it is possible "there is less loss of energy when the

electromotive forces are reversed with the rapidity of light than when they act for sensible

A times as in our experiments."

similar explanation may be given of the discordance

between the calculated and observed values of the specific inductive capacity. Prof. J. J,

Thomson in the Proceedings of the Royal Society, Vol. 46, has described an experiment by

which he has obtained the specific inductive capacities of various dielectrics when acted

on by alternating electric forces whose frequency is 25,000,000 per second. He finds that

under these conditions the specific inductive capacity of glass is very nearly the same as

the square of the refractive index, and very much less than the value for slow rates of reversals. In illustration of these remarks may be quoted the observations of Prof. Hertz who

has shewn that vulcanite and pitch are transparent for waves, whose periods of vibration are

about three hundred millionths of a second. The investigations of Hertz have shewn that

electro-dynamic radiations are transmitted in waves with a velocity, which, if not equal to, is

comparable with that of light, and have thus given conclusive proof that a satisfactory

theory of Electricity must take into account in some form or other the action of the

dielectric. But this does not prove that Maxwell's theory is to be accepted in every

A particular.

peculiarity of his theory is, as he himself points out in his treatise, that

the variation of the electric displacement is to be treated as part of the current as well

as the current of conduction, and that it is the total amount due to the sum of these

which flows as if electricity were an incompressible fluid, and which determines external

electrodynamic actions. In this respect it differs from the theory of Helmholtz which

also takes into account the action of the dielectric. Professor J. J. Thomson » in his

Review of Electric Theories has entered into a full discussion of the points at issue

PREFACE.

XXlll

between the two above mentioned theories, and the reader is referred to his paper for further information *. Maxwell in the memoir before us has also applied his theory to the passage of light through crystals, and gets rid at once of the wave of normal vibrations which has hitherto proved the stumbling block in other theories of light.

The electromagnetic Theory of Light has received numerous developments at the hands of Lord Rayleigh, Mr Glazebrook, Professor J. J. Thomson and others. These volumes also contain various shorter papers on Electrical Science, though perhaps the most complete record of Maxwell's work in this department is to be found in his Treatise on Electricity and Magnetism in which they were afterwards embodied.

Another series of papers of hardly less importance than those on Electricity are the

various memoirs on the Dynamical Theory of Gases. The idea that the properties of

matter might be explained by the motions and impacts of their ultimate atoms is as

old as the time of the Greeks, and Maxwell has given in his paper on " Atoms " a full

sketch of the ancient controversies to which it gave rise. The mathematical difficulties of

the speculation however were so great that it made little real progress till it was taken up by Clausius and shortly afterwards by Maxwell. The first paper by Maxwell on the

subject is entitled "Illustrations of the Dynamical Theory of Gases" and was published

in the Philosophical Magazine for January and July, 1860, having been read at a meeting

of the British Association of the previous year. Although the methods developed in this

paper were afterwards abandoned for others, the paper itself is most interesting, as it indicates

clearly the problems in the theory which Maxwell proposed to himself for solution, and so far

contains the germs of much that was treated of in his next memoir. It is also epoch-making,

inasmuch as it for the first time enumerates various propositions which ai-e characteristic

of Maxwell's work in this subject. It contains the first statement of the distribution of velo-

cities according to the law of errors. It also foreshadows the theorem that when two gases

are in thermal equilibrium the mean kinetic energy of the molecules of each system is the

same ; and for the first time the question of the viscosity of gases is treated dynamically.

In his great memoir "On the Dynamical Theory of Gases" published in the Philo-

sophical Transactions of the Royal Society and read before the Society in May, 1866, he returns to this subject and lays down for the first time the general d3niamical methods

appropriate for its treatment. Though to some extent the same ground is traversed as in his former paper, the methods are widely different. He here abandons his former h}^othesis

that the molecules are hard elastic spheres, and supposes them to repel each other with

forces varying inversely as the fifth power of the distance. His chief reason for assuming

this law of action appears to be that it simplifies considerably the calculation of the

collisions between the molecules, and it leads to the conclusion that the coefficient of
viscosity is directly proportional to the absolute temperature. He himself undertook an

experimental enquiry for the purpose of verifying this conclusion, and, in his paper on the

A Viscosity of Gases, he satisfied himself of its correctness.

re-examination of the numerical

* British Association Report, 1885.

XXIV

PREFACE.

reductions made in the course of his work discloses however an inaccuracy which materially affects the values of the coefl&cient of viscosity obtained. Subsequent experiments also seem to shew that the concise relation he endeavoured to establish is by no means so near the truth as he supposed, and it is more than doubtful whether the action between two molecules can be represented by any law of so simple a character.

In the same memoir he gives a fresh demonstration of the law of distribution of velocities, but though the method is of permanent value, it labours under the defect of assuming that the distribution of velocities in the neighbourhood of a point is the same in every direction, whatever actions may be taking place within the gas. This flaw in the argument, first pointed out by Boltzmann, seems to have been recognised by Maxwell,
who in his next paper "On the Stresses in Rarefied Gases arising from inequalities of
Temperature," published in the Philosophical Transactions for 1879, Part I., adopts a form of the distribution function of a somewhat different shape. The object of this paper was to arrive at a theory of the effects observed in Crookes's Radiometer. The results of the investigation are stated by Maxwell in the introduction to the paper, from which it would appear that the observed motion cannot be explained on the Dynamical Theory, unless it be supposed that the gas in contact with a solid can slide along the surface with a finite velocity between places whose temperatures are different. In an appendix to the paper he shews that on certain assumptions regarding the nature of the contact of the solid
and gas, there will be, when the pressure is constant, a flow of gas along the surface from the colder to the hotter parts. The last of his longer papers on this subject is one on Boltzmann's Theorem. Throughout these volumes will be found numerous shorter essays on kindred subjects, published chiefly in Nature and in the Encyclopcedia Britannica. Some of these contain more or less popular expositions of this subject which Maxwell bad himself in great part created, while others deal with the work of other writers in the same field. They are profoundly suggestive in almost every page, and abound in acute criticisms of speculations which he could not accept. They are always interesting; for although the larger papers are sometimes difficult to follow, Maxwell's more popular writings are characterized by extreme lucidity and simplicity of style.

The first of Maxwell's papers on Colour Perception is taken from the Transactions of

the Royal Scottish Society of Arts and is in the form of a letter to Dr G. Wilson dated

Jan. 4, 1855. It was followed directly afterwards by a communication to the Royal Society

of Edinburgh, and the subject occupied his attention for some years. The most important

of his subsequent work is to be found in the papers entitled "An account of Experiments

on the Perception of Colour " published in the Philosophical Magazine, Vol xiv. and " On

the Theory of Compound Colours and its relation to the colours of the spectrum " in the

We Philosophical Transactions for the year 1860.

may also refer to two lectures delivered

at the Royal Institution, in which he recapitulates and enforces his main positions in his

usual luminous style. Maxwell from the first adopts Young's Theory of Colour Sensation,

according to which all colours may ultimately be reduced to three, a red, a green and

PREFACE.

XXV

a violet. This theory had been revived by Helmholtz who endeavoured to find for it a physiological basis. Maxwell however devoted himself chiefly to the invention of accurate methods for combining and recording mixtures of colours. His first method of obtaining mixtures, that of the Colour Top, is an adaptation of one formerly employed, but in Maxwell's hands it became an instrument capable of giving precise numerical results by means which he added of varying and measuring the amounts of colour which were blended in the eye. In the representation of colours diagrammatical ly he followed Young in employing an equilateral triangle at the angles of which the fundamental colours were placed. All colours, white included, which may be obtained by mixing the fundamental colours in any proportions will then be represented by points lying within the triangle. Points without the triangle represent colours which must be mixed with one of the fundamental tints to produce a mixture of the other two, or with which two of them must be mixed to produce the third.
In his later papers, notably in that printed in the Philosophical Transactions, he
adopts the method of the Colour Box, by which different parts of the spectrum may be mixed in different proportions and matched with white, the intensity of which has been suitably diminished. In this way a series of colour equations are obtained which can be
used to evaluate any colour in terms of the three fundamental colours. These observations on which Maxwell expended great care and labour, constitute by far the most important data regarding the combinations of colour sensations which have been yet obtained, and
are of permanent value whatever theory may ultimately be adopted of the physiology of the
perception of colour.
In connection with these researches into the sensations of the normal eye, may be
mentioned the subject of colour-blindness, which also engaged Maxwell's attention, and is discussed at considerable length in several of his papers.
Geometrical Optics was another subject in which Maxwell took much interest. At an early period of his career he commenced a treatise on Optics, which however was never completed. His first paper "On the general laws of optical instruments," appeared in 1858, but a brief account of the first part of it had been previously communicated to the Cambridge Philosophical Society. He therein lays down the conditions which a perfect optical instrument must fulfil, and shews that if an instrument produce perfect images of an object, i.e. images free from astigmatism, curvature and distortion, for two different positions of the object, it will give perfect images at all distances. On this result as a basis, he finds the relations between the foci of the incident and emergent pencils, the magnifying power and other characteristic quantities. The subject of refraction through optical combinations was afterwards treated by him in a different manner, in three papers communicated to the London Mathematical Society. In the first (1873), "On the focal lines of a refracted pencil," he applies Hamilton's
characteristic function to determine the focal lines of a thin pencil refracted from one
isotropic medium into another at any surface of separation. In the second (1874), "On

—

XXVI

PREFACE.

Hamilton's characteristic function for a narrow beam of light," he considers the more general question of the passage of a ray from one isotropic medium into another, the two media being separated by a third which may be of a heterogeneous character. He finds the most general form of Hamilton's characteristic function from one point to another, the first being in the medium in which the pencil is incident and the second in the medium in which it is emergent, and both points near the principal ray of the pencil. This result is then applied in two particular cases, viz. to determine the emergent pencil (1) from a spectroscope, (2) from an optical instrument symmetrical about its axis. In the third paper (1875) he resumes the last-mentioned application, discussing this case more fully under a somewhat
simplified analysis.
It may be remarked that all these papers are connected by the same idea, which was first to study the optical efiects of the entire instrument without examining the mechanism by which these effects are produced, and then, as in the paper in 1858, to supply whatever data may be necessary by experiments upon the instrument itself.
Connected to some extent with the above papers is an investigation which was published
in 1868 " On the cyclide." As the name imports, this paper deals chiefly with the geometrical
properties of the surface named, but other matters are touched on, such as its conjugate isothermal functions. Primarily however the investigation is on the orthogonal surfaces to a system of rays passing accurately through two lines. In a footnote to this paper Maxwell describes the stereoscope which he invented and which is now in the Cavendish Laboratory.
In 1868 was also published a short but important article entitled " On the best arrangement for producing a pure spectrum on a screen."
The various papers relating to the stresses experienced by a system of pieces joined together so as to form a frame and acted on by forces form an important group connected with one another. The first in order was "On reciprocal figures and diagrams of forces,"
published in 1864. It was immediately followed by a paper on a kindred subject, "On
the calculation of the equilibrium and stiffness of frames." In the first of these Maxwell demonstrates certain reciprocal properties in the geometry of two polygons which are related to one another in a particular way, and establishes his well-known theorem in Graphical Statics on the stresses in frames. In the second he employs the principle of work to problems connected with the stresses in frames and structures and with the deflections arising from extensions in any of the connecting pieces.
A third paper " On the equilibrium of a spherical envelope," published in 1867, may
here be referred to. The author therein considers the stresses set up in the envelope by a system of forces applied at its surface, and ultimately solves the problem for two normal forces applied at any two points. The solution, in which he makes use of the principle of inversion as it is applied in various electrical questions, turns ultimately on the determination of a certain function first introduced by Sir George Airy, and called by Maxwell

PREFACE.

