Full text of "Miscellaneous papers"

See other formats

CORNELL

UNIVERSITY

LIBRARY

Cornell University Library
QC 3.H57 1896

Miscellaneous papers.

3 1924 012 500 306

Cornell University
Library

The original of tliis book is in
tine Cornell University Library.

There are no known copyright restrictions in
the United States on the use of the text.

http://www.archive.org/details/cu31924012500306

MISCELLANEOUS PAPERS

BV THE SAME AUTHOR.

ELECTRIC WAVES:

Besearclies on the Propagation of Electric Action

with Pinite Velocity through space.

Translated by D. E. JONES.

With a Preface hy LORD KELVIN.

8vo. 10s. net.

ELECTRICIAN :~'^ThQrQ 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."

m THE PRESS,

THE PRINCIPLES OP MECHANICS.

With a Preface by H. von Helmholtz. Translated by
D. E. Jones and J. T. Wallet.

^^^i^^-2,<^ /<?'?*-^

MISCELLANEOUS PAPEKS

HEINRICH HERTZ

LATE PEOFESSOR OF PHYSICS IN THE USTIVEESITY OP BONN

WITH AN" INTRODUCTION

PEOF. PHILIPP LENAKD

AUTHORISED ENGLISH TRANSLATION

D. E. JONES, B.Sc.

LATELY PROFESSOR OF PHYSICS IN THE UNIVERSITY COLLEGE OF WALES, ABERYSTWYTH

G-. A. SCHOTT, B.A., B.Sc.

DEMONSTRATOR AND ASSISTANT LECTURER IN THE UNIVERSITY COLLEGE OF WALES,
ABERYSTWYTH

ILontron
MACMILLAN AND CO., Ltd.

NEW YORK : MACMILLAN & CO.

1896

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 dif33.cult of
access, being scattered amongst various journals, and some
{e.g. his inaugural dissertation) could scarcely be obtained at
aU.. 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. LENAED.
February 1895.

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 Mectric 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

Introdxjotion . . . ix

1. Experiments to detekmine an Upper Limit to the Kinetic

Energy oe 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 fob the Kinetic Energy of Electricity in Motion

[ii], 1881 . . .137

5. On the Contact op 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 Merouey-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

r'AOB

16. On the Equilibiuum of Floating Elastic Plates, 1884 . 26(i

17. On the Relations between Maxwell's Fundamental Ki.eotjlo-

MAONETio Equations and tub Fundamental Equations ok

THE Opposing ELEOiiioMAdNETios, 1^84 . . ■ "''•i

18. On the Dimensions of Maqnetio Pole in Diffekent HYsiiiMs of

Units, 1885 . . . ... 291

19. A Geapiiioal Method of determining tne Adiadatii: Ciian(iicm

OF Moist Aik, 1884 ..... 296

20. On the Relations between Lkiht anii ELEOTuiiiri'y, 1889 313

21. On the Passage of Catiiodk Rays tiikougii Thin Mhiallk;

Layeiis, 1892 . . ... 328

22. Hermann von PIelmholtz, 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 liis 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 Novemher 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-

Vlll CONTENTS

PAGE

16. On the Equilibrium of Floating Elastic Plates, 1884 . 266

17. On the Relations between Maxwell's Fundamental Eleoteo-

MAGNETio Equations and the Fundamental Equations op

the Opposing Elbcteomagnetios, 1§84 . . . • 273

18. On the Dimensions of Magnetic Pole in Different Systems of

Units, 1885 . ...... 291

19. A Geaphioal Method of determining the Adiabatio Changes

of Moist Aie, 1884 . . . 296

20. On the Relations between Light and Electeicity, 1889 . 313

21. On the Passage of Cathode Rays theough Thin Metallic

Latees, 1892 ... .328

22. Heemann von 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
bad 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 Mmmier 1877.

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 Ihr nicht
das Leben ein, nie wird Eudi 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
iecided 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
)f twenty-one reports to his parents the starting of an in-

^^^ INTRODUCTION

vestigation which might well be taken for the work of an
experienced investigator.

Berlin, Slst October 1878.

I have been attending lectures— Kirohhofr's— 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 Hne, 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.

mh 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,i 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. 01

' A large room in liis parents' house.

INTEODUCTION XI

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 coxttses 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.

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-

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.^ I hope my work won't suffer from
this. At present it looks promising. I have already surmounted
the diflSculties 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.^

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.

12tt 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.

2lst 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-

^ The Head-Master of the Burrjersdiulc, 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 ivith 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.

31st 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 Ijecause 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 throiigh 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.

Kirchhofi' 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

i 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.

ith 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.

\lth 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 vein Helmholtz invited bim to
attack the problem ^ 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 Eotating 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 Princijiles of Mo:l,anics as haTiig
been proposed by himself in the belief that it was one in which his pupU wonM
feel an interest.

INTEODUCTION XVll

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 : —

21th November 1879. |

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.

IZth 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.

17th 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 spedynen 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

^■^"1 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 (XIY. 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 of&ce, 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.-XYI. in this volume.

He was first attracted by a theoretical investigation " On
the Contact of Elastic Solids" (T.) During the frequent
discussions on Xewton'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 hght
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

INTEODUCTION 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 : —

mh 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 jjroperty 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
was 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 INTEODUCTION

Both of these had heen 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 1851.

I aiQ 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.

10thMa,ch 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 haye
plucked up courage again ; I feel as fit as ever now, only 1 do
regret the valuable time which cannot be recovered.

13th June 1852.

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 Uttle instrument, " so simple that there '-&
scarcely anything in it," Hertz explains to his parents how
the air in a dwelling -room should be kept moist in winter.
There can he no harm in reproducing the explanation here

Ind Fcbrmry 155:'.

I may here give a httle calculation which wiU show father hoT
the air in the morning-room should be kept moist. On a"
average the atmosphere contains half as much water-vapour as i=
required to saturate it ; in other words, the averacje relatite
humidity is 50 per cent. Assume then that this proportion is
suitable for men, that it is the happy — or healthy — mean. In »'
cubic metre of air there shoidd then be definite quantities of wata;

INTEODUCTION XXI

which are different for diflferent 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 6^ 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|- gm., or nearly 1^
litres. Thus if the room were hermetically closed, 1^ 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 1 J 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 f to
^ 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
rarefied gases, in the so-called Geissler tubes — only the tubes I

^^11 INTEODUCTIOX

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
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 wiU 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 skiU 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.

nst Juit/ im.

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 lii-2.

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. Everv
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 7oO 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. lor »
while I feel quite fond of this monotonous mechanical occupation.

INTEODUCTION XXlll

20th September 1S82.

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, 2mh July 1883.

Geehrter Here 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
I 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

^XIV 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, 24tt 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 explanar
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 shaUow
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
efi'ect 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 ofi' 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.) Ji. 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

^^^^ INTKODUCTION

reap such a rich harvest. Xearly five years before he had
carried out his iavestigation " On Electric Eadiation " we find
in his day-book the notable remark — " 21 th January 1884.
Thought about electromagnetic rays/' and again, " Eeflected on
the electromagnetic theory of light." He was always fuU 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 hfe. 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 imexpected 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.

EXPEEIMENTS TO DETEEMINE AN UPPEE LIMIT
TO THE KINETIC ENEEGY OF AN ELECTEIC
CUEEENT.

[Wiedemann's Annalen, 10, pp. il4-448, 1880.)

ACCOEDING 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

%r = A — P— - ,
at

where r is the resistance and P the inductance of the circuit. ■"■
Multiplying by idt we get the equation

Aidt = ih'dt + ^dilH"^),

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
3ase we must add to the right-hand side of the second equation

^ [The notation has heen altered in accordance with English custom and
.he necessary changes in the equations made. The original has 2P. — Tk.]

- M.P. B

- KINETIC EXEEGY OF ELECTEICITY K MOTION [I] I

a tenn 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^, where m is a constant
depending on the form and size of the circuit. We thus get
in place of the above the following corrected equations

Aidt = i^rdt + lfZ(P z'2) + y/j,i -F),
-r^cli di.

Analogous conclusions apply to the case of a system of circuits
in which electromotive forces A^, A^ . . . act. When the
correction for inertia is introduced, the weU-known differential
equations which determine the currents take the form

'1 1 ^ " 1^ lit ^'-dt ^""dt

dt ^ -- -^'dt -"dt

Vr„ = A,-Pi,^- . -(P„,+ m„)%.

dt at

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 ob's'ious

1. That the electromotive force of the extra-currents ij
independent of the induction-currents simultaneously generated
in other conductors, and of the mass of the electricity moviag
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 inductiw
actions alone.^

1 ■ft'itli reference to these simple deductions the philosophical faculty of tb
Frederick- William University at Berlin in 1879 propounded to the students tb

I KINETIC ENERGY OF ELECTKICITY 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 A, units
of positive electricity, and let the mass of each unit be p
milligrammes. Let the length of the conductor be I, and its
cross-section, supposed uniform, q. Then unit length of the
conductor contains q\ electrostatic units, and the total positive
electricity in motion in the conductor has the mass pg\l 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 10®*, and is also equal to the velocity v
multiplied by q\. Thus

155,370 xlO«.

V = 1,

and the kinetic energy of the positive electricity contained
in the conductor is

1, , ri55,370x 10«1 V

¥p^-^[ ^ 1 ^

,U^ 155,370^x1012 , H^

= i — ■ P r = W — -

q X ^ q

;The quantity -^ii^/q can be expressed in finite measure. The
quantity p .155,o^0^ X 10/^% which has been denoted by //,,
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."
(t was pointed out that for such experiments extra-currents flowing in opposite
lirections 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 ELECTEICITY 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 /it, 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^l2, and also equal to {lfi/q)i^, it follows that
fi = qml2l. 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 + ?ri). 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

whence

arJ

= P + m

arJ' _

= P'

III =

-P'J

J' ■

It is well to choose one inductance P' 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' ,

—r- = P -f- m, -—- = P'

I KINETIC ENERGY OF ELECTRICITY IN MOTION [I]

m = — -V' — P, or if i' = i,
iJ

m = P -;

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 EXERGY OF ELECTEICITY m 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 successirt
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 pnt 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 aU 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, iaducing
currents. In aU 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
-^ 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-cm-rent from the complete arcs of
vibration thus obtained will now be explained.

Let T be the period of vibration of the galvanometer
needle, \ the logarithmic decrement constantly present, q = e"^
the ratio of any swing to the preceding one, and let, for
shortness,

T _2^tan-i^

Jtt^ + X^

\ = K .

8 KINETIC ENERGY OF ELECTKICITY IN MOTION [I] I

Further, let a^, a^, a^ be successive elongations right and
left of the position of rest, a^ = a^ + a^, a^ = a^ + %> 6*°., the
complete arcs of vibration, and \, \,. ■ ■ 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^ = K.\ + qa^,

ffig = k\ + qa^ = k\ + Kgk^ + q^a-^ ;

hence we get

a^ = K\ + a^{l + q),

a^ = k\ + Kk^O- + i) + (\q{}- + ?)•

If we multiply the first equation by q and subtract it from
the second, we find

and similarly

Hence we find the mean value of the impacts \, h^, . . .
which should all be equal if the apparatus worked quite exactly

2{n—l)

or, if we denote the sum of all the complete arcs of vibration
byS,

^^j^ _ (^ - ''^i) - g(S - an)
2{n-l)

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 t, the latter v, and the logarithmic decrement
during closing x', the magnitude of such an impact is

— 4 — TV .

If a^, a^ be the preceding and succeeding elongations the
needle reaches its position of rest with a velocity a^q/K and
leaves it with a velocity aja. As the increase of velocity is
nearly uniform, we must put for v the mean value {a^q + a^)/2K,
and thus the magnitude of the impact is

- ^7ir("'iS + «2) = - - («i^ + «2)-

i-K tC

By adding this increase of velocity to that caused by the
impact due to induction we obtain the equations

a^ = Jc^K + a^q - c(a^ + a^q),
^3 = \k + a^q - c{a^ + a^), etc.,
or

(1 + c)a^ = k\ + {1- c)a^q,
(1 + c)a^ = K^"^ + (1 - c)a^q, etc. ;
and by a similar calculation to that above

(1 + c)a^ - (1 - c)qa^ = K(k^ + \),
(1 + c)a^ - (1 - c)qa^ = k{\ + Tc^,

(1 + c)a„ - (1 - c)qa,,_^ = K(k^_^ + k^).

If instead of the quantities k^, k^, . . we write their
theoretical value k, we get after a simple transformation
the equations

a^ — qa^ + c{a^ + qa^ = 2kK,

ttg — qa.^ + c{a^ + qa^) = 2kK,

and from these the most probable values of the unknown
quantities k, etc. must be calculated by the method of least
squares.

KINETIC ENERGY OF ELECTKICITY 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

. (S - a,)-g(S-0 + c{(S-a,) + g(S-aJ}

kJc

2(m-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

a,„ - qa„ _ + c{a^ -f qa,^ )
K'c- ^ ^ .

we can get some notion as to how far the individual values
differ from their mean.

As in what foUows only the final results will be given,
I shall give here one series of multiplied deilections with the
individual impacts completely calculated, so that it may be
seen how far the observations agree amongst themselves.

EXTRA-CUEEENTS FEOM EeCTILINEAE WIEES (WIBES
TEAVEESED in THE SAME DIEECTION).

Steength of the Inducing Cueeent: 75"7.
g = 0-9830, c = 0-016.

Arcs of

Magnitude of Individual Impacts in

Headings
reduced to arc.

Vibration.

ii„ - 2a„_i

"■n + iO-n-^

Scale Divisions.

I. ein-qan-\ +c(an+8<i„_ i )

517-2

30-7

647-9

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

89-6

338-4

22-4

402-0 ^2*'^

222-3

39-0

405-6

22-7

371-2 '''■'

254-7

36-2

473-2

21-9

285-5

35-1

535-9

21-8

I KINETIC ENBEGY OF ELECTRICITY IN MOTION [I] 11

The mean value of kK = 22-0 5 ; the greatest difference
amounts to less than -^jj 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-cui'rent 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. Por different
patbs 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^ the resistance
of the galvanometer, and r^ 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^), which is
independent of r^

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 suf&cient 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,
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 '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
astaticaUy by twelve fibres of cocoon silk ; the time of swing
was 27' 6 6 seconds. The galvanometer was set up on an
isolated stone pQlar, 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 ELECTEICITY IN MOTION [I]

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

Fia. 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
-S, and the galvanometer
in passing over 0. 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 ELECTEICITY IN MOTION [I] j

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 ELECTEICITY 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' 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. Por 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. Tor 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 11 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} Further,
let «„ be the self -inductance of a single wire, a^ the mutual in-
ductance of two wires distant mq ; then we have

S=2s(logf-f), ..= 2s(log^^-l

By counting we find

n = 2wa„-2(27i-l>i-|-2(2M-2)a2- ... -'2cu,^_^,

and introducing the values of the as

XT .c fi , 2 1, 12''-i.3^»-3...(2ft-3)3.(2w-l)l
n = 4S.|i + log|-F-log o..-..4-.-....(2;-2y \

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

, l-;3-.5-... (2m -3f (271-1)
?iiog ^ — i

^ 2^4-... (2n- 2/

,, 2^ -4* 68 (2«-2f"-2

+ log - ^ ^

'1.3' 3'52' o^1^"\ln-3Y-\2n-lf-''

the two parts of which will be evaluated separately. The first
may be written

= ''^>^- jhf = «2.>g 1 - ~

one -Tr 1™"^^^ ^^^ number of double wires, positive and negative counting!

KINETIC ENERGY OF ELECTRICITY IN MOTION [I] 17

Since TZi<l for all the values concerned, we may expand

_1
4m'
log (1 — l/4m^), and thus obtain for the first part

C n-l 1 71-1 1

or, when we develop the sums by well-known formulee,

f . 1 1 1

= — Mi const. — 1 s

1 4:{n-iy8(n-lf 24(n-lf

The constant is clearly — log (2/77), for when n becomes
infinitely great the whole expression converges to m log (2/77).
If we develop the remaining terms in descending powers of n
and collect together like powers, we finally find the first part
to be

\7r/ * 8m 96m

An analogous calculation may be performed for the second
part, which is

11-1 2111^™ ^~^ I i\-™

71-1 1 71-1 1 71-1 1

-i:S — j-j--? --4- 1 ? — -1-

and hence, by a calculation similar to the above,

{0-577216+ log(M-l) + ^-^^, 4-...}

_ 1
4

jMl'

r» 1
C" 1

+ TTr2T

+ ...

M. P.

18 KINETIC ENERGY OE 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

1 7

0-18848 +^ log M- — -

8m. 1927i2

The sum of both parts gives the required term

*-/2V . 3

0-43848 + log -v/wf-j +j

VW 192%'

r *-/2\") 3

= log] 1-5503 Vn - I H

[ \7r/ J 192m^

C 41 2 1 3

= ™log|V5-7773..-| + jg2^-->

and hence to a considerable degree of approximation the seK-
inductance of a single layer becomes

I Ett

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 = 4Sn|i + log-^l.

^Ye 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 6 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]

log 2S — log e — 1

- 2 log 2S + 2 \ogJe-i + gi + 2

2S^ +21og2S-21og^e2 + 4^2_2

- log 2S + log ^£2 4. (2n +1)Y + 1
_ ,S , e(6H2V)(e^ + 4y)...(e^ + (2,0y)

= 2Slog-

2"- +

(271+1)2 +

e' + (27i)V

We get an approximate value in finite form of the expres-
sion behind the logarithm sign by dividing the equation

cos --

^2 \2

by this second equation,

sin. = .(l--^Jfl- ''

by putting v/J-l for z on both sides and dividing the result
by tj—l ; thus we get the equation

l+g-"

■■ limit

V
IT

o^+(-

(27l+l)2+(^

2n-^ + {'-

and thus the above inductance becomes approximately for a
large value of n

= 2Slog —,

1—e 1

or, if we neglect terms of the order e'^"''-''

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

If now one of them be displaced laterally through a dis-
tance q the inductance obviously becomes

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 8S?ie~^*
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 c[, e,
E, such as occur in ordinary cases, we easily convince ourselves
that the error considered amounts to less than the ^zo V^^^
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

P=2/|i + log^i^^|^^13.|

For the spirals used in our case

Z= 130032 mm., «.= 34, E= 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 P' 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 £ are rectilinear systems of wires of the dimensions
marked on the figure, and were stretched out on the floor of

KINETIC ENERGY OF ELECTEICITY IN MOTION [I]

iS)c-

Fio. 4.

the laboratory. In A (the secondary oircmt) 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 Q
was used, in which the
circuit A was as a rule
broken. As the action
of the rectilinear system
B on the spiral C was ^"^-
not zero, the latter was
inserted in the circuit
in both of the two pos-
sible ways. The values
of the inductive kick
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 :

In scale divisions of the galvanometer,

0-3997, 0-3955, 0-3791, 0-4006. Mean = 0-3939;

2. For the second position :

0-3034,0-3102. Mean = 0-3068.

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

X= 0-0172.

In order to compare the kick with experiments made with a
different damping we must multiply it by

s/7r^-FX\ ^tan-i^,

^ ^ TT A

which gives the value

1-1097

The remaining detached branch of the spiral C was now

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

696-0
T

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

Fig. 6.

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
the first one.

Execution of the Experi-
ments. — The arrangement of

the bridge used for measuring the extra-currents is shown in

I KINETIC ENERGY OF ELECTEICITY IN MOTION [I] 23

greater detail in Fig. 5. The current enters at A and A' ;
the galvanometer is connected at £ and £'. 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'I> and A£', 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 II. 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 T 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
in Scale Divisions.

Strength of Secondary Current
in Scale Divisions.

48-8

50-0

123-2

122-2

0-1790
0-1738
0-4621
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 A, = 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 X = 0-4396,
and the corresponding factor 3'876/T: hence the reduced kick
was

238-67
T ■

Til us 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 -1-10,696.

3. The self-inductance of the wire OP is -f 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 -f-5254. The
sum of all these corrections only amounts to -j- 9 5 7, 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 ELECTEICITY 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 l/lOO 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 Kectilineae Wires.

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 p= 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 [Ij 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 formulae 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

^=5-330.
V — 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]

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.

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,
0-1344, 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 ELECTEICITY IN MOTION [I] I

differs from the calculated ratio 5"330 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 oe Experiments with Rectilinear Wires.

A second series of observations with rectilinear wires was
made, which differed from the preceding one only in that the
G-erman-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 '441 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

5-^^=5-0367
V — p

I KINETIC ENERGY OF ELECTEICITY IN MOTION [I]

The experiments gave the following results : —

No.

Primary

Extra-Current.

Reduced to Strenath 100.

Current.

BraBches
Opposed.

Branches Like.

Branches
Opposec.

Branches Lilce.

11-3

0-0275

0-1225

2434

1-084

17-2

0-0492

0-1940

2860

1-127

18-8

0-0478

0-2082

2542

1-107

20-9

0-0582

0-2430

2784

1-162

24-1

0-0628

0-2775

2606

1-152

27-7

0-0700

0-3235

2527

1-167

33-3

0-0857

0-3792

2537

1-138

37-2

0-0957

0-4015

2572

1-080

47-7

0-1057

0-5243

2216

1-099

57-8

0-1330

2301

...

66-2

0-1478

0-7357

2234

1-112

72-7

0-1555

2139

88-7

0-2135

...

2135

108-6

0-2432

...

2240

128-7

0-2945

1-1425

2158

1-051

141-3

0-3005

1-1825

2128

1-049

172-6

0-3872

2276

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
I'lll, 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 ns 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 ISTo. 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 OP 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.

Eecapitulation of Ebsults and Inferences.

"We shall shortly recapitulate the results in order to deduce
from the experiments an upper limit to the quantity /it, the
meaning of which was explained in the introduction.

We start from the formulte

The quantities occurring here in the various experiments
had the following values : —

1. In the experiments with spirals we had

Z= 130,032, P' = 37,840,000,
q= 0-6793, J = 1-1467,
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<178,500(^^.f^-l

m<13,356, /t<0-0348mml

2. We obtain a smaller limit from the first series of
experiments with rectilinear wires. Here we had
P' p'

1 [The fraction g- 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 aelf-inductance. — Te. ]

I KINETIC ENERGY OF ELECTEICITY IN MOTION [I] 31

Z = 35,892, P= 179,960, J' = 0-7213,

^ = 0-1323, — = 5-330, J = 01348.

The calculated inductions may be regarded as exact, as their
error can hardly amount to l/lOO. The probable errors for
the quantities J and J' 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' too large, we

get

0-1 '^Pi'7
m< 179-960 (5-330. -1),

^ 0-7121 ^

m<4190, /i < 0-0077 mml

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

m< 185,240 (5-0367. - 1),

^ 1-085 ^

m<7042, /.(,< 0-0323 mml

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 eflect of mass.

