The Project Gutenberg eBook, James Clerk Maxwell and Modern Physics, by Richard Glazebrook

JAMES CLERK MAXWELL
AND MODERN PHYSICS

J. Clerk Maxwell

(From a Photograph of the Picture by G. Lowes Dickinson, Esq., in the Hall of Trinity College, Cambridge.)

The task of giving some account of Maxwell’s work—of describing the share that he has taken in the advance of Physical Science during the latter half of this nineteenth century—has proved no light labour. The problems which he attacked are of such magnitude and complexity, that the attempt to explain them and their importance, satisfactorily, without the aid of symbols, is almost foredoomed to failure. However, the attempt has been made, in the belief that there are many who, though they cannot follow the mathematical analysis of Maxwell’s work, have sufficient general knowledge of physical ideas and principles to make an account of Maxwell and of the development of the truths that he discovered, subjects of intelligent interest.

Maxwell’s life was written in 1882 by two of those who were most intimately connected with him, Professor Lewis Campbell and Dr. Garnett. Many of the biographical details of the earlier part of this book are taken from their work. My thanks are due tovi them and to their publishers, Messrs. Macmillan, for permission to use any of the letters which appear in their biography. I trust that my brief account may be sufficient to induce many to read Professor Campbell’s “Life and Letters,” with a view of learning more of the inner thoughts of one who has left so strong an imprint on all he undertook, and was so deeply loved by all who knew him.

R. T. G.

Cambridge,
December, 1895.

CHAPTER I.
EARLY LIFE.

“One who has enriched the inheritance left by Newton and has consolidated the work of Faraday—one who impelled the mind of Cambridge to a fresh course of real investigation—has clearly earned his place in human memory.” It was thus that Professor Lewis Campbell and Mr. Garnett began in 1882 their life of James Clerk Maxwell. The years which have passed, since that date, have all tended to strengthen the belief in the greatness of Maxwell’s work and in the fertility of his genius, which has inspired the labours of those who, not in Cambridge only, but throughout the world, have aided in developing the seeds sown by him. My object in the following pages will be to give some very brief account of his life and writings, in a form which may, I hope, enable many to realise what Physical Science owes to one who was to me a most kind friend as well as a revered master.

The Clerks of Penicuik, from whom Clerk Maxwell was descended, were a distinguished family. Sir John Clerk, the great-great-grandfather of Clerk Maxwell,10 was a Baron of the Exchequer in Scotland from 1707 to 1755; he was also one of the Commissioners of the Union, and was in many ways an accomplished scholar. His second son George married a first cousin, Dorothea Maxwell, the heiress of Middlebie in Dumfriesshire, and took the name of Maxwell. By the death of his elder brother James in 1782 George Clerk Maxwell succeeded to the baronetcy and the property of Penicuik. Before this time he had become involved in mining and manufacturing speculations, and most of the Middlebie property had been sold to pay his debts.

The property of Sir George Clerk Maxwell descended in 1798 to his two grandsons, Sir George Clerk and Mr. John Clerk Maxwell. It had been arranged that the younger of the two was to take the remains of the Middlebie property and to assume with it the name of Maxwell. Sir George Clerk was member for Midlothian, and held office under Sir Robert Peel. John Clerk Maxwell was the father of James Clerk Maxwell, the subject of this sketch.1

John Clerk Maxwell lived with his widowed mother in Edinburgh until her death in 1824. He was a lawyer, and from time to time did some little business in the courts. At the same time he maintained an interest in scientific pursuits, especially those of a practical nature. Professor Campbell tells us of an endeavour to devise a bellows which would give a continuous draught of air. In 1831 he11 contributed to the Edinburgh Medical and Philosophical Journal a paper entitled “Outlines of a Plan for combining Machinery with the Manual Printing Press.”

In 1826 John Clerk Maxwell married Miss Frances Cay, of North Charlton, Northumberland. For the first few years of their married life their home was in Edinburgh. The old estate of Middlebie had been greatly reduced in extent, and there was not a house on it in which the laird could live. However, soon after his marriage, John Clerk Maxwell purchased the adjoining property of Glenlair and built a mansion-house for himself and his wife. Mr. Maxwell superintended the building work. The actual working plans for some further additions made in 1843 were his handiwork. A garden was laid out and planted, and a dreary stony waste was converted into a pleasant home. For some years after he settled at Glenlair the house in Edinburgh was retained by Mr. Maxwell, and here, on June 13, 1831, was born his only son, James Clerk Maxwell. A daughter, born earlier, died in infancy. Glenlair, however, was his parents’ home, and nearly all the reminiscences we have of his childhood are connected with it. The laird devoted himself to his estates and to the education of his son, taking, however, from time to time his full share in such county business as fell to him. Glenlair in 1830 was very much in the wilds; the journey from Edinburgh occupied two days. “Carriages in the modern sense were hardly known to the Vale of Urr. A sort of double gig with a hood was the best apology for a travelling coach, and the most active12 mode of locomotion was in a kind of rough dog-cart known in the family speech as a hurly.”2

Mrs. Maxwell writes thus3, when the boy was nearly three years old, to her sister, Miss Jane Cay:—

To discover “how it doos” was thus early his aim. His cousin, Mrs. Blackburn, tells us that throughout his childhood his constant question was, “What’s the go of that? What does it do?” And if the answer were too vague or inconclusive, he would add, “But what’s the particular go of that?”

Professor Campbell’s most interesting account of these early years is illustrated by a number of sketches of episodes in his life. In one Maxwell is absorbed in watching the fiddler at a country dance; in another he is teaching his dog some tricks; in a third he is helping a smaller boy in his efforts to build a castle. Together with his cousin, Miss Wedderburn, he devised a number of figures for a13 toy known as a magic disc, which afterwards developed into the zoetrope or wheel of life, and in which, by means of an ingenious contrivance of mirrors, the impression of a continuous movement was produced.

This happy life went on until his mother’s death in December, 1839; she died, at the age of forty-eight, of the painful disease to which her son afterwards succumbed. When James, being then eight years old, was told that she was now in heaven, he said: “Oh, I’m so glad! Now she’ll have no more pain.”

After this his aunt, Miss Jane Cay, took a mother’s place. The problem of his education had to be faced, and the first attempts were not successful. A tutor had been engaged during Mrs. Maxwell’s last illness, and he, it seems, tried to coerce Clerk Maxwell into learning; but such treatment failed, and in 1841, when ten years old, he began his school-life at the Edinburgh Academy.

School-life at first had its hardships. Maxwell’s appearance, his first day at school, in Galloway home-spun and square-toed shoes with buckles, was more than his fellows could stand. “Who made those shoes?” they asked4; and the reply they received was—

He returned to Heriot Row that afternoon, says Professor Campbell, “with his tunic in rags and14 wanting the skirt, his neat frill rumpled and torn—himself excessively amused by his experiences and showing not the slightest sign of irritation.”

No. 31, Heriot Row, was the house of his widowed aunt, Mrs. Wedderburn, Mr. Maxwell’s sister; and this, with occasional intervals when he was with Miss Cay, was his home for the next eight or nine years. Mr. Maxwell himself, during this period, spent much of his time in Edinburgh, living with his sister during most of the winter and returning to Glenlair for the spring and summer.

Much of what we know of Clerk Maxwell’s life during this period comes from the letters which passed between him and his father. They tell us of the close intimacy and affection which existed between the two, of the boy’s eager desire to please and amuse his father in the dull solitude of Glenlair, and his father’s anxiety for his welfare and progress.

Professor Campbell was his schoolfellow, and records events of those years in which he shared, which bring clearly before us what Clerk Maxwell was like. Thus he writes5:—

15

And again, speaking of his school-life:—

The first reference to mathematical studies occurs, says Professor Campbell, in a letter to his father written soon after his thirteenth birthday.6

The summer holidays were spent at Glenlair. His cousin, Miss Jemima Wedderburn, was with him, and shared his play. Her skilled pencil has left us many amusing pictures of the time, some of which are reproduced by Professor Campbell. There were expeditions and picnics of all sorts, and a new toy known as “the devil on two sticks” afforded infinite amusement. The winter holidays usually found him at Penicuik, or occasionally at Glasgow, with Professor Blackburne or Professor W. Thomson (now Lord Kelvin). In October, 1844, Maxwell was promoted to the rector’s class-room. John Williams, afterwards Archdeacon of Cardigan, a distinguished Baliol man, was rector, and the change was in many ways an important one for Maxwell. He writes to his father: “I like P—— better than B——. We have lots of jokes, and he speaks a great deal, and we have not17 so much monotonous parsing. In the English Milton is better than the History of Greece....”

P—— was the boys’ nickname for the rector; B—— for Mr. Carmichael, the second master. This7 is the account of Maxwell’s first interview with the rector:—

Rector: “What part of Galloway do you come from?”

J. C. M.: “From the Vale of Urr. Ye spell it o, err, err, or oo, err, err.”

The study of geometry was begun, and in the mathematical master, Mr. Gloag, Maxwell found a teacher with a real gift for his task. It was here that Maxwell’s vast superiority to many who were his companions at once showed itself. “He seemed,” says Professor Campbell, “to be in the heart of the subject when they were only at the boundary; but the boyish game of contesting point by point with such a mind was a most wholesome stimulus, so that the mere exercise of faculty was a pure joy. With Maxwell the first lessons of geometry branched out at once into inquiries which became fruitful.”

In July, 1845, he writes:—

After a summer spent chiefly at Glenlair, he returned with his father to Edinburgh for the winter, and began, at the age of fourteen, to go to the meetings of the Royal Society of Edinburgh. At the Society of Arts he met Mr. R. D. Hay, the decorative painter, who had interested himself in the attempt to reduce beauty in form and colour to mathematical principles. Clerk Maxwell was interested in the question how to draw a perfect oval, and devised a method of drawing oval curves which was referred by his father to Professor Forbes for his criticism and suggestions. After discussing the matter with Professor Kelland, Professor Forbes wrote as follows8:—

In consequence of this, Clerk Maxwell’s first19 published paper was communicated to the Royal Society of Edinburgh on April 6th, 1846, when its author was barely fifteen. Its title is as follows: “On the Description of Oval Curves and those having a Plurality of Foci. By Mr. Clerk Maxwell, Junior. With Remarks by Professor Forbes. Communicated by Professor Forbes.”

The notice in his father’s diary runs: “M. 6 [Ap., 1846.] Royal Society with Jas. Professor Forbes gave acct. of James’s Ovals. Met with very great attention and approbation generally.”

This was the beginning of the lifelong friendship between Maxwell and Forbes.

The curves investigated by Maxwell have the property that the sum found by adding to the distance of any point on the curve from one focus a constant multiple of the distance of the same point from a second focus is always constant.

The curves are of great importance in the theory of light, for if this constant factor expresses the refractive index of any medium, then light diverging from one focus without the medium and refracted at a surface bounding the medium, and having the form of one of Maxwell’s ovals, will be refracted so as to converge to the second focus.

About the same time he was busy with some investigations on the properties of jelly and gutta-percha, which seem to have been suggested by Forbes’ “Theory of Glaciers.”

He failed to obtain the Mathematical Medal in 1846—possibly on account of these researches—but he continued at school till 1847, when he left, being20 then first in mathematics and in English, and nearly first in Latin.

In 1847 he was working at magnetism and the polarisation of light. Some time in that year he was taken by his uncle, Mr. John Cay, to see William Nicol, the inventor of the polarising prism, who showed him the colours exhibited by polarised light after passing through unannealed glass. On his return, he made a polariscope with a glass reflector. The framework of the first instrument was of cardboard, but a superior article was afterwards constructed of wood. Small lenses mounted on cardboard were employed when a conical pencil was needed. By means of this instrument he examined the figures exhibited by pieces of unannealed glass, which he prepared himself; and, with a camera lucida and box of colours, he reproduced these figures on paper, taking care to sketch no outlines, but to shade each coloured band imperceptibly into the next. Some of these coloured drawings he forwarded to Nicol, and was more than repaid by the receipt shortly afterwards of a pair of prisms prepared by Nicol himself. These prisms were always very highly prized by Maxwell. Once, when at Trinity, the little box containing them was carried off by his bed-maker during a vacation, and destined for destruction. The bed-maker died before term commenced, and it was only by diligent search among her effects that the prisms were recovered.9 After this they were more carefully guarded, and they are now, together with the wooden polariscope, the bits of unannealed glass,21 and the water-colour drawings, in one of the showcases at the Cavendish Laboratory.

About this time, Professor P. G. Tait and he were schoolfellows at the Academy, acknowledged as the two best mathematicians in the school. It was thought desirable, says Professor Campbell, that “we should have lessons in physical science, so one of the classical masters gave them out of a text-book.... The only thing I distinctly remember about these hours is that Maxwell and P. G. Tait seemed to know much more about the subject than our teacher did.”

An interesting account of these days is given by Professor Tait in an obituary notice on Maxwell printed in the “Proceedings of the Royal Society of Edinburgh, 1879–80,” from which the following is taken:—

In November, 1847, Clerk Maxwell entered the University of Edinburgh, learning mathematics from Kelland, natural philosophy from J. D. Forbes, and logic from Sir W. R. Hamilton. At this time, according to Professor Campbell10—

During this period he wrote two important papers. The one, on “Rolling Curves,” was read to the Royal Society of Edinburgh by Professor Kelland—(“it was not thought proper for a boy in a round jacket to mount the rostrum”)—in February, 1849; the other, on “The Equilibrium of Elastic Solids,” appeared in the spring of 1850.

The vacations were spent at Glenlair, and we learn from letters to Professor Campbell and others how the time was passed.

“On Saturday,” he writes11—April 26th, 1848, just after his arrival home—“the natural philosophers ran up Arthur’s Seat with the barometer. The Professor set it down at the top.... He did not set it straight, and made the hill grow fifty feet; but we got it down again.”

In a letter of July in the same year he describes his laboratory:—

Of his reading he says:—“I am reading Herodotus’ ‘Euterpe,’ having taken the turn—that is to say that sometimes I can do props., read Diff. and Int. Calc., Poisson, Hamilton’s dissertation, etc.”

In September he was busy with polarised light. “We were at Castle Douglas yesterday, and got25 crystals of saltpetre, which I have been cutting up into plates to-day in hopes to see rings.”

In July, 1849, he writes12:—

And again on March 22nd, 1850:—

It was at last settled that he was to go up to Cambridge. Tait had been at Peterhouse for two years, while Allan Stewart had joined him there in 1849, and after much discussion it was arranged that Maxwell should enter at the same college.

Of this period of his life Tait writes as follows:—

While W. D. Niven, in his preface to Maxwell’s collected works (p. xii.), says:—

Maxwell did not remain long at Peterhouse; before the end of his first term he migrated to Trinity, and was entered under Dr. Thompson December 14th, 1850. He appeared to the tutor a shy and diffident youth, but presently surprised Dr. Thompson by producing a bundle of papers—copies, probably, of those he had already published—and remarking, “Perhaps these may show that I am not unfit to enter at your College.”

The change was pressed upon him by many friends, the grounds of the advice being that, from the large number of high wranglers recently at Peterhouse and the smallness of the foundation, the chances of a Fellowship there for a mathematical man were less than at Trinity. It was a step he never regretted; the prospect of a Fellowship had but little influence on his mind. He found, however, at the larger college ampler opportunities for self-improvement, and it was possible for him to select his friends from among men whom he otherwise would never have known.

The record of his undergraduate life is not very full; his letters to his father have, unfortunately, been lost, but we have enough in the recollections of friends still living to picture what it was like. At first he lodged in King’s Parade with an old Edinburgh schoolfellow, C. H. Robertson. He attended the29 College lectures on mathematics, though they were somewhat elementary, and worked as a private pupil with Porter, of Peterhouse. His father writes to him, November, 1850: “Have you called on Professors Sedgwick, at Trin., and Stokes, at Pembroke? If not, you should do both. Stokes will be most in your line, if he takes you in hand at all. Sedgwick is also a great Don in his line, and, if you were entered in geology, would be a most valuable acquaintance.”

In his second year he became a pupil of Hopkins, the great coach; he also attended Stokes’ lectures, and the friendship which lasted till his death was thus begun. In April, 1852, he was elected a scholar, and obtained rooms in College (G, Old Court). In June, 1852, he came of age. “I trust you will be as discreet when major as you have been while minor,” writes his father the day before. The next academic year, October, 1852, to June, 1853, was a very busy one; hard grind for the Tripos occupied his time, and he seems to have been thoroughly overstrained. He was taken ill while staying near Lowestoft with the Rev. C. B. Tayler, the uncle of a College friend. His own account of the illness is given in a letter to Professor Campbell13, dated July 14th, 1853.

During this period he wrote some papers for the Cambridge and Dublin Mathematical Journal which will be referred to again later. He was also a member of a discussion society known as the “Apostles,” and some of the essays contributed by him are preserved by Professor Campbell. Mr. Niven, in his preface to the collected edition of Maxwell’s works, suggests that the composition of these essays laid the foundation of that literary finish which is one of the characteristics of Maxwell’s scientific writings.

Among his friends at the time were Tait, Charles Mackenzie of Caius, the missionary bishop of Central Africa, Henry and Frank Mackenzie of Trinity, Droop, third Wrangler in 1854; Gedge, Isaac Taylor, Blakiston, F. W. Farrar,14 H. M. Butler,15 Hort, V. Lushington, Cecil Munro, G. W. H. Tayler, and W. N. Lawson. Some of these who survived him have31 given to Professor Campbell their recollections of these undergraduate days, which are full of interest.

Thus Mr. Lawson writes16:—

The last clause is part of a quotation from a diary kept by Mr. Lawson at Cambridge, in which, under the date July 15th, 1853, he writes:—

How many who have struggled through the “Electricity and Magnetism” have realised the truth of the remark about the correctness of his physical intuitions and the deficiency at times of his analysis!

Dr. Butler, a friend of these early days, preached the University sermon on November 16th, 1879, ten days after Maxwell’s death, and spoke thus:—

The Tripos came in January, 1854. “You will need to get muffetees for the Senate Room. Take your plaid or rug to wrap round your feet and legs,” was his father’s advice—advice which will appeal to many who can remember the Senate House as it felt on a cold January morning.

Maxwell had been preparing carefully for this examination. Thus to his aunt, Miss Cay, in June, 1853, he writes:—“If anyone asks how I am getting on in mathematics, say that I am busy arranging everything so as to be able to express all distinctly, so that examiner may be satisfied now and pupils edified hereafter. It is pleasant work and very strengthening, but not nearly finished.”

Still, the illness of July, 1853, had left some effect. Professor Baynes states that he said that on entering the Senate House for the first paper he felt his mind almost a blank, but by-and-by his mental vision became preternaturally clear.

The moderators were Mackenzie of Caius, whose advice had been mainly instrumental in leading him to migrate to Trinity, Wm. Walton of Trinity, Wolstenholme of Christ’s, and Percival Frost of St. John’s.

36

When the lists were published, Routh of Peterhouse was senior, Maxwell second. The examination for the Smith’s Prizes followed in a few days, and then Routh and Maxwell were declared equal.

In a letter to Miss Cay17 of January 13th, while waiting for the three days’ list, he writes:—

His father, on hearing the news, wrote from Edinburgh:—

His friends in Edinburgh were greatly pleased.37 “I get congratulations on all hands,” his father writes,18 “including Professor Kelland and Sandy Fraser and all others competent.... To-night or on Monday I shall expect to hear of the Smith’s Prizes.” And again, February 6th, 1854:—“George Wedderburn came into my room at 2 a.m. yesterday morning, having seen the Saturday Times, received by the express train.... As you are equal to the Senior in the champion trial, you are very little behind him.”

Or again, March 5th, 1854:—

And now the long years of preparation were nearly over. The cunning craftsman was fitted with his tools; he could set to work to unlock the secrets of Nature; he was free to employ his genius and his knowledge on those tasks for which he felt most fitted.

From this time on Maxwell’s life becomes a record of his writings and discoveries. It will, however, probably be clearest to separate as far as possible biographical details from a detailed account of his scientific work, leaving this for consecutive treatment in later chapters, and only alluding to it so far as may prove necessary to explain references in his letters.

He continued in Cambridge till the Long Vacation of 1854, reading Mill’s “Logic.” “I am experiencing the effects of Mill,” he writes, March 25th, 1854, “but I take him slowly. I do not think him the last of his kind. I think more is wanted to bring the connexion of sensation with science to light, and to show what it is not.” He also read Berkeley on “The Theory of Vision” and “greatly admired it.”

About the same time he devised an ophthalmoscope.19

After the vacation he returned to Cambridge, and the letters refer to the colour-top. Thus to Miss Cay, November 24th, 1854, p. 208:—

A paper written for his own use originally some time in 1854, but communicated as a parting gift to his friend Farrar, who was about to become a master at Marlborough, gives us some insight into his view of life at the age of twenty-three.

His father was unwell in the Christmas vacation of that year, and he could not return to Cambridge at the beginning of the Lent term. “My steps,” he writes20 to C. J. Munro from Edinburgh, February 19th, 1855, “will be no more by the reedy and crooked till Easter term.... I should like to know how many kept bacalaurean weeks go to each of these terms, and when they begin and end. Overhaul the Calendar, and when found make note of.”

He was back in Cambridge for the May term, working at the motion of fluids and at his colour-top. A paper on “Experiments on Colour as Perceived by the Eye” was communicated to the Royal Society of Edinburgh on March 19th, 1855. The experiments were shown to the Cambridge Philosophical Society in May following, and the results are thus described in two letters21 to his father, Saturday, May 5th, 1855:

The British Association met at Glasgow in September, 1855, and Maxwell was present, and showed his colour-top at Professor Ramsay’s house to some of those interested. Letters22 to his father about this time describe some of the events of the meeting and his own plans for the term.

The examination referred to was that for a Fellowship at Trinity, and Maxwell was elected on October 10th, 1855.

He was immediately asked to lecture for the College, on hydrostatics and optics, to the upper division of the third year, and to set papers for the questionists. In consequence, he declined to take pupils, in order to have time for reading and doing private mathematics, and for seeing the men who attended his lectures.

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In November he writes: “I have been lecturing two weeks now, and the class seems improving; and they come and ask questions, which is a good sign. I have been making curves to show the relations of pressure and volume in gases, and they make the subject easier.”

Still, he found time to attend Professor Willis’s lectures on mechanism and to continue his reading. “I have been reading,” he writes, “old books on optics, and find many things in them far better than what is new. The foreign mathematicians are discovering for themselves methods which were well known at Cambridge in 1720, but are now forgotten.”

The “Poisson” was read to help him with his own views on electricity, which were rapidly maturing, and the first of that great series of works which has revolutionised the science was published on December 10th, 1855, when his paper on “Faraday’s Lines of Force” was read to the Cambridge Philosophical Society.

