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MISCELLANEOUS PAPEES 
 
BY THE SAME AUTHOR. 
 
 ELECTRIC WAVES: 
 
 Researches on the Propagation of Electric Action 
 
 with Finite Velocity through space. 
 
 Translated by D. E. JONES. 
 
 With a Preface by LORD KELVIN. 
 
 8vo. 10s. net. 
 
 ELECTRICIAN : — "There is not in the entire annals of scientific research a more 
 completely logical and philosophical method recorded than that which has been 
 rigidly adhered to by Hertz from start to finish. We can conceive of no more delight- 
 ful intellectual treat than following up the charming orderliness of the records in the 
 pages before us. . . . The researches are a splendid consummation of the efforts which 
 have been made since the time of Maxwell to establish the doctrine of one ether for 
 all energy and force propagation— light, heat, electricity, and magnetism. The original 
 papers, with their introduction, form a lasting monument of the work "thus achieved. 
 The able translation before us, in which we have a skilful blend of the original mean- 
 ing with the English idiom, and which is copiously illustrated, places the record of 
 these researches within the reach of the English reading public, and enables it to study 
 this important and epoch-making landmark in the progress of physical science." 
 
 IN THE PBESS. 
 THE PRINCIPLES OP MECHANICS. 
 
 With a Preface by H. von Helmholtz. Translated by 
 D. E. Jones and J. T. Walley. 
 
^-<2^^<^ /-^^ 
 
MISCELLANEOUS PAPEES 
 
 BY 
 
 HEINEICH HERTZ 
 
 LATE PROFESSOR OF PHYSICS IN THE UNIVERSITY OF BONN 
 
 WITH AN INTRODUCTION 
 
 BY 
 
 PEOF. PHILIPP LENAED 
 
 AUTHORISED ENGLISH TRANSLATION 
 
 BY 
 
 D. E. JONES, B.Sc. 
 
 LATELY PROFESSOR OF PHYSICS IN THE UNIVERSITY COLLEGE OF WALES, ABERYSTWYTH 
 
 AND 
 
 Gk A. SCHOTT, B.A., B.Sc. 
 
 DEMONSTRATOR AND ASSISTANT LECTURER IN THE UNIVERSITY COLLEGE OF WALES, 
 ABERYSTWYTH 
 
 3Lontr0tt 
 
 MACMILLAN AND CO., Ltd. 
 
 NEW YORK : MACMILLAN & CO. 
 
 1896 
 
 All rights reserved 
 
NH 
 
 EDITOR'S PREFACE 
 
 The present volume consists mainly of the earlier investiga- 
 tions which Heinrich Hertz carried out before his great 
 electrical researches. Hitherto they have been difficult of 
 access, being scattered amongst various journals, and some 
 (e.g. his inaugural dissertation) could scarcely be obtained at 
 all. Of later date are the last experimental investigation, 
 the Heidelberg lecture (published by the firm of Emil Strauss 
 in Bonn, by whose kind permission it is included in the 
 present volume), and the closing paper, which is a further 
 proof of the gratitude and admiration which Hertz cherished 
 towards his great master, who has now followed him. 
 
 The papers are for the most part arranged in the order 
 of their publication. By the kindness of Senator Dr. Gustav 
 Hertz I have been able to include in the Introduction extracts 
 from Hertz's letters to his parents, which give us an insight 
 into the course of his scientific development, and the way in 
 which he was led to attack the problems herein discussed. 
 
 P. LENAKD. 
 
 February 1895. 
 
 £90435 
 
TRANSLATORS' NOTE 
 
 Hertz's Miscellaneous Papers form the first volume of his 
 collected works, as edited by Dr. Philipp Lenard. The second 
 volume is a reprint of his Eesearches on the Propagation of 
 Electric Action (already published in English under the title 
 of Electric Waves). The third volume consists of his Principles 
 of Mechanics, of which an English translation is now in the 
 press. 
 
 Professor Lenard has shown a warm interest in the 
 translation, and we desire to express our hearty thanks to 
 him for his kind assistance. 
 
 The portrait which forms the frontispiece to this volume 
 has been specially engraved for it from a photograph by E. 
 Krewaldt of Bonn. 
 
 D. E. J. 
 
 G. A. S. 
 
 March 1896. 
 
CONTENTS 
 
 PAGE 
 
 Introduction ........ ix 
 
 1. Experiments to determine an Upper Limit to the Kinetic 
 
 Energy of an Electric Current [i], 1880 . . . 1 
 
 2. On Induction in Rotating Spheres, 1880 . . .35 
 
 3. On the Distribution of Electricity over the Surface of 
 
 Moving Conductors, 1881 ..... 127 
 
 4. Upper Limit for the Kinetic Energy of Electricity in Motion 
 
 [ii], 1881 ........ 137 
 
 5. On the Contact of Elastic Solids, 1881 .... 146 
 
 6. On the Contact of Rigid Elastic Solids and on Hardness, 
 
 1882 ......... 163 
 
 7. On a New Hygrometer, 1882 . . . . .184 
 
 8. On the Evaporation of Liquids, and especially of Mercury, 
 
 in Vacuo, 1882 ....... 186 
 
 9. On the Pressure of Saturated Mercury- Vapour, 1882 . 200 
 
 10. On the Continuous Currents which the Tidal Action of the 
 
 Heavenly Bodies must produce in the Ocean, 1883 . 207 
 
 11. Hot -Wire Ammeter of Small Resistance and Negligible 
 
 Inductance, 1883 ....... 211 
 
 12. On a Phenomenon which accompanies the Electric Discharge, 
 
 1883 ......... 216 
 
 13. Experiments on the Cathode Discharge, 1883 . . . 224 
 
 14. On the Behaviour of Benzene with respect to Insulation 
 
 and Residual Charge, 1883 . . . . .255 
 
 15. On the Distribution of Stress in an Elastic Right Circular 
 
 Cylinder 1884 ....... 261 
 
Vlll CONTENTS 
 
 PAGE 
 
 16. On the Equilibrium of Floating Elastic Plates, 1884 . 266 
 
 17. On the Relations between Maxwell's Fundamental Electro- 
 
 magnetic Equations and the Fundamental Equations of 
 
 the Opposing Electromagnetics, 1884 .... 273 
 
 18. On the Dimensions of Magnetic Pole in Different Systems of 
 
 Units, 1885 ........ 291 
 
 19. A Graphical Method of determining the Adiabatic Changes 
 
 of Moist Air, 1884 . . . . . .296 
 
 20. On the Relations between Light and Electricity, 1889 . 313 
 
 21. On the Passage of Cathode Rays through Thin Metallic 
 
 Layers, 1892 ....... 328 
 
 22. Hermann yon Helmholtz, 1891 ..... 332 
 
INTRODUCTION 
 
 In October 1877, at the age of twenty, Heinrich Hertz went 
 to Munich in order to carry on his engineering studies. He 
 had chosen this as his profession, and had already made some 
 progress in it ; for in addition to completing the usual year 
 of practical work he had thoroughly grounded himself in the 
 preliminary mathematical and scientific studies. He had 
 now to apply himself to engineering work proper, to the 
 technical details of his profession. At this point he began to 
 doubt whether his natural inclinations lay in the direction of 
 this work — whether he would find engineering as satisfactory 
 as the studies which led up to it. The study of natural 
 science had been a delight to him : now he feared lest his life- 
 work should prove a burden. He stood at the parting of the 
 ways. In the following letter he consults his parents in the 
 matter. 
 
 Munich, 1st November 1877. 
 
 My dear Parents — No doubt you will wonder why this 
 letter follows so quickly after my previous one. I had no inten- 
 tion of writing so soon again, but this time it is about an important 
 matter which will not brook any long delay. 
 
 I really feel ashamed to say it, but I must : now at the last 
 moment I want to change all my plans and return to the study of 
 natural science. I feel that the time has come for me to decide 
 either to devote myself to this entirely or else to say good-bye to 
 it ; for if I give up too much time to science in .the future it will 
 end in neglecting my professional studies and becoming a second- 
 rate engineer. Only recently, in arranging my plan of studies, 
 have I clearly seen this — so clearly that I can no longer feel any 
 doubt about it ; and my first impulse was to renounce all un- 
 
X INTRODUCTION 
 
 necessary dealings with mathematics and natural science. But 
 then, all at once, I saw clearly that I could not bring myself to do 
 this ; that these had been my real occupation up to now, and 
 were still my chief joy. All else seemed hollow and unsatisfy- 
 ing. This conviction came upon me quite suddenly, and I felt 
 inclined to sit down and write to you at once. Although I have 
 restrained myself for a day or two, so as to consider the matter 
 thoroughly, I can come to no other result. I cannot understand 
 why all this was not clear to me before; for I came here filled 
 with the idea of working at mathematics and natural science, 
 whereas I had never given a thought to the essentials of my pro- 
 fessional training — surveying, building construction, builders' 
 materials, and such like. I have not forgotten what I often used 
 to say to myself, that I would rather be a great scientific investi- 
 gator than a great engineer, but would rather be a second-rate 
 engineer than a second-rate investigator. But now when I am in 
 doubt, I think how true is Schiller's saying, " Und setzet Bit nicht 
 das Leben ein, nie wird Euch das Leben gewonnen sein," and that 
 excessive caution would be folly. Nor do I conceal from myself 
 that by becoming an engineer I would be more certain of earning 
 my own livelihood, and I regret that in adopting the other course 
 I shall probably have to rely upon you, my dear father, all the 
 longer for support. But against all this there is the feeling that 
 I could devote myself wholly and enthusiastically to natural 
 science, and that this pursuit would satisfy me ; whereas I now see 
 that engineering science would not satisfy me, and would always 
 leave me hankering after something else. I hope that I am not 
 deceiving myself in this, for it would be a great and woful piece of 
 self-deception. But of this I feel positive, that if the decision is in 
 favour of natural science, I shall never look back with regret 
 towards engineering science, whereas if I become an engineer I 
 shall always be longing for the other ; and I cannot bear the idea 
 of being only able to work at natural science for the purpose of 
 passing an examination. When I think of it, it seems to me that 
 I used to be much more frequently encouraged to go on with natural 
 science than to become an engineer. I may be better grounded in 
 mathematics than many, but I doubt whether this would be much of 
 an advantage in engineering ; so much more seems to depend, at any 
 rate in the first ten years of practice, upon business capacity, ex- 
 perience, and knowledge of data and formulae, which do not happen 
 to interest me. This and much else I have carefully considered 
 (and shall continue to think it over until I receive your reply), 
 but when all is said and done, even admitting that there are many 
 sound practical reasons in favour of becoming an engineer, I still 
 feel that this would involve a sense of failure and disloyalty to 
 myself, to which I would not willingly submit if it could be 
 
INTRODUCTION XI 
 
 avoided. And so I ask you, dear father, for your decision rather 
 than for your advice ; for it isn't advice that I need, and there is 
 scarcely time for it now. If you will allow me to study natural 
 science I shall take it as a great kindness on your part, and what- 
 ever diligence and love can do in the matter that they shall do. I 
 . believe this will be your decision, for you have never put a stone 
 in my path, and I think you have often looked with pleasure on 
 my scientific studies. But if you consider it best for me to con- 
 tinue in the path on which I have started (which I now doubt), 
 I will carry out your wish, and do so fully and freely ; for by this 
 time I am sick of doubt and delay, and if I remain in the state I 
 have been in lately I shall never make a start. ... So I hope to 
 have an early answer, and until it comes I shall continue to think 
 the matter over. Meanwhile I send my love to you' all, and re- 
 main your affectionate son, Heinrich. 
 
 Matters were decided as he had hoped, and, full of joy at 
 being able to carry out his wishes, Hertz now proceeded to 
 arrange his plan of studies. He remained altogether a year at 
 Munich. He devoted the winter-semester of 1877-78 in all 
 seclusion to the study of mathematics and mechanics, using 
 for the most part original treatises such as those of Laplace 
 and Lagrange. Most of the following summer-semester he 
 spent at practical work in the physical laboratory. By 
 attending the elementary courses in practical physics at the 
 University (under v. Jolly) and at the same time in the 
 Technical Institute (under v. Bezold), he was able to supple- 
 ment what he had already learned by means of his own 
 home-made apparatus. 
 
 Thus prepared he proceeded in October 1878 to Berlin, 
 eager to become a pupil of v. Helmholtz and Kirchhoff. When 
 he had arrived there, in looking at the notices on the black 
 notice-board of the University his eye fell on an intimation of 
 a prize offered by the Philosophical Faculty for the solution 
 of a problem in physics. It referred to the question of electric 
 inertia. To him it did not seem so hopelessly difficult as it 
 might have appeared to many of his contemporaries, and he 
 decided to have a try at it. 
 
 This brings us to the beginning of his first independent 
 research (the first paper in the present volume). We cannot 
 read without astonishment the letters in which this student 
 of twenty-one reports to his parents the starting of an in- 
 
 h 
 
Xll INTRODUCTION 
 
 vestigation which might well be taken for the work of an 
 experienced investigator. 
 
 Berlin, 31st October 1878. 
 
 I have been attending lectures — KirchhofTs — since Monday : 
 another course only begins on Wednesday next. Besides this I 
 have also started practical work ; one of the prize problems for 
 this year falls more or less in my line, and I am going to work at 
 it. This was not what I intended at first, for a course of lectures 
 on mineralogy, which I wished to attend, clashed with it ; but I 
 have now decided to let these stand over until the next semester. 
 I have already discussed the matter with Professor Helmholtz, 
 who was good enough to put me on the track of some of the 
 literature. 
 
 A week later we find him already at his experiments. 
 
 6th November 1878. 
 
 Since yesterday I have been working in the laboratory. The 
 prize problem runs as follows : If electricity moves with inertia 
 in bodies, then this must, under certain circumstances, manifest 
 itself in the magnitude of the extra-current (i.e. of the secondary 
 current which is produced when an electric current starts or stops). 
 Experiments on the magnitude of the extra-current have to be 
 made such that a conclusion can be drawn from them as to inertia 
 of the electricity in motion. The work has to be finished by 4th 
 May ; it was given out as early as 3rd August, and I am sorry 
 that I did not know of it before. I ought, however, to say that 
 at present I am only trying to work out the problem, and I may 
 not succeed in solving it satisfactorily : so I would not readily 
 have spoken of it as a prize research, indeed I would not have 
 mentioned it at all, if it were not necessary by way of explanation. 
 Anyhow I find it very pleasant to be able to attack such an in- 
 vestigation. So yesterday I informed Professor Helmholtz that 
 I had considered the matter and would like to start work. He 
 then took me to the demonstrators and very kindly remained some 
 twenty minutes longer, talking with me about it, as to how I had 
 better begin and what instruments I should require. So yesterday 
 and to-day I have begun to make my arrangements. I have a 
 room all to myself as large as our morning room, 1 but nearly 
 twice as high. I can come and go as I like, and you will easily 
 see that I have room enough. Everything else is capitally 
 arranged. . . . Nothing could be more convenient, and I can only 
 hope now that my work will come up to its environment. Of 
 
 1 A large room in his parents' house. 
 
INTRODUCTION xili 
 
 course at present I am only getting things ready, but I feel how 
 pleasant it will be to have the resources of a good laboratory 
 at my back. My galvanometer, which at home stood upon 
 the lathe, 1 now stands upon an iron bracket let into the wall. 
 The reading telescope can be adjusted in all directions by screws, 
 which is certainly more convenient than propping it up on 
 books. . . . Every morning I hear an interesting lecture, and 
 then go to the laboratory, where I remain, barring a short in- 
 terval, until four o'clock. After that I work in the library or in 
 my rooms ; up till now there has been plenty to do in hunting up 
 the literature on extra-currents. (It seems that there is a paper, 
 of which, however, I have only seen an extract, in which some one 
 shows that no such current exists ; it is to be hoped that the man 
 is quite wrong.) 
 
 nth November 1878. 
 
 My work goes on slowly but steadily. The first thing I found 
 out was that a bracket in the Dorotheenstrasse is much more 
 shaky than an ordinary table in the Magdalenenstrasse. At my 
 request I have been shifted into another room, in which there is 
 a brick pillar. . . . Helmholtz is very kind ; he comes in every 
 day for a few minutes, and has a look how things are getting on. 
 The task upon which I am engaged is rather an ungrateful one, 
 for in all likelihood the result will be negative : i.e. certain things 
 will not happen, and on the whole this is less exciting than when 
 something does happen ; but it can't be helped in this case. 
 
 2ith November 1878. 
 
 I am now thoroughly happy, and could not wish things better. 
 I spend the greater part of the day working in the laboratory, 
 and unfortunately the days are so short that when the greater 
 part is gone scarcely anything is left. Most of this greater part 
 is spent upon things which are very useless, or at any rate don't 
 teach one much, such as cutting corks and filing wires, and the 
 observations themselves are naturally not very delightful. 
 Possibly it may be doubtful whether it is quite right for me to 
 spend so much time at these things when I have still so much to 
 learn. And yet I feel that it is right; to get information for 
 myself and for others direct from nature gives me so much more 
 satisfaction than to be always learning it from others and for my- 
 self alone — so much more that I can scarcely express it. When I 
 am only studying books I am never free of the feeling that I am 
 a perfectly useless member of society. It is odd to think that I 
 am now working at a rather specialised research in electricity, 
 1 This he obtained when sixteen years old, and made good use of it. 
 
XIV INTRODUCTION 
 
 whereas only about half a year ago I scarcely knew any more 
 about it than what still remained in my memory since the time 
 when I was with Dr. Lange. 1 I hope my work won't suffer from 
 this. At present it looks promising. I have already surmounted 
 the difficulties which Helmholtz pointed out to me at the start as 
 being the principal ones ; and in a fortnight, if all goes well, I 
 shall be ready with a scanty kind of solution, and shall still have 
 time left to work it up properly. 
 
 He asks his parents to send on a tangent galvanometer 
 which he had made during the last holidays at home, without 
 having any suspicion that it would so soon be used in this 
 way. 2 
 
 A week later, in writing to report progress, he is not so 
 cheerful. " When one difficulty is overcome, a bigger one 
 turns up in its place." These were the difficulties mentioned 
 in pp. 5-6. The Christmas holidays were now at hand, and 
 while at home in Hamburg he made the commutator shown 
 on Fig. 1, p. 13, respecting which he later on reports. 
 
 11th January 1879. 
 
 The apparatus which I made works very well, even better than 
 I had expected ; so that within the last three days I have been able 
 to make all my measurements over again, and more accurately than 
 before. 
 
 Within three months after he had first turned his atten- 
 tion to this investigation he is able to report the conclusion of 
 the first part of it. 
 
 21st January 1879. 
 
 It has delighted me greatly to find that my observations are in 
 accordance with the theory, and all the more because the agree- 
 ment is better than I had expected. At first my calculations gave 
 a value which was much greater than the observed value. Then 
 I happened to notice that it was just twice as great. After a long 
 search amongst the calculations I came upon a 2 which had 
 been forgotten, and then both agreed better than I could have 
 expected. I have now set about making more accurate observa- 
 
 1 The Head-Master of the Biirgcrschulc, which he attended up to his six- 
 teenth year. 
 
 2 This is the galvanometer referred to on p. 12 (3) — a simple wooden disc 
 turned upon the lathe and wound with copper wire, with a hole in the centre 
 for the magnet. It is still in good order. 
 
INTRODUCTION XV 
 
 tions ; the first attempt has turned out badly, as generally happens, 
 but I hope in due course to pull things into shape. The apparatus 
 which I have made at home really works well, so well that I 
 wouldn't exchange it for one made out of gold and ivory in the 
 best workshop. (Mother might like to hear this, and if I find that 
 it pleases her I will try it again.) 
 
 Ten days later the experiments with rectilinear wires were 
 completed. 
 
 31 st January 1879. 
 
 I have now quite finished my research, much more quickly 
 than I had expected. This is chiefly because the more accurate 
 set of experiments have led to a very satisfactory, although 
 negative, result : i.e. I find that, to the greatest degree of accuracy 
 I can obtain, the theory is confirmed. I should much have 
 preferred some positive result ; but as there is nothing of the kind 
 here I must be satisfied. My experiments agree as well as I could 
 wish with the current theory, and I do not think that I can push 
 matters any further with the means now at my disposal. So I 
 have finished the experiments, and hope the Commission will be 
 satisfied ; as far as I can see, any further experiments would only 
 lead to the same result. I shall begin writing my paper in a few 
 days ; just at present I don't feel in the humour for it. 
 
 The paper was written during a period of military service 
 at Freiburg. 
 
 In these successive reports on his work we nowhere find signs 
 of his having encountered difficulties in developing the theory 
 of it ; and this is all the more surprising because at this time 
 he could scarcely have made any general survey of what was 
 already known. But it is clear that even at this early stage 
 he was able to find his own way through regions yet unknown 
 to him, and to do this without first searching anxiously for 
 the foot-prints of other explorers. Thus just about this time 
 he writes as follows : — 
 
 9th February 1879. 
 
 Kirchhoff has now come to magnetism in his lectures, and a 
 great part of what he tells us coincides with what I had worked 
 out for myself at home last autumn. Now it is by no means 
 pleasant to hear that all this has long since been well known ; still 
 it makes the lecture all the more interesting. I hope my know- 
 ledge will soon grow more extensive, so that I may know what has 
 
XVI INTRODUCTION 
 
 already been done, instead of having to take the trouble of finding 
 it out again for myself. But it is some satisfaction to find 
 gradually that things which are new to me make their appearance 
 less frequently ; at any rate that is my experience in the special 
 department at which I have worked. 
 
 His research gained the prize. 
 
 4th August 1879. 
 
 Happily I have not only obtained the prize, but the decision 
 of the Faculty has been expressed in terms of such commendation 
 that I feel twice as proud of it. ... I had gone with Dr. K. and 
 L. [to hear the public announcement of the decision] without 
 having said anything, but fully determined not to show any 
 disappointment if the result was unfavourable. 
 
 llth August 1879. 
 
 I have chosen the medal, in accordance with your wish, for 
 the prize. It is a beautiful gold medal, quite a large one, but by 
 a piece of incredible stupidity it has no inscription whatever on it, 
 nothing even to show that it is a University prize. 
 
 This prize research was Hertz's first investigation, and it 
 is to this he refers in the Introduction to his Electric Waves, 
 as being engaged upon it when von Helmholtz invited him to 
 attack the problem l propounded for the prize of the Berlin 
 Academy. For reasons now known to us, he gave up the idea 
 of working at the problem. He preferred to apply himself to 
 other work, which was perhaps of a more modest nature, but 
 promised to yield some tangible result. 
 
 So he turned his attention to the theoretical investigation 
 " On Induction in Botating Spheres " (II. in this volume). 
 This extensive investigation was made in an astonishingly 
 short time. The first sketch of it, which still exists, is dated 
 from time to time in Hertz's handwriting, and one sees with 
 surprise what rapid progress he made from day to day. He 
 had made preliminary studies at home during the autumn 
 vacation of 1896, and the results of these are partly contained 
 
 1 This latter seems to be the problem in electromagnetics to which von 
 Helmholtz refers in his Preface to Hertz's Princijjles of Mechanics as having , 
 been proposed by himself in the belief that it was one in which his pupil would 1 
 feel an interest. 
 
INTRODUCTION XV11 
 
 in the paper " On the Distribution of Electricity over the 
 Surface of Moving Conductors " (III. in this volume), which 
 was first published two years later. In November 1879 he 
 began to work at induction, and no later than the following 
 January this investigation was submitted as an inaugural 
 dissertation for the degree of doctor to the Philosophical 
 Faculty. We hear of this rapid progress in the letters to his 
 parents : — 
 
 27th November 1879. J 
 
 I secured a place in the laboratory and started working there 
 at the beginning of term, but do not feel much drawn in that 
 direction just now. I am busy with a theoretical investigation 
 which gives me great pleasure, so I work at this in my rooms 
 instead of going to the laboratory : indeed I wish that I had made 
 no arrangements for practical work. The investigation which I 
 now have in hand is closely connected with what I did at home. 
 Unless I discover (which would be very disagreeable) that this 
 particular problem has already been solved by some one else, it 
 will become my dissertation for the doctorate. 
 
 lMh December 1879. 
 
 There is little news to send about myself. I have been work- 
 ing away, with scarcely time to look about me, at the research 
 which I have undertaken. It is getting on as well and as 
 pleasantly as I could wish. 
 
 Ylth January 1880. 
 
 As soon as I got here [from Hamburg, after the Christmas 
 holidays] I settled down to my research, and by the end of the 
 week had it ready : I had to keep working hard at it, for it became 
 much more extensive than I had expected. 
 
 In its extent this second research differs from all of 
 Hertz's other publications ; he had clearly decided to follow 
 the usual custom with respect to inaugural dissertations. 
 Although long, it will be found to repay the most careful study. 
 The decision of the Berlin Philosophical Faculty (drawn up 
 by Helmholtz) was Acuminis et doctrince specimen laudabile. 
 Together with a brilliant examination it gained for him the 
 title of doctor, with the award magna cum laude, which is 
 but rarely given in the University of Berlin. 
 
 In the following summer of 1880 Hertz was again engaged 
 
xvili INTRODUCTION 
 
 upon an experimental investigation on the formation of residual 
 charge in insulators. He did not seem well satisfied with the 
 result ; at any rate he did not consider it worth writing out. 
 It was only by v. Helmholtz's special request that he was 
 subsequently induced to give an account of this research at a 
 meeting of the Physical Society of Berlin on 27th May 
 1881. It did not appear in Wiedemann's Annalen (XIV. in 
 this volume) until three years later, after the quantitative data 
 had been recovered by a repetition of the experiments made 
 for this purpose at Hertz's suggestion. 
 
 Soon afterwards, in October 1880, Hertz became assistant 
 to v. Helmholtz. He now revelled in the enjoyment of the 
 resources of the Berlin Institute. He was soon engaged, in 
 addition to the duties of his office, upon many problems both 
 experimental and theoretical ; and expresses his regret at not 
 being able to use all the resources at his disposal, and to 
 solve all the problems at once. At this time he sowed the 
 seeds which during his three years' term as assistant developed 
 one after the other into the investigations which appear as 
 IV. -XVI. in this volume. 
 
 He was first attracted by a theoretical investigation " On 
 the Contact of Elastic Solids " (V.) During the frequent 
 discussions on Newton's rings in the Physical Society of 
 Berlin it had occurred to Hertz that although much was 
 known in detail as to the optical phenomena which takes place 
 between the two glasses, very little was known as to the 
 changes of form which they undergo at their point of contact 
 when pressed together. So he tried to solve the problem and 
 succeeded. Most of the investigation was carried out during 
 the vacation of Christmas 1880. Its publication, at first in 
 the form of a lecture to the Physical Society (on 21st January 
 1881), was at once greeted with much interest. A new light 
 had been thrown upon the phenomena of contact and pressure, 
 and it was -recognised that this had an important and direct 
 bearing upon the conduct of all delicate measurements. For 
 example, determinations of a base-line for the great European 
 measurement of a degree were just then being calculated 
 out at Berlin. The steel measuring-rods used in these deter- 
 minations were lightly pressed against each other with a 
 glass sphere interposed between them. This elastic contact 
 
INTRODUCTION XIX 
 
 necessarily introduced an element of uncertainty depending 
 upon the pressure exerted : a method of ascertaining its 
 magnitude with certainty was wanting. Now the question 
 could be answered definitely and at once. In technical circles 
 equal interest was exhibited, and this induced Hertz to extend 
 the investigation further and to allow it to be published not 
 only in Borchardt's Journal (V. in this volume) but also in a 
 technical journal, with a supplement on Hardness (VI.) 
 About this he writes to his parents as follows : — 
 
 9th May 1882. 
 
 I have been writing a great deal lately ; for I have rewritten 
 the investigation once more for a technical journal in compliance 
 with suggestions which reached me from various directions. . . . 
 I have also added a chapter on the hardness of bodies, and hope to 
 lecture on this to the Physical Society on Friday. I have had 
 some fun out of this too. For hardness is a property of bodies of 
 which scientific men have as clear, i.e. as vague, a conception as 
 the man in the street. Now as I went on working it became quite 
 clear to me what hardness really was. But I felt that it was 
 not in itself a property of sufficient importance to make it 
 worth while writing specially about it ; nor was such a subject, 
 which would necessarily have to be treated at some length, quite 
 suitable for a purely mathematical journal. In a technical 
 journal, however, I thought I might well write something about 
 the matter. So I went to look round the library of the Gewerbe- 
 akademie, and see what was known about hardness. And I found 
 that there really was a book written on it in 1867 by a Frenchman. 
 It contained a full account of earlier attempts to define hardness 
 clearly, and to measure it in a rational way, and of many experi- 
 ments made by the writer himself with the same object, interspersed 
 with assurances as to the importance of the subject. Altogether 
 it must have involved a considerable amount of work, which was 
 labour lost — so I think, and he partly admits it — because there 
 Avas no right understanding at the bottom of it, and the measure- 
 ments were made without knowing what had to be measured. So 
 I concluded that now I might with a quiet conscience make my 
 paper a few pages longer ; and since this I have naturally had 
 much more pleasure than before in writing it out. 
 
 Whilst these problems on elasticity were engaging his 
 attention Hertz was also busy with the researches on evapora- 
 tion (VIII. and IX. in this volume) and the second investi- 
 gation on the Kinetic Energy of Electricity in Motion (IV.) 
 
XX INTRODUCTION 
 
 Both of these had been commenced in the summer of 1881. 
 In order to push on the three-fold task to his satisfaction he 
 devoted to it the greater part of the autumn vacation. Thus 
 the investigation on electric inertia was soon finished ; on the 
 other hand the evaporation problems took up much more time 
 without giving much satisfaction. 
 
 15th October 1881. 
 
 I am now devoting myself entirely to the research on evapora- 
 tion, which I began thinking of in the spring, and of which I have 
 now some hope. 
 
 10th March 1882. 
 
 The present research is going on anything but satisfactorily. 
 Fresh experiments have shown me that much, if not all, of my 
 labour has been misapplied ; that sources of error were present 
 which could scarcely have been foreseen, so that the beautiful 
 positive result which I thought I had obtained turns out to be 
 nothing but a negative one. At first I was quite upset, but have 
 plucked up courage again ; I feel as fit as ever now, only I do 
 regret the valuable time which cannot be recovered. 
 
 IMh June 1882. 
 
 I am writing out my paper on evaporation, i.e. as much of the 
 work as turns out to be correct ; I am far from being pleased with 
 it, and feel rather glad that I am not obliged to work it out com- 
 pletely, as originally intended. 
 
 In the midst of this period of strenuous exertion comes 
 the slight refreshing episode of the invention of the hygro- 
 meter (VII. in this volume). In sending a charming de- 
 scription of this little instrument, " so simple that there is 
 scarcely anything in it," Hertz explains to his parents how 
 the air in a dwelling -room should be kept moist in winter. 
 There can be no harm in reproducing the explanation here. 
 
 2nd February 1882. 
 
 I may here give a little calculation which will show father how 
 the air in the morning- room should be kept moist. On an 
 average the atmosphere contains half as much water-vapour as is 
 required to saturate it ; in other words, the average relative 
 humidity is 50 per cent. Assume then that this proportion is 
 suitable for men, that it is the happy — or healthy — mean. In a 
 cubic metre of air there should then be definite quantities of water, 
 
INTRODUCTION XXI 
 
 which are different for different temperatures — 2*45 gm. at 0° C, 
 4*70 gm. at 10° C, and 8'70 gm. at 20° C, for these amounts 
 would give the air a relative humidity of 50 per cent. Now let 
 us assume that the temperature is 0° out of doors, and 20° in the 
 (heated) room. Then in the room there would be (since the air 
 comes ultimately from the outside) only 2*45 gm. of water in each 
 cubic metre of air. In order to get the correct proportion there 
 should be 8*70 gm. of water. Hence the air is relatively very dry 
 and needs 6J gm. more of water per cubic metre. Since the room 
 is about 7 metres long, 7 metres broad, and 4 metres high, it 
 contains 7x7x4 cubic metres, and the additional amount of 
 water required in the room is 7 x 7 x 4 x 6 J gm., or nearly 1J 
 litres. Thus if the room were hermetically closed, 1J litres of 
 water would have to be sprinkled about in order to secure the 
 proper degree of humidity. Now the room is not hermetically 
 closed. Let us assume that all the air in it is completely changed 
 in n hours; then every n hours \\ litres of water would have to 
 be sprinkled about or evaporated into it. I think we may assume 
 that through window-apertures, opening of doors, etc., the air is 
 completely changed every two or three hours ; hence from § to 
 -j^- of a litre of water, or a big glassful, would have to be evapor- 
 ated per hour. All this would roughly hold good whenever rooms 
 are artificially heated, and the external temperature is below 10° 
 C. If you were to set up a hygrometer and compare the humidity 
 when water is sprinkled and when it is not, you could from this 
 find within what time the air in the room is completely changed. 
 . . . This has become quite a long lecture, and the postage of the 
 letter will ruin me ; but what wouldn't a man do to keep his dear 
 parents and brothers and sister from complete desiccation ? 
 
 As soon as the research on evaporation was finished Hertz 
 turned his attention to another subject, in which he had always 
 felt great interest — that of the electric discharge in gases. He 
 had only been engaged a month upon this when he succeeded 
 in discovering a phenomenon accompanying the spark-discharge 
 which had hitherto remained unnoticed (see XII. in this 
 volume). But he was too keen to allow this to detain him 
 long : he at once made plans for constructing a large secondary 
 battery, which seemed to him to be the most suitable means 
 for obtaining information of more importance. His letters 
 tell us how he attacked the subject. 
 
 29th June 1882. 
 
 I am busy from morn to night with optical phenomena in 
 l rarefied gases, in the so-called Geissler tubes — only the tubes I 
 
xxn INTRODUCTION 
 
 J 
 
 mean are very different from the ones you see displayed in public 
 exhibitions. For once I feel an inclination to take up a somewhat, 
 more experimental subject and to put the exact measurements V 
 aside for a while. The subject I have in mind is involved in much ] 
 obscurity, and little has been done at it ; its investigation would 
 probably be of great theoretical interest. So I should like to find 
 in it material for a fresh research ; meanwhile I keep rushing 
 about without any fixed plan, finding out what is already known 
 about it, repeating experiments and setting up others as they 
 occur to me ; all of which is very enjoyable, inasmuch as the 
 phenomena are in general exceedingly beautiful and varied. But 
 it involves a lot of glass-blowing ; my impatience will not allow 
 me to order from the glass-blower to-day a tube which would not 
 be ready until several days later, so I prefer to restrict myself to 
 what can be achieved by my own slight skill in the art. In point 
 of expense this is an advantage. But in a day one can only pre- 
 pare a single tube, or perhaps two, and make observations with 
 these under varied conditions, so that naturally it is laborious 
 work. At present, as already stated, I am simply roaming about 
 in the hope that one or other of the hundred remarkable pheno- 
 mena which are exhibited will throw some light upon the path. 
 
 31st July 1882. 
 
 I have made some preliminary attempts in the way of build- 
 ing up a battery of 1000 cells. This will cost some money and a 
 good deal of trouble ; but I believe it will prove a very efficient 
 means of pushing on the investigation, and will amply repay its 
 cost. 
 
 After devoting the first half of the ensuing autumn 
 vacation to recreation, he begins the construction of the 
 battery. 
 
 6th September 1882. 
 
 I am now back again, after having had a good rest, and as 
 there is nothing to disturb me here I have at once started fitting up 
 the battery. So I am working away just like a mechanic. Every 
 turn and twist has to be repeated a thousand times ; so that for 
 hours I do nothing but bore one hole and then another, bend one 
 strip of lead after the other, and then again spend hours in 
 varnishing them one by one. I have already got 250 cells 
 finished and the remaining 750 are to be made forthwith ; I expect 
 to have the lot ready in a week. I don't like to interrupt the 
 work, and that is why I haven't written to you before. For a j 
 while I feel quite fond of this monotonous mechanical occupation. 
 
INTRODUCTION XXlll 
 
 20th September 1882. 
 
 The battery has practically been ready since the middle of last 
 week ; since last Sunday night it has begun to spit fire and light 
 up electric tubes. To-day for the first time I have made experi- 
 ments with it — ones which I couldn't have carried out without it. 
 
 7th October 1882. 
 
 I have got the battery to work satisfactorily, and a week ago 
 succeeded in solving, to the best of my belief, the first problem 
 which I had propounded to myself (a problem solved, when it 
 really is solved, is a good deal !). But even this first stage was 
 only attained with much trouble, for the battery turned sick, and 
 its sickness has proved to be a very dangerous one. 
 
 By preventive measures the battery was kept going for 
 yet a little while, and later on he reports, " Battery doing 
 well." How the battery finally came to grief is explained 
 in the account of the investigation (XIII.) By its aid he was 
 able, in six weeks of vigorous exertion, to bring to a success- 
 ful issue most of the experiments which he had planned out. 
 The investigation was first published in April 1883 at Kiel, 
 in connection with Hertz's induction to the position of Privat- 
 docent there. It brought him recognition from one who 
 rarely bestowed such tokens, and whose opinion he valued 
 most highly. Hertz treasured as precious mementoes two 
 letters from Helmholtz. One of these notified his appoint- 
 ment as assistant at Berlin ; the other is the following : — 
 
 Berlin, 29th July 1883. 
 
 Geehrter Herr Doktor ! — I have read with the greatest 
 interest your investigation on the cathode discharge, and cannot 
 refrain from writing to say Bravo ! The subject seems to me to 
 be one of very wide importance. For some time I have been 
 thinking whether the cathode rays may not be a mode of pro- 
 pagation of a sudden impact upon the Maxwellian electromagnetic 
 ether, in which the surface of the electrode forms the first wave- 
 surface. For, as far as I can see, such a wave should be pro- 
 pagated just as these rays are. In this case deviation of the 
 rays through a magnetisation of the medium would also be 
 possible. Longitudinal waves could be more easily conceived ; 
 and these could exist if the constant k in my electromagnetic 
 researches were not zero. But transversal waves could also be 
 
XXIV INTRODUCTION 
 
 produced. You seem to have similar thoughts in your own mind. 
 However that may be, I should like you to feel free to make any 
 use of what I have mentioned above, for I have no time at present 
 to work at the subject. These ideas suggest themselves so readily 
 in reading your investigation that they must soon occur to you if 
 they have not already done so. . . . — With kindest regards, yours, 
 
 H. Helmholtz. 
 
 While still busily engaged in completing this investiga- 
 tion on the discharge Hertz began to reflect upon another 
 problem which seems to have been suggested to him by sheets 
 of ice floating upon water during the winter. 
 
 Berlin, 24th February 1883. 
 
 My researches are going on all right. From the date of my 
 last letter until to-day I have been wholly absorbed in a problem 
 which I cannot keep out of my head, viz. the equilibrium of a 
 floating sheet of ice upon which a man stands. Naturally the 
 sheet of ice will become somewhat bent, thus [follows a small 
 sketch of the bent sheet], but what form will it take, what will be 
 the exact amount of the depression, etc. ? One arrives at quite 
 paradoxical results. In the first place a depression will certainly 
 be produced underneath the man ; but at a certain distance there 
 will be a circular elevation of the ice ; after this there follows 
 another depression, and so on, somewhat in this way [another 
 sketch]. As a matter of fact the elevations and depressions 
 decrease so rapidly that they can never be perceived : but to the 
 intellectual eye an endless series of them is visible. Even more 
 paradoxical is the following result. Under certain circumstances 
 a disc heavier than water, and which would therefore sink when 
 laid upon water, can be made to float by putting a weight on it ; 
 and as soon as the weight is taken away it sinks. The explana- 
 tion is that when the weight is put on, the disc takes the form of 
 a boat, and thus supports both the weight and itself. If the load 
 is gradually removed the disc becomes flatter and flatter ; and 
 finally there comes an instant when the boat becomes too shallow 
 and so sinks with what is left of the load. This is the theoretical 
 result, and the way I explain it to myself, but meanwhile there 
 may be errors in the calculation. Such a subject has a peculiar 
 effect upon me. For a whole week I have been struggling to have 
 done with it, because it is not of great importance, and I have 
 other things to do, e.g. I ought to be writing out the research 
 which is to serve for my induction at Kiel, which is all ready in 
 my mind but not a stroke of it on paper. Still it seems impos- 
 sible to finish it off properly ; there always remains some contra- 
 
INTRODUCTION XXV 
 
 diction or improbability, and so long as anything of that sort is 
 left I can scarcely take my mind away from it. Then the 
 formulae which I have deduced for the accurate solution are so 
 complicated that it takes a lot of time and trouble to make out 
 clearly their meaning. But if I take up a book or try to do 
 .anything else my thoughts continually hark back to it. Shouldn't 
 things happen in this way or that 1 Isn't there still some contra- 
 diction here 1 All this is a perfect plague when one doesn't attach 
 much importance to the result. 
 
 Soon afterwards Hertz had to remove to Kiel. This 
 removal, his induction, and his lectures there took up much of 
 his time, so that his investigation on floating plates was not 
 published until a year later. Its place was taken by the 
 investigation on the fundamental equations of electromagnetics 
 (XVII.) At this time he kept a day-book, from which it 
 appears that in May 1884 he was alternately working at his 
 lectures, at electromagnetics, and at microscopic observations 
 taken up by way of change. On six successive days there are 
 brief but expressive entries — " Hard at Maxwellian electro- 
 magnetics in the evening," " Nothing but electromagnetics " ; 
 and then follows on the next day, the 19th of May — "Hit 
 upon the solution of the electromagnetic problem this 
 morning." This will remind the reader of v. Helmholtz's 
 remark that the solution of difficult problems came to him 
 soonest, and then often unexpectedly, when a period of 
 vigorous battling with the difficulties had been followed by 
 one of complete rest. 
 
 In close connection with this subject, and immediately 
 following it in order of time, came the paper " On the 
 Dimensions of Magnetic Pole " (XVIII. in this volume). 
 Directly after this came the meteorological paper " On the 
 Adiabatic Changes of Moist Air" (XIX.) A diagram illus- 
 trating the latter is reproduced at the end of this volume 
 from the original ; the drawing of this, as a recreation after 
 other work, seems to have given Hertz great pleasure. 
 
 We may complete our account of Hertz's scientific work 
 during his two years at Kiel by adding that at this time he 
 repeatedly, although unsuccessfully, attacked certain hydro- 
 dynamic problems, and that his thoughts already turned 
 frequently towards that field in which he was afterwards to 
 
XXVI INTRODUCTION 
 
 reap such a rich harvest. Nearly five years before he had 
 carried out his investigation " On Electric Eadiation " we find 
 in his day-book the notable remark — " 2*7 th January 1884. 
 Thought about electromagnetic rays/' and again, " Eeflected on 
 the electromagnetic theory of light." He was always full of 
 schemes for investigations, and never liked to be without some 
 experimental work. So he did his best to fit up in his house 
 a small laboratory with home-made apparatus, thus transport- 
 ing himself back to the times when chemists worked with the 
 modest spirit-lamp. But before his experiments were con- 
 cluded or any of his schemes carried out he was called to 
 Karlsruhe, and his removal thither relieved him from much 
 unprofitable exertion caused by the lack of proper experimental 
 facilities. 
 
 This brings us to the end of the series of papers around 
 which we have grouped the events of the author's life. After 
 this follow the great electrical investigations which now form 
 the second volume of his collected works. At this point we 
 have introduced the lecture which Hertz gave at Heidelberg on 
 these discoveries, and which will still be fresh in the remem- 
 brance of many who heard it. 
 
 After this follows the last experimental investigation 
 which Hertz made. Whilst his colleagues, and in Bonn his 
 pupils as well, were eagerly pushing forward into the country 
 which he had opened up, he returned to the study of electric 
 discharges in gases, which had interested him before. Again 
 he was rewarded by an immediate and unexpected discovery. 
 Early in the summer-semester of 1891 he found that cathode 
 rays could pass through metals. The investigation was soon 
 interrupted, but was published early in the ensuing year ; from 
 now on the subject-matter of his last work, the Principles 
 of Mechanics, wholly absorbed his attention. 
 
EXPERIMENTS TO DETERMINE AN UPPER LIMIT 
 TO THE KINETIC ENERGY OF AN ELECTRIC 
 CURRENT. 
 
 (Wiedemann's Annalen, 10, pp. 414-448, 1880.) 
 
 According to the laws of induction the current i in a 
 linear circuit, in which a variable electromotive force A acts, 
 is given by its initial value together with the differential 
 equation 
 
 where r is the resistance and P the inductance of the circuit. 1 
 Multiplying by idt we get the equation 
 
 Aidt = i 2 rdt + ^d(Pi 2 ), 
 
 which shows that the law expressed by the above equation is 
 in agreement with the principle of the conservation of energy 
 on the assumption that the work done by the battery on the 
 one hand, and the heat developed in the circuit and the 
 increase of potential energy on the other hand, are the only 
 amounts of energy to be considered. This supposition is not 
 true, and hence the above equations cannot lay claim to 
 complete accuracy, in case the electricity in motion possesses 
 inertia, the effect of which is not quite negligible. In this 
 case we must add to the right-hand side of the second equation 
 
 1 [The notation has been altered in accordance with English custom and 
 the necessary changes in the equations made. The original has 2P. — Tk.] 
 M.P. B 
 
2 KINETIC ENERGY OF ELECTRICITY IN MOTION [I] I ( 
 
 ) 
 
 a term corresponding to the increase in the kinetic energy of 
 the current. This is proportional to the square of the current ' 
 and may therefore be put equal to \mi 2 , where m is a constant 
 depending on the form and size of the circuit. We thus get I 
 in place of the above the following corrected equations 
 
 Aidt = i 2 rdt + ^d(Vi 2 ) + ±d(mi 2 ), 
 . ^di di 
 
 -A-(P + .0* 
 
 Analogous conclusions apply to the case of a system of circuits 
 in which electromotive forces A 1? A 2 . . . act. When the 
 correction for inertia is introduced, the well-known differential 
 equations which determine the currents take the form 
 
 V 2 = A 2 -p4-(P 22 + m2 )f .-. . .-PjJ, 
 
 at dt 
 
 Thus the only alteration which the mass of the electricity 
 has produced in these equations consists in an increase of the 
 self-inductance, and it is at once obvious 
 
 1. That the electromotive force of the extra-currents is 
 independent of the induction-currents simultaneously generated 
 in other conductors, and of the mass of the electricity moving 
 in them. 
 
 2. That the complete time-integrals of the induction- 
 currents are not affected by the mass of the electricity moved, 
 whether in the inducing or induced conductors. 
 
 3. That, on the other hand, the integral flow of the extra- 
 currents becomes greater than that calculated from inductive 
 actions alone. 1 
 
 1 With reference to these simple deductions the philosophical faculty of the 
 Frederick- William University at Berlin in 1879 propounded to the students the 
 
I KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 3 
 
 The amount of this increase depends on the quantities m, 
 whose meaning we will now consider more closely, basing our 
 investigation on Weber's view of electric currents. The 
 presence of the terms involving m is, however, independent of 
 the correctness of this view and of the existence of electric 
 fluids at all ; every explanation of the current as a state of 
 motion of inert matter must equally introduce these terms, 
 and only the interpretation of the quantities m will be 
 different. 
 
 Suppose unit volume of the conductor to contain X units 
 of positive electricity, and let the mass of each unit be p 
 milligrammes. Let the length of the conductor be /, and its 
 cross-section, supposed uniform, q. Then unit length of the 
 conductor contains qX electrostatic units, and the total positive 
 electricity in motion in the conductor has the mass pqXl mgm. 
 If the current (in electromagnetic measure) be i, the number 
 of electrostatic units which cross any section in unit time is 
 equal to 155,370 x 10H, and is also equal to the velocity v 
 multiplied by qX. Thus 
 
 155,370 xlO 6 . 
 
 v = i, 
 
 qX 
 
 and the kinetic energy of the positive electricity contained 
 in the conductor is 
 
 ^Ipqxl- 
 
 55,370 x 10' 
 
 ,2 
 
 X j q 
 
 Ai 2 155,370 2 x 10 12 ! li 2 
 q X q 
 
 The quantity \li 2 \q can be expressed in finite measure. The 
 quantity p.l55,370 2 x 10/ 12 X, which has been denoted by p, 
 is a constant depending only on the material of the conductor ; 
 for different conductors it is inversely as the density of the 
 electricity in them. Its dimensions are those of a surface ; in 
 milligramme-millimetres it gives the kinetic energy of the two 
 
 problem "to make experiments on the magnitude of extra-currents which 
 shall at least lead to a determination of an upper limit to the mass moved." 
 It was pointed out that for such experiments extra-currents flowing in opposite 
 directions through the wires of a double spiral would be especially suitable. 
 The present thesis is essentially the same as that which gained the prize. 
 
4 KINETIC ENERGY OF ELECTRICITY IN MOTION [I] i 
 
 electricities, i.e. the total kinetic energy of the current in a 
 cubic millimetre of a conductor in which the current has unit 
 magnetic density. 
 
 The object of the following experiments is to determine 
 the quantity /jl, or at any rate an upper limit to it. 
 
 Method of Experimenting. 
 
 Since we have put the kinetic energy of the total electricity 
 equal to mi 2 /2, and also equal to (l/Jb/q)i 2 , it follows that 
 fx = qm/2l. In order to determine m it would have sufficed 
 to measure the integral flow of the extra-current in a con- 
 ductor of known resistance r and self-inductance P ; m would 
 at once follow from the equation J = (i/r)(P + m). But 
 extra-currents can only be measured in branched systems of 
 conductors, and this would necessitate the measuring of a 
 large number of resistances. Hence it is preferable to 
 generate extra-currents in the same circuit by two different 
 inductions, when we obtain two equations for the quantities 
 r and m. If the current in the unbranched circuit is to that 
 current by which the extra-current is measured as a : 1, and 
 if J is the total flow measured, then the equations in 
 question are 
 
 arJ arJ f , 
 
 — — = P + m , — j- = P + m, 
 
 whence 
 
 P--P /J 
 
 i' i 
 m = - 
 
 _J / 
 
 i i' 
 
 It is well to choose one inductance P r so large that the 
 influence of mass is negligible in comparison, but the other P 
 as small as possible. The equations then take the simpler 
 form 
 
 arJ arJ f , 
 
 = P + m, — j— = P . 
 
 i i 
 
j i KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 
 
 * " tV -r» t ■/ 
 
 m = — -V' — P, or if % — i, 
 m = P -T- — 
 
 Vj'p 
 
 The experiments were carried out according to this 
 principle. The system of conductors through which the 
 currents flowed consisted in the earlier experiments of spirals 
 wound with double wires, in the later ones of two wires 
 stretched out in parallel straight lines side by side. These 
 systems of wires could without change of resistance be coupled 
 in such a way that the currents in the two branches flowed in 
 the same or in opposite directions. The inductances follow- 
 ing from the two methods of coupling were calculated and 
 the integral flows of the corresponding extra-currents were 
 determined by experiment. If these flows were proportional 
 to the calculated inductances, no effect of mass would be 
 demonstrated ; if a deviation from proportionality were observed, 
 the kinetic energy of the currents would follow by the above 
 formulae 
 
 The extra -currents were always measured by means of a 
 Wheatstone's bridge, one branch of which contained the system 
 of wires giving the currents, while the other branches were 
 chosen to have as little inductance as possible. The bridge 
 was adjusted so that a steady current flowing through it pro- 
 duced no permanent deflection of the galvanometer needle ; 
 but when the direction of the current outside the bridge was 
 reversed, then two equal and equally directed extra-currents 
 traversed the galvanometer, and their integral flow was measured 
 by the kick of the needle. As soon as the needle returned 
 from its kick the reversing could be repeated, and in this 
 manner the method of multiplication could be applied. 
 
 The chief difficulty in these measurements was to be met 
 in the smallness of the observed extra-currents, and on this 
 account the method described was impracticable in its simplest 
 form. It is true that by merely increasing the strength of 
 the inducing current the extra -currents could be made as 
 strong as desired, but the difficulties in exactly adjusting the 
 bridge increased very much more quickly than the intensities 
 
6 KINETIC ENERGY OF ELECTRICITY IN MOTION [I] I 
 
 thus obtained. With the greatest strength, which still per- 
 mitted permanently of such an adjustment, a single extra- 
 current from the two branches, when traversed in opposite 
 directions, only moved the galvanometer needle through a 
 fraction of a scale division, whilst the mere approach of the 
 hand to one of the mercury cups, or the radiation of a distant 
 gas flame falling on the spirals, sufficed to produce a deflection 
 of more than 100 scale divisions. Hence I attempted to 
 make use of very strong currents by allowing them to pass 
 for a very short time only through the bridge, which was 
 adjusted by using a weak current. But the electromotive 
 forces generated momentarily in the bridge by the heating 
 effects of the current were found to be of the same order of 
 magnitude as the extra-currents to be observed, so that it 
 was impossible to get results of any value. These experi- 
 ments only showed that at any rate there was no consider- 
 able deviation from the laws of the dynamical theory of 
 induction. 
 
 On this account, in order to obtain measurable deflections 
 with weaker currents, I passed a considerable number of extra- 
 currents through the galvanometer at each passage of the 
 needle through its position of rest. For this purpose the 
 current was at the right moment rapidly reversed twenty 
 times in succession outside the bridge, and at the same time 
 the galvanometer was commutated between every two successive 
 reversals of the current. In order to avoid any considerable 
 damping, after the bridge had been once adjusted the galvan- 
 ometer circuit remained open generally, and was only put in 
 circuit with the rest of the combination during the time 
 needed to generate the extra-currents. 
 
 The operations described were carried out by means of a 
 special commutator and occupied about two seconds, an in- 
 terval of time which was sufficiently long to allow of all the 
 extra-currents being fully developed, and which also proved to 
 be sufficiently small in comparison with the time of swing of 
 the needle. 
 
 This method possessed several advantages. In the first 
 place accurately measurable effects could be produced even 
 with very weak, and therefore also very constant, inducing 
 currents. In all of the following experiments the external 
 
I KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 7 
 
 circuit consisted of one Daniell cell and a ballast resistance of 
 from 3 to 80 Siemens units. Further, if the resistances of 
 the bridge are not exactly adjusted, and if in consequence a 
 fraction of the inducing current also passes through the 
 galvanometer, this fraction will yet be continually reversed in 
 the galvanometer ; so that, if the want of balance be only 
 small, the error due to it almost entirely vanishes. 
 
 Again, since the connection of the galvanometer with the 
 remaining wires of the combination is constantly changing its 
 direction those electromotive forces which exist in the bridge 
 or are generated by the current, and which are not reversed 
 when the current is reversed, are without influence on the 
 needle. It is a circumstance of great value, that for the 
 greater part of a swing the galvanometer is withdrawn from 
 all disturbing influences. 
 
 In consequence of these favourable conditions the observa- 
 tions agreed together very satisfactorily when we consider the 
 smallness of the quantities to be measured : the deviation of 
 the results obtained from their mean was in general less than 
 -g^- of the whole. Here also the proceeding was repeated each 
 time the needle passed through its position of rest. But the 
 multiplication could not be carried so far as to obtain a con- 
 stant deflection ; for to the constant small damping of the 
 needle due to air - resistance was added the damping which 
 was produced whenever the galvanometer was put in circuit 
 with the bridge, and which lasted only for a very short time. 
 The time during which connection was made was not always 
 exactly the same, and thus the damping produced could not 
 be exactly determined. As, however, its effect became very 
 marked with large swings, the method was limited to smaller 
 deflections, and generally only from 7 to 9 elongations were 
 measured. The method which was used to deduce the most 
 probable value of the extra-current from the complete arcs of 
 vibration thus obtained will now be explained. 
 
 Let T be the period of vibration of the galvanometer 
 needle, X the logarithmic decrement constantly present, q = e~ K 
 the ratio of any swing to the preceding one, and let, for 
 shortness, 
 
 T -^tan-IE 
 
 / 6 Tf \ = K . 
 
8 KINETIC ENERGY OF ELECTRICITY IN MOTION [I] I 
 
 Further, let a v a 2 , a 3 be successive elongations right and 
 left of the position of rest, c^ = a^ + a 2 , a 2 = a 2 + a g , etc., the 
 complete arcs of vibration, and Jc v k 2> . . . the increments of 
 velocity in the position of rest, which measure the inductive 
 effects. Then, if for the present any special damping during 
 the impact be neglected 
 
 a 
 
 Kk x + qa v 
 
 a 3 = kJc 2 -f qa 2 = ick 2 + Kq_\ + q\ ; 
 
 hence we get 
 
 a 1 = /ek 1 + a 1 (l + q), 
 
 a 2 = Kk 2 + kJc^I + q) + 0^(1 + q). 
 
 If we multiply the first equation by q and subtract it from 
 the second, we find 
 
 a 2 — a x q = /c(^ + h 2 ), 
 and similarly 
 
 Hence we find the mean value of the impacts h v k 2 , . . . 
 which should all be equal if the apparatus worked quite exactly 
 
 • K Jc = a 2 + a 3 + '•• + ^ — g( a ] + a 2 + ••• + a n-l) 
 
 2<>— 1) 
 
 or, if we denote the sum of all the complete arcs of vibration 
 by 2, 
 
 2(^-1) 
 
 The application of this formula is very easy and is always 
 advisable when the separate impacts are not regular enough to 
 produce a constant limiting value of the arcs of swing, or 
 when for other reasons only a limited number of elongations 
 has been observed. 
 
 If in addition to the constantly occurring damping a 
 further damping occur during the instant of closing of the 
 circuit, this latter may be regarded as an impact in a direction 
 opposed to the motion which is proportional to the duration 
 
I KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 9 
 
 of the closing of the circuit and to the velocity of the needle. 
 If the former be r, the latter v, and the logarithmic decrement 
 during closing V, the magnitude of such an impact is 
 
 - 4 y™ ' 
 
 If a v a 2 be the preceding and succeeding elongations the 
 needle reaches its position of rest with a velocity a^/ie and 
 leaves it with a velocity aj/c. As the increase of velocity is 
 nearly uniform, we must put for v the mean value (a^ + cc 2 )/2k, 
 and thus the magnitude of the impact is 
 
 - 2 tfW + a -i) = - - K? + s)- 
 
 L/C K 
 
 By adding this increase of velocity to that caused by the 
 impact due to induction we obtain the equations 
 
 a 2 = \tt + a x q - c{a % + ag), 
 a 3 = Je 2 fc -f a 2 q - c(a 3 + a 2 q), etc., 
 or 
 
 (1 + c)a 2 = k\ + (1 - c)a Y q, 
 
 (1 + c)a 3 = K k 2 + (l- c)a 2 q, etc. ; 
 
 and by a similar calculation to that above 
 
 ( 1 + c)a 2 — ( 1 - c)qa 1 = k(]c 1 + kj, 
 ( 1 + c)a„ - ( 1 - c)qa 9 = k{\ + kX 
 
 (1 + c)a„ - (1 - c)^u-! = «(*»-! + K)- 
 
 If instead of the quantities k v Jc 2 , ... we write their 
 theoretical value k, we get after a simple transformation 
 the equations 
 
 a 2 — qa ± + c(a 2 + qaj = 2^/c, 
 
 a 3 — ^a 2 + c(a g + #a 2 ) = 2kx, 
 
 a n ~ d a n- 1 + c ( a n + 2"n-i) = 2kfC \ 
 
 and from these the most probable values of the unknown 
 quantities k, etc. must be calculated by the method of least 
 squares. 
 
10 
 
 KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 
 
 The very complicated calculation was, however, not carried 
 through for all the observations, but from a number of them 
 the value of c was calculated, and the mean of the closely- 
 agreeing values obtained was assumed to be true for the 
 remaining observations. When c is known we get, more 
 simply 
 
 kJ , _ CZ-a 1 )-q&-a, t ) + c{(Z-a l ) + g(S-a„)} 
 2(71-1) 
 
 Since the term involving c only occurs as a correction, it is 
 not necessary to know c with absolute accuracy. By taking 
 the mean between two successive impacts, namely 
 
 k m K = 
 
 *m - g^m-i + C ( a m + qam-i) 
 
 we can get some notion as to how far the individual values 
 differ from their mean. 
 
 As in what follows only the final results will be given, 
 I shall give here one series of multiplied deflections with the 
 individual impacts completely calculated, so that it may be 
 seen how far the observations agree amongst themselves. 
 
 Extra-currents from Eectilinear Wires (Wires 
 Traversed in the same Direction). 
 
 Strength of the Inducing Current : 75*7. 
 2 = 0-9830, c = 0-016. 
 
 
 
 
 
 Magnitude of Individual Impacts 
 
 in 
 
 Readings 
 reduced to arc. 
 
 Vibration. 
 a n 
 
 O-n ~ QO-n-i 
 
 a n + qa-n-i 
 
 Scale Divisions. 
 j. K _ an - Qan- | +c(a„ +qa n - \ ) 
 2 
 
 
 517-2 
 
 30-7 
 
 
 
 
 
 547-9 
 
 72-0 
 
 41-8 
 
 102-3 
 
 21-7 
 
 
 475-9 
 
 111-3 
 
 40-5 
 
 182-5 
 
 21-7 
 
 
 587-2 
 437'8 
 
 149-4 
 
 40-0 
 
 258-8 
 
 22-1 
 
 
 186-5 
 
 39-6 
 
 338-4 
 
 22-4 
 
 
 402-0 624 ' 3 
 
 222-3 
 
 39-0 
 
 405-6 
 
 22-7 
 
 
 371-2 656 " 7 
 
 254-7 
 
 36-2 
 
 473-2 
 
 21-9 
 
 
 285-5 
 
 35-1 
 
 535-9 
 
 21-8 
 
 
I KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 11 
 
 The mean value of &« = 22*05; the greatest difference 
 amounts to less than -^ of the whole. As in each impact 
 40 extra-currents were combined, the deflection produced by 
 a single one was only 0*551 scale division. The remaining 
 series of multiplied deflections showed about the same degree 
 of agreement when the individual impacts were calculated. 
 
 Description of the Apparatus. 
 
 Before proceeding to discuss the individual experiments 
 I shall describe those arrangements which were common to all 
 the experiments. 
 
 1. If we desire the strength of the extra-current to be a 
 maximum in the galvanometer for a given strength of the 
 inducing current and given values of the inductances, we must 
 choose the resistance of the galvanometer as small as possible, 
 and the resistances of the other branches all equal. This 
 arrangement has another special advantage. For different 
 paths are open to the currents at make and break, since the 
 former can also discharge through the external circuit whilst 
 the latter cannot. In order to reduce all the experiments to 
 similar conditions a correction has in general to be made 
 which depends on the resistance of the external circuit. This 
 correction vanishes when the resistances of the four branches 
 are the same. In fact, if r be this resistance, r g the resistance 
 of the galvanometer, and r x that of the battery, we get for the 
 current in the galvanometer, when an electromotive force E 
 acts in one of the branches, the value E/2(r + r ff ), which is 
 independent of r x . 
 
 Hence, when the four branches were made equal, the 
 results obtained with different batteries could be directly 
 compared. 
 
 2. The passive resistances of the bridge had to be so 
 chosen that the part of the extra-current due to them was as 
 small as possible. In this respect columns of large diameter 
 of unpolarisable liquids would have been most suitable, since 
 the inductance of such columns is very small. But it was 
 impossible with the great delicacy of the bridge to obtain 
 
12 KINETIC ENERGY OF ELECTRICITY IN MOTION [I] I 
 
 them of sufficient constancy. Hence I employed wires of 
 German -silver, which were passed through glass tubes and 
 surrounded by distilled water, so as to guard against changes 
 of temperature. These were so arranged that those belonging 
 to different branches and traversed in opposite directions lay 
 side by side. The values of the inductances still remaining 
 were small and could be allowed for with sufficient accuracy 
 in the calculation. Since the German-silver wires were very 
 thin there was a danger that they might, when the current 
 was reversed, be subjected to small but sudden changes of 
 temperature. Such changes would, at the instant when the 
 current was started, have disturbed the balance of the bridge, 
 and so would have produced an increase or decrease of the 
 extra-current very difficult to estimate. Therefore, in a last 
 series of experiments I employed rods of Bunsen gas-carbon, 
 5 mm. in diameter, such as are used for electric lighting. 
 
 3. The strength of the inducing current was measured 
 outside the bridge ; the tangent galvanometer used consisted 
 of a single copper ring 213*2 mm. in diameter, at the centre 
 of which a needle about 25 mm. long was suspended by a 
 single silk fibre. In order to damp its vibrations as quickly 
 as possible it was placed in a vessel of distilled water. The 
 readings were taken by telescope and scale ; the distance of the 
 latter from the needle was 1295 mm., and one scale-division 
 corresponded to a current of 0*01218 in absolute electro- 
 magnetic units. The measurements were always made by 
 observing a deflection to the right, then one to the left, and 
 then again one to the right. The result is correct to y^ of 
 its value. 
 
 The extra-current was measured by a Meyerstein galvan- 
 ometer of very low resistance, such as is used for measurements 
 with the earth-inductor. The pair of needles was suspended 
 astatically by twelve fibres of cocoon silk ; the time of swing 
 was 27*66 seconds. The galvanometer was set up on an 
 isolated stone pillar, 2905 mm. from the scale and telescope, 
 and about the same distance from the bridge, and was connected 
 with the latter by thick parallel copper wires. 
 
 4. The commutator, at each passage of the needle through 
 its position of rest, had to perform the following operations 
 quickly one after the other : — 
 
KINETIC ENEEGY OF ELECTRICITY IN MOTION [I] 
 
 13 
 
 Connection of the galvanometer to the bridge. 
 Eeversal of the current. 
 Eeversal of the galvanometer. 
 . . . (Eepeated twenty times.) . . . 
 Eeversal of the current. 
 Throwing the galvanometer out of circuit. 
 Its arrangement is shown in Fig. 1. A circular disc 
 revolving about a vertical axis has attached to its edge radially 
 
 /^Lnn n n fln n i n 
 
 Fig. 2. 
 
 Fig. 1. 
 
 twenty amalgamated copper hooks of the form shown in Fig. 
 2, which just dip into the mercury contained in the vessels 
 B and C. They are 
 alternately nearer to 
 and farther from the 
 axis, so that the inside 
 ends of the farther 
 ones and the outside 
 ends of the nearer ones 
 lie on the same circle 
 about the axis. They 
 reverse the current in 
 passing over the vessel 
 B, and the galvanometer 
 in passing over C. The 
 arrangement of the vessels of mercury and the method of re- 
 versal are shown in Fig. 3. The vessel B is not exactly oppo- 
 site to C, but is displaced relatively to it through half the 
 
 Fig. 3. 
 
14 KINETIC ENERGY OF ELECTRICITY IX MOTION [I] i 
 
 distance between successive hooks, so that a reversal of the 
 galvanometer occurs between every two reversals of the cur- 
 rent. While the needle is completing its swing after the 
 induction impact, the hooks are symmetrically situated with 
 respect to the vessel C, so that one hook is above the space 
 between the two halves of the middle mercury cup, and the 
 neighbouring ones are right and left at the sides of the cups ; 
 the connection of the galvanometer with the bridge is then 
 broken. As soon as the needle reaches its position of rest the 
 disc is turned by hand and after a whole turn is stopped by 
 a simple catch, so that then the commutator performs the 
 above operations. 
 
 It may be mentioned that generally the wires of the 
 bridge, wherever possible, were soldered directly to each other ; 
 binding-screws and mercury-cups were only used where con- 
 nections had to be broken and remade repeatedly. 
 
 Experiments with Double-Wound Spirals. 
 
 I now come to the individual experiments, and first to 
 those with double-wound spirals. I had at my disposal two 
 spirals, exactly similar and very carefully wound, whose length 
 was 7 3 "9 mm. and whose external and internal diameters were 
 respectively 83*6 and 67'3 mm. They consisted of eight 
 layers, each with sixty-eight turns. The total length of wire 
 was found by comparison of its resistance with that of the 
 outer layer to be 130,032 mm. The diameter of the wire 
 was 0'93 mm., the total resistance about 3*1 Siemens units. 
 As the spirals were exactly alike, they were used together and 
 put in the diagonally opposite arms of the bridge. The extra- 
 currents produced by them were then added together in the 
 bridge. 
 
 According to the above explanations the inductive effects 
 of two inductances P and P' were to be observed whilst the 
 resistance of the circuit remained unchanged. The inductance 
 P was that of the spiral when the current in its two branches 
 flowed in opposite directions. To obtain a second inductance 
 P' a branch of one spiral was thrown out of circuit and re- 
 
I KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 15 
 
 placed by an equal ballast resistance, the magnitude of which 
 could be very exactly adjusted in the bridge. 
 
 When a current was passed through the branch thus 
 detached, and reversed in a suitable manner, the current in- 
 duced by this branch in the other could be measured. The 
 quantity P r was then the mutual inductance of the one branch 
 on the other. Of course the extra-current might also have 
 been used with the current flowing the same way through 
 both branches of the spirals, but this was too large compared 
 with that obtained from the spirals with their branches 
 traversed oppositely to be accurately observable under like 
 conditions. 
 
 We have first to calculate the numerical values of P and 
 P'. P may be determined with a sufficient accuracy from the 
 geometrical relations of the spirals and the calculation will be 
 performed immediately ; but P' can in this way be found only 
 by means of simplifying assumptions, which introduce a con- 
 siderable error. Hence I preferred to determine it directly 
 by experiment by comparing it with the known inductance of 
 straight wires. 
 
 Determination of P. — The following assumptions are made 
 as regards the arrangement of the wires and are very nearly 
 correct : — 
 
 1. In one and the same layer wires traversed positively 
 and negatively alternate with each other : the distances 
 between their central axes are equal to each other and to 
 the mean distance, which is got by dividing the length of the 
 spiral by the number of turns. 2. Two neighbouring layers 
 are laterally displaced relatively to each other through half 
 the distance between two centres. These assumptions com- 
 pletely determine the geometrical position of the wires ; but 
 whether the extreme wires near the ends of the spiral are all 
 traversed in the like direction or in opposite directions alter- 
 nately, it was impossible to decide in the case of the inner 
 layers. For this reason and because of the unavoidable 
 irregularities an accurate calculation of the inductance is 
 not possible ; we can only determine limits between which it 
 must lie and we shall see that these limits may be drawn 
 rather closely. In calculating the inductance of one layer we 
 may without appreciable error cut it open, develop it on a 
 
16 KINETIC ENERGY OF ELECTRICITY IN MOTION [1] I 
 
 plane and consider it as part of an infinitely long system of 
 straight wires, whose thickness is the same as that of the 
 layer. For the position of any element is unchanged rela- 
 tively to its neighbours, and the action of distant portions on 
 each other is zero. 
 
 We shall first determine the self-inductance IT of a single 
 layer. Let the length of the wires be S, their radius E, the 
 distance of two neighbouring ones q, their number n. 1 Further, 
 let a be the self-inductance of a single wire, a m the mutual in- 
 ductance of two wires distant mq ; then we have 
 
 a =2s(log^-f), a m =2s(log^-l 
 
 By counting we find 
 
 n = 2na - 2{2n - IX + 2{2n - 2> 2 - ... - 2a 2n _ v 
 and introducing the values of the as 
 „ ,« f'-i i 9. 1, l 2ft - 1 .3 2n - 3 ...(2^-3) 3 . (27i~l) ) 
 
 n = 4S.{ i+ iog| + -iog ^, 42 ^.. (2 ;J 2)2 n y 
 
 Hence the quotient II/S here has a perfectly determinate 
 value, which may be called the self-inductance per unit length. 
 The logarithm involved in the above expression cannot well 
 be directly calculated for large values of n ; for such values 
 we must use an approximation. For this purpose we split up 
 the expression into 
 
 , 1 2 3 2 5 2 ...(2^-3) 2 (2tz-1) 
 n log - 
 
 + log 
 
 2H 2 ...(2n-2f 
 2 2 44 6 e (2'ft-2) 2ft - 2 
 
 1.3 3*5* 5 3 7 3 (2n-3y i -\2n-iy 
 
 the two parts of which will be evaluated separately. The first 
 may be written 
 
 »-i (2m-l )(2m+l) 4~} f, 1 ' 
 = 7i2Llog — - -ri = n2 m los[ 1 
 
 2 
 1 
 
 (2m)' 2 T & \ 4m 
 
 1 [n is really the number of double wires, positive and negative counting as 
 one.— Tn.] 
 
KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 
 
 17 
 
 Since ■: — o<l for all the values concerned, we may expand 
 4m 2 
 
 log(l — l/4m 2 ), and thus obtain for the first part 
 
 [* i mr 3Z j m 4 J 
 
 or, when we develop the sums by well-known formulae, 
 
 H 
 
 const. — 
 
 + 
 
 1 
 
 + ...- 
 
 4(^-1) ' 8(w-l) 2 24(%-l) 3 
 
 1 1 
 
 + .. 
 
 96(m -iy 
 
 J 
 
 The constant is clearly — log (2/tt), for when n becomes 
 infinitely great the whole expression converges to w log (2/tt). 
 If we develop the remaining terms in descending powers of n 
 and collect together like powers, we finally find the first part 
 to be 
 
 \7T/ * 8t& 96% 
 
 An analogous calculation may be performed for the second 
 part, which is 
 
 = 2 log 
 
 2771 2 
 
 (2m-l) ?n (2??i+l)' 
 
 n-l / 1 \ -m 
 
 Slogfl-.-— 
 4m 
 
 ii - i i ii-ii % -ii 
 
 ! v/fv 1 771° " i 771° 
 
 and hence, by a calculation similar to the above, 
 = i {0-577216 + log(»-l) + ^- I ^ 2 + 
 
 + ^3 + 
 
 2(n-l)' 
 
 + .. 
 
 + 
 
 r «> i 
 
 ! 
 
 ^ToWi^H- 
 
 1 
 
 !m 7 
 
 M. P. 
 
18 KINETIC ENERGY OF ELECTRICITY IN MOTION [I] I 
 
 If here we calculate the constant term directly and expand 
 the remaining terms in descending powers of n, we find the 
 second part to be 
 
 0-18848+llog^- — - _ L_-.... 
 
 The sum of both parts gives the required term 
 
 4 - / 2\ n 3 
 = 0-43848+ log s/n[ - + 
 
 W 192^ 2 
 
 = log-! 1-5503 *M- ) I- + 
 
 \7rJ J 192n 2 
 
 {4«- 9 ~\ 3 
 
 it) 19 2n 2 
 
 and hence to a considerable degree of approximation the self- 
 inductance of a single layer becomes 
 
 For large values of n the root involved in this expression 
 rapidly converges to unity, so that for such values of n we 
 may write more simply 
 
 n = 4S»{ i + io g g}. 
 
 We get this approximation at once by calculating the 
 induction of the whole arrangement on one of the middle 
 wires, and assuming it true for all the wires. We may use 
 this simpler method in calculating the mutual inductance of 
 two different layers. 
 
 Let e be the perpendicular distance between two layers, 
 and suppose that in them the individual pairs of wires are so 
 placed that the wires traversed in the same direction are 
 opposite, with their axes in one plane perpendicular to the 
 two layers. Then the mutual inductance of one layer and a 
 median wire of the second layer is 
 
KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 
 
 19 
 
 2S<! 
 
 log 2S — log e — 1 
 
 - 2 log 2S + 2 logV eH 7 ? + 2 
 + 2 log 2S - 2 log J7+If - 2 
 
 - log 2S + log J e * + (2™ + 1)Y + 1 
 <e 2 + 2 V )(e 2 + 4V). . .Q 2 + (2w)V) 
 V + ? 2 )(e 2 + 3 V). • • Ve 2 + (2^+l)V ' 
 
 2Slosr 
 
 = 2S log 
 
 (-) 2 3 2 + / 
 
 */ 
 
 2 2 + 
 
 (2^+l) 2 + 
 
 7 2 + (2^)V 
 
 We get an approximate value in finite form of the expres- 
 sion behind the logarithm sign by dividing the equation 
 
 „2 \2 
 
 COS 2 - = ( 1 — 
 
 9. \ IV 
 
 v 2 \2 
 
 by this second equation, 
 
 sin z = z[ 1 — 
 
 X -3V X "5V 
 
 IV 
 
 by putting v\/—l for 2 on both sides and dividing the result 
 by /v /_ 1 ; thus we get the equation 
 
 1+e 
 
 = limit < 
 
 r+(-) 3* + 
 
 ,7T/ 
 
 Jif+w+Pp 
 
 2 2 + 
 
 2 . . . 
 
 2?i 2 + 
 
 and thus the above inductance becomes approximately for a 
 large value of n 
 
 2Slog 
 
 1+6 * 
 
 1—e ( i 
 
 or, if we neglect terms of the order e~ Sire,q ' 
 
 _ e7r 
 
20 KINETIC ENERGY OF ELECTRICITY IN MOTION [I] I 
 
 Multiplying this expression by 2n, we find the mutual induct- 
 ance of the two layers to be 
 
 ■ err 
 
 + 8Sne' 1 ' 
 
 If now one of them be displaced laterally through a dis- 
 tance q the inductance obviously becomes 
 
 err 
 
 -8Sne~~*' 
 
 For all intermediate positions the inductance has a value 
 between these two extremes. Hence, if we neglect altogether 
 the effect of the different layers on each other, the error for 
 each pair of layers to be considered is less than 8$ne~ enlq - 
 But any two contiguous layers may be altogether left out of 
 account, for the induction between them is in fact zero to a 
 high degree of approximation. By substituting values of q, e, 
 K, such as occur in ordinary cases, we easily convince ourselves 
 that the error considered amounts to less than the y^ part 
 of the self-inductance of the layer. In so far as we neglect 
 it we get the self-inductance of the whole spiral by adding the 
 self-inductances of the individual layers. Hence, if I denote 
 the whole length of wire contained in the spiral, we have 
 finally 
 
 = 2*{i 
 
 -flog 
 
 22x/5-m3?i 
 
 Ett 
 
 For the spirals used in our case 
 
 /-130032 mm., n = 34, K= 0*465 mm., 
 
 2= 1-087 mm., P= 178,500 mm. 
 
 A calculation of the error possibly committed on the above 
 principle gave it as less than 1200 mm. 
 
 Determination of P' . — The mutual inductance T f of the 
 two branches of the spiral, as has been already remarked, was 
 determined by comparison with the inductance of straight 
 wires. The arrangement of the experiment is shown in Fig. 4. 
 A and B are rectilinear systems of wires of the dimensions 
 marked on the figure, and were stretched out on the floor of 
 
KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 
 
 21 
 
 the laboratory. In A (the secondary circuit) the galvanometer 
 and one branch of the spiral C were introduced ; in B the 
 battery, the tangent galvanometer, and the commutator. 
 
 The induction of the circuit B on A was first determined. 
 As this was very small, the above described method of experi- 
 menting and calculating @% 
 was used, in which the 
 circuit A was as a rule 
 
 < 6iao 
 
 broken. As the action 
 of the rectilinear system 
 B on the spiral C was *«tf- 
 not zero, the latter was 
 inserted in the circuit 
 in both of the two pos- 
 sible ways. The values 
 of the inductive kick FlG - 4- 
 
 were observed for different strengths of the primary current ; 
 after reduction to the same strength (100 scale divisions of 
 the tangent galvanometer) they were found to be — 
 1. For the first position of the spiral : 
 
 k 
 
 In scale divisions of the galvanometer, 
 
 0-3997, 0-3955, 0'3791, 0-4006. Mean = 
 2. For the second position : 
 
 0-3034, 0-3102. Mean = 0^3068. 
 