Hence, using the first series of experiments with rectilinear
wires as being the best, we obtain this result : —

32 KINETIC ENEEGY 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

■ 8 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. ii v= 1 , 10 , ... 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 ELECTllICITY 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 yct into the
calculations which have been developed by Helmholtz in vol.
Ixxii. of Borchardt's Journal} It is there shown, on certain
definite assumptions there stated, including the truth of Weber's
law, that in a conducting sphere of radius R certain types
of currents, of given order a, become unstable when in our
notation ^

V 2

From this and from the limit found for ^, viz. yn = O'OOS 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-CUEEENTS IN IeON 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 metho'd 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.
^ Our quantity /i is expressed by /<./2A^ in the notation there used.
M. P. D

KINETIC ENERGY OF ELECTRICITY IN MOTION [l]

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 u-on
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.

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.

ON INDUCTION IN EOTATING SPHEEES

{Inaucjural Dissertation, Berlin, 15th March 1880.)

The interactions between magnets and rotating masses of
metals discovered by Arago were first explained by Paraday
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 sufS.ciently 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 EOTATING SPHERES u

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 hoUow 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 obtaiaed by means
of several drawings.

§ 1. Definition of the Symbols.

In this paragraph the symbols to be employed will be

defined and some formulfe will be collected together which

will be constantly required in the sequel.

Coordin- 1. The System of coordinates selected is that shown in

at«s- fig. 6. The positive directions of rotation are marked in the

figure. The axis of z wiU be taken to

coincide with the axis of rotation.

Polar coordinates p, w, 6 will be used :

a) is to correspond to geographical

-«/ longitude, is to be in the za;-plane

for positive x, and is to increase in

the direction of positive rotation; 6

is to correspond to the complement

Fig. 6. of the geographical latitude, and is to

be in the positive a-axis. We shall occasionally use the

notation

_9__ ^_^__^
dy dx Swj dco'

d d d

^ ^ --

dx dz dcoy

d d d

dz dy 9a)j,
Further, as in Lagrange's notation, differentiation with respect
to a will be denoted by a dash, e.g.

n INDUCTION IN ROTATING SPHERES 37

2. The calculations will be carried out in electromagnetic Definition

" "leelecl
I quau

units. In other respects the symbols for electrical magnitudes °(,*i';'^ '^''^'

will be those used by Helmholtz in vol. Ixxvii. of Borchardt's titie.s.
Journal} That is,

mgr=

u, V, w f

mm' sec

are the densities of the current parallel to x, y, z;

sec
are the corresponding components of the vector-potential ;

mm^mgr*

sec

is the potential of free electricity ; ^

sec

is the specific resistance of the material. The specific resist-
ance of a sheet of infinitesimal thickness S, viz. -, will, when

considered finite, be denoted by

, mm
sec
Further, let

mgr^

'^j /^> ^ 1

mm* sec

be the components of a magnetisation ; let

mgr^ mm^

L, M, N^

sec

be the potentials of A,, fj,, v, taken as masses,

6(0)

^ H. Helmlioltz, Wiss. AVhandl. vol. i. p. 545.

^ This is not electromagnetic measure. Measured in the latter units the
potential of free electricity is 0m=0A-, where 1/A denotes the velocity of light.
The above unit avoids the inconvenient factor 1/A^.

38 INDUCTION IN ROTATING SPHERES n

be the magnetic permeability, and

mm^ mgr^
'^ sec

the magnetic potential, so far as it is directly due to magnets ;
the magnetic potential of the currents is to be

_ mm^ mgr*
sec

O 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*

T^-

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 yjr increases by dijr, then df
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

mgr*

Y ; —

mm* sec

more generally as a function of p, 6, co such that

da . ir^p^a)

denotes the current-function of the layer between p = a ani

p = a + da.

For convenience the units have been given along with the

magnitudes quoted.
Dimen- 3. Let the external radius of the spherical shell considered

sphere con- be E, its internal radius r, the angular velocity of rotation 01.
Ridered. 4 "When a function ^, which throughout any given space

mmt iu satisfies the equation y\ = 0,^ is developed in a series of spher-

harmonios ^ '■'■" *^^ original A is used where we use v^- — Tb.]

II INDUCTION IN EOTATING SPHEEES 39

ical harmonics, then t^„ is to denote the term involving p" as a
factor, and this notation is, unless specially excepted, to apply
to n negative.

In the further analysis of ;^„ let this notation be used : —
for positive n

71 "\^

for negative n

Y„ = 2i(A„i cos ia + B^i sin ia})F„l6) :

for every n these equations hold : —

V'Xn=0,

The mth differential coefficients with regard to x, y, z of ;^„
are spherical harmonics of order n — m, unless a preceding
one should be of order zero. The expressions

hCn ^Xn 8%»

are spherical harmonics of order n.
Further

V'(P™YJ = (m - nXm + n+ 1>™-'Y,,
5. Let ^{r be the current-function of a spherical shell of Theorems

ft! on the flow

radius E, and let ^= y'— be the potential of a mass of ^^^^p^*"'"''^^

matter distributed over the shell with density -\|r ; then the
potential of the current sheet is

40 INDUCTION IN ROTATING SPHERES

and the

quantities U, V, W

are

^ U dz

z
~E

dy "

1 a^
EBo);

z a^

1 a^

E a*

a»

E ao,;

XKT * ^^

1 a^

w =

E 92/

Eao);

If ■^ is a homogeneous function of

x,y,z

then

av au

M+;

L a^

9« dy

a«'

3U 9W

71+1 a^

dz dx

ay'

aw av

w+1 a^

ay dz ■

dx'

And we

have always

az/

~&'

ay^

■ a^

a*'

These formulae are developed in Maxwell's Treatise on
Electricity, vol. ii. p. 276.^ 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.

6. The following are the expressions for the electromotive
forces which are produced by the components U, V, W of the
vector potential, supposed invariable, due to the element whose
component velocities are a, /8, 7 : —

^av

au\

/au

^ dx

^yf

A a. ■

dx ,

2nd ed.

p. 280.

II INDUCTION IN EOTATING SPHEEES 41

^ /aw 8V\ /BY dV\

^ fdv dwx jdw dY\

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 \, fi, v, then in this part of the induction in
the above formulae we must replace

„ ,^ ^,, , dM dN aN dL dL dM

U , V , W by , , .

dz dy' dx dz' oy 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

we have

V2L=0, V^M = 0, V2N = 0,

5% Jx

And for the elements of our sphere

a = — (cy, /3 = 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 : —

INDUCTION IN ROTATING SPHERES

«:M =

KV =

KW ==

further, since the currents are steady, we have inside

du dv dw

dx dy 9«
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

% = 2"^" •

We take each term by itself and thus put the external
potential = ^n-
Then we have

oz
^ Using the unit previously employed for 0.

INDUCTION IN EOTATING SPHERES

Hence these conditions follow for <f) :
In the matter of the shell

(a)
for p = r and p = E

(6)
A solution of these equations is

V> = 2»|--;

Deternima-
tion of the
electric
potential.

dp'

?Xn .

p\P' dz

nzxn

</>=

M+1

■ nzxn

For

vl/f -«%.

^%«. _ 9^^%"

= 2(2M+l)r^-2

(§1.0

so that the equation for the interior is satisfied. Further, (j>
is the product of /3''+^ by a function of the angles 6 and co,
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 (f> we get at once

Determiua-
tion of u, V,

M = —

-t

1 9/'2 9%«

M+1

d^V ^a

'^-nzx^)+x

W ■■

n+ldy
1 d

29%™

w + 1 3» \ 9«

'l«Xr

««X»

dz
oz

«%.

44 INDUCTION IN ROTATING SPHERES H

If we multiply these equations by x, y, z and add we get
Ma; + vy + ws = .

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 (&) is
satisfied throughout the mass and not merely at its surfaces.

Further we find

K y dxdz dxdz j
Vh = 0,

V^w = 0, since also v^%ji = 0-
In fact u, V, w are homogeneous functions of x, y, z of the
wth degree ; thus u, v, w are exhibited as spherical harmonics
of the mth order. We shall soon find simpler expressions
for v,, V, lu.
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 XJ, 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 k. For TJ, V, W the conditions hold

V^'U=0, V2V = 0, V'W=0
throughout space. At the surface of the shell

—-5 - — ^ = - 47rM ,
Op 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

* 27.+ 1 4 n+ldy\''^ ^y^^'dij'

' 2n+lk\ n+ldzV dz '^V "3^ ^^
[' U(, U,, denote respectively U internal, U external. — Tk.]

II INDUCTION IN KOTATING SPHERES 45

/T)\2«+l
/p\ 271+1

From these values of U, V, W we calculate the magnetising
forces inside, viz :

— - — etc
dx dy '

we put them

M + 1 3^ ^

= -^ — , etc.,

E 3^

and so obtain the function ^ (§ 1, 5). We find

2n+l Jc dg\ dy dx I E 9^

_ ^^^ ^ 1 ^^3^5 _ „^U _ ^+19^ir
2n+lk dx \ dy dx ) E dx '

2m + 1 /; 8?/ V 9?/ 9a' y E 9y '

whence

' (2ra+l)(w+ 1) h\ dy ^ dx

47rE'

ft)

7,% 1

{2n + 1) (n+ 1 A

and now all the remaining properties of the currents follow at
once from "9. An arbitrary constant may be added to ■^, 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 formulas.

Let

46 INDUCTION IN ROTATING SPHERES

be the inducing potential, then

i2n+l){n+l)k\R

' i2n+l)(n+l)k\pj

^ kn+1 ""

2n+lk\'Rj "

(2to+ l)(n+ 1) Z;\p
Again, from the relations

U = ^, ^ - — -' = - 4:Tru ,

and the corresponding ones for V and W, we find
1 df _ 1 CO 8Y'„

u =

E d(o^ w + 1 k a&jj.

_ 1 a^/r _ 1 a dY\,
E 3(Bj, n+1 k da)y

1 dir 1 CO dY\

E dco^ n+1 k dco^

Lastly, the expression for the electric potential in the
material of the spherical shell may be transformed. Write for
the moment p' = p sin 0, then

p sin -^
n+l'^ d0

and at the surface of the shell

- a>E . .dY^

<p = sm — ^ .

^ n+1 de

ir INDUCTION IN EOTATING 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

Xn~ \~ ) J- K )

then

(2h+1>/AE/ "'

^ = 47rR=^ oj/Ryfi ,

{2n+l)nk\p)

^ ^ 47rE(n+l) w/^Y'y,

2m+1 A'V/j

n k do},,
1 o) 3Y'

n h dcOy
n h du.

, '^ -r, ■ a "S^n

4> = — E sm p -^

Of the magnitudes here given, -i/r, u, v, w, (f) are got from

their preceding values at once by interchanging n with

~n~l.

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 positive or negative), the

induced currents are

INDUCTIOK IN EOTATING SPHERES

1 " 9%'u

u =

n+l K 3tUj,

m + 1 ' K ' 9(

'CO,,

W =

71+ 1 K do}^

and the current-function is

•ylr =

_ P

X «■■

Coustruo-
tion of the
lines of
flow.

n + l K

2. We analyse ^n further and consider the term

%»

^^~(e) '^°^^"^''

To this belongs the current-function

■^ni =

n+l

sin icoY„

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 7r/2i ; 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=l, 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 •yjr a. form which permits of summation
for all the spherical harmonics and makes the development of
the external potential in a series of them \mnecessary.

Let n be positive, then

Xn<^P =

n+l

%n ■

II INDUCTION IN EOTATINCt SPHERES 49

Secondly, let n be negative, then

jxJp = - ^-x.. ^

Hence, for a positive n,

and for a negative n,

K I 00}

KJ OCO
P

These expressions admit of summation at once, and we get Summation
the following second form for the solution :— hlmontt'

If ;!^i denote the part of the potential due to internal, and
•^^ the part due to external magnets, then

Similarly

cf>= - (i) sin 61— I jXe-dp- jxi- dp^j ■

For an infinitely thin spherical shell of radius E

R CO

^=-.srn4f^.,-fe4

Hence we get this relation between <f) and -\|r

dco dd

M. P. E

50 INDUCTION IN EOTATING SPHERES

§ 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 \E;

n J

''{2n+l)(n+l)k\pj

Calculation In the first place, if inside the spherical shell a second
of the sue- rotate infinitely close to the first and with the same velocity,

cessive in- *' ^

dnctions. the Currents of the first order (O,) will induce in it a current
system whose internal magnetic potential is

2n+lkJ \K

Secondly, if outside the first shell another rotate with the
same velocity and infinitely close to the first, the currents of
the first order (OJ will induce in it a current system whose
potential inside is

INDUCTION IN ROTATING SPHEEES 51

f 47rEw (o 47rB(w+ 1) to /pYy„

* ~ (2w+lXm+l) ^ ' (2w+l> k\Rj '"'

,2n+l k) \Rj

The two expressions for fi'^ 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

-Am-

E/ >fj V (2w+l)/v7 9^

D(U

o _ _ ." /R\"+'^ / 47rE6) \™9'"Y„

n+l\pj ^\ {2n+l)kl a&)"

47r(w + 1)4'"\ (2» + l)/t/ 9«"' '

The expressions obtained may be developed still further by
analysing Y^ further. We have

Y„ = '^IKi COS iw + B„i sin i(u)P„i .

We confine ourselves to one term only of this series.
Thus let

Y„ = A„i cos iaiP^i.
Then we have

o A (pXt> \ 4R&)i . . / 47rEa)t ■

^k = KA = P«d TT --smitB - — cos*£<)

E/ "Hcs^j + iyc \(2«+i)/t

47rEa)* \3 . . / 47rE6)i \* . , 1

' sini&) + cos %od+ .

\(2?H-l)/«y V (2m +1)^7 J

Put for shortness
47rE(B'i

(2?i+l)/«
then we find

= li (Ji is a pure number) ;

Oi = A„y I j P„j(sin ^(B - h cos i;(b).A.(1 -7i" + A* - /i^ + . . .).

52 INDUCTION IN ROTATING SPHERES n

If ^ is a proper fraction, the series involved in O^ con-
verges and we get

n , /R\"+i h , . .
n+1 \p/ 1+h''

2n+l h , . . , , ,^

If 7i > 1/ the series occurring in fij 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 formulae 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 S,
and now

fli = A J £ j sin 8 sin (ico - S)P„i .-

n, = - -^^aJ -) ■ sin S sin (ico - S)P„i ,
n+1 \p/

f=- /^ , -, . Ki sin fi sin (ia - S)P„i .
4='7r{n+ 1)

Hence the result is as follows : —

1. A simple spherical harmonic in the inducing potential
induces a current-function which is a surface harmonic of its
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 8/i in the direction of the rotation relative to the

^ 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 i = l, »=li ^
might be equal to 1. [? 62 revolutions per second. — Te.]

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 Trj2i. 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 a>jk = oo, 8 = 7r/2, and then The velo-

a =

■ %u

city is in-
finite

LK-

n+l^"'

f=.

2n+l -

47r(% + 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 H, 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 aUows 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 Theeiectnc

o -, ■, ■, ' potential.

expression tor the electric potential corresponding to ^^

A= -^^Esin^-^.
^ n+1 de

Taking self-induction into account, we shall have

4.= ^Bsing^>^-+^\

^ 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

INDUCTION IN ROTATING SPHEEES

symmetrical about the axis, <p 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.

Pianepiate. 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
iplates, 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 ^, rj, f,
whose connection with x, y, z is shown
\? in Fig. 7.

The direction of i; is the positive

direction of motion. We shall suppose

the inducing magnets inside the sphere,

i.e. on the side of 5" negative. We must examine what form

in %> '7. K is assumed by the spherical harmonic

In order to obtain finite variations we must make m and
i 00 of order E. We put

for n, wE,

for i, rE,

Fig. 7.

and further replace

0, e

^^^■IM-

INDUCTION IN EOTATING SPHERES 55

Thus

1 , COS ^(u

become

e~^, cos rrj.

P„i {6) must become such a function of ^ that its product
by e^^^cosr?; may satisfy the equation V^ = 0. Such a
function is cos s^ or sin s^, provided

n'' = r'' + s\

Hence the spherical harmonics formerly used now take the
form

A^sfi^^^cos rt) cos s^,
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

r, 27rr a

tan 6 = • -,

n k

where a denotes the velocity of the plate, and find

fl_^ = A„e~"'^ sin 8 sin (ri; - 8) cos s^,

Il_ = - A^s"^ sin S sin (r»; - S) cos sf, Solution.

y^r = -— A^s sin S sin (tt] - h) cos s^.

ZTT

By summation of all the terms we obtain the general
solution of the problem. The summation may be performed
in the case when ajk becomes infinite. Then we have

S =— , sin S= 1,
2

therefore

^+= -X'

56 INDUCTION IN ROTATING SPHERES li

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 2'n-ajk 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

XI = _ ^'^-^ f* 5X-u-i
' 2n+l k dco

We allow E to become infinite while we replace

0(0 Or)

n by mE ,
<bE by a ,

then

O = -1

k n dr]

n, = _^^ 1 ^

But we have

Hence summing for all values of n

+ k ]^r) ^

i
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 EOTATING SPHERES 57

CO 00

fl.(-f)= -fl+(?)

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 277 a Jk, as we may
show in the following way.

In the spherical shell the part of H^ corresponding to
%(-»-!> ina^y be represented in the form (p. 52)

k ■■

Second
form of the
solution.

{2n+l)lc

If we again introduce the substitutions to be made in the
case of a plane plate, develop

and put for h its value

2Tra T

k n '
we get

^ 2™ 1 3^ _ /27ra\V / W^ ''" ^X

k n drj \ k / n^ ' \ k / rr otj

Ic J m*

from which development the preceding one follows when we
use the relations

'I 7. ) ^4%ra '

^Vs_ _„2,

C7i;

and sum for all values of r and s.

dr^ '' ^""^ '

58 INDUCTION IN ROTATING SPHERES ii

In this connection it is natural to seek a development
in descending powers of ^iraJK for very large values of this
quantity.

When h>l, we have

hence

\2'ira) r* dr]
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 aU ^^5; but if we suppose p^ to be sym-
metrical with respect to the i^-axis, so that in its development
only terms in cos rr] appear, we have

9y„ 1 dX

»*-t' -pI^-^A'

and then the summation can be performed, at any rate for the
terms of the first order in «;/27ra.. If we confine ourselves to
these we get

Approxi- il^= -■^- \-^"'V>

mate solu- Z7ra J 0^

tion for

Jg^pcrg values

of the veio- and the very small resultant potential on the positive side is
city. ^

But in addition to the condition mentioned, this equation
is subject to another one.

However large 27ra/« may be, yet for certain elements for
which r vanishes, h<l, and the development employed wUl
be invalid. This circumstance restricts the validity of the
expression obtained to a limited region, which however is
larger the greater 27ra/«. On this point I refer to an investi-
gation to follow immediately (p. 61).

II INDUCTION IN EOTATING SPHERES , 59

We determine also the potential <h of the free electricity. Potential of

■j-Ti p frpp

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

The case is of interest when the velocity a becomes infinite.
If we assume ^ to be symmetrical with respect to the ly-axis,
and restrict our considerations to a limited region, we have
for a = CO

"+^=-s^/|''-

and hence

27rj d^

Thus <f) 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 '^'^°^-
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 KOTATING SPHERES II

We use p, &), z as co-ordinates, where p now denotes the
perpendicular distance from the axis of rotation. In the
general formulse we must replace

phj'R + z,

a remains a> ,

and after making this substitution we must allow E to become
infinite. Then a simple spherical harmonic takes the form

A^ie'^'costuJ/wp),

(and analogous ones), where J^ denotes the Bessel's function
of order i. The given t^ 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

5, 27r&) i

tan = ,

k n

then for the term in question the solution of the problem is

fi+ = A^jB ""^ sin h sin (ia - S) J^ (np) ,

Solution. n_ = - A„ie" sin S sin (ia - S) J^ {np) ,

i|r = — A„i sin S sin {ia - B) J^ (?ip) .

By summation we get the complete integrals.

We again attempt to obtain a development in powers of
21703 Ik by considering the successive inductions. By the
same method as above we get

CO CO

'--^1l--m 3'

Second XI _ ( - «) = - H _j. (s) ,

form of the

solution. 1 zr

ATT

n INDUCTION IN KOTATING SPHERES 61

But there is a limitation to the validity of these formulse, Remarks
which we had not to impose on the previous analogous ones, forn"^
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
condition is only satisfied for those terms for which '2Tro)i/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 71 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 2Tra)/k may always be chosen so small
that the approximation may be any desired one within a
given region. For by diminishing 27rw//c 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 27^&)/^' 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 27ri»/A ;
and, in the second place, we are only considering plates of
limited dimensions, and not infinite plates.

The equation

k j d(o

is exact, apart from self-induction. We see that, in order Possibility
that we may be allowed to neglect self-induction, it is neces- "nj^sfur*'
sary not only that 27ra>lk be small, but also that the investi- induction,
gation be limited to a certain finite region. The extent of
this region depends on 27ra)//i ; 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 § 4.

62 INDUCTION IN ROTATING SPHERES II

Approxi- A development can also be given for large values of 2wwlk

mation for ,„ jj.i j.ij. ^c ,.,. ''

largevaiues We denote by Xd that part of ■x^ which is symmetrical about
rftheveio-the axis of rotation, by Xi = X — %o ^^^ remainder. To '^(^^

corresponds for every velocity of rotation the value fl = 0.

Hence, assuming t^ to be symmetrical about the a;-axis, we get

for large values of 27rcB//i;

27r&)J oz

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 <^, the potential of the free
electricity. By the proper substitutions we get from the
general formulfe :—
Potential of 1- Neglecting self-induction,

the free

electricity. "

d> = aip\ -i^dz.
J dp

We must add to this value of ^ a constant, whose value
is such as to make ^ 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

When ft) is very large, we find, if ^ is symmetrical with
respect to the ic-axis.

^=a,p[f-dz+±[^JC^

J dp ^TTJ (

The first term increases indefinitely with a.

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 v! , v', w' induced by U, V, W are given by the
equations

dx \dx dyj

, d<p^ fdY au

KV = - -H + eoy[ — - —
dy \ dx dy

Differential
equations.