XXvii

Airy's Function of Stress. The methods which in this paper were attended with so much success, seem to have suggested to Maxwell a reconsideration of his former work, with the view of extending the character of the reciprocity therein established. Accordingly in 1870
there appeared his fourth contribution to the subject, "On reciprocal figures, frames and
diagrams of forces." This important memoir was published in the Transactions of the Royal Society of Edinburgh, and its author received for it the Keith Prize. He begins with a remarkably beautiful construction for drawing plane reciprocal diagrams, and then proceeds to discuss the geometry and the degrees of freedom and constraint of polyhedral frames, his object being to lead up to the limiting case when the faces of the polyhedron become infinitely small and form parts of a continuous surface. In the course of this work he obtains certain results of a general character relating to inextensible surfaces and certain
otjiers of practical utility relating to loaded frames. He then attacks the general problem of
representing graphically the internal stress of a body and by an extension of the meaning of "Diagram of Stress," he gives a construction for finding a diagram which has mechanical as well as geometrical reciprocal properties with the figure supposed to be under stress. It is impossible with brevity to give an account of this reciprocity, the development of which in Maxwell's hands forms a very beautiful example of analysis. It will be suflScient to state that under restricted conditions this diagram of stress leads to a solution for the components of stress in terms of a single function analogous to Airy's Function of Stress. In the remaining parts of the memoir there is a discussion of the equations of stress, and it is shewn that the general solution may be expressed in terms of three functions analogous to Airy's single function in two dimensions. These results are then applied to special cases, and in particular the stresses in a horizontal beam with a uniform load on its upper
surface are fully investigated.
On the subjects in which Maxwell's investigations were the most numerous it has
been thought necessary, in the observations which have been made, to sketch out briefly the connections of the various papers on each subject with one another. It is not however intended to enter into an account of the contents of his other contributions to science,
and this is the less necessary as the reader may readily obtain the information he may require in Maxwell's own language. It was usually his habit to explain by way of
introduction to any paper his exact position with regard to the subject matter and to give a brief account of the nature of the work he was contributing. There are however several memoirs which though unconnected with others are exceedingly interesting in themselves. Of these the essay on Saturn's Rings will probably be thought the most important as containing the solution of a diflScult cosmical problem ; there are also various papers on Dynamics, Hydromechanics and subjects of pure mathematics, which are most useful contributions on the subjects of which they treat.
The remaining miscellaneous papers may be classified under the following heads: (a) Lectures and Addresses, (b) Essays or Short Treatises, (c) Biographical Sketches, (d) Criticisms
and Reviews.

XXVIU

PREFACE.

Class (a) comprises his addresses to the British Association, to the London Mathematical Society, the Rede Lecture at Cambridge, his address at the opening of the Cavendish Laboratory and his Lectures at the Royal Institution and to the Chemical Society.
Class (6) includes all but one of the articles which he contributed to the Encyclopcedia Britanrdca and several others of a kindred character to Nature.
Class (c) contains such articles as " Fai-aday " in the Encyclopcedia Britannica and " Helmholtz " in Nature.
Class (d) is chiefly occupied with the reviews of scientific books as they were published. These appeared in Nature and the most important have been reprinted in these pages.
In some of these writings, particularly those in class (b), the author allowed himself a gi-eater latitude in the use of mathematical symbols and processes than in others, as for instance in the article " Capillary Attraction," which is in fact a treatise on that subject treated mathematically. The lectures were upon one or other of the three departments
— of Physics with which he had mainly occupied himself; Colour Perception, Action through
a Medium, Molecular Physics; and on this account they are the more valuable. In the whole series of these more popular sketches we find the same clear, graceful delineation of principles, the same beauty in arrangement of subject, the same force and precision in expounding proofs and illustrations. The style is simple and singularly free fi-om any kind of haze or obscurity, rising occasionally, as in his lectures, to a strain of subdued eloquence when the emotional aspects of the subject overcome the purely speculative.
The books which were written or edited by Maxwell and published in his lifetime but which are not included in this collection were the "Theory of Heat" (1st edition, 1871); "Electricity and Magnetism" (1st edition, 1873); "The Electrical Researches of the Honourable Henry Cavendish, F.R.S., written between 1771 and 1781, edited from the original manuscripts in the possession of the Duke of Devonshire, K.G." (1879). To these may be added a graceful little introductory treatise on Dynamics entitled "Matter and Motion" (published in 1876 by the Society for promoting Christian Knowledge). Maxwell also contributed part of the British Association Report on Electrical Units which was afterwards published in book form by Fleeming Jenkin.
The "Theory of Heat" appeai-ed in the Text Books of Science series published by Longmans, Green and Co., and was at once hailed as a beautiful exposition of a subject, part of which, and that the most interesting part, the mechanical theory, had as yet but commenced the existence which it owed to the genius and laboui-s of Rankine, Thomson and Clausius. There is a certain charm in Maxwell's treatise, due to the freshness and originality of its expositions which has rendered it a great favourite with students of Heat.
After his death an " Elementary Treatise on Electricity," the greater part of which he had written, was completed by Professor Garnett and published in 1881. The aim of this

PREFACE.

XXIX

treatise and its position relatively to his larger work may be gathered from the following
extract from Maxwell's preface.
" In this smaller book I have endeavoured to present, in as compact a form as I can, those phenomena which appear to throw light on the theory of electricity and to use them, each in its place, for the development of electrical ideas in the mind of the reader."
"In the larger treatise I sometimes made use of methods which I do not think the best in themselves, but without which the student cannot follow the investigations of the founders of the Mathematical Theory of Electricity. I have since become more convinced of the superiority of methods akic to those of Faraday, and have therefore adopted them from the first."
Of the "Electricity and Magnetism" it is difficult to predict the future, but there is no doubt that since its publication it has given direction and colour to the study of Electrical Science. It was the master's last word upon a subject to which he had devoted several years of his life, and most of what he wrote found its proper place in the treatise.
Several of the chapters, notably those on Electromagnetism, are practically reproductions of his memoirs in a modified or improved form. The treatise is also remarkable for the handling of the mathematical details no less than for the exposition of physical principles, and is
enriched incidentally by chapters of much originality on mathematical subjects touched on in the course of the work. Among these may be mentioned the dissertations on Spherical Harmonics and Lagrange's Equations in Dj-namics.
The origin and growth of Maxwell's ideas and conceptions of electrical action, culminating in his treatise where all these ideas are arranged in due connection, form an interesting chapter not only in the history of an individual mind but in the history of electrical science. The importance of Faraday's discoveries and speculations can hardly be overrated in their influence on Maxwell, who tells us that before he began the study of electricity he resolved to read none of the mathematics of the subject till he had first mastered the "Experimental Researches." He was also at first under deep obligations to the ideas contained in the exceedingly important papers of Sir W. Thomson on the analogy between Heat-Conduction and Statical Electricity and on the Mathematical Theory of Electricity in Equilibrium. In his subsequent efforts we must perceive in Maxwell, possessed of Faraday's views and embued with his spirit, a vigorous intellect bringing to bear on a subject still full of obscurity the steady light of patient thought and expending upon it all the resources of a never failing ingenuity.

Royal Navax College, Greenwich,
August, 1890.

1

TABLE OF CONTENTS.

II.
Ill
IV. V. VI. VII.
IX.
X. XI.
XII. XIII.
XIV. XV.

On the Description of Oval Curves and those having a plurality of Foci; with

remarks by Professor Forbes

On the Theory of Rolling Curves

*

On the Equilibrium of Elastic Solids

^^

Solutions of Problems

On the Transformation of Surfaces by Bending

On a paHicular case of the descent of a heavy body in a resisting medium

.

On the Theory of Colours in relation to Colour- Blindness

Experiments on Colour as perceived by the Eye, with remarks on Colour-Blindness

On Faraday's Lines of Force Description of a New Form of the Platometer, an Instrument for measuring the
areas of Plane Figures drawn on paper
On the elementary theory of Optical Instruments On a method of drawing the Theoretical Form3 of Faraday's Lines of Force

80 115 119 126
^"^^
230 238

without calculation

On the unequal sensibility of the Foramen Centrale to Light of different Colours 242

On the Theory of Compound Colours with reference to mixtures

Yellow Light

On an instrument to illustrate Poimot's TJieory of Rotation

.

of Blue

.

•

and

^^'^

.246

On a Dynamical Top, for exhibiting the phenomena of the motions of a body of invariable form about a fixed point, with s&ine suggestions as to the Earth's

motion Account of Experiments on the Perception of Colour
On the general laius of Optical Instruments On Theories of the Constitution of Saturn's Rings On the stability of the motion of Saturn s Rings

263 97 286 288

Illustrations of the Dynamical Theory of Gases

On the Theory of Compound Colours and the Relations of the Colours of the

Spectrum

On the Theory of Three Primary Colours

***^

On Physical Lines of Force

451

On Reciprocal Figures and Diagrams of Forces

°^*

.... A Dynamical Theory of the Electromagnetic Field
On the Calculation of the EquilibHum and Stiffness of Frames

526 598

ERRATA.

Page 40. In the first of equations (12), second group of terms, read

instead of

(hP

dy'

d^

d^^^d^^^d^^
with corresponding changes in the other two equations.

Page 153, five lines from bottom of page, read 127 instead of 276

Page 591, four lines from bottom of page the equation should be

d^M d2M_ldM

"^
da? db'

a da~

Page 592, in the first line of the expression for L change - K cos 26 into - ^ cosec 26.

—

[From the Proceedings of the Royal Society of Edinburgh, Vol, li. April, 1846.]

I. On the Description of Oval Curves, and those having a plurality of Foci; ivith remarks by Professor Forbes. Communicated by Professor Forbes.

Mr Clerk Maxwell ingeniously suggests the extension of the common
theory of the foci of the conic sections to curves of a higher degree of complication in the following manner :

(1) As in the ellipse and hyperbola, any point in the curve has the sum or difference of two lines drawn from two points or = foci a. constant

quantity, so the author infers, that curves to a certain degree analogous, may

be described and determined by the condition that the simple distance from

one focus pliLS a multiple distance from the other, may be = a constant quantity;

m or more generally,

times the one distance + n times the other = constant.

(2) The author devised a simple mechanical means, by the wrapping
of a thread round pins, for producing these curves. See Figs. 1 and 2. He

Fig. 1. Two FocL Katios 1,

Fig. 2. Two Foci Ratios 2, 3.

then thought of extending the principle to other curves, whose property should be, that the sum of the simple or multiple distances of any point of

;

;

DESCRIPTION OF OVAL CURVES.
the curve from three or more points or foci, should be = a constant quantity
and this, too, he has effected mechanically, by a very simple arrangement of a string of given length passing round three or more fixed pins, and constraining a tracing point, P. See Fig. 3. Farther, the author regards curves
Fig. 3. Three Foci. Eatios of Equality.

of the first kind as constituting a particular class of curves of the second kind, two or more foci coinciding in one, a focus in which two strings meet
being considered a double focus; when three strings meet a treble focus, &c.
Professor Forbes observed that the equation to curves of the first class is easily found, having the form
V^+7= a-VhJ{x- c)' + y\

which is that of the curve known under the name of the First Oval of Descartes*. Mr Maxwell had already observed that when one of the foci was at an infinite distance (or the thread moved parallel to itself, and was confined in respect of length by the edge of a board), a curve resembling an ellipse was traced ; from which property Professor Forbes was led first to infer the identity of the oval with the Cartesian oval, which is well known to have this property. But the simplest analogy of all is that derived from the method of description, r and r being the radients to any point of the curve from the two

foci
mr + nr — constant,

which in fact at once expresses on the undulatory theory of light the optical

character of the surface in question, namely, that light diverging from one

F / focus

without the medium, shall be correctly convergent at another point

* Herschel, On Light, Art. 232 ; Lloyd, On Light and Vision, Chap. vii.

;

DESCRIPTION OF OVAL CURVES.

J

— within it ; and in this case the ratio

expresses the index of refraction of

the medium*. If we denote by the power of either focus the number of strings leading
to it by Mr Maxwell's construction, and if one of the foci be removed to an
infinite distance, if the powers of the two foci be equal the curve is a parabola if the power of the nearer focus be greater than the other, the curve is an eUipse; if the power of the infinitely distant focus be the greater, the curve is a hyperbola. The first case evidently corresponds to the case of the reflection of parallel rays to a focus, the velocity being unchanged after reflection; the second, to the refraction of parallel rays to a focus in a dense medium (in which light moves slower) ; the third case to refraction into a rarer medium.
The ovals of Descartes were described in his Geometry, where he has also given a mechanical method of describing one of themt, but only in a particular
case, and the method is less simple than Mr Maxwell's. The demonstration of
the optical properties was given by Newton in the Principia, Book i., prop. 97, by the law of the sines; and by Huyghens in 1690, on the Theory of Undu-
lations in his Traite de la Lumiere. It probably has not been suspected that so easy and elegant a method exists of describing these curves by the use of a thread and pins whenever the powers of the foci are commensurable. For instance, the curve. Fig. 2, drawn with powers 3 and 2 respectively, give the
proper form for a refracting surface of a glass, whose index of refraction is 1'50,
f in order that rays diverging from may be refracted to F.
As to the higher classes of curves with three or more focal points, we cannot at present invest them with equally clear and curious physical properties, but the method of drawing a curve by so simple a contrivance, which shall

satisfy the condition

mr + nr +pr" + &c. = constant,

is in itself not a little interesting; and if we regard, with Mr Maxwell, the
ovals above described, as the limiting case of the others by the coalescence of two or more foci, we have a farther generalization of the same kind as that so highly recommended by Montucla^ by which Descartes elucidated the conic
sections as particular cases of his oval curves.

