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COLLECTED
SCIENTIFIC PAPERS
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COLLECTED
SCIENTIFIC PAPERS
BY
JOHN HENRY POYNTING, Sc.D., F.R.S.
Mason Professor of Physics in the University of Birmingham
Formerly Fellow of Trinity College, Cambridge
CAMBRIDGE
AT THE UNIVERSITY PRESS
1920
PREFACE
IN the summer of 1914 Sir Oliver Lodge summoned a meeting of the
colleagues and friends of the late Professor John Henry Poynting to
consider how best a suitable memorial could be established to perpetuate his
memory. The following committee was appointed to carry out the purpose
of the meeting : Sir Oliver Lodge (chairman), Guy Barlow, Neville Chamber
lain, P. F. Frankland, Sir R. T. Glazebrook, R. S. Heath, George Hookham,
Sir Joseph Larmor, Sir Napier Shaw, Sir J. J. Thomson, Sir Richard Threlfall,
and T. Sydney Walker, with G. H. Morley as Secretary and G. A. Shakespear
as Treasurer.
A fund was opened and subscriptions were invited, and the Committee
decided that there could be no better memorial than the publication in a
collected form of the Scientific Papers of John Henry Poynting, and the
distribution of copies to certain scientific institutions throughout the world.
The task of editing and proof correction was gladly undertaken by his
junior colleagues G. A. Shakespear and Guy Barlow who have throughout
had the benefit of the advice of Sir Oliver Lodge.
Biographical and critical notices by Sir Oliver Lodge, Sir Joseph Larmor,
Sir J. J. Thomson and G. A. Shakespear have been inserted as an introduction
to the volume.
The papers have been arranged in groups, with the object of bringing
together those dealing with kindred subjects. In each group the papers
are in chronological order.
In such collections there is inevitably a certain amount of repetition or
overlapping, since a subject is often dealt with from more than one point of
view — the strictly scientific and the popular — but since this variety of treat
ment is helpful, in cases of doubt the decision has generally been given in
favour of inclusion.
The popular discourses and general articles have for the most part been
relegated to the last section of the volume.
His books, and other publications which are easily accessible, are not
included; but a complete list of his works in chronological order is given
at the end of the volume.
The papers have been reproduced as originally published except for small
verbal corrections here and there. In certain cases mistakes have been
y PREFACE
corrected after reference to the original manuscripts which were kindly put
at our disposal by Mrs. Poynting ; but, in all cases, corrections of any import
ance have been indicated in footnotes, and editorial comments have been
included in square brackets and marked Ed.
We wish to acknowledge most warmly the unstinted generosity with which
permission to republish the several articles has been given both by joint
authors and by the original publishers, including the representatives of the
following :
The Birmingham Natural History and Philosophical Society.
The British Association.
Le Bureau International des Poids et Mesures.
The Electrician.
The Encyclopaedia of Religion and Ethics (Messrs T. and T. Clark).
The Hibbert Journal.
The Indiarubber Journal.
The Inquirer.
The Mason College Magazine.
The Manchester Literary and Philosophical Society.
Nature.
The Philosophical Magazine.
The Royal Astronomical Society.
The Royal Society.
The Royal Statistical Society.
The Royal Institution.
La Societe de Physique.
: , The Committee was fortunate in securing the services of the Cambridge
University Press as publishers, and the editors take this opportunity of
thanking them for their unfailing courtesy and for invaluable help in proof
correcting.
G. A. SHAKESPEAR.
GUY BARLOW.
The University. Birmingh.\m.
1920.
A PERSONAL NOTE
John Henry Poynting was a man admired of all who knew him, beloved
of all who knew him well.
Of somewhat less than middle height and sturdy thickset build, his general
appearance was suggestive rather of rural than academic interests, »but even
a casual observer would probably have been struck with the sense of power
indicated by his fine head. His face, which was of a meditative cast when
in repose, lighted with a genial and friendly warmth when he was conversing
with friends, and his greyblue eyes were expressive of the kindly gentleness
of his nature.
In habit he was methodical, and indeed the condition of his health was for
many years such that he could not have accomplished so much had he not
economised effort with method. (He told me once that he had never been
able to do more than six hours' useful work a day.) He had, moreover,
great power of concentration.
He was a remarkably clear thinker and had that characteristic insight
into fundamental ideas which intuitively distinguishes between hypothesis
and fact ; and it was probably for this reason that he viewed with suspicion
some of the more recent developments of mathematical physics. He withheld
his judgment when the experimental foundations were either wanting or else
inadequate to bear the superstructure erected upon them.
Himself a man of wide interests and sympathies, and with a finely balanced
sense of proportion, he was keenly alive to the danger of that tooexclusive
specialisation which so frequently makes a man incapable of conversation
except in his awn particular line of work. He felt the need of guarding against
atrophy of the spiritual side of his character in his outlook on life, and sought
in the reading, of fiction, and even more of poetry, the complement to the
intellectual stimulus of scientific work. Indeed of poetry he read much,
though he spoke of it but rarely even with his intimate friends ; but those
to whom he opened his mind on such matters knew his deep admiration
for many of the English poets. Shelley, Keats and still more Wordsworth,
appealed to him strongly, for he himself was imbued with that love of Nature
which inspired them.
He was probably never happier than when living in the beautiful
Alvechurch district of Worcestershire, where he found an unfailing source
of pleasure in the rolling landscape which stretched away into the distance
Viii A PERSONAL NOTE
around his upland home. He would walk in the fields with some friend,
here pointing out the haunt of a rare wildflower, there showing, with all the
interest of a schoolboy, the nest of an uncommon bird, or discussing the
effect of atmosphere on the landscape. The conversation would wander
from the botanical afiinities of the twayblade to the mechanism whereby
the grasshopper performs his prodigious leaps, or from the theory of the
grinding action of a ciderpress to the causes and prevention of crime in large
cities. To such occasions his equable temper and ready sense of humour
lent a rare charm.
Among his outdoor recreations cycling through the country lanes held
a high place. Indoors, in his later years, he derived much pleasure from
a pianoplayer ; and in the evenings, when too tired to read, he would often
amuse himself with a game of "patience."
In politics he was a liberal, and perhaps the thing for which he stood
more strongly than anything else was freedom and liberty of thought; the
thing of which he was most intolerant was bigotry, and indeed in an intimate
aquaintance of 20 years' duration the only time T ever heard him speak of
any man with bitterness was in reference to a case of religious intolerance.
The soundness of his judgment being well known, his advice was often
sought ; and, whether the matter were small or great, he always gave of his
best, earning thereby the gratitude of many.
He had a great sympathy with humanity in general and especially with
the poorer classes. As a magistrate he tempered judgment with mercy and
his experience in this capacity confirmed him in the opinion that delinquents
in general were as often sinned against as sinning.
The comradeship of his homelife was ideal, while to his students he was
an object of admiration and of affectionate regard in a degree which perhaps
they alone can appreciate.
His ability may to some extent be judged from his pubhshed work, but
his personal worth and charm are truly measured by the affection with which
he inspired those who knew him best. *
G. A. S.
OBITUARY NOTICES
[From Nature, vol. xciii, p. 138, with additions.]
On the evening of Monday, March 30, 1914, surrounded by his family
John Henry Poynting passed quietly away. A memorial service was held in
Birmingham on the Thursday following, and was attended by representatives
of many universities and learned societies, including Sir J. J. Thomson,
Sir Joseph Larmor, Dr Glazebrook, Sir William Tilden, Prof. W. M. Hicks,
Dr W. N. Shaw, and of course by many colleagues and councillors of the
University in which he occupied a chair, as well as by a large number of
private citizens and friends. For he was a man universally beloved.
At the memorial service, the following true words concerning him were
spoken by the Rev. Henry Gow, who knew him well :
"We remember that he did work to make him famous throughout the
world of science which gave him a high place amongst the discoverers of truth ;
but we remember much more than that. We remember how he loved life,
how interested he was in little things, how he delighted in children, in flowers,
and in birds; what confidence and affection he inspired, how free he was
from claims of self and from uneasy egotism; how much happiness he felt
and gave. We remember his wise judgments, strong character, cheerful
courage, his delightful humour, and a certain peaceful beauty and childlike
joyousness of spirit behind all his multifarious gifts. He rejoiced to be the
friend as well as the teacher of the young. He kept his heart free from all
bitterness and disillusion which come so often to us in our later years. He
knew and felt always how beautiful and great a thing it was to be alive."
He was born on September 9, 1852, at Monton, near Manchester, son
of the unitarian minister of that place. His first education was at home,
but the years 1867 to 1872 he passed at Owens College, Manchester, graduating
B.Sc. at the London University, and proceeding, in 1872, to Trinity College,
Cambridge, where he was bracketed third wrangler in 1876.
He was then appointed demonstrator at Owens College by Balfour Stewart,
and began a lifelong friendship with Sir J. J. Thomson, who was at that time
a student. In due time Poynting became a fellow of Trinity, and in 1880
was appointed to the professorship of physics at Birmingham, which he
held to the day of his death.
The four first professors of the Mason College, which was opened by
Huxley in 1880 (who delivered, on this occasion, a notable address, reprinted
X OBITUARY NOTICES
as the first of his collected essays), were Sir Wm. Tilden, Prof. M. J. M. Hill,
Dr T. W. Bridge, who died a few years ago, and Poynting. In this same year
Poynting married Miss M. A. Cropper, daughter of the Rev. J. Cropper, of
Stand, near Manchester. In 1887 he received the Sc.D. of Cambridge, and
in 1888 the fellowship of the Royal Society. In 1893 the Adams prize
was awarded to him, and in 1899 he presided over Section A of the British
Association at Dover. This meeting was memorable for the clear discovery
of the separate existence of electrons, which was announced to Section A
bv Sir J. J. Thomson on an occasion when many members of the French
Association, meeting simultaneously at Boulogne, had come over for friendly
fraternisation.
In 1905 Poynting became president of the Physical Society, and was
awarded a Royal medal by the Royal Society " for his researches in physical
science, especially in connection with the constant of gravitation and the
theories of electrodynamics and radiation." In this brief summary an
immense amount of work is referred to. The work for which he is locally
best known was his determination of the Newtonian constant of gravitation
by the very accurate use of an ordinary balance with an adjustable mass
under one or other of the arms — a determination which is popularly called
" weighing the earth." His account of it appears in the Phil. Trans, for 1891.
It is a classical memoir of its kind, and very instructive to the physical
student, but the papers on electrodynamics eclipse it in value. These were
''communicated" to the Royal Society in 1884 and 1885 respectively, their
titles being "On the Transfer of Energy in the Electromagnetic Field,"
and "On the Connection between Electric Current and the Electric and
Magnetic Inductions in the Surrounding Field."
The memoir on the transfer of energy aroused universal attention. The
paths by which energy travels from an electromotive source to various parts
of a circuit were displayed, and their intricacies unravelled, for the first time ;
idottihj of energy might legitimately be urged as a supplement to conservation
(see a paper by the present writer in Phil. Mag., June, 1885) ; and it is to these
papers that we owe that fundamental generalisation, connecting mechanical
motion with electric and magnetic forces, which is known all over the world
as "Poynting's Theorem."
The following letter from Sir Joseph Larmor to the writer expresses a
mathematician's view of the importance of this subject:
St John's College, Cambridge, 10th May, 1915.
•'Nobody before Poynting seems to have thought of tracing the flux of
energy in a medium elasticaUy transmitting it, and where the whole process
is therefore exposed to view. The line of flow is a ray in optics: thus it
includes a dynamical aspect of that conception added on to and of course
consistent with the Huygenian or rather YoungFresnelian one. The electric
OBITUARY NOTICES XX
and optical ray is implicitly in Maxwell's equations, and is only a corollary
to them. But in any other kind of elastic transmission, e.g. waves in an
elasticsolid medium, a corresponding theory can be worked out.
I take it this idea is Poynting's main contribution, and it clarified many
things, especially electrical."
A great expansion of this note is contained in a remarkable paper On the
Dynamics of Radiation which Sir Joseph Larmor communicated to the Inter
national Congress of Mathematicians meeting at Cambridge in August, 1912.
This paper is so intimately associated with Poynting's work, and so pleased
him when he saw it, that I have asked and obtained permission to include
extracts from it in this volume; they will be found at the end of the
Section dealing with the Pressure of Light.
" The essential characteristic of an electrodynamic system is the existence
of the correlated fields, electric and magnetic, which occupy the space sur
rounding the central body, and which are an essential part of the system;
to the presence of this pervading aethereal field, intrinsic to the system, all
other systems situated in that space have to adapt themselves. When a
material electric system is disturbed, its electrodynamic field becomes
modified, by a process which consists in propagation of change outward,
after the manner of radiation, from the disturbance of electrons that is
occurring in the core. When however we are dealing with electric changes
which are, in duration, slow compared with the time that radiation would
require to travel across a distance of the order of the greatest diameter of
the system — in fact in all electric manifestations except those bearing directly
on optical or radiant phenomena — complexities arising from the finite rate
of propagation of the fields of force across space are not sensibly involved :
the adjustment of the field surrounding the interacting systems can be taken
as virtually instantaneous, so that the operative fields of force, though in
essence propagated, are sensibly statical fields. The practical problems of
electrodynamics are of this nature — how does the modified field of force,
transmitted through the aether from a disturbed electric system, and thus
established in the space around and alongside the neighbouring conductors
which alone are amenable to our observation, penetrate into these conductors
and thereby set up electric disturbance in them also? and how does the
field emitted in turn by these new disturbances interact with the original
exciting field and with its core ? For example, if we are dealing with a circuit
of good conducting quality and finite cross section, situated in an alternating
field of fairly rapid frequency, we know that the penetration of the arriving
field into the conductor is counteracted by the mobility of its electrons, whose
motion, by obeying the force, in so far annuls it by Newtonian kinetic reaction ;
so that instead of being propagated, the field soaks in by diffusion, and it does
not get very deep even when adjustment is delayed by the friction of the vast
numbers of ions which it starts into motion, and which have to push their way
through the crowd of material molecules; and the phenomena of surface
currents thus arise. If (by a figure of speech) we abolish the aether in which
both the generating circuit and the secondary circuit which it excites are
immersed, in which they in fact subsist, the changing phases of the generator
could not thus establish, from instant to instant, by almost instantaneous
Xii OBITUARY NOTICES
radiant transmission, their changing fields of force in the ambient region
extending across to the secondary circuit, and the ions in and along that
circuit would remain undisturbed, having no stimulus to respond to. The
aethereal phenomenon, viz., the radiant propagation of the fields of force,
and the material phenomenon, viz., the response of the ions of material bodies
to those fields, involving the establishment of currents with new fields of their
own, are the two interacting factors. The excitation of an alternating current
in a wire, and the mode of distribution of the current across its section, depend
on the continued establishment in the region around the wire, by processes
of the nature of radiation, of the changing electromagnetic field that seizes
hold on the ions and so excites the current ; and the question how deep this
influence can soak into the wire is the object of investigation. The aspect
of the subject which is thus illustrated, finds in the surrounding region, in the
aether, the seat of all electrodynamic action, and in the motions of electrons
its exciting cause. The energies required to propel the ions, and so establish
an induced current, are radiant energies which penetrate into the conductor
from its sides, being transmitted there elastically through the aether; and
these energies are thereby ultimately in part degraded into the heat arising
from fortuitous ionic motions, and in part transformed to available energy
of mechanical forces between the conductors. The idea — introduced by
Faraday, developed into precision by Maxwell, expounded and illustrated
in various ways by Heaviside, Poynting, Hertz — of radiant fields of force,
in which all the material electric circuits are immersed, and by which all
currents and electric distributions are dominated, is the root of the modern
exact analysis of all electric activity."
Poynting's work on radiation appeared partly in the Phil. Trans, for 1904
and partly in the Phil. Mag. for 1905. In these memoirs the tangential
pressure of radiation is analysed and demonstrated; and it is shown, both
theoretically and experimentally, that a beam of light behaves essentially as
a stream of momentum, and gives all the mechanical results which may thus
be expected, though of a magnitude exceedingly minute. Nevertheless, he
goes on to show that these radiationpressures, however small, are of much
consequence in astronomy, and have many interesting and some conspicuous
results. A noteworthy part of the radiation memoirs, however, is independent
of considerations of pressure or momentum, and gives a means of determining
the absolute temperatures of sun and planets, and of other masses in space,
in a singularly clear and conclusive manner.
A complete list of his publications is given below, but special mention
must be made here of the important series of textbooks on physics,
written in conjunction with his friend. Sir J. J. Thomson.
He took great interest also in the philosophical aspects of physical science,
and his help is acknowledged by Prof. James Ward in connection with the
publication of that notable series of Gifford Lectures entitled Naturalism,
and Agnosticism. Poynting was strongly inclined, almost unduly, to limit
the province of science to description, and to regard a law of nature as nothing
but a formulation of observed correspondences. He wished to abolish the
OBITUARY NOTICES XUl
idea of cause in physics. In some of this he may have gone too far ; but his
rebellion against an excessive anthropomorphism which had begun to cling
around the notion of natural laws, as if they were really legal enactments to
be obeyed or disobeyed by inert matter almost as if it possessed willpower
and could exercise choice, some substances being praised as good radiators
while others are stigmatised as bad — most gases being admittedly unable to
reach a standard of perfection held out to them as Boyle's law, though a
few of excessive merit might surpass it, — Poynting's revolt against this kind
of attitude to laws of nature, though doubtless more than half humorous,
was in itself wholesome. Some of his philosophic views may be read, as a
Presidential Address to Section A of the British Association for 1899 (infra
p. 599) ; but I think it useful and legitimate to extract a few sentences
from that address and quote them here, as an illustration of his mode of
approaching the misty region where physics and metaphysics intertwine:
" To take an old but never wornout metaphor, the physicist is examining
the garment of Nature, learning of how many, or rather of how few different
kinds of thread it is woven, finding how each separate thread enters into
the pattern, and seeking from the pattern woven in the past to know the
pattern yet to come.... So, as we watch the weaving of the garment of Nature,
we resolve it in imagination into threads of ether spangled over with beads
of matter. We look still closer, and the beads of matter vanish; they are
mere knots and loops in the threads of ether."
And then, a few pages further on, when dealing with the interaction of
Matter and Mind :
"Do we, or do we not, as a matter of fact, make any attempt to apply
the physical method to describe and explain those motions of matter which
on the psychical view we term voluntary?
Any commonplace example, and the more commonplace the more is it
to the point, will at once tell us our practice, whatever may be our theory.
For instance, a steamer is going across the Channel. We can give a fairly
good physical account of the motion of the steamer. We can describe how
the energy stored in the coal passes out through the boiler into the machinery,
and how it is ultimately absorbed by the sea. And the machinery once
started, we can give an account of the actions and reactions between its
various parts and the water, and if only outsiders will not interfere, we can
predict with some approach to correctness how the vessel will run. All
these processes can be likened to processes already studied — perhaps on
another scale — in our laboratories, and from the similarities prediction is
possible. But now think of a passenger on board who has received an
invitation to take the journey. It is simply a matter of fact that we make
no attempt at a complete physical account and explanation of those actions
which he takes to accomplish his purpose. We trace no lines of induction
in the ether connecting him with his friends across the Channel, we seek no
law of force under which he moves. In practice the strictest physicist
abandons the physical view, and replaces it by the psychical. He admits
the study of purpose as well as the study of motion."
Xiv OBITUARY NOTICES
In other words he recognises Mind and Purpose as dominant over and
in a different category from Matter and Mechanism.
In psychical phenomena Poynting was, I judge, an agnostic, but on the
question of a materialistic or naturalistic explanation of mental phenomena
he expresses himself thus, in the Dover Section A address above referred to :
"It appears to me that the assumption that our methods do apply, and
that purely physical explanation will suffice to predict all motions and changes,
voluntary and involuntary, is at present simply a gigantic extrapolation,
which we should unhesitatingly reject if it were merely a case of ordinary
physical investigation. The physicist when thus extending his range is
ceasing to be a physicist, ceasing to be content with his descriptive methods
in his intense desire to show that he is a physicist throughout."
But I must not delay further on his scientific work ; the man himself
was even more than his work. When the Mason College became the University
of Birmingham Poynting was elected Dean of the Faculty of Science; in
that capacity his quiet wisdom and efficiency were very manifest, and keen
was the regret of all his colleagues when, some twelve years later, failing
health necessitated his yielding this office to another. His judgment was
as sound as his knowledge, and his conspicuous fairness endeared him to
colleagues and the members of his staff. By the latter it is not too much
to say that he was regarded with affectionate veneration ; one of them writes
to me as follows :
" As to his character it is impossible to give the right impression to those
who did not know him well. I consider him a man of very extraordinary
ability, which might have carried him much farther if it had been associated
with more selfassertion. But it was largely this modesty and selfsuppression
which created a very unusual degree of affection in those who had the privilege
of knowing him intimately. I always associate him in my mind with Faraday
and Stokes."
As a lecturer and teacher he was admirable, and the respect in which he
was held by his peers was noteworthy. I am glad to remember that so
recently as the 1913 meeting of the British Association, some of the greatest
physicists in the world, who were staying with me — Prof. H. A. Lorentz,
Lord Rayleigh, and Sir Joseph Larmor — went to his house one evening,
and met there in his study Sir J. J. Thomson and Dr Glazebrook, who were
staying with him; thus constituting an appropriate and representative
gathering, and giving him a pleasure which he remembered to the end of
his life.
There is much more that might be said ; but let his position in the world
of science be what it may, we in the University of his mature life knew him
well, and know him best as an admirable colleague, a staunch friend, and a
good man.
0. J. L.
OBITUARY NOTICES XV
[From the Proceedings of the Royal Society, A, vol. xcii, 1914.]
John Henry Poynting, the youngest son of the Rev. T. E. Poynting,
Unitarian Minister at Monton, near Manchester, was born there on September
9, 1852. He received his earlier education at the school kept by his father
and then went, in 1867, to the Owens College, which his elder brother, C. T.
Poynting, who was for many years Unitarian Minister at Fallowfield, near
Manchester, had just left. Poynting must have received a good grounding
in Mathematics at his father's school, as he gained a Dalton Entrance Exhibi
tion in Mathematics before entering the College. Owens College in those
days was in a modest building, once the residence of Richard Cobden, in
Quay Street, Deansgate. Neither the amenities of the locality nor the
accommodation in the building were anything to boast about, but few educa
tional institutions before or since, whatever their equipment or surroundings,
have had a more efficient staff than Owens College in the old Quay Street
days. As an old Quay Streeter, the writer can speak from personal experience.
The cramped space was not an unmixed disadvantage. We were so closely
packed that it was very easy for us to get to know each other. Arts students
and Science students jostled against each other continually; a crowd of
Mathematicians would be waiting outside the doors of a lecture room for it
to discharge a Latin or Greek class, and thus one of the chief difficulties of
nonresidential colleges, the lack of social intercourse between the students,
was almost absent.
The professors at Owens in Poynting's time were : Barker for Mathematics,
of whom Poynting always spoke in terms of the highest appreciation, a feeling
shared by all his pupils, for no abler or more conscientious teacher of Mathe
matics than Thomas Barker ever lived; Jack, another great teacher, was
Professor of Natural Philosophy; Roscoe of Chemistry, and Williamson of
Natural History; on the literary side. Greenwood, the Principal, was Pro
fessor of Classics, Ward of English History and Literature, Jevons of Logic,
and that very lovable man, Theodores, lecturer on Modern Languages. At
that time Owens had not the power of granting degrees, and most of the
students prepared for the examinations of the University of London. In
those days these covered a very wide range of subjects, and Poynting, who
took the London degree, must have attended the lectures of all these professors.
He was second at the London Matriculation in 1869, obtained Second Class
Honours in both Physics and Mathematics in the First B.Sc. examination
in 1871, and took the B.Sc. degree in 1872. In the spring of 1872 he obtained
an entrance scholarship at Trinity College, Cambridge, and came into residence
at Cambridge in October. At Cambridge he pursued the normal course of
one destined for high honours in the Mathematical Tripos. He read with
Xvi OBITUARY NOTICES
Routh, he obtained his Major Scholarship in due course, like many of the
reading men of his time at Trinity he joined the Second Trinity Boat Club
and rowed in the first boat in 1875 ; the fortunes of that once famous club
were, however, then declining and it came to an end in 1876. He took his
degree in the Mathematical Tripos of 1876 as Third Wrangler, bracketed with
Mr Trimmer, of Trinity College, a very brilliant man who suffered from
persistent illhealth and died within a few months of taking his degree. As
Dr Glazebrook and Dr Shaw both graduated in the same Tripos and Lord
Rayleigh was the additional Examiner, Physics was well represented on
this occasion.
After taking his degree Poynting came back for a short time to the Owens
College, which was now in the buildings it at present occupies, and demon
strated in the Physical Laboratory under Prof. Balfour Stewart, who had
succeeded Jack as Professor of Natural Philosophy shortly before Poynting's
departure for Cambridge.
On his election to a Fellowship at Trinity College in 1878, Poynting
returned to Cambridge and began, in the Cavendish Laboratory under Clerk
Maxwell, those experiments on the mean density of the earth which were
destined to occupy so much of his time for the next 10 years.
He remained at Cambridge until 1880, when he was elected to the Chair
of Physics in Mason College, Birmingham (now the University of Birmingham),
which had just been founded; this post he held until his death. The year
that he went to Birmingham, he married the daughter of the late
Rev. J. Cropper, of Stand, near Manchester.
He threw himself wholeheartedly into the arduous duties connected with
the starting of a new University College, the preparation of his lectures and
the equipment of the physical laboratory, and, as was his wont, without any
bustle or hurry he soon had things working efficiently. And so in the efficient
discharge of his duties as a Professor, in successful original research, in the
fulfilment of municipal duties, the time passed placidly on, the only cloud
on an almost idyllic domestic life being his somewhat indifierent health,
the first threatenings of the disease from which he ultimately died. To see
if a country life would suit his health better than a town one, the Poyn tings
moved from Edgbaston to Fox Hill, Alvechurch, a house about 12 miles
out of Birmingham. There was a small farm attached to the house and
Poynting entered into farming most heartily, though I am afraid he did
not derive much pecuniary profit from it. But even farming when the
agricultural depression was most acute could not impair his good temper or
ruffle his equanimity. If the farm did not yield money, it gave new interests
and experiences, and if something was always going wrong, at any rate it
drove away monotony. The quietness and simplicity of the life were
thoroughly to the taste of Mrs. Poynting and himself. Life in the country
OBITUARY NOTICES XVll
too gave free scope to his taste for Natural History, in which he always took
great interest ; he was a keen and excellent observer, and a favourite conten
tion of his was that physicists were somewhat too much inclined to confine
their observations to experiments made in the laboratory and did not
sufficiently avail themselves of the opportunities of studying the physical
phenomena going on in the sky, the sea, and the earth. The taste for Natural
History was a family one; his brother, the late Mr. F. Poynting, was an
excellent ornithologist, devoting himself especially to the study of the eggs
of British birds, of which he made most careful and accurate watercolour
drawings — some of these have been reproduced in his book The Eggs of
British Birds.
The Poyntings stayed at Foxhill until 1901, when, his health much
improved, they returned to Edgbaston. His life at this time was a busy
one, for in addition to the work demanded from him as the head of a large
and successful School of Physics, he acted as the Dean of the Faculty of
Science, was a Justice of the Peace, and for some time Chairman of the
Birmingham Horticultural Society. He had also to plan and superintend the
erection of a new physical laboratory when his department was transferred
from its old quarters to the new buildings of the University of Birmingham.
He went with the British Association to Canada in 1909, when it met at
Winnipeg, and gave one of the evening lectures ; his subject was the Pressure
of Light, on which he had been experimenting for several years. He went
the trip to Vancouver and back and seemed thoroughly to enjoy the visit.
The pressure of light was also the subject of a lecture which he gave in French
at Paris before the French Physical Society at Easter, 1911.
In the spring of 1912 a severe attack of influenza was followed by a
recrudescence of diabetes, a disease from which he had suffered for some time,
and he was ordered to take a long rest; he was, in consequence, away from
Birmingham for two terms. On his return to Birmingham he seemed much
better, he took an active part in the meeting of the British Association held
there in September, 1913, and he and Mrs. Poynting entertained a large party
of physicists at their house in Ampton Road, and it then seemed as if he
might hope to enjoy many years of useful work. Another attack of influenza
in the spring of 1914 brought on a very severe attack of diabetes, and he
died on March 30, 1914.
It is difficult to attempt to say what Poynting was to his friends without
using terms which must appear exaggerated to those who did not know him.
He had a genius for friendship, and a sympathy so delicate and acute that
whether you were well or ill, in high spirits or low, his presence was a comfort
and a delight. During a friendship which lasted for more than thirty years,
I never saw him angry or impatient and never heard him say a bitter or unkind
thing about man, woman or child.
p. c. w. h
Xviii OBITUARY NOTICES
He took pleasure in many things, in music, in literature, for he was a
lover of books and a collector in a modest way, in novels of all kinds, good
and bad. He was fond of the country, and especially of North Wales, where
he spent most of his vacations, but happiest of all when at home with his
family. Throughout his life he took considerable interest in Philosophy,
and a discussion of the philosophical basis of Physics formed part of his
Presidential Address to Section A at the Dover Meeting of the British
Association. Views similar to those he there expressed are now held by
many ; he had formed his years before, when but few in this country agreed
with them. The excellence of his work received many recognitions, though
not in my opinion so many as it deserved. He was elected a Fellow of the
Royal Society in 1888, received a Royal Medal in 1905, served on the Council
from 1909 to 1911 and was VicePresident in 191011. He received the
Adams Prize from the University of Cambridge in 1893, the Hopkins Prize
from the Cambridge Philosophical Society in 1903. He was President of
Section A when the British Association met at Dover in 1899 and was President
of the Physical Society in 190911.
He was in great request as an Examiner in Physics and no one excelled
him at this work, his long experience of students, his judgment and common
sense, the charitable view he took of the limitations of a student's knowledge,
and the fact that he was never afraid of setting easy papers, made him an
eminently fair and discriminating examiner. He was very successful as a
teacher of students of all kinds, those who only took Physics as a subsidiary
subject as well as those who made it their life's work, the latter he inspired
with an enthusiasm for research, with some of his own skill in accuracy of
measurement and with the desire for thoroughness in their work.
Poynting's Scientific Work.
This may be divided into four groups : {a) studies on gravitational attrac
tion, {b) on the change of state, (c) on the transfer of energy in the electro
magnetic field, and (d) on the pressure of light.
Gravitational Attraction.
His experiments on the mean density of the earth were commenced in
Cambridge in 1878 but it took twelve years' steady work before he obtained
a result with which he was satisfied. The method used was to measure the
attraction between two known masses A and B by suspending A from one
of the arms of a balance of the ordinary type and finding the increase in
weight produced when B was brought underneath it. The balance used in
the later experiments was one built specially for the experiment by Oertling
and had a beam 123 cm. long. With a balance of this size the difficulties
arising from air currents proved very formidable. Poynting fully recognised
OBITUARY NOTICES XIX
the advantage of Boys' short torsion balance method in this respect and said
that if he were designing the apparatus again, instead of using an exceptionally
large balance for the sake of being able to suspend large masses, he should
go to the other extreme and make the apparatus as small as possible. At
the same time, as he points out, the magnitude of the effects produced by the
air currents made their detection easy, whereas they might have been over
looked and not allowed for had they been smaller. The final results (Phil.
Trans., A, vol. clxxxii, p. 565, 1891) he obtained for A, the mean density
of the earth, and G, the gravitational constant, were
A = 54934,
G = 66984 X 108.
Poynting's long investigation incidentally added considerably to our know
ledge of the technique of accurate weighings.
With the cooperation of Gray he made a series of most interesting experi
ments (Phil. Trans., A, vol. cxcii, p. 245, 1899) to see if the attraction between
two quartz crystals was the same when the axes of the crystals were parallel
as when they were crossed. The method he used was a very ingenious
application of the principle of forced oscillations, which was so effective that,
though one sphere was only about 1 cm. in diameter and the other about
6 cm., the experiments showed that the attractions in the two positions
could not differ by as much as one part in 10,000. Later he made with
Phillips a series of experiments to see if weight depended on temperature,
using as in his first experiments a balance of the ordinary type; the result
of these was (Proc. Roy. Soc, A, vol. Lxxvi, p. 445, 1905) that between 15° C.
and 100° C. the change is not greater than 1 in 10^ and between 16*6° C.
and  186° C. it is not so great as 1 in 10i« per 1° C.
Change of State.
The problem of the change of state was one in which he took especial
interest, and it was the subject of one of his earliest papers (Phil. Mag. (5),
vol. XI, p. 32, 1887). His way of picturing this change was to suppose that
from the surface of a liquid or solid particles were continually breaking free,
so that through each unit of area of the surface there was a constant escape
of molecules. This loss was balanced by the passage from the vapour above
the solid of some of the gaseous particles which struck against its surface,
so that when there was equilibrium the flow out from the liquid or solid was
balanced by the flow inward from the gas. The proportion of gaseous mole
cules which after striking the surface passed across to the solid or liquid state
he assumed to be the same for a solid as for a liquid and to be independent
of the temperature, so that it could be measured by the vapour pressure.
Thus at the same temperature the flow across water would be proportional
to the vapour pressure of water, that across ice to the vapour pressure of
62
XX OBITUARY NOTICES
ice, thus ice could only be in equilibrium with water when the vapour pressure
over ice is equal to that over water.
Poynting supposed that the mobility of the molecules in liquids and solids
is increased by pressure — the pressure as it were squeezing the molecules
out : the amount of the increase depending on the density of the substance,
diminishing as the density increases. Thus, if pressure increases the escape
of the molecules from a liquid, a liquid under pressure will evaporate more
freely, and so for it to be in equilibrium with its vapour the vapour pressure
must be higher than that over the normal liquid; from the equilibrium
between water and its vapour in a capillary tube, he found that if Bp is the in
crease in the vapour pressure produced by applying a pressure P to the liquid,
Sp = Po/p, where o is the density of the vapour and p that of the liquid.
Poynting applied this conception of mobility to the case of solutions,
taking the view that the molecules of the salt formed aggregates with some
of the water molecules and thus diminished their mobility thereby diminishing
the number of water molecules which passed from the liquid state through
each unit of area of surface per second. The mobility of pure water is thus
greater than that of the solution, so that if the two are separated by a semi
permeable membrane more molecules will pass from the water to the solution
than from the solution to the water, and the water will flow into the solution.
To prevent this flow the mobility of the molecules of water in the solution
must be increased by the application of a pressure that will make the mobility
of the solution equal to that of pure water; this pressure is the osmotic
pressure. Since under this pressure the mobility of the solution is equal to
that of pure water the vapour pressure in equilibrium with the pressed
solution will be the vapour pressure over pure water, so that another definition
of osmotic pressure would be the pressure required to raise the vapour
pressure over the solution to that over pure water. On the assumption
that the presence of one molecule of salt to n of water would diminish the
mobihty of the water in the proportion of (n — l)/w, which would be the case
if a molecule of salt imprisoned one and only one molecule of water, Poynting
showed that the osmotic pressure on his theory would be the pressure exerted
by the salt molecules if they were in the gaseous state and occupying the
volume of the solution. Though this theory does not connect the electrical
properties of solutions with the properties associated with osmotic pressure
so readily as the dissociation theory, it is so simple and fundamental that
it helps to give vividness and definiteness to our picture of the processes
operative in solutions.
Transfer of Energy.
The researches by which Poynting is most widely known are those
published in the papers " On the Transfer of Energy in the Electromagnetic
Field" (Phil Trans., A, 1884), and "On Electric Currents and the Electric
OBITUARY NOTICES * XXI
and Magnetic Induction in the Surrounding Field" (Phil. Trans., A, 1888).
He says in the first paper, " The aim of this paper is to prove that there is a
general law for the transfer of energy, according to which it moves at any
point perpendicularly to the plane containing the lines of electric and
magnetic force, and that the amount crossing unit of area per second of this
plane is equal to the product of the two forces multiplied by the sine of the
angle between them divided by 47r, while the direction of the flow of energy
is that in which a righthanded screw would move if turned round from the
positive direction of the electromotive to the positive direction of the magnetic
intensity." He shows from the equation of the electromagnetic field that
the rate of increase in the energy inside a closed surface is equal to
1 1[? (R^ Qy) + m (Py  Ra) + n (Qa  PjS)] dS,
where dS is an element of the closed surface, I, m, n the direction cosines of
the normal to the surface, P, Q, R the components of the electromotive
intensity, and a, jS, y those of the magnetic force. This expression may be
regarded as showing that the energy flows across the surface, the components
of flux being
^(R^Qy), ^{PyRa). ^(OaP^);
the vector which has those components is now universally known as Poynting's
vector; it is at right angles to both the electric and magnetic forces and is
proportional to the product of these forces and the sine of the angle between
them. Thus when we can draw equipotential surfaces for both the electric
and magnetic forces the energy flows along the lines of intersection of the
two sets of surfaces. Poynting illustrates this theorem by applying it to
the following cases : a constant current flowing along a straight wire, a con
denser discharged by shortcircuiting the plates by a wire of great resistance,
a voltaic battery, a thermoelectric circuit.
The magnitude of the change in the point of view consequent on the
principles brought forward in this paper is perhaps shown most clearly in the
case of the discharge of the condenser and the transference of the energy
which before the discharge was distributed between its plates into heat in
the discharging circuit. Before the publication of this paper the general
opinion was that the energy was transferred along the wire much in the same
way as hydraulic power is carried through a pipe. On Poynting's yiew the
energy flows out from the space between the plates and then converges
sideways into the wire, where it is converted into heat, the paths of the energy
being those represented in the figure.
As shown in this figure the paths of energy near the wire are at right
angles to it. This is not so unless the wire is such a bad conductor that the
xxu
OBITUARY NOTICES
lines of electric force in its neighbourhood run parallel to it; if for example
the current through the wire were an alternating one with very high
frequency the electric force near the wire would be at right angles to it. In
this case the energy would flow parallel to the wire but outside it.
In the second paper Poynting, taking the view that the electromagnetic
field consists of distributions of lines of electric and magnetic force, discusses
the question of the transfer of energy from the point of view of the movement
of these lines. He applies the same considerations to the question of the
residual charge in Leyden jars in his fascinating and instructive paper on
"Discharge of Electricity in an Imperfect Insulator"' {Phil. Mag., vol. v, 1886,
p. 419). Poynting's vector occurs as a quantity of fundamental importance
in many theories of electromagnetic action in which the subject is approached
from a point of view somewhat different from the one he adopted. It appears,
for example, as a measure of the momentum per unit volume when the electro
magnetic field is regarded as a mechanical system and the properties of the
field as the result of the laws of motion of such a system. It appears, too,
when we regard magnetic force as the result of the motion of tubes of electric
force, the direction of motion of these tubes being parallel to Poynting's vector.
Pressure of Light.
For some years before his death Poynting devoted much attention to the
question of radiation and the pressure of light. On the theory of this subject
he published {Phil. Trans., A, vol. ecu) a very valuable paper, in the first
part of which he discusses the application of the fourthpower law of radiation
to determine the temperature of planets (in this he found afterwards he had
been anticipated by Christiansen). Among other interesting results he
arrived at the conclusion that the temperature of Mars must be so low that
life, as we know it, would be impossible on its surface, this result was criticised
OBITUARY NOTICES XXUl
by Lowell, but Poynting maintained his ground in a paper published in the
Philosophical Magazine, December, 1907. The second part of the paper in
the Philosophical Transactions contains investigations of the repulsive force
between two hot spheres which arises from the radiation from the one tending
to repel the other. He showed that if the bodies are in radiation equilibrium
with the Sun at the distance of the Earth from it, the repulsive effect will be
greater than the gravitational attraction between them if their radii are
less than 196 cm., if their density were that of water; if they were made
of lead the corresponding radius would be 178 cm. Thus if Saturn's rings
consisted of very small particles it is possible that the effect of radiation
might make them repel instead of attract each other. He considers at the
end of the paper the effect produced by radiation on the orbits of small bodies
round the Sun and shows that this would ultimately cause them to fall into
that body. To quote his own words : " The Sun cannot tolerate dust. With
the pressure of his light he drives the finest particles altogether away from
his system. With his heat he warms the larger particles. They give out this
heat again and with it some of that energy which enables them to withstand
his attraction. Slowly he draws them to himself and at last they unite with
him and end their separate existence." {Pressure of Light, "Romance of
Science" Series.)
He made important contributions to the experimental side of the subject,
thus with Dr. Barlow he established the existence of the tangential force
produced when light is reflected from a surface at which there is some absorp
tion, and also the existence of a torque when light passes through a prism.
They also succeeded in demonstrating the existence of the recoil from light
of a surface giving out radiation : an account of these experiments was given
in the Bakerian Lecture for 1910 {Proc. Roy. Soc, A, vol. lxxxiii, p. 534, 1910).
These investigations involved the detection of exceeding minute forces and
gave ample scope for Poynting's skill in devising methods and apparatus.
He had exceptionally good mechanical instincts and an excellent knowledge
of the capabilities of instruments; the result was that the apparatus he
designed was always simple and effective.
In addition to papers published in scientific journals and the Transactions
of Societies he wrote The Mean Density of the Earth : the Adams Prize Essay
for 1893, The Pressure of Light ("Romance of Science" Series) and The Earth
(Cambridge University Press). Of the Text Book of Physics written in con
junction with J. J. Thomson he wrote the whole of the volumes on Sound and
Heat and of the first volume of Electricity and Magnetism and the chapters
on Gravitation in the Properties of Matter. His writings exhibit to the
full the clearness, simplicity and thoroughness which was characteristic of
all his work.
J. J. T.
Xxiv OBITUARY NOTICES
[From the Philosophical Magazine for May, 1914, with additions.]
Although Prof. Poynting, whose loss will be universally deplored, graduated
with high distinction in mathematics at Cambridge, coming out as third
Wrangler in the Tripos of 1876, his interest seems always to have lain in
the direct elucidation of physical laws and principles rather than in the
evolution and exposition of their consequences by analysis. When he came
to Cambridge, in 1872, he was already largely trained in the niceties of
refined experimentation ; and after graduation he embraced an early chance
to resume experimental work at Manchester. The founding of the Mason
University College gave him the opportunity of organising a laboratory
of his own. Much work about this time was concerned with instrumental
improvements, such as the design of polarimeters and other optical apparatus ;
and to the same period belong studies in chemical physics, such as the eluci
dation of osmotic pressure, theories to which in later years he returned with
conviction, and which, though perhaps not yet fully appreciated, should
not be lost sight of, in view of his proved insight into fundamental problems
in other domains.
An exami^le of the latter class is the memoir on the transfer of energy
in the electromagnetic field, Phil Trans., 1884, culminating in the famous
result that will go down to posterity as Poynting's Theorem, which not only
specifies the path of transfer of electric energy from one material system to
another through the aether, but also as a very special case gives for the first
time (strange to say) the dynamical specification of a ray of light.
At about the same time 0. Heaviside, and a little later Hertz, were engaged
upon this aspect of electric transmission as an elastic effect propagated from
body to body across the aether, in place of the older aspect of electric charges
in movement, each carrying its field of disturbance along with it, — which latter,
rejuvenated ten years later by exact conceptions of the agency of electrons, and
duly modified for chano^e of acceleration, now includes the whole field of view.
But times not being yet ripe for that, he pursued his subject in 1885 in another
Phil. Trans. Memoir "On the connexion between electric current and the
electric and magnetic inductions in the surrounding field," which tracks out
the relations of a current circuit by the graphical device of the motion of
what are now known as Faraday tubes of force, a type of visualisation of
the phenomena which is at the present time once more widely in favour.
Afterwards he broke new ground in the experimental determination of
the constant of gravitation — the problem of weighing the earth — which had
been solved by Cavendish with his accustomed genius by aid of Michell's
principle of balancing by torsion. To Poynting's mind an ordinary balance
with lever and scalepans gave at least equal promise of practical accuracy,
and his long continued experimental investigations, which were summed
OBITUARY NOTICES XXV
up in a Phil, Trans. Memoir of 1892 and a Cambridge Adams Prize Essay
of 1893, were the startingpoint of a new interest in this subject which opened
up into many methods more or less cognate to his own. By this time, however,
the torsion method had renewed its power through the discovery of the pro
duction and properties of quartz fibres by C. V. Boys, whose remarkable
subsequent investigation with smallscale apparatus was generously acknow
ledged by Poynting as the last word on the subject.
The resource thus acquired in refined dynamical experimenting was to
reap further successes in a more untilled field. The ancient idea of a pressure
exerted by light, so obvious on the corpuscular view of optics, had been revived
by Maxwell on a foundation of an accompanying electric stress in the trans
mitting medium. Its mere existence, as distinct from an analysis of its
propagation to the place where it is in evidence, was already indubitably
involved in the Amperean forces on the electric currents induced in the surface
of a reflector, once the principles of the electric theory of light are admitted.
It had assumed some importance in its application, notably by FitzGerald,
to the elucidation of the mysterious phenomena of comets' tails. To Poynting,
this pressure exerted by a ray coming say from a distant star, far out of reach
of direct dynamical effect, involved that the ray carried momentum along
with it, and that the pressure effect was of the nature of a thrust exerted
along the ray arising from the transfer of this momentum. After long efforts,
the disturbances arising from gaseous convection as a whole, and from the
radiometric molecular effect, were eliminated by Lebedew, and were com
pensated by balance against each other by Nicholls and Hull, about the
same time, and the Maxwell value of the normal component of the pressure
was fully verified. But Poynting's line of thought led him straight to a
tangential component of the thrust as well as a direct component, and he
noted that the former could be investigated without much trouble from the
gaseffect. This idea led to many beautiful determinations in conjunction
with his assistants and students. His idea of convected momentum also
led him in another direction to the conclusion that the pressure on a receding
surface must be less than on one at rest; it also suggested that by reaction
a moving radiating body would be accelerated by its own radiation— an
impossible result which is corrected by recognising that its radiation is greater
towards the direction of its motion than towards other directions, which
leads to retardation on the whole. As this effect depends on extent of surface,
it is greater in proportion for small bodies. Thus he was led to consider
clouds of cosmic dust revolving orbitally round the sun, each particle heated
by his rays to an equilibrium temperature of the space where it is, retarded in
its motion by the reaction of its own exchanges of radiation, and thus gradually
sucked into the sun. This clearance of solar spaces from dust must be a
prominent feature in views of stellar cosmogony ; the calculation of the time
XXvi OBITUARY NOTICES
that would be required aptly illustrates his latent mathematical power,
which was never unduly obtruded ; and the whole Memoir is an example of
that simplification of reasoning and reduction to its lowest terms which is
suggestive of the depth of vision that belongs to genius. When the theory
of electrons came to be developed into Maxwell's channels by Lorentz and
others, it appeared at once that the stress argument, on which he had based
radiation pressure, was in default, and the natural first conclusion was
against the objective existence of the stress as thus specified in favour of
some type too complex for simple expression. But later Poincare and
Abraham introduced the idea of grouping the refractory outstanding terms
as a distribution of electric momentum, specified very simply as the vector
product of the aethereal and magnetic inductions. The stress in the medium
is thus taken to be the sole operating cause : it is unbalanced, and so reveals
itself partly in a distribution of mechanical forces exerted on the material
bodies that are present, and partly in storage and expenditure of mechanical
momentum of some latent type throughout the aether. This latter agrees
precisely with the momentum of radiation elucidated on very simple inde
pendent grounds by Poynting. Interest in the subject is thus stimulated,
and the problems now under discussion as to whether the effect is in all cases
simply momentum, and whence arises the subsidiary travelling inertia which
is implied in it, become of pressing interest. The application of the stress
method to calculation over a boundary surrounding a material system' leads
in fact to an additional result — that when the system gains energy SE of
electric type, its effective mass increases by SE/c^, where c is the velocity of
radiation : but this is less important practically, and would not for example
affect sensibly the clearance of cosmical dust above mentioned,— though the
idea that energy possesses inertia naturally assumes prominence in general
relativity theory. An experimental and theoretical incursion into the
different field of the elongation of a wire due to its torsion was probably
prompted primarily by these problems ; tiiough not perhaps strictly pertinent
to them, it opened up new views in the theory of elastic solids under stresses
so great that mere superposition of strains no longer holds good.
The formulation of an exact notion of the temperature of space, above
indicated, is but one phase of his interest in the theory of natural radiation ;
and It seems but yesterday that he was discussing, in private correspondence,
with his usual acuteness and judgment and no sign of failure of powers, the
theory of Stefan's law and its other fundamental relations.
J. L.
CONTENTS
PAGE
PREFACE V
A PERSONAL NOTE vii
OBITUARY NOTICES ix
PART I
THE BALANCE AND GRAVITATION
ART.
1. On the Estimation of Small Excesses of Weight by the Balance
from the Time of Vibration and the Angular Deflection of the
Beam . 1
[Manchester Lit. Phil. Soc. Proc. 18, 1879, pp. 3338. Read December 10, 1878.]
2. On a Method of using the Balance with great delicacy, and on its
employment to determine the Mean Density of the Earth . 7
[Roy. Soc. Proc. 28, 1879, pp. 235. Received June 21, 1878.]
3. On a Determination of the Mean Density of the Earth and the
Gravitation Constant by means of the Common Balance . 43
[Phil. Trans. A, 182, 1892, pp. 565656. Received May 13. Read June 4, 1891.]
4. An Experiment in Search of a Directive Action of one Quartz
Crystal on another. By J. H. Poynting and P. L. Gray, B.Sc. . 137
[Phil. Trans. A, 192, 1899, pp. 245256. Received September 27.
Read November 17, 1898.]
5. An Experiment with the Balance to Find if Change of Temperature
has any Effect upon Weight. By J. H. Poynting and Percy
Phillips, M.Sc 149
[Roy. Soc. Proc. A, 76, 1905, pp. 445457. Received July 12, 1905.]
6. On a Method of Determining the Sensibility of a Balance. By
J. H. Poynting and G. W. Todd, M.Sc 162
[Phil. Mag. 18, 1909, pp. 132135. Read June 25, 1909.]
PART II
ELECTRICITY
7. On the Law of Force when a Thin, Homogeneous, Spherical Shell
exerts no Attraction on a Particle within it . . . .165
[Manchester Lit. Phil. Soc. Proc. 16, 1877, pp. 168171. Read March 6, 1877.]
8. Arrangement of a Tangent Galvanometer for lecture room purposes
to illustrate the Laws of the Action of Currents on Magnets, and
of the Resistance of Wires 168
[Manchester Lit. Phil. Soc. Proc. 18, 1879, pp. 8588. Read April 1, 1879.]
XXViii CONTENTS
ART. PAGE
9. On the Graduation of the Sonometer ... . . 170
[Phil. Mag. 9, 1880, pp. 5964. Read before the Physical Society,
December 13, 1879.]
10. On the Transfer of Energy in the Electromagnetic Field . . 175
[Phil. Trans, lib, 1884, pp. 343361. Received December 17, 1883. Read
January 10, 1884.]
11. On the Connection between Electric Current and the Electric and
Magnetic Inductions in the Surrounding Field . . .194
[Phil. Trans. 176, 1885, pp. 277306. Received January 31. Read
February 12, 1885.]
12. Discharge of Electricity in an Imperfect Insulator . . . 224
[Birmingham Phil. Soc. Proc. 5, (1885), pp. 6882. Read December 10, 1885.]
13. On the Proof by Cavendish's Method that Electrical Action varies
Inversely as the Square of the Distance 235
[British Association Report, 1886, pp. 523524.]
14. On a Form of SolenoidGalvanometer 237
[Birmingham Phil. Soc. Proc. 6, (1888), pp. 162167. Read May 10, 1888.]
15. On a Mechanical Model, illustrating the Residual Charge in a
Dielectric 242
[Birmingha?n Phil. Soc. Proc. 6, (1888), pp. 314317. Read November 8, 1888.]
16. Electrical Theory. Letters to Dr Lodge 245
[Electrician, 21, 1888, pp. 829831.]
17. An Examination of Prof. Lodge's Electromagnetic Hypothesis . 250
[Electrician, 31, 1893, pp. 575577, 606608, 635636.]
18. Molecular Electricity ....... 269
[Electrician, 35, 1895, pp. 644647, 668671, 708712, 741743.]
PART III
WAVE PROPAGATION— RADIATION— PRESSURE OF
LIGHT— AND RELATED SUBJECTS
19. Note on an Elementary Method of Calculating the Velocity of Pro
pagation of Waves of Longitudinal and Transverse Disturbances
by the Rate of Transfer of Energy 299
[Birmingham Phil. Soc. Proc. 4, (1885), pp. 5560. Read November 8, 1883.]
Radiation in the Solar System : its Effect on Temperature and its
Pressure on Small Bodies 3Q4
[Phil. Trans. A, 202, 1903, pp. 525552. Received June 16. Read
June 18, 1903.]
20
CONTENTS XXIX
ART. PAGE
21. Note on the Tangential Stress due to Light incident obliquely on
an Absorbing Surface 332
[Phil. Mag. 9, 1905, pp. 169171. Read at Section A, British Association,
Cambridge, August, 1904.]
22. RadiationPressure 335
[Phil. Mag. 9, 1905, pp. 393406. Presidential Address, delivered at the
Annual General Meeting of the Physical Society, February 10, 1905.]
23. On Prof. Lowell's Method for Evaluating the SurfaceTemperatures
of the Planets ; with an Attempt to Represent the Effect of Day
and Night on the Temperature of the Earth .... 347
[Phil. Mag. 14, 1907, pp. 749760.]
24. The IVEomentum of a Beam of Light 357
[Atti del IV Congresso internazionale dei Matematici (Rome), 3, 1909,
pp. 169174.]
25. On Pressure Perpendicular to the ShearPlanes in Finite Pure
, Shears, and on the Lengthening of Loaded Wires when Twisted 358
[Roy. Soc. Proc. A, 82, 1909, pp. 546559. Read June 24, ]909.]
26. The WaveMotion of a Revolving Shaft, and a Suggestion as to the
Angular Momentum in a Beam of Circularly Polarised Light . 372
[Boy. Soc. Proc. A, 82, 1909, pp. 560567. Read June 24, 1909.]
27. Preliminary Note on the Pressure of Radiation against the Source :
The Recoil from Light. By J. H. Poynting and Guy Barlow,
D.Sc 380
[British Association Beport, 1909, p. 385.]
28. Bakerian Lecture. The Pressure of Light against the Source : The
Recoil from Light. By J. H. Poynting and Guy Barlow, D.Sc. 381
[Boy. Soc. Proc. A, 83, 1910, pp. 534546. Read March 17, 1910.]
29. On Small Longitudinal Material Waves accompanying Light
Waves 394
[Boy. Soc. Proc. A, 85, 1911, pp. 474476.]
30. On the Changes in the Dimensions of a Steel Wire when Twisted,
^ and on the Pressure of Distortional Waves in Steel . . . 397
[Boy. Soc. Proc. A, 86, 1912, pp. 534561. Read March 21, 1912.]
31. The Changes in the Length and Volume of an IndiaRubber Cord
when Twisted 424
[The IndiaBubber Journal, October 4, 1913.]
Appendix by Sir J. Larmor on the Momentum of Radiation . 426
XXX CONTENTS
PART IV
LIGHT
ART. PAGE
32. On a Simple Form of Saccharimeter 435
[Phil. Mag. 10, 1880, pp. 1821.]
33. On the Law of the Propagation of Light. By J. H. Poynting
and E. F. J. Love, B.A. 438
[Birmingham Phil. Soc. Proc. 5 (1887), pp. 354363. Read March 31, 1887,]
34. Haze 446
[Nature, 39, 1889, pp. 323324.]
35. A Graphical Method of Explaining the Diffraction Bands at the
Edge of a Shadow ......... 449
[Birmingham Phil. Soc. Proc. 7 (1890), pp. 210219. Read November 5, 1890.]
36. On a Parallel Plate DoubleImage Micrometer .... 455
[Roy. Astr. Soc. Monthly Notices, 52, 1892, pp. 556560.]
37. Historical Note on the ParallelPlate DoubleImage Micrometer 460
[Eoyal Astr. Soc. Monthly Notices, 53, 1893, p. 330.]
38. A Method of Making a Half Shadow Field in a Polarimeter by two
inclined Glass Plates ......... 462
[British Association Report, 1899, pp. 662663.]
PART V
MISCELLANEOUS
39. Change of State: SolidLiquid 454
[Phil. Mag. 12, 1881, pp. 3248, 232.]
40. Note on a Method of Determining Specific Heat by Mixture . 481
[Birmingham Phil. Soc. Proc. 4 (1883), pp. 4754. Read November 8, 1883.]
41. Osmotic Pressure 4gg
[Phil. Mag. 42, 1896, pp. 289300.]
42. Musical Sands 4gg
[Nature, 77, 1908, p. 248.]
PART VI
STATISTICS
43. The Drunkenness Statistics of the Large Towns in England and
Wales ^g^
[Manchester Lit. and Phil. Soc. Proc. 16, 1877, pp. 211218. Read
April 3, 1877.]
CONTENTS XXXI
ART. PAGE
44. The Geographical Distribution of Drunkenness in England and
Wales. By J. H. Poynting and John Dendy, Jun. . . 504
{^Fourth Report from the Select Committee of the House of Lords on
Irdemperance 1878. Appendix R, pp. 580591.]
45. A Comparison of the Fluctuations in the Price of Wheat and in the
Cotton and Silk Imports into Great Britain .... 506
[Statistical Society Journal, 1884. Read before the Statistical Society,
15 January, 1884.]
PART VII
ADDRESSES AND GENERAL ARTICLES
46. Change of State: Fusion and Solidification .... 538
[Birmingham, Phil. Soc. Proc. 2 (1881), pp. 354372. Read May 12, 1881.]
47. Overtaking the Rays of Light 552
[Mason College Magazine, 1, 1883, pp. 107111.]
48. University Training in our Provincial Colleges. An Address
delivered at the Mason Science College, Birmingham, Oct. 2,
1883 557
49. The Growth of the Modern Doctrine of Energy. Address to the
Mason College Physical Society, March 26, 1884 ... 565
50. The Electric Current and its Connection with the Surrounding Field 576
[Birmingham Phil. Soc. Proc. 5 (1887), pp. 337353. Read March 10, 1887.]
51. The Foundations of our Belief in the Indestructibility of Matter
and the Conservation of Energy. A Criticism of Spencer's
'First Principles,' Part II, Chaps. IV, V, and VI . . . 588
[Midland Naturalist, 12, 1889. Read before the Sociological Section of the
Birmingham Natural History and Microscopical Society, November 22, 1888.]
52. Presidential Address to the Mathematical and Physical Section
of the British Association (Dover), 1899 .... 599
[British Association Report, 1899, pp. 615624.]
53. A History of the Methods of Weighing the Earth. Presidential
Address delivered to the Birmingham Philosophical Society,
October 19, 1893 613
[Birmingham Phil. Soc. Proc. 9 (1894), pp. 123.]
54. The Mean Density of the Earth [Letter] 628
[Nature, 48, 1893, p. 370.]
55. Recent Studies in Gravitation. Address: Royal Institution of
Great Britain, February 23, 1900 629
[Roy. Inst. Proc. 16, 190002, pp. 278294.]
XXXll
CONTENTS
ART.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
284300.
Le mode de propagation de I'Energie et de la tension Electrique
dans le champ Electromagnetique ....
[Rapport presente au Congres International de Physique de 1900, 3, pp.
Paris, GauthierVillars.]
The Transformation and Dissipation of Energy .
[The Inquirer, 1902, pp. 627628.]
Molecules, Atoms and Corpuscles ....
[The Inquirer, 1902, pp. 740741, 772773.]
The Pressure of Light
[The Inquirer, 1903, pp. 195196.]
Mysteries of Matter. Radium at the British Association
[The Inquirer, 1903, pp. 635636.]
A City University [Letter]
[The Inquirer, 1903, p. 660.]
The Universities and the State
[The Inquirer, 1903, p. 779.]
Physical Law and Life
[Hibhert Journal, 1, 1903, pp. 728746.]
Radiation in the Solar System. Afternoon address delivered at
the Cambridge meeting of the British Association, August 23,
1904
[Nature, 70, 1904, pp. 512515.]
RadiationPressure [Letter in correction to above address]
[Nature, 71, 1904, pp. 200201.]
RadiationPressure. Presidential Address to the Physical Society
of London, February 1905. See Part III, Art. 22 . . .
Some Astronomical Consequences of the Pressure of Light. Dis
course delivered at the Royal Institution on May 11, 1906
[Nature, 75, 1906, pp. 9093.]
George Gore, 1826—1908
[Roy. Soc. Proc. 84, 1911, pp. xxixxii.]
Atomic Theory (Mediaeval and Modern)
[Encyclopaedia of Religion and Ethics, 2, 1909, pp. 203210.]
Quelques experiences sur la Pression de la lumiere. Address to
the French Physical Society, March 31, 1910 ....
[Bulletin des seances de la Societe frangaise de Physique, 1, 1910.]
POSTSCRIPT (1918). Retardation by Radiation Pressure: A
correction. By Sir Joseph Larmor, F.R.S
BIBLIOGRAPHY
INDEX
PAGE
645
658
664
673
677
682
683
686
699
708
711
712
722
724
742
754
758
764
PART I.