The next term found him back in Cambridge at work on his lectures, full of plans for a new colour top and other matters. Early in February he received a letter from Professor Forbes, telling him that the Professorship of Natural Philosophy in Marischal College, Aberdeen, was vacant, and suggesting that he should apply.

He decided to be a candidate if his father approved. “For my own part,” he writes, “I think the sooner I get into regular work the better, and that the best way of getting into such work is to profess one’s readiness by applying for it.” On the45 20th of February he writes: “However, wisdom is of many kinds, and I do not know which dwells with wise counsellors most, whether scientific, practical, political, or ecclesiastical. I hear there are candidates of all kinds relying on the predominance of one or other of these kinds of wisdom in the constitution of the Government.”

The second part of the paper on “Faraday’s Lines of Force” was read during the term. Writing on the 4th of March, he expresses the hope soon to be able to write out fully the paper. “I have done nothing in that way this term,” he says, “but am just beginning to feel the electrical state come on again.”

His father was working at Edinburgh in support of his candidature for Aberdeen, and when, in the middle of March, he returned North, he found everything well prepared. The two returned to Glenlair together after a few days in Edinburgh, and Maxwell was preparing to go back to Cambridge, when, on the 2nd of April, his father died suddenly.

Writing to Mrs. Blackburn, he says: “My father died suddenly to-day at twelve o’clock. He had been giving directions about the garden, and he said he would sit down and rest a little, as usual. After a few minutes I asked him to lie down on the sofa, and he did not seem inclined to do so; and then I got him some ether, which had helped him before. Before he could take any he had a slight struggle, and all was over. He hardly breathed afterwards.”

Almost immediately after this, Maxwell was appointed to Aberdeen. His father’s death had frustrated some at least of the intentions with which46 he had applied for the post. He knew the old man would be glad to see him the occupant of a Scotch chair. He hoped, too, to be able to live with his father at Glenlair for one half the year; but this was not to be. No doubt the laboratory and the freedom of the post, when compared with the routine work of preparing men for the Tripos, had their inducements; still, it may be doubted if the choice was a wise one for him. The work of drilling classes, composed, for the most part, of raw untrained lads, in the elements of physics and mechanics was, as Niven says in his preface to the collected works, not that for which he was best fitted; while at Cambridge, had he stayed, he must always have had among his pupils some of the best mathematicians of the time; and he might have founded some ten or fifteen years before he did that Cambridge School of Physicists which looks back with so much pride to him as their master.

Leave-taking at Trinity was a sad task. He writes23 thus, June 4th, to Mr. R. B. Litchfield:—

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The summer of 1856 was spent at Glenlair, where various friends were his guests—Lushington, MacLennan, the two cousins Cay, and others. He continued to work at optics, electricity, and magnetism, and in October was busy with “a solemn address or manifesto to the Natural Philosophers of the North, which needed coffee and anchovies and a roaring hot fire and spread coat-tails to make it natural.” This was his inaugural lecture.

In November he was at Aberdeen. Letters24 to Miss Cay, Professor Campbell, and C. J. Munro tell of the work of the session. The last is from Glenlair, dated May 20th, 1857, after work was over.

Another letter,25 June 5th, 1857, also to Munro, refers to the work of the University Commission and the new statutes.

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It was more than twenty years before the scheme outlined in the above letter came to anything; but, at the time of Maxwell’s death in 1879, another Commission was sitting, and the plan suggested by Maxwell became the basis of the statutes of nearly all the colleges.

For the winter session of 1857–58 he was again at Aberdeen.

The Adams Prize had been established in 1848 by some members of St. John’s College, and connected by them with the name of Adams “in testimony of their sense of the honour he had conferred upon his College and the University by having been the first among the mathematicians of Europe to determine from perturbations the unknown place of a disturbing49 planet exterior to Uranus.” Professor Challis, Dr. Parkinson, and Sir William Thomson, the examiners, had selected as the subject for the prize to be awarded in 1857 the “Motions of Saturn’s Rings.” For this Maxwell had decided to compete, and his letters at the end of 1857 tell of the progress of the task. Thus, writing26 to Lewis Campbell from Glenlair on August 28th, he says:—

And again27 to Miss Cay on the 28th of November:—

He devised a model (now at the Cavendish Laboratory) to exhibit the motions of the satellites in a disturbed ring, “for the edification of sensible image-worshippers.”

The essay was awarded the prize, and secured for its author great credit among scientific men.

In another letter, written during the same session, he says: “I find my principal work here is teaching my men to avoid vague expressions, as ‘a certain force,’ meaning uncertain; may instead of must; will be instead of is; proportional instead of equal.”

The death, during the autumn, of his College friend Pomeroy, from fever in India, was a great blow to him; his letters at the time show the depth of his feelings and his beliefs.

The question of the fusion of the two Colleges at Aberdeen, King’s College and the Marischal College, was coming to the fore. “Know all men,” he says, in a letter to Professor Campbell, “that I am a Fusionist.”

In February, 1858, he was still engaged on Saturn’s rings, while hard at work during the same time with his classes. He had established a voluntary class for his students of the previous year, and was reading with them Newton’s “Lunar Theory and Astronomy.” This was followed by “Electricity and Magnetism,”51 Faraday’s book being the backbone of everything, “as he himself is the nucleus of everything electric since 1830.”

In February, 1858, he announced his engagement to Katherine Mary Dewar, the daughter of the Principal of Marischal College.

The wedding took place early in June. Professor Campbell has preserved some of the letters written by Maxwell to Miss Dewar, and these contain “the record of feelings which in the years that followed were transfused in action and embodied in a married life which can only be spoken of as one of unexampled devotion.”

The project for the fusion of the two Colleges, to which reference has been made, went on, and the scheme was completed in 1860.

The two Colleges were united to form the University of Aberdeen, and the new chair of Natural Philosophy thus created was filled by the appointment of David Thomson, Professor of Natural Philosophy in King’s College, and Maxwell’s senior. Mr. W. D. Niven, in his preface to Maxwell’s works, when dealing with this appointment, writes:—

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The question whether Maxwell was a great teacher has sometimes been discussed. I trust that the following pages will give an answer to it. He was not a prominent lecturer. As Professor Campbell says,29 “Between his students’ ignorance and his vast knowledge it was difficult to find a common measure. The advice which he once gave to a friend whose duty it was to preach to a country congregation, ‘Why don’t you give it them thinner?’ must often have been applicable to himself.... Illustrations of ignotum per ignotius, or of the abstruse by some unobserved property of the familiar, were multiplied with dazzling rapidity. Then the spirit of indirectness and paradox, though he was aware of its dangers, would often take possession of him against his will, and, either from shyness or momentary excitement, or the despair of making himself understood, would land him in ‘chaotic53 statements,’ breaking off with some quirk of ironical humour.”

But teaching is not all done by lecturing. His books and papers are vast storehouses of suggestions and ideas which the ablest minds of the past twenty years have been since developing. To talk with him for an hour was to gain inspiration for a year’s work; to see his enthusiasm and to win his praise or commendation were enough to compensate for many weary struggles over some stubborn piece of apparatus which would not go right, or some small source of error which threatened to prove intractable and declined to submit itself to calculation. The sure judgment of posterity will confirm the verdict that Clerk Maxwell was a great teacher, though lecturing to a crowd of untrained undergraduates was a task for which others were better fitted than he.

In 1860 Forbes resigned the chair of Natural Philosophy at Edinburgh. Maxwell and Tait were candidates, and Tait was appointed. In the summer of the same year Maxwell obtained the vacant Professorship of Natural Philosophy at King’s College, London. This he held to 1865, and this period of his life is distinguished by the appearance of some of his most important papers. The work was arduous; the College course extended over nine months of the year; there were as well evening lectures to artisans as part of his regular duties. His life in London was useful to him in the opportunities it gave him for becoming personally acquainted with Faraday and others. He also renewed his intimacy with various Cambridge friends.

He was at the celebrated Oxford meeting of the British Association in 1860, where he exhibited his colour-box for mixing the colours of the spectrum. In 1859, at the meeting at Aberdeen, he had read to Section A his first paper on the “Dynamical Theory of Gases,” published in the Philosophical Magazine for January, 1860. The second part of the paper, dealing with the conduction of heat and other phenomena in a gas, was published in July, 1860, after the Oxford meeting.

A paper on the “Theory of Compound Colours”55 was communicated to the Royal Society by Professor Stokes in January, 1860. It contains the account of his colour-box in the form finally adopted (most of the important parts of the apparatus are still at the Cavendish Laboratory), and a number of observations by Mrs. Maxwell and himself, which will be more fully described later.

In November, 1860, he received for this work the Rumford medal of the Royal Society.

The next year, 1861, is of great importance in the history of electrical science. The British Association met at Manchester, and a Committee was appointed on Standards of Electrical Resistance. Maxwell was not a member. The committee reported at the Cambridge meeting in 1862, and were reappointed with extended duties. Maxwell’s name, among others, was added, and he took a prominent part in the deliberations of the committee, which, as their Report30 presented in 1863 states, came to the opinion, “after mature consideration, that the system of so-called absolute electrical units, based on purely mechanical measurements, is not only the best system yet proposed, but is the only one consistent with our present knowledge both of the relations existing between the various electrical phenomena and of the connection between these and the fundamental measurements of time, space, and mass.”

Appendix C of this Report, “On the Elementary Relations between Electrical Measurements,” bears the names of Clerk Maxwell and Fleeming Jenkin, and is the foundation of everything that has been done in56 the way of absolute electrical measurement since that date; while Appendix D gives an account by the same two workers of the experiments on the absolute unit of electrical resistance made in the laboratory of King’s College by Maxwell, Fleeming Jenkin, and Balfour Stewart. Further experiments are described in the report for 1864. The work thus begun was consummated during the year 1894 by the legalisation throughout the civilised world of a system of electrical units based on those described in these reports.

Meanwhile, Maxwell’s views on electro-magnetic theory were quietly developing. Papers on “Physical Lines of Force,” which appeared in the Philosophical Magazine during 1861 and 1862, contain the germs of his theory—expressed at that time, it is true, in a somewhat material form. In the paper published January, 1862, the now well-known relation between the ratio of the electric units and the velocity of light was established, and his correspondence with Fleeming Jenkin and C. J. Munro about this time relates in part to the experimental verification of this relation. His experiments on this matter were published in the “Philosophical Transactions” for 1868.

This electrical theory occupied his mind mainly during 1863 and 1864. In September of the latter year he writes31 from Glenlair to C. Hockin, who had taken Balfour Stewart’s place during the second series of experiments on the measurement of resistance.

This work resulted in his greatest electrical paper, “A Dynamical Theory of the Electromagnetic Field,” read to the Royal Society December 8th, 1864.

But the molecular theory of gases was still prominently before his mind.

In 1862, writing32 to H. R. Droop, he says:—

And again33 in the same year, 21st April, 1862, to Lewis Campbell:—

The experiments on the viscosity of gases, which formed the Bakerian Lecture to the Royal Society read on February 8th, 1866, were the outcome of this work. His house in 8, Palace Gardens, Kensington, contained a large garret running the complete length.

“To maintain the proper temperature a large fire was for some days kept up in the room in the midst of very hot weather. Kettles were kept on the fire and large quantities of steam allowed to flow into the room. Mrs. Maxwell acted as stoker, which was very exhausting work when maintained for several consecutive hours. After this the room was kept cool for subsequent experiments by the employment of a considerable amount of ice.”

Next year, May, 1866, was read his paper on the “Dynamical Theory of Gases,” in which errors in his former papers, which had been pointed out by Clausius, were corrected.

Meanwhile he had resigned his London Professorship at the end of the Session of 1865, and had been succeeded by Professor W. G. Adams.

For the next four years he lived chiefly at Glenlair, working at his theory of electricity, occasionally, as we shall see, visiting London and Cambridge, and59 taking an active interest in the affairs of his own neighbourhood. In 1865 he had a serious illness, through which he was nursed with great care by Mrs. Maxwell. His correspondence was considerable, and absorbed much of his time. Much also was given to the study of English literature; he was fond of reading Chaucer, Milton, or Shakespeare aloud to Mrs. Maxwell.

He also read much theological and philosophical literature, and all he read helped only to strengthen that firm faith in the fundamentals of Christianity in which he lived and died.

In 1867 he and Mrs. Maxwell paid a visit to Italy, which was a source of great pleasure to both.

His chief scientific work was the preparation of his “Electricity and Magnetism,” which did not appear till 1873; the time was in the main one of quiet thought and preparation for his next great task, the foundation of the School of Physics in Cambridge.

In 1868 the principalship of the United College in the University of St. Andrews was vacant by the resignation of Forbes, and Maxwell was invited by several of the professors to stand. He, however, declined to submit his name to the Crown.

During his retirement at Glenlair from 1865 to 1870 Maxwell was frequently at Cambridge. He examined in the Mathematical Tripos in 1866 and 1867, and again in 1869 and 1870.

The regulations for the Tripos had been in force practically unchanged since 1848, and it was felt by many that the range of subjects included was not sufficiently extensive, and that changes were urgently needed if Cambridge were to retain its position as the centre of mathematical teaching. Natural Philosophy was mentioned in the Schedule, but Natural Philosophy included only Dynamics and Astronomy, Hydrostatics and Physical Optics, with some simple Hydrodynamics and Sound.

The subjects of Heat, Electricity and Magnetism, the Theory of Elastic Solids and Vibrations, Vortex-Motion in Hydrodynamics, and much else, were practically new since 1848. Stokes, Thomson, and Maxwell in England, and Helmholtz in Germany, had created them.

Accordingly in June, 1868, a new plan of examinations was sanctioned by the Senate to come into force in January, 1873, and these various subjects were explicitly included.

Mr. Niven, who was one of those examined by Maxwell in 1866, writes in the preface to the collected works:—

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But the University possessed no means of teaching these subjects, and a Syndicate or Committee was appointed, November 25th, 1868, “to consider the best means of giving instruction to students in Physics, especially in Heat, Electricity and Magnetism, and the methods of providing apparatus for this purpose.”

Dr. Cookson, Master of St. Peter’s College, took an active part in the work of the Syndicate. Professor Stokes, Professor Liveing, Professor Humphry, Dr. Phear, and Dr. Routh were among the members. Maxwell himself was in Cambridge that winter, as Examiner for the Tripos, and his work as Moderator and Examiner in the two previous years had done much to show the necessity of alterations and to indicate the direction which changes should take.

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The Syndicate reported February 27th, 1869. They called attention to the Report of the Royal Commission of 1850. The Commissioners had “prominently urged the importance of cultivating a knowledge of the great branches of Experimental Physics in the University”; and in page 118 of their Report, after commending the manner in which the subject of Physical Optics is studied in the University, and pointing out that “there is, perhaps, no public institution where it is better represented or prosecuted with more zeal and success in the way of original research,” they had stated that “no reason can be assigned why other great branches of Natural Science should not become equally objects of attention, or why Cambridge should not become a great school of physical and experimental, as it is already of mathematical and classical, instruction.”

And again the Commissioners remark: “In a University so thoroughly imbued with the mathematical spirit, physical study might be expected to assume within its precincts its highest and severest tone, be studied under more abstract forms, with more continual reference to mathematical laws, and therefore with better hope of bringing them one by one under the domain of mathematical investigation than elsewhere.”

After calling attention to these statements the Report of the Syndicate then continues:—

“In the scheme of Examination for Honours in the Mathematical Tripos approved by Grace of the Senate on the 2nd of June, 1868, Heat, Electricity and Magnetism, if not introduced for the first time, had a63 much greater degree of importance assigned to them than at any previous period, and these subjects will henceforth demand a corresponding amount of attention from the candidates for Mathematical Honours. The Syndicate have limited their attention almost entirely to the question of providing public instruction in Heat, Electricity and Magnetism. They recognise the importance and advantage of tutorial instruction in these subjects in the several colleges, but they are also alive to the great impulse given to studies of this kind, and to the large amount of additional training which students may receive through the instruction of a public Professor, and by knowledge gained in a well-appointed laboratory.”

“In accordance with these views, and at an early period in their deliberations, they requested the Professors34 of the University, who are engaged in teaching Mathematical and Physical Science, to confer together upon the present means of teaching Experimental Physics, especially Heat, Electricity and Magnetism, and to inform them how the increased requirements of the University in this respect could be met by them.”

“The Professors, so consulted, favoured the Syndicate with a report on the subject, which the Syndicate now beg leave to lay before the Senate. It points out how the requirements of the University might be “partially met,” but the Professors state distinctly that they “do not think that they are able to meet the want of an extensive course of lectures on Physics64 treated as such, and in great measure experimentally. As Experimental Physics may fairly be considered to come within the province of one or more of the above-mentioned Professors, the Syndicate have considered whether now or at some future time some arrangement might not be made to secure the effective teaching of this branch of science, without having resort to the services of an additional Professor. They are, however, of opinion that such an arrangement cannot be made at the present time, and that the exigencies of the case may be best met by founding a new professorship which shall terminate with the tenure of office of the Professor first elected. The services of a man of the highest attainments in science, devoting his life to public teaching as such Professor, and engaged in original research, would be of incalculable benefit to the University.”

The Report goes on to point out that a laboratory would be necessary, and also apparatus. It is estimated that £5,000 would cover the cost of the laboratory, and £1,300 the necessary apparatus. Provision is also made for a demonstrator and a laboratory assistant, and the Report closes with a recommendation that a special Syndicate of Finance should be appointed to consider the means of raising the funds.

The Professors in their Report to the Syndicate point out that teaching in Experimental Physics is needed for the Mathematical Tripos, the Natural Sciences Tripos, certain Special examinations, and the first examination for the degree of M.B. It appeared to them clear that there was work for a new Professor.

In May, 1869, the Financial Syndicate65 recommended by the above Report was appointed “to consider the means of raising the necessary funds for establishing a professor and demonstrator of Experimental Physics, and for providing buildings and apparatus required for that department of science, and further to consider other wants of the University, and the sources from which those wants may be supplied.”

The Syndicate endeavoured to meet the expenditure by inquiry from the several Colleges whether they would be willing to make contributions from their corporate funds, but without success.

“The answers of the Colleges indicated such a want of concurrence in any proposal to raise contributions from the corporate funds of Colleges by any kind of direct taxation that the Syndicate felt obliged to abandon the notion of obtaining the necessary funds from this source, and accordingly to limit the number of objects which they should recommend the Senate to accomplish.”

External authority was necessary before the colleges would submit to taxation for University purposes, and it was left to the Royal Commission of 1877 to carry into effect many of the suggestions made by the Syndicate. Meanwhile they contented themselves with recommending means for raising an annual stipend of £660 for the professor, demonstrator, and assistant, and a capital sum of £5,000, or thereabouts, for the expenses of a building.

The Syndicate’s Report was issued in an amended form in the May term of 1870, and before any decision was taken on it the Vice-Chancellor, Dr. Atkinson, on66 October 13th, 1870, published “the following munificent offer of his grace the Duke of Devonshire, the Chancellor of the University,” who had been chairman of the Commission on Scientific Education.

By his generous action the University was relieved from all expense connected with the building. A Grace establishing a Professorship of Experimental Physics was confirmed by the Senate February 9th, 1871, and March 8th was fixed for the election.

Meanwhile who was to be Professor? Sir W. Thomson’s name had been mentioned, but he, it was known, would not accept the post. Maxwell was then applied to, and at first he was unwilling to leave Glenlair. Professor Stokes, the Hon. J. W. Strutt67 (Lord Rayleigh), Mr. Blore of Trinity, and others wrote to him. Lord Rayleigh’s letter35 is as follows:

On the 15th of February, Maxwell wrote to Mr. Blore:—

However, on the 24th of February, Mr. Blore wrote to the Electoral Roll:—

“I am authorised to give notice that Mr. John (sic) Clerk Maxwell, F.R.S., formerly Professor of Natural Philosophy at Aberdeen, and at King’s College, London, is a candidate for the professorship of Experimental Physics.”

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Maxwell was elected without opposition. Writing36 to his wife from Cambridge, 20th March, 1871, he says:—

While in a letter37 from Glenlair to C. J. Munro, dated March 15th, 1871, he writes:—“The Experimental Physics at Cambridge is not built yet, but we are going to try. The desideratum is to set a Don and a Freshman to observe and register (say) the vibrations of a magnet together, or the Don to turn a watch and the Freshman to observe and govern him.”

In October he delivered his Introductory Lecture. A few quotations will show the spirit in which he approached his task.

Another expression of his views on the position of Physics at the time will be found in his address to Section A of the British Association, when President at the Liverpool meeting of 1870.

But the laboratory was not yet built. A Syndicate, of which Maxwell was a member, was appointed to consider the question of a site, to take professional advice, and to obtain plans and estimates. Professor Maxwell and Mr. Trotter visited various laboratories at home and abroad for the purpose of ascertaining the best arrangements. Mr. W. M. Fawcett was appointed architect; the tender of Mr. John Loveday, of Kebworth, for the building at a cost of £8,450, exclusive of gas, water, and heating, was accepted in March, 1872, and the building38 was begun during the summer.

In the meantime Maxwell began to lecture, finding a home where he could.

It was not till June, 1874, that the building was complete, and on the 16th the Chancellor formally presented his gift of the Cavendish Laboratory to the University. In the correspondence previous to this time it was spoken of as the Devonshire Laboratory. The name Cavendish commemorated the work of the great physicist of a century earlier, whose writings74 Maxwell was shortly to edit, as well as the generosity of the Chancellor.

In their letter of thanks to the Duke of Devonshire the University write:—

“Unde vero conventius poterat illis artibus succurri quam e tua domo quæ in ipsis jam pridem inclaruerat. Notum est Henricum Cavendish quem secutus est Coulombius primum ita docuisse, quæ sit vis electrica ut eam numerorum modulis illustraret; adhibitis rationibus quas hodie veras esse constat.” And they suggest the name as suitable for the building. To this the Chancellor replied, after referring to the work of Henry Cavendish: “Quod pono in officinâ ipsâ nuncupandâ nomen ejus commemorare dignati sitis, id grato animo accepi.”

The building had cost far more than the original estimate, but the Chancellor’s generosity was not limited, and on July 21st, 1874, he wrote to the Vice-Chancellor:—

“It is my wish to provide all instruments for the Cavendish Laboratory which Professor Maxwell may consider to be immediately required, either in his lectures or otherwise.”

Maxwell prepared a list, but explained while doing it that time and thought were necessary to secure the best form of instruments; and he continues, writing to the Vice-Chancellor: “I think the Duke fully understood from what I said to him that to furnish the Laboratory will be a matter of several years’ duration. I shall consider myself, however,” he says, “at liberty to contribute to the Laboratory any instruments which I have had constructed in former years, and75 which may be found still useful, and also from time to time to procure others for special researches.”