This was perfectly well shewn by Hnyghens in his Traite de la Lumiere, p. 111. + Edit. 1683. Geometria, Lib. ii. p. 54. X Histoire dea Mathematiqties. First Edit IL 102.

(1690.)

—
[From the Transactions of the Royal Society of Edinburgh, Vol. xvi. Part v.]
II. On the Theory of Rolling Curves. Communicated by the Eev. Professor Kelland.
There is an important geometrical problem which proposes to find a curve
having a given relation to a series of curves described according to a given law. This is the problem of Trajectories in its general form.
The series of curves is obtained from the general equation to a curve by the variation of its parameters. In the general case, this variation may change the form of the curve, but, in the case which we are about to consider, the curve is changed only in position.
This change of position takes place partly by rotation, and partly by transference through space. The roUing of one curve on another is an example of this compound motion.
As examples of the way in which the new curve may be related to the series of curves, we may take the following :
1. The new curve may cut the series of curves at a given angle. When
this angle becomes zero, the curve is the envelope of the series of curves.
2. It may pass through correspondiug points in the series of curves. There are many other relations which may be imagined, but we shall confine our attention to this, partly because it aSbrds the means of tracing various curves, and partly on account of the connection which it has with many
geometrical problems. Therefore the subject of this paper will be the consideration of the relations
of three curves, one of which is fixed, while the second rolls upon it and traces the third. The subject of rolling curves is by no means a new one. The first idea of the cycloid is attributed to Aristotle, and involutes and evolutes have been long known.

—

THE THEORY OF ROLLING CURVES.

In the Histmy of the Royal Academy of Sciences for 1704, page 97, there is a memoir entitled "Nouvelle formation des Spirales," by M. Varignon, in which he shews how to construct a polar curve from a curve referred to rectangular co-ordinates by substituting the radius vector for the abscissa, and
a circular arc for the ordinate. After each curve, he gives the curve into which it is " unrolled," by which he means the curve which the spiral must
be rolled upon in order that its pole may trace a straight line; but as this
18 not the principal subject of his paper, he does not discuss it very fully. There is also a memoir by M. de la Hire, in the volume for 1706, Part ii.,
page 489, entitled "Methode generale pour r^duire toutes les Lignes courbes ^ des Roulettes, leur generatrice ou leur base ^tant donnde telle qu'on voudra."
M. de la Hire treats curves as if they were polygons, and gives geometrical constructions for finding the fixed curve or the rolling curve, the other two being given; but he does not work any examples.
In the volume for 1707, page 79, there is a paper entitled, "Methode generale pour determiner la nature des Courbes form^es par le roulement de toutes sortes de Courbes sur une autre Courbe quelconque." Par M. Nicole.
M. Nicole takes the equations of the three curves referred to rectangular
co-ordinates, and finds three general equations to connect them. He takes the
tracing-point either at the origin of the co-ordinates of the rolled curve or not.
He then shews how these equations may be simplified in several particular

cases. These cases are

(1) When the tracing-point is the origin of the roUed curve.

(2) When the fixed curve is the same as the rolling cxirve.

When (3)

both of these conditions are satisfied.

When (4)

the fixed line is straight.

He then says, that if we roll a geometric curve on itself, we obtain a new geometric curve, and that we may thus obtain an infinite number of geometric

curves.
The examples which he gives of the application of his method are all taken from the cycloid and epicycloid, except one which relates to a parabola, rolling on itself, and tracing a cissoid with its vertex. The reason of so small a number of examples being worked may be, that it is not easy to eliminate the co-ordinates of the fixed and rolling curves from his equations.
The case in which one curve roUing on another produces a circle is treated of in Willis's Principles of Mechanism. Class C. Boiling Contact.

6

THE THEORY OP ROLLHiTO CURVES.

He employs the same method of finding the one curve from the other
which is used here, and he attributes it to Euler (see the Acta Petropolitana,
Vol. v.).
Thus, nearly all the simple cases have been treated of by different authors; but the subject is still far from being exhausted, for the equations have been
applied to very few curves, and we may easily obtain new and elegant properties from any curve we please.
Almost all the more notable curves may be thus linked together in a great
variety of ways, so that there are scarcely two curves, however dissimilar, between which we cannot form a chain of connected curves.
This will appear in the list of examples given at the end of this paper.

Let there be a curve KAS, whose pole is at C.

——

THE THEORY OF ROLLING CURVES.

7

Let the angle DCA = 6, and CA=r, and let

Let this curve remain fixed to the paper.

Let there be another curve BAT, whose pole is B.

MBA Let the angle

= 0t,

and

BA=r^,

and

let

KAS Let this curve roll along the curve

without slipping.

B Then the pole

will describe a third curve, whose pole is C.

DCB Let the angle

= 0^, and CB = r„ and let

We have here six unknown quantities 0,dAr,r^r^; but we have only three
equations given to connect them, therefore the other three must be sought for
in the enunciation.
But before proceeding to the investigation of these three equations, we must premise that the three curves will be denominated as follows :
The Fixed Curve, Equation, e^ = ^^{r^. The Rolled Curve, Equation, = 0. <f>,{r,).
Tlie Traced Curve, Equation, = 6^ 4>.,{r^.

When it is more convenient to make use of equations between rectangular

We co-ordinates, we shall use the letters x^^, x^^, x^ij^.

shall always employ the

letters s^s^^ to denote the length of the curve from the pole, p.p^p^ for the per-

pendiculars from the pole on the tangent, and q^q/i^ for the intercepted part of

the tangent.
Between these quantities, we have the following equations:

r = ^/^T?,

^ = tan-|,

= a? r cos ^,

y = r sin 6,

r"
jm'S

ydx — xdy
""^w+w'

—

THE THEORY OF ROLLING CURVES.

rdr
dS
2=-r=7x!fi'

_ xdx + ydy r- J{dxy + (dyY'

W ' "^

'^d^

daf

We come now to consider the three equations of rolling which are involved
in the enunciation. Since the second curve rolls upon the first without slipping, the length of the fixed curve at the point of contact is the measure of the length of the rolled curve, therefore we have the following equation to connect the fixed curve and the rolled curve
= «! Sj.
Now, by combining this equation with the two equations

it is evident that from any of the four quantities 6{r^6^r^ or x^^x^^, we can obtain the other three, therefore we may consider these quantities as known

functions of each other.

Since the curve rolls on the fixed curve, they must have a common tangent.

Let PA be this tangent, draw BP, CQ perpendicular to PA, produce CQ,

BR and draw

BA perpendicular to it, then we have CA=r^,

= r^, and CB = r,;

CQ=p„ PB=p,, and BN=p,; AQ = q„ AP = q„ and CN=q,.

Also

r,'=CR=CR + RR = (CQ + PBY+(AP-AQf

+ + + - =p,' 2p,p, +p,' r,' -p,' 2q,q, r," -p,'

fz = n' + n' + 2piPa - 2q,q^.

Since the first curve is fixed to the paper, we may find the angle 6,.

Thus

e, = DCB = DCA + ACQ + RCB

= e?. + tan-| + tan-|§

^, = ^, + tan--^ + tan-^ ^^^^

TjdO^

Pi +pi

THE THEORY OF ROLLING CURVES.

»

Thus we have found three independent equations, which, together with the equations of the curves, make up six equations, of which each may be deduced
from the others. There is an equation connecting the radii of curvature of the
three curves which is sometimes of use. The angle through which the rolled curve revolves during the description of
the element ds„ is equal to the angle of contact of the fixed curve and the
rolling curve, or to the sum of their curvatures,
ds^ ds^ ds.

But the radius of the rolled curve has revolved In the opposite direction through an angle equal to dO,, therefore the angle between two successive posi-

tions of r, is equal to -^-dd,. Now this angle is the angle between two

successive positions of the normal to the traced curve, therefore, if

be the

centre of curvature of the traced curve, it is the angle which ds^ or ds^ subtends

at 0. Let OA^T, then

_ ds^ r4d^ ds, ,^ ds^ ds, ,.

^J__J_ 1 _^

T~ •*• '^'ds,

R, R, ds/

-tAt^tJ RJR.'

As an example of the use of this equation, we may examine a property

of the logarithmic spiral.

R —m In this curve, ^p = mr, and

= , therefore if the rolled curve be the

logarithmic spiral

/I 1\ 1 ^m "^[t^tJ-r^v/
m_ 1
t~r:,*

^0 AO therefore

-^ = in the figure ?ni2i, and

= m.

Let the locus of 0, or the evolute of the traced curve LYBH, be the

KZAS curve OZY, and let the evolute of the fixed curve

be FEZ, and let

FEZ OZF us consider

as the fixed curve, and

as the traced curve.

10

THE THEORY OF ROLLING CURVES.

^y Then in the triangles BPA, AOF, we have OAF=PBA, and ^='^ =

FOA APB OF therefore the triangles are similar, and

=

= - , therefore

is perpen-

OF dicular to OA, the tangent to the curve OZY, therefore

is the radius of

the curve which when roUed on FEZ traces OZY, and the angle which the

OFA=PAB curve makes with this radius is

= %mr^m, which is constant, there-

fore the curve, which, when rolled on FEZ, traces OZY, is the logarithmic

spiral. Thus we have proved the following proposition : " The involute of the

curve traced by the pole of a logarithmic spiral which rolls upon any curve,

is the curve traced by the pole of the same logarithmic spiral when rolled on

the involute of the primary curve."

It follows from this, that if we roll on any curve a curve having the

— property _2:»i

Wjri,

and

roll

another

curve

having = Pi 'm^r^ on

the

curve

traced,

and so on, it is immaterial in what order we roll these curves. Thus, if we

roll a logarithmic spiral, in which jp = mr, on the nth involute of a circle whose

radius is a, the curve traced is the w+lth involute of a circle whose radius

is Jl-m\

m Or, if we roll successively

logarithmic spirals, the resulting curve is the

n + mth involute of a circle, whose radius is

aJl—m^ sll- m/, Jkc.

We now proceed to the cases in which the solution of the problem may
be simplified. This simplification is generally effected by the consideration that the radius vector of the rolled curve is the normal drawn from the traced
curve to the fixed curve.
In the case in which the curve is rolled on a straight line, the perpendicular on the tangent of the rolled curve is the distance of the tracing point from the straight line ; therefore, if the traced curve be defined by an equation in iCg and y„

'^.°p.= / "'„...

(1)'

and

'••=^'^©^

^'^-

X

THE THEORY OF ROLLING CURVES.

11

By substituting for r, in the first equation, its value, as derived from the second, we obtain
-©[©-]=©'

If we know the equation to the rolled curve, we may find (-7-^') in

terms of r,, then by substituting for r, its value in the second equation, we

dx

(1

have

an

equation

containing

x^

and

-^,

from

which

we

find

the

value

of

'
-t—

dy,

du,

in terms of x^; the integration of this gives the equation of the traced curve.

As an example, we may find the curve traced by the pole of a hyperbolic
spiral which rolls on a straight line.

The equation of the rolled curve is 6^ = a

fdrA' _ rl

,ddj

~ a'

- •©-[(IJ-]'
dx^ _ ^3
'* dy,~Ja'-x,''
This is the differential equation of the tractory of the straight line, which is the curve traced by the pole of the hyperbolic spiral.
By eliminating x^ in the two equations, we obtain dr^_ /dxA
This equation serves to determine the rolled curve when the traced cuive
is given.
As an example we shall find the curve, which being rolled on a straight line, traces a common catenary.
Let the equation to the catenary be
'l(e' + e-^.

12
Then

THE THEORY OF ROLLING CURVES.
dy,~N a' '

dr

then by integration

^ =cos'^ (

1

j

r = 2a
1+COS0'

This is the polar equation of the parabola, the focus being the pole ; there-
fore, if we roll a parabola on a straight line, its focus will trace a catenary. The rectangiilar equation of = this parabola is af Aay, and we shall now
consider what curve must be rolled along the axis of y to trace the parabola.
By the second equation (2),

V n = ^9 /-4- + l> but x^^Pi, ^»

+ .-. r/=^/ 4a",
.-. 2a = Vr/-jp/ = g'„
but q^ is the perpendicular on the normal, therefore the normal to the curve always touches a circle whose radius is 2a, therefore the curve is the involute
of this circle.
Therefore we have the following method of describing a catenary by con-
tinued motion. Describe a circle whose radius is twice the parameter of the catenary; roll a
straight line on this circle, then any point in the line will describe an involute

THE THEORY OF ROLLING CURVES.

13

of the circle ; roll this curve on a straight line, and the centre of the circle will describe a parabola ; roll this parabola on a straight line, and its focus will trace

the catenary required.

We come now to the case in which a straight line rolls on a curve.