THE BALANCE AND GRAVITATION.
1.
ON THE ESTIMATION OF SMALL EXCESSES OF WEIGHT BY
THE BALANCE FROM THE TIME OF VIBRATION AND THE
ANGULAR DEFLECTION OF THE BEAM.
[Manchester Lit. Phil Soc. Proc. 18, 1879, pp. 3338.]
[Read Dec. 10, 1878.]
While working last year on an experiment to determine the mean density
of the earth by the balance, I had to measure such an exceedingly small
difference of weight that I could not at that time estimate it by means of
a rider, but was obliged to adopt the method described in this paper. Stated
generally, it consists in treating the balance as a pendulum. Knowing the
nature of the pendulum (that is, its moment of inertia) and its time of vibration,
we can calculate what force acting at the end of one arm of the beam will
produce a given angular deflection. It is, in fact, an application to the common
balance of the method which has always been used with the torsionbalance
when it has been necessary to calculate the forces measured in absolute
measure. I cannot find any record of a previous application of the method ;
and as it might be of use in very delicate weighings or in verifying the small
weights in a laboratory, I have thought it worth while to give a full account
of it.
When small quantities of the second order are neglected and the
oscillations are of the first order, it will easily be found that the equation
of motion of the beam of the balance is
(mP + ^^) 6 + (2Ph + Mgk) d = ap, (1)
p. o. w. 1
2 ON THE ESTIMATION OF SMALL EXCESSES OF WEIGHT
where MP = moment of inertia of beam about central knifeedge,
M = mass of beam,
a = half length of beam,
p = weight of either pan and the mass in it,
h = distance of Hne joining terminal knifeedges below the central
knifeedge,
k = distance of centre of gravity of beam below central knifeedge,
p = small excess in one pan,
6 = angular deflection in circular measure produced by p,
9 = gravity.
If 6* = 0, we have the position of equihbrium given by
ft ^P (2)
2Ph + Mgk ^ ^
The semiperiodic time is
MI
^^^\/ mrrwk ^^^
From equations (2) and (3) we can eliminate 2Ph f Mgk, obtaining
^MgP + 2Pa^d ...
P^TT^ ^ ^ (4)
■^ ag t^ ^
From this expression it appears that, if we know the moment of inertia
of the beam, its length, and the weight at each end, we can find the excess
p from the time of vibration and deflection.
The results given in this paper were obtained with a 16inch chemical
balance by OertUng. The exact length of the half beam {a) measured by
a dividingengine is 202484 centimetres.
To find the Moment of Inertia MP of the Beam. The simplest way
theoretically would appear to be this. Find the times of vibration t^, t^,
and the deflections 6^, d^, due to the same excess p with two different loads
Pj, P2 in each pan. Equating the values of p given for the two by equation
(4) we have
MgP + 2P^a^ d^ti"
MgP + 2P2a2 d^t^^ '
an equation which will give MgP in terms of known quantities ; but on trial
it was found that a very small proportional error in the observed time made
a large error in the value of MgP ; and the following method, that usually
adopted in magnetic observations, was employed in preference. A stirrup
was suspended by a platinum wire, and its time of vibration (^1) against the
force of torsion (/x) of the wire was observed. The moment of inertia of the
stirrup being S, we have
tj
fX
ON THE ESTIMATION OF SMALL EXCESSES OF WEIGHT 3
The time of vibration (^2) was then observed when a cylindrical brass bar of
known moment of inertia (B) was inserted in the stirrup. We now have
The bar was then removed and the balancebeam inserted in its place ; and
the time of vibration (/^) gives
t^^=(S+MI^).
From these three equations, eliminating S and /j,, we obtain
3JP Bjh't^)
Now Eg was calculated from the weight and dimensions of the bar to be
633283 (in centimetres and grammes). The observed times were
t^ = 36792 sees., t^ = 4495 sees., ^3  71483 sees.
From these values we find
M^/2 = 356516*.
To measure 9. The angle of deflection was measured by the number of
divisions of the scale which the pointer moved over. As the length of the
pointer is 321006 centimetres, while 20 divisions of the scale measure 25658
centimetres, a tenth of a division, in terms of which the deflection was
measured, corresponds to an angle of 00003996. The oscillations were
observed from a distance of six or eight feet by a telescope. The resting
point (i.e. the point where the balance would be in equilibrium) was found in
the usual way by observing three successive extremities of two swings and
taking the mean of the second and the mean of the first and third. Five
determinations of the restingpoint were usually made with the excess to be
measured alternately added and removed. From these five, three values of
the deflection {n) due to the excess were calculated in a manner which will be
seen from the example below.
The Time of Vibration. This was found from several determinations of
the time of ten oscillations. The method will be seen from the example.
No correction was needed for the resistance of the air as long as the vibrations
did not exceed two divisions of the scale. When, however, they were much
more than that, the time of vibration was found to increase with the arc.
As the time of vibration frequently changes slightly, probably through
variations of temperature, it was usually observed before and after the
determination of the deflection (n) and the mean of the two taken as the true
time.
* To this a small correction should be added if the adjustingbob is not in its lowest position.
This amounts to 76 for each turn of the screw, and may therefore in general be neglected.
1—2
4 ON THE ESTIMATION OF SMALL EXCESSES OF WEIGHT
The following example of the determination of the value of a centigramme
rider by placing it halfway along the beam will sufficiently explain the details
of the method.
Time of Vibration at Commencement.
No. of
vibration
Observed time of
passage of pointer
through resting
point
No. of
vibration
Observed time of
passage of pointer
through resting
point
1
Time of 10
vibrations
Pointer apparently moving
from left to right
2
4
6
h. m. s.
11 15 36
11 16 1
11 16 265
11 16 52
10
12
14
16
h. m. s.
11 17 43
11 18 8
11 18 335
11 18 59
s.
127
127
127
127
Mean value of 10 vibrations
127
Pointer ajiparently moving
from right to left
1
3
5
7
11 15 49
11 16 14
11 16 395
11 17 5
11
13
15
17
11 17 56
11 18 21
11 18 46
11 19 115
127
127
1265
1265
Mean value of 10 vibrations
12675
Mean of means = 126875 ;
t^ = 126875 sees.
Determination of Deflection (n).
Excess
weight
Extremities
of oscillation
Resting
point
Mean of preceding
and succeeding
restingpoints
Deflection
due to
excess
Added
109
96
109
1025
Removed
93
40
92
6625
10225
3525
Added
152
53
150
102
6675
36
Removed
80
6725
1025
3525
Added
147
60
145
103
Mean value of n = 3583.
ON THE ESTIMATION OP SMALL EXCESSES OF WEIGHT
Time of Vibration at End.
I
No. of
vibration
Observed time of
passage of pointer
through resting
point
No. of
vibration
Observed time of
passage of pointer
through resting
point
Time of 10
i vibrations
Pointer apparently moving
from left to right
2
4
6
h. m. s.
11 26 19
11 26 445
11 27 10
11 27 355
10
12
14
16
h. m. s.
11 28 27
11 28 53
11 29 18
11 29 44
s.
128
1285
128
1285
Mean value of 10 vibrations
12825
Pointer apparently moving
from right to left
1
3
5
7
11 26 325 ;
11 26 58 i
11 27 235
11 27 49
11
13
15
17
11 28 39
11 29 5
11 29 305
11 29 565
1265
127
127
1275
Mean value of 10 vibrations .
..
127
Mean of means = 127625
; ^2 = 127625 sees.
Remembering that onetenth of a division of the scale is an angle of
•0003996 in circular measure, formula (4), expressed in milligrammes, becomes
f = '% 03996 — {Mgl^ + 2Pa2).
^ t^ ag ^
In our present example'
n = 3583,
^ = ^^t^2= 12725 sees.,
Mgl^ = 35651,
2Pa^ = 94704,
p = 5724 milligrammes.
The length of time occupied in this determination was not quite a quarter
of an hour.
* For this, as for several other cases, I removed the pans and hung the weights directly by
fine wires from the suspendingpieces. By this means the resistance of the air was very much
diminished.
6 ON THE ESTIMATION OF SMALL EXCESSES OF WEIGHT
The following table contains a series of results which I have obtained of
the weight of two centigrammeriders, the first of which was accidentally
destroyed after the conclusion of the fourth determination. As the rider
was always placed at division 5 on the beam, the values given in the table
are double those actually obtained.
No. of
experi
ment
Mgl^ + 2Pa2
t in
seconds
n
Weight of
rider in milli
grammes
Mean value
1
145364
8921
13458
978^
2
3
309356
519769
1765
20435
2549
1912
1005 1
955 1
996 milligrammes
4
130355
1310
3471
1047 j
5
130355
1287
366
1144)
6
130355
1272
355
1135
7
8
130355
130355
12725
1281
3583
355
1145
1120
1135 milligrammes
9
130355
12903
3637
1131
10
454405
19406
2208
1058J
2.
ON A METHOD OF USING THE BALANCE WITH GREAT DELICACY,
AND ON ITS EMPLOYMENT TO DETERMINE THE MEAN
DENSITY OF THE EARTH.
[Roy. Soc. Proc. 28, 1879, pp. 235.]
[Received June 21, 1878.]
In the ease and certainty with which we can determine by the balance
a relatively small difference between two large quantities, it probably excels
all other scientific instruments.
By the use of agate knifeedges and planes, even ordinary chemical balances
have been brought to such perfection that they will indicate onemilHonth
part of the weight in either pan, while the best bullionbalances are still more
accurate. The greatest degree of accuracy which has yet been attained was
probably in Professor Miller's weighings for the construction of the standard
pound, and its comparison with the kilogramme, in which he found that the
probable error of a single comparison of two kilogrammes, by Gauss's method,
was TiWWoo*^ P^^^ ^^ ^ kilogramme*. (Phil. Trans. 1856.)
But, though the balance is peculiarly well fitted to detect the relatively
small differences between large quantities, it has not hitherto been considered
so well able to measure absolutely small quantities as the torsionbalance.
The latter, for instance, was used in the Cavendish experiment, when the
force measured by Cavendish was the attraction of a large lead sphere upon
a smaller sphere, weighing about 1 Jibs., the force only amounting to 5 000^0000" ^^
part of this weight, or about 50^0^^ P^^^ ^^ ^ grain.
The two great sources of error, which render the balance inferior to the
torsionbalance in the measurement of small forces, are :
1. Greater disturbing effects produced by change of temperature, such as
convectioncurrents and an unequal expansion of the two arms of the balance.
2. The errors arising from the raising of the beam on the supporting
frame between each weighing, consisting of varying flexure of the beam and
inconstancy of the points of contact of the knifeedges and planes.
* Even so far back as 1787, Count Rumford used a balance which would indicate one in a
million and measure one in seven hundred thousand. {Phil. Trans. 1799.)
8 ON A METHOD OF USING THE BAIjANCE
The disturbances due to convectioncurrents interfere with the torsion
balance as well as with the ordinary balance, though they are more easily
guarded against with the former, by reason of the nature of the experiments
usually performed with it. They might, perhaps, as has been suggested by
Mr. Crookes, be removed from both by using the instruments in a partial
vacuum, in which the pressure is lowered to the 'neutral point,' where the
convection currents cease, but the radiometer effects have not yet begun.
But a vacuumbalance requires such comphcated apparatus to work it, that
it is perhaps better to follow the course which Baily adopted in the Cavendish
experiment. He sought to remove the disturbing forces as much as possible,
and to render those remaining as nearly uniform as possible in their action
during a series of experiments, so that they might be detected and eliminated.
For this purpose the instrument was placed in a darkened draughtless room,
and was protected by a thick wooden casing gilded on its outer surface.
Most of the heat radiated from the surrounding bodies was reflected from the
surface of the case by the gilding. The heat absorbed only slowly penetrated
to the interior, and was so gradual in its action, that, for a considerable time,
the effect might be supposed nearly uniform. Under this supposition it was
then eliminated by the following method of taking the observations. The
restingpoint (that is, the central position of equilibrium, about which the
oscillations were taking place) of the torsionrod, at the ends of which were the
small attracted weights, was first observed when the two large masses pulled
it in one direction. The masses were then moved round to the opposite side,
when they pulled the rod in the opposite direction and the restingpoint was
again observed. The masses were then replaced in their original position
and the restingpoint was observed a third time. These three observations
were made at equal intervals of time; if, then, the disturbing effect was
uniform during the time, the mean of the first and third observations gave
what the restingpoint would have been, had the rod been pulled in that one
direction at the same time that it was actually observed when pulled in the
opposite direction. The difference between the second restingpoint and the
means of the first and third might, therefore, be considered as due to the
attractions of the masses alone.
In the experiments of which this paper contains an account, I have en
deavoured to apply this method of introducing time as an element to the
ordinary balance. But, before it could be properly applied, it was necessary
to remove the errors due to the raising of the beam between successive
weighings, as they could not be considered to vary in any uniform way
with the time. I think I have effected this satisfactorily, by doing away
altogether with the raising of the beam by the supporting frame, between
the weighings. For this purpose I have introduced a clamp underneath one
of the pans, which the observer can bring into action at any time, to fix that
pan in whatever position it may be. The weight can then be removed from
WITH GREAT DELICACY, ETC. 9
the pan, and another, which is to be compared with it, can be inserted in its
place without altering the relative positions of the planes and knifeedges.
The counterpoise in the other pan, meanwhile, keeps the beam in the same
state of flexure. The pan is then undamped and the new position about
which it oscillates is observed. The only changes are due to the change in
the weight and the effect of the external disturbing forces ; the latter we may
consider as proportional to the time, if sufficient precautions have been
taken, and by again changing the weights and again observing the position
of the balance, we may eliminate their effects.
Though the method when applied to the balance does not yet give such
good results as Baily obtained from the torsionbalance — partly, I believe,
because I have not yet been able to apply all his precautions to remove
external disturbing forces — ^it still gives better results than would have been
obtained without it. This may be seen by the numbers recorded in the tables,
where a progressive motion of the restingpoint may be noticed, in most cases
in the same direction, during a series of experiments. Even when this is
not the case, the method at once shows when the disturbing forces are
irregular, and when we are justified in rejecting an observation on that account.
I give in this paper two applications of the method, one to the comparison
of two weights, the other to the determination of the mean density of the
earth. The latter is given only as an example of the method, but I hope
shortly to continue the experiments with a large bullionbalance, for the
construction of which I have had the honour to obtain a grant from the
Society. The balance is now in course of construction, by Mr. Oertling, of
London.
Description of the Apparatus.
The balance which I have employed is one of Oertling's chemical balances,
with a beam of nearly 16 inches, and fitted with agate planes and knifeedges.
It will weigh up to a little more than 1 lb. To protect it from sudden changes
of temperature, the glass panes of the case are covered with flannel, on both
sides of which is pasted gilt paper, with the metallic surface outwards. This
case is enclosed in another outer case, a large box of inch deal, lined inside
and out with gilt paper. The experiments have been conducted in a darkened
cellar under the chemical laboratory at Owens College, which was kindly
placed at my disposal by Professor Roscoe. As the ceilings and floors of the
building are of concrete, any movement near the room causes a considerable
vibration of the floor and walls. It was necessary, therefore, to support the
balance independently of the floor. For this purpose, six wooden posts
(A, B, C, D, Ey F, Fig. 1) were erected resting on the ground underneath and
passing freely through the floor to a height of 6 feet 6 inches above it. They
are connected at the top by a frame like that of the table, and stayed against
each other to give firmness. The wider part of the frame, near the posts
10
ON A METHOD OF USING THE BALANCE
E and F, is boarded over to form a table for the telescope {t, Fig. 1) and scale
(s), by which the oscillations of the balance are observed. The box con
taining the balance rests on two crosspieces, on the narrower part, ABCD,
of the frame, with the beam parallel to AD, and its right end towards the
telescope.
In order to observe the position of the beam, a mirror, 1 J inches by f inch,
is fixed in the centre of the beam, and the reflection of a vertical scale (s,
Fig. 1) in this is viewed with a telescope (t) placed close to the scale. The
light from the scale passes through two small windows cut in each of the
cases of the balance and glazed with plate glass. The position of the beam
Fig. 1.
is given by the division of the scale upon the crossline on the eyepiece of the
telescope. The scale, which was photographed on glass, and reduced from
a large scale, drawn very carefully, has 50 divisions to the inch. These are
ruled diagonally with ten vertical crosslines. It is possible to read, with
almost certainty, to a tenth of a division, or ^i^th of an inch. Since the
mirror is about 6 feet from the scale, a tenth of a division means an angular
deflection of the beam of about 3"*.
The scale is illuminated from behind by a mirror (m), several inches in
diameter, which reflects through it a parallel beam from a paraffin lamp (Q.
* The numbers on the scale run from below upwards, so that an increase in the weight in the
righthand pan is indicated by a lower number on the scale.
WITH GREAT DELICACY, ETC.
11
A plate of ground glass between the scale and mirror diffuses the light evenly
over the scale and, by altering the position of the mirror, any desired degree
of brilliancy may be given to the illumination of the scale. A screen (not
shown in Fig. 1) prevents stray light from striking the balancecase.
This method of reading — which, of course, doubles the deflection — has
been so far sufficiently accurate for my purpose; that is to say, the errors
arising from other sources are far greater than those arising from imperfections
of reading. But in a long series of preliminary experiments I used the
following plan to multiply the deflection still further. A rather smaller
fixed mirror, ah, is placed opposite to and facing the beammirror, AB, fixed
on the beam, and a few inches from it. Suppose the beammirror to be
deflected from the position BL, parallel to ah, through an angle, 6, to the
position AB. If a ray, PQ, perpendicular to ah strikes AB at Q, it will make
an angle 6 with QM, the normal at Q, and will be reflected along QR, making
an angle 29 with its original direction, and therefore with the normal RO,
at R, when it strikes it. If it be reflected again to AB at S, it will make an
angle 3^ with the normal SN, and the reflected ray, ST, will make an angle
4^ with the original direction, PQ, of the ray. It may be still further
reflected between the two mirrors, if desirable, each reflection at the mirror,
AB, adding 2^ to the deflection of the ray. I have, for instance, employed
three reflections from the beammirror, so multiplying the deflection six times.
In this case, one division of my scale, at the distance at which it was placed
from the beam, corresponded to a deflection of 7" in the beam, and this could
be subdivided to tenths by the eye. The only limit to the multiplication
arises from the imperfection of the mirrors and the decrease in the illumination
of the successive reflections*.
The chair of the observer is placed on a raised platform, and a small table
rising from the platform and free from the frame on which the instruments
rest, is between the observer and the telescope. On this he can rest his note
book during an experiment. As the differences of weight observed are some
times exceedingly minute, the balance is made very sensitive— usually
* This method was used in the seventh and eighth series here recorded. Two reflections
from the beammirror were employed, giving four times the actual deflection.
12
ON A METHOD OF USING THE BALANCE
vibrating in periods between 30 sees, and 50 sees. The value of a division of
the scale cannot be determined by adding known small weights to one pan,
as the deflection would usually be too great. Any approach of the observer
to the case causes great disturbances, so that the ordinary method of moving
a rider an observed distance along the beam is inapplicable. In some experi
ments made last year I calculated the force equivalent to the small differences
in weight, in absolute measure, by observing the actual angular deflection
and the time of vibration. With a knowledge of the moment of inertia of
the beam and treating it as a case of small oscillations, it was possible to
calculate the value of the scale. But the observations and subsequent
calculations were so complicated that the following method of employing
riders was ultimately adopted.
{a)
(h)
(c)
id)
A small bridge about an inch long (Fig. 2 a) is fitted on to the beam. The
sides of the bridge are prolonged about half an inch above the arch which
fits on to the beam, as shown in the end view (Fig. 2b). In each of these
sides are cut two Vshaped notches directly opposite to each other, one of
the opposite pairs being 6654 millims. (about J inch) distant from the other
pair. Two equal riders of the shape shown in Fig. 2 c are placed across the
bridge, and are of such a size that they will just fit into the bottom of the
notches. When one of these rests across the bridge the other is raised up
WITH GREAT DELICACY, ETC.
13
from it. The lowering of one rider and the raising of the other corresponds
therefore to a transference of a single rider from one pair of notches to the
other. The length of the half beam being 202716 millims. and the distance
between the notches 6654 millims., this transference will be equivalent to
the addition to one pair of 003282 of the weight of the rider used. As I have
generally used a centigrammerider this means 0*3282 mgm.
Two levers I, I' (Fig. 2d), with hooks h, h' are used to raise one rider while
the other is lowered. These levers are worked by two cams c, c' on a rod R,
which is prolonged out of the balancecase to the observer. By turning this
rod round, the one lever is raised while the other is depressed. The hook
at the end of the raised lever picks up its rider while the other hook deposits
its rider on the bridge, and then sinks down between the raised sides (as
shown in Fig. 2d), leaving the rider resting freely on the bridge.
The levers are so adjusted that the beam even in its greatest oscillations
never comes in contact with the hooks.
This arrangement might probably be still further perfected by introducing
two small frames for the riders to rest upon, the frames resting on the beam
by knifeedges. It would then be certain that the movement of the riders
was equivalent to a transference from one knifeedge to the other, whereas
the rider at present may not rest exactly over the centre of the notch. But
I find that I get fairly consistent results by lowering the rider somewhat
suddenly so as to give it sufficient impetus to go to the bottom of the notch,
and have not therefore thought it necessary as yet to introduce more com
plicated apparatus.
In place of the righthand pan of the usual shape, another of the shape
shown in Fig. 3 a is employed. To the centre of the pan underneath is
I
^ n ///
(a)
(&)
3.
(c)
attached a vertical brass rod which passes downwards through the bottom
of the inner case of the balance. To the under side of this case is attached
14
ON A METHOD OF USING THE BAI^ANCE
the clamping arrangement before referred to. This consists of two sliding
pieces (Fig. ia, s, s) working horizontally in a slot cut in a thick brass plate
which is fastened to the case. Through a circular aperture in this plate (the
slot is not cut through the whole thickness of the plate, but only as shown in
Fig. 46) and about the middle of the slot hangs the rod r attached to the
scalepan.
(a^
By means of right and left handed screws on a rod R, which is prolonged
out of the case to the observer, these two sliding pieces can be made to
approach, and clamp the rod, or to recede and free it. By having the opposite
surfaces of the sliding pieces and the rod polished and clean, it is possible
to clamp and unclamp without producing any disturbance. The clamp is
of great use also to lessen the vibrations when they are too large, as it may be
brought into action at any moment, and on releasing carefully the beam will
start again from rest without any impetus. It may be used too to increase
the vibrations by releasing suddenly, when the beam will have a slight impetus
in one direction or the other.
The weights which I have compared are two brass pounds avoirdupois,
made for me by Mr. Oertling, and marked A and B respectively. They are
WITH GREAT DELICACY, ETC.
15
of the usual cylindrical shape with a knob at the top (Fig. 36). Two small
brass pans (Fig. 3 c) with a wire arch by which they can be suspended, are
used to carry them; these are called respectively X and Y. I found on
beginning to use them that there was too great a difference between A and
B. I therefore adjusted them by putting a very small piece of wax upon A,
the lighter. But the difference between them increased by 00782 mgm. in
two days, which I thought was probably due to the wax. After the fourth
series I therefore removed it and scraped B till it was more nearly equal
to A. The weighings I — IV have, however, been retained, for though the
differences on different days vary they are fairly constant on the same day.
The weights are changed by the following apparatus which has been
designed to effect the change as simply and quickly as possible.
A horizontal ' siderod ' or Hnk {ss, Fig. 5) is worked by two cranks (c, c,
Fig. 56), which are attached to the axles of two equal toothed wheels {t, t)
16 ON A METHOD OF USING THE BALANCE
with a pinion (p) connecting them. A second pinion (q), on a rod prolonged
out of the case to the observer, gears with one of the toothed wheels. By
turning this rod the toothed wheels are set in motion, both in the same
direction, moving the horizontal 'siderod' from the right say upwards and
over to the left. A pin (pn) stops its motion downwards further than is shown
in Fig. 6 a. Near each end of the rod is cut a notch, and across these are
hung the pans carrying the weights. The apparatus is fastened to the floor
of the case between the central upright, supporting the beam, and the scale
pan, *the siderod being perpendicular to the direction of the beam, and
exactly over the centre of the pan. In Fig. 5 a, one of the weights B is sup
posed to be resting on the scale pan (the wires suspending the pan from the
beam not being shown), the siderod having moved down so far below the
wire of the smaller pan carrying the weight that it leaves it quite free. If,
now, it is desired to change the weights the rod R is turned, setting the wheels
in motion, the siderod moves up, picks up B — the notch catching the wire —
then travels over round to the extreme right, when A will be just over and
nearly touching the scalepan. By continuing the motion slightly A will
be gently deposited on the pan, and the siderod will move slightly down
leaving the weight quite free. On the scalepan are four pins, turned slightly
outwards, acting as guides for the small pan, and ensuring that it shall always
come into the same position. The wheels and pinions are of such a size that
two revolutions of the rod just suffice to change one weight for the other.
It will be seen that all the manipulation required from the observer during
a series of weighings is the simple turning of three rods, which are prolonged
out of the balance case to where he is stationed at the telescope. By turning
one of these he can change the position of the rider on the beam by a known
amount, and so find the value of his scale. By turning a second he clamps
the scalepan, and so steadies the balance while the weights are changed by
turning a third rod. I have made this arrangement not only because it
seems as simple as possible to secure the end required, but also because it
seemed more applicable to a vacuumbalance (with which I hope ultimately
to test it).
I take this opportunity of expressing my thanks to Mr. Thomas Foster,
mechanician of Owens College, for his aid in the construction of the apparatus,
and in the planning of many of its details.
Method of conducting a Series of Weighings.
After the counterpoise has been adjusted so that the beam swings nearly
about its horizontal position, the frame is lowered so that the balance is
ready for use. The pan is then clamped and the balance is left to come to
a nearly permanent state of flexure if possible, sometimes for the night or
even longer. The lamp is lighted usually halfanhour or more before begin
ning to observe, so that its effect on the balance may attain a more or less
WITH GREAT DELICACY, ETC. 17
steady state. It is necessary also to wait some time after coming into the
room, for the opening of the door will always cause a considerable and
immediate deflection of the beam. When a sufficient time has elapsed, the
observations are commenced with a determination of the value of one scale
division by means of the riders. The three extremities of two successive
oscillations are observed with one of the riders resting on the beam. These
are then combined as follows : The mean of the first and third is taken, and
the mean again of this and the second, this constituting the 'resting point,'
that is, the position of equilibrium of the beam at the middle of the time.
For instance, in weighing No. I (see tables at the end) the three extremities
of successive oscillations were 2805, 3120, and 2860 (column 2). The resting
point was taken as
280 5 + 2860 + 2x31 20 _ ^97.62
the rider on the beam being the righthand one denoted by R (column 1).
The balance is then clamped, and the other rider is brought on to the beam
while the first is taken up. The resting point is again observed. In No. I
it was 27005. The balance is again clamped, and the first rider again brought
on to the beam, and, on unclamping, the restingpoint again observed. In the
same weighing it was 29675. These three are sufficient to give one deter
mination of the deflection due to the transference of a rider. This will be
the difference between the second restingpoint and the mean of the first
'^9762 + 29675
and third. For instance, '^ ^ 27005 = 2713 divisions. This
number is found in the fifth column.
This process is continued, the resting points being combined in threes till
several values of the deflection due to the rider have been obtained, and the
mean of these is taken as the true value. This plan of combining the resting
points requires that the observations should be taken at nearly equal intervals.
After a little practice it will always take the observer about the same time
to go through the same operations of clamping, changing the riders, unclamping,
clamping again to lessen the vibrations about the new restingpoint, and then
beginning to observe, and I have considered that this was a sufficiently correct
method of timing the observations.
When a series has been taken it will at once be seen whether they were
begun too soon after entering the room, or whether any irregular disturbing
force has acted. For instance, in weighing No. II, determination of one
scaledivision, the first restingpoint is so much lower than the succeeding
with the same rider that evidently the balance was still affected by my
entrance into the room. It was, therefore, rejected. Again, in weighing
No. Ill, determination of the difference between the weights, the fourth
restingpoint was much lower than the others with the same weight in the pan.
p. o. w. 2
18 ON A METHOD OF USING THE BALANCE
The restingpoints, when the other weight was in the pan, showed no similar
sudden drop of such magnitude. This observation was, therefore, rejected
as being affected by some irregular disturbance.
When the value of the deflection is determined, the value of one scale
division is at once found by dividing 3282 mgm. by the number of divisions
of the deflection, since the change of the sides is equivalent to the addition
of 3282 mgm. to one pan.
The determination of the difference between the weights is then begun.
This is carried on in a precisely similar manner, the only difference being
that the rod changing the weights is now turned round in place of the rod
changing the riders. I have usually taken a greater number of observations
of the difference between the weights than of the deflection due to the riders,
as the former is somewhat more irregular than the latter. This irregularity
I believe to arise from slight differences of temperature of the two weights,
and perhaps from air currents caused by their motion inside the case. They
do not seem to be due to any fault in the clamping arrangement, since that
is employed equally in both, and the changing of the weights, if effected gently,
does not move the beam at all.
When the deflection has been determined, it is multiplied by the number
of milligrammes corresponding to one scaledivision, and this, of course, gives
the difference between the weights. I have interchanged the weights in the
two pans X and Y, between the series of weighings, in order to make the
experiments like those conducted in the weighings for the standard pound.
But my object has not been to show at all that the method gives consistent
results day after day, and, in fact, the difference between the weights has
varied. For instance, according to weighings I and II, A — B = 0446,
while, according to weighings III and YV, A — B = 0116. There is a greater
difference between these than can be accounted for by errors of experiment,
and it probably arose from the small piece of wax with which I made A nearly
equal to B. The difference between the weights when measured to such
a degree of accuracy as that which I have attempted, will, no doubt, vary
from time to time, partly with deposits of dust, partly with changes in the
moisture in the atmosphere, and so on.
But I think the numbers which are given in the tables are sufficient to
show that the difference between two weights in any one series of weighings
can be measured with a greater degree of accuracy than has hitherto been
supposed possible. I give in the tables a full account of the weighings, each
series containing a determination of the value of one scaledivision and
a determination of the difference between the weights. The greatest deviation
of any one of a series from the mean of that series of differences is always
given. This I consider a better test of accuracy of weighing than the probable
error. What is wanted in weighing is rather a method which will give at
I
WITH GREAT DELICACY, ETC. 19
once a good determination of the difference between two weights. But
I may state, that if the error of any one of a series be taken as its difference
from the mean of that series, the probable error of a single determination of
the difference between the weights in the first four series is 4344 of a division,
or 0054 mgm., that is, s4ituuuwu^^ ^^ ^^^ *otal weight, while the greatest
error is 18 divisions, or 0224 mgm., that is, ^ ooo^oooo ^^ ^^ *^® ^otal weight.
It may be remarked that these weighings were all made during peculiarly
unfavourable weather when there were frequent heavy showers, causing
sudden changes of temperature, and thus seriously affecting the working of
the balance. In the series V — VIII the greatest error is only ^ ooo^oooo ^^
the total weight, the weather having improved considerably.
On the Employment of the Balance to determine the Mean Density
of the Earth.
In the Cavendish experiment, the attraction of a large sphere of lead of
known mass and dimensions upon another smaller sphere, also of known mass
and dimensions, is measured when the two are an observed distance apart.
Comparing this attraction with the weight of the small sphere — that is the
attraction of the earth upon it — and knowing the dimensions of the earth,
we can deduce the mass of the earth in terms of the mass of the large lead
sphere, and so obtain its mean density. The torsionbalance, which was
invented for the purpose by Mitchell, the original contriver of the experiment,
has hitherto been used to determine the force exerted by the mass upon the
small sphere. In the arrangement here described, I have replaced the torsion
balance by the ordinary balance, and have so been able to compare the
attraction of a lead sphere with that of the earth upon the same mass
somewhat more directly. The results which I have obtained have no value in
themselves, but they serve as an example of the employment of the balance for
more delicate work than any which it has as yet been supposed able to perform.
The method is shortly this: A lead weight (called ''the weight') weighing
45292 grms. (nearly 1 lb.) hangs down by a fine wire from one arm of a balance,
from which the pan has been removed, at a distance of about six feet below it,
and is accurately counterpoised in the other pan, suspended from the other
arm. A large lead mass (called 'the mass') weighing 154,2206 grms.
(340 lbs.) is then introduced directly under the hanging ' weight.' The
attraction of this mass increases the weight slightly and the beam is deflected
through an angle which is observed. The value of this deflection in milH
grammes is measured by the employment of riders in the manner described
above, and so the attraction of the ' mass ' is known. The increase of the
weight caused by the 'mass' has been in my experiments about 01 of a
milligramme, or 4500^0000 ^^ ^^ *^® whole weight.
The balance which I have used is that which I have described above.
It was placed in the same room and in the same position as in the weighing
2—9.
20 ON A METHOD OF USING THE BALANCE
experiments. The same method was used to observe the oscillations with a
single mirror on the beam. The scale was a simple one etched on glass and
not diagonally ruled. It had about 50 divisions to the inch, and the numbers
increased from above downwards, so that an increase in the weight hanging
from the left arm was indicated by a lower number on the scale.
The ' weight' which is suspended by a very fine brass wire from the left arm,
passing through a hole in the bottom of the balancecase, hangs in a double
tin tube, 4 inches in diameter, to protect it from aircurrents. At the bottom
of the tube is a window, through which can be seen the bottom of the 'weight'
as it hangs. The 'weight' is 4248 centims. in diameter and is gilded. The
' mass ' is a sphere of an alloy of lead and antimony. It was cast with a ' head '
on and then accurately turned. Its vertical diameter is 30477 centims.
(about 1 foot). The specific gravity of a specimen of the metal was found to
be 10422. Its weight given by a weighingmachine is 340 lbs. about, and this
agrees very nearly with the weight calculated from the specific gravity. I am
obliged to accept this as the true weight provisionally, until it is found more
correctly by the large balance referred to above and now being constructed.
This mass (Fig. 1, M) is placed in a shallow wood cup at one end of a
2inch plank, 8 inches wide and 6 feet 1 1 inches long, mounted on four flanged
brass wheels, and serving as a carriage for it (Fig. 1). A plank about 12 feet
long nailed to the floor in a direction perpendicular to the beam of the balance,
as shown in Fig. 1, pp, acts as a railway for the carriage, and a firm stop at
each end prevents the carriage from running off the rail. The distance be
tween the stops is rather less than twice the length of the carriage, and the
' weight ' hangs down from the balance exactly midway between the stops.
The 'mass' is placed on the carriage so that it is exactly under the 'weight'
when the carriage is at one end of its excursion against one of the stops. An
empty cup (c. Fig. 1) of the same dimensions as that in which the ' mass '
rests is placed at the other end of the carriage, and is just under the 'weight'
when the carriage is against the other stop. By this arrangement no correction
is needed for the attraction of the carriage upon the ' weight ' or counterpoise,
and the eflect caused by the removal of the carriage from one end of its
excursion to the other is entirely due to the difference of attractions of the
'mass' upon the 'weight' and counterpoise in its two positions. The position
of the 'mass' when directly under the 'weight' is called its 'in' position, and
that when it is at the other end of its excursion is called the ' out ' position.
The length of the excursion is 5 feet 73 inches.
To draw the carriage along the rail a vertical iron shaft with a wood
cyhnder at the lower end pivots on the floor, and is prolonged up to the
level of the observer as he sits at the telescope with a handle by which he
can turn it. The two ends of a rope which winds round the cylinder pass
through pulleys on the stops, and are attached to the ends of the carriage.
WITH GREAT DELICACY, ETC.
21
The observer can then move the 'mass' with great ease by turning the handle,
even while looking through the telescope.
When a series of observations is made, the general method is this. The
deflection (r) due to the transference of a rider from one notch to the other
on the beam is first observed exactly in the manner before described, the
mean of four or five values being taken as the true value. Then the deflection
(n) due to the difference of attraction of the 'mass' in its two positions is found
in exactly the manner in which the difference between two weights is found,
except that now when three successive extremities of oscillations have been
observed for a restingpoint the ' mass ' is moved from one position to the other
where the weights were changed in the former experiments, the clamp not
being brought into action. The second extremity of the oscillation which
is proceeding while the 'mass' is moved, is observed as the first of the next
three. When nine or more restingpoints have been observed they are com
bined in threes, and the mean of the resulting values of the deflection n is
used in the subsequent calculation.
This deflection is, of course, less than that which would be observed were
there no attraction on the counterpoise, and were the 'out' position of the
A
^ B
'mass' at an infinite distance. To find the factor/ by which the deflection n
due to the change of position of the ' mass ' must be multiplied in order to
reduce it to the deflection which would be observed under these conditions,
let AB be equal and parallel to the beam of the balance at the level of the
counterpoise of which B is the centre. Let C be the centre of the ' weight,'
D that of the 'mass' in its 'in' position, E that of the 'mass' in its 'out' posi
tion. Draw BF, DF, parallel to AD, AB. Let ju, = the mass of the 'mass.'
The vertical attraction of the 'mass' in its 'in' position will be
jM fiBF
CD^ BD^ '
The vertical attraction in its ' out ' position will be
^CD ixBF
CE^ BE^'
22 ON A METHOD OF VSINQ THE BALANCE
The difference between these is actually observed, viz. :
^ ( CD^.BF CD^ CD^ . BF
C&\ BD^ CE^ BE^
The factor by which we must multiply the observed difference to reduce it
to the attraction of the 'mass' on the 'weight' in its 'in' position is therefore
CB^ . BF CD^qJDKBF
*^~ "^ BD^ '^CE^ BE^
= 10185
since CD = 2213 centimetres.
BD = 19203
BF = 18770
CE = 17236
BE = 25709
The values of r and n being observed, the distance d between the centres
of the 'mass' and 'weight' is then measured by adding to 17362 centims. (the
sum of their radii) the distance from the top of the 'mass' to the bottom of
the 'weight' as measured by a cathetometer.
It now remains to explain the calculation of the mean density A from the
observed values of r, n, and d. We have
/ X increase in weight observed _ Attraction of 'mass' on 'weight' when 'in'
Weight of 'weight' Attraction of earth on 'weight'
But the increase in weight is ^„^.^.,^ mgms., since the distance between
^ r X 202716 ^
the notches is 6654 milHms., and the half beam 202716 millims. The weight
of the 'weight' is 45329 grammes.
The attraction of the 'mass' per gramme of the 'weight'
_ Volume X density
(distance d between centres of 'mass' and 'weight')^'
_ Mass in grammes
~ d^ '
_ 1542206
~ I^ •
The attraction of the earth is similarly
A X inR {1 + M (f M  e) cos2 A},
where A = mean density of the earth,
R = earth's polar radius in centimetres,
j^ _ centrifugal force at the equator
Equatorial gravity '
€ = ellipticity,
A = latitude.
WITH GREAT DELICACY, ETC.
23
The logarithm of the coefficient of A when R is in inches is 90209985
(Astron. Soc. Mem., vol. 14, p. 118), or if R is in centimetres it is 94258322.
Inserting these values in the equation we obtain
1542206
453290
A =
£
^ttR {1 + M  (f M  e) cos2 A}
fx
n 6654
r 202716
^""^^^
where
and
C =
1542206 X 453290 x 202716
iTvR {1 + M  (f M  e) cos2 A} X / X 6654
log C = 18951337,
.. log A = 18951337
+ log r
— log n
 2 log d.
The following table is an account of an experiment made on May 30th,
1878, and will serve as a specimen of the method of making the observations.
It is the best which I have yet made in the closeness with which all the values
of n agree with each other.
VII. May
30, 1878. Determination of
r.
Eider on
beam
Extremities of
oscillations
Resting 
point
Mean of pre
ceding and
succeeding
resting points
Differences
R
2609
2509
2606
25582
—
—
L
"2146
2050
2123
20922
25674
4752
R
2719
2447
2694
25767
21031
4736
L
2143
2092
2129
21140
25762
4622
R
2497
2638
2530
25757
21223
4534
L
2049
2207
2060
21307
—
Mean r = 4661.
24
ON A METHOD OF USING THE BALANCE
Determination of n.
Position
of mass
Extremities of
oscillations
Resting 
point
Mean of pre
ceding and
succeeding
restingpoints
Differences
= n
In
2168
2081
2159
2109
2161
2110
21222
—
—
Out
21352
21227
125
In
2139
2108
2138
21232
21361
129
Out
2127
2146
2129
21370
21241
129
In
2115
2136
2113
21250
21378
128
Out
2159
2118
2160
21387
21260
127
In
2102
2150
2106
21270
21393
123
Out
2167
2116
2161
21400
21273
127
In
2099
2154
2104
21277
21408
131
Out
2170
2114
2169
21417
21320
21298
M9
In
2098
2162
2106
—
—
Mean n = 1 26.
WITH GREAT DELICACY, ETC. 25
At the close of the experiment d was found to be 22226 centimetres.
We have therefore
log A = log
+ log 4661
 log 126
2 log 22226
= 18951337 I
' +16684791 I
i 01003705
26937226
= 07695197.
.. A =5882.
I have made in all eleven experiments with this method. The resulting
values of A are
1 May 20 5393.
2 ...
. . . ,,
23 ...
... 5570.
3 ...
5)
24 ...
... 4415.
4 ...
5,
28 ...
... 7172.
5 ...
. . 5)
29 ...
... 5109.
6 ...
,5
29 ...
... 6075.
7 ...
,,
30 ...
... 5882.
8 ...
,,
30 ...
... 6336.
9 ...
June 5 ...
... 5977.
10 ...
. . . 5j
5 ...
... 5580.
11 ...
6 ...
... 5100.
The resulting mean value of the mean density of the earth is 569.
If the eleven determinations be supposed to have equal weight, the
probable error of their value is 015.
The various determinations differ very much among themselves, but they
seem to me sufficiently close to justify the hope that with a large balance
and a large weight, which will not be so easily affected by aircurrents, and
with greater precautions to prevent those aircurrents, a good determination
of the mean density of the earth may ultimately be obtained by this method.
26
ON A METHOD OP USING THE BALANCE
I. June 12. Determination of 1 Scale Division.
Rider on
beam
Extremities of
three successive
oscillations
Resting
point
Mean of pre
ceHing and
succeeding
resting points
DifiEerence
due to R  L
L
2805
3120
2860
29762
—
—
R
2531
2849
2573
27005
29718
2713
L
3062
2885
3038
29675
26868
2807
R
2588
2750
2605
26732
29481
2749
L
2858
2987
2883
29287
26679
2608
R
2729
2606
2710
26627
29288
2661
L
2961
2900
2955
29290
—
—
Mean R  L = 2708 divisions.
03282
1 division = '  = 001212 milligramme.
WITH GREAT DELICACY, ETC.
27
Determination of {B { X) — (A \ Y).
Weight in
pan
Extremities of
oscillations
Resting 
point
Mean of pre
ceding and
succeeding
resting points
Difference
A+ Y ,
2545
2729
2573
26440
—
—
B + X
2675
2498
2648
25797
26327
530
^ + F
2585
2654
2593
26215
25704
511
B + X
2593
, 2536
2580
25612
26187
575
A+ Y
2535
2685
2559
26160
25563
597
B + X
2446
2645
2470
25515
26187
672
A+Y
2736
2520
2710
26215
25585
630
B + X
2527
2600
2535
25655
26292
637
A+Y
2750
2537
2724
26370
—
Mean
difiference
593
.. {B + X)  {A+ Y) = 001212 X 593 = 00718 milligramme.
Greatest deviation from the mean = 082 division = 00099 milligramme.
The weather during this series of weighings was very unfavourable, with
frequent heavy showers.
ON A METHOD OF USING THE BALANCE
II. June 13. Determination of 1 ScaleDivision.
Rider on
beam
Extremities of
oscillations
Resting 
point
Mean of pre
ceding and
succeeding
resting points
Difference
due to R  L
L
3028
2999
3029
30137
This is rejected as it is so
much lower than the
succeeding
R
3218
3357
3268
33000
—
—
L
2996
3087
3010
30450
33068
2618
R
3229
3375
3276
33137
30510
2627
L
2995
3109
3015
30570
33177
2607
R
3252
3371
3293
33217
30590
2627
L
2904
3194
2952
30610
—
Mean R  L = 2620 divisions.
.*. 1 division = 001252 milligramme.
The weather was as unfavourable as on the previous day.
The weights were changed shortly before the commencement of this series
and the balance then worked so irregularly that for some time I was unable
to begin the rider determination. Even then the first restingpoint had to
be rejected.
WITH GREAT DELICACY, ETC.
29
Determination of (A+ X)— {B+ Y).
Weight in
pan
Extremities of
oscillations
Resting 
point
Mean of pre
ceding and
succeeding
resting points
Diflference
B+ Y
3078
3260
3116
31785
These are all rejected, as the
motion was so irregular.
The weights had been
\ changed a short time be
fore, and had probably not
reached an uniform tem
perature
A + X
2937
3078
2958
30127
B+ Y
3096
3223
3126
31670
A + X
2955
3096
2976
30307
B+ Y
3041
3227
3081
31440
A + X
2893
3044
2915
29740
—
—
B+ Y
3055
3150
3075
31075
29735
1340
A + X
2942
3001
2948
29730
30995
1265
B+ Y
3046
3127
3066
30915
29651
1264
A + X
2900
3007
2915
29572
—
—
Mean {A + X)  {B + Y) = 1289 divisions.
.. {A + X)  (B + Y) = 01614 milligramme.
Greatest deviation from the mean = 051 division = 00062 milligramme.
30
ON A METHOD OF USING THE BALANCE
III. June 13. Determination of 1 Scale Division.
Rider
on beam
Extremities of
oscillations
Resting 
point
Mean of pre
ceding and
succeeding
restingpoints
Difference
due to R  L
R
3013
2906
2997
29555
—
—
L
3137
3287
3181
32230
29638
2592
R
2936
3004
2945
29722
32247
2525
L
3110
3316
3164
32265
29788
2477
R
2816
3132
2862
29855
32357
2502
I.
3101
3358
3163
32450
1 29862
1
2588
R
2878
3081
' 2908
29870
—

Mean R  L = 2537 divisions.
03282
1 division = o^qy = 001293 milligramme.
WITH GREAT DELICACY, ETC.
31
Determination of {A\ X) — {B^Y).
Weight in
pan
Extremities of
oscillations
Resting 
point
Mean of pre
ceding and
succeeding
resting.points
DiflEerence
A\ X
3033
2945
3018
29852
—
B+ Y
3036
3158
3062
31035
29707
1328
A + X
2889
3015
2906
29562
—
—
B+ Y
3002
3116
3025
30647
This is evidently due to some
irregular and short disturb
ing cause, and is rejected
A^ X
28S8
3042
2911
29707
—
—
B+ Y
3015
3193
3039
31100
29716
1384
A + X
2917
3020
2932
29722
31033
1311
B+ Y
3010
3168
3041
30967
—
—
Mean {A + X)  {B + Y) = 1340 divisions.
.. (A + X)  (B + Y) = 01732 milligramme.
Greatest deviation from the mean = 044 division = 00057 milligramme.
32
ON A METHOD OF USING THE BALANCE
IV. June 14. Determination of 1 ScaleDivision.
Rider on
beam
Extremities of
oscillations
Resting 
point
Mean of pre
ceding and
succeeding
restingpoints
DifEerence
due to R  L
R
2525
2534
2517
25225
This was taken soon after enter
ing the room. It is so much
lower than the succeeding
that it is rejected
L
2908
2737
2888
28175
—
—
R
2424
2667
2451
25522
28196
2674
L
2726
2897
2767
28217
25589
2628
R
2598
2539
2587
25657
28267
2610
L
2862
2800
2865
28317
—
—
Mean R  L = 2637 divisions.
03282
,. 1 division = = 001244 milligramme.
Being interrupted, I could not continue the series of rider determinations
further.
WITH GREAT DELICACY, ETC.
33
Determination of (B + X) — (A+ Y).
Weight in
pan
Extremities of
oscillations
Resting 
point
Mean of pre i
resting points
1
A+ Y
2917
2750
2895
28280
—
—
B+ X
2610
2803
2634
27125
28285
1160
A+ Y
B + X
2857
2804
2851
28290
27166
1124
2802
2653
2775
27207
28390
1183
A+ Y
2949
2761
2925
28490
27343
1147
B + X
2760
2739
2754
27480
28640
1160
A+ Y
2673
3054
2735
28790
275 n 1279
B+ X
2785
2729
2774
27542
28928 i 13S6
A+ Y
2966
2855
2951
29067
— —
Mean {B + X)  [A + Y) ^ 1206 divisions.
.. (5 + Z)  (^ + 7) = 01500 milligramme.
Greatest deviation from the mean =18 divisions = 00224 mgm. .
The previous determination oi{B+ X)— (A+ Y) was 0718 mgm. The
difference is too great, 0782 mgm., to be accounted for by errors of experiment.
There must have been some deposit on one of the weights, either of dust or
moisture. I therefore took them out, cleaned and adjusted them by scraping
B till nearly equal to A, and removing the wax from A.
p. c. w.
34
ON A METHOD OF USING THE BALANCE
V. June 14. Determination of 1 ScaleDivision.
Rider on
beam
Extremities of
oscillations
Resting 
point
Mean of pre
ceding and
succeeding
resting points
Difference
due to R  L
R
2126
2303
2147
22197
—
1
L
2523
2425
2512
24712
22216
2496
R
L
2250
2205
2236
22240
24667
2427
2386
2528
2407
24622
22187
2435
R
2249
2187
2233
22140
24598
1
2458
L
2497
2422
2489
24575
22126
2449
R
2268
2167
2243
22112
!
Mean R  L = 2453 divisions.
Q.Q902
1 division =  ^ = 001339 milligramme.
WITH GREAT DELICACY, ETC.
35
Determination of {B+ Y)— (A+ X).
Weight in
pan
Extremities of
oscillations
Resting 
point
Mean of pre
ceding and
succeeding
restingpoints
DifiEerenoe
A + X
2063
2127
2068
20962
This was rejected as being so
much higher than the rest
B+ Y
2090
2064
2086
20760
—
A + X
2124
2040
2113
20792
20771
21
B+ Y
2068
2088
2069
20782
20788
06
A + X
2118
2044
2108
20785
20764
•21
B+ Y
2091
2061
2086
1
20747 20786
i
1
39
A + X
2107
2055
2098
20787
207 11
76
B+ Y
2085
2053
2079
20675 20753
•78
A + X
2097
2051
2089
1
20720 20645
i
75
B+ Y
2033
2087 20615
2039
—
—
Mean {B + Y)  {A + X) = 045 division = 000602 milligramme.
Greatest deviation from the mean = 039 division = 000522 mgm.
A and B had here been cleaned and B readjusted by scraping. A small
vessel containing calcium chloride was put inside the balance to dry the air.
This improved the action of the clamp, diminishing the cohesion.
3—2
36
ON A METHOD OF USING THE BALANCE
VI. June 17. Determination of 1 ScaleDivision.
Mean R  L = 2302 divisions.
1 division = k^^;^ = 001425 milligramme.
WITH GREAT DELICACY, ETC.
37
Determination of (5+ X)— (A{ Y).
Weight in
pan
Extremities of
oscillations
Resting
point
Mean of pre
ceding and
succeeding
resting points
Difference
B+X
2169
2146
2166
21567
—
—
A+Y
2567
2338
2539
24455
21608
2847
B + X
2201
2137
2185
21650
24490
2840
A+Y
2548
2367
2528
24525
21765
21707
2818
B + X
2215
2145
2201
24548
2783
A+Y
2575
2353
2548
24572
21737
2835
B+X
2209
2140
2195
21710
24568
2858
A+Y
2468
2445
2468
24565
21683
2882
B + X
2237
2105
221 6
21657
—
—
Mean {B + X)  {A + Y) = 2838 divisions = 04043 milligramme.
Greatest deviation from the mean = 055 division = 000784 milligramme.
38
ON A METHOD OF USING THE BALANCE
VII. June 18. Determination of 1 ScaleDivision.
Rider on
beam
R
Extremities of
oscillations
Resting
point
Mean of pre
ceding and
succeeding
resting points
Difference
due to R  L
1716
1907
1727
18142
—
—
L
2103
2253
2124
21832
18047
3785
1 R
1892
1717
1855
17952
21754
3802
L
2232
2108
2223
21677
17904
3773
1
R
1850
1732
1829
17857
21617
3760
1
1 L
2202
2113
2195
21557
17822
3735
1
1 
1
1891
1688
1848
17787
1
—
—
Mean R  L = 3771 divisions.
Q.Q2Q9
■. 1 division ^ 07^ = 000870 milligramme.
WITH GREAT DELICACY, ETC.
Determination of (B{ Y)— {A\ X).
Weight in
pan
Extremities of
oscillations
Resting 
point
Mean of pre
ceding and
succeeding
resting points
Difference
B+ Y
2085
2217
2095
21535
—
—
A + X
2469
2144
2441
22995
21622
1373
B+ Y
2192
2156
2180
21710
23091
1381
A + X
2408
2240
2387
23187
21712
1475
B+ Y
2261
2094
2237
21715
23224
1.509
A + X
2218
2423
2241
23262
21696
1566
B+ Y
2115
2218
2120
21677
23221
1544
A + X
2190
2430
2222
23180
—
—
B+ Y
2083
2290
2099
21905
This sudden change of resting 
point must be due to some
irregular disturbance. It is
therefore rejected. It was
slowly returning to nearly its
former values
Mean {B + Y)  {A + X) = 1475 divisions = 012831 milligramme.
Greatest deviation from the mean = 102 division = 00089 mgm.
The great difference between the result here and that in series V is probably
due to deposit of dust. The new mirrors had to be fixed up just before the
experiment began, and the doors were open for some time. At the conclusion
of the weighing I found a good deal of dust on the weights.
40
ON A METHOD OF USING THE BALANCE
VIII. June 19. Determination of 1 Scale Division.
Rider on
beam
Extremities of
oscillations
Resting
point
Mean of pre
ceding and
succeeding
resting points
Difference
due to R  L
L
2358
2223
2333
22842
This is so much higher than
the rest, probably through
being observed soon after I
entered the room, that it is
rejected
R
1970
1812
1939
18832
—
—
L
2282
2227
2280
22540
18824
3716
R
1974
1807
1939
18817
i
22546 1 3729
L
2336 1
2180 22552
2325 j
i
18763 ' 3789
i
R
1917
1833 18710
1901
22528
3818
L
2302
2203
2294
22505
18735 3770
R
1938
1825
1918
18760
—
Mean R  L = 3764 divisions.
0*3282
1 division = ^  = 000872 milligramme.
WITH GREAT DELICACY, ETC.
41
Determination of {B \ X) — (A \ Y).
Weight in
pa,n
Extremities of
oscillations
Resting,
point
Mean of pre
ceding and
succeeding
resting points
Difference
A\ Y
2453
2387
2442
24172
In one observation not recorded
just before this the clamp
had been loose, and the
scalepan had slipped, and
the restingpoint was thereby
changed. The disturbance
had apparently not subsided
when this was taken, it is
therefore rejected
B + X
1930
1857
1914
18895
—
—
A+ Y
2266
2433
2292
23560
18825
4735
B+ X
1822
1928
1824
18755
23522
4767
A+ Y
2294
2392
2316
23485
18772
4713
B+ X
1944
1825
1922
18790
23471
4681
A+ Y
2398
2297
2391
23457
18775
4682
B + X
1947
1816
1925
18760
23451
4691
A+ Y
2306
2374
2324
23445
18750
4695
B + X
1860
1890
1856
18740
—
Mean (B + X)  {A + Y) = 4709 divisions = 04100 miUigramme.
Greatest deviation from the mean = 58 division = 000506 mgm.
42 ON A METHOD OF USING THE BALANCE WITH GREAT DELICACY, ETC.
Summary.
Greatest deviation
Series Mgms. from mean in
milligrammes
II
III
IV
I (B+Z)(^+y) = 0n8 00991
I (A+X){B+Y) = \U4: 0062 (^ ^ ^
 (^ + Z)(B+r) = 1732 •0057) ^^^^^g
(B+X)~(A+Y) = \m) 0224 ( ^ °^ ^
V (5+y)M + Z) = 0060 0052)
VI (B+X)(A+Y) = 4043 0078 / ^ " ^ + ^051 mgm.
VII (B+y)M + Z) = 1283 00891
VIII (,B+X)(A+Y) = 4100 0051 ( ^ " ^ + ^^^^ ""8™
The greatest error — that is the greatest deviation of any one value from
the mean of its series — in the first four series is yoooVttoo^^ ^^ ^ pound. The
greatest error in the four series V — VIII is n oWoooo^^ ^^ ^ pound.
3.
ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH
AND THE GRAVITATION CONSTANT BY MEANS OF THE
COMMON BALANCE.
[Phil. Trans. A, 182, 1892, pp. 565656.]
[Received May 13. Read June 4, 1891.]
I. Account of Apparatus and Method.
In a paper printed in the Proceedings of the Royal Society, No. 190, 1878
(vol. 28, pp. 235)*, I gave an account of some experiments undertaken in
order to test the possibihty of using the Common Balance in place of the
TorsionBalance in the Cavendish Experiment. The success obtained seemed
to justify the intention expressed in that paper to continue the work, using
a large bullionbalance, instead of the chemical balance with which the pre
liminary experiments were made.
As I have had the honour to obtain grants from the Royal Society for
the construction of the necessary apparatus, I have been able to carry out
the experiment on the larger scale which appeared likely to render the
method more satisfactory, and this paper contains an account of the results
obtained.
At the time I was making the preliminary experiments the late Professor
von Jolly was already employing the balance for gravitation investigations
(W iedemann'' s Annalen, vol. 5, p. 112), though I was not aware of the fact.
Later he published an account {Wied. Ann., vol. 14, p. 331) of a determination
of the Mean Density of the Earth by the use of the Balance. Still more
recently Drs. Koenig and Richarz have devised a method of using the balance
for the same purpose {Nature, vol. 31, pp. 260 and 475), and I believe that
their work is still in progress. It might appear useless to add another to
the list of determinations, especially when, as Mr. Boys has recently shown,
the torsionbalance may be used for the experiment with an accuracy quite
unattainable by the common balance. But I think that in the case of such
a constant as that of gravitation, where the results have hardly as yet begun
* [Collected Papers, Art. 2.]
44 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
to close in on any definite value, and where, indeed, we are hardly assured
of the constancy itself, it is important to have as many determinations as
possible made by different methods and different instruments, until all the
sources of discrepancy are traced and the results agree.
The apparatus for the experiments described in this paper was first set
up in the Cavendish Laboratory at Cambridge through the kindness of
Professor Clerk Maxwell. After spending some months in working at the
experiment, but without much success beyond the detection of some sources
of error, I left Cambridge, and ultimately the apparatus was again set up at
the Mason College, Birmingham. The difficulties in carrying out the work
with any approach to exactness have been far greater than were anticipated,
and many times work has been begun and results have been obtained, but
examination has shown them to be affected by large errors which could be
traced and eliminated by further improvements in the apparatus.
At the beginning of 1890, however, the apparatus was brought into fair
working order, and during the course of the year I made a number of experi
ments with the results recorded in this paper.
The Princifle of the Experiment.
The object of the experiment, in common with all of its class, may be
regarded, primarily, as the determination of the attraction of one known
mass M on another known mass M' a known distance d away from it. The
law of universal gravitation states that when the masses are spheres with
centres d apart this attraction is GMM'/d^, G being a constant — the gravita
tion constant — the same for all masses. Astronomical observations fully
justify the law as far as M'jd^ is concerned. They do not, however, give the
value of G, but only that of the product GM for various members of the solar
system.
To determine G we must measure GMM'/d^ in some case in which both
M and M' are known, whether they be a mountain and a plumbbob, as in
Maskelyne's experiment, the surface strata and a pendulumbob, as in Airy's
experiment, or two spheres of known mass and dimensions, as in all the
various forms of Cavendish's experiment.
Knowing the gravitation constant G, we may at once find the mean
density of the earth A. For if 7 be the volume of the earth— regarded as
a sphere of radius i?— the weight of any mass M', being the attraction of the
earth on it, is
GVAM'/R^
But if g is the acceleration of gravity the weight is also expressible as M'g.
Equating these we get
A = gR^jGV.
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 45
Method of Using the Common Balance.
In using the common balance to find the attraction between two masses,
perhaps the most direct mode of proceeding would consist in suspending
a mass from one arm of a balance by a long wire, and counterpoising it in
the other pan. Then bringing under it a known mass, its weight would be
slightly increased by the attraction of this mass. The increase would be the
quantity sought if the attracting mass had no appreciable effect before its
introduction beneath the hanging mass, and if, when beneath it, the effect
on the balance could be neglected. This is very nearly the principle of the
method used by von Jolly, and it is that of the method used in the prehminary
experiments referred to above, in which a mass of 453 grms. of lead was hung
from one arm of a chemical balance (about 40 centims. beam) by a wire
18 metres long, and was attracted by a mass of 154 kilogrms. of lead. But
the attraction to be measured was exceedingly small, rather less than 001
milligrm., and it therefore appeared advisable to use a much larger balance
with a larger hanging mass so that the attraction might be made comparable
with the weight of exactly determined riders. Other anticipations as to
proportionate increase of sensibility and diminution of effect of aircurrents,
have hardly been justified in the way I expected, though, by the ultimate
form of the apparatus, they have, I think, been more than realised.