 0-3939; 
 
 The mean of the two values, namely 0'3502, gives the induc- 
 tion of the circuit B on A. The corresponding logarithmic 
 decrement was that of the needle swinging freely, namely 
 
 \=0-0172. 
 
 In order to compare the kick with experiments made with a 
 different damping we must multiply it by 
 
 s/tt 2 + X 2 X 
 
 ff tan_1 v 
 
 which gives the value 
 
 1-1097 
 
 The remaining detached branch of the spiral C was now 
 
22 
 
 KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 
 
 put in the circuit B, so that the actions of the spiral and of 
 the rectilinear circuit aided each other, and the action of the 
 one branch on the other was observed by simply reversing the 
 current B each time the needle passed through its position of 
 rest. After reduction to the strength 100 the inductive kick 
 was found to be 
 
 164-3, 164'7. Mean = 164*5 scale divisions. 
 
 The corresponding decrement was \= 0'6362, and the reduc- 
 tion to an unresisted needle gives the value of the kick 
 
 We may here and in what follows neglect the effect of 
 damping in altering the period. Hence the mutual inductance 
 of the branches of the spiral is to that of the rectilinear cir- 
 cuits in the ratio of 
 
 696-0-1-1 : 1-1097 = 694-9 : 1-1097. 
 
 The latter inductance was 
 easily calculated from the 
 geometrical relations of the 
 wires, and was found to be 
 60428 mm., whence it 
 follows that the mutual 
 inductance of the branches 
 of the spiral =37,840,000 
 mm. 
 
 I had already found for 
 the same quantity the 
 value 
 
 38,680,000 mm. 
 
 by a different but more 
 
 uncertain method. The two 
 
 values agree sufficiently 
 
 , well, but we shall only use 
 
 ^A^\^_^= ======= ^ =====:=== ^ the first one. 
 
 ^T Execution of the Bxperi- 
 
 fig. 5. merits. — The arrangement of 
 
 the bridge used for measuring the extra-currents is shown in 
 
I KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 23 
 
 greater detail in Fig. 5. The current enters at A and A! ; 
 the galvanometer is connected at B and B r . The bridge is 
 adjusted by moving the connection A! of the battery with the 
 thick copper wire EF. The spirals are inserted in the diag- 
 onally opposite branches A'B and AB\ the passive German- 
 silver resistances are placed in the other two branches and lie 
 close together side by side, with the currents flowing through 
 them in opposite directions. The wires from the battery and 
 those to the galvanometer pass through the commutator G, 
 which is placed so that the observer can set it in motion 
 while observing with the telescope H. In every one of its 
 revolutions 20 double extra-currents from each of the two 
 spirals, in all 8 simple extra-currents, pass through the galvan- 
 ometer. The tangent galvanometer is at I and is read by the 
 telescope K. 
 
 The value of the extra- current was first determined when 
 the branches of the spirals were traversed in opposite direc- 
 tions. The following values were obtained : — 
 
 Strength of Primary Current Strength of Secondary Current 
 
 in Scale Divisions. in Scale Divisions. 
 
 48-8 0-1790 
 
 50-0 0-1738 
 
 123-2 0-4621 
 
 122-2 0-4417 
 
 By reduction to strength 100 we get the values 
 
 0-3664, 0-3476, 0'3750, 0-3600. Mean = 0-3622. 
 
 The logarithmic decrement of the needle for these experiments 
 was \ = 0*0172. If we therefore multiply the above kick by 
 the factor 3-168/T, we get the reduced value 
 
 1-1476 
 
 In accordance with previous explanations, one of the 
 branches of one spiral was next thrown out of circuit ; the 
 bridge was readjusted, and the inductive action of the detached 
 branch on the other was observed. When in the former a 
 current 100 ceased to flow, the kick observed in the galvan- 
 ometer was 
 
 61-50, 61-66. Mean = 61-58 scale divisions. 
 
24 KINETIC ENERGY OF ELECTRICITY IN MOTION [I] i 
 
 The logarithmic decrement in this experiment was \= 0*4396, 
 and the corresponding factor 3'876/T : hence the reduced kick 
 was 
 
 238-67 
 T ' 
 
 Thus for the resistances of the bridge and the arrangement of 
 circuits used this kick corresponded to the mutual inductance 
 
 P'= 37,840,000. 
 
 When the branches of the spirals were traversed in 
 opposite directions, the corresponding electromotive force of 
 the extra-current was for each of the spirals 
 
 P=178,500. 
 
 But we have to correct for the induction of the rest of the 
 bridge as follows : — 
 
 1. The self-inductance of each of the German-silver wires 
 (diameter = 0*246 mm.) is 12,790; twice this must be sub- 
 tracted, so that the correction is —25,580. 
 
 2. This action is in part compensated by the action of the 
 neighbouring German-silver wire. Their mutual inductance 
 is 5348 ; this value is to be taken twice, so the correction 
 is +10,696. 
 
 3. The self-inductance of the wire OP is +9028. 
 
 4. The mutual inductance of the wire OP and the nearer 
 German-silver wire is +2789. 
 
 5. The same for the further one is — 1230. 
 
 6. The mutual inductance of ES and EA', taken twice 
 because of the double strength in ES, gives +5254. The 
 sum of all these corrections only amounts to +957, of which 
 one half goes to the account of each spiral, namely 
 + 478 mm. 
 
 Hence, finally, the inductive electromotive force is 
 measured by 178,978, which is almost exactly that due to 
 the spirals alone. The error in this value, due to neglecting 
 portions of the induction of the spirals, according to what 
 precedes, at most amounts to 2400 mm. The error due 
 to neglecting portions of the bridge will presumably be of the 
 same order. 
 
I KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 25 
 
 According to theory, the extra current from the spirals 
 with branches traversed oppositely should have the value 
 
 238-67 178,978 _ 1-1351 
 
 ~ T ' 37,840,000 " T ' 
 
 The value actually observed was 1'1476/T. The differ- 
 ence between the two amounts to little more than 1/100 of the 
 total value, while the errors of experiment and of calculation 
 at most may amount to 1/30. Hence, though the great 
 agreement between the calculated and observed values is 
 merely fortuitous, yet the experiment shows that at most 
 1/20 to 1/30 of the very small extra-current from doubly 
 wound spirals can owe its existence to a possible mass of the 
 moving electricity; and the formula given above does in fact 
 represent the induction of such a spiral to a high degree of 
 accuracy. 
 
 I took it to be unnecessary to make further experiments 
 with spirals ; for even if the observations could be carried to a 
 higher degree of accuracy, still it was impossible to calculate 
 exactly the values of the inductances which entered into the 
 experiments. 
 
 First Series of Experiments with Eectilinear Wires, j 
 
 With a view to getting conditions of experiment more 
 favourable in this respect, I attempted to measure the 
 strength of the extra-current obtained from rectilinear double 
 wires traversed in opposite directions, and to compare its value 
 with theory. 
 
 For these experiments the bridge was arranged as follows. 
 Three or four resistances, as before all equal, were made of 
 thin German-silver wires. Two of these were formed exactly 
 alike, so that the extra-currents from them neutralised each 
 other. The third consisted of a wire folded back on itself, 
 whose self-inductance was small and could be easily calcu- 
 lated; it was found to be ^=13-194 mm. In opposition 
 to this acted the fourth resistance of the bridge, the system of 
 wires under observation. This was stretched out on the floor 
 
26 KINETIC ENERGY OF ELECTRICITY IN MOTION [I] r 
 
 of the laboratory and connected by vertical wires with the 
 experimental tables. In form it was a rectangle 7229 mm. 
 long by 946 mm. wide. Its sides were formed of two parallel 
 wires close together, each of which formed one branch of the 
 system. A commutator enabled the currents to flow through 
 them in the same or in opposite directions. The wire used 
 was hard copper wire, its diameter was determined at several 
 places by microscope and micrometer screw, and, with small 
 deviations, was found in the mean =0*4104 mm. To keep 
 the distance between the two branches everywhere exactly the 
 same, the wires were passed over wooden supports with nicks 
 cut in them and fitting exactly. These supports were pre- 
 pared by the aid of two brass stencil plates. The distance 
 between the wires was measured on these latter by microscope 
 and micrometer screw and thus found to be in the mean 
 2*628 mm. from centre to centre. The wires were covered 
 throughout their length with a layer of cotton wool, so as 
 to guard them from rapid temperature changes due to air- 
 currents. 
 
 The inductance of the whole arrangement, which con- 
 sisted partly of parallel, partly of perpendicular wires, could be 
 easily calculated with accuracy by the formulas already given. 
 It was found that — 
 
 1. When the current flowed the same way through both 
 branches 
 
 P' = 972,400 mm.; 
 
 2. When the currents flowed in opposite directions 
 
 P= 193,160 mm. 
 
 Hence we find the ratio of the strengths of the extra- 
 currents to be expected in the two cases 
 
 I -~^= 5*330. 
 Y-p 
 
 In this calculation only the action of the commutator, of 
 the movable arrangement used to adjust the bridge, and of the 
 external circuit on the parts of the bridge has been neglected. 
 These actions have only a very small effect, and the error 
 
KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 
 
 27 
 
 caused by them certainly vanishes in comparison with the 
 error due to the observations themselves. 
 
 The observations and calculations performed in the manner 
 already explained gave the following results : — 
 
 1. Branches traversed in like Directions. 
 
 Primary Current in 
 
 Simple Extra- 
 
 Primary Current in 
 
 Simple Extra- 
 
 Scale Divisions of 
 
 Current in Scale 
 
 Scale Divisions of 
 
 Current in Scale 
 
 tangent Galvan- 
 
 Divisions of Galvan- 
 
 tangent Galvan- 
 
 Divisions of Galvan- 
 
 ometer. 
 
 ometer. 
 
 ometer. 
 
 ometer. 
 
 152-7 
 
 1-121 
 
 78-9 
 
 0-561 
 
 75-7 
 
 0-551 
 
 78-4 
 
 0-548 
 
 93-6 
 
 0-673 
 
 74-2 
 
 0-549 
 
 116-4 
 
 0-831 
 
 145-2 
 
 1-065 
 
 67-6 
 
 0-478 
 
 
 
 By reduction to a current of 100 scale divisions we get 
 the values '743, '728, '720, '714, '707, '711, "701, '737, '733. 
 Mean = '7213, with a mean error of 0-0137 or 1/50 of the 
 whole value. 
 
 2. Branches traversed Oppositely. 
 
 Primary Current. 
 
 Extra-Current. 
 
 Primary Current. 
 
 Extra-Current. 
 
 152-7 
 152-7 
 140-1 
 139-0 
 
 0-2088 
 0-2051 
 0-1872 
 0-1817 
 
 150-5 
 116-1 
 228-9 
 
 0-2025 
 0-1443 
 0-3992 
 
 By reduction to a current 100 we get the values 0*1367, 
 01344, 0-1337, 0-1307, 0'1345, 0'1243, 0'1382. 
 
 The large deviation of the sixth observation from the 
 others is evidently due to some special error, and it will be 
 rejected. The others give the mean value 0'1348, with a 
 mean error of 0'0028 or about 1/50 of the whole value. 
 The ratio of the two extra-currents observed, 
 
 -7213 
 •1348 
 
 5-352, 
 
28 KINETIC ENERGY OF ELECTRICITY IN MOTION [I] I 
 
 differs from the calculated ratio 5 '3 30 only by 1/250. The 
 difference certainly is less than the unavoidable errors of 
 observation. 
 
 It is to be remarked that in the above results all observa- 
 tions without exception have been included. 
 
 Second Series of Experiments with Eectilinear Wires. 
 
 A second series of observations with rectilinear wires was 
 made, which differed from the preceding one only in that the 
 German-silver resistance A'B was replaced by a resistance of 
 Bunsen gas carbon, and in that a thicker copper wire was 
 chosen for the circuit giving the current. This change was 
 intended to remove a danger to be apprehended in the preceding 
 experiment, viz. that the small changes of temperature attending 
 the reversal itself might produce an apparent alteration in the 
 strength of the extra-current, and so might hide any deviation 
 of it from the induction law. In order to render these and 
 similar disturbances observable, the experiments were made 
 with as many different values as possible of the primary 
 current. The deviations of the integral flow of the extra- 
 currents from proportionality to the strength of the primary 
 necessarily had their origin in such disturbances. 
 
 The diameter of the wire used was 0*6482 mm.; the 
 distance between the two branches was 3 '4 41 mm. The 
 inductances were calculated exactly as above, and were found 
 to be 
 
 P' = 920,956 mm., P= 185,282 mm. 
 The self-inductance of the opposing carbon resistance was 
 P=2997 mm., 
 and hence the calculated ratio of the extra-currents was 
 
 3 — ^=5-0367 
 
KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 29 
 
 The experiments gave the following results :— 
 
 No. 
 
 Primary 
 Current. 
 
 Extra-Current. 
 
 
 Reduced to Strength 100. 
 
 Branches 
 Opposed. 
 
 Branches Like. 
 
 Branches 
 Opposet'. 
 
 Branches Like. 
 
 1 
 
 11-3 
 
 0-0275 
 
 0-1225 
 
 0-2434 
 
 1-084 
 
 2 
 
 17-2 
 
 
 
 0492 
 
 0-1940 
 
 0-2860 
 
 1-127 
 
 3 
 
 18-8 
 
 
 
 0478 
 
 0-2082 
 
 0-2542 
 
 1-107 
 
 4 
 
 20-9 
 
 
 
 0582 
 
 0-2430 
 
 0-2784 
 
 1-162 
 
 5 
 
 24-1 
 
 
 
 0628 
 
 0-2775 
 
 0-2606 
 
 1-152 
 
 6 
 
 27-7 
 
 
 
 0700 
 
 0-3235 
 
 0-2527 
 
 1-167 
 
 7 
 
 33-3 
 
 
 
 0857 
 
 0-3792 
 
 0-2537 
 
 1-138 
 
 8 
 
 37-2 
 
 
 
 0957 
 
 0-4015 
 
 0-2572 
 
 1-080 
 
 9 
 
 477 
 
 
 
 1057 
 
 0-5243 
 
 0-2216 
 
 1-099 
 
 10 
 
 57-8 
 
 
 
 1330 
 
 
 0-2301 
 
 
 11 
 
 66-2 
 
 
 
 1478 
 
 0-7357 
 
 0-2234 
 
 1-112 
 
 12 
 
 72-7 
 
 
 
 1555 
 
 
 0-2139 
 
 
 13 
 
 88-7 
 
 
 
 2135 
 
 
 0-2135 
 
 
 14 
 
 108-6 
 
 
 
 2432 
 
 
 0-2240 
 
 
 15 
 
 1287 
 
 
 
 2945 
 
 1-1425 
 
 0-2158 
 
 1-051 
 
 16 
 
 141-3 
 
 
 
 3005 
 
 1-1825 
 
 0-2128 
 
 1-049 
 
 17 
 
 172-6 
 
 
 
 3872 
 
 
 0-2276 
 
 
 18 
 
 . 192-1 
 
 0-4105 
 
 
 0-2138 
 
 
 We get the mean value of the extra-current from branches 
 traversed in like directions and with strength 100 to be 
 rill, with a mean error of 0*038. From this and the cal- 
 culated value of the ratio we get for the extra-current from 
 branches oppositely traversed the value 0*2203. If we 
 compare this with the observed values we see that the 
 observed values for very weak currents are very much larger ; 
 and the mean of all the observations, viz. 0*2379, differs 
 considerably from the calculated value. At the same time we 
 easily see that this difference in no wise justifies us in drawing 
 any inference as to mass ; for it only occurs in the case of 
 those observations which were already uncertain because of 
 the smallness of the effects, and which agree only badly 
 amongst themselves. If we use only the better part of the 
 experiments from No. 8 onwards we get for the extra-current 
 from oppositely traversed branches the mean value 0*2197, 
 with a mean error 0*0060, and for the ratio of the two extra- 
 currents the value 5*054, a value which differs from 5*037, 
 the calculated one, by a quantity much less than the un- 
 avoidable errors of experiment. The deviation of the values 
 
30 KINETIC ENERGY OF ELECTRICITY IN MOTION [I] I 
 
 observed for smaller strengths may easily and in various ways 
 be reduced to thermal, magnetic, or diamagnetic causes, whose 
 effect does not increase in proportion to the current, but quickly 
 reaches a maximum. 
 
 Kecapitulation of Eesults and Inferences. 
 
 We shall shortly recapitulate the results, in order to deduce 
 from the experiments an upper limit to the quantity //,, the 
 meaning of which was explained in the introduction. 
 
 We start from the formulae 
 
 m = P —i— - 1 , 11 = — . 
 \J'P / P 21 
 
 The quantities occurring here in the various experiments 
 had the following values : — 
 
 1. In the experiments with spirals we had 
 
 1 = 130,032, P'= 37,840,000, 
 
 q= 0-6793, J = 11467, 
 
 P = 178,500, J' = 238-67. 
 
 The probable error of the various values cannot be stated 
 exactly, as the corresponding measurements and the sources of 
 error present were so various. Nevertheless it is certain that 
 in none of the measurements was an error committed greater 
 than 1/20 of the whole ; and the errors of the calculation, 
 by which P was determined, cannot, according to what was 
 said above, reach this amount. Hence, if we assume that the 
 quantity JP'/J'P, which is compounded from the results, is in 
 error by 1/20 of its value, in such a way as to hide any 
 effect of mass perhaps present, we shall obtain a limit which 
 is most unlikely to be exceeded, namely, 
 
 m<l78,500^.fl-l 
 
 m< 13,356, fju< 0-0348 mm 2 . 
 
 2. We obtain a smaller limit from the first series of 
 experiments with rectilinear wires. Here we had 
 
 P' . P' 
 
 1 [The fraction p- is in the original replaced by — since P' denotes a mutual 
 
 inductance, and P there stands for a "potential on itself," which Hertz defines as 
 half a self-inductance. — Tn.l 
 
I KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 31 
 
 1 = 35,892, P= 179,960, J ; = 0*7213, 
 
 P' 
 
 2=0'1323, — = 5-330, J = 01348. 
 
 The calculated inductions may be regarded as exact, as their 
 error can hardly amount to 1/100. The probable errors for 
 the quantities J and J 7 may be deduced from the experiments, 
 and are found to be 
 
 for J' = 0-0092, for J = 0-0019. 
 
 If we here assume that both strengths are measured 
 wrongly to the full extent of the probable error, and both in 
 the unfavourable sense, that is J too small, J 7 too large, we 
 get 
 
 m< 179-960 (5'330 . °^ 367 _ i) 
 v 0-7121 ; 
 
 ?tt<4190, fi< 0-0077 mm 2 . 
 
 3. In the second series of experiments with rectilinear 
 wires we had 
 
 Z = 35,892, P= 185,240, J' = 1-111, 
 
 P' 
 
 2 = 0*3300, -=5-0367, J = 0"2196. 
 
 In determining J only experiments from No. 8 onwards have 
 been used. The probable errors resulting from the experi- 
 ments are 
 
 for J' 0-026, for J = 0-0040. 
 
 The same assumption as above gives here 
 
 0-2236 
 m< 185,240 (5-0367. - 1), 
 
 1*085 
 
 m< 7042, ^< 0-0323 mm 2 . 
 
 The limit here is not so close as in the preceding experi- 
 ments, chiefly because there q, the cross-section of the wire, 
 was less, and thus the conditions were more favourable for 
 rendering prominent the effect of mass. 
 
 Hence, using the first series of experiments with rectilinear 
 wires as being the best, we obtain this result : — 
 
32 KINETIC ENERGY OF ELECTRICITY IN MOTION [I] I 
 
 The kinetic energy of the electric flow in one cubic milli- 
 metre of a copper wire, which is traversed by a current of 
 density equal to 1 electromagnetic unit, amounts to less than 
 
 0*008 milligramme-millimetre. 
 
 As the kinetic energy is half the product of the mass by 
 the square of the velocity, the mass of the positive electricity 
 in 1 cubic millimetre 
 
 0-008 mg 
 
 e.g. if v = 1 , 1 ■ — - , . . . the mass of positive electricity 
 
 sec sec 
 
 < 0-008 mg., < -00008 mg., etc. 
 
 But we must bear in mind the possibility of the kinetic 
 energy of the current exceeding the limit here marked out, 
 without the observations necessarily being erroneous on that 
 account. For if the conductivity of the metals is in the ratio 
 of the densities of the electricity in them, then the electro- 
 motive forces arising from inertia will be equal in two wires 
 of equal resistance, whatever be the material, length, and cross- 
 section of the wires. In this case the extra -currents from 
 the four branches of the bridge, in so far as they were due to 
 mass, would also be equal, and would thus neutralise each 
 other. It is only on the assumption that the above propor- 
 tionality does not exist, but that the density of electricity is 
 approximately the same in different conductors, that it is 
 allowable to neglect, as we have done, the effect of the mass 
 moved in the short branches of German -silver and of gas- 
 carbon. 
 
 Vice versa, if by some other method we succeeded in prov- 
 ing that the kinetic energy of the electric current exceeds the 
 limit stated, the above experiments would show that the 
 densities of electricity in the materials used are in the ratio 
 of their conductivities. 
 
 A decision as to the possibility mentioned could theo- 
 retically be obtained by dynamometric experiments, or by 
 observing the values at different times of induction and extra- 
 currents ; but in practice all the arrangements of experiment 
 
I KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 33 
 
 I have been able to find out only offer a hope of success if the 
 inert mass exceeds the limit here determined many thousand 
 times. 
 
 In conclusion, excluding the assumption last discussed, I 
 shall introduce the limit found for the quantity //, into the 
 calculations which have been developed by Helmholtz in vol. 
 lxxii. of Borchardt's Journal. 1 It is there shown, on certain 
 definite assumptions there stated, including the truth of Weber's 
 law, that in a conducting sphere of radius E certain types 
 of currents, of given order a, become unstable when in our 
 notation 2 
 
 E< 
 
 a lam. 
 
 V 2 
 
 From this and from the limit found for /jl, viz. p = 0'008 mm., 
 it follows that on the assumptions made, the first, the funda- 
 mental, type -current would become unstable in a sphere of 
 O'll mm. radius, and that in a sphere of 1 cm. radius the first 
 90 component currents, almost the whole current, might in- 
 crease indefinitely. 
 
 Extra-Currents in Iron Wires. 
 
 If the wire conveying the current be capable of being 
 magnetically polarised, this circumstance will lead to an in- 
 crease of the self-induction like that due to any existing inert 
 mass of electricity ; hence the magnitude of the capacity for 
 polarisation may be determined by the method we have 
 employed in attempting to discover the existence of electric 
 inertia. 
 
 I have made several experiments with iron wires, partly 
 to convince myself of the applicability of the method for this 
 purpose, partly to obtain an estimate as to how far magnetic 
 properties in other metals might give rise to disturbances. In 
 these the resistance of the- bridge under examination consisted 
 of a soft iron wire 0*66 mm. in diameter and 14,070 mm. 
 in total length. This, like the copper wire previously used, 
 
 1 H. Helmholtz, Wiss. Abhandl. vol. i. p. 589. 
 2 Our quantity n is expressed by /x/2A" in the notation there used. 
 M. P. D 
 
34 
 
 KINETIC ENERGY OF ELECTRICITY IN MOTION [i] 
 
 consisted of two branches which could be connected up in two 
 different ways. The wire was again rectangular in form, so 
 that the self-inductance could be exactly calculated for both 
 arrangements. From the extra-currents obtained with these 
 two self- inductances it should be possible to calculate the 
 increase of self-induction due to magnetisation ; in practice 
 this was found to be inadvisable, for the effect of magnetisa- 
 tion was not small but very large compared with the purely 
 inductive effect. Hence the iron wire was replaced by a 
 branch of one of the above-mentioned spirals with the neces- 
 sary ballast resistance, and the extra -current produced by 
 this known inductance was compared with that from the iron 
 wire. 
 
 The observations were made by the method above described : 
 their details are of no interest, but they supply the data from 
 which the magnetic forces acting in the iron wire and the 
 polarisations attained may be determined in absolute measure. 
 The results are given in the small table which follows. The 
 first column gives in absolute measure the values of the 
 magnetising force K at the surface of the cylindrical wire 
 (whence it diminishes towards the axis in proportion to the 
 distance from the axis). The second column gives the values 
 of 6, the so-called constant of polarisation, calculated from the 
 corresponding observations. 
 
 K 
 
 e 
 
 K 
 
 e 
 
 0-96 
 
 8-12 
 
 1-98 
 
 8-83 
 
 1-17 
 
 8-42 
 
 2-94 
 
 9-67 
 
 1-47 
 
 9-02 
 
 3-12 
 
 9-67 
 
 1-62 
 
 8-92 
 
 3-99 
 
 9-96 
 
 1-74 
 
 8-65 
 
 7-20 
 
 11-60 
 
 These values of K and 6 can of course be only roughly 
 considered as corresponding values. Apart from several irre- 
 gularities, we see that 6 increases with K within the limits 
 given, a phenomenon which has already been frequently ob- 
 served under different circumstances. It was impossible to 
 extend the observations to stronger currents, owing to the con- 
 siderable generation of heat in the iron wire. 
 
II 
 
 ON INDUCTION IN EOTATING SPHEEES 
 
 {Inaugural Dissertation, Berlin, 15th March 1880.) 
 
 The interactions between magnets and rotating masses of 
 metals discovered by Arago were first explained by Faraday 
 as phenomena of electromagnetic attraction, and attributed to 
 currents induced by the magnets in the masses of metal. 
 Faraday succeeded in demonstrating the existence of such 
 currents, and in placing the inductive nature of the phenomenon 
 beyond doubt. 
 
 In 1853 Felici made the first attempt to apply the theory 
 since developed to some phenomena of this class. He succeeded 
 under simplified conditions in obtaining approximate solutions 
 agreeing with experiment sufficiently for a first approximation. 
 
 Great progress was made in 1864 by Jochmann. Start- 
 ing from Weber's laws he deduced the complete differential 
 equations of the problem, and integrated them for the case 
 when the rotating body is an infinitely extended plane plate 
 or a sphere. His calculations agreed most beautifully with 
 the observations. But he had to make the assumption, for 
 purposes of simplification, that the velocity of rotation was 
 very small, for he was unable to determine the effect of self- 
 induction. 
 
 Finally, in 1872 Maxwell gave a very elegant exposition 
 of the theory of the induction in an infinitely extended very 
 thin plate, and showed how it could be applied to the case of 
 Arago's disc. 
 
 In the present paper the problem is completely solved for 
 
36 INDUCTION IN ROTATING SPHERES n 
 
 the case when the body considered is a solid or hollow sphere 
 rotating about a diameter. The inducing magnets may be 
 outside, or in the case of the hollow sphere in the inside space. 
 The solution is also extended to the case where the material 
 of the sphere is capable of magnetisation. Clearly this problem 
 involves those previously solved as special cases. 
 
 I have attempted to exhibit the results obtained by means 
 of several drawings. 
 
 § 1. Definition of the Symbols. 
 
 In this paragraph the symbols to be employed will be 
 defined and some formulae will be collected together which 
 will be constantly required in the sequel. 
 
 1. The system of coordinates selected is that shown in 
 
 Fig. 6. The positive directions of rotation are marked in the 
 
 *» figure. The axis of z will be taken to 
 
 'T coincide with the axis of rotation. 
 
 ^- -? Polar coordinates p, to, 6 will be used : 
 
 co is to correspond to geographical 
 
 >'X / longitude, is to be in the 2^-plane 
 
 for positive x, and is to increase in 
 
 the direction of positive rotation ; 
 
 is to correspond to the complement 
 
 fig. g. of the geographical latitude, and is to 
 
 be in the positive 2-axis. We shall occasionally use the 
 
 notation 
 
 # 
 
 
 
 a 
 
 d 
 
 d 
 
 d 
 
 
 
 
 X 
 
 dy 
 
 ~ y dx 
 
 dco z 
 
 dco' 
 
 
 
 
 z d 
 
 d 
 
 d 
 
 
 
 
 
 dx 
 
 -x — = 
 
 dz 
 
 dCOy 
 
 
 
 
 
 d 
 
 d 
 
 d 
 
 
 
 
 
 dz 
 
 -z — = 
 
 dco,' 
 
 
 
 Further, 
 
 as in Lagrange's 
 
 notation, differentiation with 
 
 respect 
 
 to 
 
 w will be denoted by a 
 
 dash, 
 
 e.g. 
 
 
 
 
 
 
 dco 
 
 = %'• 
 
 
 
ii INDUCTION IN ROTATING SPHERES 37 
 
 2. The calculations will be carried out in electromagnetic Definition 
 
 units. In other respects the symbols for electrical magnitudes ° ical ie q e U a n ." 
 
 will be those used by Helmholtz in vol. lxxvii. of Borchardt's tities. 
 
 Journal. 1 That is, 
 
 mgr* 
 u, v, w f — 
 
 are the densities of the current parallel to x, y, z ; 
 
 sec 
 are the corresponding components of the vector-potential ; 
 
 , mm* mgr* 
 
 9 
 
 sec 
 
 is the potential of free electricity ; 2 
 
 mm 2 
 /c 
 
 sec 
 
 is the specific resistance of the material. The specific resist- 
 ance of a sheet of infinitesimal thickness B, viz. - will, when 
 
 o 
 
 considered finite, be denoted by 
 
 sec 
 
 Further, let 
 
 mgr 2 
 
 \, a, V ; 
 
 mm 2 sec 
 
 be the components of a magnetisation ; let 
 
 sec 
 be the potentials of A, p, v, taken as masses, 
 
 0(0) 
 
 1 H. Helmholtz, Wiss. Abhandl. vol. i. p. 545. 
 
 2 This is not electromagnetic measure. Measured in the latter units the 
 potential of free electricity is <f> m = 0A 2 , where 1/A denotes the velocity of light. 
 The above unit avoids the inconvenient factor 1/A :! . 
 
 £90435 
 
38 INDUCTION IN ROTATING SPHERES 
 
 be the magnetic permeability, and 
 
 X 
 
 mm 2 m g T l 
 
 sec 
 
 the magnetic potential, so far as it is directly due to magnets 
 the magnetic potential of the currents is to be 
 
 n 
 
 mm* mgr* 
 sec 
 
 12 has no meaning inside the matter of the hollow sphere, and 
 therefore cannot be analytically continued through it, and is 
 one-valued in external and internal space. 
 We shall denote by 
 
 mm* mgr* 
 
 ilr- 
 
 sec 
 
 the current function in an infinitely thin spherical shell. To 
 avoid all doubt as to sign we give this as the definition of i/r : 
 when on traversing the space ds i/r increases by dyfr, then dty 
 is the quantity of electricity which in unit time crosses the 
 element traversed from left to right. In traversing the path 
 the feet are supposed directed to the centre of the shell, the 
 face forwards in the direction of motion. 
 
 Since in what follows we shall only deal with currents in 
 concentric spherical shells about the origin, we define 
 
 mar* 
 
 im* sec 
 
 more generally as a function of p, 0, co such that 
 
 da . T|r (p=a) 
 
 denotes the current-function of the layer between p = a and 
 
 p = a + da. 
 
 For convenience the units have been given along with the 
 
 magnitudes quoted. 
 Dimen- 3_ L e t the external radius of the spherical shell considered 
 
 sphere° con^ be K, its internal radius r, the angular velocity of rotation w. 
 sidered. 4 when a function ^, which throughout any given space 
 
 men? in" satisfies the equation v 2 % = 0, 1 is developed in a series of spher- 
 
 sP neric al i r In tllc or igi na i A is used where we use v 2 .— Tk.] 
 
 harmonics. L ° 
 
ii INDUCTION IN ROTATING SPHERES d9 
 
 ical harmonics, then % n is to denote the term involving p n as a 
 factor, and this notation is, unless specially excepted, to apply 
 to n negative. 
 
 In the further analysis of % n let this notation be used : — 
 for positive % 
 
 %n = P %Y n> 
 
 for negative n 
 
 n 
 
 Y n = XlA ni cos ia> + B ni sin io>)V n i6) ; 
 o 
 
 for every n these equations hold : — 
 
 The mth differential coefficients with regard to x, y, z of v^ 
 are spherical harmonics of order n — m, unless a preceding 
 one should be of order zero. The expressions 
 
 d Xn d Xn d Xn_ 
 
 dcoj do)y da) z 
 
 are spherical harmonics of order n. 
 Further 
 
 VV Y J = (™ - »X™ + *> + 1K"*Yi 
 
 5. Let -^ be the current-function of a spherical shell of Theorems 
 
 C , i on the flow 
 
 = ^ be the potential of a mass of *? s P herical 
 
 radius E, and let M* = r be the potential of a mass v 
 
 matter distributed over the shell with density -ty ; then the 
 potential of the current sheet is 
 
40 INDUCTION IN ROTATING SPHERES 
 
 and the quantities U. 
 
 V, W are 
 
 U = 
 
 y dV z cV ia^ 
 
 e a7 " I ~cy = E ?^ ' 
 
 V = 
 
 2! a^ a a^ i a^ 
 
 e"S~e¥~e5T; 
 
 w= 
 
 z a^ ?/ a^ i a^ 
 
 E cy E 9a; E dco 2 ' 
 
 If ^ is a homogeneous function of x, y, z of the ?ith degree, 
 
 then 
 
 
 ay 
 
 cT n+ldy 
 
 Cx 
 
 cy E dz ' 
 
 au 
 
 3w w + 1 a^ 
 
 dz 
 
 9# E dy' 
 
 3W 
 
 av Ti + i a^ 
 
 ¥ 
 
 r- " E dx 
 
 And we have always 
 
 
 
 9V au an 
 
 
 a# a?/ ^ ' 
 
 
 au aw an 
 
 
 dz dx dy' 
 
 
 aw av an 
 
 
 cy cz dx 
 
 These formulae are developed in Maxwell's Treatise on 
 
 Electricity, vol. ii. p. 276. 1 The signs in part are different, 
 
 because the system of coordinates there employed is not ours, 
 
 but a symmetrical one. The system used here is that to 
 
 which Helmholtz's formulae apply. 
 
 Formula? 6. The following are the expressions for the electromotive 
 
 electro forces which are produced by the components U, V, W of the 
 
 motive- vector potential, supposed invariable, due to the element whose 
 
 orces ' component velocities are a, /3, y : — 
 
 „ jdv am /au aw\ 
 
 1 2nd ed. p. 280. 
 
II INDUCTION IN ROTATING SPHERES 41 
 
 M /aw av\ /av am 
 
 \ dz ox J \dy dz J 
 
 These are the expressions given by Jochmann. The change 
 produced in the results by the formulae of the dynamical 
 theory will be discussed in § 8. 
 
 If, in addition to the currents u, v, w, there are components 
 of magnetisation X, fju, v, then in this part of the induction in 
 the above formulas we must replace 
 
 TT T7 __ . dM dJX dN dL dL dM. 
 
 U, V, W by — — — . 
 
 dz dy dx dz dy dx 
 
 The formulae thus obtained still hold when the magnets 
 are on the inside of the rotating mass. If the magnets are 
 all outside, since inside the mass 
 
 V 2 L=0, V 2 M = 0, V 2 N = 0, 
 
 we have 
 
 *-<*t- 
 
 d'y 
 
 Hr- 
 
 dz 
 
 z-A- 
 
 dy 
 
 dy 
 
 And for the elements of our sphere 
 
 a = — coy, j3 = cox, 7=0. 
 
 § 2. Solution Neglecting Self-Induction. 
 
 In this paragraph the problem will be solved in the case 
 where the effect of self-induction may be neglected. For the 
 components of current u, v, w these equations hold : — 
 
42 INDUCTION IN ROTATING SPHERES n 
 
 a$ 
 
 oz 
 
 further, since the currents are steady, we have inside 
 
 dv* dv div 
 
 ox dy dz 
 and for p = E and p = r, 
 
 ux + vy + wz = 0. 
 
 Hence we have these conditions for <£ : — 
 inside, 
 
 and at the surface, 
 
 which determine <£, with one additive constant arbitrary. 
 
 The potential of the magnets in internal and external 
 space we assume to be developed in the series of spherical 
 harmonics 
 
 X = 2 M * B 
 
 We take each term by itself and thus put the external 
 potential = % n . 
 Then we have 
 
 l=a>x^ } 
 
 dz 
 
 2 = _ coy^ 1 - ©a$K» = ©^ - tony w 
 
 oy dx oz 
 
 1 Using the unit previously employed for <p. 
 
INDUCTION IN EOTATING SPHERES 
 
 43 
 
 Hence these conditions follow for cj> : — 
 In the matter of the shell 
 
 (a) 
 
 V 2 </> = 2< 
 
 dz 
 
 Determina- 
 tion of the 
 electric 
 potential. 
 
 for p = r and p = E 
 
 A solution of these equations is 
 
 &-&%-* 
 
 For 
 
 2/ „2 
 
 VI /> 
 
 = 
 
 80 
 
 ^ _\ 
 
 + i\ r & 
 
 ^x« 
 
 2(2w+l) 
 
 Pit~- nZ Xn 
 
 r 
 
 d Xn _ 9 JXn 
 
 dz 
 
 dz 
 
 (§1,4) 
 
 2(^+1) 
 
 dXn 
 
 dz 
 
 so that the equation for the interior is satisfied. Further, cf> 
 is the product of p n+1 by a function of the angles 6 and w, 
 whence it follows that </> satisfies the boundary conditions. 
 
 The value of the constant which must be added to the 
 above expression to give the general solution depends in each 
 case on the electrostatic influences to which the shell is 
 subject. We may in any case charge the sphere with so 
 much free electricity as will just make the constant zero, and 
 this we shall in the sequel suppose to have been done. 
 
 From cf> we get at once 
 
 ityn 
 + ldx\ r dz 
 
 nz Xn +x 
 
 dz 
 
 Determina- 
 tion of u, v, 
 w. 
 
 >.-t 
 
 1 
 
 W 
 
 n+ldy 
 
 i a 
 
 u. \p 
 
 n-\- 1 dz 
 
 $Xn 
 
 dz 
 
 $Xn 
 
 nz Xn) +y 
 
 ?Xn\ 
 
 dz 
 
 - nz K 
 
 + z d ^ 
 
 dz 
 
 n Xn 
 
44 INDUCTION IN ROTATING SPHERES n 
 
 If we multiply these equations by x, y, z and add. we get 
 
 ux + vy + wz = . 
 
 Thus the current is everywhere perpendicular to the 
 radius and flows in concentric spherical shells about the 
 origin. This is a consequence of the fact that equation (b) is 
 satisfied throughout the mass and not merely at its surfaces. 
 
 Further we find 
 
 re [ dxdz dxdz j 
 V 2 v = 0, 
 Vhv = 0, since also y 2 ^ % = 0. 
 
 In fact u, v, w are homogeneous functions of x, y, z of the 
 nth degree ; thus u, v, w are exhibited as spherical harmonics 
 of the nth order, We shall soon find simpler expressions 
 for u, v, iv. 
 Determina- Since the currents are similar in concentric spherical 
 function *. shells, they are also similar to those which occur in an 
 infinitely thin spherical shell ; therefore we first consider such 
 a shell and determine the values of the integrals U, V, W for 
 internal space when n is positive, for external space when n is 
 negative. We shall only work through the first case, k we 
 replace by Jc. For U, V, W the conditions hold 
 
 V 2 U=0, V 2 V=0, V 2 W=0 
 
 throughout space. At the surface of the shell 
 
 dp dp 
 
 and corresponding equations for V and W ; and in addition we 
 have the usual conditions of continuity. All these conditions 
 are satisfied by putting 
 
 w *-^+ik\ ^idz[ p ~dz nzx ") +z ^ ™ X A' 
 
 I 1 Vi, U e denote respectively U internal, U external. — Tr.] 
 
ii INDUCTION IN ROTATING SPHERES 45 
 
 From these values of U, V, W we calculate the magnetising 
 forces inside, viz : 
 
 av au 
 
 -5— - 5— > etc. 
 
 we put them 
 
 71 + 1 8^ 
 
 etc., 
 
 K dz 
 and so obtain the function M* (§ 1, 5). We find 
 
 47rE CO d I dXn_^Xn\ _ ^ + * 9 ^ ? 
 
 2n + 1 & 02 \ dy dx ) H dz 
 
 whence 
 
 2n + 1 k dx \ dy dx J E dx l ' 
 
 4?rE ^ L( x d xn_ v hu\ = _ n+ld y 
 
 2n + 1 k dy \ dy J dx J E dy l ' 
 
 1 (2w+l)(n+l) &\ 3y a^ 
 
 4ttE 2 6> , 
 
 (271+1) (71+ 1 fc' 
 
 and now all the remaining properties of the currents follow at 
 once from *P. An arbitrary constant may be added to M*, but 
 is of no importance. 
 
 We thus obtain the solution of our problem for a spherical Summary 
 shell in the following form (§ 1, 5) :— of formula. 
 
 Let 
 
46 INDUCTION IN EOTATING SPHERES n 
 
 be the inducing potential, then 
 
 * = 47rE2 »MV 
 
 • (2w+l)(w+l)A\B/ "' 
 ¥ = 4ttE 2 ^V+iy' 
 
 (271+1) (71+1) A-V/3 
 
 2w.+ 1Z;\K/ " 
 
 fl ^ttEt?. ®/?\»+iy/ 
 
 n> 
 
 (2w+l)(?i+l) &\/> 
 Again, from the relations 
 
 E dco x dp dp 
 
 and the corresponding ones for V and W, we find 
 ldyjr _ 1 co dY' n 
 
 u = 
 
 E8&), n + 1 /j 3^ ' 
 E 9a>„ ?i + 1 h day.. 
 
 n = ^H 1 8Y' 5 
 
 E 8&)- /^ + 1 h da) z 
 
 Lastly, the expression for the electric potential in the 
 material of the spherical shell may be transformed. Write for 
 the moment p' = p sin 6, then 
 
 = - _2_ p tine?** 
 
 and at the surface of the shell 
 
 T n+l d6 
 
ii INDUCTION IN ROTATING SPHERES 47 
 
 Similar calculations may be performed for the case of n 
 negative, where the inducing magnets are inside. They lead 
 to this result : — 
 
 If the inducing potential is 
 
 An = ( ~ / * n , 
 
 then 
 
 47rE L toi p 
 
 % = ~( L Y' 
 
 (2n+l)nJc\Rj n 
 
 _ 4ttE 2 b/EW , 
 c (2n+l)nk\p } 
 
 Qi 
 
 (2rc+l> /j\E/ 
 
 2?i+l k\pj 
 
 CO R„, 
 
 k n 
 
 1 to dY' n 
 
 u= - , 
 
 n k dco x 
 
 1 co dY' 
 
 w = 
 
 n k dco y 
 
 1 co dY' n 
 n k dco„ 
 
 Of the magnitudes here given, ty, u, v, w, c\> are got from 
 their preceding values at once by interchanging n with 
 
 -71-1. 
 
 On the solution obtained I make the following remarks : — 
 1. When a spherical shell of finite thickness rotates under 
 
 the influence of the potential % n (n positive or negative), the 
 
 induced currents are 
 
48 
 
 Construc- 
 tion of the 
 lines of 
 flow. 
 
 INDUCTION IN ROTATING SPHERES 
 
 II 
 
 n + 1 k dco x 
 
 n + 1 k dco z , 
 and the current-function is 
 
 w 
 
 ^ir 
 
 
 Transfor- 
 mation of 
 the solu- 
 tion. 
 
 71+1 /C 
 
 2. We analyse ^ further and consider the term 
 
 Xm = A r 
 
 To this belongs the current-function 
 
 p coi 
 
 !- ) cos iw¥. 
 
 ^ni 
 
 Mb 
 
 sin icoV ni . 
 
 Hence we get the following simple construction for the lines 
 of flow due to such a simple potential : — 
 
 Construct on any spherical sheet the equipotential lines 
 and turn the sheet through an angle irj2i ; the lines now 
 represent the lines of flow produced by that potential. 
 
 For instance, when the sphere is rotating under the action 
 of a constant force perpendicular to the axis of rotation, the 
 external potential satisfies the required conditions and we 
 have n = 1, i = 1. The equipotential lines on the sphere are 
 circles, and so also are the lines of flow. The planes of the 
 former are parallel to the axis of rotation and perpendicular to 
 the direction of the force, so that the planes of the latter are 
 parallel both to the axis of rotation and the direction of the 
 force. 
 
 3. We may give to -v^ a form which permits of summation 
 for all the spherical harmonics and makes the development of 
 the external potential in a series of them unnecessary. 
 
 Let n be positive, then 
 p 
 
 P 
 
 Xn d P = 
 
 71+1 
 
 %n 
 
ii INDUCTION IN ROTATING SPHERES 49 
 
 Secondly, let n be negative, then 
 
 p 
 
 Xn d P= -%>• 
 
 lb 
 
 Hence, for a positive n, 
 
 and for a negative n, 
 
 K / Oft) 
 
 /£ Oft) 
 
 These expressions admit of summation at once, and we get Summation 
 
 of spherical 
 harmonics. 
 
 the following second form for the solution : — 
 
 If ^ denote the part of the potential due to internal, and 
 % e the part due to external magnets, then 
 
 &) 
 
 = ~«l{\ x °- dp - 
 
 o 
 Similarly 
 
 </> = - ft) sin 6 
 
 del 
 
 p CO 
 
 ixe- d P~ \Xi- d Pj- 
 
 For an infinitely thin spherical shell of radius E 
 
 ylr 
 
 fc[jdco J dco J 
 
 R 
 
 R 
 
 ♦ --..in*ffe*-(2**l 
 
 { W^ w H f 
 
 Hence we get this relation between <£ and ^ 
 
 j¥ + «.dn*g-o. 
 
 dco d0 
 
 M. P. E 
 
50 INDUCTION IN ROTATING SPHERES 
 
 S 3. Complete Solution for Infinitely thin Spherical 
 
 Shells. 
 
 We shall now take into account the effect of self-induction, 
 but in this paragraph we shall confine our considerations to 
 the case of infinitely thin shells. For simplicity n will be 
 supposed positive in the calculations. 
 
 In accordance with usual views we regard the total in- 
 duction as compounded of an infinite series of separate 
 inductions ; the current induced by the external magnets 
 induces a second system of currents, this a third, and so on 
 ad infinitum. We calculate all these currents and add them 
 together to form a series which, so long as its sum converges 
 to a finite limit, certainly represents the current actually 
 produced. 
 
 Let 
 
 represent a part of the external potential. The potential in- 
 duced by this part is 
 
 2n+l k \RJ n 
 
 ■ i2n+lXn+l)k\p) 
 
 Calculation In the first place, if inside the spherical shell a second 
 of the sue- ro tate infinitely close to the first and with the same velocity, 
 
 cessive in- ^ J 
 
 auctions, the currents of the first order (Q { ) will induce in it a current 
 system whose internal magnetic potential is 
 
 Secondly, if outside the first shell another rotate with the 
 same velocity and infinitely close to the first, the currents of 
 the first order (I2 e ) will induce in it a current system whose 
 potential inside is 
 
INDUCTION IN ROTATING SPHERES 51 
 
 ~, _ Air'Rn co 47rE(> + 1) col p V*,, 
 
 " (2tt+l)(w+l) & ' (2rc+l> & \E/ " 
 
 £Yy'' 
 
 2^+1 &/ \E 
 
 The two expressions for O^ are the same. Hence this is 
 the potential of the current system, which is induced in the 
 spherical shell itself by the currents of the first order. If in 
 the same way we calculate the succeeding inductions and add 
 them together, we get for the whole inductive action 
 
 2 1 "SlTTWUl 
 
 n 
 
 i -*> 
 
 
 n+l\pj ^J \ (2n+l)kJ d<o m ' 
 .2n+l ^ / _ 47rKa, W 
 T 4tt(> + 1)^\ (2» + 1)7;/ 9*> m " 
 
 The expressions obtained may be developed still further by 
 analysing Y n further. We have 
 
 n 
 
 Y n = ^i(Ki cos ico + B ni sin ico)? ni . 
 
 i 
 
 We confine ourselves to one term only of this series. 
 Thus let 
 
 Y w = A ni cos icoT ni . 
 Then we have 
 
 n 4 = A J £ P ni J — — -— siniffl - ( T 
 
 E/ ni l(2n+l)Z; \(2w+l)* 
 
 47rEo)t \ 3 . . , / 47rEo)^ V . ) 
 
 smico + cos ito + • • • r- 
 
 (2rc+ 1X7 \ (271+1)*/ J 
 
 Put for shortness 
 47rEa)^ 
 
 (2^ + iyj 
 
 then we find 
 
 7& (h is a pure number) ; 
 
 fl. = A^f £ ) P ni (sin io) - ft cos ia>).7i.(l—h 2 + ft 4 - 7^ 6 + . . .)■ 
 
Soluti 
 
 52 INDUCTION IX ROTATING SPHERES n 
 
 If h is a proper fraction, the series involved in O^ con- 
 verges and we get 
 
 n i =A »\Q i+// sin ico ~ h cos i(o ^ ni ' 
 
 n+1 '"\pj 1+h* 
 
 J2 C = — - — - A ni ( — ) — — - 2 (sin ico — h cos ico)V ni , 
 
 2n-\-l h , . . , . ._ 
 yjr= — A • sm ico — h COS lay)r ni . 
 
 If 7t>l, 1 the series occurring in 12 ,- diverges and it is no 
 longer allowable to regard the phenomenon as a series of suc- 
 cessive inductions, since each one would be larger than the 
 preceding one. 
 
 Nevertheless the formulas given hold for every value of h, 
 as we may easily verify a posteriori, and deduce by the same 
 considerations that we shall have to employ in the case of 
 spherical shells of finite thickness. Since I propose again to 
 deduce the above formulae from the general ones. I shall not 
 now consider them further. 
 
 We write 
 
 h = tan 8, 
 and now 
 
 o ; = A,/£y'sinSsin(/o)-8)P,, ; 
 
 _Q ( = - _ * A J - ) ■ sin $ sin {ico - B)T ni , 
 n+1 \pj 
 
 ir = ~ ~f , n A «i sin $ sin (ico - $)P ni . 
 
 4:7T()l+ 1) 
 
 Hence the result is as follows : — 
 
 Construe- 1. A simple spherical harmonic in the inducing potential 
 
 iines°of 6 induces a current-function which is a surface harmonic of its 
 
 flow. own type. Hence we may here also retain the construction 
 
 previously given (§ 2, 2) for the lines of flow, but we must 
 
 suppose the spherical layer considered to be turned through a 
 
 certain angle B/i in the direction of the rotation relative to the 
 
 1 A copper spherical shell of 50 mm. radius and 2 mm. thickness would 
 have to make about 87 revolutions per second in order that for t=l, »=1, h 
 might be equal to 1. [? 62 revolutions per second. — Tr.] 
 
II INDUCTION IN ROTATING SPHERES 53 
 
 position previously determined. For small velocities of rota- 
 tion this angle is proportional to the velocity ; for large ones 
 it approaches to the limit irj2i. The intensity, which at first 
 increases in proportion to the velocity of rotation, increases 
 more slowly for larger values and approaches a fixed limit. 
 
 2. Finally, when co/k = oo, 8 = ir/2, and then The velo- 
 
 city is in- 
 finite 
 
 a = 
 
 a = 
 
 n 
 
 n+1 
 
 Y 4ttO + 1) 
 
 This result does not hold for those terms of the develop- 
 ment which are symmetrical about the axis of rotation. For 
 these i, and therefore also h and 12, vanish for every velocity 
 of rotation. These terms produce no currents, but merely a 
 distribution of free electricity in the sphere. 
 
 Hence a spherical shell, rotating with infinite velocity, 
 only allows those portions of the external potential which are 
 symmetrical about the axis to produce an effect in its interior. 
 If such terms are absent, the interior of the shell is com- 
 pletely screened from outside influences. If the potential is a 
 spherical harmonic, the current flows along the equipotential 
 lines. 
 
 3. We found, neglecting self-induction, the following Tneelec tnc 
 
 . . ,. potential. 
 
 expression lor the electric potential corresponding to % n 
 
 £= --^-Esin^. 
 ^ n+1 dQ 
 
 Taking self-induction into account, we shall have 
 
 d> = K sin 0-^ ■ . 
 
 Y n+1 de 
 
 Hence it follows that the form of the equipotential lines (for 
 each inducing spherical harmonic) is unaltered by self-induction, 
 but these lines are turned through the same angle as the lines 
 of flow. For the parts of the external potential which are 
 
54 
 
 INDUCTION IN ROTATING SPHERES 
 
 II 
 
 symmetrical about the axis, <fi increases indefinitely as the 
 velocity increases ; for the other parts it approaches a finite 
 limit which is easily calculated. 
 
 Limiting Forms of the Spherical Shell. 
 
 Plane plate, 
 
 Plates 
 moving in 
 a straight 
 line. 
 
 "We now make the radius of the spherical shell infinite, 
 but keep the variations of the inducing potential finite, and 
 then we examine more closely the electric currents at the 
 equator and at the pole. We thus obtain the theory of plane 
 plates, both rotating and moving in a straight line. The 
 latter may be considered as a special case of the former, but 
 
 for several reasons it is advisable to treat 
 
 these cases separately. 
 
 A. Plates moving in a Straight Line. 
 
 We introduce the co-ordinates £, 77, f, 
 whose connection with x, y, z is shown 
 in Fig. 7. 
 
 The direction of 77 is the positive 
 direction of motion. We shall suppose 
 the inducing magnets inside the sphere, 
 i.e. on the side of f negative. We must examine what form 
 in f, 77, f is assumed by the spherical harmonic 
 
 'R\" +1 . p 
 
 cos ico . r ni . 
 
 In order to obtain finite variations we must make n and 
 i 00 of order E, We put 
 
 for n, nR, 
 
 for i. rR. 
 
 Fig. 
 
 and further replace 
 
 by 
 
 <o, e 
 
 
ii INDUCTION IN ROTATING SPHERES 55 
 
 Thus 
 
 become 
 
 P 
 
 I , COS %CO 
 
 ~ ni =, cos rrj. 
 
 Y ni (6) must become such a function of £ that its product 
 by e~^cosr?7 may satisfy the equation V 2 = 0. Such a 
 function is cos s% or sin s%, provided 
 
 9 9 i 9 
 
 n z = r z + s . 
 
 Hence the spherical harmonics formerly used now take the 
 form 
 
 A rs €~ n ^ cos rr) cos sf , 
 
 and related forms. 
 
 The external potential ^ must be represented as a series 
 of such forms. This is to be done by means of Fourier's 
 integrals. 
 
 For every term (element) of the development the solution 
 is at once obtained from those found before. We now put 
 
 Z7TT a 
 
 tan 8 = 
 
 n k 
 
 where a denotes the velocity of the plate, and find 
 Xl + = A rs e — ?lf sin 8 sin (rrj - 8) cos s%, 
 I2_ = - A r /^ sin 8 sin (rrj - 8) cos s£, Solution. 
 
 ty = — A rs sin 8 sin (rrj - 8) cos s J . 
 
 27T 
 
 By summation of all the terms we obtain the general 
 solution of the problem. The summation may be performed 
 in the case when a/k becomes infinite. Then we have 
 
 8 =— , sin 8= 1, 
 
 2 
 
 therefore 
 
56 INDUCTION IN ROTATING SPHERES n 
 
 On the opposite side to the magnets the potential is 
 zero ; the currents flow everywhere along the equipotential 
 lines of the inducing distribution. 
 
 Apart from this limiting case the application of the above 
 solution is very cumbersome ; we therefore seek approximate 
 methods. In the first place we find such methods by intro- 
 ducing successive inductions. That this method may be 
 permissible it is necessary that 2irajk be a proper fraction ; if 
 this condition be satisfied the calculation leads to a convergent 
 series, as we have already shown in the general case. 
 
 We again start from the infinite spherical shell. The 
 inducing potential X-n-i produced in the space outside the 
 induced potential 
 
 a= - 
 
 47rR CO 9%_ n 
 
 2n + 1 h dco 
 We allow R to become infinite while we replace 
 
 - ty E a i , 
 
 dco drj 
 
 n by TiR , 
 
 o)R by a , 
 
 X-n-ity % ?l = A/-^cosr77Coss£ 5 
 then 
 
 _ 2ira 1 d Xn 
 k n or) 
 But we have 
 
 / 
 
 n 
 
 Hence summing for all values of n 
 
 CO 
 
 Now from this potential we can get in the same way the 
 potential of the second order ; and proceeding in this way we 
 obtain finally 
 
INDUCTION IN ROTATING SPHERES 57 
 
 This series leads to as accurate a result as we please, if 
 only it be carried far enough ; in fact it is only the develop- 
 ment of the result in ascending powers of 2iraJK J as we may 
 show in the following way. 
 
 In the spherical shell the part of D. e corresponding to 
 X(- n -i)i mav De represented in the form (p. 52) 
 
 1 /A3v 
 
 form of the 
 solution. 
 
 h 
 
 \-h 2 \i dco 
 
 If we again introduce the substitutions to be made in the 
 case of a plane plate, develop 
 
 1 
 
 and put for h its value 
 we set 
 
 l-h 2 + h 4 -h Q +... 
 
 2 it a r 
 
 k 7i 
 
 n = _ 2™ 1 dxrs _ / W\ V , / WV 8 jf ?& 
 + k n d v \ k ) ™ 2%rs \ k ) n 3 d v 
 
 k J n' Xrs "" 
 
 from which development the preceding one follows when we 
 use the relations 
 
 CO 
 
 and sum for all values of r and s. 
 
58 INDUCTION IN EOTATING SPHERES n 
 
 In this connection it is natural to seek a development 
 in descending powers of 2iraJK for very large values of this 
 quantity. 
 
 When h>l, we have 
 
 1 U l-W 
 
 1+A 2 lr\ h 2 A 4 
 hence 
 
 O 
 
 nd XrsJ t 
 
 h Yn 
 
 lira r Or) \ Ztt a ) r 
 
 / 7 
 
 \2iraj r~ Oy 
 
 It is true that the terms of this equation cannot, as is 
 shown by trial, be arranged in such a way as to at once permit 
 of summation for all % rs ; but if we suppose ^ to be sym- 
 metrical with respect to the -q-axis, so that in its development 
 only terms in cos ttj appear, we have 
 
 o 
 and then the summation can be performed, at any rate for the 
 terms of the first order in KJ2ira. If w T e confine ourselves to 
 these we get 
 
 s 
 
 Approxi- + = - X ~ „ \^~ dy > 
 
 mate solu- Lira J GC, 
 
 tion for 
 
 l^tr^G VtlltIGS 
 
 of the velo- and the very small resultant potential on the positive side is 
 city. , 
 
 k fd x 
 
 lira] d£ 
 
 
 But in addition to the condition mentioned, this equation 
 is subject to another one. 
 
 However large 27ra//c may be, yet for certain elements for 
 which r vanishes, h<l, and the development employed will 
 be invalid. This circumstance restricts the validity of the 
 expression obtained to a limited region, which however is 
 larger the greater 2 r 7ra//c. On this point I refer to an investi- 
 gation to follow immediately (p. 61). 
 
ii INDUCTION IN ROTATING SPHERES 59 
 
 We determine also the potential <£ of the free electricity. Potential of 
 This follows from the expression for the spherical shell by electricity, 
 means of the very same substitutions that we have used all 
 along. We thus find : — 
 
 1. Neglecting self-induction, 
 
 
 
 2. Taking it into account 
 
 00 
 
 ♦-.jags* 
 
 The case is of interest when the velocity a becomes infinite. 
 If we assume % to be symmetrical with respect to the ?7-axis, 
 and restrict our considerations to a limited region, we have 
 for a = oo 
 
 Jo fd X 
 
 ^ + X = -o \& dr > 
 
 'lira 
 
 *t 
 
 and hence 
 
 ») 
 
 ♦-**/*$** 
 
 o o 
 ■o — 
 
 2ttJ d% 
 
 
 
 Thus cf> approaches a definite finite limit when the velocity 
 increases. 
 
 B. Rotating Discs. 
 
 We next consider the neighbourhood of the pole, and thus Rotating 
 obtain the theory of an infinite rotating disc, We again dlscs - 
 suppose the inducing magnets to be inside the sphere. The 
 propositions we must employ are quite analogous to those of 
 the previous case. 
 
60 
 
 INDUCTION IN ROTATING SPHERES 
 
 We use p, co, z as co-ordinates, where p now denotes the 
 perpendicular distance from the axis of rotation. In the 
 general formulae we must replace 
 
 p by R + z , 
 
 6 by E . 
 J E 
 
 co remains co , 
 
 and after making this substitution we must allow E to become 
 infinite. Then a simple spherical harmonic takes the form 
 
 A^e-'^ cos icoJ^np), 
 
 (and analogous ones), where J { denotes the Bessel's function 
 of order i. The given % is to be analysed into terms of this 
 form by means of integrals analogous to those of Fourier. 
 
 "We treat each term separately. 
 
 If we put 
 
 Solution. 
 
 tan 8 = 
 
 .77 co i 
 
 h n 
 
 then for the term in question the solution of the problem is 
 Q + = A l{[ c- nz sin 8 sin (ico - 8) J-- (np) , 
 Q_ = - A ni e nz sin 8 sin (ico - 8) J { (np) , 
 1 
 
 + 
 
 A,,- sin 8 sin (ico - 8) J; (np) . 
 
 By summation we get the complete integrals. 
 
 We again attempt to obtain a development in powers of 
 \Litco k by considering the successive inductions. By the 
 same method as above we get 
 
 n 
 
 c X,n 
 
 2ir& fdx 
 
 k \cco 
 
 dz + 
 
 '.TTCO 
 
 h 
 
 "X cw 
 
 ceo 
 
 Second 
 form of the 
 solution. 
 
 i 
 
 
11 INDUCTION IN ROTATING SPHERES 61 
 
 But there is a limitation to the validity of these formula?, Remarks 
 which we had not to impose on the previous analogous ones. fo rnK Ub 
 For their deduction presupposes that for each separate term 
 of the development of % it is allowable to regard the total 
 induction as the sum of a series of successive inductions. 
 According to the results which we obtained for spheres this 
 rendition is only satisfied for those terms for which 27rcoi/kn 
 is a proper fraction. Now n may have any value from zero 
 to infinity ; thus for a number of terms the necessary con- 
 dition is not satisfied, and the result can therefore only be 
 approximate. With reference to this point I remark : — 
 
 1. At a finite distance the terms for which n is very 
 small vanish relatively to those for which n is finite. The 
 error committed in the above formula must have an appreciable 
 value for large values of p. 
 
 2. The quantity Zirw/k may always be chosen so small 
 that the approximation may be any desired one within a 
 given region. For by diminishing 27rco/k we diminish the 
 number of those terms which do not satisfy the required con- 
 dition : a suitable diminution diminishes their number in any 
 desired degree. 
 
 There may possibly be difficulties in determining exactly 
 the region of validity for a given value of 2irwjk and a given 
 degree of approximation. For practical applications this de- 
 termination is of no importance : because, in the first place, 
 we are only concerned here with very small values of 27T(o/k ; 
 and, in the second place, we are only considering plates of 
 limited dimensions, and not infinite plates. 
 
 The equation M 
 
 a- -^(^dz 
 
 dco 
 
 is exact, apart from self-induction. We see that, in order Possibility 
 that we may be allowed to neglect self-induction, it is neces- j^*-^" 
 sary not only that 2ir(ojk be small, but also that the investi- induction, 
 gation be limited to a certain finite region. The extent of 
 this region depends on 2ir(a/k ; beyond it not even an approxi- 
 mate determination of the current is possible without taking 
 self-induction into account. We shall meet with an exactly 
 analogous result at the end of S 4. 
 
62 INDUCTION IN ROTATING SPHERES II 
 
 Approxi- A development can also be given for large values of 27rco/k. 
 
 arge values We denote by ^ that part of ^ which is symmetrical about 
 
 of the velo- the axis of rotation, by %\ = X — Xo ^ ne remainder. To ^ 
 corresponds for every velocity of rotation the value 11 = 0. 
 Hence, assuming % to be symmetrical about the ^-axis, we get 
 for large values of 27ra)jk 
 
 
 27rft)J dz 
 
 
 
 The formula is deduced in the same way as above. The 
 series may also be completely developed ; and this too for 
 forms of x which are not symmetrical with respect to the 
 a?-axis. I shall not here enter into further detail on the 
 point. 
 
 In conclusion let us determine <j>, the potential of the free 
 electricity. By the proper substitutions we get from the 
 general formulas : — 
 Potentialof 1. Neglecting self-induction, 
 
 the free 
 
 electricity. *? 
 
 
 
 We must add to this value of ^> a constant, whose value 
 is such as to make <f> vanish at infinity. The formula which 
 we have found has already been given by Jochmann for the 
 case in which % is symmetrical with respect to the axis of x. 
 It is seen to be generally true. 
 
 2. Taking self-induction into account we have 
 
 00 
 
 
 
 When w is very large, we find, if % is symmetrical with 
 respect to the x-axis, 
 
 
 
 The first term increases indefinitely with co. 
 
ii INDUCTION IN ROTATING SPHERES 63 
 
 We have in the treatment of plane plates all along assumed 
 that inducing magnets exist only on one side of the plate. 
 This assumption is unnecessary. If it is not true, we divide 
 the total potential into two parts according to its origin, and 
 treat each part separately, as we have shown above for one of 
 the parts. 
 
 § 4. Complete Solution for Spheres and Spherical 
 Shells of Finite Thickness. 
 
 We now turn to the consideration of the induction in a 
 spherical shell of finite thickness. To avoid complication we 
 shall at first suppose inducing magnets to exist only outside 
 the shell. 
 
 Let U, V, W be the components of a vector potential due 
 to closed currents, wholly or partly inside the shell. The 
 currents u f , v f , w' induced by U, V, W are given by the 
 
 equations 
 
 Kid = 
 
 d(j> 
 
 dx 
 
 KV' = 
 
 d(j) 
 dy 
 
 KW l = 
 
 d<f> 
 
 ~dz 
 
 \dx dy) 
 \dx dy)' 
 
 COx(^ - dY ) - C07 (^ - dW 
 
 \dy dz ) \dz dx 
 
 Differential 
 equations. 
 
 Further, inside we have 
 
 ^ + — + — =0 
 
 dx dy dz 
 
 and at the surface 
 
 
 
 v!x + v r y + w r z = . 
 
 
 
 We write for shortness 
 
 
 
 = x( d ^- dY ) + y( d ^- d ^) + z { 
 \dy dz) \dz dx) 
 
 (dV 
 \dx ' 
 
 _axj 
 
 " dy 
 
64 INDUCTION IX ROTATING SPHERES 
 
 If we remember that 
 
 ex aj dz 
 
 we get for </> these conditions : — 
 In the material of the shell 
 
 V 2 6 = 2 co — - ?5 ) + oA V 2 Y - ?/V 2 U) : 
 9a; < 
 
 and at the boundaries 
 
 P 
 
 
 Theorem We shall first demonstrate the following theorem : — 
 
 forms the ^ ^7, V, W have the forms 
 
 v=p-%-^-y 
 
 w-W«&s-»^- 
 
 which forms satisfy the equation 
 
 C SB C^ c Z 
 
 then the solutions of the preceding differential equations are 
 
 ? X 
 
 
 cop— 1 sin 6 % . 
 
 -"piM-^X 
 k vz cy 
 
 K \ ox dz 
 
ii INDUCTION IN KOTATING SPHEEES 65 
 
 k \ dy ox 
 
 To verify this we first express the conditions for </> in Proof, 
 terms of Xn . We have (§ 1, 4) 
 
 V 2 U = m(m+ 2n + l)o m " 2 ( y^ - z ?&) , 
 
 \ dz dy J 
 
 V 2 V = m(m + 2n + l)p"" 8 ( « -^ - a? ?** ) . 
 
 r \ a« dz J 
 
 V 2 W = m(m+2n+l)p m - 2 ( x d ^-y^ 
 
 \ dy dx 
 
 yV 2 U - xV 2 Y = m(m + 2n+l)p m - 2 (p 2 ^2 - w% 
 
 ^V 2 V - 7/V 2 W = m(m + 2w + l)p m ~ 2 (p 2 ^ - nx Xn 
 
 \ dx 
 
 xV 2 W - zV 2 V = m(m + 2™ + l)/o m_2 ( p 2 ^ - ny Xn 
 And again — 
 
 Hence we get 
 
 o=-»(» + iV"x»- 
 
 The conditions for (/> become 
 
 v > = - »m(m + 2n + 3/p 2 ^ - w% V" 2 
 
 -2co(n+l)p md ^, 
 
 dz 
 
 M. P. F 
 
tions. 
 
 66 INDUCTION IX ROTATING SPHERES 
 
 and at the boundary 
 
 cp p \ dz 
 
 Xow (j) satisfies these conditions. For we have, firstly, 
 
 = - Jp m §p {(?/z + 2)(wi + 2?i+ 1) - 2w} 
 
 -^ m " 2 X»{<m+2%+l) + 2m} 
 
 = - Jm(™ + 2n + 3)p— "(/»%- «%>)+ 2(ti + 1> §& ' 
 
 so that the first condition is satisfied ; secondly, <p is the pro- 
 duct of p w+n+1 by a function of the angles, thus 
 
 d4> m + W + 1 
 
 so that the second condition also is satisfied. From this 
 correct value of eft the values of u' , r f , v:' follow by the 
 original differential equations, at first in a more complicated 
 form. But the same form has already occurred on p. 43, 
 and has already been shown to be identical with the one given 
 above. 
 Deduc- This theorem leads to the following propositions : — 
 
 1. In the theorem we may replace p m by a series of 
 powers of p, each power multiplied by an arbitrary constant, 
 that is, for p m we may substitute any arbitrary function of p. 
 And again, we may replace ^ n by a series of spherical har- 
 monics of different degrees with arbitrary coefficients ; for n is 
 without effect on the final result. Hence we get the following 
 generalisation of the theorem : — 
 
 If % is an arbitrary function, and if 
 
 
 ceo 
 
 
 V 
 
 
 w 
 
 COO; 
 
 
 then the cur 
 
 rents u f , i J , 
 
 w 
 
 induced by 
 
 u, 
 
 V, w 
 
 are 
 
 u' = 
 
 K CCO, 
 
 <v' 
 
 = 
 
 K CtOy 
 
 % 
 
 '= _ 
 
 CO CJ^ 
 
 k dco z 
 
II INDUCTION IN ROTATING SPHERES 67 
 
 It is not difficult to see the connection between this 
 theorem and the results obtained in preceding paragraphs. 
 
 2. The U, V, W which are of the above form are due 
 to currents in concentric spherical shells. For we have 
 
 xV 2 \J + ijV 2 Y + zV 2 W=Q. 
 
 And vice verscl the U, V, W of such currents may always 
 be expressed in the above form. For if % n f (p) is the term 
 involving the nth spherical harmonic in the development of 
 the current function, then the U, V, W belonging to this 
 term are at once seen to have the above form. 
 
 On the other hand, the induced currents also flow in con- 
 centric spherical shells. For we have 
 
 ux+vy+ivz = 0. 
 
 Hence we deduce the following conclusion : — The flow is 
 
 always in 
 concentric 
 
 A current which flows in concentric spherical shells in- 
 
 duces a current system possessing the same property. Further- spherical 
 more, the currents which are induced by magnets at rest in a shells * 
 rotating spherical shell always flow in concentric spherical 
 shells about the origin. 
 3. We find that 
 
 cj> = co(xY-yV) 
 
 whenever U, V, W have the above form, and the inducing 
 currents the property discussed. This we shall have to make 
 use of in § 8. 
 
 There is now no further difficulty in calculating the Calculation 
 successive inductions produced by a given external potential. °eJ^ e suc " 
 Let % n denote the nth term in its development. We found for inductions. 
 the currents of the first induction 
 
 1 (o dy f 1 o) dy f 1 a) dy f 
 
 u x = -^- } v 1 = ^- ft>! = ^ . 
 
 n-\-l kcco x n+1 k d(o y ' n-\-l/cd(o z 
 
 The corresponding values of U, V, W are 
 
 2 2r 2n+3 
 
 U _ 2tt cod x '/ B 2 I 
 
solution. 
 
 68 INDUCTION IN ROTATING SPHERES 
 
 v 2tt co a % 7 E 2 I 2r^ \ 
 
 1 n+lKdG> y \2n+l 2n + 3 (2rc + l)(2rc + 3)p 2n+l ) 
 
 Wi = _2tt o,a % 7 E 2 p 2 2r 2 *+ 3 
 
 Ti+1 kSo> \2^+1 2^+3 (2n+lX2n+3)p 2n+1 
 
 These values are got by a simple integration ; for u, v, w are 
 products of p n by spherical surface harmonics. The potential 
 of each infinitely thin layer is known inside and outside it, 
 and an integration with respect to p leads to the given values. 
 Hence follow the currents of the second induction 
 
 2tt /q>yyy r 2 p 2 2^+ 3 
 
 V = 
 
 n+l\/cj da) x \2n+l 2n+3 (2n+lX2n+3)p 2n+1 
 
 27r?/aAW7 E 2 o 2 2r 2n+s 
 
 n + l\tc/ dco y \27i+l 2n+3 (2n+lX2n+3)p 2n+1 
 2tt /gAWV E 2 a 2 2r 2n+ * 
 
 W = ~ 
 
 X * P 
 
 2 n+l\icj d<» z \2n+l 2n+3 (2n+l)(2n+3)p 2n+1 
 
 In this way the calculation may be continued as far as 
 may be desired, but the results continually increase in compli- 
 cation ; hence we now proceed to the exact solution of the 
 problem. 
 General We have seen that the currents are always perpendicular 
 
 to the radius, and may therefore again make use of the 
 current-function. 
 
 Let / (p) = / be any function of p whatever, and let 
 
 ^ = P-f-Xn 
 be the current-function of a system of currents flowing in the 
 sphere. 
 
 The current-densities are 
 
 *^f%, v-f%, «-/<&. 
 
 0(O x VCD y <JCO z 
 
 Let F (p) = F be a second function of p, which is given in 
 terms of/ by the equation 
 
 p R 
 
 4tt 1 
 
 FW 
 
 2n+l p 
 
 2/i + l 
 
 a 2n+2 f(a)da+ I p 2n+l af(a)da 
 
ii INDUCTION IN ROTATING SPHERES 69 
 
 From this we get, by differentiating, 
 
 The values of U, V, W corresponding to u, v, w are 
 
 TJ = -pj^ } y = F-2^ , W = F^- 71 . 
 d co x do)y dco z 
 
 Hence follow these currents induced by the system -ty 
 
 yj = - -F-J^ , v' = - -F^^ , w ' = - -F-^ 1 . 
 
 K V(O r K 0(0 K 0(0 
 
 d, y z 
 
 The disturbing function belonging to this system is 
 
 * = - -p*& 
 
 CO 
 
 ~l 
 
 K 
 
 Hence the function 
 induces the second function 
 
 K 
 
 Now let ijr belong to the current-system actually existing 
 in the sphere under the influence of the external potential % n ; 
 let ^ belong to the current-system directly induced by ex- 
 ternal magnets. Then clearly the condition for the stationary 
 state is 
 
 ^ = ^o + i r '- 
 
 To develop this equation further we analyse % n and con- 
 sider each term by itself. Let the one considered be 
 
 Xni = A (Q cosico.T ni . 
 We have then (p. 48) 
 
 to = ' -M £ ) "4r Sin *® • ?ni- 
 
 k \RJ n + 1 
 If we write 
 
 yfr= -"Apt?-) -?— {f^p) sin i(o+f 2 (p) cos ia)}? ni> 
 k \ K/ n + 1 
 
INDUCTION IN ROTATING SPHERES n 
 
 we get 
 
 / p \ n 
 
 f'=[- K ) Ap(J j ^j{ F^) cos m. - F,(p) sin i<»}P n{ . 
 
 The equation -fy = yjr + yjr f is satisfied if f Y and / 2 satisfy 
 the equations 
 
 / 1 (p) = i+^r ( P ) ; 
 
 K 
 
 by which / x and / are completely determined. 
 
 If we regard f Y and / as known, the result of the investi- 
 gation may be expressed in the following form : — 
 
 Self-induction leaves the form of the lines of flow un- 
 altered (for each separate term of the development). Its effect 
 is : — 
 
 Firstly, to turn the system in the direction of rotation 
 through an angle 8/i, where the angle 8 is different for different 
 layers and is given by the equation tan 8 =f 2 /f r 
 
 Secondly, to change the intensity of the current differently 
 in different layers. The ratio of the intensity actually 
 occurring to that found without taking account of self-indue- 
 
 tion is x//i 2 +/ 2 2 :l. 
 
 We shall have to occupy ourselves for some time with the 
 determination of the functions f 1 and f 2 . 
 Treatment We introduce the following contractions. Let 
 
 of the equa- 
 tions for . 
 the/'s. 47n&) 2 
 
 = p, 
 
 K 
 
 /XT = S, fJbR = S, 
 
 /i(/0 = <£iO/>) = 0i°"> 
 
 /2W = faipp) = 02 "- 
 
 In the equations which give f x and/ 2 we P 11 ^ for I\ and 
 F 2 their values, transform the equations so as to involve <f> 
 and o\ and thus obtain 
 
INDUCTION IN ROTATING SPHERES 71 
 
 s 
 f C ( 
 
 0!<7= 1 
 
 (^ZTi-h L)<T"™^\ J 
 
 o- S 
 
 09°" = 
 
 cr S 
 
 Ip^j J ah+»^i . da + (o* n+1 a<t> 2 a .da\, 
 
 l-j—j |a*H-^ fl . da+L^a^a. dcX 
 
 These give by differentiation 
 
 da\ dc 
 
 The form of the functions </> l5 </> 2 depends only on w; in 
 the constant of integration p, s, S are involved. 
 The above equations may be written 
 
 „ 2n+2 , 
 
 9 i + 9 i = - 92 > 
 
 „ , 2ri+2 
 
 9 2 + </> 9 = 9i- 
 
 As differential equations these are exactly equivalent to 
 the following : — 
 
 c 
 
 </> 2 =±<W-i, 
 
 For all solutions of the latter system satisfy the former, 
 and the general solution of the latter involves 2x2 arbitrary 
 constants, and is therefore also the general solution of the 
 former system. 
 
 Let us put \ i = — 1, where X is that root whose real part 
 is positive, so that 
 
 \= «/f.(i+«/^T), 
 \=VJ.(l-<v/^T). 
 
72 INDUCTION IN ROTATING SPHERES n 
 
 Then our equations become 
 
 ar 
 
 The two particular integrals are 
 
 + 1 co 
 
 1(1- v 2 ) n e* Kv dv , 1(1- *; 2 ) w e " ^dv , 
 
 -l i 
 
 which hold for real positive values of a. 
 