In 1877 in his annual report Professor Maxwell announced that the Chancellor39 had now “completed his gift to the University by furnishing the Cavendish Laboratory with apparatus suited to the present state of science.”

The stock of apparatus, however, was still small, although Maxwell in the most generous manner himself spent large sums in adding to it; for the Professor was most particular in procuring only expensive instruments by the best makers, with such additional improvements as he could himself suggest.

In March, 1874, a Demonstratorship of Physics had been established, and Mr. Garnett of St. John’s College was appointed.

Work began in the laboratory in October, 1874. At first the number of students was small. Only seventeen names appear in the Natural Sciences Tripos40 list for 1874, and few of those did Physics.

The fear alluded to by the Professor in his introductory lecture, that men reading for the Mathematical76 Tripos would not find time for attendance at the laboratory, was justified. One of the weaknesses of our Cambridge plan has been the divorce between Mathematics and experimental work, encouraged by our system of examinations. Experimental knowledge is supposed not to be needed for the Mathematical Tripos; the Mathematics permitted in the Natural Sciences Tripos are very simple; thus it came about that few men while reading for the Mathematical Tripos attended the laboratory, and this unfortunate result was intensified by the action of the University in 1877–78, when the regulations for the Mathematical Tripos were again altered.41

Still there were pupils eager and willing to work, though they were chiefly men who had already taken their B.A. degree, and who wished to continue Physical reading and research, even though it involved “a considerable amount of dull labour not altogether attractive.” My own work there began in 1876, and it may be interesting if I recall my reminiscences of that time.

The first experiments I can recollect related to the measurement of electrical resistance. I well remember77 Maxwell explaining the principle of Wheatstone’s bridge, and my own wish at the time that I had come to the laboratory before the Tripos, instead of afterwards. Lord Rayleigh had, during the examination, set an easy question which I failed to do for want of some slight experimental knowledge, and the first few words of Maxwell’s talk showed me the solution.

I did not attend his lectures regularly—they were given, I think, at an hour which I was obliged to devote to teaching; besides, there was his book, the “Electricity and Magnetism,” into which I had just dipped before the Tripos, to work at.

Chrystal and Saunder were then busy at their verification of Ohm’s law. They were using a number of the Thomson form of tray Daniell’s cells, and Maxwell was anxious for tests of various kinds to be made on these cells; these I undertook, and spent some time over various simple measurements on them. He then set me to work at some of the properties of a stratified dielectric, consisting, if I remember rightly, of sheets of paraffin paper and mica. By this means I became acquainted with various pieces of apparatus. There were no regular classes and no set drill of demonstrations arranged for examination purposes; these came later. In Maxwell’s time those who wished to work had the use of the laboratory and assistance and help from him, but they were left pretty much to themselves to find out about the apparatus and the best methods of using it.

Rather later than this Schuster came and did some of his spectroscope work. J. E. H. Gordon was busy with the preliminary observations for his78 determination of Verdet’s constant, and Niven had various electrical experiments on hand; while Fleming was at work on the B. A. resistance coils.

My own tastes lay in the direction of optics. Maxwell was anxious that I should investigate the properties of certain crystals. I think they were the chlorate of potash crystals, about which Stokes and Rayleigh have since written; but these crystals were to be grown, a slow process which would, he supposed, take years; and as I wished to produce a dissertation for the Trinity Fellowship examination in 1877, that work had to be laid aside.

Eventually I selected as a subject the form of the wave surface in a biaxial crystal, and set to work in a room assigned to me. The Professor used to come in on most days to see how I was getting on. Generally he brought his dog, which sometimes was shut up in the next room while he went to college. Dogs were not allowed in college, and Maxwell had an amusing way of describing how Toby once wandered into Trinity, and by some doggish instinct discovered immediately, to his intense amazement, that he was in a place where no dogs had been since the college was. Toby was not always quiet in his master’s absence, and his presence in the next room was somewhat disturbing.

When difficulties occurred Maxwell was always ready to listen. Often the answer did not come at once, but it always did come after a little time. I remember one day, when I was in a serious dilemma, I told him my long tale, and he said:—

“Well, Chrystal has been talking to me, and79 Garnett and Schuster have been asking questions, and all this has formed a good thick crust round my brain. What you have said will take some time to soak through, but we will see about it.” In a few days he came back with—“I have been thinking over what you said the other day, and if you do so-and-so it will be all right.”

My dissertation was referred to him, and on the day of the election, when returning to Cambridge for the admission, I met him at Bletchley station, and well remember his kind congratulations and words of warm encouragement.

For the next year and a half I was working regularly at the laboratory and saw him almost daily during term time.

Of these last years there really is but little to tell. His own scientific work went on. The “Electricity and Magnetism” was written mostly at Glenlair. About the time of his return to Cambridge, in October, 1872, he writes42 to Lewis Campbell:—

The book was published in 1873. The Text-book of Heat was written during the same period, while “Matter and Motion,” “a small book on a great subject,” was published in 1876.

In 1873 and 1874 he was one of the examiners for the Natural Sciences Tripos, and in 1873 he was the80 first additional examiner for the Mathematical Tripos, in accordance with the scheme which he had done so much to promote in 1868.

Many of his shorter papers were written about the same time. The ninth edition of the Encyclopædia Britannica was being published, and Professor Baynes had enlisted his aid in the work. The articles “Atom,” “Attraction,” “Capillary Action,” “Constitution of Bodies,” “Diffusion,” “Ether,” “Faraday,” and others are by him.

He also wrote a number of papers for Nature. Some of these are reviews of books or accounts of scientific men, such as the notices of Faraday and Helmholtz, which appeared with their portraits; others again are original contributions to science. Among the latter many have reference to the molecular constitution of bodies. Two lectures—the first on “Molecules,” delivered before the British Association at Bradford in 1873; the second on the “Dynamical Evidence of the Molecular Constitution of Bodies,” delivered before the Chemical Society in 1875—were of special importance. The closing sentences of the first lecture have been often quoted. They run as follow:—

This was criticised in Nature by Mr. C. J. Munro, and at a later time by Clifford in one of his essays.

Some correspondence with the Bishop of Gloucester and Bristol on the authority for the comparison of molecules to manufactured articles is83 given by Professor Campbell, and in it Maxwell points out that the latter part of the article “Atom” in the Encyclopædia is intended to meet Mr. Munro’s criticism.

In 1874 the British Association met at Belfast, under the presidency of Tyndall. Maxwell was present, and published afterwards in Blackwood’s Magazine an amusing paraphrase of the president’s address. This, with some other verses written at about the same time, may be quoted here. Professor Campbell has collected a number of verses written by Maxwell at various times, which illustrate in an admirable manner both the grave and the gay side of his character.

BRITISH ASSOCIATION, 1874.

Notes of the President’s Address.

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MOLECULAR EVOLUTION.

Belfast, 1874.

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TO THE COMMITTEE OF THE CAYLEY PORTRAIT FUND.

1874.

IN MEMORY OF EDWARD WILSON,

Who repented of what was in his mind to write after section.

Rigid Body (sings).

Another task, which occupied much time, from 1874 to 1879, was the edition of the works of Henry Cavendish. Cavendish, who was great-uncle to the Chancellor, had published only two electrical papers, but he had left some twenty packets of manuscript on Mathematical and Experimental Electricity. These were placed in Maxwell’s hands in 1874 by the Duke of Devonshire.

Niven, in his preface to the collected papers dealing with this book, writes thus:—

During the last years of his life Mrs. Maxwell had a serious and prolonged illness, and Maxwell’s work was much increased by his duties as sick nurse. On one occasion he did not sleep in a bed for three weeks,89 but conducted his lectures and experiments at the laboratory as usual.

About this time some of those who had been “Apostles” in 1853–57 revived the habit of meeting together for discussion. The club, which included Professors Lightfoot, Hort and Westcott, was christened the “Eranus,” and three of Maxwell’s contributions to it have been preserved and are printed by Professor Campbell.

After the Cavendish papers were finished, Maxwell had more time for his own original researches, and two important papers were published in 1879. The one on “Stresses in Rarefied Gases arising from Inequalities of Temperature” was printed in the Royal Society’s Transactions, and deals with the Theory of the Radiometer; the other on “Boltzmann’s Theorem” appears in the Transactions of the Cambridge Philosophical Society. In the previous year he had delivered the Rede lecture on “The Telephone.” He also began to prepare a second edition of “Electricity and Magnetism.”

His health gave way during the Easter term of 1879; indeed for two years previously he had been troubled with dyspeptic symptoms, but had consulted no one on the subject. He left Cambridge as usual in June, hoping that he would quickly recover at Glenlair, but he grew worse instead. In October he was told by Dr. Sanders of Edinburgh that he had not a month to live. He returned to Cambridge in order to be under the care of Dr. Paget, who was able in some measure to relieve his most severe suffering but the disease, of which his mother had died at the90 same age, continued its progress, and he died on November 5th. His one care during his last illness was for those whom he left behind. Mrs. Maxwell was an invalid dependent on him for everything, and the thought of her helplessness was the one thing which in these last days troubled him.

A funeral service took place in the chapel at Trinity College, and afterwards his remains were conveyed to Scotland and interred in the family burying-place at Corsock, Kirkcudbright.

A memorial edition of his works was issued by the Cambridge University Press in 1890. A portrait by Lowes Dickinson hangs in the hall of Trinity College, and there is a bust by Boehm in the laboratory.

After his death Mrs. Maxwell gave his scientific library to the Cavendish Laboratory, and on her death she left a sum of about £6,000 to found a scholarship in Physics, to be held at the laboratory.

The preceding pages contain some account of Clerk Maxwell’s life as a man of science. His character had other sides, and any life of him would be incomplete without some brief reference to these. His letters to his wife and to other intimate friends show throughout his life the depth of his religious convictions. The high purpose evidenced in the paper given to the present Dean of Canterbury when leaving Cambridge, animated him continually, and appears from time to time in his writings. The student’s evening hymn, composed in 1853 when still an undergraduate, expresses the same feelings—

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His views on the relation of Science to Faith are given in his letter47 to Bishop Ellicott already referred to—

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And again in his letter48 to the secretary of the Victoria Institute giving his reasons for declining to become a member—

Professor Campbell and Mr. Garnett have given us the evidence of those who were with him in his last days, as to the strength of his own faith. On his death bed he said that he had been occupied in trying to gain truth; that it is but little of truth that man can acquire, but it is something to know in whom we have believed.

Fifteen years only have passed since the death of Clerk Maxwell, and it is almost too soon to hope to form a correct estimate of the value of his work and its relation to that of others who have laboured in the same field.

Thus Niven, at the close of his obituary notice in the Proceedings of the Royal Society, says: “It is seldom that the faculties of invention and exposition, the attachment to physical science and capability of developing it mathematically, have been found existing in one mind to the same degree. It would, however, require powers somewhat akin to Maxwell’s own to describe the more delicate features of the works resulting from this combination, every one of which is stamped with the subtle but unmistakable impress of genius.” And again in the preface to Maxwell’s works, issued in 1890, he wrote: “Nor does it appear to the present editor that the time has yet arrived when the quickening influence of Maxwell’s mind on modern scientific thought can be duly estimated.”

It is, however, the object of the present series to attempt to give some account of the work of men of science of the last hundred years, and to show how each has contributed his share to our present stock of knowledge. This task, then, remains to be done.94 While attempting it I wish to express my indebtedness to others who have already written about Maxwell’s scientific work, especially to Mr. W. D. Niven, whose preface to the Maxwell papers has been so often referred to; to Mr. Garnett, the author of Part II. of the “Life of Maxwell,” which deals with his contributions to science; and to Professor Tait, who in Nature for February 5th, 1880, gave an account of Clerk Maxwell’s work, “necessarily brief, but sufficient to let even the non-mathematical reader see how very great were his contributions to modern science”—an account all the more interesting because, again to quote from Professor Tait, “I have been intimately acquainted with him since we were schoolboys together.”

Maxwell’s main contributions to science may be classified under three heads—“Colour Perception,” “Molecular Physics,” and “Electrical Theories.” In addition to these there were other papers of the highest interest and importance, such as the essay on “Saturn’s Rings,” the paper on the “Equilibrium of Elastic Solids,” and various memoirs on pure geometry and questions of mechanics, which would, if they stood alone, have secured for their author a distinguished position as a physicist and mathematician, but which are not the works by which his name will be mostly remembered.

The work on “Colour Perception” was begun at an early date. We have seen Maxwell while still at Edinburgh interested in the discussions about Hay’s theories.

His first published paper on the subject was a95 letter to Dr. G. Wilson, printed in the Transactions of the Royal Society of Arts for 1855; but he had been mixing colours by means of his top for some little time previously, and the results of these experiments are given in a paper entitled “Experiments on Colour,” communicated to the Royal Society of Edinburgh by Dr. Gregory, and printed in their Transactions, vol. xxi.

In the paper on “The Theory of Compound Colours,” printed in the Philosophical Transactions for 1860, Maxwell gives a history of the theory as it was known to him.

He points out first the distinction between the optical properties and the chromatic properties of a beam of light. “The optical properties are those which have reference to its origin and propagation through media until it falls on the sensitive organ of vision;” they depend on the periods and amplitudes of the ether vibrations which compose the beam. “The chromatic properties are those which have reference to its power of exciting certain sensations of colour perceived through the organ of vision.” It is possible for two beams to be optically very different and chromatically alike. The converse is not true; two beams which are optically alike are also chromatically alike.

The foundation of the theory of compound colours was laid by Newton. He first shewed that “by the mixture of homogeneal light colours may be produced which are like to the colours of homogeneal light as to the appearance of colour, but not as to the immutability of colour and constitution of light.” Two96 beams which differ optically may yet be alike chromatically; it is possible by mixing red and yellow to obtain an orange colour chromatically similar to the orange of the spectrum, but optically different to that orange, for the compound orange can be analysed by a prism into its component red and yellow; the spectrum orange is incapable of further resolution.

Newton also solves the following problem:—

In a mixture of primary colours, the quantity and quality of each being given to know the colour of the compound (Optics, Book 1, Part 2, Prop. 6), and his solution is the following:—He arranges the seven colours of the spectrum round the circumference of a circle, the length occupied by each colour being proportional to the musical interval to which, in Newton’s views, the colour corresponded. At the centre of gravity of each of these arcs he supposes a weight placed proportional to the number of rays of the corresponding colour which enter into the mixture under consideration. The position of the centre of gravity of these weights indicates the nature of the resultant colour. A radius drawn through this centre of gravity points out the colour of the spectrum which it most resembles; the distance of the centre of gravity from the centre gives the fulness of the colour. The centre itself is white. Newton gives no proof of this rule; he merely says, “This rule I conceive to be accurate enough for practice, though not mathematically accurate.”

Maxwell proved that Newton’s method of finding the centre of gravity of the component colours was confirmed by his observations, and that it involves97 mathematically the theory of three elements of colour; but the disposition of the colours on the circle was only a provisional arrangement; the true relations of the colours could only be determined by direct experiment.

Thomas Young appears to have been the next, after Newton, to work at the theory of colour sensation. He made observations by spinning coloured discs much in the same way as that which was afterwards adopted by Maxwell, and he developed the theory that three different primary sensations may be excited in the eye by light, while the colour of any beam depends on the proportions in which these three sensations are excited. He supposes the three primary sensations to correspond to red, green, and violet. A blue ray is capable of exciting both the green and the violet; a yellow ray excites the red and the green. Any colour, according to Young’s theory, may be matched by a mixture of these three primary colours taken in proper proportion; the quality of the colour depends on the proportion of the intensities of the components; its brightness depends on the sum of these intensities.

Maxwell’s experiments were undertaken with the object of proving or disproving the physical part of Young’s theory. He does not consider the question whether there are three distinct sensations corresponding to the three primary colours; that is a physiological inquiry, and one to which no completely satisfactory answer has yet been given. He does show that by a proper mixture of any three arbitrarily chosen standard colours it is possible to match any98 other colour; the words “proper mixture,” however, need, as will appear shortly, some development.

We may with advantage compare the problem with one in acoustics.

When a compound musical note consisting of a pure tone and its overtones is sounded, the trained ear can distinguish the various overtones and analyse the sound into its simple components. The same sensation cannot be excited in two different ways. The eye has no such corresponding power. A given yellow may be a pure spectral yellow, corresponding to a pure tone in music, or it may be a mixture of a number of other pure tones; in either case it can be matched by a proper combination of three standard colours—this Maxwell proved. It may be, as Young supposed, that if the three standard colours be properly selected they correspond exactly to three primary sensations of the brain. Maxwell’s experiments do not afford any light on this point, which still remains more than doubtful.

When Maxwell began his work the theory of colours was exciting considerable interest. Sir David Brewster had recently developed a new theory of colour sensation which had formed the basis of some discussions, and in 1852 von Helmholtz published his first paper on the subject. According to Brewster, the three primitive colours were red, yellow and blue, and he supposed that they corresponded to three different kinds of objective light. Helmholtz pointed out that experiments up to that date had been conducted by mixing pigments, with the exception of those in which the rotating disc was used, and that it is99 necessary to make them on the rays of the spectrum itself. He then describes a method of mixing the light from two spectra so as to obtain the combination of every two of the simple prismatic rays in all degrees of relative strength.

From these experiments results, which at the time were unexpected, but some of which must have been known to Young, were obtained. Among them it was shown that a mixture of red and green made yellow, while one of green and violet produced blue.

In a later paper (Philosophical Magazine, 1854) Helmholtz described a method for ascertaining the various pairs of complementary colours—colours, that is, which when mixed will give white—which had been shown by Grassman to exist if Newton’s theory were true. He also gave a provisional diagram of the curve formed by the spectrum, which ought to take the place of the circle in Newton’s diagram; for this, however, his experiments did not give the complete data.

Such was the state of the question when Maxwell began. His first colour-box was made in 1852. Others were designed in 1855 and 1856, and the final paper appeared in 1860. But before that time he had established important results by means of his rotatory discs and colour top. In his own description of this he says: “The coloured paper is cut into the form of disc, each with a hole in the centre and divided along a radius so as to admit of several of them being placed on the same axis, so that part of each is exposed. By slipping one disc over another we can expose any given portion of each colour.100 These discs are placed on a top or teetotum, which is spun rapidly. The axis of the top passes through the centre of the discs, and the quantity of each colour exposed is measured by graduations on the rim of the top, which is divided into 100 parts. When the top is spun sufficiently rapidly, the impressions due to each colour separately follow each other in quick succession at each point of the retina, and are blended together; the strength of the impression due to each colour is, as can be shown experimentally, the same as when the three kinds of light in the same relative proportions enter the eye simultaneously. These relative proportions are measured by the areas of the various discs which are exposed. Two sets of discs of different radius are used; the largest discs are put on first, then the smaller, so that the centre portion of the top shows the colour arising from the mixture of those of the smaller discs; the outer portion, that of the larger discs.”

In experimenting, six discs of each size are used, black, white, red, green, yellow and blue. It is found by experiment that a match can be arranged between any five of these. Thus three of the larger discs are placed on the top—say black, yellow and blue—and two of the smaller discs, red and green, are placed above these. Then it is found that it is possible so to adjust the amount exposed of each disc that the two parts of the top appear when it is spun to be of the same tint. In one series of experiments the chromatic effect of 46·8 parts of black, 29·1 of yellow, and 24·1 of blue was found to be the same as that of 66·6 of101 red and 33·4 of green; each set of discs has a dirty yellow tinge.

Now, in this experiment, black is not a colour; practically no light reaches the eye from a dead black. We have, however, to fill up the circumference of the top in some way which will not affect the impression on the retina arising from the mixture of the blue and yellow; this we can do by using the black disc.

Thus we have shown that 66·6 parts of red and 33·4 parts of green produce the same chromatic effect as 29·1 of yellow and 24·1 of blue. Similarly in this manner a match can be arranged between any four colours and black, the black being necessary to complete the circumference of the discs.

Thus using A, B, C, D to denote the various colours, a, b, c, d the amounts of each colour taken, we can get a series of results expressed as follows: a parts of A together with b parts of B match c parts of C together with d parts of D; or we may write this as an equation thus:—

a A + b B = c C + d D,

where the + stands for “combined with,” and the = for “matches in tint.”

We may also write the above—

d D = a A + b B - c C,

or d parts of D can be matched by a proper combination of colours A, B, C. The sign - shows that in order to make the match we have to combine the colour C with D; the combination then matches a mixture of A and B.

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In this way we can form a number of equations for all possible colours, and if we like to take any three colours A, B, C as standards, we obtain a result which may be written generally—

x X = a A + b B + c C,

or x parts of X can be matched by a parts of A, combined with b parts of B and c parts of C. If the sign of one of the quantities a, b, or c is negative, it indicates that that colour must be combined with X to match the other two.

Now Maxwell was able to show that, if A, B, C be properly selected, nearly every other colour can be matched by positive combinations of these three. These three colours, then, are primary colours, and nearly every other colour can be matched by a combination of the three primary colours.

Experiments, however, with coloured discs, such as were undertaken by Young, Forbes and Maxwell, were not capable of giving satisfactory results. The colours of the discs were not pure spectrum colours, and varied to some extent with the nature of the incident light. It was for this reason that Helmholtz in 1852 experimented with the spectrum, and that Maxwell about the same time invented his colour box.

The principle of the latter was very simple. Suppose we have a slit S, and some arrangement for forming a pure spectrum on a screen. Let there now be a slit R placed in the red part of the spectrum on the screen. When light falls on the slit S, only the red rays can reach R, and hence conversely, if the103 white source be placed at the other end of the apparatus, so that R is illuminated with white light, only red rays will reach S. Similarly, if another slit be placed in the green at G, and this be illuminated by white light, only the green rays will reach S, while from a third slit V in the violet, violet light only can arrive at S. Thus by opening the three slits at V, G and R simultaneously, and looking through S, the retina receives the impression of the three different colours. The amount of light of each colour will depend on the breadth to which the corresponding slit is opened, and the relative intensities of the three different components can be compared by comparing the breadths of the three slits. Any other colour which is allowed by some suitable contrivance to enter the eye simultaneously can now be matched, provided the red, green and violet are primary colours.

By means of experiments with the colour box Maxwell showed conclusively that a match could be obtained between any four colours; the experiments could not be carried out in quite the simple manner suggested by the above description of the principle of the box. An account of the method will be found in Maxwell’s own paper. It consisted in matching a standard white by various combinations of other colours.

The main object of his research, however, was to examine the chromatic properties of the different parts of the spectrum, and to determine the form of the curve which ought to replace the circle in Newton’s diagram of colour.

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Maxwell adopted as his three standard colours: red, of about wave length 6,302; green, wave length 5,281; and violet, 4,569 tenth metres. On the scale of Maxwell’s instrument these are represented by the numbers 24, 44 and 68.