With increase in the length of beam, a differential method became
appHcable, by means of which the attraction of the mass on the beam w^as
eliminated, and the necessity for prolonging the case to allow of a long
suspending wire was removed. This will be seen from a consideration of
Fig. 1. Let A, B represent equal masses suspended from the two arms of the
balance, and let M be the attracting mass put first under A, the position of
the beam being noted. If M is then placed under B its attraction is not only
taken away from A but added to B, so that the tilting of the beam is that due
to nearly double the attraction to be measured. Of course there are what
we may term crossattractions, in the first position, of M on B, and in the
second position, of ilf on ^, but these may be allowed for in the calculations.
We cannot give any mathematical expression for the attraction of M on the
beam and suspending wires, owing to their irregularity of shape. But this
attraction is eliminated if a second experiment is made in which A and B
are raised equal known distances to A' and B' . For the difference between
the two increments of weight on the right, is due solely to the alteration of
the positions of A and B relative to M, the attraction on the beam remaining
the same in each. From the observed effect of a known alteration of distance
the attraction at any distance can be found.
This is, shortly, the method adopted. The arrangement was ultimately
complicated by the addition of a second mass m. Originally the mass M
46 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
was alone on a turntable which revolved about a vertical axis immediately
under the central knifeedge of the balance. And some experiments which
I made led me to suppose that mere change of position of the mass did not
affect the level of the balance. However, after a complete determination
in 1888 of the mean density, when I supposed that the work was finished,
an examination of the results showed some curious anomahes, which I could
only ascribe to a tilting of the whole floor on the displacement of the mass.
Making new tests as to the effect of removal of the mass, I found that the
Fig. 1. Elevation of balanoeroom and observingroom. The front of the case is removed,
and the front pillar is not shown. The pointer and mirrors are at the back.
previous tests had been quite wrong in principle, and that there was a very
appreciable effect quite visible in the telescope when the masses A and B
were removed, and M was removed from one side to the other, the slope of
the floor changing by an angle comparable with a third of a second. If this
had been absolutely constant in amount, the differential method would have
eUminated it ; but, probably, it varied sKghtly in successive motions of the
turntable, and the results showed that there was also a secular change, the
amount of tilt gradually increasing. This secular change was probably due
to increasing rigidity of the floor, so that it tilted over bodily, moving the
supports of the balance with it, an increase partly due, perhaps, to the
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 47
pressure of the building, which had only been erected ten or twelve years,
but chiefly, I think, to a gasengine recently erected next door. When this
was doing heavy work, the vibrations were very plainly felt, and no doubt
they greatly aided the floor in 'settling down.' A second balancing mass
m was therefore added, half as great as M, and on the opposite side of the
turntable, but twice as far from the axis. The resultant pressure was now
always through the axis, and I could detect no tilting of the floor when the
turntable was moved. Of course the balancing mass acted somewhat to
reduce the effect of the larger attracting mass, but in a calculable ratio.
Finally, in order to eliminate or reduce the effect of any want of symmetry
in the moving parts or in the masses, a second set of experiments was made
with all the masses turned over and moved from left to right, and the mean
of the first and second sets was taken.
I now proceed to a detailed description of the various parts of the apparatus
and the mode of experiment.
The BalanceRoom. The balanceroom is in the basement of the Mason
College, immediately under my room, and about 20 metres from the street.
On one side were three windows looking on to a small courtyard, entirely
surrounded by high buildings, but the windows have been bricked up. On
the two adjacent sides are two other rooms, and on the opposite side a closely
fitting door opening on a short corridor with doors at each end. There is no
chimney in the room, and only an opening in the ceihng through which the
balance was observed from the room above. The floor is of brick, resting on
earth, and is very firmly laid.
The temperature of the room was taken by means of a thermometer with
a protected bulb at the end of a long wooden rod hanging down from the
room above. The thermometer was about 6 feet from the floor, near one
end of the case, and it could be rapidly pulled up into the room above and
read by the observer before its temperature sensibly varied. The tempera
ture never appeared to vary so much as 01° C. in the course of two or three
hours.
The BalanceCase and its Supports. The case (Fig. 1) is a large cabinet of
IJ inch wood, 194: metres high, 163 metres wide, 61 metre deep, with three
large doors in front giving access to the hanging masses and riders, and a small
door at the back near the mirror hereafter described. It is lined inside and
out with tinfoil, and under each of the suspended masses is a double bottom
with a layer of wool between, making a total thickness of about 1 J inches or
4 centims. At the top is a small window about 10 centims. square, through
which the oscillations of the beam were observed. On each side within the
case are placed three horizontal partitions, like shelves, to hinder circulation
of the air.
The larger attracting mass and the attracted masses are gilded, and it is
48 03Sr A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
possible that some advantage may arise from having the surface of the case
of different metal. For if it, too, were gilded, it would readily absorb
radiation from the large mass, and when the inside temperature changed,
the suspended masses would readily absorb radiation from the inner surface
of the case. But gold probably absorbs considerably less of tin radiation
than it absorbs of gold radiation, and so temperature changes are probably
lengthened out more than if the case were gilded.
Plan of turntable, girders, pillars, and balancecase. w. Window
in case. c. Usual position of cathetometer.
It was necessary to support the case so that the attracting masses could
be moved about underneath it, and also to make it independent of the floor.
Two brick pillars, 58 centims. x 36 centims. and 56 centims. high, were
therefore built on thick beds of concrete under the floor, and about 3J metres
apart. They rise up free from the bricked floor. Stretching between them
are two parallel iron girders (g, g), about 30 centims. apart, and with their
under side 56 centims. above the floor. The balancecase is placed across
the middle of these girders (see plan, Fig. 2), with its under surface level with
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 49
that of the girders. The square baseplate of the balance is placed on the
girders on three levelling screws. Two horizontal screws attached to the
girders bear against each edge of the baseplate, so that it can be adjusted
and fixed in any position.
To lessen vibration one tier of bricks is removed from each pillar, and in
its place are inserted eight cyhndrical blocks of indiarubber (^, i, Fig. 1),
originally 75 centims. diameter and 75 centims. high. These crushed down
almost 1 centim. at once, but have not shown any further measurable con
traction in the coursfe of several years. Their effect in deadening vibration
has been surprisingly great.
The Turntable. On a bed of concrete, and quite free from the brickwork
of the floor, is a circular rail of cast iron, 13 metres in diameter. On this, on
conical brass wheels and pivoted at the centre, runs the turntable, about
15 metres in diameter. This is made of wood and covered with tinfoil. It
is like a wheel with a flat circular rim, and with four flat spokes arranged as
a cross. It is as nearly symmetrical as possible, and at opposite ends of a
diameter are placed two shallow cups, in either of which the large attracting
mass may rest. The centres of these cups are a distance apart, equal to the
length of the balancebeam. There are cut slots through the bottom of each
cup, so that the bottom of the mass can be seen for the purpose of measuring
the vertical diameter.
Two beams, 274 metres long, run across the turntable 26 centims. apart,
with the cups between them, and across the ends are two boards, each with
a circular hole 12 centims. in diameter, and in either of these the smaller,
or balancing mass, may rest. These beams are braced by brass rods to brass
uprights at their middle points to diminish bending.
The turntable is moved by an endless gut rope passing round it, and
fixed at one point of the rim. The two sides of the rope pass over pulleys
on to a drum in the room above. There are stops on the circular rail, against
which come brass pieces on the turntable when the masses are in position
at either end of the motion. The drum can be turned easily by the observer
at the telescope. Since the knifeedges and planes of the balance are of steel,
all other moving parts of the apparatus were made free from iron. As an illus
tration of the necessity of this, I may mention that for some time I used what
I supposed to be a brass wire rope to move the turntable, but on looking out
for the explanation of some irregularities, I found that the brass was wrapped
round a core of steel wire, which acquired poles at the highest and lowest
points in the position in which it always rested between different sets of
weighings. These poles had quite an appreciable action on the balancebeam.
The Balance. This is of the large bullionbalance type, with gunmetal
beam and steel knifeedges and planes. It was made specially for the
experiment by Mr. Oerthng, with extra rigidity of beam. Its performance
p c. w. 4
50 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
has shown the great excellence of the design. The central knifeedge is
supported on a steel plane by a framework rising 107 centims. above the
baseplate, and the usual moveable frame can be raised or lowered from out
side the case, fixing the beam or setting it free to oscillate. The beam has
often been left free to oscillate for months at a time, with the full load of
20 kilogrms. on each side, but I have no reason to suppose that the knife
edges have suffered at all.
The length of the beam was measured by taking the length of each half
separately by a beamcompass, and the mean of several measurements gave
123329 centims. as the total length. The standard scale used throughout
was that of a cathetometer made by the Cambridge Scientific Instrument
Company. This scale has been verified at the Standards Office, and taking
its coefficient of expansion as ewoo' ^^ ^^J ^® regarded for our purpose as
perfectly correct at 18°, any errors being at that temperature much less than
the errors of experiment. Comparing the beamcompass with this scale, it was
found that 06 centim. must be subtracted, reducing the length to 123269
centims. Now both beam and scale are of gunmetal and may, therefore,
without serious error, be assumed to have the same coefficient of expansion,
so that this is the length of the beam at 18°. At 0° it is 123232 centims.
Mirrors, Telescope, and Scale. At first a mirror was attached to the
centre of the beam and the reflection of a scale in it was observed, either
in the ordinary method or in the method described in the former paper
(Roy. Soc. Proc, vol. 28, 1879)*, where a second fixed mirror is used to throw
the ray of light a second, or even a third time back on to the moving mirror,
each return increasing the deflection of the ray. But it was then necessary
to make the time of vibration very long, and even when the time was three
minutes, the tilt due to the attraction, i.e. the change of restingpoint, did
not amount to more than two or three scaledivisions. Now certain
irregularities observed when the apparatus was first set up at Cambridge,
led to experiments on the time taken by heat to get through the case in
sufficient quantity to affect the balance, and I found that a coil of copper
wire placed close under the case on one side (the bottom of the case being
then solid, 1 inch thickness), heated by a current yielding 100 calories per
minute, began to produce an appreciable disturbance on the balance in about
10 minutes, doubtless by the creation of aircurrents from the heated floor
of the case. It appeared advisable, therefore, to reduce the time of a complete
experiment to less than this if possible, and, consequently, the time of a single
swing very much below 3 minutes. This could only be done if at the same
time the optical sensibility were very greatly increased.
The employment of what may be termed the doublesuspension mirror
method due I beUeve to Sir WilHam Thomson, and used by Messrs. G. H. and
* [Collected Papers, Art. 2.]
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 51
Horace Darwin in their experiments on the Lunar Disturbance of Gravity
{Brit. Assoc. Rep., 1881), has very satisfactorily solved the problem, giving
a greatly increased deflection on the scale, even when the time of oscillation
is as short as twenty seconds.
This method, which deserves to be more generally known and applied for
the detection of small motions, consists in suspending a mirror by two threads,
Microscope sijage
Bnrad^xt
1
Tir^
«q
Varus ivorkUn^
im dashpot
Fig. 3. DoubleSuspension Mirror (half size).
one from a fixed point, the other from the point which moves. The angle
through which the mirror turns for a given motion of the latter point is
inversely as the distance between it and the fixed point, so that by diminishing
this distance the sensibility of the arrangement may be almost indefinitely
increased.
To apply it to the balance, a small bracket (Fig. 3) is fixed to the ordinary
pointer of the balance, about 60 centims. below the central knifeedge. This
projects horizontally at right angles to the axis of the beam, and it is bevelled
at the edge. Close to it is another bevelled edge attached to a microscope
4—2
52 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
stage movement which is fixed on to the central pillar of the balance.
A thread of silk (as supphed for the Kew magnetometer) is fastened to the
stage, passes over the bevelled edge, through two eyes (e, e) on a Hght frame
holding the mirror, up over the bevelled edge of the bracket, and is fastened
to the bracket. The microscopestage movement allows the distance between
the threads to be adjusted, and also enables the azimuth of the mirror to
be altered.
Of course, if the mirror were weightless, it would not affect the sensibihty
of the balance, and the threads might be brought very close together. But
the weight of the mirror — it is silver on glass, 56 milhms. x 38 miUims. x 10
miUims. — has a considerable effect on the sensibility, diminishing it with de
crease of distance between the points of suspension. In practice it has been
found convenient to work with the threads parallel, and from 3 to 4 miUims.
apart, the time of swing one way being adjusted to about 20 seconds.
A less time hardly suffices for a correct determination and record of the scale
reading. Taking 4 milHms. as the distance, and supposing the bracket to
be 600 millims. below the knifeedge of the balance, the mirror evidently turns
through an angle 150 times as great as that through which the beam turns.
The drawback to this method of magnification is that the mirror has its
own time of swing and is easily disturbed. The swings of the mirror and the
disturbances are, however, effectually damped by having four light copper
vanes attached to the end of a thin wire, projecting down from the mirror
and working in a dashpot with four radial partitions not quite meeting in
the centre, one vane being in each compartment. I found that mineral lubri
cating oil is very suitable for the dashpot, as the surface keeps quite clean and
there is little evaporation. The swings of the balance are also very greatly
damped by this arrangement, but the effect of this will be discussed later.
The telescope and scale are in the room over the balanceroom (see Fig. 1),
a hole being cut through the floor, and a small glass window being fixed in
the top of the case. As the suspended mirror is in a vertical plane it is
necessary to have an inclined mirror fixed in front of it to direct the light
from the scale horizontally on to it and back again to the telescope. With
the magnification used it was necessary, for good definition, to have an ex
ceedingly good inclined mirror, and several were rejected before a suitable
one was obtained. That finally used is a silver on glass oval mirror,
60 milHms. x 40 milhms., by Browning. The glass window in the case is
optically worked and carefully adjusted to be normal to the path of the light.
The telescope has a 3inch objectglass of about 4 feet focal length. It is
fixed on a brick pillar, on one of the brick arches which form the ceiHng
of the balanceroom, and it rises free from the floor of the observingroom.
To destroy vibration one course of bricks is replaced by blocks of india
rubber. The scale has 50 divisions to the inch (say J milhm.), ruled diagonally,
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 53
and divided to tenths by crosslines. It is photographed on glass from a scale
drawn on paper with very great care, 50 inches long (say 127 centims.), and
with 500 divisions. The photograph is ^^^th of this length, and only the
central part of the scale, about 60 divisions in length, has been used. The
diagonal ruling enables a tenth of a division to be read with certainty, and
the readings recorded in the Tables, pp. 98128, are in tenths. Though the
lines appear somewhat coarse, I have not been able to find another scale
equal to it in distinctness and in ease of reading. As all the results depend
on the ratio of measurements, taken almost simultaneously, of deflection
due to attraction and rider respectively, in the same part of the scale, I have
not thought it necessary to calibrate it.
The scale is fixed horizontally on the end of the telescope close to the
objectglass with a piece of ground glass over it. It was illuminated in
general by an incandescent lamp placed above it, once by an Argand burner.
The distance from the scale to the mirror and back is about 5 metres.
It follows that 1 division of the scale corresponds to an angular motion of
the mirror through 0001 radian. But this is at least 150 times the angle
through which the beam turns for the same defiection. So that 1 scale
division implies an angular motion of 0000006 radian, or ^" , in the beam.
As the total length of swing in Table III is never more than 12 divisions, the
angular vibrations of the beam are at the most about l"6, and the hnear
vibrations of the masses, since the half beam is about 60 centims., are at
the most about 005 millim. This shows that it is quite unnecessary to
consider any change of distance due to vibration. The greatest deviation
from the mean in any of the series of weighings recorded is about 1 per cent,
of the ridervalue, corresponding to about ^^th. of a division, or an angle
of ^" in the beam, and a distance of 00004 miUim., say goo'ooo inch, in the
motion of the masses. This seems to show that the method is accurate as
well as sensitive.
Determination of the Value of the ScaleDivisions by means of Riders. This
was done by means of centigrammeriders (Fig. 4), these being the least
weights which appeared capable of sufficiently accurate determination.
Instead of transferring the same rider from point to point, it was much
easier to use two equal riders, and to take one up while the other was
being let down a given distance from it. The distance selected was about
25 centims., since the deflection due to the transfer of one centigramme
so far along the beam was nearly equal to that due to the greatest attraction
to be measured.
At first the riders when on the beam rested in Vnotches in a pair of parallel
brass strips fixed on and parallel to the beam. But this plan was soon
abandoned, as there was no certainty about the position of the rider in the
notches. The riders were then supported in little wire frames, each hung
54 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
by two cocoonfibres from the edges of a plate fixed to the beam, the edges
being parallel to the central knifeedge. The only objection to this method
was the very considerable time spent in replacing the fibres after the breakages
which occurred on dusting or any readjustment of the balance.
Fig. 4. Rider, actual size, and end of liftingrod,
r^
^JX
Fig. 5 a. Subsidiary riderbeam, 66, attached to centre Fig. 5 6. Wire frames depending like
of balancebeam, BB, by plate jp just above central scalepans from ends of 66, Fig. 5 a,
knifeedge, k (half size). side and end views (half size).
Ultimately a small subsidiary beam, about 25 centims. long, was attached
to the centre of the balancebeam just above the knifeedge (Fig. 5a), the
scalepans being represented by small wire frames in which the riders could
rest (Fig. 56). These frames depend from agate pieces resting on steel points
at the extremities of the subsidiary beam in the way now usually adopted
in delicate assaybalances. This mode of supporting the riders appears to
be perfectly satisfactory.
To raise or lower the riders two short horizontal liftingrods parallel to the
beam move up and down within the supporting wire frames with a nearly
parallel motion, and on them are two metal pieces with their upper surfaces
shaped so that the riders rest on them without swinging (Fig. 4, r). They
are the extremities of Lshaped projections from a jointed parallelogram
framework (Fig. 6), supported on an upright in front of the subsidiary beam.
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 56
The framework is moved by a tongue engaging with it, and projecting from
a horizontal rod, which rotates about its axis in bearings, one within the case
and the other outside. The rod is turned through an angle of about 30°
between stops by an endless string passing upwards and round a wheel in
the observingroom.
The parallelogramframework and the bearing of the rotating rod within
the case are both supported, independently of the case, from the ceihng. At
first they were supported respectively on the central pillar of the balance
and on the case ; but when the increase of optical sensitiveness enabled me
to detect small irregularities, I realised how essential it was for accurate
weighing that all parts of the apparatus moved from the outside should be
supported quite independently of the balance. Even the string moving the
Fig. 6. Liftingrods to raise or lower riders (half size).
rod transmitted great and continual vibration. The rod and the framework
with the liftinglevers were, therefore, supported by iron rods coming down
from the ceiling through holes in the top of the case, large pieces of cardboard
stretching from these rods over the holes to hinder the passage of dust into
the case. Once or twice in the course of prehminary experiments irregularities
were traced to accidental contact of outside bodies with the case.
It appeared just possible that there might be electrification of the riders
by friction with the liftingrods, especially when they were supported by
cocoonsilk. It was, therefore, advisable that the surface of the liftingrods
should be of the same kind as that of the riders. As the latter are silver wire
gilded, the liftingrods are also gilded. It may not be uninteresting to note
here a curious phenomenon which occurred during some early preliminary
experiments. The shaped pieces on the Hftingrods were then of wood
covered with gold leaf, put on with ordinary paste. After they had been on
for some months, I obtained some very various results for the deflection due
to the riders, and on examining the liftingrods I found that a number of long
56 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
needlegrowths projected from the wood pieces and interfered with the
supporting wire frames. At first I thought these were organic, but my
colleague, Professor Hillhouse, examined them and found that they were
crystalhne. Doubtless, the hygroscopic paste set up electric action between
the gold leaf and the brass to which the wood pieces were attached, and the
crystals were probably zinc sulphate. The wood was then replaced by brass
gilded, and no further difficulty of the kind was experienced.
The length of the subsidiary beam was kindly determined for me by
Mr. Glazebrook at the Cavendish Laboratory. The steel points are hardly
sharp enough to determine the distance to 1 in 10,000, but the mean of the
results is sufficiently exact. The following are Mr. Glazebrook's determina
tions ; the four points being denoted by a, h,c,d\
Date
1889 July 4
Julv 11
July 12
Temperature
Number of
readings
, , Number of , , ,
«t°^ readings ^ ^ ^^ ^
22^5
215
23
6
3
3
inches \ inches
•9985 i 6 ^9979
•9990 i 3 j ^9982
•9988 3 ^9979
These are in terms of a gunmetal standard of which the error is only 3 in
100,000 at 0°, and, therefore, for my purpose negUgible. The beam is of
brass, and we may assume with sufficient exactness that it has the same
expansion as the standard. The temperature may, therefore, be left out of
account. The mean value of \{ah + cd) is therefore 9983375 inch, or taking
2539977 centims. to the inch we obtain
Length of beam at 0°, 253575 centims.
There is an advantage in fixing this beam at the centre, which should be
noted here. Suppose the riders are not quite equal, but have values w and
w + S. Let the two ends of the subsidiary beam be distant a and a f I from
the central knifeedge. Then the effect of picking up the rider w from the
nearer, and letting down the rider ii; + S on the further end, is equivalent to
putting at unit distance
{w + h)(a\l)wa==wl\h{a\l) = wl (l + ^^ ^\ ,
or the error hjw is multiplied by {a + 1)11, and, if the beam is not central,
(a f l)jl may be greater than 1, so that the error is magnified.
If, however, the small beam is central, I is equal to — 2a, and the error is
multipHed by + J.
If the riders are interchanged and the weighings are then repeated, the
mean result is the same as if riders with the mean value were used, for
w[a + l) {w\Z)a = wl~ Sa
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 57
and the mean of this and the above is
(1)
The Attracting and Attracted Masses. These are all made of an alloy of
lead and antimony, for the sake of hardness, the specific gravity in each case
being about 104. They were made at various times and places, the large
attracting mass M being made more than 12 years ago by Messrs. Storey, of
Manchester. The smaller balancing mass m was made in 1889 by Messrs.
Heenan, of Manchester and Birmingham. These were both cast with a
'head' on, and then turned. The attracted masses A and B were made by
Messrs. Whitworth, and subjected to hydraulic pressure before turning. The
dimensions have been measured from time to time, and there is no evidence
of any sensible change of shape.
The larger mass M and the attracted masses A and B were weighed at
the Mint through the kindness of the Deputy Master and Professor Koberts
Austen. For the weight of the balancing mass m, I am indebted to Messrs.
Avery, of Birmingham. The large mass M has suffered two accidents since
it was weighed, once being slightly cut into by a saw during some alteration
of the case, and once being scratched by coming into contact with a piece
of metal fixed to the turntable in taking it out of its place. The sawcut was
carefully filled in with lead, and the scratch removed only a fraction of a
gramme, as was determined by taking a mould of the hollow. I should be
glad to think that the determination of the attraction was sufficiently exact
to make reweighing necessary, but I am afraid that the alteration in weight
is very far beyond the important figures, and I therefore take the original
weight as sufficiently near the truth. The masses A and B have been gilded
since the original weighing, but I carefully determined their increase of weight
by the balance used in the gravitation experiment.
The values given below in the second column are the true masses. In
the third column are the masses of M and m, less the air displaced by them,
this being taken as 184:1 and 92 grms. respectively. It will be shown later
that the true masses of A and B and the reduced masses of M and m may be
used in the calculation of the result.
True mass in
grammes
Mass less that of air
displaced
M
m
A
B
15340726
764974
2158233
2156621
15338885
764882
58 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Suspension of the Attracted Masses. Each of the attracted masses is
drilled through along a diameter, the hole being 6215 centim. in diameter,
and a brass rod (Fig. 7) terminating in an eye e below, is passed through the
hole. The mass is secured in position by a nut n working in a screwthread
cut for a short distance in the rod. An exactly
similar rod terminating in a similar eye e', and
with a similar nut n', is fastened end to end to
this by a union u. The nuts and the inner sides
of the enlargements for the eyes are hollowed
out so as to fit exactly on to the spheres.
From the ends of the balance beam hang
down stout brass wires terminating in hooks.
If these hooks are passed through the eyes e'
the attracted masses are close to the floor of
the balancecase, and their centres are adjusted
to be about 32 centims. from the centre of the
large attracting mass when under either of them.
If the masses are turned over so that the hooks
pass through the eyes e, they are about
30 centims. higher or at nearly double the
distance, the length ee' being about 48 centims.
The rods being perfectly symmetrical about the
union u, the attraction on them is the same in
either position. The weight of each is about
212 grms., or about jJ^ of the attracted mass,
so that any small variation in their position
would produce a negligible variation in the
total attraction. By the differential method,
the attraction on them entirely disappears from
the results.
The Mode of Support of the Attracting Masses
M and m. This has already been described
when describing the turntable.
The Riders. Four centigrammeriders, A, B,
C, D, of silver wire gilt were made by Mr. Oerthng
of the form shown in Fig. 4. These were weighed
in 1886 at the Bureau International des Poids
et Mesures, by M. Thiesen. The following is an extract from the certificate :
' Densite et volume. Comme densite on a accepte celle de 1' argent, et par
consequent comme volume de chacun des cavahers, 00010 miUihtre.
' Deterynination des poids des cavaliers. L'etude des poids de ces quatre
cavaliers a ete faite par M. le Dr. Thiesen, adjoint du Bureau International,
Fig.
7. Suspender for Attracted
Mass (onefourth size).
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 59
charge de la section des pesees. M. Thiesen au moyen de la balance Stuchrath,
destinee a des poids au dessous du gramme, a d'abord determine les differences
entre les quatre cavaliers pris deux a deux dans les six combinaisons possibles,
et ensuite la difference entre 1' ensemble des quatre cavaliers et le poids de
40 milligrms. de la serie du Bureau, serie en platine iridie recemment
etalonnee par M. Thiesen. Les comparaisons ont ete faites du 19 au 29 Mars,
1886.
' Resultats. De I'ensemble de ces comparaisons resultent les poids :
A = 101247 milligrms.
B = 100615
0=101196
Z)= 101262
' L'incertitude de ces determinations ne depasse pas 0001 milhgrm.'
A and D were selected for use as being the nearest to each other in value.
B and C were kept untouched in boxes till 1890. In the various experiments
made between 1886 and the final weighings, A and D had necessarily been
handled to some extent, especially through the frequent breaking of the silk
fibre suspension used before the subsidiary beam described above, and it
appeared possible that their weights might be altered. It was also necessary
to determine whether an appreciable amount of dust was deposited on them
in the course of several weeks as it was inconvenient to dust them frequently.
The riders B and C might be assumed to have the same weight as in 1886,
and could be taken as standards.
To make the weighings a 16inch chemical balance was arranged with a
doublesuspension mirror on exactly the principle already described for the
large balance. The apparatus was put together quickly with materials at
hand, and might easily be greatly improved. It is only described here to
show how accurate the method is, even with such rough apparatus, and that
it is applicable to a small as well as a large balance.
A cork sliding on the pointer with a horizontal needle stuck in it, served
to support one thread of the mirror; a stand with a projecting arm — one
made to hold platinum wires in a Bunsen flame — served to support the other
thread. A wire with a small copper vane depended from the mirror and was
immersed in an oil dashpot. The telescope and a miUimetrescale were on
a level with the mirror about 2 metres distant on one side of the balance.
Two brass strips, parallel to each other and the beam, were fixed on the top
of one arm of the beam, and in each of these were two Vnotches in which
centigrammeriders could rest. Two levers, worked by cams on a rod rotated
by the observer, picked one rider up and let down the other, so that the effect
was equivalent to the transfer of 1 centigrm. from one notch to the other.
Their distance apart was such that this was equivalent to the addition of
60 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
•3284 mimgrm. to one pan of the balance. This was the arrangement
described in my former paper. Attached to one pan was a pair of brass
strips parallel to each other, and such that the riders A, B, C, or D, would
just rest across them. Two liftingrods worked up and down between these
strips, so that of the two riders to be compared, one could be picked up at the
instant the other was let down. The liftingrods were worked by a rod
rotated by the observer and supported quite independently of the balance,
and of the slab on which it rested. By this plan the value of the scaledivisions
and the shifting of the centre of swing on changing the weights to be compared,
could all be determined without raising the beam of the balance between the
successive weighings, an essential condition, I beheve, for exact work.
The weighings were made in the large room of the Physical Laboratory,
and no precaution was taken to protect the balancecase beyond placing
a board in front of it. The room is draughty and subject to great variations
of temperature, so that the weighings were made under very disadvantageous
circumstances. One result of this was a rapid and sometimes very great
change of restingpoint in the course of a few hours, so that the scale passed
out of the field of view. In order to bring it back without opening the case,
two glass tubes passed through the top of the case, almost down to the scale
pans, and small bits of wire could be dropped through these on to either pan
as needed. Caps fitted on to the tubes to prevent draughts. This plan
appears worthy of mention, as it suggests a mode of determining the value
of a scaledivision when a balance is either too sensitive for riders or has no
special arrangement for their accurate use. If a piece of wire weighing, say,
1 milligrm. is cut into say ten nearly equal parts, and if these are dropped
on to the two pans alternately the shiftings of the centre of swing will be to
and fro, about equal distances, due to about 1 milligrm., but the sum of the
shiftings will be that due to 1 miUigrm., and the balance at the end will be
nearly in the same position as at the beginning.
The following is an abstract of the comparisons of the riders. They were
made soon after the first determinations of attraction on February 4, when
A and D had not been dusted for three months.
In each case three extremities of swing were observed, and the centre of
swing was determined from these by the graphic construction described later
(p. 72).
The centres of swing were combined in consecutive threes in the usual
way to give the differences in scaledivisions.
Thus, in the first series, the successive centres of swing with D and A
alternately in the scalepan were
D
A
D
A
D
!31
223
217
2119
208
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 61
whence (D  A)^ = ^  223 = + 10 division,
fn A^ 017 223 + 2119 ^ .^ .. . .
(D — ^)2 = 217— = — 045 division,
(D  A)^ = ^1+^ _ 2119 = + 06 division.
Mean D — A = SS division.
Successive values of the differences alone are given below.
The time of swing one way was about 16 seconds.
February 16, 1890.
(1) Comparison of A and D, undusted.
Deflection due to 328 milligrm., 8345, 8245, 8445 divisions. Mean 8345
divisions.
D A = 10,  45, + 06 division. Mean 38 division;
therefore D = .4 + 0015 milligrm.
(2) Comparison of A undusted, D dusted.
Value of scaledivision taken as in the last.
D A = '6, + 3,  1,  4, + 25. Mean  09 division ;
therefore D = A — 0004 milligrm.
Februanj 17, 1890.
(3) Comparison of A and D, both dusted.
Value of scaledivision taken as below (4).
B A = 'I,  2, + 3,  3,  8. Mean  22 division;
therefore B = A — 0008 milhgrm.
(4) Comparison of C and B.
Beflection due to 328 milligrm., 8535, 854, 8465. Mean 8513 divisions.
BC = + 15, 00, + 05,  15, + 05, + 3, + 05,  05,  35,
05, 35, 45, 50, 2. Mean 114 division;
therefore D = C + 00044 milligrm.
February 18, 1890.
(5) Comparison of C and B repeated.
Beflection due to 328 milligrm., 9275, 923, 9165. Mean 9223 divisions.
BC= 35,  05,  8,  95,  1, + 05, 0,  15,  1, + 05.
Mean — 17 division;
therefore B = C — 0006 milligrm.
62 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Combining this with the last, and weighting them in the ratio of the
numbers of determinations in each,
D = C + (00044 X 14  0006 x 10) h 24 =  0000 milHgrm.
(6) Comparison of A and D.
Value of scaledivision taken as above, 328 milHgrm. = 9223 divisions.
DA = 45, 25, 1,  2,  1, 35, 25, 45, 60, 5, 5, 55, 5, 75, 55,
•1, 05, 45, 10, 30, 8, 9, 35, 2, 30, 55, 50, 35, 45, 45.
Mean 378 division;
therefore D = A + 00134 milligrm.
Examining the values obtained in (1), (2), and (3), it will be seen that
no trustworthy evidence is given of a difference due to dusting. Any existing
difference was probably under 002 milHgrm., and since the weighings on
February 4, before dusting, were made with the attracted masses in the upper
position, when the attraction was only onefourth of that on which the final
results depend, we may safely neglect the effect. After this the riders were
dusted more frequently, so that we may probably assume their values more
constant.
The comparisons of C and D, and of A and D, in (4), (5), and (6), were
made more carefully. That of A and D in (6) is much the best of the series,
the air in the laboratory happening to be steadier while it was made. The
range between the greatest and least values of the difference is one scale
division, or 0036 miUigrm., and the different results are grouped fairly
closely about the mean.
The centres of swing and the differences are plotted in Diagram VIII
(p. 136). I do not claim that these results show any remarkable accuracy
when compared with those obtained at the Bureau International des Poids
et Mesures, but remembering how rough the apparatus was, and how little
precaution was taken to ward off aircurrents, I have not the slightest doubt
that, with special design of apparatus and more suitable locality, the results
could be very greatly improved, and the accuracy carried far beyond anything
hitherto reached. As they stand, they seem to show the value of the com
bination of a short time of swing with optical magnification.
The results of comparisons (4), (5), and (6), is, that if C has its Paris value,
viz., C = 101196 miUigrms., then, A = 101183 milligrms., and D = 101196
milUgrms. ; whence 1{A ^ D) ^ 10119 milhgrms. This value may be used
in calculating the result, since the riders were interchanged before Set II was
taken.
The losses experienced since 1886 by A and D are respectively, by A
0064 miUigrm., and by D 0066 milligrm., i.e., they have diminished by
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 63
practically equal amounts. This was to be expected as they have probably
received equal amounts of rough usage.
The substitution of the subsidiary beam for the cocoonfibre suspension
of the riders having greatly diminished the handling to which they were
subjected, I have not thought it necessary to weigh them again during the
work.
Linear Measurements.
In the mathematical theory it will be shown that the lengths required
are those marked in Fig. 14, viz., the horizontal distances, L and I, and the
vertical distances, D^ D^ , d^d^, H^ H^ , h^h^.
The Horizontal Distances. Except when estimating the moment of the
rider, the distance L is really that between the
verticals through the centre of M and the centre
of the more distant attracted mass. But the
verticals through the centre of M in each position
so nearly passed through the centre of the mass
above it, and, therefore, through the knifeedge
from which it hung, that L was taken as equal
to the length of the beam (p. 50).
The accuracy of this adjustment was secured
as follows. A horizontal crosspiece was fixed on
the top of each attracted mass, with two horizon
tal cards at its two ends, each with a portion of
a circular arc on it, with radius equal to that of
the large mass M, and with centre over that of
the attracted mass (Fig. 8). A plumbline was
then hung just in front of the case, and the
balance was moved by the horizontal screws
bearing against the baseplate until the plumb
line always appeared to touch the circular arc
above, when it appeared to touch the large mass
below. The adjustment was not quite perfect,
but the error in the worst case was probably
not more than 1 millim., and certainly less than
2 millims. Such an error in the horizontal distance is negligible.
The distance I had different values for the two positions occupied by m
on the turntable. Calling these values l^ and l^ respectively, l^ + l^ was found
by measuring a, the inside distance between M and m, arranged as in Set II,
and 6, the inside distance between them, when m was put on the same side
of the turntable as M, and adding to a + h the sum of the diameters of M
and m in the radial direction of the turntable as taken by square calipers.
Fig. 8. Plumbline
Adjustment of Masses.
64 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
The following are the values in terms of the cathetometerscale already
referred to, the temperature being 15° C. :
a = 15701
h= 3395
Diameter of M = 3052
m= 2423
therefore h + k 24571
The value of l^  l^. was found by measuring the shortest distance of m
from the wall when respectively in the first and the second positions on the
turntable. It was found that
whence h = 12291o
l^ = 122795.
We may obtain from these measures an independent value of the radius
of the circle in which the centre of M moves. With perfect adjustment this
should be IL = 6166 at 18°.
It is equal to a + radius of M + radius of m  l^ , or, by the above
measures,
= 15701 + 1526 + 12115  122795
= 6159,
which is only 07 centim. less than \L.
Inasmuch as the wood probably expanded less than the cathetometer
scale, while the metal expanded more, I have assumed as a rough approxi
mation that the total expansion equalled that of the scale, so that the values
of /j and I2 are correct at 18° (see p. 50). No importance is, however, to be
attached to this temperaturecorrection.
The Vertical Distances. At the conclusion of each set of weighings with
the attracted masses in a given position, the vertical distances between the
top of the attracting masses and the bottom or top of the attracted masses
(accordingly as they were in the upper or lower position) were measured by
the cathetometer already referred to.
This instrument is of the wellknown design of the Cambridge Scientific
Instrument Company, and is especially adapted for measuring differences of
level at different distances in different vertical planes. It reads to 002 centim.
The account of these measurements will be found in Table II (p. 89, et seq.).
To find the distances D, f/, H, h (Fig. 14), it was necessary to add to the
actual distances measured the sum or difference of the vertical radii of the
attracting and attracted masses, and, therefore, the vertical diameters of all
the masses were measured.
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 65
For this purpose I used a cathetometer which has lately been constructed
for me by Messrs. Bailey, of Bennett's Hill, Birmingham. I have to thank
Mr. Potts, of that firm, for his care in
its construction, and also for the trouble
which he has taken in the construction
and alteration of much of the apparatus
used throughout the work recorded in
this paper. As the cathetometer is, I
beheve, new in design and satisfactory
in its performance, it appears worthy of
description.
The Cathetometer used to measure
Vertical Diameters (Fig. 9). There are
two telescopes, one to sight the upper
the other to sight the lower of the
points between which the vertical height
is required. There is no scale on the
instrument, but after the telescopes are
fixed to sight the two points the instru
ment is turned round a vertical axis, so
that the telescopes sight a vertical scale
at the same distance from them as the
points. In general, of course, the cross
wire will appear to lie between two
divisions, but by means of the fine
adjustment, to be described below, the
two nearest scaledivisions are brought
in succession on to the crosswire, and
by interpolation the reading correspond
ing to the point first sighted by the
telescope is determined.
The telescopes are fixed on collars
running up and down the main pillar,
which has a section of the form shown in Fig. 10 (shaded).
The guides consist of three knobs, k, k, on the inside of the collar, two
sliding in a vertical Vgroove and one on a plane, both groove and plane
being at the back of the pillar. A screw, s, clamps the collar in any position.
Gut strings running up over pulleys and supporting counterpoises, sliding
on the thinner pillars (see Fig. 9), are attached to the collars so that these
move easily. At first springs were used to keep the knobs always in contact,
but I found it much better to remove these and trust merely to handpressure
to keep the collars in the proper position before clamping with the screw s.
p. c. w. 6
Fig. 9.
Cathetometer used to measure
Vertical Diameters.
66
ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
The fine adj ustment is secured by the use of a piece of plateglass {g, Fig. 10),
placed in the front of each objectglass and capable of rotation about a
Fig. 10. Section of pillar and collar of new Cathetometer.
5, clamping screw, k, k, guiding knobs, g, glass plate for fine
adjustment, turning on axis hh, with pointer at %> perpendicular
to plane of figure.
horizontal axis, hh. A pointer is fixed on the end of this axis at f, and at
its end is a small glass plate with a scratch on it moving close against a straight
scale. If the plate is initially normal to the optic axis of the telescope, on
turning it through an angle </>, the ray which now comes along the optic
axis has been shifted, by transmission through the plate, parallel to itself, a
distance t sin ((/>  j/f)/cos xjj, where t is the thickness of the plate and ifj is
the angle of refraction within it (see Fig. 11).
Fig. 11. Section of fine adjustment plate.
For small angles this shifting happens to be nearly proportional to tan</),
and, therefore, to the reading on the straight scale. To show how nearly
this is the case the following table gives the shifting for angles of 5°, 10°,
and 20°, with a thickness oit = I centim. and a refractive index a = I:
ngle
Shifting*
5=
itan 5°(1 + 00042)
10°
itanlO°(l + 00160)
20°
itan20°(l + 00526)
The error in taking the shifting as proportional to tanc/) is, up to 20°,
quite negUgible in ordinary telescopecathetometer work. If it is desirable
to have greater accuracy, it is probably best to use a table of corrections
to the tangent ; but it is possible to get an exact scale thus :
* [These expressions are given with the wrong sign in the original paper. A sHght correction
has also been made in the values of the numerical coefiicients. Ed.]
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 67
Let OP, Fig. 12, represent the pointer of length /*, making an angle </>
with a line MN. Let a pointer PM jointed to this at P be of length /ur,
and let its extremity M move on the Hne MN. Drawing OD at right
angles to MN, if s is the shifting, we have
sin (<^ — ijj)
or
OD^OP
_ rs
s = ~OD.
r
cos l/j
Probably the practical difficulties in the use of such an arrangement would
render it troublesome and uncertain.
The plate is used as follows: Adjust it normal to
the optic axis of the telescope, and move the telescope
till the required point is brought as near to the cross
wire as is possible by the hand. Clamp the telescope,
and then turn the plate till the point is exactly on the
crosswire. Read on its scale the position of the pointer
attached to the plate. Repeat these operations with
the other telescope on the other point, then turn the
instrument about its vertical axis till the telescopes
sight the vertical scale placed at the same distance
away as the two points. Looking through one of the
telescopes the cross wire is in general not exactly on a
division. Turn the plate so that first the nearest
division above, and next the nearest division below, is
on the crosswire. Reading the position of the pointer
in each case, interpolation gives us the reading on the
vertical scale corresponding to the position of the
pointer when the cross wire was between the two scale
divisions. Doing this for each telescope the difference
between the two points is found in terms of the vertical scale.
The plates I have used are about 9 millims. thick, and the pointers about
9 centims. long. They move over scales such that 25 to 27 divisions corre
spond to a shifting of 1 milhm. The lower scale is graduated from to 50,
the upper from 50 to 100, to prevent confusion. The 50 divisions occupy
a distance of 66 millims.
It will be observed that in this form of instrument the 'level error' is
practically entirely obviated. It can only come in if the scale is not at the
same distance as the height to be measured, and may then be made neghgible
in practice by levelling the telescopes. Indeed, the uncertainty of measure
ment appears only to depend on the uncertainty with which the crosswire
5—2
Fig. 12.
68 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
can be brought to the proper point, that is, it depends only on the magnifying
power and definition of the telescopes used.
To illustrate the use of the instrument, a full account of the determinations
of the vertical diameters is given in Table II. Below are the results, and for
the sake of showing that there has certainly been no great change in shape,
I give results obtained with a cathetometer more than 10 years earlier at the
Cavendish Laboratory at Cambridge.
1890 1880
centims. centims.
Large attmcting mass J/ ... 30526 305192
Small „ „ m ... 24176
Attracted mass .4 158203 158166
B 157829 157842
The diameters of M and m in a horizontal direction parallel to a radius
of the turntable measured by square calipers were
M = 3052 centims.
m = 2423 „
TemferatureCorrection. Though the expansion of the masses was to be
expected of an unimportant amount, I thought it advisable to attempt to
measure it, in case there might be anything anomalous. One of the attracted
spheres, B, was for this purpose placed between two vertical levers, in a tank
through which could be run a continuous stream of cold or warm water.
These levers depended from horizontal rods which could rock or slightly
rotate on fine pointsuspensions. This was, in fact, a kind of double Lavoisier
and Laplace apparatus. The motion of each lever was shown by another
lever of about the same length, rising vertically up from each horizontal axis,
and serving as the moving support for a doublesuspension mirror in which
was viewed the reflection of a millimetrescale. Two telescopes and one scale
were used for the two mirrors, though it would not have been difficult to
arrange one telescope and two scales. The value of one scaledivision was
determined by inserting a piece of thin glass between the sphere and each
lever in turn. The method is exceedingly sensitive, but I have not been
able to make it exact, owing to the warping produced in the rods due to
unequal temperatures.
The measures of the expansion varied between 0000214 and 0000277,
both vertical and horizontal diameters (in the position in the balance) being
tested. The true value is probably nearly 000025 or 1/40000. It will,
therefore, lead to no appreciable error if we take the expansion as equal to
that of the scale of the cathetometer, say 1/60000 (see p. 92, Table II).
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 69
Determination of the Attraction by the Balance.
When the balance is used to measure such small forces and weights as
those with which we are here concerned, it must be left swinging on its knife
edge throughout any set of weighings in which the deflections are to be com
pared one with another. For there is not the slightest reason to suppose
that if the beam is Hfted up and let down again, its new position of equilibrium
will coincide with the old. And again, the beam, especially with such loads
as the attracted masses, is put into a state of considerable strain, and continues
to alter its shape sensibly for hours, and probably even days, after the masses
are put on to it. I have, therefore, always left the beam free for at least
two or three days before commencing work with the balance, and it has of
course remained free during the course of each day's work. The balance
room was never entered just before any weighing, as it took many hours
for the disturbance due to entrance and interference with the case to die
away.
When the turntable supporting the attracting masses is moved half round,
from one stop to the other, the bulk of the attraction is taken away from one
attracted mass and put on to the other. The balance, being free, is slightly
tilted over to the side on which is the larger attracting mass. But the
deflection in the apparatus as arranged is so very small — at the most only
10 scaledivisions — that errors of reading can only be neutralised by making
a great number of successive measures.
Probably other errors are also largely eliminated, such as those due to
the deposition of dust particles, shaking, change of ground level, and varying
aircurrents. Of such errors I have found those due to varying aircurrents
by far the worst. Sometimes — especially in autumn and winter — the balance
will move quite irregularly through more than a scaledivision, and continue
to move to and fro in this way for days or weeks. When in such an unsteady
condition it is useless for accurate work. In spring and summer, however,
it is much more steady as a rule, and frequently the scale can hardly be seen
to move. I have never worked when on looking into the telescope for some
time the irregular movements appeared to be more than a fraction of a tenth,
i.e., a fraction of one of the diagonal divisions, though, doubtless, irregularities
comparable with a tenth of a whole division have often made their appearance
in the work. It is perhaps not safe to ascribe these always to aircurrents.
I have always found the air steadiest in warm quiet weather, with a slowly
rising temperature in the balanceroom, and most unsteady after a sudden
fall of temperature. As the alteration of temperature spreads downwards,
this is fully in accord with Lord Rayleigh's observation that when the air is
steady the ceiling is warmer than the floor, and that when it is unsteady the
70 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
floor is the warmer of the two. In the observingroom I had a gas stove
often kept burning day and night, in the hope that the higher temperature
it produced in the ceihng of the balanceroom below might steady the air.
But the vertical walls of the balanceroom interfered with the action of the
ceihng, and often produced unsteadiness.
A door opening or shutting anywhere in the building had a visible though
transient effect, doubtless through an air wave. In a high wind the balance
was always unsteady, partly, I suspect, through rushes of air into and out
of the case with sudden pressurechanges, and partly through changes of
grouDdlevel, with variations of windpressure against the building.
At all times there was a march, in one direction or the other, of the centre
of swing. This was especially marked soon after the frame was lowered and
the beam left free. As already remarked, readings were not taken till changes
due to change in strain of the beam had subsided. But the march was very
appreciable at other times, as will be seen from the diagrams. Perhaps the
change was sometimes due to tilting of the ground, with barometric variation,
since the balance was a very dehcate level, and sometimes due to the change
in buoyancy of the air affecting the two sides unequally, though I have not
been able to make out any direct connection between barometric height and
position of centre of swing. I believe that the explanation is to be sought
for the most part in unsymmetrical effect on the beam of sHght changes of
temperature, for I have frequently noticed that a rising temperature produced
an upward march, and a falling one a downward march. This explanation
is supported by the following table (p. 71) of observations of the centre of
swing, extending from May 9 to May 22, 1890, the balance being free, and the
balanceroom undisturbed meanwhile.
The relation between temperature and centre of swing is represented in
Diagram IX (p. 136).
Of course, after a change in the position of the attracting masses or of the
riders, the balance does not at once settle in a new position of equilibrium,
but oscillates about it. Inasmuch as the balance never rests in this position,
it is better to term it the centre of swing rather than the equihbrium position
or restingpoint. The dashpot used to damp the vibrations of the mirror
reflecting the scale serves also to damp those of the balancebeam, and they
die down rapidly. Instead of waiting, however, to observe directly the point
on which they are closing in, it is much more exact, and also saves much time,
to find the centre of swing, as with an undamped balance, from the extremities
of the swmgs. I have always observed and recorded four extremities of three
successive swings, occupying in all a httle more than a minute.
Notwithstanding the very considerable damping, the successive lengths
of swing are still in geometrical progression, but the rate of reduction is too
great to allow the ordinary approximation, in which the geometrical is assumed
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 71
I
I
Temperature
Date, 1890
Time
Centre of
Barometer
swing
Balance
room
Observing
room
May 9
11.5 a.ni.
1360
12°0
134
7398
12.55 p.m.
1330
120
150
7392
„ 12
11.15 a.m.
1338
1205
145
7386
1.15 p.m.
1319
1205
158
7383
2.40 p.m.
1337
1205
166
7381
Stove left on all night of 12th13th
„ 13
11.0 a.m.
1817
126
175
7402
12.35 p.m.
1815
126
184
7403
3.15 p.m.
1850
127
186
7403
Stove turned off
5.25 p.m.
1894
127
165
7405
„ 14
11.20 a.m.
1674
126
143
7458
1.10 p.m.
1655
126
144
7460
„ 15
11.5 a.m.
1568
124
138
7495
2.45 p.m.
1600
124
140
7493
„ 16
1.25 p.m.
1583
124
140
7443
„ 17
8.50 p.m.
1710
1255
137
7419
„ 19
10.30 a.m.
1743
1255
140
7435
6.5 p.m.
1818
126
140
7410
„ 20
11.30 a.m.
1752
1275
140
7398
1.5 p.m.
1737
1275
141
7400
5.20 p.m.
1723
127
138
7417
,, 21
11.10 a.m.
1727
126
137
7519
„ 22
11.30 a.m.
192 about
1285
141
7576
to be an arithmetical progression. The exact method of determining the
centre of swing is as follows :
Let a, h, c, d be four successive readings of extremities of swing, and let
X be the reading of the required centre.
Let the constant ratio of each swing length to the next be A.
Then ax^X(x b), (1)
X — 6 = A (c — x), (2)
cx = X{xd) (3)
Eliminating A from (1) and (2), we may readily obtain x in the form
and from (2) and (3)
x=b+, — i'~y ., (5)
(c — 6) + (c — d)
72 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
With no disturbances and no errors of reading, the values of x in (4) and
(5) will coincide ; but usually there is some small difference, the result of error
or disturbance, and it is better to find both and take the mean. A third
value might be obtained from (1) and (3); but it appears unadvisable to
combine directly observations so far separated in time.
These formulae lend themselves to easy arithmetical treatment, especially
with the aid of a shderule; but the following graphic method of finding the
centre of swing is much less tiring and quite sufficiently exact.
Let the line OA, Fig. 13, represent the scale; its zero, and A, B, C, D
the points distant respectively a, b, c, d from 0.
Let O'C be a parallel line, B\ C, D' being points
opposite to B, C, D respectively. Let AB' and BC
intersect in iiTi . Draw X^K^X^^' perpendicular to OA.
Then Zj is the centre of swing given by equation (4).
For
AX^ AX, K,X, ^ X^^X.B
X.B'X.'B'" K,X,' Z/C ZiC
i.e., Zj is the point dividing AB and BC in the same
ratio. Similarly if BC and CD' intersect in K2, and
X2K2X2 be drawn perpendicular to OA, X2 is the
point given by equation (5).
The third point given by equations (1) and (3) is
obtained fronj the intersection of AB' and CD', but
evidently a small error in C or D' may considerably
alter the position of this point, and it is better not to
use it.
The construction was carried out thus: a large
opal glass plate, 10 in. x 11 in., was etched with cross Fig. 13.
lines 10 to the inch, so as to present the appearance
of ordinary sectionpaper. The glaze was taken of! so that pencilmarks could
be made. A diagonal line ran at 45° across the plate through the corners of
the inch squares, and this was always taken as the line BC in the figure.
Taking any convenient horizontal line, usually, of course, far below the plate,
as zero, each inch represented a scaledivision, each tenth a diagonal division.
The values of b and c fixed the hues to be taken as OA, O'C, and on these were
marked the points A, C, B', U. A long glass shp, with a straight scratch
on it, was then laid across from A to B' so that the scratch passed through
A and B' , and its intersection A\ with the diagonal BC w^as x^ from the zero
hue. The slip was then laid with the scratch passing through C and U , and
its intersection K^ with BC gave x^. It will be observed that all the actual
construction for a set of readings of the balanceswings consisted in marking
four points on the plate.
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 73
The following cases, the first of very regular, the second of very disturbed
swing, will serve to compare the results by this exact method with those
obtained from the ordinary arithmeticmean method. At the same time they
will show how nearly constant is the ratio of swing decrease.
Date and
Number, 1890
Scaleread
ings in
diagonal
divisions
Length of
swing
Ratio of
each to
preceding
Centre of swing,
exact
Centre of swing,
approximate
a 4 26 H c
4
May 4, No. 3
865
939
896
921
74
43
25
•581
•581
9118
91b8
90975
9130
Mean 911^8
Mean 911375
Sept. 17, No. 45
1118
1053
1093
1068
65
40
25
615
625
10778
10776*
107925
107675
Mean 10777 Mean 10780
In finding the attraction the observations were always made in the same
order, the determination of the scalevalue of rider and attraction being
sandwiched so that each might be equally affected by any comparatively
slow changes. Starting with the initial position, the attracting masses and
riders were so arranged that, on moving either, the balance was deflected in
the same direction and over the same part of the scale.
The following was the order of proceeding always observed, the column
headed ' Centre of swing ' being supposed to contain the values of the position
in each case determined from four swing extremities as just explained :
Centre of swing.
(1) Initial position *i
(2) Riders moved r^
( 3 ) Riders moved back to initial position i^
(4) Masses moved round m^
(5) Masses moved back to initial position i^
(6) Riders moved ^2
(7) Riders moved back to initial position i^
(8) Masses moved round Mg
and so on.
To minimise the effect of progressive changes these observations were
always combined in threes in the following way. Denoting the scalevalue
of rider by R, and of attraction by M :
* [The original has 10786 which is evidently a slip ; hence this is not really a good example
of a 'very disturbed' swing. Ed.]
74 ON A DETEEMINATION OF THE MEAN DENSITY OF THE EAETH AND
From (1), (2), (3) Ri = ri'^^,
„ (3), (4), (5) M, = m,'^\
„ (5), (6), (7) R2 = r^~'^\
and so on.
These again were combined in threes, so that (the notation being continued)
the successive values of attraction /rider are
i?2 + jB2 ' 2i?2 ^2 "^ ^3
The successive centres of swing i^, r^, i^, m^, etc., correspond to instants
of time following each other at intervals of about 2 minutes, rather more than
1 minute being taken up in making and recording the four readings for each,
and the rest in making the change of position in rider or mass and waiting
for the next readings. It will be seen that each value of M or i^ is based
on three successive centres of swing, the w^eighings extending over about
6 minutes, while each value of M/R is based on seven successive centres of
swing determined in about 14 minutes.
A series of readings was usually continued for about 2 or 3 hours. The
temperature in both observing and balance rooms was read at the beginning
and end of the series, and the barometric height was also observed. As soon
as possible after the desired number of determinations was completed with
the attracted masses in one of the two positions, the vertical distances between
attracting and attracted masses were measured by the cathetometer in the
manner explained in Table II, and the position of the attracted masses was
then altered.
A full account of all the weighings is given in Table III, and the results
are represented in Diagrams IVI (pp. 130135). The three upper rows of
points in each diagram represent the centres of swing, those in the initial
position being marked • . After movement of the rider they are marked x ,
and after movement of the masses they are marked o. The baselines for
the different rows are altered to save space, as described on the diagrams, for
on the scale adopted the rider series would always be about 10 inches
above or below the initial series. In Diagram I the rider and mass series
are also brought down and superposed on the initial series, so that each of
the three has the same average height. It will be seen that all three are
affected by the same disturbances. The advantage of the short time of
swing and the mode of combining the results in threes will be realised more
easily from this superposition.
The basehne may be regarded as a timescale, as the instants corresponding
to successive centres of swing were almost exactly equidistant.
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 75
In each case, under the representation of the centres of swing, are plotted
the resulting values of MjR, and at the side will be found a representation of
the distribution of results about the mean.
Assuming that each day's mean value is correct, and that the differences
for different days are to be set down to variation of distance, etc., we can
find the distribution of all the values about the mean by simply superposing
the marginal curves at the side of the figures. The result fairly shows the
accuracy as far as the weighing alone is concerned. It is represented in
Diagram VII, where A is the mean value of the attraction in the lower, and
a that in the upper position. A and a are brought near together to save
space, but really they should be 40 inches apart. It will be seen that the
range is about 2 per cent, of ^ — a on each side of the mean, or taking the
value of ^ — a in milHgrammes weight as about \ milHgrm., and the load
on each side as 20 kilogrms., the range is about 1/3 x 10^ of this load on each
side of the mean.
A comparison of the values of MjR in Diagrams I and II, shows a very
curious similarity in the fluctuations, and at first I was inchned to think
there was some common external disturbance producing these fluctuations.
But an analysis of the two sets of values appeared to show that the resemblance
is merely accidental. When the values of M and R are set out separately,
it is seen that the fluctuations depend chiefly on M, of which the fluctuations
are shghtly hke each other for the two series, while those of R are quite
different, but such that they make the fluctuations in MjR resemble each
other more closely than those in M alone. Further, it is not easy to see
how fluctuations due to some external source would aflect the values of M
equally in the upper and lower positions and not have any effect on R. Some
periodic change of level might be suspected, but this ought certainly to be
traced in R. I have examined all the other diagrams and plotted out the
component values of M and R, but have found no trace of resemblance, so
that I think the curious likeness in I and II must be set down to accident.
There is a curious step by step descent of the centre of swing in the initial
position on September 23, Diagram VI, which I cannot explain. It may be
due to some error in the method of finding the centre of swing which comes
in with a rapid march of that centre. The effect on the result is probably
only small, for the value of MjR obtained with a march in the reverse direction
on September 25 is very nearly the same, the two values being
September 23 2112753.
25 2112533.
The following is a hst of the weighings recorded, with the distances
measured and the mean values of the attraction:
76 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Date
1890
Position of
attracted
Feb. 4 Upper
April 30 and May 4 Lower
May 25 Upper
Date
1890
Position of
attracted
masses
July 28 Lower
Sept. 17 Lower
Sept. 23 and 25 ... Upper
Set I.
No. of values
oiM/R
50
100
50
Set IL
No. of values
oiM/E
25
25
52
Mean value
of M/R
•2142212
10109685
•2157379
Mean value
otM/R
•9973168
•9984148
•2112647
Dord
in centims.
62318
31783
62308
D or d
in centims.
32106
32116
62708
Horh
in centims.
61416
30824
61373
H ork
in centims.
30965
30954
61566
On the completion of Set I the four masses were inverted, and changed
over from right to left or left to right, and the initial position was after this
always arranged so that movement of rider or mass decreased the reading.
This was done in order to lessen errors due to want of symmetry. If reversal
had no effect, Set II should, with the increased distance recorded above, give
a value of M/R in the lower position of about 990, instead of 998. The
larger value actually found is no doubt chiefly due to a want of symmetry
in the large attracting mass M, The effect of this want of symmetry will
be discussed after the investigation of the mathematical formula, and an
account will be given of an independent method of detecting it. I think
there is still outstanding a small difference, due, perhaps, to want of symmetry
in the turntable or in the attracted masses. The result of the reversal shows
how necessary it was to make it. I should have Hked to have in Set II as
many determinations as in Set I, so that the mean should be based on values
of equal weight. During June and July, 1890, a complete set of 100 in each
position, upper and lower, was made; but, owing to the pressure of other
work, I was unable to calculate the results till the completion of the set.
I then found that the value of M/R was still more than in Set II, and, on
plotting out the results, it appeared that occasionally the ridervalue fell very
considerably, and in an irregular way. On examination, there was Httle
doubt that the rider came in contact occasionally with the suspending frame,
when it was raised and should have been clear from it. Very likely tempera
turechanges had brought about a displacement of the leverapparatus.
Comparison with Set I seemed to show that during that set no such contact
had taken place, for there was no comparable irregularity. As it appeared
dangerous to attempt to disentangle the good from the bad, the set of June
and July was rejected, and Set II was taken as recorded. When I had made
the weighings giving 50 and 52 values in the two positions respectively, the
balance became so irregular, through the cooler weather, that it was useless
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 77
to continue work. Rather than carry over the experiment into another
season, when it might be necessary to repeat the whole of the work, I have
preferred to take Set II as it stands, and give it the same weight as Set I.
The final results are calculated from the means of Sets I and II, as explained
hereafter. I may here state the results obtained :
^ r ■ ^ 66984
Constant of attraction "^ = — tt^— .
Mean density of the Earth ... A = 54934.
General Remarks on the Method.