When the tracing-point is in the straight line, the problem becomes that
of involutes and evolutes, which we need not enter upon ; and when the tracmgpoint is not in the straight line, the calculation is somewhat complex; we shall
therefore consider only the relations between the curves described in the first

and second cases.
— Definition. The curve
given radius whose centres

which
are in

cuts at a given angle all the circles of a a given curve, is called a tractory of the

given curve.

Let a straight line roll on a curve A, and let a point in the straight

line describe a curve B, and let another point, whose distance from the first

point is b, and from the straight line a, describe a curve C, then it is evident

B that the curve

cuts the circle whose centre is in C, and whose radius is b,

5 at an angle whose sine is equal to r, therefore the curve

is a tractory of

the curve C.

B When a = b, the curve

is the orthogonal tractory of the curve C. If

tangents equal to a be drawn to the curve B, they will be terminated in

the curve C; and if one end of a thread be carried along the curve C, the

other end will trace the curve B.

B When a = 0, the curves

C and

are both involutes of the curve A,

they are always equidistant from each other, and if a circle, whose radius is

6, be rolled on the one, its centre will trace the other.

A If the curve

is such that, if the distance between two points measured

along the curve is equal to 6, the two points are similarly situate, then the

B curve

A is the same with the curve C. Thus, the curve

may be a re-

entrant curve, the circumference of which is equal to 6.

B When the curve -4 is a circle, the curves

and C are always the same.

The equations between the radii of curvature become

1 1_ r

—

14

THE THEORY OF ROLLING CURVES.

B When a = 0, T=0, or the centre of curvature of the curve

is at the

C point of contact. Now, the normal to the curve

passes through this point,

therefore

"The normal to any curve passes through the centre of curvature of its

tractory,"

In the next case, one curve, by rolling on another, produces a straight line. Let this straight line be the axis of y, then, since the radius of the rolled curve is perpendicular to it, and terminates in the fixed curve, and since these curves have a common tangent, we have this equation,

If the equation of the rolled curve be given, find -j-^ in terms of r^, sub-

stitute Xi for r^, and multiply by x^, equate the result to -^ , and integrate.

Thus, if the equation of the rolled curve be
d = Ar-"" + &c. + Kr-^ + Lr'^ + if log r + iVr + &c. + Zr"",

^ N+ = - n^r-(»+^) - &c. - 2Kr-' - I/p-' + Mr'' +

&c. + wZr"-^

dr

M+ a-rx-= - nAx~'* - &c. - 2Kx~"- - Lx~^ +

Nx + &c. + nZx",

-^ -^ y =

Aa^-"" + &c. + 2Kx-' -L\ogx + Mx + ^Naf + &c. +

Zx""^',

which is the equation of the fixed curve. If the equation of the fixed curve be given, find -^ in terms of cc, sub-

stitute r for X, and divide by r, equate the result to -t-, and integrate.

Thus, if the fixed curve be the orthogonal tractory of the straight line, whose equation is

y

=

a

log

a

+

.
\la^

—

x^

+

Ja^

dy _ Jo' — af dx~ X

THE THEORY OF ROLUNG CURVES.

15

de _ Ja?-7*

dr

r*

= cos"^

this is the equation to the orthogonal tractory of a circle whose diameter is equal to the constant tangent of the fixed curve, and its constant tangent

equal to half that of the fixed curve.

This property of the tractory of the circle may be proved geometrically,

— P CD thus Let

be the centre of a circle whose radius is PD, and let

be

BCP a line constantly equal to the radius. Let

be the curve described by

C D the point

when the point

is moved along the circumference of the circle,

CD then if tangents equal to

be drawn to the curve, their extremities will

ACH BCP OPE be in the circle. Let

be the curve on which

rolls, and let

CDE be the straight line traced by the pole, let

be the common tangent,

let it cut the circle in D, and the straight line in E.

Then CD = PD, .'. LDCP^ LDPC, and CP is perpendicular to OE, .'. L CPE= LDCP+ LDEP. Take away LDCP-^ L DPC, and there remains DPE=DEP, .-. PD=^DE, .-. CE=2PD.

16

THE THEORY OF ROLLING CURVES.

ACH Therefore the curve

haa a constant tangent equal to the diameter of

ACH the circle, therefore

is the orthogonal tractorj of the straight line, which

is the tractrix or equitangential curve.

The operation of finding the fixed curve from the rolled curve is what Sir John Leslie calls " divesting a curve of its radiated structure."

The method of finding the curve which must be rolled on a circle to trace a given curve is mentioned here because it generally leads to a double result, for the normal to the traced curve cuts the circle in two points, either of which may be a point in the rolled curve.

Thus, if the traced curve be the involute of a circle concentric with the given circle, the rolled curve is one of two similar logarithmic spirals.
If the curve traced be the spiral of Archimedes, the rolled curve may be
either the hyperbolic spiral or the straight line.

In the next case, one curve rolls on another and traces a circle.

Since the curve traced is a circle, the distance between the poles of the fixed curve and the rolled curve is always the same; therefore, if we fix the rolled curve and roll the fixed curve, the curve traced will still be a circle,
and, if we fix the poles of both the curves, we may roU them on each other
without friction.

Let a be the radius of the traced circle, then the sum or difference of the radii of the other curves is equal to a, and the angles which they make
with the radius at the point of contact are equal,

.. n-=±(a±r,)andn^^ = r,^\

dO, _ ±(a±r^ dS,

drt~ r,

dvi'

If we know the equation between ^j and r,, we may find ^— in terms of r„

substitute ± (a ± r,) for r„ multiply by ^

\ and integrate.

Thus, if the equation between 6^ and r^ be
= r, a sec $,,

—

TEU: THEORY OF ROLLING CURVES.

17

which is the polar equation of a straight line touching the traced circle whose
equation is r = ay then

dd _ a dr, ~ r, -Jr.'-a'
a {r,±a)Jr,'±2r,a

dO^ r^±a

a

dr, r, (r,±a) Jrf±2r^

a

_ 2a _ 2a

Now, since the rolling curve is a straight line, and the tracing point is

not in its direction, we may apply to this example the observations which

have been made upon tractories.

^ Let,

therefore,

the

curve ^ =

2a
7

be denoted by A, its involute by B, and

B the circle traced by C, then

is the tractory of C; therefore the involute

— of the

curve

^

=

2a
^

r

is

the

tractory

of

the

circle,

the

equation

of which

is

^ = cos"'

— /— — I. The curve whose equation is ^'=s ; seems to be among

spirals what the catenary is among curves whose equations are between rectangular co-ordinates ; for, if we represent the vertical direction by the radius vector, the tangent of the angle which the curve makes with this line is
proportional to the length of the curve reckoned from the origin ; the point at the distance a from a straight line rolled on this curve generates a circle,
and when rolled on the catenary produces a straight line ; the involute of this curve m the tractory of the circle, and that of the catenary is the tractory of the straight line, and the tractory of the circle rolled on that of the straight
line traces the straight line ; if this curve is rolled on the catenary, it produces the straight line touching the catenary at its vertex ; the method of drawing

.

18

THE THEORY OF ROLLING CURVES.

tangents is the same as in the catenary, namely, by describing a circle radius is a on the production of the radius vector, and drawing a tangent to the circle from the given point.

In the next case the rolled curve is the same as the fixed curve. It is

evident that the traced curve wiU be similar to the locus of the intersection

of the tangent with the perpendicular from the pole ; the magnitude, however,

of the traced curve will be double that of the other curve; therefore, if we

n = = call

<^o^o the equation to the fixed curve, r,

that of the traced curve,
<f>,6,

we have

also,

£^ = f.

A^ ^ SimUarly, r, = 2p, = 2r,f =

Ar,

0,^6,-2 cos- .

(^J,

^ 2^ Similarly, r„ = 2p„., = 2r„_,

&c. =

,
(^^J

and

^^f.

^„ = ^„-7lC0S-f-\
'o
0n = 6. — ncos~^ -V^

^ ^. Let e, become 6^'; 0„ 6,' and

,

Let ^„^-^„ = a,

^„^ = ^;-ncos- ^, » «. ^ a = ^„^- e„ = ^.^-^o-ncos-^ ^' +n cos-^

— = ~ \

-1
cos ^

^P^-n^

-1
COS *

P-^n—

-O-

^0

4,

^0
.

;

THE THEORY OF ROLLING CURVES.

19

— Now, cos"^

is the complement of the angle at which the curve cuts the

'n

— radius vector, and cos"' —cos"' -^ is the variation of this angle when 6^ varies

by an angle equal to a. Let this = variation (^ ; then if — = 6^ 6J fi,

^n n
Now, if n increases, will diminish ; and if n becomes infinite, <f>
^ ^ = <^ + = when a and )8 are finite.
Therefore, when n is infinite, <}> vanishes ; therefore the curve cuts the radius
vector at a constant angle ; therefore the curve is the logarithmic spiral. Therefore, if any curve be rolled on itself, and the operation repeated an
infinite number of times, the resulting curve is the logarithmic spiral Hence we may find, analytically, the curve which, being rolled on itself,
traces itself.
For the curve which has this property, if rolled on itself, and the operation repeated an infinite number of times, will still trace itself.
But, by this proposition, the resulting curve is the logarithmic spiral therefore the curve required is the logarithmic spiral. As an example of a curve rolling on itself, we will take the curve whose equation is
n=2"a(cos|)".

-1=2". (sing (oosf-;

2"a'(cos^")'"
= .'. r^ 2p,= 2 ^2-a'(cosg%2-a^(sing (cosg"^'^

2"a cos — r, = 2
^^cos-j+(sm-j

/

n\ „+i

20

THE THEORY OF ROLLING CURVES.

Now ^1-^0= -cos-^^"= -cos-' cos -" = -^,

" n+1
substituting this value of 6^ in the expression for r^,
r. = 2-'a^cos--J ,

similarly, if the operation be repeated ni times, the resulting curve is

*a\fcosn—+^m^jy

When n=l, the curve is

r = 2a cos 9,

the equation to a circle, the pole being in the circumference.

When n = 2, it is the equation to the cardioid

r = 4a (cos -J .

In order to obtain the cardioid from the circle, we roll the circle upon itself, and thus obtain it by one operation ; but there is an operation which, bei6g performed on a circle, and again on the resulting curve, will produce a cardioid, and the intermediate curve between the circle and cardioid is

=>

/

r2

20\i

As the operation of rolling a curve on itself is represented by changing n into (n + 1) in the equation, so this operation may be represented by changing n + into (w i).
Similarly there may be many other fractional operations performed upon
the curves comprehended under the equation
r = 2"a(cos-j.

We may also find the curve, which, being rolled on itself, will produce a
given curve, by making 7i= — 1.

THE THEORY OF ROLLING CURVES.

21

We may likewise prove by the same method as before, that the result of
performing this inverse operation an infinite number of times is the logarithmic
spiral.
As an example of the inverse method, let the traced line be straight, let
its equation be
= r<, 2a sec d^,
P^^p,^2a^2a_
then

therefore suppressing the suflSx,

= ar,

* • \d0j a

'

dr
r 7i-''
&-')

- 2a ^~l-cos^'
the polar equation of the parabola whose parameter is 4rt.
The last case which we shall here consider affords the means of constructing two wheels whose centres are fixed, and which shall roll on each other, so that the angle described by the first shall be a given function of the angle described by the second.

= + = — = Let 0^

then r^
(f}0i,

r^

a, and -j^

;

d0^ a-r^' Let us take as an example, the pair of wheels which will represent the angular motion of a comet in a parabola.

,

'

Here

THE THEORY OF ROLLING CURVES.

6^ = tan -^

. ^_
2 cos' -^

a + 2 cos ^1

therefore the first wheel is an ellipse, whose major axis is equal to | of the distance between the centres of the wheels, and in which the distance between the foci is half the major axis.

Now since

= ^i 2 tan"' B^ and r^ = a - r„

'•
a

1+^2(2-1 ^)'

'-'-±;'

a

which is the equation to the wheel which revolves with constant angular velocity.

Before proceeding to give a list of examples of rolling curves, we shall state a theorem which is almost self-evident after what has been shewn pre-
viously.

Let there be three curves. A, B, and C. Let the curve A, when rolled

on itself, produce the curve B, and when rolled on a straight line let it

C B produce the curve C, then, if the dimensions of

be doubled, and

be

rolled on it, it will trace a straight line.

A Collection of Examples of Rolling Curves.
First. Examples of a curve rolling on a straight line.
Ex. 1. When the rolling curve is a circle whose tracing-point is in the
circumference, the curve traced is a cycloid, and when the point is not in the
circumference, the cycloid becomes a trochoid.
Ex. 2. When the rolling curve is the involute of the circle whose radius
is 2a, the traced curve is a parabola whose parameter is 4a.

THE THEORY OF ROLLING CURVES.