, 9(4 /sw 8V\ /au aw

Kiv = - -i - axi — — - -— — win — - —
a« \dy dzj ^\de dx

Further, inside we have

aw/ a?/ aw/^Q

dx dy a«

and at the surface

u'x + v'y + w'z = .
We write for shortness

\dy dzJ -"Xdz dx J \dx dy

64: INDUCTION IN ROTATING SPHERES

If we remember that

— — + r — = ^

dx dy dz

we get for ^ these conditions : —
In the material of the shell

and at the boundaries

dp p\ \dx dy J J

Theorem "VVe shall first demonstrate the following theorem : —

forms the If U"> ^- W have the forms

basis of

what

follows. U = /J"'l y

dz dy J

V = pVz^^ - x-^

^ dx dz
W = p^lx^-^-y^-^Y

which forms satisfy the equation

au ay aw^Q

dx dy dz
then the solutions of the preceding differential equations are

„m/ „2a%»

= 03p^+'' sin e 1^ ,

u'^-^-p-(y%-^-z^-f),
K \ oz dy J

v'= -'^p4z%^-x^i^
K \ Ox dz

II INDUCTION IN EOTATING SPHEBES 65

K \ ^y Oi.:

To verify this we first express the conditions for ^ in Proof,
terms of ;^„. We have (§ 1, 4)

V^U = m(?ft+ 2m+ l)p™-2( y^^ - « ^

V Oz dy

V-V = m(m -\-2n+ l)p'"-1 2 -^-x ^^-™ ) ,

\ dy dx

yVJJ - xV^Y = m(m + 2n+ l)p'^-'^(p' '%^ - nzx,.

xV'W - zW = m(m +2n+ l)p"'-4 p^^~ mjx„
And again —

Hence we get

The conditions for ^ become

V> = - com{m + 2n + 3)(p^ ^ - nzx\'^-^

-2cc(n+iy

M. P.

66 IXDUCTIOX IX ROTATING SPHERES

and at the boundary

Now satisfies these conditions. For we have, firstly,

= -a)L™^''{(m+2)(m+2w+l)-2M}

- nzp^-\^{m{m+2n+l) + 2m.}

so that the first condition is satisfied ; secondly, ^ is the pro-
duct of p™+''+i by a function of the angles, thus

9"^ m + n+l
3^ = -p '^'

so that the second condition also is satisfied. From this
correct value of cj} the values of u', v', vJ 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.

This theorem leads to the following propositions : —
1. In the theorem we may replace p™ by a series of
powers of p, each power multiplied by an arbitrary constant,
that is, for p^ we may substitute any arbitrary function of p.
And again, we may replace ^,j by a series of spherical har-
monics of different degrees with arbitrary coefiicients ; for n is
without effect on the final result. Hence we get the following
generalisation of the theorem : —

If T^ is an arbitrary function, and if

9a)^' dwy d<i>^

then the currents u', v', w' induced by U, V, W are

, (0 dy' , CO dy' , CO dy'

II INDUCTION IN EOTATING 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

a;V2U + 2/V2V + 2V'W=0.

And vice versd the U, V, W of such currents may always
be expressed in the above form. For if ^,^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

A current which flows in concentric spherical shells in- ''I'^^ays m

^ concentric

duces a current system possessing the same property. Further- spherical
more, the currents which are induced by magnets at rest in a '^^''^^^'
rotating spherical shell always flow in concentric spherical
shells about the origin.

3. We find that

(j) = w(xY - yJJ)

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. °^ ^^'^ ^^'^'
Let ■y^j^ denote the wth term in its development. We found for inductions,
the currents of the first induction

1 adv 1 GJ dy' 1 o) dy'

% = ^ ^ Vi= i- «! = ^ .

n+1 icdcoj n+lKdaSy n+lKdwz

The corresponding values of U, V, W are

U = 277 ad// E^ p^ 2r^''+^

^ n+1 Kd^\2n + 1 2n+B (2m-MX2m-t- Sy+i

solution.

68 INDUCTION IN ROTATING SPHERES li

Y =^'^bl(^ p1 2^-"^+^ \

' n-\-lKd(o\2n+l 2w+3 (2%+ lX2w+ Sy+V

^ 71+1 «:aa),\2«,+ l 2w+3 (2w+lX2«+3y+7'

These values are got by a simple integration ; for u, v, vj are
products of p" 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

n+l\K) da,^\2n+l 2w+3 (2«+l)(27i+3y+7'

n + l\K) do>X2n+l 2w+3 (2m+lX2«+3y+7'

2,r fcoV^x"/ '^' P' 27-— 3

Wo — — — — -- — —

n+l\Kj do,,\2n+l 2n+3 (2m+1X2«+3)p^"+'

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 f(p)=f 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

u^fp, ,=fp, ^=fp.
oai^ OWy CO),

Let F (p) = F be a second function of p, which is given in
terms of/ by the equation

^^''^ = 2-^,-^1 f^^'^^^^^^^ + J^^-'^^

.f{a)da

II INDUCTION IN EOTATING SPHEEES 69

From this we get, by differentiating,

The values of U, V, W corresponding to u, v, w are

XJ = ri2^ y = p^» "w = pr^™ .
3 &)„ du>,, day.

Hence follow these currents induced by the system -v/r
The disturbing function belonging to this system is

Hence the function

i^ = P-f-Xn
induces the second function

t'=-Vx',

Now let yjr belong to the current-system actually existing
in the sphere under the influence of the external potential ^^ ;
let i/rg belong to the current-system directly induced by ex-
ternal magnets. Then clearly the condition for the stationary
state is

•^ = "^0 + ■^'•
To develop this equation further we analyse ^^ and con-
sider each term by itself Let the one considered be

We have then (p. 48)

to = - -^p( B ) -7T ®"^ ^^ ■ ^'»-
K \K/ n + 1

If we write

f=- -Ap{ ^ — _-{/i(p) sin ia ^flp) cos iuy)^^^,

70 INDUCTION IN ROTATING SPHERES n

we get

The equation -\/r = i/to + -«|r' is satisfied if /j and/2 satisfy
the equations

K
K

by. which /j and /^ are completely determined.

If we regard /i 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 hji, where the angle S is different for different
layers and is given by the equation tan S =fjf-i-

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-induc-
tion is s/f^+f} : 1.

We shall have to occupy ourselves for some time with the
determination of the functions /^ and f^.

We introduce the following contractions. Let

=/>

fi/r = s, fjM = S,

flip) = ^i(w) = <^lO".

In the equations which give/j and/2 ^6 put for Fj and
F2 their values, transform the equations so as to involve <j>
and a, and thus obtain

INDUCTION IN EOTATING SPHERES 71

<io-= 1+

<j s

o- S

These give by differentiation

The form of the functions <^i, 4>^ depends only on n; in
the constant of integration fi, s, S are involved.
The above equations may be written

9 1 + 9i= -92'

,,271+2
9 2 + '/'a^^i-

As differential equations these are exactly equivalent to
the following : —

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 X*= — 1, where X is that root whose real part
is positive, so that

INDUCTION IN EOTATING SPHERES

Then our equations become

The two particular integrals are

+ 1 CO

(1 - vye'^'^^dv , 1(1- v-)"e-'"'''dv

which hold for real positive values of a:

We shall prove farther on that these integrals satisfy the
equations. Since in our case w 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

i;,(o-)= (l-vje^dv,

Definition „

of ^ and q. f

1
then clearly the solutions of the differential equations are: —

^1 = M^n(.\a-) + Bi7„(\20-) + C2„(Xia-) + D2/X20-) ,

These solutions must be substituted in the integral equa-
tions so that the constants may be determined. The following
formulae will serve to evaluate the integrals which occur : —

p,(\a)=[(l-v're"^'--dv,

-1
+1

^-2.J^^^2.+i^j^ [(^l^^:"-f(^Xv+2n+l)e^'"dv
dcr J

-1

= 2nj7„.i(Xcr)/

1 The last members of the equations are got by transforming the preceding
integrals, especially by integration by parts.

cl
da

INDUCTION IN EOTATING SPHERES
+1

X I (1 - vy(a\v^ + 2(« + l)v)e^'"dv

= \'ap.^(\ay

The, last equation shows that ^„ is a solution of the
equation under discussion.

The preceding equations when integrated give

a2:>,,(\a)da = ~p,,_-^(\a) ,

and

a-y„_,(\a)da=la'-+'p^(\a),
2n

whence by differentiating we get these recurring formulse

, 2np'„_,(\a)

Integral
and recurr-
ing for-
mulse for
p and. q.

PnO^^)

X a

. , 2n+l ., s , Xo- , . ,
2n In

whence

271+1 ,

XV^

Pn.+1

2«, ^" 4m(w+l)
2
A. a

Exactly similar calculations may be performed for the q's.
The results are got by replacing ^ by g' ; hence

aqj\a)da = ^^^^-iCXo-) ,
A,

a-'\,_,{-Ka)da= ~a'-+\l\a), etc.
2n

With the use of these formulae it is easy to perform the
necessary integrations; for instance, we get by help of an
integration by parts

^ See footnote on p. 72.

74 INDUCTION I>? ROTATING SPHERES

ja'"'+'p„(\a)da + <r^''+^ j ap^{\a)da

A, A, A,

A, ^[7b -|- XJ A,

"When we substitute these and the similar expressions for q
in the equations, and remember that ^2 ^ ~ ■^^^i *^*^ ^* "" ~ ■'-'
the ^'s and ^'s cancel as they must do, and we are left with
equations of the form

„ , , - , (const),
= (constX+i-^2,

which are the solutions of the equation

da\_ da ^ '

>^«+V)]^

The constants occurring here must vanish separately;
remembering that — = — 5 we thus obtain the foUowiug four
equations for A, B, C, D

^-^ = Ap,_,{X^S) + Ba.i(X,S) + C^,_i(\S) + Bq^.^iX^) ,

= Ap,^,(\S) - Ba_i(X^S) + C?,.i(\S) - D2„./X,S) ,
= Ap,+i(\s) + Bp,+^(\s) + Cq^+fi^s) + I)qn+,i\s) ,

These equations are easily solved and give

A = -

».+ i y^+iCV)

in Pn-i(XiS')qn+i(\s) - iJ„+i(XiS)^„_i(\iS)
C= - -^+^ i^«+i(V)

4m 'pa-liXiS)qn+l(X-,s) - Pn+i(\s)q^_.^{\S) '

B =

INDUCTION IN ROTATING SPHERES 75

2n+l g^+i(V)

4« >«-i(^2S)?u+i(V) - A+i(V)&- 1(^28) '

^^_2»+l Pn+l(.\s)

4m A-i(>-2S)?„+i(X2S) - A+i(X2s)?„_i(X2S)

We get the complete solution by substituting these values
in (p^ and (j)^. It may be more simply exhibited thus : Since
Xj and X^ are conjugate, p {\<t) andp (\cr) also are conjugate ;
in the same way A and B are conjugate, as is easily seen, and
hence

is equal to twice the real part of either expression.
In the same way

-^Pn(\'^) - Bi5„(X2<r),

which expression occurs in ^^, 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

2n+l P^i\<y)qn+i(\s) - q,,i\a-)Pn+i(\s) Solution of

= <^l + <^2 V/~1 =/, +/2 V/"1. f3std

This equation is especially simple when s = 0, that is in the
case of a solid sphere. Then q^^^ (s) is infinite, and thus
our equation becomes

The quantities, a knowledge of which is of special interest
to us, are the angle o- = tan-yj/Zi, and the ratio of increase of

current, v /1+/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 ^fXe^'^s'
for integral values of n, and thus p and q may be expressed and q's.

The first
qs.

' INDUCTION IX EOTATING SPHERES ii

in a finite form. We may and will denote by jp and q the
functions thus calcidated. Then also ^'s with negative argu-
ments are admissible, and this equation holds

whence follows

2^nip)=rn(-p)-
For let the indefinite integral

then we have

1 1

= Y(-cr,l)-V(-a,0)
+ T(<r,l)-V(<7,0).
But for integral values of n

Y(CT,00)=Q, V(-^,0)=-V(<7,0),

and thus the statement made follows.

Hence the simplest integrals of the original equations
are

q„{a-) and q^{ - a).

The following are the expressions for the first few q's;
the first one has been determined directly, the others by the
recurring formula.

cr \ (7

2^|2.e-Y 3 3 ,
cr \ <T a

■2'.\3.e-y 6 15 15\ ^

qs=- = 1 + -+ -2 +-3 h etc-

cr \ cr £7 cr

II INDUCTION IX ROTATING SPHERES

Hence follow the equations

Pi-

i'o = -

e' + e'"

e" - e

e" -e-

, etc.

a~\ cr

For large values oi a- p and q approach the values

For very small values of a we get

{-2r.\n.l.Z...{2n-l) _^
?«(°") = — ^a„+i e

The equation - p{cr) = C[(cr)+q( - a) here has no longer any
meaning; for when o-=0,2 = ±o3- In order to find ^ for
very small values of t also, we expand it in a series of ascend-
ing powers of cr. 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.

Pni'^) =

1 . 3 ...(2W-I-1)

1 +

i(2?i + 3)

2.4.(2«+3X2w-|-5)

+ .

•J

The following formula is of importance for our further
investigations. "We have

?«(o-)&-i( - o") - SJi - o-)2b-i(o-)

(«)

4»(m —

\n An— 1

-Un-li°-)ln-2(- <^) - Sn-li - 0-)qn-:

i'r)\

which equality is easily demonstrated by means of the recur-
ring formula found for q.

In all the properties of p,^ and q^ considered we notice

' "5 DCDUCTIOX rs- EOTATING SPHEEES II

their close relation to Bessel's functions ; in fact we may
express J^ in terms of j?^+j and q^^^.

We may now remove p from the formula expressing our
result, and thus we obtain ^

Fmalform -^+1 g.+l(>^)g.( " ^<^) - g«+l( " ^)gr/^o-)

ofthesoln- 2n ■2,^i(X^)2',_i(-XS)-g',+i(-X5)2„_ASj

tion.

=01+<^y-l-

Its appiica- We shall apply this formida, which gives the exact solution,

to some special cases which admit of simplifications.
Thinspher- 1. In the first place, let the spherical shell be verr thin,

ical shell - ^

and let d be its thickness. Then S is only slightly different
from s. Let

S = s+S, where now S = fid.

For cr we may put any convenient value between s and S :
suppose o- = S.

We substitute these values in the above formula. In the
denominator we employ the substitution

2 71 + 1 X-a-

and divide Ijy the numerator. Thus we obtain

20' + 1)(2« + 1) g,^i(Xs)2„( - XS) - 3„+i( - Xs:2,(XS,
We develop and put

g.+l( - ^S; = ?„^/ - X^) + ' I ,_l - \S\

-.« + -;

' From this point onwards we write X instead of \i ; thus —

x=v'i;i+V~i).

II INDUCTION IN EOTATING SPHERES 79

Using formula («) (p. 77) we may divide out the qs and
thus obtain

</.i + </.,V-l=-

''^ 1+ ^^^^

2n+l

But now

have

47rEifa j^
'(2n+l)k ^'

according

our

previous notation, and hence

cj>i + 4>,>r-

-j

1+hJ-l

'^ /^

1 + h^ 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 formulfe hold for all values
of h, which proof still remained to be given.

2. Secondly, we apply our formulae to the case where we Small
need only retain the first power of the angular velocity in /^ ^f rotation.
and /,. For simplicity we restrict the investigation to a solid
sphere. In this case we found

... /-— , 2n+l ff„(W)
Expanding the ^'s and retaining only first powers we get

24-

2w+l

A closer consideration of this equation shows that the
values of f^, /, thence found when substituted in i^ merely
give the inductions of the first and second order, which we
have already calculated on p. 68. Here we only consider the

80 IXDUCTIOX IX ROTATING SPHERES n

angle through which the lines of flow are turned. Eetaining
only lowest powers we find

tan-^=S=-^Y-^^ '—

/i K \2n+l 2n+-6

so that the angle in question is
S _ 27ra/ E^

K \2?i+l 2/1+3

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

2/ = a; tan- p=Jor + y\

or very approximately

_ 27ra)

2w+l 2m + 3

In Fig. 8 these curves are drawn for a copper sphere for
which E = 50 mm., n=\, when it makes 1, 2, 3, 4 revolutions
per second.

Fig. 8.

Large 3. Thirdly, let us assume that fi is so large that for ^(SX)

ofrototion ^'^^ 2(«^) ^^ °^*y P^* *^®^^ approximate values. Further,
assume that the ratio r/E 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

'I'l + '/'s^^^l^

/ — - _2n+l S" gH<--») + 6-^''-^>

^n+l gA(S-s)_g-X(8-s)-

II INDUCTION IN ROTATING SPHEEES 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

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

2W+1/EV+1

A-S \p /

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 formulte 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

without performing the separation into real and imaginary
parts we easily find

H' P

Substituting these values in t/t we find

2m +1 /ico --^(E-p) . /, TT

^ 2(«+l)^V/<:7r V 4

which is the current-function produced for very large velocities
of rotation by the external potential

%^ = ^(£) COStcoP,,,..
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/4t ; 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 He
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 ia the
outer than in the inner layers. In the outermost layer it
constantly increases, ultimately as v &> ; 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 o) 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 lq im) ;
for this I refer to the formulae.

An apparent contradiction between the theory of an in-
finitely thia 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 J shall deal shortly with the case where the inducing
nets are magnets are inside the spherical shell, so that spherical har-
inside the nionics of negative degree occur.

shell.

INDUCTION IN ROTATING SPHERES 83

Let

Xn = M-) COS icoV^i,

then

If we write

i^o = --^pI - - sin icoV^
K \p/ n

i|r = -Ap[ - -(/ sin iw +f cos i&))P^
K \pj n

the function i^' induced by -^ becomes

w\\ /rV+V

f'= -l-)Ap{-) -(Fi cos ia> - F sin ia))P„i.
\k/ \p/ n

But here the connection between / and F is somewhat
different from what it was before, for we have

p E

r p

The condition
leads to the same equations as before,

Using the same contractions as before we get

<r s

s <J

84 IXDUCTIOX IN' EOTATIXG SPHEEES

On differentiation these become

da da I

da da /

In
u

A. " -'"a ' A

92 - — 9" =9i-
a

If we jiut ^ = o-^""*"^^, we get for ^ the equations

^1 H ~ ^1 = " ^2'

- „ 2 » + 2 - -
^" + — ^^ — <^/ = ^1'

which equations again lead to the p's and g's.

The constants may be determined in the same way as
before, and then we find

The formula becomes especially simple for the case where
S = X , that is, when we have to deal with a spherical hollow
in an infinitely extended mass.

Then ^^^./XS^ = 0, and we get

</,, y^ -</,,= - 2^n + 1X27. + V^ . l"^^^

If we may neglect /xp = cr in comparison with unity, on
account of the small value of a, we may put for the g's their
values for smaU arguments (p. 77). "We then obtain

<^iv/-l-</., = X2e -<--'>,

II INDUCTION IN EOTATING SPHERES 85

or since we have in part already neglected quantities of order a

^1 V — 1 — (j}.-i=\' = V ~ 1

as must be the case.

On the other hand, if fip be large compared with unity,
and we put for the q's their approximate values for large
arguments (p. 77), we get

\?7 As

Hence result phenomena similar to those for the spherical
shell; the rotation is 7r/4'i at the innermost layer, and thence
increases indefinitely as p increases to infinity.

For -^ we find

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-
hmited space that the approximation made may be permissible : Neglecting
hence even for the smallest values of &> the induction will pass tf^'™'^"'''
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 EOTATING SPHERES II

A. Potential of the Indiwed, 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 . /«\''+l

exten^al ^f2^ = ^(« ^J«)^«,

space. 2n+l\p/

when we consider the term ■yjr^^ of the whole current-function ■\jr.
Now

ir^lojda = - A"a( ^ ) —^ (/i(a) sin ia +fla) cos ■ia))P„A-
K \K/ n-\-l

Substituting this value in dD,, and attempting to perform
the integrations, we meet with the integrals

a'-^+y {ct)da.

But we have

a^-^y,{a)da = ^^"^l^^'' -F,(E)

according to the definition of F (p. 68), and the equations
satisfied by/j, U Fi, Fj (p. 70) ; and similarly

|a-+y,(a)cZa= - (i|+l>(l -/,(E)). K-+\

Using these expressions we find

a = -^ A^ ^y^ACE) sin iu>+{l -/i(E)} cos -io,]?,,
n+1 \pj

For' very small angular velocities /2 = 0,/i = 1, and thus
D,^ = 0. For very large ones /j =/2 = 0, and thus at the
surface of the spherical shell

— n

Xl = — -

n-\-

-^Xn-

11 INDUCTION IN ROTATING 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 spLe!^^

n,= - (|)"A[/,(r) sin ^., + { 1 -Mr)} cos ioyJP,^

Hence for the whole potential

^i + Xn = M~) [/i(r)cosio) -/2(r)sinHPm-

For vanishing angular velocities this expression reduces
to p^„, for large ones to zero ; more exactly for large values of
fi we find by means of a formula which we have employed
previously (p. 81)

/.>j_/.> /— r 2(27^+1) /By 1

\r J Xfj/r
Hence it follows that

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 of th^Zg-

the forces conditioned by the potential take up successively ?etic forces
all the directions of the compass ; and this can be repeated side space,
any number of times as the velocity goes on increasing.

B. Seat Generated.

Let E be the radius of a very thin spherical shell, and
suppose that in it exists the current-function

88 INDUCTION IN ROTATING SPHERES II

Let the resistance of the shell be k : required the heat W
generated in it.

We determine the values of u, v, w belonging to yjr, and in
particular to the term

Heatgener- ^^. = A sin tcoP^j.

ated m a

spherical "When u, V, w have been found, the heat generated is

shell. '8

the integral being extended over the whole surface of the sphere.
Introducing O and © to denote currents parallel to circles
of latitude and to the meridians in the direction of increasing
6 and to, we have

E de'

@= -

1 8f

E sin 6 da
M = @ cos ^ cos o) + fl sin w,
v = ® cos ^ sin (B — O cos co,
w= — @ sin ^.

Substituting for i|r„i the value given we get

A
M = — { — cos i<B cos tu . ■iP„j . cot ^ — sin ico sin fuP'^j}
E

M = — { — COS ico sin a . iT.^^ . cot 6 + sin ico cos &)P'„j},
E

A. . ^
w = —i cos ^o) . P„,-.
E

But now

*P,, cot e = i{P,,i+i + {n + iXn - i + l)P„,,.i},
P',, = i{ - P„,,+i + (n + i){n - i + l)P„,,.i}.

Substituted in u, v, w these give
w = - ^{cos (i+ 1>. P^_i+i

+ (m + i){n — i + 1) cos C^ - 1)« . P„,i-i}-

II INDUCTION IN EOTATING SPHEEES 89

- (n + iXn -i+l).sm(i- !)&> . P„,i_i},
A. . ^
E

Thus u, 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^ + v^ + iu'^)ds.