Let us take three points A, B, C at the corners of an equilateral triangle to represent on a diagram these three colours. The position of any other colour on the diagram will be found by taking weights proportional to the amounts of the colours A, B, C required to make the match between A, B, C and the given colour; these weights are placed at A, B, C respectively; the position of their centre of gravity is the point required. Thus the position of white is given by the equation—

W = 18·6 (24) + 31·4 (44) + 30·5 (68)

which means that weights proportional to 18·6, 31·4 and 30·5 are to be placed at A, B, C respectively, and their centre of gravity is to be found. The point so found is the position of white. Any other colour is given by the equation—

X = a (24) + b (44) + c (68).

Again, the position on the diagram for all colours for which a, b, c are all positive lies within the triangle A B C. If one of the coefficients, say c, is negative the same construction applies, but the weight applied at C must be treated as acting in the opposite direction to those at A and B. A mixture of the given colour and C matches a mixture of A and B. It is clear that the point corresponding to X will then lie outside the triangle105 A B C. Maxwell showed that, with his standards, nearly all colours could be represented by points inside the triangle. The colours he had selected as standards were very close to primary colours.

Again, he proved that any spectrum colour between red and green, when combined with a very slight admixture of violet, could be matched, in the case of either Mrs. Maxwell or himself, by a proper mixture of the red and green. The positions, therefore, of the spectrum colours between red and green lie just outside the triangle A B C, being very close to the line A B, while for the colours between green and violet Maxwell obtained a curve lying rather further outside the side B C. Any spectrum colour between green and violet, together with a slight admixture of red, can be matched by a proper mixture of green and violet.

Thus the circle of Newton’s diagram should be replaced by a curve, which coincides very nearly with the two sides A B and B C of Maxwell’s figure. Strictly, according to his observations, the curve lies just outside these two sides. The purples of the spectrum lie nearly along the third side, C A, of the triangle, being obtained approximately by mixing the violet and the red.

To find the point on the diagram corresponding to the colour obtained by mixing any two or more spectrum colours we must, in accordance with Newton’s rule, place weights at the points corresponding to the selected colours, and find the centre of gravity of these weights.

This, then, was the outcome of Maxwell’s work on106 colour. It verified the essential part of Newton’s construction, and obtained for the first time the true form of the spectrum curve on the diagram.

The form of this curve will of course depend on the eye of the individual observer. Thus Maxwell and Mrs. Maxwell both made observations, and distinct differences were found in their eyes. It appears, however, that a large majority of persons have normal vision, and that matches made by one such person are accepted by most others as satisfactory. Some people, however, are colour blind, and Maxwell examined a few such. In the case of those whom he examined it appeared as though vision was dichromatic, the red sensation seemed to be absent; nearly all colours could be matched by combinations of green and violet. The colour diagram was reduced to the straight line B C. Other forms of colour blindness have since been investigated.

In awarding to Maxwell the Rumford medal in 1860, Major-General Sabine, vice-president of the Royal Society, after explaining the theory of colour vision and the possible method of verifying it, said: “Professor Maxwell has subjected the theory to this verification, and thereby raised the composition of colours to the rank of a branch of mathematical physics,” and he continues: “The researches for which the Rumford medal is awarded lead to the remarkable result that to a very near degree of approximation all the colours of the spectrum, and therefore all colours in nature which are only mixtures of these, can be perfectly imitated by mixtures of three actually attainable colours, which are the red, green107 and blue belonging respectively to three particular parts of the spectrum.”

It should be noticed in concluding our remarks on this part of Maxwell’s work that his results are purely physical. They are not inconsistent with the physiological part of Young’s theory, viz., that there are three primary sensations of colour which can be transmitted to the brain, and that the colour of any object depends on the relative proportions in which these sensations are excited, but they do not prove that theory. Any physiological theory which can be accepted as true must explain Maxwell’s observations, and Young’s theory does this; but it is, of course, possible that other theories may explain them equally well, and be more in accordance with physiological observations than Young’s. Maxwell has given us the physical facts which have to be explained; it is for the physiologists to do the rest.

Maxwell in his article “Atom,” in the ninth edition of the Encyclopædia Britannica, has given some account of Modern Molecular Science, and in particular of the molecular theory of gases. Of this science, Clausius and Maxwell are the founders, though to their names it has recently been shown that a third, that of Waterston, must be added. In the present chapter it is intended to give an outline of Maxwell’s contributions to molecular science, and to explain the advances due to him.

The doctrine that bodies are composed of small particles in rapid motion is very ancient. Democritus was its founder, Lucretius—de Rerum Naturâ—explained its principles. The atoms do not fill space; there is void between.

According to Boscovitch an atom is an indivisible point, having position in space, capable of motion, and possessing mass. It is also endowed with the power of exerting force, so that two atoms attract or repel each other with a force depending on their distance109 apart. It has no parts or dimensions: it is a mere geometrical point without extension in space; it has not the property of impenetrability, for two atoms can, it is supposed, exist at the same point.

In modern molecular science according to Maxwell, “we begin by assuming that bodies are made up of parts each of which is capable of motion, and that these parts act on each other in a manner consistent with the principle of the conservation of energy. In making these assumptions we are justified by the facts that bodies may be divided into smaller parts, and that all bodies with which we are acquainted are conservative systems, which would not be the case unless their parts were also conservative systems.

“We may also assume that these small parts are in motion. This is the most general assumption we can make, for it includes as a particular case the theory that the small parts are at rest. The phenomena of the diffusion of gases and liquids through each other show that there may be a motion of the small parts of a body which is not perceptible to us.

“We make no assumption with respect to the nature of the small parts—whether they are all of one magnitude. We do not even assume them to have extension and figure. Each of them must be measured by its mass, and any two of them must, like visible bodies, have the power of acting on one another when they come near enough to do so. The properties of the body or medium are determined by the configuration of its parts.”

These small particles are called molecules, and a110 molecule in its physical aspect was defined by Maxwell in the following terms:—

We are to look upon a gas as an assemblage of molecules flying about in all directions. The path of any molecule is a straight line, except during the time when it is under the action of a neighbouring molecule; this time is usually small compared with that during which it is free.

The simplest theory we could formulate would be that the molecules behaved like elastic spheres, and that the action between any two was a collision following the laws which we know apply to the collision of elastic bodies. If the average distance between two molecules be great compared with their dimensions, the time during which any molecule is in collision will be small compared with the interval between the collisions, and this is in accordance with the fundamental assumption just mentioned. It is not, however, necessary to suppose an encounter between two molecules to be a collision. One molecule may act on another with a force, which depends on the distance between them, of such a character that the force is insensible except when the molecules are extremely close together.

It is not difficult to see how the pressure exerted111 by a gas on the sides of a vessel which contains it may be accounted for on this assumption. Each molecule as it strikes the side has its momentum reversed—the molecules are here assumed to be perfectly elastic.

Thus each molecule of the gas is continually gaining momentum from the sides of the vessel, while it gives up to the vessel the momentum which it possessed before the impact. The rate at which this change of momentum proceeds across a given area measures the force exerted on that area; the pressure of the gas is the rate of change of momentum per unit of area of the surface.

Again, it can be shown that this pressure is proportional to the product of the mass of each molecule, the number of molecules in a unit of volume, and the square of the velocity of the molecules.

Let us consider in the first instance the case of a jet of sand or water of unit cross section which is playing against a surface. Suppose for the present that all the molecules which strike the surface have the same velocity.

Then the number of molecules which strike the surface per second, will be proportional to this velocity. If the particles are moving quickly they can reach the surface in one second from a greater distance than is possible if they be moving slowly. Again, the number reaching the surface will be proportional to the number of molecules per unit of volume. Hence, if we call v the velocity of each particle, and N the number of particles per unit of volume, the number which strike the surface in one second will be N v;112 if m be the mass of each molecule, the mass which strikes the surface per second is N m v; the velocity of each particle of this mass is v, therefore the momentum destroyed per second by the impact is N m v × v, or N m v², and this measures the pressure.

Hence in this case if p be the pressure

p = N m v².

In the above we assume that all the molecules in the jet are moving with velocity v perpendicular to the surface. In the case of a crowd of molecules flying about in a closed space this is clearly not true. The molecules may strike the surface in any direction; they will not all be moving normal to the surface. To simplify the case, consider a cubical box filled with gas. The box has three pairs of equal faces at right angles. We may suppose one-third of the particles to be moving at right angles to each face, and in this case the number per unit volume which we have to consider is not N, but ⅓ N. Hence the formula becomes p = ⅓ N m v².

Moreover, if ρ be the density of the gas—that is, the mass of unit volume—then Nm is equal to ρ, for m is the mass of each particle, and there are N particles in a unit of volume.

Hence, finally, p = ⅓ ρ v².

Or, again, if V be the volume of unit mass of the gas, then ρ V is unity, or ρ is equal to 1/V.

Hence pV = ⅓v².

Formulæ equivalent to these appear first to have been obtained by Herapath about the year 1816 (Thomson’s “Annals of Philosophy,” 1816). The113 results only, however, were stated in that year. A paper which attempted to establish them was presented to the Royal Society in 1820. It gave rise to very considerable correspondence, and was withdrawn by the author before being read. It is printed in full in Thomson’s “Annals of Philosophy” for 1821, vol. i., pp. 273, 340, 401. The arguments of the author are no doubt open to criticism, and are in many points far from sound. Still, by considering the problem of the impact of a large number of hard bodies, he arrived at a formula connecting the pressure and volume of a given mass of gas equivalent to that just given. These results are contained in Propositions viii. and ix. of Herapath’s paper.

In his next step, however, Herapath, as we know now, was wrong. One of his fundamental assumptions is that the temperature of a gas is measured by the momentum of each of its particles. Hence, assuming this, we have T = m v, if T represents the temperature: and

p = ⅓ N m v² = ⅓ (N/m) (m v)².

Or, again—

p = ⅓ N·T·v = ⅓·(N/m)·T².

These results are practically given in Proposition viii., Corr. (1) and (2), and Proposition ix.49 The114 temperature as thus defined by Herapath is an absolute temperature, and he calculates the absolute zero of temperature at which the gas would have no volume from the above results. The actual calculation is of course wrong, for, as we know now by experiment, the pressure is proportional to the temperature, and not to its square, as Herapath supposed. It will be seen, however, that Herapath’s formula gives Boyle’s law; for if the temperature is constant, the formula is equivalent to

p V = a constant.

Herapath somewhat extended his work in his “Mathematical Physics” published in 1847, and applied his principles to explain diffusion, the relation between specific heat and atomic weight, and other properties of bodies. He still, however, retained his erroneous supposition that temperature is to be measured by the momentum of the individual particles.

The next step in the theory was made by Waterston. His paper was read to the Royal Society on March 5th, 1846. It was most unfortunately committed to the Archives of the Society, and was only disinterred by Lord Rayleigh in 1892 and printed in the Transactions for that year.

In the account just given of the theory, it has been supposed that all the particles move with the same velocity. This is clearly not the case in a gas. If at starting all the particles had the same velocity, the collisions would change this state of affairs. Some particles will be moving quickly, some slowly. We may,115 however, still apply the theory by splitting up the particles into groups, and, supposing that each group has a constant velocity, the particles in this group will contribute to the pressure an amount—p₁—equal to ⅓ N₁ m v₁², where v₁ is the velocity of the group and N₁ the number of particles having that velocity. The whole pressure will be found by adding that due to the various groups, and will be given as before by p = ⅓ N m v², where v is not now the actual velocity of the particles, but a mean velocity given by the equation

N v² = N₁ v₁² + N₂ v₂² + .....,

which will produce the same pressure as arises from the actual impacts. This quantity v² is known as the mean square of the molecular velocity, and is so used by Waterston.

In a paper in the Philosophical Magazine for 1858 Waterston gives an account of his own paper of 1846 in the following terms:—“Mr. Herapath unfortunately assumed heat or temperature to be represented by the simple ratio of the velocity instead of the square of the velocity, being in this apparently led astray by the definition of motion generally received, and thus was baffled in his attempts to reconcile his theory with observation. If we make this change in Mr. Herapath’s definition of heat or temperature—viz., that it is proportional to the vis-viva or square velocity of the moving particle, not to the momentum or simple ratio of the velocity—we can without much difficulty deduce not only the primary laws of elastic fluids, but also the other116 physical properties of gases enumerated above in the third objection to Newton’s hypothesis. [The paper from which the quotation is taken is on ‘The Theory of Sound.’] In the Archives of the Royal Society for 1845–46 there is a paper on ‘The Physics of Media that consist of perfectly “Elastic Molecules in a State of Motion,”’ which contains the synthetical reasoning on which the demonstration of these matters rests.... This theory does not take account of the size of the molecules. It assumes that no time is lost at the impact, and that if the impacts produce rotatory motion, the vis viva thus invested bears a constant ratio to the rectilineal vis viva, so as not to require separate consideration. It does, also, not take account of the probable internal motion of composite molecules; yet the results so closely accord with observation in every part of the subject as to leave no doubt that Mr. Herapath’s idea of the physical constitution of gases approximates closely to the truth.”

In his introduction to Waterston’s paper (Phil. Trans., 1892) Lord Rayleigh writes:—“Impressed with the above passage, and with the general ingenuity and soundness of Waterston’s views, I took the first opportunity of consulting the Archives, and saw at once that the memoir justified the large claims made for it, and that it marks an immense advance in the direction of the now generally received theory.”

In the first section of the paper Waterston’s great advance consisted in the statement that the mean square of the kinetic energy of each molecule measures the temperature.

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According to this we are thus to put in the pressure equation—½ m v² = T, the temperature, and we have at once—p V = ⅔ N · T.

Now this equation expresses, as we know, the laws of Boyle and Gay Lussac.

The second section discusses the properties of media, consisting of two or more gases, and arrives at the result that “in mixed media the mean square molecular velocity is inversely proportional to the specific weights of the molecules.” This was the great law rediscovered by Maxwell fifteen years later. With modern notation it may be put thus:—If m₁, m₂ be the masses of each molecule of two different sets of molecules mixed together, then, when a steady state has been reached, since the temperature is the same throughout, m₁ v₁² is equal to m₂ v₂². The average kinetic energy of each molecule is the same.

From this Avogadros’ law follows at once—for if p₁, p₂ be the pressures, N₁, N₂ the numbers of molecules per unit volume—

p₁ = ⅓ N₁ m₁ v₁², p₂ = ⅓ N₂ m₂ v₂².

Hence, if p₁, is equal to p₂, since m₁ v₁² is equal to m₂ v₂², we must have N₁ equal to N₂, or the number of molecules in equal volumes of two gases at the same pressure and temperature is the same. The proof of this proposition given by Waterston is not satisfactory. On this point, however, we shall have more to say. The third section of the paper deals with adiabatic expansion, and in it there is an error in calculation which prevented correct results from being attained.

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At the meeting of the British Association at Ipswich, in 1851, a paper by J. J. Waterston of Bombay, on “The General Theory of Gases,” was read. The following is an extract from the Proceedings:—

The author “conceives that the atoms of a gas, being perfectly elastic, are in continual motion in all directions, being constrained within a limited space by their collisions with each other, and with the particles of surrounding bodies.

“The vis viva of these motions in a given portion of a gas constitutes the quantity of heat contained in it.

“He shows that the result of this state of motion must be to give the gas an elasticity proportional to the mean square of the velocity of the molecular motions, and to the total mass of the atoms contained in unity of bulk” (unit of volume)—that is to say, to the density of the medium.

“The elasticity in a given gas is the measure of temperature. Equilibrium of pressure and heat between two gases takes place when the number of atoms in unit of volume is equal and the vis viva of each atom equal. Temperature, therefore, in all gases is proportional to the mass of one atom multiplied by the mean square of the velocity of the molecular motions, being measured from an absolute zero 491° below the zero of Fahrenheit’s thermometer.”

It appears, therefore, from these extracts that the discovery of the laws that temperature is measured by the mean kinetic energy of a single molecule, and that in a mixture of gases the mean kinetic energy of119 each molecule is the same for each gas, is due to Waterston. They were contained in his paper of 1846, and published by him in 1851. Both these papers, however, appear to have been unnoticed by all subsequent writers until 1892.

Meanwhile, in 1848, Joule’s attention was called by his experiments to the question, and he saw that Herapath’s result gave a means of calculating the mean velocity of the molecules of a gas. For according to the result given above, p = ⅓ ρ v²; thus v² = 3 p/ρ, and p and ρ being known, we find v². Thus for hydrogen at freezing-point and atmospheric pressure Joule obtains for v the value 6,055 feet per second, or, roughly, six times the velocity of sound in air.

Clausius was the next writer of importance on the subject. His first paper is in “Poggendorff’s Annalen,” vol. c., 1857, “On the Kind of Motion we call Heat.” It gives an exposition of the theory, and establishes the fact that the kinetic energy of the translatory motion of a molecule does not represent the whole of the heat it contains. If we look upon a molecule as a small solid we must consider the energy it possesses in consequence of its rotation about its centre of gravity, as well as the energy due to the motion of translation of the whole.

Clausius’ second paper appeared in 1859. In it he considers the average length of the path of a molecule during the interval between two collisions. He determines this path in terms of the average distance between the molecules and the distance between the centres of two molecules at the time when a collision is taking place.

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These two papers appear to have attracted Maxwell’s attention to the matter, and his first paper, entitled “Illustrations of the Dynamical Theory of Gases,” was read to the British Association at Aberdeen and Oxford in 1859 and 1860, and appeared in the Philosophical Magazine, January and July, 1860.

In the introduction to this paper Maxwell points out, while there was then no means of measuring the quantities which occurred in Clausius’ expression for the mean free path, “the phenomena of the internal friction of gases, the conduction of heat through a gas, and the diffusion of one gas through another, seem to indicate the possibility of determining accurately the mean length of path which a particle describes between two collisions. In order, therefore, to lay the foundation of such investigations on strict mechanical principles,” he continues, “I shall demonstrate the laws of motion of an indefinite number of small, hard and perfectly elastic spheres acting on one another only during impact.”

Maxwell then proceeds to consider in the first case the impact of two spheres.

But a gas consists of an indefinite number of molecules. Now it is impossible to deal with each molecule individually, to trace its history and follow its path. In order, therefore, to avoid this difficulty Maxwell introduced the statistical method of dealing with such problems, and this introduction is the first great step in molecular theory with which his name is connected.

He was led to this method by his investigation into the theory of Saturn’s rings, which had been121 completed in 1856, and in which he had shown that the conditions of stability required the supposition that the rings are composed of an indefinite number of free particles revolving round the planet, with velocities depending on their distances from the centre. These particles may either be arranged in separate rings, or their motion may be such that they are continually coming into collision with each other.

As an example of the statistical method, let us consider a crowd of people moving along a street. Taken as a whole the crowd moves steadily forwards. Any individual in the crowd, however, is jostled backwards and forwards and from side to side; if a line were drawn across the street we should find people crossing it in both directions. In a considerable interval more people would cross it, going in the direction in which the crowd is moving, than in the other, and the velocity of the crowd might be estimated by counting the number which crossed the line in a given interval. This velocity so found would differ greatly from the velocity of any individual, which might have any value within limits, and which is continually changing. If we knew the velocity of each individual and the number of individuals we could calculate the average velocity, and this would agree with the value found by counting the resultant number of people who cross the line in a given interval.

Again, the people in the crowd will naturally fall into groups according to their velocities. At any moment there will be a certain number of people whose velocities are all practically equal, or, to be122 more accurate, do not differ among themselves by more than some small quantity. The number of people at any moment in each of these groups will be very different. The number in any group, which has a velocity not differing greatly from the mean velocity of the whole, will be large; comparatively few will have either a very large or a very small velocity.

Again, at any moment, individuals are changing from one group to another; a man is brought to a stop by some obstruction, and his velocity is considerably altered—he passes from one group to a different one; but while this is so, if the mean velocity remains constant, and the size of the crowd be very great, the number of people at any moment in a given group remains unchanged. People pass from that group into others, but during any interval the same number pass back again into that group.

It is clear that if this condition is satisfied the distribution is a steady one, and the crowd will continue to move on with the same uniform mean velocity.

Now, Maxwell applies these considerations to a crowd of perfectly elastic spheres, moving anyhow in a closed space, acting upon each other only when in contact. He shows that they may be divided into groups according to their velocities, and that, when the steady state is reached, the number in each group will remain the same, although the individuals change. Moreover, it is shown that, if A and B represent any two groups, the state will only be steady when the numbers which pass from the group A to the group B are equal to the numbers which pass back from the group B to the group A. This condition, combined123 with the fact that the total kinetic energy of the motion remains unchanged, enables him to calculate the number of particles in any group in terms of the whole number of particles, the mean velocity, and the actual velocity of the group.

From this an accurate expression can be found for the pressure of the gas, and it is proved that the value found by others, on the assumption that all the particles were moving with a common velocity, is correct. Previous to this paper of Maxwell’s it had been realised that the velocities could not be uniform throughout. There had been no attempt to determine the distribution of velocity, or to submit the problem to calculation, making allowance for the variations in velocity.

Maxwell’s mathematical methods are, in their generality and elegance, far in advance of anything previously attempted in the subject.

So far it has been assumed that the particles in the vessel are all alike. Maxwell next takes the case of a mixture of two kinds of particles, and inquires what relation must exist between the average velocities of these different particles, in order that the state may be steady.

Now, it can be shown that when two elastic spheres impinge the effect of the impact is always such as to reduce the difference between their kinetic energies.

Hence, after a very large number of impacts the kinetic energies of the two balls must be the same; the steady state, then, will be reached when each ball has the same kinetic energy.

Thus if m₁, m₂ be the masses of the particles in124 the two sets respectively, v₁, v₂ their mean velocities we must have finally—

½ m₁ v₁² = ½ m₂ v₂²

This is the second of the two great laws enunciated by Waterston in 1845 and 1851, but which, as we have seen, had remained unknown until 1859, when it was again given by Maxwell.

Now, when gases are mixed their temperatures become equal. Hence we conclude, in Maxwell’s words, “that the physical condition which determines that the temperature of two gases shall be the same, is that the mean kinetic energy of agitation of the individual molecules of the two gases are equal.”

Thus, as the result of Maxwell’s more exact researches on the motion of a system of spherical particles, we find that we again can obtain the equations—

T = ½ mv² p = ⅓ N mv² = ⅔ NT = ⅔ ρ T/m

From these results we obtain as before the laws of Boyle, Charles and Avrogadro.

Again if σ be the specific heat of the gas at constant volume, the quantity of heat required to raise a single molecule of mass m one degree will be σ m.