23

Ex. 3. When the rolled curve is the parabola whose parameter is 4a, the
traced curv^e is a catenary whose parameter is a, and whose vertex is distant a from the straight line.
Ex. 4. "When the rolled curve is a logarithmic spiral, the pole traces a straight line which cuts the fixed line at the same angle as the spiral cuts
the radius vector.
Ex. 5. When the rolled curve is the hyperbolic spiral, the traced curve
is the tractory of the straight line.
Ex. 6. When the rolled curve is the polar catenary
r 2a

the traced curve is a circle whose radius is a, and which touches the straight
line.
Ex. 7. When the equation of the rolled curve is

the traced curve is the hyperbola whose equation is
= + y' d' a^.

Second. In the examples of a straight Hne I'olling on a curve, we shall
C use the letters A^ B, and to denote the three curves treated of in page 22.

^ B Ex. 1. When the curve

is a circle whose radius is a, then the cui-ve

C is the involute of that circle, and the curve

is the spiral of Archimedes, r = ad.

^ Ex. 2. When the curve

is a catenary whose equation is

B the curve

is the tractory of the straight line, whose equation is

X

I

= + — y a log

,

JcL' -f^,

+ - a V a' ar"

C and is a straight line at a distance a from the vertex of the catenary.

,.

24

THE THEORY OF ROLLING CURVES.

A Ex. 3. When tKe curve is the polar catenaxy

B the curve

is the tractory of the circle

and the curve (7 is a circle of which the radius is -

Third. Examples of one curve rolling on another, and tracing a straight
line.

Ex. 1.

The curve whose equation is
= Ar-"* + &c. + Kr-' + Lr'^ + Jflog r + iVr + &c. + Zt^,

when rolled on the curve whose equation is

n— 1

71+ L

traces the axis of y.
Ex. 2. The circle whose equation is r = a cos ^ rolled on the circle whose
radius is a traces a diameter of the circle.

Ex. 3. The curve whose equation is

^=J'i- 1 — versm

a

rolled on the circle whose radius is a, traces the tangent to the circle.
Ex. 4. If the fixed curve be a parabola whose parameter is 4a, and if we roll on it the spiral of Archimedes r = ad, the pole will trace the axis of the
parabola.

Ex. 5. If we roll an equal parabola on it, the focus will trace the directrix
of the first parabola.

Ex. 6. If we roll on it the curve ^ = t^ t^® P^^® "^^ ^^^^ ^^® tangent
at the vertex of the parabola.

THE THEORY OF ROLLING CURVES.

25

Ex. 7. If we roll the curve whose equation is r = a cos (t^)
on the ellipse whose equation is

the pole will trace the axis h.

K Ex. 8.

we roll the curve whose equation ia

on the hyperbola whose equation is
the pole will trace the axis h.
Ex, 9. If we roll the lituus, whose equation is

on the hyperbola whose equation is

the pole will trace the asymptote.

Ex. 10.

The cardioid whose equation is
r = a(H- cos ^),

rolled on the cycloid whose equation is

1^2

=

a

versin"'

a

+

J2ax

-

ic*,

traces the base of the cycloid.

Ex. 11. The curve whose equation is

= versm-'- + 2^/

1,

rolled on the cycloid, traces the tangent at the vertex.

26

THE THEORY OF ROLLING CURVES.

Ex. 12. The straight line whose equation is
r = a sec B,
rolled on a catenary whose parameter is a, traces a line whose distance from the vertex is a.
Ex. 13. The part of the polar catenary whose equation is

rolled on the catenary, traces the tangent at the vertex. Ex. 14. The other part of the polar catenary whose equation is

rolled on the catenary, traces a line whose distance from the vertex is equal to 2a.

Ex. 15. The tractory of the circle whose diameter is a, rolled on the tractory of the straight line whose constant tangent is a, produces the straight
line.

Ex. 16.

The hyperbolic spiral whose equation is a
'=5'

rolled on the logarithmic curve whose equation is
2/ = al1 og-^,
traces the axis of y or the asymptote.

Ex. 17. The involute of the circle whose radius is a, rolled on an orthogonal trajectory of the catenary whose equation is

traces the axis of y.
Ex. 18. The curve whose equation is

THE THEORY OF ROLLING CURVES.

27

rolled on the witch, whose equation is

traces the asymptote.
Ex. 19. The curve whose equation is
r — a tan Q,
rolled on the curve whose equation is

traces the axis of y.
Ex. 20. The curve whose equation is
e= 2r

rolled on the curve whose equation is

y= /

= , or r a tan $,

traces the axis of y.

Ex. 21. The curve whose equation is
r = a (sec d — tan 0),
rolled on the curve whose equation is

traces the axis of y.

2/ = alogg+l),

Fourth. Examples of pairs of rolling curves which have their poles at a fixed
distance = a.

straight line whose equation is
Ce
The polar catenary w.h.„ose equation is

^=sec"'2a
0, = ±fj I ±. r

Ex. 2. Two equal ellipses or hyperbolas centered at the foci.

Ex. 3. Two equal logarithmic spirals.

Ex. 4.

(Circle whose equation is Curve whose equation is

r = 2a cos 6.

— ^-/J^

l + versin"^-.

'

28

THE THEORY OF ROLLING CURVES.

Ex. 5.

fCaxdioid whose equation is [Curve whose equation is

r=2a(l+co8^).
— ^ = sin"*- + log ,— — .

Ex. 6.

(Conchoid, Icurve,

Ex. 7.

Spiral of Archimedes, Curve,

r

=

a

secg-
(

1).

>A-? ^ =

+ sec"^ -
a

r = a0.

^

=

T a

+

lo°g

T a

Ex. 8.

fHyperbolic spiral,
-!
ICurve,

whose equation is
Cpse

r=-Q a
e'+l
1
^"^^2+ ~Q'

(Involute of circle, Ex. 10.
'curve,

^~Ja^^^ ®®^"^ a e^J^±2l±log(-±l+J^.±2'^.

Fifth. Examples of curves rolling on themselves.

When Ex. 1.

the curve which rolls on itself is a circle, equation

r = a cos 6,

the traced curve is a cardioid, equation r = a(l+cos^).

Ex. 2. When it is the curve whose equation is

r = 2"a (cos-j ,

the equation of the traced curve is

When Ex. 3.

it is the involute of the circle, the traced curve is the spiral

of Archimedes.

THE THEORY OF ROLLING CURVES.

29

Ex. 4. When it is a parabola, the focus traces the directrix, and the vertex
traces the cissoid.

When Ex. 5.

it is the hyperbolic spiral, the traced curve is the tractory of

the circle.

Ex. 6.

When it is the polar catenary, the equation of the traced curve is

J — 2a

,
1

r . ., versin - .

r

a

Ex. 7. When it is the curve whose equation is

the equation of the traced curve is — = r a (e' €~").

This paper commenced with an outline of the nature and history of the problem of rolling curves, and it was shewn that the subject had been discussed previously, by several geometers,
amongst whom were De la Hire and Nicolfe in the Memoires de I'Academie, Euler, Professor
Willis, in his Principles of Mechanism, and the Rev. H. Holditch in the Cambridge Philosophical
Transactions.
None of these authors, however, except the two last, had made any application of their methods ; and the principal object of the present communication was to find how far the general equations could be simplified in particular cases, and to apply the results to practice.
Several problems were then worked out, of which some were applicable to the generation of curves, and some to wheelwork ; while others were interesting as shewing the relations which exist between different curves ; and, finally, a collection of examples was added, as an illustration of the fertihty of the methods employed.

—

—

[From the Transactions of the Royal Society of Edinburgh, Vol. XX. Part i,]
III. On the Equilibrium of Elastic Solids.
There are few parts of mechanics in which theory has differed more from
experiment than in the theory of elastic sohds. Mathematicians, setting out from very plausible assumptions with respect to
the constitution of bodies, and the laws of molecular action, came to conclusions which were shewn to be erroneous by the observations of experimental philosophers. The experiments of (Ersted proved to be at variance with the mathe-
matical theories of Navier, Poisson, and Lame and Clapeyron, and apparently
deprived this practically important branch of mechanics of all assistance from mathematics.
The assumption on which these theories were founded may be stated thus :
Solid bodies are composed of distinct ^molecules, which are kept at a certain
distance from each other by the opposing principles of attraction and heat. When
the distance between two molecules is changed, they act on each other with a force whose direction is in the line joining the centres of the molecules, and whose magnitude is equal to the change of distance multiplied into a function of the distance which vanishes when that distance becomes sensible.
The equations of elasticity deduced from this assumption contain only one coefficient, which varies with the nature of the substance.
The insufficiency of one coefficient may be proved from the existence of
bodies of different degrees of solidity.
No effort is required to retain a liquid in any form, if its volume remain unchanged; but when the form of a solid is changed, a force is called into
action which tends to restore its former figure ; and this constitutes the differ-

— ;

THE EQUILIBRITJM OF ELASTIC SOLIDS.

31

ence between elastic solids and fluids. Both tend to recover their vohirne, but fluids do not tend to recover their shape.
Now, since there are in nature bodies which are in every intermediate state from perfect soHdity to perfect liquidity, these two elastic powers cannot exist in every body in the same proportion, and therefore all theories which assign to them an invariable ratio must be erroneous.
I have therefore substituted for the assumption of Navier the following axioms as the results of experiments.
If three pressures in three rectangular axes be applied at a point in an
elastic solid,
1. TTie sum of the three pressures is proportional to the sum of the com-
pressions ichich they produce.
2. The difference between two of the pressures is propo7'tional to the difference of the compressions which they produce.
The equations deduced from these axioms contain two coefficients, and differ from those of Navier only in not assuming any invariable ratio between the cubical and linear elasticity. They are the same as those obtained by Professor Stokes from his equations of fluid motion, and they agree with all the laws of elasticity which have been deduced from experiments.
In this paper pressures are expressed by the number of units of weight to the unit of surface ; if in English measure, in pounds to the square inch, or in atmospheres of 15 pounds to the square inch.
Compression is the proportional change of any dimension of the solid caused by pressure, and is expressed by the quotient of the change of dimension divided by the dimension compressed'".
Pressure will be understood to include tension, and compression dilatation pressure and compression being reckoned positive.
Elasticity is the force which opposes pressure, and the equations of elasticity are those which express the relation of pressure to compression f.
Of those who have treated of elastic solids, some have confined themselves to the investigation of the laws of the bending and twisting of rods, without
* The laws of pressure and compression may be found in the Memoir of Lam6 and Clapeyrou. St^t-
note A. t See note B.

32

THE EQUIUBRIUM OF ELASTIC SOLIDS.

considering the relation of the coefficients which occur in these two cases; while others have treated of the general problem of a solid body exposed to any forces.
The investigations of Leibnitz, Bernoulli, Euler, Varignon, Young, La Hire, and Lagrange, are confined to the equilibrium of bent rods; but those of Navier, Poisson, Lam^ and Clapeyron, Cauchy, Stokes, and Wertheim, are principally directed to the formation and application of the general equations.
The investigations of Navier are contained in the seventh volume of the Memoirs of the Institute, page 373; and in the AnnoUes de Chimie et de Physique, 2^ Sdrie, xv. 264, and xxxviii. 435 ; L'AppUcati(m de la Micanique, Tom. I.
Those of Poisson in Mem. de I'lnstitut, vm. 429 ; Annales de Chimie, 2" S^rie, XXXVI, 334 ; xxxvii. 337 ; xxxvtil 338 ; xlu. Journal de VEcole
Polytechnique, cahier xx., with an abstract in Annales de Chimie for 1829.
The memoir of MM. Lam^ and Clapeyron is contained in Crelle's Mathe-
matical Journal, Vol. vii. ; and some observations on elasticity are to be found in Lamp's Cours de Physique,

M. Cauchy's investigations are contained in his Exercices d!Analyse, Vol. in.
p. 180, published in 1828.

Instead of supposing each pressure proportional to the linear compression

which it produces, he supposes it to consist of two parts, one of which is pro-

portional to the linear compression in the direction of the pressure, while the

other is proportional to the diminution of volume. As this hypothesis admits

two coefficients, it differs from that of this paper only in the values of the

K coefficients selected. They are denoted by

and h, and K^fi — ^m, k = m.

The theory of Professor Stokes is contained in Vol. vin. Part 3, of the Cambridge Philosophical Transactions, and was read April 14, 1845.
— He states his general principles thus : " The capability which solids possess
of being put into a state of isochronous vibration, shews that the pressures called into action by small displacements depend on homogeneous functions of those displacements of one dimension. I shall suppose, moreover, according to the general principle of the superposition of small quantities, that the pressures due to different displacements are superimposed, and, consequently, that the
pressures are linear functions of the displacements."