Comparing the common balance with the torsionbalance, there is no
doubt that the former labours under the great disadvantage that the dis
turbances due to aircurrents are greatest in the vertical direction, that of
the displacement to be measured. But even with this disadvantage the
common balance may, I beheve, be made to do much more than has hitherto
been supposed possible. As an instrument in itself, apart from the external
disturbances of aircurrents, dust, etc., I believe its accuracy would be far
beyond anything approached when these external disturbances are, as they
always are, present to interfere with its action. I have always found that
every precaution to ward off aircurrents and external disturbance has been
accompanied by a corresponding increase in steadiness ; and I have seen no
sign of a limit of accuracy depending on the instrument itself.
Besides the protection from aircurrents, there are two conditions essential
above all others for accurate work :
1st. That during any set of weighings in which the deflections are to be
compared with each other, the beam should be supported on its knifeedge,
and should be under constant strain.
2nd. That all moving parts, such as apparatus for changing riders or
weights, should be supported quite independently of the balance or its case.
With regard to the first condition, it seems impossible to make the
supporting frame move so truly and with so little disturbance that the knife
edge shall return exactly to the same line. Even were it possible, the beam
after raising and lowering would be practically a different beam, for, as my
observations show, the condition of strain changes considerably after the
load is first put on, and it would be merely a chance coincidence if the mean
state of strain were the same during successive weighings. I have, in my
former paper {Proceedings of the Royal Society, vol. 28, 1879)*, described one
method of comparing weights of nearly equal value with the beam throughout
on its knifeedge and equally strained f, and I should now only modify that
* [Collected Papers, Art. 2. J
t I am glad that Dr. Thiesen urges the importance of this condition {Travaux et M^moires du
Bureau International des Poida et Mesures, vol. 5, 'ilfitudes sur la Balance').
78 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
method in having regard to the second condition, of which I have since
reahsed the importance when working with the large balance and with
increased optical sensitiveness. It is surprising to find how much disturbance
is produced by having the moving parts of the apparatus connected with the
balance or its case.
As to aircurrents there is no doubt that, as Professor Boys has shown,
the greater the apparatus the greater the errors produced by them. At the
time my apparatus was designed I did not know this, and there seemed
to be a great advantage in making it large, as riders could be used of weight
large enough to be measured accurately. Were I about to start with a new
design I should certainly go towards the other extreme and make the apparatus
small, attempting to get over the riderdifficulty by some such method as that
explained on p. 60. For not only is a smaller apparatus kept more easily
at a uniform temperature, and, therefore, freer from the source of aircurrents,
but it is much more handy to adjust, and even if the adjustments are not
more accurate they will at least take much less time to make.
At the same time it is only fair to say, on behalf of the large apparatus,
that some errors have been magnified on a like scale till they have become
observable, and so could be investigated and eliminated. Starting with
a small apparatus they would probably never have been detected, and would,
therefore, have appeared in the final result.
II. Mathematical Investigation.
The Value of the Attraction Expressed in Terms of the Masses and Distances,
and the Investigation of the Effect of Want of Syynmetry in the Masses.
Let us suppose that initially the attracting masses are in the positions
Ml, »?i, Fig. U, the larger on the left, the smaller on the right, and that the
attracted masses are in the lower positions A, B. When the turntable is
moved round so that the positions of the masses are M^, m^, the greater
attraction is taken from the left and put on to the right. Let the centre of
swing of the balance alter by an amount corresponding to a total change of
vertical pull of n dynes. Assuming that a spherical mass M attracts another
spherical mass M' when their centres are D centimetres apart with a force of
GMM'/D" dynes, we can express the change of vertical pull due to the change
of position of the masses as 6^ x a function F of the masses and distances.
There is also a change of pull on the suspendingrods and the balancebeam
which we may denote by E.
Then n = GF + E.
In order to eliminate E let the attracted masses be moved into their
upper positions A', B\ and let the change on moving round the attracting
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 79
masses be n' dynes. If / is the function of the masses and new distances
corresponding to F,
Subtracting
whence
n — n'
G =
Ff
and knowing G, the mean density of the earth may be at once found in the
manner shown later.
Fig. 14.
We have then to find the form of the functions F, f, and as a prehminary
step it is necessary to find the effect of the holes bored through the attracted
masses A, B. This may be made to take the form of a correcting factor to
the attraction which would be exercised on them if they were spheres.
The piece bored out in each case has radius 31 centim. This we denote
by c. It may be taken as practically a cylinder with plane ends and length
equal to 158 centims., the diameter 2r of the spheres. The intensity due to
such a cylinder of mass /x at Z) from its centre is (Todhunter's An. Stat.,
Ed. 5, p. 292),
2r  V{{D + r)2 + c2} + ^{(D  r)2 + c^}
GfJL
c^r
which equals, to a sufficient approximation,
GfJL
80 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
If the mass remaining after fx is removed is A, and if the centre of the mass
Mis D below that of A, the attraction of M on A is
GMJJ^ji) GMji
GMA , rM (^ ^ \
= ^{l ^(J + ^ig^er powers of JJl.
1
and the greatest value of
N f = ^J^^r = I % ^'^^'^ =  ^^'^ = ■'^''''^
' _ / 79 \2 _ .Afil
Then the higher powers may be neglected, and the attraction may be
written
GMA /, 3 c2 \ GMA
IMAf. 3 C^\ brMA
When A and B are in the lower position, D = 32, and 1  6 ^ 99986.
When they are in the upper position, D = 62 and 1  ^ = 99996, a value
so near 1 that we shall in this position omit the correction, since it is only
applied to onefourth of the final result.
In the crossattractions we shall also omit the correction.
Referring to Fig. 14 let the vertical differences of level between the centres
of the various spheres be denoted as follows, the suffixes to M and m denoting
their first and second positions respectively :
AM^ = D^, BM^ = D^,
B
B
M^=D^,
AM2 = D^,
nil  H^,
A  mi = i?/.
m^ = H2,
B  m^^ H^.
When the masses A, B are placed in their upper positions, let the corre
sponding distances be denoted by small letters.
Let the horizontal distance between the centres of A and B be L, being
within sensible limits equal to that between the centres of M in its two
positions, and to the length of the beam, and let the radius of the circle in
which m moves be I.
Then we have the following horizontal distances :
AM^^BM^=^L,
A — m^ = B — m^ = I + \L,
A — 7)12 = B — m^ ^ I — \L.
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 81
We may now write the change in vertical pull on the left by the motion
of M from left to right, and of m from right to left, as follows — the first four
terms representing the vertical attractions on A and Bhj M and m in their
first position, the next four their attractions when moved round, and the last
term E representing the change in attraction on the beam and suspending
rods :
( MA (1  6) _ MBD,' mBH^ , mAH^'
\ Di' {D^^ + L^f
MB(\d) MAD^ mAH^ mBH:
H{'
'r
«."+('+ 1)
LV)^
+
D,
{D,'^ + L^)^
H,+ il^f
+
2)]
H''^
'!)
LV)^
+ E.
We may arrange all but the last term in nearly equal pairs.
Thus the first and fifth go together, and if we put B^^ 3^ = 2D and
Di + S = D2 — S = D, their sum is
A B
(?M(l^)(^^+^^,)
GM(ld)\'^,{l
= GM(l d)
382
2
Z)2
D^V^ D^ D^^ '"J^ D^V D^ D
(^1 + ^ + higher powers of ^^j
^ A + B d[ ^ D^
Now (S/Z))2 is negligible, as will be seen by reference to the table of
distances, p. 92, and {A — B)I(A + B) is less than jjjiQjj, or less than S/D.*
To a sufficiently close approximation then the sum of the two terms is
GM (A \B)(l d)
2)2
The second and sixth terms may also be taken together, and putting
D^ + D^ = 2B' and Z)/ + 8' = D/  8' = Z)',
we may show that to a sufficient approximation
BD^ , .4Z)./ I _ GM {A + B)U^
(Z)'2+ 2,2)1
GM
The two pairs with m give similar results with
H = i{H^ + H2) and
H' = i {H,'
H,').
* [A slight correction of the original, obviously required, has been made here. Ed.]
P. c w. 6
82 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Now 2D=D^ + D, = AM^hBM^
= BM, + AM,=D,'\D,' = 2D',
and similarly 2H = 2H' , so that we may put the expression in the form
(M(A^B\i\e) _M{A + B)D m(A + B)H m(A + B)H \
{ E=GF + E say.
It is evident that we may combine experiments at different distances on
different occasions in the same way by taking D and H to represent the mean
values of these distances, so long as there is only a small variation from the
mean.
If the attracted masses are now moved into their upper positions the
expression for the change in attraction may be at once deduced from that
in the lower position by replacing D and H hj d and h, and omitting the
factor 1  e. Let it be denoted by Gf + E.
Subtracting one expression from the other E is ehminated, and we have
0{Ff)
[M^ + B){le) M{A + B )D m{A + B)H ^ m(A + B)H
M(A + B) , M(A + B)d , m{A + B)h m{A + B)h ^
d^ "^ {d^ + L^)'^
MWf MW
This is to be equated to the difference in the values of the change in attrac
tion in the two positions, as determined by the rider.
Let
b = the length of the small riderbeam,
ir = the mass of each rider,
A = mass deflection ^ rider deflection in lower position*,
« = » ^, „ „ upper „ *,
gjj = acceleration of gravity, or dynes weight per unit mass at
Birmingham.
Then . G(F j^ _ i^  f bwgj,^
Whence we may find the gravitationconstant
^ 2hwgj^ {A a)
where all the quantities on the righthand are given in the tables at the end.
* [It will be noted that this symbol is used with two distinct significations. Ed.]
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 83
The value of gjg may be found sujSiciently nearly from the formula
(Everett's Units, p. 21): g = 9806056  25028 cos 2A  000003/i, where A
is the latitude = 52° 28' at Birmingham, and h is the height above sealevel,
which may be taken as 450 feet, or 13,725 centims. Whence g^ = 98121.
Since all the operations are conducted in air, the effective masses should
throughout be less by the mass of air each displaces. But since they all have
nearly the same densities, and w and A ^ B appear respectively in numerator
and denominator, it is sufficient to take their true masses, and to correct for
air displaced in the case of M and m only.
To obtain the mean density of the earth A, we must express the acceleration
of gravity in terms of G and the mass and dimensions of the earth.
The ordinary formula (Pratt, Figure of the Earth, 4th ed., p. 119) is based
on the assumption that the earth is a spheroid. It is sufficiently correct for
our purpose, the departure of the assumed spheroid from the actual shape
being very small. Adding a term — 3 x 10~^h, or approximately, — 41 x 10~®,
since the balanceroom is taken as 13,725 centims. above sealevel (see above),
the value of gravity at Birmingham may be written
g^ = ^^l +^^^+(1^^) sin^ 52° 28'  41 X 10«
where
F = volume of the earth = 10832 x 10^7 (Everett's Units, p. 57),
a = mean radius of the earth = 63709 x 10^ {loc. oil.),
A = mean density of the earth,
m = equatorial ' centrifugal force ' ^ gravity = ^^ ,
€ = ellipticity of the earth = ^b 2 •
The value of the ellipticity is taken to make the formula agree with that
quoted above from Everett's Units. The uncertainty in the value is quite
unimportant, for were e as low as 2^5, the error in A, introduced by taking
it as 2^2 ' would be less than 1 in 50,000.
Substituting for G, the value of the mean density of the earth is
a^L{Ff)
2bwV
l + I  I m + f m  e") sin^ 52° 28'  41 x 104 (A  a)
Here, as in the value of G, w and A + B may have their true values,
M and m their values less the mass of air displaced.
In the foregoing investigation we have supposed that all the masses are
homogeneous and spherical, with the exception of the borings through A and
B. We have supposed, also, that the turntable is exactly symmetrical about
a vertical plane through its axis, so that its motion through two right angles
is without effect. Doubtless, these suppositions and the formula based on
6—2
84 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
them are not quite true. But, if we invert all the masses and change their
sides, or pervert the whole arrangement of them, on taking the mean of the
results obtained in the original and inverted and perverted positions we ought
to greatly reduce the errors. Indeed, those due to want of symmetry in the
turntable should evidently be quite ehminated, and those due to want of
homogeneity in the masses should certainly be lessened.
To show this, we shall calculate the effect of a spherical ' blowhole,' or
gascavity in M, in the first and most important term of F. This we shall
take as being
GM (A + B)
on the supposition that M is homogeneous and spherical.
If the mass of metal which would fill the blowhole is A, supposing it to be
placed there, the sphere is completed and its attraction is
G(M + X){A + B) ^
2)2
but the vertical attraction is less than this in reahty by the vertical component
of the attraction of A.
Let B be the centre of the cavity,
P the centre of the attracted mass,
the centre of the attracting mass,
8 the distance of B from the centre of M,
e the angle BOP.
The vertical component of the attraction of A is
GX (A +B) cos BPO
PB^
but 5P2 = Z)2 + §2 _ 22)3 cos d,
1 7? on DScosd
and . cos BFU = „„ — ,
Br
whence the attraction of A may be put
GX(A + B){DScosd) GX{A + B) "^ ^
( D2 + §2 _ 22)8 cos d)^ ' ' dD VZ)2 + 8^ _ 2Z)8 cos 6
GX(A + B)f^ , ,„ 8 , ,„ 8^
A + B)( 6 6^
2)2 ^^ + ^12) + ^^2 2)2
where P^, P^, ... are zonal harmonics. The attraction of the sphere with the
cavity is therefore
GM(A + B)L A / „ 8 , ,„ 8^
2)2 i MV^^D^^^' D^^ 'D^
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 85
If the mass is inverted, the vertical component is obtained by changing the
sign of 8, and the mean of the two values is
6MiA + B)i^_X_j^pJ^^^^p^8^
2)2 M\ ^D^ " D*
■)i
the first power of hjD being ehminated.
If ^ = 0, Pg ^^^ ^11 ^h^ other harmonics = 1.
If ^ = 90°, Pg =  1, P4 = I, etc.
Now, with the actual dimensions of the apparatus, (S/Z))^ cannot be so
great as (J)^ or J, and may, of course, be much smaller. The first term of
those involving A, therefore, is the most important, and it lies between
+ f (A/M) (8VI>2) and  3 (XjM) {h^jB^), changing sign for the value of 6
given by Pg == 0.
The second set of experiments recorded in this paper was taken after
inversion and change of side of all the masses, and the final result obtained
from this set differs by a little more than 1 per cent, from that obtained from
the first set, the observed attraction being slightly greater at the same distance.
The difierence may be due to irregularities in any or all of the masses and in
the turntable, and to other undetected effects, such as change of level on
rotating the turntable. It would be a very long task to disentangle these,
and I have contented myself with trying to find how much must be set down
to irregularity in the large mass M, by taking a set of weighings with it alone
inverted.
After the weighings on July 28, and the subsequent measures of distances,
M was inverted only, and the other masses remained as in Set II. Some
weeks later, on September 14, 25 values of MjR ^ A were obtained, the
mean being 9926. The distances were B = 32118, H = 30978. The mass
M was then put in its original position, as in Set II, and on September 17,
as will be seen on referring to the tables, the value of MjR obtained was
•9984, the distances being B = 32117 and H = 30955, practically the same
as on September 14.
Assuming that the difference in attraction is due to cavities in various
places, and that, for each, the term ^tP^^h'^jB^ is negligible, we have, approxi
mately,
2AP,8
" MP _ 9926
^ .SAPi8~9984'
Whence, approximately, since B = 32,
,>r— = 0464 centim.
M
86 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
This result may be tested by independent experiment. For, let the
centre of gravity be x below the horizontal plane through the point bisecting
the vertical diameter (i.e., the centre of figure), in the position of Septem
ber 14. The distance of any missing particle A from the horizontal plane is
8 cos ^ = Pi§. Completing the sphere by the addition of all such particles,
the centre of gravity is brought to the centre of figure, so that we have
Mx = SAPiS,
SAP^S
and X = — ^^ .
We have, therefore, to determine the vertical distance of the centre of
gravity from the centre of figure.
In order to do this, a large flatbottomed scalepan (one belonging to the
balance used in the gravitationexperiment) was suspended by two parallel
wires about 8 centims. apart and 3 metres long. In the middle of the pan
was a shallow cup about 75 centims. internal diameter, arranged so that it
could turn freely but truly about a vertical axis. The mass, M, was placed
on this cup with the diameter, which had been vertical, arranged horizontal,
and perpendicular to the plane of the suspending wires. A vertical flat plate,
worked by a horizontal micrometerscrew, could be brought just in contact
with the end of the diameter, and the reading of the micrometer gave the
position of the point of contact. The position of the scalepan was deter
mined by a plumbline hanging over one edge in front of a horizontal scale.
On turning the cup and mass through 180°, and repeating the readings,
knowing the weight of the scalepan, and the position of its centre of gravity,
X could at once be found.
Two separate experiments gave
X = 0536 centim.,
and X  0516 centim.,
not very different from the value 0464 obtained from the attractionexperi
ments. The agreement is, I think, very close when it is noted that a difference
of 1 in 1000 in the attraction in one of the sets of weighings would make x
either 038 or 054.
This result appears to justify the rejection of all terms in the expansion
above the first, and so supports the belief that the reversal largely ehminates
errors due to irregularity of shape. For it is in the case of M that there is
the greatest danger of a large value for S/D, and the above experiments seem
to indicate that even in this case it is small.
It is, perhaps, noteworthy that the largest term rejected in the attraction
of M, viz., ^XP^S^/MD^ is, if we give P^ its maximum value 1,
3AS S^^3x S
MB' D~ D'D'
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 87
which is not greater than
since the radius of the mass is 15.
This is in a term about 5/4 of the final result, so that the greatest error
which can be introduced by neglecting this term is 0025, or 1 in 400.
In calculating the results of the experiments the means of Sets I and II
have been taken. Equal weights have been given to each set. It would
have been more satisfactory if the number of experiments had been the
same in each set ; but I should have had to wait for another season to obtain
more, and then it would, probably, have been necessary to repeat the whole
series in both arrangements, as it is not safe to assume that the various
disturbing causes remain the same over a wide interval of time. The second
set, though fewer in number, are, in some respects, I beheve, better ; partly
owing to the additional experience gained when they were taken.
In order that the various terms in F —f may be compared, I give below
their numerical values, as determined from the values of the masses and
distances given in the tables. The meaning of each term in the first column
will be seen on referring to Fig. 14. The second column contains the actual
values ; the third column the values in terms of the lowest term, the fourth.
Value oi F f.
M{A{B){le)
2)2
M(A + B)D
(Z)2 + L^)^
miA + B)H
m(A + B)H
M(A + B)
M{A + B)d
m(A + B)h
+ 64839388 416
 1024163 66
 3162433 20
+ 155799 1
16936872
109
1567280
10
3106950
20
^iA + B)h _ 27597.7 ].7
{'('ST
Whence F f= 48269972.
88 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
The mean value oi A  a (see Table III) is
^_a = 791295;
substituting these values of J /and ^  a in the formula for G (p. 82), we
obtain
66984^
substituting them in the formula for A we obtain
A = 54934.
The values given by Sets I and II, treated separately, are to two figures
of decimals :
Set I. A = 552
Set 11. A = 546.
i
III. Tables.
Table I. Constants of the Apparatus and Dimensions of the Earth.
Masses
Attracting mass M, in vacuo
Less air displaced, say . . .
Attracting mass m, in vacuo
Less air displaced, say ...
Attracted mass A, in vacuo
55 55 ^5 ?5
,, ,, A + B, in vacuo
Riders each, in vacuo
grms.
15340726
15338885
764974
764882
2158233
2156621
4314854
0010119
Vertical Diameters of Masses in terms of Catheto meter Scale correct at 18°.
The masses are taken as having the same coefficient of expansion as the
scale.
centims.
M = 30526
m = 24176
A = 158203
B  157829
The diameters of the masses A and B are taken between the nuts securing
them on the suspending wires.
centims.
Balance beam at 0°, L _ =123232
Rider beam at 0°, 6 ^ 253575
L/b (as occurring exphcitly in G and A, independent of tempera
ture, assuming them to have the same coefficient of expansion) = 4859775
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 89
Latitude of Birmingham
Height of balanceroom above sealevel
Gravity at Birmingham, g^ ...
Mean radius of earth ...
Volume of earth
Equatorial ' centrifugal force '/gravity
Ellipticity of earth
f]sin2 52° 28' 41x106
3 , /5
= 52° 28'
= 13725 centims.
= 98121 centims./sec.2
= 63709 X 108 centims.
= 10832 X 1027 cub. centims,
= 999161.
Table II. Vertical and Horizontal Distances.
Vertical Diameters of Masses taken by the Cathetometer, described p. 65.
In the tables below p.s. signifies divisions on the scale over which moves
the pointer, which is attached to the small adjustmentplate, v.s. signifies
divisions on the vertical millimetrescale.
Diameter of Large Attracting Mass M.
Reading on pointerscale
Upper telescope sighting top of mass ... 732, 734, 732
Lower „ „ bottom „ ... 230, 230, 232
Mean
7327 P.s.
2307 p.s.
Turning round to the Vertical Scale.
Reading on pointerscale
Upper telescope sighting 459 millims. v.s. ... 946, 949, 944, 950, 940
458 „ ... 688, 688, 697, 690, 687,
700, 696, 694
Therefore 2533 p.s. divisions = 1 millim. v.s.,
Mean
9458 P.s.
6925 p.s.
and scalereading for top of mass = 458
= 458158 millims. v.s.
7327  6925
25^33
Lower telescope sighting 153 millims. v.s.
„ 152
Reading on pointerscale Mean
273, 274, 277, 270 2735 r.s.
00, 03,  5, 00  005 p.s.
Therefore 2740 p.s. divisions = 1 milhm. v.s.,
2307 f 005
and scalereading for bottom of mass = 152 +
= 152844 milhms. v.s.
The difference = 305314 centims.
2740
90 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
This IS rather greater than the diameter of the mass, as the cross wire was
made to touch the image of the mass in each case. A series of measures of
1 milUm. on the scale, in which the crosswire was on the centre of each
division, and of 1 miUim. between the jaws of a wiregauge, in which the wire
touched the images of the jaws, showed that at the distance at which the scale
was, 005 centim. must be subtracted, leaving
Diameter of if = 30526 centims.
Vertical Diameter of Small Attracting Mass m.
Reading on pointerscale Mean
Upper telescope sighting top of mass ... 756, 756, 750 7540 p.s.
Lower „ „ bottom „ ... 265, 263, 268 2653 p.s.
Turning round to the Vertical Scale.
Reading on pointerscale Mean
Upper telescope sighting 388 millions, v.s. ... 100, 999, 997 9987 p.s.
387 „ ... 739,734,740 7377 p.s.
Therefore 2610 p.s. divisions = 1 millim. v.s.,
7540 — 7377
and scalereading for top of mass = 387 + — ^r^^r.
= 387062 millims. v.s.
Reading on pointerscale Mean
Lower telescope sighting 146 millims. v.s. ... 459, 459, 450 4560 p.s.
145 „ ... 204,194,200 1993 p.s.
Therefore 2567 p.s. divisions = 1 millim. v.s.,
and scalereading for bottom of mass = 145 + "^'^^ ~ ^^'^^
2567
= 145257 milhms. v.s.
The difference = 241805 centims.
Subtracting the same correction as in the last case for the crosswire,
Diameter of m = 24176 centims.
Vertical Diameters of Attracted Masses A and B taken between the
Junctions of the Securing Nuts ivith the Sphere.
A.
Reading on pointerscale Mean
Upper telescope sighting top of mass ... 827, 830, 829 829 p.s.
^'''''^'' •' » bottom „ ... 310,' 315,! 313 313 p.s.
I
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 91
• Turning round to the Vertical Scale.
Reading on pointerscale Mean
Upper telescope sighting 429 millims, v.s. ... 950, 952, 953 952 p.s.
428 „ ... 690, 690, 689 690 p.s.
Therefore 262 p.s. divisions = 1 millim. v.s.,
and scalereading for top of mass = 428 H ^r^^
= 428531 millims. v.s.
Reading on pointerscale Mean
Lower telescope sighting 271 millims. v.s. ... 480, 478, 484 481 p.s.
„ 270 „ ... 234, 230, 228 231 p.s.
Therefore 250 p.s. divisions == 1 millim. v.s.,
31*3 — 23*1
and scalereading for bottom of mass = 270 H ^r^^r
^ 250
= 270328 millims. v.s.
The difference gives the diameter since the middle of the crosswire was
used, so that
Diameter of ^ = 158203 centims.
B.
Reading on pointerscale Mean
Upper telescope sighting top of mass ... 720, 710, 710 713 p.s.
Lower „ „ bottom „ ... 246, 250, 252 249 p.s.
Turning round to the Vertical Scale.
Reading on pointerscale Mean
Upper telescope sighting 430 millims. v.s. ... 940, 946, 940 942 p.s.
429 „ ... 680, 681, 681 681 p.s.
Therefore 261 p.s. divisions == 1 milHm. v.s.,
and scalereading for top of mass = 429 H ^^ ,,
= 429123 minims, v.s.
Reading on pointer scale Mean
Lower telescope sighting 272 millims. v.s. ... 433, 430, 430 431 p.s.
„ 271 „ ... 171, 173, 174 173 p.s.
Therefore 258 p.s. divisions = 1 milhm. v.s.,
249 — 173
and scalereading for bottom of mass = 271 f ^„ „
= 271294 milhms. v.s.
And diameter oi B = 157829 centims.
92 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Vertical Distances between the Levels of the Centres of the Attracting
and Attracted Masses Measured by Cathetometer.
The measurements were made as soon as possible after the completion
of a set of weighings, usually on the following day.
It was necessary to fix the attracted masses in the position occupied
during the weighings, and with the beam of the balance in the same strained
condition. This was done in some cases by gripping the left suspending wire
by a pair of jaws ; in others, by adding a small weight to one side, and placing
a block of the right thickness under the mass on that side.
The cathetometer was placed in front of the left side of the balancecase,
from which position all the masses could be viewed by turning the telescope
round the central pillar (Fig. 2). It was read when sighting the top of each
attracting mass and the top of each attracted mass when in the lower position,
the bottom of each attracted mass when in the upper position, the top and
bottom being taken at the junctions of the securing nuts with the masses.
It is therefore necessary to add to the distances measured by the cathetometer
the difference of the radii of attracting and attracted masses in the lower
position, and their sum in the upper position (see p. 88). The work is shown
in full for February 5 and May 5.
Tests were made at various times, showing that there was no change in
the distances (at least within errors of reading), either through moving the
turntable or in the course of a few days (see February 5 and May 5 for
examples).
TemveratureCorrtction. The cathetometerscale is taken as correct at
18°, and its coefficient of expansion is assumed to be 1/60000. That of the
masses is probably about 1/40000, but, for simphcity, is taken as equal to
that of the scale, the difference, 1/120000, never amounting to as much as
the errors of reading, since the greatest length concerned is 23 centims.
The temperature was estimated to be about 1° above that observed during
the immediately preceding weighings, the presence of the observer and the
lights used tending to raise it.
The cathetometer rested always on the brick floor of the room. Its vernier
reads to 002 centim.
Set I.
Attracted masses A on the left, B on the right. Attracting mass M
moving round from left to right in front of the balancecase.
February 5, 1890. Attracted masses in upper position. Assumed
temperature 11°.
Halfway through the measurements the cathetometer was accidentally
moved, and could not be exactly replaced. Repeating the reading of A it
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 93
was found that 197 must be added to the previous readings to compare with
the following ones. This addition is made where the numbers have an asterisk.
23448
A
64999*
65001*
^1
25895*
25889*
B
65284
65282
26070
26064
mi
23947*
Differences: ^  m, = 41552
A Mt^ = 39108
B  M^ = 39216
B  m^ = 41336
Table I
P
88, the sums of the radii of the masses are
' Rm + Ra =
23173,
Rm+ ^B =
23154,
Rm^RA =
19998,
Rm+RB =
19979,
d
= 1 {39108 + 23173 + 39216 + 23154}
= 62326,
h
= 1 {41336 + 19979 + 41552 + 19998}
= 61433.
and
These are in terms of a scale correct at 18°, so that the value is too great
by about 7/60000. We take as true values
Corrected d = 62318,
h = 61425.
Test Experiment. At the conclusion, the distance A — M^ was measured
again and found to be 39110.
May 28, 1890. Attracted masses in upper position.
Assumed temperature 14°.
A
64674
64674
B
65286
65288
mo
23422
23424
25726
25724
25920
25920
m^
23766
23756
Differences: J.
 ma = 41251
A M^ = 38949
J5  ¥2 = 39367
5  m.i = 41526
whence
d = 62312
5
h = 61377
94 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Subtracting temperaturecorrection 004,
Corrected d = 62308,
/i 61373.
Mean values in Set I, d = 62313,
h = 61399.
May 5, 1890. Attracted masses in lower position.
Assumed temperature 13°.
A
B
50324
50622
50324
50634
50328
50630
7m,
M,
M,
Ml
23672
25972
26138
23998
23674
25970
26138
24008
25972
26132
Differences : A  m.^ = 26652
A  Mi = 24354
B M^ = 24493
B  m^ = 26626
From Table I, p. 88,
^31 Ra = 7353,
Rjyj Rb= 7372,
R,, i?^== 4178,
R,, i?5^ 4197,
whence
and
D
l_ {24354 + 7353 + 24493 + 7372}
= 31786,
H = \ {26626 + 4197 + 26652 + 4178}
 30827.
Subtracting temperaturecorrection 0025,
Corrected values for Set I,
D = 31783,
H = 30824.
Test Experiment. The balance was set free at the end of these measures,
and two days later, on May 7, it was again fixed, and the distance D was
determined by the cathetometer described on p. 65. The value obtained
was D = 31786.
Note. If the apparatus were perfectly rigid and constant in its dimensions
we should expect D  H = d~h = constant. The values actually given by
the above experiments are
February 5 ... ... 892,
May 5 .959^
May 28 .935.
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 95
There is apparently a slight increase during the course of the spring,
probably due to the warping of the wood supporting the mass m. But
there was some uncertainty in sighting the top of the mass m, especially when
in the distant position on the right.
Set II.
Attracted masses A on the right, B on the left. Attracting mass M
moving round from left to right behind the balancecase. All the masses
inverted.
July 29, 1890. Attracted masses in lower position.
Assumed temperature 16°.
mi
22434
22436
B
49014
49014
M^
24584
24586
A
49846
49844
M,
24788
24782
^2
22868
22864
Differences : B  m^ = 26579
B M^ = 24429
A M^ = 25060
A  m^^ 26979
whence D  32107,
H = 30967.
Subtracting temperaturecorrection 001 ,
Corrected D  32106,
H = 30966.
September 18, 1890. Attracted masses in lower position.
Assumed temperature 16°.
B
A
49076
49768
49074
49766
mi
J/2
Ml
^2
22467
24576
24756
22840
24576
24758
Differences : B  m^ = 26608
B M^ = 24499
A  M^ = 25010
A  m^ = 26927
whence D  32117
H = 30955
Subtracting temperaturecorrection 001,
Corrected D = 32116
H = 30954
Mean values in Set II,
Z) = 32111
H = 30960
96 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
September 27, 1890. Attracted masses in upper position.
Assumed temperature 16°.
)
B
A
63880
63876
64540
64544
Ml
22450
22448
24570
24572
24756
24758
Mo
228^10
22816
Differences: B 
 Wj = 41429
B
 M^ = 39307
A 
 M, = 39785
A
 mo = 41729
whence d  62710,
h= 61568.
Subtracting temperaturecorrection 002,
Corrected values for Set II,
d = 62708,
h = 61566.
Note. The values oi D — H and d — h, which should be constant, are
from the above, and from another set of measures (not here recorded, see
p. 85) on September 15, as follows. (We have no reason to expect the same
value as in Set I, as the masses M, m have changed sides.)
July 29 1140,
September 15 1110,
September 18 1162,
September 27 1142.
From July 29 to September 15 inclusive, the balance was swinging freely
without alteration. The values of H should, therefore, be the same on those
dates. They were
July 29 30967,
September 15 30978,
equal almost within errors of reading for the top of m.
Means of Sets I and II :
Z) = 1(31783 + 32111)
= 31947.
H = \ (30824 + 30960)
 30892.
d=\ (62313 + 62708)
 62511.
h = \ (61399 t 61566)
= 61483.
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 97
At 18°
and
whence
Horizontal Distances,
Set I.
L = 123269 centims.
Zi = 122915
^= 61635
h + %= 184550
h^= 61280 „
Taking the mean temperature of the Set as 12°, and assuming 1/60000 as
the coefficient of expansion, on correcting to 12°,
l^ + ^ = 184532 centims.
L
At 18'
h
±J
~ 2^
61274
Set XL
55
\ =
122795 centims.
L
2
61635
55
h
*\'
184430
"
h
L
2 ~
61160
55
Whence
Taking the mean temperature of the Set as 15°, and correcting to 15°,
^2 +  = 184421 centims.
''2 9
61157
Mean values for the two Sets
L = 123260
l\%= 184477
l~= 61216
p. c. w.
98
ON A DETERMmATION OF THE MEAN DENSITY OF THE EARTH AND
Table III. Determination of Attraction by the Balance.
Determinations of the Attraction in terms of the Riders by the Balance.
In each case four turningpoints of three successive swings are recorded
in tenths of a division, i.e., in divisions on the diagonal lines. In the columns
headed i the masses and riders are in the initial position, in those headed r
the riders are moved, and in those headed m the masses are moved. Under
each set of four readings is the calculated centre of swing (see p. 71). In the
next hne are the deflections due to movements of riders and masses, each
placed under the middle one of the three centres of swing from which
it is calculated. In the next Hne are the values of deflection due to
mass ^ deflection due to rider, or M/R (see p. 74).
Set I.
Attracted Masses in Upper Position. Feb. 4, 1890, 7.59 p.m. to 10.49 p.m.
Temperature: in Observing Koom, 15°716°5; in Balance Eoom,
10°05. Barometer, 75227520 miUims. Weather mild and still, after
shght frost on the two previous nights. Time between successive
passages of centre about 20 seconds.
I.
i
r
i
771
r
m
Scalereadings . . .
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
725
912
725
804
764
913
726
804
798
838
799
787
779
838
800
787
759
878
759
796
771
879
759
797
780
856
781
791
776
857
781
791
Centre of swing
77255
86385
77300
79280
77390
86460
77340
79320
Deflection due to
rider or mass...
...
91075
...
19350
90950
19700
Mass deflections
rider deflection
...
...
212608
214688
...
217110
/
r
i
m
i
r
i
m
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
Scalereadings...
763
913
724
804
764
914
725
805
779
837
801
787
779
838
801
789
771
879
759
796
771
880
760
796
Centre of swing
Deflection due to
775
857
782
792
776
857
783
792
77360
86425
77385
79305
77390
86505
77450
79365
rider or mass...
Mass deflections
90525
19175
90850
...
18950
rider deflection
214720
1
211440
...
•209824
•208758
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 99
Table III (continued).
i
r
i
m
i
r
i
m
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
Scalereadings...
765
916
726
803
765
914
725
805
780
839
803
788
779
838
800
787
772
881
762
797
771
879
759
796
111
859
784
792
776
857
782
791
Centre of swing
77490
86630
77630
79365
77400
86450
77360
79285
Deflection due to
rider or mass...
...
90700
...
18500
90700
...
19225
Mass deflection ^
rider deflection
...
•206174
•203966
...
207966
...
•211438
i
r
i
m
i
r
i
m
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
Scalereadings...
763
914
121
805
764
913
725
804
780
838
800
789
779
838
800
786
770
880
760
797
771
879
760
796
776
857
782
792
115
857
781
791
Centre of swing
77365
86505
774 15
79390
77365
86465
77380
79250
Deflection due to
rider or mass...
...
91150
...
20000
...
90925
19275 i
Mass deflections
rider deflection
...
•215167
...
219690
...
215975
•211494
i
r
i
m
^•
r
•
i m
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
Scalereadings...
763
913
724
803
763
912
726
802
778
838
801
789
778
837
799
787
770
879
759
796
769
879
759
797
774
857
782
792
774
857
780
792
Centre of swing
772^65
86460
77385
79350
77230
86420
77295
79330
Deflection due to
rider or mass...
...
91350
...
20425
...
91575
...
19875
Mass deflections
rider deflection
217296
...
•223316
...
220038
...
•216503
7—2
100 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Table III (continued).
i
r
i
m
i
r
i
m
(41)
(42)
(43)
(44)
803
(45)
(46)
(47)
(48)
Scalereadings...
764
915
725
763
913
726
803
779
839
800
786
778
838
799
787
771
880
759
795
770
879
759
796
776
858
780
791
774
857
781
792
Centre of swing
77390
86555
77315
79225
77270
86460
77315
79295
Deflection due to
rider or mass...
92025
...
19325
91675
...
19200
Mass deflection ^
rider deflection
•212986
...
210397
...
•210117
...
209693
*
r
i
m
i
r
i
m
1
(49)
(50)
(51)
(52)
(53)
(54)
(55)
(56)
Scalereadings...
764
914
724
803
762
912
721
801
779
839
802
787
778
836
798
785
772
880
759
795
769
877
755
794
776
858
781
790
774
855
779
788
Centre of swing
77435
86555
77385
79210
77220
86255
77035
79055
Deflection due to
rider or mass...
91450
19075
91275
20275
Mass deflections
rider deflection
209267
208784
215557
...
222161
(57)
(58)
(59)
m
(60)
Scalereadings,
760
911
724
776
836
799
767 1
877
758
772
855
779
803
785
795
789
(61)
762
777
769
773
(62)
(63)*
911
836
877
855
Centre of swing
Deflection due to
rider or mass...
Mass deflection s
rider deflection
7020
722
800
758
780
86250 77220 79135 , 77170 86250 ; 77250
91250 ... 19425  ... 90400
•217534 ... 213873
•217506
(63 a)
725
799
759
780
m
(64)
803
786
796
790
77290 I 79230
19900
217873
* After 63 the riders were moved by mistake instead of the masses, therefore it was necessary
to return to the initial position, and take the readings in (63 a).
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 101
Table III (continued).
i
r
i
m
i
r
i
m
(65)
(66)
(67)
(68)
(69)
(70)
(71)
(72)
Scalereadings...
762
913
725
802
759
909
722
802
in
838
800
785
776
834
797
784
769
879
758
794
767
875
757
794
774
857
780
789
772
854
779
789
Centre of swing
77190
86460
112'lb
79080
77015
86080
77105
79055
Deflection due to
rider or mass...
92275
19350
90200
19450
Mass deflection :
,
rider deflection
•213170
...
212084
...
215078
214947
*
r
i
m
i
r
i
m
(73)
(74)
(75)
(76)
{11)
(78)
(79)
(80)
Scalereadings...
760
911
724
803
762
911
721
800
777
835
798
785
111
835
797
784
768
877
151
795
769
877
756
793
773
854
779
790
773
854
778
788
Centre of swing
77115
86205
77140
79150
77170
86205
77030
78975
Deflection due to
rider or mass...
90775
19950
...
91050
...
20150
Mass deflection ^
rider deflection
...
•217020
...
219442
220209
...
221064
i
r
.
m
i
r
i
m
(81)
(82)
(83)
(84)
(85)
(86)
(87)
(88)
Scalereadings . . .
759
910
722
801
759
910
723
802
774
833
796
783
115
834
797
783
766
876
757
793
767
876
757
793
771
854
778
787
771
854
778
787
Centre of swing
76890
86095
77050
78930
76960
86125
77090
78935
Deflection due to
rider or mass...
91250
19250
...
91000
19300
Mass deflection ^
rider deflection
...
•215990
...
211248
...
•211813
212995
102 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Table III
i
r
i
m
i
r
i
m
(89)
(90)
(91)
(92)
(93)
(94)
(95)
(96)
Scalereadings . . .
759
908
719
800
760
910
723
800
775
831
795
783
775
835
798
785
766
874
754
792
767
876
756
793
771
852
776
788
772
854
779
789
Centre of swing
76920
85900
76835
78905
76990
86160
77090
79030
Deflection due to
, rider or mass...
...
90225
...
19925
...
91200
19350
Mass deflection ^
rider deflection
217373
•219650
•215323
...
•212462
i
r
i
m
i
r
*
m
i
Scalereadings . . .
(97)
(98)
(99)
(100)
(101)
(102)
(103)
(104)
(105)
761
910
721
798
759
909
721
800
759
111
835
796
783
775
833
796
783
115
768
876
756
791
765
874
756
793
767
772
853
111
787
770
852
777
787
771
i
Centre of swing
77100
86135
76980
78835
76860 85965
76980
78935
76960
Deflection due to
rider or mass...
90950
19150
90450
19650
Mass deflections
rider deflection
211655
211136
...
214483
Feb. 4, 1890. Mean of 50 determinations of M/R = a
Attracted masses in upper position
•21422122.
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 103
Table III (continued).
11. Attracted Masses in Lower Position. April 30, 1890, 7.45 p.m. to
10.32p.m. Temperature: in Observing Room, 17°16°1 ; in Balance
Room, 11°1. Barometer, 74867492 millims. Weather clear; S.E.
wind; sunny during day. Time between successive passages of centre
not quite 20 seconds.
i
r
i
m
i
r
i
m
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Scalereadings . . .
1046
1133
951
1127
955
1134
952
1123
969
1055
1024
1062
1025
1059
1028
1069
1012
1098
984
1099
986
1102
985
1099
988
1075
1007
1078
1007
1077
1009
1082
Centre of swing
99660
108285
99835
108550
99980
108625
100040
108825
Deflection due to
rider or mass...
...
85375
86425
86150
87350
Mass deflections
rider deflection
...
...
100772
...
100856
...
101437
i
r
i
m
i
r
i
m
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
Scalereadings . . .
962
1136
955
1129
961
1136
956
1133
1023
1060
1029
1067
1027
1064
1031
1069
989
1104
987
1102
989
1105
989
1103
1009
1079
1012
1083
1012
1081
1013
1085
Centre of swing
100140
108800
100245
108955
100315
109000
100415
109125
Deflection due to
rider or mass...
86075
86750
86350
86350
Mass deflection^
rider deflection
101133
...
100623
...
100232
100101
i
r
i
m
i
r
i
m
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
1134
Scalereadings...
967
1141
957
1135
965
1143
958
1027
1064
1034
1070
1031
1066
1036
1073
994
1108
990
1106
994
1110
993
1108
1012
1083
1015
1086
1015
1085
1017
1089
Centre of swing
100565
109200
100600
109320
100740
109400
100835
109540
Deflection due to
rider or mass...
...
86175
...
86500
86125
86825
Mass deflection H
rider deflection
...
100290
...
100406
100624
...
101106
104 ON^ A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Table HI (continued).
i
r
*
m
i
r
i
m
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
Scalereadings...
976
1143
963
1141
966
1145
960
1141
1027
1069
1037
1073
1037
1070
1040
1075
998
1110
996
1112
996
1112
995
1113
1016
1087
1019
1090
1019
1089
1020
1092
Centre of swing
100880 109535
101065 109790
101085
109705
101115
109930
Deflection due to
rider or mass...
...
85625
87150
86050
87575
Mass deflections
rider deflection
...
101591
101529
101264
101713
i
r
i
m
i
r
i
1
m 1
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
Scalereadings...
973
1147
962
1137
975
1146
965
1138
1034
1071
1041
1079
1034
1075
1042
1080
1000
1114
996
1112
1002
1114
999
1113
1020
1090
1022
1094
1020
1091
1021
1094
Centre of swing
101230
109855
101250
110020
101340
109980
101400
110100
Deflection due to
rider or mass...
86150
87250
86100
...
86650
Mass deflections
rider deflection
101465
...
101306
100987
...
100858
i
r
i
m
i
r
i
m
'
(41)
(42)
(43)
(44)
(45)
(46)
(47)
(48)
Scalereadings... ,
977
1150
964
1089
976
1153
968
1
1148 1
i
1035
1073
1043
1 1110
1038
1074
1044
1078
1003
1116
1000
1098
1004
1118
1001
1118
1022
1093
1025
1104
1023
1094
1025
1095
Centre of swing
101470
110080
1015451 110220
101615
110235
101645
110340
Deflection due to
rider or mass...
...
85725
86400
86050
86475
Mass deflections
rider deflection
...
100933
j 100597
100450
100625
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 105
Table III (continued).
i
r
i
m
i
r
i
m
(49)
(50)
(51)
(52)
(53)
(54)
(55)
(56)
Scalereadings...
978
1153
969
1149
976
1153
968
1145
1039
1076
1045
1080
1043
1079
1047
1085
1005
1119
1002
1118
1005
1121
1004
1118
1025
1094
1027
1097
1026
1096
1028
1099
Centre of swing
101740
110335
101765
110445
101860
110555
101925
110615
Deflection due to
rider or mass...
...
85825
...
86325
86625
86875
Mass deflection ^
rider deflection
...
100670
100116
...
99971
...
100973
i
r
i
m
i
.
i
m
(57)
(58)
(59)
(60)
(61)
(62)
(63)
(64)
Scalereadings . . .
984
1155
971
1151
977
1157
972
1152
1039
1078
1048
1083
1046
1081
1051
1087
1008
1120
1004
1122
1007
1123
1007
1123
1026
1097
1029
1100
1030
1100
1031
1102
Centre of swing
101930
110510
102000
110785
102125
110810
102260
110995
Deflection due to
rider or mass...
...
85450
87225
...
86175
86850
Mass deflection ^
rider deflection
101872
...
101646
...
101011
100798
i
r
I
m
i
r
i
m
(65)
(66)
(67)
(68)
(69)
(70)
(71)
(72)
Scalereadings...
983
1159
976
1153
983
1161
978
1158
1046
1082
1051
1088
1049
1083
1053
1088
1011
1125
1008
1126
1012
1127
1011
1127
1031
1102
1033
1104
1032
1102
1034
1106
Centre of swing
102360
110980
102370
111205
102515
111105
102595
111315
Deflection due to
rider or mass...
...
86150
87625
85500
...
86700
Mass deflections
rider deflection
...
101262
...
102097
...
101944
...
101226
106 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Table III (continued).
i
r
i
m
*
r
*
m
(73)
(74)
(75)
(76)
(77)
(78)
(79)
(80)
Scalereadings...
985
1163
980
1156
990
1163
982
1161
1051
1086
1056
1092
1051
1087
1057
1093
1014
1129
1013
1129
1017
1131
1015
1132
1033
1104
1036
1108
1036
1107
1039
1109
Centre of swing
102695
111340
102825
111550
102920
111520
103015
111765
Deflection due to
rider or mass...
...
85880
...
86775
85525
...
87100
Mass deflection ^
rider deflection
101093
...
101299
...
101652
...
101782
^
r
i
m
i
r
i
m
(81)
(82)
(83)
(84)
(85)
(86)
(87)
(88)
Scalereadings...
991
1167
984
1158
992
1169
984
1161
1054
1090
1059
1098
1056
1092
1063
1100
1018
1133
1018
1132
1021
1136
1018
1135
1038
1108
1041
1112
1041
1111
1042
1115
Centre of swing
103095
111740
103260
111955
103355
112000
103400
112220
Deflection due to
rider or mass...
85625
86475
86225
87600
Mass deflection ^
rider deflection
101358
100640
100942
101890
i
r
i
m
i
r
i
m
(89)
(90)
(91)
(92)
(93)
(94)
(95)
(96)
, Scalereadings...
996
1171
987
1165
995
1172
989
1169
1058
1094
1064
1100
1061
1097
1066
1099
1022
1137
1022
1137
1024
1137
1023
1140
1043
1114
1045
1117
1045
■ 1116
1046
1117
Centre of swing
103520
112175
103685
112375
103735
112315
103820
112505
Deflection due to
rider or mass...
85725
86650
85375
86575
Mass deflection ^
rider deflection
101633
...
101286
...
101450
...
101065
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 107
Table III (continued).
i
r
i
m
i
r
i
m
i
(97)
(98)
(99)
(100)
(101)
(102)
(103)
(104)
(105)
Scalereadings...
998
1175
992
1174
995
1176
995
1169
1001
1062
1097
1066
1102
1064
1098
1067
1105
1066
1026
1141
1025
1141
1027
1143
1026
1143
1029
1045
1116
1047
, 1119
1048
1118
1049
1121
1049
Centre of swing
103875
112505
103945
112715
104015
112670
104075
112890
104220
Deflection due to
rider or mass...
85950
87350
86250
87425
Mass deflection ^
rider deflection
...
101178
...
101452
101319
April 30. Mean of 50 determinations of M/R = A
Attracted masses in lower position
1010905.
May 4, 1890, 11.11 to 11.50 a.m. Temperature: in Observing Koom, 13°5
to 13°8; in Balance Room, ll°7. Barometer, 7420 to 7417 millims.
Weather inclined to rain; a little cooler than previous day; wind S.
to S.W.
*
r
i
m
^•
r
*
m
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Scalereadings . . .
875
1045
865
1044
865
1045
861
1041
936
969
939
970
938
967
940
971
900
1013
896
1014
897
1013
894
1012
920
988
921
989
920
986
921
989
Centre of swing
91310
99695
91180
99775
91160
99600
91095
99710
Deflection due to
rider or mass...
84500
86050
...
84725
...
86275
Mass deflection h
rider deflection
...
101699
...
101697
...
101950
108 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Table III (continued).
*
r
i
m
i
r
i
m
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
Scalereadings...
869
1046
862
1036
867
1045
860
1040
936
966
939
972
936
964
937
968
896
1012
894
1009
896
1011
893
1009
919
985
920
988
t
918
984
919
986
Centre of swing
910700
99510
91045
99555
91040
99380
90920
99420
Deflection due to
rider or mass...
...
84525
...
85125
...
84000
...
85450
Mass deflections
rider deflection
101390
...
101024
101533
101454
i
r
i
m
i
r
.
m
i
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
Scalereadings .. .
863
1043
859
1033
863
1040
857
1035
862
934
964
936
971
932
963
936
966
930
894
1009
892
1006
892
1007
889
1006
891
916
983
917
985
915
982
915
983
914
Centre of swing
90830
99260
90800
99315
90660
99100
90610
99140
90525
Deflection due to
rider or mass...
84450
85850
84650
85725
Mass deflections
rider deflection
101421
...
101538
101344
May 4, morning. Mean of 10 determinations of MjR = A
Attracted masses in lower position
1015050.
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 109
Table III (continued).
Same Day. 2.40 to 4.54 p.m. Temperature: in Observing Koom, 13°9
14°1 ; in Balance Room, ll°7ll°75. Barometer, 74037397 millims.
i
r
i
m
i
r
i
m
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Scalereadings...
847
1035
853
1031
864
1035
853
1026
933
957
930
961
925
960
931
965
883
1003
886
1002
890
1003
885
1001
912
977
911
979
909
977
912
980
Centre of swing
90140
98620
90200
98705
90255
98710
90200
98765
Deflection due to
rider or mass...
...
84500
...
84775
...
84825
85300
Mass deflection ^
rider deflection
...
...
...
100133
100251
100783
1
^•
r
i
m
i
r
.
m
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
Scalereadings...
866
1035
853
1023
864
1036
854
1024
924
959
932
968
925
960
931
968
891
1004
886
1000
889
1004
886
1000 1
909
977
912
982
910
978
912
982
Centre of swing
90270
98720
90280
98835
90230
98775
90250
98845
Deflection due to
rider or mass...
...
84450
...
85800
...
85350
...
85925
Mass deflection ^
rider deflection
101303
101060
100601
...
100940
i
r
i
m
i
r
i
m
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
Scalereadings . . .
864
1039
855
1025
866
1040
855
1011
925
958
931
969
925
958
932
977
890
1005
887
1001
891
1005
888
996
909
978
913
982
911
978
913
985
Centre of swing
90255
98780
90325
98920
90350
98790
90400
98910
Deflection due to
rider or mass...
...
84900
...
85825
...
84150
...
85225
Mass deflection f
rider deflection
...
101148
...
101538
101634
...
101232
110 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Table III (continued).
i
r
i
m
i
r
i
m
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
Scalereadings...
864
927
1036
961
854
933
1024
970
865
926
1037
962
854
934
1031
967
890
1004
887
1001
892
1005
888
1004
912
979
914
984
912
980
915
983
Centre of swing
90375
98810
90400
98985
90435
98925
90490
99055
Deflection due to
rider or mass...
84225
...
85675
84625
85625
Mass deflections
rider deflection
101454
...
101481
...
101211
...
101182
*
r
m
i
r
^
m
,
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
Scalereadings...
864
1039
855
1024
864
1041
856
1025
928
961
934
972
929
961
934
971
892
1006
888
1002
891
1006
888
1002
912
980
914
985
913
980
915
984
Centre of swing
90495
98950
90480
99M0
90500
98965
90500
99065
Deflection due to
rider or mass...
84625
86200
84650
...
85500
Mass deflections
rider deflection
...
101521
...
101846
...
101418
...
100796
i
r
i
m
.
r
.
m
(41)
(42)
(43)
(44)
(45)
(46)
(47)
(48)
Scalereadings...
866
1038
857
1030
865
1043
858
1035
927
962
934
960
930
962
934
966
893
1007
889
1004
893
1008
891
1006
913
981
915
984
915
981
915
984
Centre of swing
Deflection due to
90530
99040
90550
99130
90660
99115
90655
99155
rider or mass...
Mass deflections
...
85000
...
85250
84575
...
8495
rider deflection
...
100441
100546
...
100621
100741
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 111
Table III (continued).
i
r
i
m
i
r
i
m
(49)
(50)
(51)
(52)
(53)
(54)
(55)
(56)
Scalereadings...
869
1041
858
1037
867
1041
856
1029
i
927
961
935
965
929
963
936
972
895
1008
891
1007
894
1007
890
1004
914
982
916
984
914
982
916
986
Centre of swing
90665
99090
90700
99180
90660
99105
90670
99250
Deflection due to
rider or mass...
84075
...
85000
84440
85750
Mass deflections
rider deflection
101070
...
100905
...
101155
...
10] 509
i
r
i
m
^
r
i
m
(57)
(58)
(59)
(60)
(61)
(62)
(63)
(64)
Scalereadings...
871
1041
859
1035
869
1044
860
1030
928
963
936
969
931
963
937
973
895"
1008
890
1008
895
1010
892
1005
913
982
917
985
916
983
917
986
Centre of swing
90680
99150
90710
99350
90820
99285
90825
99330
Deflection due to
rider or mass...
84550
85850
84625
...
84775
Mass deflections
rider deflection
101478
101493
100812
...
100162
i
r
i
m
i
r
i
m
(65)
(66)
(67)
(68)
(69)
(70)
(71)
(72)
Scalereadings . . .
863
1042
863
1039
840
1042
861
1037
935
965
935
969
949
965
937
971
894
1010
894
1008
886
1009
893
1009
917
984
917
985
923
985
918
987
Centre of swing
90880
99345
90880
99375
90915
99325
90905
99505
Deflection due to
rider or mass...
84650
...
84775
84150
85750
Mass deflection s
rider deflection
...
100148
100444
...
101322
101449
112 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Table III (continued).
i
r
i
m
i
r
i
m
(73)
(74)
(75)
(76)
.(77)
(78)
(79)
(80)
Scalereadings...
865
1045
860
1038
868
1045
863
1041
935
965
938
969
934
965
937
969
895
1011
893
1010
895
1012
894
1011
918
985
919
987
918
985
919
988
Centre of swing
90955
99430
90925
99500
90950
99475
90980
99580
Deflection due to
rider or mass...
84900
...
85625
...
85100
85625
Mass deflection^
rider deflection
...
100928
...
100735
...
100617
100765
i
r
i
m
i
(81)
(82)
(83)
(84)
(85)
Scalereadings...
864
1044
860
1036
867
938
967
940
974
936
895
1012
894
1010
896
919
986
920
989
919
Centre of swing
91055
99545
91065
99680
91060
Deflection due to
rider or mass...
84850
86175
Mass deflections
rider deflection
...
101238
May 1, afternoon. Mean of 40 determinations of MIR = A]
r 1 '01 on'?7ft
Attracted masses in lower position J ^ *
April 30 and May 4. Mean of 100 determinations of MIR = A]
\4.. . 1 . . . r 10109685.
Attracted masses m upper position j
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 113
Table III (continued).
III. Attracted Masses in Upper Position. May 25, 1890, 11.20 to 12.53
noon. Temperature : in Observing Room, 15°416° ; in Balance Room,
13°3. Barometer, 74857481 millims. Weather, E. wind, warm, very
bright. Time of swing not recorded.
i
r
i
m
i
r
i
m
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Scalereadings...
1071
1175
960
1049
1003
1173
956
1049
986
1085
1047
1028
1021
1085
1047
1028
1033
1134
998
1041
1010
1134
996
1040
1005
1108
1025
1034
1017
1107
1024
1034
Centre of swing
101590
111690
101550
103620
101425
111655
101425
103580
Deflection due to
rider or mass...
101200
...
21325
...
102300
21500
Mass deflection ^
rider deflection
...
...
...
209582
209311
•210320
i
r
i
m
i
r
i
m
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
Scalereadings...
1001
1173
958
1049
1002
1173
957
1049
1021
1085
1046
1028
1020
1084
1046
1028
1011
1134
996
1038
1010
1134
995
1040
1016
1106
1024
1033
1015
1105
1023
1032
Centre of swing
101435
111635
101405
103470
101350
111580
101325
103590
Deflection due to
rider or mass...
102150
20925
102425
...
22850
Mass deflection ^
rider deflection
...
207660
...
204571
...
213688
...
223297
i
r
i
m
i
r
i
m
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
Scalereadings...
1003
1173
956
1048
1001
1172
958
1048
1019
1082
1043
1025
1019
1081
1042
1026
1009
1133
994
1038
1008
1131
994
1038
1015
1104
1023
1032
1014
1103
1021
1030
Centre of swing
101285
111460
101190
103360
101205
111310
101135
103350
Deflection due to
rider or mass...
102225
21625
...
101400
...
22375
Mass deflections
rider deflection
...
217535
212400
216962
...
•220172
p. c. w.
114 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Table III (continued).
i
r
i
m
i
r
I
m
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
Scalereadings...
1000
1171
953
1047
1000
1171
955
1046
1017
1081
1044
1025
1016
1080
1041
1025
1007
1130
992
1037
1007
1130
992
1036
1014
1103
1021
1031
1013
1103
1021
1030
Centre of swing
101090
111270
101080
103290
101045
111240
101000
103215
Deflection due to
rider or mass...
101850
...
22275
...
102175
22050
Mass deflection ^
rider deflection
219195
•218356
...
•216907
...
•215885
i
r
;
m
i
r
i
m
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
Scalereadings...
999
1168
952
1046
999
1170
955
1046
1017
1080
1043
1024
1018
1082
1043
1026
1006
1131
992
1037
1007
1130
993
1039
1013
1102
1021
1030
1012
1102
1021
1030
Centre of swing
101020
111240
101040
103235
101070
111265
101105
103375
Deflection due to
rider or mass...
...
102100
21800
101775
22100
Mass deflection ^
j rider deflection
...
•214740
•213857
...
•215672
216858
i
r
i
m
i
r
i
m
(41)
(42)
(43)
(44)
(45)
(46)
(47)
(48)
Scalereadings...
998
1171
955
1046
1000
1173
956
1048
1019
1082
1043
1027
1019
1082
1046
1028
1009
1132
994
1038
1009
1131
995
1039
1014
1104
1022
1031
1014
1104
1023
1032
Centre of swing
Deflection due to
101225
111400
101165
103385
101235
111380
101320
103485
rider or mass...
Mass deflections
102050
...
21850
...
101025
21900
rider deflection
...
•215388
...
215197
...
•216531
216350
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 115
Table III (continued).
i
r
i
m
i
r
i
(49)
(50)
(51)
(52)
(53)
(54)
(55)
Scalereadings . . .
1001
1172
956
1048
1000
1173
956
1019
1083
1046
1027
1020
1082
1044
1009
1132
996
1039
1009
1133
995
1015
1105
1023
1032
1016
1105
1023
Centre of swing
101270
111460
101365
103460
101315
111470
101265
Deflection due to
rider or mass...
...
101425
...
21200
101800
Mass deflection ^
rider deflection
...
•212472
...
208636
May 25, morning. Mean of 25 determinations of MjR = a
Attracted masses in upper position
•21446168.
Same Day. 3.15 to 4.50 p.m. Temperature: in Observing Room, 16°0 to
16°25; in Balance Room, 13°3 to 13°35. Barometer, 74777474millims.
i
r
i
m
i
r
i
m
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Scalereadings . . .
1001
1205
990
1081
1034
1207
991
1083
1069
1116
1077
1061
1055
1120
1080
1062
1031
1165
1029
1073
1044
1168
1030
1076
1052
1138
1057
1066
1049
1139
1059
1068
Centre of swing
104455
114760
104630
106855
104760
115050
104835
107065
Deflection due to
rider or mass...
...
102175
...
21600
102525
21675
Mass deflection ^
rider deflection
...
211041
...
•211046
212162
i
r
i
m
i
r
i
m
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
Scalereadings . . .
1037
1209
995
1086
1039
1213
994
1088
1056
1119
1082
1066
1058
1121
1086
1069
1046
1169
1030
1078
1048
1172
1034
1078
1052
1141
1059
1071
1054
1145
1064
1073
Centre of swing
104960
115110
1049^00
107350
105160
115410
1052^90
107490
Deflection due to
rider or mass...