On integration, terms involving products of spherical harmonics
of different orders vanish, so that we may determine W^ sepa-
rately for each -\^„, and add the results. A closer consideration
then shows that we may also determine the heat separately
for each i/r^^ and again add the results. It is true that all
the integrals do not vanish which correspond to combinations

of different i/r^/s ; but those integrals in I ti^ds for which this

occurs will be destroyed by corresponding ones in | vds.
We now get for the -\|r„j above quoted

W^i = ki(v} + v' + w')ds

"^^ I (7,,,^rfds

+ (n + i)\n -i+l)A (P»,i_i)'ffe + 4^' [(cos ia)P J^cZsl,

which gives, by weU-known formula3 and simple reductions

W,, = kK^ '^'^''^'' +^\ n-i + l){n - ^ + 2) . . . (« + ^)
2n+ 1

= kA\n,i),

where (n,i) is an easily intelligible abbreviation. Since we
have further

90 INDUCTION m ROTATING SPHERES ii

our problem is solved. It is easily seen that the result may
be expressed in the forms

W = ^tn{n+l)U\ds.

-Km J

Analytical
formulae.

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 formulae are
implicitly involved

^-(sin ifflP J = -^cos{i+ l)co'F^,i+i

- -Km + i)(n - i + 1) cos (i - l)G)P^_i_i,
-- (sin ia)P„i) = - ^ sin (i + l)a)T^,i+i

+ K« +'i'Xn - * + 1) sin (i - l)a)P^_j.i,

9
-— ( sm ia)P„j) = I cos t&)P„i,

003^

and to these similar equations may easily be added. If w, 6
and a', 6' refer to systems of polar co-ordinates with different
axes, the equations last quoted enable us to deduce iategrals
of the form

cos ia>P„,(^) cosj(o''£„f6')ds',

(in which the integrations are to be performed with respect
to a', 6') from the well-known integral

P,,o(^) cosya,'P,/0^s' = -1— cos/tuP,/^),

Generation but, it is true. Only by laborious calculation.

of heat
the rota
sphere.

of heat in j ^^^ proceed to determine the heat generated in the

therotatmg -^ °

11 INDUCTION IN EOTATING SPHERES 91

rotating sphere by the term ^„^ of the inducing potential.
To ')(^i corresponds

■^ni = - -M ^ ) ^-rC/i sill ^" +/2 cos i(o)'P^i,
K \K/ w+ 1

and hence the heat

(•}'' A2 'i'X'n,i) [fp\ ^^2_L/2>„2„

The integration can be performed for small and for large Small velo-
angular velocities. For the former /^ = 1, /^ = 0, and thus °*^^-
the heat generated becomes in this case

For very large angular velocities we had

cities.

/1+/2 -2 T2^H^^ •

The integral W^j may be taken from r = 0, and becomes

for E/A may be regarded as infinite. Hence we get for very
large values of tu

^ J^2n+l)Xn,i) n^i

and W depends on E, in so far as A involves E.

Hence the heat generated increases indefinitely as co in-
creases/ and indeed proportionately to ^m. 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

' As regards the apparent contradiction with the result got for infinitely thin
spherical shells the remark on p. 82 holds.

92 INDUCTIOX IX EOTATIXG SPHEEES II

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 oo, then the couple of moment
D, which maintains the rotation, does work 27rajD per unit
time, and this work is equal to the heat generated Thus —

iTTOa

Couple ex- But this couple is equal to that with which in the reverse

j^^^?^*'^® case the rotating sphere acts on the magnets at rest. It is
magnets, easy to See that for small values of m D increases proportion-
ately to &)//c, but for large values decreases proportionately to
Vx/^j^^^'^^t^^y becoming zero. (This does not prevent work
being done of the order /^Ju^) On the other hand we have
seen already that for infinitely large values of a 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 ;\[AGXETIC SpHEKES.

I now assume that the material of the sphere is capable
of magnetisation, but that it is without coercive force.

"\Ye 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

TT 1 ?^^

^ by — -

Wby|^-
dy

dx'

INDUCTION IN KOTATING SPHERES 93

We must accordingly replace —

dx dy dz

9i^_^byV^M-H|% (1),

d« ox dy

since

But now

dy dz dx

dx dy dz

V^M = - 47r/^ (3),

Vm = -4

TTV

and

^ 82 3?/ 8a;

H'_dW_^Jl_dx
6 ?« 9^ dy

V _dX] 3V 8x
^ dy dx dz

(4).

If by these equations we eliminate L, M, JST from the
expressions (1), and add the forces directly exerted by the

3V_3U
dx dy

currents, that is - — - - — , etc., we get for the expressions,

which are now to be substituted for —---—, etc., the follow-

ox dy

ing:—

,^^,^e)ifJ^^f) (5).

\ oz dx dy J
\ dy dz ox

94: INDUCTION IN KOTATING SPHERES II

Here -^^ denotes the total potential ; but this consists —

(a) Of the given external potential of the inducing
magnets.

(6) Of the potential ^^ of the magnetised sphere itself.

This last satisfies the conditions : —

Inside

V'X<.= (6),

as follows from equation (2) and (4); at the surface —

47r^lf, = (1 + ^^0)l^-^y - (^^^ , (7),

where Np 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 + 4:776: 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

neekcted ^° ^® remarked that this is allowable only when

-P /47r&)(l + 47r6l)

is very small. ^Vhen 6 and E are large, to 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

Since this potential consists of two spherical harmonics,
the currents may by what precedes be regarded as known.
For the current-function we get

Under like conditions, only with = 0, we should have got
the current-function

By division we get

■^ = {1+ iTrO^U - '!L±lBf-y^\^.

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 -yjr and i^Tg The
quantities A, B are given by equations (6), (7) ; if we write

— = e, they are found to be

A =

(2w + l){(2w -I- 1X1 + 47r6l) - 4.77071}
n{n + 1)1677^6X1 - e'"'+') + (2n + 1)^1 + 4:770)'

^^ 4:77en(2n+l)

%{% -h l)167r'^6l\l - e'"+i) -f (2w + 1/(1 -(- 4.776)

As the interpretation of these expressions is not very special

obvious, we shall apply them to some simple cases. ''^'^^•
1. is very small. Expanding we get

A=l !^47r^, B = — ^^4776/, ^-^^,7

271 -fl 2w-f-l «'"'*"•

96 IXDUCTION Es EOTATIXG SPHERES ii

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 9.
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.

2. Let be very great and e not nearly equal to unity.

Then we have

■2n + l

B =

and hence

A =

■2n + l

iif

Thus the current in the innermost layer is here zero:
thence it increases rapidly outwards and becomes (2ft+l)/«
times as great at the outer surface as for the neutral sphere.
If 6 is at aU large the increase of current is almost independent

of its absolute value.
,j,,^ ^ ,^_ 3. Let e be infinitely nearly equal to unity.

ical shells. Then

_'2n + l/l+i7re^ -iirdn

B =

47rdn

^2n + 1X1 + ^-6)

Thus

In infinitely thin spherical shells the magnetic penne-

II INDUCTION IN ROTATING SPHEEES 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 6=0, which is the case of a solid sphere. The Solid
term with a negative power of p vanishes, and we get '^^^ ^''^'''

A = -

. 4fn-0n'

271+1

, l + 4:7re
Y = f'o-

2ft +1

For large values of 6 we have

2n+l
n

The quantity 'In + ljn lies betwean 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 '' ^'^^^'

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 + 47rg
^ 2 -h 27rr "■

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 IN ROTATING SPHEEES it

hold; the calculations are easily performed, but since they
give no very simple results they have been omitted here.

We shall now take into account self-induction, but shall
only perform the calculations for a solid sphere. Spherical
shells do not offer analytical difficulties of any special kind,
but the calculations become exceedingly complicated.

We find the currents by the following reasoning : —

Let the inducing potential be

Xni = M^j cosiwP^i.

Let t/tq be the current -function directly induced by ■^^^
then we have

1+4:176 CO. (pV i . . ^

2n+l
Let the actual current-function be

■^= ---^/'(b-) — ^T'^/l^^^^"+/2°°s'^")P™•
We have to find the currents induced by this. For this

purpose it is necessary first to know the potential ^^ induced

by i/r in the magnetic mass.
The current-function

t = /=(|)/(p)Y.

produces a radial magnetic force

p\dy dz J p\dz ox J p\dx dy I

= 5 (p. 63)
P

= -'<!!±1)f(p).(|)\ (p. 65),
where F, / are connected by the equation of p. 68.

II INDUCTION IN ROTATING SPHEKES 99

From these equations and from the equations determining
•^g (equations (6) and (7), p. 94) we get p^;^ in the general
case, and therefore in our particular case we have

4:Tren(n+l)

2n+l+4:Tr0n

^A-^/^) . {F/E)sini<« + P/E)costa,!-P,

If now by ■^' we mean that current-function which i/r and
^g together would produce in the unmagnetised sphere, then
in the magnetic mass they will produce the cxirrent-function

f , = (1 + 4^6) f,

and the condition for the stationary state becomes

•v^' is to be formed in exactly the same way as before, so that
\jr'g also is known. If we substitute the values of ■yfr, i/tq, -yjr'g
in the last equation and equate coefficients of cos ico and
sin ico, we get for/i,/^, F^, and F^ these equations

(2n+ l)(l + 47r6i) _ coi 4:-rren(l + 4:170)
2m + 1 + ^iren Y 2n + 1 + iirdn

+ ^(l+4,r^)F,(p) ,

^^'^^ k 2n + l + 4:Tr6n k

If we put

— — (l + 47r6') = /i^
k

flip) = 4>i(P-P) = </'i(°") '

fiip) = 4>2(P-P) = 4>2(°')'

then ^1, (j)^ are given by precisely the same differential equa-
tions as before (p. 71). Since we are dealing with a solid

100 INDUCTION IN EOTATING SPHERES li

sphere, we must only retain those solutions which are finite
at the centre, and may put

^i(o-) = Ap,l\(r) + BpJX^cr),

<f>la) = - XlApJX^a) - J^BpJX^a-) ,

^ = \= ^{1 + s/^l),

The constants are determined in precisely the same way
as above. 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,
however, is comparatively simple, namely

The Rolu- fJn') 4- f.(o') ^/ - 1 =^ ■ ■ ■

tion. JAPJt-J-APJ',/ -L 2np„.i(\p,B) + 4:7ren23^{XfjIi)

We first verify this result. For vanishing 6 it gives

Compari- . . /— r_ 2w+l j?„(Vp)

^™^"t^ /1+/2V -L 2n v„ i(XwE)'

previous m n i.\ r /

resu s. .^yj^jjg]^ agrees with the result already obtained for a non-
magnetic solid sphere (p. 75).

Further, for vanishing a it gives, since

91+iL
A(0) =

3...(2?i+l)

{2n+l){l + 4.176)

/i +/ W - 1 = ~2W+l + A^i^H~ '

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 -f 4:ird) 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.

Small vei- I apply the formula to two special cases.

rotation* 1- ^'^'^ ^'^^ ^® ^^^^ S^^^*' ^^* " Sufficiently small that

II INDUCTION IN KOTATING SPHERES 101

fi^B/' may be neglected in comparison with unity. We must
expand the expression, retaining only the first power of that
quantity. We have

M.sTi-'''^''^^''''^

2n+l 2(2»+3) + /^Vx/^
n ■2(2«+3) + ;ti^EV^

(p- m

We get for the angle of rotation, neglecting higher powers
than the first

S = tan-i^= 1^ (E'-p'),

A 2(2,1 + 3)^ P^'

i 2(2m+3)k ' P''

Hence the rotation vanishes at the outer surface ; ^ gener-
ally it is considerably increased, compared with that for a
non-magnetic sphere nearly in the ratio 4:Trd : 1.

In Fig. 9 are given the curves for an iron sphere cor-
responding to those for a copper sphere represented on p. 80.

Fio. 9.

The resistance of iron is taken to be six times that of copper,
and AttO 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 6, p. 123).

2. If CO become very great, whilst retains a finite but Large
otherwise arbitrary value, the phenomenon becomes very '''^i°<'*"=^-
similar to that in non-magnetic spheres, as may be easily
deduced from the formulae. Here also the angle of rotation

' A consequence of the fact tliat at this surface for large values of 6, according
to the equations for xe,

N -?^
Or

102 IXDUCnON IX ROTATING SPHEEES n

becomes TrjAi at the outer surface. The phenomenon is
identical with that which occurs in a non-magnetic sphere
with (1 + 4:7r6) times the velocity. The heat generated is
^l+4:Tr9 times that generated in a non-magnetic sphere
rotating with equal Telocity.

§ 7. Eelated Problems.

In this paragraph we shall consider some problems which
stand in very close relation to those already treated.

Any solid If we neglect seK-induction we may apply our knowledge

of revoiu- q£ ^j^g 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 hounding the soHd, n
its inward normal Describe about it a sphere of any radius.
Let %, fj, 'a\ be the currents which would flow in the latter,
and let

X = % cos a + I'l cos i + Wj^ cos e

be the current in the direction of n at the surface S. If we
determine u v , w so that

«'./„ =

KV.

then clearlv

ex
by '

Cx Oy Iz
u „cos a + «„ cos & + w., cos c = - X

2 ^ -

U^-VV-^, l'l+'t"2. ''•■!+ 'A

II INDUCTION IN KOTATING SPHEKES 103

are the currents sought in S. Hence the problem is reduced
to this simpler one : —

To determine such a function (f>^ that inside S V^^^ = 0,
and at its surface di^jdn = k'S, a given function.

1. As an example, suppose a plate bounded by the straight Plate
line f = 6 to move parallel to a given straight line. Suppose a,°straigiit^
the external potential expanded, and let a term of it be edge.

Ae"^** cos rrj cos s^

Then we found for the current in the infinite plate

Y^ = A~ — sm TT] . cos sf .
n K

Thus the current perpendicular to the boundary is

^ = - A— . - . cos 7'?? . cos so.

drj n K

Hence we get for ^, the conditions

and for f = &

We have

-Jl? I :iz? =

d4> f"^ a

-^ = A- ■ - -cos Tr] . cos sb.

0^ n K

d>„ = A — e"^^ *' cos r« . cos si.

'• UK

To <^2 corresponds the cm-rent-function

-v^„= -A- - . e'^^'^^sinr?;. COSS&,

'■ UK

and thus the total current-function becomes

t/tj + -i/r = A- . - . 6"''* sin r'q{e^^ . cos s^ - e''^ . cos sb) .
n K

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 IX ROTATING SPHEEES II

Limited 2. In Order to determine the currents in a limited

' ''^°' rotating disc, let a term of the external potential be

Then we had

i/tj = A sin ia)J^(np).

K n

Thus the inward radial current is at the boundary where
p = E

^^^ _

E9&)

= A^?
K n

. J
cos ICO -

Hence

find

above

i>.

= -A

'.>M

1 cos im

Determining the corresponding current-function ■^^ we get
for the total current-function

1,1 a" isin-ift),--^ . , ,

ti + ^2 = ^- • ^ {EiJi(wp) - p'J/»E)}.

K n xx

"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 ^2J i* is sufficient to determine -v/tj so that
inside the plate

dx oy

and at its boundary -^2= ~ i^v Some simple examples will be
given in § 9.

II.

Dielectric In conductors electromotive forces of electromagnetic

spheres. origin produce the same effects as nmnerically 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.

INDUCTION IN ROTATING SPHERES 105

Let

^ '■ sec

be the components

of the polarisation,

6 (number)^

the dielectric constant.

For X, g, J we

have the equations

s-f+'i-

i'-'t*'--

for /3 = E

dx'^d^+dz-4^^''f''

Hence we have for </>

and for p = E

(l + i^e)^fl-^p = ^-^ix% + y^ + .Z).
op Op p "^

In external space we must have V^^ = 0.
If T^^ again be the nth term of the external potential we
have, as above (p. 43)

Sx dy dz 8s

x% + y^ + zZ = co(p'^^-nzx^

' The units are again such that 1/A, the velocity of light, does not occur.
The corresponding magnitudes in magnetic measure are Ah, A^g, A^J, A%.

Earth in
dielectric
space.

106 INDUCTION IN ROTATING SPHERES

To satisfy the equations of condition we put

4776 a

</>?^

1 + 4716 n+1

■ «%

J.0
9e =

4776

1 +4776 71+ 1

1+2

n + l\2n+l dz

1=-%

■^V/o/ 2m+1 32.

<^° satisfies the partial differential equation which ^ is to
satisfy. ^^ is so formed that (1) it satisfies the equation

'(2) at the surface of the sphere it is equal to ^°. That the
first condition is satisfied is seen when we notice that the
expressions under the straight lines are spherical surface
harmonics of degrees (n+1) and (n—1), as is easily proved.
Substituting cj)" + tf>' in the equations for (p, we get for 0' these
equations

V^(j)' = everywhere, cp' continuous,

when p = E

dp dp dp

to satisfy which is not difficult, as we have already expressed
^° as a series of spherical harmonics.

A case of especial interest is that in which a spherical
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 Y = Syto is the earth's potential the problem reduces to
this : —

INDUCTION IN EOTATING SPHERES 107

To determine ^ so that in external space

^A- ^^^ oJx

v^<^= , :; -2

CO-

a«

and at the surface

1 + 4776^"^^ 2n + l dz

Hence follows the rate of increase of potential at the
earth's surface

^i = _ ^^^ 2Ka,>^-J— ^^- .
dp 1 + 4776 -^2n+l dz

Much the greater part of the earth's magnetic force is due
to terms for which «= - 2, or at any rate is small. There-
fore we may write approximately

-!- = S-E(U . -^ .

dp 1 + 47re" dz

^%/9* 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 m., that is, very small values.
However, a term of the form const/p may have to be added
to the above value of <p. 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 ™iwl
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 EOTATING SPHERES II

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

Fro. 10.

meridional section. The form of the lines of flow does not
depend upon the resistances of the magnet and the Uquid,
But the intensity vanishes when either resistance becomes
infinitely great.

§ 8. Solution foe the FoEMUL.ffl 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. Ixxviii. of
Borchardt's Journal}

If X, ^, Z denote the electromotive forces hitherto assumed,
X', 'W, Z' those which follow from the potential law, we have

^ Helmholtz, Wiss. Abhandl. vol. i. p. 702.

II INDUCTION IN ROTATING SPHERES 109

But we saw on p. 67, that for all U, V, W occurring in
the investigation

^ = wiYx - Uy).

We see at once that we may retain the previous solutions
unaltered as regards u, v, ro, yjr, D,. The only alteration which
must be made is to put for <f>', the potential of the free
electricity,

(j)' = const,

and, when free electricity was not present originally,

On an iniinite 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 aJJ + ^Y + 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 formulfe obtained will be applied to
some particular cases.

1. A single magnetic pole of strength 1 moves in a straight Maguetic
line parallel to an infinitely thin plane plate. Let the origin a°^iane°^^
of ^, Tj, f be taken at the foot of the perpendicular from the plate,
pole on the plate, and let the negative 7;-axis be parallel to

110 INDUCTION IN EOTATING SPHERES u

the direction of its motion.-' 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 ^

Hence we get for the potential of the second order

+ c-r)-^|.

In the same way the calculation may he continued.
We get for the current-functions of the first and second
orders

k^'^ + 7]\ r) kr(r + c)'

■x|r„= - 2„, ,

^' \k){f-c^){r + c)r

where now r^ = f ^ + 1;^ + cl

In the 7;-axis we have (since ^ = 0)

t,= -2.f^V «

x^k) (r + c)r'
thus

, 1,1 * 1 / 2-Trac\
A(r + c)r\ A /

Dis lace- ^® "^^^ regard the point |:= 0, i/r = as the centre of

ment of the the distribution ; thus it appears displaced through a distance

induced

distribu- ' a then becomes positive.

tion. ^ This result agrees exactly with that obtained by Jochmann.

II INDUCTION IN ROTATING SPHEKES 111

iiTcUijh 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

"^ 27rr'

for very large values of 27ra/A;

h fdy

_1_^ /.• v(^+c)

k rjC

t=-;^ 1-

27rrV 27ra'fHc';
Here also the abscissa of the point f = 0, t/t = is

277 ac

" = -!-''

but since this value is very large, and our formula holds only
for finite values of t?, the value of the distance must be regarded
as only an approximation.

The potential of the free electricity in the plate is

so that for smaU velocities

^ = afe?-=

9| ^(jj^c'^

112

INDUCTION IN ROTATING SPHERES

Fig. 11, a and h. — Pole moving in a straight line, | nat. size.

INDUCTION IN EOTATING SPHERES

113

M. P.

Pig. llj c and fL — Pole moving in a straight line, | nat. size.
I

114 INDUCTION IN EOTATING SPHERES II

Thus in this case the equipotential lines have the same
form as the lines of flow. For very large velocities we have

<h= — (^v = cl-— ^^

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 k= 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 i/r marked give absolute
measure when the strength of the pole is 13,700 mm'^mgrYsec.
In Fig. 11, a and &, 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 l-Kajk. It is true that for the value chosen the approxima-
tion is not very close. Fig. 11 (^, p. 113, corresponds to an
infinite velocity of the pole. The electric equipotential hnes
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

aro1;rtin™ 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 ?; f,

of which the origin is the foot of the perpendicular let fall
from the pole on the disc. Further, let

^ = x-a, 'n=y, ^=z;
thus

0(0 Ot] Orj Of
a is the distance of the pole from the axis of rotation ; let c be
its distance from the plate. Then

1 1

^~ \/e + rf + {K+cf r'

INDUCTION IN ROTATING SPHEEES 115

Hence

or since

k J 0(0
i

k ]dv h f + 97^ r /'

k rir + c)

Hence the form of the lines of flow is independent of the
distance of the pole from the axis.^ For the induction of the
second order we get

'''^' — '^8«i r+.^

^--Kf«a^(^:

= _ 27r(^?'*^^-'' ''^^ + '"?^

\k/ [(r^-OC + c)

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 ^ = 0,
-v/r = as the centre of the distribution. Its ordinate is found
to be

2'7r(oac

Hence in the neighbourhood of the pole the distribution Rotation of
is turned through the angle distribu-"^

27r6)c *^°"-

in the direction of rotation of the disc.

^ As already found by Jochmaim.

116 INDUCTION IX EOTATIXG SPHERES

Rectilinear 3. I shall now apply the formulfe to another example,

and uniim- Suppose that abovB the rotating disc two wires are stretched
ited disc, parallel to the a;-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 = tan ^ tan ^- j- .

^ z + c z + c'

Hence it follows, by means of the formula used before, if
r, r^ denote perpendicular distances from the wires, that

„ 2'7rco , fr'
Oi = — - X log -
k \rj

t. = ^.log0^^

For the potential of the free electricity in the plate we
get

' r

4> = mj 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
"^^ °" case where

c = c' = 1 mm, « = - a' = 2 mm.