Thus, when a molecule is heated, the kinetic energy must increase by this amount. But the increase of temperature, which in this case is 1°, is measured by the increase of kinetic energy of the125 single molecule. Hence the amount of heat required to raise the temperature of a single molecule of all gases 1° is the same. Thus the quantity σ m is the same for all gases; or, in other words, the specific heat of a gas is inversely proportional to the mass of its individual molecules. The density of a gas—since the number of molecules per unit volume at a given pressure and temperature is the same for all gases—is also proportional to the mass of each individual molecule. Thus the specific heats of all gases are inversely proportional to their densities. This is the law discovered experimentally by Dulong and Petit to be approximately true for a large number of substances.

In the next part of the paper Maxwell proceeded to determine the average number of collisions in a given time, and hence, knowing the velocities, to determine, in terms of the size of the particles and their numbers, the mean free path of a particle; the result so found differed somewhat from that already obtained by Clausius.

Having done this he showed how, by means of experiments on the viscosity of gases, the length of the mean free path could be determined.

An illustration due to Professor Balfour Stewart will perhaps make this clear. Let us suppose we have two trains running with uniform speed in opposite directions on parallel lines, and, further, that the engines continue to work at the same rate, developing just sufficient energy to overcome the resistance of the line, etc., and to maintain the speed126 constant. Now suppose passengers commence to jump across from one train to the other. Each man carries with him his own momentum, which is in the opposite direction to that of the train into which he jumps; the result is that the momentum of each train is reduced by the process; the velocities of the two decrease; it appears as though a frictional force were acting between the two. Maxwell suggests that a similar process will account for the apparent viscosity of gases.

Consider two streams of gas, moving in opposite directions one over the other; it is found that in each case the layers of gas near the separating surface move more slowly than those in the interior of the streams; there is apparently a frictional force between the two streams along this surface, tending to reduce their relative velocity. Maxwell’s explanation of this is that at the common surface particles from the one stream enter the other, and carry with them their own momentum; thus near this surface the momentum of each stream is reduced, just as the momentum of the trains is reduced by the people jumping across. Internal friction or viscosity is due to the diffusion of momentum across this common surface. The effect does not penetrate far into the gas, for the particles soon acquire the velocity of the stream to which they have come.

Now, the rate at which the momentum is diffused will measure the frictional force, and will depend on the mean free path of the particles. If this is considerable, so that on the average a particle can penetrate a considerable distance into the second gas before a127 collision takes place and its motion is changed, the viscosity will be considerable; if, on the other hand, the mean free path is small, the reverse will be true. Thus it is possible to obtain a relation between the mean free path and the coefficient of viscosity, and from this, if the coefficient of viscosity be known, a value for the mean free path can be found.

Maxwell, in the paper under discussion, was the first to do this, and, using a value found by Professor Stokes for the coefficient of viscosity, obtained as the length of the mean free path of molecules of air 1/447000 of an inch, while the number of collisions per second experienced by each molecule is found to be about 8,077,200,000.

Moreover, it appeared from his theory that the coefficient of viscosity should be independent of the number of molecules of gas present, so that it is not altered by varying the density. This result Maxwell characterises as startling, and he instituted an elaborate series of experiments a few years later with a view of testing it. The reason for this result will appear if we remember that, when the density is decreased, the mean free path is increased; relatively, then, to the total number of molecules present, the number which cross the surface in a given time is increased. And it appears from Maxwell’s result that this relative increase is such that the total number crossing remains unchanged. Hence the momentum conveyed across each unit area per second remains the same, in spite of the decrease in density.

Another consequence of the same investigation is that the coefficient of viscosity is proportional to the128 mean velocity of the molecules. Since the absolute temperature is proportional to the square of the velocity, it follows that the coefficient of viscosity is proportional to the square root of the absolute temperature.

The second part of the paper deals with the process of diffusion of two or more kinds of moving particles among one another.

If two different gases are placed in two vessels separated by a porous diaphragm such as a piece of unglazed earthenware, or connected by means of a narrow tube, Graham had shewn that, after sufficient time has elapsed, the two are mixed together. The same process takes place when two gases of different density are placed together in the same vessel. At first the denser gas may be at the bottom, the less dense above, but after a time the two are found to be uniformly distributed throughout.

Maxwell attempted to calculate from his theory the rate at which the diffusion takes place in these cases. The conditions of most of Graham’s experiments were too complicated to admit of direct comparison with the theory, from which it appeared that there is a relation between the mean free path and the rate of diffusion. One experiment, however, was found, the conditions of which could be made the subject of calculation, and from it Maxwell obtained as the value of the mean free path in air 1/389000 of an inch.

The number was close enough to that found from the viscosity to afford some confirmation of his theory.

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However, a few years later Clausius criticised the details of this part of the paper, and Maxwell, in his memoir of 1866, admits the calculation to have been erroneous. The main principles remained unaffected, the molecules pass from one gas to the other, and this constitutes diffusion.

Now, suppose we have two sets of particles in contact of such a nature that the mean kinetic energy of the one set is different from that of the other; the temperatures of the two will then be different. These two sets will diffuse into each other, and the diffusing particles will carry with them their kinetic energy, which will gradually pass from those which have the greater energy to those which have the less, until the average kinetic energy is equalised throughout. But the kinetic energy of translation is the heat of the particles. This diffusion of kinetic energy is a diffusion of heat by conduction, and we have here the mechanical theory of the conduction of heat in a gas.

Maxwell obtained an expression, which, however, he afterwards modified, for the conductivity of a gas in terms of the mean free path. It followed from this that the conductivity of air was only about 1/7000 of that of copper.

Thus the diffusion of gases, the viscosity of gases, and the conduction of heat in gases, are all connected with the diffusion of the particles carrying with them their momenta and their energy; while values of the mean free path can be obtained from observations on any one of these properties.

In the third part of his paper Maxwell considers130 the consequences of supposing the particles not to be spherical. In this case the impacts would tend to set up a motion of rotation in the particles. The direction of the force acting on any particle at impact would not necessarily pass through its centre; thus by impact the velocity of its centre would be changed, and in addition the particles would be made to spin. Some part, therefore, of the energy of the particles will appear in the form of the translational energy of their centres, while the rest will take the form of rotational energy of each particle about its centre.

It follows from Maxwell’s work that for each particle the average value of these two portions of energy would be equal. The total energy will be half translational and half rotational.

This theorem, in a more general form which was afterwards given to it, has led to much discussion, and will be again considered later. For the present we will assume it to be true. Clausius had already called attention to the fact that some of the energy must be rotational unless the molecules be smooth spheres, and had given some reasons for supposing that the ratio of the whole energy to the energy of translation is in a steady state a constant. Maxwell shows that for rigid bodies this constant is 2. Let us denote it for the present by the symbol β. Thus, if the translational energy of a molecule is ½ m v², its whole energy is ½ β m v².

The temperature is still measured by the translational energy, or ½ m v²; the heat depends on the whole energy. Hence if H represent the amount of131 heat—measured as energy—contained by a single molecule, and T its temperature, we have—

H = βT

From this it can be shewn50 that if γ represent the ratio of the specific heat of a gas at constant pressure to the specific heat at constant volume, then—

β = ⅔ 1/(γ-1)

For air and some other gases the value of γ has been shown to be 1·408. From this it follows that132 β = 1·634. Now, Maxwell’s theory required that for smooth hard particles, approximately spherical in shape, β should be 2, and hence he concludes “we have shown that a system of such particles could not possibly satisfy the known relation between the two specific heats of all gases.”

Since this statement was made many more experiments on the value of γ have been undertaken; it is not equal to 1·408 for all gases. Hence the value of β is different for various gases.

It is of some importance to notice that the value of β just found for air is very approximately 1·66 or 5/3.

For mercury vapour the value of γ has been shown by Kundt to be 1·33 or 1⅓, and hence β is equal to 1. Thus all the energy of a particle of mercury vapour is translational, and its behaviour in this respect is consistent with the assumption that a particle of mercury vapour is a smooth sphere.

The two results of this theory which seemed to lend themselves most readily to experimental verification were (1) that the viscosity of a gas is independent of its density, and (2) that it is proportional to the square root of the absolute temperature. The next piece of work connected with the theory was an attempt to test these consequences, and a description of the experiments was published in the “Philosophical Transactions” for 1865, in a paper on the “Viscosity or Internal Friction of Air and other Gases,” and forms the Bakerian lecture for that year.

The first result was completely proved. It is133 shewn that the value of the coefficient51 of viscosity “is the same for air at 0·5 inch and at 30 inches pressure, provided that the temperature remains the same.”

It was clear also that the viscosity depended on the temperature, and the results of the experiments seemed to show that it was nearly proportional to the absolute temperature. Thus for two temperatures, 185° Fah. and 51° Fah., the ratio of the two coefficients found was 1·2624; the ratio of the two temperatures, each measured from absolute zero, is 1·2605.

This result, then, does not agree with the hypothesis that a gas consists of spherical molecules acting only on each other by a kind of impact, for, if this were so, the coefficient would, as we have seen, depend on the square root of the absolute temperature. But Maxwell’s result, connecting viscosity with the first power of the absolute temperature, has not been confirmed by other investigators. According to it we should have as the relation between μ, the coefficient of viscosity at t° and μ₀, that at zero the equation—

μ = μ₀ (1 + .00365 t).

The most recent results of Professor Holman (Philosophical Magazine, Vol. xxi., p. 212) give—

μ = μ₀ (1 + .00275 t - .00000034 t²).

And results similar to this are given by O. E. Meyer,134 Puluj, and Obermeyer. Maxwell’s coefficient ·00365 is too large, but ·00182, the coefficient obtained by supposing the viscosity proportional to the square root of the temperature, would be too small.

It still remains true, therefore, that the laws of the viscosity of gases cannot be explained by the hypothesis of the impact of hard spheres; but some deductions drawn by Maxwell in his next paper from his supposed law of proportionality to the first power of the absolute temperature require modification.

It was clear from his experiments just described that the simple hypothesis of the impact of elastic bodies would not account for all the phenomena observed. Accordingly, in 1866, Maxwell took up the problem in a more general form in his paper on the “Dynamical Theory of Gases,” Phil. Trans., 1866.

In it he considered the molecules of the gas not as elastic spheres of definite radius, but as small bodies, or groups of smaller molecules, repelling one another with a force whose direction always passes very nearly through the centre of gravity of the molecules, and whose magnitude is represented very nearly by some function of the distance of the centres of gravity. “I have made,” he continues, “this modification of the theory in consequence of the results of my experiments on the viscosity of air at different temperatures, and I have deduced from these experiments that the repulsion is inversely as the fifth power of the distance.”

Since more recent observation has shown that the numerical results of Maxwell’s work connecting viscosity and temperature are erroneous, this last135 deduction does not hold; the inverse fifth power law of force will not give the correct relation between viscosity and temperature. Maxwell himself at a later date, “On the Stresses in Rarefied Gases,” Phil. Trans., 1879, realised this; but even in this last paper he adhered to the fifth power law because it leads to an important simplification in the equations to be dealt with.

The paper of 1866 is chiefly important because it contains for the first time the application of general dynamical methods to molecular problems. The law of the distribution of velocities among the molecules is again investigated, and a result practically identical with that found for the elastic spheres is arrived at. In obtaining this conclusion, however, it is assumed that the distribution of velocities is uniform in all directions about any point, whatever actions may be taking place in the gas. If, for example, the temperature is different at different points, then, for a given velocity, all directions are not equally probable. Maxwell’s expression, therefore, for the number of molecules which at any moment have a given velocity only applies to the permanent state in which the distribution of temperature is uniform. When dealing, for example, with the conduction of heat, a modification of the expression is necessary. This was pointed out by Boltzmann.52

In the paper of 1866, Maxwell applies his generalised results to the final distribution of two gases136 under the action of gravity, the equilibrium of temperature between two gases, and the distribution of temperature in a vertical column. These results are, as he states, independent of the law of force between the molecules. The dynamical causes of diffusion viscosity and conduction of heat are dealt with, and these involve the law of force.

It follows also from the investigation that, on the hypotheses assumed as its basis, if two kinds of gases be mixed, the difference between the average kinetic energies of translation of the gases of each kind diminishes rapidly in consequence of the action between the two. The average kinetic energy of translation, therefore, tends to become the same for each kind of gas, and as before, it is this average energy of translation which measures the temperature.

A molecule in the theory is a portion of a gas which moves about as a single body. It may be a mere point, a centre of force having inertia, capable of doing work while losing velocity. There may be also in each molecule systems of several such centres of force bound together by their mutual actions. Again, a molecule may be a small solid body of determinate form; but in this case we must, as Maxwell points out, introduce a new set of forces binding together the parts of each molecule: we must have a molecular theory of the second order. In any case, the most general supposition made is that a molecule consists of a series of parts which stick together, but are capable of relative motion among each other.

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In this case the kinetic energy of the molecule consists of the energy of its centre of gravity, together with the energy of its component parts, relative to its centre of gravity.53

Now Clausius had, as we have seen, given reasons for believing that the ratio of the whole energy of a molecule to the energy of translation of its centre of gravity tends to become constant. We have already used β to denote this constant. Thus, while the temperature is measured by the average kinetic energy of translation of the centre of gravity of each molecule, the heat contained in a molecule is its whole energy, and is β times this quantity. Thus the conclusions as to specific heat, etc., already given on page 130, apply in this case, and in particular we have the result that if γ be the ratio of the specific heat at constant pressure to that at constant volume, then—

β = ⅔ 1/(γ-1)

Maxwell’s theorem of the distribution of kinetic energy among a system of molecules applied, as he gave it in 1866, to the kinetic energy of translation of the centre of gravity of each molecule. Two years later Dr. Boltzmann, in the paper we have already138 referred to, extended it (under certain limitations) to the parts of which a molecule is composed. According to Maxwell the average kinetic energy of the centre of gravity of each molecule tends to become the same. According to Boltzmann the average kinetic energy of each part of the molecule tends to become the same.

Maxwell, in the last paper he wrote on the subject (“On Boltzmann’s Theorem on the Average Distribution of Energy in a System of Material Points,” Camb. Phil. Trans., XII.), took up this problem. Watson had given a proof of it in 1876 differing from Boltzmann’s, but still limited by the stipulation that the time, during which a particle is encountering other particles, is very small compared with the time during which there is no sensible action between it and other particles, and also that the time during which a particle is simultaneously within the distance of more than one other particle may be neglected.

Maxwell claims that his proof is free from any such limitation. The material points may act on each other at all distances, and according to any law which is consistent with the conservation of energy; they may also be acted on by forces external to the system, provided these are consistent with that law.

The only assumption which is necessary for the direct proof is that the system, if left to itself in its actual state of motion, will sooner or later pass through every phase which is consistent with the conservation of energy.

In this paper Maxwell finds in a very general manner an expression for the number of molecules139 which at any time have a given velocity, and this, when simplified by the assumptions of the former papers, reduces to the form already found. He also shows that the average kinetic energy corresponding to any one of the variables which define his system is the same for every one of the variables of his system.

Thus, according to this theorem, if each molecule be a single small solid body, six variables will be required to determine the position of each, three variables will give us the position of the centre of gravity of the molecule, while three others will determine the position of the body relative to its centre of gravity. If the six variables be properly chosen, the kinetic energy can be expressed as a sum of six squares, one square corresponding to each variable. According to the theorem the part of the kinetic energy depending on each square is the same. Thus, the whole energy is six times as great as that which arises from any one of the variables. The kinetic energy of translation is three times as great as that arising from each variable, for it involves the three variables which determine the position of the centre of gravity. Hence, if we denote by K the kinetic energy due to one variable, the whole energy is 6 K, and the translational energy is 3 K; thus, for this case—

β = 6K/3K = 2

Or, again, if we suppose that the molecule is such that m variables are required to determine its position relatively to its centre of gravity, since 3 are needed to fix the centre of gravity, the total number140 of variables defining the position of the molecule is m + 3, and it is said to have m + 3 degrees of freedom. Hence, in this case, its total energy is (m + 3) K and its energy of translation is 3 K, thus we find—

β = (m + 3)/3

Hence γ = 1 + 2/(m + 3) = 1 + 2/n

if n be the number of degrees of freedom of the molecule.

Thus, if this Boltzmann-Maxwell theorem be true, the specific heat of a gas will depend solely on the number of degrees of freedom of each of its molecules. For hard rigid bodies we should have n equal to 6, and hence γ = 1·333. Now the fact that this is not the value of γ for any of the known gases is a fundamental difficulty in the way of accepting the complete theory.

Boltzmann has called attention to the fact that if n be equal to five, then γ has the value 1·40. And this agrees fairly with the value found by experiment for air, oxygen, nitrogen, and various other gases. We will, however, return to this point shortly.

There is, perhaps, no result in the domain of physical science in recent years which has been more discussed than the two fundamental theorems of the molecular theory which we owe to Maxwell and to Boltzmann.

The two results in question are (1) the expression for the number of molecules which at any moment will have a given velocity, and (2) the proposition141 that the kinetic energy is ultimately equally divided among all the variables which determine the system.

With regard to (1) Maxwell showed that his error law was one possible condition of permanence. If at any moment the velocities are distributed according to the error law, that distribution will be a permanent one. He did not prove that such a distribution is the only one which can satisfy all the conditions of the problem.

The proof that this law is a necessary, as well as a sufficient, condition of permanence was first given by Boltzmann, for a single monatomic gas in 1872, for a mixture of such gases in 1886, and for a polyatomic gas in 1887. Other proofs have been given since by Watson and Burbury. It would be quite beyond the limits of this book to go into the question of the completeness or sufficiency of the proofs. The discussion of the question is still in progress.

The British Association Report for 1894 contains an important contribution to the question, in the shape of a report by Mr. G. H. Bryan, and the discussion he started at Oxford by reading this report has been continued in the pages of Nature and elsewhere since that time.

Mr. Bryan shows in the first place what may be the nature of the systems of molecules to which the results will apply, and discusses various points of difficulty in the proof.

The theorem in question, from which the result (1) follows as a simple deduction, has been thus stated by Dr. Larmor.54

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“There exists a positive function belonging to a group of molecules which, as they settle themselves into a steady state—on the average derived from a great number of configurations—maintains a steady downward trend. The Maxwell-Boltzmann steady state is the one in which this function has finally attained its minimum value, and is thus a unique steady state, it still being borne in mind that this is only a proposition of averages derived from a great number of instances in which nothing is conserved in encounters, except the energy, and that exceptional circumstances may exist, comparatively very few in number, in which the trend is, at any rate, temporarily the other way.”

This theorem, when applied to cases of motion, such as that of a gas at constant temperature enclosed in a rigid envelope impermeable to heat, appears to be proved. For such a case, therefore, the Maxwell-Boltzmann law is the only one possible.

But whether this be so or not, the law first introduced by Maxwell is one of those possible, and the advance in molecular science due to its introduction is enormous.

We come now to the second result, the equal partition of the energy among all the degrees of freedom of each molecule. Lord Kelvin has pointed out a flaw in Maxwell’s proof, but Boltzmann showed (Philosophical Magazine, March, 1893) how this flaw can easily be corrected, and it may be said that in all cases in which the Boltzmann-Maxwell law of the distribution of velocities holds, Maxwell’s law of the equal partition of energy holds also.

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Three cases are considered by Mr. Bryan, in which the law of distribution fails for rigid molecules: the first is when the molecules have all, in addition to their velocities of agitation, a common velocity of translation in a fixed direction; the second is when the gas has a motion of uniform rotation about a fixed axis; while the third is when each molecule has an axis of symmetry. In this last case the forces acting during a collision necessarily pass through the axis of symmetry, the angular velocity, therefore, of any molecule about this axis remains constant, the number of molecules having a given angular velocity will remain the same throughout the motion, and the part of the kinetic energy which depends on this component of the motion will remain fixed, and will not come into consideration when dealing with the equal partition of the energy among the various degrees of freedom.

Such a molecule has five, and not six, degrees of freedom; three quantities are needed to determine the position of its centre of gravity, and two to fix the position of the axis of symmetry.

In this case, then, as Boltzmann points out, in the expression for the ratio of the specific heats, we must have n equal to 5, and hence

γ = 1 + 2/n = 1 + 2/5 = 1·4

agreeing fairly with the value found for air and various other permanent gases.

For cases, then, in which we consider each atom as a single rigid body, the Boltzmann-Maxwell144 theorem appears to give a unique solution, and the Maxwell law of the distribution of the energy to be in fair accordance with the results of observation.55

If we can never go further—and it must be admitted that the difficulties in the way of further advance are enormous—it may, I think, be claimed for Maxwell that the progress already made is greatly due to him. Both these laws, for the case of elastic spheres, are contained in his first paper of 1860; and while it is to the genius of Boltzmann that we owe their earliest generalisation, and in particular the proof of the uniqueness of the solution under proper restrictions, Maxwell’s last paper contributed in no small degree to the security of the position. Not merely the foundations, but much of the superstructure of molecular science is his work.

The difficulties in the way of advance are, as we have said, enormous. Boltzmann, in one of his papers, has considered the properties of a complex molecule of a gas, consisting maybe of a number of atoms and possibly of ether atoms bound with them, and he concludes that such a molecule will behave in its progressive motion, and in its collisions with other molecules, nearly like a rigid body. But to quote from Mr. Bryan: “The case of a polyatomic molecule, whose atoms are capable of vibrating relative to one another, affords an interesting field for investigation and speculation. Is the Boltzmann distribution still unique, or do other permanent distributions exist in which the kinetic energy is unequally divided?”

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Again, the spectroscope reveals to us vibrations of the ether, which are connected in some way with the vibrations of the molecules of gas, whose spectrum we are observing. It seems clear that the law of equal partition does not apply to these, and yet, if we are to suppose that the ether vibrations are due to actual vibrations of the atoms which constitute a molecule, why does it not apply? Where does the condition come in which leads to failure in the proof? Or, again, is it, as has been suggested, the fact that the complex spectrum of a gas represents the terms of a Fourier Series, into which some elaborate vibration of the atoms is resolved by the ether? or is the spectrum due simply to electro-magnetic vibrations on the surface of the molecules—vibrations whose period is determined chiefly by the size and shape of the molecule, but in which the atoms of which it is composed take part? There are grave difficulties in the way of either of these explanations, but we must not let our dread of the task which remains to be done blind our eyes to the greatness of Maxwell’s work.

One other important paper, and a number of shorter articles, remain to be mentioned.

The Boltzmann-Maxwell law applies only to cases in which the temperature is uniform throughout. In a paper published in the Philosophical Transactions for 1879, on “Stresses in Rarefied Gases Arising from Inequalities of Temperature,” Maxwell deals, among other matters, with the theory of the radiometer. He shows that the observed motions will not take place unless gas, in contact with a solid, can slide along146 the surface of the solid with a finite velocity between places where the temperature is different; and in an appendix he proves that, on certain assumptions regarding the nature of the contact of the solid and the gas, there will be, even when the pressure is constant, a flow of gas along the surface from the colder to the hotter parts.