...
101800
...
23200
...
101850
...
21675
Mass deflection ^
rider deflection
...
•220408
...
227842
...
•220299
...
•216738
8—2
116 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Table III {continued).
i
r
*
m
i
r
i
m
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
Scalereadings...
1045
1212
996
1088
1043
1215
1002
1094
1059
1126
1088
1071
1063
1126
1088
1072
1050
1175
1037
1080
1053
1177
1039
1083
1056
1147
1066
1076
1058
1148
1066
1077
Centre of swing
105355
115720
105530
107710
105630
115850
105655
107925
Deflection due to
rider or mass...
...
102775
21300
...
102075
...
22350
Mass deflection ^
rider deflection
...
•209073
...
•207957
...
•213813
...
•219575
i
r
^
m
i
r
i
m
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
Scalereadings . . .
1044
1217
999
1095
1048
1221
1002
1096
1064
1127
1093
1073
1067
1131
1094
1076
1053
1178
1041
1086
1056
1180
1042
1088
1061
1150
1069
1079
1061
1152
1071
1081
Centre of swing
105725
115980
105935
108135
1059^70
116250
1060^70
108355
Deflection due to
rider or mass...
101500
21825
102300
22800
Mass deflection ^
rider deflection
...
•217611
•214181
•218112
•223147
(33)
r
(34)
(35)
m
(36)
(37)
(38)
(39)
m
(40)
Scalereadings... 1049
1067
1057
1064
1221
1131
1182
1154
1004
1095
1045
1072
1097
1079
1089
1082
1053
1069
1059
1066
1224
1134
1183
1156
1007
1096
1047
1075
1101
1079
1091
1085
Centre of swing 106080
Deflection due to
rider or mass...
Mass deflections
rider deflection
116375
102050
221583
106260
108520
22425
219907
106295
116570
101900
217983
106465
108690
22000
215898
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 117
Table III
{continued).
*
r
i
m
i
r
i
m
(41)
(42)
(43)
(44)
(45)
(46)
(47)
(48)
Scalereadings . . .
1054
1226
1009
1101
1054
1225
1008
1102
1072
1135
1096
1081
1073
1135
1099
1081
1061
1185
1048
1093
1064
1187
1048
1094
1068
1157
1076
1086
1068
1159
1080
1089
Centre of swing
106515
116715
106535
108855
106685
116840
106700
108970
Deflection due to
rider or mass...
...
10190
22450
101475
21625
Mass deflection ^
rider deflection
•218106
...
220774
217172
...
•213607
i
r
i
m
i
r
i
(49)
(50)
(51)
(52)
(53)
(54)
(55)
Scalereadings . . .
1058
1228
1014
1104
1059
1229
1019
1075
1138
1102
1086
1076
1141
1103
1066
1189
1053
1097
1068
1192
1055
1072
1161
1080
1091
1074
1163
1081
Centre of swing
106915
117080
1070^45
109300
107095
117340
107215
Deflection due to
rider or mass...
...
101000
22300
...
101850
Mass deflection f
rider deflection
•217458
...
•219867
May 25, afternoon. Mean of 25 determinations of MjR = a
Attracted masses in upper position
Mean of 50 determinations, morning and afternoon, '2157379
21701412.
Summary of Set I.
February 4 ... a= 2142212
May 25
Mean value of
April 30
May 4
Mean value of
a = 2157379
a = 2149791
A = 1010905
A = 1011032
A = 10109685
therefore
a = 7959894.
118 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Table III (continued).
Set II.
All Attracting and Attracted Masses inverted and changed over, each to the other
side. The Suspending Rods also reversed and Riders interchanged. The
initial position always the higher reading on the scale.
I. Attracted Masses in Lower Position. July 28, 1890, 8.10 to 9.43 p.m.
Temperature: in Observing Room, 17°16°9; in Balance Room, 15°4.
Barometer, 7476748 millims. Weather fine and calm; wind W.
r
i
m
i
r
i
m
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Scalereadings...
1099
912
1130
917
1126
914
1131
922
1051
1007
1034
1005
1036
1008
1035
1005
1081
951
1093
952
1091
951
1093
954
1063
985
1057
985
1058
986
1057
984
Centre of swing
106965
97195
107055
97210
107020
97260
107095
97315
Deflection due to
rider or mass...
...
98150
98275
97975
97575
Mass deflections
rider deflection
...
100217
99949
99541
i
r
i
m
i
r
i
m
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
Scalereadings...
1128
913
1134
924
1130
915
1137
919
1035
1010
1035
1006
1038
1013
1034
1012
1092
951
1095
956
1094
953
1098
955
1058
987
1058
987
1061
989
1060
989
Centre of swing
107050
97330
107225
97500
107305
97565
107380
97640
Deflection due to
rider or 7nass...
98075
97650
91775
97700
Mass deflections
rider deflection
...
•99528
...
99719
...
99898
...
99719
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 119
Table III (continued).
i
r
i
m
i
r
i
m
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
Scalereadings...
1132
917
1140
924
1134
916
1136
960
1040
1014
1036
1009
1042
1016
1041
991
1095
954
1099
958
1098
955
1098
972
1062
989
1060
990
1064
993
1064
983
Centre of swing
107440
97655
107505
97740
107690
97820
107670
97905
Deflection due to
rider or mass...
...
98175
98575
...
98600
...
98100
Mass deflections
rider deflection
...
•99962
...
100191
...
99734
...
99506
i
r
i
m
i
r
i
m
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
Scalereadings...
1133
916
1142
925
1134
918
1143
925
1044
1018
1039
1013
1045
1019
1042
1018
1098
956
1103
960
1101
957
1103
961
1065
994
1064
994
1068
997
1066
996
Centre of swing
107760
97955
107865
98030
107980
98100
108000
98265
Deflection due to
rider or mass...
...
98575
...
98925
98900
97075
Mass deflection ^
rider deflection
...
99937
100190
...
•99090
99031
i
r
i
m
i
r
i
m
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
Scalereadings...
1136
924
1145
930
1140
918
1143
928
1046
1019
1042
1016
1046
1022
1045
1018
1099
961
1104
964
1104
959
1104
962
1068
996
1067
995
1069
997
1068
996
Centre of swing
107945
98295
108075
98350
108200
98280
108180
98330
Deflection due to
rider or mass...
97150
97875
...
99100
...
98875
Mass deflections
rider deflection
...
100335
...
99745
...
•99268
...
100051
120 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Table III (continued).
i
r
i
m
i
r
*
m
(41)
(42)
(43)
(44)
(45)
(46)
(47)
(48)
Scalereadings...
1138
926
1146
928
1142
927
1144
937
1047
1020
1045
1022
1047
1021
1047
1015
1104
963
1107
964
1106
963
1107
967
1071
997
1069
998
1071
999
1070
996
Centre of swing
108255
98445
108345
98575
108370
98510
108405
98510
Deflection due to
rider or mass...
...
98550
...
97825
...
98775
98900
Mass deflections
rider deflection
...
•99797
99151
99582
100139
i
r
i
m
i
r
i
(49)
(50)
(51)
(52)
(53)
(54)
(55)
Scalereadings...
1140
923
1148
932
1141
924
1144
1049
1024
1045
1021
1050
1024
1048
1105
962
1108
966
1106
963
1107
1072
999
1071
998
1072
1001
1072
Centre of swing
108395
98540
108435
98660
108480
98625
108475
Deflection due to
rider or mass...
...
98750
97975
98525
Mass deflection s
rider deflection
...
99684
...
99328
July 28, 1890. Mean of 25 determinations of MjR
Attracted masses in lower position
•9973168.
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 121
Table III (continued).
September 17, 1890, 8.0 to 9.31 p.m. Temperature: in Observing Room,
17°17°5; in Balance Room, 15°8. Barometer, 74627464 millims.
Weather warm, cloudy.
i
r
i
m
i
r
i
m
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Scalereadings . . .
1085
908
1118
921
1109
905
1126
921
1051
1004
1029
995
1036
1006
1026
996
1073
945
1085
949
1081
944
1087
951
1058
981
1050
978
1053
981
1050
978
Centre of swing
106420
96735
106340
96670
106375
96735
106395
96790
Deflection due to
rider or mass...
96450
96875
96500
96450
Mass deflection ^
rider deflection
...
...
...
100415
...
100168
•99613
i
r
i
m
i
r
i
m
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
Scalereadings . . .
1113
907
1126
929
1110
910
1131
934
1034
1006
1027
993
1038
1007
1027
993
1084
944
1088
953
1083
947
1092
956
1053
982
1052
978
1056
984
1052
979
Centre of swing
106475
96775
106505
96840
106590
96990
106710
97020
Deflection due to
rider or mass...
97150
97075
96600
96850
Mass deflection ;
rider deflection
...
99601
...
100206
...
100375
...
100026
i
r
i
m
i
r
i
m
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
Scalereadings . . .
1104
910
1129
924
1116
909
1121
927
1044
1008
1030
1001
1040
1009
1036
1000
1081
947
1091
953
1086
947
1088
956
1059
985
1054
983
1057
986
1056
984
Centre of swing
106700
97044
106790
97145
106670
97085
106705
97280
Deflection due to
rider or mass...
97050
95850
96025
...
95550
Mass deflections
rider deflection
...
•99279
99288
99662
...
•99105
122 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Table III (continued).
i
r
*
m
i
r
*
m
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
Scalereadings...
1112
914
1131
929
1114
916
1132
934
1043
1009
1033
1002
1044
1011
1035
1002
1086
951
1093
957
1087
952
1094
960
1060
987
1056
984
1061
988
1058
986
Centre of swing
106965
97310
107015
97400
107070
97450
107170
97600
Deflection due to
rider or mass...
96800
...
96425
96700
96550
Mass deflection ^
rider deflection
99161
...
•99664
99780
99690
[ 
i
r
i
m
^•
r
.
m
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
Scalereadings...
1097
916
1135
942
1119
919
1136
935
1058
1015
1037
999
1048
1017
1039
1006
1083
954
1098
965
1093
957
1099
963
1067
991
1061
986
1066
993
1063
989
Centre of swing
107340
97710
107480
97785
107580
97970
107625
97910
Deflection due to
rider or mass...
97000
97450
96325
97025
Mass deflections
rider deflection
100000
...
100815
100947
100362
(41)
(42)
(43)
m
(44)
i
(45)
(46)
(47)
m
(48)
Scalereadings . . .
1122
1048
1093
1065
917
1018
956
994
1141
1038
1101
1062
929
1011
962
993
1118
1053
1093
1068
921
1019
958
996
1141
1041
1103
1065
941
1009
966
993
Centre of swing
Deflection due to
rider or mass...
M;iss deflections
rider deflection
107600
97945
97025
99948
107695
98065
96925
•99704
107820
98140
97400
99538
107940
98265
96975
•99628
THE GBAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 123
Table III {continued).
i
r
i
m
i
r
*
(49)
(50)
(51)
(52)
(53)
(54)
(55)
Scalereadings . . .
1134
920
1143
932
1134
925
1140
1047
1022
1041
1016
1048
1021
1045
1100
958
1104
964
1102
962
1104
1067
998
1065
996
1068
997
1068
Centre of swing
107985
98265
108000
98385
108115
98420
108155
Deflection due to
rider or mass...
...
97275
96725
97150
Mass deflection h
rider deflection
...
•99563
...
99499
September 17, 1890. Mean of 25 determinations of M/R = A]
Attracted masses in lower position J
July 28 and September 17. Mean of 50 determinations of M/R = A, 9978658.
II. Attracted Masses in Upper Position. September 23, 1890, 7.52 to
9.30 p.m. Temperature: in Observing Eoom, 15°315°4; in Balance
Room, 15°05. Barometer, 74987502 millims. Weather, light S.W.
wind and clear after heavy showers. Scalereadings between about 1100
and 1300; 1000 omitted.
i
r
i
m
^•
r
^•
?/i
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Scalereadings . . .
307
113
329
235
281
112
326
233
248
210
235
257
261
208
232
256
285
151
293
243
273
149
290
241
263
186
257
251
265
185
256
249
Centre of swing
27100
17310
27095
24825
26835
17145
26825
24660
Deflection due to
rider or mass...
97875
21400
96850
21175
Mass deflection h
rider deflection
...
...
...
219797
219799
...
•218581
124 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Table III {continued).
i
r
i
m
i
r
i
m
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
Scalereadings . . .
279
110
331
232
277
110
324
230
260
207
228
255
258
205
229
254
272
148
290
239
271
147
288
239
264
183
253
248
262
182
252
247
Centre of swing
26730
17015
26680
24515
26580
16890
26555
24450
Deflection due to
rider or mass...
...
96900
21150
...
96775
...
20225
Mass deflection ^
rider deflection
•218395
...
218407
...
213769
209179
i
r
^
m
^•
r
^■
m
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
Scalereadings...
275
108
323
228
276
107
328
226
256
204
228
253
255
203
224
252
269
145
286
237
268
145
287
236
261
181
251
247
260
179
249
245
Centre of swing
26390
16740
26410
24315
26300
16665
263^5
24185
Deflection due to
rider or mass...
96600
20400
96475
20225
Mass deflections
rider deflection
...
•210274
211317
•210547
...
209652
i
r
i
m
i
r
i
m
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
Scalereadings...
274
106
320
232
271
100
317
222
254
199
224
245
252
197
221
247
265
143
283
237
262
138
281
232
258
176
246
241
255
175
243
241
Centre of swing
26090
16390
26040
23985
25825
16050
25790
23760
Deflection due to
rider or mass...
96750
19475
97575
19100
Mass deflection ^
rider deflection
205323
200437
197668
196730
THE GBAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 125
Table III (continued).
i
r
i
m
i
r
i
m
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
Scalereadings . . .
266
98
317
221
265
97
311
218
249
197
219
244
246
193
218
242
259
136
279
228
257
135
275
226
254
173
241
237
251
171
240
235
Centre of swing
25550
15910
25590
23420
25305
15700
25330
23210
Deflection due to
rider or mass...
...
96600
...
20275
96175
20150
Mass deflection ^
rider deflection
...
203804
210351
...
210164
...
209271
i
r
i
m
i
r
i
m
(41)
(42)
(43)
(44)
(45)
(46)
(47)
(48)
Scalereadings...
264
95
310
215
261
91
311
212
243
191
216
239
241
188
210
236
256
133
273
223
253
128
272
220
249
168
238
232
246
166
234
229
Centre of swing
25120
15490
25140
22910
24855
15110
24840
22610
Deflection due to
rider or mass...
96400
...
20875
...
97375
20675
Mass deflection ^
rider deflection
...
212785
...
215456
...
213350
...
213585
i
r
i
m
i
r
^•
m
(49)
(50)
(51)
(52)
(53)
(54)
(55)
(56)
Scalereadings . . .
257
90
306
211
256
88
303
208
237
186
208
234
236
184
209
232
250
127
269
218
248
125
265
216
243
162
232
227
242
160
231
225
Centre of swing
24515
14925
24580
22410
24425
14720
24395
22210
Deflection due to
rider or mass...
96225
20925
96900
20350
Mass deflections
rider deflection
216160
216699
•212977
...
•209956
126 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Table III (continued).
(57)
(58)
(59)
m
(60)
(61)
Scalereadings...
253
233
246
238
86
180
122
157
301
205
263
227
203
228
213
222
293
204
280*
245
Centre of swing
Deflection due to
rider or mass...
24095
14400
96950
24095
September 23, 1890. Mean of 27 determinations of M/R = a) .2112753
Attracted masses in upper position J
September 25, 1890, 7.108.43 p.m. Temperature: in Observing Koom,
15°15°2; in Balance Room, 15°. Barometer, 7608 millims., steady.
Weather cloudy, with westerly airs. Time of swing 21 seconds. 1000
omitted in scalereadings.
I
r
i
m
i
r
*■
m
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Scalereadings...
246
84
301
206
248
82
297
202
238
179
205
229
233
178
204
228
243
121
263
215
243
119
260
212
239
156
228
224
236
156
226
222
Centre of swing
24090
14290
24095
22040
23895
14160
23895
21810
Deflection due to
rider or mass...
...
98025
19550
97350
20850
Mass deflections
rider deflection
•200128
•207499
•213163
* This is a considerable rise, showing either a sudden disturbance or a displacement of the
apparatus; possibly the telescope was touched. The rise was maintained and therefore the
observations were discontinued.
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 127
Table III (continued).
i
r
i
m
*
r
i
m
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
Scalereadings...
248
83
300
204
252
84
303
207
233
176
203
228
233
182
206
232
243
119
261
214
245
122
265
217
236
155
226
224
239
158
228
226
Centre of swing
23895
14080
23920
•21950
24070
14460
24245
22260
Deflection due to
rider or mass...
...
98275
20450
...
96975
20825
Mass deflection ^
rider deflection
...
210125
...
209475
...
212813
...
•214718
i
r
.
m
*■
r
i
m
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
Scalereadings...
255
87
307
210
257
90
271
215
237
184
207
234
238
184
233
233
249
125
267
219
251
127
255
222
242
162
232
228
244
163
241
229
Centre of swing
24440
14755
24470
22465
24605
14875
24670
22615
Deflection due to
rider or mass...
97000
20725
97625
...
20725
Mass deflection:
rider deflection
...
214175
212974
...
212292
...
212564
i
r
•
m
i
r
^•
m
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
Scalereadings...
258
90
307
213
262
93
307
215
241
186
213
237
242
189
215
239
251
129
270
223
253
131
272
225
244
164
236
232
246
167
237
233
Centre of swing
24705
15045
24860
22830
24885
15300
25025
23005
Deflection due to
rider or mass...
97375
20425
...
96550
...
19750
Mass deflection^
rider deflection
211297
•210649
...
•208053
204425
128 ON A DETERMINATION OF THE MEAN DENSITY OF THE EARTH AND
Table III (continued).
i
r
i
m
i
r
*
m
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
Scalereadings...
261
243
93
189
312
214
215
241
263
245
96
192
312
217
217
243
253
132
273
225
257
135
275
227
247
167
237
235
250
168
240
237
Centre of swing
24935
15340
25080
23110
25240
15605
25305
23305
Deflection due to
rider or mass...
...
96675
20500
96675
...
20525
Mass deflection f
rider deflection
...
•208172
•212051
...
•212180
...
•212254
i
r
.
m
i
r
i
m
(41)
(42)
(43)
(44)
(45)
(46)
(47)
(48)
Scalereadings...
264
98
314
220
267
100
321
224
247
194
219
243
249
197
223
246
259
136
277
231
260
138
281
233
251
171
242
238
255
174
246
242
Centre of swing
25410
15785
25505
23520
25625
16035
25930
23805
Deflection due to
rider or mass...
...
96725
20450
...
97425
...
21250
Mass deflection ^
rider deflection
•211812
...
210662
...
•214011
•218650
i
r
i
m
i
r
i
(49)
(50)
(51)
(52)
(53)
(54)
(55)
Scalereadings...
271
102
321
224
271
102
314
252
200
221
247
251
198
226
264
139
282
233
264
140
280
1
256
176
245
242
256
174
247
' Centre of swing
25930
16220
25900
23840
258^95
16160
25950
Deflection due to
rider or mass...
96950
20575
...
97625
Mass deflections
rider deflection
...
215704
...
•211487
September 25, 1890. Mean of 25 determinations of MIR = a\ .01125332
Attracted masses in upper position J
' ^September 25^ } ^^^^ ^^ ^^ determinations of MjR = a, 2112647.
THE GRAVITATION CONSTANT BY MEANS OF THE COMMON BALANCE 129
Table III (continued).
Summa/ry of Set II.
July 28
September 17
Mean value of
September 23
25
Mean value of
A = 9973168
A = 9984148
A = 9978658
a  2112753
a = 2112533
a =2112647
Therefore Aa= 7866011.
Mean value, giving equal weights to Sets I and II,
Aa = 791295.
[The experimental data in these tables have been verified from the
original MS. Certain slips in calculations from the data occur in Table III,
but it was not considered advisable to correct these. Ed.1
p. c. w.
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130
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.

Diagram IX. Relation between temperature
and scaleieading. May 9 — 22.
4.
AN EXPERIMENT IN SEARCH OF A DIRECTIVE ACTION
OF ONE QUARTZ CRYSTAL ON ANOTHER.
By J. H. PoYNTiNG and P. L. Gray, B.Sc.
[Phil Trans. A, 192, 1899, pp. 245256.]
[Received September 27. Read November 17, 1898.]
Since so many of the physical properties of crystals differ along the
different axes, our ignorance of the nature and origin of gravitation allows
us to imagine that the gravitative field of crystals may also differ along those
axes. Dr. A. S. Mackenzie {Phys. Rev., vol. 2, 1895, p. 321) has described
an experiment in which he failed to find any such difierence. Using Boys's
form of the Cavendish apparatus, he showed that the attraction of calcspar
crystals on lead and on other calcspar crystals was independent of the
orientation of the crystalline axes, within the limits of experimental error —
about onehalf per cent, of the total attraction. He further showed that
the inversesquare law holds in the neighbourhood of a crystal, the attractions
at distances 3714 centims., 5565 centims., and 7421 centims. agreeing with.
the law" to onefifth per cent.
One of the authors of this paper had already pointed out (The Mean
Density of the Earth, 1894, p. 7) that if the attraction between two crystal
spheres were different for a given distance, according as their like axes were
parallel or crossed, such difference should show itself by a directive action
on one sphere in the field of the other. This directive action is suggested
by the growth of a crystal from solution, where the successive parts are laid
down in parallel arrangement — a fact which we might perhaps interpret on
the molecular hypothesis as showing that, within molecular range at least,
there is directive action.
The experiment now to be described is a modification of one indicated in
the work above referred to, carried out for two quartz spheres, and we may
say at once that we have certainly not succeeded in proving the existence of
a directive action of the kind sought for.
To bring out the principle of the method, let us suppose that the law of
the attraction between two spheres w^ith their hke axes parallel, as in Fig. 1 (a),
138
AN EXPERIMENT IN SEARCH OF A DIRECTIVE ACTION
is GMM'jr^, where M, M' are the masses, r the distance between the centres,
and G a constant for this arrangement. Let us further suppose that the law
of attraction when the axes are crossed, as in Fig. 1 (6), is G'MM'/r^, where
G' is a constant for this arrangement, and different from G.
Let us start with the spheres r apart, as
in Fig. 1 {a). The work done in removing
M' to an infinite distance, in a Hne perpen
dicular to the parallel axes, is GMM'jr.
Now turn M' through 90° to cross the axes,
and bring it back to the original position, but
with the axes crossed.
The force will do work G'MM'jr. Then
turn M' through 90° into its original orienta ^^S ^■
tion. Assuming that the forces are conserva
tive, the total work vanishes, so that there must be a couple acting during
the last rotation, which does work equal to the difference between the works
done on withdrawal and approach.
If we take the average value of the couple as L, then
Our suppositions as to the law of force are doubtless arbitrary, but they
serve to show the probability of the existence of a directive couple accom
panying any axial difference in the gravitative field.
In the absence of any distinction between the ends of an axis we may
assume that the couple is 'quadrantal,' that is, that it goes through its
range of values with the rotation of the sphere through 180° and vanishes
in every quadrant, and we shall suppose that it is zero when the crystals are
in the positions shown in Fig. 1 (a), and Fig. 1 (h).
Taking the couple as a sinefunction of amplitude F, we have
F sin 26 cW
whence
F=^{GG'
MM'
But it is conceivable that the two ends of an axis are different, having
polarity of the magnetic type. The couple would then be 'semicircular,'
going through its range of values once and vanishing twice in the revolution.
We shall suppose that the couple is zero when the axes are parallel. We
should now have G and G' constants for the axes parallel, the one when Hke
OF ONE QUARTZ CRYSTAL ON ANOTHER 139
ends are in the same direction, the other when they are in opposite directions,
and we have
MM'
r
But if F is the amphtude of the couple
ttL = [" J sin QdB = 2F,
Jo
MM'
and 2F=(GG')^^.
r
To seek for the directive action we have made use of the principle of
forced oscillations, thereby obtaining to some extent a cumulative effect,
and at the same time largely eliminating the errors due to accidental dis
turbances.
Briefly the method was as follows : A small quartz sphere, about 09
centim. in diameter, was carried in a frame to which a light mirror was
attached, and suspended by a quartz fibre inside a brass case, the position
being determined by the reflection of a scale in the usual way. The complete
time of torsional vibration was about 120 seconds.
Outside the case was a larger quartz sphere, about 66 centims. in diameter,
its centre being level with that of the suspended sphere, and 59 centims.
from it. The larger sphere could be rotated about a vertical axis through
its centre at any desired rate. The crystalline axes of both were horizontal,
that of the smaller sphere being perpendicular to the hne joining the centres.
To test for the quadrantal couple, the larger sphere was rotated once in
230 seconds — a period nearly double that of the smaller sphere. To test for
the semicircular couple, the larger sphere was rotated once in 115 seconds, or
nearly the period of the smaller sphere.
Assuming that a couple exists, a continuous rotation of the larger sphere
would set up a forced oscillation in the smaller sphere of the same period as
the couple, and since the damping was very considerable, this forced oscillation
would soon rise to approximately its full value. Meanwhile, any natural
vibrations of the suspended system would be rapidly damped out. Though
continually renewed by disturbances due to convectioncurrents and tremors,
they would be irregularly distributed, and there was no reason to suspect
that their maximum amplitude would recur at any particular phase of the
period of the apphed couple. To secure the distribution of successive maxima
of natural vibrations of the smaller sphere over all phases of the forced period,
the latter was made sensibly different from the natural period in the ratio
23 : 24 ; and though the cumulative effect of the forced oscillations was
reduced by the largeness of this difference, we did not think it advisable
to make the periods more nearly coincident, lest the distribution of the
disturbances, which were sometimes large, should not be sufficient. This
140 AN EXPERIMENT IN SEARCH OE A DIRECTIVE ACTION
conclusion was arrived at from the results of preliminary experiments with
more nearly equal periods.
During each complete period of the supposed appHed couple, the position
of the smaller sphere was read ten times at equidistant intervals of time,
and the scalereadings were entered in ten parallel columns, one horizontal
line for each period. The observations were continued usually for 70 or 80
periods. Adding up the columns and dividing by the number of periods,
any forced oscillation would be indicated by a periodicity in the quotients.
The periodicities found were too irregular to be taken as evidence of the
existence of a couple.
Bescriftion of the Afj)aratus.
The quartz spheres were placed in a cellar at Mason College, Birmingham,
below the room in which the observing telescope and rotatingapparatus
were fixed.
The smaller sphere, 09 centim. diameter and weighing 1004 grams, was
held in an aluminium wire cage, and was suspended by a long, fine quartz
fibre in a brass case from a torsionhead at the top of the case.
A light plane mirror was fixed to the cage, and opposite this mirror was
a glass window in the case; in front of the window was a plane mirror at
45°, by means of which the light from the scale was reflected into the case
and back again to the telescope, as shown in Fig. 2.
The case was surroimded by a doublesided wooden box, lined within and
without with tinfoil, and with cottonwool between its inner and outer walls.
The box was supported on indiarubber blocks to lessen tremors.
The larger sphere, 66 centims. diameter and weighing 3999 grams, was
held at the lower end of a vertical brass tube which terminated in a very
carefully turned shallow brass bell, in which the sphere was held by tapes.
The tube passed upwards through the top of the wooden casing without
contact, a kind of air stuffingbox indicated in the figure serving to prevent
currents through the hole. The tube came into the room above, and was
there connected with a train of wheels, driven by an electromotor, the
rotation of the motor being geared down from 1000 to 1. The observing
telescope was fixed to a heavy stone slab resting on indiarubber blocks,
standing on a brickpillar, which was built on the brick arches forming the
cellarroof. A diagonal scale (of halfmilUmetre graduations, divided into
tenths by the diagonal ruhng) was clamped to the telescopetube and
illuminated by an incandescent lamp, aided by a concave mirror. A tenth
of a division could be read with certainty, and as the distance from scale
to mirror was 358 centims., the position of the suspended sphere could be
determined within a Httle more than one second of arc.
The steady rotation of the larger sphere was maintained by a regulator,
OF ONE QUARTZ CRYSTAL ON ANOTHER
141
for which we are indebted to Mr. R. H. Housman. It consisted of two parts :
(1) the governor proper, which automatically maintained approximate
steadiness, and (2) a fine handadjustment, by which the motion could be
accelerated or retarded when it got 'out of time.'
One lead to the motor went through two mercurycups, and the circuit
was completed by a fork of platinum wire dipping into the cups. This wire
7b Accumf
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rm it,n"inr
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Fig. 2. Diagrammatic sketch of the apparatus.
was fastened to one end of a wooden lever, the other end of which was attached
to a sliding collar on the axle of the motor. To this collar were fastened the
upper ends of the loaded springs of the governor, as shown in the figure. If
the speed increased, the loads flying out pulled the collar down and so raised
the wire out of the mercurycups, and broke the circuit. As the speed
diminished, the wire again dipped into the mercury and reestabhshed the
current. To diminish sparking the mercury was covered with alcohol, and
the two cups were permanently connected by a high resistance shunt.
142 AN EXPERIMENT IN SEARCH OF A DIRECTIVE ACTION
The fine handadjustment consisted of a small wooden plunger working
in a tube connected with one of the mercurycups ; by means of a screw the
plunger could be raised or lowered, and the level of the mercury in the cup
varied accordingly.
If the revolving sphere was found to be gaining or losing, it was quite
easy to bring it 'up to time' again by working the screw of the plunger.
The last of the train of drivingwheels was fixed on the tube supporting
the larger sphere ; its rim was divided into equal parts by numbered marks,
the use of which will be explained directly. There were 20 numbered marks,
at intervals of 18° ; of these only 10 alternate ones were used for the quicker
rotation, while the whole 20 were used for the slower speed.
The Observations.
Two observers were required, one at the telescope to note the position of
the smaller sphere, the other to regulate the speed of rotation of the larger
sphere, and to notify when readings were to be taken by the first observer.
The motion having been started, and brought to about the right speed, a
timetable was rapidly prepared, showing the times, on the chronometer used,
at which each of the numbered marks above mentioned should pass a fixed
mark throughout the whole set of observations for one occasion. A signal
was given at each passage of a mark past the fixed point, the observer at the
telescope putting down the simultaneous scalereading in a manner which
will be understood from Table I, which may serve as a typical record. It
does not appear to be necessary to give the full details in other cases. If
the motion did not keep to the timetable, it was easily corrected by the
handadjustment already described.
Every reading in the same column is taken at the same phase in the
rotation of the larger sphere, and therefore the mean readings of the columns
should preserve any periodicity in the motion of the smaller sphere equal to
that of the larger sphere, and more or less ehminate all others. These mean
readings are given at the foot of Table I, and appear to indicate a sKght
periodic vibration, but this might be due to a want of symmetry in the larger
sphere and its attachments about its axis of rotation, since the system
supporting the smaller sphere and mirror was necessarily not symmetrical.
The observations for each couple were on this account divided into two sets :
for the semicircular couple the larger sphere was, in the second set, turned
through 180° about a vertical axis from its position in the first set ; for the
quadrantal couple the rotation was 90°. For the final results the means of
the results of the two sets were taken, in each case after the second set had
been advanced by an amount corresponding to the change of position of the
sphere.
OF ONE QUARTZ CRYSTAL ON ANOTHER 143
Table II contains all the mean results obtained in the same way as the
figures at the foot of Table I, the greatest range being given in the last column
as an indication of the magnitude of the disturbances.
In Table III are given the means for each azimuth of the larger sphere in
its support, the B and D series being advanced as mentioned above.
In combining the results it appeared useless to attempt to weight them
according to the number of periods taken, since no accurate conclusion could
be expected. It will be seen that in each case there is an outstanding
periodicity, but the amplitude is less when the disturbances (as indicated
by the greatest range during a period) are less, and it diminishes when the
results are combined so as to lessen the effect of want of symmetry.
In the 'quadrantal' observations (Series C, D), where the effect of want
of symmetry of the apparatus should almost be eliminated, since it is
approximately semicircular, the mean range is much smaller than in Series
A and B.
For these reasons we do not think that our observations can be taken as
indicating the existence of a couple of the kind sought, but only as giving
a superior limit to its value, should it exist.
We now proceed to the Calculation of the Superior Limit of the Couple.
Equation of Motion of the Smaller Sphere.
Let / be the moment of inertia of sphere and cage.
„ jjL „ torsioncouple per radian.
,, A ,, damping couple per unit angular velocity.
,, F cos 'pt be the supposed couple due to the larger sphere, having
period Stt/^.
Then Id + XO + yi^ = F cos ft.
Putting K^XjI; n^ = ^II', E = F/I
we have . S + k6 + n^d = E cos ft (1)
The solution of this is
d = — ^^ cos (pt €) + Ae^'^ cos {Vin^  ^k^) t  a}, (2)
where tan e = ij^ — ^ and A, a are constants.
n^ — f^
The first term in the value of 6 in (2) gives the forced, and the second
term the natural vibrations, the period of the latter being
= T, say.
Vin^  \k^)
144 AN EXPERIMENT IN SEARCH OF A DIRECTIVE ACTION
The value of T was always very near to 120 sees., and the mean of various
determinations during the observations gave
r=/v^i^, = 1208 sees (3)
Value of K. When there are only natural vibrations
any complete swing ^ i^ ly
next complete swing
The value of this ratio was usually near 14. The mean of a number of
determinations taken at various times was 13953. Putting
g3o2. _ 13953,
,veget K' = 0011033.
Value of n. Substituting for k in the value of T in (3) we get
^2  00027359,
and n = 0052306.
Value of e. The forced period 27r/p was always 115 sees., whence
tan 6 =,^^ = 2420,
and e ^ 67° 33',
sin e = 09242.
From equation (1) it will be seen that the steady deflection due to F is
n
while from (2) the amphtude of the forced oscillations is
E sin c n^ sin e E
or . ^ .
Using the values found for uk and e we have
or the forced oscillations give a cumulative effect, about four times the steady
deflection due to the couple at its maximum value.
Value of Moment of Inertia, I. This was found by vibrating the cage
hung by a short quartz fibre, (1) when empty, (2) when containing the sphere,
the times of vibration being respectively 838 sees, and 1122 sees. The
sphere weighs 1004 grams, and its radius is 045 centim., so that its moment
of inertia jMr^  08132.
From this, and the times of vibration, we get
7 = 01821.
Value of F. The vibrations were observed in scaledivisions, each
005 centim., the distance between mirror and scale being 358 centims. If
OF ONE QUARTZ CRYSTAL ON ANOTHER 145
N is the number of scale divisions in the ampHtude of vibration, i.e., in half
the range, we have from (2)
Eainc _ 6N
pK " 2 X 35800'
whence F = EI = 08293i\^ x lO^,
using the values already found for e, k, I.
Taking the Hmiting values of the amplitudes as half the mean ranges
given in Table III, the vibration due to the quadrantal couple has amplitude
not greater than 0033 div., and that due to the semicircular couple, amplitude
not greater than 0095 div. Whence
F (quadrantal) is not greater than 2737 x 10^°,
and F (semicircular) is not greater than 7878 x 10"^°.
Perhaps some idea of these values may be obtained by noticing that the
times of vibration of the small sphere under couple F per radian would be
respectively 32 hours and 25 hours. But it is probably best to interpret the
value in terms of the assumptions we made as to the force in the introduction.
We found for the quadrantal couple
F = (G G') MM'jf,
GG' GMM'
'^ G ' r '
where M, M' are the masses of the spheres, r the distance between their centres,
G, G' the parallel and crossed gravitationconstants.
Now M, the mass of the larger sphere, is 3999, say 400 grams,
M' „ ,, smaller „ 1004 grams,
r is 59 centims.,
G and G' are exceedingly near 666 x 10"^,
GG' Ft
whence ^ = GMM'^^^^'
On the assumed law of force this imphes that the attractions between the
two spheres, with distance 59 centims. between their centres, do not differ in
the parallel and crossed positions by as much as yeioo ^^ ^^^ whole attraction.
We may compare this result with Rudberg's values of the refractive indices
of quartz for the mean D line
^,^„ 155328  154418 _ ,
^/x— r54418 TT^^bout.
For the semicircular couple
^^ GG' GMM'
2F = —^ ,
G r
whence — __ = 3^j^^.
p. c. w. ^^
146
AN EXPERIMENT IN SEARCH OF A DIRECTIVE ACTION
On the assumed law of force, this imphes that the attractions between
the two spheres, with distance 59 centims. between their centres, with their
axes parallel and respectively in Hke and unhke directions, do not differ by
as much as ^^^^ of the whole attraction.
This hmit is large, undoubtedly owing to the want of axial symmetry in
the apparatus which produced a semicircular couple as already pointed out.
This couple was large, and though we attempted to ehminate it by the two
sets of observations with the different azimuths of the larger sphere, in all
probability w^e failed.
Table I. Showing Scale Readings in Tenths of a Division at Phases at
Heads of Columns. Time of Revolution of Larger Sfhere 115 sees.
1
2
3
4
5
6
7
8
9
55
61
61
64
61
42
25
26
31
40
50
55
60
53
52
45
44
40
45
50
54
51
49
49
48
49
52
57
52
54
57
52
40
33
28
25
30
44
57
66
70
64
52
40
38
36
39
46
55
60
63
61
52
44
44
45
44
43
49
50
52
48
42
30
32
37
45
50
62
71
69
62
52
42
39
38
44
55
58
65
65
66
61
51
45
45
41
38
40
49
56
61
62
60
56
50
48
42
39
37
40
42
58
69
69
68
58
48
41
38
38
42
48
54
60
57
55
50
43
41
41
42
47
49
55
57
63
60
58
49
46
47
46
44
51
52
50
54
48
45
44
40
36
40
50
60
67
67
62
54
44
33
35
35
38
50
57
62
68
62
52
45
36
36
39
44
51
56
59
53
47
48
53
51
50
49
50
49
52
50
50
51
53
52
54
55
48
47
44
41
44
52
55
58
60
56
49
41
41
42
43
47
50
55
60
60
60
56
58
47
43
47
49
50
50
50
54
54
54
48
50
51
49
52
52
45
42
43
48
49
bQ
56
52
52
52
57
56
51
46
42
42
43
49
51
55
55
55
52
49
57
50
50
50
44
43
50
50
50
43
43
46
50
58
54
55
50
49
49
49
48
50
51
54
53
5.6
56
57
58
56
56
51
43
40
38
41
51
60
60
60
58
52
48
48
48
52
57
58
60
57
47
41
41
51
62
63
59
53
46
40
40
40
43
49
51
61
60
60
56
51
48
42
42
43
51
59
63
62
61
55
52
51
50
51
51
52
56
58
58
53
45
40
41
49
60
70
70
60
52
48
48
50
50
54
55
53
51
50
50
1 47
50
50
52
53
53
50
48
48
46
48
50
51
51
50
50
52
52
49
46
44
44
49
50
55
59
57
58
51
49
46
43
44
51
59
68
64
56
50
42
40
49
57
68
71
70
59
50
OF ONE QUARTZ CRYSTAL ON ANOTHER
147
Table I (continued).
Mean of 80 in
divisions . . .
1
2
3
4
5
6
7
8
9
47
41
43
56
65
66
56
45
39
35
33
40
48
59
68
70
64
51
43
42
48
51
60
64
70
67
56
42
39
38
40
47
52
65
60
61
59
51
51
50
48
47
50
54
53
60
52
50
41
39
40
44
51
58
66
70
71
63
50
38
35
38
41
43
50
56
70
75
70
59
50
46
45
51
61
70
71
70
62
52
41
40
40
40
47
54
60
71
71
68
60
50
48
45
39
39
42
49
51
60
64
61
50
46
47
49
52
60
72
75
70
62
57
32
23
21
30
42
57
77
84
79
63
51
42
33
34
38
49
60
66
64
57
51
49
44
47
49
52
52
55
55
52
58
59
56
57
51
42
40
43
47
55
61
66
64
60
52
50
45
41
45
49
59
67
67
56
50
49
43
38
45
48
53
55
56
57
55
54
56
53
49
42
42
51
61
69
70
65
54
45
41
40
47
51
56
61
59
55
49
48
52
60
60
60
58
50
46
44
43
45
51
53
60
63
67
62
60
55
48
44
46
49
50
52
54
53
50
50
52
60
62
63
61
51
41
39
38
42
50
55
59
54
51
48
47
42
47
48
55
58
61
62
60
59
54
52
52
50
50
58
54
55
55
58
56
56
50
51
51
56
58
51
52
48
48
54
55
50
51
52
51
51
50
45
44
42
46
51
55
56
53
56
59
59
60
58
59
59
54
49
46
46
49
50
52
58
56
57
53
51
50
50
46
49
51
58
66
67
69
65
62
51
46
39
39
39
45
51
56
62
61
53
48
40
38
47
62
67
63
55
52
57
56
56
53
49
42
38
41
51
60
65
71
73
72
60
52
50
40
42
49
52
62
71
73
73
65
59
44
39
38
40
43
51
51
59
61
61
50
49
41
49
51
52
58
58
52
52
50
53
56
57
51
50
49
49
49
51
52
49
52
53
52
...
...
...
...
5175
5163
5143
min.
5186
5246
5294
5355
max.
5284
5300
5216
Mean range 5355  5143  0212 division.
Greatest range in one period 75 — 35 =^ 40 divisions.
10—2
148 DIRECTIVE ACTION OF ONE QUARTZ CRYSTAL ON ANOTHER
Table II.
1
Series
Azimuth of large
sphere
Mean readings at phases (whole numbers omitted)
Greatest range in
a period in
Scaledivisions
1
2
3 \ 4
5
6
7
8
9
A 1
o
sees.
115
80
•175
•163
•143
•186 ^246
•294
•355
•284
•300
•216
•212
40
A2
115
80
•653
•558
•566
•653 I813
•950
1030
1008
•929
•769
•472
31
Bl
180
115
80
•485
•590
•624
•648 556
•464
•379
328
•284
•364
•364
30
B2
180
115
70
•423
■503
•650
•836 941
•961
•843
•714
•540
•464
•538
75
B3
180
115
54
•650
•632
•619
•648 ^656
•680
•717
•739
785
•739
•166
2^7
CI
°0
sees.
230
72
•708
•731
•708
•739 717
•711
•676
•678
642
•688
097
1^3
(72
230
80
•370
•400
•358
•326
•271
•214
•173
•158
•253
•310
242
34
C3
230
80
•616
•654
•686
•673
663
•627
•571
•560
•566
584
126
2^0
D\
90
230
50
b024
1042
1031
b004
988
•920
•926
•954
994
1010
•122
' 22
D2
90
230
70
•031
•090
150
•210
220
•230
•223
•176
•126
•096
•199
31
Table III.
Series
Mean readings at phases
Mean
1
2
3 4 ' 5 6 7
8
9
•493
range
•338
A
•414
361
355
420 530 622 ' 693 646
•615
5 (advanced 180°)
•702
•646 ^594 536 522 519
•575 ^631
•711
•663
•718
•199
Means of A and B
•558
503 474 478 526 570
•634 ^638
•605 ^189
C
•565
1 '
•595 584 579 550 517
•473 ^465
•487
527
•130
D (advanced 90°)
•575
575 565 560 553 528
•566 ^592
•607
•604
079
Means of C and D
570
1 i
585 575 570 552 523
•520 ^529
•547
•566
065 i
1
AN EXPERIMENT WITH THE BALANCE TO FIND IF CHANGE OF
TEMPERATURE HAS ANY EFFECT UPON WEIGHT.
By J. H. PoYNTiNG and Percy Phillips, M.Sc.
[Roy. Soc. Proc. A, 76, 1905, pp. 445457.]
[Received July 12, 1905.]
In all the experiments hitherto made to determine the gravitative attrac
tion between two masses, the temperature has not varied more than a few
degrees, and there are no results which would enable us to detect with
certainty any dependence of attraction upon temperature even if such
dependence exists. It is true, as Professor Hicks has pointed out*, that
Baily's results for the Mean Density of the Earth, if arranged in the order
of the temperature of the apparatus when they were obtained, show a fall in
value as the temperature rises. But this is almost certainly some secondary
effect, due to errors in the measurements of the apparatus, or to the seasons
at which different attracted masses were usedf .
The ideal experiment to find if temperature has an effect on gravitation
would consist in one determination of the gravitative attraction between two
masses at, say 15° C, and another determination at, say, the temperature of
boiling liquid air. But the difficulties of exact determination at ordinary
temperatures are not yet overcome, and at any very high or very low tempera
tures, they would be so much increased that the research seems at present
hopeless.
The question can, however, be attacked in a somewhat less direct method
by examining whether the weight of a body— the gravitative attraction of
the earth upon it — varies when the temperature of the body varies. The
various parts of one of the attracting masses — the Earth — remain, each
part, at the same temperature throughout, and this is, no doubt, a weak
ness of the method. For it is perhaps conceivable that in the expression
for the attraction a temperature factor might exist of some such form as
1 + AC {Mt + mt')l(M + m), where M and m are the two masses, and t and
t' are their temperatures. If m/M is negligible, this reduces to 1 + kI, and
* Proc. Camb. Phil. Soc, vol. 5, p. 156.
t Poynting, Mean Density of the Earth, p. 56.
150 AN EXPERIMENT WITH THE BALANCE TO FIND IF
is independent of the temperature t' of the smaller mass. But it seems more
hkely that each mass would have a separate temperature factor. If such
a factor exists, and if its variation is appreciable, then we ought to be able
to detect a change of weight with change of temperature.
Observations on pendulums suffice to show that at the most any such
effect must be small. The nearly constant period of vibration with the
nearly constant length of a compensated or an 'invar' pendulum shows
constancy of weight of the bob to a considerable degree of exactness. Again,
the agreement of weightmethods and volumemethods of measuring the
expansion of liquids with rise of temperature shows, though less conclusively,
that there is no great variation.
It appeared to us that it would be possible to go much further in testing
constancy of weight by a direct weighing experiment, in which the weight
on one side of a balance should be subjected to great changes of temperature
while the counterpoise should remain at a uniform temperature. We give
an account in this paper of a series of experiments carried out on the following
principle. A brass cylinder weighing 266 grammes was hung by a wire from
one arm of a balance so as to be near the bottom of a tube depending from
the floor of the balancecase, the tube being closed at the bottom and opening
at the top into the case, the wire passing down through the opening. The
brass cylinder was counterpoised by an equal cylinder hung by a short wire
from the other arm inside the case. To this short wire was attached a finely
divided scale on which the swings of the balance could be read by a microscope
looking through a window in the case. The balance was released and left
free to swing. Then the case was exhausted till the pressure was not more
than a small fraction of a milHmetre of mercury. Steam was passed round
the lower part of the tube where the weight hung, and after a time the weight
was allowed to cool again. In other experiments the lower part of the tube
was cooled by liquid air and again brought up to the temperature of the room.
While the changes of temperature were in progress there were considerable
apparent variations in weight. But ultimately, when the temperature became
steady, the weight, too, became steady. At 100° C. it was slightly less than at
the temperature of the room. This difference was partly due to a rise in the
temperature of the case, such a rise being always accompanied by an apparent
diminution of the weight in the tube, whether steam was apphed or the
balance was merely left to follow the temperature of the room. Probably this
effect was due to some change in the balancebeam. But the difference was
partly due to convectioncurrents, or at any rate to the residual air in the
case, for it varied with the disposition of diaphragms in the tube. There
were no doubt convectioncurrents, as there was always a tendency for the
case to rise in temperature when steam was apphed, and this could hardly
be accounted for by conduction or radiation, under the conditions of the
CHANGE OF TEMPERATURE HAS ANY EFFECT UPON WEIGHT 151
apparatus. As effects due to residual air should depend upon surface and
not upon volume, similar experiments were made with hollow weights, each
about 58 grammes, and of the same size and form as the solid weights. There
was again an apparent diminution in weight when steam was applied. Any
true diminution due to change of temperature should be shown by a difference
in the diminution with the solid and with the hollow weights, the surface
effects being eliminated, and this diminution should be that of 266 — 58 = 208
grammes.
The net result of all the experiments was that there was not a greater
change in 208 grammes between 15° C. and 100° C, than 0003 milligramme.
But an inspection of the detailed account given later shows that this result
is probably accidentally small — within the limits of experimental error. It
would imply that there is not a change greater than 1 in 6 x 10^ per 1° C.
But the experiments hardly justify us in saying more than that there is not
a change greater than 1 in 10^ per 1°.
When liquid air was used, aircurrents were absent, and the temperature
variations of the case were much less. The net result of these experiments
was that there is not a change of weight in 208 grammes between 16° C. and
— 186° C. greater than 0002 milHgramme. This would imply that there is
not a change greater than 1 in 13 x 10^^ per 1° change of temperature. We
may probably assert that the change is not greater than 1 in 10^^ per 1° C.
We now proceed to a detailed account of the apparatus and of the mode
of using it.
The Balance,
The balance has a 6inch beam and was specially constructed for the
experiment by Mr. Oertling. The general arrangement will be seen from
Fig. 1. The baseplate is of gunmetal, as are also two sides and the top of the
case. The front and back of the case are of thick plateglass fixed to the
metal by marine glue. In the experiments the baseplate was supported on
levellingscrews on a slate slab, and between it and the slab was a gaspipe
with pinhole burners so that it could be warmed. When the case was to be
fixed in position the jets were lighted and seahngwax was smeared on to the
area of contact of plate and case. When the wax was quite liquid the case
was put down on the plate and the gas was turned off. When the metal was
cool the joint was perfectly airtight.
The tube T in which the weight W hung was of brass, 41 cm. internal
diameter and 625 cm. long. It consisted of three parts. The topmost was
brazed to the baseplate and the two lower parts were attached to it and to
each other by flanged joints ff. Between the flanges was placed a circular
lead washer of diamondshaped section. When the flanges were pressed
together by bolts the joint was quite airtight. Round the middle section of
152
AN EXPERIMENT WITH THE BALANCE TO FIND IF
the tube was a waterjacket wj through which water flowed while an experi
ment was in progress, and round the lowest section was a steamjacket sj
through which either water or steam could be passed. This jacket could be
removed and could be replaced by a vacuum vessel 30 cm. long containing
liquid air.
Fig. 1.
W, weight of which the temperature is to be raised, Tf counterpoise.
T, tube in which it hangs, with a number of diaphragms with inch holes.
sj, steamjacket, replaced by hquidair jacket.
//, flanged joint with lead washer.
wj, waterjacket.
p, pipe to the exhausting pump.
sc, scale read by a microscope not shown.
rr, riderrod passing through stuffingboxes, sb, enlarged in Fig. 2.
GG, gasburners to heat the baseplate before seahng up.
The weights W, W were turned from the same gunmetal bar. The length
of each was 445 cm., the diameter 3 cm., and the solid weights were each
26617 grammes, while the hollow ones were 5786 grammes. They were hung
directly from the endplates of the balance by platinum wires, and any
residual mequality was compensated by moving a centigrammerider along
CHANGE OF TEMPERATURE HAS ANY EFFECT UPON WEIGHT
153
the beam by the riderrod rr. This rod passed through stuffingboxes sh,
designed for us by Mr. G. 0. Harrison, the mechanical assistant in the labora
tory, to whom we are much indebted for this and many other valuable
suggestions, and for the careful construction of all the apparatus except the
balance. These stuffingboxes were perfectly airtight when screwed up and
the rod could still be rotated without any leakage. But to draw it in or out
it was necessary to loosen the screws slightly, and in one case when this was
done some leakage occurred. As the construction appears to give an efficient
mode of moving apparatus inside a vacuum from without, we give in Fig. 2
a section of a stuffingbox.
Fig. 2.
rr, riderrod.
CO, side of case.
ivw, two or three circular washers punched out of soft leather and soaked in oil.
p, plunger driven in by screws, ss.
oh, oilhole through which valvoline, a thick lubricating oil, was inserted.
The position of the balancebeam was read by a microscope viewing a glass
scale sc, Fig. 1, interposed in the suspension of W . The scale was divided to
01 mm. and numbered in millimetres. The objective of the microscope was
placed inside the case and the eyepiece with crosshairs was fixed outside it.
The axis of the microscope was horizontal and a lamp at the back illuminated
the scale. The case was surrounded by felt and a tin cover was placed over
the whole, small windows through the felt then allowing the scale to be seen.
A thermometer placed between the felt and the case was taken to give the
temperature of the case.
A brass pipe f from the floor of the case led to the pumping apparatus.
This pipe was connected to a branched glass tube, one branch going to a
Fleuss pump and the other to a 4fall Sprengel, made continuous in its action
154 AN EXPERIMENT WITH THE BALANCE TO FIND IF
by a steel pump which was worked by a motor, and which raised the mercury
again from the cistern at the base to the reservoir at the top. When the case
was to be exhausted the Fleuss pump was first used and then sealed off and
the exhaustion was carried on by the Sprengel. The degree of exhaustion
was estimated by sending a discharge through a vacuumbulb 10 cm. diameter
connected with the tube to the pump, and usually the pumping was continued
till the negative dark space was of the order of 4 cm. As a rule the vacuum
held without serious change for days or even for weeks.
Mode of Experiment.
A large number of preliminary experiments was made with a pair of brass
weights each about 187 grammes. These were only useful in bringing to the
front the difficulties in obtaining good results and in suggesting means for
overcoming them. We shall only record the final results with the 266
grammes and 58 grammes weights.
The weights and the lower section of the tube were first cleaned by boiling
in caustic potash solution and washing in distilled water. They were then
suspended, being handled with gloves only, and the lowest section of the tube
was screwed on.
Stea?n Heating.
The jacket sj (Fig. 1) was fixed on the lower section of the tube and the
balance was set free to vibrate, being left free during a whole series of
experiments. The case was then sealed on and the value of a scaledivision
was determined by the rider. Any change in the value during a series could
be determined from the change in period of the swing. The time of swing in
different series ranged from 24 to 42 seconds. After the stuffingboxes were
tightened the case was exhausted till the pressure was estimated to be not
more than j^jj mm. of mercury. The weight of air displaced by a weight was
then of the order 0001 milHgramme and the change in this with change of
temperature was quite neghgible.
Cold water was passed through the waterjacket tvj, and sometimes, while
steam was being got up in a boiler at some distance well screened from the
balance, through sj also. The centre of swing and the temperature of the case
were observed, and before any heating occurred the balance was usually quite
steady. Steam was then blown through sj, water still flowing through wj.
After considerable changes, which will be described later, the centre of swing
in the course of five or six hours settled down to a steady march which
appeared to correspond to change in temperature of the case. KSometimes
steam was turned off after eight or nine hours, but in some cases it was kept
on for 24 and 48 hours and even longer. Then it was turned off and the
CHANGE OF TEMPERATURE HAS ANY EFFECT UPON WEIGHT 155
jacket was allowed to cool. The centre of swing was observed when steady,
several hours later or next day.
The results in the first few heatings and cooHngs, after an exhaustion of the
case, were rejected, as there was evidence that the earlier heatings drove gas
from the weight. Only after successive heatings gave fairly consistent values
were these taken into account.
One effect of the steamheating was always to raise the temperature of the
case, probably through convection of the residual air. A rise of temperature
in the case was always accompanied by a lowering of the scalereading,
corresponding to a diminution in weight. The effect was somewhat irregular,
but an average value of the lowering per 1° rise was determined by observing
the centre of swing of the balance at intervals through several days, when
the balance was left to follow the varying temperature of the room and no
steam was flowing. The value thus obtained was used to correct all the
readings to 15° C.
As an example of the method pursued, we give in Table I the series of
readings used to obtain the temperaturecorrection for the hollow weights.
Table I. Change of Centre of Swing with Change of Temperature of Case.
Date
Time
Centre of swing in
Temperature
millimetres of scale
of case
6.12.04
1.0 p.m.
1434
1475
,,
3.0 „
14355
1455
,,
5.40 „
1435
1465
7.12.04
12.35 „
14ai
1270
jj
5.5 „
14465
1360
8.12.04
11.5 a.m.
1443
1470
,,
12.55 p.m.
1420
1495
9.12.04
■ 9.51 a.m.
14555
1280
>>
2.27 p.m.
14525
133
10.12.04
9.37 a.m.
1446
150
The temperaturecorrection deduced from these numbers by the method of
least squares is a decrease of 013 division per 1° C. rise, and as the sensibility
was 1 division for 0248 milligramme, there was an apparent decrease of
weight of 0032 miUigramme per 1° rise.
Two similar series with the soHd weights gave a decrease of 0044 division
per 1° C. rise, and as the sensibility was now 1 division per 0803 milligramme,
there was an apparent decrease of weight of 0035 milhgramme per 1° rise.
Another series with the solid weights when steam was passing all the time
for several days, gave a decrease of 0052 division per 1° rise, but as the
156
AN EXPERIMENT WITH THE BALANCE TO FIND IF
values were more irregular, the series giving 0044 division were used. This
series with steam sufficed to show that very nearly the same temperature
correction apphed when the weight was hot as when it was cold.
The irregularity of the observations is only to be expected when it is
remembered that the balance was subjected to some considerable vibration
at times through machinery running in the same building, and that the
observations extended over several days. Indeed it is remarkable that there
was not more irregularity, and the fair consistency of the observations illus
trates once more the marvellous accuracy of a wellmade balance.
The following Table II will serve as an example of a complete experiment
in which one of the hollow weights was cold initially, was then surrounded
with steam for 24 hours, and was then allowed to get cold again. The obser
vations recorded are at about hourly intervals, but intermediate ones, not
used, were frequently taken to be sure that there were no sudden changes.
Table II. Experiment with Hollow Weight raised to 100° C. and then
cooled, 1 mm. = 0248 milligramme.
Correction for temperature of case — 013 division per 1°.
Condition
Centre of
Tempera
Centre of
Date
Time
of
weight
swing,
1 = 1 mm.
ture
of case
corrected
to 15° C.
Remarks
17.11.04
9.25 a.m.
Cold
14905
14°75
14872
Steam put on just after 9.25
,,
4.0 p.m.
Hot
1440
15 0
14400
and kept on till 10 a.m.
,,
5.20 „
,,
1439
15 0
14390
next day
>?
6.12 „
,,
14375
15 0
14375
^,
7.8 „
?j
14365
15 05
14372
,,
7.56 „
1436
15 1
14373
18.11.04
9.5 a.m.
,,
1430
15 05
14307
„
9.55 „
.,
14295
15 1
14308
Steam turned off just after
,,
5.36 p.m.
Cold
1455
16 00
14680
9.55
,,
6.39 „
"
1456
1595
14684
,,
7.56 „
j»
1458
I 15 75
14678
19.11.04
9.40 a.m.
1467
14 30
14579
1
11.38 „
"
14665
1420
14561
Initial reading, cold at 15° 14872 divisions
Final mean reading, cold at 15° 14636 „
Mean reading, cold 14754 „
hot 14360
Coldhot 0394 division
The following Table III gives the results of the various experiments with
the hollow weight, treated as in Table II, the readings of the centre of swing
being at about hourly intervals when on the same day.
CHANGE OF TEMPERATURE HAS ANY EFFECT UPON WEIGHT
157
Table III. Exferiments with Hollow Weight raised to 100° C. and then
cooled, 1 mm. = 0248 milligramme.
Correction for temperature of case — 0044 division per 1°.
Date
Condi
tion of
weight
Centre
of swing
corrected
to 15°
Number
of readings
from which
centre
of swing is
found
Greatest
deviation
from
the mean
Excess
of cold
above
hot
Remarks
16.11.04 ...
17.11.04 ...
Hot
Cold
14749
14872
4
1
00201
0123
Temperatures 14°75 to 16°3
The initial cold reading was
rendered useless by a subse
quent shift of scalereading,
probably due to slight dis
placement of the eye piece
17.11.04 ...
1718.11.04
1819.11.04
Cold
Hot
Cold
14872
14360
14636
1
7
5
0058 i
0075J
0394
Temperatures 14°2 to 16°
The set given in full in Table
II. The last of the preceding
used as the first of this
21.11.04 ...
2223.11.04
23.11.04 ...
Cold
Hot
Cold
14218
14091
14294
1
6
3
0095 i
0040j
0165
Temperatures 10° 45 to 14° 8
25.11.04 ...
1
26.11.04 ...
Cold
Hot
Cold
14421
14029
14010
1
2
2
0001
0016
0187
Temperatures 10°95 to 14°5
1.12.04 ...
2.12.04 '.'.'.
Cold
Hot
Cold
14500
14100
14367
1
3
1
0007
0334
Temperatures 16° to 16° 9
2.12.04 ...
23.12.04
3.12.04 ...
Cold
Hot
Cold
14367
14218
14400
1
3
1
0048
0166
Temperatures 16°9 to 17°45
12.12.04 ...
13.12.04 ...
Cold
Hot
Cold
14379
14106
14390
1
3
1
0018
0279
Temperatures 13°65 to 15°75
Mean value cold  hot = 0235 division = 0058 milligramme.
The following Table IV gives the results with the sohd weight. They are
not so consistent as those with the hollow weight, probably because they were
spread over a longer time on the average. This was done to secure that the
weight should be more nearly at the temperature of its surroundings. A
rough estimate shows that if heat be gained by radiation alone and the brass
is taken as a full radiator, three hours will be required to bring it within
1° of the temperature of the steam. The last two experiments were incom
plete in that no final cold weighing was taken, but the results obtained were
regarded as probably sufficient.