THE EQUILIBRIUM OF ELASTIC SOLIDS.

33

Having assumed the proportionality of pressure to compression, he proceeds

-^8 to define his coefficients.— "Let

be the pressures corresponding to a uniform

linear dilatation 8 when the solid is in equilibrium, and suppose that it becomes

mA8, in consequence of the heat developed when the solid is in a state of rapid

vibration. Suppose, also, that a displacement of shifting parallel to the plane
xy, for which 8x = kx, Sy= - hj, and hz = 0, calls into action a pressure - Bk

on a plane perpendicular to the axis of x, and a pressure Bk on a plane

perpendicular to the axis of y; the pressure on these planes being equal and

of contrary signs; that on a plane perpendicular to z being zero, and the tan-

A gential forces on those planes being zero." The coefficients

and B, thus

^ B defined, when expressed as in this paper, are = 3/x,, = -.

Professor Stokes does not enter into the solution of his equations, but gives

their results in some particular cases.

A 1.

body exposed to a uniform pressure on its whole surface.

A 2.

rod extended in the direction of its length.

A 3.

cylinder twisted by a statical couple.

A B He then points out the method of finding

and

from the last two cases.

While explaining why the equations of motion of the luminiferous ether are
the same as those of incompressible elastic solids, he has mentioned the property of jylasticity or the tendency which a constrained body has to relieve itself from a state of constraint, by its molecules assuming new positions of equilibrium. This property is opposed to Hnear elasticity ; and these two properties

exist in all bodies, but in variable ratio.

M. Wertheim, in Annales de Chimie, 3« Sdrie, xxiii., has given the results

of some experiments on caoutchouc, from which he finds that K=k, or fi = ^m;

K and concludes that k =

in all substances. In his equations, fi is therefore

made equal to f m. The accounts of experimental researches on the values of the coefficients
are so numerous that I can mention only a few.
Canton, Perkins, (Ersted. Aime, CoUadon and Sturm, and Regnault, have determined the cubical compressibilities of substances; Coulomb, Duleau, and Giulio, have calculated the linear elasticity from the torsion of wires; and a great many observations have been made on the elongation and bending of beams.
^ VOL. I.

34

THE EQUILIBRIUM OF ELASTIC SOLIDS.

I have found no account of any experiments on the relation between the doubly refracting power communicated to glass and other elastic solids by compression, and the pressure which produces it^^" ; but the phenomena of bent glass seem to prove, that, in homogeneous singly-refracting substances exposed to pressures, the principal axes of pressure coincide with the principal axes of double refraction ; and that the diflference of pressures in any two axes is proportional to the difference of the velocities of the oppositely polarised rays
whose directions are parallel to the third axis. On this principle I have
calculated the phenomena seen by polarised light in the cases where the solid is bounded by parallel planes.

In the following pages I have endeavoured to apply a theory identical with that of Stokes to the solution of problems which have been selected on account of the possibility of fulfilling the conditions. I have not attempted to extend the theory to the case of imperfectly elastic bodies, or to the laws of permanent bending and breaking. The solids here considered are supposed not to be compressed beyond the limits of perfect elasticity.

The equations employed in the transformation of co-ordinates may be found
in Gregory's Solid Geometry.

I have denoted the displacements by Zx, By, Bz. They are generally denoted

by

y a, /8,

;

but as I had employed these letters

to denote

the principal axes

at any point, and as this had been done throughout the paper, I did not alter

a notation which to me appears natural and intelligible.

The laws of elasticity express the relation between the changes of the dimensions of a body and the forces which produce them.
These forces are called Pressures, and their effects Compressions. Pressures are estimated in pounds on the square inch, and compressions in fractions of the dimensions compressed.
Let the position of material points in space be expressed by their co-ordinates X, y, and z, then any change in a system of such points is expressed by giving to these co-ordinates the variations Bx, By, Bz, these variations being functions of
X, y, 2.
* See note C.

—

THE EQUILIBRIUM OF ELASTIC SOLIDS.

35

The quantities Sx, Sy, 8z, represent the absolute motion of each point in the directions of the three co-ordinates ; but as compression depends not on absolute, but on relative displacement, we have to consider only the nine
quantities
dSx

—

36

THE EQUILIBRIUM OF ELASTIC SOLIDS.

By resolving the displacements 8a, h/S, By, B6„ B9.„ Z6„ in the directions

of the axes x, y, z, the displacements in these axes are found to be

hx = a,8a + h,Bp + c3y -Be^ + Bd,y,

aM By =

+ - + h,Bl3 -f c,By Bd,x Bd.z,

hM Bz = a,Ba +

+ CsBy - BO^ + Bd,x.

But

Sa

.^ ^Si8
B^^rf,

and 8y = y^,

and

+ + + = + = + a^ = Q. a^x

h^ a.^, /3 b,x

h.^, and y c,x c,y -h c^z.

Substituting these values of Sa, Sy8, and By in the expressions for Bx, By, Bz, and differentiating with respect to x, y, and z, in each equation, we obtain

the equations

dBx Ba, ,. 8/8,2 ^y ,

dy a

^

y

dBz _ Ba

dz a

p

y

dBx Ba dy a '

B^ T

J

By

^

y

,5s/,

dBx Ba

BI3

Be,

dz a

J'

Ba dz a
dBy Ba dx a

8^ p BB T ^
p

+ c.f^ Bdi
y
By
Be,
y

— dZz
-dJx—

=

8^ a

ctjCti

+

-8^^

6361

+

-^ r

C3C1

+

8^2

dBz Sa

S/8

Be,

(1)-
Equations of compression.
{2).

Equations of the equilibnum of an element of the solid. The forces which may act on a particle of the solid are : 1. Three attractions in the direction of the axes, represented by X, Y, Z. 2. Six pressures on the six faces.

—

THE EQUILIBRIUM OF ELASTIC SOLIDS.

37

3. Two tangential actions on each face.

Let the six faces of the small parallelopiped be denoted by x^, 3/,, z„ x^ y„ and z,, then the forces acting on x^ are :

A 1.

normal pressure jp, acting in the direction of x on the area dydz,

A 2.

tangential force g, acting in the direction of y on the same area.

A 3.

tangential force q^ acting in the direction of z on the same area,

and so on for the other five faces, thus :

Forces which act in the direction of the axes of

a;

2/

z

On the face a:,

38

THE EQUILIBRIUM OF ELASTIC SOLIDS.

The resistance which the sohd opposes to these pressures is called Elasticity, and is of two kinds, for it opposes either change of volume or change of Jigure. These two kinds of elasticity have no necessary connection, for they are possessed in very different ratios by different substances. Thus jelly has a cubical elasticity little different from that of water, and a linear elasticity as small as we please ; while cork, whose cubical elasticity is very small, has a much greater Imear elasticity than jelly.

Hooke discovered that the elastic forces are proportional to the changes that excite them, or as he expressed it, " Ut tensio sic vLs."

To fix our ideas, let us suppose the compressed body to be a parallelepiped,

and let pressures Pi, Pj, P3 act on its faces in the direction of the axes

A a>

y, which will become the principal axes of compression, and the com-

So. 8^ Sy
pressions will be a' ^' y

The fundamental assumption from which the following equations are deduced is an extension of Hooke's law, and consists of two parts.

I. The sum of the compressions is proportional to the sum of the pressures.

II. The difference of the compressions is proportional to the difference of
the pressures.

These laws are expressed by the following equations

f ^ I. (P. + P, + P.) = 3,(^ + +

(4).

rv
(P,-P,) = m

^rts T Equations of Elasticity.

„,g_^ II.

(P._p.) =

h

(5).

7

(P.-P,) = m By Ba

The quantity

m is the coefiicient of cubical elasticity, and

that of linear

fj.

elasticity.

'

;

THE EQUILrBRIUM OF ELASTIC SOLmS.

39

By solving these equations, the values of the pressures P„ P,, P„ and the

—8a 8^

compressions

' ~S

Sy

^^7

,
^^

rfoundJ.

(6).

a \9/x 3m/ ^

^m

! (!_ M(p. + + lp, =

j3

\9/x

3 m/ ^ * P, + p.^)

?7i

'

(7).

?r y

=

(_L_
\9/z

i\(P_+P_+P_)

3m/ ^

^

+

ip_
m

From these values of the pressures in the axes a, )8, y, may be obtained.. the equations for the axes x, y, z, by resolutions of pressures and compressions*.

For

and

q = aaP^ + hhP, + ccP,

,

. . IdZx d%y d8z\

d8x'

,

,

.

,

. V IdZx d8y d8z\

.

,

dBy

,

, , fdSx d8y rfSj\

dSz

,

,

,

(8)-

m /c?Sz
2 Vo?a;

c?Sx
c?2

See the Memoir of Lame and Clapeyron, and note A.

.(9).

40

THE EQUIUBRIUM OP ELASTIC SOLIDS.

d$X /I

1\,

,

,

N, 1

(10).

dy

* ax

' m^

dz

dy

m^

d^

dx

dz

m^

(11).

By substituting in Equations (3) the values of the forces given in Equa-
tions (8) and (9), they become

(12).

These are the general equations of elasticity, and are identical with those of M. Cauchy, in his Exercices d'Analyse, Vol. ni., p. 180, published in 1828,

K where h stands for m, and

for

-

ft

o" >

and

those

of Mr

Stokes,

given in the

Cambridge Philosophical Transactions, Vol. viii., part 3, and numbered (30);

^ B — = in his equations = 3/x,

.

If the temperature is variable from one part to another of the elastic soHd, the compressions -y- , -r^, -J^ , at any point will be diminished by a quantity proportional to the temperature at that point. This prmciple is applied in Cases X. and XI. Equations (10) then become

THE EQUILIBRIUM OF ELASTIC SOLIDS.

—
41

(13).
dy

^ = fe - 3mj (P^-^P^+P^) + '^^^^P^
CfV being the linear expansion for the temperature v.
Having found the general equations of the equilibrium of elastic solids, I proceed to work some examples of their application, which afford the means of determining the coefficients /t, m, and o), and of calculating the stiffness of solid figures. I begin with those cases in which the elastic soHd is a hollow cylinder exposed to given forces on the two concentric cylindric surfaces, and the two parallel terminating planes.

In these cases the co-ordinates x, y, z are replaced by the co-ordinates

x = x, measured along the axis of the cylinder.

= 2/ r, the radius of any point, or the distance from the axis.

z — rd, the arc of a circle measured from a fixed plane passing

through the axis.

dZx dx

dSx dx

Px = o, are the compression and pressure in the direction of the
axis at any point.

-^ = -J— , Pi =p, are the compression and pressure in the direction of the

radius.

m dBz dhrd Br
~dz~'db¥~l^'

= JP8 ?,

are

.
the

.

_

compression and

pressure

.

,

,.

.

-1

the direction of the

tangent.

Equations (9) become, when expressed in terms of these co-ordinates
m doO

m dB0

.(14).

m dSx *=2 dr

The length of the cylinder is h, and the two radii a, and a, in every

VOL. I.

G

,

42

THE EQUIUBRnJM OF ELASTIC SOLIDS.

Case I.

The first equation is applicable to the case of a hollow cylinder, of which

the outer surface is fixed, while the inner surface is made to turn through

a small angle Bd, by a couple whose moment is M.

M The twisting force

is resisted only by the elasticity of the solid, and

therefore the whole resistance, in every concentric cylindric surface, must be equal

to M.

The resistance at any point, multiplied into the radius at which it acts, is

expressed by

m „ dhd

Therefore for the whole cylindric surface

ar

Whence

8,=_^^ (1,_1.)

^^

"' = 2^&-i)

The optical effect of the pressure of any point is expressed by

I=<oq,b = <o.^^

('«>
(15).

Therefore, if the solid be viewed by polarized light (transmitted parallel to the axis), the difference of retardation of the oppositely polarized rays at any point in the solid will be inversely proportional to the square of the distance fi-om the axis of the cylinder, and the planes of polarization of these lays will be inclined 45" to the radius at that point.
The general appearance is therefore a system of coloured rings arranged
oppositely to the rings in uniaxal crystals, the tints ascending in the scale as they approach the centre, and the distance between the rings decreasing towards
the centre. The whole system is crossed by two dark bands inclined 45* to the plane of primitive polarization, when the plane of the analysing plate is perpen-
dicular to that of the first polarizing plate.

THE EQUILIBRIUM OF ELASTIC SOLIDS.