Among his less important papers bearing on molecular theory must be mentioned a lecture on “Molecules” to the British Association at its Bradford meeting; “Scientific Papers of Clerk Maxwell,” vol. ii., p. 361; and another on “The Molecular Constitution of Bodies,” Scientific Papers, vol. ii., p. 418.

In this latter, and also in a review in Nature of Van der Waals’ book on “The Continuity of the Gaseous and Liquid States,”56 he explains and discusses Clausius’ virial equation, by means of which the variations of the permanent gases from Boyle’s law are explained. The lecture gives a clear account, in Maxwell’s own inimitable style, of the advances made in the kinetic theory up to the date at which it was delivered, and puts clearly the difficulties it has to meet. Maxwell thought that those arising from the known values of the ratio of the specific heats were the most serious.

In the articles, “Atomic Constitution of Bodies” and “Diffusion,” in the ninth edition of the Encyclopædia Britannica, we have Maxwell’s later views on the fundamental assumptions of the molecular theory.

The text-book on “Heat” contains some further developments of the theory. In particular he shows147 how the conclusions of the second law of thermo-dynamics are connected with the fact that the coarseness of our faculties will not allow us to grapple with individual molecules.

The work described in the foregoing chapters would have been sufficient to secure to Maxwell a distinguished place among those who have advanced our knowledge; it remains still to describe his greatest work, his theory of Electricity and Magnetism.

Clerk Maxwell’s first electrical paper—that on Faraday’s “Lines of Force”—was read to the Cambridge Philosophical Society on December 10th, 1855, and Part II. on February 11th, 1856. The author was then a Bachelor of Arts, only twenty-three years in age, and of less than one year’s standing from the time of taking his degree.

The opening words of the paper are as follows (Scientific Papers, vol. i., p. 155):—

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The idea was a bold one: for a youth of twenty-three to explain, by means of the motions of an incompressible fluid, some of the less complicated phenomena of electricity and magnetism, to show how the laws of the attractions of magnets and currents may be clearly conceived without making any assumption as to the physical nature of electricity, or adding anything to that which has already been proved by experiment.

It may be useful to review in a very few words the position of electrical theory57 in 1855.

Coulomb’s experiments had established the fundamental facts of electrostatic attraction and repulsion, and Coulomb himself, about 1785, had stated a theory based on these experiments which could “only be attacked by proving his experimental results to be inaccurate.”58

Coulomb supposes the existence of two electric fluids, the theory developed previously by Franklin, but says—

Cavendish was working in England about the same time as Coulomb, but he published very little,155 and the value and importance of his work was not recognised until the appearance in 1879 of the “Electrical Researches of Henry Cavendish,” edited by Clerk Maxwell.

Early in the present century the application of mathematical analysis to electrical problems was begun by Laplace, who investigated the distribution of electricity on spheroids, and about 1811 Poisson’s great work on the distribution of electricity on two spheres placed at any given distance apart was published. Meanwhile the properties of the electric current were being investigated. Galvani’s discovery of the muscular contraction in a frog’s leg, caused by the contact of dissimilar metals, was made in 1790. Volta invented the voltaic pile in 1800, and Oersted in 1820 discovered that an electric current produced magnetic force in its neighbourhood. On this Ampère laid the foundation of his theory of electro-dynamics, in which he showed how to calculate the forces between circuits carrying currents from an assumed law of force between each pair of elements of the circuits. His experiments proved that the consequences which follow from this law are consistent with all the observed facts. They do not prove that Ampère’s law alone can explain the facts.

Maxwell, writing on this subject in the “Electricity an Magnetism,” vol. ii., p. 162, says—

The experimental evidence for Ampère’s theory, so far, at least, as it was possible to obtain it from experiments on closed circuits, was rendered unimpeachable by W. Weber about 1846, while in the previous year Grassman and F. E. Neumann both published laws for the attraction between two elements of current which differ from that of Ampère, but lead to the same result for closed circuits. In a paper published in 1846 Weber announced his hypothesis connecting together electrostatic and electro-dynamic action. In this paper he supposed that the force between two particles of electricity depends on the motion of the particles as well as on their distance apart. A somewhat similar theory was proposed by Gauss and published after his death in his collected works. It has been shown, however, that Gauss’ theory is inconsistent with the conservation of energy. Weber’s theory avoids this inconsistency and leads, for closed circuits, to the same results as Ampère. It has been proved, however, by Von Helmholtz, that, under certain circumstances, according to it, a body would157 behave as though its mass were negative—it would move in a direction opposite to that of the force.59

Since 1846 many other theories have been proposed to explain Ampère’s laws. Meanwhile, in 1821, Faraday observed that under certain circumstances a wire carrying a current could be kept in continuous rotation in a magnetic field by the action between the magnets and the current. In 1824 Arago observed the motion of a magnet caused by rotating a copper disc in its neighbourhood, while in 1831 Faraday began his experimental researches into electro-magnetic induction. About the same period Joseph Henry, of Washington, was making, independently of Faraday, experiments of fundamental importance on electro-magnetic induction, but sufficient attention was not called to his work until comparatively recent years.

In 1833 Lenz made some important researches, which led him to discover the connection between the direction of the induced currents and Ampère’s laws, summed up in his rule that the direction of the induced current is always such as to oppose by its electro-magnetic action the motion which induces it.

In 1845 F. E. Neumann developed from this law the mathematical theory of electro-magnetic induction, and about the same time W. Weber showed how it might be deduced from his elementary law of electrical action.

The great name of Von Helmholtz first appears in connection with this subject in 1851, but of his writings we shall have more to say at a later stage.

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Meanwhile, during the same period, various writers, Murphy, Plana, Charles, Sturm, and Gauss, extended Poisson’s work on electrostatics, treating the questions which arose as problems in the distribution of an attracting fluid, attracting or repelling according to Newton’s law, though here again the greatest advances were made by a self-taught Nottingham shoemaker, George Green by name, in his paper “On the Application of Mathematical Analysis to the Theories of Electricity and Magnetism,” 1828.

Green’s researches, Lord Kelvin writes, “have led to the elementary proposition which must constitute the legitimate foundation of every perfect mathematical structure that is to be made from the materials furnished by the experimental laws of Coulomb.”

Green, it may be remarked, was the inventor of the term Potential. His essay, however, lay neglected from 1828, until Lord Kelvin called attention to it in 1845. Meanwhile, some of its most important results had been re-discovered by Gauss and Charles and Thomson himself.

Until about 1845, the experimental work on which these mathematical researches in electrostatics were based was that of Coulomb. An electrified body is supposed to have a charge of some imponderable fluid “electricity.” Particles of electricity repel each other according to a certain law, and the fluid distributes itself in equilibrium over the surface of any charged conductor in accordance with this law. There are on this theory two opposite kinds of electric fluid, positive and negative, two charges of the same kind repel, two charges of opposite kinds attract; the repulsion or159 attraction is proportional to the product of the charges, and inversely proportional to the square of the distance between them.

The action between two charges is action at a distance taking place across the space which separates the two.

Faraday, in 1837, in the eleventh series of his “Experimental Researches,” published his first paper on “Electrostatic Induction.” He showed—as indeed Cavendish had proved long previously, though the result remained unpublished—that the force between two charged bodies will depend on the insulating medium which surrounds them, not merely on their shape and position. Induction, as he expresses it, takes place along curved lines, and is an action of contiguous particles; these curved lines he calls the “lines of force.”

Discussing these researches in 1845, Lord Kelvin writes60:—

In the pages which follow, Lord Kelvin develops the consequences of an analogy between the conduction of heat and electrostatic action, which he had pointed out three years earlier (1842), in his paper on “The Uniform Motion of Heat in Homogeneous Solid Bodies,” and discusses its connection with the mathematical theory of electricity.

The problem of distributing sources of heat in a given homogeneous conductor of heat, so as to produce a definite steady temperature at each point on the conductor is shewn to be mathematically identical with that of distributing electricity in equilibrium, so as to produce at each point an electrical potential having the same value as the temperature.

Thus the fundamental laws of the conduction of heat may be made the basis of the mathematical theory of electricity, but the physical idea which they suggest is that of the propagation of some effect by means of the mutual action of contiguous particles, rather than that of material particles attracting or repelling at a distance, which naturally follows from the statement of Coulomb’s law.

Lord Kelvin continues:—

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Lord Kelvin’s papers on “The Mathematical Theory of Electricity,” published from 1848 to 1850, his “Propositions on the Theory of Attraction” (1842), his “Theory of Electrical Images” (1847), and his paper on “The Mathematical Theory of Magnetism” (1849), contain a statement of the most important results achieved in the mathematical sciences of Electrostatics and Magnetism up to the time of Maxwell’s first paper.

The opening sentences of that paper have already been quoted. In the preface to the “Electricity and Magnetism” Maxwell writes thus:—

Now, Maxwell saw an analogy between electrostatics and the steady motion of an incompressible fluid like water, and it is this analogy which he develops in the first part of his paper. The water flows along definite lines; a surface which consists wholly of such lines of flow will have the property that no water ever crosses it. In any stream of water we can imagine a number of such surfaces drawn, dividing it up into a series of tubes; each of these will be a tube of flow, each of these tubes remain always filled with water. Hence, the quantity of water which crosses per second any section of a tube of flow perpendicular to its length is always the same. Thus, from the form of the tube, we can obtain information as to the direction and strength of the flow, for where the tube is wide the flow will be proportionately small, and vice versâ.

Again, we can draw in the fluid a number of surfaces, over each of which the pressure is the same; these surfaces will cut the tubes of flow at right angles. Let us suppose they are drawn so that the difference of pressure between any two consecutive163 surfaces is unity, then the surfaces will be close together at points at which the pressure changes rapidly; where the variation of pressure is slow, the distance between two consecutive surfaces will be considerable.

If, then, in any case of motion, we can draw the pressure surfaces, and the tubes of flow, we can determine the motion of the fluid completely. Now, the same mathematical expressions which appear in the hydro-dynamical theory occur also in the theory of electricity, the meaning only of the symbols is changed. For velocity of fluid we have to write electrical force. For difference of fluid pressure we substitute work done, or difference of electrical potential or pressure.

The surfaces and tubes, drawn as the solution of any hydro-dynamical problem, give us also the solution of an electrical problem; the tubes of flow are Faraday’s tubes of force, or tubes of induction, the surfaces of constant pressure are surfaces of equal electrical potential. Induction may take place in curved lines just as the tubes of flow may be bent and curved; the analogy between the two is a complete one.

But, as Maxwell shows, the analogy reaches further still. An electric current flowing along a wire had been recognised as having many properties similar to those of a current of liquid in a tube. When a steady current is passing through any solid conductor, there are formed in the conductor tubes of electrical flow and surfaces of constant pressure. These tubes and surfaces are the same as those formed by the flow of164 liquid through a solid whose boundary surface is the same as that of the conductor, provided the flow of liquid is properly proportioned to the flow of electricity.

These analogies refer to steady currents in which, therefore, the flow at any point of the conductor does not depend on the time. In Part II. of his paper Maxwell deals with Faraday’s electro-tonic state. Faraday had found that when changes are produced in the magnetic phenomena surrounding a conductor, an electric current is set up in the conductor, which continues so long as the magnetic changes are in progress, but which ceases when the magnetic state becomes steady.

Maxwell shows how to obtain a mathematical expression for Faraday’s electro-tonic state. In his “Electricity and Magnetism,” this electro-tonic state receives a new name. It is known as the Vector Potential,61 and the paper under consideration contains,166 though in an incomplete form, his first statement of those equations of the electric field which are so indissolubly bound up with Maxwell’s name.

The great advance in theory made in the paper is the distinct recognition of certain mathematical functions as representing Faraday’s electrotonic-state, and their use in solving electro-magnetic problems.

The paper contains no new physical theory of electricity, but in a few years one appeared. In his later writings Maxwell adopted a more general view of the electro-magnetic field than that contained in his early papers on “Physical Lines of Force.” It must, therefore, not be supposed that the somewhat gross conception of cog-wheels and pulleys, which we are about to describe, were anything more to their author than a model, which enabled him to realise how the changes, which occur when a current of electricity passes through a wire, might be represented by the motion of actual material particles.

The problem before him was to devise a physical theory of electricity, which would explain the forces exerted on electrified bodies by means of action between the contiguous parts of the medium in the space surrounding these bodies, rather than by direct action across the distance which separates them. A similar question, still unanswered, had arisen in the case of gravitation. Astronomers have determined the forces between attracting bodies; they do not know how those forces arise.

Maxwell’s fondness for models has already been alluded to; it had led him to construct his top to illustrate the dynamics of a rigid body rotating about167 a fixed point, and his model of Saturn’s rings (now in the Cavendish Laboratory) to illustrate the motion of the satellites in the rings. He had explained many of the gaseous laws by means of the impact of molecules, and now his fertile ingenuity was to imagine a mechanical model of the state of the electro-magnetic field near a system of conductors carrying currents.

Faraday, as we have seen, looked upon electrostatic and magnetic induction as taking place along curved lines of force. He pictures these lines as ropes of molecules starting from a charged conductor, or a magnet, as the case may be, and acting on other bodies near. These ropes of molecules tend to shorten, and at the same time to swell outwards laterally. Thus the charged conductor tends to draw other bodies to itself, there is a tension along the lines of force, while at the same time each tube of molecules pushes its neighbours aside; a pressure at right angles to the lines of force is combined with this tension. Assuming for a moment this pressure and tension to exist, can we devise a mechanism to account for it? Maxwell himself has likened the lines of force to the fibres of a muscle. As the fibres contract, causing the limb to which they are attached to move, they swell outwards, and the muscle thickens.

Again, from another point of view, we might consider a line of force as consisting of a string of small cells of some flexible material each filled with fluid. If we then suppose this series of cells caused to rotate rapidly about the direction of the line of force, the cells will expand laterally and contract longitudinally; there will again be tension along the lines168 of force and pressure at right angles to them. It was this last idea, as we shall see shortly, of which Maxwell made use—

Lord Kelvin had in 1847 given a mechanical representation of electric, magnetic and galvanic forces by means of the displacements of an elastic solid in a state of strain. The angular displacement at each point of the solid was taken as proportional to the magnetic force, and from this the relation between the various other electric quantities and the motion of the solid was developed. But Lord Kelvin did not attempt to explain the origin of the observed forces by the effects due to these strains, but merely made use of the mathematical analogy to assist the imagination in the study of both.

Maxwell considered magnetic action as existing in the form of pressure or tension, or more generally, of some stress in some medium. The existence of a medium capable of exerting force on material bodies and of withstanding considerable stress, both pressure and tension, is thus a fundamental hypothesis with him; this medium is to be capable of motion,169 and electro-magnetic forces arise from its motion and its stresses.

Now, Maxwell’s fundamental supposition is that, in a magnetic field, there is a rotation of the molecules continually in progress about the lines of magnetic force. Consider now the case of a uniform magnetic field, whose direction is perpendicular to the paper; we are to look upon the lines of force as parallel strings of molecules, the axes of these strings being perpendicular to the paper. Each string is supposed to be rotating in the same direction about its axis, and the angular velocity of rotation is a measure of the magnetic force. In consequence of this rotation there will be differences of pressure in different directions in the medium; the pressure along the axes of the strings will be less than it would be if the medium were at rest, that in the directions at right angles to the axes will be greater, the medium will behave as though it were under tension along the axes of the molecules under pressure at right angles to them. Moreover, it can be shown that the pressure and the tension are both proportional to the square of the angular velocity—the square, that is, of the magnetic force—and this result is in accordance with the consequences of experiment.

More elaborate calculation shows that this statement is true generally. If we draw the lines of force in any magnetic field, and then suppose the molecules of the medium set in rotation about these lines of force as axes, with velocities which at each point are proportional to the magnetic force, the distribution of170 pressure throughout is that which we know actually to exist in the magnetic field.

According to this hypothesis, then, a permanent bar magnet has the power of setting the medium round it into continuous molecular rotation about the lines of force as axes. The molecules which are set in rotation we may consider as spherical, or nearly spherical, cells filled with a fluid, or an elastic solid substance, and surrounded by a kind of membrane, or sack, holding the contents together.

So far the model does not give any account of electrical actions which go on in the magnetic field.

The energy is wholly rotational, and the forces wholly magnetic.

Consider, however, any two contiguous strings of molecules. Let them cut the paper as shown in the two circles in Fig. 1:—

Fig. 1. Fig. 2.

Then these cells are both rotating in the same direction, hence at C, where they touch, their points of contact will be moving in opposite directions, as shown by the arrow heads, and it is difficult to imagine how such motion can continue; it would require the surfaces of the cells to be perfectly smooth, and if this were so they would lose the power of transmitting action from one cell to the next.

The cells A and B may be compared to two171 cog-wheels placed close together, which we wish to turn in the same direction. If the cogs can interlock, as in Fig. 2, this is impossible: consecutive wheels in the train must move in opposite directions.

Fig. 3.

But in many machines the desired end is attained by inserting between the two wheels A and B a third idle wheel C, as shewn in Fig. 3. This may be very small, its only function is to transmit the motion of A to B in such a way that A and B may both turn in the same direction. It is not necessary that there should be cogs on the wheels; if the surfaces be perfectly rough, so that no slipping can take place, the same result follows without the cogs.

Guided by this analogy Maxwell extended his model by supposing each cell coated with a number of small particles which roll on its surface. These particles play the part of the idle wheels in the machine, and by their rolling merely enable the adjacent parts of two cells to move in opposite directions.

Consider now a number of such cells and their idle wheels lying in a plane, that of the paper, and suppose each cell is rotating with the same uniform angular velocity about an axis at right angles to that plane, each idle wheel will be acted on by two equal and opposite forces at the ends of the diameter in which172 it is touched by the adjacent cells; it will therefore be set in rotation, but there will be no force tending to drive it onwards; it does not matter whether the axis on which it rotates is free to move or fixed, in either case the idle wheel simply rotates. But suppose now the adjacent cells are not rotating at the same rate. In addition to its rotation the idle wheel will be urged onward with a velocity which depends on the difference between the rotations, and, if it can move freely, it will move on from between the two cells. Imagine now that the interstices between the cells are fitted with a string of idle wheels. So long as the adjacent cells move with different velocity there will be a continual stream of rolling particles or idle wheels between them. Maxwell in the paper considered these rolling particles to be particles of electricity. Their motion constitutes an electric current. In a uniform magnetic field there is no electric current; if the strength of the field varies, the idle wheels are set in motion and there may be a current.

These particles are very small compared with the magnetic vortices. The mass of all the particles is inappreciable compared with the mass of the vortices, and a great many vortices with their surrounding particles are contained in a molecule of the medium; the particles roll on the vortices without touching each other, so that so long as they remain within the same molecule there is no loss of energy by resistance. When, however, there is a current or general transference of particles in one direction they must pass from one molecule to another, and in doing so may experience resistance and generate heat.

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Maxwell states that the conception of a particle, having its motion connected with that of a vortex by perfect rolling contact, may appear somewhat awkward. “I do not bring it forward,” he writes, “as a mode of connection existing in Nature, or even as that which I would willingly assent to as an electrical hypothesis. It is, however, a mode of connection which is mechanically conceivable and easily investigated, and it serves to bring out the actual mechanical connections between the known electro-magnetic phenomena, so that I venture to say that anyone who understands the provisional and temporary character of this hypothesis will find himself rather helped than hindered by it in his search after the true interpretation of the phenomena.”

The first part of the paper deals with the theory of magnetism; in the second part the hypothesis is applied to the phenomena of electric currents, and it is shown how the known laws of steady currents and of electro-magnetic induction can be deduced from it. In Part III., published January and February, 1862, the theory of molecular vortices is applied to statical electricity.

The distinction between a conductor and an insulator or dielectric is supposed to be that in the former the particles of electricity can pass with more or less freedom from molecule to molecule. In the latter such transference is impossible, the particles can only be displaced within the molecule with which they are connected; the cells or vortices of the medium are supposed to be elastic, and to resist by their elasticity the displacement of the particles within174 them. When electrical force acts on the medium this displacement of the particles within each molecule takes place until the stresses due to the elastic reaction of the vortices balance the electrical force; the medium behaves like an elastic body yielding to pressure until the pressure is balanced by the elastic stress. When the electric force is removed the cells or vortices recover their form, the electricity returns to its former position.

In a medium such as this waves of periodic displacement could be set up, and would travel with a velocity depending on its electric properties. The value for this velocity can be obtained from electrical observations, and Maxwell showed that this velocity, so found, was, within the limits of experimental error, the same as that of light. Moreover, the electrical oscillations take place, like those of light, in the front of the wave. Hence, he concludes, “the elasticity of the magnetic medium in air is the same as that of the luminiferous medium, if these two coexistent, coextensive, and equally elastic media are not rather one medium.”

The paper thus contains the first germs of the electro-magnetic theory of light. Moreover, it is shown that the attraction between two small bodies charged with given quantities of electricity depends on the medium in which they are placed, while the specific inductive capacity is found to be proportional to the square of the refractive index.

The fourth and final part of the paper investigates the propagation of light in a magnetic field.

Faraday had shown that the direction of vibration175 in a wave of polarised light travelling parallel to the lines of force in a magnetic field is rotated by its passage through the field. The numerical laws of this relation had been investigated by Verdet, and Maxwell showed how his hypothesis of molecular vortices led to laws which agree in the main with those found by Verdet.

He points out that the connection between magnetism and electricity has the same mathematical form as that between certain other pairs of phenomena, one of which has a linear and the other a rotatory character; and, further, that an analogy may be worked out assuming either the linear character for magnetism and the rotatory character for electricity, or the reverse. He alludes to Prof. Challis’ theory, according to which magnetism is to consist in currents in a fluid whose directions correspond with the lines of magnetic force, while electric currents are supposed to be accompanied by, if not dependent upon, a rotatory motion of the fluid about the axis of the current; and to Von Helmholtz’s theory of a somewhat similar character. He then gives his own reasons—agreeing with those of Sir W. Thomson (Lord Kelvin)—for supposing that there must be a real rotation going on in a magnetic field in order to account for the rotation of the plane of polarisation, and, accepting these reasons as valid, he develops the consequences of his theory with the results stated above.

His own verdict on the theory is given in the “Electricity and Magnetism” (vol. ii., § 831, first edition, p. 416):—

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In studying this part of Maxwell’s work, it must clearly be remembered that he did not look upon the ether as a series of cog-wheels with idle wheels between, or anything of the kind. He devised a mechanical model of such cogs and idle wheels, the properties177 of which would in some respects closely resemble those of the ether; from this model he deduced, among other things, the important fact that electric waves would travel outwards with the velocity of light. Other such models have been devised since his time to illustrate the same laws. Prof. Fitzgerald has actually constructed one of wheels connected together by elastic bands, which shows clearly the kind of processes which Maxwell supposed to go on in a dielectric when under electric force. Professor Lodge, in his book, “Modern Views of Electricity,” has very fully developed a somewhat different arrangement of cog-wheels to attain the same result.