Since, moreover, at infinity the currents become infinite, we
must suppose iirajk 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 KOTATING SPHERES

117

therefore assume that the perpendicular distance of the wires Rectilinear
conveying the currents is at a distance from the centre large ™"'^?'*': ^^
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'.
If we develop

in powers of the coordinates, and neglect higher powers of
the expression

c^ + a^'

we get

•^1 =

2axy 2ay^x('ic^ - a^)

ap^sin2(u (3c^ - ffi )a ,, ■ „ i „■„ ^ ^
= - -L. L > — ^p* (sm 2oj - -A- sm 4a)) .

c +a 6{c +ay

118

INDUCTION IN EOTATING SPHEEES

See Fig.
12 b.

The corresponding i/tj is (§ 7, I., conclusion)

, ap^ am 2a) (2,0^ - a?)a „,-^„ . „ i 2 • . n
t. - ^r^:^ - '^(^A^' -- 2. - ip^ sm 4.).

Hence we have

f^V'-ME'-f^

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^2>, it vanishes. If a<c^Z, the direction of
the current is the same as in the unlimited plate; if a>c Ji,
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. 12 6.

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 6. — 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

-v^ =

2/> COS (U(E^ - /3^) •

a(3c2 - a')

f" sin 2a)(E2 - p').

The connection with the previous result is easily seen.

INDUCTION IN ROTATING SPHERES

119

/6.

Pia. 13, a and i.— Rotating disc and rectilinear current, f nat. size.

120

mDUCTION IN ROTATING SPHERES

See Fig. 13. 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 5 mm.,
its thickness 2 mm. ; since ?i= 1, *= 1, we have

tan 6 = — . — ,
3 k

and if T be the inducing force, we find the moment of the
shell to be

Rotating
spherical
shells.

Execution
of experi-
ments.

If q is the number of revolutions per second,
CO — 2irq,
and since k= 113,500, we find

tan S= 0-0116 ^.

From the above the following table has been calculated: —

3°19'

3,614

42°61'

42,500

6=27'

7,178

46°13'

45,100

13-3'

14,110

100

49°13'

47,310

19°10'

20,520

200

66°40'

57,360

24°53'

26,290

500

80°15'

61,570

30°6'

31,340

34°49'

35,680

90»

62,500

39°4'

39,380

,1 INDUCTION IN ROTATING SPHEEES 121

Figs. 14 and 15, pp. 122 and 123, are intended to illustrate ^f^s
the distribution of current in solid spheres rotating under spheres and
the influence of a constant force perpendicular to the axis of j°^'|,';*^°*
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 u= 0, v= in gee Fig.g.
the 2;?/-plane, and thus the current-density = w. The diagrams ^* ^°'^ ■^•'•
represent the density of the current in the plane in question
by means of the curves

IV = const.

The values of lo marked give absolute values when the
influencing force

mgr-

T=289-

The size of the spheres is that drawn (E = 5 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. 1 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 47r^= 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 eiectro-
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

■^= - Tp sin 6 cos a> ;
thus

•v/r = — Tp^ sin 6 sin m,

122

INDUCTION" IN KOTATING SPHEEES

AXy.

Fig. 14, a and &. — Rotating copper sphere, five turns per second, \ nat. size.

INDUCTION IN ROTATING SPHERES

123

Fig. 15 a. — 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)

27rE5 TW

W =

15 K

If F be the moment of inertia of the sphere, wq its velocity
at time ^ = 0, and if it rotate under no external forces, the
equation of its motion is

1 iaidt = - °

15 Fk
CO = Q)n. e

If q be the mass of 1 cub. cm. of the material.

15^

thus

CO = a)n

4:C[K

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 motion and brought to rest with equal velocity. This in

expen- ^ . ,

ment. fact corresponds with an experiment made by Matteucci.

The angle which the sphere traverses after excitation of
the electromagnet amounts to

AqK

For strongly magnetic spheres we find

9T2

^ "Wiedemann, Galvanismus, § 878 ; Lehre von der EhktricitSi, 1885, vol.
iv. § 386, p. 322.

INDUCTION IN EOTATING SPHEEES

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
(27r) per second. The angles described are given in turns.
The relative values hold for every T and every wq.

Material.

Aluminium

Iron .

Silver

Copper . ...

German silver .

Graphite

Cone. sol. of copper sulphate

0-14
0-16
0-27
0-31
3-90
27-2
about 544,000

7. Damping in a galvanometer.

Consider a magnet swinging inside a conducting spherical Damping i»
shell ; and suppose it to be very small, or to have approxi- ^ s^^^™-
mately the form of a uniformly magnetised sphere. If M be
its moment, then in the spherical shell

Thus

M . .
P(;= — ^sm ycos 0).

^= sm 6/ sm o) ;

K p

and the heat generated per second is

-^^SttMV/I 1

3 K \r E/'

where, as before, r denotes the inner and E the outer radius of
the spherical shell.

Let now <f> 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

cl'cj> MT_, , .^ cU ^
^ + -i=^</' + 26— =0

dtj)

126 INDUCTION IN ROTATING SPHERES

which may be written

\dtj
so that the rate at which the heat is generated is

and thus we have

If 6 be small, we thence obtain for the logarithmic decre-
ment of the needle

47r^ E-r ImF

Sk Er V TF '

Aperiodic j^ Order that the aperiodic state may occui', we must have

R-r 3/c /TF

^eT^I^Vms'

from which equation, for given values of T, F, 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 DISTEIBUTION OF ELECTEICITY OVEE
THE SUEFACE OF MOVING CONDUCTOES

{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 DISTEIBUTIOX OF ELECTRICITY OX ilOVIXG CONDUCTORS ill

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 exphcit mention
of this assumption is not superfluous, because it contradicts
another, not unreasonable, assumption. Allien 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 ta favour of the one above mentioned. Further, we
leave out of account aR 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; (f) the total potential, h the surface-density, k the
specific resistance of a conductor, all measured in absolute
electrostatic units. Thus /c is a time, in fact the time in
which a charge arbitrarily distributed through the conductor
diminishes to l/e*"' of its original value. If now we refer
everything to coordinates fixed in the conductor and consider -
the motion in this conductor, we have

(1).

> <-,i; cy tz I

(111 , . „ .

- — = M cosa-F-y cos o-f ('.- cose (3),

o. z

-4^kJ^ + ^' (4),

in which equation W;, n^ denote respectively the internal and

KU =

dx'

KV = --1- , Kir =
dy

_c±

, fdu dv cir\
\ C-:c cy cz /

in DISTRIBUTION OF ELECTRICITY ON MOVING CONDUCTORS 129

external normal, and a, i, c denote the angles which n^ makes
with the axes. From (1) and (2) we get

— L.= - — V^c^, or V^rf) = (V^rf))„e ic''
dt K

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

VV=0 (5).

Further, from (1) and (3),

dh d<pi

K- —

dt on,-

(6),

or by using equation (4)

Air dt\dn^ dnj dn^

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 weU-known conditions of
continuity and the initial value of <^. In the differential co-
efficient dlijdt, h relates to a definite element of the surface ;
if the velocities of this element relative to any system of co-
ordinates be a, /3, 7, then the above equations will refer to
these coordinates, provided dh/dt be replaced by

dh , dh , _aA , dh

ot dx dy OS v

AVe get for the heat generated in time ht

h'W = UjKi:u^ + v'' + io')dT,

K J dn^
= - J'cpShds ,

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 in

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 <^^ 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 h-^ be the density cor-
responding to <^j. Then let ^j be determined so that V^^^ = 0,
that at the surfaces of the inductors 8^2/9% = kdh^/dt, 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 <p2 is formed from (j)^ let ^3 be formed from (p^,
^4 from ^3, and so on ; then clearly = <^^ + <^2 + ^3+ • • ■ ^^-
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 due to electrical resistance
are here inappreciable, and the form of the currents alone is
of interest. Since cfi^ is constant inside a conductor, and (p^
vanishes in comparison with (f)^, all the currents flow along
the lines of force of the potential (f)^, and we have

^u=^-^-h, ^v=-±-\ ^w=-^±\

9a; 9y 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
a-axis is the axis of rotation. In addition, we employ polar
coordinates p, a>, 6 with the same axis. Let T be the time
of one turn. The conditions which <p must satisfy in the
conductor are_ in this case: (1) inside V^^ = ; (2) at the

Ill DISTKIBUTION OF ELECTRICITY ON MOVING CONDUCTORS 131

surface, d^,jdni= K(7^hjdt)+{2'irKlT){dhldco), 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 dhjdt = ; and thus
a(^i/a», = (27r/<:/T)/(a/!,/3a)).

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 ?\ Suppose the
external potential $, 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 $ = A^jCp/E)" cos iwY^lO). Denote by </> the
potential of the electrical charge itself, which is induced on
the spherical shell ; in particular denote it by j)^ in the inside
space, by j)^ in the substance of the shell, by ^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

dp dp 2TSa)\3p 3/3/

All these requirements are fulfilled when we put

i^ JiAcosio + Bsint«)P„i(^) + [ ^ )(A'costM + B'sini(o)V^l0),
'^ \Acosico + 'Bsmico)T^ld) + { - )(A'cosia> + B'smico)V„Jie),

E/ \p/

4>s =

''E

(Acosico + Bsiniaj)P„i(6l) + - (A'cos^m + B'sini&))P„/6').
'>P/ \p/

For the general conditions are at once satisfied, and the

132 DISTBIBUTIOX OF ELECTfilCITY ON JIOVIXG CONDUCTOES in

two boundary conditions give, when we equate factors of
cosift) and simtu, four linear equations for the four constants
A, B, A', B'. If these latter are satisfied, so also will be the
former. Using the contractions /c/2T = a, rjH = e, we get for
these equations —

A„,/i = -MA-(2n+l)aiB+(»+l)6''+iA' * ,
=(2n+l)aiA -wB * +(m+l)e''+^B',

A„i.Me" = - we"A * +{n+l)M +(2w+l)aiB',
= * -we"B-(2w+l)azA'+(m+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^^, 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^^ and A' are
of order a-, B, B' 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 7r/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' are of order l/a. A, A' of order
ija?. 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 OX 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 6=0. If we
put tan S=(2n+ Vjaijn, then hji 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 S : 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 Hues 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 = 3a = -f/c/T. For a rotating cylinder the circumstances
are quite similar ; the angle of rotation is here found to be
2a = KJT.

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 kJcI = k, the specific

superficial resistance, may be a finite quantity. With this

assumption the tangent of the angle of rotation becomes

^ {2n+l)i liR

generally tanS = —^ 7T-7?r> and in the particular case of

^ ■' 2n{n+l) T' ^

a uniform force tanS = J/iE/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 S : 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 ON 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
kJT, or when for very
thin shells the quantity
^•E/T, has a finite value.
Here E denotes the mean
distance of the sheU from
the axis of rotation. Since
T cannot well become less
than yJo^ second, k must reach at least several hundredths of
a second. Thus it is obvioiis 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 DISTEIBUTIOX OF ELECTKICITY OX ilOVIXG COXDUCTOKS 135

duction w-ithout resistance. For very thin shells these dis-
turbing influences disappear-.

I know of no previous experiments wliich 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 10 cm. long was suspended by a wire
and allowed to execute torsional \T.brations ; 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. ^Mien 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 DanieU 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 DISTEIBUTIOX OF ELECTRICITY OX MOTIXG COXDUCTOES iii

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.

UPPEE LIMIT FOE THE KINETIC ENEEGY OF
ELECTEICITY IN MOTION

(IViedemann' s AnnaUn, 14, pp. 581-590, 1881.)

In a previous paper'- 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. Toxa^jsec^ 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

1 See I. p. 1.

138 KINETIC EXERGY OF ELECTRICITY IX MOTIOX [II] iv

from the direction of the meridian. The consequence of this
tendency will be a difference of potential between the points
C and B, 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 B to C, as shown by
the arrows.

An action of the kind mentioned must occur, whatever be
thie 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. Xo 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 fi, the kinetic energy of a current of magnetic strength 1
in a cubic millimetre of a sUver conductor, cannot greatly
exceed 0"00002 mg. mm.^'set".

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
CB lb 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 oi Siemens units in the direction AB, and 3 "5
Siemens units in the direction CB. From some unexplained
causes these resistances diminished in time, and after some
weeks were found to be -i^S 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 ELECTEICITY IN MOTION [II]

139

K3@^

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 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 thia-
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 £ 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 J5 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 IIOTION [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 I) and C gave a deflection of 32 scale divisions.
The motion of the needle was aperiodic ; a second position of
rest was reached in aljout 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 imtil 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 defl.ectiou ; 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 OE ELECTEICITY 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.

(6) 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 ACjBC and
ADiBB 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 ENEEGY OF ELECTRICITY IN MOTION [II] iv

8. A part whose sigu 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.

(6) 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 parafiin 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^/see magnetic units ; the velocity of rotation, according
to what has been said above, was on the average 290/8|-= 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 i) 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

IT KINETIC ENERGY OF ELECTEICITY IN MOTION [II] 143

scale divisions, obtained for the part of the deflection mentioned
under the head (3 a) : —

+ 3-6, -1-0, -0-0, -27, -1-1, +0-1, -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 +0'23. 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 to 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'mvo} + C, where C is the centrifugal force at the
position of the unit. The imit of opposite sign in the same
position is subject to a force — 27nva)+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 = Amvco, 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]

density, measured electrostatically, divided by the velocity v ■
hence m = M.vji and X = 4:u>.M.v^li = 4:ia . Mv^/i^. Now with-
out altering the equation we may use magnetic units on both
sides ; if we do so, Mv^/i^ = Mvl/il is that quantity which in
the introduction is denoted by /a, and thus X = 4/[i^(u. 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 cj), the differ-
ence of potential between the points C and D, by h, the breadth
of the plate ; if we call its mean thickness d, then we get
<ji = AfiJcohfq = -i^Jco/d, or, as we require fi,

_ (j)q _ (fid

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 witli 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 = 5"1 Siemens
units, we get the required cross-section q= 0'00014 mm^ and
the corresponding thickness d=Q'6 x 10"" 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 mg*^ mm* sec"\
(u = 27rx34 sec"\ (f)=l scale di\dsion= 1/32 x 10" of a
Daniell = 3300 mg* mm" sec"^; and thus find yii= 0-0000185
mm^; 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 <p= 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 NO3 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 /a=0'0078/w mm^
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.

ON THE CONTACT OF ELASTIC SOLIDS

(JourTialfur die reine und angewandte Matlumatilc, 92, pp. 156-171, 1881.)

Ix the theory of elasticity the causes of the deformatioDS are
assumed to be partly forces acting throughout the volume of
the body, partly pressures appHed to its surface. For both
classes of forces it may happen that they become infinitely
great in one or more infiinitely 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,^
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 Lehre von der Elasticitdt uiid Festigkeit, vol. i. p. 43 (Prag.
1867) ; and Graahqf, Theorie der Elasticitdt uiid Festigkeit, pp. 49-54 (Berlin,
1878).

^ CONTACT OF ELASTIC SOLIDS M7

acts between the parts in contact. The portion of the surface
which durinc; 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 ?■'
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 e, 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 7/ so that in the last-named function the term involving
xy is absent.

' 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. o{ 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

sj, = Aia;^ + Gxy + ^^\ z^ = A^x^ + Cxij + B^,

and we have for the distance between corresponding points
of the two surfaces s^ — j;^ = Aa;^ + By^ where A = Ai-A„
B = Bj — Bj, and A, B, C are all infinitesimal.^ From the
meaning of the quantity z^ — z^ 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 a-axis as the direction of the
normal to this surface.

Let f , rj, f be the component displacements parallel to the
axes of x,y,z; let Y,. denote the component parallel to 0«/ of
the pressure on a plane element whose normal is parallel to
<dx, 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-

^ Let pji, pj2 ^s the reciprocals of the principal radii of curvature of the sur-
face of the first body, reckoned positive when the corresponding ce'nti-es of
curvature lie inside this body ; similarly let Psj, pga ^^ ^^ principal curvatures
of the surface of the second body ; lastly, let a be the angle which the planes
of the cxurvatures pu and p2i make with each other. Then

2(A + B) = pu + pi3 + P21 + P22,

2(A - B) = V'(pii - pi„)2 + 2(pii - pi2)(p2i - P22) cos 2u + (p2i - p^K

If we introduce an auxiliary angle t by the equation cos t = ( A - B)/( A + B), then

2A = (pu + P12 + P21 + P22) cos^^, 2B = (pu + P12 + P21 + P22) sin^^.

V CONTACT OP ELASTIC SOLIDS 149

ponents of pressiu-e ; lastly let Kj^j and K^Oi^ 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

0=v'^ + (l + 2^& O = v''7 + (Hh20&
ox oy

oz 9a; dy oz

and in 1 we have to put 6^ for 6, in 2 6^ for 9.

2. At the boundaries the following conditions must hold : —
(a) At infinity ^, ■/;, f vanish, for our systems of co-
ordinates are rigidly connected with the bodies there.

(6) For a = 0, i.e. at the surface of the bodies, the tan-
gential stresses which are perpendicular to the «-axis must
vanish, or

---KM)-' ---KS40-'

(c) For « = 0, outside a certain portion of this plane, viz.
outside the surface of pressure, the normal stress also must
vanish, or

Z,= 2K('|+0.) = O.

Inside that part

^21 - 2z2-

We do not know the distribution of pressure over that part,
but instead we have a condition for the displacement ^ 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^ + ~Ry^ + ^j^ — ^^ — a, and since this
distance vanishes inside the surface of pressure we have

fj - ^2 = * ~ ^*^ - B«/2 = a - »j + «2.

(e) To the conditions enumerated we must add the con-

^ [Kirclilioff's notation, ilechanilc, p. 121. — Tr.]

150 CONTACT OF ELASTIC SOLIDS T

dition that inside the surface of pressure Z^ is everywhere
positive, and the condition that outside the surface of pressure
fj — ^^a — Ax^ — By^, otherwise the one body would overflow
into the other.

(/) . Lastly the integral / 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 bo\mdary condition (2 «!), 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 2=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 v^P = ; in 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= -- + — ^--ri\vdz-A,

K^K(l + 2^)\j J

where % is an infinitely great quantity, and J is a constant so
chosen as to make 11 ■ 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.
From the definition of 11 it follows that

. 2(1 + 61)
Introducing the contraction 3- = „, „^. we put

an an an „„^

2 ap 2 ap

This system of displacements is easily seen to satisfy the

V CONTACT OF ELASTIC SOLIDS 151

differential equations given for ^, rj, ^, and the displacements
vanish at infinity. For the pressure components we find

" dxdy

7- oTfi^'" I 2(2 + 3ff) aP

ra^n api a^p

The last two formulae show that for the given system the
stress-components perpendicular to the 2-axis vanish through-
out the plane « = 0. We determine the displacement t, and
the normal pressure Z^ at the plane « = 0, and find

f=^P, Z,= -2|?.

The density of the electricity producing the potential P is
— (l/27r)(9P/98), hence we have the following theorem. The
displacement f in the surface, which corresponds to the normal
pressure Z^, is equal to 5-/47r times the potential due to an
electrical density numerically equal to the pressure 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 2 = 0; we make IIj and n„ equal to
the expressions derived from the given expression for 11 by
giving to the symbols K and the index 1 and 2, and put

^-_9n, ^__9n, .__9n,_

152 CONTACT OF ELASTIC SOLIDS V

whence we have for s =

8P c'P

Oz dz

/
This assumption satisfies the conditions (1), (2 a), and ('2 b)
according to the explanations given. Since dTjciz has on the
two sides of the plane s = 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. Prom the fact that P is continuous across
the plane z= 0, it follows that for s = 0, 5-2^1 + ^1^2 ~ 0- But
according to the condition (2 d) we have for the surface of
pressure, ?i — ^2 = * ~ ^1 + ~2 j ^^re 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 2 = a^ + fj = Sj + ?2) o^"
(5-1 + 3-2)2 = S-^Zi + 3-i22- 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.T_.

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 6^coincide with the axes of

2,p I %2 ^
X and y, with a density — ^— / 1 — — — ^ , so that it can be

regarded as a charge which fills an infinitely flattened ellipsoid
with uniform volume density. Then

P = JZ. f /'l _ ^^ ^ _ r^^=±=_

167rJ\ «' + X l" + \ \/x/(a' + A,)(62 + X)X'

V CONTACT OF ELASTIC SOLIDS 153

where u, the inferior limit of integration, is the positive root
of the cubic equation

„2

Inside the surface of pressure, which is bounded by the given
ellipse, we have u=Q, P = L — Ma;^ — Nj'^ ; where L, M, N
denote certain positive definite integrals. The condition (2 d)
is satisfied by choosing a and 6 so that

which is always possible. The unknown a which occurs in
the condition is then determined by the equation

(^i + .9-2)L = a.

It follows directly from the equation

that the first of the conditions (2 «) is satisfied.

To show that the second also is satisfied is to prove that
when 2=0 and x^ja? + y^lh^>l, (^^ + S-,)? > a - Ak^ - Byl
For this purpose we observe that here

P = L - Ma;' - 'Sf

iy h _ x" y" \ d\

and hence P > L — Mcc^ — 'Ny^, for the numerator of the ex-
pression under the sign of integration is negative throughout
the region considered. Multiplying by 5-^^ + 3-, 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 a solution which satisfies all the
conditions.

154

CONTACT OF ELASTIC SOLIDS

The equations for the axes of the ellipse of pressin-e
written explicitly are

IQtt

\/(a'' + uf(b'' + u)u ^j+^^ 3p

B IGtt

or introducing the ratio k = ajh, and transforming,

iir

«'J v/(l + /^V/( 1 + z-") 3p ^^+\ '

iTT

By division we obtain a transcendental equation for the
ratio k} 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

^ The solution of this equation and the evaluation of the integrals required
for the determination of a and 6 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

^ 8(/)ll+ft2 + p2i+p22) V 8(/)ll + Pi2 + p2i + p22)

where /i, v are transcendental functions of the angle t. The table gives the
values of these functions for ten values of the argument t expressed in degrees.

1-0000

1-1278

1-2835

1-4858

1-7542

2-1357

2-7307

3-7779

6-6120

1-0000

0-8927

0-8017

0-7171

0-6407 0-5673

0-4930

0-4079

0-3186

0-0000

V CONTACT OF ELASTIC SOLIDS 155

the cube root of the total pressure and as the cube root of
the quantity 3-^ + 3-^. By the preceding tlie distance through
which the bodies approach each other under the action of the
given pressure is

Stt' a J v/(l + /.-VXl+»2)'

If we perform the multiplication by 3-^ + 3^, a splits up into
two portions which have a special meaning. 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 3-^ + 3-^. 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 I

finding the three displacements t^ , i?- , tt ■ In the first

Ox oy dz

place we have at the origin

_ 2 aP_ 3p 1

K(l + 26l) dz 2K{1 + 26)77 ab

d^_ 1 9P_ 3p 1^

dz~K{l-i-2e)dz 4K(l + 26l)7ra&'
Further, at the plane z =

an ^an

dx dy

n = — — vciz = — = Ych .