43

A jelly of isinglass poured when hot between two concentric cylinders forms,
when cold, a convenient solid for this experiment ; and the diameters of the rings may be varied at pleasure by changing the force of torsion appUed to the interior
cylinder.
By continuing the force of torsion while the jeUy is allowed to dry, a hard plate of isinglass is obtained, which still acts in the same way on polarized light, even when the force of torsion is removed.
It seems that this action cannot be accounted for by supposing the interior parts kept in a state of constraint by the exterior parts, as in, unannealed and heated gla^s ; for the optical properties of the plate of isinglass are such as would indicate a strain preserving in every part of the plate the direction of the original strain, so that the strain on one part of the plate cannot be maintained by an opposite strain on another part.
Two other uncrystallised substances have the power of retaining the polarizing structure developed by compression. The first is a mixture of wax and resin
pressed into a thin plate between two plates of glass, as described by Sir David Brewster, in the Philosophical TransoLctions for 1815 and 1830.
When a compressed plate of this substance is examined with polarized light,
it is observed to have no action on light at a perpendicular incidence ; but when inclined, it shews the segments of coloured rings. This property does not belong to the plate as a whole, but is possessed by every part of it. It is therefore
similar to a plate cut from a uniaxal crystal perpendicular to the axis.
I find that its action on light is like that of a jpositive crystal, while that of a plate of isinglass similarly treated would be negative.
The other substance which possesses similar properties is gutta percha. This substance in its ordinary state, when cold, is not transparent even in thin films; but if a thin film be drawn out gradually, it may be extended to more than double its length. It then possesses a powerful double refraction, which it retains so strongly that it has been used for polarizing light""'. As one of its refractive indices is nearly the same as that of Canada balsam, while the other is very different, the common surface of the gutta percha and Canada balsam will transmit one set of rays much more readdy than the other, so that a film of extended gutta percha placed between two layers of Canada balsam acts like

* By Dr Wright, I believe.

44

THE EQUILIBRIUM OF ELASTIC SOLIDS.

a plate of nitre treated in the same way. That these films are in a state of constraint may be proved by heating them slightly, when they recover their
original dimensions.
As all these permanently compressed substances have passed their limit of
perfect elasticity, they do not belong to the class of elastic solids treated of in this paper ; and as I cannot explain the method by which an imcrystallised body maintains itself in a state of constraint, I go on to the next case of twisting, which has more practical importance than any other. This is the case of a
cylinder fixed at one end, and twisted at the other by a couple whose moment is M.

Case II.

In this case let hB be the angle of torsion at any point, then the resistance to torsion in any circular section of the cylinder is equal to the twisting force M,
The resistance at any point in the circular section is given by the second Equation of (14).
?2 = 1^^ dx '
This force acts at the distance r from the axis ; therefore its resistance to torsion will be q.r, and the resistance in a circular annulus will be
q^r^Ttrdr = mirr' -r- dr

and the whole resistance for the hollow cylinder will be expressed by

„, mn dS6 , ^

,.

/,^v

720 M^(-1-]

(17).

m In this equation,

is the coefl&cient of linear elasticity; a^ and a^ are the

M radii of the exterior and interior surfaces of the hollow cyUnder in inches ;

is

the moment of torsion produced by a weight acting on a lever, and is expressed

'

THE EQUILIBRIUM OF ELASTIC SOLIDS.

45

bj the product of the number of pounds in the weight into the number of inches in the lever; b is the distance of two points on the cylinder whose angular motion is measured by means of indices, or more accurately by small mirrors attached to the cylinder ; n is the difference of the angle of rotation of the two
indices in degrees.
m This is the most accurate method for the determination of independently
of /x, and it seems to answer best with thick cylinders which cannot be used with the balance of torsion, as the oscillations are too short, and produce a vibration of the whole apparatus.

Case III.

A hollow cylinder exposed to normal pressures only. When the pressures
parallel to the axis, radius, and tangent are substituted for p^, p^, and pt, Equations (10) become

S ^ = (i-34)(^+^-^^) +

(^«)-

^^t^(±-±]io+p + q) + :^q

(20).

By multiplying Equation (20) by r, differentiating with respect to r, and
comparing this value of —j— with that of Equation (19),

^ ^ p-q rm

_"(J__ \9/x

_1\ 3m/

/^
\dr

.

dr

.

^\ _ i
drj m dr

The equation of the equilibrium of an element of the solid is obtained by
considering the forces which act on it in the direction of the radius. By equating the forces which press it outwards with those pressing it rnwarde, we
find the equation of the equiHbrium of the element,

ir£ = 4

r

dr

(21).

46

THE EQUILIBRIUM OF ELASTIC SOLIDS.

By comparing this equation witli the last, we find

Integrating,

\9fi Zmj dr \9/i ^ 3m/ \dr ^ drj

Since o, the longitudinal pressure, is supposed constant, we may assume

Therefore

c -(^-^]o

12 '
= c.

\9u,

3m/

.

.

,

=(^ + g)-

9/x,

3m

q—p = c^ — 2p, therefore by (21),

a linear equation, which gives

1 ^c,
^ = ^3^ + 2-

The coefficients Cj and Cj must be found from the conditions of the surface of the soHd. If the pressure on the exterior cylindric surface whose radius is a, be denoted by A,, and that on the interior surface whose radius is a^ by A,,

then p = h^ when r = ai and p = h.j when r = a^

and the general value of p is _a^h^ — a^\
^" a,' -a,'

^a^a^ — h^ h^ oT^^

/22\

^

^'

2-i'=2i^ ^73^- ''y (21).

*= «.'-«.' +^^57::^'

(^^^

/=5<.(^-2)=-26<.^"A^.

(24).

This last equation gives the optical eflfect of the pressure at any point. The law of the magnitude of this quantity is the inverse square of the radius, as in

—

THE EQUILIBRIUM OF ELASTIC SOLIDS.

47

Case I. ; but the direction of the principal axes ia different, as in this case they are parallel and perpendicular to the radius. The dark bands seen by polarized Ught wiU therefore be parallel and perpendicular to the plane of polarisation, instead of being inclined at an angle of 45", as in Case I.

By substituting in Equations (18) and (20), the values of p and q given in (22) and (23), we find that when r = a,.

hx

(l\(

^aX-ct'h-X 2 / .

a,%-a,%\ ]

X \9/x

1/1 = o(^ + ~] + 2{Ka,^-Ka,^)

1

3m/ ,9/x

'

^''

' 'Ui,'-a,'\9fj, 3mJ

.(25).

- ^ ^4-^) When r = a.,

^ fo4-2

r

9/x \

—+

-

a/ a/ / 3^?^^ (

^._^.

-o '
'

(26).

- ^ 3m - ~ VSft 3my "^ ' a; a,' \ 9/x

/

^ cv

3m/ "^

a,' 1,9/x

J

From these equations it appears that the longitudinal compression of cylin-
dric tubes is proportional to the longitudinal pressure referred to unit of surface
when the lateral pressures are constant, so that for a given pressure the com-
pression is inversely as the sectional area of the tube.

These equations may be simplified in the following cases :

1. When the external and internal pressures are equal, or = h^ h^.

2. When the external pressure is to the internal pressure as the square of tlie interior diameter is to that of the exterior diameter, or when = a^-h^ a^-h^.

3. When the cylinder is soHd, or when = a. 0.

4. When the solid becomes an indefinitely extended plate with a cylindric
hole in it, or when a^ becomes infinite.

When 5.

pressure is applied only at the plane surfaces of the solid cylinder,

and the cylindric surface is prevented from expanding by being inclosed in a

strong case, or when — = 0.

6. When pressure is applied to the cylindric surface, and the ends are

— retained at an invariable distance, or when

= 0.

X

48

THE EQUILIBRIUM OF ELASTIC SOLIDS.

1. When = ^ji A„ the equations of compression become

3m \9fi'*"3mj"'"^ '\9ij.
7 = i('>+2^) + 3i(^-<')

(27).

When hi = hi = o, then

Zx X

_hr ~r

_"

\
Sfi'

The compression of a cylindrical vessel exposed on all sides to the same hydrostatic pressure is therefore independent of m, and it may be shewn that the same is true for a vessel of any shape.

2. When a,% = a^%

Bx X ^ \9yx "^ 3m/

|w 7 =

+ 3l(3^--»)^

(28).

In this case, when o = 0, the compressions are independent of /x. = 3. In a solid cylinder, aj 0,

— — The expressions for

and

are the same as those in the first case, when

h^ — hf

When the lon^tudinal pressure o vanishes,

Bx
X

'
r ' \9/x 3m/

'
THE EQUILIBRIUM OF ELASTIC SOLIDS.
When the cylinder ia pressed on the plane sides only,
8x

r

\9fi dmj

When 4.

the solid is infinite, or when a, is infinite,

p = K--._a-(\-K)

I=<o{p-q)=-^a.;{h,-h,)

49
(29).

r

9/x ^

' 3m ^

'

5. When 8r = in a solid cylinder,

6. When

Zx

Zo

X 2m + 3/A

_ _ So;

hr

2>h

x~ m ~ + * r

6iM

Since the expression for the efiect of a longitudinal strain is

B-x=o(— + —)

X

\9/i, 3m/

if we make
VOL. I.

E ^ — = = r,

9mu,

^,

8x

1

, then

o ^^

m E + 6/x

cc

.(30). (31).

50

THE EQUILIBRIUM OF ELASTIC SOLIDS.

E The quantity may be deduced from experiment on the extension of wires

m E or rods of the substance, and /x is given in terms of

and

by the equation,

„ = _^!!L_

(32),

^^^

^=S

(^^)'

P being the extending force, h the length of the rod, s the sectional area,
and Bx the elongation, which may be determined by the deflection of a wire,
as in the apparatus of S' Gravesande, or by direct measurement.

Case IV.
The only known direct method of finding the compressibihty of liquids is that employed by Canton, (Ersted, Perkins, Aime, &c.
The liquid is confined in a vessel with a narrow neck, then pressure is applied, and the descent of the liquid in the tube is observed, so that the difference between the change of volume of liquid and the change of internal capacity of the vessel may be determined.
Now, since the substance of which the vessel is formed is compressible, a change of the internal capacity is possible. If the pressure be applied only to the contained liquid, it is evident that the vessel will be distended, and the compressibihty of the liquid will appear too great. The pressure, therefore, is commonly applied externally and internally at the same time, by means of a hydrostatic pressure produced by water compressed either in a strong vessel or
in the depths of the sea.
As it does not necessarily follow, from the equality of the external and
internal pressures, that the capacity does not change, the equilibrium of the vessel must be determined theoretically. (Ersted, therefore, obtained from Poisson his solution of the problem, and applied it to the case of a vessel of lead. To find the cubical elasticity of lead, he appUed the theory of Poisson to the
numerical results of Tredgold. As the compressibility of lead thus found was
greater than that of water, (Ersted expected that the apparent compressibility
of water in a lead vessel would be negative. On making the experiment the
apparent compressibihty was greater in lead than in glass. The quantity found

——

THE EQUILIBRrcrM OF ELASTIC SOLIDS.

51

by Tredgold from the extension of rods was that denoted by E, and the value

E of ft deduced from

alone by the formulae of Poisson cannot be true, unless

— = |-; and as — for lead is probably more than 3, the calculated compressi-

bility is much too great.
A similar experiment was made by Professor Forbes, who used a vessel of
caoutchouc. As in this case the apparent compressibility vanishes, it appears
that the cubical compressibihty of caoutchouc is equal to that of water.
Some who reject the mathematical theories as unsatisfactory, have conjec-
tured that if the sides of the vessel be sufficiently thin, the pressure on both sides being equal, the compressibility of the vessel will not affect the result. The following calculations shew that the apparent compressibility of the liquid depends on the compressibility of the vessel, and is independent of the thickness
when the pressures are equal.
A hollow sphere, whose external and internal radii are a^ and a,, is acted
on by external and internal normal pressures h^ and K, it is required to determine the equilibrium of the elastic solid.

The pressures at any point in the solid are :

A 1.

pressure p in the direction of the radius.

A 2.

pressure q in the perpendicular plane.

These pressures depend on the distance from the centre, which is denoted by r.

— The compressions at any point are -.— in the radial direction, and

in

the tangent plane, the values of these compressions are :

fr=[h-^^P^''i)*h^

('")•

T = fe-3fJ(^ + 2,) + l5

(35).

Multiplying the last equation by r, differentiating with respect to r, and equating the result with that of the first equation, we find

52

THE EQUILIBRITTM OF ELASTIC SOLIDS.

Since the forces whicli act on the particle in the direction of the radius must balance one another, or
2qdrde +p (rdey =(^p + ^d7^(r + dry 6,

therefore

_r dp ^""-^ = 2 37

^^^^'

-p Substituting this value of q

in the preceding equation, and reducing,

therefore Integrating,

^ + 2^ = 0.
dr dr
p-\-2q = c,.

But

r dp ,

and the equation becomes

dp + 3^-^-i = 0,
dr

therefore

1 c.