158
AN EXPERIMENT WITH THE BALANCE TO FIND IF
Table IV. Exferiments with Solid Weight raised to 100° C. and then
cooled, 1 mm. = 0803 milligramme.
Correction for temperature of case — 0044 division per 1°.
Date
Condi
tion of
weight
Centre
of swing
corrected
to 15°
Number
of readings Greatest
from which deviation
centre from
of swing is ; the mean
found
j
Excess
of cold T> 1
above I Remarks
hot
1
26.12.04 ...
2728.12.04
29.12.04 ...
Cold
Hot
Cold
16295
16032
16046
1
2
1
0017
0139
Temperatures 9°9 to ll°0
30.12.04 ...
3031.12.04
2.1.05
Cold
Hot
Cold
16016
15914
16035
1
4
1
0021
0112
Temperatures 9°l to 12°05
2.1.05
3.1.05
4.1.05
Cold
Hot
Cold
16035
16022
16047
1
2
1
0018
0019
Temperatures 9°l to 12° 1
5.1.05
>9
Cold
Hot
15987
15919
1
3
0002
Temperatures ll°95 to 12°65
0068 Experiment interrupted by
stoppage of steam tubes
9.1.05
10.1.05
Cold
Hot
16048
16039
1
4
0020
0009
Temperatures 13°7 to 15°65
Heating continued several days
after this to obtain tempera
turecorrection. No final cold
reading taken
Mean value cold hot = 0069 division = 0055 milligramme.
From Tables III and IV we have :
Solid weight, 266 grammes: cold — hot ... = 0055 milligramme.
Hollow weight, 58 grammes: cold — hot ... = 0058 ,,
For the difference, 208 grammes : hot — cold = 0003 ,,
Taking the rise in temperature as 85°, this gives a change of the order
of 1 in 6 X 10^ per 1° rise. But evidently the smallness of the result is
accidental, and probably all we can assert from the work is that any change
of weight with change of temperature between 15° C. and 100° C. is not
greater than 1 in 10^.
Cooling with Liquid Air.
Experiments were made in which heating by steam was replaced by
coohng with liquid air. This was supphed to us by Sir Wilham Ramsay,
and we desire to express our hearty thanks to him for his ready kindness
in helping us to increase the temperature range so considerably. In these
experiments the steamjacket was removed and replaced by a vacuum vessel
30 cm. deep and 6 cm. inside diameter, kept full of Hquid air. After the
CHANGE OF TEMPERATURE HAS ANY EFFECT UPON WEIGHT 159
steady state was reached the hquid air was removed, the jacket was replaced
and cold water was again passed round the tube.
Owing to the evaporation of the air the experiments had to be carried out
more rapidly than those with steam, but through the absence of convection
currents a steady state was more rapidly reached, and the variation in the
temperature of the case was very small.
The temperaturecorrection was not observed, but as in the subsequent
observations with both sohd and hollow weights it was found to be about
003 milHgramme per 1°, this value was assumed to hold here. In any case
its effect is very small, as the temperature varied so little.
The centre of swing was observed nearly continuously from the time when
the hquid air was apphed and again after it was removed. After a time in
each case it became steady and only these steady values are recorded in the
following Tables.
Table V. Experiment with Solid Weight cooled by Liquid Air,
1 mm. = 0'315 milligramme.
Correction for temperature of case — 01 division per 1°.
Date
Time
Condition
of
weight
Centre of
swing
Tempera
ture
of case
Centre of
swing
corrected
to 16°6
Remarks
28.7.04
3.25 p.m.
5.50 „
6.0 „
6.10 „
8.45 „
9.15 „
Normal
Cold
Normal
11085
1107
1107
1107
11095
11095
16°6
16 65^
16 65 \
16 •65j
16 4 \
16 4 j
11085
11075
11075
Liquid air applied just after
3.25
Liquid air removed and
water applied just after
6.10
Normal cold = 0005 division = 00016 milligramme.
Table VI. Experiment with Hollow Weight cooled by Liquid Air,
1 mm. = 0343 milligramme.
Correction for temperature of case — 01 division per 1°.
Date
Time
Condition
of
weight
Centre of
swing
Tempera
ture
of case
Centre of
swing
corrected
to 16°6
Remarks
9.9.04
9.40 a.m.
11.40 „
11.50 „
5.25 p.m.
Normal
Cold
Normal
14485
14480
14480
14480
16°3
16 4 1
16 4 /
16 4
14455
14460
14460
Liquid air applied at 9.43
Removed at 11.52
Steady
The balance next morning
read 1448 at 16°3
Normal  cold =  0002 division =  00007 milligramme.
160 AN EXPERIMENT WITH THE BALANCE TO FIND IF
From Tables V and VI we have :
Solid weight, 266 grammes: normal  cold ... = 00016 milhgramme.
Hollow weight, 58 grammes: normal — cold... = — 00007 „
For the difference, 208 grammes : normal — cold = 0002 „
Taking the fall in temperature as 200°, this gives a change of the order
of 1 in 2 X 1010 per 1° fall.
These hquidair experiments were not repeated. But the conditions are
probably much less disturbed than with the steam experiments, and we may
safely say that if there is any change of weight with change of temperature
between 16°6 C. and  186° C, it is not so great as 1 in lO^^ per 1° C.
Note on the Change of Af parent Weight on First Heating or Cooling.
We have mentioned that while the changes in the temperature of the
weight were in progress there were considerable apparent variations in
weif^ht. These, in a few cases, amounted to as much as 06 milligramme.
They were almost certainly due to radiometric forces or to other gasaction,
for they were very dependent on the disposition of the diaphragms in the
tube T (Fig. 1), and also on the way in which the steam was blown through
the jacket.
In the preliminary experiments with solid weights the lowest diaphragm
was 5 to 6 inches above the weight, and the steam was blown into the top
of the jacket. Under these circumstances the following variations occurred
when the steam was turned on :
At first the weight apparently increased, until in 15 to 20 minutes it
reached a maximum, which was in some cases as much as 06 milligramme
above the real weight. After reaching this maximum the weight apparently
decreased, till in four hours it had reached a nearly steady value, which was
a little less than the value at the temperature of the laboratory.
If, now, the jacket was filled with cold water, the apparent weight first
increased for about one minute and then decreased for about two hours to
a minimum, which was a little lower than the final weight at 100°. After
this the weight very slowly increased, till in five to six hours it had recovered
the value which it had before the experiment.
These changes did not vary very much with the pressure, but at lower
pressures they took place more rapidly than at higher ones.
On coohng the weight with liquid air, changes occurred exactly similar
to those which occurred when the weight was cooled from 100° to the
temperature of the laboratory ; and when the weight was warmed up from
the temperature of liquid air to the temperature of the laboratory, the
changes were similar to those when the weight was warmed from the
temperature of the laboratory to 100° C.
CHANGE OF TEMPERATURE HAS ANY EFFECT UPON WEIGHT 161
So long as the arrangement of the diaphragms and the weight remained
the same, and so long as the steam was blown through in the same way,
bhese changes were exactly similai:, but as soon as any alteration was made
in these arrangements the character of the changes altered.
In one series of experiments a sealed glass bulb containing mercury was
used in place of the brass weight. In this case, immediately after the steam
was turned on there was a rapid decrease in weight, and a minimum was
reached in less than one minute. After this the changes were very similar
to those occurring with the brass weight. On cooling, however, the changes
were almost exactly the reverse of the changes on heating, and were not at
all like the changes with the brass weight.
In the final experiments, those recorded, the lowest diaphragm was within
Jinch of the top of the brass weight. With this arrangement and with the
steam blown into the top of the jacket, the following changes occurred :
The apparent weight first increased rapidly, reaching a maximum in about
one minute, then it rapidly decreased, reaching a minimum in about four
minutes, and again increased to another and lower maximum in 8 to 10
minutes. After this it slowly decreased to a nearly steady value a little
lower than the original value.
On passing cold water through the jacket the apparent weight rapidly
increased for about one minute, and then slowly decreased to its original
value.
Still another variation was arranged by blowing the steam in at the
bottom of the jacket instead of at the top, all the other things remaining as
in the last experiment.
In this case, on turning on the steam, the apparent weight first decreased
to a minimum in about one minute, then increased to a maximum in about
six minutes, and finally decreased slowly to a nearly steady value a little
below the original value.
The cooling and the consequent changes were exactly similar to those in
the last experiment.
It is somewhat difficult to follow out exactly the changes which w^ould be
caused by radiometeraction and by convectioncurrents in these different
arrangements of the apparatus, but the fact that these changes depend
entirely on the arrangement is sufficient evidence that they are caused by
gasaction, and, as we have before said, we have some reason to believe that
even the small final difference is due to aircurrents.
p. c w. 11
6.
ON A METHOD OF DETERMINING THE SENSIBILITY
OF A BALANCE.
By J. H. PoYNTiNG and G. W. Todd, M.Sc.
[Phil. Mag. 18, 1909, pp. 132135.]
[Read June 25, 1909.]
In the method, as we have arranged it, a small frame (Fig. 1, endview)
is fixed at the centre of the beam of a 16inch Oerthng balance. This carries
two Vs about 2 cm. apart, and in the Vs hes a straight wire or fibre about
3J cm. long, parallel to the beam and level with the central knifeedge. This
wire takes the place of the ordinary rider, and we shall call it 'the rider.'
Its weight is determined before use as accurately as possible by weighing on
an assaybalance. The sensibility is determined by moving the rider either
to right or left a measured distance. If this distance is d, if the halflength
of beam is b, and if the weight of the rider is R, the movement is equivalent
to an addition of weight to one pan, Rd/b.
In order to move the rider a definite distance a stout horizontal rod
(Fig. 2) passes through the balancecase from side to side without contact
with the case, and is supported at its ends outside, and independent of, the
case. It is parallel to the beam and a little lower than the V frame. On
the rod are fixed horizontally two Brown and Sharp micrometerscrews
divided to 001 mm. and allowing an estimate of 0001 mm. Their axes are
in one fine coinciding with the axis of the rider, and they are fixed so that
one can bear against one end and the other against the other end of the rider.
Their ends are plane and the ends of the rider are bluntly pointed. Each
micrometerscrew head has a crosspiece fixed on it ; and a fork, which can be
rotated about an axis in the continuation of the axis of the screw by a pulley
outside the case, can engage with the crosspiece and so advance or withdraw
the screw. The pulley is worked by an endless string passing to a pulley at
the side of the observer, who is about 2 metres in front of the balance. The
micrometerdivisions are illuminated and each micrometer is viewed by its
own telescope. The position of the balancebeam is read by a double
suspension mirror, telescope and scale. The scale is divided to miUimetres
ON A METHOD OF DETERMINING THE SENSIBILITY OF A BALANCE 163
and is about 3 metres from the mirror. The doublesuspension mirror is fully
described in the Phil. Trans. A, vol. 182, p. 572*. It is of course not essential
to the method, but was chosen because of the great magnification of the
deflection which it gives.
Let us suppose that the value of the scaledivisions of the deflection is to
be determined by a movement of the rider from right to left. The two
micrometerscrews are withdrawn so that neither is in contact with the rider,
that on the left so far that the rider will not touch it in its subsequent travel.
The beam is lowered and allowed to swing. Then the righthand screw is
advanced till it bears against the end of the rider and pushes it some small
distance. The contact is seen to have occurred by the interference with
freedom of swing, as watched in the telescope. The micrometer is then
y/ qULJi> " I
Fig. I. Endview of V frame fixed to balancebeam.
Fig. 2. Arrangement of righthand micrometerscrew.
read. Let its reading be m^. It is then withdrawn a httle so as to leave
the rider free, and the centre of swing C^ is determined in the usual way
from three successive turningpoints. Then the micrometer is advanced
again so as to push the rod a little further, and its reading m^ is taken. It
is then withdrawn and the new centre of swing C^ is taken. If m^ — m^ = d,
Ci — C2 divisions of deflection are due to an addition of Rdjh to the left pan.
The righthand micrometer may then be withdrawn and the lefthand
micrometer may be brought into action in a similar manner, and so on, the
two screws being used alternately.
The balancecase was fixed on a shelf and was enclosed in a tinfoiled
wooden box with wool loosely packed between box and case. The case and
* [Collected Papers, Art. 3.]
11—2
164 ON A METHOD OF DETERMINING THE SENSIBILITY OF A BALANCE
box were provided with plateglass windows to view the mirror and the
micrometerdivisions. The following abstract of some determinations of
sensibihty will serve to show what accuracy may be attained:
I. Rider German silver wire, 735 mgm.
Halflength of beam, 20272 cm.
10 determinations alternately left and right.
Mean travel of rider ... ... 24850 mm.
Mean deflection ... ... ... 2126 divisions.
Mean value for 20 divisions . . . 00848 mgm.
The separate determinations range between
20 divisions  00877
and 20 „ = 00824.
II. The same rider.
10 determinations alternately left and right.
Mean travel of rider ... ... 52713 mm.
Mean deflection 4547 divisions.
Mean value for 40 divisions ... 01681 mgm.
The separate determinations range between
40 divisions = 01722
and 40 „ = 01632.
III. Rider German silver wire, 18905 mgm.
7 determinations alternately left and right.
Mean travel of rider 01764 mm.
Mean deflection 3870 divisions.
Mean value for 40 divisions ... 01691 mgm.
The separate determinations range between
40 divisions = 01709
and 40 „ = 01654.
IV. The same rider.
7 determinations alternately left and right.
Mean travel of rider 03004 mm.
Mean deflection 6484 divisions.
Mean value for 60 divisions ... 02578 mgm.
The separate determinations range between
60 divisions = 02612
and 60 „ = 02518.
PAET II.
ELECTRICITY.
ON THE LAW OF FOKCE WHEN A THIN, HOMOGENEOUS,
SPHEKICAL SHELL EXEKTS NO ATTRACTION ON A
PARTICLE WITHIN IT.
[Manchester Lit. Phil. Soc. Proc. 16, 1877, pp. 16817L]
[Read March 6, 1877.]
If a homogeneous, thin, spherical shell of uniform thickness exert no
attraction on a particle within it, then the law of the force is the law of
nature.
Professor Maxwell uses this proposition {Electricity, vol. 1, § 74) to deduce
the law of the force between electrified bodies, and shows that it proves, far
more conclusively than any direct measurements of electrical forces, that
the law is that of the inverse square. It would therefore be an advantage
to have a simpler proof of such an important proposition than that given
by Laplace {Mec. Celeste, liv. ii, ch. 2) and followed by Maxwell. The following
seems more simple, as it requires neither integration nor the solution of a
functional equation :
Let P be any point inside the spherical shell,
C the centre of the sphere, DPCE the diameter
through P, and APB perpendicular to CD. In
Newton's proof of the proposition that, if the law
of attraction be that of the inverse square, the
force at P is zero, the surface is divided into an
indefinitely great number of opposite elements by
small cones having their vertices at P, and the
attractions of each of these pairs of elements are
shown to balance each other. We shall first show
Fig. 1.
that if the attraction at P is zero, then it follows inversely that, for at least
166 ON THE LAW OF FORCE WHEN A THIN, HOMOGENEOUS, SPHERICAL
one position (if not for all positions) of the cone MPm besides the position
APB, the attractions of the opposite elements balance each other ; and we
shall thence prove that the law of attraction must be that of the inverse
square.
Let us suppose the cone, with vertex at P, to move round from the position
where AB is its axis to any other position MPm. At AB the attractions of
the opposite sections on P are e({ual, whatever the law of the force. As the
cone leaves AB let us suppose the resultant attraction of the two opposite
elements to be no longer zero, but to act, say, towards the centre side of
APB. Then it will either continue towards that side as the cone moves all
the way round from APB to BPA, or it will vanish at some position, and then
act in the opposite direction. In the first case we should have a number of
forces all acting from P towards the same side of APB, whose resultant is
zero ; then each separate force must be zero. In the second case the resultant
attraction of the opposite sections vanishes somewhere between AP and
EP\ then for at least one position of the cone, besides the position APB,
the resultant attraction of the opposite sections vanishes.
Since this is true for any position of P, we can show that the law of the
force must be that of the inverse square.
In the position where the two opposite sections exert equal attractions,
two sections of the same thickness perpendicular to the axis of the cone
would also exert equal attractions ; for they would bear to each other the
same ratio as the two oblique sections made by the sphere, since these two
obhque sections make equal angles with the axis of the cone. Then what
we have proved is, that for every position of P there are two different distances
for which the attractions of the sections of a small cone of equal thickness
on a point at its vertex are equal.
N
Fig. 2.
Let us take VMX to represent the axis of a cone of very small angle of
which F is the vertex.
At any point M draw an ordinate MN to represent the attraction of a
section of the cone at M of small given thickness on a point at the vertex.
Then N will trace out a curve as M moves along VX.
Now take a spherical shell of thickness equal to the thickness of the
sections of the cone, and of radius nearly equal to VM, where M is any
arbitrary point in VX. Take a point near the centre of this sphere. As
a cone moves round with this as vertex, its sections by the sphere must
be always at distances very nearly equal to the radius from the vertex;
SHELL EXERTS NO ATTRACTION ON A PARTICLE WITHIN IT 167
and, by what we have proved above, for some position of the cone the
attractions of the opposite sections must be equal. Therefore (in Fig. 2),
for two distances very nearly equal to VM the ordinates must be equal to
one another. Then the tangent to the curve near N must be parallel to VX.
But M is arbitrary, for we can take the sphere of any size. Therefore at
all points the tangent to the curve is parallel to VX ; and therefore the curve
must be a straight line parallel to VX; or, the attractions by sections of
the cone of equal thickness are constant, wherever the sections be taken.
But the sections are proportional to the direct square of the distance; and
therefore the law of the attraction must be that of the inverse square of the
distance.
8.
ARRANGEMENT OF A TANGENT GALVANOMETER FOR LECTURE
ROOM PURPOSES TO ILLUSTRATE THE LAWS OF THE
ACTION OF CURRENTS ON MAGNETS, AND OF THE RESIS
TANCE OF WIRES.
[Manchester Lit. Phil Soc, Proc. 18, 1879, pp. 8588.]
[Read April 1, 1879.]
Three coils of similar wire are arranged round the circumference of a circle
with a compassneedle at the centre, each wire going only once round the
circle. The six ends of the three wires are connected by thick copper w'ltes
Avith the six binding screws A, B, C, D, E, F.
On a concentric circle of twice the
radius is arranged a coil of the same
wire going twice round the circle and
having its ends connected by thick
wires with the binding screws G, H.
On the sam.e circle is a single wire of
twice the diameter, making only one
turn round the circle, and having its
ends connected with the binding screws
K, L.
The coils are denoted respectively
by the numbers 15.
The null method is adopted in each
case. That is two forces acting on
the needle and arising from different
arrangements of the circuit are shown
to balance each other, and from the
arrangements necessary to produce this
equihbrium the desired laws are deduced.
I. If the current be reversed the force is reversed. Introduce the current
at A. Join BD and lead away at C. Then the same current goes in opposite
B A
GALVANOMETER TO ILLUSTRATE ACTION OF CURRENTS ON MAGNETS 169
directions round the coils (1) and (2), and since the needle is not deflected the
reversal of the force when the current is reversed is proved.
II. The force is frofortional to the length of current acting. This is proved
by the last, for the two coils (1) and (2) exert equal forces on the needle.
If the current went round them in the same direction we should have twice
the force which each exerted singly, with twice the length of current. This
assumes that the current in different parts of the circuit is the same, which
might be shown by sHghtly modifying I, thus : introduce between B and B
various resistances and the equilibrium is not disturbed.
III. The force is proportional to the strength of the current. Introduce at
A, and connect A with C, and B with D; and connect D with F, and lead off
from E. There will be two equal currents in the same directions in coils
(1) and (2), for they are exactly similar to each other and similarly situated.
These two currents unite to give a double current in the opposite direction
in coil (3), and the double current in a single wire exerts twice the force
exerted by each single current, since there is no deflection.
IV. The force is inversely proportional to the square of the distance. Intro
duce at A, connect B with H, and lead off at G. Then since we have two
turns to the coil (4) we have a current of four times the length at twice the
distance acting in opposition to the same current through (1). Since there
is no deflection, the two exert equal forces, and therefore the force is inversely
proportional to the square of the distance.
Resistance.
I. The resistance is proportional to the length. Connect B with C, Z) with
E, A with F. Introduce at A and lead off at E. We have then a divided
circuit joining A and E, one branch consisting of the two coils (1) and (2),
and the other the coil (3) only onehalf the length and going in the opposite
direction. But as there is no deflection the current through the first circuit
of twice the length must be only onehalf that through the other to exert
an equal force, i.e., the resistance is doubled when the length is doubled.
II. The resistance is inversely proportional to the crosssection. Introduce
at A, connect A with L, connect B with C and D with K, and lead off at A'.
Then we have two circuits connecting A with K, the first consisting of the
two coils (1) and (2), the second of the coil (5) of the same length but of four
times the crosssection and going round in the opposite direction. Since the
needle is not deflected the two currents exert equal forces. But the coil (5)
is at twice the distance and must therefore have four times as great a current
through it as that through (1) and (2). That is, with equal lengths of wire
of the same material connecting two points the currents conveyed are pro
portional to the crosssections.
9.
ON THE GKADUATION OF THE SONOMETER.
[Phil Mag. 9, 1880, pp. 5964.]
[Read before the Physical Society, December 13, 1879.]
It seems likely that such valuable results will be obtained by means of
Professor Hughes's sonometer, that it is desirable that some method should
be employed to turn its at present arbitrary readings into absolute measure,
so that, for instance, the induced currents caused by different metals in the
inductionbalance may be measured and compared with each other.
In Maxwell's Electricity, vol. 2, chap, xiv, the general formula is given
for the coefficient of induction of one circular circuit on another. Adapting
this to the case where two equal circular circuits are on the same axis at a
distance apart greater than the radius of the coils, the following formula is
obtained.
Let a = distance between centres,
h = radius of either circle,
c = distance of either circumference from centre of other,
M = coefficient of induction.
a^ (2 2a^^l6a^ 32 a^^ 256 a^ etc. . ...^^j
Of these the latter uses directly the distance between the centres, the observed
quantity — but is not nearly so convergent as the former, in which c may be
at once deduced from c = Va^^b^
To obtain formulae which might be strictly applied to the sonometer, we
should have to consider the more general case of two coils of unequal radii
6 and ^, for which I have found the formula corresponding to (2), viz.
^^ _ 47762^2 ji ^ 3 52^__ ^2 15 54 ^ 352^2 _!_ ^4
+
16 a4
_35 6e+66_^^66^^_^^^^^ (3)
ON THE GRADUATION OF THE SONOMETER 171
We should then have to take the finite integrals of each term between the
limiting values of b and ^. But this would be exceedingly complicated and
would require a knowledge of all the details of construction; and we may
at least get a first approximation to the true result by replacing the coils by
a single one of a radius intermediate between the greatest and least radii.
In Prof. Hughes's paper {Phil. Mag. July 1879) he gives the internal and
external radii of his coils as 15 miUims. and 275 millims. respectively. I have
considered, then, that 25 millims. will give results not very far from the truth ;
and as it makes the calculations considerably easier, I have taken that as the
value of b and applied the formulae to the numbers given in the paper. The
resultant current in the middle coil was zero when it was distant 47 millims.
from one end and 200 from the other. This enables us to find the ratio between
the number of turns in the two ends at least sufficiently nearly to apply to
some of the results.
Let M^ be the coefficient of induction of the larger coil on the moveable
one, ilf 2 that of the smaller, the former having m turns, the latter n. When
the moveable coil was 200 millims. from the large and 47 millims. from the
small coil, since there was no induced current,
mMi = nM^ .
Applying formula (1), we have for the larger coil
c = V2002+ 252 = 2015,
and for the smaller coil
c = \/472+ 252 = 532,
b being the same for both. Then
_jri__ jl _ 3 / 25 y 15 / 25 y _
(2015)3 [2 4 12015/ ^ 8 V2015y ~ ^*^'
1 3/25 \2 15 / 25 \4 35 / 25 \6 2835 / 25
(532)3 (2 4
/ 25 \2 15 / 25 y 35 / 25 \« 2835 / 25 y
1532/ + 8 V532J 8 V532/ "^ 256^ 1532;
Multiplying each side by 2 and finding the successive terms,
122
m X ^3 {1  02308 + 00088  etc.}
= nx ^ {1  33123 + 18286  09422 + 02633  etc.},
^ = 436.
n
I have applied the formula to the results for various metals given by
Prof. Hughes in a table in his paper. In the table below, in the second column
are Prof. Hughes's numbers, i.e. distances from the point of no induction.
In the third are numbers proportional to mM^ — nM^ ; where M^ , M2, are
the coeiB&cients of induction of two simple coils calculated on the above
172
ON" THE GRADUATION^ OF THE SONOMETER
hypothesis, m and n the number of turns in the two respectively. In the
fourth column are the resistances for bars of the metal 100 miUims. long and
1 millim. in diameter (Jenkin, p. 249). In the last column are the products
of the numbers in the two preceding columns.
Metal
Distance
from point of
no induction
7nMj^  nM., ,
proportional to
E
(mif 1  nM^) R
Silver
125
178
•21
374
Gold
117
135
27
365
Aluminium
112
116
•375
435
Copper
100
84
21
176
Zinc
80
501
•72
361
Tin
74
446
170
758
Iron ...
45
2246
125
281
Lead
38
1887
25
472
Antimony
35
1735
45
781
Bismuth
10
575
168
966
Mercury has been omitted, as it gives a very much higher value than any
of the others. Were the induced currents in the inductionbalance pro
portional to the resistances given in the table, the numbers in the last column
would of course be all the same. The deviations from equality are far greater
than could be accounted for by errors in the approximations I have adopted,
especially for the metals not at the beginning or end of the list. Hence we
are driven to conclude, either that the resistances of the metals given in the
tables are not the same as the resistances of the metals used by Prof. Hughes,
or that the induced current is not proportional to the conductivity of the
metal.
It should be noticed that the method of measuring currents by the
sonometer assumes that the telephone integrates, as it were, the current;
i.e. the loudness of the sound depends only on the total current, not on the
time during which the current is passing, provided that the time be very
short. I do not know whether this point has been investigated ; but if not,
it would probably be easy to examine it by means of the sonometer. It
would be advisable to modify the instrument in such a way that the formulae
might be more easily employed, and that the approximations might be nearer
to the truth.
The formulae used in this paper may be obtained as follows, the method
being adapted from that given by Maxwell.
The potential of a circular unit current at any point is the same as that
of a magnetic shell of unit strength bounded by the circuit. This, again,
is the same as the attraction of a thin plate of matter of unit surfacedensity
ON THE GRADUATION OF THE SONOMETER 173
in a direction perpendicular to the plane of the plate. If co be the attraction
of a plate of radius b, at a point distant c from the plate along its axis,
= 277
Cv'i)
^ (162 1.3 6* 1.3.5 6« )
= ^" 12 ^271 0^ + 27176 c«"*"t
If we introduce zonal harmonics as coefficients, this becomes
^ fl 62 1 . 3 6* ^ 1 . 3 . 5 6« ^
" = ^" 12 ^^^"274 0^^3+ 27476 B^5etc.
This is now the potential at any point in space where 6 < c.
If there be a second circular circuit of radius j8 on the same axis, we may
suppose it replaced by a magnetic shell bounded by the current and lying on
the sphere, with centre at the centre of the first current, the radius of the
sphere being c.
This shell may be considered to consist of two layers of matter of equal
and opposite densities, fi and — /x, at distances c and c + dc from the centre.
The potential on the second layer is
II'
II
fJLO) dS,
where the integration is taken over the shell. The potential on the first
layer is
jLt f CO + , dc) dS,
the sum being — j I [jl ^ dcdS ;
but since the strength = 1, fjidc = 1, and we have the mutual potential
Replacing the element dS by cH^zdcj), the Umits will be for ^ from to 277,
and for jjl from 1 to fx.
Integrating with respect to (f>, and remembering that c is constant in
integrating for jjl, we have
But we have the relation for zonal harmonics,
174 ON THE GRADUATION OF THE SONOMETER
Substituting, we obtain
The following are the values for the coefficients (Ferrers's Spherical Har
monies : p. 23), both in terms of /jl and when we substitute /x^ = 1 — ^ :
djji
dl^
4 „,._.,. l(4_4:).
dp,
'dl^
.i?2V'lV+l)f(828f + ='«'.)^
dp,
3003/x«  3465^4 + 945^^ _ 35 ^^^ " ^^^4 ^ + etc.
di.
16 16
dPs
109395/x8  180180/x6 + 90090^*  13860/x2 + 315
d^
128
5760 + etc.
128
Substituting these values and putting c^ = a^ + ^^^ ^g obtain
, 62/3M1 3 62 + R2 15 6*+ 362^2 _^ ^4
M = — 477 — ^  — I     
a3 ^^2 4 a2 +16 ^4
_ 35 66jf664^2 _!_ 652^4 __ ^6 I
32 a« +etc..
The more useful form is obtained by retaining c. If we take the two circles
of equal radius (i.e. 6 = ^), we obtain
,. , o &M1 3 62 15 64 35 ¥ 2835 6^ ]
^^^""c3]24c2 + ¥c4¥c«+^5^cS^'4'
10.
ON THE TRANSFER OP ENERGY IN THE
ELECTROMAGNETIC FIELD.
[Phil. Trans. 175, 1884, pp. 343361.]
[Received December 17, 1883. Read January 10, 1884.]
A space containing electric currents may be regarded as a field where
energy is transformed at certain points into the electric and magnetic kinds
by means of batteries, dynamos, thermoelectric actions, and so on, while in
other parts of the field this energy is again transformed into heat, work done
by electromagnetic forces, or any form of energy yielded by currents.
Formerly a current was regarded as something travelling along a conductor,
attention being chiefly directed to the conductor, and the energy which
appeared at any part of the circuit, if considered at all, was supposed to be
conveyed thither through the conductor by the current. But the existence
of induced currents and of electromagnetic actions at a distance from a
primary circuit from which they draw their energy has led us, under the
guidance of Faraday and Maxwell, to look upon the medium surrounding
the conductor as playing a very important part in the development of the
phenomena. If we believe in the continuity of the motion of energy, that is,
if we believe that when it disappears at one point and reappears at another
it must have passed through the intervening space, we are forced to conclude
that the surrounding medium contains at least a part of the energy, and
that it is capable of transferring it from point to point.
Upon this basis Maxwell has investigated what energy is contained in the
medium, and he has given expressions which assign to each part of the field
a quantity of energy depending on the electromotive and magnetic intensities
and on the nature of the matter at that part in regard to its specific inductive
capacity and magnetic permeabihty. These expressions account, as far as
we know, for the whole energy. According to Maxwell's theory, currents
consist essentially in a certain distribution of energy in and around a con
ductor, accompanied by transformation and consequent movement of energy
through the field.
Starting with Maxwell's theory, we are naturally led to consider the
problem : How does the energy about an electric current pass from point to
176 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD
point — that is, by what paths and according to what law does it travel from
the part of the circuit where it is first recognisable as electric and magnetic
to the parts where it is changed into heat or other forms ?
The aim of this paper is to prove that there is a general law for the transfer
of energy, according to which it moves at any point perpendicularly to the
plane containing the Hnes of electric force and magnetic force, and that the
amount crossing unit of area per second of this plane is equal to the product
of the intensities of the two forces, multiphed by the sine of the angle between
them, divided by 477 ; while the direction of flow of energy is that in which
a righthanded screw would move if turned round from the positive direction
of the electromotive to the positive direction of the magnetic intensity.
After the investigation of the general law several appHcations will be given
to show how the energy moves in the neighbourhood of various current
bearing circuits.
The following is a general account of the method by which the law is
obtained.
If we denote the electromotive intensity at a point (that is, the force per
unit of positive electrification which would act upon a small charged body
placed at the point) by @, and the specific inductive capacity of the medium
at that point by K, the magnetic intensity (that is, the force per unit pole
which would act on a small northseeking pole placed at the point) by ^ and
the magnetic permeability by yi, Maxwell's expression for the electric and
magnetic energies per unit volume of the field is
K^^I^TT + ilSyI^TT (1)
If any change is going on in the supply or distribution of energy the
change in this quantity per second will be
/.ef/4. + ,^f/4. (2)
According to Maxwell the true electric current is in general made up of
two parts, one the conductioncurrent ^T, and the other due to change of
electric displacement in the dielectric, this latter being called the displace
mentcurrent. Now, the displacement is proportional to the electromotive
intensity, and is represented by /i(S/47r, so that when change of displacement
takes place, due to change in the electromotive intensity, the rate of change,
that is, the displacementcurrent, is K~ Utt, and this is equal to the difference
between the true current (5 and the conductioncurrent >T. Multiplying this
difierence by the electromotive intensity ik the first term in (2) becomes
— ^^=m^.^ (3)
ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 177
The first term of the right side of (3) may be transformed by substituting
for the components of the total current their values in terms of the com
ponents of the magnetic intensity, while the second term, the product of the
conductioncurrent and the electromotive intensity, by Ohm's law, which
states that ^ = 6'@, becomes ^^/C, where C is the specific conductivity.
But this is the energy appearing as heat in the circuit per unit volume according
to Joule's law. If we sum up the quantity in (3) thus transformed, for the
whole space within a closed surface, the integral of the first term can be integrated
by parts, and we find that it consists of two terms — one an expression
depending on the surface alone to which each part of the surface contributes
a share depending on the values of the electromotive and magnetic intensities
at that part, the other term being the change per second in the magnetic
energy (that is, the second term of (2)) with a negative sign. The integral
of the second term of (3) is the total amount of heat developed in the con
ductors within the surface per second. We have then the following result.
The change per second in the electric energy within a surface = (a
quantity depending on the surface) — (the change per second in the magnetic
energy) — (the heat developed in the circuit).
Or rearranging :
The change per second in the sum of the electric and magnetic energies
within a surface together with the heat developed by currents is equal to
a quantity to which each element of the surface contributes a share depending
on the values of the electric and magnetic intensities at the element. That
is, the total change in the energy is accounted for by supposing that the
energy passes in through the surface according to the law given by this
expression.
On interpreting the expression it is found that it implies that the energy
flows as stated before, that is, perpendicularly to the plane containing the
lines of electric and magnetic force, that the amount crossing unit area per
second of this plane is equal to the product
electromotive intensity x magnetic intensity x sine included angle
while the direction of flow is given by the three quantities, electromotive
intensity, magnetic intensity, flow of energy, being in righthanded order.
It follows at once that the energy flows perpendicularly to the fines of
electric force, and so along the equipotential surfaces where these exist. It
also flows perpendicularly to the lines of magnetic force, and so along the
magnetic equipotential surfaces where these exist. If both sets of surfaces
exist their lines of intersection are the lines of flow of energy.
The following is the full mathematical proof of the law :
V. a. w. 12
178 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD
The energy of the field may be expressed in the form (Maxwell's Electricity,
vol. 2, 2nd ed., p. 253)
1 1 j(P/+ Qg + Ith) dxdydz + g  U(aa + 6^ + cy) dxdydz,
the first term the electrostatic, the second the electromagnetic energy.
But since f = — P, with corresponding values for g and h, and a = fia,
h = /x^, c ^ /xy, substituting, the energy becomes
^ \\\{P^ + Q^ + i?2) ^:r%6?0 + £ [J(a2 + ^^ + ^2) dxdydz. ...(1)
Let us consider the space within any fixed closed surface. The energy
within this surface will be found by taking the triple integrals throughout
the space.
If any changes are taking place the rate of increase of energy of the electric
and magnetic kinds per second is
Now Maxwell's equations for the components of the true current are
, df dg dh
where f, q, r are components of the conductioncurrent.
But we may substitute for ^ its value y , and so for the other two,
and we obtain
KdP_
477 dt ~
K dQ
4^dt^'^
K dR
in dt
Taking the first term in (2) and substituting from (3) we obtain
,(3)
477
= i i {^ i^* ~P)^Q{vq) + R {w  r)} dxdydz
= \\\(Pu + Qv + Rw) dxdydz  1 1 j(P^ + §^ + Er) dxdydz. ...(4)
Now the equations for the components of electromotive force are (Maxwell
vol. 2, p. 222) :
ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 179
Q = azcx^^ ^r = azcx + Q'^, (5)
„ , . . dH dih J . . ,,\
R = hx — ay — J — ^ = bx — ay + R ]
where P', Q', R' are put for the parts of P, Q, Ji which do not contain the
velocities.
Then
Pu + Qv + Rw = (cij — h'z) u h {az — ex) v + (bx — ay) iv + P'u + Q'v + R' w
= — {{vc — wb) X + {wa — uc)y { (ub — va) z) + P'u + Q'v + R'w
= (Xx+ Yy + Z'z) + P'u + Q'v + R'w,
where X, Y, Z are the components of the electromagnetic force per unit of
volume (Maxwell, vol. 2, p. 227).
Now substituting in (4) and putting for u, v, w their values in terms of
the magnetic force (Maxwell, vol. 2, p. 233) and transposing we obtain
+ I \j{{Xx +Yy + Z'z) + [Pv + Qq + Rr)} dxdydz
= \\\(P'u + Q'v + R'w) dxdydz
(Integrating each term by parts)
= ^ \\(R'^  Q'y) dydz + ^ [[(P^  R'a) dzdx + ^ \\(Q'a  P'^) dxdy
I [(({^dR' dQ' ^ dP' dR' ^ dQ' r,dP'\. . .
(The double integral being taken over the surface)
1
^^ ^{l (R'p  Q'y) + m (P'y  R'a) + n (Q'a  P'^)} dS
 Mk S  f )  * (f  f )  r (f  SI "'"■■■■^"
where 2, m, n are the direction cosines of the normal to the surface outwards.
12—2
180 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD
But from the values of P' , Q\ R' in (5) we see that
dQ'
dz
dR' d^G d^ dm ^ d^
dy ~ dtdz dxdz dtdy dzdx
d /dH dG\
dt \ dy dz )
"Tt^^Tt (^^^^^11' ^^^ 2> P 216)
dR' dP' db dp
dx dz dt ^ dt'
dP' dQ' _dc _ dy
dy dx dt ^ dt'
similarly
Whence the triple integral in (6) becomes
Transposing it to the other side we obtain
(Xx + Yy + Zz) dxdydz + I \(Pf + Qq+ Rr) dxdydz
= ~ j {/ (R'P  Q'y) + m {P'y  R'a) + n (Q'a  P'^)} dS. ...(7)
477.' .'
The first two terms of this express the gain per second in electric and
magnetic energies as in (2). The third term expresses the work done per
second by the electromagnetic forces, that is, the energy transformed by the
motion of the matter in which currents exist. The fourth term expresses
the energy transformed by the conductor into heat, chemical energy, and so
on ; for P, Q, R are by definition the components of the force acting at a point
per unit of positive electricity, so that Pp dxdydz or Pdxpdydz is the work
done per second by the current flowing parallel to the axis of x through the
element of volume dxdydz. So for the other two components. This is in
general transformed into other forms of energy, heat due to resistance,
thermal effects at thermoelectric surfaces, and so on.
The left side of (7) thus expresses the total gain in energy per second
within the closed surface, and the equation asserts that this energy comes
through the bounding surface, each element contributing the amount ex
pressed by the right side.
This may be put in another form, for if (S' be the resultant of P', Q\ R\
and 9 the angle between its direction and that of ^, the magnetic intensity,
ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 181
the directioncosines L, M, N of the Une perpendicular to the plane containing
@' and «§ are given by
@'§sin<9' r§sin6>' ($'^ sin ^ '
so that the surfaceintegral becomes
^ [fd'^ sin ^ (Z? + Mm + iV^n) cZ/Sf.
If at a given point (?/S be drawn to coincide with the plane containing
(S' and t§, it then contributes the greatest amount of energy to the space ;
or in other words the energy flows perpendicularly to the plane containing
($' and ^, the amount crossing unit area per second being ^'^ sin ^/477. To
determine in which way it crosses the plane take ^' along Oz, § along Oy.
Then
P' = 0, g' = o, !=1,
a=0, 1 = 1, y = 0,
and if sin ^ = 1 X  1, M = 0, iV = 0.
If now the axis Ox be the normal to the surface outwards, ? = 1, m = 0,
n = 0, so that this element of the integral contributes a positive term to the
energy within the surface on the negative side of the yz plane ; that is, the
energy moves along xO, or in the direction in which a screw would move if
its head were turned round from the positive direction of the electromotive
to the positive direction of the magnetic intensity. If the surface be taken
where the matter has no velocity, (S' becomes equal to d, and the amount
of energy crossing unit area perpendicular to the flow per second is
electromotive intensity x magnetic intensity x sine included angle
477
Since the surface may be drawn anywhere we please, then wherever there is
both magnetic and electromotive intensity there is flow of energy.
Since the energy flows perpendicularly to the plane containing the two
intensities, it must flow along the electric and magnetic level surfaces, when
these exist, so that the lines of flow are the intersections of the two surfaces.
We shall now consider the appHcations of this law in several cases.
Applications of the Law of Transfer of Energy.
(1) A straight wire conveying a current.
In this case very near the wire, and within it, the lines of magnetic force
are circles round the axis of the wire. The Hnes of electric force are along
the wire, if we take it as proved that the flow across equal areas of the cross
182 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD
section is the same at all parts of the section. If AB, Fig. 1, represents the
wire, and the current is from A to B, then a tangent plane to the surface
at any point contains the directions of both the electromotive
and magnetic intensities (we shall write e.m.i. and m.i. for
these respectively in what follows), and energy is therefore
flowing in perpendicularly through the surface, that is, along
the radius towards the axis. Let us take a portion of the
wire bounded by two plane sections perpendicular to the axis.
Across the ends no energy is flow^ing, for they contain no
component of the e.m.i. The whole of the energy then enters
in through the external surface of the wire, and by the general
theorem the amount entering in must just account for the heat
developed owing to the resistance, since if the current is steady
there is no other alteration of energy. It is, perhaps, worth
while to show independently in this case that the energy moving
in, in accordance with the general law, will just account for
the heat developed.
Let r be the radius of the wire, i the current along it, a
the magnetic intensity at the surface, P the electromotive
intensity at any point within the wire, and V the difference of potential
between the two ends. Then the area of a length I of the wire is 27Trl,
and the energy entering from the outside per second is
c.
_)
Fig. 1.
area x e.m.i. x m.i. ^Trrl.P.a
Att
477
27rm . PI
4i7T
for the hneintegral of the magnetic intensity 277m round the wire is 47r x current
through it, and PI = V.
But by Ohm's law 7 = iR and iV = i^R, or the heat developed according
to Joule's laAv.
It seems then that none of the energy of a current travels along the wire,
but that it comes in from the nonconducting medium surrounding the wire,
that as soon as it enters it begins to be transformed into heat, the amount
crossing successive layers of the wire decreasing till by the time the centre
is reached, where there is no magnetic force, and therefore no energy passing,
it has all been transformed into heat. A conductioncurrent then may be
said to consist of this inward flow of energy with its accompanying magnetic
ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 183
and electromotive forces, and the transformation of the energy into heat within
the conductor.
We have now to inquire how the energy travels through the medium on
its way to the wire.
(2) Discharge of a condenser through a wire.
We shall first consider the case of the slow discharge of a simple condenser
consisting of two charged parallel plates when connected by a wire of very
great resistance, as in this case we can form an approximate idea of the actual
path of the energy.
Fig. 2.
Let A and B, Fig. 2, be the two plates of the condenser, A being positively
and B negatively electrified. Then before discharge the sections of the equi
potential surfaces will be somewhat as sketched. The chief part of the energy
resides in the part of the dielectric between the two plates, but there will
be some energy wherever there is electromotive intensity. Between A and B
the E.M.I, will be from A to B, and everywhere it is perpendicular to the level
surfaces. Now connect A and 5 by a fine wire LMN of very great resistance,
following a line of force and with the resistance so adjusted that it is the same
for the same fall of potential throughout. We have supposed this arrange
ment of the resistance so that the level surfaces shall not be disturbed by the
flow of the current. The wire is to be supposed so fine that the discharge
takes place very slowly.
While the discharge goes on a current flows round LMN in the direction
indicated by the arrow, and there is also an equal displacementcurrent from
B to A due to the yielding of the displacement there. The current will be
184 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD
encircled by lines of magnetic force, which will in general form closed curves
embracing the circuit. The direction of these round the wire will be from
right to left in front, and round the space between A and B from left to right
in front. The e.m.i. is always from the higher level surfaces — those nearer
A, to the lower— those nearer B, both near the wire and in the space between
A and B.
Now, since the energy always moves perpendicularly to the hues of e.m.i.
it must travel along the equipotential surfaces. Since it also moves perpen
dicularly to the hnes of m.i. it moves, as we have seen in case No. (1), inwards
on all sides to the wire, and is there all converted into heat — if we suppose the
discharge so slow that the current is steady during the time considered. But
between A and B the e.m.i. is opposed to the current, being downwards,
while the m.i. bears the same relation to the current as in the wire. Eemem
bering that e.m.i., m.i., and direction of flow of energy are connected by the
righthanded screw relation, we see that the energy moves outwards from the
space between A and B. As then the strain of the dielectric between A and
B is gradually released by what we call a discharge current along the wire
LMN, the energy thus given up travels outwards through the dielectric,
following always the equipotential surfaces, and gradually converges once
more on the circuit where the surfaces are cut by the wire. There the energy
is transformed into heat. It is to be noticed that if the current may be con
sidered steady the energy moves along at the same level throughout.
(3) A circuit containing a voltaic cell.
When a circuit contains a voltaic cell we do not know with certainty what
is the distribution of potential, but most probably it is somewhat as follows* :
— Suppose we have a simple copper, zinc, and acid cell producing a steady
current. There is probably a considerable sudden rise in passing from the
zinc to the acid, the place where the chemical energy is given up, a fall through
the acid depending on the resistance, a sudden fall on passing from the acid
to the copper, where some energy is absorbed with evolution of hydrogen,
* It seems probable that the only legitimate mode of measuring the difference of potential
between two points in a circuit consisting of dissimilar conductors carrying a steady current,
consists in finding the total quantity of energy given out in the part of the circuit between the
two points while unit quantity of electricity passes either point. If this is the case, it seems
impossible that the surface of contact of dissimilar metals can be the chief seat of the electro
motive force, for we have only the very slight evolution or absorption of energy there due to the
Peltier effect. I have therefore adopted the theory of the voltaic circuit in which the seat of
at least the chief part of the electromotive force is at the contact of the acid and metals. The
large differences of potential found by electrometer methods between the air near two different
metals in contact are, in this theory, to be accounted for by the supposition that the air acts in
a similar manner to an oxidising electrolyte. A short statement of the theory is given in a letter
by Professor Maxwell in the Electrician for April 26th, 1879, quoted in a note on page 149 of his
Elementary Treatise on Electriciiy. (See also § 249, vol. 1, Maxwell's Electricity and Magnetism.)
June 19, 1884.
ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 185
and then a gradual fall through the wire of the circuit round to the zinc
again. There will be a sHght change of potential in passing from copper to
zinc, but this we shall neglect for simphcity. The equipotential surfaces will
probably then be somewhat as sketched in Fig. 3*, all the surfaces starting
from where the acid comes in contact with the zinc, some of the highest
potential passing through the acid, others passing between the acid and
copper, and crowding in there, the rest lower than these cutting the circuit
at right angles in points at intervals representing equal falls of potential.
Fig. 3.
If this be the actual arrangement, then it is seen that the current, which
travels round the circuit from zinc through acid to copper, is in opposition
to the E.M.I, between the zinc and acid, while the m.i. is related to the current
in the ordinary way. The energy will therefore pass outwards from there
along the level surfaces. In fact, the medium between the zinc and acid
behaves Kke the medium between the plates of the condenser in case No. (2j,
and it seems possible that the chemical action produces continually fresh
* electric displacement' from acid towards zinc which yields as rapidly as it
is formed, the energy of the displacement moving out sideways.
Some of this energy which travels along the highest level surfaces will
converge on the acid, and there be, at any rate ultimately, converted into
heat. Some of it will move along those surfaces which crowd in between
the acid and copper and there converge to supply the energy taken up by the
escaping hydrogen. The rest spreads out to converge at last at different
parts of the circuit, and there to be transformed into heat according to
Joule's law.
It may be noticed that if the level surfaces be drawn with equal differences
* In this and the succeeding cases the circuit is alone supposed to cause the distribution of
potential. In actual cases the surfaces would probably be very much deflected from their normal
positions in the dielectric through the presence of conductors, electrified matter, and so on.
186 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD
of potential, equal amounts of energy travel out per second between successive
pairs of surfaces. For the amount transformed in the circuit in a length
having a given difference of potential V between its ends will be 7 x current,
and therefore the amount transformed between each pair of surfaces drawn
with the same potential difference will be the same. But since the current
and the field are steady, the energy transformed will be equal to the energy
moving out from the cell between the same surfaces — the energy never
crossing level surfaces. This admits of a very easy direct proof, but the
above seems quite sufficient.
This result has a consequence which, though already well known, is worth
mentioning here. Let F^ be the difference of potential between the zinc and
acid, V2 that between the acid and copper. If i be the current, V^i is the
total energy travelling out per second from the zinc surface. Of this Fg^
is absorbed at the copper surface, the rest, viz., (Fi — Fg) ^, being trans
formed in the circuit. The fraction, therefore, of the whole energy sent out
which is transformed in the circuit is  ^w — , a result analogous to the
expression for the amount of heat which can be transformed into work in
a reversible heatengine.
One or two interesting illustrations of this movement of energy may be
mentioned here in connection with the voltaic circuit.
Suppose that we are sending a current through a submarine cable by a
battery with, say, the zinc to earth, and suppose that the sheath is every
where at zero potential. Then the wire will everywhere be at higher potential
than the sheath, and the level surfaces will pass from the battery through
the insulating material to the points where they cut the wire. The energy
then which maintains the current, and which works the needle at the further
end, travels through the insulating material, the core serving as a means to
allow the energy to get in motion.
Again, when the only effect in a circuit is the generation of heat, we have
energy moving in upon the wire, there undergoing some sort of transformation,
and then moving out again as heat or Ught. If Maxwell's theory of light be
true, it moves out again still as electric and magnetic energy, but with a
definite velocity and intermittent in type. We have in the electric Hght,
for instance, the curious result that energy moves in upon the arc or filament
from the surrounding medium, there to be converted into a form which is
sent out again, and which, though still the same in kind, is now able to affect
our senses.
(4) Thermoelectric circuits.
Let us first take the case of a circuit composed of two metals, neither of
which has any Thomson effect. Let us suppose the current at the hot junction
flows from the metal A to the metal B, Fig. 4. According to Professor Tait's
ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 187
theory it would appear that the e.m.i. at the hot junction is to that at the cold
as the absolute temperature at the hot is to that at the cold junction. If
the current is steady there is probably then a sudden rise in potential from
A to B at the hot junction, a gradual fall along B, a sudden fall at the cold
junction — less, however, than the sudden rise at the other — and a gradual
fall along A. The level surfaces will then all start from the hot junction,
the higher ones cutting the circuit at successive points along B, several con
verging at the cold junction, and the rest cutting the circuit at successive
points along A. The heat at the hot junction is converted into electric and
magnetic energy, which here moves outwards, since the current is against
the E.M.I. Some of this energy converges upon B and A, to be converted
Fig. 4.
into heat, according to Joule's law, and some on the cold junction, there
producing the Peltier heating effect.
Let us now suppose that we have a circuit of the same two metals, now all
at the same temperature, but with a battery interposed in B, which sends
a current in the same direction as before (Fig. 5). Then if C be the junction
which was hot, and D that which was cold in the last case, we know that the
current will tend to cool C and to heat Z). In going from A to B at C there
will be a sudden rise of potential, and in going from B to A at D there will
be a sudden fall. Then, since the potential falls, as we go with the current
along A, there will be a point on A near C which has the same potential as
B at the junction. From this point to C, A will have lower potentials, and
points with the same potentials will exist on B between C and the battery.
Then either the level surfaces passing through C are closed surfaces, cutting
A or B, and not passing through the battery at all, or, as seems much more
188 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD
probable, the surfaces from the battery which pass through C cut the circuit
in three points in all outside the battery: once somewhere along A, once at
C, and once somewhere along B. I have drawn and numbered the surfaces
in the figure on this supposition. The heat developed in the parts of the
circuit near C will thus be partly supphed from the junction C, where the
Fig. 6.
current is against the e.m.i. The energy therefore moves out thence, giving
a cooling effect.
The Thomson effect may be considered in somewhat the same way. Let
us suppose that a metal BC of the iron type, and with temperature falhng
from B to C, forms part of a circuit between two neutral metals of the lead
type AB and CD, Fig. 6, and let us further, for simphcity, suppose that these
1
ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 189
metals are each at the neutral temperatures with respect to BC, so that there
is no E.M.I, at the junction. If we drive a current from A to D hj means
of some external e.m.i., say at a junction elsewhere in the circuit, the potential
will tend to fall from A to D. But a current in iron from hot to cold cools
the metal, that is, the e.m.i. appears to be in opposition to the current, so
that the energy moves outwards. The potential, therefore, tends to rise
from B to C, and actually will do so if the resistance of BC is negligible
compared with that of the rest of the circuit. In this case the level
surfaces will probably be somewhat as indicated in Fig. 6, where they are
numbered in order, each surface which cuts BC also cutting AB and CD,
and the energy moving outwards will come into the circuit again at the
parts of AB and CD near the junctions, where it will be transformed once
more into heat. If the resistance of BC be gradually increased the fall of
potential, according to Ohm's law, will tend to lessen the rise, and fewer
surfaces will cut BC. It would seem possible so to adjust matters that the
two exactly neutralised each other so that no energy either entered or left
BC. In this case we should only have hues of magnetic force round BC,
and no other characteristic of a current in that part of the circuit*.
If this is the true account of the Thomson effect it would appear that it
should be described not as an absorption of heat or development of heat by
the current but rather as a movement of energy outwards or inwards,
according as the e.m.i. in the unequally heated metal opposes or agrees with
the direction of the current.
(5) A circuit containing a motor.
This case closely resembles the third case of a circuit containing a copper
zinc cell, the motor playing a part analogous to that of the surface of contact
of the acid with the copper. Let us, for simplicity, suppose that the motor
has no internal resistance. When it has no velocity all the level surfaces
cut the circuit, and the energy leaving the dynamo or battery is all transformed
into heat due to resistance. But if the motor is being worked the current
diminishes, the level surfaces begin to converge on the motor and fewer cut
the circuit. Some of the energy therefore passes into the motor, and is there
transformed into work. As the velocity increases the number cutting the
rest of the circuit decreases, for the current diminishes, and, therefore, by
Ohm's law, the fall of potential along the circuit is less ; and ultimately when
the velocity of the motor becomes very great the current becomes very small.
In the limit no level surface cuts the circuit, all converging on the motor.
* Perhaps this is only true of the wire as a whole. If we could study the effects in minute
portions it is possible that we should find the seat of the e.m.i. due to difference of temperature
not the same as that which neutrahses it, which is according to Ohm's law. One, for instance,
might be between the molecules, the other in their interior, so that there might be an interchange
of energy still going on, though no balance remained over to pass out of the wire.
190 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD
That is, all the energy passes into the motor when it is transformed into work,
and the efficiency of the arrangement is perfect, though the rate of doing
work is infinitely slow.
(6) Induced currents.
It is not so easy to form a mental picture of the movement of energy
which takes place when the field is changing and induced currents are created.
But we can see in a general way how these currents are accounted for. When
there is a steady current in a field there is corresponding to it a definite dis
tribution of energy. If there is a secondary circuit present, so long as the
primary current is constant, there is no e.m.i. in the secondary circuit for it
is all at the same potential. The energy neither moves into nor out of it,
but streams round it somewhat as a current of liquid would stream round
a solid obstacle. But if the primary current changes there is a redistribution
of the energy in the field. While this takes place there will be a temporary
E.M.I, set up in the conducting matter of the secondary circuit, energy will
move through it, and some of the energy will there be transformed into heat
or work, that is, a current will be induced in the secondary circuit.
(7) The electromagnetic theory of light.
The velocity of plane waves of polarised light on the electromagnetic
theory may be deduced from the consideration of the flow of energy. If
the waves pass on unchanged in form with uniform velocity the energy in
any part of the system due to the disturbance also passes on unchanged in
amount with the same velocity. If this velocity be v, then the energy con
tained in unit volume of cubical form with one face in a wavefront will all
pass out through that face in l/t'th of a second. Let us suppose that the
direction of propagation is straightforward, while the displacements are up
and down; then the magnetic intensity will be right and left. If (S be the
E.M.I, and «5 the m.i. within the volume, supposed so small that the intensities
may be taken as uniform through the cube, then the energy within it is
/i(£'/87r f [mS^^/Stt. The rate at which energy crosses the face in the wave
front is l^"«§/477 per second, while it takes 1/vth of a second for the energy in
the cube to pass out.
Then p>^m^^f^^
4.7TV 877 ^ 877 ^ ^
Now, if we take a face of the cube perpendicular to the direction of dis
placement, and therefore containing the m.i., the fineintegral of the m.i.
round this face is equal to 477 x current through the face. If we denote
distance in the direction of propagation from some fixed plane by z, the fine
integral of the M.I. is — J , while the current, being an alteration of dis
, ^ . K di^
placement, is j" "T~ •
ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 191
Therefore f = ^f (2)
But since the displacement is propagated unchanged with velocity v,
the displacement now at a given point will alter in time dt to the displace
ment now a distance dz behind, where dz == vdt.
Therefore W^'^Tz (^^)
Substituting in (2) ^ = Kv ^ ,
whence § = Kv^, (4)
the function of the time being zero, since S^ and (S* are zero together in the
parts which the wave has not yet reached.
If we take the lineintegral of the e.m.i. round a face perpendicular to
the M.I. and equate this to the decrease of magnetic induction through the
face, we obtain similarly
^ = iJ^v^ (5)
It may be noticed that the product of (4) and (5) at once gives the value
of V, for dividing out ($'*& we obtain
1 = ixKv''
1
or _ v= =^ .
But using one of these equations alone, say (4), and substituting in (1)
K for <^ and dividing by (E^, we have
K^K^ IJiKV
47r " Stt Stt
or 1 = p.Kv'^,
whence v = ^=^ . 
This at once gives us the magnetic energy equal to the electric energy, for
877 877 877
It may be noted that the velocity is the greatest velocity with which
the two energies can be propagated together, and that they must be equal
when travelhng with this velocity. For if v be the velocity of propagation
and 6 the angle between the two intensities, we have
^^ sin^ _ m^ fj.^
4:7TV StT 877 '
2 sin d
43 (i
192 ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD
The greatest value of the numerator is 2 when ^ is a right angle, and the
least value of the denominator is 2 V]JiK, when the two terms are equal to
each other and to VijlK.
1 TT
The maximum value of v therefore is . — , and occurs when 6 = ^ and
The preceding examples will suffice to show that it is easy to arrange
some of the known experimental facts in accordance with the general law of
the flow of energy. I am not sure that there has hitherto been any distinct
theory of the way in which the energy developed in various parts of the
circuit has found its way thither, but there is, I believe, a prevaihng and
somewhat vague opinion that in some way it has been carried along the
conductor by the current. Probably Maxwell's use of the term 'displace
ment' to describe one of the factors of the electric energy of the medium
has tended to support this notion. It is very difficult to keep clearly in mind
that this 'displacement' is, as far as we are yet warranted in describing it,
merely a something with direction which has some of the properties of an
actual displacement in incompressible fluids or solids. When we learn that
the ' displacement ' in a conductor having a current in it increases continually
with the time, it is almost impossible to avoid picturing something moving
along the conductor, and it then seems only natural to endow this something
with energycarrying power. Of course it may turn out that there is an
actual displacement along the lines of electromotive intensity. But it is
quite as likely that the electric 'displacement' is only a function of the true
displacement, and it is conceivable that many theories may be formed in
which this is the case, while they may all account for the observed facts.
Mr Glazebrook has already worked out one such theory in which the com
ponent of the electric displacement at any point in the direction of x is
^ V^^, where  is the component of the true displacement {VMl. Mag. June
1881). It seems to me then that our use of the term is somewhat unfortunate,
as suggesting to our minds so much that is unverified or false, while it is so
difficult to bear in mind how little it really means.
I have therefore given several cases in considerable detail of the apphcation
of the mode of transfer of energy in currentbearing circuits according to the
law given above, as I think it is necessary that we should reaUse thoroughly
that if we accept Maxwell's theory of energy residing in the medium, we
must no longer consider a current as something conveying energy along the
conductor. A current in a conductor is rather to be regarded as consisting
essentially of a convergence of electric and magnetic energy from the medium
upon the conductor and its transformation there into other forms. The
current through a seat of socalled electromotive force consists essentially
ON THE TRANSFER OF ENERGY IN THE ELECTROMAGNETIC FIELD 193
of a divergence of energy from the conductor into the medium. The magnetic
Hnes of force are related to the circuit in the same way throughout, while
the lines of electric force are in opposite directions in the two parts of the
circuit — with the socalled current in the conductor, against it in the seat
of electromotive force. It follows that the total e.m.i. round the circuit
with a steady current is zero, or the work done in carrying a unit of positive
electricity round the circuit with the current is zero. For work is required
to move it against the e.m.i. in the seat of energy, this work sending energy
out into the medium, while an equal amount of energy comes in in the rest
of the circuit where it is moving with the e.m.i. This mode of regarding the
relations of the various parts of the circuit is, I am aware, very different
from that usually given, but it seems to me to give us a better account of
the known facts.