 We shall prove farther on that these integrals satisfy the 
 equations. Since in our case n is a whole number the integra- 
 tions can be performed, and the solution expressed in a finite 
 form ; but for simplicity we may retain the integral form. 
 Let us write 
 
 PM= f(l -v 2 ) n e* v dv, 
 
 Definition 
 of p and q. 
 
 q,M= |(l-^)»e— «fo, 
 
 CO 
 
 f(l - J 
 
 then clearly the solutions of the differential equations are — 
 
 These solutions must be substituted in the integral equa- 
 tions so that the constants may be determined. The following 
 formulas will serve to evaluate the integrals which occur : — 
 
 +i 
 
 P n fr<r)=((l-v 2 Te^dv, 
 
 -l 
 +i 
 
 ln j-_{cr 2n+l l\)= [(1 -v 2 y\a\v+ 2n+iy° v dv 
 
 da 
 
 -i 
 
 i 
 
 1 The last members of the equations are got by transforming the preceding 
 integrals, especially by integration by parts. 
 
d 
 
 da 
 
 2 ' l U°- n+1 Pn) 
 
 INDUCTION IN ROTATING SPHERES 73 
 
 +1 
 X | (1 - v 2 )\a\v 2 + 2{n + l)v)e K,TV dv 
 
 = X 2 ap n (\a)} 
 
 The last equation shows that p n is a solution of the 
 equation under discussion. 
 
 The preceding equations when integrated give 
 
 2n 
 
 ap n (\a)da = —p n _ 1 (\ ( r) , Integral 
 
 A and recurr- 
 
 and in s for ; 
 
 raiilpe for 
 
 / 
 
 2n 
 
 23 and q. 
 
 whence by differentiating we get these recurring formulae 
 
 2np f n _ 1 (Xcr) 
 
 Pn(M) = 
 
 A (T 
 
 2% Z71 
 
 whence 
 
 _ 2ti+1 AV 
 
 2ll 4%(?i+l) 
 
 2 
 
 ^ = T2~2{ 2 ^- 1K-2-K2W- IK-]}- 
 A <T 
 
 Exactly similar calculations may be performed for the qs. 
 The results are got by replacing p by q ; hence 
 
 *&' 
 
 2 71 
 A 
 
 f«* , ft.-iC*«0*r- ^^ n+1 ?«(^). etc. 
 
 With the use of these formulas it is easy to perform the 
 necessary integrations ; for instance, we get by help of an 
 integration by parts 
 
 1 See footnote on p. 72. 
 
74 INDUCTION IN" ROTATING SPHERES n 
 
 xx S 
 
 ja 2n+2 p n (\a)da + a 2n+1 J ap n (Xa)da 
 
 S <T 
 
 A 
 
 A 2(71+1) A 
 
 When we substitute these and the similar expressions for q 
 in the equations, and remember that <j> 2 = - A 2 ^ and A 4 = - 1, 
 the p's and ^'s cancel as they must do, and we are left with 
 equations of the form 
 
 _ N (const)., 
 
 r 
 
 which are the solutions of the equation 
 
 d 
 da 
 
 d (cr 2 ' l+1 </))] = . 
 
 da 
 
 The constants occurring here must vanish separately ; 
 remembering that — = = we thus obtain the following four 
 
 ° a; a: ° 
 
 equations for A, B, C, D 
 
 2 A±± = A A _i(\S) + Bp^S) + ? .. 1 (\S) + Dj..,(X^) , 
 = A^.^S) - B^.^S) + C^.^S) - D^.^S) , 
 
 o = AiVH(V) + B iWV) + C 2V +1 (V) + d&+i(V) . 
 
 = Ap n+1 (V) ~ Bp*+i(V) + C^ +i (\s) - D 2ft+1 (A 2 s) . 
 These equations are easily solved and give 
 
 A- 2 ™" 1 " 1 . ^n+l(V) 
 
 4:n ' p n . 1 (\S)q n+l (\s) - ^ n+1 (X 1 s)^_ 1 (A 1 S) ' 
 
 c= _ 2^+1 ^ ff, t +i(V) 
 
ii INDUCTION IN ROTATING SPHERES 75 
 
 B _ 2n + 1 gn+i(V) 
 
 4ti > TO _i(X 2 S)^ +1 (V) - ^ +1 (X 2 s)^_ 1 (X 2 S) ' 
 
 D= - 27l+1 . ffn+l(V) _ 
 
 We get the complete solution by substituting these values 
 in cj) l and <j> . It may be more simply exhibited thus : Since 
 \ and \ are conjugate, _p (\°") an d i? (^9°0 a l so are conjugate ; 
 in the same way A and B are conjugate, as is easily seen, and 
 hence 
 
 4P»(V)+,BPn(V) 
 
 is equal to twice the real part of either expression. 
 In the same way 
 
 which expression occurs in <£ 2 , is twice the imaginary part of 
 the first term. Eemembering this, and also the values of A 
 and C, we easily recognise the truth of the equation 
 
 2^+1 Pn(\<r)qn+l(\s) ~ giAl^K+lC V) Solution of 
 
 S's. 
 
 This equation is especially simple when s = 0, that is in the 
 case of a solid sphere. Then q n + 1 (s) is infinite, and thus 
 our equation becomes 
 
 271+1 p n (\a) .— 
 
 The quantities, a knowledge of which is of special interest 
 to us, are the angle a = tan" 1 /^, and the ratio of increase of 
 
 current, v/i +/ 2 . These have a very simple analytical mean- 
 ing : they are the amplitude and modulus of the complex 
 quantity on the left-hand side. 
 
 The calculation may be carried still further by means of 
 the following remarks : — Further 
 
 The indefinite integrals defining p and q may be evaluated ^X^? 
 for integral values of n, and thus p and q may be expressed and q's. 
 
The first 
 q'a. 
 
 I INDUCTION IN ROTATING SPHERES n 
 
 in a finite form. We may and will denote by p and q the 
 functions thus calculated. Then also qs with negative argu- 
 ments are admissible, and this equation holds 
 
 whence follows 
 
 For let the indefinite integral 
 
 then we have 
 
 2 , 1 (<t) = V( - ) co)-V(<t,1), 
 
 1 1 
 
 p n ( a ) = j(l _ tf)* e -° v €h> + f(l - v 2 ) n e° v clv 
 
 
 
 = V(-<r,l)-V(-<7,0) 
 + V(o-,1)-V(cr,0). 
 
 But for integral values of n 
 
 V(«r, oo ) = 0, V( - a, 0) = - V(<r, 0), 
 
 and thus the statement made follows. 
 
 Hence the simplest integrals of the original equations 
 are 
 
 qja) and q n ( - a). 
 
 The following are the expressions for the first few ^'s; 
 the first one has been determined directly, the others by the 
 recurring formula. 
 
 
 
 
 
 
 2*.|2..-/ 3 3\ 
 
 
 % S \ + * ] a>) 
 
 c h = 
 
 2 8 .|3.€"7 i 6 15 15 
 
 - — I 1 1 X 1 
 
 a* \ a a a 
 
 etc. 
 
II INDUCTION IN ROTATING SPHERES 
 
 Hence follow the equations 
 
 77 
 
 Po = — » 
 
 , etc, 
 For large values of a p and q approach the values 
 
 Vn = (-2) n .\n.±l, 
 
 For very small values of a we get 
 
 (-2)\|».1.3...(2*-1) _ a 
 
 &W = ~ gSn + 1 — 6 • 
 
 The equation - p(cr) = g'M+S'C - cr) here has no longer any 
 meaning ; for when <x = 0, q = ± °°- In order to find p for 
 very small values of cr also, we expand it in a series of ascend- 
 ing powers of a. This is easily done by substituting for e™ 
 its expansion in the integral representing p ; integrating each 
 term we obtain 
 
 Them's and 
 q's for large 
 and small 
 values of 
 the argu- 
 ment. 
 
 tn+1 
 
 PnW 
 
 1.3. ..(271+1) 
 
 1 + 
 
 2(2ti + 3) 
 
 + 
 
 2.4.(2w+3)(2w + 5) 
 
 + 
 
 The following formula is of importance for our further 
 
 investigations. We have 
 
 (a) = ~ — ^ — \ 2»-i0O?»-2( - o-) - ^-i( - oOfiW-sOO r 
 
 71 . 71— 1 
 
 4 \» 
 cr 
 
 which equality is easily demonstrated by means of the recur- 
 ring formula found for q. 
 
 In all the properties of p n and q n considered we notice 
 
78 INDUCTION IN ROTATING SPHERES II 
 
 their close relation to Bessel's functions ; in fact we may 
 express J m in terms of p m+i and g m+| . 
 
 We may now remove p from the formula expressing our 
 result, and thus we obtain 1 
 
 Final form 2w+l g w+ i(Xs)g w ( - A<x) - g n+1 ( - \s)q n (\cr) 
 
 ofthe solu- 2n ' ^ +1 (^)g TO _ 1 ( - AS) - q n+1 ( - As)g n - AS) 
 
 = <f> l + ci> 2 J-l. 
 
 its appiica- We shall apply this formula, which gives the exact solution, 
 
 tlon ' to some special cases which admit of simplifications. 
 
 Thinspher- 1. In the first place, let the spherical shell be very thin, 
 
 and let d be its thickness. Then S is only slightly different 
 
 from s. Let 
 
 S = s+8, where now 8 = fid. 
 
 For cr we may put any convenient value between s and S ; 
 suppose a = S. 
 
 We substitute these values in the above formula. In the 
 denominator we employ the substitution 
 
 _2n+l XV 
 
 and divide by the numerator. Thus we obtain 
 
 ! + ^ 2 S 2 g»+i( Xs )^+i( ~ xS ) ~ g*+i( ~ *s)g B +iPlS) ' 
 
 2(n+ 1)(2^+ 1) q )l+1 (\s)q n ( - XS) - q n+1 ( - Xs)g tt (XS) 
 
 We develop and put 
 
 g. + i(^s) = ^ +1 {x( S +a)} 
 
 = g ft +A s ) + W % +As) 
 
 = g»+i(^)+ 2 3 2 g n+2 (Xs), 
 
 8X 2 s 
 g n+1 ( - XS) = g ft+1 ( - Xs) + A + 2 ( - *s). 
 
 1 From this point onwards we write X instead of Xi ; thus — 
 
 \=Vi(i+v~i). 
 
li INDUCTION IN ROTATING SPHERES 79 
 
 Using formula (a) (p. 77) we may divide out the q's and 
 thus obtain 
 
 , 1 
 
 2n+l 
 But now we have 
 
 s8 AiirRiw 7 
 
 2n+l (2n+l)k 
 according to our previous notation, and hence 
 
 1 h . — 
 
 which result agrees with the one previously obtained. 
 
 Thus we have on the one hand tested our formula by 
 means of a result already known ; on the other hand we have 
 proved that the previously given formulae hold for all values 
 of h, which proof still remained to be given. 
 
 2. Secondly, we apply our formulas to the case where we Small 
 need only retain the first power of the angular velocity in f l f rotation. 
 and / . For simplicity we restrict the investigation to a solid 
 sphere. In this case we found 
 
 Expanding the p's and retaining only first powers we get 
 
 2+ u I 
 
 Q2 
 
 2 + i£l^ 
 
 A closer consideration of this equation shows that the 
 values of / , / 2 thence found when substituted in yjr merely 
 give the inductions of the first and second order, which we 
 have already calculated on p. 68. Here we only consider the 
 
80 INDUCTION IN ROTATING SPHERES II 
 
 angle through which the lines of flow are turned. Eetaining 
 only lowest powers we find 
 
 ta-i-*- -!=*/'_* I- 
 
 / x k \2»+l 2ti+3 
 
 so that the angle in question is 
 
 8 2ir(of E p~ 
 
 1~ V~\2n+l~ 2n+Z, 
 
 Thus all the layers appear rotated : the rotation is least at 
 the surface of the sphere and increases continuously inwards. 
 If we imagine a plane section taken through the equator of 
 the sphere and join corresponding points of the different 
 layers we get a system of congruent curves, which is very 
 suitable for representing the state of the sphere. The equa- 
 tion of one of these curves clearly is 
 
 ^ 
 
 y = «tan- p = Jx 2 + y 2 , 
 
 V 
 
 or very approximately 
 
 2ttco ( E 2 x 2 
 
 2n+l 2n + 3 
 
 In Fig. 8 these curves are drawn for a copper sphere for 
 which E = 50 mm., n = 1, when it makes 1, 2, 3, 4 revolutions 
 per second. 
 
 Fig. 8. 
 
 Large 3. Thirdly, let us assume that //, is so large that for q(BX) 
 
 ofrotatTon. and %( sX ) we ma 7 P ut their approximate values. Further, 
 assume that the ratio r/K is neither very nearly = 1 nor very 
 nearly = 0. The former case has been considered already ; 
 the latter requires special consideration. Substituting the 
 approximate values in the exact formula we find 
 
 /— - 2n+l S 71 €**-»> + €-*<*-« > 
 
 </>l + 9 2 V 1 - 3+TXS-s)_ e -A(S-s)' 
 
2»+l/R\»+ 1 _ v(R _ p) 
 
 e 
 
 II INDUCTION IN ROTATING SPHERES 81 
 
 Since S is not nearly equal to s, and both are very great, 
 the second term in the denominator vanishes compared with 
 the first, and we get 
 
 ft + ft V^l = ^QyV^-Xe^ + e-^). 
 
 The second term in the bracket vanishes in comparison 
 with the first except when p = r; if then we are content with 
 an approximate knowledge of the current at the inner surface 
 we may write 
 
 ft + ftV^T = ^p(JJ 
 
 Since s or r has disappeared from this equation, we may 
 assume that it holds also for a solid sphere. In fact it is 
 easily deduced from the exact formulas which hold for a solid 
 sphere if we make similar approximations to those used above, 
 and do not require an exact knowledge of the currents at the 
 centre (where, as a matter of fact, the current intensity is 
 very small). 
 
 In the expressions obtained 
 
 \= */J(i+ «/-!); 
 
 without performing the separation into real and imaginary 
 parts we easily find 
 
 -£--s->-» 
 
 tan 
 
 r P 
 
 Substituting these values in ty we find 
 
 271+1 t hw --^(R-P) • / • 7T 
 
 A. / — e V2 sin [ico — - 
 
 2(n+l)^V kit \ 4 
 
 which is the current-function produced for very large velocities 
 of rotation by the external potential 
 
 = A( B. ) cos i(o¥„ 
 
 M. P. G 
 
82 INDUCTION IN ROTATING SPHERES II 
 
 The meaning of the above formula is easily grasped. If 
 we collect together its result and the results previously obtained 
 we may describe the phenomena, which would be presented by 
 a spherical shell rotating with constantly increasing velocity 
 under the influence of an inducing spherical harmonic function, 
 in the following terms : — 
 Summary When self-induction begins to be appreciable, it does not 
 
 result. alter the form of the lines of flow in the various spherical 
 layers, but these latter commence to undergo an apparent 
 rotation in the direction of rotation ; and then the inner 
 layers gain on the outer ones. There is no limit to the 
 rotation of the inner layers ; it may increase indefinitely. 
 The angle of rotation of the outermost layer converges to the 
 value 7r/4:i ; moreover, for spherical shells it may in the first 
 instance have exceeded this value. If the velocity of rotation 
 be very great, corresponding points of the different layers lie 
 on spirals of Archimedes, and the number of turns which 
 these make in the sphere increases indefinitely with the velocity 
 of rotation. 
 
 At first the intensity increases with the velocity of rota- 
 tion, but nowhere proportionately to it ; more quickly in the 
 outer than in the inner layers. In the outermost layer it 
 constantly increases, ultimately as v co ; in the other layers 
 it reaches a maximum for some definite velocity and then 
 decreases. For large velocities it decreases inwards from the 
 surface in proportion to an exponential, whose argument has 
 V co for a factor. 
 
 It is of interest to note also the dependence of the pheno- 
 menon on the order i (whose square root is involved in fi) ; 
 for this I refer to the formulae. 
 
 An apparent contradiction between the theory of an in- 
 finitely thin spherical shell and that of one of finite thickness 
 may excite notice ; it is easily explained when we consider 
 that every spherical shell, however thin, may only be regarded 
 as infinitely thin up to a certain value of the velocity of 
 rotation. 
 Case where I shall deal shortly with the case where the inducing 
 
 the mag- . n . i-i-i 
 
 nets are magnets are inside the spherical shell, so that spherical har- 
 niside the m onics of negative degree occur. 
 
ii INDUCTION IN ROTATING SPHERES 83 
 
 Let 
 
 / r \n+l 
 Xn = M~] COSlQ)P mi , 
 
 then 
 
 f = ~ A P - - Sm ^ P m- 
 
 AC \/3/ n 
 
 If we write 
 
 ^ = -Apf - ) -(/ sin i© +/ cos ^)P^, 
 k \pj n 
 
 the function yjr f induced by i|r becomes 
 
 y? = - ( - ) Ap( - ) -(Fi cos ico - F sin i<o)¥ ni . 
 \kJ \pj n 
 
 But here the connection between / and F is somewhat 
 different from what it was before, for we have 
 
 P R 
 
 F(p) = *rh [\ af {a)da + j P ^ f {a)d( j 
 
 r P 
 
 The condition 
 
 leads to the same equations as before, 
 /x=l + ^F 2 , 
 
 K 
 
 /2 K r 
 
 Using the same contractions as before we get 
 
 &0)='i + 
 
 2n 
 
 a- a 
 
 8 
 
 )da 
 
84 INDUCTION IX ROTATING SPHERES 
 
 On differentiation these become 
 
 or 
 
 
 >i - — 9i = ~ 9-» 
 
 _:/? 
 
 <£ 2 " - —</>./ = </>!• 
 
 If we put = tr 2 *" 1 " 1 ^, we get for <£ the equations 
 -„ 2rc+2- 
 
 -„ 2// + 2-, - 
 £/ + — - — 0/ = 4>i> 
 
 which equations again lead to the p's and q's. 
 
 The constants may be determined in the same way as 
 before, and then we find 
 
 a-- -1 
 
 s 
 
 0— l(*S)?»+l( - XS) - &-1 ( - xs)& +1 (h 
 
 The formula becomes especially simple for the case where 
 S = oo , that is, when we have to deal with a spherical hollow 
 in an infinitely extended mass. 
 
 Then $ ft _i(XS)= 0, and we get 
 
 If we may neglect fip = a in comparison with unity, on 
 account of the small value of co, we may put for the q's their 
 values for small arguments (p. 77). "We then obtain 
 
 fa s/ ~ 1 - (£,, = \-6 
 
 2 -A(<r-s) 
 
ii INDUCTION IN ROTATING SPHERES 85 
 
 or since we have in part already neglected quantities of order a 
 
 <f>l\/ — 1 — ^ ) 2 = ^~ ** ~ ^ 
 
 as must be the case. 
 
 On the other hand, if pup be large compared with unity, 
 and we put for the qs their approximate values for large 
 arguments (p. 77), we get 
 
 fc + <t> 2 v/^T = (2n + l)f y"le-^- s ). 
 
 Hence result phenomena similar to those for the spherical 
 shell ; the rotation is irj^i at the innermost layer, and thence 
 increases indefinitely as p increases to infinity. 
 
 For -\fr we find 
 
 2m V / 
 
 ]£-&-*m*{«.-l-Jfe P -r: 
 
 which expression is quite analogous to that obtained for the 
 spherical shell. 
 
 Moreover, we easily see that, even for the smallest 
 velocities of rotation, p can always be chosen so large in un- 
 limited space that the approximation made may be permissible : Neglecting 
 hence even for the smallest values of co the induction will pass ^ on " m uc ~ 
 through all possible angles, at distances, it is true, where the 
 intensity is very small. 
 
 I here wish to draw attention to the remark I have already 
 made on p. 61 in regard to neglecting self-induction. 
 
 It would be very easy to extend the results obtained for 
 spherical shells to plane plates of finite thickness ; but in 
 order to avoid complicated calculations I omit the investiga- 
 tion. The chief part of the phenomena can, in fact, be deduced 
 without calculation from what has been already discussed. 
 
 § 5. Forces which are exerted by the Induced Currents. 
 
 We shall now calculate the forces exerted by the induced 
 currents and the heat generated by them. The latter is 
 equivalent to the work which must be done in order to main- 
 tain the rotation. 
 
86 INDUCTION IN ROTATING SPHERES n 
 
 A. Potential of the Induced Currents. 
 
 1. We first calculate it for external space. The part of 
 it due to the spherical layer between p = a and p = a + da is 
 
 Its value in , /„\n+l 
 
 space. 27i+l\p/ 
 
 when we consider the term yjr ni of the whole current-function yjr. 
 
 Now 
 
 ty n ia)da = - A™ at ^ ) —?— (f^a) sin ico +f 2 (a) cos iw)V ni da. 
 k \K/ 7i+ 1 
 
 Substituting this value in dCl, and attempting to perform 
 the integrations, we meet with the integrals 
 
 R 
 
 I 
 
 a 2n+2 f(a)da. 
 
 But we have 
 L^,.„.. (2»+l)R 2 » +1 
 
 a " + y i (a>to = ^" T 4 ^ F l( R) 
 
 according to the definition of F (p. 68), and the equations 
 satisfied hy f lt f 2) F 1? F 2 (p. 70); and similarly 
 
 I 
 
 a 2n+ %(a)da = - (2 " + . 1)ig (l -// E)) . K 2w+1 . 
 
 47T^ft) 
 
 Using these expressions we find 
 O c = ^- Af? Y l+1 [/ 2 (R) sin *» + { 1 -/i(K)} cos *• 
 
 71+1 \/3/ 
 
 For very small angular velocities f 2 = 0,/ x = 1, and thus 
 H e = 0. For very large ones f Y =/ 2 = 0, and thus at the 
 surface of the spherical shell 
 
 n — 
 
 n = ^iX«- 
 
II INDUCTION IN EOTATING SPHERES 87 
 
 2. In precisely the same way we may perform the investi- Value in 
 gation for the space inside the spherical shell ; we find space 
 
 O* = - (0 A[/ 2 (0 sin ico + {1 -AO)} cos H*\«- 
 
 Hence for the whole potential 
 
 &i + Xn = A ' £■ ) LAM C0S ^ ".AM Sin *®] P «i- 
 
 \±c/ 
 
 For vanishing angular velocities this expression reduces 
 to % n , for large ones to zero ; more exactly for large values of 
 fju we find by means of a formula which we have employed 
 previously (p. 81) 
 
 2(2» + l)/E\* 1 
 
 /^_l_ ,-t 2(2k+1) /B\ 
 
 Hence it follows that 
 
 r/ e A(S-s) _ e -A(S-s) 
 
 «.+* 
 
 = d&±lW. ■%" cos (fa-* ' (E - r) W 
 
 /^ W V 4 V2 / 
 
 Thus the internal potential diminishes with exceedingly 
 great rapidity as the velocity increases. At the same time its 
 equipotential surfaces exhibit the peculiarity of appearing 
 turned through an angle proportional to the [square root of Peculiar 
 the] angular velocity. As the velocity gradually increases J tnTnSg- 
 the forces conditioned by the potential take up successively netic forces 
 all the directions of the compass ; and this can be repeated S i c i e space . 
 any number of times as the velocity goes on increasing. 
 
 B. Heat Generated. 
 
 Let E be the radius of a very thin spherical shell, and 
 suppose that in it exists the current-function 
 
88 INDUCTION IX ROTATING SPHERES ir 
 
 Let the resistance of the shell be h : required the heat "W 
 generated in it. 
 
 We determine the values of u, v, w belonging to ty, and in 
 particular to the term 
 
 Heatgener- ^ — X sin ico?, ,. 
 
 atedina ^ T hi 
 
 spherical When u, v, w have been found, the heat generated is 
 
 shell. 
 
 the integral being extended over the whole surface of the sphere. 
 Introducing H and to denote currents parallel to circles 
 of latitude and to the meridians in the direction of increasing 
 6 and co, we have 
 
 K c6' 
 
 = - 
 
 1 df 
 
 E sin 6 ceo 
 u = cos 6 cos co + H sin co, 
 v = cos 6 sin co — H cos co, 
 
 v: = — sin 6. 
 
 Substituting for yjr 7li the value given we get 
 u =— { — cos ico cos co . £P ni . cot 6 — sin ico sin fflP^} 
 
 u = — { — cos ico sin a) . £P wi . cot # + sin z'o) cos ©P'-}, 
 A. . ^ 
 
 W-— l COS ZO) . ±\-;. 
 
 R 
 
 But now 
 
 2'P ?U - cot 6 = U V n,i + i + (n + iXn - i+l)Pn,i-i} 3 
 
 Substituted in m, r, wi these give 
 u= --— {cos(i+l)©.P M+1 
 
 + (71 + l)(7l — *+ 1) COS (i - 1)© . P^i-l}, 
 
INDUCTION IN ROTATING. SPHERES 89 
 
 ?- -2^{ sin (^+ 1 >-I > M+i 
 
 A. . _, 
 «? = ^ cos ico . P ni . 
 
 - (ti + i)(rc - i + 1) . sin .(* - 1> . P^.J 
 
 Thus w, v, w are developed in terms of spherical harmonics. 
 We suppose the u, v, w expanded in a similar form for all the 
 terms, and then form the expression 
 
 {u 2 + v 2 + w 2 )ds. 
 
 On integration, terms involving products of spherical harmonics 
 of different orders vanish, so that we may determine W n sepa- 
 rately for each yjr n , and add the results. A closer consideration 
 then shows that we may also determine the heat separately 
 for each yfr ni and again add the results. It is true that all 
 the integrals do not vanish which correspond to combinations 
 
 of different yfr ni 'a ; but those integrals in I u 2 ds for which this 
 
 occurs will be destroyed by corresponding ones in I v 2 ds. 
 We now get for the -\jr ni above quoted 
 
 W ni = k((u 2 + v 2 + w 2 )ds 
 
 + O + i)\n -i+lf ((Pn^fds + U 2 J (cos i<o? ni fds\, 
 which gives, by well-known formulae and simple reductions 
 
 W ni = kA 2 27r<n+1 \ n-i+ l)(n-i+2)... (n + i) 
 2n+ 1 
 
 = kA\n,i) f 
 
 where (n,i) is an easily intelligible abbreviation. Since we 
 have further 
 
 W = 22W„ 
 
 n i 
 
90 INDUCTION IN ROTATING SPHERES n 
 
 our problem is solved. It is easily seen that the result may 
 be expressed in the forms 
 
 or 
 
 W=l2»(» + l)xff 2 „ i rf s , 
 
 w=As»(»+i)Lvfe 
 
 -K n J 
 
 Analytical 
 formulas. 
 
 The preceding theorem might perhaps have been more 
 simply demonstrated by means of reasoning based on Green's 
 theorem. In the proof here given the following formulas are 
 implicitly involved 
 
 /{(£H$ ,+ (s£)>-* + '>J** 
 
 --(sin ico? ni ) = - i cos (i + l)a>P tt>i+1 
 
 — ^(n + i){n — i + 1) cos (i — tyaiP^j, 
 ^- (sin icoY m ) = - \ sin (*+ 1)©P M+1 
 
 + 4( w +*X™ - * + 1) sin (» - l>P W)i _ ls 
 
 a 
 
 — ( sm io)P Ml .) = * cos i(*? ni , 
 
 VC0 2 
 
 and to these similar equations may easily be added. If w, 
 and &/, 6 f refer to systems of polar co-ordinates with different 
 axes, the equations last quoted enable us to deduce integrals 
 of the form 
 
 / 
 
 cos i(o? ni {6) cos jWP^O^ds', 
 
 (in which the integrations are to be performed with respect 
 to a/, 6 f ) from the well-known integral 
 
 |p, !]O (0) anjtftJpjV = -i— cos>P^), 
 
 J 2/i-\-l 
 
 Generation but, it is true, only by laborious calculation. 
 
 thero^atSie -^ now P rocee d to determine the heat generated in the 
 
 sphere. 
 
II INDUCTION IN ROTATING SPHERES 91 
 
 rotating sphere by the term % ni of the inducing potential. 
 To % ni corresponds 
 
 fni = ~ -A/>( £ ) — ^-{fi sin ico +/ 2 cos ico)? ni , 
 k \E/ n+ 1 
 
 and hence the heat 
 
 R 
 
 i 2 (n,i) f f p 
 
 W„-^-l^ £) (ff+ftyap. 
 
 r 
 
 The integration can be performed for small and for large Small velo- 
 angular velocities. For the former f 1 = 1, f 2 = 0, and thus Clties * 
 the heat generated becomes in this case 
 
 W„ = A 2 ?^ *£2 (l-(^) ' 
 
 k (ti+1) 2 (2^+3)V \B/ 
 
 For very large angular velocities we had 
 
 2 2 (2n+lf E 2}i ,- Large velo- 
 
 /l+/2 = 2 2^+2 6 P « CltieS ' 
 
 The integral W ni may be taken from r = 0, and becomes 
 
 it 
 
 / 
 
 (2n+iy yn + iy 
 
 f € dp - ^s/2 
 
 for E^u, may be regarded as infinite. Hence we get for very 
 large values of co 
 
 I2n+l)\n,i) J Km 
 
 VV - ^ 8(7.+ I) 2 V27T 3 ' 
 
 and W depends on E, in so far as A involves E. 
 
 Hence the heat generated increases indefinitely as <o in- 
 creases, 1 and indeed proportionately to ^/u. The same holds 
 good for the work which has to be done in order to maintain 
 the rotation. If the inducing magnets form a rigidly con- 
 nected system, they are subject to a couple about the axis of 
 
 1 As regards the apparent contradiction with the result got for infinitely thin 
 spherical shells the remark on p. 82 holds. 
 
92 INDUCTION IN ROTATING SPHERES n 
 
 rotation which can be calculated from the heat generated. 
 For if we imagine the shell at rest and the inducing magnets 
 rotating with angular velocity co, then the couple of moment 
 D, which maintains the rotation, does work 2itcoD per unit 
 time, and this work is equal to the heat generated. Thus — 
 
 W 
 D = — — • 
 
 Couple ex- But this couple is equal to that with which in the reverse 
 
 inducing 116 case ^ ne rotating sphere acts on the magnets at rest. It is 
 magnets, easy to see that for small values of w D increases proportion- 
 ately to co/k, but for large values decreases proportionately to 
 s/x/<», ultimately becoming zero. (This does not prevent work 
 being done of the order ^w.) On the other hand we have 
 seen already that for infinitely large values of co the forces 
 exerted on the inducing magnets are finite, and since now they 
 produce no couple about the axis of rotation their resultant 
 must act in a plane through the axis. 
 
 In fact for infinite velocities the sphere behaves as regards 
 the external magnets as a conducting sphere does with regard 
 to electric charges ; but a conducting sphere cannot impart 
 to inducing charges any rotation about an axis through its 
 centre. 
 
 § 6. Eotation of Magnetic Spheres.' 
 
 I now assume that the material of the sphere is capable 
 of magnetisation, but that it is without coercive force. 
 
 We must first form the expressions for the electromotive 
 forces in this case. According to the precedent of § 1, 6, in 
 order to find the effect of magnetisation, we must in the 
 general expressions for the electromotive forces replace 
 
 U by , 
 
 J dz dy' 
 
 V by — - — - 
 
 W by . 
 
 dy dx 
 
INDUCTION IN ROTATING SPHERES 93 
 
 We must accordingly replace — 
 
 (i). 
 
 since 
 
 and 
 
 
 ox dy dz 
 
 
 au aw - _ . _ , a % 
 a« a» ay 
 
 
 aw av . __ 2T , 3y 
 
 __ by V"L + ■£ 
 
 dy dz dx 
 
 
 aL aM aisr _ 
 
 hc + 'dy + ~dz~" X 
 
 now 
 
 
 
 V 2 L = - 4tt\ 
 
 
 V 2 M = - 4cTTfl 
 
 
 V 2 N = -4wv 
 
 
 x av aw dx 
 
 
 ~e~Tz~~dhi~dx 
 
 
 fi aw au a % 
 
 aa; dz dy 
 
 
 v au av d x 
 
 
 6 dy dx dz 
 
 (2). 
 
 (3), 
 
 (4). 
 
 If by these equations we eliminate L, M, N from the 
 expressions (1), and add the forces directly exerted by the 
 
 currents, that is - — - - — , etc., we get for the expressions, 
 
 dx dy 
 
 7\XT PTT 
 
 which are now to be substituted for -— - — , etc., the follow- 
 
 dx dy 
 
 (l+^eC-lJV + h) Thediffe, 
 
 V dx dy dz entialequa. 
 
 x ° J tions. 
 
94 INDUCTION IN ROTATING SPHERES n 
 
 Here % denotes the total potential ; but this consists — 
 
 («) Of the given external potential of the inducing 
 magnets. 
 
 (b) Of the potential % e of the magnetised sphere itself. 
 
 This last satisfies the conditions : — 
 
 Inside 
 
 V 5 X. = <> (6), 
 
 as follows from equation (2) and (4); at the surface — 
 
 4^, -(l + 4^&) -•(&). (7), 
 
 where Xp is the radial force exerted by the external magnetism 
 and the induced currents. 
 
 In words we may thus express the effect of the magnetis- 
 ability of the medium : — 
 
 The magnetisation firstly alters the internal magnetising 
 force in the manner shown by the general theory of magnetism, 
 and secondly increases the effects of the magnetising force in 
 the ratio 1 + 47r(9 : 1. The two effects are opposed, and the 
 result is that the action is found to be increased in only a 
 finite degree even for large values of 6. 
 Self- Let us again, to begin with, neglect self-induction. It is 
 
 to be remarked that this is allowable only when 
 
 induction 
 neglected, 
 
 /47T&)(1+47T<9) 
 V ~ K 
 
 is very small. When 6 and E are large, co must be very 
 small absolutely to satisfy this condition. 
 If the external potential be 
 
 the potential of the spherical shell itself may be expressed in 
 the form 
 
 and the total potential therefore in the form 
 
 \) 
 
II INDUCTION IN ROTATING SPHERES 95 
 
 According to what precedes, the magnetic spherical shell 
 is in exactly the same condition as a non-magnetisable one 
 of equal resistance which is subject to the influence of a 
 potential 
 
 (1 + 4*0) A + B - X» 
 
 >N + 1 
 
 Since this potential consists of two spherical harmonics, 
 the currents may by what precedes be regarded as known. 
 For the current-function we get 
 
 •- • ■ -/ A B/rV" +1 \ d Xn 
 
 + --_(l + 4^_-y y dw 
 
 Under like conditions, only with 6 = 0, we should have got 
 the current-function 
 
 __ ft) p d X n 
 
 k n + 1 ceo 
 
 By division we get 
 
 yfr = (1 + 47iWa - ^- B (-) 2 " ^o- 
 
 The form of the currents in the various layers is unaltered, 
 but the intensity is differently distributed. It is convenient 
 to describe the phenomenon by comparison of -fy and ijr The 
 quantities A, B are given by equations (6), (7) ; if we write 
 
 T 
 
 — = e, they are found to be 
 B 
 
 A _ (2n + 1){(2m + 1)(1 + ±7T0) - kirOn) 
 
 n(n + 1)16tt 2 6> 2 (1 - e 2 ' l+1 ) + (2w + 1) 2 (1 + 4tt<9)' 
 
 47r07i(2w + l) 
 
 ~ 71(71 + 1)16tt 2 6' 2 (1 - e 2w+1 ) + (2w + 1) 2 (1 + 4,ir6) 
 
 As the interpretation of these expressions is not very Special 
 obvious, we shall apply them to some simple cases. cases - 
 
 1. 6 is very small. Expanding we get 
 
 A=l--^— 4tt0 3 B = -^— AitO, ^7 
 
 2n+l 2n + l sma11 - 
 
96 INDUCTION IN ROTATING SPHERES n 
 
 and hence 
 
 Thus the current intensity is unaltered at the inner surface 
 of the spherical shell ; in other portions it is always increased 
 when 6 is positive. The increase is directly proportional to 6. 
 In diamagnetic spheres the intensity is everywhere less than in 
 neutral ones. The rotation of magnetic spheres absorbs more 
 work, that of diamagnetic ones less work than that of neutral 
 ones. 
 
 very 2. Let 6 be very great and e not nearly equal to unity. 
 
 arge " Then we have 
 
 B 
 
 4tt^t<1-€ 2w+1 ) j 
 2ti+1 
 
 
 -47r6>(/i+l)(l-e 2 " 
 
 and hence 
 
 
 
 
 
 ' " -(iT 
 
 f/V 
 
 Thus the current in the innermost layer is here zero : 
 thence it increases rapidly outwards and becomes (2n+l)/n 
 times as great at the outer surface as for the neutral sphere. 
 If 6 is at all large the increase of current is almost independent 
 of its absolute value. 
 Thinspher- 3 ' Let e be in fi nitel 7 nearly equal to unity. 
 
 ical shells. Then 
 
 k _ (271 + 1)(1 + ±ir0) - 47T071 
 
 (2^+1X1+477(9) ' 
 4:7r0n 
 
 B = 
 
 (2n +1)(1 + 4tt0)' 
 
 Thus 
 
 In infinitely thin spherical shells the magnetic perme- 
 
ii INDUCTION IN ROTATING SPHERES 97 
 
 ability is without effect on the induced currents (though the 
 magnetisation is not zero, and the magnetic forces in the shell 
 are altered). It may here be noted that this result holds also 
 when self-induction is taken into account. 
 
 4. Let e = 0, which is the case of a solid sphere. The Solid 
 term with a negative power of p vanishes, and we get hp 
 
 A = 
 
 4:7T0?l 
 
 2n+l 
 
 , 1 + 4tt0 
 yfr = — — to- 
 
 1 + — — -47T0 
 
 2ra+l 
 
 For large values of 6 we have 
 
 2^+1 
 n 
 
 The quantity 2n-\-l/n lies betwesn 2 and 3. 
 
 Hence in iron spheres the currents are from two to three 
 times as strong as in a non-magnetic metal of equal resistance ; 
 the heat generated, the work used up, and the damping pro- 
 duced are from four to nine times as great as in such a metal. 
 
 5. Plane plates. 
 
 A very thin plane plate may be looked upon as portion of Plane 
 a very thin spherical shell, hence for such a one p a es * 
 
 A very thick plate may be regarded as portion of an in- 
 finitely large solid sphere ; since n is to be put very large we 
 have for such a plate 
 
 1+4tt6> 
 
 ^ 2 + 2ird^' 
 
 In both limiting cases the total current-function remains 
 unaltered; in the last case for large values of 6 the intensity 
 is doubled by the permeability. 
 
 For medium thicknesses of the plates intermediate values 
 
 M. P. H 
 
98 INDUCTION IX ROTATING SPHERES i[ 
 
 hold ; the calculations are easily performed, but since they 
 
 give no very simple results they have been omitted here. 
 
 Self-indue- We shall now take into account self-induction, but shall 
 
 into only perform the calculations for a solid sphere. Spherical 
 
 account, shells do not offer analytical difficulties of any special kind, 
 
 but the calculations become exceedingly complicated. 
 
 Limitation We find the currents by the following reasoning : — 
 
 to solid , . _ . . , , 
 
 spheres. -Let the inducing potential be 
 
 Let i{r be the current -function directly induced by % ni 
 then we have 
 
 1+47T0 co, fp\ l i . . _ 
 
 to = • ~M p — TT sm l<oV <«' 
 
 271+1 
 
 Let the actual current- function be 
 
 k \RJ n+1 
 
 We have to find the currents induced by this. For this 
 purpose it is necessary first to know the potential % e induced 
 by -v/r in the magnetic mass. 
 
 The current-function 
 
 * = />(0/(/>)Y„ 
 
 produces a radial magnetic force 
 
 _x/dW dY\ y/aU dW\ z/dY dU 
 p\dy dz J p\dz dx J p\dx dy 
 
 = 5 (p . 63) 
 P 
 
 = -'^±1)f(,).(0Y, 1 (p. 65), 
 where F,/ are connected by the equation of p. 68. 
 
II INDUCTION IN ROTATING SPHERES 99 
 
 From these equations and from the equations determining 
 % 9 (equations (6) and (7), p. 94) we get % in the general 
 case, and therefore in our particular case we have 
 
 4:7r0n(n+l) 
 
 2n+l+4:Tr0n 
 
 ' l A ^iff ' < Fl(R) sin ico + Fa(R) cos iw ^ ni ' 
 
 If now by yjr f we mean that current-function which -ty and 
 % g together would produce in the unmagnetised sphere, then 
 in the magnetic mass they will produce the current-function 
 
 and the condition for the stationary state becomes 
 
 ^ = ^0 + ^6 • 
 
 ^ is to be formed in exactly the same way as before, so that 
 i|r' fl also is known. If we substitute the values of ifr, yjr , ^ f e 
 in the last equation and equate coefficients of cos ico and 
 sin ico, we get for f l} f 2 , F 2 , and F 2 these equations 
 
 (2^+l)(l + 47r6 >)_^ 47r6>7<l + 47r( 9) F 
 JlW 2n+l + 4:7r6n ' k 2n + 1 + kirOn 2 ^ } 
 
 + ^(1+4tt0)F 2 O>) , 
 
 „ „ v G)i47T0>l(l +477-0)-^ ,-r,. Coi , A , . ^-^ , s 
 
 If we put 
 
 ^(1 + 4^) = A 
 
 /J 
 
 AGO = Ww) = 4>i<» > 
 
 f,(p) = Uw) = ^(o-), 
 
 then (/>!, </> 9 are given by precisely the same differential equa- 
 tions as before (p. 71). Since we are dealing with a solid 
 
100 
 
 INDUCTION IX ROTATING SPHEEE3 
 
 The solu- 
 tion. 
 
 Compari- 
 son with 
 
 previous 
 results. 
 
 Small vel- 
 ocities of 
 rotation. 
 
 sphere, we must only retain those solutions which are finite 
 at the centre, and may put 
 
 ^ 1 (o-) = Ap w (X 1< r) + Bp n (X 2< r) 3 
 
 X 2 = x/f(l- n/^T). 
 
 The constants are determined in precisely the same way 
 as aboye. The integrals to be formed are not different from 
 those got before, but the calculation is somewhat more in- 
 tricate, owing to the complicated constants. The result, 
 howeyer, is comparatively simple, namely 
 
 (2 % +l)(l + 4^K(Vp) 
 
 /iW+/*)V-i = 
 
 "We first verify this result. For vanishing 6 it gives 
 
 2rc+l pJX/jup) 
 
 /i+/ 2 x/-l = 
 
 2?l Pn-i(kftB,) 
 
 which agrees with the result already obtained for a non- 
 magnetic solid sphere (p. 75). 
 
 Further, for vanishing co it gives, since 
 
 PnW = 
 
 /i-f/V-i 
 
 2 n+1 \ 
 
 ...(2»+l) 
 
 (2tt+l)(l+47rfl) 
 
 ~2n + 1 + 4tt07i 
 
 which result also we have found (p. 97). 
 
 In general it appears that the form of the currents in a 
 magnetic sphere is the same as for a non-magnetic sphere of 
 equal resistance which is rotating (1 + 47T0) times as fast 
 as the magnetic sphere. But in addition the two current- 
 systems differ in that they are turned as a whole through a 
 certain angle relatively to one another, and that their in- 
 tensities are different. 
 
 I apply the formula to two special cases. 
 
 1. Let ±ir6 be very great, but co sufficiently small that 
 
n INDUCTION IN ROTATING SPHERES 101 
 
 <u, 2 K 2 may be neglected in comparison with unity. We must 
 expand the expression, retaining only the first power of that 
 quantity. We have 
 
 Zn+l.pJiwX) 
 
 9«.4-1 9/<).<n. -L 5n_l- i, 2 n 2 . / - 1 
 
 (p. m 
 
 n pJji'BX) 
 2rc+l 2(2n+3) + fjuy s /~^l 
 
 n 2(2w+3) + // 2 ItV- 1 
 
 We get for the angle of rotation, neglecting higher powers 
 than the first 
 
 S^tan- 1 ^ _ ^ (K 2 -p 2 ), 
 
 f x 2(2^ + 3)^ PJ ' 
 
 B = _ 16^0 2 _ 2 
 i 2(2ra + 3)/c 
 
 Hence the rotation vanishes at the outer surface; 1 gener- 
 ally it is considerably increased, compared with that for a 
 non-magnetic sphere nearly in the ratio 47r# : 1. 
 
 In Fig. 9 are given the curves for an iron sphere cor- 
 responding to those for a copper sphere represented on p. 80. 
 
 Fig. 9. 
 
 The resistance of iron is taken to be six times that of copper, 
 and AlttQ is put =200. The velocities represented are exceed- 
 ingly small ones, namely one revolution in five and one in ten 
 seconds ; even here the effect of self-induction is well marked 
 (c/. Fig. 15 b, p. 123). 
 
 2. If a* become very great, whilst 6 retains a finite but Large 
 otherwise arbitrary value, the phenomenon becomes very ve 0Clties - 
 similar to that in non-magnetic spheres, as may be easily 
 deduced from the formulae. Here also the angle of rotation 
 
 1 A consequence of the fact that at this surface for large values of 0, according 
 to the equations for xe> 
 
 Nr - dr ' 
 
102 INDUCTION IX ROTATING SPHERES n 
 
 becomes irj^i at the outer surface. The phenomenon is 
 identical with that which occurs in a non-magnetic sphere 
 with (1 + 473-0) times the velocity. The heat generated is 
 fjl + 47T0 times that generated in a non-magnetic sphere 
 rotating with equal velocity. 
 
 § 7. Belated Problems. 
 
 In this paragraph we shall consider some problems which 
 stand in very close relation to those already treated. 
 
 Any solid If we neglect self-induction we may apply our knowledge 
 
 of the currents in a sphere to find those in a solid of revolution 
 of any form whatever, or at least to reduce their determination 
 to a simpler problem. 
 
 Let S be the surface of revolution bounding the solid, n 
 its inward normal. Describe about it a sphere of any radius. 
 Let u 1} v 1} w x be the currents which would flow in the latter, 
 and let 
 
 N = Ui C0S a + v l C0S ^ + W\ cos c 
 
 be the current in the direction of n at the surface S. If we 
 determine u 2 , v 2 , w 2 so that 
 
 of revolu- 
 tion. 
 
 
 ku ^~i: 
 
 
 3y 
 
 
 KlU = -~--, 
 
 1 dz 
 
 
 . ox Oy dz 
 
 
 u 2 cos a + v cos I + w 2 cos c = 
 
 then clearly 
 
 
 u Y + u v T + v 2 , w 1 + w 2 
 
ii INDUCTION IN KOTATING SPHERES 103 
 
 are the currents sought in S. Hence the problem is reduced 
 to this simpler one : — 
 
 To determine such a function </> that inside S V 2 </> 9 = 0, 
 and at its surface dcjy/dn = k~N, a given function. 
 
 1. As an example, suppose a plate bounded by the straight Plate 
 line £ = b to move parallel to a given straight line. Suppose ^straight* 7 
 the external potential expanded, and let a term of it be edge. 
 
 Ae~^ 1 cosr?;cossf . 
 
 Then we found for the current in the infinite plate 
 
 y 1 = A- — sin T7] . cos sf . 
 n k 
 
 Thus the current perpendicular to the boundary is 
 <tyi a ^ a 7 
 
 i-i = - A- • - • COS T7J . COS SO. 
 
 drj n k. 
 
 Hence we get for </> 2 the conditions 
 
 af + H 
 
 and for £ = o 
 
 We have 
 
 3</> 9 . r 2 a 7 
 
 ~- 2 = A- • - -cos rrj . cos so. 
 
 Og 71 K 
 
 6 = A e r( ^ b) cos rrj . cos sb. 
 
 n k 
 
 To </> 2 corresponds the current-function 
 
 -v/r = - A- . - . e r ^' h) sin rr\ . cos sb, 
 n k 
 
 and thus the total current-function becomes 
 
 ^ + ty 2 = A- . - . e~ rb sin r^(e r& . cos s£ - e r ^ . cos s&) 
 
 By summing for all the terms we get the complete solu- 
 tion. The solution for a band bounded on both sides is 
 similar. 
 
104 INDUCTION IN ROTATING SPHERES n 
 
 Limited 2. In order to determine the currents in a limited 
 
 rotating disc, let a term of the external potential be 
 
 Ae"*^ cos lea J i(np). 
 Then we had 
 
 i/tj = A sin icoJ^np). 
 
 k n 
 
 Thus the inward radial current is at the boundary where 
 
 dslr . co i 2 . J/riR) 
 — ^- =A — cosico— . 
 
 Hence we find as above 
 
 <t> = - A- - J £ (wE)( £ ) cos ico . 
 k n • \±v/ 
 
 Determining the corresponding current-function i/r 2 we get 
 for the total current-function 
 
 ^ + 1 „ = A? . I ^"{RJinp) - A(»R)}- 
 
 We again get the complete solution by summing for the 
 various terms. In the same way the currents may be deter- 
 mined in rings bounded by concentric circles. 
 
 In general the solution of the problem requires neither the 
 development in a series of separate terms nor the determination 
 of the potential (f> 2 ; it is sufficient to determine i/r 2 so that 
 inside the plate 
 
 !!+?+!!*?= 0, 
 
 dx 2 dy 2 
 
 and at its boundary i/r 2 = - yjr v Some simple examples will be 
 given in § 9. 
 
 II. 
 
 Dielectric In conductors electromotive forces of electromagnetic 
 
 spheres. origin produce the same effects as numerically equal electro- 
 static forces. If this is true also of dielectrics, then spheres 
 of dielectric material must become polarised when rotating in 
 a magnetic field. 
 
ii INDUCTION IN ROTATING SPHERES 105 
 
 Let 
 
 mm* nigr* 1 
 ^ c sec 
 be the components of the polarisation, 
 
 e (number) 1 
 
 the dielectric constant. 
 
 For X, g> J we have the equations 
 
 dx 
 d<j> 
 
 9r ^ 
 
 ax a g aj l 
 
 for o = E 
 
 * 4 4;7r\dp dp 
 Hence we have for <£ 
 
 4tt6 /as aj dz 
 
 and for p = E 
 
 dp dp p ** 
 
 In external space we must have V 2 </> = 0. 
 If ^ TC again be the %th term of the external potential we 
 have, as above (p. 43) 
 
 dx dy dz dz 
 
 a£+yg+«-4^-«% . 
 
 ,a? 
 
 1 The units are again such that 1/A, the velocity of light, does not occur. 
 The corresponding magnitudes in magnetic measure are A 2 i, A 2 g, A 2 $, A 2 e. 
 
106 INDUCTION IX EOTATIXG SPHERES 
 
 To satisfy the equations of condition we put 
 
 <$>=ft+ft, 
 = 4 776 CO 
 
 4776 CO 
 
 Vf-^j. 
 
 1 + 4776 n+l\ 
 
 r~/p\7i-f2 ^ 
 
 i ( E r r: ^ 
 
 p) 2/1 + 1 b 
 
 L\P/ »+r 
 
 1+4776 /*+ 1 
 
 </>? satisfies the partial differential equation which </> is to 
 satisfy. <f>° e is so formed that (1) it satisfies the equation 
 
 (2) at the surface of the sphere it is equal to ft. That the 
 first condition is satisfied is seen when we notice that the 
 expressions under the straight lines are spherical surface 
 harmonics of degrees (/z + 1) and (n — 1), as is easily proved. 
 Substituting ft -{-ft in the equations for <\>, we get for ft these 
 equations 
 
 V 2 ft = everywhere. ft continuous. 
 
 when p = E 
 
 (1 +4776 -^— = -+1+^-, 
 cp C^ Cp 
 
 to satisfy which is not difficult, as we have already expressed 
 
 ft as a series of spherical harmonics. 
 Earth in A case of especial interest is that in which a spherical 
 
 space. ' magnet rotates in a surrounding dielectric. For the earth is 
 a rotating magnet, and according to many physicists inter- 
 planetary space is a dielectric. To determine the electric 
 potential in this case we must remember that the earth is a 
 conductor ; hence in it a distribution will form which will 
 react on the dielectric and make the potential constant at the 
 earth's surface. 
 
 If ^ = 2v„ is the earth's potential the problem reduces to 
 this : — 
 
INDUCTION IN ROTATING SPHERES 10 1 
 
 To determine (j> so that in external space 
 
 and at the surface 
 We easily find 
 
 1 + 4-Tre dz 
 
 cf> = const. 
 
 y l + 4;7re C °^2n + l ' dz ' 
 
 Hence follows the rate of increase of potential at the 
 earth's surface 
 
 d(j)_ 4.7T6 OT > ^j 1 dxn 
 
 2Eo)2" 
 
 dp 1 + 4tt6 ^2n+l dz 
 
 Much the greater part of the earth's magnetic force is due 
 to terms for which n= - 2, or at any rate is small. There- 
 fore we may write approximately 
 
 dcf> 47re a d x 
 
 IE 
 
 tOJ . 
 
 a^ 1 + 4:716* dz 
 
 d%/dz is the component of the earth's magnetic force in 
 the direction of the north pole of the heavens. 
 
 If we assume that for interplanetary space 47re/(l + 47re) 
 is very nearly 1, we get for the electromotive forces values of 
 the order of 1 Daniell in 50 in., that is, very small values. 
 However, a term of the form const/p may have to be added 
 to the above value of <£. Its value depends on the quantity 
 of free electricity on the earth, although it does not vanish 
 with this quantity ; but the order of magnitude of the cal- 
 culated forces is not altered by the presence of this term. 
 
 III. 
 
 When a sphere of any arbitrary magnetic properties rotates Spherical 
 in a liquid, which is itself a conductor, and makes electric ^f^ 
 contact with the surface of the sphere, the sphere will induce 
 currents in the liquid. In general these no longer flow in 
 concentric spherical shells, but traverse the magnet. 
 
108 
 
 INDUCTION IN ROTATING SPHERES 
 
 The determination of these currents presents no further 
 difficulty apart from self-induction. I shall not enter in detail 
 into the calculations. Fig. 10 represents the simplest case. 
 A homogeneous magnetic sphere rotates about its magnetic 
 axis. The figure drawn represents the lines of flow in a 
 
 Fig. 10. 
 
 meridional section. The form of the lines of flow does not 
 depend upon the resistances of the magnet and the liquid. 
 But the intensity vanishes when either resistance becomes 
 infinitely great. 
 
 § 8. Solution for the Formulae of the Potential Law. 
 
 So far we have assumed for the induced electromotive 
 forces the expressions which Jochmann has deduced for them 
 from Weber's fundamental law. We shall now inquire what 
 changes the results undergo when we use the formulae which 
 follow from the potential law and are given in vol. lxxviii. of 
 Borchardt's Journal} 
 
 If 3E, |f, % denote the electromotive forces hitherto assumed, 
 %', J?', %' those which follow from the potential law, we have 
 
 1 Helmholtz, Wiss. Abhandl. vol. i. p. 702. 
 
II INDUCTION IN ROTATING SPHERES 109 
 
 ox 
 
 S'=g-<4(v*-%), 
 
 cz 
 
 But we saw on p. 67, that for all U, V, W occurring in 
 the investigation 
 
 4> = co(Vx - Uy). 
 
 We see at once that we may retain the previous solutions 
 unaltered as regards u, v, w } -*Jr, H. The only alteration which 
 must be made is to put for <//, the potential of the free 
 electricity, 
 
 <// = const, 
 
 and, when free electricity was not present originally, 
 
 </>'=0. 
 
 On an infinite sphere or plane plate we must have always 
 
 (//=0. 
 
 Maxwell obtained the same result, starting from the 
 formulae of the potential law for conductors at rest. If we 
 reject the terms aU -f @V + 7W in the expressions for the 
 electromotive forces in conductors in motion, the equations for 
 conductors at rest must also be altered, and the equation 
 
 </> = 
 then no longer holds. 
 
 § 9. Special Cases and Applications. 
 
 In conclusion, the formulae obtained will be applied to 
 some particular cases. 
 
 1. A single magnetic pole of strength 1 moves in a straight Magnetic 
 line parallel to an infinitely thin plane plate. Let the origin pol ]r above 
 of £ V, ? be taken at the foot of the perpendicular from the plate, 
 pole on the plate, and let the negative 77-axis be parallel to 
 
110 INDUCTION IN ROTATING SPHERES n 
 
 the direction of its motion. 1 Let the coordinates of the pole 
 be 0, 0, - c ; then its potential is 
 
 1 _1 
 
 Thus the induced potential of the first order becomes for 
 positive f 
 
 1 h J d v b h P + y\ r 
 
 Hence we get for the potential of the second order 
 
 k J p + v [Z' + V r J 
 
 In the same way the calculation may be continued. 
 We get for the current-functions of the first and second 
 orders 
 
 *=t~4-/i A a 
 
 
 hj (r* - c)(r -f c)r 
 where now r 2 = £ 2 -f v 2 + <?• 
 
 In the ?7-axis we have (since f = 0) 
 
 hj (r + c)r 
 thus 
 
 We may regard the point f=0, i^=0 as the centre of 
 
 .Displace- m m 
 
 ment of the the distribution : thus it appears displaced through a distance 
 
 induced 
 
 distribu- * a then becomes positive. 
 
 tion. 2 This result agrees exactly with that obtained by Jochmann. 
 
ii INDUCTION IN EOTATING SPHERES 111 
 
 ZTracjk in consequence of self-induction, and in fact lags 
 behind the moving pole by this distance. The same is true of 
 the whole distribution near the pole. 
 For infinite velocities we get 
 
 1 
 
 ^ = ~2^' 
 
 for very large values of 2irajk 
 
 °--x-d=||f* 
 
 1 k vU + c) ^ 
 r lira r(f - if) ' 
 
 2irr\ 2ira'!?+c 2 J' 
 Here also the abscissa of the point f = 0, -ty = is 
 
 lirac 
 
 7] = 
 
 but since this value is very large, and our formula holds only 
 for finite values of rj, the value of the distance must be regarded 
 as only an approximation. 
 
 The potential of the free electricity in the plate is 
 
 CO 
 
 
 
 so that for small velocities 
 
 0-«J&(*f- 
 
 3f r(r + c) 
 
112 
 
 INDUCTION IN ROTATING SPHERES 
 
 OC. 
 
 ZS -2 -4 
 
 OS 1. t.S 
 
 Fig. 11, a and h. — Pole moving in a straight line, * nat. size. 
 
INDUCTION IN ROTATING SPHERES 
 
 113 
 
 Fig. 11, c and d.— Pole moving in a straight line, f nat. size. 
 M. P. I 
 
114 INDUCTION IN ROTATING SPHERES II 
 
 Thus in this case the equipotential lines have the same 
 form as the lines of flow. For very large velocities we have 
 
 <t>=JLf d x v = d -S- ft., , 
 
 Y 27rjd£ 2irr{f~ v 2 ) 
 
 o 
 
 which formula is not applicable at infinity. 
 See Fig. 11. The formulae here developed are illustrated by Fig. 11, 
 pp. 112 and 113. The assumptions on which the diagrams 
 are based are the following : — 
 
 The plate is made of copper (thus tc= 227,000) and has 
 a thickness 2 mm. (thus k= 113,500). The distance of the 
 pole from it is 30 mm. The values of ty marked give absolute 
 measure when the strength of the pole is 13,700 mm-mgr-/sec. 
 In Fig. 11, a and l, p. 112, the velocity of the pole is 
 5 m/sec(a = 5000); here a represents the phenomenon when 
 self-induction is neglected, h when it is taken into account. 
 
 Fig. 11 c represents the phenomenon for a velocity of 
 100 m/sec, calculated by means of the formula for large values 
 of 2Trajk. It is true that for the value chosen the approxima- 
 tion is not very close. Fig. 11 d, p. 113, corresponds to an 
 infinite velocity of the pole. The electric equipotential lines 
 are also shown in this diagram. The values of the electric 
 potential marked are in millions of the units employed by us. 
 The connection between the various states is clearly shown 
 by the diagrams themselves. 
 Magnetic 2. A magnetic pole at rest is placed above a rotating 
 
 a rotating 6 infinite disc. Let the a?2-plane be taken so as to pass through 
 disc. the pole. In addition to xyz we introduce coordinates f 77 f , 
 
 of which the origin is the foot of the perpendicular let fall 
 from the pole on the disc. Further, let 
 
 f; = x-a, 7]=y, ?=z; 
 thus 
 
 day drj drj 3f 
 
 a is the distance of the pole from the axis of rotation ; let c be 
 its distance from the plate. Then 
 
 1 1 
 
ii INDUCTION IN EOTATING SPHERES 115 
 
 Hence 
 
 or since 
 
 CO 
 
 k J oco 
 
 £-<!-•■ 
 
 TO 
 
 _ 2-rra>a (d x 2ir(oa V ( 
 
 S + c 
 
 
 h r(r + c) 
 
 Hence the form of the lines of flow is independent of the 
 distance of the pole from the axis. 1 For the induction of the 
 second order we get 
 
 " 2 v */ m e+v 2 J' 
 
 +--KiK(4-<) 
 
 \ & y \( r a _ c 2)( r + c ) r + a ( r + C ) J ' 
 
 which formulae are meaningless at infinity. 
 
 When the angular velocity is small, if the inducing pole 
 is not very close to the axis, we may regard the point f = 0, 
 -v/r = as the centre of the distribution. Its ordinate is found 
 to be 
 
 2nrcoaG 
 
 Hence in the neighbourhood of the pole the distribution Rotation of 
 
 . . , ., , ,, , the induced 
 
 is turned through the angle distribu- 
 
 2tion. 
 7TG)C 
 
 k 
 
 in the direction of rotation of the disc. 
 
 1 As already found by Jochmann. 
 
116 INDUCTION IN ROTATING SPHERES II 
 
 Rectilinear 3. I shall now apply the formulas to another example. 
 
 currents 
 
 and nnlin 
 
 ited disc. 
 
 and tmlim- Suppose that above the rotating disc two wires are stretched 
 
 parallel to the aj-axis and are traversed in opposite directions 
 by equal currents of unit intensity. For a single current the 
 currents induced in the unlimited disc would become infinite. 
 
 Let the coordinates of the points in which the wires meet 
 the plane yz be 0, a, - c, and 0, a', -c'\ we then have for 
 positive values of z 
 
 , y — a _ , y — a 1 
 
 Y = tan -1 - tan ^ r . 
 
 A z + c z + c' 
 
 Hence it follows, by means of the formulas used before, if 
 r, r denote perpendicular distances from the wires, that 
 
 O x - — - x log - 
 
 * i= if log (i 
 
 For the potential of the free electricity in the plate we 
 get 
 
 ' r 
 
 <j> = toy log 
 
 so that the equipotential lines are straight lines parallel to 
 See Fig. the wires. In Fig. 12a the lines of flow are drawn for the 
 12 a - case where 
 
 c = c' = 1 mm, a = - a! = 2 mm. 
 
 Since, moreover, at infinity the currents become infinite, we 
 must suppose 2ir(ojh to be exceedingly small in order to get a 
 sufficient approximation in a finite region. 
 
 Further, as all the currents are closed at infinity we cannot, 
 from the case of an unlimited disc, directly draw inferences as 
 to a limited one. 
 
 Hence I shall calculate, by the method developed in § 7, 
 the currents in a limited disc under like conditions. Let the 
 radius of the disc be E. 
 
 The exact solution of the problem requires us to develop 
 rather complicated functions in series of sines and cosines. I 
 
INDUCTION IN ROTATING SPHERES 
 
 117 
 
 therefore assume that the perpendicular distance of the wires Rectilinear 
 conveying the currents is at a distance from the centre large c ^ ?? tf L d 
 compared with the radius of the disc, so as to simplify the disc, 
 calculations. 
 
 Fig. 12 a.— Rotating disc and rectilinear currents. 
 
 In the first place, suppose again 
 
 c = c' \ a = - a'. 
 
 ft) IT 
 
 Y\ = t x log 
 
 If we develop 
 
 in powers of the coordinates, and neglect higher powers of 
 the expression 
 
 ir 
 
 we get 
 +1 = 
 
 c' + a? 
 
 2axy 2cmfx{3c 2 - a 2 ) 
 
 + 
 
 c 2 + a 2 ^ 3(c 2 + a 2 ) 3 
 
 + ... 
 
 ap i sin 2&> (3c 2 - a )a ., . -i ■ a \ 
 — _i_ ^ — ^— p 4 sin 2(o - -A- sin 4w) . 
 
 r /?/„2 i ~2\3~ v * ' 
 
 c' + a" 
 
 6(r + a7 
 
118 
 
 INDUCTION IX ROTATING SPHERES 
 
 The corresponding ^r 2 * s (§ %> !•> conclusion) 
 
 See Fig. 
 12 6. 
 
 ft 
 
 ap- sin 
 
 cr + a 
 
 Hence we have 
 
 (3c 2 - cr)a 0/T) o . -, o • a \ 
 — .— /3"(E - sm 2w - j/r sm 4©). 
 
 6(c 2 + « 2 ) 8 
 
 <3r - cr) 
 6(c 2 + rr) 3, 
 
 sin 2o\B 2 - p 2 ) 
 
 Hence the form of the lines of flow is independent of the 
 ratio a : c, but the current is greatly dependent upon it. If 
 a = or a = c^/Sj it vanishes. If c.Kc^/3, the direction of 
 the current is the same as in the unlimited plate; if a>c /Jo, 
 it is the opposite. When we consider closely the distribution 
 of the forces which act, this at first sight astonishing result is 
 explicable. The form of the distribution is shown in Fig. 1 2 b. 
 
 In the same way the problem may be solved for any de- 
 sired position of the wires. "When one of them moves off to 
 
 Fig. 12 5.— Rotating disc and rectilinear currents, § nat. size. 
 
 infinity, the currents remain finite in the limited disc, and we 
 find on retaining the first two powers of the dimensions of 
 the disc 
 
 2 2 /o 9 *>\ 
 
 . cr - a , „ a{oc - or) 2 . ._, .->. 
 
 * " 8(7^ C ° S W(E " " P) - 12</ + cr/ ~ Sm 2&,(E - P) - 
 
 The connection with the previous result is easily seen. 
 
INDUCTION IN EOTATING SPHERES 
 
 119 
 
 A 
 
 Fig. 13, a and &.— -Rotating disc and rectilinear current, § nat. size. 
 
120 
 
 INDUCTION IN EOTATING SPHEEES 
 
 See Fig. 13, 
 
 Kotating 
 spherical 
 shells. 
 
 Execution 
 of experi- 
 ments. 
 
 In Fig. 1 3 two particular cases are represented. In a the 
 straight wire cuts the axis of rotation at a sufficient distance 
 from the disc, and in this case the second term above vanishes. 
 In h the wire lies in the plane of the disc, and in fact at the 
 distance from the disc at which it is represented in the figure 
 itself. 
 
 4. If measurements are to be made in experiments on the 
 rotatory phenomena of induction, very thin spherical shells 
 should be used; for in their case the calculations can be 
 easily and exactly performed. The simplest form of experi- 
 ment would be one in which such a spherical shell is made 
 to rotate under the influence of a constant force. The 
 rotation of the current planes might be demonstrated either 
 by the effect of the currents on a very small magnet, or better 
 by a galvanometric method. 
 
 As an example I shall calculate the angle of rotation and 
 the magnetic moment of the rotating spherical shell. 
 
 Suppose the shell to be of copper, let its radius be 50 mm., 
 its thickness 2 mm. ; since n = 1, i = 1, we have 
 
 ~ 4-7T Eft) 
 
 tan b = — . — , 
 3 h 
 
 and if T be the inducing force, we find the moment of the 
 shell to be 
 
 M _ T K 3 sinS 
 
 If q is the number of revolutions per second, 
 ft) = 2irq, 
 and since Jc = 113,500, we find 
 
 tan 8= 0-0116 0. 
 
 From the above the following table has been calculated :- 
 
 
 
 M 
 
 
 
 M 
 
 Q 
 
 6 
 
 T 
 
 Q 
 
 8 
 
 T 
 
 5 
 
 3°19' 
 
 3,614 
 
 80 
 
 42°51' 
 
 42,500 
 
 10 
 
 6°27' 
 
 7,178 
 
 90 
 
 46°13' 
 
 45,100 
 
 20 
 
 13°3' 
 
 .14,110 
 
 100 
 
 49°13' 
 
 47,310 
 
 30 
 
 19°10' 
 
 20,520 
 
 200 
 
 66°40' 
 
 57,360 
 
 40 
 
 24°53' 
 
 26,290 
 
 500 
 
 80°15' 
 
 61,570 
 
 50 
 
 30°6' 
 
 31,340 
 
 
 
 
 60 
 
 34°49' 
 
 35,680 
 
 00 
 
 90° 
 
 62,500 
 
 70 
 
 39°4' 
 
 39,380 
 
 
 
 
ii INDUCTION IN ROTATING SPHERES 121 
 
 Figs. 14 and 15, pp. 122 and 123, are intended to illustrate Rotating 
 the distribution of current in solid spheres rotating under sp heres and 
 the influence of a constant force perpendicular to the axis of constant 
 
 r r force. 
 
 rotation. 
 
 Here the closed circuits are all circles whose planes are 
 parallel to the axis of rotation. Hence if we know the 
 current-density in the equatorial plane it is very easy to deter- 
 mine it at all other points. But in our case ^ = 0, v = in See Figs. 
 the £?/-plane, and thus the current-density = w. The diagrams u and l5, 
 represent the density of the current in the plane in question 
 by means of the curves 
 
 w = const. 
 
 The values of w marked give absolute values when the 
 influencing force 
 
 T=289 T* ■ 
 
 The size of the spheres is that drawn (E = 50 mm.). 
 
 In Fig. 14 a copper sphere is illustrated making five 
 turns a second (in a neglecting self-induction). 
 
 In Fig. 15 a the same sphere is illustrated when making 
 fifty turns a second. 
 
 Fig. 15 5 shows the currents in an iron sphere making five 
 turns a second. Here the resistance of iron is taken to be six 
 times that of copper, and 4:ir6 = 200. We see that even with 
 the very moderate speed chosen an approximate representation 
 could not be obtained if we neglected self-induction. 
 
 6. There is a well-known experiment in which a conduct- stoppage of 
 ing sphere rotating between the poles of an electromagnet is spheres by 
 brought to rest by suddenly exciting the latter. The theory ele ^ tr0 " 
 of this experiment is very simple if we assume the magnetic 
 field to be uniform, neglect self-induction, and at every instant 
 treat the currents as steady. If T be the magnetic force 
 parallel to Ox, the external potential is 
 
 X = - Tp sin cos co ; 
 
 thus 
 
 i/r = — Tp 2 sin 6 sin co, 
 
122 
 
 INDUCTION IN ROTATING SPHERES 
 
 ii 
 
 yCL/. 
 
 Fig. 14, a and b.— Rotating copper sphere, five turns per second. § nat. size. 
 
INDUCTION IN EOTATING SPHERES 
 
 123 
 
 Fig. 15 o.— Rotating copper sphere, 50 turns a second, § nat. size. 
 
 Fig. 15 6.— Rotating iron sphere, five turns a second, § nat. size. 
 
124 INDUCTION IN ROTATING SPHERES 
 
 and hence the heat generated is (§ 6) 
 
 2ttE 5 TV 
 
 W = 
 
 15 
 
 If F be the moment of inertia of the sphere, co its velocity 
 at time t = 0, and if it rotate under no external forces, the 
 equation of its motion is 
 
 Fo> 2 2- t2t?5 
 
 15 k J 2 
 
 or 
 
 2tt T 2 E 5 
 
 15 F*r 
 
 CO = CO n . € 
 
 t. 
 
 If q be the mass of 1 cub. cm, of the material, 
 
 F = -§-£7rK 5 > 
 15* 
 
 thus 
 
 CO — COq ; € * 
 
 An analogous law holds when the sphere is set in motion 
 by the action of rotating magnets. 
 
 Spheres of different radii and spherical shells are set in 
 Matteucci's mo tion and brought to rest with equal velocity. This in 
 ment. fact corresponds with an experiment made by Matteucci. 1 
 
 The angle which the sphere traverses after excitation of 
 the electromagnet amounts to 
 
 oo 
 
 I codt = 
 
 4zOk 
 
 For strongly magnetic spheres we find 
 
 / 
 
 codt = ^ K co () 
 
 9T 2 
 
 1 Wiedemann, Galvanismus, § 878 ; Lehre von der EWktricitdt, 1885, vol. 
 iv. § 386, p. 322. 
 
INDUCTION IN ROTATING SPHERES 
 
 125 
 
 From the above the following table is calculated. In it 
 T is taken = 5000, which corresponds to an electromagnet of 
 medium strength. The initial velocity is taken: to be one turn 
 (2tt) per second. The angles described are given in turns. 
 The relative values hold for every T and every co . 
 
 
 CO 
 
 Material. 
 
 fudt 
 
 
 
 Aluminium .... 
 
 0-14 
 
 Iron ...... 
 
 0-16 
 
 Silver 
 
 0-27 
 
 Copper 
 
 0-31 
 
 German silver .... 
 
 3-90 
 
 Graphite ..... 
 
 27-2 
 
 Cone. sol. of copper sulphate . 
 
 about 544,000 
 
 7. Damping in a galvanometer. 
 
 Consider a magnet swinging inside a conducting spherical Damping in 
 shell ; and suppose it to be very small, or to have approxi- a 
 mately the form of a uniformly magnetised sphere, 
 its moment, then in the spherical shell 
 
 If M be 
 
 ometer. 
 
 M 
 
 X 
 
 sin 6 cos co. 
 
 Thus 
 
 »M-. ' . 
 
 Y= - sin sin w , 
 
 K p 
 
 and the heat generated per second is 
 w= 8ttMV/1 
 
 3 k \r ~R 
 
 where, as before, r denotes the inner and E the outer radius of 
 the spherical shell. 
 
 Let now <£ be the deflection of the needle from its position 
 of rest, and F its moment of inertia ; then its vibrations are 
 determined by the equation 
 
 ^i. ^6-r-^ CU 
 
 o ; 
 
126 INDUCTION IN EOTATING SPHERES 
 
 which may be written 
 
 ^j+y^w^Wo, 
 
 so that the rate at which the heat is generated is 
 
 \dt 
 and thus we have 
 
 4ttMV1 1 
 
 3 /cF\r E 
 
 If e be small, we thence obtain for the logarithmic decre- 
 ment of the needle 
 
 4tt 2 K - r /M 3 
 A/ = 
 
 3k Br \J TF ' 
 Aperiodic j n or( j er that the aperiodic state may occur, we must have 
 
 . MT 
 
 £> 1T' 
 or 
 
 E-r 
 
 Er 
 
 3* /TF 
 
 4^V M 3 ' 
 
 from which equation, for given values of T, Y, M, k, it is easy 
 to calculate the thickness of the damper necessary to ensure 
 that the aperiodic state may be attained. 
 
Ill 
 
 ON THE DISTRIBUTION OF ELECTRICITY OVER 
 THE SURFACE OF MOVING CONDUCTORS 
 
 (Wiedemann's Annalen, 13, pp. 266-275, 1881.) 
 
 If conductors charged with electricity are in motion re- 
 latively to one another, the distribution of free electricity at 
 the surface varies from instant to instant. This change pro- 
 duces currents inside the conductors which, on their part 
 again, presuppose differences of potential, unless the specific 
 resistance of the conductors be vanishingly small. Hence we 
 may draw these inferences : — 
 
 1. That the distribution of electricity at the surface of 
 moving conductors is at each instant different from that at the 
 surface of similar conductors at rest in similar positions. In 
 particular, the potential at the surface and inside is no longer 
 constant, so that a hollow conductor does not entirely screen 
 its interior from external influence when it is in motion. 
 
 2. That the motion of charged conductors is attended by 
 a continual development of heat. Hence continual motions of 
 such conductors are possible only by a supply of external work, 
 and under the sole action of internal forces a system of such 
 conductors must come to rest. 
 
 The changes which the motion of conductors compels us 
 to make in the conclusions of electrostatics are especially 
 noticeable in those cases where the geometrical relations be- 
 tween the surfaces are invariable, that is, for surfaces of revolu- 
 tion rotating about their axes. Such bodies will have a 
 tendency to drag with them in their motion electrically- 
 
128 DISTRIBUTION OF ELECTRICITY OX MOVING CONDUCTORS HI 
 
 charged bodies near them, and the same is true of charged 
 liquid jets. 
 
 The nature and magnitude of the phenomena indicated 
 will in the following be submitted to calculation. 
 
 In forming the differential equations we assume that the 
 only possible state of motion of electricity in a conductor is the 
 electric current. Hence if a quantity of electricity disappears 
 at a place A and appears again at a different place B, we 
 postulate a system of currents between A and B, not a motion 
 of the free electricity from A to B. The explicit mention 
 of this assumption is not superfluous, because it contradicts 
 another, not unreasonable, assumption. When an electric pole 
 moves about at a constant distance above a plane plate the 
 induced charge follows it, and the most obvious and perhaps 
 usual assumption is that it is the electricity considered as a 
 substance which follows the pole ; but this assumption we 
 reject in favour of the one above mentioned. Further, we 
 leave out of account all inductive actions of the currents 
 generated. This is always permissible, unless the velocity of 
 the moving conductors be comparable with that of light. 
 
 Let u, v, w be components of current parallel to the axes 
 of x, y, z ; cj) the total potential, h the surface-density, k the 
 specific resistance of a conductor, all measured in absolute 
 electrostatic units. Thus k is a time, in fact the time in 
 which a charge arbitrarily distributed through the conductor 
 diminishes to l/e 47r of its original value. If now we refer 
 everything to coordinates fixed in the conductor and consider 
 the motion in this conductor, we have 
 
 d(f> dcf) d(j) /1N 
 
 ox dy cz 
 
 dA<j) _ fdu dv die 
 dt \3a; dy dz 
 
 (2), 
 
 - — = ucosa-\-v cos b + w cose (3), 
 
 dt 
 
 -4*h = d fi + p (4), 
 
 c ni dn e 
 
 in which equation n t , n e denote respectively the internal and 
 
in DISTRIBUTION OF ELECTRICITY ON MOVING CONDUCTORS 129 
 
 external normal, and a, b, c denote the angles which n { makes 
 with the axes. From (1) and (2) we get 
 
 -7T. 
 
 C 1T = -^ V2 <MrV¥=(VW 
 
 4rr 
 
 Hence, if the density inside is not zero initially, it still 
 continually approaches this value, and cannot be again pro- 
 duced by electrostatic influences. Hence we have here 
 
 (5). 
 (6), 
 
 V 2 (£=0 
 
 
 Further, from (1) and (3), 
 
 
 dh d(j>i 
 dt dn i 
 
 
 by using equation (4) 
 
 
 4-7T dt\dni dnj 
 
 d 4>t 
 
 The equations (5) and (7) involve </> alone. Equation (5) 
 must be satisfied throughout space ; equation (7) at the surfaces 
 of all conductors. (/> is determined for all time by these equations 
 — which no longer involve a reference to any particular system 
 of coordinates — together with the well-known conditions of 
 continuity and the initial value of </>. In the differential co- 
 efficient dh/dt, h relates to a definite element of the surface ; 
 if the velocities of this element relative to any system of co- 
 ordinates be a, ft, 7, then the above equations will refer to 
 these coordinates, provided dh/dt be replaced by 
 
 dh dh dh dh 
 
 dt dx dy dz 
 
 We get for the heat generated in time St 
 
 SW = St/K(u 2 + v 2 + w 2 )dT, 
 
 \*l 
 
 ^ 
 
 = -f(j)8hds, 
 
 where ds denotes an element of surface, and the integrals are 
 to be taken, the first throughout the interior, the others over 
 M. P. K 
 
130 DISTRIBUTION OF ELECTRICITY ON MOVING CONDUCTORS III 
 
 the surface, of all the conductors. It is easy to prove in our 
 particular case that the equations used agree with the principle 
 of the conservation of energy, which has, however, been proved 
 true of them in general. 
 