Maxwell’s predictions as to the propagation of electric waves have in recent days received their full verification in the brilliant experiments of Hertz and his followers; it remains for us, before dealing with these, to trace their final development in his hands.

The papers we have been discussing were perhaps too material to receive the full attention they deserved; the ether is not a series of cogs, and electricity is something different from material idle wheels. In his paper on “The Dynamical Theory of the Electro-magnetic Field,” Phil. Trans., 1864, Maxwell treats the same questions in a more general manner. On a former occasion he says, “I have attempted to describe a particular kind of motion and a particular kind of strain so arranged as to account for the phenomena. In the present paper I avoid any hypothesis of this kind; and in using such words as electric momentum and electric elasticity in reference to the known phenomena of the induction of currents178 and the polarisation of dielectrics, I wish merely to direct the mind of the reader to mechanical phenomena, which will assist him in understanding the electrical ones. All such phrases in the present paper are to be considered as illustrative and not as explanatory.” He then continues:—

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In his introduction to the paper, he discusses in a general way the various explanations of electric phenomena which had been given, and points out that—

These general laws of dynamics, applicable to the motion of any connected system, had been developed by Lagrange, and are expressed in his generalised equations of motion. It is one of Maxwell’s chief claims to fame that he saw in the electric field a connected system to which Lagrange’s equations could be applied, and that he was able to deduce the mechanical and electrical actions which take place by means of fundamental propositions of dynamics.

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The methods of the paper now under discussion were developed further in the “Treatise on Electricity and Magnetism,” published in 1873; in endeavouring to give some slight account of Maxwell’s work, we shall describe it in the form it ultimately took.

The task which Maxwell set himself was a double one; he had first to express in symbols, in as general a form as possible, the fundamental laws of electro-magnetism as deduced from experiments, chiefly the experiments of Faraday, and the relations between the various quantities involved; when this was done he had to show how these laws could be deduced from the general dynamical laws applicable to any system of moving bodies.

There are two classes of phenomena, electric and magnetic, which have been known from very early times, and which are connected together. When a piece of sealing-wax is rubbed it is found to attract other bodies, it is said to exert electric force throughout the space surrounding it; when two different metals are dipped in slightly acidulated water and connected by a wire, certain changes take place in the plates, the water, the wire, and the space round the wire, electric force is again exerted and a current of electricity is said to flow in the wire. Again, certain bodies, such as the lodestone, or pieces of iron and steel which have been treated in a certain manner, exhibit phenomena of action at a distance: they are said to exert magnetic force, and it is found that this magnetic force exists in the neighbourhood of an electric current and is connected with the current.

Again, when electric force is applied to a body, the181 effects may be in part electrical, in part mechanical; the electrical state of the body is in general changed, while in addition, mechanical forces tending to move the body are set up. Experiment must teach us how the electrical state depends on the electric force, and what is the connection between this electric force and the magnetic forces which may, under certain circumstances, be observed. Now, in specifying the electric and magnetic conditions of the system, various other quantities, in addition to the electric force, will have to be introduced; the first step is to formulate the necessary quantities, and to determine the relations between them and the electric force.

Consider now a wire connecting the two poles of an electric battery—in its simplest form, a piece of zinc and a piece of copper in a vessel of dilute acid—electric force is produced at each point of the wire. Let us suppose this force known; an electric current depending on the material and the size of the wire flows along it, its value can be determined at each point of the wire in terms of the electric force by Ohm’s law. If we take either this current or the electric force as known, we can determine by known laws the electric and magnetic conditions elsewhere. If we suppose the wire to be straight and very long, then, so long as the current is steady and we neglect the small effect due to the electrostatic charge on the wire, there is no electric force outside the wire. There is, however, magnetic force, and it is found that the lines of magnetic force are circles round the wire. It is found also that the work done in travelling once completely round the wire against the magnetic force182 is measured by the current flowing through the wire, and is obtained in the system of units usually adopted by multiplying the current by 4π. This last result then gives us one of the necessary relations, that between the magnetic force due to a current and the strength of the current.

Again, consider a steady current flowing in a conductor of any form or shape, the total flow of current across any section of the conductor can be measured in various ways, and it is found that at any time this total flow is the same for each section of the conductor. In this respect the flow of a current resembles that of an incompressible fluid through a pipe; where the pipe is narrow the velocity of flow is greater than it is where the pipe is broad, but the total quantity crossing each section at any given instant is the same.

Consider now two conducting bodies, two spheres, or two flat plates placed near together but insulated. Let each conductor be connected to one of the poles of the battery by a conducting wire. Then, for a very short interval after the contact is made, it is found that there is a current in each wire which rapidly dies away to zero. In the neighbourhood of the balls there is electric force; the balls are said to be charged with electricity, and the lines of force are curved lines running from one ball to the other. It is found that the balls slightly attract each other, and the space between them is now in a different condition from what it was before the balls were charged. According to Maxwell, Electric Displacement has been produced in this space, and the electric displacement at each183 point is proportional to the electric force at that point.

Thus, (i) when electric force acts on a conductor, it produces a current, the current being by Ohm’s law proportional to the force: (ii) when it acts on an insulator it produces electric displacement, and the displacement is proportional to the force; while (iii) there is magnetic force in the neighbourhood of the current, and the work done in carrying a magnetic pole round any complete circuit linked with the current is proportional to the current. The first two of these principles give us two sets of equations connecting together the electric force and the current in a conductor or the displacement in a dielectric respectively; the third connects the magnetic force and the current.

Now let us go back to the variable period when the current is flowing in the wires; and to make ideas precise, let the two conductors be two equal large flat plates placed with their faces parallel, and at some small distance apart. In this case, when the plates are charged, and the current has ceased, the electric displacement and the force are confined almost entirely to the space between the plates. During the variable period the total flow at any instant across each section of the wire is the same, but in the ordinary sense of the word there is no flow of electricity across the insulating medium between the plates. In this space, however, the electric displacement is continuously changing, rising from zero initially to its final steady value when the current ceases. It is a fundamental part of Maxwell’s theory that this variation of electric184 displacement is equivalent in all respects to a current. The current at any point in a dielectric is measured by the rate of change of displacement at that point.

Moreover, it is also an essential point that if we consider any section of the dielectric between the two plates, the rate of change of the total displacement across this section is at each moment equal to the total flow of current across each section of the conducting wire.

Currents of electricity, therefore, including displacement currents, always flow in closed circuits, and obey the laws of an incompressible fluid in that the total flow across each section of the circuit—conducting or dielectric—is at any moment the same.

It should be clearly remembered that this fundamental hypothesis of Maxwell’s theory is an assumption only to be justified by experiment. Von Helmholtz, in his paper on “The Equations of Motion of Electricity for Bodies at Rest,” formed his equations in an entirely different manner from Maxwell, and arrived at results of a more general character, which do not require us to suppose that currents flow always in closed circuits, but permit of the condensation of electricity at points in the circuit where the conductors end and the non-conducting part of the circuit begins. We leave for the present the question which of the two theories, if either, represents the facts.

We have obtained above three fundamental relations—(i) that between electric force and electric current in a conductor; (ii) that between electric185 force and electric displacement in a dielectric; (iii) that between magnetic force and the current which gives rise to it. And we have seen that an electric current—i.e. in a dielectric the variation of the strength of an electric field of force—gives rise to magnetic force. Now, magnetic force acting on a medium produces “magnetic displacement,” or magnetic induction, as it is called. In all media except iron, nickel, cobalt, and a few other substances, the magnetic induction is proportional to the magnetic force, and the ratio between the magnetic induction produced by a given force and the force is found to be very nearly the same for all such media. This ratio is known as the permeability, and is generally denoted by the symbol μ.

A relation reciprocal to that given in (iii) above might be anticipated, and was, in fact, discovered by Faraday. Changes in a field of magnetic induction give rise to electric force, and hence to displacement currents in a dielectric or to conduction currents in a conductor. In considering the relation between these changes and the electric force, it is simplest at first not to deal with magnetic matter such as iron, nickel, or cobalt; and then we may say that (iv) the work which at any instant would be done in carrying a unit quantity of electricity round a closed circuit in a magnetic field against the electric forces due to the field is equal to the rate at which the total magnetic induction which threads the circuit is being decreased. This law, summing up Faraday’s experiments on electro-magnetic induction, gives a fourth principle, leading to a fourth series186 of equations connecting together the electric and magnetic quantities involved.

The equations deduced from the above four principles, together with the condition implied in the continuity of an electric current, constitute Maxwell’s equations of the electro-magnetic field.

If we are dealing only with a dielectric medium, the reciprocal relation between the third and fourth principle may be made more clear by the following statement:—

(A) The work done at any moment in carrying a unit quantity of magnetism round a closed circuit in a field in which electric displacement is varying, is equal to the rate of change of the total electric displacement through the circuit multiplied by 4 π.62

(B) The work done at any moment in carrying a unit quantity of electricity round a circuit in a field in which the magnetic induction is varying, is equal to the rate of change of the total magnetic induction through the circuit.

From these two principles, combined with the laws connecting electric force and displacement, magnetic force and induction, and with the condition of continuity, Maxwell obtained his equations of the field.

Faraday’s experiments on electro-magnetic induction afford the proof of the truth of the fourth principle. It follows from those experiments that when the number of lines of magnetic induction187 which are linked with any closed circuit are made to vary, an induced electromotive force is brought into play round that circuit. This electromotive force is, according to Faraday’s results, measured by the rate of decrease in the number of lines of magnetic induction which thread the circuit. Maxwell applies this principle to all circuits, whether conducting or not.

In obtaining equations to express in symbols the results of the fourth principle just enunciated, Maxwell introduces a new quantity, to which he gives the name of the “vector potential.” This quantity appears in his analysis, and its physical meaning is not at first quite clear. Professor Poynting has, however, put Maxwell’s principles in a slightly different form, which enables us to see definitely the meaning of the vector potential, and to deduce Maxwell’s equations more readily from the fundamental statements.

We are dealing with a circuit with which lines of magnetic induction are linked, while the number of such lines linked with the circuit is varying. Now, let us suppose the variation to take place in consequence of the lines of induction moving outwards or inwards, as the case may be, so as to cut the circuit. Originally there are none linked with the circuit. As the magnetic field has grown to its present strength lines of magnetic induction have moved inwards. Each little element of the circuit has been cut by some, and the total number linked with the circuit can be found by adding together those cut by each element. Now, Professor Poynting’s statement of Maxwell’s fourth principle is that the electrical force in the direction of any element of the circuit is found by188 dividing by the length of the element the number of lines of magnetic induction which are cut in one second by it.

Moreover, the total number of lines of magnetic induction which have been cut by an element of unit length is defined as the component of the vector potential in the direction of the element; hence the electrical force in any direction is the rate of decrease of the component of the vector potential in that direction. We have thus a physical meaning for the vector potential, and shall find that in the dynamical theory this quantity is of great importance.

Professor Poynting has modified Maxwell’s third principle in a similar manner; he looks upon the variation in the electric displacement as due to the motion of tubes of electric induction,63 and the magnetic force along any circuit is equal to the number of tubes of electric induction cutting or cut by unit length of the circuit per second, multiplied by 4π.

From the equations of the field, as found by Maxwell, it is possible to derive two sets of symmetrical equations. The one set connects the rate of change of the electric force with quantities depending on the magnetic force; the other set connects in a similar manner the rate of change of the magnetic force with quantities depending on the electric force.189 Several writers in recent years adopt these equations as the fundamental relations of the field, establishing them by the argument that they lead to consequences which are found to be in accordance with experiment.

We have endeavoured to give some account of Maxwell’s historical method, according to which the equations are deduced from the laws of electric currents and of electro-magnetic induction derived directly from experiment.

While the manner in which Maxwell obtained his equations is all his own, he was not alone in stating and discussing general equations of the electro-magnetic field. The next steps which we are about to consider are, however, in a special manner due to him. An electrical or magnetic system is the seat of energy; this energy is partly electrical, partly magnetic, and various expressions can be found for it. In Maxwell’s theory it is a fundamental assumption that energy has position. “The electric and magnetic energies of any electro-magnetic system,” says Professor Poynting, “reside, therefore, somewhere in the field.” It follows from this that they are present wherever electric and magnetic force can be shown to exist. Maxwell showed that all the electric energy is accounted for by supposing that in the neighbourhood of a point at which the electric force is R there is an amount of energy per unit of volume equal to KR²/8π, K being the inductive capacity of the medium, while in the neighbourhood of a point at which the magnetic force is H, the magnetic energy per unit of volume is μH²/8π, μ being the permeability. He supposes, then, that at each point of190 an electro-magnetic system energy is stored according to these laws. It follows, then, that the electro-magnetic field resembles a dynamical system in which energy is stored. Can we discover more of the mechanism by which the actions in the field are maintained? Now the motion of any point of a connected system depends on that of other points of the system; there are generally, in any machine, a certain number of points called driving-points, the motion of which controls the motion of all other parts of the machine; if the motion of the driving-points be known, that of any other point can be determined. Thus in a steam engine the motion of a point on the fly-wheel can be found if the motion of the piston and the connections between the piston and the wheel be known.

In order to determine the force which is acting on any part of the machine we must find its momentum, and then calculate the rate at which this momentum is being changed. This rate of change will give us the force. The method of calculation which it is necessary to employ was first given by Lagrange, and afterwards developed, with some modifications, by Hamilton. It is usually referred to as Hamilton’s principle; when the equations in the original form are used they are known as Lagrange’s equations.

Now Maxwell showed how these methods of calculation could be applied to the electro-magnetic field. The energy of a dynamical system is partly kinetic, partly potential. Maxwell supposes that the magnetic energy of the field is kinetic energy, the electric energy potential. When the kinetic energy of a191 system is known, the momentum of any part of the system can be calculated by recognised processes. Thus if we consider a circuit in an electro-magnetic field we can calculate the energy of the field, and hence obtain the momentum corresponding to this circuit. If we deal with a simple case in which the conducting circuits are fixed in position, and only the current in each circuit is allowed to vary, the rate of change of momentum corresponding to any circuit will give the force in that circuit. The momentum in question is electric momentum, and the force is electric force. Now we have already seen that the electric force at any point of a conducting circuit is given by the rate of change of the vector potential in the direction considered. Hence we are led to identify the vector potential with the electric momentum of our dynamical system; and, referring to the original definition of vector potential, we see that the electric momentum of a circuit is measured by the number of lines of magnetic induction which are interlinked with it.

Again, the kinetic energy of a dynamical system can be expressed in terms of the squares and products of the velocities of its several parts. It can also be expressed by multiplying the velocity of each driving-point by the momentum corresponding to that driving-point, and taking half the sum of the products. Suppose, now, we are dealing with a system consisting of a number of wire circuits in which currents are running, and let us suppose that we may represent the current in each wire as the velocity of a driving-point in our dynamical system. We can also express192 in terms of these currents the electric momentum of each wire circuit; let this be done, and let half the sum of the products of the corresponding velocities and momenta be formed.

In maintaining the currents in the wires energy is needed to supply the heat which is produced in each wire; but in starting the currents it is found that more energy is needed than is requisite for the supply of this heat. This excess of energy can be calculated, and when the calculation is made it is found that the excess is equal to half the sum of the products of the currents and corresponding momenta. Moreover, if this sum be expressed in terms of the magnetic force, it is found to be equal to μ H²/8 π, which is the magnetic energy of the field. Now, when a dynamical system is set in motion against known forces, more energy is supplied than is needed to do the work against the forces; this excess of energy measures the kinetic energy acquired by the system.

Hence, Maxwell was justified in taking the magnetic energy of the field as the kinetic energy of the mechanical system, and if the strengths of the currents in the wires be taken to represent the velocities of the driving-points, this energy is measured in terms of the electrical velocities and momenta in exactly the same way as the energy of a mechanical system is measured in terms of the velocities and momenta of its driving-points.

The mechanical system in which, according to Maxwell, the energy is stored is the ether. A state of motion or of strain is set up in the ether of the field. The electric forces which drive the currents, and also193 the mechanical forces acting on the conductors carrying the currents, are due to this state of motion, or it may be of strain, in the ether. It must not be supposed that the term electric displacement in Maxwell’s mind meant an actual bodily displacement of the particles of the ether; it is in some way connected with such a material displacement. In his view, without motion of the ether particles there would be no electric action, but he does not identify electric displacement and the displacement of an ether particle.

His mechanical theory, however, does account for the electro-magnetic forces between conductors carrying currents. The energy of the system depends on the relative positions of the currents which form part of it. Now, any conservative mechanical system tends to set itself in such a position that its potential energy is least, its kinetic energy greatest. The circuits of the system, then, will tend to set themselves so that the electro-kinetic energy of the system may be as large as possible; forces will be needed to hold them in any position in which this condition is not satisfied.

We have another proof of the correctness of the value found for the energy of the field in that the forces calculated from this value agree with those which are determined by direct experiment.

Again, the forces applied at the various driving-points are transmitted to other points by the connections of the machine; the connections are thrown into a state of strain; stress exists throughout their substance. When we see the piston-rod and the shaft194 of an engine connected by the crank and the connecting-rod, we recognise that the work done on the piston is transmitted thus to the shaft. So, too, in the electro-magnetic field, the ether forms the connection between the various circuits in the field; the forces with which those circuits act on each other are transmitted from one circuit to another by the stresses set up in the ether.

To take another instance, consider the electrostatic attraction between two charged bodies. Let us suppose the bodies charged by connecting each to the opposite pole of a battery; a current flows from the battery setting up electric displacement in the space between the bodies, and throwing the ether into a state of strain. As the strain increases the current gets less; the reaction resulting from the strain tends to stop it, until at last this reaction is so great that the current is stopped. When this is the case the wires to the battery may be removed, provided this is done without destroying the insulation of the bodies; the state of strain will remain and shows itself in the attraction between the balls.

Looking at the problem in this manner, we are face to face with two great questions—the one, What is the state of strain in the ether which will enable it to produce the observed electrostatic attractions and repulsions between charged bodies? and the other, What is the mechanical structure of the ether which would give rise to such a state of strain as will account for the observed forces? Maxwell gives one answer to the first question; it is not the only answer which could be given, but it does account for the195 facts. He failed to answer the second. He says (“Electricity and Magnetism,” vol. i. p. 132):—

Faraday had pointed out that the inductive action between two bodies takes place along the lines of force, which tend to shorten along their length and to spread outwards in other directions. Maxwell compares them to the fibres of a muscle, which contracts and at the same time thickens when exerting force. In the electric field there is, on Maxwell’s theory, a tension along the lines of electric force and a pressure at right angles to those lines. Maxwell proved that a tension K R²/8 π along the lines of force, combined with an equal pressure in perpendicular directions, would maintain the equilibrium of the field, and would give rise to the observed attractions or repulsions between electrified bodies. Other distributions of stress might be found which would lead to the same result. The one just stated will always be connected with Maxwell’s name. It will be noticed that the tension along the lines of force and the pressure at right angles to them are each numerically equal to the potential energy stored per unit of volume in the field. The value of each of the three quantities is K R²/8 π.

In the same way, in a magnetic field, there is a state of stress, and on Maxwell’s theory this, too,196 consists of a tension along the lines of force and an equal pressure at right angles to them, the values of the tension and the pressure being each equal to that of the magnetic energy per unit of volume, or μH²/8π.

In a case in which both electric and magnetic force exists, these two states of stress are superposed. The total energy per unit of volume is KR²/8π + μH²/8π; the total stress is made up of tensions KR²/8π and μH²/8π along the lines of electric and magnetic force respectively, and equal pressures at right angles to these lines.

We see, then, from Maxwell’s theory, that electric force produced at any given point in space is transmitted from that point by the action of the ether. The question suggests itself, Does the transmission take time, and if so, does it proceed with a definite velocity depending on the nature of the medium through which the change is proceeding?

According to the molecular-vortex theory, we have seen that waves of electric force are transmitted with a definite velocity. The more general theory developed in the “Electricity and Magnetism” leads to the same result. Electric force produced at any point travels outwards from that point with a velocity given by 1/√(Kμ). At a distant point the force is zero, until the disturbance reaches it. If the disturbance last only for a limited interval, its effects will at any future time be confined to the space within a spherical shell of constant thickness depending on the interval; the radii of this shell increase with uniform speed 1/√(Kμ).

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If the initial disturbance be periodic, periodic waves of electric force will travel out from the centre, just as waves of sound travel out from a bell, or waves of light from a candle flame. A wire carrying an alternating current may be such a source of periodic disturbance, and from the wire waves travel outwards into space.

Now, it is known that in a sound wave the displacements of the air particles take place in the direction in which the wave is travelling; they lie at right angles to the wave front, and are spoken of as longitudinal. In light waves, on the other hand, the displacements are, as Fresnel proved, in the wave front, at right angles, that is, to the direction of propagation; they are transverse.

Theory shows that in general both these waves may exist in an elastic solid body, and that they travel with different velocities. Of which nature are the waves of electric displacement in a dielectric? It can be shewn to follow as a necessary consequence of Maxwell’s views as to the closed character of all electric currents, that waves of electric displacement are transverse. Electric vibrations, like those of light, are in the wave front and at right angles to the direction of propagation; they depend on the rigidity or quasi-rigidity of the medium through which they travel, not on its resistance to compression.

Again, an electric current, whether due to variation of displacement in a dielectric or to conduction in a conductor, is accompanied by magnetic force. A wave of periodic electric displacement, then, will be also a wave of periodic magnetic force travelling at198 the same rate; and Maxwell shewed that the direction of this magnetic force also lies in the wave front, and is always at right angles to the electric displacement. In the ordinary theory of light the wave of linear displacement is accompanied by a wave of periodic angular twist about a direction lying in the wave front and perpendicular to the linear displacement.

In many respects, then, waves of electric displacement resemble waves of light, and, indeed, as we proceed we shall find closer connections still. Hence comes Maxwell’s electro-magnetic theory of light.

It is only in dielectric media that electric force is propagated by wave motion. In conductors, although the third and fourth of Maxwell’s principles given on page 185 still are true, the relation between the electric force and the electric current differs from that which holds in a dielectric. Hence the equations satisfied by the force are different. The laws of its propagation resemble those of the conduction of heat rather than those of the transmission of light.

Again, light travels with different velocities in different transparent media. The velocity of electric waves, as has been stated, is equal to 1/√(μK); but in making this statement it is assumed that the simple laws which hold where there is no gross matter—or, rather, where air is the only dielectric with which we are concerned—hold also in solid or liquid dielectrics. In a solid or a liquid, as in vacuo, the waves are propagated by the ether. We assume, as a first step towards a complete theory, that so far as the electric waves are concerned the sole effect199 produced by the matter shews itself in a change of inductive capacity or of permeability. It is not likely that such a supposition should be the whole truth, and we may, therefore, expect results deduced from it to be only approximation to the true result.