It may seem at first sight that we ought to have new experimental indica
tions of this sort of movement of energy, if it really takes place. We should
look for proofs at points where the energy is transformed into other modifi
cations, that is, in conductors. Now in a conductor, when the field is in
a steady state, there is no electromotive intensity, and therefore no motion
and no transformation of energy. The energy merely streams round the
outside of the conductor, if in motion at all in its neighbourhood. If the
field is changing, energy can pass into the conductor, as there may be
temporary e.m.i. set up within it, and there will be transformation. But
we already know the nature of this transformation, for it constitutes the
induced current. Indeed, the fundamental equation describing the motion
of energy is only a deduction from Maxwell's equations, which are formed
so as to express the experimental facts as far as yet known. Among these
are the laws of induction in secondary circuits, and they must therefore
agree with the law of transfer. We can hardly hope, then, for any further
proof of the law beyond its agreement with the experiments already known
until some method is discovered of testing what goes on in the dielectric
independently of the secondary circuit.
p. c. w.
13
11.
ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND
THE ELECTRIC AND MAGNETIC INDUCTIONS IN THE
SURROUNDING FIELD*.
[Phil Trans. 176, 1885, pp. 277306.]
[Received January 31. Read February 12, 1885.]
In a paper published in the Philosophical Transactions for 1884 (Part ii,
pp. 343361 )t, I have deduced from Maxwell's equations for the electromagnetic
field the mode in which the energy moves in the field. The result there
obtained is that the energy moves at any point perpendicularly to the plane
containing the directions of the electric and magnetic intensities, and in the
direction in which a righthanded screw would move if turned round from
the positive direction of the electric intensity to the positive direction of the
magnetic intensity. The quantity crossing the plane per unit area per
second is equal to the product of the two intensities, multipHed by the sine
of the included angle, divided by 47r t.
Hence it follows that the energy moves along the intersections of the two
sets of level surfaces, electric and magnetic, where they both exist, their
intersections giving, as it were, the lines of flow. In the particular case of
a steady current in a wire where the electrical level surfaces cut the wire
* [Added July 15. Since the reading of the paper I have found a remarkable passage in
Faraday's Experimental Researches, vol, 1, p. 529, § 1659, which I give below. The words I have
put in italics might be regarded as the startingpoint of the views which I have attempted to
develop in this paper. '§ 1659. According to the beautiful theory of Ampere, the transverse
force of a current may be represented by its attraction for a similar current and its repulsion of
a contrary current. May not then the equivalent transverse force of static electricity be repre
sented by that lateral tension or repulsion which the lines of inductive action appear to possess
(1304)? Then, again, when current or discharge occurs between two bodies, previously under
inductrical relations to each other, the lines of inductive force will weaken and fade away, and, as
their lateral repulsive tension diminishes, will contract and ultimately disappear in the live of
discharge. May not this be an effect identical with the attractions of similar currents, i.e., may
not the passage of static electricity into current electricity, and that of the lateral tension of the
lines of inductive force into the lateral attraction of Hnes of similar discharge, have the same
relation and dependence, and run parallel to each other?']
t [Collected Papers, Art. 10.]
J I here adopt the simpler term 'Electric Intensity,' denoted by e.i., instead of 'Electro
motive Intensity,' for the force which would act on a small body charged with unit of positive
electrification. The magnetic intensity, i.e., the force which would act on a unit northseeking
Pole, will be denoted by m.i.
ON THE CONNECTION BETWEEN ELECTRIC CURRENT, ETC. 195
perpendicularly to the axis, it appears that the energy dissipated in the wire
as heat comes in from the surrounding medium, entering perpendicularly to
the surface.
In that paper I made no assumption as to the transfer of the electric and
magnetic inductions — the electric and magnetic conditions — through the
medium, merely considering the movement of energy. I now propose to
develop a hypothesis as to the transfer of the inductive condition in the
medium, and its movement inwards upon currentbearing wires.
The value of the electric induction at any point in an isotropic medium is
equal to K x E.I./477, and the direction of the induction coincides with that
of the intensity. Maxwell terms this electric induction 'displacement,' but
I think that 'induction' is preferable, as it impUes no hypothesis beyond
that of some alteration in the medium, which can be described by a vector.
The value of the magnetic induction is equal to /x x m.i., and its direction
coincides with that of the magnetic intensity.
If we symbolise the electric and magnetic conditions of the field by
inductiontubes running in the directions of the intensities, the tubes being
supposed drawn in each case so that the total induction over a crosssection
is unity, then we have reason to suppose that the electric tubes are con
tinuous except where there are electric charges, while the magnetic tubes
are probably in all cases continuous and reentrant.
In the neighbourhood of a wire containing a current, the electric tubes
may in general be taken as parallel to the wire while the magnetic tubes
encircle it. The hypothesis I propose is that the tubes move in upon the
wire, their places being supplied by fresh tubes sent out from the seat of the
socalled electromotive force. The change in the point of view involved in
this hypothesis consists chiefly in this, that induction is regarded as being
propagated sideways rather than along the tubes or lines of induction. This
seems natural if we are correct in supposing that the energy is so propagated,
and if we therefore cease to look upon current as merely something
traveUing along the conductor carrying it, and in its passage affecting the
surrounding medium. As we have no means of examining the medium, to
observe what goes on there, but have to be content with studying what takes
place in conductors bounded by the medium, the hypothesis is at present
incapable of verification. Its use, then, can only be justified if it accounts
for known facts better than any other hypothesis.
The basis of MaxwelVs Electromagnetic Theory.
Maxwell's Electromagnetic Theory rests on three general principles.
I. The first principle consists in the assumption that energy has position,
i.e., that it occupies space. The electric and magnetic energies of an electro
magnetic system reside therefore somewhere in the field. It is an inevitable
13—2 *
196 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE
conclusion that they are present wherever the electric and magnetic intensities
can be shown to exist. For instance, suppose a small electrified body placed
in a field where there is electric intensity ; then the body will be acted on by
force and will receive energy which appears as the energy of motion, the
electric energy at the same time decreasing. If energy has position, that
which is now in the body must have come into it through the surrounding
space, or it was present in that space before the body took it up. The
alternative that it appeared in the body without passing through the space
immediately surrounding the body need not be discussed. Hence the
existence of electric intensity impUes the existence of electric energy in the
place where the electric intensity is capable of manifestation. Similarly
magnetic energy accompanies magnetic intensity. The inductive condition
of the medium imagined by Faraday is due then to its modification when
containing energy. Maxwell has shown that all the energy is accounted for
on the supposition that the electric energy per unit volume at any point is
K{B.i.yj87T, and that the magnetic energy is ju, (m.i.)2/877. He has given in
his Elementary Treatise on Electricity, p. 47, another way of describing the
distribution of energy which will be more useful for my purpose. If the
field be mapped out by unit inductiontubes — either electric or magnetic —
i.e., tubes drawn so that the total induction over every crosssection of a tube
is unity, and if these tubes be divided into cells of length such that the
difference of potential or the lineintegral of the intensity between the two
ends of each cell is unity, then each cell contains, if electric, half a unit of
enero^v, if magnetic 5^ of a unit, the divisor 47r being introduced bv the
077 ''^ "
difference in definition of the two inductions. Maxwell terms these unit
cells.
II. The second principle is in part experimental, viz. : — that the line
integral of the electric intensity round any closed curve is equal to the rate
of decrease of the total magnetic induction through the curve. This is
verified by experiment when the curve is drawn through conducting material.
Maxwell supposes it to be true in all cases, that is, he supposes that electric
induction can be produced in insulators by means of magnetic changes,
without the presence of charges on conductors, and is therefore led to identify
the growth and decrease of electric induction with current.
III. The third principle is also in part experimental, viz. : — that the
hneintegral of the magnetic intensity round any closed curve is equal to
iv X current through the curve. This is verified by experiment when the
current is in a wire, and Maxwell supposes it to be also true in the case where
there is change of electric induction in an insulator. The supposition is
justified by Prof. Rowland's wellknown experiment.
From these three principles Maxwell deduces his general equations of the
ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 197
Electromagnetic Field. I have stated them in full as I propose to modify
the second and third principles, and I wish to make quite clear the nature
of the proposed changes.
Modification of the Second Principle.
I propose to replace the second principle by the following : Whenever
electromotive force is produced by change in the magnetic field, or by motion of
matter through the field, the e.m.f. per unit length or the electric intensity is equal
to the number of tubes of magnetic induction cutting or cut by the unit length per
second, the e.m.f. tending to produce induction in the direction in which a right
handed screw would move if turned round from the direction of motion relatively
to the tubes towards the direction of the magnetic induction*.
In order that the results obtained from this should agree with those
obtained from Maxwell's statement of the principle, it is necessary that
change in the total quantity of magnetic induction passing through a closed
y curve should always be produced by the passage of inductiontubes through
" the curve inwards or outwards. In some instances this is undoubtedly the
case, as, for instance, where a part of a circuit moves so as to cut a fixed
magnetic field, or where a magnet moves in the neighbourhood of a circuit.
Here the e.m.f. is equal to the number of tubes cut by the wire per second,
and its seat is that part of the wire cutting the tubes. In other cases, as,
for instance, where the wire is between the poles of an electromagnet whose
magnetising current is changing, we have no direct experimental evidence
of the movement of the induction in or out. But the induction tubes are
closed, and to make them thread a circuit we might expect that they would
have to cut through the boundary. The alternative seems to be that they
should grow or diminish from within, the change in intensity being propagated
along the tubes. This would be inconsistent with their closed nature, unless
the energy were instantaneously propagated along the whole length, and is
further negatived by the theory of the transfer of energy, which implies that
the energy flows transversely to the direction of the tubes. I shall suppose,
then, that alteration in the quantity of magnetic induction through a closed
curve is always produced by motion of inductiontubes inwards or outwards
through the bounding curve.
* Taking the electric intensity as always perpendicular to the plane of motion of the magnetic
tubes through a point, and equal to the number cut per second by unit length of the normal
to the plane of motion, we can easily show that the component of the intensity in any other
direction will be equal to the number of tubes cut by a Hne of unit length in that direction. For
let OA represent a small length drawn perpendicular to the plane of motion, and let OP represent
a line drawn in any direction making an angle 6 with OA. Draw AP perpendicular to OA, and
meeting OP in P. Then the same number of tubes will cut both OA and OP, since AP is parallel to
their plane of motion. If the number cutting OA he Ex OA, where E is the number cutting unit
length, and therefore equal to the resulting intensity, the number cutting unit length of OP will
OA
be E . Yyp = E cos d, or the component of the intensity along OP.
198 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE
Modification of the Third Principle.
The third principle admits of similar analysis, according to which we may
regard the magnetic intensity along a closed curve as due to the cutting of
the curve by tubes of electric induction. If we regard the lineintegral of
the magnetic intensity round a tube of induction as measuring the magneto
motive force— employing a useful term suggested by Mr. Bosanquet — we may
put the modification in the following form :
Whenever magnetomotive force is produced by change in the electric field, or
by motion of matter through the field, the magnetomotive force per unit length is
equal to iir x the number of tubes of electric induction cutting or cut by unit
length per second, the magnetomotive force tending to produce induction in the
direction in which a righthanded screw would move if turned round from the
direction of the electric induction towards the direction of motion of the unit
length relatively to the tubes of induction.
This is the most general form of the principle, but we shall only require
the more special statement which immediately follows from it : that the
lineintegral of the m.i. round any curve is equal to 477 x the number of tubes
passing in or out through the curve per second.
We have reasons exactly similar to those given in the last case for supposing
that any change in the total electric induction through a curve is caused by
the passage of inductiontubes in or out across the boundary. The alternative,
that change takes place by propagation from the ends, seems inconsistent
with the theory of the transverse flow of energy.
I shall postpone the discussion of the modifications of the general equations
of the electromagnetic field following from these changes in the fundamental
principles, and proceed to discuss the bearing which they have upon the nature
of currents in conductors.
A straight wire carrying a steady current.
Let AB represent a wire in which is a steady current from A to B. The
direction of the electric induction in the surrounding field near the wire, if
the field be homogeneous, is parallel to AB.
Let E be the value of the electric intensity, or the difference of potential
per unit length perpendicular to the level surfaces, and let R be the resistance
E
of the wire per unit length. Then = ^ where C is the current, and C is
uniform throughout the circuit. The magnetic intensity in the immediate
2C
neighbourhood of the wire at a distance r from the axis of the wire is — .
The hypothesis proposed as to the nature of the current is that C electric
ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 199
c^)
inductiontubes close in upon the wire per second. The wire is not capable
of bearing a continuallyincreasing induction, and breaks the tubes up, as it
were, their energy appearing finally as heat*.
Let us see how this hypothesis accounts for known facts,
when aided by the two principles just laid down.
It accounts at once for the constancy of the current at all
parts of the wire in the steady state, in so far as it reduces
this constancy to a particular case of the law according to
which there is the same total induction over all crosssections
of a tube. If, for instance, there were more induction entering
at A than at B, then more tubes must be entering at A, and
so there would be an increase in the number of tubes left in
the medium about B, or the field would not be steady.
Further, if we draw any closed curve embracing the wire
once, we may apply the third principle to give us the fine
integral of the magnetic intensity round the curve. For this ^ .
is a case where change is certainly going on in the electric field. ^jg, i,
and the magnetomotive force is due to this change. The field
being steady, if C tubes enter the wire and are there broken up, C tubes
must cross through any encirchng curve to supply their place, or the line
integral of the magnetic intensity round the curve is equal to iir x number of
tubes passing through the boundary per second, i.e., iirC. If the curve be
a circle of radius r, with its centre in the axis and plane perpendicular thereto,
the intensity at any point of this circle will be tangential to it, and equal to
477(7 ^ 2C
277r r
The known constancy of the lineintegral of the magnetic intensity round
the wire, which the hypothesis thus accounts for, almost seems to force the
hypothesis upon us, if we regard the field as caused by the inward flowing of
the energy rather than by something propagated out from the wire.
Assuming that the inductiontubes bring in their energy, the quantity is
easily found. The number of unit cells per unit length is equal to the difference
of potential per unit length, or E. Hence the energy per unit length of each
E .
tube is ^ , since each cell contains a half unit. If C tubes disappear in the
CE
Now the total
energy dissipated per unit length is CE per second. Or the movement
wire per second, they yield up ^ of energy per unit length.
* May we not say that the tubes are dissolved ? The term seems to suggest that the induction
is not destroyed, but only loses its continuity. Probably this is the case; for on the electro
magnetic theory of radiant energy, when the wire is heated, it sends out the energy it received,
again in the electromagnetic form.
200 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE
inwards of the electric induction will only account for half of the energy.
The other half must be accounted for by the movement inwards of the
magnetic induction. This movement of the magnetic induction is suggested
by the existence of electric induction, which cannot be ascribed to statical
charges.
The electric intensity is E. Hence E tubes of magnetic induction must
move in per second, cutting unit length parallel to the axis of the wire, in
accordance with the second principle, and it will easily be seen that the inward
motion gives the right direction of the electric intensity. The hneintegral
of the magnetic intensity round a tube is ^ttO, the tubes being closed rings.
Hence there are 4:7rC unit cells in the length. Since each of these contains
5— of energv, the quantitv per tube = ^ — = o • ^ tubes entering the wire
07T " " 077" Z
OF'
per second will carry in ^ of energy, the other half to be accounted for.
We can in a similar manner trace the dissipation of the energy, which we
must suppose taking place within the wire. The lineintegral of the magnetic
intensity round a circle, with its centre in the axis of the wire, is constant
up to the wire, and equal to irrC. Within the wire it gradually diminishes
as the circle contracts. At a distance r from the centre it is ^ttC ^ where
a is the radius of the wire. If we assume this intensity to be still due to the
passage inwards of the tubes of electric induction only, — ^ cross inwards
per second at a distance r, the difference between this number and the C tubes
entering the outer boundary being destroyed and their energy dissipated.
The energy thus dissipated per unit length between the outer boundary and
a coaxal cylinder of radius r will be ^ [l A per second. H r = the
whole of the electric energy is dissipated. It would appear, then, that we
may represent the dissipation of the electric energy by the total destruction
of the tubes all through their length.
The value of the electric intensity being E throughout the wire the number
of tubes of magnetic induction cutting unit length parallel to the axis is the
same at all parts, viz., E per second. Hence the magnetic tubes are not
destroyed as the electric tubes are. But the Hneintegral of the magnetic
mtensity round the tubes diminishes as they approach the axis, being ^tt — ^
round that at distance r. The number of unit cells diminishes, and, therefore,
the energy per^tube is less, the decrease being due to that dissipated. Thus
4 OF OF
the energy entering in the E tubes at the outer boundary is "^ or ^ .
877 2i
ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 201
That crossing in E tubes at a distance ri&^ E = ^ —  . The difference
877 a^ 2 a^
^(1
gj has been dissipated.
Hence it appears that the energy dissipated per second may be repre
sented as half electric half magnetic, the electric energy being dissipated by
the breaking up of the tubes and their disappearance, while the magnetic
energy is dissipated by the shortening of the tubes and their final disappear
ance by contraction to infinitely small dimensions of the diameters of the
rings by which we may represent them. At all points therefore outside and
inside the energy crossing any surface may be represented as equally divided
between the two kinds.
As we know the value of the induction at any point, or the number of
tubes passing through unit area, and as we also know the number of tubes
cutting the boundary it is easy, on the assumption that the tubes move on
unchanged, to calculate their velocity. Of course this velocity is purely
hypothetical, as we cannot examine minutely into the medium and observe
what goes on there. Probably, if we could observe with sufficient minuteness
we should find unevennesses in the induction. If the velocity of the tubes
has any physical meaning it is that these unevennesses are carried forward
with that velocity. To illustrate this let us suppose that we have water
flowing through a glass tube at a steady rate. We have nothing to show
that the water is moving past any point in the tube beyond its disappearance
at the entrance and its appearance at the exit, but knowing the crosssection
of the tube, i.e., the quantity of water in any part of it, and the quantity
entering and leaving, it is easy to assign a velocity to the water in the tube
which shall account for the observed amount entering and leaving. This
velocity is to a certain extent hypothetical. But if we examine the tube
with a sufficient magnifying power to show particles of dust in the water
the existence of the velocity receives a more direct proof. I do not know
whether we should have any right to expect a similar proof of the motion of
induction even if we had the means of observation.
To find the hypothetical velocity of the electric inductiontubes let us
calculate the number of tubes passing through a circular band with radii
r and r \ dr and centre in the axis of the wire, and lying in a plane perpen
KE
dicular to the axis. The intensity being E the induction is r— , and theretore
4:7T . ....
the area of crosssection of each tube is fv^, since area x induction is unity.
The number passing through the circular band is therefore
^ , KE KErdr
ZTvrdr . j— = — ^ — .
4:77 2
202 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE
Since C tubes move in through the inner circle per second, — ^ — tubes
move in in of a second, i.e., all the tubes passing through the band will
have just moved in in this time. The outermost tubes therefore describe
the space dr in time — ^r^:^ , or the velocity is ^^ . Now we know that if
E
R be the resistance per unit length, C = ^. Hence we may put the velocity
in the form
2 1
KR' r'
which is independent of the current.
To take a special case, let us calculate the velocity just outside the
boundary of a copper wire, the specific resistance of copper being 1642 in
electromagnetic measure. Then if a be the radius of the wire
,2
and K = ^ where v is the ratio of the units, which in air may be taken as
3 X IQio.
2'U^77tt^
Then the velocity = v/> /^~
^ 1642a
2 X 9 X IO^Ott^
1642
 345 X lO^S.
At greater distances the velocity will be less, diminishing according to the
inverse distance.
The hypothetical velocity of propagation of the magnetic induction may
be calculated in a similar manner. The intensity at a distance r from the
2C ^llC
axis is' — and the induction is ^— . The area of each tube is therefore
r r
T
^— ^ , and the number lying in a ring of rectangular section with depth unity
and interna] and external radii r and r + dr, will he 1 x dr ^ ^^ = — .
2/xC r
But E tubes move in per second through the inner face of the ring, so that
"^yiCdr . . . 2uiCdr
—~ — tubes move m m time ^^  , or this is the time taken by the outer
r Er ^
most tubes to move across the ring describing a distance dr. The velocity is
therefore
Er Rr
'2iiC~~ 2jLt'
which is again independent of the current.
ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 203
If the currentbearing wire is copper, R = , and with ft = 1 the
7Ta'
velocity becomes
1642r
277^2
We cannot assign a velocity to the electric tubes within the wire since
the number is diminishing as their energy dissipates. But the magnetic
tubes crossing unit length parallel to the axis are still unchanged in number,
so that we may assign a velocity to them. This velocity means that with
the known value of the magnetic induction this velocity will give the number
crossing inwards required to produce electric intensity E,
The velocity will be found equal to
2/xCr ^^ 2/xr'
In the case of a copper wire this becomes
1642
2/x77r *
Discharge of a condenser through a fine wire.
Let us suppose that we have a condenser consisting of two parallel plates
A and B and charged with equal and opposite charges. Then we know that
there will be electric induction between the two plates, and that according
to Maxwell's theory the energy of the system is stored there. We may form
an idea of the distribution of the energy by drawing the unit inductiontubes,
each starting from and ending in unit quantity of electricity, and dividing
these into unit cells by the level surfaces, drawn at unit difference of potential
(Fig. 2). If the dimensions of the plates be great compared with their distance
apart, then nearly all the cells will be between the two plates, and since each
cell contains half a unit of energy, nearly all the energy is there. There will,
204 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE
however, be slight induction, and therefore some small quantity of energy in
the surrounding space.
Now let the two plates be connected by a wire. Discharge takes place,
and we are fairly justified, from the heat in the wire and the transient magnetic
effects, in saying that a current has been in the wire from the positive to the
negative plate, or the wire was for the time being in the same relation to the
surrounding medium as the wire in the case just considered, the condition of
affairs, however, not being steady.
Let us suppose the wire to have a very great resistance, in order that, at
least in imagination, we may lengthen out the time of discharge. On the
ordinary current theory, combined with Maxwell's 'displacement' theory,
the medium between the plates has returned from the strained condition,
denoted by 'displacement' from the positive to the negative plate, causing
displacement through the plates and along the wire, the displacement being
in the same direction all round the circuit. This is generally, I think,
supposed to take place by the recovery of the medium between the plates
causing displacement in the metal immediately in front of it, the displacement
being analogous to the forcing of water along a pipe corresponding to the
plates and wire, by the recovery from strain of some substance placed in
a chamber corresponding to the space between the plates.
According to the hypothesis here advanced we must suppose the lessening
of the induction between the plates — induction being used with the same
physical meaning as Maxwell's displacement — to take place by the divergence
outwards of the inductiontubes. We may picture them as taking up the
positions of successive Hues of induction further and further away from the
space between the plates, their ends always remaining on the plates. They
finally converge on the wire, and are then broken up and their energy dissi
pated as heat. At the same time some of the energy becomes magnetic, this
occurring as the difference of potential between the plates lowers, so that the
tubes contain fewer unit cells.
The magnetic energy will be contained in ringshaped tubes which will
expand from between the plates and then contract upon some other part of
the circuit. To illustrate the movement of the electric inductiontubes let
us suppose them to be represented by elastic strings stretched between the
two plates. Then the motion of the tubes outwards would be roughly repre
sented by pulhng the elastic strings outwards and doubhng them back close
against the wire, their ends being still attached to the plates. It is evident
that if any ring surround the wire each of the strings must break through it
in order to reach the wire. Hence the total number of strings cutting any
ring surrounding the wire is the same wherever the ring be placed. Similarly
the total number of tubes of electric induction cutting any curve encirchng
the wire is the same, and therefore the fineintegral of the magnetic intensity
ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 205
round the curve integrated throughout the time of discharge is the same, or
the total magnetic effect is the same at all parts of the circuit. It is not
necessary to suppose that a tube enters the wire at the same moment through
out its whole length ; indeed, the experiments of Wheatstone on the socalled
velocity of electricity prove clearly that this is not the case, for in those
experiments the tubes reached airbreaks near the two ends of the wire before
they reached a break in the middle.
We cannot by this general reasoning show that the energy entering any
length of the wire will be proportional to the resistance of that length — the
result obtained by Kiess. Indeed, this cannot always be the case. For
instance, imagine a condenser discharged by two wires connected to the two
plates of another condenser of greater capacity, whose plates are again
connected by a fine wire of enormous resistance, through which the discharge
can only take place slowly. Then the energy dissipated in the wires will not
to a first approximation depend on their resistances but on the ratios of the
capacities, that in the wire of high resistance bearing to that in the other
wires the ratio of the less capacity to the greater. Probably Riess's results
only hold when the discharge takes place in such a way that it may be looked
upon at any one moment as approximately in the steady state.
We have shown that the magnetic measure of the total current is the
same all along the wire. Probably also the chemical measure is the same —
meaning by the chemical measure whatever interchanging or turning round
of molecules may occur when induction takes place in a conductor. For
even if a tube does not enter the wire at the same time throughout its length,
an end part, say, entering first, the point of attachment of the tube to the
conductor being transferred from the plate to somewhere along the wire, this
transference of the point of attachment from molecule to molecule imphes
the same amount of chemical change within the wire as if the tube entered
all at the same moment. It will not, however, take place equally throughout
the crosssection as it does in the steady state.
Probably we only have the simultaneous disappearance of all parts of
a tube when the wire follows a line of electric induction, and has its resistance
per unit length proportional to the intensity which would exist there if the
wire were removed.
The hypothesis here advanced is in accordance with Maxwell's doctrine
of closed currents. For the induction dissipated at one part of the circuit
has come there from another part where relatively to the circuit it ran in
the opposite direction. The total result is equivalent to the addition of so
many closed inductiontubes to the circuit, the induction running the same
way relatively to the circuit throughout.
If the two plates of the condenser are not connected by a wire but are
discharged gradually by the imperfect insulation of the dielectric, then we
206 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE
must suppose that the tubes of induction in this case are dissipated in situ,
the induction simply decaying at a rate depending on its amount and upon
the conductivity of the dielectric. We may still represent this process by
a closed current by regarding the loss of induction (Maxwell's — j.) and the
quantity of induction dissipated (Maxwell's f) as two different quantities.
We have then p{ J = ov we have two equal and opposite currents. But
this seems artificial. It is more natural to look upon the process merely as
a decay of electric induction without movement inwards of fresh induction
tubes, and therefore without the formation of magnetic induction.
I have discussed the case of discharge of a condenser at some length, as
we can here reahse more easily what goes on at the source of energy. The
results obtained suggest that a similar action occurs at the source of energy
or seat of the electromotive force in other cases where we do not know the
distribution of induction, and are obhged to guess at the action.
A circuit containing a voltaic cell.
We may pass on from the discharge of a condenser to the consideration
of the current in a circuit containing a voltaic cell. The chemical theory of
the cell will be here adopted — in fact, the hypothesis I am endeavouring to
set forth has no meaning on the voltaic metalcontact theory.
Let us suppose the cell to consist of zinc and copper plates, a vessel of
dilute sulphuric acid, and copper wires attached to each plate which on
junction complete the circuit. For simplicity I shall disregard the effect
of the air and suppose that it is a neutral gas causing no induction.
We shall begin by supposing the circuit open. Then we know that on
immersion there will be temporary currents in the wires, the quantities of
these currents depending on the electrostatic capacity of the system composed
of the wires. The currents last till the wires have received charges such that
they are, say, at difference of potential V. If the terminals are connected
to a condenser the temporary currents may be easily detected by a galvano
meter in the circuit. They are in no way to be distinguished in kind from the
permanent current which will be estabhshed when the circuit is complete,
except that they are of short duration and in general very small. There is
no reason then to suppose that the action in the cell is different from that
which takes place when the current is permanent, and I think we may safely
assume that Faraday's law of electrolysis holds according to which the
quantity of electricity flowing along either wire is proportional to the quantity
of chemical action — or, in the form appropriate here, the number of tubes of
induction produced is proportional to the quantity of chemical action.
ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 207
Let Q be the total quantity of electricity upon the positive terminal ; then
QV
^r is the total energy thrown out into the dielectric.
Let z be the quantity of zinc consumed per unit of electricity, then Qz is
the total quantity consumed in the charging of the terminals. Let E be the
energy set free by each quantity z of zinc consumed, after all actions in the
cell have been provided for. E then is the e.m.f. which the cell will have
on the closure of the circuit, as long as the chemical actions remain the same,
for z corresponds to the passage of a unit of electricity or the production of
one tube, and we know that the energy set free by C units is CE.
Now while the charges are gathering and while the potential difference
of the terminals is gradually increasing, the energy required to add equal
increments of charge will also increase, and the charging will cease when the
amount of energy given up by a given amount of chemical action in the cell
is equal to the amount required to add the corresponding charge to the
terminals. For to suppose the action to go beyond this is to suppose that
the energy thrown out into the space between the terminals is greater than
that yielded by the battery.
Let dQ be the last quantity of charge added to the terminals. This
requires energy VdQ.
The corresponding quantity of zinc consumed is zdQ, giving up energy EdQ.
The condition of equilibrium is that
VdQ = EdQ
or V^E,
which agrees with the result of experiment that the difference of potential
of the terminals in open circuit is equal to the e.m.f. of the cell immediately
after closure.
It may be noticed that the total quantity of energy extracted from the
batterv is
QE = QV,
while the electric energy left in the medium is
QV
2 '
or half the energy has been converted into heat in the wires.
We will now consider the distribution of level surfaces in the field while
the circuit is still open. There will be F — 1 surfaces between the terminals,
dividing each tube into V cells. None of these will cut the homogeneous
parts of the circuit, since the whole of each of these must be at one and the
same potential.
They can only cut the circuit by passing through the regions where there
208 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE
is contact of dissimilar bodies. We will neglect the contact of the zinc and
copper, as the difference of potential there is insignificant compared with
that at the two surfaces, zincacid and copperacid.
Now we know that the energy of the cell is put out at the zincacid contact,
but the amount is greater than that obtained from a consideration of the
E.M.F. of the cell, for some energy is absorbed again, probably, at the copper
acid contact in the evolution of hydrogen. There is probably, then, induction
between the acid and the zinc, and between the acid and the copper, these
resembUng the spaces between the plates of two condensers, the acid being
at a higher potential than either. But if a given amount of induction dis
appears from the zincacid contact and appears at the terminals, more energy
is lost at the former than appears at the latter. Hence all the cells have not
been transferred from one to the other, or the difference of potential zincacid
Fig. 3.
is greater than 7. Then more than V — I level surfaces pass between the
zinc and the acid, the excess over 7—1 going round and passing between
the copper and the acid, somewhat as in Fig. 3, where A, B are the metal
plates. The surfaces are roughly sketched and numbered, on the supposition
that the zinc terminal is at 0, the copper at 5, and the acid at 8. They have
probably the same shape as those which would be produced by condensers
at A and B with the wires attached, respectively, to one terminal of each,
the other terminals being connected together and the charges adjusted so
that the difference of potential of the two terminals at A was 3, while that
at B was 8.
Let us now suppose the circuit closed. Then the level surface will 'cut
the circuit at various points, somewhat as in Fig. 4.
The energy being dissipated in the wire, the cell will continually send out
fresh energy, the inductiontubes, which proceed from the acid to the zinc.
ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 209
diverging outwards in the same way as described in the discharge of a con
denser. They bend round, and finally go into the circuit, the energy they
carry being used for the necessary molecular changes, and finally appearing
as heat in the circuit — except at the copperacid contact where there is a
crowding in of level surfaces, and therefore a convergence of more energy,
which is required to set the hydrogen free.
Fig. 4.
At the same time magnetic ringshaped tubes will be continually sent out
from the zincacid contact, expanding for a time and then contracting again
on various parts of the circuit and also giving up their energy.
There is, therefore, a convergence of tubes of electric induction on the
circuit, running in the same direction throughout, viz., from copper to zinc
outside the cell, and from zinc to copper inside, except between the zinc
and acid, where there is a divergence of tubes in which the induction runs
in the opposite way. But a divergence of negative tubes causes magnetic
intensity in the same direction as, and may therefore be considered as equiva
lent to, a convergence of positive tubes. The current may therefore be said
to go round the circuit in the same way throughout.
The tendency to a steady state in which the current or the number of
inductiontubes broken up per second is the same at all parts of the circuit,
admits of simple explanation. We know, as the result of experiment given
P
by Ohm's law, that C' = p where R is the resistance per unit length and E
the electric intensity. Until we can explain the molecular working of the
current, i.e., the mode in which the inductiontubes are broken up, we must
accept Ohm's law as a simple fact. Let us suppose that we have not yet
arrived at the steady state, so that in some part of the circuit the electric
intensity is less than in the steady state, while in another part it is equal
p. c. w. 14
210 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE
to it or greater. Let the steady value of the intensity be E, the actual value
in the former part E' , and in the latter E" , By Ohm's law the number of
tubes absorbed by the wire per second is given by C = E' jR, and C" = E" jR,
in the two parts respectively, so that C"< C" since E' < E" or less tubes are
being destroyed in the first than in the second part. But all the tubes are
sent out from the source of the energy, and are only destroyed in the circuit,
being otherwise continuous and with their two ends in the circuit. Hence,
if more tubes are destroyed at one part than another, the parts of the tubes
not yet destroyed will gather in the medium surrounding the part where
fewer are destroyed, increasing the induction there, and so raising the intensity
in the wire and therefore the number of tubes destroyed. The field can
evidently only be steady when the number of tubes destroyed in all parts
of the circuit is the same.
But it does not follow that in the steady state each tube enters the wire
along its whole length at the same moment. This would imply that the axis
of the wire is a line of electric induction perpendicular everywhere to the
level surfaces. If we draw the level surfaces due to the seats of induction
at the contacts of acid and metal, they will probably be somewhat as drawn
in Fig. 4. If now the wire is not so arranged as to follow with properly
adjusted resistances a line of induction for these surfaces, but pursues an
irregular course, then the level surfaces will be much distorted, and the
distribution of the induction will be greatly altered.
We may ascribe this alteration to a distribution of electricity along the
wire, the quantity in any small area on the surface of the wire being equal
to the difference between the number of tubes which have entered and the
number which have left that area since the beginning of the system. We
have a famihar example of this in the charging of deepsea cables. Another
example is afforded by a condenser with terminals connected to two points
in the circuit. The plates of the condenser are then virtually parts of the
circuit.
The effect of a junction of two wires, say of the same diameter, but of
different specific resistances, upon the level surface will resemble that of a
charge upon the separating surface. This can be seen in a general way from
the fact that the level surfaces must cut the wire with the higher specific
resistance at intervals shorter than those at which it cuts the other wire.
If there be an insulated conducting body, say a metal sphere, near the
circuit, we know that in the steady state there is no electric intensity, and
therefore no current within it ; consequently there is no movement of energy
and no movement of induction through it. We can see how this condition
is arrived at. As the first tubes of electric and magnetic induction come
up to the sphere they will enter it, and the parts of the electric induction
tubes thus entering will be broken up, causing a transient current in the
ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 211
sphere. The parts of the tubes left in the medium will end on the sphere
giving a negative charge on the end nearer the regions of higher potential,
and a positive charge on the end nearer the regions of lower potential. This
will go on until such charges have accumulated that the sphere becomes
itself a level surface. When this point is reached no more energy can enter
the sphere, and the parts of the magnetic tubes within it cease to move.
The charges formed on the wire or on neighbouring conductors are to
be distinguished from ordinary statical charges in this : that their existence
depends on the existence of the current, and therefore on the motion of
magnetic induction. If the current is stopped by a break in the circuit, so
that the motion of the magnetic induction ceases, the electric induction
ceases and the charges are all lost. We should expect, therefore, to find
that these charges can be described in terms of the magnetic motions which
have occurred and are occurring in the system.
Current produced by motion of a conductor iyi a magnetic field.
We may explain by general reasoning the production of a current by
motion of a part of a circuit so as to cut the tubes of magnetic induction.
We will consider the simple case of a sHder
AB, Fig. 5, running on two parallel rails, ^ ^
AC, BD, with a fixed crosspiece CD, the
tubes of magnetic induction running from
above downwards through the paper. Let
AB move so as to enlarge the circuit. We
know from experiment that this tends to Fig. 5.
cause a current in the direction ACDB.
As AB moves through the field its motion tends to cause electric intensity
in the direction BA. At the same time its kinetic energy is being continually
converted into electric and magnetic energy which travels to the rest of the
circuit there to be dissipated, that is, there must be a divergence of energy
from AB. Instead then of a convergence of positive tubes running from
B to A, we shall have what is magnetically equivalent — a divergence of
negative tubes or tubes running from A to B, their motion outwards being
accompanied by tubes of magnetic induction running round in the same
way as if there were an ordinary current from B to A. These magnetic
tubes must be supposed to move outwards in order to account for the direction
of the electric intensity*.
When these electric and magnetic tubes converge upon the rest of the
circuit they will evidently form a current running in the direction ACDB.
* Note added July 15: The above must not be regarded as an attempt to explain the
production of electric induction by the motion of a conductor in a magnetic field, but merely as
an attempt to show how the induction arising in the moving part of a circuit finds its way into
the rest of the circuit.
14—2
212 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE
We have here taken, just as in the case of the condenser and the voltaic cell,
the lessening of negative induction by its motion outwards, as equivalent to
the increase of positive induction by its motion inwards, and we have con
sidered both of them to indicate the apphcation of electric intensity in the
same direction in the conductor.
If instead of considering AB as a whole we break it up into elements,
each element will be a source of diverging negative tubes, and the remainder
of AB will, to that element, be a part of the rest of the circuit. Hence some
of the energy sent out from the element will converge on and be dissipated
in AB, or AB will be heated just as the rest of the circuit.
The general equations of the electromagnetic field.
We can easily obtain equations corresponding to and closely resembling
those of Maxwell by means of the principles upon which this paper is founded.
The assumption that if we take any closed curve the number of tubes of
magnetic induction passing through it is equal to the excess of the number
which have moved in over the number which have moved out through the
boundary since the beginning of the formation of the field, suggests a historical
mode of describing the state of the field at any moment.
Let a, b, c be the components of magnetic induction at any point 0.
Consider a small area dy dz close to the point, then the number of tubes
passing through the area dy dz will be adydz. This will be equal to the
difference between those which have come in and those which have gone out.
Let Ldx, Mdy, Ndz denote the numbers of tubes which have cut the
lengths dx, dy, dz since the beginning of the system, those being positive
which have tended to produce electric intensity in the positive direction
along the axes, and those being negative, and therefore subtracted, which
have tended to produce intensity in the opposite direction.
Let us consider the number which has come into the area
OB DC = dydz (Fig. 6). The number which has come in
across OB is  Mdy ( because the movement of tubes
passing through dydz in the positive direction must be
outwards to produce e.i. along OB). The number which
has passed out across CD is  (m + j dz) dy. The differ ^ig ^■
dM
^nce is ^ dydz. The number which has come in across OC is + Ndz
{+ because the movement of tubes passing through dydz in the positive
direction must be inwards to produce e.i. along OC). The number which
has passed out across BD is (n + ^ dy] dz. The difference is  ^
ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 213
The number still passing through dydz is therefore (^ ^^ dydz.
dy.
Equating this to the actual induction through the area, viz.,
adydz
and performing the same process for the corresponding areas dzdx, dxdy, we
obtain
dM dN\
dz dy
dN_dL
dx dz
a =
c =
dL
dy
dM
dx
(1)
Comparing these with Maxwell's equations (vol. 2, p. 216) we see that
dM_dN^dH_dG
dz dy dy dz '
with two similar equations, F, G, H being the components of the vector
potential. We should obtain Maxwell's equations if we defined F, G, H to
be the number of tubes which would cut the axes per unit length if the system
were to be allowed to return to its original unmagnetic condition, the tubes
now moving in the opposite direction. According to Maxwell, the vector
whose components are F, G, and H 'represents the timeintegral of the
electromotive force which a particle placed at the point {x, y, z) would
experience if the primary current were suddenly stopped' (vol. 2, 2nd ed.,
p. 215). If the electric intensity is produced by the motion of magnetic
induction, then our definition of F, G, H will by the second fundamental
principle agree with Maxwell's statement.
If u, V, w be the components of current — including, of course, under
currents, growth of induction — we have from the third principle Maxwell's
equations E (vol. 2, p. 233), which on multiplying by /x become when /x is
constant
, dc dh\
dy
da do
4:7TIJLV = ^ ^
4:7TfJLW
db
dx
dx
da
dy
>•
(2)
Combining these with equations (1) (as in Maxwell, vol. 2, pp. 2367),
d^
and writing
V^ for i^ + i
dx^ dy
2 + ^^2' we obtain
214 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE
d /dL dM , dN\\
dx \dx
d /dL
„,,^ d (dL , dM , dJS\
dM^
dy
dM
dz J
dN^
y.
dN\
4^ ^ = _ vw   r + "^ +  1
^^ dz \dx dy dz J
(3)
These equations only differ in sign from Maxwell's, and are therefore to be
solved in the same way.
It is easy to see by substitution that if we assume
L' = — ^\\\ dxdydz
M' = — ii\\\ dxdydz
iV' = — jLt I  dxdydz
dx dy
then the following will be solutions
L = U 
M = M'
N = N' ■
dz
dH
dx
dH
dy
dH
dz ,
J  dxdydz
(4)
,(5)
It is evident that we may add to the righthand side of equations (5)
7 ' 77 ' 77 respectively, where </> is any function of x, y, z, since these
will disappear from (3) and also from (1).
The electric intensity, in so far as it depends upon magnetic motions, will
consist of two terms, one depending upon the motion of the material at the
point (its components being found as in Maxwell, vol. 2, p. 227, note), the
other upon the motion of magnetic induction about the point. We may add
a third term, arising from any electrical distribution with a potential ijj.
If there is no material motion we shall have
d^\
dx
dM di/j
dt dy
^_#
dt dz /
dL
dt
Q
y.
R
(6)
dx dy
dR ,
dz = '^
 dxdydz,
'ffdu
 dxdydz
ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 215
Substituting from (4) and (5) we get
r> (ddu 1 ■, 7 7 Id fff/ d dL , d dM , d dN\ 1 ^ , , #
^^^\\\dt'r^''^y^'^Tx\j\[Txlt+ryW^dz^^^^
\du \ . . . Id {[[(dP dQ dR\l ^ ^ ^
_ . _ dxdydz  ^^^j\\ (^ + ^ + ^)  dxdydz
substituting for ^ , etc., from (6).
The last two terms cancel each other, and we get
du 1 . , . Id [[{(dP dQ dR\l , , . ,„,
or if we put
and
jj.fr " '
with similar equations for Q and' 7?.
If the system is steady 777^ jT^ ;7r are all zero, and then
dx' dy' dz'
The quantity />, of which 7 is the potential, will be zero within non
conducting homogeneous parts of the field, for there
,_ZP _KQ ._KR
^ 477 ' ^ " 477 ' '^ ~ 477 '
nd ^ ^ dR_i7Tfdl dg dJi\ _
dx dy dz K \dx dy dz) '
since no charges can reside within a homogeneous nonconducting medium.
Or, stating it in another way, all the inductiontubes brought into any part
of such a medium remain there without dissipation, a charge in a nonhomo
geneous medium being due to unequal amounts of dissipation of induction in
different parts of the medium.
But p will have value at surfaces separating dissimilar substances either
in the insulating or conducting parts of the medium. For in the former the
induction is continuous, while the intensity is discontinuous, and in the latter
the current or rate of destruction of induction may be continuous, but the
relation between intensity and current changes discontinuously with the
216 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE
conductivity. At surfaces separating insulators from conductors p may have
value, as, for instance, at the surfaces of the plates of a condenser with its
terminals connected with two points in a circuit, or at the surface of an
insulated conductor near the circuit. It is also to be noted that p will have
value at the seat of electromotive force.
The values of the components of magnetic induction a, b, c are not in
any way dependent on p. For taking the first of equations (1) and sub
stituting from (5) we have
_dM _dN ^dM^_dN^_ d^ d^^dM;__dN^
~ dz dy ~ dz dy dydz dzdy dz dy ' '"^ '
where M' and N' depend on the currents in the system and not on the charges.
Comparing our equations with Maxwell's we see that the important point
of difference is that we can no longer put the quantity corresponding to his
7 1 . 7 K • • . dF ^dG ^dH
J equal to zero, J bemg given by , — h 7 — ^ ~J~ '
This does not affect the determination of velocity of propagation of dis
turbance in a homogeneous nonconducting medium.
For in such a medium we shall have
df K d^
dt 477 dt '
with corresponding values for v and w.
Substituting in (3) the first equation becomes
dt dx\ dx dy dz
differentiating with respect to t
^ dt^ dt dx [dx dt dy dt dz dt .
and putting ^7 = ? + ^,
since d^^dQ dR^i^^df dg dj.^
dx^ dy^ dz K \dx ^ dy^ dz)~
within a homogeneous nonconductor.
This gives the velocity of propagation of electric induction equal to
We can also obtain the corresponding equation for the magnetic induction.
(10)
ELECTBIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 217
Substituting in (3) for u, v, and w in terms of P, Q, and R, as above,
differentiating the second with respect to z, and the third with respect to y,
and subtracting
dfdQ_dR\^_ fdM _ dN\
^dt\dz dy)~ ^ \dz dyl'
then from (6)
^ d/d^dMd^ddN d^ \_ fdM dN\
dt\dt dz dzdy dt dy dydzj~ \dz dyj'
^dt^Kdz dy) \dz dyj'
or from (1) Kyi'^^ = V^a, (]])
whence the velocity of propagation of magnetic induction is also equal to
It would seem that in some cases, such as that of the field surrounding
a straight wire with a steady current, the electric intensity may be regarded
as entirely due to the motion of magnetic induction, and its components will
^. , . dL dM dN
thereiore be ^ , ^r , ^r 
dt' dt ' dt
But in other cases it would seem that the electric induction cannot be
wholly due to the motion of magnetic induction, and we must therefore
introduce the terms involving ifj. If, for instance, the electric and magnetic
intensities were inchned at an angle 6, we should have to suppose the electric
intensity E to be produced by the motion of the component of magnetic
induction I perpendicular to E, viz., /xZsin^, the other component [jlI cos 6
being at rest. To produce intensity E, E tubes must cut unit length in the
direction of E per second ; and since the value of the magnetic induction is
^I sin 6, this requires a velocity v, given hj v . [jlI sind = E oi v = E/fil sin 6.
Now we can easily imagine a case where E and I coincide, as, for instance,
a condenser with its planes parallel to the axis of a wire carrying a current,
and its terminals connected with two points in the wire. Here 7 sin ^ = 0,
and V is infinite. Or we have to suppose the electric intensity to be produced
by the movement of tubes of induction of no intensity with infinite velocity,
a statement without physical meaning.
But it is, perhaps, worth noting that if we suppose that the electric
intensity is produced by the motion of magnetic induction, and that the
magnetic intensity is produced by the motion of the electric induction, each
carrying its energy with it, the right quantity of energy crosses the unit
area.
218 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE
For E magnetic tubes, with / sin d unit cells per unit length, will carry
p n IT ^., EI sind 1 ,£ j_,
across unit area m the plane of E and / a quantity — ^ , or halt the energy
which actually crosses the plane. If I sin 9 is due to the motion of electric
tubes, then I sin d/in tubes must cut unit length in the direction of I sin 6
per second. The number of unit cells per unit length is E, and therefore
the motion of the tubes will carry a quantity ot energy — ^ , or the other
half actually crossing.
The equations which have been obtained in the foregoing manner by the
aid of the hypothesis of movement of magnetic induction may also be
obtained without any special hypothesis as to the motion of the induction
tubes, merely assuming that growth of induction through a curve is accom
panied by electric intensity round the curve. Instead of connecting L, M, N
with the number of tubes which have cut the axes, we start with the following
definitions :
Let L, M, N denote the timeintegrals of the components of the electric
intensity parallel to the axes since the origin of the system, so that
L = \pdt, M = JQdt, N = ^Rdt,
,. J. dL ^ dM ^ dN
then p = Q = ,^ , R = ^^ ,
dt ^ dt ' dt
If a, b, c be components of magnetic induction, since the growth of in
duction through a curve is equal to the Hneintegral of the electric intensity
round a curve in the negative direction, we have
(h^dQ_dR^d /dM _ ^\
dt dz dy dt \ dz dy j
with corresponding equations for j and j .
Integrating with respect to t from the origin of the system, when all the
quantities were zero
^dM_dN
dz dy
'SS^ <>■)
_ dL dM
dy dx
equations the same in form as equations (1).
ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 219
As before we obtain equations (3), (4), and (5), while instead of (6) we have
the simple equations P = ^rr and the two others.
dt
dL
Substituting for j we obtain an equation of the same form as (7), which
may also be put into the form (8). Equations (9) and (10) will also follow.
Just as we have obtained equations by considering the growth of the
magnetic induction to its present state so we may obtain corresponding
equations by considering the growth of the electric induction.
Let
Adx
Bdy
477 =
Cdz
be the algebraic sum of the number of electric induc
477 ' 477 ' 477
tiontubes which have cut dx, dy, dz drawn from a point in such a way as to
create magnetic intensities in the positive direction along dx, dy, dz.
The excess of the number of tubes which have passed in over those which
have passed out through the boundary of any area will be equal to the time
integral of the total current through the area.
The components of the total current are
V +
de
dg
dt'
w
r +
dh
dt'
f, q, and r being the components of the conductioncurrent or the number of
tubes dissipated per second, and/, g, h the components of the induction actually
existing.
As in the last case, if we put/'== \udt, etc., we at once obtain the equations
477/
dC
dy
dJB
dz
^ dz dx
^h' =
dB
dx
dA
dy \
,(12)
Corresponding to the currentequations (2) we have three equations obtained
from the condition that the rate of increase of magnetic induction through
an area is equal to the integral of the electric intensity round it in the negative
direction. These are
da _ dQ dR
dt dz dy
dh _dR _dP
dt dx dz
dc_dP_dQ
dt dy dx
.(13)
220
ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE
If C^ is the specific conductivity we may by Ohm's law put the current
equations after integrating in the form
KQ
4:77 ■
KR
g'= C.JQdt
J 47r
whence in media where K is constant
dg' dh' _ ^(dQ dR
dz dy ' }\dz dy
dt
K(dQ_
477 \dz
dR
dy.
K da
with two similar equations.
Finding the values of the lefthand side from (12) we obtain
47rC,a + iiC^ = VM
' dt
d fdA
dx \dx
dB dC\\
dy dz)
' dt dy\dx dy dz)
irrCn + K
dc
dt
V^C
dy
d fdA
dy
dB
+
dC
If we assume
477
///
dz\dx ' dy ' dz
da\ 1
iirC .a + X^j dxdyd
with corresponding values for B' and C and
J^ [[[fdA dB dC\\
4:7TjJJ\dx dy dz
M
1  dxdydz,
then
A'
B'
C
dL^
dx
dM
dy
dN
dz
(14)
(15)
are solutions of (13).
We may obtain by substitution from (15) in (12) values for/, g', h'
corresponding to the values of the magnetic induction in (9), viz. :
dC _ dR
dy dz '
w
ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 221
and two others ; where A', B\ C are given in terms of the magnetic induction
as above.
It is only in special cases, such as that of a straight wire with a steady
current, that the magnetic intensity will be equal to 47r times the number of
electric induction tubes passing through unit length per second. In all cases
the Uneintegral of the magnetic intensity round a closed curve is equal to
47r times the number of electric tubes passing through the boundary, but the
electric tubes may be more crowded in some parts than in others, while the
magnetic intensity is not altered in a corresponding manner. For instance,
the magnetic tubes will be continued through an insulated conductor in the
field, while in the steady state no electric tubes pass through it. But each
element adds to the hneintegral the quantity which, after Mr. Bosanquet,
I have called the magnetomotive force, this being equal to hr times the
number of electric tubes passing through the element. But it only adds it
on integrating round the whole of the closed curve.
The intensity at any point will therefore be the resultant of the intensities
produced by the magnetomotive forces in the various elements. Perhaps
the simplest mode of finding it is as follows.
The components of the magnetomotive force produced in a cube dx, dy, dz
parallel to the three edges will be
dA . dB . dC .
W^^' W^^' Tt^'^
for T , . T— , r T are by definition the rates at which electric tubes
47r dt ' i^ dt ' 477 dt ^
are cutting unit lengths parallel to the axes.
But these magnetomotive forces would be produced by currents round the
cube in planes perpendicular to the axes respectively, and equal to
I dA , ^ dB . 1 ^C ,
4.rW^^' SrW^^' i^Tt^'^
for the Hneintegral of the intensity round a curve threading a current is
47r X current. But the magnetic intensity at any point due to a current is
equal to that of a magnetic shell of strength (i.e., intensity x thickness)
equal numerically to the current bounding the shell.
If we suppose the thickness of the shell equal to that of the cube, the
effect is the same as if the cube were magnetised with intensity having com
ponents
]^dA l^dB \_dC
47r dt ' ^ dt' 47r dt '
222 ON THE CONNECTION BETWEEN ELECTRIC CURRENT AND THE
^m
The potential of such a distribution of magnetisation is (Maxwell, vol. 2,
p. 29, equation (23))
dA dp dBdjp dCdjp\
Wdx^dtdy^ dt dz) ^''^^^'''
where p = , and the magnetic intensity is given by
""' dx' ^ dy' '^ dz'
It may be noticed that in a steady field ~j, ^, rr are all zero, so that
47rjj.
d dM dp d^dM dp d^dM dp\,
dx dt dx dy dt dy dz dt dz) ^ '
We may obtain equations of the same form as those given in (14) without
any hypothesis as to the movement of electric induction tubes, merely assuming
that the total current through a curve is equal to 477 x lineintegral of magnetic
intensity round the curve.
We start with the following definitions. Let A, B,ChQ the timeintegrals
of the components of magnetic intensity since the origin of the system.
Then
A=^jadt, B=l^dt, C^jydt,
and
a
dA. n_dB _dC
dt' ^~lt' '^~Tt'
We have the equation ^ttu = ^ ^
dy dz
and two others.
Integrating with respect to t we have
. r ,, , ., dC dBs
J ^ dy dz \
also 4^.' = ^ _ ^ I
dz dx I
4 y ._ ^^ ^^
dx dy
which are of the same form as (12).
Hence exactly as before we obtain equations (14) and their solutions (15).
The equations for the magnetic intensity are now
^_dA dB dC
"" dt^ p = iu' y^it'
ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD 223
If we differentiate (14) with respect to t, and substitute from these equations
for magnetic intensity, we obtain
with corresponding equations for ^ and y.
Differentiating the second of these with respect to z, and the third with
respect to y, and subtracting, we obtain
with corresponding equations for v and w.
These correspond to Maxwell's equations (7), p. 395.
In conclusion it may be remarked that the equations found in this paper
give the same expression for the rate of Transfer of Energy as that in my
previous paper derived from Maxwell's equations involving F, G, and H.
12.
DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR.
[Birmingham Phil. Soc. Proc. 5, 1885, pp. 6882.]
[Read December 10, 1885.]
Maxwell has shown that the phenomenon known as the Residual Discharge
may be accounted for on the supposition that the dielectric is an imperfect
insulator in which the conductivity varies in different parts. His theory is
really quite simple and straightforward and free from any hypothesis beyond
the fundamental one of electric displacement. But its very generality makes
it, I beheve, difficult to grasp. The idea of a yielding of displacement in the
dielectric, accompanied by a conductioncurrent in the opposite direction,
gives us no help in forming a mental picture of the process actually going on
in the dielectric. A hypothesis as to the nature of electric current, which
will shortly be pubHshed in the Philosophical Transactions, seems to me to
render the theory easier to follow, and I propose in this paper to arrange
Maxwell's account of the Residual Discharge in accordance with it.
I shall first give some account of the hypothesis referred to in the special
case of the discharge of a condenser. Let us imagine that we have two
conductors, A and B, which we may suppose to be the two plates of a con
denser, charged with equal and opposite amounts of electricity, that of A
being positive. Then the lines of force will run from A to B through the
medium, the condition of the medium being described by saying that there
is 'electric displacement' from A to B. Or we may describe it without
introducing the confusing term 'displacement' by returning to Faraday's
term 'induction.' We may then say that tubes of electric induction pass
through the medium, each tube starting from + 1 of electricity on A, and
ending in — 1 on B. The total induction across any section of a tube is then
always equal to L If we draw the level surfaces at unit differences of
potential the tubes will be divided up into cells, and if we suppose each cell
to contain half a unit of energy then the whole energy of the electrified system
is accounted for. Maxwell has called these unit cells {Elementary Treatise on
Electricity, p. 47). According to the views of Faraday and Maxwell, the
charges on the conductors bounding the dielectric are to be regarded as the
surfacemanifestations of the altered state of the dielectric corresponding to
DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR
225
the energy put into it, somewhat as the pressure on a piston in the wall of
a closed vessel of compressed water might be regarded as the surfacemani
festation of the strained condition of the water.
In order to follow out the process of discharge in the medium, i.e., the
mode in which it is relieved from its strained condition, we will first take
a simpler ease in which we connect the two plates, A and B (Fig. 1), of
one condenser to the two plates, C and D, of another condenser previously
uncharged, and so far from A and B that there is no appreciable direct induc
tive action on C and D. When equilibrium is again restored the + charge
is shared between A and C, the — charge between B and D, while the difference
of level has decreased. There is the same total number of tubes of induction,
but each contains fewer unit cells than before, the energy corresponding to
the decrease having been transferred to the wires, where it has been dissipated
as heat. I shall use the term energylength to indicate the lineintegral of
the electric intensity along its axis, this being the same as the difference of
WIRE
lllllllllimiTTTT
""™=n
■■""■ i""iijij f
imimiijiiiiimirn
potential when there is equilibrium. We may say then that the energy
length of the tubes has decreased. During the change some of the electric
energy was converted into magnetic energy in the medium. This might be
observed if sufficiently delicate means were used.
If we confine our attention to the charges on the conductors we must say
that equal quantities of + and — have moved respectively from ^ to C and
from B to D along the wires.
But taking into account the condition of induction in the medium,
described by the inductiontubes, we must say that the induction tubes move
sideways out from the space between A and B into the space between C and
D, the motion of the charges along the wires being really the motion of the
ends of the induction tubes. (See Fig. 1, where 16 may be taken as
successive positions of a tube.) During the motion of the tubes some of
their energy was converted into the magnetic form, the coexistence of the
two forms, electric and magnetic, being a necessary condition of motion.
We may illustrate this from the analogous case of a strained incompressible
solid which can be sheared. If there is any mode of escape given to the strain
15
p.c.w
226 DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR
energy by a slipping of the surface against the constraint, then the state of
strain will be propagated outwards from the interior of the sohd, but some
of the strain energy will be converted into kinetic energy, and the presence
of the two is a necessary condition for the propagation of the strain.
Since the energylength of a tube diminishes as its ends move along the
connecting wires, we may represent this by supposing that parts of the tubes
move into the wire. If a similar motion of electric induction took place into
a dielectric it would remain, and the dielectric would become electrically
strained, but in the wire the strain breaks down rapidly, the energy being
converted into heat. I think there is good reason tO suppose that it is the
electric energy which thus breaks down, the magnetic only being dissipated
after it has been reconverted into the electric form.
We may now consider the case in which total discharge of a condenser
takes place through a connecting wire. Considering merely the conducting
plates and the wire, we say that the charges move along them towards each
other and finally unite, neutralising each other and producing heat in the
wire. Regarding the medium we must suppose the tubes of induction to
move sideways towards the wire, shortening as their ends, which are repre
sented by the charges, approach each other, and finally disappearing into the
wire. Faraday describes the process by saying that 'when current or dis
charge occurs between two bodies, previously under inductrical relations to
each other, the lines of inductive force will weaken and fade away, and, as
their lateral repulsive tension diminishes, will contract and ultimately dis
appear in the line of discharge.' [Exf. Res. vol. 1, p. 529, § 1659.) The
socalled velocity of electricity is merely the velocity of the ends of the tubes,
and this may evidently vary according to the nature of the circuit. It is
quite conceivable that if the wire be in a neutral medium, i.e., one in which
there is no surfacedifference of potential, say gold in air, and if it follow the
direction of a tube of induction, then a tube may move into the wire throughout
its whole length at once. In this case the 'velocity of electricity' would be
infinite.