 If k be very small, <£ may be expanded in ascending powers 
 of k. The individual terms of this expansion may be found 
 in the following way, if we regard the ordinary electrostatic 
 problem as solved. 
 
 Let <p x be for all time the potential corresponding to the 
 state of equilibrium for the existing charges and the positions 
 of the bodies at each instant, and let /^ be the density cor- 
 responding to </> . Then let cp 2 be determined so that V 2 <£ 2 = 0, 
 that at the surfaces of the inductors d^/d^ = kdhjdt, that the 
 conditions of continuity are satisfied, and that the sum of the 
 free electricity may vanish for each conductor. In the same 
 way in which (f> 2 is formed from <£ p let <£ 3 be formed from <£ 2 , 
 <j) i from cj) 3 , and so on ; then clearly <£ = ^ + <£ 2 + </> 3 + . . . re- 
 presents exactly the potential, provided the series converges. 
 The convergence of the series depends on the relation between 
 k, the dimensions of the conductors, and their velocities ; for 
 any value of k we can imagine velocities sufficiently small to 
 ensure convergence. For metallic conductors and terrestrial 
 velocities each term vanishes in comparison with the preced- 
 ing one. The special phenomena clue to electrical resistance 
 are here inappreciable, and the form of the currents alone is 
 of interest. Since cf> 1 is constant inside a conductor, and <£ 3 
 vanishes in comparison with (f> 2 , all the currents flow along 
 the lines of force of the potential cj). 2> and we have 
 
 dd> 2 (b 2 dd>o 
 
 KU = - —' , KV = - -!—= , KIV = ~ — — . 
 
 dx dy dz 
 
 We shall now confine ourselves to the case in which only 
 one conductor is in motion, and shall assume this to be a solid 
 of revolution rotating about its axis. We refer our investiga- 
 tion to a system of coordinates fixed in space, of which the 
 2-axis is the axis of rotation. In addition, we employ polar 
 coordinates p, co, with the same axis. Let T be the time 
 of one turn. The conditions which </> must satisfy in the 
 conductor are in this case: (1) inside V 2 ^> = ; (2) at the 
 
ill DISTRIBUTION OF ELECTRICITY ON MOVING CONDUCTORS 131 
 
 surface, d<l> i /dn i = K(dh/dt)+(2'ine/TXdh/dco), where h now refers 
 to a point fixed in space. When the conductor rotates with 
 uniform velocity under the influence of a potential independent 
 of the time, after the lapse of a certain time a stationary state 
 is reached, the condition for which is dh/dt = ; and thus 
 d(f> i /dn i = (27r K /T)/(dh/d(o). 
 
 As an example we shall consider the case of a spherical 
 shell rotating with constant velocity about a diameter. Let 
 its external radius be E, its internal radius r. Suppose the 
 external potential <J>, under whose influence the motion takes 
 place, developed in a series of spherical harmonics inside the 
 spherical shell. The actions produced by the separate terms 
 may be added, so that we may limit the investigation to one 
 term. Let O = A ni (p/ll) n cos icoT ni (0). Denote by <j> the 
 potential of the electrical charge itself, which is induced on 
 the spherical shell ; in particular denote it by cj) 1 in the inside 
 space, by <j> 2 in the substance of the shell, by <f> 3 in the outside 
 space. In addition to the general conditions for the potential 
 of electrical charges, </> must satisfy the condition that for 
 p = r and p = E 
 
 d® d(j>._ /c_a/a^_a^ 
 
 dp dp~ Wfa\ty dp 
 All these requirements are fulfilled when we put 
 
 <*>! = 
 
 £ j(Acosio> + Bsint<i>)P ni (60 + IX J(A'costa> + B'sim'a>)P„4(0), 
 
 £ J(Acosia> + Bsinia))P Mi (5) + ( - )(A'cosi<» + B'sim'<»)P„i(0): 
 
 </>3 = 
 
 ■r\«-+i /r\ n+1 
 
 (Acos^co + Bsin*©)P wi (0) + - VA'cosi© + B'siiUG>)P Bi (0). 
 
 krJ \p/ 
 
 For the general conditions are at once satisfied, and the 
 
132 DISTRIBUTION OF ELECTRICITY ON MOVING CONDUCTORS ill 
 
 two boundary conditions give, when we equate factors of 
 cosi'o) and sim co, four linear equations for the four constants 
 A, B, Af 3 B'. If these latter are satisfied, so also will be the 
 former. Using the contractions k/2T = a, r/K = e, we get for 
 these equations — 
 
 A ni .n = - ?iA-(2wH-l)a2B + (n+l)e ,l+1 A / * 
 
 = (2n+l)aiA - riB * + (?i+l)e n+1 B', 
 
 A ni .ne n = - ne n A * +(»+ 1)A' +(2ra + l)aiB', 
 
 = * -/ze ,i B-(2/i+l)azA / +(?i+l)B / . 
 
 These equations determine the four constants uniquely. 
 "Without actually performing the somewhat cumbrous solu- 
 tion it is easy to recognise the correctness of the following 
 remarks : — 
 
 1. When a = 0, A = - A ni , A' = B' =B = 0, as must be the 
 case for a sphere at rest. 
 
 2. If a be finite but very small, then A+A ni and A f are 
 of order or, B, B r are of order a. Hence it follows that the 
 chief points of the phenomenon are these. The distribution 
 of the charge on the outer surface (the form of the lines of 
 constant density) is not changed by the rotation (of course 
 only for the separate terms of the development) ; but the 
 charge appears rotated in the direction of the rotation of the 
 sphere through an angle of order a, and the density has 
 diminished by a small quantity of order a". In addition a 
 charge makes its appearance on the sphere forming the inner 
 boundary, and its type is similar to that of the first charge ; 
 its density is of order a, and it is turned relatively to the first 
 charge through a small angle w/2i. In the substance of 
 the shell as well as inside we get differences of potential of 
 order a. 
 
 3. If a be large, B, B r are of order lja, A, A ; of order 
 l/a 2 . As the velocity increases the charge on the external 
 surface finally appears turned through the angle 7r/2i; its 
 density is small, of order l/a, and the charge on the inner 
 spherical surface is like it as regards type, position, and density. 
 In the ultimate state </> = everywhere, and then we have 
 the external potential in the substance and the interior of the 
 spherical shell ; the currents everywhere flow in the lines of 
 
Ill DISTRIBUTION OF ELECTRICITY ON MOVING CONDUCTORS 133 
 
 force of that potential. The free electricity, which by the 
 currents is brought to the boundary, is by the rotation of the 
 sphere carried back to its starting-point so quickly that the 
 density remains infinitely small. A screening of the internal 
 space no longer takes place. 
 
 In particular cases the calculation itself becomes very 
 simple. In the first place, for a solid sphere e = 0. If we 
 put tan 8 =(2n-\- l)ai/n, then 8/i is the angle through which 
 the distribution appears to be turned, and the density of the 
 charge is to that induced on the sphere at rest as cos 8 : 1. 
 When the sphere rotates under the influence of a uniform 
 force perpendicular to the axis of rotation, the distribution on 
 it is represented by a spherical harmonic of the first degree. 
 The lines of flow are parallel straight lines whose direction for 
 small velocities of rotation is perpendicular to the axis and 
 to the direction of the force, but for large velocities appears 
 turned from the latter direction through an angle whose 
 tangent = 3 a = %/e/T. For a rotating cylinder the circumstances 
 are quite similar ; the angle of rotation is here found to be 
 2a = fc/T. 
 
 Secondly, suppose e nearly unity, that is, the thickness d 
 
 of the spherical shell infinitely small. We must then suppose 
 
 the specific resistance k to be so small that KJd = k, the specific 
 
 superficial resistance, may be a finite quantity. With this 
 
 assumption the tangent of the angle of rotation becomes 
 
 (2n+l)i JcR 
 
 generally tan 8 = r— ; tt-t^, and in the particular case of 
 
 6 J 2n(n+l) T r 
 
 a uniform force tanS = |X'K/T. 
 
 Under similar circumstances we find for a thin hollow 
 cylinder tanS = /jE/T, so that in this case the rotation is 
 greater for the cylinder than for the sphere, although for a 
 solid cylinder it was less. The density in the last case also 
 is to that for the sphere in the ratio cos 8:1. 
 
 As an illustration of the results of the calculation I have 
 in the accompanying diagram represented the flow of electricity 
 in a rotating hollow cylinder, whose internal is one-half its 
 external radius. The time of one turn is twice the specific 
 resistance of the material. The arrow A marks the direction 
 of the external inducing force, the arrow B that of the force 
 in the inside space ; the remaining two arrows indicate the 
 
134 DISTRIBUTION OF ELECTRICITY OX MOVING CONDUCTORS ill 
 
 position of the charges on the outer and inner curved surfaces. 
 The lines occupying the substance of the cylinder represent 
 the lines of flow. 
 
 It remains to inquire in what practically realisable cases 
 the effects discussed could become appreciable. Clearly they 
 
 attain a measurable value 
 when the angle of rota- 
 tion becomes measurable, 
 and this occurs when for 
 solid bodies the quantity 
 tc/T, or when for very 
 thin shells the quantity 
 &K/T, has a finite value. 
 Here E denotes the mean 
 distance of the shell from 
 the axis of rotation. Since 
 T cannot well become less 
 than t Jq- second, k must reach at least several hundredths of 
 a second. Thus it is obvious that in metallic conductors, for 
 which k is of the order of trillionths of a second, the rotation 
 phenomenon can never be appreciable. On the other hand, it 
 is obvious that even at moderate velocities no measurable 
 charge can be formed upon insulators such as shellac and 
 paraffin, for which k is many thousand seconds. But for 
 certain other substances, which lie at the boundary between 
 semi-conductors and bad conductors, the phenomenon should 
 be capable of complete demonstration ; e.g. for ordinary kinds 
 of glass, for mixtures of insulators with conductors in form of 
 powder, for liquids of about the conductivity of petroleum, oil 
 of turpentine, or mixtures of these with better conducting 
 ones, etc. As the specific resistance k is connected in a simple 
 way with the angle of rotation, measurements of the latter 
 might serve to determine the former. However, in bodies of 
 the necessary resistance the phenomena of residual charge 
 occur, and our differential equations only hold roughly for 
 these. The effect of a residual charge will always be to make 
 the constant k appear less than it is found to be from observa- 
 tions on steady currents, and less too by an amount increasing 
 with the velocity of rotation. The dielectric displacement 
 acts in the same sense, since it is equivalent to partial con- 
 
in DISTRIBUTION OF ELECTRICITY ON MOVING CONDUCTORS 135 
 
 duction without resistance. For very thin shells these dis- 
 turbing influences disappear. 
 
 I know of no previous experiments which might serve to 
 illustrate the effects investigated. Hence I have performed 
 the following one. Above a plate of mirror glass, of relatively 
 high conductivity (by a different method k had been found to 
 be = 4 seconds), a needle 1 cm. long was suspended by a wire 
 and allowed to execute torsional vibrations ; the moment of 
 inertia of the needle was sufficiently increased by means of 
 added weights, and at its ends it carried two horizontal brass 
 plates, each 3 cm. long and 2 cm. broad. Their distance a 
 from the glass plate could be varied. When the needle was 
 electrically charged the brass plates acted on the opposing 
 glass surface as condensers ; the bound electricity was com- 
 pelled to follow the motion of the needle, and ought, according 
 to the preceding, to damp the vibration of the needle. Now 
 such a damping actually showed itself. The needle was con- 
 nected with a Leyden jar, of which the sparking distance was 
 0'5 mm., whilst a was 2 mm. The needle was found to return 
 to its position of rest without further oscillation, though pre- 
 viously it had vibrated freely ; even when a was increased to 
 3 5 mm., the increase of the damping at the instant of charging 
 was perceptible to the naked eye. And when I charged the 
 needle by a battery of only 50 Daniell cells, while a was 
 2 mm., I obtained an increase of damping which could be 
 easily perceived by mirror and scale. It was impossible to 
 submit the experiment to an exact computation, but by making 
 some simplifying assumptions I was able to convince myself 
 that theory led to a value of the logarithmic decrement of the 
 order of magnitude of that observed. 
 
 As we have shown, we possess, in a conductor rotating 
 under the influence of external forces, a body at the surface 
 of which the potential has different values, which it again 
 resumes after a slight disturbance. Hence if we connect two 
 points of the surface by a conductor, a current flows through 
 the connection ; if we connect the points with two conductors, 
 these may as often as we please be raised to different potentials. 
 If we use metallic discs as the rotating bodies the differences 
 of potential obtainable by means of possible velocities of rota- 
 tion are indefinitely small ; but if we use very bad conductors 
 
136 DISTRIBUTION OF ELECTRICITY ON MOVING CONDUCTORS in 
 
 the differences of potential even for moderate velocities are 
 of the order of the inducing differences. Induction machines 
 without metallic rubbers are based on this principle. The 
 theoretically simplest of such machines consists of a cylinder 
 rotating under the influence of a constant force. How far, 
 however, the explanation here indicated is a complete one 
 must for the present remain a moot question. 
 
IV 
 
 UPPEB LIMIT FOE THE KINETIC ENEBGY OF 
 ELECTEICITY IN MOTION 
 
 (Wiedemann's Annalen, 14, pp. 581-590, 1881.) 
 
 In a previous paper 1 I have deduced, from experiments on the 
 strength of extra -currents, the conclusion that the kinetic 
 energy of an electric current of magnetic strength 1 in a 
 copper conductor is less than 0*008 mg. mm. 2 /sec. 2 This con- 
 clusion, however, could only be drawn on the supposition that 
 a certain relation did not exist between the specific resistance 
 of metals and the density of electricity in them. In the 
 present paper I propose to describe an experiment which I 
 have made with a view to demonstrating kinetic energy in 
 electrical flow, but equally with a negative result. This 
 experiment, however, has advantages over the previous ones : 
 for, in the first place, it is more direct ; secondly, it gives a 
 smaller value of the upper limit ; and thirdly, it gives it 
 without limitation of any kind. 
 
 Suppose a thin metal plate of the form shown in Fig. 1 7 
 to be traversed by as strong a current as possible between the 
 electrodes A and B ; further, let the points G and D be con- 
 nected with a delicate galvanometer, and let the system be so 
 adjusted that no current flows through the galvanometer. Let 
 the plate be made to rotate about an axis through its centre 
 and perpendicular to its plane. The current will now tend to 
 deviate laterally from the direction AB in case electricity in 
 motion exhibits inertia, for the same mechanical reason that 
 the rotation of the earth causes the trade winds to deviate 
 
138 KINETIC ENERGY OF ELECTRICITY IN MOTION [II] iv 
 
 from the direction of the meridian. The consequence of this 
 tendency will be a difference of potential between the points 
 C and D, and a current through the galvanometer. This 
 current must be reversed when the direction of rotation is 
 reversed ; when the rotation is clockwise and the current flows 
 in the plate from A to B, then the current through the galvan- 
 ometer outside the plate must flow from D to G, as shown by 
 the arrows. 
 
 An action of the kind mentioned must occur, whatever be 
 the nature of the electric current, provided only that with it 
 a motion of an inert mass is connected, which changes its 
 direction when the current is reversed. The difficulty of the 
 experiment consists in preparing four connections, sufficiently 
 certain and steady, even with rapid rotations ; this difficulty 
 I have overcome to such an extent that one of the most deli- 
 cate galvanometers could be used when the velocity was 30 
 turns per second, and the difference of potential between A 
 and B was that of 1 Daniell. No deflection of the needle 
 could be detected which would indicate the existence of electric 
 inertia. Basing my calculations on Weber's hypothesis, I am 
 able from my experiments to infer by the method given below 
 that fju, the kinetic energy of a current of magnetic strength 1 
 in a cubic millimetre of a silver conductor, cannot greatly 
 exceed 0*00002 mg. mm. 2 /sec 2 . 
 
 As regards the method of experimenting I may mention 
 the following. The metal plate used was the silvering of a 
 glass plate, produced by Liebig's process. Its form is shown 
 in Fig. 17 ; the distance AB was about 45 mm., the distance 
 GD 25 mm. The leads were soldered to small platinum 
 plates, and these were pressed into contact with the silvering 
 by small screws penetrating the glass plate ; a layer of gold- 
 leaf was introduced between the silvering and the plates, so as 
 to produce a more uniform contact. The electrical resistance 
 was at first 5*4 Siemens units in the direction AB, and 3*5 
 Siemens units in the direction GD. From some unexplained 
 causes these resistances diminished in time, and after some 
 weeks were found to be 4*8 and 3*1 Siemens units respectively. 
 From the ratio of these resistances and from special experi- 
 ments, it followed that the resistances of the contacts at the 
 leads did not amount to any appreciable fraction of the whole 
 
KINETIC ENERGY OF ELECTRICITY IN MOTION [II] 
 
 139 
 
 ^a® 
 
 Fig. 17. 
 
 resistance. The system was adjusted to bring the needle to 
 zero by scraping off the silver at various points of the edge ; 
 but as a sufficiently accurate adjustment from various causes 
 could not be permanently obtained, shunts 
 of several hundred Siemens units resistance 
 were introduced between A and C and 
 between C and B, and by their adjustment 
 the needle could always be brought to zero, 
 in so far as that seemed desirable. 
 
 The glass plate was fastened to a brass 
 disc so as to permit of a rapid rotation ; 
 the silvered surface faced the disc, and 
 was only separated from it by the thin- 
 nest possible air film. The disc itself was 
 at the end of a horizontal steel spindle, 
 which was set in two bearings in such a way that its two 
 ends were free. The connection to the galvanometer was 
 made at the glass plate itself; that to the battery, which 
 supplied the current, at the other end of the spindle ; the 
 connections to the points A and B were formed by the spindle 
 itself and by a wire lying in a canal bored through the spindle. 
 The arrangement by which the last connection was effected 
 between the moving and fixed parts is shown in Fig. 18. A 
 fine platinum wire passes through a piece of glass tube drawn 
 out to a very fine point and very exactly centred. A second 
 platinum wire is wound round the tube ; and the latter, 
 together with the wires, passes through one vessel of mercury 
 and enters a second in such a way that the first-mentioned 
 wire rotates in the mercury of the last vessel, and the second 
 wire in the mercury of the first vessel. The glass tube was 
 
 fastened at one end of the 
 spindle by sealing-wax to 
 the glass plate ; at the 
 other end to the spindle 
 itself. As the diameter of 
 the windings of the wire B was only about ^ mm., the 
 platinum moved relatively to the surrounding mercury at 
 a speed of only 160 mm/sec, even with a velocity of rota- 
 tion of 100 revolutions per second. The result was good, 
 for even with the latter velocity there was no appreciable 
 
 Fig. 18. 
 
140 KINETIC ENERGY OF ELECTRICITY IN MOTION [II] IV 
 
 transition-resistance ; and the disturbances due to heating were 
 only just perceptible, and small compared with other unavoid- 
 able ones. The spindle was rotated by a cord, which con- 
 nected it with the quickest spindle of a Becquerel's phosphor- 
 oscope, so that it revolved at double the speed of the latter. 
 The crank of the phosphoroscope was turned by hand, one 
 turn of it corresponding to 290 revolutions of the spindle. 
 As the whole apparatus was built as lightly as possible, even 
 large velocities could be rapidly generated and again annulled. 
 The galvanometer used was of Siemens' pattern, with an 
 astatic system of two bell magnets and four coils, with a total 
 resistance of about 7 Siemens units. By aid of external 
 magnets the arrangement could be made as astatic as desired ; 
 in the final experiments the sensitiveness was such that a 
 difference of potential of one-millionth of a Daniell between 
 the points D and C gave a deflection of 32 scale divisions. 
 The motion of the needle was aperiodic ; a second position of 
 rest was reached in about 8 seconds with an accuracy sufficient 
 for the experiments described. The current was supplied from 
 a Daniell cell and measured by a common tangent galvan- 
 ometer. A commutator was placed in the connections to both 
 galvanometer and battery. 
 
 After the current had been allowed to flow through the 
 plate until no further heating took place, the needle was 
 brought nearly to its natural position of rest by adjusting the 
 external resistances between A, C, and B. Then the crank of 
 the phosphoroscope was made to turn once round as uniformly 
 as possible, an operation which on the average required 8 to 9 
 seconds, and was terminated by an automatic catch. But 
 after the rotation ceased the needle hardly ever returned to 
 the original position of rest, but to a new position of rest. 
 As soon as this was attained (i.e. after 6 to 8 seconds) it was 
 read off. The deviation from the original position of rest I 
 shall call the permanent deflection ; by the instantaneous de- 
 flection will be meant the distance of the needle at the end of 
 the rotation from the mean of the initial and final resting 
 points. We regard the instantaneous deflection as a measure 
 of the current whose causes act only during the rotation, e.g. 
 the influence of inert mass ; while we ascribe the permanent 
 deflection to disturbances which continue to act after rotation 
 
iv KINETIC ENERGY OF ELECTRICITY IN MOTION [II] 141 
 
 has ceased. This method of calculation could only lay claim 
 to accuracy if the rotation were uniform and the permanent 
 deflection small; but the disturbances were too various and 
 the deflections too irregular to permit of fuller discussion. 
 
 The first experiments already showed that if there was 
 any effect of inertia, it did not much exceed the errors un- 
 avoidably introduced by disturbing causes. In order to detect 
 such an effect, and to find as small as possible a value of its 
 upper limit, I took a set of eight observations together, in 
 which the direction of rotation was changed between every 
 two observations : the connection to the galvanometer was 
 reversed every other observation, and the current in the plate 
 was reversed between the first four and the last four observa- 
 tions. Such a set of eight observations I call an experiment. 
 By suitably combining the observations it would be possible 
 to calculate the mean effect of the various disturbing causes 
 for each experiment. For the deflections must include, and 
 we should be able to eliminate from them : — 
 
 1. A part which changes sign only when the connection 
 with the galvanometer is reversed, but not when the direction 
 of rotation or the connection to the battery is changed. It 
 could only be due to an electromotive force generated by the 
 rapid rotation at the point of contact of the galvanometer 
 circuit. In so far as this force was thermoelectric the cor- 
 responding deflection must have been permanent. 
 
 2. A part whose sign depended on the direction of the 
 galvanometer and battery connections, but not on the direction 
 of rotation. This could be due to various causes : — 
 
 (a) The straining of the plate by the considerable centri- 
 fugal force, whose effect could only appear in the momentary 
 deflection. 
 
 (b) An uniform change of temperature of the whole plate 
 owing to rotation, whose effect would be felt in the permanent 
 deflection. 
 
 (c) A change in the ratios of the resistances AC/BC and 
 AD/BD during the experiment, due to external causes. In 
 fact the resting-point of the needle changed slowly even when 
 there was no rotation, but continuously and so much that the 
 error produced was of the order of the others. The effect was 
 felt in the permanent deflection. 
 
142 KINETIC ENERGY OF ELECTRICITY IN MOTION [II] iv 
 
 3. A part whose sign depended, on the direction of rotation 
 as well as on the connections. Thus : — 
 
 (a) If such a part occurred in the momentary deflection 
 no other cause perhaps could be assigned except the inertia of 
 the electricity moved. 
 
 (&) In the permanent deflection such a part might be 
 produced, because during rotation two diagonally opposite 
 branches of the bridge moved in front of the other two, and 
 thus were more strongly cooled by the air-currents than the 
 latter. As the conducting layer of silver was very close to 
 the brass disc, I had not anticipated such an effect ; but it 
 proved to be very large, and was especially inconvenient, since 
 it only differed from the effect of inertia in lasting for a time 
 after the rotation ceased. By surrounding the plate and brass 
 disc by cotton wool and by a drum of paper I was able to 
 diminish this disturbance considerably ; and still further 
 by hermetically sealing the paper drum by a coating of 
 paraffin. But even then the disturbance did not completely 
 disappear. 
 
 I performed two series, each of twenty, of the experiments 
 described. They differed in the strength of the current 
 employed, in the sensitiveness of the galvanometer, and especi- 
 ally in this, that in the first series the paraffin coating 
 mentioned was wanting. The second series was by far the 
 better, and what follows refers to it alone. To it also refers 
 the statement made above respecting the sensitiveness of the 
 galvanometer. The strength of the current was 1*17 mg* 
 mm^/sec magnetic units ; the velocity of rotation, according 
 to what has been said above, was on the average 290/8j= 34 
 turns per second. The galvanometer deflection at the end of 
 the rotation amounted on the average to 10 to 15 scale divi- 
 sions, and in the succeeding seconds changed mostly by only 
 a few divisions. The greater part of this deflection cor- 
 responded to the causes (2 V) and (2 c), which could no longer be 
 separated: the effect of disturbances (1) and (3 b) was found 
 to be 2 to 4 scale divisions ; the disturbance (2 a) was small. 
 The practicability of the method followed from the fact that 
 the separate disturbances were found to be of the same sign and 
 of the same order of magnitude in all the experiments, almost 
 without exception. The following are the twenty values, in 
 
IV KINETIC ENERGY OF ELECTRICITY IN MOTION [II] 143 
 
 scale divisions, obtained for the part of the deflection mentioned 
 
 mder the head (3 a) : — 
 
 
 
 
 
 + 3-6, -1-0, -0-0, 
 
 -2-7, 
 
 -M, 
 
 + o-i, 
 
 -0-6, 
 
 + 0-8, -1-1, +0-2, 
 
 -0-4, 
 
 + 0-5, 
 
 + 0-7, 
 
 + 0-5, 
 
 + 0-8, +1-2, +1-1, 
 
 + 0-7, 
 
 + 0-6, 
 
 + 0-7. 
 
 
 The mean of these values is +0*2 3. The difference from 
 zero is somewhat larger than the probable error of the result, 
 but perhaps the cause of the difference is to be looked for in 
 the somewhat arbitrary calculation of the momentary deflection 
 rather than in any physical phenomenon. The effect of inertia 
 should have been a negative deflection, according to the cir- 
 cumstances of the experiment and the sign used ; thus such 
 an effect could not be detected at all. If we attribute the 
 constant deflection 0*23 to some other cause, and calculate the 
 error of the experiments from zero, we still find that the odds 
 are 14 to 1, that no deflection exceeding ^ a scale division, 
 and 3480 : 1, that no deflection exceeding 1 scale division 
 existed, which could be attributed to an inert mass. 
 
 In calculating the experiment on the basis of Weber's 
 hypothesis, for simplicity I assume that the mass of a positive 
 unit is the same as that of a negative unit, and that both 
 electricities flow in the current with equal and opposite velo- 
 cities. Let m be the mass of the electrostatic unit, v the 
 velocity with which it is compelled to move in the axis of the 
 plate AB or in a parallel straight line ; and let co be the 
 velocity of rotation of the plate. Then the apparent force due 
 to rotation, which acts on the unit perpendicular to its path, 
 is equal to 2?nvco + C, where C is the centrifugal force at the 
 position of the unit. The unit of opposite sign in the same 
 position is subject to a force — 2mvco-\-C The sum of the 
 two forces, 2C, represents a ponderomotive force, namely, the 
 increase in the amount of the centrifugal force acting on the 
 material of the conductor, due to increase of its mass by that 
 of the electricity ; but the difference, X = 4mvw, is in fact the 
 electromotive force which we tried to detect by the galvan- 
 ometer. Now m is equal to M, the mass of all the positive 
 and negative electricity contained in one cubic millimetre, 
 divided by the number of electrostatic units contained in one 
 cubic millimetre ; this number again is equal to i, the current- 
 
144 KINETIC ENERGY OF ELECTRICITY IN MOTION [II] IV 
 
 density, measured electrostatically, divided by the velocity v ; 
 hence m = Mv/i and X = 4co.Mv 2 /i = 4:ico . Mv 2 /i 2 . Now with- 
 out altering the equation we may use magnetic units on both 
 sides ; if we do so, Mv 2 /i 2 = M^/*o is that quantity which in 
 the introduction is denoted by [i, and thus X = 4//i&). Here 
 we put for the current-density i the quotient of the total 
 current-strength J by q, the cross-section of the conductor, and 
 for the electromotive intensity X the quotient of <£, the differ- 
 ence of potential between the points C and D, by b, the breadth 
 of the plate ; if we call its mean thickness cl, then we get 
 cp = 4:/jLj(ob/q = 4:jj.Ja>/d, or, as we require fi, 
 
 cj>q <pd 
 
 4:Jbco 4J&) 
 
 We may approximately calculate the cross-section q or 
 the thickness d from the amount of silver deposited ; but it is 
 more rational, as well as more accurate, to determine it from 
 the electrical resistance of the plate, for this resistance depends 
 directly on the mean velocity with which the electricity flows 
 through the plate, and we are concerned with just this velocity 
 and only indirectly with the cross-section. As the conduction 
 was doubtless metallic we must take for the specific resistance 
 of the conducting material that of silver ; from the length of 
 the plate = 45 mm., and its mean resistance = o'l Siemens 
 units, we get the required cross-section ^ = 0"00014 mm 2 , and 
 the corresponding thickness cZ=0 # 6xlO~ 6 mm. It is true 
 this thickness is only about one-tenth of that deduced from 
 the amount of silver deposited ; but this only shows what 
 was very probable before, namely, that the silver is very un- 
 equally distributed over the glass. Employing the value thus 
 obtained for the thickness, we put J = 1*17 nig* mm* sec -1 , 
 co = 2-7T x 34 sec -1 , (/> = 1 scale division = 1/32 x 10 6 of a 
 Daniell = 3300 mg* mm 1 sec ; 1 ; and thus find yLt=0-0000185 
 mm 2 . Thus \x appears as an area, namely, energy divided by 
 the unit of the square of a magnetic current-density and by 
 the unit of volume. Since the value (/> = 1 scale division was 
 found to be extremely improbable, the statement made in the 
 introduction is justified. Even if the assumptions made in 
 calculating the experiments were only very rough approxima- 
 
iv KINETIC ENERGY OF ELECTRICITY IN MOTION [II] 145 
 
 tions, it would still remain unlikely that even a much narrower 
 limit should be exceeded. 
 
 It is worth noting that we do know electric currents, 
 which certainly possess kinetic energy [of matter] considerably 
 exceeding in magnitude the limit determined, namely, currents 
 in electrolytes. From the chemical equivalent of a current 
 of strength 1 in magnetic measure, and from the migration 
 number of silver nitrate, it is easy to calculate the velocities 
 with which the atomic groups Ag and N0 3 move in a solu- 
 tion of this salt of given concentration, when a current of 
 unit density flows through the solution. Hence the kinetic 
 energy of this motion follows, and in fact we find approximately 
 for solutions of average concentrations fi=Q'00*78/n mm 2 , 
 when there are n parts by weight of salt to 1 of water. Thus 
 if the experiment described could be performed with an elec- 
 trolyte under the same conditions as with a metal, it would 
 give a positive result ; but as a matter of fact, the resistance 
 and decomposition of the electrolyte prevent our obtaining 
 anything like equally favourable conditions of experiment. 
 
 M. P. 
 
OX THE COXTACT OF ELASTIC SOLIDS 
 
 {Journal fur die reine und angewandtc Mathematik, 92, pp. 156-171, 1881.) 
 
 In the theory of elasticity the causes of the deformations are 
 assumed to be partly forces acting throughout the volume of 
 the body, partly pressures applied to its surface. For both 
 classes of forces it may happen that they become infinitely 
 great in one or more infinitely small portions of the body, but 
 so that the integrals of the forces taken throughout these 
 elements remain finite. If about the singular point we describe 
 a closed surface of small dimensions compared to the whole 
 body, but very large in comparison with the element in which 
 the forces act, the deformations outside and inside this surface 
 may be treated independently of each other. Outside, the 
 deformations depend upon the shape of the whole body, the 
 finite integrals of the force-components at the singular point, 
 and the distribution of the remaining forces ; inside, they 
 depend only upon the distribution of the forces acting inside 
 the element. The pressures and deformations inside the sur- 
 face are infinitely great in comparison with those outside. 
 
 In what follows we shall treat of a case which is one of 
 the class referred to above, and which is of practical interest, 1 
 namely, the case of two elastic isotropic bodies which touch 
 each other over a very small part of their surface and exert 
 upon each other a finite pressure, distributed over the common 
 area of contact. The surfaces in contact are imagined as 
 perfectly smooth, i.e. we assume that only a normal pressure 
 
 1 Cf. Winkler, Die Lchre von der Elastic itdt unci Festigkeit, vol. i. p. 43 (Prag. 
 1867) ; and Grashof, Tfieorie der Elasticitdt und Festigkeit, pp. 49-54 (Berlin, 
 
 1878). 
 
v CONTACT OF ELASTIC SOLIDS 147 
 
 acts between the parts in contact. The portion of the surface 
 which during deformation is common to the two bodies we 
 shall call the surface of pressure, its boundary the curve of 
 pressure. The questions which from the nature of the case 
 first demand an answer are these : What surface is it, of 
 which the surface of pressure forms an infinitesimal part l l 
 What is the form and what is the absolute magnitude of the 
 curve of pressure ? How is the normal pressure distributed 
 over the surface of pressure ? It is of importance to determine 
 the maximum pressure occurring in the bodies when they are 
 pressed together, since this determines whether the bodies will 
 be without permanent deformation ; lastly, it is of interest to 
 know how much the bodies approach each other under the 
 influence of a given total pressure. 
 
 We are given the two elastic constants of each of the 
 bodies which touch, the form and relative position of their 
 surfaces near the point of contact, and the total pressure. We 
 shall choose our units so that the surface of pressure may be 
 finite. Our reasoning will then extend to all finite space ; 
 the full dimensions of the bodies in contact we must imagine 
 as infinite. 
 
 In the first place we shall suppose that the two surfaces 
 are brought into mathematical contact, so that the common 
 normal is parallel to the direction of the pressure which one 
 body is to exert on the other. The common tangent plane 
 is taken as the plane xy, the normal as axis of z, in a rect- 
 angular rectilinear system of coordinates. The distance of 
 any point of either surface from the common tangent plane 
 will in the neighbourhood of the point of contact, i.e. through- 
 out all finite space, be represented by a homogeneous quad- 
 ratic function of x and y. Therefore the distance between 
 two corresponding points of the two surfaces will also be 
 represented by such a function. We shall turn the axes of x 
 and y so that in the last-named function the term involving 
 xy is absent. 
 
 1 In general the radii of curvature of the surface of a body in a state of strain 
 are only infinitesimally altered ; but in our particular case they are altered by 
 finite amounts, and in this lies the justification of the present question. For 
 instance, when two equal spheres of the same material touch each other, the 
 surface of pressure forms part of a plane, i.e. of a surface which is different in 
 character from both of the surfaces in contact. 
 
148 CONTACT OF ELASTIC SOLIDS V 
 
 Then we may write the equations of the two surfaces 
 
 z, = A,x 2 + Qxy + B^ 2 , z 2 = A/ + Qxy + B/, 
 
 and we have for the distance between corresponding points 
 of the two surfaces z Y — z 2 = Ax 2 + By 2 , where A = A 1 — A 2 , 
 B = B x — B 2 , and A, B, C are all infinitesimal. 1 From the 
 meaning of the quantity z 1 — z 2 it follows that A and B have 
 the like sign, which we shall take positive. This is equivalent 
 to choosing the positive 2-axis to fall inside the body to 
 which the index 1 refers. 
 
 Further, we imagine in each of the two bodies a rect- 
 angular rectilinear system of axes, rigidly connected at 
 infinity with the corresponding body, which system of axes 
 coincides with the previously chosen system of xyz during the 
 mathematical contact of the two surfaces. When a pressure 
 acts on the bodies these systems of coordinates will be shifted 
 parallel to the axis of z relatively to one another ; and their 
 relative motion will be the same in amount as the distance by 
 which those parts of the bodies approach each other which 
 are at an infinite distance from the point of contact. The 
 plane z = in each of these systems is infinitely near to the 
 part of the surface of the corresponding body which is at a 
 finite distance, and therefore may itself be considered as the 
 surface, and the direction of the 2-axis as the direction of the 
 normal to this surface. 
 
 Let £ , 7), f be the component displacements parallel to the 
 axes of x, y, z; let Y x denote the component parallel to Oy of 
 the pressure on a plane element whose normal is parallel to 
 Ox, exerted by the portion of the body for which x has 
 smaller values on the portion for which x has larger values, 
 and let a similar notation be used for the remaining com- 
 
 1 Let p u , p 12 be the reciprocals of the principal radii of curvature of the sur- 
 face of the first body, reckoned positive when the corresponding centres of 
 curvature lie inside this body ; similarly let p 21 , /> 22 be the principal curvatures 
 of the surface of the second body ; lastly, let co be the angle which the planes 
 of the curvatures p n and p 21 make with each other. Then 
 
 2(A + B)=p u + p 12 +/) 21 +/> 22 , 
 
 2( A - B) = V( Pll - p^) 2 + 2(p 11 - Pv2 )(p 2l - p 22 ) cos 2w + (p 21 - p 22 )~. 
 If we introduce an auxiliary angle r by the equation cos r= (A - B)/(A + B), then 
 
 2 A = ( Pn + Pl2 + p n + p 22 ) cos 2 ^, 2B = ( Pn + Pn + p 2l + p 22 ) sin 2 ^. 
 
v CONTACT OF ELASTIC SOLIDS 149 
 
 ponents of pressure ; lastly let K^ and K 2 2 l be the respective 
 coefficients of elasticity of the bodies. Generally, where the 
 quantities refer to either body, we shall omit the indices. We 
 then have the following conditions for equilibrium: — 
 
 1. Inside each body we must have 
 
 = V 2 £ + (l + 20)^, = v 2 ^ + (l + 20& 
 
 Ox oy 
 
 oz ox t oy oz 
 
 and in 1 we have to put 1 for 0, in 2 # 2 for 0. 
 
 2. At the boundaries the following conditions must hold : — 
 
 (a) At infinity f, 77, f vanish, for our systems of co- 
 ordinates are rigidly connected with the bodies there. 
 
 (b) For 2=0, i.e. at the surface of the bodies, the tan- 
 gential stresses which are perpendicular to the 2-axis must 
 vanish, or 
 
 (c) For z = 0, outside a certain portion of this plane, viz. 
 outside the surface of pressure, the normal stress also must 
 vanish, or 
 
 :(|f+^ 
 
 Inside that part 
 
 %zl - %z2- 
 
 We do not know the distribution of pressure over that part, 
 but instead we have a condition for the displacement f over it. 
 
 (d) For if a denote the relative displacement of the two 
 systems of coordinates to which we refer the displacements, 
 the distance between corresponding points of the two surfaces 
 after deformation is Ax 2 + By 2 -f ?i — f 2 "" a > anc ^- s i Rce this 
 distance vanishes inside the surface of pressure we have 
 
 fj - f 2 = a - Ax 2 - B?/ 2 = a - z } + « 2 . 
 
 (e) To the conditions enumerated we must add the con- 
 
 1 [Kirchhoff's notation, MechaniJc, p. 121. — Tp,.] 
 
150 CONTACT OF ELASTIC SOLIDS v 
 
 clition that inside the surface of pressure Z z is everywhere 
 positive, and the condition that outside the surface of pressure 
 f x — % 2 >a — Ax 2 — By 2 , otherwise the one body would overflow 
 into the other. 
 
 (/) Lastly the integral Z z ds, taken over the part of the 
 surface which is bounded by the curve of pressure, must be 
 equal to the given total pressure, which we shall call p. 
 
 The particular form of the surface of the two bodies only 
 occurs in the boundary condition (2 d), apart from which 
 each of the bodies acts as if it were an infinitely extended 
 body occupying all space on one side of the plane z— 0, and 
 as if only normal pressures acted on this plane. We there- 
 fore consider more closely the equilibrium of such a body. 
 Let P be a function which inside the body satisfies the 
 equation y 2 ? = \ m particular, we shall regard P as the 
 potential of a distribution of electricity on the finite part of 
 the plane 2=0. Further let 
 
 n= - 
 
 % 
 
 where i is an infinitely great quantity, and J is a constant so 
 chosen as to make II finite. For this purpose J must be 
 equal to the natural logarithm of i multiplied by the total 
 charge of free electricity corresponding to the potential P. 
 Prom the definition of II it follows that 
 
 2TT 28P 
 
 2(1 + 0) 
 Introducing the contraction 3- = „, 9 ^ we put 
 
 an an an 
 
 dP 2 SP 
 
 This system of displacements is easily seen to satisfy the 
 
V CONTACT OF ELASTIC SOLIDS 151 
 
 differential equations given for f, rj, £ and the displacements 
 vanish at infinity. For the pressure components we find 
 
 X = -2K-^ + 
 
 ja 2 n 
 
 X„, = - 2K 
 
 n 
 
 [dx 2 ' K(l + 20)<fe 
 
 a 2 n 
 
 dxdy ' 
 
 !• 
 
 -»v 
 
 a 2 n 29 ap\ 
 a 2 n api a 2 P 
 
 2 "" \ 9a?3« d%) dxdz' 
 
 (dm 2(2 + 30) 3P) 
 A ~~ ^t'^ 2 K(l + 2^) a*J 5 
 
 T ra 2 n api a 2 p 
 lfya« a^ J a^ 
 
 The last two formulas show that for the given system the 
 stress-components perpendicular to the 2-axis vanish through- 
 out the plane z = 0. We determine the displacement f and 
 the normal pressure Z z at the plane 2=0, and find 
 
 £ = £P, Z 2 =-2^. 
 oz 
 
 The density of the electricity producing the potential P is 
 — (l/27r)(3P/8s), hence we have the following theorem. The 
 displacement f in the surface, which corresponds to the normal 
 pressure Z z , is equal to &/47T times the potential due to an 
 electrical density numerically equal to the pressure Z z . 
 
 We now consider again both bodies : we imagine the 
 electricity whose potential is P to be distributed over a finite 
 portion only of the plane z = ; we make IT^ and IT 2 equal to 
 the expressions derived from the given expression for II by 
 giving to the symbols K and 6 the index 1 and 2, and put 
 
 l-_OTj „- 8n i r - 8n > + ^P 
 an 2 an 2 dn, 9ap 
 
 ft = - -t— . % = - -5- » & = - ~ - 2£ 2 P ; 
 dx dy dz 
 
152 CONTACT OF ELASTIC SOLIDS v 
 
 whence we have for z = 
 
 ?, = *,P, ft = - S S P, Z, x = - 2±, Z zl = 2^ . 
 
 oz oz 
 
 This assumption satisfies the conditions (1), (2 a), and (2 &) 
 according to the explanations given. Since dVjdz has on the 
 two sides of the plane z = values equal but of opposite sign, 
 and since it vanishes outside the electrically charged surface 
 whose potential is P, the conditions (2 c) also are fulfilled, pro- 
 vided the surface of pressure coincides with the electrically 
 charged surface. From the fact that P is continuous across 
 the plane z = 0, it follows that for z = 0, S- 2 ^ + S-^ 2 = 0. But 
 according to the condition (2 d) we have for the surface of 
 pressure, f x — f 2 = a — z l + z 2 ; here therefore 
 
 Apart from a constant which depends on the choice of the 
 system of coordinates, and need therefore not be considered, the 
 equation of the surface of pressure is z = z 1 + ^ 1 =z 2 + f 2 > or 
 (-9-! + S- 2 )z = S- 2 z x -f 3-iZ 2 . Thus the surface of pressure is part 
 of a quadric surface lying between the undeformed positions of 
 the surfaces which touch each other ; and is most like the 
 boundary of the body having the greater coefficient of elasticity. 
 If the bodies are composed of the same material it is the 
 mean surface of the surfaces of the two bodies, since then 
 2z — Zi -\- z 2 . 
 
 We now make a definite assumption as to the distribution 
 of the electricity whose potential is P. Let it be distributed 
 over an ellipse whose semi-axes a and b coincide with the axes of 
 
 y 
 
 x and y, with a density — — / 1 — — _ — -^ , so that it can be 
 87r 2 ab\J a 1 V 
 
 regarded as a charge which fills an infinitely flattened ellipsoid 
 
 with uniform volume density. Then 
 
 op 
 
 1 - 
 
 1 6ttJ \ cr + \ b 2 + \ \ ) J (a 2 + \)(b 2 + \)\ 
 
 dX 
 
v CONTACT OF ELASTIC SOLIDS 153 
 
 where u, the inferior limit of integration, is the positive root 
 of the cubic equation 
 
 a 2 -\-u b 2 + u u 
 
 Inside the surface of pressure, which is bounded by the given 
 ellipse, we have u = 0, P = L — Mx 2 — N?/ 2 ; where L, M, N 
 denote certain positive definite integrals. The condition (2d) 
 is satisfied by choosing a and b so that 
 
 (.»! + £ 2 )M = A, (^ + £ 2 )N = B, 
 
 which is always possible. The unknown a which occurs in 
 the condition is then determined by the equation 
 
 It follows directly from the equation 
 
 2irah 
 
 V a 2 b 2 
 
 that the first of the conditions (2 e) is satisfied. 
 
 To show that the second also is satisfied is to prove that 
 when z = and x 2 ja 2 + y 2 jb 2 >l ) (^ + £ 2 )P > a - Ax 2 - Bif. 
 For this purpose we observe that here 
 
 P = L - Mx 2 - % 
 
 gg /•/ _ x 2 y 2 \ d\ 
 
 16tt )\ a 2 + \ ~ 6 2 + V J (a 2 + \)(b 2 + \)\ 
 o 
 
 and hence P > L — Mx 2 — N?/ 2 , for the numerator of the ex- 
 pression under the sign of integration is negative throughout 
 the region considered. Multiplying by 3- + 3- 2 we get the 
 inequality which was to be proved. Finally, a simple integra- 
 tion shows that the last condition ( 2 /) also is satisfied ; 
 therefore we have in the assumed expression for P and the 
 corresponding system f, r\, f a solution which satisfies all the 
 conditions. 
 
154 
 
 CONTACT OF ELASTIC SOLIDS 
 
 The equations for the axes of the ellipse of pressure 
 written explicitly are 
 
 l 
 
 du 
 
 A 16tt 
 
 \/(a 2 + u)Xb 2 + u)u ^+^ 2 3p 
 
 OO 
 
 J 
 
 rZw 
 
 B 16tt 
 
 v/(ft 2 + m) (& 2 + ufu ^+£ 9 3p ' 
 
 
 
 or introducing the ratio h = a/b, and transforming, 
 
 <j v/(l + 
 
 tfe 
 
 &V) 3 (l+z 2 ) 3^+V 
 
 a 8 ]* 8 
 
 ^ 
 
 ;tt 
 
 B 
 
 V^(1+&V)(1+* 2 ) 3 3p^+V 
 
 By division we obtain a transcendental equation for the 
 ratio /j. 1 This depends only on the ratio A : B, and it follows 
 at once, from the meaning we have attached to the forces and 
 displacements, that the ellipse of pressure is always more 
 elongated than the ellipses at which the distance between the 
 bodies is constant. As regards the absolute magnitude of the 
 surface of pressure for a given form of the surfaces it varies as 
 
 1 The solution of this equation and the evaluation of the integrals required 
 for the determination of a and b may be performed by the aid of Legendre's 
 tables without necessitating any new quadratures. The calculation, usually 
 somewhat laborious, may in most cases be avoided by the use of the following 
 small table, of which the arrangement is as follows. If we express A and B in 
 the equations for a and b in terms of the principal curvatures and the auxiliary 
 angle r introduced in a previous note, the solutions of these equations are 
 expressible in the form 
 
 ^Vi 
 
 Wjh+P) 
 
 
 8(P11 + Pl2 + P21 + P22) V *KPll + Pl2 + P21+P&) 
 
 where fx, v are transcendental functions of the angle r. The table gives the 
 values of these functions for ten values of the argument r expressed in degrees. 
 
 T 
 
 90 
 
 80 70 
 
 60 
 
 50 ! 40 
 
 30 
 
 20 10 
 
 
 
 ix 1-0000 
 
 1-1278 
 
 1-2835 
 
 1-4858 
 
 1-7542 2-1357 
 
 2-7307 
 
 3-7779 6-6120 
 
 CO 
 
 1 
 
 v ,1-0000 
 
 0-8927 
 
 0-8017 
 
 0-7171 
 
 0-6407 0-5673 
 
 0-4930 
 
 0-4079 0-3186 
 
 o-oooo 
 
v CONTACT OF ELASTIC SOLIDS 155 
 
 the cube root of the total pressure and as the cube root of 
 the quantity S- l + & 2 . By the preceding the distance through 
 which the bodies approach each other under the action of the 
 given pressure is 
 
 _ 3p ^+£ 2 
 
 dz 
 
 8tt' a J v/(l+&V)(H-2 2 )' 
 o 
 
 If we perform the multiplication by 3- + & 2 , a splits up into 
 
 two portions which have a special meaning. J They denote the 
 
 distances through which the origin approaches the infinitely 
 
 distant portions of the respective bodies ; we may call them 
 
 the indentations which the respective bodies have undergone. 
 
 With a given form of the touching surfaces the distance of 
 
 approach varies as the pressure raised to the power -| and 
 
 also as the same power of the quantity S- x + 3- 2 . When A and 
 
 B alter in magnitude while their ratio remains unchanged, the 
 
 dimensions of the surface of pressure vary inversely as the 
 
 cube roots of the absolute values of A and B, and the distance 
 
 of approach varies directly as these roots. When A and B 
 
 become infinite, the distance of approach becomes infinite ; 
 
 bodies which touch each other at sharp points penetrate into 
 
 each other. 
 
 In connection with this we shall determine what happens 
 
 to the element at the origin of our system of coordinates by 
 
 of* 7) oo 
 
 finding the three displacements ^ , t^- , ~ . In the first 
 
 OX Oy CZ 
 
 place we have at the origin 
 
 2 ap_ _ 3p 1 
 
 °"~K(l + 20) Tz~ 2K(1 + 20)7T^ ; 
 
 ag = i ap_ 3p i_ 
 
 dz~K(l + 26)dz ~~ 4K(l + 20)7ra&" 
 Further, at the plane z = 
 
 an = an 
 
 dx ' dy 
 
 11= lTdz= lTdz 
 
 K(1 + 20)J 
 
 1 
 2K(i~+2<9) 
 
 i) 
 
156 CONTACT OF ELASTIC SOLIDS v 
 
 We see that in the said plane f and rj are proportional to 
 the forces exerted by an infinitely long elliptic cylinder, which 
 stands on the surface of pressure and whose density increases 
 inwards, according to the law of increase of the pressure in the 
 surface of pressure. In general then, £ and 77 are given by 
 complicated functions ; but for points close to the axis they 
 can be easily calculated. Surrounding the axis we describe a 
 very thin cylindrical surface^ similar to the whole cylinder ; 
 this [small] cylinder we may treat as homogeneous, and since 
 the part outside it has no action at points inside it, the com- 
 ponents of the forces in question, and therefore also f and 77, 
 must be equal to a constant multiplied respectively by x/a and 
 by y/b. Hence 
 
 al- 
 
 dx 
 
 dy 
 
 a? 
 
 3p 1 
 
 dz 
 
 4K(l + 20)7ra& 
 
 On the other hand we have 
 
 3f d v _ 
 
 dx dy 
 
 From these equations we find for the three quantities 
 which we sought 
 
 dj__ _ 3p 1 
 
 dx 
 
 drj 
 
 4K(l + 26>)7r a(a+b) 
 
 3p 
 
 1 
 
 4K(1 + 20> 
 
 b{a + b) 
 
 dp 
 
 1 
 
 dj 
 
 dz 4K(1 + 20)tt«& 
 
 The negative sign of these three quantities shows that the 
 element in question is compressed in all three directions. 
 The compressions vary as the cube root of the total pressure. 
 It is easy to determine from them the pressures at the origin. 
 These pressures "are the most intense of all those occurring 
 throughout the bodies pressed together ; we may therefore 
 say that the limit of elasticity will not be exceeded until 
 these pressures become of the order of magnitude required for 
 transgressing the elastic limit. In plastic bodies, e.g. in the 
 
v CONTACT OF ELASTIC SOLIDS 157 
 
 softer metals, this transgression will at first consist in a lateral 
 deformation accompanied by a permanent compression ; so that 
 it will not result in an infinitely increasing disturbance of 
 equilibrium, but the surface of pressure will increase beyond 
 the calculated dimensions until the pressure per unit area is 
 sufficiently small to be sustained. It is more difficult to de- 
 termine what happens in the case of brittle bodies, as hard 
 steel, glass, crystals, in which a transgression of the elastic 
 limit occurs only through the formation of a rent or crack, i.e. 
 only under the influence of tensional forces. Such a crack 
 cannot start in the element considered above, which is com- 
 pressed in every direction ; and with our present-day knowledge 
 of the tenacity of brittle bodies it is indeed impossible exactly 
 to determine in which element the conditions for the production 
 of a crack first occur when the pressure is increased. However, 
 a more detailed discussion shows this much, that in bodies 
 which in their elastic behaviour resemble glass or hard steel, 
 much the most intense tensions occur at the surface, and in 
 fact at the boundary of the surface of pressure. Such a dis- 
 cussion shows it to be probable that the first crack starts at 
 the ends of the smaller axis of the ellipse of pressure, and 
 proceeds perpendicularly to this axis along that ellipse. 
 
 The formulas found become especially simple when both 
 the bodies which touch each other are spheres. In this case 
 the surface of pressure is part of a sphere. If p is the recip- 
 rocal of its radius, and if p and p 9 are the reciprocals of the 
 radii of the touching spheres, then we have the relation 
 (S- 1 + S- 2 )p = S- 2 p 1 + S- 1 p 2 ; which for spheres of the same material 
 takes the simpler form 2p = p l + p 2 . The curve of pressure is 
 a circle whose radius we shall call a. If we put 
 
 
 then will 
 
 16ttJ\ « 2 + w «/0 2 +w) n A'' 
 
 u 
 
 which may also be expressed in a form free of integrals. 
 
 We easily find for a, the radius of the circle of pressure, 
 and for a, the distance through which the spheres approach 
 
158 CONTACT OF ELASTIC SOLIDS v 
 
 each other, and also for the displacement f over the part of 
 the plane z — inside the circle of pressure : — 
 
 
 16a 
 
 _6p 2«'-r- 
 
 fe 32^ a 3 " 
 
 Outside the circle of pressure f is represented by a some- 
 what more complicated expression, involving an inverse tangent. 
 Very simple expressions may be got for f and 77 at the plane 
 z = 0. For the compression at the plane z = we find 
 
 3^> \Ar — r 
 
 tF ~" 2K(1 + 20>r a 3 
 
 inside the circle of pressure ; outside it <r = 0. For the 
 pressure Z z inside the circle of pressure we obtain 
 
 27r a 3 
 
 at the centre we have 
 
 z - 3 ff X = Y - 1 + 46> 5?. 
 
 3 27m 2 ' ^ ' 4(1 + 20) 7m 2 ' 
 
 The formulae obtained may be directly applied to particular 
 cases. In most bodies 6 may with a sufficient approximation 
 be made equal to 1. Then K becomes f- of the modulus of 
 elasticity ; .9- becomes equal to - 3 ^ 2 - times the reciprocal of that 
 modulus ; in all bodies 3- is between three and four times this 
 reciprocal value. If, for instance, we press a glass lens of 100 
 metres radius with the weight of 1 kilogramme against a 
 plane glass plate (in which case the first Xewton's ring would 
 have a radius of about 5*2 millimetres), we get a surface of 
 pressure which is part of a sphere of radius equal to 200 
 metres. The radius of the circle of pressure is 2 "6 7 millimetres ; 
 the distance of approach of the glass bodies amounts to only 
 7 1 millionths of a millimetre. The pressure Z z at the centre 
 of the surface of pressure is 0'0669 kilogrammes per square 
 millimetre, and the perpendicular pressures X x and Y have 
 
v CONTACT OF ELASTIC SOLIDS 159 
 
 about |- that value. As a second example, consider a number 
 of steel spheres pressed by their own weight against a rigid 
 horizontal plane. We find that the radius of the circle of 
 pressure in millimetres is very approximately a = 1( j L ()0 B*. 
 Hence for spheres of radii 
 
 1 mm., 1 m., 1 km., 1000 km., 
 
 a becomes about 
 
 3-^ mm., 10 mm. 100 m., 1000 km. 
 
 or 
 
 _i l _i i 
 
 1000' 100' 10' 1 
 
 of the radius. For spheres whose radius exceeds 1 km. 
 the radius of the circle of pressure is more than -^ of the 
 radius of the sphere. Our calculations do not apply to such 
 ratios, for we presupposed the ratio to be a small fraction. 
 But the very fact that for such large spheres equilibrium is 
 no longer possible with small deformations shows that equili- 
 brium is altogether impossible. Consider further two steel 
 spheres of equal radius touching one another and pressed 
 together only by their mutual gravitational attraction. In 
 millimetres we find l the radius of the circle of pressure to be 
 p = 0-000000378K 1 . If the radius of the two spheres is 4*3 
 kilometres, then p = ^bo^ > ^ ^ * s ■"• ^ 6 kilometres, then 
 p = yqR. That value of E, for which the elastic forces cease 
 to be able to equilibrate gravitational attraction, will lie 
 between the above values and nearer to the greater. If steel 
 spheres of greater radius be placed touching each other, they 
 will break up into pieces whose dimensions are of the order of 
 the values of R just mentioned. 
 
 Finally, we shall apply the formulas we have obtained to 
 the impact of elastic bodies. It follows, both from existing 
 observations and from the results of the following considera- 
 tions, that the time of impact, i.e. the time during which the 
 impinging bodies remain in contact, is very small in absolute 
 value ; yet it is very large compared with the time taken by 
 waves of elastic deformation in the bodies in question to 
 traverse distances of the order of magnitude of that part of 
 their surfaces which is common to the two bodies when in 
 
 1 In these calculations the modulus of elasticity of steel is taken to be 20,000 
 kg/mm 2 , its density 77, and the mean density of the earth 6. 
 
160 CONTACT OF ELASTIC SOLIDS V 
 
 closest contact, and which we shall call the surface of impact. 
 It follows that the elastic state of the two bodies near the 
 point of impact during the whole duration of impact is very 
 nearly the same as the state of equilibrium which would be 
 produced by the total pressure subsisting at any instant 
 between the two bodies, supposing it to act for a long time. 
 If then we determine the pressure between the two bodies by 
 means of the relation which we previously found to hold 
 between this pressure and the distance of approach along the 
 common normal of two bodies at rest, and also throughout the 
 volume of each body make use of the equations of motion of 
 elastic solids, we can trace the progress of the phenomenon 
 very exactly. \Ye cannot in this way expect to obtain general 
 laws ; but we may obtain a number of such if we make the 
 further assumption that the time of impact is also large com- 
 pared with the time taken by elastic waves to traverse the 
 impinging bodies from end to end. \Vhen this condition is 
 fulfilled, all parts of the impinging bodies, except those infinitely 
 close to the point of impact, will move as parts of rigid bodies ; 
 we shall show from our results that the condition in question 
 may be realised in the case of actual bodies. 
 
 We retain our system of axes of wyz. Let a be the 
 resolved part parallel to the axis of z of the distance of two 
 points one in each body, which are chosen so that their 
 distance from the surface of impact is small compared with 
 the dimensions of the bodies as a whole, but large compared 
 with the dimensions of the surface of impact ; and let a' denote 
 the differential coefficient of a with regard to the time. If 
 dJ is the momentum lost in time dt by one body and gained 
 by the other, then it follows from the theory of impact of 
 rigid bodies that da! = — Jc^U, where Jc x is a quantity depending 
 only upon the masses of the impinging bodies, their principal 
 moments of inertia, and the situation of their principal axes of 
 inertia relatively to the normal at the point of impact. 1 On 
 
 1 See Poisson, TraiU de mecanique, II. chap. vii. In the notation there 
 employed we have for the constant Jc\ 
 
 _ 1 (b cos y- c cos j3) 2 (c cos a - a cos y) 2 (a cos /3 - b cos a) 2 
 
 1 (&'cos7'-c'cos/3') 2 (c' cos a - a' cosy') 2 (a' cos /3' - b' cos a') 2 
 + M' + ~ A' ~ + W +_ C~ 
 
V CONTACT OF ELASTIC SOLIDS 161 
 
 the other hand, dJ is equal to the element of time clt, multi- 
 plied by the pressure which during that time acts between the 
 bodies. This is k 2 a? } where h 2 is a constant to be determined 
 from what precedes, which constant depends only on the form 
 of the surfaces and the elastic properties quite close to the 
 point of impact. Hence dJ = h 2 a-dt and da = — kjc 2 cfidt ; 
 integrating, and denoting by a the value of a just before 
 impact, we find 
 
 a — a I + ^Jcjc 2 a^ = 0, 
 
 which equation expresses the principle of the conservation of 
 energy. When the bodies approach as closely as possible a 
 vanishes ; if a m denote the corresponding value of a, then 
 
 a m = I j , and the simultaneous maximum pressure is 
 
 Vm = &2 a i From this we at once obtain the dimensions of 
 the surface of impact. 
 
 In order to deduce the variation of the phenomenon with 
 the time, we integrate again and obtain 
 
 m 
 
 da 
 
 a 
 
 The upper limit is so chosen that t = at the instant of nearest 
 approach. For each value of the lower limit a, the double 
 sign of the radical gives two equal positive and negative values 
 of t. Hence a is an even and a an odd function of t ; im- 
 mediately after impact the points of impact separate along 
 the normal with the same relative velocity with which they 
 approached each other before impact. And the same tran- 
 scendental function which represents the variation of a between 
 its initial and final values, also represents the variations of all 
 the component velocities from their initial to their final values. 
 In the first place, the bodies touch when a = ; they 
 separate when a again = 0. Hence the duration of contact, 
 that is the time of impact, is 
 
 T=2| *** : =2 V / 25 = o "j 
 
 
 M. P. M 
 
162 CONTACT OF ELASTIC SOLIDS 
 
 V= , = 1-4716. 
 
 o 
 Thus the time of impact may become infinite in various ways 
 without the time, with which it is to be compared, also 
 becoming infinite. In particular the time of impact becomes 
 infinite when the initial relative velocity of the impinging 
 bodies is infinitely small ; so that whatever be the other 
 circumstances of any given impact, provided the velocities 
 are chosen small enough, the given developments will have 
 any accuracy desired. In every case this accuracy will be 
 the same as that of the so-called laws of impact of perfectly 
 elastic bodies for the given case. For the direct impact of 
 two spheres of equal radius E and of the same material of 
 density q the constants l\ and l\ 2 are 
 
 
 8 /E 
 
 hence in the particular case of two equal steel spheres of 
 radius E, taking the millimetre as unit of length, and the 
 weight of one kilogramme as unit of force, we have 
 
 log \ = 8-78 -3 log R, 
 
 logh = 4-03+1 log E. 
 
 Thus for two such -spheres impinging with relative velocity v : 
 
 the radius of the surface of impact . a m = - 0020E#mm, 
 the time of impact . . . T = 0-000024E?T i sec, 
 
 the total pressure at the instant of 
 
 nearest approach . . . p m = 0'00025RVkg, 
 
 the simultaneous maximum pressure 
 
 at the centre of impact per unit 
 
 area . . . . p' m = 29'l^kg/mm 2 . 
 
 For instance, when the radius of the spheres is 25 mm., the 
 velocity 10 mm/sec, then a m =-- 013 mm., T = 0-00038 sec, 
 jt? m =2 , 47kg. J p f m = 73'0 "kg/mm. 2 For two steel spheres as 
 large as the earth, impinging with an initial velocity of 10 
 mm/sec, the duration of contact would be nearly 2 7 hours. 
 
VI 
 
 OX THE COXTACT OF EIGID ELASTIC SOLIDS 
 AND OX HAEDXESS 
 
 ( Verhancllungen des Vcrcins zur Bcforderung des Gewerbcfleisses, November 1882.) 
 
 When two elastic bodies are pressed together, they touch each 
 other not merely in a mathematical point, but over a small 
 but finite part of their surfaces, which part we shall call the 
 surface of pressure. The form and size of this surface and 
 the distribution of the stresses near it have been frequently 
 considered ("Winkler, Lehre von der Elasticitdt unci Festigkeit, 
 Prag. 1867, I. p. 43 ; Grashof, Thcoric der Mastic licit unci 
 Festiglccit, Berlin, 1878, pp. 49-54); but hitherto the results 
 have either been approximate or have even involved unknown 
 empirical constants. Yet the problem is capable of exact 
 solution, and I have given the investigation of the problem in 
 vol. xcii. of the Journal fur reine unci angevjandte Mathcmatik, 
 p. 156. 1 As some aspects of the subject are of considerable 
 technical interest, I may here treat it more fully, with an 
 addition concerning hardness. I shall first restate briefly the 
 proof of the fundamental formulae. 
 
 We first imagine the two bodies brought into mathematical 
 contact ; the common normal coincides with the line of action 
 of the pressure' which the one body exerts upon the other. 
 In the common taugent plane we take rectangular rectilinear 
 axes of xy, the origin of which coincides with the point of 
 contact; the third perpendicular axis is that of z. We can 
 confine our attention to that part of each body which is very 
 close to the point of contact, since here the stresses are 
 extremely great compared with those occurring elsewhere, and 
 
 1 See Y. p. 146. 
 
164 OX HAEDXESS vi 
 
 consequently depend only to the very smallest extent on the 
 forces applied to other parts of the bodies. Hence it is suffi- 
 cient to know the form of the surfaces infinitely near the point 
 of contact. To a first approximation, if we consider each 
 body separately, we may even suppose their surfaces to coin- 
 cide with the common tangent plane z = 0, and the common 
 normal to coincide with the axis of z\ to a second approxima- 
 tion, when we wish to consider the space between the bodies, 
 it is sufficient to retain only the quadratic terms in xy in the 
 development of the equations of the surfaces. The distance 
 between opposite points of the two surfaces then becomes a 
 homogeneous quadratic function of the x and y belonging to 
 the two points ; and we can turn our axes of x and y so that 
 from this function the term in xy disappears. After com- 
 pleting this operation let the distance between the surfaces 
 be given by the equation c = Arc 2 + By' 2 . A and B must of 
 necessity have the same sign, since e cannot vanish ; when we 
 construct the curves for which e has the same value, we obtain 
 a system of similar ellipses, whose centre is the origin. Our 
 problem now is to assign such a form to the surface of pressure 
 and such a system of displacements and stresses to its neigh- 
 bourhood, that (1) these displacements and stresses may satisfy 
 the differential equations of equilibrium of elastic bodies, and 
 the stresses may vanish at a great distance from the surface of 
 pressure ; that (2) the tangential components of stress may 
 vanish all over both surfaces; that (3) at the surface the 
 normal pressure also may vanish outside the surface of pressure, 
 but inside it pressure and counterpressure may be equal ; 
 the integral of this pressure, taken over the whole surface of 
 pressure, must be equal to the total pressure p fixed before- 
 hand ; that, lastly (4) the distance between the surfaces, which 
 is altered by the displacements, may vanish in the surface of 
 pressure, and be greater than zero outside it. To express the 
 last condition more exactly, let £ lt tj 1} ^ be the displacements 
 parallel to the axes of x, y, z in the first body, f 2 , rj 2i f 2 those 
 in the second. In each let them be estimated relatively to 
 the undeformed parts of the bodies, which are at a distance 
 from the surface of pressure; and let a denote the distance 
 by which these parts are caused by the pressure to approach 
 each other. Then any two points of the two bodies, which 
 
vi ON HARDNESS 165 
 
 have the same coordinates x, y, have approached each other 
 by a distance a — ^ + f 2 under the action of the pressure ; 
 this approach must in the surface of pressure neutralise the 
 original distance Ax 2 + By 2 . Hence here we must have 
 f x — f 2 = a — Ax 2 — By 2 , whilst elsewhere over the surfaces 
 fj — J 2 ><z — Ax 2 — By 2 . All these conditions can be satisfied 
 only by one single system of displacements ; I shall give this 
 system, and prove that it satisfies all requirements. 
 
 As surface of pressure we take an ellipse, whose axes 
 coincide with those of the ellipses e = constant, but whose 
 shape is more elongated than theirs. We reserve the deter- 
 mination of the lengths of its semi-axes a and b until later. 
 First we define a function P by the equation 
 
 CO 
 
 P= 3 JL|Yi_^I t z l\ _^_ 
 
 167rJ\ cr + X b' + X \)y/(a* + \)(V + \)\ 
 
 U 
 
 where the lower limit of integration is the positive root of the 
 cubic equation 
 
 v 2 ii 2 z 2 
 
 a + u u" -\-u u 
 The quantity u is an elliptic coordinate of the point xyz ; it 
 is constant over certain ellipsoids, which are confocal with 
 the ellipse of pressure, and vanishes at all points which are 
 infinitely close to the surface of pressure. The function P has 
 a simple meaning in the theory of potential. It is the poten- 
 tial of an infinitely flattened gravitating ellipsoid, which would 
 just fill the surface of pressure ; in that theory it is proved 
 that P satisfies the differential equation 
 
 „ 9T , d 2 B a 2 p a 2 p A 
 
 ox vy dz 
 
 Now from this P we deduce two functions IT, one of which 
 refers to the one body, the second to the other, and we make 
 
 CO 
 
 n 2 = -—(zv- — - 
 
 K 2 \ l + 2© 2 ' 
 
166 ON HARDNESS vi 
 
 Here K, © denote the coefficients of elasticity in Kirchhoff's 
 notation. Young's modulus of elasticity is expressed in terms 
 of these coefficients by the equation 
 
 1 + 3© 
 
 E = 2K 
 
 1 + 2® 
 
 The ratio between lateral contraction and longitudinal exten- 
 sion is 
 
 © 
 
 ^ 1 + 2© 
 
 For bodies like glass or steel, this ratio is nearly -J-, or © 
 nearly 1, and K is nearly |- E. For slightly compressible 
 bodies the ratio is nearly ^ ; here then © = <x> , K = -JE. As 
 a matter of fact a particular combination of K and © will 
 play the principal part in our formulae, for which we shall 
 therefore introduce a special symbol. We put 
 
 2(1+©) 
 K(l + 2©)' 
 
 In bodies like glass, &=4/3K=32/9E; in all bodies 3- lies 
 between 3/E and 4/E, since © lies between and co . In 
 regard to the ITs we must note that calculated by the above 
 formulae they have infinite values ; but their differential co- 
 efficients, which alone concern us, are finite. It would only 
 be necessary to add to the ITs infinite constants of suitable 
 magnitude to make them finite. By a simple differentiation, 
 remembering that \7 2 P =0, we find 
 
 2 ap . 2 ap 
 
 1 K 2 oz K 2 dz 
 
 We now assume the following expressions for the displace- 
 ments in the two bodies : — 
 
 an, an, 8 n 
 
 whence follow 
 
 ^,^,%_ nm , n sp 2— sp 
 
ON HARDNESS 167 
 
 2 8P 
 
 K 2 (l + 2® 2 ) d: 
 
 In the first place, this system satisfies the equations of 
 equilibrium, for we have 
 
 and similar equations hold for f 2 , rj ly t) 2 ; for the f s we get the 
 same result, remembering that y 2 P = 0. For the tangential 
 stress components at the surface (2 = 0) we find, leaving out 
 the indices : — 
 
 \dx dz) \ dzdx ^ dx J ^"dxdz 
 
 as the second condition requires. 
 
 It is more troublesome to prove that the third condition 
 is satisfied. We again omit indices, as the calculation applies 
 equally to both bodies. We have generally 
 
 3 f r, \ \ dm 2(2 + 3@ 9P] 
 
 2K ^ + 0o- = -2KW +~ 
 
 2Ki 
 
 dz t OT ; = - ^W * K(l + 2@) dz~ ( 
 
 a 2 p ap 
 
 = 2z dz° : ~ 2 T Z ' 
 
 ap 
 
 therefore at the surface Z z = — 2^— . Now, using the equa- 
 tion for u, we have generally 
 
 ap Sp f d\ 
 
 'dz = ~~ 8?j xV(a 2 + XX6 2 + ^)x' 
 
 ap 
 
 and therefore at the surface -~- vanishes, as it must do, and 
 
 dz ' 
 
 with it Z 3 , at any rate outside the surface of pressure. In 
 the compressed surface, where u=0, the expression takes the 
 
168 ON HARDNESS vi 
 
 form . co ; the ordinary procedure for 
 
 the evaluation of 
 
 such an indeterminate form gives 
 
 
 dV Sp Z \z 
 
 
 dz~ Swabus/u' 
 
 
 that is, since for u = we have 
 
 
 / X 2 lf\ 
 
 
 o ap sp 1 x 2 
 
 dz 2irab\! a 2 
 
 6-" 
 
 Here no quantity occurs which could be affected by an 
 index. Hence in the surface of pressure Z z is the same for 
 both bodies ; pressure and counter-pressure are equal. Lastly, 
 the integral of Z z over the surface of pressure is 3p/4z7rah 
 times the volume of an ellipsoid whose semi-axes are 1, a, b; 
 i.e. it equals p, and therefore the total pressure has the 
 required value. 
 
 It remains to be shown that the fourth condition can be 
 satisfied by a suitable choice of the semi-axes a and b. .For 
 this purpose we remark that 
 
 an, z ap ft 
 
 so that at the surface £i = ^P and f 2 = & 2 P. Since inside 
 the surface of pressure the lower limit u of the integral is 
 constantly zero, inside that surface P has the form P = L — 
 Mx 2 — ~Ny 2 ; and therefore it is necessary so to determine a, b 
 and a that (^ + £ 2 )M = A, (^ + S 2 )N = B, (^ + £ 2 )L = a, so 
 as to satisfy the equation ^ — f = a — Ax 2 — B?/ 2 , and this 
 determination is always possible. Written explicitly the 
 equations for a and b are 
 
 du A 16tt 
 
 du B 16tt 
 
 *J(a 2 + u)(b 2 + uYu ~ S~+^ 2 ~Sp' 
 
 (I) 
 
VI ON HARDNESS 169 
 
 Finally, it is easily shown that the very essential in- 
 equality, which must he fulfilled outside the surface of 
 pressure, is actually satisfied ; but I omit the proof, since it 
 requires the repetition of complicated integrals. 
 
 Thus our formulae express the correct solution of the 
 proposed problem, and we may use them to answer the chief 
 questions which may be asked concerning the subject. It is 
 necessary to carry the evaluation of the quantities a and b a 
 step further ; for the equations hitherto found for them cannot 
 straightway be solved, and in general not even the quantities 
 A and B are explicitly known. I assume that we are given the 
 four principal curvatures (reciprocals of the principal radii of 
 curvature) of the two surfaces, as well as the relative position 
 of their planes ; let the former be p n and p 12 for the one 
 body, p 21 and p 22 for the other, and let co be the angle 
 between the planes of p u and of p 21 . Let the p's be reckoned 
 positive when the corresponding centres of curvature lie 
 inside the body considered. Let our axes of xy be placed so 
 that the ajz-plane makes with the plane of p n the angle &/, so 
 far unknown. Then the equations of the surfaces are 
 
 2z 1 = p n (x cos co + y sin a/) 2 + p l2 {y cos co — x sin co') 2 , 
 
 2z 2 — — p 2l {x cos (&/ — co)-\-y sin (&/ — co)} 2 
 
 — p 22 {y cos (g/ — co) — x sin (a/ — co)} 2 . 
 
 The difference z x — z 2 gives the distance between the surfaces. 
 Putting it = Ax 2 + ~By 2 , and equating coefficients of x 2 , xy, y 2 
 on both sides, we obtain three equations for co ' , A and B ; 
 their solution gives for the angle co ', which evidently de- 
 termines the position of the axes of the ellipse of pressure 
 relatively to the surfaces, the equation 
 
 , , (/> 21 -/> 22 )sin2a> 
 
 tan 2 co = 
 
 for A and B 
 
 2(A-B) = 
 
 Pn ~ pu + (P21 ~ /> 22 ) cos 2 * 
 2(A + B) = p n + p l2 + p 2l + p 22 , 
 
 - s/(pn - Pvi) 2 + 2 (f ii - P12XP21 - p 22 )cos 2o) + (p 21 - p 22 J 
 
170 ON HARDNESS VI 
 
 For the purpose of what follows it is convenient to 
 introduce an auxiliary angle r by the equation 
 
 A-B 
 
 COS T = 
 
 A + B' 
 
 and then 
 
 2 A = ( Pll + p 12 + p 21 + p2^s™2 ' 
 
 2B = (p u + p 12 + p 21 + p 22 )cos 2 - • 
 
 We shall introduce these values into the equations for 
 a and b, and at the same time transform the integrals 
 occurring there by putting in the first u = b 2 z 2 , in the second 
 u = a 2 z 2 . Denoting the ratio b/a by k we get 
 
 If dz 4tt p n + p 12 + 
 
 a 3 J V( 1 + & V) 3 ( 1 + r ) 3^ " ^ + 
 
 £ 2 
 
 I I ^ _ 4ir p n + pig + p 2 i + P22 cos 2^ 
 
 J 1 + s (!+«") 
 
 Dividing the one equation by the other we get a new one, 
 involving only h and t, so that h is a function of r alone ; 
 and the same is true of the integrals occurring in the equa- 
 tions. If we solve them by writing 
 
 Vs(o n 
 
 3^i + ^2) 
 
 Kpll + Pl2 + p-n + pw) ' 
 
 'V KP11 + P12 + P21 + P22)' 
 
 then yL6 and v depend only on r, that is on the ratio of the 
 axes of the ellipse e = constant. The integrals in question 
 may all be reduced to complete elliptic integrals of the first 
 species and their differential coefficients with respect to the 
 
ON HARDNESS 
 
 171 
 
 modulus, and can therefore be found by means of Legendre's 
 tables without further quadratures. But the calculations are 
 wearisome, and I have therefore calculated the table given 
 below, 1 in which are found the values of /x and v for ten 
 values of the argument r ; presumably interpolation between 
 these values will always yield a sufficiently near approxima- 
 tion. We may sum up our results thus : The form of the 
 ellipse of pressure is conditioned solely by the form of the 
 ellipses e = constant. With a given shape its linear di- 
 mensions vary as the cube root of the pressure, inversely as 
 the cube root of the arithmetical mean of the curvatures, and 
 also directly as the cube root of the mean value of the elastic 
 coefficients & ; that is, very nearly as the cube root of the 
 mean value of the reciprocals of the moduli of elasticity. It 
 is to be noted that the area of the ellipse of pressure in- 
 creases, other things being equal, the more elongated its form. 
 If we imagine that of two bodies touching each other one be 
 rotated about the common normal while the total pressure is 
 kept the same, then the area of the surface of pressure will be 
 a maximum and the mean pressure per unit area a minimum 
 in that position in which the ratio of the axes of the ellipse 
 of pressure differs most from 1. 
 