Now, electro-magnetic experiments show that, excluding magnetic substances, the permeability of all bodies is very nearly the same, and differs very slightly from that of air. The inductive capacity, however, of different bodies is different, and hence the velocity with which electro-magnetic waves travel differs in different bodies.

But the refraction of waves of light depends on the fact that light travels with different velocities in different media; hence we should expect to have waves of electric displacement reflected and refracted when they pass from one dielectric, such as air, to another, such as glass or gutta-percha; moreover, for light the refractive index of a medium such as glass is the ratio of the velocity in air to the velocity in the glass.

Thus the electrical refractive index of glass is the ratio of the velocity of electric waves in air to their velocity in glass.

Now let K₀ be the inductive capacity of air, K₁ that of glass, taking the permeability of air and glass to be the same, we have the result that—

Electrical refractive index = √(K₁/K₀).

But the ratio of the inductive capacity of glass to that of air is known as the specific inductive capacity of glass.

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Hence, the specific inductive capacity of any medium is equal to the square of the electrical refractive index of that medium.

Since Maxwell’s time the mathematical laws of the reflexion and refraction of electric waves have been investigated by various writers, and it has been shewn that they agree exactly with those enunciated by Fresnel for light.

Hitherto we have been discussing the propagation of electric waves in an isotropic medium, one which has identical properties in all directions about a point. Let us now consider how these laws are modified if the dielectric be crystalline in structure.

Maxwell assumes that the crystalline character of the dielectric can be sufficiently represented by supposing the inductive capacity to be different in different directions; experiments have since shewn that this is true for crystals such as Iceland Spar and Aragonite; he assumes also, and this, too, is justified by experiment, that the magnetic permeability does not depend on the direction. It follows from these assumptions that a crystal will produce double refraction and polarisation of electric waves which fall upon it, and, further, that the laws of double refraction will be those given by Fresnel for light waves in a doubly refracting medium. There will be two waves in the crystal. The disturbance in each of these will be plane polarised; their velocity and the position of their plane of polarisation can be found from the direction in which they are travelling by Fresnel’s construction exactly.

Maxwell’s theory, then, would appear to indicate201 some close connection between electric waves and those of light. Faraday’s experiments on the rotation of the plane of polarisation by magnetic force shew one phenomenon in which the two are connected, and Maxwell endeavoured to apply his theory to explain this. Here, however, it became necessary to introduce an additional hypothesis—there must be some connection between the motion of the ether to which magnetic force is due and that which constitutes light. It is impossible to give a mechanical account of the rotation of the plane of polarisation without some assumption as to the relation between these two kinds of motion. Maxwell, therefore, supposes the linear displacements of a point in the ether to be those which give rise to light, while the components of the magnetic force are connected with these in the same way as the components of a vortex in a liquid in vortex motion are connected with the displacements of the liquid. He further assumes the existence of a term of special form in the expression for the kinetic energy, and from these assumptions he deduces the laws of the propagation of polarised light in a magnetic field. These laws agree in the main with the results of Verdet’s experiments.

We have endeavoured in the preceding pages to give some account of Maxwell’s contributions to electrical theory and the physics of the ether. We must now consider very briefly what evidence there is to support these views. At Maxwell’s death such evidence, though strong, was indirect. His supporters were limited to some few English-speaking pupils, young and enthusiastic, who were convinced, it may be, in no small measure, by the affection and reverence with which they regarded their master. Abroad his views had made very little way.

In the last words of his book he writes, speaking of various distinguished workers—

Let us see, then, what were the experimental grounds in Maxwell’s day for accepting as true his views on electrical action, and how since then, by the genius of Heinrich Hertz and the labours of his followers, those grounds have been rendered so sure that nearly the whole progress of electrical science during the last twenty years has consisted in the development of ideas which are to be found in the “Treatise on Electricity and Magnetism.”

The purely electrical consequences of Maxwell’s theory were of course in accord with all known electrical observations. The equations of the field accounted for the electro-magnetic forces observed in various experiments, and from them the laws of electro-magnetic induction could be correctly deduced; but there was nothing very special in this. Similar equations had been obtained from the theory of action at a distance by various writers; in fact, Helmholtz’s theory, based on the most general form of expression for the force between two elements of current consistent with certain experiments of Ampère’s, was more general in its character than Maxwell’s. The destructive features of Maxwell’s theory were:

(1) The assumption that all currents flow in closed circuits.

(2) The idea of energy residing throughout the electro-magnetic field in consequence of the strains and stresses set up in the electro-magnetic medium by the actions to which it was subject.

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(3) The identification of this electro-magnetic medium with the luminiferous ether, and the consequent view that light is an electro-magnetic phenomena.

(4) The view that electro-magnetic forces arise entirely from strains and stresses set up in the ether; the electrostatic charge of an insulated conductor being one of the forms in which the ether strain is manifested to us.

(5) A dielectric under the action of electric force is said to become polarised, and, according to Maxwell (vol. i. p. 133), all electrification is the residual effect of the polarisation of the dielectric.

Now it must, I think, be admitted that in Maxwell’s day there was direct proof of very few of these propositions. No one has even yet so measured the displacement currents in a dielectric as to show that the total flow across every section of a circuit is at any given moment the same, though there are other experiments of an indirect character which have now completely justified Maxwell’s hypothesis. Experiments by Schiller and Von Helmholtz prove it is true that some action in the dielectric must be taken into consideration in any satisfactory theory; they therefore upset various theories based on direct action at a distance, “but they tell us nothing as to whether any special form of the dielectric theory, such as Maxwell’s or Helmholtz’s, is true or not.” (J. J. Thomson, “Report on Electrical Theories,” B.A. Report, 1885, p. 149.)

When Maxwell died there had been little if any experimental evidence as to the stresses set up in a206 body by electric force. Fontana, Govi, and Duter had all observed that changes take place in the volume of the dielectric of a condenser when it is charged. Quincke had taken up the work, and the first of his classic papers on this subject was published in 1880, the year following Maxwell’s death. Maxwell himself was fond of shewing an experiment in which a charged insulated sphere was brought near to the surface of paraffin; the stress on the surface causes a heaping up of the paraffin under the sphere.

Kerr had shewn in 1875 that many substances become doubly refracting under electric stress; his complete determination of the laws of this action was published at a later date.

As to direct measurements on electric waves, there were none; the value of the velocity with which, if Maxwell’s theory were true, they must travel had been determined from electrical observations of quite a different character. Weber and Kohlrausch had measured the value of K for air, for which μ is unity, and from their observations it follows that the value of the wave velocity for electro-magnetic waves is about 31 × 10⁹ centimetres per second. The velocity of light was known, from the experiments of Fizeau and Foucault, to have about this value, and it was the near coincidence of these two values which led Maxwell to write in 1864:—

“The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electro-magnetic disturbance propagated through the field according to electro-magnetic laws.”

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By the time the first edition of the “Electricity and Magnetism” was published, Maxwell and Thomson (Lord Kelvin) had both made determinations of K, and had shewn that for air at least the resulting value for the velocity of electro-magnetic waves was very nearly that of light.

For other substances at that date the observations were fewer still. Gibson and Barclay had determined the specific inductive capacity of paraffin, and found that its square root was 1·405, while its refractive index for long waves is 1·422. Maxwell himself thought that if a similar agreement could be shewn to hold for a number of substances, we should be warranted in concluding that “the square root of K, though it may not be the complete expression for the index of refraction, is at least the most important term in it.”

Between this time and Maxwell’s death enough had been done to more than justify this statement. It was clear from the observations of Boltzmann, Silow, Hopkinson, and others that there were many substances for which the square root of the specific inductive capacity was very nearly indeed equal to the refractive index, and good reason had been given why in some cases there should be a considerable difference between the two.

Hopkinson found that in the case of glass the differences were very large, and they have since been found to be considerable for most solids examined, with the exception of paraffin and sulphur. For petroleum oil, benzine, toluene, carbon-bisulphide, and some other liquids the agreement between Maxwell’s theory and experiment is close. For the fatty oils,208 such as castor oil, olive oil, sperm oil, neatsfoot oil, and also for ether, the differences are considerable.

It seems probable that the reason for this difference lies in the fact that, in the light waves, we are dealing with the wave velocity of a disturbance of an extremely short period. Now, we know that the substances mentioned shew optical dispersion, and we have at present no completely satisfactory theory from which we can calculate, from experiments on very short waves, what the velocity for very long waves will be. In most cases Cauchy’s formula has been used to obtain the numbers given. The value of K, however, as found by experiment, corresponds to these infinitely long waves, and to quote Professor J. J. Thomson’s words, “the marvel is not that there should not be substances for which the relation K = μ² does not hold, but that there should be any for which it does.”66

It has been shewn, moreover, both by Professor J. J. Thomson himself and by Blondlot, that when the value of K is measured under very rapidly varying electrifications, changing at the rate of about 25,000,000 to the second, the value of the inductive capacity for glass is reduced from about 6·8 or 7 to about 2·7; the square root of this is 1·6, which does not differ much from its refractive index. The values of the inductive capacity of paraffin and sulphur, which it will be remembered agree fairly with Maxwell’s theory, were found to be not greatly different in the steady and in the rapidly varying field.

On the other hand, some experiments of Arons209 and Rubens in rapidly varying fields lead to values which do not differ greatly from those given by other methods. The theory, however, of these experiments seems open to criticism.

To attempt anything like a complete account of modern verifications of Maxwell’s views and modern developments of his theory is a task beyond our limits, but an account of Maxwell written in 1895 would be incomplete without a reference to the work of Heinrich Hertz.

Maxwell told us what the properties of electro-magnetic waves in air must be. Hertz67 in 1887 enabled us to measure those properties, and the measurements have verified completely Maxwell’s views.

The method of producing electrical oscillations in a conductor had long been known. Thomson and Von Helmholtz had both pointed it out. Schiller had examined such oscillations in 1874, and had determined the inductive capacity of glass by their means, using oscillations whose period varied from ·000056 to ·00012 of a second.

These oscillations were produced by discharging a condenser through a coil of wire having self-induction. If the electrical resistance of the coil be not too great, the charge oscillates backwards and forwards between the plates of the condenser until its energy is dissipated in the heat produced in the wire, and in the electro-magnetic radiations which leave it.

The period of these oscillations under proper conditions is given by the formula T = 2π√(CL) where210 L, the coefficient of self induction, and C the capacity of the condenser. These quantities can be calculated, and hence the time of an oscillation is known. From such an arrangement waves radiate out into space. If we could measure by any method the length of such a wave we could determine its velocity by dividing the wave length by the period. But it is clear that since the velocity is comparable with that of light the wave length will be enormous, unless the period is very short. Thus, a wave, travelling with the velocity of light, whose period was ·0001 second, such as the waves Schiller worked with, would have a length of ·0001 × 30,000,000,000 or 3,000,000 centimetres, and would be quite unmeasurable. Before measurements on electric waves could be made it was necessary (1) to produce waves of sufficiently rapid period, (2) to devise means to detect them. This is what Hertz did.

The wave length of the electrical oscillations can be reduced by reducing either the electrical capacity of the system, or the coefficient of self-induction of the wire. Hertz adopted both these expedients. His vibrator, in some of his more important experiments, consisted of two square brass plates 40 cm. in the side. To each of these is attached a piece of copper wire about 30 cm. in length, and each wire ends in a small highly-polished brass ball. The plates are placed so that the wires lie in the same straight line, the brass balls being separated by a very small air gap. The two plates are then charged, the one positively the other negatively, until the insulation resistance of the air gap breaks down and a discharge passes across. Under these conditions the discharge211 is oscillatory. It does not consist of a single spark, but of a series of sparks, which pass and repass in opposite directions, until the energy of the original charge is radiated into space or dissipated as heat; the plates are then recharged and the process repeated. In Hertz’s experiments the oscillator was charged by being connected to the secondary terminals of an induction coil.

In 1883 Professor Fitzgerald had called attention to this method of producing electric waves in air, and had given two metres as the minimum wave length which might be attained. In 1870 Herr von Bezold had actually made observations on the propagation and reflection of electrical oscillations, but his work, published as a preliminary communication, had attracted little notice. Hertz was the first to undertake in 1887 in a systematic manner the investigation of the electric waves in air which proceed from such an oscillator with a view to testing various theories of electro-magnetic action.

It remained, however, necessary to devise an apparatus for detecting the waves. When the waves are incident on a conductor, electric surgings are set up in the conductor, and may, under proper conditions, be observed as tiny sparks. Hertz used as his detector a loop of wire, the ends of which terminated in two small brass balls. The wire was bent so that the balls were very close together, and the sparks could be seen passing across the tiny air gap which separated them. Such a wire will have a definite period of its own for oscillations of electricity with which it may be charged, and if the frequency of the electric waves212 which fall on it agrees with that of the waves which it can itself emit, the oscillations which are set up in the wire will be stronger than under other conditions, the sparks seen will be more brilliant.68 Hertz’s resonator was a circle of wire thirty-five centimetres in radius, the period for such a resonator would, he calculated, be the same as that of his vibrator.

There is, however, very considerable difficulty in determining the period of an electric oscillator from its dimensions, and the value obtained from calculation for that of Hertz’s radiator is not very trustworthy. The complete period is, however, comparable with two one hundredth millionths of a second; in his original papers, Hertz, through an error, gave a value greater than this.

With these arrangements Hertz was able to detect the presence of electrical radiation at considerable distances from the radiator; he was also able to measure its wave length. In the case of sound waves the existence of nodes and loops formed under proper conditions is well known. When waves are directly reflected from a flat surface, interference takes place between the incident and reflected waves, stationary vibrations are set up, and nodes and loops—places, that is, of minimum and of maximum motion respectively—are formed. The position of these nodes and loops can be determined by the aid of suitable apparatus, and it can be shewn that the distance between two consecutive nodes is half the wave length.

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Similarly when electrical vibrations fall on a reflector, a large flat surface of metal, for example, stationary vibrations due to the interference between the incident and reflected waves are produced, and these give rise to electrical nodes and loops. The position of such nodes and loops can be found by the use of Hertz’s apparatus, or in other ways, and hence the length of the electrical waves can be found. The existence of the nodes and loops shews that the electric effects are propagated by wave motion. The length of the waves is found to be definite, since the nodes and loops recur at equal intervals apart.

If it be assumed that the frequency is known, the velocity of wave propagation can be determined. Hertz found from his experiments that in air the waves travelled with the velocity of light. It appears, however, that there were two errors in the calculation which happened to correct each other, so that neither the value of the frequency given in Hertz’s paper nor the wave length observed is correct.

By modifying the apparatus it was possible to measure the wave length of the waves transmitted along a copper wire, and hence, again assuming the period of oscillation, to calculate the velocity of wave propagation along the wire. Hertz made the experiment, and found from his first observations that the waves were propagated along the wire with a finite velocity, but that the velocity differed from that in air. The half-wave length in the wire was only about 2·8 metres; that in air was about 4·5 metres.

Now, this experiment afforded a crucial test between the theories of Maxwell and Von Helmholtz.214 According to the former, the waves do not travel in the wire at all; they travel through the air alongside the wire, and the wave length observed by Hertz ought to have been the same as in air. According to Von Helmholtz, the two velocities observed by Hertz should have been different, as, indeed, they were, and the experiment appeared to prove that Maxwell’s theory was insufficient and that a more general one, such as that of Von Helmholtz, was necessary. But other experiments have not led to the same result. Hertz himself, using more rapid oscillations in some later measurements, found that the wave length of the electric waves from a given oscillator was the same whether they were transmitted through free space or conducted along a wire.69 Lecher and J. J. Thomson have arrived at the same result; but the most complete experiments on this point are those of Sarasin and De la Rive.

It may be taken, then, as established that Maxwell’s theory is sufficient, and that the greater generality of Von Helmholtz is unnecessary.

In a later paper Hertz showed that electric waves could be reflected and refracted, polarised and analysed, just like light waves. In his introduction to his “Collected Papers” he writes (p. 19):—

Since Maxwell’s death volumes have been written on electrical questions, which have all been inspired by his work. The standpoint from which electrical theory is regarded has been entirely changed. The greatest masters of mathematical physics have found, in the development of Maxwell’s views, a task that called for all their powers, and the harvest of new truths which has been garnered has proved most rich. But while this is so, the question is still often asked, What is Maxwell’s theory? Hertz himself concludes the introduction just referred to with his most interesting answer to this question. Prof. Boltzmann has made the theory the subject of an important course of lectures. Poincaré, in the introduction to his “Lectures on Maxwell’s Theories and the Electro-magnetic Theory of Light,” expresses the difficulty, which many feel, in understanding what the theory is. “The first time,” he says, “that a French reader opens Maxwell’s book a feeling of uneasiness, often even of distrust, is mingled with his admiration. It is only after prolonged study, and at the cost of many efforts, that this feeling is dissipated. Some great minds retain it always.” And again he writes: “A French savant, one of those who have most completely fathomed Maxwell’s meaning, said to me once, ‘I understand everything in the book except what is meant by a body charged with electricity.’”

In considering this question, Poincaré’s own217 remark—“Maxwell does not give a mechanical explanation of electricity and magnetism, he is only concerned to show that such an explanation is possible”—is most important.

We cannot find in the “Electricity” an answer to the question—What is an electric charge? Maxwell did not pretend to know, and the attempt to give too great definiteness to his views on this point is apt to lead to a misconception of what those views were.

On the old theories of action at a distance and of electric and magnetic fluids attracting according to known laws, it was easy to be mechanical. It was only necessary to investigate the manner in which such fluids could distribute themselves so as to be in equilibrium, and to calculate the forces arising from the distribution. The problem of assigning such a mechanical structure to the ether as will permit of its exerting the action which occurs in an electro-magnetic field is a harder one to solve, and till it is solved the question—What is an electric charge?—must remain unanswered. Still, in order to grasp Maxwell’s theory this knowledge is not necessary.

The properties of ether in dielectrics and in conductors must be quite different. In a dielectric the ether has the power of storing energy by some change in its configuration or its structure; in a conductor this power is absent, owing probably to the action of the matter of which the conductor is composed.

When we are said to charge an insulated conductor we really act on the ether in the neighbourhood of the body so as to store it with energy; if there be another conductor in the field we cannot store energy in the218 ether it contains. As, then, we pass from the outside of this conductor to its interior there is a sudden change in some mechanical quantity connected with the ether, and this change shows itself as a force of attraction between the two conductors. Maxwell called the change in structure, or in property, which occurs when a dielectric is thus stored with electrostatic energy, Electric Displacement; if we denote it by D, then the electric force R is equal to 4πD/K, and hence the energy in a unit of volume is 2πD²/K, where K is a quantity depending on the insulator.

Now, D, the electric displacement, is a quantity which has direction as well as magnitude. Its value, therefore, at any point can be represented by a straight line in the usual way; inside a conductor it is zero. The total change in D, which takes place all over the surface of a conductor as we enter it from the outside measures, according to Maxwell, the total charge on the conductor. At points at which the lines representing D enter the conductor the charge is negative; at points at which they leave it the charge is positive; along the lines of the displacement there exists throughout the ether a tension measured by 2πD²/K; at right angles to these lines there is a pressure of the same amount.

In addition to the above the components of the displacement D must satisfy certain relations which can only be expressed in mathematical form, the physical meaning of which it is difficult to state in non-mathematical language.

When these relations are so expressed the problem of finding the value of the displacement at all points219 of space becomes determinate, and the forces acting on the conductors can be obtained. Moreover, the total change of displacement on entering or leaving a conductor can be calculated, and this gives the quantity which is known as the total electrical charge on the conductor. The forces obtained by the above method are exactly the same as those which would exist if we supposed each conductor to be charged in the ordinary sense with the quantities just found, and to attract or repel according to the ordinary laws.

If, then, we define electric displacement as that change which takes place in a dielectric when it becomes the seat of electrostatic energy, and if, further, we suppose that the change, whatever it be mechanically, satisfies certain well-known laws, and that in consequence certain pressures and tensions exist in the dielectric, electrostatic problems can be solved without reference to a charge of electricity residing on the conductors.

Something such as this, it appears to me, is Maxwell’s theory of electricity as applied to electrostatics. It is not necessary, in order to understand it, to know what change in the ether constitutes electric displacement, or what is an electric charge, though, of course, such knowledge would render our views more definite, and would make the theory a mechanical one.

When we turn to magnetism and electro-magnetism, Maxwell’s theory develops itself naturally. Experiment proves that magnetic induction is connected with the rate of change of electric displacement, according to the laws already given. If, then, we knew the nature of the change to which the name220 “electric displacement” has been given, the nature of magnetic induction would be known. The difficulties in the way of any mechanical explanation are, it is true, very great; assuming, however, that some mechanical conception of “electric displacement” is possible, Maxwell’s theory gives a consistent account of the other phenomena of electro-magnetism.

Again, we have, it is true, an electro-magnetic theory of light, but we do not know the nature of the change in the ether which affects our eyes with the sensation of light. Is it the same as electric displacement, or as magnetic induction, or since, when electric displacement is varying, magnetic induction always accompanies it, is the sensation of light due to the combined effect of the two?

These questions remain unanswered. It may be that light is neither electric displacement nor magnetic induction, but some quite different periodic change of structure of the ether, which travels through the ether at the same rate as these quantities, and obeys many of the same laws.

In this respect there is a material difference between the ordinary theory of light and the electro-magnetic theory. The former is a mechanical theory; it starts from the assumption that the periodic change which constitutes light is the ordinary linear displacement of a medium—the ether—having certain mechanical properties, and from those properties it deduces the laws of optics with more or less success.

Lord Kelvin, in his labile ether, has devised a medium which could exist and which has the necessary mechanical properties. The periodic linear221 displacements of the labile ether would obey the laws of light, and from the fundamental hypotheses of the theory, a mechanical explanation, reasonably satisfactory in its main features, can be given of most purely optical phenomena. The relations between light and electricity, or light and magnetism, are not, however, touched by this theory; indeed, they cannot be touched without making some assumption as to what electric displacement is.

In recent years various suggestions have been made as to the nature of the change which constitutes electric displacement. One theory, due to Von Helmholtz, supposes that the electro-kinetic momentum, or vector potential of Maxwell, is actually the momentum of the moving ether; according to another, suggested, it would appear originally in a crude form by Challis, and developed within the last few months in very satisfactory detail by Larmor, the velocity of the ether is magnetic force; others have been devised, but we are still waiting for a second Newton to give us a theory of the ether which shall include the facts of electricity and magnetism, luminous radiation, and it may be gravitation.71

Meanwhile we believe that Maxwell has taken the first steps towards this discovery, and has pointed out the lines along which the future discoverer must direct his search, and hence we claim for him a foremost place among the leaders of this century of science.

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