E:(1 + 26)} 2K(1 + 26)]

156 CONTACT OF ELASTIC SOLIDS V

We see that in the said plane ^ and 77 are proportional to
bhe 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 r) 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 ^ and r),
must be equal to a constant multiplied respectively by xja and
by yjh. Hence

Ox oy
On the other hand we have

d^ dr, S^ 3» 1

dx ' dy dz 4K(1 + 2e)ir ab

From these equations we find for the three quantities
which we sought

3? _ 3^ 1

dr)

4K(1 + 26l)7r

a{a+h)'

4K(1 + 26')7r

b{a + b)'

dz 4K(1 + 26l)7r ab '

The negative sign of these three quantities shows that the
element in question is compressed in aU three directions.
The compressions vary as the cube root of the total pressure.
It is easy to determine from them the x^ressures 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 otir 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. I 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 formula 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^ ^"^^ ^^^ reciprocals of the
radii of the touching spheres, then we have the relation
(-9-J -1- B-^p = 3-^p^ + 3-^p^ ; which for spheres of the same material
takes the simpler form 2p = p^ + p^. The curve of pressure is
a circle whose radius we shall call a. If we put

then will

a +u u

^,Ift(i-^-'-V. *'

ICttJ V a' + u uJ(a^j^u)J'.

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 ^ over the part of
the plane s = inside the circle of pressure : —

\/l6(p, + p,)' 16a '

Outside the circle of pressure ^ is represented by a some-
what more complicated expression, involving an inverse tangent.
Very simple expressions may be got for ^ and r) at the plane
a = 0. For the compression at the plane « = we find

3p V a^ •

2K(l + 2^)7r

inside the circle of pressure ; outside it o- = 0. For the
pressure Z^ inside the circle of pressure we obtain

y _ip V a? — r'^ .

at the centre we have

7=^ y _v - 1 + 4^ ^P
^ 2-^0?' " ' 4(l + 2^)7ra^'

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 ^ of the modulus of
elasticity ; 5- becomes equal to ^^- times the reciprocal of that
modulus ; in all bodies -9- 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 Newton'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^ at the centre
of the surface of pressure is 0"0669 kilogrammes per square
millimetre, and the perpendicular pressures X^ and Y have

V CONTACT OF ELASTIC SOLIDS 159

about -g- 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— io\ ) -^^-

Hence for spheres of radii

1 mm., 1 m.,

1 km., 1000 km..

« becomes about

TinJD mm., 10 mm.

100 m., 1000 km..

or a= ^-^j^,

100' 10' T

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. Om- 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 fm'ther two steel
spheres of equal radius touching one another and pressed
together only by their mutual gravitational attraction. In
millimetres we find ■^ the radius of the circle of pressure to be
p = 0-000000378E>. If the radius of the two spheres is 4-3
kilometres, then p = Ymj^ > ^^ ^* ^^ -'- "^ ^ kilometres, then
p = -YjjR. 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 E just mentioned.

Finally, we shall apply the formulae 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

^ In these calculations the modulus of elasticity of steel is taken to be 20,000
kg/mm^, 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. We 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. When 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 xyz. 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 coefl&cient of a with regard to the time. If
cZJ is the momentum lost in time At by one body and gained
by the other, then it follows from the theory of impact of
rigid bodies that dcJ = — k^dJ, where Jc^ 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.-' On

^ See Poisson, Traiti de m/canique, II. chap. vii. In the notation there
employed we have for the constant ki

. _ 1 (5 cos 7 - c cos /3)2 (c cos a - a cos 7)' {a cos /3 - 5 003 a)"
*i-M+ A + B + C

1 (y cos 7' -c' cos ;87 (c' cos 11' -a' cos 77 (g' cos /3' - 6' cos g'f
'*'M' A' B' "*" C ■

V CONTACT OF ELASTIC SOLIDS 161

the other hand, dJ is equal to the element of time dt, multi-
plied by the pressure which during that time acts between the
bodies. This is Ji\a\ where k„ 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 = k^a^dt and da = — JcJc.^ahU ;
integrating, and denoting by a^ the value of a' just before
impact, we find

a " — a g + ^JiJiM' — 0,

which equation expresses the principle of the conservation of
energy. When the bodies approach as closely as possible a
vanishes ; if a„^ denote the corresponding value of a, then

/ 5a'2\*
a„ = I — ) , and the simultaneous maximum pressure is

2}^i = k^al^. Prom this we at once obtain the dimensions of
the surface of impact.

In order to deduce the variation of the phenomenon witlr
the time,, we integrate again and obtain

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

am 5

j v/ aV - ^k,k,a^ V 1 6 a/^/4 a\'

M. P. M

162 CONTACT OF ELASTIC SOLIDS V

.= [-^. = 1-4716.

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 j the constants k^ and k.2 are

/• - ^ k-^ F-

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 7^1 = 8-78 -3 log E,

logi-2 = 4-03+1 log E.

Thus for two such spheres impinging with relative velocity v :

the radius of the surface of impact . am,= 0'0020E#mm,

the time of impact . T=0-000024E»" *sec,

the total pressure at the instant of

nearest approach . . . ^^= 0-00025EVkg,

the simultaneous maximum pressure

at the centre of impact per unit

area . . . y^= 29'l'y%g/mml

For instance, when the radius of the spheres is 25 mm., the
velocity 10 mm/sec, then a,„=0-13 mm., T= 0-00038 sec,
/»^=2-47kg., y^= 73-0 kg/mm.^ 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 27 hours.

ON THE CONTACT OF EIGID ELASTIC SOLIDS
AND ON HAKDNESS

{ Verhandlungen des Vereins zur Beforderung des Geweriefleisses, 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, Zehre von der Elasticitcit und Festigkeit,
Prag. 1867, I. p. 43 ; Grashof, Theorie der Elasticitdt und
Festigkeit, 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 filr reine und angewandte Mathematik,
p. 15 6.'' 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 tangent plane we take rectangular rectilinear
axes of xy, the origia 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 V. p. 146.

164 ON HAEDNESS 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 2=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 ia xy disappears. After com-
pleting this operation let the distance between the surfaces
be given by the equation c = Ax" + By'. 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 fi, 171, fj be the displacements
parallel to the axes of x, y, z in the first body, ^2. Vi> ^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 — ^^ + ^o under the action of the pressure ;
this approach must in the surface of pressure neutralise the
original distance Ka? + 1 \y- Hence here we must have
^1 — ^2 = « ~ ^'^ ~ B?/^, whilst elsewhere over the surfaces
fi — f2>a — ^y? — B?/^. 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 c = constant, but whose
shape is more elongated than theirs. We reserve the deter-
mination of the lengths of its semi-axes a and h until later.
First we define a function P by the equation

^^h^ii-i-A. y^ ^'\ _J^

1 6 TT j \ ((- -I- X V-^X \js/ (ft2 + X)lb-' + X)X'

where the lower limit of integration is the positive root of the
cubic equation

a? -{■ u ly' + It to
The quantity lo is an elliptic coordinate of the point xye ; 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

92p 92p g2p

dir oij- dz

ISTow from this P we deduce two functions 11, one of which
refers to the one body, the second to the other, and we make

n, = -— f«p — ^- fp&

n, = - —(zV -^ I I'd:

' KX 1 + 26;

166 ON HARDNESS

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©

The ratio between lateral contraction and longitudinal exten-
sion is

For bodies like glass or steel, this ratio is nearly -^, or ©
nearly 1, and K is nearly f E. For slightly compressible
bodies the ratio is nearly ^ ; here then ® = oo , K = -^E. 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;(i + 2©)'

In bodies like glass, ^ = 4/3K= 32/9E ; in all bodies S- lies
between 3/E and 4/E, since © lies between and oo . In
regard to the II's 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 II's infinite constants of suitable
magnitude to make them finite. By a simple differentiation,
remembering that v^P = 0, we find

2 3P „ 2 3P
V2TT t v^TT

^^^'- K,dz- ^^^'~ K,dz

We now assume the following expressions for the displace-
ments in the two bodies : —

an,

ri =

^"^ + 2^P
dz ^^^'^'

^2 =

an„

dy'

^2 =

an„

2^,P,

whence follow

8?i

9'7i S^'i _
dy dz

= v'ni + 2^

2
Ki(l +

2©i) dz'

VI ON HARDNESS 167

2 dV

0-9 =

K2(l + 2e2) dz

In the first place, this system satisfies the equations of
equilibrium, for we have

and similar equations hold for fj, Vi> V2 > for the ^s we get the
same result, remembering that v^P = 0- For the tangential
stress components at the surface (2 = 0) we find, leaving out
the indices : — •

- \dx dz) \dzdx dx ) ''" dxdz

Y^=_Kf?^ + if)._Kf2^+2^^U2.^^0,
\dz^dyj V 9y9« 3*7 32/9«

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

^.r(^^ ^ \ (9'n 2(2 + 3© 8P

dz^ ) -^U^^ ^K(l + 2©)a8

32P 3P

= 2z -^ - 2-g^ ;

3P
therefore at the surface Z^ = — 2^ . Now, using the equa-
tion for w, we have generally

ap 3j? r ^ ^

9P . .

and therefore at the surface -^ vanishes, as it must do, and

with it Z^, at any rate outside the surface of pressure. In
the compressed surface, where m = 0, the expression takes the

168 ON HAKDNESS VI

form . 00 ; the ordinary procedure for the evaluation of
such an indeterminate form gives

dT Sp " dz

tliat is, since for ?t = we have

3P 3« / X- if

Here no quantity occurs wliich could be affected by an
index. Hence in the surface of pressure Z- is the same for
both bodies ; pressure and counter-pressure are equal. Lastly,
the integral of Z_ over the sm-face of pressure is Spl^irah
times the volume of an ellipsoid whose semi-axes are 1, a, i;
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 h. For
this purpose we remark that

so that at the surface ^1=-^!? and ^2 = 5-2?. Since inside
the surface of pressure the lower limit m of the integral is
constantly zero, inside that surface P has the form P = L —
Ma;^ — 'Nf ; and therefore it is necessary so to determine a, i
and a that (3-^ + 3-^)^ = A, (S-^ + \)lii = B, (9-^ + \)L = a, so
as to satisfy the equation fj — f, = a — Ax^ — By'', and this
determination is always possible. Written explicitly the
equations for a and b are

du A 16

Jla'

'■ + u)%h^ + io)u

^1 + ^2

■a

IGtt

J(a'

■■ + uXb^ + uyu

^1 + ^2

(I)

VI ON HARDNESS 169

Finally, it is easily shown that the very essential in-
equality, which must be fulfilled outside the surface of
pressure, is actually satisfied ; but I omit the proof, since it
requires the repetition of complicated integrals.

Thus our formulse 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 h 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^^ and p-^^ for the one
body, P21 and p.2.2 for the other, and let a> be the angle
between the planes of p^ and of p^^ . 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 rc^-plane makes with the plane of pu the angle &>', so
far unknown. Then the equations of the surfaces are

2^1 = pij(x' cos co' + y sin w')^ + p-y^{y cos a! — x sin aJf,

2^2 = — p2i{*^ cos (&)' — (u) + y sin iw' — &))}^

— p.2«\y cos {J — od) — x sin (w' — ro)}^.

The difference Sj — »2 gives the distance between the surfaces.
Putting it = Ax^ + By^, and equating coefficients of a;^, xy, y"^
on both sides, we obtain three equations for at', A and B ;
their solution gives for the angle m, which evidently de-
termines the position of the axes of the ellipse of pressure
relatively to the surfaces, the equation

^ o / (^21-^22) sin 2(u

for A and B

3(A-B) =

Pii - P12 + (^21 - P22)COS 2a)'

2(A 4- B) = pii -I- P12 + P21 + ^22 >

J{pn - Puf + 2(pii - PuXpn - P22)cos 2&) + (P21 - Piif

170 ON HARDNESS VI

For the purpose of what follows it is convenient to
introduce an auxiliary angle r by the equation

A-B

COST ■■

A + B'

and then

2 A = (pn + Pi2 + P21 + /'22)sin^2 '

2B = (pii + P12 + P21 + p22)cos^2 ■

We shall introduce these values into the equations for
a and h, and at the same time transform the integrals
occurring there by putting in the first u = bh^, in the second
u = aV. Denoting the ratio h/a by k we get

1 f ^^ _ ^ PU + P12 + P21 + P22 gjj^2;^

d^ _ ^ P1I+P12 + P2I + P22 j.Qg2^

Dividing the one equation by the other we get a new one,
involving only k and t, so that Z; is a function of t alone ;
and the same is true of the integrals occurring in the equa-
tions. If we solve them by writing

V8(/P„ + /3i2-

-J

+ ^2)

+ P21 + P22) '

KPll + P12+ P2I+ P22)'

then /x and v depend only on t, 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 HAEDNESS

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,^ in which are found the values of /i and v for ten
values of the argument t ; 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 -9- ; 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 {S-^ + B-^Ij. Transforming the
integral L a little, we get

Zp 5i -I- S-.

J{i+k'z'){i+z'y

The distances by which the origin approaches the distant

1-000

1-128

1-284

1-486

1-754

2-136

2-731

3-778

6-612

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-^ + 3-2 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
I power of the mean value of the coefficients 3-, 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 fj — ^2 ^ ^^ ~^^^ ""B?/" = a — ;i+,ro,
again ^^ = rS-^P, ^^ = 3-2?, or ^^ : ^2 = -^! •'^2' ^^^ lastly, the
equation of the deformed surface is » = «i + ?i = -2 + ^'2 ; whence
neglecting a constant, we easily deduce (3-^ + 32)s = 3^i + ^iZ^.
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 = ^' /1 *'_^'

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

ON HARDNESS 173

equally distributed over the surface of pressure. Besides 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^,
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, in the ratio of (1 + 4@)/2(l + 26) : 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 pressm'e 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 pressm-e.

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 s-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 HAEDNESS

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 sc-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 a;-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 ;e-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 k= fi = v = 1, pu= pi2 = pi, Pa. = P22 = Pi >
hence

V l&{p^ + p,) ' 16«

The formulEe 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

ON HAEDNESS 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 = -2-(/3j + p^. Further, we may neglect u compared with
a?, take a outside the sign of integration, and put for the
indeterminate quantity ^/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

V -rrip^ + p,)
For the pressure 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 HARDNESS

I first examined whether the dimensions of the surface of

pressure increased as the cube root of the pressure. Tij this

end a glass lens of 28"0 mm. radius was fastene'd 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". According to Wertheim,

and 3-= 0005790 mmY^^g- Hence our formula gives for the

diameter of the circle of pressure in mm., d= "3 6 5 Op*, where

p is measm-ed 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 measm-ed

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.

0-2

0-4

0-6

0-s

1-0

1-5

2-0

2-5

3-0

3-5

1-56

2-03

2-19

2-59

2-68

3-13

3-52

3-69

3-97

4-02

d-.'^p

2 '67

2-75

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 a, using
former equations we get pn = p^^ = p, p^^ = poo = 0, A + B = p,
A - B = - p cos CO, and therefore the auxiliary angle t = w
Hence if we determine the large and small axes of the ellipse
of pressure for one and the same pressure but different inclina-

ON HAKDNESS

177

tions, divide the major axes by the function jj, 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{Sp3-/8p)^. The
following table gives in the first column the inclination a in
degrees, in the next two the values of 2 a and 2 6 as measured
in parts of the scale of the micrometer eye-piece, of which
96 equal one millimetre, and in the last two the quotients 2a/yu,
and 2&/i' : —

2 a

2 b

40'6

40-6

45-4

36-6

40-2

41-0

52-8

31-0

41-3

38-7

59-6

27-6

40-0

38-5

72-2

26-4

41-2

90-4

23-8

42-2

42-0

110-0

21-0

40-3

42-6

156-2

18-4

41-3

45-3

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 3- 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
41-88, 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 HAEDNESS

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 formula 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 versd; 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 horn-. According to a book published in
1865,^ 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.
From the results obtained we can draw no conclusions at all

^ F. Hugueny, Rechcrchcs cxperimentales sur la dureti des corps.

ON HARDNESS 179

as to the force necessary to drive in anotlier 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 Grace -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^, Yj^, Z^ in any element lie
so that the limit of elasticity may not be exceeded ? If we
represent X^,, Y^,, 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^., Yj,, 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 -»|r (X^,, Y^,,
Z^) = 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

point outside the surface yfr (X^, Y^^, ZJ = 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 ])Osition.
This object we arttain by the following definition, — Hardness
is the strength of a body relative to the kind of deformation
which corresponds to contact ivith a circular surf rax 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
2}ressure per unit area ivhich 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 ( if pressure. We should from both cases get the
same value for the ha-rdness, 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

Yi ON HAEDXESS 181

sui'face 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
one. 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 li. If now
we increase the pressure between them until the normal
pressure at the origin just exceeds h, the body of hardness li
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 l)y assuming for it any definite ellipticity. The
hardnesses thus diversely defined wiU. show slight nimierical
variations. Xow the order of the bodies in the different
scales of hardness is either the same, or it is not. In the first
ease, 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.

Xow 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 HABDNESS VI

this value wo can make this assertion, that all contacts with a
circular surface (if pressure for which

or for which

can he borne, and only these.

The force which is just sullicient to drive a, pohit with
spherical end into the plane surface of a softer body, is pro-
portional to the cube of the liardness ol' this latter body, to
the square of the radius of curvature of the cud of the i)oint,
and also to the square of the mean of the coefficients^ Inr the
two bodies. To l)rinn' this assertion into better accord with
the usual determinations of hardness we might be tempted
to measure the latter not by th(( normal iireKM\u'e itself, l)uL
rather by its cube. Apart i'rom the I'aet that the analogy
thus produced would ))e fictitious (for the ionxi necessary to
drive one and the same point into dilferent bodies wouhl not
even then be proportionate to the hardness of the bodies), thi.s
proceeding would be irrational, since it would roiiiove hardness
from its place in the series ol' strengths of matiuial.

Though (lur deductions rest on results which are satis-
factorily verified by exjierience, 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 whicli we have nia,de our basis. Indeed, it is
sufldciently well known that the conditions as to strength near
the surface, with which we are licre concerned, arc quite different
from those inside the bodies. 1 liave made only a few i^.\i)eri-
ments on glass. In glass and all similar bodies the first trans-
gressiiiu beyond the elastic limit shows itself as a circular crack
which arises in the surlacc at the edge ol' the conqiressed surface,
and is pr(ipa,gated inwards ali)Ug a surface conical outwards when
the pressure inerca,ses. 'When the pressure inerc^ases still
further, a second crack encircles the first and .siniiliirly pro-
pagates itself inwards; then a third ai)pears, and ko on, the
phenomenon naturally becoming more and mon^ 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^ for the hardness of the latter.
From the phenomena accompanying the impact of two glass
spheres, 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 HYGEOMETEE

(Verliandlmujen der physiTcalischen 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 satui-ated
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

HYGKOMETER 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
latter was supported on a very thin silver wire stretched
horizontally, 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 eifect 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 EVAPOKATION OF LIQUIDS, AND
ESPECIALLY OF MEECUEY, IN VACUO

(Wiedemann s AnnaUn, 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

VIII 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.
Either (a) evaporation has no limit beyond that which is
involved iu 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 (&) 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 (6). 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 viil

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 simviltaneous 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 boilinc. The
first experiments were carried out with the apparatus shown

VIII

EVAPORATION OF LIQUIDS

189

Fig. 20.

in Kg. 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 £, which was immersed
in a fairly large cooling vessel, and could
thus 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 £, 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

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, 1 60° 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. 21 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 beo-an'

Fio. 21.

VIII 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.^
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
sought. 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 ^ ; at a given temperature the depth
of the layer which evaporated in unit time was proportional
to the difference Pj — P ; when this difference was 1 mm. the
depth of the layer which evaporated per minute was 0'5 mm.

^ 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 tliis

paper. The title, indeed,- has only been used for brevity in place of a more
precise one.

192

EVAPORATION OF LIQUIDS

VIII

at 120°, 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^ 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
plate. The open arm communicates with
the cold receiver ]3 ; the communicating
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

^ For all data as to the pressure of saturated mercury vapour here used see
the determinations given in the next paper (IX. p. 200).

Fig. 22.

VIII EVAPOEATION OF LIQUIDS 193

function has been determined. In order to find this, special
experiments were made with the apparatus shown in Fig. 22, 5.
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 0°'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°'0 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 0'105/3'66 = 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 72'8 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 ll7 mm. high through 56-4/117 = 0°-48.
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 viii

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°, orO'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 O'Ol 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.^ For, if they had been correct,
deviations of pressure of 0-10 to 0'20 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.

VIII 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 Tj and Tg 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 f 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,
ET=_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, = vd.

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. Por 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 EVAPOEATIOX OF LIQUIDS viil

the space u and have the velocity w. Hence its kinetic energy
is ^mu^jg; this is attained by the force 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? =
gd{-p-p).

Xow 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^, T^, T, p, d,
u, m, and P, namely T^ and To, 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, Tj and P as the independent variables,
and consider only evaporation in the narrower sense, we are
no longer interested in To, and the problem resolves itself into
representing two of the quantities T, p, d, u, in as functions
of Tj 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 ve
can imagine such evaporation taking place as if we allowed a
piston to rest upon the surface at temperature Tj, and at a
given instant removed it from the surface with velocity ii.
The result of this experiment must be singly determined by
Tj and v.. 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

VIII

EVAPOEATION OF LIQUIDS

197

can only be diminished, not increased. (2) The vapour arising
from an evaporating surface is either saturated or unsaturated,
never supersaturated. Por 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 'i^<Pi, the second
that d<(l^„ if we denote by p^ the pressure of the saturated
vapour at the temperature T^, and by cZj, the density of the
saturated vapour at the pressure p. Now m = s/od(V —p),
and therefore in<,^gcl^{^^—p). But the right-hand side of
this inequality is zero when pi = Q and when p=Pi; between
these it attains a maximum value which m cannot under any
circumstances exceed for a surface-temperature Tj. 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. Prom the above equation I have calculated the
limits for in, 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^ = 10-59271 - 0-847 log T- 3342/T, which is
established elsewhere.-' 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 =
h<
u<
P>

dldj>

100

0-70

2110

0-046

0-0034

110
1-11
2192
0-07
32

120
1-86
2294
0-09
30

130
3-01
2400
0-14
28

140
4-50
2522
0-20

150
6-73
2668
0-27

160
9-82
2823
0-38

170
14-31
29S0

0-53

180

20-42

3145

0-71

mm.

min.

secT
mm.

mm.
mm.
m.
sec.