 Our next inquiry concerns the indentations experienced 
 by the bodies and the distance by which they approach each 
 other in consequence of the pressure ; the latter we called a 
 and found its value to be (^ + 3- 2 )L. Transforming the 
 integral L a little, we get 
 
 a = 
 
 3p &! + S- 2 
 
 Sir a 
 
 dz 
 
 V(l+£V)(l + s 2 )' 
 
 The distances by which the origin approaches the distant 
 
 T 
 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 I 
 
 /* 
 
 1-000 
 
 1-128 
 
 1-284 
 
 1-486 
 
 1-754 
 
 2-136 
 
 2731 
 
 3-778 
 
 6-612 
 
 OO 
 
 V 
 
 1-000 
 
 0-893 
 
 0-802 
 
 0-717 
 
 0-641 
 
 0-567 
 
 0-493 
 
 0-408 
 
 0-319 
 
 
 
172 ON HARDNESS VI 
 
 parts of the bodies may be suitably denoted as indentations. 
 Their values are easily found by multiplying by ^ + S- 2 and 
 thus separating a into two portions. Substituting for a its 
 value, we see that a involves a numerical factor which depends 
 on the form of the ellipse of pressure ; and that for a given value 
 of this factor a varies as the % power of the pressure, as the 
 |- power of the mean value of the coefficients S-, and as the cube 
 root of the mean value of the curvatures. If one or more of these 
 curvatures become infinitely great, then distance of approach 
 and indentations become infinitely great — a result sufficiently 
 illustrated by the penetrating action of points and edges. 
 
 We assumed the surface of pressure to be so small that 
 the deformed surfaces could be represented by quadric sur- 
 faces throughout a region large compared with the surface of 
 pressure. Such an assumption can no longer be made after 
 application of the pressure ; in fact outside the surface of 
 pressure the surface can only be represented by a complicated 
 function. But we find that inside the surface of pressure the 
 surface remains a quadric surface to the same approximation 
 as before. Here we have ^ — f 2 = a — Ax 2 — ~hy 2 — a — z 1 + z 2 , 
 again £i = 3~i~P> £2 = ^2^> or £1 : £2 = ^"1 : ^2> anc ^ lastly, the 
 equation of the deformed surface is z = % + £1 = %2 + f 2 5 whence 
 neglecting a constant, we easily deduce (^-^^z = S- 2 z 1 + S- 1 z 2 . 
 This equation expresses what we wished to demonstrate ; it 
 also shows that the common surface after deformation lies 
 between the two original surfaces, and most nearly resembles 
 the body which has the greater modulus of elasticity. When 
 spheres are in contact the surface of pressure also forms part 
 of a sphere : when cylinders touch with axes crossed it forms 
 part of a hyperbolic paraboloid. 
 
 So far we have spoken of the changes of form, now we 
 will consider the stresses. We have already found for the 
 normal pressure in the compressed surface 
 
 7 3 ^ /1 ?ZZ 
 
 z 2ahirSJ L a 2 b 2 ' 
 
 This increases from the periphery to the centre, as do the 
 ordinates of an ellipsoid constructed on the ellipse of pressure ; 
 it vanishes at the edge, and at the centre is one and a half 
 times as great as it would be if the total pressure were 
 
vi ON HARDNESS 173 
 
 equally distributed over the surface of pressure. Besides Z z 
 the remaining two principal tensions at the origin can be 
 expressed in a finite form. It may be sufficient to state that 
 they are also pressures of the same order of magnitude as Z z> 
 and are of such intensity that, provided the material is at 
 all compressible, it will suffer compression in all three direc- 
 tions. When the curve of pressure is a circle, these forces 
 are to Z 2 in the ratio of (1 + 4©)/2(l + 2©) : 1 ; for glass 
 about as 5/6 : 1. The distribution of stress inside depends 
 not only on the form of the ellipse of pressure, but also 
 essentially on the elastic coefficient © ; so that it may be 
 entirely different in the two bodies which are in contact. 
 When we compare the stresses in the same material for the 
 same form but different sizes of the ellipse of pressure and 
 different total pressures, we see that the stresses at points 
 similarly situated with regard to the surface of pressure are 
 proportional to each other. To get the pressures for one case 
 at given points we must multiply the pressures at similarly 
 situated points in the other case by the ratio of the total 
 pressures, and divide by the ratio of the compressed areas. 
 If we suppose two given bodies in contact and only the 
 pressure between them to vary, the deformation of the 
 material varies as the cube root of this total pressure. 
 
 It is desirable to obtain a clear view of the distribution 
 
 of stress in the interior ; but the formulae are far too compli- 
 cated to allow of our doing this directly. But by considering 
 the stresses near the z-axis and near the surface we can form 
 a rough notion of this distribution. The result may be 
 expressed by the following description and the accompanying 
 diagram (Fig. 19), which represents a section through the axis of 
 
174 ON HARDNESS 
 
 VI 
 
 z and an axis of the ellipse of pressure ; arrow-heads pointing 
 towards each other denote a tension, those pointing away from 
 each other a pressure. The figure relates to the case in which 
 = 1. The portion ABDC of the body, which originally 
 formed an elevation above the surface of pressure, is now 
 pressed into the body like a wedge ; hence the pressure is 
 transmitted not only in the direct line AE, but also, though 
 with less intensity, in the inclined directions AF and AG. 
 The consequence is that the element is also powerfully com- 
 pressed laterally ; while the parts at F and G are pressed apart 
 and the intervening portions stretched. Hence at A on the 
 element of area perpendicular to the ^-axis there is pressure, 
 which diminishes inwards, and changes to a tension which 
 rapidly attains a maximum, and then, with increasing distance, 
 diminishes to zero. Since the part near A is also laterally 
 compressed, all points of the surface must approach the origin, 
 and must therefore give rise to stretching in a line with the 
 origin. In fact the pressure which acts at A parallel to the 
 axis of x already changes to a tension inside the surface of 
 pressure as we proceed along the as-axis ; it attains a maximum 
 near its boundary and then diminishes to zero. Calculation 
 shows that for ® = 1 this tension is much greater than that 
 in the interior. As regards the third principal pressure which 
 acts perpendicular to the plane of the diagram, it of course 
 behaves like the one parallel to the a>axis ; at the surface it 
 is a pressure, since here all points approach the origin. If 
 the material is incompressible the diagram is simplified, for 
 since the parts near A do not approach each other, the tensions 
 at the surface disappear. 
 
 We shall briefly mention the simplifications occurring in 
 the formulae, when the bodies in contact are spheres, or are 
 cylinders which touch along a generating line. In the first 
 case we have simply Jc= fi = v = 1, p n = p 12 = p v p 2i = P22 — Pi 5 
 hence 
 
 V 16(^4-^) 16a 
 
 The formulas for the case of cylinders in contact are not 
 got so directly. Here the major semi-axis a of the ellipse 
 becomes infinitely great ; we must also make the total pressure 
 
vi ON HARDNESS 175 
 
 p infinite, if the pressure per unit length of the cylinder is 
 to be finite. We then have in the second of equations (I) 
 B = Kpj -f- p 2 ). Further, we may neglect u compared with 
 a 2 , take a outside the sign of integration, and put for the 
 indeterminate quantity p/a = co /co an arbitrary finite constant, 
 say ^p' ; then, as we shall see directly, p' is the pressure per 
 unit length of the cylinder. The integration of the equation 
 can now be performed, and gives 
 
 5= [FK±E>. 
 
 V wfo + ft) 
 
 For the pressure Z z we find 
 
 and it is easy to see that p' has the meaning stated. The 
 distance of approach a, according to our general formula, be- 
 comes logarithmically infinite. This means that it depends 
 not merely on what happens at the place of contact, but also 
 on the shape of the body as a whole ; and thus its determina- 
 tion no longer forms part of the problem we are dealing with. 
 I shall now describe some experiments that I have per- 
 formed with a view to comparing the formulae obtained with 
 experience ; partly that I may give a proof of the reliability 
 of the consequences deduced, and their applicability to actual 
 circumstances, and partly to serve as an example of their 
 application. The experiments were performed in such a way 
 that the bodies used were pressed together by a horizontal 
 one-armed lever. From its free end were suspended the 
 weights which determined the pressure, and to it the one body 
 was fastened close to the fulcrum. The other body, which 
 formed the basis of support, was covered by the thinnest 
 possible layer of lamp-black, which was intended to record the 
 form of the surface of pressure. If the experiment succeeded, 
 the lampblack was not rubbed away, but only squeezed flat ; 
 in transmitted light the places of action of the pressure could 
 hardly be detected ; but in reflected light they showed as 
 small brilliant circles or ellipses, which could be measured 
 fairly accurately by the microscope. The following numbers 
 are the means of from 5 to 8 measurements. 
 
176 
 
 ON HAEDNESS 
 
 I first examined whether the dimensions of the surface of 
 
 pressure increased as the cube root of the pressure. To this 
 
 end a glass lens of 28*0 mm. radius was fastened to the lever; 
 
 the small arm of the lever measured 114*0 mm., the large one 
 
 930 mm. The basis of support was a plane glass plate ; the 
 
 Young's modulus was determined for a bar of the same glass 
 
 and found to be 6201 kg/mm 2 . According to Wertheim, 
 
 Poisson's ratio for glass is 0'32,whence ® = |,K = 2349kg/mm, 2 
 
 and $ = 0005790 mm 2 /kg. Hence our formula gives for the 
 
 diameter of the circle of pressure in mm., d — 0*3650^, where 
 
 p is measured in kilogrammes weight. In the following 
 
 table the first row gives in kilogrammes the weight suspended 
 
 from the long arm of the lever, the second the measured 
 
 diameter of the surface of pressure in turns of the micrometer 
 
 screw of pitch 0*2737 mm. Lastly, the third row gives the 
 
 3 _ 
 quotient d ' \/p, which should, according to the preceding, be 
 
 a constant. 
 
 p 
 
 0-2 
 
 0-4 
 
 0-6 
 
 0-8 
 
 1-0 
 
 1-5 
 
 2-0 ! 2-5 
 
 3-0 
 
 3-5 
 
 d 
 
 1-56 
 
 2-03 
 
 2-19 
 
 2-59 
 
 2-68 
 
 3-13 
 
 3-52 
 
 3'69 
 
 3-97 
 
 4-02 
 
 3 
 
 2-67 
 
 275 
 
 2-60 
 
 2-79 
 
 2-68 
 
 2-73 
 
 2-79 2-71 
 
 2-70 
 
 2-65 
 
 The ratio in question does indeed remain constant, apart 
 from irregularities, though the weights vary up to fifteen times 
 their initial value. To get the theoretical value of the ratio 
 we must divide the factor *3650 calculated above by the pitch 
 in millimetres of the screw, and multiply by the cube root of 
 the ratio of the long to the short arm of the lever ; we thus 
 obtain 2'685, a number almost exactly coincident with 2'707, 
 the mean of the experimental numbers. 
 
 Secondly, I have tested the laws relating to the form of 
 the curve of pressure by pressing together two glass cylinders, 
 of equal diameter 7*37 mm., with their axes inclined at dif- 
 ferent angles to each other. If this angle be called co, using 
 former equations we get p n = p 12 = p, p 2l = p 22 = 0, A + B = p, 
 A - B = - p cos co, and therefore the auxiliary angle r = co 
 Hence if we determine the large and small axes of the ellipse 
 of pressure for one and the same pressure but different inclina- 
 
ON HAEDNESS 
 
 177 
 
 tions, divide the major axes by the function //, belonging to 
 the inclination used and the minor axes by the corresponding 
 function v, the quotient of all these divisions must be one and 
 the same constant, namely, the quantity 2(3pS-/8p)K The 
 following table gives in the first column the inclination &> in 
 degrees, in the next two the values of 2 a and 2 b as measured 
 in parts of the scale of the micrometer eye-piece, of which 
 9 6 equal one millimetre, and in the last two the quotients 2a/ fi 
 and 2b /v : — 
 
 w 
 
 2 a 
 
 2 b 
 
 2a 
 
 2b 
 
 V 
 
 90 
 
 40-6 
 
 40-6 
 
 40-6 
 
 40-6 
 
 80 
 
 45-4 
 
 36-6 
 
 40-2 
 
 41-0 
 
 70 
 
 52'8 
 
 31 
 
 41-3 
 
 387 
 
 60 
 
 59-6 
 
 27-6 
 
 40-0 
 
 38-5 
 
 50 
 
 72-2 
 
 26-4 
 
 41-2 
 
 41-2 
 
 40 
 
 90-4 
 
 23-8 
 
 ' 42-2 
 
 42-0 
 
 30 
 
 110-0 
 
 21-0 
 
 40-3 
 
 42-6 
 
 20 
 
 156-2 
 
 18-4 
 
 41-3 
 
 45-3 
 
 10 
 
 274-6 
 
 15-0 
 
 41-6 
 
 47-0 
 
 The quotients are fairly constant, excepting those for the 
 minor axes at small inclinations. But at such an inclination 
 it is extremely difficult to bring the cylinders together so as 
 to make the common tangent plane exactly horizontal ; and in 
 any other position a slight slipping of one cylinder on the 
 other occurs, which unduly magnifies the minor axis. In all 
 these measurements the pressure was 12 kg. weight. Taking 
 for & the value "0005790 already used, we get from the given 
 values the value of the constant to be 40'80, which agrees 
 almost exactly with 40'97, the mean resulting from the values 
 for a ; whilst it differs slightly, for the reasons explained, from 
 41 '8 8, the mean resulting from the value for b. 
 
 Lastly, I have attempted to examine the effect of the 
 moduli of elasticity by pressing a steel lens against planes of 
 different metals. But here I encountered difficulties in the 
 observation. In the first place, it is not so easy to obtain 
 quite plane and smooth surfaces as for glass ; secondly, the 
 metallic surfaces cannot so easily be covered with lamp-black ; 
 thirdly, we have to confine ourselves to very small pressures 
 M. P. n 
 
178 ON HARDNESS 
 
 VI 
 
 so as not to exceed the elastic limits. All these causes together 
 preclude our obtaining any but very imperfect curves of pressure, 
 and in measuring these there is room for discretion. I obtained 
 values which were always of the order of magnitude of those 
 calculated, but were too uncertain to be of use in accurately 
 testing the theory. However, the numbers given show con- 
 clusively that our formulae are in no sense speculations, and 
 so will justify the application now to be made of them. The 
 object of this is to gain a clearer notion and an exact measure 
 of that property of bodies which we call hardness. 
 
 The hardness of a body is usually defined as the resist- 
 ance it opposes to the penetration of points and edges into it. 
 Mineralogists are satisfied in recognising in it a merely com- 
 parative property ; they call one body harder than another 
 when it scratches the other. The condition that a series of 
 bodies may be arranged in order of hardness according to this 
 definition is that, if A scratches B and B scratches C, then A 
 should scratch C and not vice versa ; further, if a point of A 
 scratches a plane plate of B, then a point of B should not 
 penetrate into a plane of A. The necessity of the concurrence 
 of these presuppositions is not directly manifest. Although 
 experience has justified them, the method cannot give a 
 quantitative determination of hardness of any value. Several 
 attempts have been made to find one. Muschenbroek measured 
 hardness by the number of blows on a chisel which were 
 necessary to cut through a small bar of given dimensions of 
 the material to be examined. About the year 1850 Crace- 
 Calvert and Johnson measured hardness by the weight which 
 was necessary to drive a blunt steel cone with a plane end 
 1*25 mm. in diameter to a depth of 3 "5 mm. into the given 
 material in half an hour. According to a book published in 
 I860, 1 Hugueny measured the same property by the weight 
 necessary to drive a perfectly determinate point 0*1 mm. deep 
 into the material. More recent attempts at a definition I 
 have not met with. To all these attempts we may urge the 
 following objections : ( 1 ) The measure obtained is not only not 
 absolute, since a harder body is essential for the determination, 
 but it is also entirely dependent on a point selected at random. 
 Erom the results obtained we can draw no conclusions at all 
 
 1 F. Hugueny, Rechcrchcs experiment ales sur la durete des corps. 
 
vi ON HARDNESS 179 
 
 as to the force necessary to drive in another point. (2) Since 
 finite and permanent changes of form are employed, elastic 
 after-effects, which have nothing to do with hardness, enter 
 into the results of measurement to a degree quite beyond 
 estimation. This is shown only too plainly by the introduc- 
 tion of the time into the definition of Crace- Calvert and 
 Johnson, and it is therefore doubtful whether the hardness of 
 bodies thus measured is always in the order of the ordinary 
 scale. (3) We cannot maintain that hardness thus measured 
 is a property of the bodies in their original state (although 
 without doubt it is dependent upon that state). For in the 
 position in the experiment the point already rests upon per- 
 manently stretched or compressed layers of the body. 
 
 I shall now try to substitute for these another definition, 
 against which the same objections cannot be urged. In the 
 first place I look upon the strength of a material as determined, 
 not by forces producing certain permanent deformations, but 
 by the greatest forces which can act without producing de- 
 viations from perfect elasticity, to a certain predetermined 
 accuracy of measurement. Since the substance after the action 
 and removal of such forces returns to its original state, the 
 strength thus defined is a quantity really relating to the original 
 substance, which we cannot say is true for any other definition. 
 The most general problem of the strength of isotropic bodies 
 would clearly consist in answering the question — Within what 
 limits may the principal stresses X x , Y y , Z 2 in any element lie 
 so that the limit of elasticity may not be exceeded ? If we 
 represent X,., Y y , Z z as rectangular rectilinear coordinates of a 
 point, then in this system there will be for every material a 
 certain surface, closed or in part extending to infinity, round 
 the origin, which represents the limit of elasticity ; those values 
 of X x , Y y , Z z which correspond to internal points can be borne, 
 the others not so. In the first place it is clear that if we 
 knew this surface or the corresponding function -v/r (X^,, Y y , 
 ZJ = for the given material, we could answer all the 
 questions to the solution of which hardness is to lead us. For 
 suppose a point of given form and given material pressed 
 against a second body. According to what precedes we know 
 all the stresses occurring in the body ; we need therefore only 
 see whether amongst them there is one corresponding to a 
 
180 ON HARDNESS 
 
 VI 
 
 point outside the surface yjr (X x , Y y , Z z ) = 0, to be enabled to 
 tell whether a permanent deformation will ensue and, if so, in 
 which of the two bodies. But so far there has not even been 
 an attempt made to determine that surface. We only know 
 isolated points of it : thus the points of section by the positive 
 axes correspond to resistance to compression ; those by the 
 negative axes to tenacity ; other points to resistance to torsion. 
 In general we may say that to each point of the surface of 
 strength corresponds a particular kind of strength of material. 
 As long as the whole of the surface is not known to us, we 
 shall let a definite discoverable point of the surface correspond 
 to hardness, and be satisfied with finding out its position. 
 This object we attain by the following definition, — Hardness 
 is the strength of a body relative to the kind of deformation 
 which corresponds to contact with a circular surface of pressure. 
 And we get an absolute measure of the hardness if we decide 
 that — The hardness of a body is to be measured by the normal 
 pressure per unit area which must act at the centre of a circular 
 surface of pressure in order that in some point of the body the 
 stress may just reach the limit consistent with perfect elasticity. 
 To justify this definition we must show (1) that the neglected 
 circumstances are without effect ; (2) that the order into 
 which it brings bodies according to hardness coincides with 
 the common scale of hardness. To prove the first point, 
 suppose a body of material A in contact with one of material 
 
 B, and a second body made of A in contact with one made of 
 
 C. The form of the surfaces may be arbitrary near the point 
 of contact, but we assume that the surface of pressure is 
 circular, and that B and C are harder or as hard as A. Then 
 we may simultaneously allow the total pressures at both con- 
 tacts to increase from zero, so that the normal pressure at the 
 centre of the circle of pressure may be the same in both cases. 
 We know that then the same system of stresses occurs in both 
 cases, therefore the elastic limit will first be exceeded at the 
 same time and at points similarly situated with respect to 
 the surface of pressure. We should from both cases get the 
 same value for the hardness, and this hardness would cor- 
 respond to the same point of the surface of strength. It is 
 obvious that the elements in which the elastic limit is first 
 exceeded may have very different positions relatively to the 
 
VI ON HARDNESS 181 
 
 surface of pressure in different materials, and that the positions 
 of the points of hardness in the surface of strength may be 
 very dissimilar. We have to remark that the second body 
 which was used to determine the hardness of A might have 
 been of the same material A ; we therefore do not require a 
 second material at all to determine the hardness of a given 
 rtie. This circumstance justifies us in designating the above 
 as an absolute measurement. To prove the second point, 
 suppose two bodies of different materials pressed together ; let 
 the surface of pressure be circular ; let the hardness, defined as 
 above, be for one body H, for the second softer one h. If now 
 we increase the pressure between them until the normal 
 pressure at the origin just exceeds h, the body of hardness h 
 will experience a permanent indentation, whilst the other one 
 is nowhere strained beyond its elastic limit ; by moving one 
 body over the other with a suitable pressure we can in the 
 former produce a series of permanent indentations, whilst the 
 latter remains intact. If the latter body have a sharp point 
 we can describe the process as a scratching of the softer by 
 the harder body, and thus our scale of hardness agrees with 
 the mineralogical one. It is true that our theory does not 
 say whether the same holds good for all contacts, for which 
 the compressed surface is elliptical ; but this silence is justifi- 
 able. It is easy to see that just as hardness has been defined 
 by reference to a circular surface of pressure, so it could have 
 been defined by assuming for it any definite ellipticity. The 
 hardnesses thus diversely defined will show slight numerical 
 variations. Now the order of the bodies in the different 
 scales of hardness is either the same, or it is not. In the first 
 case, our definition agrees generally with the mineralogical one. 
 In the second case, the fault lies with the mineralogical 
 definition, since it cannot then give a definite scale of hardness 
 at all. It is indeed probable that the deviations from one 
 another of the variously defined hardnesses would be found 
 only very small ; so that with a slight sacrifice of accuracy we 
 might omit the limitation to a circular surface of pressure 
 both in the above and in what follows. Experiments alone 
 can decide with certainty. 
 
 Now let H be the hardness of a body which is in contact 
 with another of hardness greater than H. Then by help of 
 
182 ON HARDNESS vi 
 
 this value we can make this assertion, that all contacts with a 
 circular surface of pressure for which 
 
 Z.= ^2 
 
 I 'JPJPu + Pu + Pa. + P22) 2 s h 
 
 or for which 
 
 (S-. + ^f - 24 
 
 can be borne, and only these. 
 
 The force which is just sufficient to drive a point with 
 spherical end into the plane surface of a softer body, is pro- 
 portional to the cube of the hardness of this latter body, to 
 the square of the radius of curvature of the end of the point, 
 and also to the square of the mean of the coefficients -9- for the 
 two bodies. To bring this assertion into better accord with 
 the usual determinations of hardness we might be tempted 
 to measure the latter not by the normal pressure itself, but 
 rather by its cube. Apart from the fact that the analogy 
 thus produced would be fictitious (for the force necessary to 
 drive one and the same point into different bodies would not 
 even then be proportionate to the hardness of the bodies), this 
 proceeding would be irrational, since it would remove hardness 
 from its place in the series of strengths of material. 
 
 Though our deductions rest on results which are satis- 
 factorily verified by experience, still they themselves stand much 
 in need of experimental verification. For it might be that 
 actual bodies correspond very slightly with the assumptions of 
 homogeneity which we have made our basis. Indeed, it is 
 sufficiently well known that the conditions as to strength near 
 the surface, with which we are here concerned, are quite different 
 from those inside the bodies. I have made only a few experi- 
 ments on glass. In glass and all similar bodies the first trans- 
 gression beyond the elastic limit shows itself as a circular crack 
 which arises in the surface at the edge of the compressed surface, 
 and is propagated inwards along a surface conical outwards when 
 the pressure increases. When the pressure increases still 
 further, a second crack encircles the first and similarly pro- 
 pagates itself inwards ; then a third appears, and so on, the 
 phenomenon naturally becoming more and more irregular. 
 
VI ON HARDNESS 183 
 
 From the pressures necessary to produce the first crack 
 under given circumstances, as well as from the size of this 
 crack, we get the hardness of the glass. Thus experiments in 
 which I pressed a hard steel lens against mirror glass gave the 
 value 130 to 140 kg/mm 2 for the hardness of the latter. 
 From the phenomena accompanying the impact of two glass 
 f neres ; I estimated the hardness at 150; whilst a much 
 larger value, 180 to 200, was deduced from the cracks pro- 
 duced in pressing together two thin glass bars with natural 
 surfaces. These differences may in part be due to the defici- 
 encies of the methods of experimenting (since the same method 
 gave rise to considerable variations in the various results) ; 
 but in part they are undoubtedly caused by want of homogeneity 
 and by differences in the value of the surface-strength. If 
 variations as large as the above are found to be the rule, then 
 of course the numerical results drawn from our theory lose 
 their meaning ; even then the considerations advanced above 
 afford us an estimate of the value which is to be attributed to 
 exact measurements of hardness. 
 
VII 
 ON A NEW HYGKOMETEK 
 
 {Verhandlungen cler physikalischen Gesellschaft zu Berlin, 20th January 1882.) 
 
 In this hygrometer, and others constructed on the same 
 principle, the humidity is measured by the weight of water 
 absorbed from the air by a hygroscopic inorganic substance, 
 such as a solution of calcium chloride. Such a solution will 
 absorb water from the air, or will give up water to the air, 
 until such a concentration is attained that the pressure of the 
 saturated water-vapour above it at the temperature of the air 
 is equal to the pressure of the (unsaturated) water -vapour 
 actually present in the air. If the temperature and humidity 
 change so slowly as to allow the state of equilibrium to be 
 attained, the absolute humidity can be deduced from the 
 temperature and the weight of the solution. But it appears 
 that for most salts, and at any rate for calcium chloride (and 
 sulphuric acid), the pressure of the saturated vapour above 
 the salt solution at the temperatures under consideration is 
 approximately a constant fraction of the pressure of saturated 
 water-vapour. Hence the relative humidity can be deduced 
 directly from the weight with sufficient accuracy for many 
 purposes. And if great accuracy is required, the effect of 
 temperature can be introduced as a correcting factor, which 
 need only be approximately known. 
 
 The idea suggested can be realised in two ways. The 
 instrument may either be adapted for rapidly following changes 
 of humidity, when great accuracy is not required, as in balance- 
 rooms ; or it may be adapted for accurate measurements, if we 
 
Vii HYGROMETER 185 
 
 only require the average humidity over a lengthened period 
 (days, weeks, or months), as in meteorological investigations. 
 An instrument of the first kind was exhibited to the Society. 
 The hygroscopic substance was a piece of tissue-paper of 1 
 sq. cm. surface, saturated with calcium chloride, and attached 
 to one arm of a lever (glass fibre) about 10 cm. long. The 
 V jter was supported on a very thin silver wire stretched 
 norizontally, so that the whole formed a very delicate torsion 
 balance. The hygrometer was calibrated by means of a series 
 of mixtures of sulphuric acid and water by Eegnault's method. 
 In dry air the fibre stood about 45° above the horizontal. In 
 air of relative humidity 10, 20, . . . 90 per cent it sank 
 downwards through 18, 31, 40, 47, 55, 62, 72, 86, 112 de- 
 grees. In saturated water- vapour it naturally stood vertically 
 downwards. The only thing ascertained as to the effect of 
 temperature was that it is very small. For equal relative 
 humidities the pointer stood 1 to 2 degrees lower at 0° than 
 at 25°. When brought into a room of different humidity, the 
 instrument attained its position of equilibrium so rapidly that 
 it could be read off after 10 to 15 minutes. The instrument 
 has the disadvantage that when the humidity is very great 
 (85 per cent and upwards), visible drops are formed on the 
 paper, and if it be carelessly handled these may be wiped or 
 even shaken off. 
 
 In instruments of the second kind the calcium chloride 
 would be contained in glass vessels of a size adapted to the 
 interval of time for which the mean humidity is required. 
 These vessels would be weighed from time to time, or placed 
 on a self-registering balance. 
 
VIII 
 
 ON THE EVAPOEATION OF LIQUIDS, AND 
 ESPECIALLY OF MEECUBY, IN VACUO 
 
 {Wiedemann 's Annalen, 17, pp. 177-193, 1882.) 
 
 When" a liquid evaporates into a gas whose pressure is greater 
 than the pressure of the saturated vapour of the liquid, the vapour 
 near the surface is always exceedingly near the state of satura- 
 tion ; and the rate of evaporation is chiefly determined by the 
 rate at which the vapour formed is removed. The removal of 
 the vapour, at any rate through the layers nearest the surface, 
 takes place by diffusion. Starting with this conception, the 
 evaporation of a liquid into a gas has been frequently discussed. 
 But hitherto no attention seems to have been paid to the con- 
 ditions which determine the rate of evaporation in a space 
 which contains nothing but the liquid and its vapour. In 
 this paper evaporation under these conditions will be considered. 
 In the first place evaporation in vacuo is affected by the rate at 
 which the vapour formed can escape, in so far as this escape may 
 under certain circumstances be greatly retarded by viscosity ; 
 but clearly this is a matter of very little importance. For if 
 we imagine the evaporation to take place between two plane 
 parallel liquid surfaces, then as far as this is concerned 
 the rate of evaporation might be infinite. Again we may 
 specify the rate at which heat is supplied to the surface 
 of the liquid as the condition of evaporation. When the 
 stationary state is attained, the amount of liquid which 
 evaporates is just so much that its latent heat is equal 
 to the amount of heat supplied. But this explanation 
 
vin EVAPORATION OF LIQUIDS 187 
 
 is incomplete, since we might equally well regard the supply 
 of heat, conversely, as being determined by the evaporation. 
 For both depend upon the temperature of the outermost layer 
 of liquid ; and this again is determined by the relation between 
 the possible supply of heat by conduction and the possible loss 
 of heat by evaporation. Now one of two things must happen. 
 F'ther (a) evaporation has no limit beyond that which is 
 mvolved in the supply of heat ; so that if sufficient heat is 
 supplied, an unlimited amount of liquid can evaporate from a 
 given surface in unit time, and the temperature, density, and 
 pressure of the vapour produced will not differ perceptibly 
 from that of saturated vapour. In this case all liquid surfaces 
 in the same space must assume the same temperature ; and 
 this temperature as well as the amounts of liquid which 
 evaporate are determined by the relation between the possible 
 supply of heat and the different areas. Or (b) only a limited 
 quantity of liquid can evaporate from a liquid surface at a 
 given temperature. In this case there may be surfaces at 
 different temperatures in the same space, and the pressure and 
 density of the vapour arising must differ by a finite amount 
 from the pressure and density of the saturated vapour of at 
 least one of these surfaces : the rate of evaporation will depend 
 upon a number of circumstances, but chiefly upon the nature 
 of the liquid ; so that there will be for every liquid a specific 
 evaporative power. It will be seen that the alternative (a) 
 can be regarded as a limiting case of (&). Hence in the 
 absence of any hypothesis or experimental information we 
 should have to assume the latter, which is the more general, 
 to be correct. But we shall presently show by a more detailed 
 discussion that the first-mentioned alternative is an extremely 
 improbable one. 
 
 I have made a number of experiments on evaporation in 
 vacuo in the hope of arriving at an experimental decision 
 between these two alternatives, if possible by exact measure- 
 ments of the evaporative power of any liquid under different 
 conditions. The experiments have only partly achieved their 
 aim : nevertheless I describe them here, because they throw 
 light upon the problem, and may clear up the way for better 
 methods. The experiments are described in the first section : 
 in the second section is given a theoretical discussion, which 
 
188 EVAPORATION OF LIQUIDS vm 
 
 justifies the view adopted and establishes limits for the quan- 
 tities under consideration. 
 
 I. I started the experiments on the assumption that the 
 rate of evaporation of a liquid is at all events determined by 
 the temperature of the surface and the pressure exerted upon 
 it by the vapour which arises. In the course of the investiga- 
 tion I began to doubt, not whether these magnitudes were 
 necessary conditions, but whether they were sufficient condi- 
 tions for determining the amount of liquid which evaporates : 
 in the second section it will be shown that there was no reason 
 for this doubt. Hence I first set to work at the following 
 problem : — To find for any fluid simultaneous values of the 
 temperature t of the surface, the pressure P upon it, and the 
 height h of the layer of liquid which evaporates from it 
 in unit time. The difficulty experienced in solving this 
 apparently simple problem arises in the determination of t 
 and P. Even when the evaporation only goes on at a moderate 
 rate very considerable quantities of heat are required to keep 
 it up ; the result of which is that the temperature increases 
 very rapidly from the surface towards the interior. Hence if 
 we dip a thermometer the least bit into the liquid it does not 
 show the true surface temperature. The experiments further 
 showed that at moderate rates of evaporation there was only 
 a slight difference between the pressure and the pressure of 
 saturated vapour. As it is just this difference that we wish 
 to examine, it follows that both pressures must be very 
 accurately measured. Lastly, the interior of the liquids in 
 these experiments is necessarily in the superheated state ; and 
 since boiling with bumping would render the experiments 
 impracticable, one is restricted to a very narrow range of 
 temperature and pressure. 
 
 I pass over certain experiments made with water, for I 
 soon observed that water, on account of its high latent heat 
 and low conductivity, was ill suited for my purpose. Mercury 
 appeared to be the most suitable liquid, for it has a relatively 
 small latent heat and a conductivity similar to that of metals ; 
 and on account of its cohesion and the low pressure of its 
 vapour, it can be superheated strongly without boiling. The 
 first experiments were carried out with the apparatus shown 
 
EVAPORATION OF LIQUIDS 
 
 Fig. 20. 
 
 in Fig. 20. Into the retort A, placed inside a heating vessel, 
 was fused a glass tube open above and closed below ; inside 
 this and just under the surface of the mercury was the ther- 
 mometer which indicated the temperature. To the neck of the 
 retort was attached the vertical tube B, which was immersed 
 in a fairly large cooling vessel, and could 
 tHis be maintained at 0° or any other 
 temperature. By brisk boiling and simul- 
 taneous use of a mercury pump all per- 
 ceptible traces of air were removed from 
 the apparatus. The rate of evaporation 
 was now measured by the rate at which the 
 mercury rose in the tube B. The pressure 
 P was not to be directly measured. I sup- 
 posed, as is frequently done, that P could 
 not exceed the pressure of the saturated 
 vapour at the lower temperature, viz. that 
 of B ; and assumed that it would suffice to vary the latter 
 temperature only in order to obtain corresponding values of 
 the pressure. It soon became clear that this assumption was 
 erroneous; for when the temperature began to exceed 100°, 
 and the evaporation became fairly rapid, the vapour did not 
 condense in the cold tube B, but in the neck or connecting 
 tube at C. This became so hot that one could not touch it ; its 
 temperature was at least 60° to 80°. This cannot be explained 
 on the assumption that the vapour inside has the exceedingly 
 low pressure corresponding to 0° ; for in that case it could 
 only be superheated by contact with a surface at 60°, and 
 could not possibly suffer condensation. In order to measure 
 the pressure I introduced at C the manometer tube shown in 
 the figure. But this did not show any change from its initial 
 position when the rate of evaporation was increased. It was 
 certain that the vapour moved with a certain velocity, so that 
 its pressure upon the surface from which it arose must be 
 different from the pressure which it would naturally possess. 
 It could easily be seen that this velocity was very considerable ; 
 for when the drops of mercury on the glass attained a certain 
 size they did not fall downwards from their weight, but were 
 carried along nearly parallel to the direction of the tube. In 
 order to see whether the vapour exerted a pressure upon the 
 
190 
 
 EVAPORATION OF LIQUIDS 
 
 VIII 
 
 evaporating surface (for this pressure is really the interesting 
 point), I now fused the manometer on at A, as shown in the 
 figure, so that the retort itself formed the open limb. It 
 turned out that there was a very perceptible pressure ; it 
 amounted to 2 to 3 mm. when the thermometer stood at 160° to 
 170°, and the evaporation went on at such a rate that a layer 
 0"8 mm. deep evaporated per minute. Hence there was no 
 difficulty in seeing that in its condensation the vapour might 
 produce a temperature exceeding 100°; however, it became 
 clear that the simple method which had been tried would not 
 lead to the desired result, but that direct measurements would be 
 necessary. The apparatus shown in Fig. 2 1 was therefore used. 
 A is again the retort. The heating vessel (only indicated in 
 the diagram) in which it was contained consisted of a hollow brass 
 cylinder 1*5 cm. thick, closely surrounding the retort and covered 
 over with asbestos. It was heated by a ring gas-burner, and 
 had in it a vertical slit through which the level of the mercury 
 could be observed. B is again the tube in which the condensa- 
 tion takes place ; the manometer tube is shown in perspective 
 at C. The magnifying power of the cathetometer telescope 
 used was such that it could be set with 
 certainty to within 0*02 mm. The pres- 
 sure, i.e. the difference of level between 
 the two surfaces, was measured by a 
 micrometer eye-piece with two threads : 
 the absolute height of the surface, i.e. 
 the rate of evaporation, was read off 
 on the scale of the instrument. The 
 temperature was varied by altering the 
 gas supply. The apparatus was at 
 first quite free from air : by admitting 
 varying, but always small, quantities of 
 air different pressures could be obtained 
 at the same temperature. If the pressure of the air 
 introduced amounted, say, to 1 mm. no evaporation in the 
 sense here considered could take place so long as the pressure 
 of the saturated vapour above the surface did not exceed 1 
 mm., i.e. so long as the temperature of the surface did not 
 exceed 120° ; but when this temperature was exceeded the air 
 retreated into the condensing tube, and evaporation began ; 
 
 Fig. 21. 
 
vin EVAPORATION OF LIQUIDS 191 
 
 but of course it now took place under greater pressure than it 
 did at the same temperature before the air was introduced. 1 
 Three quantities, h, P, and t, had to be measured. In deter- 
 mining the first there was no difficulty. The determination 
 of P was not simply a case of measuring accurately the 
 difference of level ; large corrections on account of the ex- 
 pansion of the mercury, etc. had to be applied, and some of 
 these were much larger than the quantity whose value was 
 sou \ tit. But by a careful application of theory and by special 
 experiments these corrections could be so far determined that 
 the final measurement could be relied upon to about 0*1 mm. 
 The outstanding error was so small that the greater part of 
 the observations would not be injuriously affected by it. The 
 most uncertain element was the determination of t. I thought 
 it was safe to assume that the true mean temperature of the 
 surface could not differ by more than a few degrees from the 
 temperature indicated by the thermometer when the upper 
 end of its bulb (about 18 mm. long) was just level with the 
 surface ; and it seemed probable that of the two the true 
 temperature would be the higher. For the bulk of the heat 
 was conveyed by the rapid convection currents ; these seemed 
 first to rise upwards from the heated walls of the vessel, then to 
 pass along the surface, and finally, after cooling, down along the 
 thermometer tube. If this correctly describes the process, the 
 bulb of the thermometer was at the coolest place in the liquid. 
 With this apparatus I carried out a large number of experi- 
 ments at temperatures between 100° and 200°, and at nine 
 different pressures (i.e. with nine different admissions of air). 
 The separate observations naturally showed irregularities ; but 
 unless some constant error was present, they undoubtedly point 
 to the following result : — The observed pressure P was always 
 smaller than the pressure P^ of the saturated vapour corre- 
 sponding to the temperature t ; at a given temperature the depth 
 of the layer which evaporated in unit time was proportional 
 to the difference P^ — P ; when this difference was 1 mm. the 
 depth of the layer which evaporated per minute was 0*5 mm. 
 
 1 During the observations there was no air in the retort or the connecting 
 
 tube. Thus the introduction of the air does not invalidate the title of this 
 
 paper. The title, indeed, has only been used for brevity in place of a more 
 precise one. 
 
192 EVAPORATION OF LIQUIDS vm 
 
 at 120° J 0'35 at 150°,and 0-25 mm. at 180° to 200°. As an 
 example may be given the case in which the highest rate of 
 evaporation was observed. In this case the vessel was quite 
 free from air, the temperature was 183°'3, the pressure 3*32 
 mm., and the level of the mercury sank uniformly at the rate 
 of 1*80 mm. per minute. Now, since the pressure of the 
 saturated vapour 1 is 10*35 mm. at 183*3°, and 3*32 mm. at 
 153°*0, we must assume that there was an error of 7 mm. in 
 the measurement of pressure, or of 30° in the measurement 
 of temperature, if we are unwilling to admit that this proves 
 the existence of a limited rate of evaporation peculiar to the 
 liquid. The first-mentioned error could not have occurred ; 
 nor do I believe that the second could. But I could not 
 conceal from myself that the results, from the quantitative 
 point of view, were very uncertain ; and so I endeavoured to 
 support them by further experiments. For this purpose I 
 made observations with the apparatus shown in Fig. 22, a. 
 The glass vessel A, shaped like a mano- 
 meter and completely free from air, is 
 contained in a thick cast-iron heating 
 vessel in a paraffin bath. The level of 
 the mercury in both limbs is observed 
 from the outside through a plane glass 
 I 1 J plate. The open arm communicates with 
 the cold receiver B ; the communicating 
 FlG - 22 - tube is not too wide, in order that the 
 
 evaporation may take place slowly. The small condenser inside 
 the heating vessel is intended to prevent condensed mercury from 
 flowing back into the retort. There is now no difficulty in 
 observing the rate of evaporation or the pressure, at any rate 
 if we regard the pressure of the saturated vapour in the closed 
 limb as known ; the uncertainty comes in again in determining 
 the temperature of the evaporating surface. This temperature 
 is equal to that of the bath, less a correction which for a given 
 apparatus is a function of the convection current only which 
 supplies heat to the surface. The known rate of evaporation 
 gives us the required supply of heat ; from this again we can 
 deduce the difference of temperature when the above-mentioned 
 
 1 For all data as to the pressure of saturated mercury vapour here used, see 
 the determinations given in the next paper (IX. p. 200). 
 
vni EVAPORATION OF LIQUIDS 193 
 
 function has been determined. In order to find this, special 
 experiments were made with the apparatus shown in Fig. 22, b. 
 A piece of the same tube from which the manometer was 
 made, was bent at its lower end into the shape of the manometer 
 limb. This was filled with mercury to the same depth as 
 the manometer tube ; above the mercury was a layer of water 
 about 1 cm. deep, and in this a thermometer and stirrer were 
 placed. This tube was immersed up to the level of the mercury 
 in a warm linseed-oil bath, the temperature of which was 
 indicated by a second thermometer. A steady flow of heat 
 soon set in from the bath through the mercury to the water. 
 The difference between the two thermometers gave the differ- 
 ence between the temperatures of the bath and of the mercury 
 surface ; the increase of the temperature gave the corresponding 
 flow of heat. Of course a number of corrections were neces- 
 sary ; after applying these it was found that the flow of heat 
 increased somewhat more rapidly than the difference of tem- 
 perature. For example, a difference of 10°*0 was necessary in 
 order to convey to the surface per minute sufficient heat to 
 warm a layer of water 117 mm. high (lying above the surface) 
 through o, 48. I shall make use of these data for calculating 
 out an experiment made with the evaporation apparatus. 
 When the temperature of the bath was 118 o, and the differ- 
 ence of level was 0*26 mm., it was found that in 3*66 minutes 
 the mercury in both limbs sank 0*105 mm. (this was the mean 
 of measurements in both limbs). As the evaporation took 
 place only in one limb, the depth of the layer removed from 
 this in a minute was 2 x 0*1 05/3 '66 = 0*057 mm. In order 
 to vaporise unit weight of mercury at 118° under the pressure 
 of its saturated vapour, an amount of heat is required which 
 would raise 7 2 '8 units of water through 1°. This value may 
 be used with a near approach to accuracy in calculating the 
 results of our experiment. Thus there must have been con- 
 veyed to the surface per minute enough heat to raise a layer 
 of water 0*057 X 13*6 x 72*8 = 56*4 mm. high through 1°, or 
 a layer of water 117 mm. high through 56*4/117 = 0°*4 8. 
 For this, according to what precedes, there must have been a 
 difference of temperature of 10°*0 between the bath and the 
 surface ; so that the true temperature of the evaporating sur- 
 face was 108°*0. Since the mercury in the open limb was 
 M. P. 
 
194 EVAPORATION OF LIQUIDS vm 
 
 colder than that in the closed limb, the measured difference 
 of level (0 - 26 mm.) was somewhat smaller than it would 
 have been if both limbs were at the same temperature. An 
 examination of the distribution of heat in the interior gives 
 0'03 mm. as the necessary correction; thus the difference of 
 pressure in the two limbs was equal to 0*29 mm. of mercury at 
 118°, or 0*28 mm. of mercury at 0°. If we subtract from this 
 pressure the difference between the saturation-pressures at 118° 
 and 108°, we obtain the divergence between the pressure upon 
 the evaporating surface and the saturation -pressure. The 
 difference to be subtracted amounts to 0*27 mm.; so that 
 only 0*01 mm. is left. This shows that the pressure of the 
 vapour does not differ perceptibly from the saturation-pressure ; 
 and the same result follows from all the observations made by 
 this method. At lower temperatures (90° to 100°) deviations 
 of a few hundredths of a millimetre, in the direction anti- 
 cipated, were found ; but at high temperatures, on the other 
 hand, pressures were calculated which slightly exceeded the 
 saturation-pressure. Clearly there must have been slight 
 errors in the corrections, as indeed might have been expected 
 from the method of determination. But the experiments 
 undoubtedly prove two things. In the first place, that the 
 method is not well adapted for giving quantitative results, 
 because the constant errors of experiment are of the same 
 order as the quantities to be observed. In the second place, 
 that the positive results obtained by the earlier method had 
 their origin partly, if not entirely, in the errors made in 
 measuring the temperature. 1 For, if they had been correct, 
 deviations of pressure of 0'10 to 0'2 mm. must have mani- 
 fested themselves in the last experiments, and these could not 
 have escaped observation. 
 
 Thus the net result of the experiments is a very modest 
 one. They show that the pressure exerted upon the liquid 
 by the vapour arising from it is practically equal to the 
 saturation-pressure at the temperature of the surface ; and 
 hence that of the two alternatives mentioned in the intro- 
 duction, the first is to be regarded as correct. But they do 
 not show definitely the existence of the small deviation from 
 
 1 That very large errors are possible can be easily seen by calculating those 
 which would arise if the surface were only supplied with heat by conduction. 
 
/Hi EVAPORATION OF LIQUIDS 195 
 
 this rule which probably occurs, and which is of interest from 
 the theoretical point of view. 
 
 II. Let us now consider a process of steady evaporation 
 taking place between two infinite, plane, parallel liquid surfaces 
 kept at constant, but different temperatures. We shall sup- 
 pose that the liquid which evaporates over can return to its 
 starting-point by means of canals or similar contrivances. 
 All the particles of vapour will move from the one surface to 
 the other in the direction of the common normal and, neglecting 
 radiation, we may with sufficient accuracy assume that in 
 passing over they neither absorb nor give out heat. On this 
 assumption it follows from the hydrodynamic equations of 
 motion that during the whole passage from the one surface 
 to the other, whatever the distance between them may be, 
 the pressure, temperature, density, and velocity of the vapour 
 must remain constant. From this it follows that the process 
 is completely known to us when we know the following 
 quantities : — 
 
 1. The temperatures T 1 and T 2 of the two surfaces. 
 
 2. The temperature T, the pressure p, and the density d 
 of the vapour which passes over. We must suppose the 
 temperature to be measured by means of a thermometer 
 which moves forward with the vapour and with the same 
 velocity. In the same way the pressure p is to be supposed 
 measured by a manometer moving with the vapour, or deter- 
 mined by the equation of condition of the vapour. We may 
 approximately take as the latter the equation of a perfect gas, 
 KT=_p/d 
 
 3. The velocity u and the weight m which passes over in 
 unit time from unit area of the one surface to the other. 
 Clearly m = ud. 
 
 4. The pressure P which the vapour exerts upon the 
 liquid surfaces. This is necessarily the same for both surfaces, 
 and is different from the proper pressure p of the vapour 
 itself. But we can calculate P if the other quantities men- 
 tioned are known. For let us suppose the quantity m spread 
 over unit surface, the pressure upon one side of it being P 
 and on the other side p, and its temperature T maintained 
 constant. It will evaporate just as before ; after unit time 
 it will be completely converted into vapour, which will occupy 
 
196 EVAPOEATION OF LIQUIDS vni 
 
 the space u and have the velocity u. Hence its kinetic energy 
 is ^mu 2 jg- y this is attained by the force P — p acting upon its 
 centre of mass through the distance uj2, so that an amount of 
 work (P—p)u/2 is done by the external forces. From this 
 follows the equation V —p = mujg \ or, since m = ud,m 2 = 
 gd(P -p). 
 
 Now the problem which evaporation places before us is to 
 find the relations between these quantities for all possible 
 values of them. Two of the eight quantities T l5 T 2 , T, p, d, 
 u, m, and P, namely T^ and T 2 , are independent variables ; so 
 also are any two of the others. The other six are connected 
 with these by six equations. Of these we have already given 
 three ; in order to solve the problem completely we have to 
 find, from theory or experiment, three more. But if we choose, 
 as in the experiments, T 1 and P as the independent variables, 
 and consider only evaporation in the narrower sense, we are 
 no longer interested in T 2 , and the problem resolves itself into 
 representing two of the quantities T, p, d, u, m as functions 
 of T 1 and P. But now the functions to be determined do not 
 apply only to the case of evaporation between parallel walls ; 
 they hold good for any vapour which arises from a plane 
 element of a liquid, and exerts upon it a pressure P. For we 
 can imagine such evaporation taking place as if we allowed a 
 piston to rest upon the surface at temperature T 1? and at a 
 given instant removed it from the surface with velocity u. 
 The result of this experiment must be singly determined by 
 T-l and u. But the two above-mentioned functions give us one 
 possible result, and hence this result is the only possible one. 
 
 Thus the quantities relating to an element of the evap- 
 orating surface are completely determined by two of them, 
 and the assumption upon which the experiments were based 
 is justified ; on the other hand, our discussion shows that the 
 experiments, even if they had been successful, would not have 
 completely solved the problem. 
 
 We can assign limits to the quantities in question if we 
 make use of the two following assertions which, according to 
 general experience, are at any rate exceedingly likely to be 
 correct. (1) If we lower the temperature of one of several 
 liquid surfaces in the same space while the others remain at 
 the original temperature, the mean pressure upon these surfaces 
 
EVAPORATION OF LIQUIDS 
 
 197 
 
 can only be diminished, not increased. (2) The vapour arising 
 from an evaporating surface is either saturated or unsaturated, 
 never supersaturated. For it appears perfectly transparent, 
 which could not be the case if it carried with it substances in a 
 liquid state. The first statement asserts that P<Px, the second 
 that d<d p , if we denote by p 1 the pressure of the saturated 
 vapour at the temperature T 1} and /by d p the density of the 
 saturated vapour at the pressure p. Now m = Jgd{¥ —p), 
 and therefore m<s/gd p (p 1 —p). But the right-hand side of 
 this inequality is zero when ^>=0 and when p =p 1 ; between 
 these it attains a maximum value which m cannot under any 
 circumstances exceed for a surface-temperature T v But if in 
 spite of an adequate supply of heat the evaporation cannot 
 exceed a finite limit, the hindrance can only lie in the nature 
 of the fluid ; and hence every fluid must have a specific 
 evaporative power. The existence of such a constant is there- 
 fore as probable as the assumptions on which our reasoning 
 is based. From the above equation I have calculated the 
 limits for m, assuming the Gay - Lussac - Boyle law to be 
 applicable to the vapour, and taking for the relation between 
 the pressure and temperature of the saturated vapour the 
 equation log^?= 10'5927l - 0-847 log T- 3342/T, which is 
 established elsewhere. 1 By dividing the values of m by the 
 density of mercury we get values for the maximum depth of 
 the layer of liquid which can evaporate in unit time from a 
 surface at the given temperature. 
 
 T = 
 
 100 
 
 110 
 
 120 
 
 130 
 
 140 
 
 150 
 
 160 
 
 170 
 
 180 
 
 °C 
 
 h< 
 
 070 
 
 I'll 
 
 1-86 
 
 3-01 
 
 4-50 
 
 6-73 
 
 9-82 
 
 14-31 
 
 20-42 
 
 mm. 
 
 min. 
 
 u< 
 
 2110 
 
 2192 
 
 2294 
 
 2400 
 
 2522 
 
 2668 
 
 2823 
 
 2980 
 
 3145 
 
 m. 
 
 sec. 
 
 P> 
 
 0-046 
 
 0-07 
 
 0-09 
 
 0-14 
 
 0-20 
 
 0-27 
 
 0-38 
 
 0-53 
 
 0-71 
 
 mm. 
 
 d/d 1 > 
 
 0-0034 
 
 32 
 
 30 
 
 28 
 
 26 
 
 24 
 
 22 
 
 20 
 
 18 
 
 mm. 
 min. 
 
 h> 
 
 o-os 
 
 0-13 
 
 0-21 
 
 0-32 
 
 0-47 
 
 0-65 
 
 0-88 
 
 1-21 
 
 1-67 
 
 u> 
 
 7-5 
 
 7-4 
 
 7-3 
 
 7-1 
 
 6-9 
 
 6-8 
 
 6-6 
 
 6-5 
 
 6-2 
 
 m. 
 
 <?-P)IPi> 
 
 0-0034 
 
 32 
 
 30 
 
 28 
 
 26 
 
 24 
 
 22 
 
 20 
 
 18 
 
 
 1 See IX. p. 204. 
 
198 EVAPOKATION OF LIQUIDS viii 
 
 These values, reckoned in mm./min., are given in the second 
 row of the above table ; they are about ten times greater than 
 the highest values observed at the corresponding temperatures. 
 The latter are given in the sixth row as lower limits. They 
 are not lower limits for evaporation in general, for this can 
 fall to zero ; but they are lower limits for the greatest possible 
 rate of evaporation. The other limits given in the table hold 
 good also for the case in which the evaporation has reached 
 its greatest value. Those given in the third, fourth, and fifth 
 rows also hold good in general ; for we may assume that the 
 maximum of u and the minimum of P and d occur simultane- 
 ously with the maximum of m. In deducing these limits we 
 have first u = (P — p)/ing = m/d ; and since w>m min#> P —p<Pi, 
 and^<^, it follows that jp/m min >M>flt min Jd v Again P =p + m 2 jd, 
 and since m>m min and d<d p , it follows that ¥>p-{-m 2 min Jd p . 
 But the expression on the right hand has a minimum, since it 
 becomes infinite when p = 0, and when p = co ; this minimum 
 value is given in the table. .Finally d = m 2 J(P — p) and P— p 
 = m 2 /d. Hence d/d-^m^^/d^, and (P — i?)/Pi>^ 2 m i n ./^ii?i- 
 
 The meaning of the table may be illustrated by an example 
 of what it asserts, such as the following. From a mercury 
 surface at 100° C. we cannot cause a layer of more than 0*7 
 mm. to evaporate per minute ; its vapour will not issue from 
 the surface with a greater velocity than 2110 m./sec. ; the 
 pressure upon the surface will not be less than 4 to 5 
 hundredths of a millimetre, nor will the density of the vapour 
 which issues from it be less than -g-l^ of the density of the 
 saturated vapour. On the other hand, we can in any case 
 cause the evaporation to exceed 0*08 mm. per minute; the 
 velocity of the vapour to exceed 7*3 m./sec. ; and the pressure 
 of the issuing vapour to differ from the saturation-pressure by 
 more than 3^- of the latter. 
 
 In conclusion, I would further point out that the existence 
 of a limited rate of evaporation, peculiar to each fluid, is also 
 in accordance with the kinetic theory of gases ; and that with 
 the aid of this conception a fairly reliable upper limit for this 
 rate can be deduced. Let T, p, and d denote the temperature, 
 pressure, and density of the saturated vapour. Then the 
 weight which impinges in unit time upon unit area of a 
 solid surface bounding the vapour is m= /s /pdg/2ir. And in 
 
viii EVAPORATION OF LIQUIDS 199 
 
 greatly rarefied vapours nearly the same amount will impinge 
 upon the liquid boundary-surface, for the molecules at their 
 mean distance from the surface will be removed from the 
 influence of the latter. Now, as the amount of the saturated 
 vapour neither increases nor decreases, we may conclude that 
 an equal amount is emitted from the liquid into the vapour. 
 The amount thus emitted from the liquid will be approximately 
 independent of the amount absorbed ; thus evaporation, i.e. 
 diminution of the amount of liquid, takes place when for any 
 cause a smaller amount than that above mentioned returns 
 from the vapour to the liquid. In the extreme case in which 
 no single molecule is returned to the liquid, the latter jnust 
 lose the above amount in unit time from unit surface. This 
 amount is therefore an upper limit for the rate of evaporation. 
 It is somewhat narrower than the one first deduced. Calcu- 
 lation shows that for mercury at 100° this limit is 0*54 
 mm./min., whereas from our earlier assumptions we could 
 only conclude that the rate of evaporation must be less than 
 0*70 mm./min. Similar reasoning can be applied to the 
 maximum amount of energy which can proceed from an 
 evaporating surface ; we thus find that the velocity of the 
 issuing vapour can never exceed the mean molecular velocity 
 of the saturated vapour corresponding to the temperature of 
 the surface, e.g. for mercury at 100° it cannot exceed 215 
 m./sec. Finally, since the pressure of a saturated vapour upon 
 its liquid arises half from the impact of the molecules entering 
 the liquid and half from the reaction of those which leave the 
 surface, and since the number and mean velocity of the latter 
 approximately retain their original values, it follows that the 
 pressure upon an evaporating surface cannot be much smaller 
 than half the saturation-pressure. 
 
 These considerations enable us to fix limiting values, but 
 they will not carry us further unless we are willing to accept 
 the assistance of very doubtful hypotheses. 
 
IX 
 
 ON THE PEESSUEE OF SATURATED 
 MEECUEY-VAPOUE 
 
 {Wiedemann's Annalen, 17, pp. 193-200, 1882.) 
 
 The following determinations of the pressure of saturated 
 mercury -vapour suggested themselves as a continuation of 
 previous experiments 1 on evaporation. In working out the 
 latter I at first used the data given by Eegnault ; but these 
 did not prove suitable, as the following will show. I plotted 
 out the results of the experiments made by the second method, 2 
 taking as abscissse the amounts which evaporated in unit time 
 from a surface at a given temperature, and as ordinates the 
 corresponding pressures, and thus obtained series of points 
 lying approximately on straight lines. By prolonging these 
 straight lines a very little beyond the observed interval, I 
 found the pressures which corresponded to zero evaporation, 
 and which must therefore have represented the saturation- 
 pressures. The numbers thus found were always smaller than 
 Eegnault's. That this might be explained by errors in the 
 latter was first suggested to me by Hagen's experiments ; 3 
 but his data, again, did not agree well with my results. 
 Hagen himself suspected that his values were too small at 
 temperatures above 100° ; and as these were just the tempera- 
 tures which interested me, I decided to investigate the matter 
 myself. 
 
 The experiments were first carried on as a continuation 
 of the experiments on evaporation. The measurements were 
 
 1 See VIII. p. 186. 2 See p. 191. 3 See Wied. Ann. 16, p. 610, 1882. 
 
ix VAPOUR-PRESSURE OF MERCURY 201 
 
 made with the U-shaped manometer of the evaporation 
 apparatus shown in Fig. 21 (p. 190); but as there was now 
 no evaporation, the condenser and connecting tube were not 
 required. By boiling and pumping out with a mercury pump, 
 all air was removed from both limbs of the apparatus. The 
 temperature of the heated limb was indicated by a ther- 
 mometer dipping right into the mercury ; the thermometer 
 was calibrated and its readings were reduced to those of an 
 air-thermometer. In determining the pressure the difference 
 of level between the two limbs was read off, and then a con- 
 siderable correction had to be applied. The major part of 
 this depended upon the expansion of the mercury with heat. 
 In calculating this, care was taken to ascertain the distri- 
 bution of temperature, as determined by the law of conduction, 
 in the tube connecting the two limbs ; and the constants 
 required for ascertaining this distribution were determined by 
 special experiment. A smaller part of the correction arose 
 from the difference in the capillary depressions in the two 
 limbs. It seemed safe to assume that this correction would 
 be constant for all the temperatures under consideration ; so 
 that it was simply determined by measuring the difference 
 of level when both limbs were at the same temperature. Of 
 the pressures measured by this method, only those which 
 relate to temperatures above 150° were retained for the final 
 calculations : these were reduced to three mean values, which 
 are given in the table below and are marked by asterisks. 
 The observations below 150° were rejected because the correc- 
 tions were here much larger than the quantities to be observed, 
 so that the results were uncertain. For example, at 137°'4 
 the pressure was found to be 1*91 mm.; but here the cor- 
 rection was +2*49 mm. and the amount directly observed 
 only — 0*58 mm. Allowing for these unfavourable condi- 
 tions the rejected observations are found to agree sufficiently 
 well with the values obtained by the second method and 
 given as correct. They never differed from the latter by more 
 than 0*2 or 0*3 mm. They lay between these and Eegnault's 
 values ; but were twice or three times as far from Eegnault's 
 values as from my own final ones. 
 
 The following method was adopted as much more suitable 
 for measuring the lower pressures. The open limbs of two 
 
202 VAPOUR-PRESSURE OF MERCURY ix 
 
 manometers A and B (Fig. 23) communicate with one another. 
 
 They contain air of low pressure, 
 B — about 10 to 20 mm. The closed 
 ^ limbs are quite free of air. The 
 
 manometer A is kept in a water- 
 bath at the temperature of the room. 
 The manometer B was heated in a 
 FlG - 23 - vessel of thick cast-iron in a paraffin 
 
 bath, but never so far as to allow the mercury in the closed 
 limb to sink below the level in the open limb. Thus the 
 pressure of the mercury-vapour was smaller than the pressure 
 of the air present (at the time) in the open limb ; so that no 
 evaporation, excepting by diffusion, could take place. Hence 
 the pressure in the open limbs of both manometers was the 
 same ; and the difference of the readings of the two mano- 
 meters, reduced to mercury at 0°, gave the difference between 
 the saturation -pressure at the temperature of the hot mano- 
 meter and that of the cold one. But, according to the results 
 of this investigation, the pressure of the mercury-vapour in 
 the latter can be put equal to zero. The temperature of the 
 bath was read off on a very good Geissler thermometer ; and 
 I compared the indications of this with a Jolly air thermo- 
 meter. The difference of level was measured by means of a 
 micrometer eye -piece in the cathetometer microscope. The 
 adjustment of the cross-wires upon the top of the meniscus 
 was facilitated by means of a wire grating placed behind 
 it, the wires being inclined at 45° to the horizontal. The 
 manometer tube had a clear bore of 20 mm. The pressure of 
 the air in the open limbs was varied. Lastly, after each heating, 
 I convinced myself afresh of the absence of air in the closed 
 limbs by producing electrte discharges in them ; the tubes 
 then exhibited a green phosphorescence, and only this, so that 
 the pressure of the air in them could not have exceeded one to 
 two hundredths of a millimetre. The result of the experi- 
 ments was as follows. Up to 50° I could perceive no pres- 
 sure exceeding the limits of error (0*02 mm.) of a single 
 experiment. At 60° the pressure was about 0*03 mm., at 
 70° 0*05 mm., at 80° 0*09 mm. From here on the errors 
 were small compared with the whole values. From 120° to 
 130° the observed pressures can be taken as correct, since 
 
IX 
 
 VAPOUR-PRESSURE OF MERCURY 
 
 203 
 
 their errors were negligible compared with those which arose 
 in determining the temperatures. Groups of eight to twelve 
 separate observations, lying sufficiently close to each other, 
 were then formed, the mean temperature being simply associ- 
 ated with the mean pressure. The six principal values thus 
 obtained, together with the three determined by the first 
 method, are given in the first two columns of the following 
 table. The subsequent calculations are based upon the results 
 given in these columns. 
 
 t 
 
 P 
 
 Ap 
 
 At 
 
 t 
 
 P 
 
 Ap 
 
 A* 
 
 89'4 
 
 0-16 
 
 o-oo 
 
 o-o 
 
 *184-7 
 
 11-04 
 
 + 0-15 
 
 + 0-4 
 
 117'0 
 
 0-71 
 
 + 0-04 
 
 +1-1 
 
 190-4 
 
 12-89 
 
 -0-37 
 
 -0-8 
 
 154-2 
 
 3-49 
 
 + 0-01 
 
 + o-i 
 
 203-0 
 
 20-35 
 
 + 0-23 
 
 + 0-3 
 
 *165'8 
 
 5-52 
 
 + 0-04 
 
 + 0-2 
 
 *206'9 
 
 22-58 
 
 -0-20 
 
 -0-3 
 
 177-4 
 
 8-20 
 
 -0-22 
 
 -0-7 
 
 
 
 
 
 In calculating out the experiments I have made use of a 
 formula which has not hitherto been employed for the same 
 purpose. 1 It can be theoretically justified and must be 
 correct to the same degree of approximation that the laws of 
 Gay-Lussac and Boyle, which apply to very dilute vapours, are 
 correct for saturated vapours. On the assumption that this 
 law holds good, the vapour possesses a constant specific heat 
 at constant volume. Let this be denoted by c ; further let s 
 denote the specific heat of the liquid, and p T the internal heat 
 of vaporisation at the absolute temperature T. Then it 
 necessarily follows from our assumption that p T = const — 
 (s — c)T. This can be proved as follows. Let a quantity of 
 the liquid at temperature T be brought to any other tempera- 
 ture. At this temperature it is converted into vapour with- 
 
 1 An analogous formula, deduced by similar reasoning, lias indeed been 
 used by Kolacek ( Wiccl. Ann. 15, p. 38, 1882) for representing the pressure of 
 unsaturated water-vapour upon salt solutions. In that case the theoretical 
 justification of the formula is much stronger than in ours, where its appli- 
 cability is only really proved by comparing it with the results of experiment. 
 With regard to Kolacek's investigation, I may remark that all the experimental 
 data are known for applying the above formula to the pressure of vapour above 
 ice and above water cooled below its freezing-point down to the absolute zero. 
 Such an application would have to be justified by proving that the formula 
 obtained represents with satisfactory approximation the pressure of the vapour 
 for a considerable interval above 0°. For if the formula holds good for a given 
 interval of temperature, it must hold good for all temperatures below this interval, 
 inasmuch as a saturated vapour approximates more and more to a perfect gas as 
 the temperature diminishes. 
 
204 VAPOUR-PRESSURE OF MERCURY ix 
 
 out any external work. The vapour, again without external 
 work, is brought back to the temperature T and reduced to 
 liquid. During these processes the fluid can neither have 
 absorbed nor given out heat. Now according to the laws of 
 the mechanical theory of heat p T = Au(Tdp/dT — p), where p 
 denotes the pressure of the saturated vapour and % its specific 
 volume. Hence we can put u = "RT/p. If we eliminate p T 
 and u from the above three equations, we obtain for the curve 
 of the vapour pressure a differential equation which gives the 
 following integral 
 
 P = K\ I AR6 T. 
 
 For mercury s is known. From his own experiments, and 
 from a result given by Eegnault, Winkelmann 1 finds that this 
 quantity decreases slightly as the temperature increases ; the 
 mean value of s between and 100° is 0*0330. Experiments 
 made by Dr. Eonkar of Liege in the Berlin Physical Institute 
 have shown that the change between —20° and +200° is 
 exceedingly small. These experiments give 0*0332 as the 
 mean value of s, and I shall use this value in the calculation. 
 Kundt and Warburg have shown that the ratio of the specific 
 heats for mercury is |-: hence it follows that the quantity c 
 is equal to 0*0149. From this it follows that the exponent of 
 T is equal to — 0*847. The two remaining constants are to be 
 determined from the observations. Two of them are sufficient 
 for this : if we choose from the first series the observation at 
 206°, and from the second series the observation at 154°, 
 we obtain a formula which represents all the observations 
 satisfactorily. The constants thus determined can be im- 
 proved by applying the method of least squares. In doing 
 this we naturally assume the pressures to be correct, and 
 therefore make the sum of the squares of the temperature- 
 errors a minimum. In this way I find that 
 
 log k x = 10*59271, log 7c 2 = 3*88623. 
 
 Introducing these constants into the formula, and throwing 
 it into a form more convenient for calculation, we get 
 
 log p = 10*59271 - 0*847 log T - 3342/T. 
 
 1 See Poggendorff's Ann. 159, p. 152, 1876. 
 
VAPOUR-PRESSURE OF MERCURY 
 
 205 
 
 In the above table the third and fourth columns are added 
 so as to make it possible to compare the values calculated by 
 the formula with the observed values. The third column 
 gives the errors which must have occurred in the pressure 
 measurements if the observed temperatures are correct. The 
 fourth column gives the errors which must be attributed to 
 the temperature measurements if the pressures are to be 
 regarded as correct. It will be seen that the formula repre- 
 sents the observations completely, if we admit an uncertainty 
 of 0*02 mm. in the pressure measurements and of 0°*6 in the 
 temperature measurements ; and the disposition of the devia- 
 tions shows that such uncertainties must be admitted. The 
 measurements made below 89° agree perfectly with the formula, 
 as far as a comparison is possible. The following table is 
 calculated by means of the formula, and gives the pressure of 
 the vapour for every 10° between 0° and 220° — 
 
 t 
 
 P 
 
 t 
 
 V 
 
 t 
 
 • 
 
 I 
 
 t 
 
 P 
 
 0° 
 
 0-00019 
 
 60° 
 
 0-026 
 
 120° 
 
 0-779 
 
 180° 
 
 9-23 
 
 10 
 
 0-00050 
 
 70 
 
 0-050 
 
 130 
 
 1-24 
 
 190 
 
 13-07 
 
 20 
 
 0-0013 
 
 80 
 
 0-093 
 
 140 
 
 1-93 
 
 200 
 
 18-25 
 
 30 
 
 0-0029 
 
 90 
 
 0-165 
 
 150 
 
 2-93 
 
 210 
 
 25-12 
 
 40 
 
 0-0063 
 
 100 
 
 0-285 
 
 160 
 
 4-38 
 
 220 
 
 34-90 
 
 50 
 
 0-013 
 
 110 
 
 0-478 
 
 170 
 
 6-41 
 
 
 
 It should be noted that p = when t= - 273° ; and that 
 the formula gives for the internal latent heat of the vapour 
 the value p T = 76*15- 0*0183 T. The values given above 
 differ considerably from Eegnault's as well as from Hagen's. 
 They are always smaller than Eegnault's, but approach the 
 latter as the temperature rises, and almost coincide with them 
 at 220°. Compared with Hagen's they are smaller below 
 80°, nearly coincide between 80° and 100°, and above this are 
 larger. 
 
 The most interesting point is the pressure of the vapour 
 at the ordinary temperature of the air. According to the 
 results of our investigation this amounts to less than a 
 thousandth of a millimetre. 1 Hence no correction need be 
 
 1 It might be objected that this value is only calculated ; whereas both the pre- 
 vious observers made observations at the temperature of the air, and both believed 
 that they perceived a pressure of a few hundredths of a millimetre. But the 
 
206 VAPOUR-PRESSURE OF MERCURY ix 
 
 applied on account of this pressure to readings of barometers 
 and manometers. And it is the smallness of this pressure, 
 and not any special property of mercury itself, that explains 
 why the influence of mercury- vapour is negligible in discharge- 
 phenomena, although it is always present in Geissler tubes. 
 
 formula used appears to be satisfactorily established, and to be sufficiently tested 
 as far as the single hypothesis contained in it is concerned ; so that it merits 
 at least as much confidence as an observation of such small quantities, which 
 must be difficult and deceptive. In addition to this, I may add that up to 50° I 
 could discover no perceptible pressure ; whereas 0*10 mm., as given by Regnault, 
 or even 0'0-i mm., as given by Hagen, could not have escaped observation. 
 
ON THE CONTINUOUS CUEEENTS WHICH THE 
 TIDAL ACTION OF THE HEAVENLY BODIES 
 MUST PEODUCE IN THE OCEAN. 
 
 {VerhandliLngcn der pliysikaliselicn Gcscllschaft zu Berlin, 5th January 1883.) 
 
 In consequence of the friction of the water of the sea, internal 
 as well as against its bed, the tidal skin whose axis in the 
 absence of friction would lie in the direction of the tide -gener- 
 ating body or in a perpendicular direction, will be turned 
 through a certain angle out of the positions named. Hence 
 the attraction of the tide-generating body on the protuberances 
 of the tidal ellipsoid gives rise to a couple opposed to the 
 earth's rotation. The work done by the earth against this 
 couple as it keeps rotating is that energy at whose expense 
 the tidal motion is continually maintained in spite of the 
 friction. It would be impossible to transfer to the solid 
 nucleus of the earth this couple, which directly acts on the 
 liquid, if the motion of the liquid relative to the nucleus were 
 purely oscillatory, and if the mean ocean level coincided with 
 the mean level surface. The transference becomes possible 
 only because the mass of liquid constantly lags a little behind 
 the rotating nucleus ; or because there is a continual elevation 
 above the mean level at the western coasts of the ocean ; or 
 because both phenomena occur together. I have attempted to 
 deduce from the theory of the motion of a viscous fluid an 
 estimate of the character and order of magnitude of the 
 currents generated in this way. The results of the investiga- 
 tion are as follows. 
 
 Consider a closed canal. Let I be the distance along it 
 from the origin, L its whole length, h its depth, t the time, 
 
208 OCEANIC CURRENTS x 
 
 T the length of a day. Let f denote the elevation of the 
 water above the mean level, and let 
 
 t— fnCOS 4:7r{ 
 
 b bo y L T 
 
 be a bidiurnal tidal wave which would traverse the canal 
 under the action of a heavenly body, on the equilibrium 
 theory. Then the tidal wave which is actually produced is 
 given by the equation 
 
 ?= £i cos47r( 
 
 where 
 
 and 
 
 tan 47re = 
 
 L T 
 JcAL 
 
 2wfjJb 2 (gh-A 2 ) 
 
 f 1 = ^— f Sill 47T6 . 
 
 bl &AL b0 
 
 L 
 Here h denotes the coefficient of viscosity of water, and A = - 
 
 denotes the velocity of propagation of the wave, /ul the density 
 of water, and g the acceleration of gravity. In the calculation 
 squares and products of small quantities are neglected. For 
 instance, at the free surface the tangential component of 
 pressure is taken to be zero for the mean level ; whilst in 
 reality it is zero for the actual level. We find that this error 
 of the second order may be compensated by supposing a tension 
 t to act at the surface in the direction of propagation of the 
 wave, of which the magnitude is the mean of the values of 
 ^X. at different times, where X denotes the component of 
 gravitational attraction along the canal. For the tidal wave 
 considered above, we have 
 
 4wWfc 8 gg . 2A Aha 
 
 This tension corresponds to a current flowing along the canal 
 in the direction of the tidal wave, and increasing in velocity 
 uniformly from the bed of the canal to the velocity 
 
 a = — , o £o sin 2 47re = A^- 
 ^ . k 2 AI? b0 A 2 
 
 at the surface. 
 
x OCEANIC CURRENTS 209 
 
 If we apply this result to the case of the earth we see that 
 generally the tidal wave in its progress must be followed by a 
 current in the same direction. In a canal encircling the earth 
 along a parallel of latitude the current would flow everywhere 
 from east to west ; in a canal situated in any way whatever 
 it would be from east to west near the equator, in the opposite 
 direction at a distance from it. In general the current is very 
 small, but it may become very appreciable when the length and 
 depth of the canal are such that the period of the oscillation 
 of the water in it is one day, in which case without friction 
 the tides would be infinitely great. The formulae given are 
 not suitable for getting numerical values, as the differential 
 equations used are not applicable to the motion of deep seas. 
 In fact, if we substitute for the coefficient of viscosity the very 
 small value obtained from experiments with capillary tubes, 
 we get ridiculously high tides and ridiculously violent currents. 
 On the other hand we get currents of only about 100 metres 
 per hour if we use the formula 
 
 and substitute for £ values corresponding to actually occurring 
 tides. 
 
 A posteriori, we can from the magnitude of tidal friction 
 as approximately known draw a conclusion as to the order 
 of magnitude of the currents caused by gravitation. In one 
 century the earth lags twenty-two seconds behind a correct 
 chronometer. 1 To produce such a retardation a force must be 
 constantly applied at the equator equal to 530 million kilo- 
 grammes' weight and acting from east to west. If we imagine 
 this force distributed along a system of coast-lines which run 
 parallel to the meridian, bound the ocean on the west, and 
 have a total length of one earth-quadrant, then we get a 
 pressure of 53 kilogrammes' weight for each metre length of 
 coast. To produce this pressure the sea must at these western 
 coasts be elevated 0'3 metre above the level surface with which it 
 coincides at the eastern coasts. In so far then as the retardation 
 mentioned of the earth's rotation has its origin in tidal friction, 
 we can conclude that in consequence of the tide-generating 
 
 1 Thomson and Tait, Natural Philosophy, § 830. 
 M. P. P 
 
210 OCEANIC CURRENTS x 
 
 action of the heavenly bodies we get deviations of the mean 
 sea-level from the mean level surface amounting to ^ to ^ 
 metre, and currents of such magnitude as can be produced by 
 these differences of level. Though we are unable to state the 
 magnitude of these currents, yet we can conclude that they 
 are about equal in magnitude to those which are due to differ- 
 ences of temperature. For the differences of temperature may 
 indeed cause variations of the sea-level from the mean level 
 surface up to several metres ; but only a small fraction of 
 this height will give rise to currents at all, and only a small 
 part of this fraction will cause currents flowing from east to 
 west. 
 
XI 
 
 HOT-WIEE AMMETEE 1 OF SMALL EESISTANCE 
 AND NEGLIGIBLE INDUCTANCE 
 
 (Zeitschrift fur Instrumentenkunde, 3, pp. 17-19, 1883.) 
 
 All the forms of the. electro - dynamometer invented by 
 Wilhelm Weber which are intended for weak currents suffer 
 from two defects which are very inconvenient in many 
 investigations. In the first place, the resistance is high, 
 usually amounting to many hundred Siemens units ; in the 
 second place, the self-inductance is large. In many respects 
 the second defect restricts the use of the instrument more than 
 the first ; for it causes the instrument to offer an apparently 
 increased resistance to alternating currents, and in the case of 
 very rapidly alternating currents this increase can be very 
 considerable. If r is the resistance of the instrument, P its 
 self-inductance, and T the period of the alternating current, 
 the apparent resistance to this current is to the actual 
 resistance r as x/l + PV/TV : 1. For the instrument 
 described by Wilhelm Weber, and similar ones which are 
 actually in use, the self-inductance P can be estimated as 
 being of the order of one to two earth-quadrants. If we take 
 r as 200 Siemens units, or approximately 200 earth-quadrants 
 per second, it follows that for a current which alters its 
 direction 50 times per second the resistance is apparently 
 increased in the ratio of ^/2 : 1 ; and a current which altered 
 its direction 500,000 times per second would encounter in the 
 instrument an apparent resistance of 20,000 S.U. As to the 
 
 1 [Dynamomctrische Vorrichtung.] 
 
212 
 
 HOT-WIRE AMMETER 
 
 XI 
 
 presence or absence of currents alternating more rapidly than 
 10,000 times per second, the dynamometer could tell us 
 nothing ; for its introduction into the circuit would prevent 
 the establishment of such currents. For example, it could 
 not be used for investigating the discharge of a Leyden jar 
 through a short metallic circuit. 
 