K Since p = h, when = r a.,, and p = when r = a,, the value of p at any
distance is found to be

^~ a^-af r' a^-a,'

(37).

9-

a,'-ai

"^^
7^

<-a/

.(38).

When r = a„ -y = -^r:^^ - + t ^^ ^^737^3 ^

U ~ a,' - a/

- 2»i/

a/ «/ \jx

2wi/ _

When the external and internal pressures are equal
K SV
h^ = h.,=p = q, and -y-

.(39). .(40),

—

THE EQUILIBRIUM OF ELASTIC SOLIDS.

53

the change of internal capacity depends entirely on the cubical elasticity of the vessel, and not on its thickness or linear elasticity.
When the external and internal pressures are inversely as the cubes of the
radii of the surfaces on which they act,
aX = a,%, p = ^ K q= -i^K

(41).

when = r r-

—
V

^

'

2 ^^

In this case the change of capacity depends on the linear elasticity alone.

M. Regnault, in his researches on the theory of the steam engine, has given an account of the experiments which he made in order to determine with accuracy the compressibility of mercury.

He considers the mathematical formulae very uncertain, because the theories
of molecular forces from which they are deduced are probably far from the truth ; and even were the equations free from error, there would be much uncertainty in the ordinary method by measuring the elongation of a rod of the substance, for it is diflScult to ensure that the material of the rod is the same as that of the hollow sphere.

He has, .therefore, availed himself of the results of M. Lam6 for a hollow
sphere in three different cases, in the first of which the pressure acts on the interior and exterior surface at the same time, while in the other two cases the pressure is applied to the exterior or interior surface alone. Equation (39) becomes in these cases,

— 1. When = ^1 /ij, -^ = and the compressibility of the enclosed liquid being

/x,, and the apparent diminution of volume S'F,

v-.£-;)

«

When 2.

= /i, 0,

;

54

THE EQUILIBRIUM OF ELASTIC SOLIDS.

3. When h,^0,

K 8V_ h

9^\

,

m V a^-a^ \ii ^

^'

V2 J

M. Lamp's equations differ from these only in assuming that = fi, |-m. If

this assumption be correct, then the coefficients /u,, m, and jMj, may be found

from two of these equations ; but since one of these equations may be derived

from the other two, the three coefficients cannot be found when /u, is supposed

independent of m. In Equations (39), the quantities which may be varied at

\ pleasure are

and h^, and the quantities which may be deduced from the

apparent compressions are,

'=G+4)^°<^S-i)=^"

therefore some independent equation between these quantities must be found, and this cannot be done by means of the sphere alone; some other experiment must be made on the liquid, or on another portion of the substance of which

the vessel is made.

The value of /x^, the elasticity of the liquid, may be previously known.

m The linear elasticity

of the vessel may be found by twisting a rod of

the material of which it is made

E Or, the value of

may be found by the elongation or bending of the

audi:
,

We have here five quantities, which may be determined by experiment.

on sphere.

THE EQUILIBRIUM OF ELASTIC SOLIDS.

55

When the elastic sphere is solid, the internal radius a, vanishes, and fh=p = q, and -y = ^-
When the case becomes that of a spherical cavity in an infinite solid, the
external radius a^ becomes infinite, and
P=K-f{K-K)

r-

—

66

THE EQUILIBRIUM OF ELASTIC SOLIDS.

Let a rectangular elastic beam, whose length is 2irc, be bent into a circular form, so as to be a section of a hollow cylinder, those parts of the beam which lie towards the centre of the circle will be longitudinally compressed, while the opposite parts will be extended.

The expression for the tangential compression is therefore

Br _ r — c

~ r

c'

—Sr

Comparing this value of

with that of Equation (20),

r

V=(^-4)<''+-p+«)+^'''

and by (21),

q=p + r
dr

^) + ,,.

,

/I

By substituting for q its value, and dividing by r (q-

2\ .,
the equat:ion
•

becomes

— — 2m + _ dp

{m 3/x j9 9?n/i.

3/x) o

9m/x c

m dr

r~ + 6fx

(m + 6fi) r

+ (m * 6/x) r'

a linear differential equation, which gives

^

^

m — 3fir

2m + 3/x

Ci may be found by assumiQg that when r^a^, p = \, and q may be found

from p by equation (21).

As the expressions thus found are long and cumbrous, it is better to use

the following approximations :
_/_9m^\ y

()

l^\llcl^ \ (48).

In these expressions a is half the depth of the beam, and y is the distance of any part of the beam from the neutral surface, which in this case is a cylindric surface, whose radius is c.
These expressions suppose c to be large compared with a, since most sub-
stances break when - exceeds a certain small quantity.

THE EQUILIBRIUM OF ELASTIC SOLIDS.

57

Let b be the
M = resists flexure

breadth of the beam, then the force
M=lhyq = ^^^-^ = Ef

with

which

the beam
(49),

which is the ordinary expression for the stiffness of a rectangular beam. The' stiffness of a beam of any section, the form of which is expressed by
an equation between x and y, the axis of x being perpendicular to the plane of flexure, or the osculating plane of the axis of the beam at any point, is expressed by

Mc = E{ifdx

(50),

M being the moment of the force which bends the beam, and c the radius of

the circle into which it is bent.

Case YI.
At the meeting of the British Association in 1839, Mr James Nasmyth
described his method of making concave specula of silvered glass by bending.
A circular piece of silvered plate-glass was cemented to the opening of an
iron vessel, from which the air was afterwards exhausted. The mirror then became concave, and the focal distance depended on the pressure of the air.
Buffon proposed to make burning- mirrors in this way, and to produce the partial vacuum by the combustion of the air in the vessel, which was to be effected by igniting sulphur in the interior of the vessel by means of a burning-glass. Although sulphur evidently would not answer for this purpose, phosphorus might; but the simplest way of removing the air is by means of the air-pump. The mirrors which were actually made by Buffon, were bent by means of a screw acting on the centre of the glass.
To find an expression for the curvature produced in a flat, circular, elastic plate, by the difference of the hydrostatic pressures which act on each side of it,—
Let t be the thickness of the plate, which must be small compared with
its diameter. Let the form of the middle surface of the plate, after the curvature is
produced, be expressed by an equation between r, the distance of any point from the axis, or normal to the centre of the plate, and x the distance of the point from the plane in which the middle of the plate originally was, and let

ds=-^{dxY + {dr)\

VOL I.

8

58

THE EQUILIBRIUM OF ELASTIC SOLIDS.

Let A, be the pressure on one side of the plate, and h^ that on the other.
Let p and q be the pressures in the plane of the plate at any p point,
acting in the direction of a tangent to the section of the plate by a plane passing through the axis, and q acting in the direction perpendicular to that

plane.
By equating the forces which act on any particle in a direction parallel to

the axis, we find

^ drdx ^ dpdx ^

d^x ^

,,

,

,

j^dr

By making p = when r = in this equation, when integrated,

p-l^l^^--'^-)

The forces perpendicular to the axis are [drV . dpdr ^ d^r .^
,

i\dx

^

.

("^-

Substituting for p its value, the equation gives

^"_

(^1 - h^ idr dr dx\ (h^ - h^

/dr ds^d^^ds ^r\

,

.

t ''[d'sdi'^d^)'^ 2t "^^[didxd^ dxd^)""^ ^'

The equations of elasticity become

dSs (\

1\/ ^

h, + h\^p

^ ^^ Differentiating = -j- -^ (""''')' ^^^

*^^^®

dhr
dr ~

dr dr dSs
'
ds ds ds

By

a

comparison

of these

values

of

-t— ds

,

dtr^\rwl
,
ds) \9iJ,

1\/ ,K + h\,qdrp^ (I ,

l\fdp,dq\
w dr as

THE EQUILIBRIUM OF ELASTIC SOUDS.

59

To obtain an expression for the curvature of the plate at the vertex, let a be the radius of curvature, then, as an approximation to the equation of the

plate, let

— — r»

x

.

2a

By substituting the value of a: in the values of p and q, and in the equa-
tion of elasticity, the approximate value of a is found to be

= + m-+ a

18m/x,

\-\-h^

3/x

— lOm lOw ^i-A,

A,-^2 . 1 c 1

"T" ' T

51/x

7~ ~T~z

; TT" 51/t

'

.(53).

Since the focal distance of the mirror, or -, depends on the difference of
pressures, a telescope on Mr Nasmyth's principle would act as an aneroid baro-
meter, the focal distance varying inversely as the pressure of the atmosphere.

Case VIL

To find the conditions of torsion of a cylinder composed of a great number

of parallel wires bound together without adhering to one another.

Let X be the length of the cylinder, a its radius, r the radius at any point,

M hS the angle of torsion,

the force producing torsion, hx the change of length,

P and the longitudinal force. Each of the wires becomes a helix whose radius

— IJ is r, its angular rotation Zd, and its length along the axis x-Zx.

Its length is therefore

{rZey

and the tension is

- V - = - 1 jE; 1

/[ 1

] r^ (-]'] .

This force, resolved parallel to the axis, is

60

THE EQUIUBRTCM OF ELASTIC SOUDS.

—XX— and since

and r are small, we may assume

-"-{-l-n?)'}

<">

The force, when resolved in the tangential direction, is approximately

"-^m'i-m

'">

— By eliminating

between (54) and (55) we have

X

M: ^^^'ip.E.^m

(56).

X

24 \ a?/

P M When = 0,

depends on the sixth power of the radius and the cube

of the angle of torsion, when the cylinder is composed of separate filaments.

Since the force of torsion for a homogeneous cylinder depends on the

fourth power of the radius and the first power of the angle of torsion, the

torsion of a wire having a fibrous texture will depend on both these laws.
The parts of the force of torsion which depend on these two laws may be

found by experiment, and thus the difference of the elasticities in the direction

of the axis and in the perpendicular directions may be determined.
A calculation of the force of torsion, on this supposition, may be found in

Young's Mathematical Principles of Natural Philosophy; and it \s introduced here to account for the variations from the law of Case II., which may be

observed in a twisted rod.

Case VIII.
It is well known that grindstones and fly-wheels are often broken by the centrifugal force produced by their rapid rotation. I have therefore calculated the strains and pressure acting on an elastic cylinder revolving round its axis, and acted on by the centrifugal force alone.

THE EQUILIBBIUM OF ELASTIC SOLIDa.

61

The equation of the equilibrium of a particle [see Equation (21)], becomes dp Air'k
,

where q and p are the tangential and radial pressures, k is the weight in
pounds of a cubic inch of the substance, g is twice the height in inches that a body falls in a second, t is the time of revolution of the cylinder in seconds.

^ By substituting the value of q and

in Equations (19), (20), and neglect-

ing 0,

-(i-3^)(«|-?-g)-M^S-f-^.^)

which gives

1

TT^k

2+^K 2gt^\

+ ^«

1 , Tj'k
(-"?)

(57).

TT'k
^=-V + 22g^f»(-2 + f)^ + c.
If the radii of the surfaces of the hollow cylinder be a, and cu„ and the pressures actmg on them h^ and h^, then the values of c^ and c, are

(58).
-f^'-(«--.')S(^-S.
When o, = 0, as in the case of a solid cylinder, = c, 0, and

*'+0 « =

{2('^ + «.') + |(3'^-«,')}

When = A, 0, and r^a^,

(59).

^ = ^U-2)

(60).

When q exceeds the tenacity of the substance in pounds per square inch,

the cylinder will give way; and by making q equal to the number of pounds

which a square inch of the substance will support, the velocity may be found

at which the bursting of the cylinder will take place.

g2

THE EQUILIBRIUM OP ELASTIC SOLIDS.

'^ Since I=ho>(q-p) =

(^-2\br', a transparent revolving cylinder, when

polarized light is transmitted parallel to the axis, will exhibit rings whose diameters are as the square roots of an arithmetical progression, and brushes
parallel and perpendicular to the plane of polarization.

Case IX.

A hollow cylinder or tube is surrounded by a medium of a constant
temperature while a liquid of a different temperature is made to flow through it. The exterior and interior surfaces are thus kept each at a constant tem-
perature till the transference of heat through the cylinder becomes uniform. Let V be the temperature at any point, then when this quantity has

reached its limit,

rdv _

v = Ci\ogr + Ci

(61).

Let the temperatures at the surfaces be 0^ and 0^, and the radii of the

surfaces a, and a^, then

^ 0^-0^

loga,0^-logaA

^'""logaj-loga/ '~ loga^-loga^

Let the coeflBcient of linear dilatation of the substance be c,, then the proportional dilatation at any point will be expressed by c,v, and the equations of elasticity (18), (19), (20), become

m r

^ \,9/x 3m/ ^

^'

The equation of equHibrivuu is

2-P+r'^

and since the tube is supposed to be of a considerable length
-J— =c^ a constant quantity.
CL2C

(21),