{'P-P)IPi>

0-OS

7-5

'0-0034

0-13
7-4
32

0-21
7-3
30

0-32
7-1
28

0-47
6-9
26

0-65
6-8
24

0-88
6-6
22

1-21
6-5
20

1-67
6-2
18

1 See IX. p. 204.

198 EVAPOEATION 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 ia 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 hmits we
have first m = (P —p)jmg = mfd ; and since m>m^i^^^ P — ^<Pi,
Sindd<di, it follows that^/m„^i^ >M>m,^i^ /f?i. Again P =j; + vi^jd,
and since m>v\^^^^_ and d<d^„ it follows that P>i' + wi^min./^p-
But the expression on the right hand has a minimum, since it
becomes infinite when p = 0, and when ^=00; this minimum
value is given in the table. Finally d = m^jCP —p) and P -p
= m^jd. Hence djd^vi^^^^Jd^p-^, and (P -p)/Pi>'>n^jnmJdiPi-

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 ^^j 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= Jpdgjlir. 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 must
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 tlius 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.

ON THE PEESSUEE OF SATUEATED
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^ 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,^
taking as abscissse the amounts which evaporated in unit time
from a surface at a given temperature, and as ordinates the
corresponding pressm^es, 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;^
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. " See Wied. Ann. 16, p. 610, 1882.

IX VAPOUE-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 +2-4:9 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 VAPOUE-PEESSURE OF MEECUEY

manometers A and B (Fig. 23) communicate with one another.

Tliey contain air of low pressure,
B — about 10 to 20 mm. The closed
=5p^\ I 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

^"^- ^2- 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 JoUy 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 electric 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 O'OS mm., at
70° 0-05 mm., at 80° 0-09 mm. Prom here on the errors
were small compared with the whole values. From 1 20" to
130° the observed pressures can be taken as correct, since

VAPOUE-PEESSUEE OF MEEOUEY

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 cohimns.

(

89-4

0-16

0-00

0-0

*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

+ 0-1

203-0

20-35

+ 0-23

+ 0-3

*165-8

5-62

+ 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.! jl gg^jj^ ]3g 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^ the internal heat
of vaporisation at the absolute temperature T. Then it
necessarily follows from our assumption that p^ = 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-

^ An analogous formula, deduced by similar reasoning, has indeed been
used by Kolacek ( Wied. 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 Kolafiek'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 ajjproximation the pressure of the vapour
for a considerable interval above 0°. For if the formula holds good for a given
interval of temjierature, 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-PEESSTJEE OF MEECUEY

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^ = Au{TdpldT -p), where p
denotes the pressure of the saturated vapour and u its specific
volume. Hence we can put u = KT/p. If we eliminate p^
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 = /CjT "ARC'^.

For mercury s is known. From his own experiments, and
from a result given by Eegnault, Winkelmann ^ 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 formida 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 /ji= 10-59271, log /fc2= 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.

^ See Poggendorff's Ann. 159, p. 152, 1876.

VAPOUE-PEESSURE OF MEECUEY

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 vapom' for every 10° between 0° and 220° —

0»

0-00019

60»

0-026

120°

0-779

180"

9-23

0-00050

0-050

130

1-24

190

13-07

0-0013

0-093

140

1-93

200

18-25

0-0029

0-165

150

2-93

210

25-12

0-0063

100

0-285

160

4-38

220

34-90

0-013

110

0-478

170

6-41

It should be noted that p = when t= - 2^2i° ; and that
the formula gives for the internal latent heat of the vapour
the value p-r= 76"15 - 0-0183T. 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.^ Hence no correction need be

^ 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 VAPOUE-PRESSURK OF MERCURY

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 O'lO mm., as given by Regnault,
or even 0'04 mm., as given by Hagen, could not have escaped observation.

ON THE CONTINUOUS CURRENTS WHICH THE
TIDAL ACTION OF THE HEAVENLY BODIES
MUST PRODUCE IN THE OCEAN.

{Verhandlungeii der physikaKscAen GesellscAaft 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 ^ denote the elevation of the
water above the mean level, and let

^=?'ocos47r(

\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

where

and

^ = ^-. cos AttI — 6

tan iwe = -

2Triih?{gli-A'')'

t. = ^^— -to Sm 4776 .

Here li denotes the coefficient of viscosity of water, and A = ^

denotes the velocity of propagation of the wave, /a the density
of water, and g the acceleration of gravity. In the calculation
squares and products of small quantities are neglected. Tor
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
/Lt^X at different times, where X denotes the component of
gravitational attraction along the canal. Por the tidal wave
considered above, we have

47rVyA'fo • 2. ^-..1.

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

at the surface.

X OCEANIC CUEEENTS 209

If we apply this result to the case of the earth we see that
generally the tidal wave in its progress must be fonowed 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 fj 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.-' 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

^ 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 -l
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.

HOT-WIEE AMMETEE^ OF SMALL EESISTANCE
AND NEGLIGIBLE INDUCTANCE

{Zeitsehrift filr InstrumentenJcwnde, 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 ^1 + VV/T'^'r : 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 [Dyiiamometrische VorricMung.']

212

HOT-WIRE AMMETER

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^ 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 httle
solder to the vertical steel wire ah
and twisted round this, as shown in
Fig. 24, K The steel wire ah has a
diameter of 0'8 mm., and is as smooth
and round as possible ; the twisting
of the silver wire can easily te
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, ab is held in position by the torsion
which it produces in the thinner steel wires ac and Id; these

' See XIII. p. 224.

Pio. 24.

HOT-WIEE AJIMETER 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 hd and cause the wire ah 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 miUionths of
its length, so that each half of it would expand by 760
miUionths 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 \ 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 xi

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 6 the
square root of this deflection multiplied by the total resistance
of the circuit (consisting of r S.U. together with 0'85 S.TJ. 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 IVS'S
I = 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 DanieU 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 difiiculty
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 dif&culty
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. ITurthermore,
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 instriunent 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 ab,
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 PHENOMEXOJSr WHICH ACCOMPANIES THE
ELECTEIC DISCHAEGE

{Wiedemann's Amuilen, 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 Leyden 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 ELECTPJC 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. Tor 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 flickering 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 ^'°' ^^'

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 strise 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 ELECTEIC DISCHARGE xii

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 em.
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 waU.
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 (hi 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. when 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 ELECTPJC DISCHARGE 219

first discharge. Eadiometer-like vanes of various kinds may
be set revolving eontimiously 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 ^V 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 ELECTKIC DISCHARGE xii

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 Euhmkorff 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 ELECTEIC DISCHAEGB 221

the luminosity to impurities/ 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.

10. 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 jpa^ssed 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 biirns 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-

' The hydrogen was prepared from pure zinc and dilute sulphuric acid.

222

A PHENOMEXOX ACCOMPAXYIXG THE ELECTRIC DISCHARGE xii

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
interesting information as to the produc-
tion of the bulge. First the bright
straight spark appears, and during its
presence the yellow is stiU absent or
cannot be seen owing to the dazzling of the eye ; it is followed
by the aureole (Fig. 2 7, i, /S), 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

b^ ^JJ;|,ViVii."i"^a^ ff _

•5

Fio. 27.

XII A PHENOMENON ACCOMPANYING THE ELECTPJC 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 ^'y- 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 yeUow light.

XIII
EXPEEIMENTS ON THE CATHODE DISCHAEGE

{Wiedemann s Annalen, 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 Euhlmann,
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.'' 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.

^ I was first induced to undertake these experiments by conversations whicli
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 EXPBEIMENTS ON THE CATHODE DISCHARGE 225

Dbsceiption of the Battery

The battery was based on the principle employed by
Poggendorff in his polarisation battery, and applied by Plaute
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, 10 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 constru.ction 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 OX THE CATHODE DISCHARGE iiii

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 capdlary)
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 merciuy in the commutator-cups. The
mercury then amalgamated the wire along its whole length,

XIII 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 sufiiciently 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,^ 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 Wied. Ann. 10, p. 244, 1880.

228 EXPERIMENTS ON THE CATHODE DISCHARGE xili

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,^ Goldstein,^ and Warren de la Eue.^ 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 ^V to yL 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 Tried. Ann. 10, p. 245, 1880. = Zoc. eit. 12, p. 101, 1881.

^ Ann. dc Ghim. et de Phys., series 5, 24, p. 461, 1881 ; and Phil. Trms.
169, p. 225, 1878.

XIII EXPERIMENTS ON THE CATHODE DISCHAEGE 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-cm-rent, 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 Eue and MuUer. 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. In addition to the tube and a large liquid resistance,

' Ann. de Ghim. et de Phys., series 5, 24, p. 461, 1881 : and Phil. Trans
169, p. 225, 1878.

^ 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 AUenkunrj," in which case the galvanometer would be acting
as a very delicate dynamometer.

230 EXPERIMENTS ON THE CATHODE DISCHARGE

xni

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 Wheatstone
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 G-eissler 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

EXPEEIMENTS 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
liquidresistancesof 700,000 S.TJ.each.
These were made by iilling thin glass

■' o o Pjq^ 28.

tubes 3 cm. long with a dilute solution

of zinc sulphate. The mean potential difference at the points
a and 5 — 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 6 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 milUon
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

XIII 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 9 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 DISCHAEGE xiii

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. coidd be perceived ; the current produc-
ing this rise was equal to j^ Daniell/S.U. The wire thus
formed a sort of dynamometer without any coil, and wdl in
future be referred to as such.'- Now the battery-current, which
in these experiments was as strong as ^^ 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 cm-rent 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
' See description of the apparatus in XI. p. 211.

XIII EXPERIMENTS ON THE CATHODE DISCHARGE 235

of the cathode could not have exceeded ^-^^ of a Daniell ;
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 -^-'g- 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 strife, 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.

(6) 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 Kuhmkorff 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 OX THE CATHODE DISCHARGE

XIII

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
]^ o\) (J 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-
sists at the value which enables the weakest
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,
1^ — and at any rate in those under consideration,

I — replace a gas-tube as a conductor of electricity.

Fig. 29. and 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 EXPEEIMENTS 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 /3 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 /3 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 pjr. 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 (Stromfdden) 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 strife

238 EXPERIMENTS ON THE CATHODE DISCHAEGE xiii

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 Eats indicate the Path
OF THE Current ?

As is weU 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 cui-rent. As a fuller knowledge of
the facts was attained this A-iew 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.^ 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, "\ATiat is the path

1 See, e.g. W. Spottiswoode and J. Fletcher Moulton, Phil. Trans. 171,
p. 649, 1880.

XIII EXPEKIMENTS 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

EXPERI.MEXTS OS THE CATHODE DISCHAEGE

XIII

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 e^en one-tenth of a
scale-division in the telescope. The
strength of the current was from j^
to ^-J-Q 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
3-^ part of the 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 paraUel-
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 fi-ame. The
plates sustained safely the powerful pres-

Fia. 31.

XIII 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 a ^^

free use of any fatty sub- ^^^ I "■ I

stance would have been ^
fatal. Fig. 31 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. Tlris 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 tlie 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 wlrich
has been already described, at such a height that the exhausted
case could be moved about underneath without touching it.
M. p. B

2i2 EXPERDIEXTS OX THE CATHODE DISCHAEGE xai

It was only 2 Trim, 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
Y^ to -:tj)X) 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 hnes,
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 Bot
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

' Vol. ii. p. 264, 18/3.

XIII EXPERIMENTS ON THE CATHODE DISCHAEGE 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, h, and c.

244

EXPERIilEXTS OX THE CATHODE DISCHARGE

XIII

In these a denotes the blue cathode light, /3 the positive
strife; 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 h 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 constructinw

each figure some fifty

Fig. 32, a (^ nat. size). , • . j i3 ..■

to Sixty deuections
were u^ed ; these were not distributed unifoiinly 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

1 . 1 1 Fig. 32, 6 (A nat. size).

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 cleai

XIII

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. Eoughly
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 '''°- ''■ " <^ ""'■ ''''^■

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.^ De la
Eive'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 Eive's experiment the
action is undoubtedly electromagnetic ; but it takes place at a

^ See Goldstein, Vber eine Form der elektr. Abstossung, iii. Teil.

246 EXPERIMENTS ON THE CATHODE DISCHARGE xiii

speed which is very easily measurable. And in every actual
electromagnetic eifect the ponderable substratum of the
current is set in motion ; which is not the case with the
deflection of the discharge.-^ 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 he 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
accoimt 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, TFied. Ann. 12, p. 262, 1881.

XIII 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 EXPERIJIEXTS OX THE CATHODE DISCHAEGE xiil

neighbourhood of the cathode, was not due to the direct action
of the ciirrent, 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 striee 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
strife 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 dhect
effect of the current, but arises indirectly through an absorption
of the cathode rays ^ 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
strise, 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.

' 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 sti'iffl.

EXPEEIMENTS ON THE CATHODE DISCHARGE

249

III. Have the Cathode Eays Electrostatic Properties ?

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 resiilts are not so sharp as they otherwise might have

_^c^^^^^^^^

been ; but the conclusion to
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.

Flo. 33.

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 protectiug 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 cotdd be estimated at two to three thou-
sand scale -divisions. When the cm-rent was stopped the
electrometer needle went back to about its old position; and
this could be repeated at will Xow 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. Allien the connection between the quadrants
was broken, the needle, in consequence of iiTCgularities 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 imelectrified.
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 xiii

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, actiag perpendicular to the dii-ection 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 wu-e which produced the
shadow, the rays had to travel a distance of 12 cm. between
these strips. The latter were fii'st 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 liy an electromotive force of one Daniell per mUlimetre 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 Eulamkorff 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.

XIII EXPERIMENTS ON THE CATHODE DISCHAKGE 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 xiii

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.'-

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^ and E. Gold-
stein.' 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.

^ 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 IVied. Ann. 10, p. 249, 1880.

s Loc. cit. 12, p. 265, 1881.

XIV

ON THE BEHAVIOUE OF BENZENE WITH EESPECT
TO INSULATION AND EESIDUAL CHAEGE

(Wiedemann' s Annalen, 20, pp. 279-284, 1883.)

EOWLAND and Nichols have shown^ 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.^ The benzene is contained in a zinc canister {B, Fig. 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 PUl. Mag. (Series 5) 11, p. 414. 1881.
2 Wied. Ann. 9, p. 160, 1880.

256

RESIDUAL CHAEGE

XIV

Fig. 3i.

formed the inner coating, and the zinc canister the outer
coating of a Leyden jar, of wliich 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 ^ 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 /3, and open at 7, the
needle of the electrometer will clearly
stand at zero, and the circuit wUl 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: 1.^ The whole potential of the
battery would have deflected the needle 5500 scale-divisions
from its position of rest. ISTow suppose that the connection
to the electrometer at 8 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 j 5500 of its value. In the absence of the electrometer it
would have fallen (4.5 -i- 1) times as rapidly, i.e. through

its value in 47r.lOOO/«
be divided by the specific

x/1000 in a second, or to l/e*" of

a/iUUU m a secona, or to i/e'" 01 its value m
seconds. This latter time is to be divided by
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

^ The large ajiparent capacity of the quadrants is due to the sti'ong charge on
the electrometer needle.

XIY

KBSIDUAL CHABCtE

257

::>

-o

"r-J

1 )

"^ the residual charge the outer coating
was disconnected by the key /3 from
the battery, and connected by 7 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 oi¥. 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 10 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 valtie ; ■'
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. Fig. 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.

M. P.

258 EESIDUAL CHAEGE xiv

conducthaties 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.-'

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 ciirrent. 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.

^ 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
Leyden 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 EESIDUAL CHARGE XIV

this corresponded to a portion of the residual charge which other-
wise would have developed upon the plates A and B. Xow, 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 ii-regular : 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.

ON THE DISTRIBUTION OF STRESS IN AN
ELASTIC RIGHT CIRCULAR CYLINDER

(SchUmilch's Zeitschrift fur Math. u. Physik, 28, pp. 123-128, 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„ 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 (771, 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

M,,= - p cos in,m) + lh ^°^ ^^''^ '"' ^"'^'^ ''' ^'''''\ls

""J '' (1).

262 STRESSES IN A EIGHT CIECULAE CYLINDER xT

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 pressui-es X„ Yj,, Y^., which are
independent of the third coordinate, form a possible system if
they satisfy these differential equations

0=?^ + "^, qJ1i + ^, ^!Ii + %=2^ (2).
dx dy Cx dy "dy- dor oydx

Let p, (o be polar coordinates, p cos co = x, p sin (o=y, and
in the stress-components P^, P„, £1^ let a> denote the direction
perpendicular to a> ; then the system of stress

-p^^co^^ P„ = 0, O„ = (3)

satisfies the equations (2). This is proved by calculating the
values of X„ Y^, Y^., 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 (m.p)

A sum of such M„ with different poles and multiphed by
arbitrary constants will represent a system which satisfies the
differential equations (2). Xow the integral which occui-s 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 foUows 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„ coincides with the
component of F in the dii-ection ??i ; so that M,^ = F cos {m,n).
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

XT STRESSES IN A EIGHT CIRCULAE CYLINDER 263

o-D / ^ COS (p,r) COS (n,r) 1

r = 2E cos (p,r), ^^i^^ = ^-^-^ = -— ;

r r ZK

so that this part of the integral

= — I F cos (f,r) cos (m,r)ds
Ett I

= I- ({F cos (/,m) + r cos [(f,p)-(m,p)])ds,

-iXlTT J

since (/,p) + (m,r) = (/,m) ; (/,r) - (m,r) = (f,p) - (m,p).

Now since the forces P must not produce a displacement
of the cylinder in the direction m, nor a rotation round the
axis,

Jf cos (/,m)& = , |F sin (/,p)cfe = 0.
Hence the part of the integral examined is equal to

cos {m,p) F cos {f,p)ds,

2E7;

and thus cancels the first term in M„; and M„ 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,7'). Hence

^2lS;^ds = ^^^^P^ds = d(p,r),
r r

and as F, / may be regarded as constant over the smal.
portion of surface considered, we have

M„ = - F I cos (/,r) cos {m,r)d{p,r)

_ 77

= - F I cos [(r,p) - (/,/>)] cos [(r,p) - (m,p)]d(p,r)

TT J

264 STEESSES IN A EIGHT CIRCULAE CYLINDEE xv

2 2

F f FT

= - cos [2{r,p) - (f,p)']d(p,r) +- cos (/, m) rf(r,p)

TT I TT I

= F COS (f,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 i\, fj
the distances of the element considered from the points of
contact. Then the component of stress normal to the element
considered is

_ P 2P J cos (r^*) cos^ (''I'O , cos (j^x) cos^ {r^n)
Ett TT i 1\ r^

If we determine the direction n so that N„ becomes a
maximum or minimum, keeping r^, r^ the same, we get the
values and directions of the principal stresses at the point
(r , r ). 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

2 3E^' ,. ^ _^
' - "' " ■ ^ Ett'

XV STRESSES IN A EIGHT CIRCULAR CYLINDER 265

and for the axis of y

Ett E* + 2Ey + if " EttVE- + if

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 G/tt times what it
would he if the pressure P were distributed uniformly over the
whole section 2E. Even in this simplest case it appears that
the distribution of stress is extremely complicated.

XYI

OX THE EQUILIBEIUM OF FLOATIXG ELASTIC

PLATES

{TTiedeiruinns Annalen, 22, pp. 449-455, 1S84.)

Suppose an infinitely extended elastic plate, e.g. of ice, to float
on an infinitely extended heavy liqiiid, 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 eifects 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 so, y, of which the
plane coincides with the plate, supposed infinitely thin.
Further we write v" = '^' , ^■'-'^ + c'", ci/-, p^ = or + y", 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 {E/(Yl2(l - /u.^)}v"- is
the upward pressure per unit area due to the elastic stresses,^
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 siace
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

^ Clebsct, Thmrie der Elastidtat, § 73, 1S62.

XVI FLOATING ELASTIC PLATES 267

12(1 -/x>_ 4_ 1

E/l«

our problem may be stated mathematically thus : Eequired
an integral of the equation y*z + ci^z = 0, which vanishes at
infinity, is finite at the origin, and is such that the integral of
sa*fy*«dw taken throughout the neighbourhood of the origin
may be equal to P.

With Heine ^ we write

then K(p) is a solution of the equation y's = - s, and therefore
K(p ^ — a*) is a solution of our equation. And

z= ^A K[ap ^/i(l + i)] - K[ap v/4(l - i)]\ (1)

is also a solution. It is real, and if we bring it into a real
form we get by transforming the integral

z = ^ (e-''i'^^sm ap \/lvdv (2)

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,^
first in a series of Bessel's functions, then of powers of p,
and thus obtain

a'Yjay ay ,, ,. ^

+ T 1

4\ 2^4^ 2^4^ 6^8^ ■■■/ (3>

-a+.og2-o,(f-,^+.

Heine, Haiidluch der Kugelfunktionen, vol. i. p. 192, 1878.
^ I.e. p. 244. The sign of C is wi'ongly 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 % remains
finite when p = ; further, the integral over a small circular
area surrounding the origin is

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

z =

o — A. / ^ — 7= 1 sm ap Vi^ + -

'ap

-g-sinUpVi + ^) +

(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 = Zq = a^PjSs. 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 = '64680 ; for p = 2a, s = ■2082:0 ; for p = 3a,
z = •066»o- Near the distance p = -g-Tr \l^a = 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 tt \/2 . 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= lO'^kg/mm^; ^ can betaken as^; and.

XVI

FLOATING ELASTIC PLATES 269

according to Eeusch/ E is equal to 236 kg/mm^. We thus
obtain for different thicknesses h —

h= 10 20 50 100 200 mm.
a =0-38 0-64 1-27 214 3-60 m.,

whence we easily get the depression produced by 100 kg.

2o=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 s 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 maximrmr
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 y^^o ^^ ^^^ origin. If we call the distance from the
origin of the element at which dV rests p, then the portion of
V% due to this element is, by equation (3),

^ (log cip - log 2 + C), -
ztts

where the terms which vanish with p have been neglected. A
simple integration now gives

V%=2g=2^^ = |^^(log«E-i-log2 + C)

= — (log aE- 0-65 19).
27rs

The maximum tension at the centre of the curved plate
is p = (E/i/2)9^»/9a;^ ; by forming the expression for f and
substituting for a* its value we find

3(l-/x^)P
V= \^[2 (logftE-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 Reusoh, TFied. Ami. 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 E equal iA in
the preceding formula ; thus we get for the greatest tension
which the weight P can produce at all in the plate

P= \IJ (log 3 -1-3090

For example, in the case of the plates of ice just con-
sidered, we get for a weight of 100 kg. the values

^=221, 53, 8-1, 1-9, 0-47 kg/cml

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

27rs , ,
27r szpdp = - —4" V « ■ P^/s = - P>

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 ^ must
hold

At the centre we have the same condition as before. The
solution is to be compounded of the three integrals of the
equation y*» + a*2 = 0, 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 » = at
the edge, is a limiting case. When we enquire under what
conditions the equations (a) (&) (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 2'5a and 2-8a. It is equal to 2-5a when /a=0,
and to 2'8a when /i = -g-. 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

' Clebsch, Theorie der Masticitdt, § 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 AiinaUn, 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
their magnetic actions. Whether Ampere actually started
from this principle or not, he certainly stated it at the close
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.