 In pursuing an investigation 1 which depended upon 
 detecting unusually rapid alternating currents, I found it 
 necessary to have a fairly delicate instrument of small resist- 
 ance and negligible self-inductance ; and it occurred to me to 
 use the heating effect of the current in thin metallic wires as a 
 means of detecting it. The attempt succeeded much better 
 than was to be expected, and I may here be allowed to 
 describe the simple instrument which I used. For a given 
 current it certainly gives a much smaller deflection than the 
 
 usual dynamometers. But it is much 
 more delicate than any instrument of 
 comparable resistance, its self-induct- 
 ance is negligible, and it is as easily 
 handled as any other instrument 
 which gives equally accurate results. 
 The apparatus is shown in Fig. 
 24. The essential part of it consists 
 of a very thin silver wire, 80 mm. 
 long and 0'06 mm. in diameter, 
 stretched between the screws A and 
 B ; the wire does not run right 
 across from the one screw to the 
 other, but is attached by a little 
 solder to the vertical steel wire db 
 and twisted round this, as shown in 
 Fig. 24, b. The steel wire db has a 
 diameter of 0*8 mm., and is as smooth 
 and round as possible ; the twisting 
 of the silver wire can easily be 
 managed by first stretching it loosely and then turning the 
 steel wire in the direction of the arrow. The silver wire 
 being now well stretched, db is held in position by the torsion 
 which it produces in the thinner steel wires ac and bd ; these 
 
 1 See XIII. p. 224. 
 
 Fig. 24. 
 
xr HOT-WIRE AMMETER 213 
 
 are 0*1 to 0*2 mm. in diameter, and 25 mm. long. It is now 
 clear that any warming of the silver wire must tend to 
 untwist the wires ac and M and cause the wire ab to turn 
 around its axis ; by means of a mirror attached to the axis this 
 motion is read off through a telescope on a scale at a distance 
 of about 2 metres. In order to prevent any deflection of the 
 mirror through a general change of temperature, the screws A 
 and B are not fixed directly upon the wooden frame, but upon 
 a strong strap of brass (from which they are of course 
 insulated). Since brass and silver have very nearly the same 
 expansion, changes of temperature of the whole apparatus 
 have but a very slight effect upon the position of rest. The 
 instrument is protected from air-currents by a case, which 
 is not shown in the figure. The apparatus can either stand 
 on a table or hang by a hook from a wall ; in the former case 
 levelling-screws are unnecessary. 
 
 If we suppose the wire to be warmed 1° above its 
 surroundings, its expansion would amount to 19 millionths of 
 its length, so that each half of it would expand by 760 
 millionths of a millimetre. On the scale this expansion 
 appears magnified in the ratio of 2 x 2000/0*4 :1 = 10, 000:1, 
 and therefore causes a deviation of 7*6 mm. Hence an 
 elevation of temperature of ^° C. would correspond to a 
 deviation of about J mm. which should be clearly perceptible. 
 
 The following are the results of my observations : — 
 
 1. The resistance of the instrument is 0*85 S.U. 
 
 2. The instrument can be used in any position and requires 
 no special care in adjustment. The image of the scale remains 
 perfectly quiet, even in a place where a delicate galvanometer or 
 dynamometer keeps continually moving on account of ground- 
 tremors. When the mirror is thrown into vibration, the 
 vibrations are so rapid that the motion of the image of the 
 scale cannot be followed : but the air-damping is sufficient to 
 bring the image completely to rest in a second or less. 
 
 3. When a current of suitable strength is passed through 
 the silver wire, the image moves with a jerk into its new 
 position of rest, and the latter can be read off after 1 or 2 
 seconds. When the current is stopped the image jerks back 
 again to its first position of rest. If the deflection is large, 
 there remains a certain amount of after-effect, but this appears 
 
214 HOT-WIRE AMMETER xr 
 
 to be an elastic rather than a thermal after-effect, and is not 
 greater than in other instruments in which forces are measured 
 by the elasticity of wires. After a few minutes, at the outside, 
 the image returns to the original position of rest. 
 
 4. The following data indicate the sensitiveness of the 
 instrument. It was included in a circuit containing a Daniell 
 cell and a resistance of r Siemens units. In the following 
 table a denotes the deflection in scale-divisions, and h the 
 square root of this deflection multiplied by the total resistance 
 of the circuit (consisting of r S.U. together with 0'85 S.U. for 
 the instrument, and 0*77 S.U. for the Daniell cell) and divided 
 by 10. 
 
 r = 100 50 30 20 10 5 3 2 
 a= 0-25 0-9 2-2 4*9 16'9 521 106*8 173*8 
 b = 4-94 4-89 4-68 4'77 4*77 4*78 4*77 4*77 
 
 The numbers in the third row, excepting those corre- 
 sponding to the smallest deflections, are all equal : this shows 
 that the deflections are proportional to the square of the 
 current, and that the instrument is well adapted for measure- 
 ments. The current sent by 1 Daniell through 100 to 150 S.U. 
 can be easily detected : currents sent by 1 Daniell through 
 30 S.U. and, by means of shunts, all stronger currents, can be 
 measured. 
 
 5. When currents alternating a few hundred times per 
 second are sent through the instrument, there arises a difficulty 
 which is due to the small period of vibration of the mirror. 
 The wire absorbs and emits heat very rapidly, and the mirror 
 oscillates in accordance, following every impulse. In itself 
 this is an advantage : but as the eye cannot follow the 
 oscillations, the image of the scale becomes indistinct and the 
 mean deflection cannot be accurately read off. This difficulty 
 is much reduced by using the objective instead of the subjective 
 method of observation ; the scale then remains at rest, and 
 although the spot of light oscillates backward and forward, its 
 mean position can be accurately determined. Furthermore, 
 without diminishing the sensitiveness, the period of vibration 
 can be increased at will by increasing the moment of inertia 
 about the axis. 
 
 It appeared that the sensitiveness of the instrument was 
 
XI HOT-WIRE AMMETER 215 
 
 only limited by the accuracy with which the rotation of the 
 axis could be read off. I therefore made experiments with 
 the object of rendering visible even smaller extensions of the 
 wire by further magnification. This was done partly by 
 applying to the axis of the instrument a lever which rotated 
 other axes ; and partly by quite different arrangements of the 
 stretched wire. In this way I succeeded in obtaining 
 deflections ten times as large as those given above : but I 
 cannot recommend those modifications, because they do not 
 admit of the same ease in handling and the same certainty 
 of adjustment. The sensitiveness is best increased by using a 
 thinner silver wire, diminishing the diameter of the axis ah, 
 and increasing the length of the silver wire ; for it is rarely 
 that one requires a dynamometer of such small resistance as 
 the one here described. 
 
 If we further investigate the theory of the instrument, 
 assuming that ceteris paribus the amount of heat emitted by 
 the wire is proportional to its surface but approximately 
 independent of the nature of the metal, we obtain the 
 following rule for the most appropriate construction of the 
 instrument : — Of the metals which appear to be suitable, 
 choose that which expands most on heating : use as thin a 
 wire as can be procured, and choose its length so that the 
 internal resistance of the instrument is equal to the external 
 resistance for which the maximum sensitiveness is required. 
 
XII 
 
 ON A PHENOMENON WHICH ACCOMPANIES THE 
 ELECTEIC DISCHAEGE 
 
 (Wiedemann's Annalen, 19, pp. 78-86, 1883.) 
 
 In the following a phenomenon is described which often 
 accompanies the electric discharge, and in particular the 
 Leyden jar spark, in air and other gases, when the density 
 is not too small. It is true that in most circumstances it is 
 so trivial as not to have appeared worthy of mention, but the 
 first time I noticed it its appearance was so striking 
 as to induce me to make several investigations as 
 to its nature. I remark at once that in the experi- 
 ments a somewhat large induction coil was used, 
 which in the open air gave sparks 4 to 5 cm. long ; 
 the Ley den jar mentioned had a coating of some 
 two square feet in area, and it was simply joined 
 up with one coating connected to each pole of the 
 induction coil, without making any other alteration 
 whatever in the circuit. 
 
 1. Fig. 25 represents a discharging apparatus, 
 - which consists of a glass tube, not too finely drawn 
 out, and of two electrodes, one inside the tube, the 
 other attached to it outside near the opening. 
 When this apparatus is placed under the receiver 
 of an air pump, the receiver filled with well-dried 
 air and exhausted down to 30 to 50 mm. pressure, 
 and the discharge from the induction coil then sent 
 through, the following phenomenon is observed : 
 
 Near 
 
 the cathode is the blue glow 
 
 it is succeeded towards 
 
xii A PHENOMENON ACCOMPANYING THE ELECTRIC DISCHARGE 217 
 
 the anode by the dark space, one or more millimetres 
 wide, and from its end to the anode the path of the current 
 is marked by a red band 1 to 2 mm. in diameter. For both 
 directions of the current this band occupies the greater 
 part of the length of the glass tube, and at its opening 
 bends round sharply towards the electrode outside. But in 
 addition I observed a jet, brownish -yellow in colour, and 
 sharply defined, which projected in a straight line from the 
 mouth of the tube ; it was some 4 cm. long, and its form was 
 like that shown in the drawing, Fig. 25. The greater portion 
 of the jet appears to be at rest, and only at the tip does it 
 split into a few nickering tongues. The jet 
 does not change its shape appreciably when 
 the current is reversed. But when a Leyden 
 jar is joined up, an important change occurs : 
 the jet becomes brighter, and is straight for a 
 distance of only 1 to 2 cm. ; then it splits up 
 into a brush of many branches, which are 
 violently agitated and separate in all directions, 
 in the way shown in Fig. 26. 
 
 2. If we increase or diminish the pressure 
 of the air, neglecting for the present the 
 effect of the jar, then in both cases the FlG> 26, 
 
 jet becomes less striking, but in different ways. If the 
 pressure be increased, the path of the spark no longer 
 completely fills the cross-section of the mouth of the tube, 
 neither does the escaping jet do so, but it only emerges at 
 that side of the mouth where the spark appears ; it becomes 
 narrower, shorter, and assumes a darker, reddish-brown tint. 
 If the pressure be diminished, the jet is again shortened, but 
 at the same time it widens out, and assumes a lighter yellow 
 tint and becomes less bright. When the first striae form in 
 the tube, it is only just perceptible, and then occupies a small 
 hemispherical space just outside the mouth of the tube. 
 When a Leyden jar is used, a similar succession of appearances 
 is observed, but the greatest development occurs at smaller 
 pressures, and it is advisable to choose a wider-mouthed tube. 
 I obtained the most striking forms in air with the follow- 
 ing arrangement. The glass tube was 5 mm. wide and 3 cm. 
 long, and without any contraction at the mouth : the air was 
 
218 A PHENOMENON ACCOMPANYING THE ELECTRIC DISCHARGE xn 
 
 exhausted down to 10 to 20 mm. pressure, and was kept well 
 dried by placing under the receiver a small dish with sulphuric 
 acid or phosphorus pentoxide ; a large Leyden jar was joined 
 up, and the glare of the discharge itself screened off by using 
 as outside electrode a metal tube placed round the glass tube, 
 and projecting slightly beyond it. Under these conditions 
 the jet was in form like a tree, which reached up to 12 cm. 
 in height ; the part corresponding to the stem projected up 
 straight from the tube a distance of 1 to 5 cm., while the top 
 consisted of flames, which shot violently apart in all directions. 
 The brightness may be judged from the fact that the appear- 
 ance was still visible in a lighted room, but all details could 
 be observed only in a darkened room. 
 
 3. When the wall opposite to the jet is too close, so that 
 the jet cannot be fully developed, it spreads out over the wall. 
 When it meets it perpendicularly, it forms a circular mound 
 round the point of impact; but when it is inclined at an 
 angle, it creeps along the wall in the direction in which a 
 body would be reflected after impinging on the wall (in the 
 direction of the flame). The phenomena which here occur 
 may be most simply described by saying that the jets behave 
 as liquid jets would do if they emerged from the mouth of 
 the tube. 
 
 4. A magnet has no action on the jet. Neither have 
 conductors, when brought near, not even when they are 
 charged, e.g. w T hen they are connected with one of the two 
 electrodes. 
 
 5. The jet generates much heat in the bodies which it 
 encounters. A thermometer brought into the jet shows a rise 
 of ten or more degrees according to circumstances. When the 
 jet encounters the glass receiver it heats it perceptibly ; small 
 objects are melted off from wires on which they have been 
 stuck by wax. When the jet is produced in the open air (see 
 § 10) the heat generated may be felt directly. On the other 
 hand it was found impossible to cause a platinum wire, 
 however thin, to glow when hung in the current. 
 
 6. The jet exerts considerable mechanical force. A wire 
 suspended in it is set in violent oscillation, so also is a mica 
 plate, used to deflect the jet. A mica plate, placed on the 
 mouth of the tube, is violently thrown to a distance by the 
 
XII A PHENOMENON ACCOMPANYING THE ELECTRIC DISCHARGE 219 
 
 first discharge. Eadiometer-like vanes of various kinds may 
 be set revolving continuously by the jet. But the impulse 
 does not act in one direction only — away from the mouth. A 
 mica plate set up before the mouth, so as to be only movable 
 towards the mouth, is also set in vibration, which fact shows 
 that each impulse directed away from the mouth is followed 
 by a return impulse, though one of less strength. 
 
 7. The jet does not appear instantaneously, but takes a 
 conveniently measurable time to develop. I have examined 
 its time -changes, first with a rotating mirror, and secondly 
 with an apparatus specially constructed for the purpose ; this 
 has, however, been already described by others, and is arranged 
 as follows : A disc with a narrow radial slit is fixed to the 
 axis of a Becquerel's phosphoroscope ; at every revolution of 
 the disc in one particular position of it the apparatus breaks 
 the primary circuit. When the disc is rapidly rotated it 
 appears to be transparent, but if we look through it at 
 different places, we see the phenomena as they occur at certain 
 definite different times after break. This apparatus usually 
 gives better results than the rotating mirror, but in this case 
 the latter is sufficient. Both methods of observation lead to 
 the following results. The phenomenon is not instantaneous, but 
 lasts about -^ sec. The different parts of the jet do not all 
 appear at once ; the lower portions emit light before the upper 
 ones commence ; the upper parts are visible after the lower 
 ones have gone out. Thus the phenomenon is a jet only to 
 the unaided eye ; in reality it consists of a luminous cloud, 
 which is emitted from the tube with a finite velocity. When 
 no Leyden jar is used, this velocity is for the whole path of 
 the order of 2 m. per second, but it appears to be much greater 
 at the commencement of the phenomenon : so also it seemed 
 much greater for Leyden jar sparks ; for such sparks it may be 
 that often only the after-glow of the gas and not the develop- 
 ment of the jet was observed. 
 
 8. Analogous phenomena to those described occur in other 
 gases, but the jets show characteristic differences as regards 
 colour, form, effect of density, etc. In oxygen the jet is very 
 beautiful, much like that in air, but the tint is a purer yellow. 
 The appearance in nitrous oxide resembles that in oxygen 
 almost exactly. In nitrogen it was possible to produce only 
 
220 A PHENOMENON ACCOMPANYING THE ELECTRIC DISCHARGE xn 
 
 very faint jets ; the colour was nearest to a dark red. In 
 hydrogen the jets are best developed at about 100 mm. pres- 
 sure with the help of red Leyden jar sparks : the tint is a 
 fine blue indigo ; the brightness is not great. But the size is 
 much larger than in air, so that even in a glass receiver 20 
 cm. high the jet cannot fully develop itself, but spreads out 
 along the top. In the vapour of turpentine, and of ether, and 
 in coal gas, the jets are greenish-white, short, sharply defined. 
 The spectrum of the light is in air and oxygen continuous, 
 especially bright in the red, yellow, and green; in vapours 
 containing carbon it is a band spectrum, which could with 
 certainty be recognised as one of carbon ; in hydrogen it was 
 difficult to observe, owing to the faint light, yet at various 
 times I recognised several bands with certainty, of which the 
 most conspicuous was at any rate very close to the greenish- 
 blue hydrogen line, the others being situated more towards 
 the violet ; in nitrogen a spectrum could not be obtained. 
 
 9. In the gases mentioned it is always possible to detect 
 the presence of a jet by its mechanical effects, but the jet is 
 by no means clearly visible under all conditions, and its visi- 
 bility seems to depend on very curious conditions. The air 
 of a room when moist gives a very much weaker appearance 
 than when it has been dried. When we place a dish contain- 
 ing sulphuric acid or phosphorus pentoxide or calcium chloride 
 under the receiver of the air pump, we see the appearance 
 become more distinct as the air becomes drier. The behaviour 
 of hydrogen is still more incomprehensible. When the receiver 
 was filled with this gas the discharges of the Buhmkorff coil 
 did not at once produce the appearance, — Leyden jar sparks 
 were necessary ; but when the jet had once been rendered 
 visible, it could be maintained without using the Leyden jar. 
 But it lasted only a few minutes and then went out, without 
 my being able to reproduce it. I have not succeeded in finding 
 out the conditions necessary for visibility. Greater or less 
 humidity seemed without effect ; equally without effect was the 
 presence of a small quantity of oxygen. When the hydrogen 
 was kept for several hours under the air pump without being 
 used, it did not lose its power of becoming luminous ; but 
 when this power had once been destroyed by the discharges, 
 it was not restored even after hours of rest. I should attribute 
 
xii A PHENOMENON ACCOMPANYING THE ELECTRIC DISCHAEGE 221 
 
 the luminosity to impurities, 1 did I not feel confident that I 
 had recognised the spectrum of the emitted light to be a 
 hydrogen spectrum. However, the vibrations of a plate of 
 mica placed across the jet are just as lively in moist as in dry 
 air, in freshly prepared hydrogen as in that which has ceased 
 to become luminous ; so that the visibility of the jet seems to 
 be only an accidental property. 
 
 1 0. The jets may also be produced in gases at atmospheric 
 pressure ; it is advisable for this purpose to use a discharge 
 apparatus similar to but smaller than that used before. The 
 appearance, it is true, is only a few millimetres long and not 
 very striking, but further experiments may conveniently be 
 made on it. The heating effect and impact of the jet can be 
 directly felt. The jet scatters smoke and small flames at a 
 distance 2 to 3 cm. from the mouth of the glass tube. A strong 
 current of air bends the jet and drives it to one side. When 
 we blow through the opening at which the jet is formed, it 
 lengthens out ; when we suck in air, it shortens. When we 
 pass another gas through the opening and invert a test-tube 
 over it, we get the appearance corresponding to that gas ; so 
 with hydrogen we may obtain a very distinct blue jet, only 
 a few millimetres long. If coal gas is passed through and 
 lighted, the flame oscillates violently when the sparks pass ; 
 the apparatus described in § 7 shows that each spark drives out 
 a small cloud of gas, which burns above the mouth of the tube 
 and apart from the remaining portion of gas. 
 
 11. According to all that has been said, there can hardly 
 be any doubt that the jet is formed by a luminous portion of 
 gas escaping from the tube, and it is natural to assume that 
 the projective impulse is the force of expansion occasioned by 
 the rise in the temperature of the gaseous content. But if 
 we place the electrode, which previously was outside the tube, 
 close to the mouth of the tube inside it, or if we allow sparks 
 to pass inside a glass tube sealed at both ends and possessing 
 a lateral opening, in these cases also jets escape from the mouth 
 of the tube ; but they are much weaker than those which would 
 be produced if the spark also passed through the opening. If 
 rise of temperature were the cause of the emission, such a 
 difference could not exist. The above assumption is con- 
 
 1 The hydrogen was prepared from pure zinc and dilute sulphuric acid. 
 
222 A PHENOMENON ACCOMPANYING THE ELECTRIC DISCHAEGE xn 
 
 tradicted, more directly than by these somewhat ambiguous 
 experiments, by the shapes which the jet takes up when the 
 discharge apparatus is completely altered. 
 
 12. By shortening the tube more and more, and changing 
 the distance and form of the electrodes, we may continuously 
 change the discharge apparatus so far used into any other form 
 we please ; the jet then changes its shape, but does not dis- 
 appear, rather passes continuously into other forms. It is to 
 be observed that the discharge apparatus hitherto used has the 
 advantage of all others, only because it separates the appear- 
 ance considered from the mass of the luminous effects of the 
 discharge. The forms which occur are very various and often 
 very elegant ; my observations do not suffice to represent them 
 in order. In general their shape appears to depend on the 
 direction of the current, and it is clearly seen that the portions 
 of gas set in motion have velocities along the path of the 
 current, of which the cause cannot be sought merely in the 
 rise of temperature. A sufficient confirmation is afforded by 
 the single example which I will mention 
 here. When we allow the jar discharge 
 to pass between spherical electrodes not 
 too far apart, the appearance analogous 
 to the jet is a bulge surrounding the 
 centre of the spark path (Fig. 27 a, a). 
 Its colour, like that of the jet, is yellow at 
 low pressure, reddish-brown at atmospheric 
 pressure. With this last tint the bulge 
 can, with some care, be seen on every 
 spark which passes between the electrodes 
 of a Holtz machine, when its condensers 
 (which must not be too small) are used. 
 The apparatus described in § 7 gives 
 ,^\ f~2 interesting information as to the produc- 
 
 f^^ ^^ ^ on °^ ^ ie bulge. First the bright 
 ^^y straight spark appears, and during its 
 
 Fia - 27 - presence the yellow is still absent or 
 
 cannot be seen owing to the dazzling of the eye ; it is followed 
 by the aureole (Fig. 27, b, /3), which proceeds from the positive 
 electrode as a red band surrounded by the yellow light a ; the 
 latter, somewhat more than halfway, banks itself up into a wall 
 
 <E 
 
xn A PHENOMENON ACCOMPANYING THE ELECTRIC DISCHARGE 223 
 
 opposite the cathode, and forms a vortex (Fig. 27, c), and this 
 vortex maintains itself for some time in the air between the 
 electrodes (Fig. 27, d) after the rest of the appearance has died 
 away, but the whole only lasts about -^ second. 
 
 I have not found any mention of the above phenomena in 
 the literature of the subject. Dr. Goldstein has often noticed 
 analogous appearances in his numerous experiments on the 
 discharge in rarefied gases, and he was also the first to bring 
 to my notice the favourable effect which a careful drying of 
 the air has on the brightness of the yellow light. 
 
XIII 
 EXPEEIMENTS ON THE CATHODE DISCHAEGE 
 
 (Wiedemann's Annalcn, 19, pp. 782-816, 1883.) 
 
 As sources of electricity for experiments on the cathode 
 discharge in gases under diminished pressure, induction 
 machines, induction coils and batteries of many cells have 
 usually been employed. G. Wiedemann and Eiihlmann, 
 E. Wiedemann and Spottiswoode in many researches preferred 
 the induction machine ; Pliicker, Hittorf in his earlier ex- 
 periments, Goldstein and Crookes used mainly the induction 
 coil. In addition to the early experiments of Gassiot with 
 large batteries we have the more recent ones carried out by 
 Hittorf with his chromic-acid battery, — the silver chloride 
 battery of Warren de la Eue and Miiller, and the researches 
 carried out with it are the most famous of all. 1 It appeared 
 to me that certain experiments, which are of importance for a 
 proper understanding of the nature of the cathode discharge, 
 could only be successfully performed with a battery : I 
 therefore set up for these experiments a battery of 1000 
 secondary Plante cells. The battery as set up did not last 
 well ; but it sufficed for carrying out part of the experiments 
 which I had in mind. These experiments will now be 
 described. 
 
 1 I was first induced to undertake these experiments by conversations which 
 I had with Dr. E. Goldstein as to the nature of the cathode discharge, which he 
 had so frequently investigated. My best thanks are due to Dr. Goldstein for 
 the ready way in which he placed at my disposal his knowledge of the subject 
 and of its literature while I was carrying out the experiments. 
 
XIII EXPERIMENTS ON THE CATHODE DISCHARGE 225 
 
 Description of the Battery 
 
 The battery was based on the principle employed by 
 Poggendorff in his polarisation battery, and applied by Plante 
 to the cells which bear his name. The cells are arranged in 
 parallel, charged by a battery of comparatively small electro- 
 motive force, and then arranged in series. In this way very 
 high electromotive forces may be attained. It is not 
 necessary to deal singly with each cell thus : groups of five or 
 ten or more cells can be set up permanently in series ; and 
 during the charging only these groups need be placed in 
 parallel. The larger the number of cells in each group, the 
 simpler does the commutating mechanism become ; of course 
 the electromotive force required for charging increases at the 
 same time. I arranged my cells in groups of five in series. 
 The cells were made of test-tubes, 125 mm. high, 14-15 mm. 
 in diameter, and were filled two-thirds deep with sulphuric 
 acid diluted with nine times its volume of water. The 
 electrodes were strips of lead of suitable length, 1 mm. broad 
 and 1 mm. thick, varnished at the top with asphalt varnish. 
 The neighbouring pairs of electrodes within each group of five 
 cells were formed by bending a single strip of lead (so that no 
 connecting wire was necessary) ; copper wires soldered to the 
 outer electrodes led to two glass mercury cups which formed 
 the poles of the group. The cells were cemented in fifties on 
 boards, of which five (or 250 cells) went in a box 84 cm. long, 
 12 cm. wide, and 17 cm. high. The 100 glass cups forming 
 the corresponding poles lay in a row on the front side of the 
 box. The commutation was effected by two interchangeable 
 commutators, of which one was used in charging and the other 
 in discharging. These were made of bent wires attached to 
 a strip of wood ; the construction was simple and does not 
 require special explanation. 
 
 So long as the battery remained in good condition, it 
 worked as follows. Ten Bunsen or Grove cells were required 
 to charge it. When these had been in action for an hour the 
 battery was charged sufficiently for a day's work. The 
 difference of potential between its poles was about equal to 
 that of 1800 Daniell cells. Its internal resistance, as 
 M. p. Q 
 
226 EXPERIMENTS ON THE CATHODE DISCHARGE xin 
 
 deduced partly from the behaviour of single cells and partly 
 from experiments in which the current from the whole 
 battery was sent through very high resistances, was about 
 600 Siemens units. This potential difference kept up, when 
 the battery was on open circuit or only very slightly used, for 
 twelve to fourteen hours ; after which there was loss of charge 
 (mainly through chemical action, but partly also through 
 short-circuiting), and the potential difference sank rapidly to 
 lower values. If the circuit was only closed from time to 
 time through rather large resistances, as was usually the case 
 in these experiments, the battery remained in good working 
 order for about six hours. It could supply for two or three 
 hours the current required for continuously lighting a Geissler 
 tube : but if it was closed through a small resistance or short- 
 circuited, it became exhausted in a few minutes or even in a 
 fraction of a minute. It then exhibited the well-known 
 partial recovery of charge. On closing the circuit in free air 
 the battery gave a spark nearly half a millimetre long. It lit 
 up Geissler tubes of the usual form (without capillary) 
 through an interval of pressure from 1^- mm. to a few 
 hundredths of a millimetre ; at the former limit the blue 
 glow-light surrounded the cathode as a thin layer; at the 
 latter the cathode rays attained a length of 120-150 mm. 
 In general the connecting wires should not be attached to the 
 electrodes of a Geissler tube without introducing a resistance 
 of several thousand Siemens units ; otherwise the cathode 
 discharge passes into an arc discharge, and generally the tube 
 breaks and the battery becomes exhausted in a few moments. 
 
 This battery came to grief in the following way. The 
 sulphuric acid crept up in the capillary space between the 
 lead strips and the layer of varnish, and went on spreading 
 farther and farther in this space. If the apparently uninjured 
 varnish was scraped away at any point, it was easy to detect 
 the presence of the acid by its taste. Thus the acid worked 
 its way to the copper wires of the end strips and produced 
 upon these growths of copper sulphate which spread along the 
 wires. After the battery had been in use for three or four 
 weeks, these growths on the wires of the front row of end 
 strips reached to the mercury in the commutator-cups. The 
 mercury then amalgamated the wire along its whole length, 
 
xin EXPERIMENTS ON THE CATHODE DISCHARGE 227 
 
 and as the inner end of the wire lay lower than the level of 
 the mercury in the cups, the mercury flowed along the 
 amalgamated wire just as if it were a siphon, and emptied 
 each newly-filled cup in a few hours. This emptying could 
 be prevented by heating the wires to redness and coating them 
 for some distance with melted shellac ; but the destruction of 
 the copper wires went on, and after four or five months a 
 number of them broke off at the soldered joints. A few of 
 the wires remained quite unattacked, probably because they 
 happened to have been tinned for some distance from the joint. 
 The general nature of the battery discharge in gases under 
 diminished pressure is now sufficiently well known ; I shall, 
 therefore, pass this by and proceed to the description of 
 certain special experiments. 
 
 I. Is the Battery Discharge in Gases under diminished 
 
 PRESSURE CONTINUOUS OR DISCONTINUOUS ? 
 
 When Gassiot first produced the cathode discharge by 
 means of a large battery, and examined its appearance — 
 apparently quite continuous — in a rotating mirror, he found 
 that it could be decomposed into a number of partial dis- 
 charges following each other very rapidly. On this result is 
 based the view held by physicists, that the cathode discharge 
 is of a disruptive nature, and that every apparently continuous 
 discharge must consist of a series of separate disruptive 
 discharges. Most physicists approved of this view until 
 Hittorf, in 1879, showed that Gassiot's experiments do not 
 warrant any such general conclusions ; that with a battery 
 of sufficiently small resistance a cathode discharge can be 
 produced which cannot be decomposed into partial discharges, 
 /at any rate by a rotating mirror ; and that various circum- 
 stances indicate that a mirror, however rapidly rotated, could 
 not effect such a resolution. On the other hand, according to 
 a calculation made by E. Wiedemann, 1 the rotating mirror 
 would fail to perform its function if the number of successive 
 discharges in a second were to attain to even a hundred 
 thousand. Hence certain physicists, who for other reasons 
 
 1 Wicd. Ann. 10, \\ 244, 1880. 
 
228 EXPERIMENTS ON THE CATHODE DISCHARGE xm 
 
 felt compelled to assume a discontinuity, were not convinced 
 by Hittorf, although they were willing to admit that the 
 current might be made up of hundreds of thousands, or even 
 millions, of separate discharges per second. Among these 
 were E. Wiedemann, 1 Goldstein, 2 and Warren de la Eue. 3 The 
 latter had also described experiments by which he had 
 demonstrated the discontinuity of an apparently continuous 
 discharge otherwise than with a rotating mirror ; but this 
 demonstration could only be carried out under special condi- 
 tions, and these conditions appeared to be just those under 
 which the rotating mirror would have proved discontinuity. 
 
 The point in question may therefore be regarded as still 
 an open one. The question is not whether an apparently 
 continuous discharge may under certain circumstances be 
 shown to be discontinuous ; there is no doubt that this would 
 have to be answered in the affirmative. The question should 
 rather be put in the following form : — Can we establish the 
 existence of a discharge which is undoubtedly a cathode 
 discharge, but in which, nevertheless, no trace of discontinuity 
 can be detected, even by the most delicate methods ? 
 
 The discharge, which was tested by the following methods, 
 was produced in a tube of length 340 mm., and clear width 
 20 mm., between a steel plate (serving as the cathode) 18 mm. 
 in diameter and a steel wire. It took place in air under such 
 a pressure that the blue glow-light extended to a distance of 
 50 to 60 mm. from the cathode; furthermore, from six to 
 nine positive red strise were formed in the tube. The 
 current used lay between ^-J-^- and ^-^ of that sent by a 
 Daniell cell through a Siemens unit, and was regulated by 
 introducing a large liquid resistance. Only in the method 
 which will last be described was a stronger current necessary, 
 and this was T ^ to -fa Dan./S.U. In this case it was found 
 advisable to use a somewhat wider and shorter tube, so that 
 only one positive stria was visible, and this only indistinctly. 
 But at the same time there could be no doubt that the 
 discharge was of the nature of a cathode discharge. Of course 
 the discharges investigated showed none of the ordinary 
 
 1 JVied. Ann. 10, p. 245, 1880. 2 Loc. cit. 12, p. 101, 1881. 
 
 3 Ann. de Chim. et de Phys., series 5, 24, p. 461, 1881 ; and Phil. Trans. 
 169, p. 225, 1878. 
 
xin EXPERIMENTS ON THE CATHODE DISCHARGE 229 
 
 symptoms of intermittence. They were in no way affected by 
 the approach of a conductor ; a telephone introduced into the 
 circuit did not sound ; the tubes themselves gave out no 
 sound, nor could the image of the discharge be decomposed by 
 a rotating mirror into separate images. 
 
 1. The above-mentioned experiments of Warren de la Eue * 
 were first repeated. The battery-current, in addition to pass- 
 ing through the gas-tube, was sent through the primary or 
 secondary coil of various small induction-coils, the free coil 
 being closed through a dynamometer or galvanometer. In 
 no case did I obtain a deflection of these instruments, such as 
 would indicate a surging induction current due to inter- 
 mittence of the battery current. However, this does not 
 prove much. Consider first the dynamometric effect of the 
 induced currents. At first this certainly increases with the 
 number of interruptions of the inducing current ; but if this 
 number becomes very large, the dynamometric effect does not 
 become infinite. Since the separate induction impulses are 
 impeded by self-induction, the dynamometric effect approaches 
 a fixed limit ; but even this maximum effect could scarcely be 
 perceived with the dynamometer which I used. And as far 
 as the effect on the galvanometer is concerned, the accepted 
 theory of induction does not indicate that any effect should be 
 expected, even if the current at each separate discharge sinks 
 more rapidly than it rises. I was only induced to perform 
 these experiments by the fact that results to the contrary had 
 been obtained by Warren de la Kue and Mtiller. Unfor- 
 tunately I did not succeed in reproducing the phenomenon 
 observed by them. When the galvanometer had been removed 
 from the direct magnetic action of the coil through which the 
 current flowed, no permanent deflection could be perceived 
 after the battery current was closed, although the induction 
 impulse on opening and closing the current drove the needle 
 beyond the visible scale. 2 
 
 2. In addition to the tube and a large liquid resistance, 
 
 1 Ann. de Chim. et de Phys., series 5, 24, p. 461, 1881 ; and Phil. Trans. 
 169, p. 225, 1878. 
 
 2 It is certain that any deflection of the needle produced cannot be regarded 
 as due to any normal galvanometric action. More probably it was of the nature 
 of " doppelsinnige Ablenkung" in which case the galvanometer would be acting 
 as a very delicate dynamometer. 
 
230 EXPERIMENTS ON THE CATHODE DISCHARGE xm 
 
 a galvanometer and a dynamometer were simultaneously 
 introduced into the circuit, and the deflections produced in 
 both instruments were read off. The battery, gas-tube, and 
 liquid resistance were then separated from the two measuring 
 instruments, and replaced by a Daniell cell, and such a 
 metallic resistance as gave the same galvanometer deflection as 
 before. It was found that the dynamometer reading also was 
 precisely the same as before. But if the current sent by the 
 large battery through the gas-tube had been an intermittent 
 current, it would for a given magnetic effect have produced a 
 much larger dynamometric effect. Suppose, for example, that 
 the duration of one of the partial discharges was equal to a 
 fourth of the time from the beginning of such a discharge to 
 the beginning of the next. While this current lasted it 
 would be four times as strong as a continuous current capable 
 of exerting an equal magnetic effect. While it lasted, its 
 dynamometric effect would be sixteen times as great, or, on an 
 average over the whole time, four times as great as that of the 
 continuous current. Hence this experiment indicates that the 
 discharge is continuous. 
 
 3. The current was led to the tube through a Wheat stone 
 bridge arrangement. One arm of this consisted of the 
 secondary of a small induction-coil, having a resistance of 
 1700 S.U., and a coefficient of self-induction of ten earth- 
 quadrants. The other three arms consisted of equivalent 
 metallic resistances of negligible self-induction. In the actual 
 bridge a dynamometer was introduced, and the arrangement 
 was so adjusted that when a continuous current flowed 
 through it there was no deflection of the dynamometer. It 
 was then found that no deflection was produced by the battery 
 current flowing through the Geissler tube, although this was 
 strong enough to produce a very marked effect as soon as 
 the equilibrium was destroyed by inserting a resistance of 
 100 S.U. in one of the branches. This experiment tells 
 against discontinuity. For we may regard an intermittent 
 current as composed of a part which flows continuously, and 
 another part which continually changes its direction. The 
 bridge was only adjusted for the former : to the latter the 
 coil, on account of its high self-induction, would certainly 
 offer a far higher apparent resistance than the other branches. 
 
xiii EXPERIMENTS ON THE CATHODE DISCHARGE 231 
 
 Hence if an alternating part had been present, an oscillating 
 current would have flowed through the dynamometer, and would 
 have been strong enough to produce a perceptible deflection. 
 
 The preceding experiments prove that the current flows 
 continuously through the greater part of the metallic circuit 
 even when an air-gap is introduced. They only enable us to 
 form a conclusion as to the current in the tube itself if we 
 assume that the current is uniform in all its parts. But if 
 the number of the partial discharges amounts to 100,000 or 
 more per second, the assumption is unsafe ; indeed there is no 
 doubt that the current-variations can only penetrate a small 
 distance into the coil, on account of its large self-induction, 
 and that inside it they must be effaced. Thus only a fraction, 
 and probably a very small fraction, of the effect under con- 
 sideration would actually occur. On this account coils are 
 avoided in the following experiments. 
 
 4. The current was sent through 
 a Wheatstone bridge (Fig. 28) of 
 which the four arms consisted of equal 
 liquid resistances of 700,000 S.U.each. 
 These were made by filling thin glass 
 tubes 3 cm. long with a dilute solution 
 
 of zinc sulphate. The mean potential difference at the points 
 a and b — more accurately the mean square of this difference — 
 could be observed by means of a gold-leaf electroscope : this was 
 enclosed in a metal case connected to the point a, while the 
 leaves themselves were connected to the point b. In this and 
 the following experiments the gold leaves were observed under 
 a microscope. The difference of potential which could just be 
 observed was about one-tenth of that which existed between a 
 and c when the current was flowing. By means of short metallic 
 conductors the points a and c could be connected with the two 
 coatings of a condenser of very large capacity. The resistances 
 were so adjusted that the gold leaves showed no divergence 
 when the current was allowed to pass in the absence of the 
 condenser. On introducing the condenser it was again found 
 that not the slightest divergence could be perceived. This 
 result again tells against discontinuity. For let us suppose 
 that a very rapid intermittent current flows through the 
 apparatus, and let us, as before, conceive of this as being 
 
232 EXPERIMENTS ON THE CATHODE DISCHARGE xm 
 
 composed of a part which flows continuously and an alternating 
 part. Our bridge is only adjusted for the former ; for the 
 latter the arm ac has apparently a vanishing resistance, for 
 the condenser is capable of taking in and giving out the 
 quantities of electricity conditioned by the alternating current 
 without any appreciable change of the potential difference 
 between its coatings. It follows that for the alternating part 
 the potential difference between a and c must be very small, 
 and that between a and b must become large enough to be 
 detected. It seemed advisable to test the correctness of this 
 conclusion by experiment. Into the external circuit was 
 introduced a toothed wheel having a large number of teeth, 
 by which the current could be broken artificially up to 2000 
 times per second. When the current was thus interrupted 
 the gold leaves still remained at rest, provided that the 
 condenser was not in action. When the condenser was 
 introduced they diverged immediately ; the divergence in- 
 creased with the rate of interruption, and was very considerable 
 at the above-mentioned rate. A single opening and closing of 
 the current could be recognised, when the condenser was 
 introduced, by a slight twitching of the gold leaves. I 
 estimate that the number of partial discharges per second 
 would have to amount to many hundred thousand before the 
 method of testing here used would become ineffective. Of 
 course it would become ineffective if the intermittence was so 
 rapid that in the intervals the electric waves could only 
 travel along a small fraction of the lengths of the liquid 
 resistances used. 
 
 5. The leaves of a gold-leaf electroscope were connected by 
 a short copper wire to the negative electrode ; these leaves 
 were suspended in a metallic case which could either be 
 connected to the positive electrode by a metallic wire or to the 
 negative electrode through a large resistance of a few million 
 S.U. When the current was passed through the tube and the 
 metal case connected to the positive electrode, the gold leaves 
 diverged strongly ; they showed no trace of divergence when 
 the metal case was connected to the negative electrode through 
 the above-mentioned resistance. This result tells against 
 discontinuity of the discharge. For if the potential at the 
 cathode fluctuated very rapidly between that necessary for the 
 
xtii EXPERIMENTS ON THE CATHODE DISCHARGE 233 
 
 discharge and a much lower ^potential, the potential of the 
 gold leaves would be able to follow these fluctuations, but the 
 potential of the metal case would not ; the quantity of 
 electricity upon the latter would always be that corresponding 
 to the mean potential value, and the divergence of the gold 
 leaves would therefore be proportional to the square of the 
 difference between the potential and its mean value. That an 
 intermittent discharge does actually produce such divergence 
 was shown as follows. The resistance through which the 
 battery-current flowed was gradually increased more and more ; 
 when it had attained a certain very high value the discharge 
 began to exhibit the criteria of discontinuity given by 
 Hittorf; and at the same time the gold leaves began to 
 diverge distinctly. The same result was attained by artificial 
 interruption of the battery-current. Here again the method 
 of testing used must fail as soon as the number of interrup- 
 tions per second reaches a certain value, but this value can be 
 approximately calculated. The partial discharges, if any such 
 were present, could only consist in discharges of the electricity 
 accumulated on the cathode and the gold leaves in metallic 
 connection with it. The capacity of this system was certainly 
 not greater than that of a sphere of 20 mm. radius. The 
 fluctuation of potential at each discharge could not exceed the 
 value of 90 Daniells, for it was found that such a difference of 
 potential between the gold leaves and the surrounding case 
 could be recognised by a perceptible divergence. Now a 
 thousand discharges per second of a sphere of 20 mm. radius, 
 charged each time to a potential of 90 Daniells, would just 
 correspond to the current produced by 1 Daniell through 
 about 5,000,000 S.U. But the currents used in the experi- 
 ments were about equal to that sent by a Daniell through 
 100 S.U. Hence if they consisted of partial discharges, the 
 frequency of the latter must have amounted to at least 50 
 millions per second. 
 
 6. The anode of the gas-tube used was connected by a 
 thick metallic wire with one plate of a Kohlrausch condenser, 
 and the cathode was connected with the other plate by a very 
 thin silver wire of 8 cm. length and about 0*8 S.U. resistance. 
 To the latter was attached an arrangement with mirror and scale, 
 by means of which an exceedingly small extension, and there- 
 
234 EXPERIMENTS ON THE CATHODE DISCHARGE xm 
 
 fore a very slight elevation of temperature produced in the 
 wire by a current traversing it, could be detected. A rise of 
 temperature of ^° C. could be perceived ; the current produc- 
 ing this rise was equal to t J-q DanielL/S.U. The wire thus 
 formed a sort of dynamometer without any coil, and will in 
 future be referred to as such. 1 Now the battery- current, which 
 in these experiments was as strong as ^o Daniell/S.U., could 
 be conducted to the cathode in either of two ways. Either it 
 flowed in between the condenser and the dynamometer, in 
 which case it flowed through the latter, and produced in it a 
 deflection of four to five scale-divisions ; or else it flowed in 
 between the dynamometer and the gas-tube, in which case not 
 the slightest deflection of the dynamometer could be perceived, 
 — certainly not a deflection of quarter of a scale -division. 
 Now if the current had been composed of partial discharges, a 
 continual charging and discharging of the condenser would 
 have taken place, and therefore an alternating current would 
 have flowed through the dynamometer. The deflection pro- 
 duced by this alternating current would certainly have 
 amounted to a half of that produced by the whole current. 
 Here again I caused intermittence by an artificial interruption 
 of the external circuit. The result was that the dynamometer 
 was deflected whether the current flowed in the one way or 
 the other ; and the deflections were even larger (six to eight 
 scale-divisions) than when the current was not interrupted. 
 The explanation of this paradox is that the artificial interrup- 
 tion produces condenser-discharges which act more strongly upon 
 the dynamometer. The criterion here used only ceases to be 
 applicable when the separate partial discharges follow each 
 other so rapidly that the electric waves corresponding to 
 them are no longer able to traverse the silver wire of the 
 dynamometer in the interval. The requisite rapidity can be 
 estimated in various ways ; even the lowest estimate gives 
 many thousand millions per second. The following is perhaps 
 the simplest way. If the electric wave does not traverse the 
 dynamometer wire, then each partial discharge consists simply 
 of a discharge of the electricity stored upon the cathode. The 
 capacity of the cathode was less than that of a sphere of 2 cm. 
 radius. During a single discharge the variation of potential 
 
 1 See description of the apparatus in XL p. 211. 
 
EXPERIMENTS ON THE CATHODE DISCHARGE 
 
 1 
 TOO 
 
 for there was a perceptible deflection when the mean potential 
 difference of the terminals of the dynamometer attained this 
 value. Thus in order that the successive discharges should be 
 equivalent to a current of -fa Daniell/S.U., they must have 
 amounted to two billions per second. This mode of estimat- 
 ing is open to criticism, and I do not wish to insist strongly 
 on the large number to which it leads. But I would ask 
 whether it is likely that an electric current could traverse a 
 gas-tube 20 cm. long as a fully formed partial discharge with 
 all its striae, in a time which would not allow of its traversing 
 as a steady current 8 cm. of a metallic conductor ? 
 
 7. I have not discovered any more decisive methods of 
 testing. But a few further experiments may be mentioned 
 which, although not in themselves decisive, tend in the same 
 direction as those already described. 
 
 (a) If the observer closes the circuit, containing the gas- 
 tube and a sufficient liquid resistance, through his own body, 
 he feels a shock on closing and a much weaker shock on open- 
 ing. By frequent opening and closing, the sensation can be 
 heightened until it becomes unbearable. But while the tube 
 glows uniformly nothing is perceived beyond a burning at the 
 points where the current enters and leaves. 
 
 (b) The battery-discharge never gives rise to the auxiliary 
 phenomenon of oscillatory currents, even under conditions which 
 are very favourable for their production, and under which 
 the Buhmkorff discharge produces very powerful currents of 
 this kind. 
 
 (c) The following phenomena have already been mentioned 
 by Hittorf : When a sufficiently large resistance is introduced, 
 the discharge is certainly discontinuous. The tube then fre- 
 quently gives out a note, the pitch of which indicates the rate 
 at which the discharges succeed each other. When the 
 resistance is diminished, the note becomes higher and the 
 tube less bright. But it does not gradually pass over 
 into the quiet indifferent discharge ; when the resistance is 
 reduced to a certain value the note stops suddenly, the tube 
 becomes three times as bright, and no further indications 
 whatever of discontinuity can be obtained. The sudden 
 change is still more striking when the electrodes of the tube 
 
236 EXPERIMENTS ON THE CATHODE DISCHARGE xm 
 
 are connected with the coatings of a large condenser ; for it 
 then often takes place from a state of things in which the 
 separate discharges can be distinguished by the eye. Once 
 the change has taken place, the switching of the condenser in 
 or out of circuit has not the slightest effect upon the appear- 
 ance of the discharge. 
 
 The general conclusion which I draw from the experiments 
 described is that the discharges tested were continuous : from 
 this I further conclude that the battery-discharge in general 
 is to be regarded as continuous, excepting when it exhibits 
 the known criteria of discontinuity ; further, that the dis- 
 charges of an induction-coil, whose period may^ be between 
 xooo and -^ of a second according to the size of the 
 apparatus, are to be regarded as being continuous during this 
 interval. 
 
 In order to establish these conclusions fully, it is necessary 
 to show that the considerations which lead to the opposite 
 conclusion cannot be regarded as decisive. These conclusions 
 appear chiefly to depend upon the following experimental 
 results: (1) that a weak current (e.g. such as an induction- 
 machine gives) is always discontinuous, and does not become 
 continuous even when the partial discharges succeed each 
 other at the rate of several thousand per second ; (2) that the 
 heating effect in a gas-tube is not proportional to the square 
 of the current, but to the current itself; and (3) that in 
 accordance with this the potential difference at the ends of 
 the tube does not increase with increasing current, but per- 
 I sists at the value which enables the weakest 
 
 N current to traverse the tube. In order to show 
 
 that these results do not necessarily prove discon- 
 tinuity, I shall make use of a simple mechanical 
 analogue. The arrangement which I shall de- 
 scribe is such that it might in many respects, 
 — and at any rate in those under consideration, 
 — replace a gas-tube as a conductor of electricity, 
 fig. 29. an( j nevertheless a current would flow continu- 
 ously through it under certain conditions. Let A (Fig. 29) 
 represent the anode, and let this be connected by a metallic 
 spring or other elastic good conductor with the weight a 
 
XIII EXPERIMENTS ON THE CATHODE DISCHARGE 237 
 
 which lies close to the cathode B. If there is a difference of 
 potential between A and B, a will be attracted by B. But 
 a is prevented from coming directly into contact with B by 
 an under layer fi of relatively high resistance r. Between A 
 and B there may exist any number more of these spring- 
 carriers, only differing from the first in that the resistance 
 corresponding to ft is for each very large. A certain potential 
 difference must exist between A and B in order to bring the 
 weight a of the first spring-carrier up against B, and so for 
 all the other spring-carriers. Suppose this potential difference 
 to be very nearly the same for all of them, and equal to p. 
 The whole arrangement may now replace a gas-tube as follows. 
 It does not allow any current to pass unless the potential 
 difference between A and B attains the value p. It allows 
 an intermittent current to pass when A and B are connected 
 with a source of electricity which can produce a potential 
 difference p, but is not at the same time able to produce a 
 current of strength p/r. If, however, the source is capable of 
 yielding such a current, then a and B remain permanently in 
 contact and the current flows continuously. Whatever the 
 strength of the current may be, the potential" difference cannot 
 exceed p, for more and more of the spring-carriers would come 
 into action. The whole heating effect would therefore be 
 proportional to the current itself, and not to its square. This 
 proves our point. 
 
 Another circumstance seems to have influenced the 
 opinions of previous writers. The position and develop- 
 ment of each stria of the glow -light depends upon the 
 preceding stria (in the direction of the cathode) : upon this 
 is founded the legitimate view that from the cathode out- 
 wards there must be a time-development from one stria to 
 the next. But such a time-development is not conceivable, 
 if the discharge in all parts persists continuously. Per- 
 haps we shall form a correct conception of the circumstances 
 in question if we admit that the discharge as a whole is 
 continuous, but assume that its course along the separate 
 current-lines (Stromfciden) is a function of the time. For 
 example, if the contact of a gas-molecule with the cathode 
 gave rise to an electric disturbance travelling in waves 
 through the medium, the successive production of striae 
 
238 EXPERIMENTS ON THE CATHODE DISCHARGE xm 
 
 would be easily intelligible without necessitating any split- 
 ting up of the discharge into partial discharges. This would 
 still be a continuous discharge in the sense in which we have 
 used the word. 
 
 II. Do the Cathode Eays indicate the Path 
 of the Current ? 
 
 As is well known, the cathode rays spread outwards in 
 straight lines, approximately perpendicular to the cathode 
 and without reference to the position of the anode. Accord- 
 ing to the density of the gas, they proceed in the medium for 
 a few millimetres, centimetres, or even up to lengths of the 
 order of a metre. In air they are blue, but at low densities 
 their luminosity is exceedingly feeble ; they are then most 
 noticeable on account of the phosphorescence which they 
 excite when they strike the glass. If a magnet is brought 
 near the tube they appear bent, much as an elastic wire 
 attached to the cathode and traversed by a current would 
 become bent under the influence of the magnet. This is 
 universally regarded as an electromagnetic action, and, ex- 
 cepting that passing doubts were expressed, the view that 
 used to be held by physicists was as follows : The cathode 
 rays indicate the path of the current, and their blue light 
 arises from the glowing or phosphorescence of the gas-particles 
 under the action of the current. As a fuller knowledge of 
 the facts was attained this view appeared less probable, and 
 more recent experimenters express themselves very reservedly 
 as to the relation between the cathode rays and the actual 
 process of discharge. 1 Under these circumstances it appeared 
 advisable to obtain by experiment a decisive answer to the 
 question — Does the current travel along the cathode rays 
 before it turns towards the anode ? If this question was to 
 be answered in the negative it would become clear that the 
 path of the current could not be recognised by the naked eye, 
 and a fresh question would arise, namely, "What is the path 
 
 1 See, e.g. W. Spottiswoode and J. Fletcher Moulton, Phil. Trans. 171, 
 p. 649, 1880. 
 
xiii EXPERIMENTS ON THE CATHODE DISCHARGE 239 
 
 of the current in a space in which various paths are open to 
 it ? I have tried to answer both questions for a space con- 
 taining gas and traversed by a current by determining 
 experimentally the current-lines from the deflections produced 
 by the discharge in a small magnet in its neighbourhood. 
 
 Before attacking this problem it was necessary to solve a 
 preliminary one. Whether the cathode rays form the path of 
 the current or not, there is no doubt that they are affected 
 by a magnet. Conversely, it was not improbable that the 
 cathode rays would in any case produce a deflection of the 
 magnet : and this effect might be other than an electro- 
 magnetic effect. If such an effect existed, the proposed ex- 
 periment would be useless. The following experiments show 
 that no such effect occurs. 
 
 In a tube 300 mm. long and 28 mm. wide was intro- 
 duced a cathode consisting of a circular turned brass disc 
 which just filled the cross-section of the tube. Through a 
 hole bored in the centre of the disc was passed a thermometer 
 tube ; inside this, and quite centrically with reference to the 
 disc, was a wire of non-magnetic metal. The ends of the 
 wire, projecting but little beyond the disc in the gas-space, 
 formed the anode. The wires used to carry the current in and 
 out were twisted around one another. Now the current-lines 
 must at all events be symmetrical with reference to the axis 
 of the tube ; if we suppose the currents replaced by magnetic 
 surfaces, these would be closed ring-magnets which would have 
 no external action. But the cathode rays were fully developed 
 and, according to the density, filled either the whole tube or 
 a part of it with blue light. If they have any action peculiar 
 to themselves upon a magnet outside the tube, it would here 
 exhibit itself apart from any electromagnetic effect. In order 
 to avoid any electrostatic effects the tube was surrounded with 
 tinfoil which was connected to earth ; without this precaution 
 the experiments could not have been carried out. The magnet 
 upon which the cathode rays were to act was the one which 
 was used in the subsequent experiments ; it was a strongly 
 magnetised piece of watch-spring, 12 mm. long, and was 
 attached to a small mirror of very thin glass. This was 
 hung by a single spider-thread in a very narrow space between 
 two plates of plate-glass. Thus the arrangement was the 
 
240 
 
 EXPERIMENTS ON THE CATHODE DISCHARGE 
 
 XIII 
 
 ft 
 
 same as in a Thomson galvanometer. 
 In all the following experiments it was 
 made strongly astatic by external mag- 
 nets ; this, with the air-damping, made 
 it dead-beat, and in all other respects 
 its behaviour was most satisfactory. 
 The tube was now brought as near as 
 possible to the magnet, first in such a 
 position that the magnet would indicate 
 a force tangential to the tube, then 
 radial, and lastly, parallel to the tube. 
 But there was never any deflection, — 
 none amounting to even one-tenth of a 
 scale-division in the telescope. The 
 strength of the current was from T ^ 
 to 2-J-o Daniell/S.U. By using a second 
 anode the same current could be made 
 to traverse the length of the tube ; it 
 then produced deflections of thirty to 
 forty scale-divisions. Similar deflections 
 were obtained when the first anode was 
 retained and portions of the circuit 
 outside the tube were brought within a 
 few centimetres of the magnet. It was 
 thus proved that, if there was any 
 specific action of the cathode rays upon 
 the magnet, this could not amount to 
 3TTo P ar ^ °f ^ ne effect produced by the 
 cathode rays as current-carriers. 
 
 In the principal experiments the 
 discharge was investigated in an air- 
 space of the form of a flat parallel- 
 epipedon, 12 cm. long, 12 cm. broad, 
 and 1 cm. deep. The case enclosing it 
 is shown in Fig. 30. It was made of 
 a strong brass casting, which formed 
 the side walls and framing, and of two 
 sheets of plate-glass 4 to 5 mm. thick, 
 applied air-tight to this frame. The 
 plates sustained safely the powerful pres- 
 
1, ' 
 
 
 V 
 
 I' — r ^l 
 
 <—<^ 1 1 
 
 ^^ ml 
 
 i 
 
 n 1 
 
 
 ) 
 
 
 I 
 
 
 r^ 
 
 xin EXPERIMENTS ON THE CATHODE DISCHARGE 241 
 
 sure of the air, and could be heated while this pressure was on ; 
 but they bent under it so strongly that the curvature could 
 easily be observed on looking at them sideways. Through the 
 brass frame were inserted a tube with stopcock for pumping 
 out, and also several aluminium electrodes ; the latter were 
 cemented in glass tubes so as to be insulated from the frame. 
 It was only after several fruitless attempts that the case was 
 made air-tight. One difficulty arose from the bending of the 
 glass ; this made any accurate grinding impossible, and every 
 solid cement cracked on pumping out. Another difficulty 
 arose from the fact that no trace of any decomposable organic 
 substance could be allowed 
 inside the case, so that a 
 free use of any fatty sub- 
 stance would have been 
 fatal. Tig. 3.1 explains 
 how a tight joint was 
 at last secured. On the 
 
 ground projecting inner rim a of the frame was laid a thin 
 strip of rolled gutta-percha, which was kept about a millimetre 
 away from its inner edge. The glass plates were then heated 
 and applied, and the case was exhausted as far as the leakage 
 would admit ; at the same time a mixture of four parts of 
 rosin and one part of olive oil was poured into the hollow 
 space /3. This mixture proved, after cooling, sufficiently fluid 
 to follow the movements of the glass plates, and was yet so 
 tough that only after several months did it begin to flow out 
 of the hollow space on account of its own weight. The case 
 could be kept exhausted for days together. If the current 
 was sent through it for a considerable time, the pressure of 
 the gas increased slightly, but not so much as to interfere 
 with the experiments. The case was next enclosed in a tin- 
 foil covering, connected to earth but insulated from the brass 
 frame and the electrodes. After exhausting it to a pressure 
 of a few hundredths of a millimetre, it was placed upon a 
 board covered with co-ordinate paper (squared paper) and 
 provided with levelling screws. Exactly over the zero point 
 of this co-ordinate system hung the magnetic needle which 
 has been already described, at such a height that the exhausted 
 case could be moved about underneath without touching it. 
 M. p. R 
 
242 EXPERIMENTS ON THE CATHODE DISCHARGE xm 
 
 It was only 2 mm. above the surface of the upper glass 
 plate, and therefore 12 mm. above the mean stria of gas 
 through which the current passed. When the current was 
 turned on, the magnet was deflected, the deflection depending 
 upon the strength of the current and the direction of its path 
 with reference to the needle. The total current used was from 
 Ttro" t° 2TTo Daniell/S.U. In favourable positions of the needle 
 this gave deflections up to eighty scale-divisions. As one- 
 tenth of a division could be read off, the measurements could 
 be made accurately. With the help of the squared paper the 
 position of the plate with reference to the magnet could be 
 altered and accurately read off. Such an arrangement enables 
 us to determine with considerable accuracy the distribution of 
 magnetic force which the current in the stria of air pro- 
 duces just above and parallel to itself. But what we have to 
 do is to deduce from this distribution the distribution of the 
 current in the air-stria itself. 
 
 This can be done with the aid of the following pro- 
 position : The current- function of the electric current in a 
 plane stria is equal to the potential function of the mag- 
 netic force excited by the current in the immediate neighbour- 
 hood of the stria, multiplied by a constant. The current- 
 lines therefore coincide with the magnetic equipotential lines, 
 and the current-strengths between every two equipotential 
 lines, between which the potential increases by the same 
 amount, are equal. A proof of this proposition may be found 
 in Maxwell's Treatise on Electricity and Magnetism} But it 
 can easily be seen to be true if we consider the case of a 
 magnetic pole brought infinitely near to a plate traversed by 
 a current ; for only those parts of the current which are in its 
 immediate neighbourhood can exert upon the pole a force 
 parallel to the plate. 
 
 The current-stria which we have to investigate is not 
 infinitely thin, and the testing magnet does not lie in the 
 immediate neighbourhood of its mean plane, but 1 2 mm. above 
 it. Hence it only enables us to investigate the distribution 
 of the potential in a plane which is 12 mm. above the mean 
 plane of the stria of air. But the magnetic force in this 
 plane will be approximately the same as that inside the stria 
 
 1 Vol. ii. p. 264, 1873. 
 
xin EXPERIMENTS ON THE CATHODE DISCHARGE 243 
 
 of air ; and hence the equipotential lines on the plane in 
 which the magnet moves will be very similar to the current- 
 lines. The most elegant way of investigating these equi- 
 potential lines would be to move the plate beneath the magnet 
 in such a way that the latter always remains undeflected. 
 The curve then described by the projection of the magnet on 
 the plate is an equipotential line, and therefore a current-line. 
 But as the deflection of the magnet had to be read off from a 
 distance with mirror and scale, this method could not be 
 carried out without elaborate mechanism. Hence the follow- 
 ing method was adopted. The case was moved under the 
 magnet in such a way that the projection of the latter upon 
 it described a parallel to one side of the square case, and so 
 that the magnet in its undeflected position was perpendicular 
 to this parallel. The current being maintained constant, the 
 deflections for a series of points along this straight line were 
 determined. These were proportional to the differential co- 
 efficients of the potential along this straight line. These 
 differential coefficients were plotted graphically and carefully 
 interpolated ; the area of the curve, obtained by a mechanical 
 quadrature, gave the changes of potential along the 
 straight line examined. The same process was carried 
 out for a series of straight lines parallel to the first, and 
 for one straight line perpendicular to them. Thus the 
 potential for all points in the plane investigated could be 
 specified, and it was easy to connect the equipotential 
 points and draw the connecting lines at such distances that 
 ; the potential increased by a constant amount in passing from 
 , each one to the next. In consequence of the method followed 
 there was bound to be some uncertainty as to the values 
 obtained, and it was necessary to get an estimate of this. For 
 this purpose the potential was measured along several straight 
 lines perpendicular to the parallel ones, instead of along one 
 only. Thus the value of the potential at every point could be 
 determined in a corresponding number of independent ways. 
 By adjusting these we obtain not only a trustworthy result, 
 but a measure of the uncertainty attaching to the method. It 
 turned out that this was not large enough to interfere much 
 with the results. 
 
 These results can best be represented by Fig. 3 2, a, b, and c. 
 
244 
 
 EXPERIMENTS ON THE CATHODE DISCHARGE 
 
 In these a denotes the blue cathode light, /3 the positive 
 striae ; the curved lines are equidistant equipotential lines. 
 In a and c the pressure amounted to one-tenth of a millimetre ; 
 
 hence in these the 
 cathode rays have free 
 ends. In b the pressure 
 was reduced so much 
 that the battery could 
 only barely keep up a 
 continuous discharge ; 
 hence in this the cathode 
 rays end perpendicularly 
 upon the opposite side. 
 With regard to these 
 equipotential curves we 
 have to remark: (1) 
 That in constructing 
 each figure some fifty 
 
 Fio. 32, a (* nat. size). ^ ^^ deflections 
 
 were used ; these were not distributed uniformly over the 
 whole surface, but were mostly taken in the places which 
 seemed most important. 
 (2) The uncertainty 
 which remained is in- 
 dicated by the number 
 of equipotential curves 
 drawn. For just so 
 many were drawn that 
 the uncertainty as to 
 the position of any 
 single one was equal 
 to the interval between 
 two neighbouring ones. 
 
 (3) In order to obtain 
 
 the actual current-lines 
 
 from the equipotential 
 
 lines here drawn, we 
 
 must imagine the end-points of the latter joined to the electrodes, 
 
 and the lines themselves compressed somewhat more together 
 
 towards those places in which they are closest. It is clear 
 
 Fig. 32, b (J nat. size). 
 
EXPERIMENTS ON THE CATHODE DISCHARGE 
 
 245 
 
 that the actual current-lines could never cut the sides of the 
 vessel, as the lines in our drawings do. 
 
 The figures show without any doubt that the direction of 
 the cathode rays does not coincide with the direction of the 
 current. In some places the current-lines are almost perpen- 
 dicular to the direction of the cathode rays. Some parts of 
 -the gas-space are lit up brilliantly by the cathode light, 
 although the current in them is vanishingly small. Koughly 
 speaking, the distribution of the current in its flow from pole 
 to pole is similar to what it would be in a solid or liquid 
 conductor. From this it follows that the cathode rays have 
 nothing in common with the path of the current. 
 
 1. Against the preliminary experiment the objection may 
 be raised that since a 
 magnet deflects the 
 cathode rays, conversely 
 the cathode rays must 
 deflect the magnet. But 
 when we come to con- 
 sider the expression " a 
 magnet deflects this or 
 that ray," and the com- 
 parison thus set up with 
 the deflection of an 
 elastic wire traversed by 
 a current, we may well 
 doubt whether these are 
 so suitably chosen as at 
 first sight they appear FlG - 32 > c ( * nat size) - 
 
 to be. Such a wire when the current starts would be 
 straight, and would only be brought into its deflected 
 position after a finite time. But we know that cathode 
 rays, even when the corresponding discharges last less than a 
 millionth of a second, appear completely bent. 1 De la 
 Bive's experiment in which the discharge is made to rotate 
 about a magnetic pole tells against the supposition that 
 electromagnetic action can set gaseous discharges in motion 
 with such speed as this. In De la Bive's experiment the 
 action is undoubtedly electromagnetic ; but it takes place at a 
 
 1 See Goldstein, Vber eine Form der elektr. Abstossung, iii. Teil. 
 
246 EXPERIMENTS ON THE CATHODE DISCHAEGE xm 
 
 speed which is very easily measurable. And in every actual 
 electromagnetic effect the ponderable substratum of the 
 current is set in motion ; which is not the case with the 
 deflection of the discharge. 1 Hence this deflection corresponds 
 much more nearly to Hall's phenomenon. But this analogy 
 again is seen to be defective when we recollect that the 
 cathode rays are not to be regarded as the path of the current. 
 Lastly, it is known that the battery-discharge can be ex- 
 tinguished by bringing a powerful magnet near it ; and after 
 the magnet is removed the discharge immediately starts off again. 
 This shows that the action of the magnet upon the discharge 
 cannot be purely electromagnetic. The action of the magnet, 
 which prevents the current from starting, certainly cannot be 
 an action upon the current itself; it can only be an action 
 upon the medium through which the current has to pass. On 
 account of these difficulties, and the fact that the cathode rays 
 do not react upon the magnet, it seems to me probable that 
 the analogy between the deflection of the cathode rays and 
 the electromagnetic action is quite superficial. Without 
 attempting any explanation for the present, we may say that 
 the magnet acts upon the medium, and that in the magnetised 
 medium the cathode rays are not propagated in the same way 
 as in the unmagnetised medium. This statement is in accord- 
 ance with the above-mentioned fact, and avoids the diffi- 
 culties. It makes no comparison with the deflection of a wire 
 carrying a current, but rather suggests an analogy with the 
 rotation of the plane of polarisation of light in a magnetised 
 medium. 
 
 2. E. Wiedemann and Goldstein have expressed the 
 opinion that the discharge consists of an ether-disturbance, of 
 itself invisible, and only converted into light by imparting its 
 energy to the gas-particles. This view seems to me to be 
 based upon convincing arguments. I should, however, like to 
 see the word ' discharge ' replaced by ' cathode rays ' : the two 
 things are quite distinct, although the physicists referred to 
 do not observe the distinction. If we consider carefully the 
 following experiment, it will be difficult to resist the view that 
 the cathode rays themselves are invisible, and that they only 
 produce light by their absorption in the gas. The tube 
 
 1 See Goldstein, Wicd. Ann. 12, p. 262, 1881. 
 
xin EXPERIMENTS ON THE CATHODE DISCHARGE 247 
 
 already described, which was used in the preliminary experi- 
 ment of this section, was exhausted so far that the discharges 
 of a large induction coil could only just traverse it : under the 
 action of such discharges there was a brilliant phosphorescence 
 at the end opposite the cathode. After what we have already 
 said, there can be no doubt that the current-paths are re- 
 stricted to the immediate neighbourhood of the electrodes, 
 which are quite near to one another, and that only the cathode 
 rays traverse the length of the tube. Now at the phosphor- 
 escing end of this tube there happened to be a drop of mercury. 
 When that part of the tube was heated, so as to vapourise the 
 mercury and produce there a gas of comparatively high density, 
 the end of the tube became filled with crimson light, which 
 showed the spectrum of mercury. The green phosphorescence 
 of the glass then faded away, and ceased entirely when the 
 stria of mercury vapour attained a certain thickness. By 
 means of a magnet the cathode rays could be made to follow 
 a path in which they had not to traverse the vapour ; the 
 luminescence of the latter then ceased, and was replaced by a 
 green phosphorescence on the glass at the side of the tube, 
 where the rays now fell. In this way one could at will pro- 
 duce a luminescence of the glass or of the mercury vapour. 
 By further heating and distilling, a larger portion of the tube 
 could be filled with the heavy vapour ; it was then found that 
 the luminescence only extended to the 5 or 6 cm. of the 
 portion which lay nearest to the cathode, the part of the tube 
 behind it remaining dark. Finally, when the whole tube was 
 filled with the heavy vapour, the luminescence — in the form of 
 the ordinary cathode light — filled the space about the cathode 
 for a distance of a few centimetres. Thus the cathode rays 
 first excite luminescence when they enter a denser medium 
 and are themselves absorbed by it. For this absorption an 
 infinitely thin stria of a solid suffices, but a finite stria of a 
 gas is requisite. The denser the gas the shorter the distance 
 through which the cathode rays can penetrate into it. This 
 is probably one reason why the cathode light in comparatively 
 dense gases is restricted to the immediate neighbourhood of 
 the cathode. 
 
 3. There can be no doubt that in the preceding experi- 
 ment the luminescence of the gas, even in the immediate 
 
248 EXPERIMENTS ON THE CATHODE DISCHARGE xm 
 
 neighbourhood of the cathode, was not due to the direct action 
 of the current, but to the action of the cathode rays. For 
 without any sudden change it could be gradually transformed 
 into a quite similar luminescence, situated at a great distance 
 from the cathode and in a space where the current was zero. 
 And if we admit that in this special case the cathode light is 
 not directly produced by the current, we can scarcely assume 
 that in the general case it is so produced. According to 
 Goldstein's researches, the cathode light has so many analogies 
 with the separate positive striae that it can be regarded as a 
 degenerated form of such a stria. It is therefore very 
 improbable that the luminescence of the gas in the positive 
 striae is due to any causes other than those which produce 
 the luminescence in the cathode light. We are thus led to 
 the assumption, which at first seemed hazardous, that the 
 luminescence of the gas in the glow discharge is not a direct 
 effect of the current, but arises indirectly through an absorption 
 of the cathode rays 1 which are produced by the current. If 
 we could prevent the production of the cathode rays, the gas 
 would everywhere be as dark as it is in the dark intervals 
 between the striae (although the current flows through these 
 intervening spaces). Conversely, if we could produce the 
 cathode rays in some other way than by the discharge, we 
 could get luminescence of the gas without any current. For 
 the present such a separation can only be carried out ideally. 
 
 4. A number of phenomena, which otherwise can only be 
 explained with difficulty, are seen to follow almost as a matter 
 of course when we regard the cathode rays as a disturbance 
 which is quite independent of the actual discharge, and no 
 more connected with it than the light which radiates from 
 the discharge. I shall only mention the penetration of the 
 striae, the reflection of cathode rays from the anode, and the 
 way in which these rays pass out through anodes consisting 
 of close metal gratings completely surrounding the cathode. 
 With respect to the latter, I may say that I have seen fully 
 developed cathode rays pass through wire-gauze containing not 
 less than thirty-six meshes to the square millimetre. 
 
 1 i.e. of rays which in their nature are identical with the cathode rays. The 
 name obviously becomes unsuitable if it has also to include the rays of the 
 positive striae. 
 
 
EXPEEIMENTS ON THE CATHODE DISCHARGE 
 
 249 
 
 III. Have the Cathode Eays Electeostatic Peopeeties ? 
 
 If we admit that cathode rays are only a subsidiary pheno- 
 menon accompanying the actual current, and that they do not 
 - exert electromagnetic effects, then the next question that arises is 
 as to their electrostatic behaviour. For the experiments relating 
 to this the battery, unfortunately, was no longer available, and 
 I had to make use of the discharges of a small induction coil. 
 On account of their irregularity and suddenness these are very 
 ill adapted for electrostatic measurements. Hence the experi- 
 mental results are not so sharp as they otherwise might have 
 been ; but the conclusion to ^ -^ dinnr^ 
 which they lead may certainly 
 be regarded as correct. The 
 question at the head of this 
 section may be split up into 
 two simpler ones. Firstly : 
 Do the cathode rays give 
 rise to electrostatic forces in 
 their neighbourhood ? Sec- 
 ondly : In their course are 
 they affected by external 
 electrostatic forces ? By 
 cathode rays are here meant 
 such as are separated from 
 the path of the current 
 which produces them : to 
 prevent confusion we shall 
 call these pure cathode rays. 
 
 A. In seeking an answer 
 to the first question I made 
 use of the apparatus shown in Fig. 33. AB is the glass tube, 
 25 mm. wide and 250 mm. long, in which the rays were 
 produced, a is the cathode. All the parts marked /3 are in 
 good metallic connection with each other, and such of them as 
 lie inside the tube form the anode. They consist, in the first 
 place, of a brass tube which nearly surrounds the cathode, and 
 only opposite it has a circular opening 10 mm. in diameter, 
 
250 EXPERIMENTS ON THE CATHODE DISCHARGE xin 
 
 through which the cathode rays can pass ; secondly, of wire- 
 gauze, about 1 sq. mm. in mesh, through which the cathode 
 rays have to pass ; thirdly, of a protecting metallic case, which 
 completely surrounds the greater part of the tube and screens 
 that part of the gas-space which lies beyond the wire-gauze 
 from any electrostatic forces which might be produced by 
 induction from without, e.g. from the cathode. If the results 
 which we have already obtained have any meaning, the cathode 
 rays are to be regarded as pure after they have passed through 
 the opening in the metal cylinder and the wire-gauze beyond 
 it. They are none the less vivid ; at low densities they cause 
 the glass at B to shine with a brilliant green phosphorescence, 
 upon which the shadow of the wire-gauze is plainly marked. 
 The part of the glass tube which lay within the protecting 
 case was now enclosed in a metallic mantle 7, which was 
 connected with one pair of quadrants of a delicate electrometer ; 
 the protecting case and the other quadrants were connected to 
 earth. When even a small quantity of electricity was brought 
 inside this mantle, it attracted by induction electricity of the 
 opposite sign from the electrometer, so that a deflection was 
 produced. The electricity could, e.g., be introduced by re- 
 placing the tube AB inside the protected space and the mantle 
 7 by a metal rod which had about the same size and position 
 as the cathode rays. This was placed in metallic connection 
 with the cathode, while the current from the induction coil 
 passed, as it did in the actual experiments, through the tube. 
 The deflection then produced in the electrometer was too great 
 to be measured, but could be estimated at two to three thou- 
 sand scale -divisions. When the current was stopped the 
 electrometer needle went back to about its old position ; and 
 this could be repeated at will. Now if the cathode rays 
 consisted of a stream of particles charged to the potential of 
 the cathode, they would produce effects quantitatively similar 
 to the above, or qualitatively similar if they produced any 
 electrostatic forces whatever in their neighbourhood. On 
 trying the experiment the following results were obtained. 
 When the quadrants of the electrometer were connected 
 together and the induction coil started, the needle naturally 
 remained at rest. When the connection between the quadrants 
 was broken, the needle, in consequence of irregularities in the 
 
xiii EXPERIMENTS ON THE CATHODE DISCHARGE 251 
 
 discharge, began to vibrate through ten or twenty scale- 
 divisions from its position of rest. When the induction coil 
 was stopped, the needle remained at rest in its zero-position, 
 and again began to vibrate as above when the current was 
 started. As far as the accuracy of the experiment allows, we 
 can conclude with certainty that no electrostatic effect due to 
 the cathode rays can be perceived ; and that if they consist of 
 streams of electrified particles, the potential on their outer 
 surface is at most one-hundredth of that of the cathode. And 
 this conclusion remains correct even if we now find that there 
 are complications in the part of the tube beyond the wire- 
 gauze, viz. that this part of it is by no means unelectrified. 
 If we start the induction coil after the apparatus has been 
 long at rest, and is therefore free from electricity, a consider- 
 able deflection (150 to 200 scale-divisions), showing a negative 
 charge on the tube, is produced in the electrometer. But this 
 charge and deflection remain constant, however often the coil 
 is put in and out of action. They remain for an hour after the 
 discharge has been stopped. But while the discharge is on, the 
 position of the needle changes instantaneously when a magnet 
 is brought near the tube, and the needle remains constant in 
 its new position so long as the magnet is not moved. As a 
 matter of fact, then, electricity does penetrate through the 
 wire -gauze into the protected part of the tube until its 
 entrance is prevented by the rise of potential. We shall not 
 here establish the laws which underlie this penetration of the 
 electricity ; it is enough that it has nothing to do with the 
 cathode rays. For the passage of these latter is in no way 
 influenced when the further penetration of the electricity is 
 prevented ; nor, as the first experiment shows, is the amount 
 of electricity in the tube appreciably increased when the 
 cathode rays again begin to enter it. 
 
 B. In order to find out whether pure cathode rays are 
 affected by electrostatic forces, the following experiments were 
 made. The rays were produced in a glass tube 26 cm. long, 
 provided with a circular aluminium cathode 5 mm. in 
 diameter. As in the preceding experiments, the cathode was 
 almost completely surrounded by the anode, and the cathode 
 rays had to pass out through the wire-gauze. Further on in 
 their path was a fine wire ; the sharp shadow of this, appear- 
 
252 EXPERIMENTS ON THE CATHODE DISCHARGE xin 
 
 ing on the phosphorescent patch at a distance of 1 2 cm., served 
 as an accurate indicator of any deflection. A magnetic force 
 only half as strong as the horizontal intensity of the earth's 
 magnetism, acting perpendicular to the direction of the ray, 
 was sufficient to change quite notably the position of this 
 shadow. The tube was now placed between two strongly 
 and oppositely electrified plates : no effect could be observed 
 in the phosphorescent image. But here there was a doubt 
 whether the large electrostatic force to which the tube was 
 subjected might not be compensated by an electrical distribution 
 produced inside it. In order to remove this doubt, two 
 metallic strips were placed inside the tube at a distance of 2 
 cm. from one another, and were connected to external con- 
 ductors by which they could be maintained at different 
 potentials. After passing the wire which produced the 
 shadow, the rays had to travel a distance of 12 cm. between 
 these strips. The latter were first connected with the poles 
 of a battery of twenty small Daniell cells. Opening and 
 closing this connection produced not the slightest effect upon 
 the phosphorescent image. Hence no effect is produced upon the 
 ray by an electromotive force of one Daniell per millimetre acting 
 upon it perpendicular to its length. 240 Plante cells of the 
 large battery were next charged and connected with the two 
 metallic strips. By themselves these 240 cells were not able to 
 discharge across the strips ; but as soon as the induction coil 
 was set to work and the cathode rays filled the space between 
 the strips, the battery also began to discharge between them ; 
 and, as there was no liquid resistance in the circuit, this at 
 once changed into an arc discharge. The same phenomenon 
 could be produced with a much smaller number of cells — down 
 to twenty or thirty. This is in accordance with Hittorfs 
 discovery that very small electromotive forces can break through 
 a space already filled with cathode rays. The 240 cells were 
 next connected up through a large liquid resistance : during 
 each separate discharge of the induction coil there was now 
 only a weak battery-discharge lasting for an equally short 
 time. The phosphorescent image of the Buhmkorff discharge 
 appeared somewhat distorted through deflection in the neigh- 
 bourhood of the negative strip ; but the part of the shadow in 
 the middle between the two strips was not visibly displaced. 
 
xt ii EXPERIMENTS ON THE CATHODE DISCHARGE 253 
 
 The result may therefore be expressed as follows. Under the 
 conditions of the experiment the cathode rays were not 
 deflected by any electromotive force existing in the space 
 traversed by them, at any rate not by an electromotive force 
 of one to two Daniells per millimetre. Upon this we may 
 make the following remarks : — 
 
 1. As far as the imperfect experiments described under 
 III. enable us to decide, the cathode rays cannot be recognised 
 as possessing any electrostatic properties. Under II. we 
 have partly proved, and partly shown it to be probable, that 
 they do not produce any strictly electromagnetic effects. 
 Thus the question arises : are we justified in regarding the 
 cathode rays as being in themselves an electrical phenomenon ? 
 It does not appear improbable that, as far as their nature is 
 concerned, they have no closer relation to electricity than has 
 the light produced by an electric lamp. 
 
 2. The experiments described under II. can quite well be 
 reconciled with the view, which has received support in many 
 directions, that the cathode rays consist of streams of electri- 
 fied material particles. But the results described under III. 
 do not appear to be in accordance with such a view. For we 
 find that the cathode rays behave quite unlike a rod of the 
 same shape connected with the cathode, which is pretty well 
 the opposite of what one would expect, according to this 
 conception. We may also ask with what speed electrified 
 particles would have to move in order that they should be 
 more strongly deflected by a magnetic force of absolute strength 
 unity, acting perpendicularly to their path, than by an 
 electrostatic force of 1 Daniell per millimetre. The requisite 
 speed would exceed eleven earth-quadrants per second, — a 
 speed which will scarcely be regarded as probable. But unless 
 we assume such a speed, the conception here referred to cannot, 
 in accordance with the experiments described under B, account 
 for the action of the magnet upon the rays. 
 
 Conclusion 
 
 By the experiments here described I believe I have 
 proved : — 
 
 1. That until stronger proofs to the contrary are adduced, 
 
254 EXPERIMENTS ON THE CATHODE DISCHARGE xm 
 
 we may regard the battery discharge as being continuous, and 
 therefore the glow discharge as not being necessarily dis- 
 ruptive. 
 
 2. That the cathode rays are only a phenomenon accom- 
 panying the discharge, and have nothing directly to do with 
 the path of the current. 1 
 
 3. That the electrostatic and electromagnetic properties 
 of the cathode rays are either nil or very feeble. 
 
 I have also endeavoured to bring forward a definite 
 conception as to how the glow discharge takes place. The 
 following are the principal features of this : — 
 
 The luminescence of the gas in the glow discharge is not 
 a phosphorescence under the direct action of the current, but 
 a phosphorescence under the influence of cathode rays produced 
 by the current. These cathode rays are electrically indifferent, 
 and amongst known agents the phenomenon most nearly allied 
 to them is light. The rotation of the plane of polarisation of 
 light is the nearest analogue to the bending of cathode rays by 
 a magnet. 
 
 If this conception is correct, we are forced by the pheno- 
 mena to assume that there are different kinds of cathode rays 
 whose properties merge into each other and correspond to the 
 colours of light. They differ amongst themselves in respect of 
 exciting phosphorescence, of being absorbed, and of being 
 deflected by a magnet. 
 
 The views which most nearly coincide with these are those 
 which have been expressed by E. Wiedemann 2 and E. Gold- 
 stein. 3 By comparing this paper with those below referred to, 
 it will be easy to recognise the points of agreement and differ- 
 ence. The experiments here described were carried out in the 
 Physical Institute of the University of Berlin. 
 
 1 Since the presence of cathode rays in a gas-space modifies considerably the 
 possibility of .passing a discharge through it, there can scarcely be any doubt 
 that the position and development of the cathode rays do indirectly affect the 
 path of the current. 
 
 2 See Wied. Ann. 10, p. 249, 1880. 
 
 3 Loc. cit. 12, p. 265, 1881. 
 
XIV 
 
 ON THE BEHAVIOUE OF BENZENE WITH EESPECT 
 TO INSULATION AND EESIDUAL CHABGE 
 
 (Wiedemann's Annalen, 20, pp. 279-284, 1883.) 
 
 Eowland and Nichols have shown 1 that in certain insulating 
 crystals dielectric polarisation is not accompanied by any 
 electric after-effect or formation of a residual charge. They 
 interpret this result as supporting the view that the forma- 
 tion of a residual charge is simply a necessary consequence 
 of imperfect homogeneity in an insulator. Some years ago I 
 wanted to find whether the formation of a residual charge 
 could be detected in a conductor undoubtedly homogeneous ; 
 and with this object I tested various liquids. The conductivity 
 of most of these proved to be too high for such experiments ; 
 but commercial pure benzene exhibited a sufficiently high 
 resistance, and also a distinct residual charge. A closer 
 investigation disclosed certain peculiarities in the behaviour 
 of benzene which are described below, and these can be inter- 
 preted in the same way as the behaviour of crystals. I had 
 not kept the numerical results of my experiments ; but Herr 
 E. Heins has been good enough to repeat the experiments, and 
 to allow me to make use of his results. The numerical data 
 given below are taken from Herr Heins' experiments! 
 
 1. The method adopted is copied from that of Herr W. 
 Giese. 2 The benzene is contained in a zinc canister (B, Eig. 34). 
 In this, and entirely surrounded by it, a zinc plate about 12 
 cm. long and 8 cm. broad was hung by two wires. This plate 
 
 1 Phil. Mag. (Series 5) 11, p. 414. 1881. 
 2 Wied. Ann. 9, p. 160, 1880. 
 
256 
 
 KESIDUAL CHARGE 
 
 Fig. 34. 
 
 formed the inner coating, and the zinc canister the outer 
 coating of a Leyden jar, of which the benzene formed the 
 dielectric. The inner coating was connected with one pair of 
 quadrants — not earthed — of an electrometer ; and by means 
 
 of the key a this coating could be con- 
 nected to earth. The outer coating 
 could either be connected by 7 to the 
 earth as well, or by ft to one pole of a 
 constant battery of 100 small Daniell 
 cells, the other pole of which was kept 
 at zero potential by an earth-connection. 
 If we now suppose the circuit to be 
 closed at a and j3, and open at 7, the 
 needle of the electrometer will clearly 
 stand at zero, and the circuit will be traversed by a current 
 whose strength will depend upon the resistance of the benzene. 
 If the circuit is now broken at a, the inner coating strives to 
 charge itself to the potential of the outer, and hence the 
 electrometer needle is deflected in the direction in which it 
 would move if the unearthed quadrants were directly connected 
 with the insulated pole of the battery. This we shall call 
 the positive direction. The rate of deflection of the needle 
 enables us to measure the resistance of the benzene. The 
 capacity of the electrometer was to that of the benzene con- 
 denser in the ratio of 4.5 : l. 1 The whole potential of the 
 battery would have deflected the needle 5500 scale-divisions 
 from its position of rest. Now suppose that the connection 
 to the electrometer at S was broken one second after opening 
 the circuit at a, and that the electrometer was found to give 
 a deflection of a scale-divisions. The difference of potential 
 of the coatings would then have sunk in a second through 
 a/5500 of its value. In the absence of the electrometer it 
 would have fallen (4.5 + 1) times as rapidly, i.e. through 
 a/1000 in a second, or to l/e in of its value in 4-7T. 1000/<x 
 seconds. This latter time is to be divided by the specific 
 inductive capacity of benzene in order to obtain the specific 
 resistance in absolute electrostatic measure. In this way, 
 then, we can measure the resistance. In order to observe 
 
 1 The large apparent capacity of the quadrants is due to the strong charge on 
 the electrometer needle. 
 
RESIDUAL CHARGE 
 
 257 
 
 T 
 
 J 
 
 I 
 
 J 
 
 J> 
 
 ^ the residual charge the outer coating 
 was disconnected by the key (3 from 
 the battery, and connected by y with 
 the earth ; just a second after closing 
 7 the current was broken at a. The 
 residual charge present then pro- 
 duced a negative deflection of the 
 electrometer ; this rapidly increased, 
 reached a maximum, and then, owing to loss 
 of charge by conduction, slowly fell off. An 
 exact measure of the residual charge could 
 only be deduced by complicated calculations 
 from the course of the deflections; but an 
 estimate of its magnitude can be obtained 
 directly from the maximum deflection. 
 
 2. The canister was filled with com- 
 mercial benzene, and the current was closed, 
 excepting when the resistance and residual 
 charge were from time to time tested. The 
 results were as follows. At first the resist- 
 ance was so small that in a few moments 
 after opening a the scale moved quite out of 
 the field of view. The residual charge was 
 fairly considerable; its maximum value was 
 more than 1 per cent of the original charge, 
 but in consequence of the high conductivity 
 it soon disappeared. Twenty to thirty minutes 
 later the resistance was found to have in- 
 creased to a conveniently measurable value ; 1 
 at the same time the residual charge had 
 become much smaller. The same changes 
 went on without interruption ; after twenty- 
 four hours the benzene had become almost a 
 perfect insulator, and scarcely any residual 
 charge could be detected. Tig. 35 represents 
 correctly the numerical results of one of the 
 experiments. The abscissae give the time in 
 hours and minutes from the beginning of the 
 experiment. The vertical ordinates give the 
 
 1 It had increased for both directions of the current. 
 
258 RESIDUAL CHARGE xiv 
 
 conductivities measured at the corresponding times. The pro- 
 gress of formation of the residual charge, as far as it was 
 followed, is shown by the curves. The measurements are 
 only relative ; the conductivity at time was too great to be 
 measured. 1 
 
 3. The conductivity and residual charge at the beginning 
 are certainly due to impurities. For, in the first place, they 
 could be reproduced, after they had once disappeared, by any 
 action which introduced fresh impurities, e.g., by pouring the 
 benzene into other vessels, by stirring it up, blowing in moist 
 air, dipping in a wire of oxidisable metal, or mixing any 
 powder with it. The effects thus produced could again be 
 destroyed as before. In the second place, if a sample of benzene 
 having a high conductivity was carefully distilled over calcium 
 chloride, and only allowed to come in contact with vessels 
 which had been rinsed out with purified benzene, its con- 
 ductivity was very much reduced. But the highest grade of 
 insulation could never be attained in this way. 
 
 4. The reduction of both of these effects is due, at any rate 
 in part, to the action of the current. They certainly fell oft 
 even when the benzene was simply allowed to stand ; but 
 neither so rapidly nor so far as when the coatings of the jar 
 were connected to the poles of , a battery. In this respect 
 different samples of benzene seem to behave differently. The 
 resistance of the sample which I investigated only changed 
 very slightly when no current was used, whereas the con- 
 ductivity of that examined by Herr Heins fell to very low 
 values simply by standing. It may have been that the former 
 contained chiefly soluble impurities and the latter matter in 
 suspension. Experiments made with the intention of finding out 
 the nature of the active impurities were unsuccessful. I shall 
 only remark that in such experiments the benzene can be 
 tested between glass electrodes ; for the resistance of the latter 
 is negligible in comparison with that of the benzene. 
 
 5. In its behaviour electrically purified benzene comes 
 extraordinarily near to that of an ideal liquid insulator. 
 
 1 In Fig. 35 it will be noticed that after every considerable interruption of the 
 current the flow to the condenser becomes greater than it had been before. This 
 is only very slightly, if at all, due to a decrease in the resistance ; it rather 
 expresses the fact that for a short time after each interruption the flow proper is 
 reinforced by that due to the residual charge. 
 
XIV 
 
 RESIDUAL CHARGE 259 
 
 Scarcely a trace of residual charge can be detected, and its 
 insulating power is not far short of that of our best insulators. 
 On account of the evaporation which took place, the experi- 
 ments could not be extended much beyond twenty-four hours, 
 and at the end of this time a definite limit to the resistance 
 had not yet been reached. But the insulating power may be 
 judged from the fact that in a minute after breaking the 
 earth-connection at a, the electrometer needle had only moved 
 six scale divisions from its position of rest. From this, and 
 from the ratio of the capacities given above, it follows that a 
 Ley den jar containing this benzene as dielectric would require 
 two hours to lose half its charge. 
 
 6. The residual charge exhibited by impure benzene arises 
 from polarisation, which produces some kind of after-effect, 
 and not from an absorption of free electricity. This can be 
 proved by the following experiments. After breaking the 
 circuit at a, just at the instant when the residual charge 
 begins to exhibit itself, the benzene is allowed to run out 
 through an opening in the bottom as quickly and quietly 
 as possible ; the residual charge now makes its appearance 
 suddenly, and its sign is the same as it would have been if 
 the benzene had not been run out. If the residual charge 
 were due to an absorption of electricity by the dielectric, the 
 removal of this dielectric would certainly be followed by a 
 sudden deflection ; but the sign of this would have been 
 opposite to that of the electricity removed with the dielectric, 
 and therefore opposite to that of the original residual charge. 
 In another experiment I polarised impure benzene in a large 
 vessel between two plates, A and B, which were several centi- 
 metres apart. A and B were then brought to the potential 
 zero ; and a system of three other plates, 1, 2, and 3, was 
 immediately introduced into the space between them. The 
 outer plates 1 and 3 were connected to earth and to one pair 
 of quadrants of an electrometer. Plate 2, which was connected 
 to the other pair of quadrants, was closely attached to 3, but 
 insulated from it. Thus no benzene could enter between 2 
 and 3, whereas there was a layer a few centimetres thick 
 between 1 and 2. On introducing this system there was an 
 immediate deflection in the galvanometer, the direction of 
 which changed with the direction of polarisation of the benzene ; 
 
260 RESIDUAL CHARGE xiv 
 
 this corresponded to a portion of the residual charge which other- 
 wise would have developed upon the plates A and B. Now, it is 
 easily seen that if the residual charge is an after-effect of polari- 
 sation, the sign of this deflection must be opposite to what it 
 would be if it were due to electric absorption from the plates ; 
 for the electrical double stria produced between plates 1 and 2 
 would have opposite signs in these two cases. The result of 
 the experiment indicated that the effect was due to polarisation. 
 It is not surprising that the results of such experiments are 
 somewhat irregular ; for it is impossible to prevent friction 
 and irregular motions which disturb the polarised elements. 
 As a matter of fact, the magnitude of the deflection varied 
 very considerably, and now and again an experiment even gave 
 a result in the opposite direction. But the results of the large 
 majority of the experiments were such as to justify the state- 
 ment above made. 
 
XV 
 
 ON THE DISTRIBUTION OF STRESS IN AN 
 ELASTIC RIGHT CIRCULAR CYLINDER 
 
 (Schlomilch's Zeitschrift fur Math. u. Physik, 28, pp. 125-128, 1884.) 
 
 A homogeneous elastic right circular cylinder is bounded by 
 two rigid planes perpendicular to its axis. Let pressures be 
 applied to its curved surface at any desired inclination to it, 
 and let them be independent of the coordinate parallel to the 
 axis and act perpendicularly to that axis. Then the distribu- 
 tion of stress in the interior can be expressed in a finite form 
 so remarkably simple that it may be of interest in spite of the 
 narrow limits of the problem. 
 
 Let F be the pressure on the element ds of the curved 
 surface, and / its direction. Further, let M n be the component 
 in direction n of the pressure on a plane element parallel to 
 the axis of the cylinder, and the normal to which has the 
 direction n. Let (m, n) denote the angle between the direc- 
 tions m and n ; r the radius vector joining the element 
 considered to the element ds of the curved surface ; p the 
 perpendicular let fall on the axis of the cylinder from the 
 element ds ; and R the radius of the cylinder. With this 
 notation 
 
 ti/t / N , 2Lcos(/,r) cos (n.r) cos (m.r) 
 M n = -p cos (w, m) + - F ^^ ^^ —^ds 
 
 V = mrr F C ° S (f' r ) d8m 
 
262 STRESSES IN A EIGHT CIRCULAR CYLINDER xv 
 
 The integrations extend all round the circumference of the 
 cylinder. 
 
 Proof. — We shall first show that the expression (1) 
 represents a possible system of stress. Let x and y be 
 rectangular coordinates in a plane perpendicular to the axis 
 of the cylinder ; then the pressures ~K X , Y y , Y x , which are 
 independent of the third coordinate, form a possible system if 
 they satisfy these differential equations 
 
 0= 3X, + 3Y^ JJ_* + d J £> ?%+?%= £l- (2). 
 dx dy dx dy dy 2 dx 2 dydx 
 
 Let p, co be polar coordinates, p cos co = x, p sin co — y, and 
 in the stress-components P p , P w , fl M let co denote the direction 
 perpendicular to co ; then the system of stress 
 
 p p = — , p„ = o, n„ = o (3) 
 
 p 
 
 satisfies the equations (2). This is proved by calculating the 
 values of X x , Y y , Y x , which follow from (3), and substituting 
 them in (1). The three equations (3) may be replaced by the 
 one equation 
 
 M _ cos co cos (n,p) cos (m,p) cos (x,p) cos (n,p) cos (rn,p) 
 
 P P 
 
 A sum of such M n with different poles and multiplied by 
 arbitrary constants will represent a system which satisfies the 
 differential equations (2). Now the integral which occurs in 
 (1) is such a sum, and as the expression in front of the 
 integral merely represents a constant pressure p uniformly 
 distributed through the cylinder, it follows that the system 
 expressed by the equation (1) is a possible one. 
 
 We shall prove secondly that when we approach infinitely 
 near to the curved surface and make the direction n coincide 
 with that of the radius p, then M n coincides with the 
 component of F in the direction m ; so that M n = F cos (m,ri). 
 For this purpose we separate the integral into two parts, one 
 relating to the portion of the boundary infinitely near the 
 element considered, the other to the remaining more distant 
 portion. For this latter, and for it alone, we have 
 
xv STRESSES IN A RIGHT CIRCULAR CYLINDER 263 
 
 O-O / N C0S (P> T ) C0S ( 7l > r ) 1 
 
 r = 2E cos (p,r), ^L-L = ^L-Z = ; 
 
 r v ZK 
 
 so that this part of the integral 
 
 = — I F cos (f,r) cos (m,r)ds 
 
 = ^ f {F cos (/771) + F cos [(f,p) - (m,,,)]}*, 
 
 since (/,/)) + (m,r) = (/,m) ; (/» - (m,r) = (/,/o) - (m,p). 
 
 Now since the forces F must not produce a displacement 
 of the cylinder in the direction m, nor a rotation round the 
 axis, 
 
 JF cos (f,m)ds =-- , |F sin (/,/>)* = 0. 
 
 Hence the part of the integral examined is equal to 
 
 cos (ra,jo) I F cos {f,p)ds, 
 
 2Ett 
 
 and thus cancels the first term in M n ; and M. tt reduces to 
 that part of the integral which is due to the part of the 
 curved surface near the element considered. Here we have 
 rd(p,r) = ds cos (p,r). Hence 
 
 cos (n } r) , cos (p,r) , ,, * 
 
 \^_Z ds = - — ^— - ds = d(p,r), 
 
 r r 
 
 and as F, / may be regarded as constant over the smal. 
 portion of surface considered, we have 
 
 --H 
 
 M n = - F cos (/» cos (m,r)d(p,r) 
 
 n 
 
 - F I cos [(r,p) - (/,/>)] cos [(r,p) - (m,p)]d(j> t r) 
 
 IT 
 
 ~2 
 
264 STRESSES IN A RIGHT CIRCULAR CYLINDER xv 
 
 +E +E 
 
 2 2 
 
 F f F f 
 
 = - cos [2(r,p) - (/,/>)M/>,r) +- cos (/, m) tffV,^) 
 
 7Tj 7T J 
 
 _1T _7T 
 
 2 2 
 
 = F cos (/,m), 
 
 which was to be proved. 
 
 In calculating the first of the parts into which we 
 separated the integral we ought strictly to have excluded from 
 the integration the portion of the curved surface lying close 
 to the element considered ; but a simple investigation shows 
 that the error thus committed is infinitesimal. 
 
 Example. — Particular applications of our formula may be 
 made to cases where pressures are applied at isolated points of 
 the curved surface. Imagine, for instance, a cylinder placed 
 between two parallel plane plates which are pressed together 
 with a pressure P. This is approximately the position of the 
 rollers which frequently form the basis of support of iron 
 bridges. We take the axis of x to be the line joining the 
 points of contact of the cylinder with the planes; its inter- 
 section with the axis of the cylinder we take as origin. The 
 coordinate perpendicular to x we call y, and denote by r 1} r 2 
 the distances of the element considered from the points of 
 contact. Then the component of stress normal to the element 
 considered is 
 
 ±V7T IT 
 
 cos (r x) cos 2 (r^) cos (r 2 x) cos 2 (r 2 n)\ 
 r, r 9 J ' 
 
 If we determine the direction n so that N n becomes a 
 maximum or minimum, keeping r , r 9 the same, we get the 
 values and directions of the principal stresses at the point 
 (r v r 2 ). This calculation can be performed. For the axes of 
 x and y the principal stresses are parallel to the axes, whence 
 we easily obtain, for the axis of x 
 
 P 3K 2 + ^ 2 P 
 
xv STRESSES IN A RIGHT CIRCULAR CYLINDER 265 
 
 and for the axis of y 
 
 P 3R 4 -2EV-y 4 P / R 2 -?/ 
 
 ^~R^rR 4 +2Ry + ^ ' ' y R^VRMY 
 
 All the elements of the axes suffer compression in the direction 
 of x, and extension in the perpendicular direction. At the 
 centre the pressure in the direction of x is Qjir times what it 
 would be if the pressure P were distributed uniformly over the 
 whole section 2R. Even in this simplest case it appears that 
 the distribution of stress is extremely complicated. 
 
XVI 
 
 ON THE EQUILIBEIUM OF FLOATING ELASTIC 
 
 PLATES 
 
 (Wiedemann's Annalen, 22, pp. 449-455, 1884.) 
 
 Suppose an infinitely extended elastic plate, e.g. of ice, to float 
 on an infinitely extended heavy liquid, e.g. water ; on the plate 
 rest a number of weights without production of lateral tension; 
 the position of equilibrium of the plate is required. The 
 solution of this problem leads to certain paradoxical results, on 
 account of which it is given here. 
 
 If we confine ourselves to small displacements, we may 
 regard the effects of the separate weights as superposed, and 
 need only consider the case of a single weight P. We suppose 
 it placed at the origin of coordinates of x, y, of which the 
 plane coincides with the plate, supposed infinitely thin. 
 Further we write y 2 = d 2 /dx 2 + d 2 /dy 2 , p 2 = x 2 + y 2 , and denote 
 by E and fi in the usual notation the elastic constants of the 
 material of the plate, by h its thickness, and by s the density 
 of the liquid. Let z denote the vertical displacement of the 
 deformed plate from the plane of x, y, reckoned positive when 
 downwards; then on the one hand {Wi 3 / 12(1 - f/)}\7 4 z is 
 the upward pressure per unit area due to the elastic stresses, 1 
 on the other hand sz is the upward hydrostatic pressure per 
 unit area. The sum of both pressures must vanish every- 
 where except at the origin. Here the integral of that sum 
 taken over a very small area must be equal to P. But since 
 the integral of the hydrostatic pressure over such an area is 
 infinitesimal, that condition must be satisfied by the integral 
 of the elastic reaction alone. If we write for shortness 
 1 Clebsch, Theorie cler Elasticity, § 73, 1862. 
 
FLOATING ELASTIC PLATES 267 
 
 12(1 -^>_ , 1 
 
 E/r 
 
 our problem may be stated mathematically thus : Eequired 
 an integral of the equation ^ 4 z + a 4 z = 0, which vanishes at 
 infinity, is finite at the origin, and is such that the integral of 
 sa 4 frfzdw taken throughout the neighbourhood of the origin 
 may be equal to P. 
 
 With Heine 1 we write 
 
 K(p)= fe^ cosiu du, 
 o 
 
 then K(/o) is a solution of the equation rfz = - z, and therefore 
 K(/o ^/^V) is a solution of our equation. And 
 
 z = _ J K[ap x/i(l + i)] - K[ap */$(l - i)] \ (1) 
 
 4:7TSl [ J 
 
 is also a solution. It is real, and if we bring it into a real 
 form we get by transforming the integral 
 
 CO 
 
 z = f5 (V apA ^sin ap s/^vdv , 2 \ 
 
 4tt S J - " V^-l 
 i 
 
 which form shows that the solution assumed vanishes at 
 infinity. In order to examine its value near the origin, we 
 employ an expansion of the function K given by H. Weber, 2 
 first in a series of Bessel's functions, then of powers of p, 
 and thus obtain 
 
 z 
 
 £{*'-"- ?5>"-*> + 
 
 <x 4 /> 4 a 8 p* 
 
 + T 1 
 
 4\ 2 2 .4 2 2 2 .4 2 .6 2 .8 2 / (3> 
 
 _ (1+log2 _c ) (f- F ^ !+ ...)} 
 
 1 Heine, Handbuch der KugelfunTctioiien, vol. i. p. 192, 1878. 
 2 I.e. p. 244. The sign of C is wrongly quoted here. 
 
268 FLOATING ELASTIC PLATES xvi 
 
 C is equal to '57721. The rows are so arranged that each 
 horizontal row by itself represents a particular integral of the 
 given differential equation. This form shows that z remains 
 finite when p = ; further, the integral over a small circular 
 area surrounding the origin is 
 
 i^zdio = 2tt [p^zclp = 2ttU (p d -^\ = -P. 
 
 Hence the integral considered is the required one ; at the 
 same time the form (3) is one very suitable for the numerical 
 calculation of z for small values of p. For large values we 
 use a semi-convergent series which one gets from (2) by 
 expanding the root and integrating the separate terms, and 
 whose first terms are 
 
 a 2 F jnre-^h / w \ 
 
 = ~ — . / - — ==-< sin [ dp V Jr + - 
 
 "P 
 
 sm ( aps/\ + — I + . . . 
 
 (4). 
 
 The solution can be expressed in several additional forms. 
 We shall interpret the above in the following remarks. 
 
 1. At the place where the weight is put, the indentation 
 of the plate has its greatest value z = z = a 2 ~P/8s. The plate 
 rises from there in all directions towards the level zero ; at 
 first slowly, then faster, then again more slowly. At the 
 distance p = a, z = 'Q4:Qz ; for p = 2a, z = '2582 ; for p = 3a, 
 z = *066% Near the distance p = ^ir v f 2a = 3 - 887a, z changes 
 sign and thus the plate appears raised into a ridge round the 
 central depression. But it is extraordinary that at further 
 distances from the origin ridges and hollows follow each other 
 with the period irslz.a. The plate is thrown into a series of 
 circular waves ; it is true that they diminish so rapidly in 
 height as we go outwards that we need not wonder at being 
 unable to see them without special arrangements. The 
 quantity a, which is characteristic of the system of waves, is a 
 length. To calculate it for the case of ice floating on water, 
 we notice that s = 1 " 6 kg/mm 3 ; p, can be taken as \ ; and, 
 
xvi FLOATING ELASTIC PLATES 269 
 
 according to Eeusch, 1 E is equal to 236 kg/mm 2 . We thus 
 obtain for different thicknesses h — 
 
 h = 10 20 50 100 200 mm. 
 a=0-38 0-64 1*27 2*14 3*60 m., 
 
 whence we easily get the depression produced by 100 kg. 
 
 2 = 86'4 30-5 7-72 2-73 0'96 mm. 
 
 2. The strain produced in the plate depends on the 
 second differential coefficients of z with respect to x and y ; 
 hence it becomes infinite at the origin. This shows that the 
 greatest strain cannot be found without a knowledge of the 
 distribution of the weight. We shall calculate the maximum 
 tension in the simple case when the weight P is uniformly 
 distributed over a circular area of radius E, where E is 
 supposed small compared with a. For this purpose we 
 calculate v% at tne origin. If we call the distance from the 
 origin of the element at which dF rests p, then the portion of 
 V% due to this element is, by equation (3), 
 
 — (log«p-lo g 2 + C), 
 
 aits 
 
 where the terms which vanish with p have been neglected. A 
 ; simple integration now gives 
 
 ^o=2^=2_ = _(lo ga E-i-log2 + C) 
 = ~ P (log aK- 0*6519). 
 
 2-7TS 
 
 The maximum tension at the centre of the curved plate 
 is p = (Eh/2)d 2 z/dx 2 ', by forming the expression for p and 
 substituting for a 4 its value we find 
 
 3(1 -V)P 
 
 P = — 2 ^ (log «R- 0-6519). 
 
 It would be a mistake to attempt to apply this formula 
 even when E is of the order of the thickness of the plate or 
 
 1 Reusch, Wiecl. Ann. 9, p. 329, 1880. 
 
270 FLOATING ELASTIC PLATES xvi 
 
 smaller still. In this case the pressure inside the plate will 
 still be distributed over a circular area whose diameter is 
 approximately equal to the thickness of the plate. We may 
 roughly represent the case when the weight is as far as 
 possible concentrated at a point by making R equal ^h in 
 the preceding formula ; thus we get for the greatest tension 
 which the weight P can produce at all in the plate 
 
 For example, in the case of the plates of ice just con- 
 sidered, we get for a weight of 100 kg. the values 
 
 2^=221, 53, 81, 1-9, 0"47 kg/cm 2 . 
 
 The plate 100 mm. thick would certainly bear the weight, 
 that 50 mm. thick probably not. 
 
 3. The force with which the water buoys up the weight 
 owing to its deformation is 
 
 2tts, 
 27r szpdp = — 4 v 4 2 . pdp 
 
 and is therefore equal to the load applied. However great 
 the load, it will always be supported ; the force with which 
 the plane unloaded plate is buoyed up is immaterial. If we 
 place a small circular disc of stiff paper on the surface of 
 water, we may put at its centre a load of several hundred 
 grammes, although the force buoying up the paper alone is 
 but a few grammes. Hence when a man floats on top of a 
 large sheet of ice, it is in strictness more correct to say that 
 he floats because by his weight he has hollowed the ice into 
 a very shallow boat, than to say that he floats because the 
 ice is light enough to support him in addition to its own 
 weight. For he would float just as well if the ice were no 
 lighter than the water ; and if instead of the man we placed 
 weights as large 'as we pleased upon the ice, they might break 
 through, but could never sink with the ice. The limit of the 
 load depends on the strength, not on the density of the ice. 
 The case is different when men or weights are uniformly 
 
xvi FLOATING ELASTIC PLATES 271 
 
 distributed over the surface, or when the radius of the plate 
 is not very large, that is, not many times a. 
 
 4. If we consider the latter case, that of a finite plate, 
 more closely, we get the above-mentioned paradoxical result. 
 For the free edge of the circular plate these conditions x must 
 hold 
 
 1 — a dz • d 
 
 At the centre we have the same condition as before. The 
 solution is to be compounded of the three integrals of the 
 equation \/ 4 z -f a 4 z = , which are finite at the origin; and 
 is completely determined by the given conditions. According 
 as at the edge of the plate z is negative or positive, that is, 
 according as the edge is above or below the surface of the 
 water, the plate will or will not float without further aids 
 (without buoyancy of its own). The case (c) when z = at 
 the edge, is a limiting case. When we enquire under what 
 conditions the equations (a) (b) (c) are simultaneously possible, 
 we are led to the vanishing of a determinant involving the 
 radius of the plate as the unknown. This determinant in 
 fact vanishes for certain values of E, and with a little patience 
 we find that the least value of E for which this occurs lies 
 between 2*5<z and 2 - 8a. It is equal to 2 - 5<z when /x=0, 
 and to 2'8a when fju = J. If we suppose the plate to be of 
 the same density as water, then, so long as the radius is less 
 than the above value, for every load the edge must be below 
 the surface of the water in the position of equilibrium, and 
 the plate will be unable to support even the slightest load. 
 When the radius just has the critical value, then for every 
 load the edge is at the surface of the water, and thus the 
 plate suddenly becomes able to support every weight which 
 does not exceed the elastic limit. When the radius is still 
 larger, z is negative at the edge and we can now distribute a 
 certain additional weight uniformly over the plate without 
 lowering the edge below the surface of the water ; i.e. we 
 may suppose the plate somewhat heavier than water to begin 
 with. If we suppose such a plate, which by itself could not 
 float, to be loaded at the centre with a sufficient weight and 
 
 1 Clebsch, Theorie der Elasticitat, § 73. 
 
272 FLOATING ELASTIC PLATES XVI 
 
 then placed on the surface of the water, it will float. The 
 certainty with which it floats is increased by adding more 
 weight at the centre, it is diminished by removing weights 
 from the centre ; and if in this removal we exceed a certain 
 limit, floating will cease to be possible and the plate will sink 
 together with the remaining weights. 
 
XVII 
 
 ON THE EELATIONS BETWEEN MAXWELL'S FUN- 
 DAMENTAL ELECTEOMAGNETIC EQUATIONS 
 AND THE FUNDAMENTAL EQUATIONS OF THE 
 OPPOSING ELECTEOMAGNETICS. 
 
 (Wiedemann's Annalen, 23, pp. 84-103, 1884.) 
 
 When Ampere heard of Oerstedt's discovery that the electric 
 current sets a magnetic needle in motion, he suspected that 
 electric currents would exhibit moving forces between them- 
 selves. Clearly his train of reasoning was somewhat as 
 follows : — The current exerts magnetic force, for a magnetic 
 pole moves when submitted to the action of the current ; and 
 the current is set in motion by magnetic forces, for by the 
 principle of action and reaction a current -carrier will also 
 move under the influence of a magnet. Unless we make the 
 improbable assumption that different kinds of magnetic force 
 exist, a current-carrier must also move under the action of the 
 magnetic forces which a second current exerts, and thus the 
 interaction between currents follows. 
 
 The essential step in this reasoning is the assumption that 
 only one kind of magnetic force exists ; that therefore the 
 magnetic forces exerted by currents are in all their effects 
 equivalent to equal and equally directed forces produced by 
 magnetic poles. But this assumption is well known to be 
 sufficient to deduce not only the existence but also the precise 
 magnitude of the electromagnetic actions of closed currents from 
 
 i their magnetic actions. Whether Ampere actually started 
 from this principle or not, he certainly stated it at the close 
 
 j of his investigations when he reduced the action of magnets 
 directly to the action of supposed closed currents. At a 
 M. p. T 
 
274 FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS XVII 
 
 later stage the principle was hardly mentioned, but was taken 
 for granted as self-evident. After the discovery of the electric 
 forces exerted by variable currents or moving magnets, a similar 
 principle was added relative to these electric forces, and this, 
 too, was not definitely expressed. It has perhaps nowhere 
 been explicitly stated that the electric forces, which have their 
 origin in inductive actions, are in every way equivalent to 
 equal and equally directed electric forces of electrostatic origin ; 
 but this principle is the necessary presupposition and conclusion 
 of the chief notions which we have formed of electromagnetic 
 phenomena generally. According to Faraday's idea the electric 
 field exists in space independently of and without reference to 
 the method of its production ; whatever therefore be the cause 
 which has produced an electric field, the actions which the 
 field produces are always the same. On the other hand, by 
 those physicists who favour Weber's and similar views, electro- 
 static and electromagnetic actions are represented as special 
 cases of one and the same action-at-a-distance emanating from 
 electric particles. The statement that these forces are special 
 cases of a more general force would be without meaning if we 
 admitted that they could differ otherwise than in direction 
 and magnitude, that is, according to nature and method of 
 action. But, apart from all theory, the assumption we are 
 speaking of is implicitly made in most electric calculations ; 
 it has never been directly rejected, and may thus be regarded 
 as one of the fundamental ideas of all existing electromagnetics. 
 Nevertheless, to my knowledge no one has yet drawn attention 
 to certain consequences to which it leads, and which will be 
 developed in what follows. As premises we in the first place 
 employ the two principles referred to, which we might desig- 
 nate as the principle of the unity of electric force and that 
 of the unity of magnetic force. These may be regarded as 
 generally accepted, even if not as self-evident. In the second 
 place, we use the principle of the conservation of energy ; that 
 of action and reaction as applied to systems of closed currents ; 
 that of the superposition of electric and magnetic actions ; 
 and lastly, the well-known laws of the magnetic and electro- 
 motive actions of closed currents and of magnets. The in- 
 vestigation throughout refers to closed currents, even where 
 this is not specially stated. 
 
xvn FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS 275 
 
 1. Suppose a ring-magnet, whose cross - section we shall 
 for simplicity take as small compared with its other dimen- 
 sions, to lose its magnetism. Then it will exert a force on all 
 electricity in its neighbourhood which causes this electricity 
 to circulate round the body of the magnet. The magnitude 
 of this force is proportional to the rate of loss of magnetisa- 
 tion, and may be constant during a short but finite time, if 
 during this time the magnetisation diminishes at a constant 
 rate. The distribution of force in space is precisely the same 
 as that which would be produced by a current flowing in the 
 body of the magnet. Like the latter the electric force con- 
 sidered has a potential which is many- valued, and, apart from 
 its multiplicity, is the same as that due to an electric double 
 layer of uniform moment bounded^by the axis of the magnet. 
 The potential of the ring-magnet on an electric pole can, apart 
 from its multiplicity, be represented by the potential of the 
 double layer on the pole ; or, taking the multiplicity into 
 account, it can be represented by the solid angle subtended by 
 the magnet at the pole, multiplied by a suitable constant. 
 
 Now this potential determines the action of the magnet 
 on the pole as well as of the pole on the magnet. If we have 
 not a single pole but a whole system of electric charges, the 
 potential of the diminishing magnetisation on it can be found 
 by a simple summation. In particular, when the electric 
 forces which act on the ring-magnet are due, not to electric 
 charges, but to a second ring-magnet of diminishing moment, 
 their distribution is the same as if they were due to an electric 
 double layer. Hence, according to our assumption of the unity 
 of electric force, interaction occurs between our two ring- 
 magnets of diminishing moment ; and the potential determin- 
 ing this interaction is the mutual potential of two electric 
 double layers which are bounded by the bodies of the magnets. 
 As in electromagnetics the mutual potential of two magnetic 
 double layers is reduced to an integral to be taken along their 
 boundaries, so here we can bring into the same form the 
 potential of the electric layers, that is, of the two magnets of 
 diminishing moment. We thus find that this potential is the 
 product of the factor 1 A 2 of the rates of diminution of the 
 
 1 A is, as usual, the reciprocal of the velocity of light. We get this factor 
 by a quantitative investigation of the case which above is only considered quali- 
 tatively. Cf. in this respect paragraph 2, p. 278. 
 
276 FUNDAMENTAL EQUATIONS OF ELECTEOMAGNETICS xvn 
 
 moments of the magnets per unit length measured in electro- 
 magnetic units, and of the integral /(cos e/r)dede f , where de, 
 de r denote elements of the axes of the magnets, e the inclina- 
 tion of these elements to one another. 
 
 The potential thus determined is of the same form as the 
 mutual potential of electric currents, and therefore represents 
 the same actions. Two ring-magnets, which are placed close 
 together and side by side, will attract each other at the moment 
 when they both lose their magnetism, if they are magnetised 
 in the same direction ; they will repel each other if oppositely 
 magnetised. In the usual 1 electromagnetics this action is 
 missing. To describe it more simply we shall introduce a 
 new name. "We call the change of magnetic polarisation a 
 magnetic current, and take as unit that magnetic current in- 
 tensity which corresponds to unit change per unit time of the 
 polarisation per unit volume measured in absolute magnetic 
 units. So far as we can conclude from the phenomena of 
 unipolar induction as yet known, magnetic poles distributed 
 continuously along a closed curve and moving along it exert 
 the same electromagnetic action at outside points as a ring- 
 magnet coinciding with that curve and of suitably changing 
 moment. If this relation can be looked upon as true in 
 general, the name " magnetic current " includes all the different 
 cases of magnetism in motion ; and we may speak of constant 
 magnetic currents just as we speak of constant electric cur- 
 rents. But here that name is only to be regarded as a simple 
 contraction for " changing polarisation." Our result may now 
 be stated in this form : — Magnetic currents act on each other 
 according to the same laws as electric currents ; in absolute 
 magnitude the action between magnetic currents of S magnetic 
 units is equal to that between electric currents of S electrical 
 units. This theorem may not be capable of experimental 
 verification. It may be possible to show that electrically 
 charged bodies are set in motion by a ring-magnet whose 
 moment is diminishing ; perhaps even that the magnet itself 
 is turned by electrostatic force so that its plane sets itself 
 
 1 By usual I mean, here and in what follows, that electromagnetics which 
 regards the forces deduced from Neumann's laws of the potential as exactly 
 applicable, even when we consider the attraction of variable currents. Every 
 such system of electromagnetics is necessarily opposed to Maxwell's. 
 
xvn FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS 277 
 
 normally to that force ; but even with very powerful electro- 
 static forces these actions will lie at the limits of observation, 
 and hence it is hopeless to expect to see a ring-magnet set itself 
 under the action of the weak forces produced by a second ring- 
 magnet when the moments of both are diminishing. 
 
 But our premises permit of our drawing further inferences 
 It is known that a knowledge of the mutual electromagnetic 
 potential of two currents, together with the principle of the 
 conservation of energy, enables us to predict the existence and 
 absolute value of the inductive action. Similar conclusions 
 may be drawn for magnetic circuits (rings of soft iron). A 
 determinate expenditure of work is necessary to maintain in 
 such a circuit a magnetic current, which we may suppose to 
 be alternating. If the amount of this work were the same, 
 whether the magnet were at rest free from any electrical 
 influence, or did work in moving through the electric field, 
 nothing could be simpler than the infinite production of work 
 from this motion. Hence such an independence is impossible. 
 The work done must depend on the nature and velocity of the 
 circuit's motion and on the changes in the electric field ; and 
 thus the magnetic (magnetomotive) force which produces this 
 uniform current must also depend upon these circumstances. 
 This may be expressed by saying that a magnetic force, 
 produced by the motion and the changes in the field, is super- 
 imposed upon the magnetic forces due to other causes ; this 
 added force we may describe as induced. Its magnitude is 
 given by the condition that for any displacement whatever of 
 the circuit the external work done in this displacement must 
 be compensated by an equal additional amount of work which 
 in consequence of the displacement must be done in the 
 circuit. This reasoning is in form the same as that used to 
 deduce the inductive actions in electric circuits ; and since 
 also the forces between magnetic circuits are of the same form 
 as those between electric circuits, the final result must in form 
 be the same in both cases. In the laws of electric induction 
 we need only interchange the words " electric " and " magnetic " 
 throughout in order to obtain the inductive actions in 
 magnetic circuits. Thus we find that a plane magnetic 
 circuit, e.g., a plane ring of soft iron, whose plane is perpen- 
 dicular to the lines of force in an electric field, is traversed by 
 
278 FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS XVII 
 
 a magnetising force at the instant when the intensity of the 
 field is reduced to zero ; and that the same ring is subject to 
 an alternating polarisation when we turn it about an axis 
 which is perpendicular to the direction of the electric force. 
 It does not appear impossible that such actions may become 
 capable of experimental detection. Again, a ring-magnet 
 whose polarisation is continually changing its direction must 
 by induction call forth alternating polarisations in all neigh- 
 bouring iron rings ; but this action is certainly too small to 
 reach an observable value. 
 
 2. It may at first sight seem as if the actions here 
 deduced from generally accepted premises permitted of 
 incorporation without disturbance into the usual system of 
 electromagnetics ; but this is not the case. In fact, suppose 
 that in place of the ring-magnets so far considered we have 
 endless electric solenoids, in which the current-intensity is 
 variable ; then the induced electric forces produced by these 
 solenoids are certainly quite analogous to those exerted by the 
 variable magnets. From these latter forces we deduced 
 magneto -dynamic attractions, and we must therefore infer 
 corresponding electrodynamic attractions between the variable 
 solenoids. But as long as the currents in them are constant 
 no action takes place. Hence in general the electromagnetic 
 attraction between currents must depend on their variations 
 and not merely on their momentary intensities. This state- 
 ment is in opposition to an assumption uniformly accepted in 
 the usual electromagnetics. 1 The correction which must be 
 made in the laws of the magnetic actions of constant currents 
 to make them applicable to variable currents may be 
 calculated from our premises. But this correction requires, 
 on account of the principle of the conservation of energy, a 
 correction in the induced electrical forces as well. This again 
 requires a second correction in the magnetic forces, and so on ; 
 so that we obtain an infinite series of successive approxima- 
 tions. We shall now calculate these separate terms. We 
 assume that they are simply to be added to the total result, 
 and that, if only the infinite sums converge to definite limits, 
 then these limiting values are those corresponding to the 
 
 1 Cf.v. Helmholtz, " Uber die Theorie der Eiektrodynamik," Wissenschaftliche 
 Abliandlungcn, vol. i. p. 729. 
 
xvii FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS 279 
 
 actual case. In the calculation we use the following special 
 notation, u is to denote a function U for which throughout 
 infinitely extended space V 2 U == — 4™ ; hence generally 
 
 U = u = I - dr, 
 
 I 
 
 the integral being taken throughout all space. 
 
 As regards the electric currents, let u, v, w be the com- 
 ponents. As we only consider closed circuits we have 
 
 du dv div _ 
 dx dy dz 
 
 Further let Uj = u, Y x = v, W x = w. Then the components 
 L ± M 1 N x of the magnetic force exerted by the currents 
 are, according to the usual electromagnetics, given by the 
 equations 
 
 S + fL+^-O (1). 
 
 dx dy dz 
 
 From the existence of these forces, and from the principle of 
 the conservation of energy, it may be, and has been, concluded 
 that changes of u, v, w produce electrical forces whose 
 components X x Y 1 Z l are 
 
 Xl =-A^, Yl =-A 2 ^, Z 1= -A 2 ^l (2). 
 dt dt dt 
 
 These expressions hold good inside the conductors conveying 
 the currents u, v, w as well as for the space outside. The 
 forces (2) have been deduced from the forces (1) on the 
 assumption that the latter were due to electric currents. But 
 on account of our premises we may affirm that even if the 
 forces (1) are caused by any system whatever of variable 
 currents and variable magnets, then their variation must 
 equally give rise to the forces (2). Let A denote the system 
 which produces the forces (1). We superpose a system B 
 
 h- 
 
 -M 
 
 / dY 1 
 
 \ dz 
 
 aw; 
 
 Mi- 
 
 = A( 
 
 \ dx 
 
 
 Mi- 
 
 -H 
 
 
 dY,\ 
 
 " dx J 
 
280 FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS xvn 
 
 consisting only of electric currents which still neutralises the 
 forces of system A everywhere. Such a system is possible ; 
 we need only choose as current-components u, v, w where 
 4. w u = V 2 !^, 4cirv = V 2 V X , 4:7rw = V 2 W r If we now move 
 electric currents about under the action of both systems A and 
 B, there is no work done in this motion. Hence the electro- 
 motive force necessary to maintain the currents must be 
 independent of the motion, so that the induced electromotive 
 force is zero. But the system B by itself exerts inductive 
 actions ; hence the system A must exert inductive actions 
 equal and opposite to those of B, and therefore equal to 
 those of a purely electrical system which exerts the same 
 magnetic forces as A. What is true of inductive actions due 
 to motion must also be true of those due to variations of 
 intensity ; both are most simply determined in terms of each 
 other by the principle of the conservation of energy. Hence 
 from the existence of magnetic forces of the form (1) we may 
 directly infer that of electric forces of the form (2), whatever 
 may be the origin of those magnetic forces. 
 
 Let us now consider magnetic currents. Let X, /jl, v be 
 the components of magnetisation throughout space, and let 
 
 ax. + ^ + a, = 0;andA = ^ M= . )N = . 
 
 ox dy cz 
 
 These quantities are to be measured in absolute magnetic 
 units. It follows from the forces (1) by the principle of the 
 conservation of energy, and is indeed generally accepted in 
 electromagnetics, that the electric force produced by variations 
 of X, ft, v has for components 
 
 dt\dy dzj dt\dz dx 
 
 dt\dx dyj 
 We now put, in accordance with our notation, 
 
 dX da dv 
 
 P = — > 2 ~ > r = — ? 
 dt dt dt 
 
 1 Cf. v. Helmholtz, Wissenschaftliche Abhandlungen, vol. i. p. 619. 
 
xvii FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS 281 
 
 and call p, q, r the components of the magnetic current, 
 Further we put P x =p, Q x = q, E x = r and call P x , Q v E x the 
 components of the vector-potential of this current. Then the 
 electric forces produced by the magnetic current are 
 
 \ dy dz 
 
 \d% dy 
 
 The reasoning which allows us to infer from the forces (1) 
 that the mutual potential of two electric current-systems 
 u v v lt w l and u 2 , v 2 , w 2 has the form 
 
 3 
 
 {U X U 2 + V X V 2 + W{U0 2 ^)d J T 1 dT 2 , 
 
 leads to the conclusion, using forces (3), that the magnetic 
 current-systems p v 9_i> T i an d Pi, q-i, ^ have the mutual 
 potential 
 
 A - (prf* + M2 + r^dr^r. 
 
 The same considerations which led us from that potential 
 of electric currents to the inductive forces (2) allow us from 
 the potential of magnetic currents to infer the existence of 
 induced magnetic forces of the form 
 
 L^-A 2 ^, M^-A 2 ^!, N^-A 2 ^ 1 (4) 
 1 dt 1 dt 1 dt K } ' 
 
 Here also we may affirm that these forces act inside the 
 magnetic bodies as well as in the space outside ; and we 
 easily convince ourselves that we cannot well confine the 
 connection between the force (3) and (4) to the case where 
 the forces (3) are due to magnetic currents alone. We must 
 conclude that when a system of currents or magnets gives rise 
 to electrical forces of the form (3), then a variation of this 
 system will give rise to magnetic forces of the form (4). 
 
282 FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS XVII 
 
 So far we have merely repeated in precise form the results 
 of the preceding paragraph. We now go further and conclude 
 that a system of variable currents exerts electric forces of the 
 form (2). These may be represented in the form (3). Hence 
 unless they are constant they will give rise to magnetic forces 
 of form (4). And these must be added as a correction to the 
 known magnetic forces of form (1). To arrive at the expres- 
 sion of the forces (2) in the form (3) we put 
 
 dt \ dy dz 
 
 _ A2 ^Vi =A /ap dK 
 
 ) 
 
 dt \dz dx 
 
 dW, . fdQ 8P\ 
 
 A 2— i = A -* 
 
 dt \dx dy J 
 
 Assuming for the present that 
 
 3 P + aQ + ^E = {a) 
 
 dx dy dz 
 
 we get, by differentiating the second equation with respect to 
 z, the third with respect to y, and subtracting the results, 
 
 dt\ dz dy ) 
 and thence 
 
 We get similar expressions for Q and E. It is easy to see 
 that these satisfy the equation (a), and the assumption of the 
 truth of this equation is justified. 
 
 From the values of P, Q, K follow the magnetic forces 
 produced by their variation. The ^-component is 
 
 A9 dP 1 A3 d 2 d a — 
 
 - A 2 — = - — A 3 — —V, - — Wi 
 dt 4tt df\dz 1 dy 
 
 This term we must add to the component L x of the previously 
 assumed magnetic force. Let us call the component thus 
 
xvii FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS 283 
 
 corrected L 2 and form the similarly corrected components 
 M 2 , N 2 ; then these forces may be represented by the system 
 
 \dz dy }' 
 
 . r ./aw 2 au,\ ?u 2 , av 2 aw., . 
 
 3y 9a? 
 where we have put 
 
 47r rfr 4tt d£- 
 
 w 2 = w 1 --1a 2 ^w 1 . 
 
 47T a£" 
 
 From what precedes we may at once conclude that the electro- 
 motive forces produced by a variation of the current-system no 
 longer have exactly the form (2), but have these corrected 
 values 
 
 X 2 = _A 2 ^?, Y 2 = -A 2 ^, Z,= -A 2 ^? (6). 
 dt dt dt 
 
 Exactly similar reasoning compels us to correct the actions of 
 magnetic systems presented by the equations (3) and (4). 
 The results may be represented by the following sample 
 equations 
 
 *-*($-§)•- "»' 
 
 dV 
 
 L 9 = -A 2 — 2 , etc. (8), 
 
 dt 
 
 1 /?2 _ 
 
 where P 9 = Pi - — A 2 — -P 1; etc. 
 
 If we wish to represent the forces by which the corrected 
 equations (5) and (7) differ from the usually accepted ones (1) 
 
284 FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS xvn 
 
 and (3), as distinct from these latter in nature, we need only 
 form a system of electric or magnetic currents in which the 
 forces (1) or (3), as the case may be, are zero. Any endless 
 electric or magnetic solenoid will serve as an example. 
 
 We see at once that we cannot regard the result obtained 
 as final. Indeed we deduced forces (5) from forces (2) ; but 
 now the forces (2) have been found inexact and have been 
 replaced by the forces (6). Hence we must repeat our 
 reasoning with these latter forces. The result is easily seen ; 
 we obtain it if we everywhere replace the index 2 by 3 
 and put 
 
 TT TT A 2 d 2 - A 2 d 2 - A 4 d' = 
 
 V ^^-^^ 2 ^ = ^-^d^ + l^ 2 d^^ 
 
 and similar expressions for the other components of the vector- 
 potential. The terms in A 5 which here appear in the ex- 
 pressions for the magnetic forces of electric currents, and the 
 electric forces of magnetic currents, may be perceived apart 
 from the terms of lower order. We need only take an 
 ordinary electric or magnetic solenoid, which may be called a 
 solenoid of the first order, and roll it up into a solenoid which 
 may be called a solenoid of the second order, in order to get 
 a system in which the forces here calculated are the largest of 
 those occurring. From the consideration of such solenoids we 
 may demonstrate the existence of the separate terms, inde- 
 pendently of the fact of our admitting or not admitting that 
 they are simply added together to give the final result. 
 
 Our reasoning prevents us from stopping at any stage and 
 constantly adds, as before, more and more terms, thus leading 
 to an infinite series. To represent the final result we denote 
 by L, M, N, X, Y, Z the completely corrected forces and obtain 
 
 L = A 
 
 /av dW\ 
 
 \dz dy ) 
 
 X = 
 
 -A 2 ^ 
 A dt 
 
 M = A( 
 
 'aw am 
 K ~dx~ "&7 (9) ' 
 
 Y = 
 
 -A 2 ^ 
 dt 
 
 N = A 
 
 /au av\ 
 
 \dy dx J 
 
 Z = 
 
 dt 
 
 (10). 
 
xvii FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS 285 
 
 where we now have for IT, V, W 
 
 A 2 # = A 4 <V 
 
 U = U — — IjM + TTl^" 
 
 A 2 d 2 _ A 4 d± 
 
 Y =v — -. j-oV + 
 
 A 2 d 2 _ , A 4 d* = 
 
 47T tfr 167T 2 ^ 4 
 
 3U 3V o W = 
 
 3x 9?/ 3s; 
 
 Corresponding equations hold for the magnetic currents. If 
 the series are convergent, there is no reason to doubt that they 
 give us the true values. But in general they will converge. 
 For let us consider that element of the integral U which is 
 due to the current u in a certain element of space. We resolve 
 this current into a series of simple harmonic functions of the 
 time and suppose u sin nt to be the term involving sin nt. 
 Then the element of U due to this term will be given by the 
 equation 
 
 w sin ntf 1 A 2 2 2 
 
 dU = dr- 
 
 1.2 4tt 
 
 1 A 4 
 
 + — %V 4 + . 
 
 1.2.3.4 16tt 2 
 
 This series converges to a limit easily found. If n and r are 
 not very large, then every term after the first few will be in- 
 finitesimal compared with the preceding one. Hence also the 
 integral of the elements of U will have a determinate value. 
 Since the same is true of V, W, P, Q, and E, we may expect 
 to find, in the equations (9) (10) and the corresponding ones 
 for magnetic currents, a system of forces in complete agreement 
 with all our requirements. 
 
 3. It is obvious that this system may be represented, or in 
 technical terms described, more simply than by the equations 
 (9) and (10). By these equations we have 
 
 V 2 U = - 4:iru + A 2 — u — . . . 
 at 
 
286 FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS xvn 
 
 and 
 
 A2 d 2 V K „d? 
 
 hence 
 
 V 2 U-A 2 — = -4,7ru. 
 dt 2 
 
 The other components of the vector -potentials, both 
 of electric and magnetic currents, satisfy analogous differential 
 equations. Since u, v, w, p, q, r vanish in empty space, the 
 distribution of these potentials is there given by the equations 
 
 V 2 TJ _ A 2^U =0 
 
 df 
 
 d 2 V 
 dt 2 
 
 d 2 V 
 
 df 
 
 V 2 Q - A 2 ^ = 
 dt 2 
 
 r/ 2 W 
 
 V*W-AT V -0, 
 
 dt 2 
 
 V 2 E - A 2 ~ = 
 df 
 
 au dv dw _ 
 
 dx dy dz 
 
 dx dy dz 
 
 (11). 
 
 The vector-potentials now show themselves to be quantities 
 which are propagated with finite velocity — the velocity of light 
 — and indeed according to the same laws as the vibrations of light 
 and of radiant heat. Eiemann in 1 8 5 8 and Lorenz in 1 8 6 7, with 
 a view to associating optical and electrical phenomena with one 
 another, postulated the same or quite similar laws for the pro- 
 pagation of the potentials. These investigators recognised 
 that these laws involve the addition of new terms to the forces 
 which actually occur in electromagnetics ; and they justify this 
 by pointing out that these new terms are too small to be ex- 
 perimentally observable. But we see that the addition of 
 these terms is far from needing any apology. Indeed their 
 absence would necessarily involve a contradiction of principles 
 which are quite generally accepted. 
 
 The vector-potentials of electric and magnetic currents have 
 hitherto occurred as quite separate, and from them the electric 
 and magnetic forces were deduced in an unsymmetric 
 manner. This contrast between the two kinds of forces dis- 
 
xvn FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS 287 
 
 appears as soon as we attempt to determine the propagation of 
 these forces themselves, i.e. as soon as we eliminate the vector- 
 potentials from the equations. This may be performed by 
 differentiating equations (9) with respect to t and removing 
 the differential coefficients of U, V, W with respect to t by 
 equations (10). It may also be performed by differentiating 
 equations (10) with respect to t, remembering that, e.g. 
 
 A 2 — = V 2 JJ = -( dJJ - d — \- -f— dJJ \ 
 
 dt 2 dy\dy dx J d%\dx dz ) 
 
 and removing the functions of IT, V, W in the brackets by 
 means of equations (9). In this way we get six equations 
 connecting together the values of L, M, N, X, Y, Z in empty 
 space, viz. the following : — 
 
 dL_aZ_9Y 
 
 dt dy dz 
 
 A ^m ax dz 
 .<m _ ay ax 
 
 dt dx dy 
 
 These same equations connect together the forces produced 
 by magnetic currents, for they are got by eliminating P, Q, K 
 as well as U, V, W. Hence they connect together the electric 
 and magnetic forces in empty space quite generally, whatever 
 the origin of these forces. The electric and magnetic forces are 
 now interchangeable. If we eliminate first one set and then the 
 other we obtain the following system, which, however, does not 
 completely represent the system (12): — 
 
 V 2 X~A^!?=0 
 dt 2 
 
 dX 
 
 aM 
 
 aN 
 
 ± — = 
 
 dt 
 
 ~~ dz ' 
 
 dy 
 
 ,dY 
 
 A : 
 
 dt 
 
 ax 
 
 dL 
 
 " dx ' 
 
 ~ dz 
 
 A dZ 
 
 A -aT 
 
 dL 
 
 aM 
 
 dy 
 
 dx 
 
 V 2 L- 
 
 -A 2 
 
 dt 2 
 
 = 0, 
 
 V 2 M- 
 
 -A' 
 
 2 d 2 M 
 dt 2 
 
 = o, 
 
 V 2 N- 
 
 -A 5 
 
 ,d 2 N _ 
 dt 2 
 
 = 0, 
 
 aL 
 
 dx 
 
 aM 
 
 aN 
 + aT 
 
 = 0, 
 
 r/ 2 V 
 
 V 2 Y - A 2 ~ = 
 
 dt 2 
 
 V 2 Z - A 2 -4 = 
 dt 1 
 
 * + ?* + ?5 = 
 
 dx dy dz 
 
 (13). 
 
288 FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS xvii 
 
 Now the system of forces given by the equations (12) and 
 (13) is just that given by Maxwell. Maxwell found it by 
 considering the ether to be a dielectric in which a changing 
 polarisation produces the same effect as an electric current. 
 We have reached it by means of other premises, generally 
 accepted even by opponents of the Faraday-Maxwell view. 
 The equations (12) and (13) appear to us to be a necessary 
 complement of the equations (1), (2), (3), which are usually 
 regarded as exact. From our point of view, the Faraday- 
 Maxwell view does not furnish the basis of the system of 
 equations (12) and (13), although it affords the simplest 
 interpretation of them. In Maxwell's theory the equations 
 (12) and (13) apply not merely to empty space but also to 
 any other dielectric. Starting from our premises we can also 
 show these laws to hold in every homogeneous medium. We 
 must assume the fact as experimentally demonstrated that the 
 magnetic forces which surround a current-system placed in a 
 homogeneous medium are distributed according to the equations 
 (1) in the same way as in empty space. Hence we need only 
 imagine the conductors and masses of iron which we have con- 
 sidered to be completely immersed in the given medium. In 
 this medium we must define the units of electricity and 
 magnetism in the same terms as in empty space. We must 
 then determine the constant A, which gives the absolute value 
 of the magnetic force produced by unit current in the new 
 electrostatic measure. All further forces follow from the 
 assumed experimental fact and the general premises ; and 
 since all the propositions are the same as those for empty 
 space, the final result is the same. It is true that the value 
 of the constant A will not be the same as in empty space, and 
 that it will have different values in different media. Its recip- 
 rocal is always the velocity of propagation of electric and 
 magnetic changes. It is an internal constant, but the only 
 internal electromagnetic constant of the medium. The two 
 constants of which it is generally built up, viz. the specific 
 inductive capacity and the magnetic permeability, should in 
 contrast to it be termed external constants. Not only the 
 measurement, but even the definition of these latter constants, 
 requires the specification of at least two media (one of which 
 may be empty space). 
 
xvn FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS 289 
 
 In what precedes I have attempted to demonstrate the 
 truth of Maxwell's equations by starting from premises which 
 are generally admitted in the opposing system of electro- 
 magnetics, and by using propositions which are familiar in it. 
 Consequently I have made use of the conceptions of the latter 
 system ; but, excepting in this connection, the deduction given 
 is in no sense to be regarded as a rigid proof that Maxwell's 
 system is the only possible one. It does not seem possible to 
 deduce such a proof from our premises. The exact may be 
 deduced from the inexact as the most fitting from a given 
 point of view, but never as the necessary. 1 I think, however, 
 that from the preceding we may infer without error that if 
 the choice rests only between the usual system of electro- 
 magnetics and Maxwell's, the latter is certainly to be preferred ; 
 and that for the following reasons : — 
 
 1. The system of the electromagnetic action of closed 
 currents founded on direct action-at-a-distance is in its present 
 state certainly incomplete. Either it must introduce different 
 kinds of electric force, which it has never done, or it must 
 admit the existence of actions which hitherto it has not taken 
 into account. Maxwell's system does not in the same way 
 contain within itself the proof of its incompleteness. 
 
 2. When we attempt to complete the usual system of 
 electromagnetics, we always arrive at laws which are very com- 
 plicated and very difficult to handle. And either we refuse 
 to admit the accumulated results of paragraph 2, in which 
 case we end with an unfruitful declaration of incompetence ; 
 or, as from the standpoint of the system seems more reasonable, 
 we accept them as being valid, and so arrive at forces which 
 in fact are the same as those demanded by Maxwell's system. 
 But then the latter offers by far the simplest exposition of the 
 results. 
 
 1 The mode in which we have deduced conclusions from the principle of 
 the conservation of energy clearly marks at each stage the point at which 
 our deductions are only the most fitting, and not the necessary ones. This 
 mode is the most fitting from the standpoint of the usual system of electro- 
 magnetics, for it corresponds exactly to the accepted proposition in which 
 Helmholtz in 1847 and Sir W. Thomson in 1848 deduced induction from 
 electromagnetic action. But perhaps it may not be the only possible method ; 
 for just as in that proposition, so we have in ours made tacit assumptions besides 
 the principle of the conservation of energy. That proposition also is not valid 
 if we admit the possibility that the motion of metals in the magnetic field may 
 of itself generate heat ; that the resistance of conductors may depend on that 
 motion ; and other such possibilities. 
 
 M. P. U 
 
290 FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS xvu 
 
 3. The objections which may perhaps be raised to the 
 further conclusions of paragraph 2 do not apply to the reason- 
 ing specially exhibited in paragraph 1, which proved the 
 attraction between magnetic currents. This latter depends 
 directly on the premises : it stands or falls with them alone. 
 But it is sufficient to show the superiority of Maxwell's 
 system ; for it is predicted in the latter, whereas it is un- 
 known in the opposing system. 
 
XVIII 
 
 ON THE DIMENSIONS OF MAGNETIC POLE IN 
 DIFFEKENT SYSTEMS OF UNITS 
 
 (Wiedemann's Annalen, 24, pp. 114-118, 1885.) 
 
 Two years ago this subject was discussed in the Annalen, 1 and 
 even more vigorously in the Philosophical Magazine. In 
 general we may now regard the matter as settled ; but I think 
 there is still one point which admits of a more complete ex- 
 planation, and as the question is one of principle it is worthy 
 of attention. Between the electromagnetic 2 and electrostatic 
 systems of units there appeared to be a certain discrepancy. 
 In the former there was agreement as to the starting-point, 
 viz. the magnetic pole, and also as to the electrical pole whose 
 dimensions were deduced ; whereas in the latter a difference of 
 opinion was possible, not indeed as to the starting-point, viz. 
 the electrical pole, but as to the magnetic pole deduced from 
 it. Side by side with Maxwell's electrostatic system there 
 appeared that of Clausius. Now even if it is shown that 
 neither of these is necessarily incorrect — that the only question 
 is, which of the two is preferable — there may yet remain in 
 the minds of many physicists a feeling that both of them, and 
 therefore the electrostatic system generally, are inferior to the 
 magnetic system, respecting which no doubt has been raised ; 
 
 1 Cf. Clausius, Wiecl. Ann. 16, p. 529, 1882, and 17, p. 713, 1882 ; v. 
 Helmholtz, Wied. Ann. 17, p. 42 ; also a series of papers in the Phil. Mag. 
 (Ser. 5) 13, and 14, 1882. [Phil. Mag. 13, pp. 376, 427, 429, 431, 530 ; 14, 124, 
 225, 357. Translations of Clausius' two papers will be found in the same 
 journal, 13, pp. 381-398, and 15, pp. 79-83 ; and of Helmholtz's paper in 14, pp. 
 430-440.— Tr.] 
 
 2 Called by Clausius electrodynamic. [In this volume, as in the Electric 
 Waves, " elektrodynamisch" is generally rendered as "electromagnetic." — Tr.] 
 
292 DIMENSIONS OF MAGNETIC POLE xvm 
 
 and that by using the latter one may avoid the pitfalls which 
 undoubtedly beset the former. I shall endeavour to show that 
 such a conception would be incorrect, by comparing with the 
 assumptions from which Maxwell and Clausius start two 
 others ; theoretically, although not practically, the latter are 
 as well established as the former, and by using them we can 
 make the magnetic and electrostatic systems change places. 
 If these new assumptions had originally been adopted instead 
 of the old ones, there would have been agreement as to the 
 electrostatic system, but discussion as to the magnetic system. 
 This shows clearly a posteriori (and the same can be proved a 
 priori) that neither of the two systems is in general preferable 
 to the other or more reliable than it ; only one of the two may 
 be preferable in a given department of electromagnetics, or 
 more reliable in its application to a given electromagnetic 
 calculation. In a sense it is simply a matter of chance that 
 the discussion arose in connection with the electrostatic and 
 not the magnetic system. I shall compare, as thesis and 
 antithesis, the old and the new assumptions, together with 
 the deductions from them. 
 The thesis then is : — 
 
 (a) The work which must be done in order to move a 
 magnetic pole m in a closed path once around a constant 
 electric current, which in the time t conveys the quantity e, is 
 proportional to the strength m of the pole and the strength 
 e/t of the current : it is independent of geometrical relations. 
 If then we put A = k-^me/t, Tc x is a constant whose magnitude 
 and dimensions depend only upon the system of units chosen. 
 Maxwell considers it best to connect electrical and magnetic 
 quantities in such a way that this constant becomes a number 
 of no dimensions. Thus, using the usual notation 
 
 MM^MLT- 1 (M). 
 
 (b) The moment mt> of a magnetic doublet, which for 
 purposes of calculation can be completely replaced by a small 
 circular current, is proportional to the strength e/t of the current 
 and the area / enclosed by it. Hence mS = k 2 ef/t } where again 
 k 2 is a constant which depends only upon the units chosen. 
 If &! is a pure number, h 2 in general will not be so ; and con- 
 
xvin DIMENSIONS OF MAGNETIC POLE 293 
 
 versely. Now Clausius holds that according to Ampere's theory 
 it is necessary to connect magnetic and electrical quantities in 
 such a way that k. 2 shall be a number of no dimensions ; from 
 which it follows that 
 
 N = H LT-' (C). 
 
 The consequences of the assumptions (M) and (C) are as 
 follows : — 
 
 1. In the magnetic system we start with the dimensions 
 [m] = M*L*T -1 . From this we deduce, as the dimensions of 
 electric pole, 
 
 W\J, according to (M) ; M-L-, according to (C). 
 
 Hence there is agreement. 
 
 2. In the electrostatic system we start with the dimensions 
 of electric pole [e] = M*L f T " \ From this we deduce, as the 
 dimensions of magnetic pole, 
 
 M*L* according to (M) ; M'L f T _2 , according to (C). 
 
 The two expressions are different, and this is the objection 
 urged against the electrostatic system. 
 
 In setting forth the antithesis I shall make use of the 
 expression " magnetic current." 1 A constant magnetic 
 current is represented by a wire -shaped ring -magnet which 
 gains or loses equal quantities of magnetism in equal times. 
 For sufficiently short periods we can produce such currents of 
 any desired strength, and for periods of any length if we make 
 them sufficiently weak. The electrical forces exerted by such 
 a current are known, and every system of electromagnetics 
 contains the following propositions, although they may be 
 differently expressed : — 
 
 (a) The work which must be done in order to move an 
 electric pole e in a closed path once around a constant magnetic 
 current, which in the time t conveys the quantity m, is pro- 
 portional to the strength e of the pole and the strength m/t 
 of the current ; it is independent of geometrical relations. If 
 then we put A = kie.m/t, what has been stated for k x and Je 2 
 
 1 See XVII. p. 276. 
 
294 DIMENSIONS OF MAGNETIC POLE xvm 
 
 will hold good for &/. Now we may regard it as advantageous, 
 say with respect to the theory of unipolar induction, to take 
 this very equation as the fundamental equation for the con- 
 nection and to make h[ a pure number. We thus arrive at 
 the assumption : — 
 
 [m][e] = ML 2 J- 1 (MO, 
 
 which also coincides with (M). 
 
 (b) For purposes of calculation an electrical doublet 
 can be completely replaced by a small magnetic circular 
 current, whose plane is perpendicular to the axis of the 
 doublet. Thus the moment eS of the doublet must be pro- 
 portional to the strength m/t of the current and the area 
 which it embraces. Hence we put eS = k^fm/t. Theoretically 
 there would be nothing wrong — although from the standpoint 
 of present theories and applications it would be unpractical — 
 if we started from this equation and made k% a pure number. 
 We should then have : — 
 
 W-WLT- 1 (C). 
 
 The consequences of the assumptions (M') and (C r ) are as 
 follows : — 
 
 1. In the magnetic system we still have 
 
 [m] = M*L ? T- 1 . 
 
 Hence the dimensions of electric pole would be deduced as 
 
 M*L*, according to (M ; ) ; M*L f T' 2 , according to (C). 
 
 Thus there is now the inconvenience that in the magnetic 
 system different assumptions lead to different results. 
 
 2. In the electrostatic system we still have 
 
 M = M'=L*T- 1 . 
 
 Hence in this the dimensions of magnetic pole are 
 
 M*L*, according to (M') ; WU, according to (C'), 
 
 and the electrostatic system has the advantage previously 
 assigned to the magnetic system. 
 
xviii DIMENSIONS OF MAGNETIC POLE 295 
 
 The thesis and antithesis together show that, regarded 
 purely from the standpoint of calculation, neither system has 
 any advantage over the other. From the practical point of 
 view the forms based upon (M) and (JVJ 7 ) have the advantage of 
 being most easily remembered. If we regard magnetism 
 simply as a phenomenon of electricity in motion, the electro- 
 static system in the form (C) will appear preferable ; for 
 according to this view it alone renders the physical as well as 
 the mathematical connections. For my own part I always 
 feel safest from errors of calculation when I use, according to 
 v. Helmholtz's advice, 1 none of these apparently consistent 
 systems, but adhere to what he calls Gauss's system. This 
 defines the units of electricity and magnetism separately with 
 the same dimensions [e] = [m] — M-L-T -1 , and introduces factors 
 with the dimensions whenever electrical and magnetic quantities 
 occur together. 
 
 1 Wied. Ann. 17, p. 48, 1882, [Phil. Mag. (5) 14, p. 436, 1882.] 
 
XIX 
 
 A GKAPHICAL METHOD OF DETEBMINHSTG THE 
 ADIABATIC CHANGES OF MOIST AIE 
 
 {Meteorologische Zeitschrift, 1, pp. 421-431, 1884.) 
 (With diagram at end of book. ) 
 
 In the course of theoretical discussions meteorologists fre- 
 quently have to consider the changes of state which take place 
 in moist air when it is compressed or expanded without any 
 supply of heat. They wish to obtain solutions of such 
 problems as quickly as possible, and do not care to be referred 
 to complicated thermodynamic formulae. In practice they 
 generally refer to the small but useful table published by 
 Prof. Hann in 1874. 1 But it seems possible to attain greater 
 completeness, with at least equal facility, by using the graphical 
 method, and the accompanying table constitutes an attempt in 
 this direction. It contains nothing theoretically new except 
 in so far as it takes fully into account the peculiar behaviour of 
 moist air at 0° ; this, to the best of my knowledge, has not been 
 treated of before. 2 As the exact formulae of the problem do 
 not seem to have been collected, I shall state them completely 
 under A. Under B I shall explain how the formulae are 
 represented in the diagram. Under C I shall explain fully, 
 by means of a numerical example, the use of the diagram 
 (which may be purely mechanical). By following this example 
 
 1 Hann, Zeitschrift der osterreichischen Gescllschaft fur Meteorologie, 9, 
 p. 328, 1874. 
 
 2 The editor of the Met. Zeitschr. gives a reference to Guldberg and Mohn, 
 Etudes sur les mouvcmcnts cle V atmosphere, 1, pp. 9-16, and Osterr. Zeitschrift, 
 1878, pp. 117-122. See also the supplementary note on p. 311 of this volume. 
 
xix ADIABATIC CHANGES IN MOIST AIR 297 
 
 with the diagram in hand, we shall be able to judge of its 
 utility and to see how it is used, without needing to wade 
 through the calculations in A and B. 
 
 A. Suppose that 1 kilogramme of a mixture of air and 
 water-vapour contains X parts by weight of dry air, and /z 
 parts by weight of unsaturated water-vapour. Let the pressure 
 of the mixture be p, its absolute temperature T. The question 
 is : — What changes will the mixture undergo as the pressure 
 gradually diminishes to zero without heat being supplied ? 
 We must distinguish several stages. 
 
 Stage 1. The vapour is unsaturated, and no liquid water 
 is present. We assume that the unsaturated mixture obeys 
 the laws of Gay-Lussac and Boyle. If then e be the partial 
 pressure of the water-vapour, p — e that of the dry air, v the 
 volume of 1 kilogramme of the mixture, we have 
 
 ET P 4 T 
 
 p — e = \ , e = a , 
 
 v v 
 
 where E, E x are constants of known meaning and magnitude. 
 Since the total pressure is the sum of these two values, we 
 get pv = (\E + /^Ki)T, and this is the so-called characteristic 
 equation of the mixture. Further, let c v denote the specific 
 heat at constant volume of the air, cj that of water-vapour ; 
 then in order to produce the changes dv and dT, we must 
 supply the dry air with an amount of heat 
 
 dQ 1 = \U v dT + ART~ \ 
 
 and the water-vapour with 
 
 dQ 2 = JcJdT + AR^y 
 
 or both together with the amount of heat 
 
 dv 
 dQ = (\c v + /jLC v ')dT + A(XE + /*Ri)T— • 
 
 v 
 
 But this amount of heat vanishes for the change we are 
 considering. In order to integrate the differential equation 
 
 1 Clausius, Mechanische Wdrmetheorie, vol. i. p. 51, 1876. 
 
298 ADIABATIC CHANGES IN MOIST AIR xix 
 
 resulting from putting dQ = 0, we divide it by T. We know 
 a priori from the theory of heat that this operation renders 
 the equation integrable, and we find it confirmed a posteriori. 
 Performing the integration, eliminating v by the characteristic 
 equation, and remembering that c v + AR equals c p the specific 
 heat at constant pressure, 1 we obtain 
 
 - (Xe,+ /<) log £ - A(\E + pB£ log I (I). 
 
 The right-hand member of this equation has a physical mean- 
 ing; it is the difference of the entropy of the mixture in the 
 two states defined by the quantities p, T and p , T . Clearly 
 the mixture behaves like a gas, whose density and specific 
 heat have values which are the means of those of the water- 
 vapour and of the air. 
 
 We must now calculate the limiting value of p down to 
 which equation (I) may be used. Now and in what follows 
 let e denote the pressure of saturated aqueous vapour at the 
 temperature T. e is a function of T, but of T alone. Then 
 the quantity v of saturated aqueous vapour contained in the 
 volume v at temperature T, is 
 
 ve , . 
 
 v = (a). 
 
 and this quantity must exceed yu,, as long as the vapour remains 
 unsaturated. Thus the limit is reached as soon as //, = v. If 
 we introduce the value of v from the characteristic equation, 
 this condition takes the form 
 
 XE -f ^E x . 
 
 V = ip- ' e ( h )- 
 
 As soon as p and T reach values satisfying this condition, we 
 must leave equation 1, and enter on — 
 
 Stage 2. The air is saturated with aqueous vapour, and 
 contains liquid water in addition. We neglect the volume of 
 the latter. Then we may here also consider the air by itself, 
 and the water with its vapour by themselves, in each case as 
 if the other were absent. Both have the same volume v and 
 
 1 Clausius, Meclianische TFdrmetheorie, vol. i. p. 51, 1876. 
 
xix ADIABATIC CHANGES IN MOIST AIR 299 
 
 temperature T as the mixture ; but the pressure p of the 
 
 mixture is the sum of the partial pressures P 2 = X — of the 
 
 v 
 
 air, and p 2 = e of the water vapour. The equation 
 
 PT 
 
 p = X h e, or (p — e)v = XBT 
 
 v 
 
 is now the characteristic equation of the mixture. The amount 
 of heat necessary to produce the changes clT and dv is for the 
 air as before 
 
 iT-Y 
 
 but the amount of heat to be supplied to the water in order 
 to produce the change dT, and at the same time to increase 
 the amount v of water- vapour by dv, while pressure and volume 
 suffer corresponding changes, is 
 
 VT S 
 
 dQ 2 = Td[-)+f*cdT. 
 