XVII 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 ^ A^ of the rates of diminution of the

' 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 FUXDAMEXTAL EQUATIOXS OF ELECTEOJIAGXETICS xvii

moments of the magnets per imit length measured in electro-
magnetic units, and of the integral |(cos elr)dede', where de,
cle' denote elements of the axes of the magnets, e the inchna-
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 ^ 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 cmrent 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 cm-rents 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

^ 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 OE 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 cm'rent, 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 FUXDAMEXTAL EQUATIONS OF ELECTEOIIAGNETICS xvii

a magnetising force at the instant when the intensity of the
field is reduced to zero ; and tliat 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. Prom 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.-^ The correction which must be
made in the laws of the magnetic actions of constant currents
to make them applicable to variable currents may he
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

^ Cy. V. Helmholtz, "Uber die Theorie der ElektrodyDamik,'' Wissenschaftlvh
Abhaiidlungen, 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^U = - 47rw ; hence generally

U = w = - dr,
J '■
the integral heing 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

(Sii /bv dw _ .
dx dy dz

Further let Uj = S, Y^ = v, W^ = w. Then the components
Lj M^ Nj of the magnetic force exerted by the currents
are, according to the usual electromagnetics, given by the
equations

1 \ de

dx dz J ' dx dy

M^ = A(^-^i), -;il + ^+^=0 (1).

\dy dx

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 Xj Yj Zj are

Xi=-A^'^\ Y,= -A-'^, Z,= -A''^ (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

280 FUXDA^ilENTAL EQUATIONS OF ELECTEOMAGXETICS xvn

consisting only of electric currents which still neutralises the
forces of system A eyerywhere. Such a system is possible;
we need only choose as current-components m, v, w where
4:-n-u = V-Uj, 47rr = V-V^, 47r«' = V-Wj. 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 ; heuce 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, ^, v be
the components of magnetisation throughout space, and let

^_^ + |^ + ^^=0,andA = X,M = /Z,N = 5.

ox oy 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, fjL, V has for components

"- dt\d}/ ?~J dt\dz dxj

dt\ dx dy

"We now put, in accordance with our notation,

dX da dv

^ dt ^ dt dt

^ Cf. V. Pelmholtz, IFusenschaftliche Abhandhmgen, vol. i. p. 619.

XVII FUNDAMENTAL EQUATIONS OF ELECTKOMAGNETICS 281

and call p, ([, r the components of the magnetic current.
Further we put Pi=^, Qi = q, Ei = 'r and call Pj, Qi, Ej the
components of the vector-potential of this current. Then the
electric forces produced by the magnetic current are

\ dx dy

The reasoning which allows us to infer from the forces (1)
that the mutual potential of two electric current-systems
%, ■Wi, Wj and Mj) '^2' '^2 l^S'S the form

A^ I 1 — {U^U^ + VyV^ + W-^W^dT^clT^ ,

leads to the conclusion, using forces (3), that the magnetic
current-systems p^^, q^, 7\ and p^) 2'2> ''2 have the mutual
potential

^ - (P1P2 + 2i2'2 + V2)dT^dT.

2 •

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

Li=-A2^i, Mi=-A2'^, Ni=-A2— 1 m
dt ^ dt ^ dt ^ '■

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 cui'rents 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 ELECTEOMAGXETICS xvn

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

Kdv

\dz

=Am-

\dx

Assuming for the present that

? + f^ + ^^ = (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

lA|/iv,-^W,
4cTr dtVdz oy

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, E follow the magnetic forces
produced by their variation. The ^-component is

-A=^-?=-l-A3^/^y,-iw,

dt 47r dt\dz ' dy

This term we must add to the component Lj of the previously
assumed magnetic force. Let us call the component thus

XVII FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS 283

corrected Lo and form the similarly corrected components
Mj, N2 ; then these forces may be represented by the system

^, A /9W2 aUA dXJ dY„ dW, „

\ 9a; ciz / dx dy dz

\ dy dx
where we have put

47r dr 47r at

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^^ -A^^^ Y2= -A=='il?, Z2= -A^~^ (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

x,=Ar-p--^],Bic. (7),

\ dy dz

L,= -A^^^etc. (8),

1 d^ -
where P = P _ — A^ -^ Pj , etc.

47r dt-

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 xvii

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

-,r TT A^ d' - A^ £- A* d' =

and similar expressions for the other components of the vector-
potential. The terms in A* 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, IST, X, Y, Z the completely corrected forces and obtain

/ay dW\ ^(W

\dz dy ) dt

\or/ dx J dt

XVII FUNDAMENTAL EQUATIONS OF ELECTEOMAGNETICS 285

where

now

have for U, V, W

U =

= u-

A' £

d^-

AttcW^'^

IGtt'^

•dt^'' ■

V =

= v -

W =

--W ■

A' d' ^
-A'Trdt^"^

1677'

su av 3w _

da: dy dz

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 w in a certain element of space. We resolve
this current into a series of simple harmonic functions of the
time and suppose Mq 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 nt/ ^ 1 A^ ., .,

dU = dr-^ 1 — n-r

r \ 1 . 2 47r

1 A*

+ = ~nh^ + ...

1.2.3.4 IGtt^

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^U= -47^^. + A^^^S-...

286 FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS xvii

and

hence

clr dt-

dt^

The other components of the vector -potentials, both
of electric and magnetic currents, satisfy analogous differential
equations. Since u, v, w, p, j, '/' vanish in empty space, the
distribution of these potentials is there given by the equations

V2U-A^i4=0, V2p-A2^ =

di- de

V-V - A^'^ = 0, V^Q - A^'^-!^ =

(11).

dtr ^ df

V^W - A^^ = 0, V^E - A^^ =

dr dr

dx dy dz dx dy 9»

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 he 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-

XVII FUNDAMENTAL EQUATIONS OF ELECTEOMAGNETICS 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.

dt^ dy\dy dxj d\'dx dz

and removing the functions of U, 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 : —

(12).

These same equations connect together the forces produced
by magnetic currents, for they are got by eliminating P, Q, E
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^L-A^^ =0,
dt^

d-L

dt '

^IX dU

dt ~ dz

~-dy

' dz '

dM.

, dY 31^

A-— =

dt dx

A — =
dt

" dz '

dx '

" a»

A — =

^dZ 8L

dt ~ a?/

V^M-A^'^ =0,
dt'

V^N-A^^ =0,
dt^

aL aM aN^Q x 3y az^^

dx dy dz ' dx dy dz

V^X-

^ df

V^Y-

A2^^Y_
df

V^Z-

,d^Z_
^ df

(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 changincr
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-MaxweU 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-
MaxweU 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 imit 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. Xot 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).

xvil FUNDAMENTAL EQITATIONS 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. "■ 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 elect:^ic 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.

^ 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 ELECTKOMAGNETICS xvii

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
DIFFEEENT SYSTEMS OF UNITS

{Wiedemann's Annalen, 24, pp. 114-118, 1885.)

Two years ago this subject was discussed in the Annalen} and
even more vigorously in the FhilosopJiical 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 ^ 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 ;

^ Of. Clausius, Wied. 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.]

^ Called by Clausius electrodynamic. [In this volume, as in the Electric
Waves, " elektrodynamisch" is generally rendered as "electromagnetic." — Tr.]

292 DIMENSIONS OF MAGNETIC POLE xvin

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 assum]ptions 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
ejt of the current : it is independent of geometrical relations.
If then we put A = k^me/t, k^ 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

[m][e]=MUT-' (M).

(&) The moment mS 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^efjt, where again
k^ is a constant which depends only upon the imits chosen.
If /cj is a pure number, k^ in general will not be so ; and con-

XVIII 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^ shall be a number of no dimensions ; from
which it follows that

[m] = HLT-i (C).

The consequences of the assumptions (M) and (C) are as
follows : —

1. In the magnetic system we start with the dimensions
[7ft] = M-L-T"-'. From this we deduce, as the dimensions of
electric pole,

WU, 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-T"\ Prom this we deduce, as the
dimensions of magnetic pole,

IVI^L=, according to (M) ; WU'J''^, 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." ^ 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 mjt
of the current ; it is independent of geometrical relations. If
then we put A = k-^e.mjt, what has been stated for ki and k.^

1 See Xyil. p. 276.

294 DIMENSIONS OF MAGNETIC POLE XVlii

will hold good for k^. 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 k^ a pure number. We thus arrive at
the assumption : —

[m][e] = IVIL^T-^ (MO,

which also coincides with (M).

(V) For purposes of calculation an electrical doublet
can be completely replaced by a small magnetic circidar
current, whose plane is perpendicular to the axis of the
doublet. Thus the moment eh of the doublet must be pro-
portional to the strength mji of the current and the area
which it embraces. Hence we put eS = k^fmjt. 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 munber.
We should then have : —

The consequences of the assumptions (M') and (C) are as
follows : —

1. In the magnetic system we still have

[m] = W{}J-\

Hence the dimensions of electric pole would be deduced as

M*L*, according to (M') ; M*UT-^ 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

W = M^UT-^

Hence in this the dimensions of magnetic pole are

M^L*, according to (M') ; M*L^ 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 (M') 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,^ 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] = [?)i] = M^L-T"-', and introduces factors
with the dimensions whenever electrical and magnetic quantities
occur together.

1 Wied. Ann. 17, p. 48, 1882, [Phil. Hag. (5) 14, p. 436, 1882.]

XIX

A GEAPHICAL METHOD OF DETEEMIXING THE
ADIABATIC CHANGES OF MOIST AIR

(MdcorologisAe Zeitsehrift, 1, pp. 421-431, 1884.)
(AVith diagi-am at end of book.)

Ix 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 formidas. In practice they
generally refer to the small but useful table pubhshed by
Prof Hann in 1874.^ 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.^ 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 fuUy,
by means of a numerical example, the use of the diagram
(which may be purely mechanical). By following this example

^ Hann, Zeitsehrift der osterrcicMscken Gescllschaft fiir Mctcorologie, 9,
p. 328, 1874.

^ The editor of the Met. Zeitschr. gives a reference to Guldberg and Mohn,
Etudes sur les mouvements de VatmospMre, 1, pp. 9-16, and Osterr, Zeitsehrift,
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 /x
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 — c that of the dry air, v the
volume of 1 kilogramme of the mixture, we have

ET E,T

p — e = X , c = u, ,

V V

where E, E^ are constants of known meaning and magnitude.
Since the total pressure is the sum of these two values, we
get pv = (XE -|- /iEj)T, and this is the so-called characteristic
equation of the mixture. Further, let c„ denote the specific
heat at constant volume of the air, c/ 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

^Qi = X( c,dT + AET— ],

and the water-vapour with

c^Qs = Jc^dT -H AEjT— Y'

(?r both together with the amount of heat

dq = (Xc„ -I- /jicJ)dT + A(\E + /iEi)T— .

But this amount of heat vanishes for the change we are
considering. In order to integrate the differential equation

^ Clausius, Medianische Wdrmetlieorie, vol. i. p. 51, 1876.

298 ADIABATIC CHAifGES IN MOIST AIE 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^ + AB, equals c^ the specific
heat at constant pressure,-' we obtain

--A(XE + /.P4)log^
^0 i'o

= (Xcp+ f^cj) log - - A(XE + ^T^) log ^ (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 limi ting value of p down to
which equation (I) may be used. Now and in what foUows
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

i-e , ,

EiT '

and this quantity must exceed /u., as long as the vapour remains
unsaturated. Thus the limit is reached as soon as ^ = y. If
we introduce the value of v from the characteristic equation,
this condition takes the form

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

' Clausius, Medumische TFarmetheorie, vol. i. p. 51, 1876.

XIX ADIABATIC CHANGES IN MOIST AIE 299

temperature T as the mixture ; but the pressure p of the
mixture is the sum of the partial pressures Pj = X — of the

air, and p^^^e of the water vapour. The equation
p = X 1- e, or (p — e)v = XET

is now the characteristic equation of the mixture. The amount
of heat necessary to produce the changes dT and dv is for the
air as before

cIjV
tZQj = X(c„c^T + AET-

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\
-^j+WT.

The equation is deduced by Clausius in his MechaniscJie
Warmetheorie, vol. i. part vi. § 11. c is the specific heat of
hquid 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 = xfc^dT + AUT—\+Tdr-^j+ficdT.

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

= (Xc^+ f,c) log;^ + XAE log^?^^^^«
T(, p-6

+ X^f^^— ^^ (II).

The quantity equated to zero again represents the difference

300 ADIABATIC CHANGES IX MOIST AIR xix

of entropy batween 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. ISTow 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 /a which
exists as vapour, a 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 dT = 0, we need only supply the air with
the heat XAHTdv/v, the water which is evaporated with the
heat rdv, and the water which is frozen with the heat — qda.
Hence the whole of the mixture is supplied with the heat

dv
dQ = XAET h rdv - gd<r.

As before we put cZQ = 0, divide by T, and integrate, and
thus get

= XAE log^^ + r( J, _ ^^) _ |(<, _ ^^)_

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 a- 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 temperature 0°, in which therefore ice was not
present, and where o"g = 0. The iudex 1 refers to the state
in which all the water is just frozen, in which therefore the

XIX

ADIABATIC CHANGES IN MOIST AIK 301

temperature just begins to fall below 0°. Here clearly a = fx — v,
since ice and vapour alone are present. If we insert these
values, after introducing the pressure, we get

=xAElog^-'' — + X— ^ -^ir - — : t

Pi — e El ^1 - e T Ej p„ — e i

-4 (III)-

This equation connects the pressures p^ and ^j, 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 shaU 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 fj, in all the equations. Then we substitute
the values p^ and T^ for the initial state in equation I. The
resulting equation and equation (6) 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
Pq and Tj in equation II. By putting T = 273 in the resulting

302 ADIABATIC CHANGES IN MOIST AIR Xlx

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-^ of the third stage, this pressure and the tempera-
ture 273° constitute the p^ and T,, in the equations of the
fourth stage. It wiU 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 p 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 A, which corresponds to the different states
p^ and p-^, 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 vdp,

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
fjX, 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 /i ; 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
abscissae 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 /jl is neglected in comparison with X, the adiabatics
of the first stage are given by the equation

const = Cp log T — AE log^.

The logarithms are natural ones throughout. With Clausius
we must put

calorie
^ degree C. X kilogr

1 calorie

A = ■

4 2 3 ■ 5 5 kilogramme-metre '

304 ADIABATIC CHANGES IN MOIST AIR xix

kilogramme-metre
~ degree C. x kilogr '

These adiabatics 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

0-0025:5 7i r^ •

degree (J. 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 ^

Ere
const = Cj, log T - AE log p + ^ • Tj, • - .

E/Ei is the density of aqueous vapour referred to air, i.e.
0'6219. r, according to Clausius, is equal to 607 — 0-708

pal Qfl p

fT — 273) • The values of e for the various tempera-

^ ^ kilogr

tures I have taken from the table calculated by Broch.^ 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, but yet are not quite the same ; for their
equation is got from that of the former curves by putting
r+q for r, where §'= 80 calorie/kilogr. They are marked 7

^ Even thougli /j, is neglected in comparison with X, yet it is doubtful
whether c/j. should be neglected compared with c^K ; for c is four times as gi'eat
as Cp. Though in the diagram /m is no more than -jV of X, still the ratio Cfi^jCpX
amounts to ^. 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.

^ Traveaux du Bureau international despoids et mesures, tome i.

XIX ADIABATIC CHANGES IN MOIST AIR 305

and are drawn according to the same scale as a and ^ ; but
in general they are not continuations of the curves /3.

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 = Ee/E^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
w; 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 ^q, corresponding to this inter-
section, and II, the amount of water present, determine ^j, the
pressure at which we must pass from the third to the fourth
stage. To determine p^ 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 fju of water in all the stages,
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-^ and fi, 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 Pi=Pq-
The corresponding value of fj, is not zero, but is equal to v,
the least value /i 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 fi we require
the corresponding value oip-^, we must look out the inclined line
whose highest point has the abscissa p^,, and follow it down
to the ordinate jjl. The pressure at which this ordinate is
reached is p-^, the pressure sought. With it the point of
transition to the fourth stage is found.
M, p. X

306 ADIABATIC CHANGES IN MOIST AIE 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 jx
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
S=j?/ET or log S = logjp - log T - log K. 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,
S, 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. Prom the points 1 and 2 draw two straight lines
parallel to h and cutting the isothermal for 0° C. ; and read
off the pressures p^ and p^ at these intersections. The densi-
ties at 1 and 2 are in the ratio Pi'.Pi- ^or the densities in
the states p-^, 0° and p^, 0° are by Boyle's law in the ratio
Pi'.p-i, and they are equal to the densities in 1 and 2, as they
lie on lines of equal density.

3. The difference of height A, which corresponds to the
passage from the state j?o to the state p, on the assumption of
an adiabatic equilibrium state, is given by the equation

p p

Pa Pa

= \vdp = vA'\

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 Tj = 273, the error in h will hardly ever exceed l of
its total value. If we choose to neglect this error, we get li
= const — ETg log _p, and may at once introduce the heights
as well as the pressures as abscissae. 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 -va]5our 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 iirst given by
the Line of system a which passes through the initial state.^
There is no such line actually drawn, so we must interpolate

^ The letters a, /3, 7, 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 CHAXGES IX MOIST AIE 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 '•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 /8 and follow
its course. This curve is much less inclined to the axis of
abscissae 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°'2, i.e. only
0^'51 for every 100 m. We find ourselves on the dotted Hne
8 "9, and thus see that 8 '9 grammes of water stiU exist as
vapour, so that 2"1 grammes of water per kilogramme of air
have been condensed in this first thousand metres of the cloud-
layer. We reach the temperature 0' at a pressure 472 mm.
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 tem-

XIX ADIABATIC CHANGES IN MOIST AIR 309

perature cannot fall any fui-ther, 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 Hues
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 j) = 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 S. 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 AIE 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 -^^o^ qj. _i_ {jji^n 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 7 passing through this point.
It is of some interest to inquire how high the dew-point of
our mixture may be in its initial state of pressure and

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 witli the dotted line 5-25. Thus we cannot have
more than 5-25 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°'8. This
is the required highest value of the dew-point.

[Tlie following editorial note occurs at tlie end of the number of
the Meteorohgische 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 Olausius, 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. pressiure, 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 formulas 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 woiild be required for accurately calculating Guld-
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. Heetz."

Kiel, 8th Dec. 1884.

ON THE EELATIONS BETWEEN LIGHT AND
ELECTEICITY

A Lectuke delivered at the Sixty-Secoxd Meeting of
THE German Association foe the Advancement of
Natural Science and Medicine in Heidelberg on
September 20th, 1889.

{Puilishecl by JEniil 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,
Hght 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 nmnber
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 ELECTEICITY 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 1
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 ELECTEICITY 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 XX

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 suf&ce 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 deUcate 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

32'2 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 juillionth 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

Thev are the peculiarities of the instruments with which he
has to work ; and the success of a workman depends upon
whether he properly undei-stands 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. "VMien 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 foUow each other at regular and equal
distances : from this he determines the wave-length, and, if lie
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 ^ve use
an oscillating conductor. In place of the resonator we use
our interrupted wire, which may also be called an electric
resonator. "We oliserve 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
measm-ed. Xext 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 founcL "We multiply
the measm-ed 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 calcidation is trustworthy,
there is stUl 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 ELECTEICITY 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 foUow 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 AND 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
organ — the eye. 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 fiUs 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 LAYEES

[Wiedemann' s AnnaUn, 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 CATPIODE 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 KAYS 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 ^ of the cathode
rays to pass, it will not reduce the luminescence to -J- but only
to I 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 BAYS 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 fiat ; 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
cathode 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.^

^ See Wiedeinann's Annalen, 15, p. 246, 1882.

XXII
HERMANN VON HELMHOLTZ

(Fi'om the Supplonu'ut to t\\a ilihichc^icr AHijemrinc. Zcitnini, Aii^'ust 31, 1891.)

In Germany the men who now stand uimu the thrt-shold of
old age have inaugurated and lived through a pi^iod ol' rare
felicity and success. They have seen a.ims attained nud
desires realised, and this not only in matterw political : they
have seen mighty developments in tho arts of pea,ee ; tliey
have seen our Fatherland take its plae(^ in the fvont rank of
nations, not only in t)ur own entiiiiation but in that of otlieva.
Even in the beginning of tliis century tlie natural sciences
were far from being neglected in (ieiniany: the labour.s of a
Humboldt, the undying fame of a (iauss, were sullicieut to
keep alive respect for (iernian rcHoarch. But side by side
with the wheat of true effort there sprang up the tares (if a
false philosophy which flourished so luxuriantly as to hinder
the full growth of the crop. Up to the middle of the ciMitury
sober progress along this path ol' experimental investigation
lacked the glory which aecoiiipanics internatinual success; and
the successes of a fictitious natural phihisophy were vi'ry pro-
perly not greeted with the same I'xultation abroad as in Ger-
many. Germans followed eagerly and diligently the, discoveries
made in other lands ; but they always expected the grt^at dis-
coveries and successes to cdiuc^ from Paris and London. Thither
young investigators travelled to see famous scientilic men and
to learn how great investigations were earrit'd cm : thence they
obtained the materials for th(iir own research ( ^s ; then; alone
could new discoveries be ])roperly and authentically ])ublisiicd.
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 HEEIIAXN 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. "\Ve 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 himseK
hampered by the gaps in our knowledge of inanimate nature :
so he set about fiUing 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
lie has enriched practical medicine. I mean the ophthal-
moscope. Before him no one was able to investigate the

XXII HEEMANN 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
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 iirst 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 IIEEJIANN VOX 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 tui-ned 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 defined and hidden by a luxuriant
growth of error. He left it carefuUy defined 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 oiu'-
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 Eesthetic opposition between the two depend ? By what
ideas of order within us were those codes of music, the musical
scales, developed ? JSTot 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

XXII HERMANN VON HBLMHOLTZ 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

338 HERMAHN VON HELMHOLTZ xxil'

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 stiU
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

xxn 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 result
of a designed wisdom : its general statement in the region of
pm-e 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 HEEJIANN VON HELMHOLTZ XXII

work. Here exaraple 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 things 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 wiU 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.

Printed ^y R. & R. Clark, Limited, Edhiburgh.

Internet Archive Audio

Featured

Top

Images

Featured

Top

Software

Featured

Top

Books

Featured

Top

Video

Featured

Top

Mobile Apps

Browser Extensions

Archive-It Subscription

Save Page Now

Full text of "Miscellaneous papers"

See other formats