 The equation is deduced by Clausius in his Meclianische 
 Warmetheorie, vol. i. part vi. § 11. c is the specific heat of 
 liquid water and r the external latent heat of steam, both being 
 measured in heat units. Hence the whole amount of heat to 
 be supplied is 
 
 dQ = \(c v dT + AKT— Ut^J+^T. 
 
 Here again we put dQ, = 0, divide by T, and integrate. From 
 the integral equation we eliminate v and v by means of the 
 characteristic equation and equation (a), and obtain 
 
 
 = (\c + ^c)log— + XAE log' 
 
 +X ^_1__^_A_) (ii). 
 
 The quantity equated to zero again represents the difference 
 
300 ADIABATIC CHANGES IN MOIST AIR xix 
 
 of entropy between the initial and final states. We may use 
 the equation obtained until the temperature falls to the freezing- 
 point ; then we pass to — 
 
 Stage 3, in which the air contains ice, as well as vapour 
 and liquid water. Now the temperature will not at once 
 fall any further with further expansion, for the latent heat 
 developed by the freezing water will, without any lowering of 
 temperature, yield the work necessary to overcome the external 
 pressure. But the latent heat will not be spent in this work 
 alone, but also in evaporating a portion of the water already 
 condensed. For since during the expansion the volume 
 increases without fall of temperature, at the end of the pro- 
 cess there will be more aqueous vapour than before, and the 
 weight of ice formed will be less than that of the liquid water 
 initially present. Let v again denote that portion of jju which 
 exists as vapour, o the portion existing as ice, and let q be 
 the latent heat of fusion of 1 kilogramme of ice. T, e, r are 
 constants. Since then clT = 0, we need only supply the air with 
 the heat \ARTdv/v, the water which is evaporated with the 
 heat rdv, and the water which is frozen with the heat — qdo. 
 Hence the whole of the mixture is supplied with the heat 
 
 dQ = X AKT h rdv - ado: 
 
 v 
 
 As before we put dQ = 0, divide by T, and integrate, and 
 thus get 
 
 = XAE log 1 + '-( v - „„) - |(<r - <r ). 
 
 
 
 The division by T was only necessary to make the right-hand 
 member a difference of entropy. By the aid of the character- 
 istic equation and of equation (a), we can eliminate v and v, 
 and introduce the pressure p. The equation then shows us 
 how the quantity o of ice formed varies with the change of 
 pressure. But the details of the process are of less interest 
 to us than the limits within which it takes place. Hence we 
 let the index refer to the state when the mixture just 
 reached the tempera'ture 0°, in which therefore ice was not 
 present, and where <r =0. The index 1 refers to the state 
 in which all the water is just frozen, in which therefore the 
 
xix ADIABATIC CHANGES IN MOIST AIR 301 
 
 temperature just begins to fall below 0°. Here clearly a = fi — v, 
 since ice and vapour alone are present. If we insert these 
 values, after introducing the pressure, we get 
 
 =XAElog^° + X— -^r ~ x ~ 
 
 p 1 -e K x p 1 - e T R x p — e T 
 
 -4 (III) - 
 
 This equation connects the pressures p and p lt at which re- 
 spectively the third stage is entered and quitted. 
 
 It is not necessary to furnish e and T with an index, for 
 they have the same values at the beginning and end of the 
 stage. 
 
 Stage 4. When the temperature falls still lower, we have 
 only steam and ice. The investigation is the same as for 
 Stage 2, and the final formula also is the same. But here the 
 latent heat of vaporisation has a different value. Here it is 
 r + q, for the heat necessary to evaporate the ice directly 
 must clearly be the same as that required to first melt it, and 
 then evaporate the water produced. If we wish to be quite 
 rigorous, we must not take q to be constant, but must suppose 
 it to vary slightly with the temperature ; but the differences 
 are so small that they may be neglected here. Thus in this 
 fourth stage we shall reach those temperatures at which the 
 air itself can no longer be regarded as a permanent gas. 
 
 The four stages which we have distinguished might very 
 properly be called the dry, the rain, the hail, and the snow 
 stages. 
 
 If now we are compelled to follow exactly the changes in 
 a mixture containing a considerable percentage of water, there 
 is no choice but to make use of these complicated formulae. 
 In that case we proceed as follows. We first substitute the 
 values of X and /x in all the equations. Then we substitute 
 the values p and T for the initial state in equation I. The 
 resulting equation and equation (b) we regard as two equations 
 to determine the unknown quantities p and T. Solving for 
 these we get the transition state between the first and the 
 second stage. The values obtained are then substituted for 
 p and T in equation II. By putting T = 273 in the resulting 
 
302 ADIABATIC CHANGES IN MOIST AIR xix 
 
 equation we get the p , which occurs in the equations of the 
 third stage. If now we determine from equation III the final 
 pressure p Y of the third stage, this pressure and the tempera- 
 ture 273° constitute the p and T in the equations of the 
 fourth stage. It will frequently happen that the temperature 
 down to which the first stage holds, lies below the freezing- 
 point ; then we pass on at once to the fourth stage, the second 
 and third disappearing. After we have in this way determined 
 the coefficients and limits of validity of all the equations, we 
 may employ them to determine for any desired jp the corre- 
 sponding T, and vice versd. These calculations can only be 
 performed by successive approximations,' and it will be advisable 
 to take the necessary approximate values from the table. When 
 we have determined p and T for any state, its remaining pro- 
 perties are easily deduced. The density of the mixture follows 
 from the corresponding characteristic equation. The equation 
 (a) gives the quantity of vapour, and thus also that of the 
 water condensed. We may often need to know the differ- 
 ence of height h, which corresponds to the different states 
 p and Pi, on the assumption that the whole atmosphere is in 
 the so-called adiabatic state of equilibrium. The exact solu- 
 tion of the problem is given by the laborious evaluation of the 
 integral 
 
 h = I vdjp, 
 
 i 
 
 n 
 but as in this particular respect an exact determination is 
 never of any special use, we may always use the convenient 
 diagram here given. 
 
 B. If we were here concerned with one mixture of one 
 determinate composition, i.e. with only one value of the ratio 
 /jl\ we could exactly represent the formulae deduced by a 
 diagram showing directly the adiabatic changes of the mixture, 
 starting from any state whatever. We could use pressure and 
 temperature as coordinates of a point in a plane, and could 
 cover the plane with a system of curves connecting all those 
 states which can pass adiabatically into each other. Then it 
 would only be necessary from a given initial state to follow 
 the curve passing through the corresponding point, in order 
 
xix ADIABATIC CHANGES IN MOIST AIR 303 
 
 to trace the behaviour of the mixture through all its stages. 
 But as meteorology has of necessity to deal with mixtures of 
 very various proportions, it would for this method require a 
 large number of diagrams. But we find it possible to manage 
 with only one diagram, if in the first place we confine ourselves 
 to cases in which the pressure and weight of the aqueous 
 vapour are small compared with those of air ; and if, secondly, 
 we expect no greater accuracy from the results than corresponds 
 to a neglect of the former quantities as compared with the 
 latter. For if we neglect //« in comparison with X, and e in 
 comparison with p, the form of the curves to be drawn is the 
 same for all the different absolute values of /j, ; so that the 
 same curves may be used for all the various mixtures. The 
 points where the various stages pass one into the other are 
 situated very differently for different mixtures, and hence 
 special means must be devised to determine these points. The 
 diagram is constructed in accordance with these principles. 
 
 In the net of coordinates the pressures are introduced as 
 abscissse through a range from 300 to 800 mm. of mercury, 
 and temperatures as ordinates from — 20° to +30° C. As 
 will be seen, a constant increase of the coordinate does not 
 represent a constant increase of pressure or of temperature ; 
 but the diagram is so drawn that equal increases of length 
 correspond to equal increases of the logarithms of the pressure 
 and of the absolute temperature. The advantage of this 
 arrangement is that the curves of special importance become 
 straight lines, in part exactly, in part approximately ; and 
 this is of considerable importance for the accurate construction 
 and employment of the diagram. 
 
 When //, is neglected in comparison with \, the adiabatics 
 of the first stage are given by the equation 
 
 const = c p log T — AB log p. 
 
 The logarithms are natural ones throughout. With Clausius 
 we must put 
 
 calorie 
 c =0-2375 
 
 degree C. x kilogr ' 
 
 1 calorie 
 
 4 2 3 - 5 5 kilogramme-metre ' 
 
304 ADIABATIC CHANGES IN MOIST AIR 
 
 kilogramme-metre 
 
 E= 29-27 
 
 degree C. x kilogr 
 
 These adiabaties are straight lines in our diagram. One of 
 them is marked by the letter (a), and we may denote the 
 system by this letter. The individual lines are so drawn that 
 the value of the constant, the entropy, increases from one line 
 to the next by 
 
 calorie 
 degree C. x kilogr 
 
 Thus they are equidistant. One of them is drawn through 
 the point 0° C. and 760 mm. pressure. 
 
 Now the curves of the second stage satisfy the equation 1 
 
 ~R r e 
 
 const = c p log T - AE log^> + — •-.- . 
 
 R/K-l is the density of aqueous vapour referred to air, i.e. 
 - 6219. r, according to Clausius, is equal to 607 — 0*708 
 
 (T — 273) • The values of e for the various tempera- 
 
 v ' kilogr 
 
 tures I have taken from the table calculated by Broch. 2 The 
 
 curves run from the right-hand top corner to the left-hand 
 
 bottom corner with a slight curvature. One of them is 
 
 marked /3. They also are drawn so that the entropy per 
 
 kilogramme increases from one curve to the next by 0'0025 
 
 calorie/degree C, and so that one of them passes through the 
 
 point 0°C, 760 mm. 
 
 The portions of the curves which correspond to the third 
 stage coincide with the isothermal 0° C. 
 
 Lastly, the curves of the fourth stage are very similar to 
 those of the second, bat yet are not quite the same ; for their 
 equation is got from that of the former curves by putting 
 r+q for r, where q=80 calorie/kilogr. They are marked y 
 
 1 Even though ft is neglected in comparison with X, yet it is doubtful 
 whether c/jl should be neglected compared with c p \ ; for c is four times as great 
 as c p . Though in the diagram /x is no more than ^V of X, still the ratio Cfi/c p \ 
 amounts to T V For meteorological applications we must, however, remember 
 that in these extreme cases the liquid water will not all be carried about with 
 the air. Frequently a large proportion will be deposited as rain, so that we 
 may be nearer the truth in neglecting the specific heat of the liquid water 
 altogether than in taking the whole quantity into account. 
 
 2 Travcaux du Bureau international des poids et mesurcs, tome i. 
 
Xix ADIABATIC CHANGES IN MOIST AIR 305 
 
 and are drawn according to the same scale as a and ft ; but 
 in general they are not continuations of the curves j3. 
 
 Means must now be provided for finding the points of 
 transition between the various stages. The dotted lines serve 
 to determine the end of the first stage. They give in grammes 
 the greatest amount of water which one kilogramme of the 
 mixture can just retain as vapour in the various states, 
 calculated by means of the formula v = He/T^T. Thus the 
 curve 25 connects together all those states in which 1 kilo- 
 gramme of the mixture is just saturated by 25 grammes of 
 steam. These curves are drawn for every gramme. If a 
 mixture contain n grammes of steam in every kilogramme, we 
 may follow the curve of the first stage up to the dotted line 
 n ; then we must change to the second, or the fourth stage, as 
 the case may be. 
 
 The boundary of the second and third stages is given by 
 the intersection of the corresponding adiabatic /3 with the 
 isothermal 0° C. The pressure p , corresponding to this inter- 
 section, and fi, the amount of water present, determine p x , the 
 pressure at which we must pass from the third to the fourth 
 stage. To determine p Y we must use the small supplementary 
 diagram, which is just below the larger one. It has for 
 abscissae the pressures arranged as in the large diagram, and 
 for ordinates the total quantity fi of water in all the sta"ges, 
 in grammes per kilogramme of the mixture. The inclined 
 lines of the diagram are merely the curves which correspond 
 to equation III of the third stage, when in this equation we 
 regard p as constant, but p x and [x as variable coordinates. 
 These lines are not quite straight, though on the scale of the 
 diagram they are not to be distinguished from straight lines. 
 The highest point of each line corresponds to the case p 1 = p . 
 The corresponding value of fi is not zero, but is equal to v, 
 the least value /jl must have in order that the mixture may be 
 saturated at 0° and that the supplementary diagram may be 
 required at all. When for given values of p and /x we require 
 the corresponding value of p 1} we must look out the inclined line 
 whose highest point has the abscissa p , and follow it down 
 to the ordinate yu. The pressure at which this ordinate is 
 reached is p lt the pressure sought. With it the point of 
 transition to the fourth stage is found. 
 M. p. X 
 
306 ADIABATIC CHANGES IN MOIST AIR xix 
 
 Now that in this way we have determined the whole of 
 the series of states which the mixture traverses, we may for 
 each individual state find as follows the remaining quantities 
 which interest us : — 
 
 1. The dotted line, on which we are, shows at once how 
 many grammes of water are present as vapour in the corre- 
 sponding state. If we subtract this from the total amount /a 
 of water present originally, we get the amount of water 
 already condensed. 
 
 2. The density 8 of the mixture with the approximations 
 introduced may for all states be calculated from the formula 
 3 = WRT or log 8 = logp - log T — log E. Graphically it 
 would be read off at once if the diagram were covered by a 
 system of lines of equal density. These lines are seen to be 
 a system of parallel straight lines. Only one of these lines, 
 8, is actually drawn on the diagram, so as not to overload it. 
 But we may by the help of this line alone compare the 
 densities in two states 1 and 2 according to the following 
 rule. From the points 1 and 2 draw two straight lines 
 parallel to 8 and cutting the isothermal for 0° C. ; and read 
 off the pressures p 1 and p 2 at these intersections. The densi- 
 ties at 1 and 2 are in the ratio P\'.p\. For the densities in 
 the states Pi, 0° and p 2 , 0° are by Boyle's law in the ratio 
 p x :p 2 , and they are equal to the densities in 1 and 2, as they 
 lie on lines of equal density. 
 
 3. The difference of height h, which corresponds to the 
 passage from the state p to the state p, on the assumption of 
 an adiabatic equilibrium state, is given by the equation 
 
 Po r 
 
 I vdp = E I 
 
 i dp 
 
 Here we would now find T as a function of p from the 
 diagram and then evaluate the integral mechanically. In 
 actual practice the supposition of an adiabatic equilibrium 
 will never be satisfied so nearly as to make an exact develop- 
 ment of its consequences of any importance. And again we 
 shall only commit a comparatively slight error for moderate 
 heights if we give to T a mean value and then regard it as 
 
xix ADIABATIC CHANGES IN MOIST AIR 307 
 
 constant. For within the limits of the diagram it varie 
 only from 253 to 303 ; so that if we give it the constant 
 value T = 273, the error in h will hardly ever exceed ^ of 
 its total value. If we choose to neglect this error, we get h 
 = const — ET log p, and may at once introduce the heights 
 as well as the pressures as abscissas. Everywhere indeed 
 equal increases of length of the abscissa will correspond to 
 equal increases of height. The scale of heights is marked at 
 the foot of the diagram; its zero is placed at the pressure 
 760 mm., because this is usually regarded as the normal 
 pressure at sea-level. 
 
 C. In order to explain the use of the diagram by an 
 example, let us consider the following concrete problem. We 
 are given at sea-level a quantity of air, whose pressure is 
 750 mm., temperature 27° C, and relative humidity 50 per 
 cent. Eequired to find what states this air passes through as 
 it is transferred without loss or gain of heat to higher strata 
 of the atmosphere and thus to lower pressures ; and also at 
 what heights approximately above sea-level the various states 
 are reached. 
 
 First, we look out on the diagram the point which cor- 
 responds to the initial state. It is the intersection of the 
 horizontal isothermal 27 and the vertical isobar 750. We 
 observe that it lies almost exactly on the dotted line 22. 
 This means that each kilogramme of our air would contain 
 22*0 grammes of water -vapour when saturated. But as its 
 relative humidity is only 50 per cent it contains only 11*0 
 grammes per kilogramme. This we note for further use. 
 Again we follow the isobar 750 down to the scale of heights 
 at the foot of the diagram and read off 100 metres. Thus 
 the zero of the scale of heights lies 100 m. beneath the sea- 
 level chosen by us as our starting-point ; and we must subtract 
 100 m. from all actual readings of the scale of heights in 
 order to get heights above sea-level. If now we raise up our 
 mass of air, the series of states traversed by it is first given by 
 the line of system a which passes through the initial state. 1 
 There is no such line actually drawn, so we must interpolate 
 
 1 The letters a, (3, y, which denote the systems, are given at the edge of the 
 diagram, enclosed by small circles. One line of the system which the letter 
 denotes is continued right up to it. The changes of state of our example are 
 marked by a special line of dots and dashes in the diagram. 
 
308 ADIABATIC CHANGES IN MOIST AIR xix 
 
 one. If the number of lines crossing each other appear con- 
 fusing, we may lay a strip of paper parallel to the system 
 of lines considered, when all confusion will be avoided. In 
 order to find out the state in the neighbourhood of the height 
 700 m., we seek out the point 700 + 100 = 800 in the scale 
 of heights, and go vertically upwards to meet our line a. The 
 point of intersection gives the pressure 687 mm., and the 
 temperature 19°*3. But we must only use the line a down 
 to the point at which it cuts the dotted line 11. For reach- 
 ing this line means that we reach a state in which the air can 
 only just retain 1 1 grammes of water per kilogramme in the 
 form of vapour. And as we have 1 1 grammes per kilogramme, 
 any further cooling produces condensation. The pressure for 
 the point of incipient condensation is 640 mm., the tempera- 
 ture 13°'3. This is not the temperature of the initial dew- 
 point, but is lower. The dotted line 11 cuts the isobar 750 
 at 15°"8, which is the initial dew-point. But since our air 
 has increased in volume in addition to having cooled, the 
 water has been able to keep itself in the state of vapour down 
 to 13 c, 3. The height at which we find ourselves corresponds 
 to the lower limit of cloud formation; it is about 1270 m. 
 To trace still further the changes of state we draw through 
 the point of intersection a curve of the system /3 and follow 
 its course. This curve is much less inclined to the axis of 
 abscissse than the line a previously used; so that now the 
 change of temperature with height is much less than before, 
 owing to the evolution of the latent heat of the steam. When 
 we have risen 1000 m. above the point at which condensation 
 commenced, the temperature has only fallen to 8 0, 2, i.e. only 
 o, 51 for every 100 m. We find ourselves on the dotted line 
 8 '9, and thus see that 8 "9 grammes of water still exist as 
 vapour, so that 2*1 grammes of water per kilogramme of air I 
 have been condensed in this first thousand metres of the cloud- 1 
 layer. We reach the temperature 0° at a pressure 472 mm. I 
 and at a height 3750 m., while we should have reached it at 
 a height of 2600 m. if the air had been dry and we had not 
 had to forsake the line a. 4*9 grammes of water, or 45 per 
 cent of the whole contents, are now found to have condensed ; 
 and this portion on further expansion begins to freeze to form 
 hail. But until the last trace of water has frozen, the tern- 
 
xix ADIABATIC CHANGES IN MOIST AIR 309 
 
 perature cannot fall any further, and thus we shall for a certain 
 distance keep at a constant temperature of 0°. To ascertain 
 how far, we use the small supplementary diagram between the 
 larger one and the scale of heights. We follow the isobar 
 472 down to the dotted line of this diagram. Through the 
 point of meeting we draw a line parallel to the sloping lines 
 of the diagram, and follow this line to its intersection with 
 the horizontal line characterised by the number 11, the total 
 weight of water. This latter line is easily interpolated between 
 the horizontal lines 10 and 15 drawn. As soon as we reach 
 this line we read off the pressure p = 463 mm. and return to 
 the large diagram. At the pressure thus found the process of 
 freezing is completed ; the layer within which it took place 
 has a thickness of about 150 m. It will appear strange that, 
 according to the dotted lines, the amount of water in form of 
 vapour has increased a little during the freezing. But this is 
 quite true, for the volume has increased, without any corre- 
 sponding fall in temperature. At the pressure 463 mm. we 
 leave the temperature 0°. The water which henceforth con- 
 denses passes directly into the solid state. As in a short time 
 little water is left in the form of vapour, the temperature 
 begins to fall more rapidly with increase of height. We find 
 the various states by following that one of the lines 7 which 
 passes through the point 463 on the isothermal 0°. The 
 temperature — 20°, up to which the diagram is available, is 
 reached at the height 7200 m., and at the pressure 305 mm.; 
 only 2 grammes of water per kilogramme remain as vapour, 
 the other 9 are condensed. If we wish to know how the 
 density in this state compares with the initial density, we 
 draw two lines through the corresponding points parallel to 
 the line 8. These meet the isothermal 0° at the pressures 
 330 and 680. The densities are in the ratio of these pressures, 
 i.e. as 33:68, and they are in the ratios 33 and 68 to 76 to 
 the density of air in the normal state at a pressure 760 mm. 
 and a temperature 0°. 
 
 All these results have been read off direct from the 
 diagram. Errors which could cause inconvenience are prob- 
 ably only to be found in the heights given. For these in 
 strictness relate to a rise through an atmosphere everywhere 
 at the same temperature 0°. But in most cases it may be 
 
310 ADIABATIC CHANGES IN MOIST AIR xix 
 
 assumed that the temperature of the atmosphere is everywhere 
 the same as that of the mass of air ascending through it. We 
 may considerably reduce the error due to this cause with a 
 very small amount of calculation. Thus we found the point 
 of incipient condensation to occur at a pressure 640 mm. This 
 corresponds to the height 1270 m. only when the temperature 
 is 0°. In our case it lay between 27° and 13°, so that the 
 mean was about 20°. At this temperature the height is 
 greater by -^Pg- or ^ than at 0°, since the density of the air 
 is less by the same fraction ; hence in reality the height lies 
 between 1350 and 1400 m. 
 
 We must now complete the example by mentioning some 
 special cases : — 
 
 1. We assumed that during the hail stage the whole of 
 the original quantity of water, 1 1 grammes, was still contained 
 in the air. Now this is only true when the ascent is very 
 rapid ; in other cases the greater part of the condensed water 
 will probably have been deposited as rain, and thus only a 
 fraction will become frozen. If we can form an estimate as 
 to the size of this fraction, the diagram still permits of a 
 determination of the correct amounts. If in our example we 
 had reason to suppose that one-half of the water condensed 
 down to 0° had been removed, then on reaching the isothermal 
 0° there would have been present only 8 '5 grammes of water 
 in each kilogramme of air. Then, in using the supplementary 
 diagram we would have gone not as far as the horizontal line 
 11, but only down to the line 8*5 ; and would have left the 
 temperature 0° at the pressure 466 mm. This would have 
 been the only alteration. 
 
 2. If we had in our example assumed only 10 per cent 
 relative humidity in place of 50 per cent, we could have used 
 the line a down to the dotted line 2'2. This point of inter- 
 section occurs at 455 mm. and — 13°*6, i.e. far below zero. We 
 should never have had any liquid water formed at all ; thus 
 no hail stage would have occurred, but merely a sublimation 
 of the water from the gaseous to the solid state. From the 
 point of intersection with the line 2 '2 we should at once have 
 followed the line of the system y passing through this point. 
 It is of some interest to inquire how high the dew-point of j 
 our mixture may be in its initial state of pressure and j 
 
xix ADIABATIC CHANGES IN MOIST AIR 311 
 
 temperature, so that the condensation of liquid water, i.e. 
 condensation above 0°, may just be avoided. To find the 
 answer we follow the line a as far as the isothermal 0° and 
 here meet with the dotted line 5*25. Thus we cannot have 
 more than 5 '2 5 grammes of water per kilogramme of air. 
 To find at what temperature the air would then be saturated 
 at the pressure 750 mm., we follow the line 5*25 up to the 
 isobar 750 mm. and meet it at the temperature 4 0, 8. This 
 is the required highest value of the dew-point. 
 
 [The following editorial note occurs at the end of the number of 
 the Meteorologische Zeitschrift in which this paper appeared.] 
 
 We had already begun to print off this number when a letter from 
 Dr. Hertz arrived, a part of which we take the liberty of printing. At 
 the same time we are glad to have the opportunity of publishing in our 
 journal the introductory part of his paper. It is all the more valuable 
 because its results agree satisfactorily with those of Guldberg and Mohn, 
 while the method by which they are obtained is to a certain extent 
 different and follows more closely the papers of Clausius, etc. The 
 papers by Guldberg and Mohn, to which we have drawn the attention of 
 Dr. Hertz, are not easily accessible, and the subject is of so much im- 
 portance in meteorology that an exposition of it in another journal is by 
 no means out of place. Dr. Hertz writes us : — 
 
 " My best thanks for the paper by Guldberg and Mohn 
 which you have so kindly sent me. Had I known of it 
 before I should have omitted the whole of part A of my 
 paper ; for, as a matter of fact, except in notation, it corre- 
 sponds exactly with the calculation of Guldberg and Mohn. 
 Yet in investigating with the aid of my diagram the example 
 calculated by Guldberg and Mohn I became rather alarmed. 
 Down to 0° things went all right, but on proceeding further 
 I found that, according to my diagram, the mixture reached 
 the temperature —20° at 320 mm. pressure, whereas Guld- 
 berg and Mohn with their formulae get 292*73 mm. 
 
 " An error of 2 8 mm. was too large, and I felt much afraid 
 that I had made some mistake in the construction. But it 
 appears that Guldberg and Mohn have made an error in 
 working out the numerical example, for I have repeatedly 
 made the calculation with their own formulae and constants, 
 and always find 313 mm. for the pressure in question. Thus 
 there is at most a difference of 7 mm. between the exact 
 
312 ADIABATIC CHANGES IN MOIST AIR xix 
 
 formulae and the readings of the diagram, and an error of this 
 magnitude is accounted for by the approximations which of 
 necessity have to be made. I believe that in ninety cases out 
 of a hundred such an error would be of no importance in 
 meteorology, and that it would be outweighed by the vastly 
 greater convenience. In fact I estimate that at least three or 
 four hours would be required for accurately calculating G-uld- 
 berg and Mohn's example, whereas it can be worked out on 
 the diagram in a few minutes. Besides, these 7 mm. are no 
 greater than the uncertainty introduced into the whole calcu- 
 lation by the fact that only a part of the condensed water is 
 carried along by the air. 
 
 " Is there still time for me to add a note of ten or fifteen 
 lines acknowledging the priority of Guldberg and Mohn, and 
 pointing out the cause of the above discrepancy ? I am 
 afraid others may compare the diagram with their example, 
 regard it as inaccurate to the extent of 28 mm., and hence 
 reject it. But to the priority you have yourselves re- 
 ferred. . . . 
 
 "H. Hertz." 
 
 Kiel, 8th Dec. 1884. 
 
XX 
 
 ON THE BELATIONS BETWEEN LIGHT AND 
 ELECTEICITY 
 
 A Lecture delivered at the Sixty-Second Meeting of 
 the German Association for the Advancement of 
 Natural Science and Medicine in Heidelberg on 
 September 20th, 1889. 
 
 {Published by Emil Strauss in Bonn.) 
 
 When one speaks of the relations between light and elec- 
 tricity, the lay mind at once thinks of the electric light. 
 With this the present lecture is not concerned. To the mind 
 of the physicist there occur a series of delicate mutual reactions 
 between the two agents, such as the rotation of the plane of 
 polarisation by the current or the alteration of the resistance 
 of a conductor by the action of light. In these, however, 
 light and electricity do not directly meet ; between the two 
 there comes an intermediate agent — ponderable matter. With 
 this group of phenomena again we shall not concern ourselves. 
 Between the two agents there are yet other relations — rela- 
 tions in a closer and stricter sense than those already men- 
 tioned. I am here to support the assertion that light of 
 every kind is itself an electrical phenomenon — the light of 
 the sun, the light of a candle, the light of a glow-worm. 
 Take away from the world electricity, and light disappears ; 
 remove from the world the luminiferous ether, and electric 
 and magnetic actions can no longer traverse space. This is 
 our assertion. It does not date from to-day or yesterday ; 
 already it has behind it a long history. In this history its 
 
314 LIGHT AND ELECTRICITY xx 
 
 foundations lie. Such researches as I have made upon this 
 subject form but a link in a long chain. And it is of the chain, 
 and not only of the single link, that I would speak to you. I 
 must confess that it is not easy to speak of these matters in a 
 way at once intelligible and accurate. It is in empty space, 
 in the free ether, that the processes which we have to 
 describe take place. They cannot be felt with the hand, 
 heard by the ear, or seen by the eye. They appeal to our 
 intuition and conception, scarcely to our senses. Hence we 
 shall try to make use, as far as possible, of the intuitions and 
 conceptions which we already possess. Let us, therefore, stop 
 to inquire what we do with certainty know about light and 
 electricity before we proceed to connect the one with the 
 other. 
 
 What, then, is light ? Since the time of Young and 
 Fresnel we know that it is a wave-motion. We know the 
 velocity of the waves, we know their length, we know that 
 they are transversal waves ; in short, we know completely 
 the geometrical relations of the motion. To the physicist 
 it is inconceivable that this view should be refuted ; we can 
 no longer entertain any doubt about the matter. It is 
 morally certain that the wave theory of light is true, and 
 the conclusions that necessarily follow from it are equally 
 certain. It is therefore certain that all space known to us 
 is not empty, but is filled with a substance, the ether, which 
 can be thrown into vibration. But whereas our knowledge 
 of the geometrical relations of the processes in this substance 
 is clear and definite, our conceptions of the physical nature 
 of these processes is vague, and the assumptions made as to 
 the properties of the substance itself are not altogether con- 
 sistent. At first, following the analogy of sound, waves of 
 light were freely regarded as elastic waves, and treated as 
 such. But elastic waves in fluids are only known in the 
 form of longitudinal waves. Transversal elastic waves in 
 fluids are unknown. They are not even possible ; they con- 
 tradict the nature of the fluid state. Hence men were 
 forced to assert that the ether which fills space behaves like 
 a solid body. But when they considered and tried to explain 
 the unhindered course of the stars in the heavens, they found 
 themselves forced to admit that the ether behaves like a 
 
xx LIGHT AND ELECTRICITY 315 
 
 perfect fluid. These two statements together land us in a 
 painful and unintelligible contradiction, which disfigures the 
 otherwise beautiful development of optics. Instead of trying 
 to conceal this defect let us turn to electricity ; in investigat- 
 ing it we may perhaps make some progress towards removing 
 the difficulty. 
 
 What, then, is electricity ? This is at once an important 
 and a difficult question. It interests the lay as well as the 
 scientific mind. Most people who ask it never doubt about 
 the existence of electricity. They expect a description of it 
 — an enumeration of the peculiarities and powers of this 
 wonderful thing. To the scientific mind the question rather 
 presents itself in the form — Is there such a thing as elec- 
 tricity ? Cannot electrical phenomena be traced back, like all 
 others, to the properties of the ether and of ponderable 
 matter ? We are far from being able to answer this question 
 definitely in the affirmative. In our conceptions the thing 
 conceived of as electricity plays a large part. The traditional 
 conceptions of electricities which attract and repel each other, 
 and which are endowed with actions-at-a-distance as with 
 spiritual properties — we are all familiar with these, and in a 
 way fond of them ; they hold undisputed sway as common 
 modes of expression at the present time. The period at 
 which these conceptions were formed was the period in which 
 Newton's law of gravitation won its most glorious successes, 
 and in which the idea of direct action-at-a-distance was 
 familiar. Electric and magnetic attractions followed the 
 same law as gravitational attraction ; no wonder men thought 
 the simple assumption of action-at-a-distance sufficient to 
 explain these phenomena, and to trace them back to their 
 ultimate intelligible cause. The aspect of matters changed in 
 the present century, when the reactions between electric 
 currents and magnets became known ; for these have an 
 infinite manifoldness, and in them motion and time play an 
 important part. It became necessary to increase the number 
 of actions-at-a-distance, and to improve their form. Thus 
 the conception gradually lost its simplicity and physical 
 probability. Men tried to regain this by seeking for more 
 comprehensive and simple laws — so-called elementary laws. 
 Of these the celebrated Weber's law is the most important 
 
316 LIGHT AND ELECTRICITY xx 
 
 example. Whatever we may think of its correctness, it is an 
 attempt which altogether formed a comprehensive system full 
 of scientific charm ; those who were once attracted into its 
 magic circle remained prisoners there. And if the path indi- 
 cated was a false one, warning could only come from an intellect 
 of great freshness — from a man who looked at phenomena 
 with an open mind and without preconceived opinions, who 
 started from what he saw, not from what he had heard, learned, 
 or read. Such a man was Faraday. Faraday, doubtless, 
 heard it said that when a body was electrified something 
 was introduced into it ; but he saw that the changes which 
 took place only made themselves felt outside and not inside. 
 Faraday was taught that forces simply acted across space ; 
 but he saw that an important part was played by the par- 
 ticular kind of matter filling the space across which the 
 forces were supposed to act. Faraday read that electricities 
 certainly existed, whereas there was much contention as to 
 the forces exercised by them ; but he saw that the effects 
 of these forces were clearly displayed, whereas he could per- 
 ceive nothing of the electricities themselves. And so he 
 formed a quite different, an opposite conception of the matter. 
 To him the electric and magnetic forces became the actually 
 present, tangible realities ; to him electricity and magnetism 
 were the things whose existence might be disputable. The 
 lines of force, as he called the forces independently considered, 
 stood before his intellectual eye in space as conditions of 
 space, as tensions, whirls, currents, whatever they might be — 
 that he was himself unable to state — but there they were, 
 acting upon each other, pushing and pulling bodies about, 
 spreading themselves about and carrying the action from 
 point to point. To the objection that complete rest is the 
 only condition possible in empty space he could answer — Is 
 space really empty ? Do not the phenomena of light compel 
 us to regard it as being filled with something ? Might not 
 the ether which transmits the waves of light also be capable 
 of transmitting the changes which we call electric and 
 magnetic force ? Might there not conceivably be some con- 
 nection between these changes and the light-waves. Might 
 not the latter be due to something like a quivering of the 
 lines of force ? 
 
XX LIGHT AND ELECTRICITY 317 
 
 Faraday had advanced as far as this in his ideas and 
 conjectures. He could not prove them, although he eagerly- 
 sought for proof. He delighted in investigating the connec- 
 tion between light, electricity, and magnetism. The beautiful 
 connection which he did discover was not the one which he 
 sought. So he tried again and again, and his search only 
 ended with his life. Among the questions which he raised 
 there was one which continually presented itself to him — Do 
 electric and magnetic forces require time for their propaga- 
 tion ? When we suddenly excite an electromagnet by a 
 current, is the effect perceived simultaneously at all distances ? 
 Or does it first affect magnets close at hand, then more 
 distant ones, and lastly, those which are quite far away ? 
 When we electrify and discharge a body in rapid succession, 
 does the force vary at all distances simultaneously ? Or do 
 the oscillations arrive later, the further we go from the body ? 
 In the latter case the oscillation would propagate itself as a 
 wave through space. Are there such waves ? To these 
 questions Faraday could get no answer. And yet the answer 
 is most closely connected with his own fundamental concep- 
 tions. If such waves of electric force exist, travelling freely 
 from their origin through space, they exhibit plainly to us 
 the independent existence of the forces which produce them. 
 There can be no better way of proving that these forces do 
 not act across space, but are propagated from point to point, 
 than by actually following their progress from instant to 
 instant. The questions asked are not unanswerable ; indeed 
 they can be attacked by very simple methods. If Faraday 
 had had the good fortune to hit upon these methods, his 
 views would forthwith have secured recognition. The con- 
 nection between light and electricity would at once have 
 become so clear that it could not have escaped notice even by 
 eyes less sharp-sighted than his own. 
 
 But a path so short and straight as this was not vouch- 
 safed to science. For a while experiments did not point to 
 any solution, nor did the current theory tend in the direction of 
 Faraday's conceptions. The assertion that electric forces could 
 exist independently of their electricities was in direct opposition 
 to the accepted electrical theories. Similarly the prevailing 
 theory of optics refused to accept the idea that waves of light 
 
318 LIGHT AND ELECTRICITY XX 
 
 could be other than elastic waves. Any attempt at a thorough 
 discussion of the one or the other of these assertions seemed 
 almost to be idle speculation. All the more must we admire 
 the happy genius of the man who could connect together these 
 apparently remote conjectures in such a way that they mutually 
 supported each other, and formed a theory of which every one 
 was at once bound to admit that it was at least plausible. 
 This was an Englishman — Maxwell. You know the paper 
 which he published in 1865 upon the electromagnetic theory 
 of light. It is impossible to study this wonderful theory with- 
 out feeling as if the mathematical equations had an independent 
 life and an intelligence of their own, as if they were wiser than 
 ourselves, indeed wiser than their discoverer, as if they gave 
 forth more than he had put into them. And this is not alto- 
 gether impossible : it may happen when the equations prove 
 to be more correct than their discoverer could with certainty 
 have known. It is true that such comprehensive and accurate 
 equations only reveal themselves to those who with keen in- 
 sight pick out every indication of the truth which is faintly 
 visible in nature. The clue which Maxwell followed is well 
 known to the initiated. It had attracted the attention of 
 other investigators : it had suggested to Eiemann and Lorenz 
 speculations of a similar nature, although not so fruitful in 
 results. Electricity in motion produces magnetic force, and 
 magnetism in motion produces electric force ; but both of 
 these effects are only perceptible at high velocities. Thus 
 velocities appear in the mutual relations between electricity 
 and magnetism, and the constant which governs these relations 
 and continually recurs in them is itself a velocity of exceed- 
 ing magnitude. This constant was determined in various 
 ways, first by Kohlrausch and Weber, by purely electrical 
 experiments, and proved to be identical, allowing for the 
 experimental errors incident to such a difficult measurement, 
 with another important velocity — the velocity of light. This 
 might be an accident, but a pupil of Faraday's could scarcely 
 regard it as such. To him it appeared as an indication that 
 the same ether must be the medium for the transmission of 
 both electric force and light. The two velocities which were 
 found to be nearly equal must really be identical. But in 
 that case the most important optical constants must occur in 
 
xx LIGHT AND ELECTRICITY 319 
 
 the electrical equations. This was the bond which Maxwell 
 set himself to strengthen. He developed the electrical equa- 
 tions to such an extent that they embraced all the known 
 phenomena, and in addition to these a class of phenomena 
 hitherto unknown — electric waves. These waves would be 
 transversal waves, which might have any wave-length, but 
 would always be propagated in the ether with the same 
 velocity — that of light. And now Maxwell was able to point 
 out that waves having just these geometrical properties do 
 actually occur in nature, although we are accustomed to denote 
 them, not as electrical phenomena, but by the special name 
 of light. If Maxwell's electrical theory was regarded as false, 
 there was no reason for accepting his views as to the nature 
 of light. And if light waves were held to be purely elastic 
 waves, his electrical theory lost its whole significance. But 
 if one approached the structure without any prejudices arising 
 from the views commonly held, one saw that its parts sup- 
 ported each other like the stones of an arch stretching across 
 an abyss of the unknown, and connecting two tracts of the 
 known. On account of the difficulty of the theory the number 
 of its disciples at first was necessarily small. But every one 
 who studied it thoroughly became an adherent, and forth- 
 with sought diligently to test its original assumptions and its 
 ultimate conclusions. Naturally the test of experiment could 
 for a long time be applied only to separate statements, to the 
 outworks of the theory. I have just compared Maxwell's 
 theory to an arch stretching across an abyss of unknown 
 things. If I may carry on the analogy further, I would say 
 that for a long time the only additional support that was 
 given to this arch was by way of strengthening its two abut- 
 ments. The arch was thus enabled to carry its own weight 
 safely ; but still its span was so great that we could not 
 venture to build up further upon it as upon a secure founda- 
 tion. For this purpose it was necessary to have special pillars 
 built up from the solid ground, and serving to support the 
 centre of the arch. One such pillar would consist in proving 
 that electrical or magnetic effects can be directly produced 
 by light. This pillar would support the optical side of the 
 structure directly and the electrical side indirectly. Another 
 pillar would consist in proving the existence of waves of 
 
320 LIGHT AND ELECTRICITY 
 
 electric or magnetic force capable of being propagated after 
 the manner of light waves. This pillar again would directly 
 support the electrical side, and indirectly the optical side. 
 In order to complete the structure symmetrically, both pillars 
 would have to be built ; but it would suffice to begin with 
 one of them. With the former we have not as yet been able 
 to make a start ; but fortunately, after a protracted search, a 
 safe point of support for the latter has been found. A suffi- 
 ciently extensive foundation has been laid down : a part of 
 the pillar has already been built up ; with the help of many 
 willing hands it will soon reach the height of the arch, and 
 so enable this to bear the weight of the further structure 
 which is to be erected upon it. At this stage I was so 
 fortunate as to be able to take part in the work. To this I 
 owe the honour of speaking to you to-day ; and you will 
 therefore pardon me if I now try to direct your attention 
 solely to this part of the structure. Lack of time compels 
 me, against my will, to pass by the researches made by many 
 other investigators ; so that I am not able to show you in 
 how many ways the path was prepared for my experiments, 
 and how near several investigators came to performing these 
 experiments themselves. 
 
 Was it then so difficult to prove that electric and magnetic 
 forces need time for their propagation ? Would it not have 
 been easy to charge a Leyden jar and to observe directly 
 whether the corresponding disturbance in a distant electro- 
 scope took place somewhat later ? Would it not have sufficed 
 to watch the behaviour of a magnetic needle while some 
 one at a distance suddenly excited an electromagnet ? As a 
 matter of fact these and similar experiments had already been 
 performed without indicating that any interval of time elapsed 
 between the cause and the effect. To an adherent of Max- 
 well's theory this is simply a necessary result of the enormous 
 velocity of propagation. We can only perceive the effect of 
 charging a Leyden jar or exciting a magnet at moderate dis- 
 tances, say up to ten metres. To traverse such a distance, 
 light, and therefore according to the theory electric force like- 
 wise, takes only the thirty-millionth part of a second. Such 
 a small fraction of time we cannot directly measure or even 
 perceive. It is still more unfortunate that there are no 
 
xx LIGHT AND ELECTRICITY 321 
 
 adequate means at our disposal for indicating with sufficient 
 sharpness the beginning and end of such a short interval. 
 If we wish to measure a length correctly to the tenth part 
 of a millimetre it would be absurd to indicate the beginning 
 of it with a broad chalk line. If we wish to measure a time 
 correctly to the thousandth part of a second it would be absurd 
 to denote its beginning by the stroke of a big clock. Now 
 the time of discharge of a Leyden jar is, according to our 
 ordinary ideas, inconceivably short. It would certainly be 
 that if it took about the thirty-thousandth part of a second. 
 And yet for our present purpose even that would be a thousand 
 times too long. Fortunately nature here provides us with a 
 more delicate method. It has long been known that the dis- 
 charge of a Leyden jar is not a continuous process, but that, 
 like the striking of a clock, it consists of a large number of 
 oscillations, of discharges in opposite senses which follow 
 each other, at exactly equal intervals. Electricity is able to 
 simulate the phenomena of elasticity. The period of a single 
 oscillation is much shorter than the total duration of the dis- 
 charge, and this suggests that we might use a single oscillation 
 as an indicator. But, unfortunately, the shortest oscillation 
 yet observed takes fully a millionth of a second. While such 
 an oscillation is actually in progress its effects spread out 
 over a distance of three hundred metres ; within the modest 
 dimensions of a room they would be perceived almost at the 
 instant the oscillation commenced. Thus no progress could 
 be made with the known methods ; some fresh knowledge was 
 required. This came in the form of the discovery that not 
 only the discharge of Leyden jars, but, under suitable con- 
 ditions, the discharge of every kind of conductor, gives rise to 
 oscillations. These oscillations may be much shorter than 
 those of the jars. When you discharge the conductor of an 
 electrical machine you excite oscillations whose period lies 
 between a hundred-millionth and a thousand-millionth of a 
 second. It is true that these oscillations do not follow each 
 other in a long continuous series ; they are few in number 
 and rapidly die out. It would suit our experiments much 
 better if this were not the case. But there is still the possi- 
 bility of success if we can only get two or three such sharply- 
 defined indications. So in the realm of acoustics, if we were 
 
 M. P. Y 
 
322 LIGHT AND ELECTRICITY XX 
 
 denied the continuous tones of pipes and strings, we could get 
 a poor kind of music by striking strips of wood. 
 
 We now have indicators for which the thirty-thousandth 
 part of a second is not too short. But these would be of little 
 use to us if we were not in a position to actually perceive 
 their action up to the distance under consideration, viz. about 
 ten metres. This can be done by very simple means. Just 
 at the spot where we wish to detect the force we place a con- 
 ductor, say a straight wire, which is interrupted in the middle 
 by a small spark-gap. The rapidly alternating force sets the 
 electricity of the conductor in motion, and gives rise to a 
 spark at the gap. The method had to be found by experience, 
 for no amount of thought could well have enabled one to 
 predict that it would work satisfactorily. For the sparks are 
 microscopically short, scarcely a hundredth of a millimetre 
 long ; they only last about a millionth of a second. It almost 
 seems absurd and impossible that they should be visible ; but 
 in a perfectly dark room they are visible to an eye which has 
 been well rested in the dark. Upon this thin thread hangs the 
 success of our undertaking. In beginning it we are met by a 
 number of questions. Under what conditions can we get the most 
 powerful oscillations ? These conditions we must carefully 
 investigate and make the best use of. What is the best form 
 we can give to the receiver ? We may choose straight wires 
 or circular wires, or conductors of other forms ; in each case 
 the choice will have some effect upon the phenomena. When 
 we have settled the form, what size shall we select ? We soon 
 find that this is a matter of some importance, that a given 
 conductor is not suitable for the investigation of all kinds of 
 oscillations, that there are relations between the two 
 which remind us of the phenomena of resonance in acoustics. 
 And lastly, are there not an endless number of positions in 
 which we can expose a given conductor to the oscillations ? 
 In some of these the sparks are strong, in others weaker, and 
 in others they entirely disappear. I might perhaps interest 
 you in the peculiar phenomena which here arise, but I dare 
 not take up your time with these, for they are details — 
 details when we are surveying the general results of 
 an investigation, but by no means unimportant details to 
 the investigator when he is engaged upon work of this kind. 
 
xx LIGHT AND ELECTRICITY 323 
 
 They are the peculiarities of the instruments with which he 
 has to work ; and the success of a workman depends upon 
 whether he properly understands his tools. The thorough 
 study of the implements, of the questions above referred to, 
 formed a very important part of the task to be accomplished 
 After this was done, the method of attacking the main prob- 
 lem became obvious. If you give a physicist a number of 
 tuning-forks and resonators and ask him to demonstrate to 
 you the propagation in time of sound-waves, he will find no 
 difficulty in doing so even within the narrow limits of a room. 
 He places a tuning-fork anywhere in the room, listens with 
 the resonator at various points around and observes the in- 
 tensity of the sound. He shows how at certain points this is 
 very small, and how this arises from the fact that at these 
 points every oscillation is annulled by another one which 
 started subsequently but travelled to the point along a shorter 
 path. When a shorter path requires less time than a longer 
 one, the propagation is a propagation in time. Thus the prob- 
 lem is solved. But the physicist now further shows us that 
 the positions of silence follow each other at regular and equal 
 distances : from this he determines the wave-length, and, if he 
 knows the time of vibration of the fork, he can deduce the 
 velocity of the wave. In exactly the same way we proceed 
 with our electric waves. In place of the tuning-fork we use 
 an oscillating conductor. In place of the resonator we use 
 our interrupted wire, which may also be called an electric 
 resonator. We observe that in certain places there are sparks 
 at the gap, in others none ; we see that the dead points follow 
 each other periodically in ordered succession. Thus the pro- 
 pagation in time is proved and the wave-length can be 
 measured. Next comes the question whether the waves thus 
 demonstrated are longitudinal or transverse. At a given 
 place we hold our wire in two different positions with refer- 
 ence to the wave : in one position it answers, in the other not. 
 This is enough — the question is settled : our waves are trans- 
 versal. Their velocity has now to be found. We multiply 
 the measured wave-length by the calculated period of oscilla- 
 tion and find a velocity which is about that of light. If 
 doubts are raised as to whether the calculation is trustworthy, 
 there is still another method open to us. In wires, as well as 
 
324 LIGHT AND ELECTRICITY xx 
 
 in air, the velocity of electric waves is enormously great, so 
 that we can make a direct comparison between the two. 
 Now the velocity of electric waves in wires has long since 
 been directly measured. This was an easier problem to solve, 
 because such waves can be followed for several kilometres. 
 Thus we obtain another measurement, purely experimental, of 
 our velocity, and if the result is only an approximate one it at 
 any rate does not contradict the first. 
 
 All these experiments in themselves are very simple, but 
 they lead to conclusions of the highest importance. They are 
 fatal to any and every theory which assumes that electric 
 force acts across space independently of time. They mark a 
 brilliant victory for Maxwell's theory. No longer does this 
 connect together natural phenomena far removed from each 
 other. Even those who used to feel that this conception as 
 to the nature of light had but a faint air of probability now 
 find a difficulty in resisting it. In this sense we have reached 
 our goal. But at this point we may perhaps be able to do 
 without the theory altogether. The scene of our experiments 
 was laid at the summit of the pass which, according to the 
 theory, connects the domain of optics with that of electricity. 
 It was natural to go a few steps further, and to attempt the 
 descent into the known region of optics. There may be some 
 advantage in putting theory aside. There are many lovers of 
 science who are curious as to the nature of light and are 
 interested in simple experiments, but to whom Maxwell's 
 theory is nevertheless a seven-sealed book. The economy of 
 science, too, requires of us that we should avoid roundabout 
 ways when a straight path is possible. If with the aid of our 
 electric waves we can directly exhibit the phenomena of light, 
 we shall need no theory as interpreter ; the experiments them- 
 selves will clearly demonstrate the relationship between the 
 two things. As a matter of fact such experiments can be 
 performed. We set up the conductor in which the oscillations 
 are excited in the focal line of a very large concave mirror. 
 The waves are thus kept together and proceed from the mirror 
 as a powerful parallel beam. We cannot indeed see this beam 
 directly, or feel it ; its effects are manifested in exciting sparks 
 in the conductors upon which it impinges. It only becomes 
 visible to our eyes when they are armed with our resonators. 
 
xx LIGHT AND ELECTRICITY 325 
 
 But in other respects it is really a beam of light. By rotat- 
 ing the mirror we can send it in various directions, and by 
 examining the path which it follows we can prove that it 
 travels in a straight line. If we place a conducting body in its 
 path, we find that the beam does not pass through — it throws 
 shadows. In doing this we do not extinguish the beam but 
 only throw it back : we can follow the reflected beam and 
 convince ourselves that the laws of its reflection are the same 
 as those of the reflection of light. We can also refract the 
 beam in the same way as light. In order to refract a beam of 
 light we send it through a prism, and it then suffers a deviation 
 from its straight path. In the present case we proceed in the 
 same way and obtain the same result ; excepting that the dimen- 
 sions of the waves and of the beam make it necessary for us 
 to use a very large prism. For this reason we make our 
 prism of a cheap material, such as pitch or asphalt. Lastly, 
 we can with our beam observe those phenomena which hitherto 
 have never been observed excepting with beams of light — the 
 phenomena of polarisation. By interposing a suitable wire 
 grating in the path of the beam we can extinguish or excite 
 the sparks in our resonator in accordance with just the same 
 laws as those which govern the brightening or darkening of 
 the field of view in a polarising apparatus when we interpose 
 a crystalline plate. 
 
 Thus far the experiments. In carrying them out we are 
 decidedly working in the region of optics. In planning the 
 experiments, in describing them, we no longer think electric- 
 ally, but optically. We no longer see currents flowing in the 
 conductors and electricities accumulating upon them : we only 
 see the waves in the air, see how they intersect and die out 
 and unite together, how they strengthen and weaken each 
 other. Starting with purely electrical phenomena we have 
 gone on step by step until we find ourselves in the region of 
 purely optical phenomena. We have crossed the summit of 
 the pass : our path is downwards and soon begins to get level 
 again. The connection between light and electricity, of which 
 there were hints and suspicions and even predictions in the 
 theory, is now established : it is accessible to the senses and 
 intelligible to the understanding. From the highest point to 
 which we have climbed, from the very summit of the pass, we 
 
326 LIGHT AXD ELECTRICITY xx 
 
 can better survey both regions. They are more extensive than 
 we had ever before thought. Optics is no longer restricted to 
 minute ether- waves a small fraction of a millimetre in length ; 
 its dominion is extended to waves which are measured in 
 decimetres, metres, and kilometres. And in spite of this ex- 
 tension it merely appears, when examined from this point of 
 view, as a small appendage to the great domain of electricity. 
 We see that this latter has become a mighty kingdom. We 
 perceive electricity in a thousand places where we had no 
 proof of its existence before. In every flame, in every 
 luminous particle we see an electrical process. Even if a 
 body is not luminous, provided it radiates heat, it is a centre 
 of electric disturbances. Thus the domain of electricity 
 extends over the whole of nature. It even affects ourselves 
 closely : we perceive that we actually possess an electrical 
 oman — the eve. These are the things that we see when we 
 look downwards from our high standpoint. Not less attractive 
 is the view when we look upwards towards the lofty peaks, 
 the highest pinnacles of science. We are at once confronted 
 with the question of direct actions-at-a-distance. Are there 
 such ? Of the many in which we once believed there now 
 remains but one — gravitation. Is this too a deception ? 
 The law according to which it acts makes us suspicious. In 
 another direction looms the question of the nature of electricity. 
 Viewed from this standpoint it is somewhat concealed behind 
 the more definite question of the nature of electric and mag- 
 netic forces in space. Directly connected with these is the 
 great problem of the nature and properties of the ether 
 which fills space, of its structure, of its rest or motion, of its 
 finite or infinite extent. More and more we feel that this is 
 the all-important problem, and that the solution of it will not 
 only reveal to us the nature of what used to be called im- 
 ponderables, but also the nature of matter itself and of its 
 most essential properties — weight and inertia. The quint- 
 essence of ancient systems of physical science is preserved for 
 us in the assertion that all things have been fashioned out of 
 fire and water. Just at present physics is more inclined to 
 ask whether all things have not been fashioned out of the 
 ether ? These are the ultimate problems of physical science, 
 the icy summits of its loftiest range. Shall we ever be per- 
 
XX LIGHT AND ELECTRICITY 327 
 
 mitted to set foot upon one of these summits ? Will it be 
 soon ? Or have we long to wait ? We know not : but we 
 have found a starting-point for further attempts which is a 
 stage higher than any used before. Here the path does not 
 end abruptly in a rocky wall ; the first steps that we can see 
 form a gentle ascent, and amongst the rocks there are tracks 
 leading upwards. There is no lack of eager and practised 
 explorers : how can we feel otherwise than hopeful of the 
 success of future attempts ? 
 
XXI 
 
 ON THE PASSAGE OF CATHODE EAYS THEOUGH 
 THIN METALLIC LAYEKS 
 
 (Wiedemann's Annalen, 45, pp. 28-32, 1892.) 
 
 One of the chief differences between light and cathode rays is 
 in respect of their power of passing through solid bodies. The 
 very substances which are most transparent to all kinds of 
 light offer, even in the thinnest layers which can be prepared, 
 an insuperable resistance to the passage of cathode rays. I have 
 been all the more surprised to find that metals, which are opaque 
 to light, are slightly transparent to cathode rays. Metallic 
 layers of moderate thickness are of course as opaque to cathode 
 rays as they are to light. But if a metallic layer is so thin 
 as to allow a part of the incident light to pass through, it will 
 also allow a part of the incident cathode rays to pass through ; 
 and the proportion transmitted appears to be larger in the 
 latter than in the former case. This can be demonstrated by 
 a very simple experiment. Take a plane glass plate capable 
 of phosphorescing, best a piece of uranium glass : partially 
 cover one side, which we shall call the front side, with 
 pure gold leaf, and in front of this fasten a piece of mica. 
 Expose this front side to cathode rays proceeding from a flat 
 circular aluminium cathode of 1 cm. diameter, say at a dis- 
 tance of 20 cm. from the cathode. So long as the ex- 
 haustion is but moderate the cathode rays fill the whole of 
 the discharge tube as a powerful cone of light, and the glass 
 only phosphoresces outside the patch which is covered with 
 gold. At this stage the phosphorescence is chiefly caused by 
 the light of the discharge, and only a very small part of this 
 
xxi PASSAGE OF CATHODE RAYS THROUGH METALS 329 
 
 penetrates through the gold. But as the exhaustion increases 
 there is less and less light inside the discharge tube, and the 
 rays which impinge upon the glass are more purely cathode 
 rays. The glass now begins to phosphoresce behind the layer 
 of gold leaf, and when the cathode rays have attained their 
 most powerful development, the gold leaf, when observed from 
 the back, simply looks like a faint veil upon the glass plate, 
 chiefly recognisable at its edges and by the slight wrinkles in 
 it. It can scarcely be said to throw a real shadow. On the 
 other hand the thin mica plate, which we have superposed on 
 the gold leaf, throws through this latter a marked black 
 shadow upon the glass. Thus the cathode rays seem to pene- 
 trate with but little loss through the layer of gold. I have 
 tested other metals in the same way with the same result — 
 silver leaf, aluminium leaf, various kinds of impure silver and 
 gold leaf (alloys of tin, zinc, and copper), silver chemically 
 precipitated, and also layers of silver, platinum and copper 
 precipitated by the discharge in vacuo. These latter layers 
 were much thinner than the beaten metallic leaves. I have 
 not observed any characteristic differences between the various 
 metals. Commercial aluminium leaf seems to work best. It 
 is almost completely opaque to light but very transparent to 
 the cathode rays : it is easily handled, and is not attacked by 
 the cathode rays, whereas a layer of silver leaf, for example, is 
 soon corroded by them in a peculiar manner. 
 
 It might be urged, against the assumption that the 
 cathode rays in these experiments penetrate right through the 
 mass of the metal, that such thin metallic layers are full of 
 small holes, and that the cathode rays might well reach the 
 glass through these without going through the metal. It is 
 the behaviour of the beaten metallic leaves that is most sur- 
 prising, and one is bound to admit that these contain many 
 pores : but the aggregate area of the holes scarcely amounts 
 to a few per cent of the area of the leaf, and is not sufficient 
 to account for the brilliant luminescence of the covered glass. 
 Furthermore, the covered part of the glass exhibits no lumin- 
 escence when it is viewed from the front, i.e. from the side on 
 which the cathode is. Hence the cathode rays must have 
 reached the glass by a way which the light excited by. them 
 cannot retrace ; so that they cannot have reached the glass 
 
330 PASSAGE OF CATHODE RAYS THROUGH METALS xxi 
 
 through openings in the metallic leaf which lies close against 
 it. Again, if we place two metallic leaves one on top of the 
 other, the number of coincident holes must become vanishingly 
 small. But the cathode rays are able to make glass luminesce 
 brightly under a double layer of metallic leaf; even under a 
 three or fourfold layer of gold or aluminium leaf we can per- 
 ceive the phosphorescence of the glass and the shadows of 
 objects in front of the leaf. I have been rather surprised by 
 the extent to which the rays are weakened by passing through 
 a double layer ; it is much larger than one would expect from 
 the slight weakening produced by a single layer. I think the 
 following sufficiently explains this phenomenon. The metallic 
 layer has a reflecting surface by which the phosphorescent 
 light is reflected. This reflecting surface prevents the light 
 from radiating towards the cathode, but it doubles the intensity 
 of the light in the direction away from the cathode. If we 
 assume that the metallic layer allows only -J- of the cathode 
 rays to pass, it will not reduce the luminescence to -J- but only 
 to |- of its previous value : whereas the second layer will reduce 
 it to ^, and further layers will soon cause the phosphor- 
 escence to vanish. If this conception is correct, metallic 
 layers capable of transmitting more than half of the cathode 
 rays should not weaken the luminescence at all : behind such 
 metallic layers the glass ought actually to phosphoresce more 
 strongly than in parts where it is not covered. I think I 
 have been able to verify this expectation in the case of layers 
 of silver chemically precipitated and of suitable thickness : but 
 the observation is not quite trustworthy, because in the un- 
 covered parts one cannot avoid seeing through the phosphor- 
 escing glass the greyish-blue luminescence of the gas, and it is 
 not easy to separate with any certainty the brightness of this 
 from that of the green phosphorescence light. 
 
 Lastly, if the cathode rays went right through the holes 
 in the metal they would afterwards continue their rectilinear 
 path. But this is just what they do not do ; by their passage 
 through the metal they become diffused, just as light does by 
 passing through a turbid medium such as milk glass. Let 
 part of a cylindrical discharge tube be shut off, say at a 
 distance of 20 cm. from the cathode, by a metal plate extend- 
 ing right across it but containing a circular aperture a few 
 
xxi PASSAGE OF CATHODE KAYS THROUGH METALS 331 
 
 millimetres in diameter ; let this aperture be closed by a piece 
 of aluminium leaf. If we now place a suitable glass plate 
 close behind the aperture we get, as might be expected, a 
 distinct and bright phosphorescent image of the aperture upon 
 the glass ; but if we remove the glass plate even one or two 
 millimetres, the image becomes perceptibly larger and suffers 
 a corresponding loss of brilliancy, its edge at the same time 
 becoming indistinct. When the glass plate is moved back 
 several millimetres the image of the aperture becomes very 
 indistinct, large and faint ; and when the plate is shifted still 
 further away, the tube behind the diaphragm appears quite 
 dark. That this is simply due to the feebleness of the cathode 
 rays which have been diffused from the small aperture can be 
 shown by introducing into the diaphragm several such 
 apertures closed by aluminium leaf. For this purpose the 
 [diaphragm is best made of wire gauze hammered flat ; upon 
 ■this is stretched a piece of aluminium leaf. Behind such a 
 [diaphragm the whole of the discharge tube becomes filled 
 with a uniform, moderately bright light. The phosphorescence 
 is sufficiently strong to allow of our obtaining separate beams 
 by means of further diaphragms : with these we can convince 
 ourselves that even after passing through metallic leaf the 
 sathode rays retain their properties of rectilinear propagation, 
 of being deflected by a magnet, etc. 
 
 There must be some connection between the phenomenon 
 of the diffusion of cathode rays on passing through thin layers 
 of bright metal and another phenomenon, namely, that 
 when cathode rays impinge upon such a surface the portion 
 reflected back is diffused, as E. Goldstein has shown. 1 
 
 1 See Wiedemann's Annalen, 15, p. 246, 1882. 
 
XXII 
 HEEMANN VON HELMHOLTZ 
 
 (From the Supplement to the Miinchener Allgemeinc Zeitung, August 31, 1891.) 
 
 In Germany the men who now stand upon the threshold of 
 old age have inaugurated and lived through a period of rare 
 felicity and success. They have seen aims attained and 
 desires realised, and this not only in matters political : they 
 have seen mighty developments in the arts of peace ; they 
 have seen our Fatherland take its place in the front rank of 
 nations, not only in our own estimation but in that of others. 
 Even in the beginning of this century the natural sciences 
 were far from being neglected in Germany : the labours of a 
 Humboldt, the undying fame of a Gauss, were sufficient to 
 keep alive respect for German research. But side by side 
 with the wheat of true effort there sprang up the tares of a 
 false philosophy which nourished so luxuriantly as to hinder 
 the full growth of the crop. Up to the middle of the century 
 sober progress along the path of experimental investigation 
 lacked the glory which accompanies international success ; and 
 the successes of a fictitious natural philosophy were very pro- 
 perly not greeted with the same exultation abroad as in Ger- 
 many. Germans followed eagerly and diligently the discoveries 
 made in other lands ; but they always expected the great dis- 
 coveries and successes to come from Paris and London. Thither 
 young investigators travelled to see famous scientific men and 
 to learn how great investigations were carried on : thence they 
 obtained the materials for their own researches ; there alone 
 could new discoveries be properly and authentically published. 
 They found it hard to believe that things could ever be other- 
 
xxii HERMANN VON HELMHOLTZ 333 
 
 wise. But all this has long since been changed. In science 
 Germany is no longer dependent upon her neighbours : in ex- 
 perimental investigation she is the peer of the foremost nations 
 and keeps in the main well abreast of them, sometimes leading 
 and sometimes following. This the country owes to the co- 
 operation of many eager workers ; but it naturally honours 
 most the few whose names are most closely connected with 
 the actual successes. Of these some have already left us for 
 ever : others still remain, and we hope long to have them 
 with us. 
 
 The greatest among all these, the acknowledged repre- 
 sentative of this period of progress and well-earned fame, the 
 scientific leader of Germany, is Hermann von Helmholtz, whose 
 seventieth birthday we celebrate to-day after he has for nearly 
 half a century astonished the scientific world by the number, 
 the depth, and the importance of his investigations. To the 
 countless tributes of admiration and gratitude which will this 
 day be laid at his feet we would with all modesty add our 
 own. As Germans we are glad and proud to claim as our 
 countryman one whose name we deem worthy to be placed 
 among the noblest names of all times and all nations, con- 
 fident that subsequent generations will confirm our judgment. 
 As men we cherish the same feelings of admiration and 
 gratitude. Other nations, too, will join us in paying honour 
 to him to-day, as indeed they have done in the past. For 
 although nations may appear narrow-minded in political 
 affairs, men have not wholly lost the sense of a common 
 interest in matters scientific : a Helmholtz is regarded as one 
 of the noblest ornaments of humanity. 
 
 Let us try to recall the achievements for which we to-day 
 do him honour. Here we at once feel how impossible it is to 
 make others share fully in our admiration if they are not 
 themselves in a position to appreciate his work. It is a 
 mistake to suppose that the importance of an investigator's 
 work can be gauged by stating what problem he has solved. 
 A man must see a picture, and must see it with the eyes of 
 an artist, before he can fully appreciate its value. Even so 
 scientific investigations have a beauty of their own which can 
 be enjoyed as well as understood ; but in order to enjoy it a 
 man must understand the investigation and steep himself in 
 
334 HERMANN VON HELMHOLTZ xxil 
 
 it. Take one of Helmholtz's minor researches, e.g. the theoreti- 
 cal paper in which he discusses the formation of liquid 
 jets. The problem is not one that appeals to the lay 
 mind : its solution is only attained by the aid of assumptions 
 which correspond but indifferently to the actual conditions ; 
 the influence of the investigation upon science and life can ! 
 scarcely be called other than slight. And yet the manner in 
 which the problem is solved is such that in studying even a 
 paper like this one feels the same elevation and wonder as in 
 beholding a pure work of art. Upon our comprehension of 
 the difficulties to be surmounted depends the depth of this 
 feeling. We see a man of surpassing strength spring across a 
 yawning chasm apparently without effort, but in reality strain- 
 ing every nerve. Only after the leap do we clearly see how 
 wide the chasm is. Instinctively we break out into applause. 
 But we cannot expect the same spontaneous enthusiasm of 
 spectators from whose standpoint the chasm is not visible, and 
 who can only learn from our descriptions how trying the feat 
 was. 
 
 To give a brief but fitting sketch of Helmholtz's work is 
 difficult on account of its many-sidedness as well as its pro- 
 fundity. His scientific life interests us like an Odyssey 
 through the whole region of exact investigation. He began 
 as a doctor : he had to study the laws of that life which he 
 wished to succour, and this led him to the study of physiology, 
 which is the scientific part of medicine. He found himself 
 hampered by the gaps in our knowledge of inanimate nature : 
 so he set about filling these and thus drifted more and more 
 towards physics. For the sake of physics he became a 
 mathematician, and in order to probe thoroughly the founda- 
 tions of mathematical knowledge, and knowledge in general, 
 he became a philosopher. When we look through the techni- 
 cal literature of any of these sciences we meet his name : upon 
 all of them he has left his mark. Without attending to 
 chronological order we shall here only describe briefly three of \ 
 those great achievements which constitute his title to fame. 
 
 I consider that the most beautiful and charming amongst 
 these, although not the highest, is the invention with which 
 he has enriched practical medicine. I mean the ophthal- 
 moscope. Before him no one was able to investigate the! 
 
xxn HERMANN VON HELMHOLTZ 335 
 
 living eye. Beyond the doubtful and unreliable feelings of 
 the patient there were no means of diagnosing the disease or 
 determining the defects in refraction. Before any cure was 
 possible it was absolutely necessary for the surgeon to acquire 
 an accurate knowledge of the disease ; and this, in the 
 majority of cases, was only attainable after the invention of 
 this simple instrument. Ophthalmic surgery rapidly rose to 
 its present high level. Who can say how many thousands 
 who have recovered their sight owe it to our investigator — to 
 him personally, although they are unconscious of this and 
 think that their thanks are simply due to the surgeon who 
 has treated them ! The invention of the ophthalmoscope is 
 like vaccination against smallpox, the antiseptic treatment of 
 wounds, or the sterilisation of children's food — one of those 
 great gifts which enrich all without impoverishing any, one of 
 I those advances which are gratefully acknowledged everywhere 
 by all men, and which keep alive in us the belief that there is 
 ;such a thing as progress. 
 
 Equally powerful as a protection against blindness on the 
 intellectual side are the advances which physiology owes to 
 Helmholtz, although their value may not be so easily or 
 generally recognised. Here we may remind the reader in 
 passing that he was the first to measure the speed with 
 which sensation and volition travel along the nerves : this 
 would have sufficed to establish the fame of any other man, 
 [but it is not this that we now have in mind. His chief 
 investigation in this science, the work of his mature years, is 
 the development of the physiology of the senses, especially of 
 sight and hearing. Within our consciousness we find an 
 inner intellectual world of conceptions and ideas : outside 
 our consciousness there lies the cold and alien world of 
 actual things. Between the two stretches the narrow border- 
 land of the senses. No communication between the two 
 worlds is possible excepting across this narrow strip. No 
 change in the external world can make itself felt by us unless 
 it acts upon a sense-organ and borrows form and colour from 
 this organ. In the external world we can conceive no causes 
 for our changing feelings until we have, however reluctantly, 
 assigned to it sensible attributes. For a proper understand- 
 ing of ourselves and of the world it is of the highest import- 
 
336 HERMANN VON HELMHOLTZ xxn 
 
 ance that this borderland should be thoroughly explored, so 
 that we may not make the mistake of referring anything 
 which belongs to it to one or the other of the worlds which 
 it separates. When Helmholtz turned his attention to this 
 borderland it was not in a wholly uncultivated state ; but he 
 found the richest fields in it lying fallow, and on either 
 side its limits were badly denned and hidden by a luxuriant 
 growth of error. He left it carefully denned and well 
 parcelled out, and much of it had been transformed into a 
 blooming garden. 
 
 His celebrated treatise On the Sensations of Tone is 
 known to a fairly wide circle of students. That which out- 
 side ourselves is a mere pulsing of the air becomes within our 
 minds a joyful harmony. What interests the physicist is the 
 air-pulsation, what interests the musician and the psychologist 
 is the harmony. The transition between the two is discovered 
 in the sensation which connects the definite physical process 
 with the definite mental process. What is there outside our- 
 selves which corresponds to the quality of the tones of musical 
 instruments, of human song, of vowels and consonants ? What 
 corresponds to consonance and dissonance ? Upon what does 
 the aesthetic opposition between the two depend ? By what 
 ideas of order within us were those codes of music, the musical 
 scales, developed ? Not all the questions which are prompted 
 by a thirst for knowledge can be answered; but nearly all 
 the questions which Helmholtz had to leave open thirty years 
 ago remain unanswered to the present day. In his Physiological 
 Optics he discusses similar questions relating to sight. How 
 is it possible for vibrations of the ether to be transformed by 
 means of our eyes into purely mental processes which 
 apparently can have nothing in common with the former ; and 
 whose relations nevertheless reflect with the greatest accuracy 
 the relations of external things ? In the formation of mental 
 conceptions what part is played by the eye itself, by the form 
 of the images which it produces, by the nature of its colour- 
 sensations, accommodation, motion of the eyes, by the fact 
 that we possess two eyes ? Is the manifold of these relations 
 sufficient to portray all conceivable manifolds of the external 
 world, to justify all manifolds of the internal world ? 
 
 We see how closely these investigations are connected 
 
xxn HERMANN VON HELMHOLTZ 337 
 
 with the possibility and the legitimacy of all natural know- 
 ledge. The heavens and the earth doubtless exist apart from 
 ourselves, but for us they only exist in so far as we perceive 
 them. Part of what we perceive therefore appertains to our- 
 selves : part only has its origin in the properties of the heavens 
 - and the earth. How are we to separate the two ? Helm- 
 holtz's physiological investigations have cleared the ground for 
 the answering of this question : they have supplied a firm 
 fulcrum to which a lever can be applied. His own inclina- 
 tions have led him to discuss these very questions in a series 
 of philosophical papers, and no more competent judge could 
 express an opinion upon them. Will his philosophical views 
 continue to be esteemed as a possession for all time ? We 
 should not forget that we have here passed beyond the 
 bounds of the exact sciences : no appeal to nature is possible, 
 and we have nothing but opinion against opinion and view 
 against view. 
 
 As on the one hand Helmholtz was led by the study of 
 the senses to the ultimate sources of knowledge, so on the 
 other hand the same study led him to the glories of art. 
 The rules which the painter and the musician instinctively 
 observe were for the first time recognised as necessary con- 
 sequences of our organisation, and were thereby transformed 
 into conscious laws of artistic creation. 
 
 Great and manifold as are these discoveries, they are all 
 eclipsed by another with which the name of Helmholtz will 
 ever be connected. This is a physical discovery of a more 
 abstract nature. Here the human observer with his sensations 
 retires into the background : light and colour fade away and 
 sound becomes fainter ; their place is taken by geometrical 
 intuitions and general ideas, time, space, matter, and motion. 
 Between these ideas relations have to be found, and these 
 relations must correspond to the relations between the things. 
 The value of these relations is measured by their generality. 
 As relations of the most general nature we may mention 
 the conservation of matter, the inertia of matter, the mutual 
 attraction of all matter. Of new relations discovered in this 
 century the most general is that which was first clearly 
 recognised by Helmholtz. It is the law which he called the 
 Principle of the Conservation of Force, but which is now 
 
 z 
 
338 HERMANN VON HELMHOLTZ xxn 
 
 known to us as the Principle of the Conservation of Energy. 
 It had long before been suspected that in the unending 
 succession of phenomena there was something else besides 
 matter which persisted, which could neither be created nor 
 destroyed, something immaterial and scarcely tangible. At 
 one time it seemed to be quantity of motion measured in this 
 way or that, at another time force, or again an expression 
 compounded of both. 
 
 In place of these obscure guesses Helmholtz brought 
 forward distinct ideas and fixed relations which led immedi- 
 ately to a wealth of general and special connections. Magni- 
 ficent were the views which the principle opened up into the 
 past and future of our planetary system ; in every separate 
 investigation, even the most restricted, its applications were 
 innumerable. For forty years it has been so much expounded 
 and extolled that no man of culture can be quite ignorant of 
 it. It is noteworthy that about this time other heads began 
 to think more clearly of these things ; and it came about that 
 as far as the phenomena of heat were concerned other men 
 had anticipated Helmholtz by a few years without his know- 
 ing it. It would be far from his wish to detract from the 
 fame of these men ; but it should not be forgotten that their 
 researches were almost entirely restricted to the nature of 
 heat, whereas the significance and value of the principle lie 
 precisely in the fact that it is not limited to this or that 
 natural force, but that it embraces all of them and can even 
 serve as our pole-star amongst unknown forces. 
 
 It is not generally known that in his mature years Helm- 
 holtz has returned to the work of his youth and has still 
 further developed it. The law of the conservation of energy, 
 general though it is, nevertheless appears to be only one half of 
 a still more comprehensive law. A stone projected into empty 
 space would persist in a state of uniform motion, and thus its 
 energy would remain constant : to this corresponds the con- 
 servation of energy in any system, however complicated that 
 system may be. But the stone would also tend to retain its 
 direction and to travel in a straight line : to this behaviour 
 there is a corresponding general behaviour on the part of 
 every moving system. In the case of purely mechanical 
 systems it has long been known that every system, according 
 
xxii HERMANN VON HELMHOLTZ 339 
 
 to the conditions in which it is placed, arrives at its goal 
 along the shortest path, in the shortest time, and with the 
 least effort. This phenomenon has been regarded as the resnlt 
 of a designed wisdom : its general statement in the region of 
 pnre mechanics is known as the Principle of least Action. To 
 trace the phenomenon in its application to all forces, through 
 the whole of nature, is the problem to' which Helmholtz has 
 devoted a part of the last decade. As yet the significance of 
 these researches is not thoroughly understood. An investigator 
 of this stamp treads a lonely path : years pass before even a 
 single disciple is able to follow in his steps. 
 
 It would be futile to try to enter into particulars of all 
 Helmholtz's researches. Our omissions might be divided 
 amongst several scientific men and would amply suffice to 
 make all of them famous. If one of them had carried out 
 Helmholtz's electrical researches and nothing else, we should 
 regard him as our chief authority on electricity. If another 
 had done nothing but discover the laws of vortex-motion in 
 fluids, he could boast of having made one of the most beautiful 
 discoveries in - mechanics. If a third had only produced the 
 speculations on the conceivable and the actual properties of 
 space, no one would deny that he possessed a talent for pro- 
 found mathematical thought. But we rejoice to find these 
 discoveries united in one man instead of divided amongst 
 several. The thought that one or other of them might be a 
 mere lucky find is rendered impossible by this very union : we 
 recognise them as proofs of an intellectual power far exceeding 
 our own, and are lost in admiration. 
 
 And yet these actual performances give but an inadequate 
 idea of his whole personality. How can we estimate the 
 intellectual value of the inspiration which he imparted, at first 
 to his contemporaries, and afterwards to the pupils who flocked 
 to him from far and near ? It is true that Helmholtz never 
 had the reputation of being a brilliant university teacher, as 
 far as this depends upon communicating elementary facts to the 
 beginners who usually fill the lecture-rooms. But it is quite 
 another matter when we come to consider his influence upon 
 trained students and his pre-eminent fitness for guiding them in 
 original research. Such guidance can only be given by one who 
 is himself a master in it, and its value is measured by his own 
 
340 HERMANN VON HELMHOLTZ xxn 
 
 work. Here example is of more value than precept ; a few 
 occasional hints can point out the path better than formal and 
 well-arranged lectures. The mere presence of the marvellous 
 investigator helps the pupil to form a just estimate of his own 
 efforts and of those of his fellow-students, and enables him to 
 see things sub specie ceterni instead of from his own narrow 
 point of view. Every one who has had the good fortune to work 
 even for a brief period under Helmholtz's guidance feels that 
 in this sense he is above all thiugs his pupil, and remembers 
 with gratitude the consideration, the patience, and the good- 
 will shown to him. Of "the many pupils now scattered over 
 the earth there is not one who will not to-day think of his 
 master with love as well as admiration, and with the hope 
 that he may yet see many years of useful work and of happy 
 leisure. 
 
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