We know from experiment that if a galvanometer be inserted in the
connecting wire then the same magnetic impulse is observed wherever in
the circuit the galvanometer be placed, the impulse depending on the
galvanometerconstant and on the total discharge. The same experimental
result may be stated in an equivalent form, viz., that the lineintegral of the
magnetic intensity round a closed curve encircHng the wire if integrated for
the time of discharge is the same for all positions of the curve. On the
hypothesis here described all the electric inductiontubes of the system
finally pass sideways from the medium into the wire. They must, therefore,
on their way pass inwards across any curve encircling the wire, so that the
total number of inductiontubes cutting such a curve is the same for all
DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR 227
positions of the curve. In the paper above referred to I have sought to
connect these two constants by supposing that the magnetic effect is due to,
or more correctly accompanies, the motion inwards of the condition of electric
induction. As soon as motion commences some of the electric energy is
converted into magnetic, and the magnetic induction may be represented by
ringshaped closed tubes surrounding the wire. The two inductions, electric
and magnetic, coexisting, will propagate the energy onwards till it finally
arrives in the wire and is dissipated as heat, the induction there losing its
directed condition.
The flowing of electric charges along the wire, which is usually considered
as the essential part of the phenomenon, or at least that to which attention
is to be chiefly directed, becomes on this hypothesis merely the last stage in
the process, which consists of a propagation from the surrounding dielectric
towards the wire of electric and magnetic induction, which we may symbolise
by the motion inwards of two sets of tubes, the electric tubes being, on the
whole, more or less in the direction of the wire, the magnetic tubes being
closed rings surrounding it. The wire plays the part of the refrigerator in
a heatengine, turning the energy it receives into heat — a necessary condition
for the working of the machinery.
Let us now take the case of a condenser in which the dielectric, though
homogeneous, is imperfectly insulating, so that the charge gradually dis
appears. According to Maxwell, in this case 'induction and conduction are
going on at the same time.' Though Maxwell gave no precise account of
the process of discharge, his theory and the mechanical illustration accom
panying it are based on the supposition that two processes are going on at
the same time in every part of the medium, viz. : (1) a yielding of the electric
strain or 'displacement' in the dielectric, equivalent to a displacement
current from the negative towards the positive plate, and (2) a conduction
current from the positive plate to the negative equal to (1) in amount. This
latter is accompanied by dissipation of energy. The two equal and opposite
currents being superposed have no external magnetic effect.
But it seems to me that we may equally well and more simply represent
the facts by considering the first process alone, viz., the yielding of the electric
strain, the medium being incapable of bearing it permanently. The electric
energy is gradually converted into heat in the same part of the dielectric
where it was previously electric, i.e., there is here no transfer of energy. The
decrease of induction in the medium is accompanied by a corresponding
decrease of charge on the plates, not by conduction of + or — electricity
either way through the medium, but simply because there is a decrease of
the induction in the medium of which the charges on the plates are the surface
manifestations. The induction decreases equally through the whole length
of a tube, so that the tube 'weakens' at the same rate throughout its length.
15—2
228 DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR
There will be no magnetic effect in the surrounding space for there is no
movement inwards of electric inductiontubes to supply the place of those
which decay.
Perhaps we may take the following as illustrating the two modes of
regarding the process. Suppose that a solid is submitted to some strain
and kept in the strained position, but that the energy of the strain gradually
dissipates ; then we may confine ourselves simply to the statement that owing
to some rearrangement of the molecules they cease to have molecular strain
energy, the energy in each portion of the mass being transformed into heat
in that portion, or we may imagine that there is a continual return from the
strained towards the original position, accompanied by an equal reverse flow
of the matter towards the strained position, this latter not storing up energy
but dissipating the energy given up by the yielding of the strain. The
ultimate result according to each is the same, but the latter account is purely
hypothetical.
We may at once obtain the equation giving the value of the charge at any
time in terms of the initial charge when the condenser is left insulated.
Let a be the charge per unit area, this being equal to the electric induction
across unit area in the dielectric.
Let K be the specific inductive capacity.
Let X be the electric intensity in the dielectric, i.e., force per unit electricitv
on a small electrified body.
We have ^ "^ ^ (1)
Now we know that the rate of decrease of charge on the ends is proportional
to the charge a and therefore to X.
The decrease of charge or of induction in the medium is therefore
(it r' ^^
where r is a constant, which we may term the specific resistance.
Hence from (1) ^ + ^ ^^ 0, (3)
or CT = GToC ^^'^ (4)
If we use p to denote the decrease of induction per second,
_ da _ _KdX
KX^
The energy per unit volume is ^— ; its rate of decrease is therefore
KX dX
"l^TW (^>
DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSLTLATOR 229
Substituting from (2) and (5) we get the expression which here corresponds
to Joule's law for the heating effect, viz., rate of decrease of electric energy
per unit volume = pV.
If at any moment the two endplates be connected by a wire, transfer
of induction will at once take place into the wire, and the whole system will
be completely discharged. During this discharge there will be magnetic
energy accompanying the motion of electric induction.
We will now investigate the more complicated case of a stratified dielectric
in which the different layers have different specific resistances. Before
proceeding to the mathematical account we shall consider the process generally,
taking the simple case in which K is the same throughout. Let the con
denser be charged very rapidly and then insulated. At the first moment
there will be equal and opposite charges on the two end plates, and the number
of inductiontubes running through unit area parallel to the plates will be
the same in each layer. But decay of induction, and dissipation of energy,
at once sets in, the rate of decay varying in different layers, so that after
a time the number of inductiontubes in contiguous layers will differ and there
will be charges on the separating surfaces. In those layers where the rate
of decay is most rapid there will be negative charges on the surface nearer
the + plate, and + charges on the surface nearer the — plate. But still the
induction in all is in the same direction.
Now let the two endplates be connected by a wire. At once induction
is propagated into the wire and transference takes place from the space between
the plates until they are at the same potential, i.e., until the lineintegral of
the electric intensity, or, since K is constant, that of the induction, from plate
to plate is zero. The same number of tubes must have entered all parts of
the wire, otherwise there would be charges at points along its length. Hence
the same number of tubes running in the positive direction must have passed
out from each of the layers. The result must be a reversal of the induction
in some of the layers, viz., in those in which the induction decayed most
rapidly. This, of course, means that after their positive induction has all
flowed out and they are quite discharged, tubes from the other layers have
bent round and entered them, now charging them in the opposite direction.
We may imagine the process to be somewhat as in Figs. 2 and 3, representing
a condenser with three layers, A, B, C, the decay having been most rapid
in the middle one, so that it has become completely discharged, while there
is still positive induction in A and C. 1, 2, 3, 4 (Fig. 2) represent successive
positions of a tube moving out from A towards the wire ; T, 2', 3', 4', successive
positions of a tube moving out from C. When they have taken up the positions
4, 4' they come in contact, and where they overlap they will neutraHse each
other and break up into two portions, the outer part of each forming one
230
DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR
positive tube, as 5, Fig. 3, which will move off to the wire, inner parts uniting
to form a negative tube 6 in B.
When the difference of potential between the endplates is zero, suppose
the wire to be removed. The induction still remaining decays. If it decayed
in the same proportion throughout, the difference of potential would always
remain zero. But it decays in greater proportion in the negative layers,
since in these the dissipation is, by hypothesis, most rapid. Hence in the line
integral of the induction from plate to plate the negative terms decrease more
rapidly than the positive, and so the total value becomes positive. Then on
WIRE
Fig. 2.
Fig. 3.
again connecting with a wire another positive discharge occurs,
may evidently be repeated, the discharge
always being positive, until finally it
becomes insensible.
The process
p r r A r r
r B
r r
Fig. 4.
The analogy between the residual dis
charge and the phenomenon of elastic
recovery in strained solids, pointed out
by Kohlrausch, suggests a simple illus
tration.
Suppose that we build up a cube with
successive layers of substances with the
same instantaneous rigidity but with different viscosities. Let this be
placed between two plates, A, B, Fig. 4, the lower plate being fixed. Let
rigid transverse partitions, r, r, be passed through the layers and attached
by hinges to the two plates, and then let the upper plate be acted on by
a force in a direction perpendicular to the partitions, so that a shearing strain
is given to the whole cube. The partitions, r, r, are merely put so that the
distortion from the original position shall always be the same throughout.
When a given strain has been produced let the upper plate be also fixed.
Now if the rate of dissipation of strain energy were the same throughout
the layers the stress would also be the same throughout, though gradually
DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR 231
decreasing, and on removing the constraint the upper plate would return by
a certain amount and then remain in its new position. But the dissipation
is not uniform and after a time the stress in some of the layers is greater
than in others. Hence, on removing the constraint from A and allowing it
to return, when those in which dissipation has been most rapid have become
entirely free from strainenergy, there is still some remaining in the other
layers. These latter will, therefore, strain the former, and we shall have
a reverse stress in some of the layers. Thus A will come to a new position
of equilibrium, not so far, however, as its first position. Suppose that it
is now again fixed. At first no force is necessary to keep it in position, but
the stress exerted by the negative layers decays more rapidly than that
exerted by the positive, and soon, on being released, A will return still further
towards its original position. The process may be repeated, the successive
discharges of momentum imparted to A being always in the same direction.
(Added April 16th, 1886. The supposition of stratification made by
Maxwell is, no doubt, very artificial, and was made for the sake of simplicity
in the mathematical treatment. He states that 'an investigation of the
cases in which materials are arranged otherwise than in strata would lead
to similar results, though the calculations would be more complicated, so
that we may conclude that the phenomena of electric absorption may be
expected in the case of substances composed of parts of different kinds, even
though these individual parts should be microscopically small.
' It by no means follows that every substance which exhibits this pheno
menon is so composed...' (Electricity and Magnetism, 2nd ed., vol. 1, p. 419).
Probably in the case of blown glass or any dielectric made up of hetero
geneous parts, which has been flattened by rolling, there is more or less
approach to the stratified condition, but in other cases, such as shellac or
paraffin, we might fairly expect the dielectric to be similarly constituted in
all directions. We can only, therefore, take Maxwell's investigation as
showing in a general way that heterogeneity would introduce absorption
phenomena, and we cannot expect the results obtained on the supposition
of such a special arrangement to agree with those of experiment. We may
regard the stratified arrangement as giving a superior limit, as it were, this
being the constitution most favourable to the production of the phenomena
in the way supposed. The inferior limit would be given by an arrangement
in which each portion of the substance of the same kind stretched from plate
to plate with the same crosssection throughout. In this case there would
be no residual discharge produced. Using Maxwell's notation (see below),
the resistance per unit crosssection may be shown to be
^ + '+ ...
232 DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR
instead of R = a^r^ + a^rz + ...
and R' is always less than R. In any intermediate composition in which
portions of more conducting matter are insulated from each other by less
conducting matter we shall have residual discharge.
It appears probable from experiments of Dr. SchulzeBerge {Nature,
March 4th, 1886, p. 432) that the resistance of certain dielectrics is not pro
portional to the thickness, but is much less for thin layers than might be
expected. May this not possibly arise from the size of the heterogeneous
portions being comparable with the thickness of the dielectric, so that the
more easily conducting portions may stretch in some parts from plate to
plate? If so, we approximate more nearly to the inferior limit.)
The mathematical account of the residual discharge on this hypothesis is
practically the same as Maxwell's, but it may, perhaps, be worth while to
give it with the necessary alterations, as these seem to make it somewhat
more straightforward and evident.
We shall suppose with Maxwell, ' for the sake of simplicity, that the
dielectric consists of a number of plane strata of different materials and of area
unity,' and that the induction is in the direction of the normal to the strata.
Let a^, a^, etc. be the thicknesses of the different strata.
Let Zi , Z2 , etc. be the electric intensity within each stratum.
Let 2^1, 2^2 5 ®tc. be the amount of decay of induction per second in each
stratum.
Let/1,/2, etc. be the induction in each stratum.
Let %i^, ^^2 5 ^^^ b^ ^tie total number of tubes of electric induction entering
each layer sideways, i.e., crossing in through its boundary, per second.
Let r^, r^, etc. be the specific resistance referred to unit of volume.
Let K^, K2, etc. be the specific inductive capacity.
Let ^1, ^2 5 ^tc. be the reciprocal of the specific inductive capacity.
Let E be the electromotive force due to a voltaic battery placed in the
part of the circuit leading from the last stratum towards the first, which we
shall suppose good conductors.
Let Q be the total number of inductiontubes which have left the battery
and entered the wires and dielectric up to the time t.
Then since the same number of tubes enter all parts of the circuit in
a given time,
u^ = u^^ u^= ... ^ u say (1)
These tubes tend to increase the induction in the layers. But at the same
time decay is going on so that we have
% = Vi + §, ^h = V2 + 5, etc., (2)
DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR 233
whence 2?i + ^ = 2^2 + ^^ = etc (3)
We also have by Ohm's law
Vi = ^> :P2=v^> etc., (4)
and by the relation between induction and intensity
X,= ^7rKh, (5)
whence '^=^+ir^ tt (6)
Let us suppose that at j&rst there is no charge and that suddenly the
E.M.F. E is made to act. Then if at once Q tubes enter the dielectric,
Zi= iTTk^Q, etc., (7)
and since £■ = a^Zi + 0^2X3 + ..., (8)
E ^ 4:77 (^1% +^52^2+ ...)Q.
The instantaneous capacity C which is equal to ^ is given by
C= . .. J. — . (9)
i7r{k^ai + k^a^i ...)
But dissipation at once sets in, and if the electromotive force E be continued
uniform a steady state will ultimately be reached in which the dissipation
in each layer is equal to the number of fresh tubes reaching that layer. The
number of tubes entering being the same throughout, the dissipation p is
also the same throughout.
We have then p = ^ = ^ = etc., (10)
and substituting in (8) E = (/!% + r2^2 + ••) V
Hence if R = r^^a^^ ...,
PR <^^'
In this state we have the induction given by
'^^^ ^7Tk^~ ^irkj ^irk^R ^ ^
If we now suddenly connect the extreme strata by means of a conductor
of small resistance, E will be suddenly changed from the value Eq to zero
and Q' tubes will pass out from each layer of the dielectric into the wire.
If then X' be the new value of the intensity,
O' = ^1 _ ^1
^ iTTk^ iirk^'
whence * Z/ = Zi  477^^1^' (13)
234 DISCHARGE OF ELECTRICITY IN AN IMPERFECT INSULATOR
Since then the difference of potential is zero
a^X^ ^a^X^ V .,. = 0,
substituting from (13) we get
^iZi + a^X^ + ... = 477 («!% + ajk^ + ...)Q'
or ^ %Zi + a,Z, + ... Q^^Q (14)
from (8) and (9).
Hence the instantaneous discharge is equal to the instantaneous charge.
By (10) and (11) we may put (13) in the form
Zi' = fV^  iTT^Q
= ^^i^k,Q (15)
Let us next suppose the connection broken immediately after the dis
charge. No fresh tubes enter any layer, so that putting t^ = we have
from (6)
=
Zi 1 dX^
^1 477A;i dt '
4TrA;i ,
or Zi = Z/6
where Zi is now the value of the electric intensity at any time t after the
connection is broken.
Substituting from (15) and putting Eq for the initial value of E,
X,==E,(^^i7Tk,Cy~'^ (16)
The value of E at any time is
E = ^iZi + a^X^^ ...
= Eo (f"^'^  ^^a,k,c) e'f' + (^^^^  477^2^1^26')
e
irrko
E, {^^  477;l^iCJ a,e r. + Q  477y^2C) a,e ^ ' + ..\
EM.^CnY^^e"^^' ^\ (17)
A J R
The instantaneous discharge obtained at any time t will be, as before, CE.
If the terms be arranged in descending order of magnitude of ~\ then the
exponentials are also in descending order of magnitude, or the negative terms
decrease more rapidly than the positive, and E is positive.
13.
ON THE PKOOF BY CAVENDISH'S METHOD THAT ELECTRICAL
ACTION VARIES INVERSELY AS THE SQUARE OF THE
DISTANCE.
[British Association Report, 1886, pp. 523524.]
The proof of the law of electrical action depending on the fact that there
is no electrification within a charged conductor was first given by Cavendish.
His proof was made more general by Laplace, who has been followed by other
writers, including Maxwell. Maxwell and MacAlister have also verified the
experimental fact, repeating an investigation of Cavendish only recently
pubHshed in Maxwell's edition of the Cavendish papers. The proof may be
analysed in the following way : Take the case of a uniformly charged sphere.
The action at a point within it may be considered as the resultant of the actions
of the pairs of sections of the surface by all the elementary cones, with the
point as vertex. If, then, the resultant action is zero for all points and for
all sizes of the sphere, it follows that the action of the pair of sections by each
elementary cone is zero ; and, since the sections of the surfaces are directly
as the squares of the distances, the two sections neutraHsing each other, the
force per unit area must be inversely as the squares of the distances. There
appear to be two objections to this proof. (1) That it takes no account of
the always existing opposite charges. When the sphere, for instance, is
positively charged, an equal and opposite negative charge is on the walls of
the room, and the action of this should be considered. Probably this objection
could be removed. (2) There is a solution still simpler than the inverse
square law — viz., that no element of the surface has any action within the
closed conductor. If we suppose that a conductor is a complete screen to
electrical action, then, whatever the law of the force exerted across an insulator,
there will be no action within the conductor. In any null proof it is not
sufficient merely to show that there is no action in the null arrangement, but
it is also necessary to show that on disturbing the null arrangement some
action is manifested. Now, in the case here considered it is impossible to
obtain any action within the conductor in any statical arrangement; it is
only during changes of the system while charging or discharging that we can
get a disturbance of the null arrangement. But here new phenomena come
236 ELECTRICAL ACTION VARIES INVERSELY AS THE SQUARE OF DISTANCE
in, for we have currents, and therefore electromagnetic action. But, even
disregarding the different kind of action occurring, the only experiment which
I know of on this point was that of Faraday with his electrified cube. While
the most violent charges and discharges were taking place on the outside of
the cube, so that the null arrangement was probably disturbed, he found no
action on his electroscope within. Possibly the actions were alternating, and
so rapid that no electroscope of ordinary construction would reveal them.
But he himself went into the cube, and he would probably be sensitive to
rapidly alternating electromotive forces. It appears to me, then, that we
cannot accept this proof, and must fall back upon the more direct proof of
Coulomb*. I do not know whether Maxwell was aware of this objection ; but
it is worthy of note that in the remarkable fragment published since his
death, as An Elementary Treatise on Electricity, he returned to Coulomb's
proof, and was apparently building up the mathematical theory of electricity
in a way quite different from that followed in his larger work.
* [In his lectures on electrostatics Poynting used to give an experimental proof somewhat
differing from that of Coulomb, and simpler. A brief account of Poynting's apparatus will be
found in Electricity and Magnetism by Poynting and Thomson, vol. 1, pp. 65 and 66. Ed.]
14.
ON A FORM OF SOLENOIDGALVANOMETER.
[Birmingham Phil Soc. Proc. 6, (1888), pp. 162167.]
[Read May 10, 1888.]
The instrument described in this paper is a form of solenoidgalvanometer
in which the iron core is still far from saturation, so that the attraction of
the core by the coil is nearly proportional to the square of the current. The
peculiarity consists in an arrangement by which a pointer moves over a scale
a distance not very far from proportional to the current.
The moveable core of the solenoid consists of an iron rod or bundle of
wires, and is suspended by a silk fibre, which is wrapped on to the circumference
of a small wheel with a horizontal axis turning in bearings as free from friction
as possible. The wheel has an arm (Fig. 1) with a moveable bob on it, and
the bob is so adjusted that the weight of the iron core just balances it when
the arm is horizontal. The equilibrium is of course unstable, and a stop S
is necessary jiist above the arm when in the horizontal position. The arm
ends in a pointer moving over a divided quadrant.
The solenoid is placed above the iron core so as to act against its weight,
and in the position of maximum pull. The coil is moveable up and down by
means of a screw, so that it may always be put in this position of maximum
pull. The current passing through the coil does not saturate the iron, and
the upward attraction is therefore nearly proportional to the square of the
current. Let it be equal to KC^ where K is a constant for the particular
instrument.
If W is the weight of the core, a the radius of the wheel, w the weight of
the wheel and bob, and b the distance of its centre of gravity from the axis,
the condition for equilibrium when no current passes is
Wa = wb (1)
If now a current C passes, the down pull of the core is lessened and the
arm falls into a position in which the bob has a less moment. If it moves
through an angle 9,
(W KC^) a = wb cos (2)
238 ON A FORM OF SOLENOIDGAL VANOMETEE
Substituting from (1) for Wa,
KC^a = wb{l cos 6) = 2wb sin^
e
or
From this
^ / 2wb . 6
de = 2
Ka dC
cos
which only very gradually increases with 6, and when 6 = 90° it has a value
'\/2 or 141 times its value at 0°.
Fig. L
ON A FORM OF SOLENOIDGALVANOMETER 239
It is very easy to construct an arc divided to give readings proportional to
sin ^ as follows : — Describe a quadrant, and mark off points on it with
ordinates increasing by equal amounts. On the radius from which these
ordinates are measured describe a semicircle. Drawing the radii of the
quadrant to the successive points marked, they will intersect the semicircle
in points with equally increasing values of sin ^, 6 being the angle subtended
at the centre of the semicircle.
In practice it would no doubt be better to graduate by trial, having a
standard instrument in the circuit.
The instrument shown, though faulty in several points and far from
frictionless, works fairly well. The range is limited by the fact that unless
the adjustment is very perfect, the readings cannot be trusted below 10° or
20°, but I think the principle might be usefully adopted for voltmeters of
small range, or for ammeters, to give a correct value for a current within
a small range. It might be useful to extend the range by a counterpoise
to part of the weight of the core, on the other side of the wheel. The instru
ment has the advantage that, when the current is passing, the pointer very
rapidly comes to rest.
A Suggestion for a Wattmeter.
The above instrument has suggested to me a possible form of wattmeter
which I have not seen described before. I have not yet constructed an
instrument on this plan.
A soft iron core is fixed vertically at one end of a steelyard, with a moveable
counterpoise as usual on the arm beyond the knifeedge. Two coaxial
solenoids, one of high and the other of low resistance, are fixed in the position
of maximum pull on the core when the arm is horizontal. For stability they
should be above the core. The ends of the highresistancecoil are connected
to the two ends of the circuit in which the rate of working is to be measured,
a commutator being interposed so as to reverse the current in the coil. The
lowresistancecoil forms part of the main circuit. When the currents pass
in the same way through the two coils, the pull on the core will be
{aC + hEf,
where a and h are constants for the solenoids. The counterpoise is to be
adjusted for equilibrium. The current now being reversed in the high
resistancecoil, and the counterpoise being again adjusted, the pull on the
core will be
(aC  hE)\
240
ON A FORM OF SOLENOIDGALVANOMETER
The distance through which the counterpoise has been moved will be pro
portional to the difference between these two pulls, or to
4:abCE,
i.e., to the rate of working CE.
A SquareRoot Steelyard.
Some years since another arrangement occurred to me for obtaining an
equally divided scale, giving directly the square root of the pull on a soft
iron core or on a moveable coil. After recently constructing a model, I found
it was only a particular case of the very remarkable machine for solving
equations, devised and constructed by Mr. Boys (Philosofhical Magazine,
vol. 21, 1886, p. 241). Being, however, a very special case, it is less com
plicated than the general instrument, and as the model works easily and
correctly, it may be worth while to describe it.
ABC (Fig. 2) is a lever balancing on a knifeedge at B, and the pull W,
of which the square root is to be measured, is applied at the end A. GE is
C D B A
H G F
y^^^.y^yy.^y^yf/yZ'yyy/^y^^yyy^/y^.^^
Dp
^
^ w
Fig. 2.
another lever balancing on a knifeedge at G, the arm GE being about equal
to the arm BC. The plane upon which G rests is, in the model, a plate of glass,
about equal in length to BC, and so arranged that GE may be moved until
E is under any point of BC. E is connected by a link DE with BC, and from
E , exactly under C, hangs a fixed weight P.
If the down pull of the link at I) is T, and w is its weight, the up pull
dXEi^T — w.
The equations of equilibrium of the two levers are
and
or
if GR be made equal to ^ GE.
W . AB ^ T . BD
(Tw)GE = P. GF,
T .GE^P.GF
= p(gf
= P.HF
w
(1)
.GE
ge]
(2)
ON A FORM OF SOLENOIDGALVANOMETER 241
Multiplying (1) and (2) together, T is eliminated and
W .AB.GE = P .BD.HF,
Making HE equal to BC, and keeping P always exactly under 0, BD is
equal to HF, and
W .AB.GE = P,BD^:
BD
y
AB,GE
Vw,
If then BC is equally divided, the equilibrium position of D gives a reading
proportional to the square root of W.
In the model a lever, not shown in the figure, fixes ABC, and at the same
time Hfts P up so as to release GE. GE and the link DE can then be moved
along to a new position. On moving back the lever, ABC is released and
P is dropped again into position on GE, exactly under 0.
p.o.w.
[6
15.
ON A MECHANICAL MODEL, ILLUSTRATING THE RESIDUAL
CHARGE IN A DIELECTRIC.
[Birmingham Phil. Soc. Proc. 6, 1888, pp. 314317.]
[Read November 8, 1888.]
The model is designed to exhibit a phenomenon analogous to the residual
charge which gathers in a condenser after it has been charged and then
discharged, when the dielectric is not a perfect insulator. Its mode of action
is similar to that which Maxwell supposes to occur in the dielectric. According
to his theory the residual charge is due to the breaking down of the state of
strain (or, perhaps, more correctly, of the stress) in the dielectric corresponding
to the original charge, but in an uneven manner in different parts of the
dielectric, so that just before the discharge the stress is greater in some parts
than in others. On discharging, it is impossible, from the nature of electric
discharge, to remove all the strain by connecting the two plates of the con
denser, and the condition of equilibrium which is arrived at consists in an
actual reversal of the strain in the parts where the breaking down has been
most rapid, the reversed stress in these parts balancing the remnant of the
original stress in the other parts. On insulation, the strain breaks down
again, and at the greatest rate in the same layers, now reversed. Consequently
the reversed stress is no longer able to balance the direct stress, and, on the
whole, there is a preponderance of strain in the original direction, or a gathering
of charge the same in kind as the original charge.
The model consists of a trough (see figure) of semicircular crosssection,
24 ins. long, 6 ins. diameter, and divided into eight equal compartments by
a middle partition along the axis and three crosspartitions. It is supported
at the two ends, so that it can rotate about its axis 00, a pointer P attached
to one end moving in front of a scale S. Four pipes, with taps t, t, t, t, connect
the opposite compartments when the taps are turned on. The trough is
balanced by the weights w, w, so that when empty it is in neutral equihbrium.
Turning the taps off, and pouring in water to the same depth in all the com
partments, the equilibrium at once becomes stable, and the trough, if displaced,
stores up energy. It may be considered as analogous to a 'tube of force,'
connecting charges ± q on the surfaces of two opposite conductors, the axis
A MECHANICAL MODEL, ILLUSTRATING THE RESIDUAL CHARGE 243
of the trough representing the axis of the tube of force, the angle of displace
ment the charge at either end, or the induction along the tube. A clockwise
rotation at the pointerend might signify a positive charge at that end. As
long as the taps are off, the trough represents a perfect insulator, a displacement
through a given angle, and fixture at that angle, corresponding to the com
munication of a charge and subsequent insulation. The energy remains in
the trough undissipated. Discharge, of course, corresponds to release of the
trough, and we have oscillations corresponding to the electrical oscillations
brought recently into such prominence. It may be noted that a decrease
in the quantity of water corresponds to an increase in specific inductive
capacity, while a decrease ,in the weight of the trough corresponds to an
increase in magnetic permeability. We might, perhaps, obtain an analogy
to the spark discharge by completing the cylinder, of which the trough forms
half, and carrying the partitions up through the added half. On turning
the trough through anything more than a right angle it would fall over and
oscillate about a new position 180° from the original one, the discharge of
energy occurring now with an increase of strain, not with a return to the
unstrained condition. If the taps are turned on, but all to the same extent,
the trough corresponds to a 'leaky' dielectric in which the conductivity is
uniform. Turning the trough through a given angle and holding it, the
water begins to flow back from the higher to the lower compartments, thus
dissipating the energy, and if after a short time the trough is released it
returns to a position short of the original position and remains there, the
level of the water in the two sides of the middle partition being the same.
But if the taps are turned on by different amounts — if, for example, the two
endtaps are turned off while the two middle ones are turned on, — then on
turning the trough through a given angle and holding it, the energy of the two
middle pairs of compartments is gradually lessened, and on release the trough
moves part way back. But now it is only the mean level which is the same on
the two sides. In the two pairs of compartments with no communication there
is still a positive difference of level, while in the other two there is now a
negative difference. Holding the trough in its new position for a short time,
the negative difference is reduced by leakage from one side to the other, and
on release the trough returns by another amount towards its original position
16—2
244 A MECHANICAL MODEL, ILLUSTRATING THE RESIDUAL CHARGE
—and this may be repeated several times, until finally the original position is
sensibly regained.
The first model I made, for ease of construction and without sufiicient
consideration, with rectangular instead of circular crosssection; and with
this the phenomenon of residual charge is obtained, even though all the taps
are turned on equally. For consider what happens if the trough is turned
through an angle and held. The water comes to a level after a time, but
still its centre of gravity is not in the lowest possible position, and on release
the trough returns part way, making a negative difference of level between
the two sides. Again holding it, this difference is reduced, and on release
there is another return, and so on. This suggests that possibly residual
charge may occur not only when the substance is heterogeneous, but also
when it is homogeneous, if with electric induction or strain there is both
energy of the molecules as a whole and internal energy between the parts
of each molecule. If the latter dies away after the bounding conductors
are charged, the former may still remain, and on discharge it is possible
that it may not all be dissipated, but may partly go to renew the internal
energy. If this renewal accompanies a reversal of the direction of electric
strain, we shall have the phenomena of residual charge.
It is hardly necessary to point out that the model serves equally as an
illustration of a possible explanation of elastic afteraction. It is evident
that the phenomenon of residual charge will always occur when a body
strained is such that the stress dies away unequally in different parts, while
at the same time its constitution is such that on release from strain an equal
amount of strain is taken from each part.
From this illustration of residual charge we may pass to a possible analogue
of conduction in a metal wire. Let us suppose the trough replaced by a hollow
cylinder, with its axis horizontal, ends closed, and without partitions. If the
cylinder is only partly filled with water, a small couple applied to it will
produce continuous rotation, but with a limiting angular velocity, attained
when the water is dragged up in one side so far that the moment of its weight
about the axis is equal to that of the applied couple. The energy put in by
the couple is all ultimately converted into heat in the water. Thus the
angular displacement increases indefinitely, though the stress always remains
small. Similarly, as I believe, the 'electric strain,' or 'induction,' or
' displacement ' in a wire carrying a current increases indefinitely, as induction
IS continually coming into it from the outside, although the stress always
remains small.
16
ELECTRICAL THEORY. LETTERS TO DR. LODGE.
[Electrician, 21, 1888, pp. 829831.]
to the editor of ' the electrician.*
Sir:
I have prevailed on Prof. Poynting to let me send you the enclosed
two letters, wherein he continues the discussion of electrical theory begun
in Section A at Bath. It must be understood that the letters are merely
hasty epistles, not intended for publicity ; but Prof. Poynting's ideas are so
original and weighty that one is glad to extract from him, when possible,
a casual contribution to a discussion, as well as one of his sledgehammer
communications to the Royal Society. I hope that this may be the means
of extracting a reply or a criticism more competent than anything of mine
would be.
Yours, etc.
Oliver J. Lodge.
Dear Lodge:
I thank you very much for the copy of your exceedingly interesting
account of Electrical A. Perhaps my gratitude would be best shown by
silence, but I am tempted to show my appreciation by asking you to help
me with some difficulties.
My first difficulty is as to the interpretation of Hertz. You say — though,
I think, FitzGerald is responsible for the statement — that ether is a demon
strated fact. I do not see how Hertz adds to our certainty. Is not our
belief in ether due to the fact that light takes time to travel in interplanetary
spaces, where we cannot put enough matter for it to travel by. so that we
have to imagine something else for it to use. The fact that the velocity of
light is nearly the same in vacuo and in gases, and not widely different in
denser substances, of course supports the view that on the earth it also uses
ether. Hertz shows that there is an 'interference' in electromagnetic
disturbance which we can only (at least, with our present knowledge) put
down to wavemotion traveUing with a definite velocity which he finds equal
to that of light. Hence these disturbances probably make use of the same
ether. Does this prove its existence any more? I should expect a sceptic
246
ELECTRICAL THEORY. LETTERS TO DR. LODGE
to ask why may not electromagnetic disturbance make use of air, since
Hertz carried out his experiments in air. I could only reply to the sceptic
that he was a very disagreeable person.
Secondly, Thomson's [Kelvin's] 'Simple Hypothesis' Paper* appears to be
a very serious attack on Maxwell's theory ; in fact, on reading it over carefully,
I can only come to the conclusion that it would lop off not only Maxwell's
excrescences but his whole theory. According to the concluding sentence
of § 4, (/ each component of electric current at any point is equal to the electric
conductivity multiplied into the sum of the corresponding component of
electrostatic force and the rate of decrease per unit of time of the corresponding
component of velocity of liquid in our primary') the current
pf_d^_ duj\
\ dx dtj'
which = if C the conductivity = 0, so that Maxwell's / (his ' displacement
current ') goes altogether. The x component of Maxwell's e.m.f. will contain
a term
^ ' ' ' r^ dxdydz, since Maxwell's u
477
Thomson's u{ . — ^ , where
477 dt
P = E.M.F. along X. This has no representative in Thomson. Thus with
a homogeneous but leaky condenser with no connecting wire, we have,
according to Maxwell, total current = 0, for leak is made up for by yield of
displacement ;
•• dx'
where V is potential due to electrification on the plates.
According to Thomson, u is to be taken as rate of leak, and is positive ;
P =
du
dt , , , dV
— axaydz r •
r dx
According to Maxwell there is no magnetic effect, since total current = 0.
According to Thomson^ — using his notation — x component of magnetic
dtVi dvi _ ^_2 fdw dv^
dy dz \dy dz,
d
force u
if V2 and
But
dz \dy
etc., are transposable
u = KP, V
:. u, =  477V2
from Thomson's equation (7), i.e..
dy
KQ,
w
KB;
dR dQ
dy
477V
[dy\
dz)
' dw
di
ff,
dxdydz
T " dz
* Reprinted in The Electrician, vol. 2L Sept. 14, 1888, p. 605.
— dxdydzi
ELECTRICAL THEORY. LETTERS TO DR. LODGE 247
the term in V disappearing. This is awful ; but I see no reason to suppose
that it vanishes. If it does not vanish, then the existence of magnetic effect
would decide against Maxwell.
Thirdly, I note on p. 10 of your sketch that Rowland and FitzGerald
consider that electrostatic potential is not propagated by endthrust, and you
remark that it is the magnetic potential which travels, generating the electro
static potential as it goes along. I think I remember that you have expressed
the view that potential energy must undergo a kind of conversion, and that
unless it be born again as kinetic energy it can in no wise go forward on its
journey*. With strained solid waves it looks as if it were so, though it is,
I think, possible to regard both energies as going forward linked together,
yet retaining their individuality. And if potential energy is, after all, kinetic,
but of another kind, it is conceivable that they should keep their separate
identities. But with electrostatic and electromagnetic strains, which is
potential and which kinetic? I know it is usual to call the magnetic kinetic ;
but if we had started with permanent magnets, and travelled by means of
magnetoelectric machines to our present knowledge of electric phenomena,
I expect we should now^ be discussing the propagation of magnetostatic
potential, and magnetoelectric potential, and we should, perhaps, consider
the former generated by the latter. This is really the view I take, or, rather,
I think both are true. It seems to me that the sideway propagation of
electric induction is accompanied by (let us drop 'generated by') magnetic
induction, and equally the sideway propagation of magnetic induction is
accompanied by electric induction. The two go together when a disturbance
is propagated. In a steady state they do not, i.e., if we can separate them
from each other ; at least I do not see how otherwise to interpret the results
I have obtained (see pp. 2845 of paper referred to below) f. This brings us
back to the old point whereon we have differed before. It would be better
to give in like the unjust judge for the sake of peace and quietness, and to
ward off any more such letters as this.
I have been going again through a paper ' On Connection between Electric
Current and Electric and Magnetic Induction' (Phil. Trans. 1885) {, in which
I tried to work out the equations to the magnetic field on the supposition of
this sideway propagation. "^ ceases to be troublesome, and both electric and
magnetic inductions are propagated at the same rate; indeed, they are by
Maxwell's equations, though I do not think he ever definitely worked this
out. I cannot see any point, in the assumption to begin with, or in the
subsequent reasoning, where I have gone astray. If you have any time to
* Phil. Mag. October, 1879, p. 281, § 11; June, 1881, p. 534; and June, 1885, p. 486. I do
not regard this as a 'view,' however, but as a proved truth, — 0. J. L.
t [Collected Papers, pp. 2012.] J [Collected Papers, Art. 11.]
248 ELECTRICAL THEORY. LETTERS TO DR. LODGE
spare, would you look at the Paper, pp. 277281, and 294300? The rest
is not essential, though on 301 I show that Maxwell, pure and simple, gives
the same results*.
Yours, etc.,
J. H. POYNTING.
Mason College, Birmingham,
October 12, 1888.
[The following extract from my note in answer may be inserted, in order
to make the next letter clear :
' As regards the proof of the ether, I confess I did not quite see FitzGerald's
point as to why Hertz's experiments rendered the existence of ether any more
certain; but knowing that I had felt it thoroughly estabhshed long ago,
T supposed I was not a good judge. Of course he must appreciate all the
stock arguments about air, etc., not transmitting transverse disturbances,
and about neither it nor glass transmitting anything at the speed 3 x lO^^,
etc. And Hertz's experiments only seemed to me to prove that electro
magnetic waves existed and travelled at the same speed, thus practically
proving that Ught is electromagnetic waves, and estabhshing Maxwell's
theory.
' This seems to me far more important than proving once more the existence
of ether. At the same time I feel sure FitzGerald has some point. It may
be only that an electromagnetic ether has been proved, and thus the action
atadistanceGermans confounded. He spoke as if he meant more than this.
Perhaps I have to that extent misrepresented him in my " sketch." How
does it strike you?
'With regard to Thomson's Paper, it is certainly very antiMaxwellian,
but I believe it only represents a transition stage through which he was
somewhat rapidly passing, and through which he may now have almost
passed. I certainly do not know now where he is.
' Is it not that, finding that displacementcurrents have no magnetic effect,
therefore he ignores them? But, then, have they no magnetic effect? In
some cases they cannot have, for electrostatic displacement is of the nature
of an '' expansion," as Clifford called it ; there is no '' spin " about it.
' I cannot find Thomson's Paper this minute to refer to, but I have it some
where. I will look it up again in the light of your remarks.' 0. J. L.
After referring to my reply, Prof. Poynting writes, in a second letter :]
About the ether, I should entirely accept your interpretation of Hertz.
I should think, as you say, that FitzGerald was aiming at believers in action
atadistance, and probably he knew where to have them in showing that
there is an electromagnetic medium.
You say, with regard to Thomson's Paper, 'Is it not that, finding that dis
placementcurrents have no magnetic effect, therefore he ignores them?
But, then, have they no magnetic effect? In some cases they cannot have.'
* [Collected Papers, pp. 194198, 212217, and 218 respectively.]
ELECTRICAL THEORY. LETTERS TO DR. LODGE
249
It is just here that I find the supposition of transverse propagation of
electric strain accompanied by magnetic induction so clarifying to my ideas.
If the magnetic effect is the whirling of the machinery which is sending
electric strain energy onwards, the machinery cannot know whether the
U + + + + 4) Ai
"C_ ni\\\ _J>"
' • ] r , V
' , ' ' ' ' \
/ 1 . *  . J \A
.' / / / I
HiW
energy is going to be dissipated in a Pronybrakelike wire, or whether it is
going to increase the electric strain, so producing a 'displacementcurrent.'
To use an illustration, I regard a reentrant line of magnetic force as a kind
of ring of Custom House officers registering the amount of electric strain sent
in per second (they are porters as well), and they do not know what will happen
to the energy. They will register just the same whether the imports are for
immediate consumption or go to add to stock. Of course, they will take no
account of shooting stars, balloons, or destruction of stock already within
the ring ; which, being interpreted, is that the lineintegral of the magnetic
force (47^^■, isn't it?) will not be affected by Knes of electric force other than
those coming in or going out through the boundary. For instance, if A moves
through the ring RR to A^ (see figure) conveying a charge, hardly any Unes
of electric force will cut the ring RR, yet the number of lines through it is
increased. And it will not be affected by induction which dissipates itself
in situ as in a leaky Leyden jar, where the energy changes to heat without
moving. I do not know of any other case, but, being utterly ignorant of
pyroelectricity, I imagine it might supply a case of establishment of electric
induction without motion of the energy and without magnetic effect.
Yours, etc.,
J. H. POYNTING.
October 18, 1888.
17.
AN EXAMINATION OF PROF. LODGE'S ELECTRO
MAGNETIC HYPOTHESIS.
[Electrician, 31, 1893, pp. 575577, 606608, 635636.]
The leaders in Physical Science, impressed perhaps with the responsibility
of their position, and fearing that their weaker followers will distort their
views, are, as a rule, very cautious in giving us their vaguer speculations as
distinguished from the more exact hypotheses which can at once be put into
working shape. Yet these lessformed speculations are often helpful, even
if they only arouse our minds to attempt to disprove them. Still more are
they helpful if they aid us in thinking of facts in a more connected way until
the finished working hypothesis is ready to take their place.
We owe a special debt of gratitude on this ground to Dr. Lodge for his
wellknown book on the Modern Views of Electricity, in which he describes
not only the more definite beliefs which he firmly holds, but gives us also
any suggestions rising in his mind which seem to give promise of light to
guide us in the dark ways of the science. He talks as it were confidentially
to us, and though the speculative character of the book makes it by no means
the easy reading which the absence of mathematical treatment might lead
us to expect, and even perhaps unfits it for beginners, the bold attempts at
explanation give it great value to more advanced students. Such students
will be brought face to face with many difficulties which they may hitherto
not have recognised through haziness of thought. And even where they are
unconvinced by Dr. Lodge's attempts to solve the difficulties they will be
gainers by the orderly review of their knowledge necessary before they can
form a judgment.
The book is built round a central hypothesis of the nature of electric
action, which I propose to examine. I shall first give an account of the
hypothesis as it appears to me to stand, freed from the details and the wealth
of illustration, which, though appropriate, and even necessary, in the original
work, make the argument at times rather difficult to follow. I shall then
examine the evidence for or against the hypothesis.
We start with that which everyone accepts as the result of experiment
and observation, that there are two kinds of electrification with oppositely
AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS 251
directed qualities, and that they make their appearance always in equal
amounts. Hence, on their union, the net electrification is zero. Compare
this with the case of momentum. According to our experience, as summed
up in the third law of motion, we may regard the mutual stress between two
bodies as consisting of a transfer of momentum from one to the other. A gun
at rest is fired. The momentum gained by the bullet and powder may con
veniently be regarded as derived from the gun, which, having none to begin
with, now has an amount of negative momentum equal to the positive possessed
by the charge. In other words, positive momentum is transferred from gun
to charge. Or compare with a case of material transfer, as when A lends B
a sum of money. Then A and B, after the transfer, are oppositely affected,
so that if they both assign their share in the transaction to a third person
C the effect on C is zero ; or if 5 retransfers the sum borrowed to A the net
result is zero.
Such cases as these suggest that positive and negative electrification are
merely the creditor and debtor sides of a single transaction, the sending out
and the reception of something transferred (p. 9). What kind of transfer
we must imagine is best gathered from the ' icepail ' experiment, a particular
case of the general principle that when induction occurs + and — always /ace
each other in equal quantities with an insulating medium between. We may
suppose that we have a nearly closed hollow insulated conductor, and that
through an orifice we introduce a body having on it a charge + Q. Immedi
ately — Q gathers opposite to it on the inside surface, and + Q is on the
outside surface, facing — Q on the walls of the room in which the experiment
is made. If, instead of carrying out this electrical experiment, we imagine
an indefinitely extended incompressible liquid, and think of merely mathe
matical surfaces occupying the positions of the surfaces of the conductors,
the introduction of Q of liquid within the inner surface would force Q of hquid
through each of the two surfaces, and relative to the space between the
surfaces, inwards towards the inner surface, and outwards from the outer.
Now, suppose that such an incompressible hquid, to which for the
present we need not ascribe gravitation, has an actual existence, that it fills
all space with which we are concerned, and that it permeates matter. Let
us suppose that in bodies which we term electrical conductors it is free to
flow with nothing worse than frictional loss of energy, but that in insulators
it has some kind of attachment to the matter, so that in the displacement
of one relative to the other — that is, in the strain — energy is stored, and in
such a way that it can be regained when the strain is relaxed. We shall
call this liquid Electricity. When a displacement occurs, so that some of
the liquid is pushed into or out of any conductor, the quantity flowing through
the surface from or to the insulator is that which we have hitherto called the
charge of electricity on the surface. We must now term it Electrification to
252 AN EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS
distinguisli it from the general body of the hquid, which is, according to
Lodge, all Electricity. It is important to notice this distinction in studying
Lodge's hypothesis. Electricity is the generic name for the fluid all through
space, Electrification the specific name for that part of it which happens to
have flowed in any given disturbance through the surface of a conductor.
Thus, if we imagine a conducting sphere A within a hollow conducting sphere
B, B having an orifice through which we can introduce a wire to charge the
inner sphere, say, positively, the gathering of a charge Q on ^ is to be regarded
as a flowing of some of the allextensive electricity along the wire into the
sphere, which is, however, already full of electricity. Hence Q must be
pushed through the surface of the sphere out into the insulator or dielectric.
This outward displacement manifests itself as the + electrification of ^. As
the dielectric is also initially full of electricity, Q must be pushed out through
every surface in it completely enclosing A. The displacement relative to the
airparticles stores energy — the energy of the charge. When we come to the
inner surface of B, Q is pushed into the substance of B, an inward displacement
which we term a negative charge. It is also pushed through the substance
of B, but as this substance is conducting, no energy is stored, and only a little
is dissipated by the frictional rub, or, perhaps better, the viscous cling. At
the outer surface of B there is another displacement of Q outwards into the
dielectric — i.e., another positive charge, and energystoring begins again.
The pushing out will take place through the second dielectric till we come to
the walls of the room in which the action is occurring. Here it will probably
end, for the generator of the original charge has probably sucked in the fluid
from the walls. There is therefore a confined circulation, and not an infinitely
extended pushingout.
Sources of electrification with their connecting wires are evidently to be
regarded as turbines working in pipes or channels laid in space, incompressible
and fluid electricity filHng both the pipes and the space outside them. When
the turbines work, the fluid runs along the pipes, forming what we call an
electric current. When a pipe ends in a reservoir bounded by a dielectric,
the fluid presses out into the dielectric, and there stores the energy put into
it by the turbine, minus that dissipated by the viscous resistance in the pipes
and conducting channels.
Prof. Lodge illustrates the connection between the electric incompressible
fluid (why not the electric Hquid?) and the molecules of matter by a series of
ingenious and suggestive models either with cord running through beads or with
Hquid to represent the electricity. It is hardly necessary here to describe these.
Any reader who is not yet acquainted with them should study the original
account. It is enough to say that Prof. Lodge can make his models behave Hke
Leyden jars and give the phenomena of charge, residual charge, and oscillating
discharge as perfectly as if they were thorough beHevers in his hypothesis.
AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS 253
I find it somewhat easier to form a picture of the hypothesis by supposing
matter to have a spongeHke constitution, i.e., to be permeated in every
direction by passages, the pores so dear at one time to the writers on elemen
tary science. These passages are filled with Hquid electricity. In every
passage or tube, however short, we must imagine a Httle turbine turning
round with the flow through it and never letting any fluid pass without
duly turning. In dielectrics there is a spring, Hke the mainspring of a watch,
attached to the spindle of each turbine so that when the wheel turns energy
is stored. In conductors this spring is wanting, and there is only a viscous
resistance to rotation. We may think of the source of electrification, machine
battery or induction coil, as a large turbine somewhere in the system with
a supply of energy behind it dealt out by a motor of some kind or other.
When the large turbine works, a flow takes place in the system dissipating
energy in the conductors and storing it in the dielectrics. When the charging
turbine is removed or disconnected from the motor, so that the way is clear
for a return, all the woundup mainsprings return and drive the liquid back
through the sponge. We have only to make the turbines with different
moments of inertia, with different qualities of lubricator in the conductors,
and with different strengths of spring, and different firmness of attachment
in the dielectrics, to get varying permeability, electric resistance, specific
inductive capacity, and residual charge. We may, perhaps, simphfy the
arrangement of affairs by supposing that the molecules themselves are the
turbines, and then we get a kind of inversion of the hypothesis described
hereafter, which Dr. Lodge develops to account for electromagnetism.
In the displacement hypothesis, with a single electric Hquid flowing past
matter, there is a very serious difficulty. The energy being stored by the
flow past the molecules of matter, we might reasonably expect the electricity
to pull, or to tend to pull, the molecules with it, and there should, therefore,
be a motion of matter along the lines of force in one direction. But instead
of this we have a tension, both ways as it were, along the lines of force or flow.
There is displacement of matter only in the case of electrolytes, and here it
is both ways along the lines of flow, one set of atoms going one way and another
set the opposite way. A modification of the hypothesis is suggested by
Dr. Lodge to meet this difficulty and to account for the double electrolytic
procession. He supposes that there are two constituents of the electric fluid,
each in general filling half any space, intimately mixed and evenly distributed.
The molecules of matter are made up each of two constituents in accordance
with the usual view, and one of these is attached to one kind of electric fluid
the other to the other. When electric displacement occurs it is really a double
flow, the two constituents of the fluid travelling equal distances in opposite
directions past each other. The atomic constituents of each molecule move
in opposite directions, but not in general very far, so that there is in dielectrics
no displacement of the matter as a whole. In electrolytes the displacement
254 AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS
continues till separation and re pairing occur, and thus we get a double
procession.
We see that in dielectrics, the pulls of the two constituents of electricity in
the molecules, one on one atom, the other on the other, balance each other.
These electric pulls lead to internal stresses within the molecule about which
Dr. Lodge does not say anything very definite, but it seems to me that we have
to introduce chemical forces here to account for the pull of the atoms on each
other, distinct from the electric forces or the pulls of electricity on matter.
This dualism is hardly in accordance with the late exposition of Dr. Lodge's
ideas, where he appears to identify electrical and chemical forces (p. 84).
Perhaps we might as well, while we are inventing a constitution for the ether,
make a third or neutral constituent to which all the atoms of matter are
attached. We will suppose this neutral electricity to resist extension and
compression. We then have electropositive atoms attached to positive
electricity, electronegative atoms to negative electricity, and all of them to
neutral electricity. When an electric displacement occurs, positive ether
tugs at one set of atoms, negative ether at the other set, and neutral ether
prevents their separation, so that all our forces are of one kind, insomuch
as they are forces between atoms and ether. I rather Hke this neutral ether,
but I am afraid Dr. Lodge will not adopt a strange infant into a family already
sufficiently large.
But taking the hypothesis as set forth by its author, the mere duahty
does not much affect the general notion of the nature of electric charge. We
must remember that motion of negative fluid inwards equally with that of
positive charge outwards gives a positive electrification, so that the explana
tion of the electrification of a sphere within a conductor already given has
only to be amplified by supposing that there is another ethereal fluid displaced
in the opposite direction at the same time and throughout the system.
I can imagine the agnostic in ethereal matters protesting here against
the multiplication of unknowns and unknowables. I can imagine him saying
that his senseorgans are only excited by material motions and affections,
that his instruments are all material, and only appear to undergo changes
of shape, colour, sound — i.e., affections of matter, and that he cannot with
any certainty get beyond these material affections. He will argue that, as
we have no sense affected by ether alone, we can form no adequate conception
of the ether ; we can only suppose it endowed with material properties, and
conceive of it as some form of matter. And it would appear possible to
imagine various material ethereal constitutions or connecting machineries
between the different portions of matter evident to our senses, all equally
accounting for all known facts. When a new fact turned up he would own
that probably some of the ethereal machinery would fail to account for it,
and so would have to be taken off to the lumberroom for wornout hypotheses ;
AN EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS 255
but the new fact, he would argue, would very Ukely enable us to imagine new
types of machinery to replace some, at least, of the old rejected ones. And
probably, till the whole range of physical phenomena was known, it would
always be possible to imagine more than one kind of machinery to account
for the phenomena known. Probably only when nothing remained to be
discovered would there be a single solution, and only then would it be possible
to give a single answer to the question. What is ether? And even then we
might be wrong, for the ether might have properties in its action on matter
quite different from any of which we have material types.
Though our agnostic, when following this train of thought, may be un
convinced by Dr. Lodge's preface, and may urge that the ether is and will
probably remain a hypothetical medium, he will, no doubt, adopt some form
of hypothesis for working purposes. If he is an ordinary human being, when
he studies such actions as we term actionsatadistance, he will prefer to
think of the different parts of the acting matter as connected by something
continuous, with material properties. Following Boscovich and Faraday,
he may extend the atoms throughout space, and give this extension material
properties (a special case of this type of hypothesis is presented to us in the
ring vortex theory of the Universe) ; or he may limit the atoms and put in
some new connecting machinery to fill up the vacuum he abhors, and this
he may as well call 'ether.' While, therefore, he may protest against
Dr. Lodge's ' cocksureness ' about any particular constitution for the ether,
he is bound to examine any hypothesis reasonably presented to see if it is
likely to form a good working hypothesis to account provisionally for the
observed facts. He would, no doubt, admit that Dr. Lodge's hypothesis is
reasonably presented, and it would only be a question with him whether so
complicated a constitution for the ether enables him to think sufficiently
easily of the phenomena for which it is to account. Leaving him to consider
this, we may pass on to the further development of the hypothesis.
So far, we have only been thinking of the properties of electricity at rest.
It is true that we have thought of the electricity as being pumped along
conductors and as wasting energy in the passage, but this was only a step
onwards to a final statical distribution. We are now to concentrate our
attention on the pumping stage.
When electricity is in motion in sufficient quantity and for sufficient time,
a new set of phenomena come into prominence. Among these are the heating
of the conductor, the heating or cooHng of junctions, the opposite ionic pro
cessions in electrolytes, and the creation of a magnetic field. The heating
of the conductor is to be explained, according to Dr. Lodge, as something
analogous to frictional, or rather viscous, dissipation of energy. We may
think of the two streams of electricity flowing past the atoms of matter, and
continually catching hold of them and letting them go again, as a fiddlebow
256 AN EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS
catches hold of and lets go a fiddlestring. Thus, some of the energy of flow
is converted into vibrational energy of the atoms, that is into heat. The
junctional heat phenomena still wait for complete explanation. We may
consider that the facts imply that at a junction of dissimilar metals there is
a tendency for positive electricity to move more easily in one direction than
the other, and, of course, the reverse with negative electricity. A compound
bar with free ends will thus tend to be positively electrified at one end and
negatively at the other. Suppose, further, that the tendency to separation
of electricities varies with the temperature, and we have at once the thermo
electric current in a closed circuit. Returning to the fiddlebow and string
used to illustrate the development of heat by conduction, we can see how it
ought to work to illustrate thermoelectricity. Suppose a set of parallel
strings in a horizontal plane, one half tuned to one note and the other half
to another. The one set may represent one metal with its atoms vibrating
in given modes, and the other set another metal in contact with it and with
its atoms vibrating in other given modes. Now, laying the fiddlebow across
the strings after they are set in vibration, if the motion of the bow is always
in one direction it illustrates the pushing of one of the electricities from one
metal to the other. I have found that when light bits of paper are laid across
two vibrating strings of different pitch there is frequently a movement of
translation. Unfortunately for the illustration it is sometimes in one
direction, sometimes in the other. The paper was not part of Prof. Lodge's
book, and knew nothing of the hypothesis it was expected to support. The
opposite procession of ions we may think of as the transport of the atoms
by the positive and negative electric streams respectively, the connections
between the pairs of atoms being broken down and renewed with fresh partners
all along the line and continually.
The most evident phenomenon characterising the electric current is the
magnetic field around it. This we may regard as manifesting the existence
of so much magnetic energy in the neighbourhood of the conducting wire.
Let us see how the hypothesis will account for this energy. The first step
is to reduce permanent magnetism and currentmagnetism to one species by
adopting Ampere's theory. If we consider a small closed currentbearing
circuit, observation tells us that, at a distance from the circuit, the field is
indistinguishable from that due to a small steel magnet, with centre in the
plane of the circuit and axis perpendicular to it. Starting from this, we know
that we can deduce an arrangement of permanent magnets, equivalent, as
regards the outside field, to any currentbearing circuit. The circuit differs
magnetically from the steel in two respects only, viz., that it requires a con
tinual supply of energy to maintain it, and that we can get into its inside.
This last difference may be merely due to the large size of any apparatus at
our command, and it is quite thinkable that, if we could make ourselves or
our apparatus smaller than molecules, we could explore the inside of the steel
AN EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS 257
molecules. The other difierence is possibly not one of the kind ; for suppose
the resistance to be diminished till it disappears, the rate of energysupply,
C^R, disappears also, and we have a currentcircuit — never mind how the
current was started — which is as permanent a magnet as a steel bar. It is
a short and inevitable step from this to Ampere's hypothesis that a magnetic
molecule — a molecule of steel, say — is essentially a small closed perfectly
conducting circuit with a current of electricity in it ; or, in terms of Dr. Lodge's
hypothesis, either two equal and opposite currents whirhng round at equal
speeds in opposite directions in each molecule, or a positive whirl in one
direction in one molecule, accompanied by a negative whirl in the other
direction in the next molecule. We may dismiss this duaUty for the present,
on condition that it comes up for sentence when called upon, and return to
the single circuit. Such a circuit, when placed in a magnetic field, behaves,
doubtless, Hke finite circuits, and tends to set itself perpendicular to the lines
of force, and with its own lines parallel to the lines of the field, and in the
same direction through the circuit. It tends to move from weaker to stronger
parts of the field — tends, in fact, to include as many positive and exclude
as many negative lines as possible.
But the disappearance of resistance has a pecuHar effect. Even a circuit
of the resisting kind, with which alone we have practical acquaintance, would
protest against the inclusion of foreign fines of force in addition to its own,
and the current would diminish while they were being included. An Amperean
circuit would not merely protest, but would absolutely prevent any change
in the total number included, for any increase would be accompanied by a
finite negative e.m.f. proportional to the rate of increase, and therefore by
an infinite current, since R is zero. This is not to be accepted as possible,
so that all that can happen is that some of the current's own fines of force
shall be replaced by those of the field, and there is a consequent weakening
of the current.
When a mass of iron consisting, we suppose, of such Amperean currents is
brought into a magnetic field, all the circuits tend to turn round to include
the fines of the field, and at the same time the Amperean currents decrease in
strength. The circuits thread themselves like beads on to the fines of force
of the external field, their currents falling as they thread on. But their own
surviving lines of force are added to those of the field, so that the total field
is greater, and the iron has greater permeabifity than a vacuum ; or perhaps
it will be better here to say that the iron conducts the lines of force better
than a vacuum would, for permeabifity has an exact significance, not quite
describing the property now under discussion. As new fines of force are
added to the field the iron will conduct them better than a vacuum, until
every molecule has all its lines brought into service, and its current, therefore,
reduced to zero.' After this point is reached, since the molecules either will
p. c w. 17
258 AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS
not take any more lines of force, or if they take any more will establish
negative currents to neutrahse them, they are worse conductors than a
vacuum, for they fill up part of the space uselessly or injuriously. They will
therefore tend to move from stronger to weaker parts of the field, as if
diamagnetic. Some indication of such a change in conducting power has
been detected by Ewing in the magnetisation of iron in exceedingly strong
fields, for though the permeabihty or induction produced f the field producing
it — both reckoned from zero — was always greater than 1, the value of increase
in induction ^ increase in field producing it, or the conducting power, as I have
called it, fell ultimately below 1.
For some reason, not yet explicable, the magnetic chains in iron become
unstable, and break up when the temperature is raised to the neighbourhood
of 800° C. The iron then above this temperature is practically equivalent
magnetically to any other substance.
Assuming then that we have some notion of what we mean by currents
of electricity whirhng in channels of no resistance. Ampere's hypothesis gives
us a fair explanation of iron and steel magnetism. If we accept the view
that the interior of a magnet does not differ from the exterior in kind but
only in degree and in permanence, we may attempt to extend the hypothesis
to explain the magnetic quahties of all other substances, i.e., their power of
carrying the Hnes of force, and of carrying them in sHghtly different degrees.
Let us think of the lines of force in air circling round a current or passing
from pole to pole of a magnet. We may think of these as passing through
a number of electric whirls, or at any rate through perfectly conducting rings
ready to exist as whirls. Before the passage of the lines of force these rings
are turned in all directions. After the passage they tend to set perpendicular
to the lines of force. If the medium is paramagnetic we may suppose the
rings to have initial currents in them; if it is diamagnetic we may with
Weber suppose that they have no currents initially, and when no Hnes of
force pass through. On the estabhshment of the field negative currents are
excited of such value as to make negative Hnes of force thread each ring equal
in number to the positive Hnes sent through by the field. Thus each ring
acts as a part of the field through which no lines of force pass, and the per
meabihty is thereby diminished. At the same time the diamagnetic substance
will tend to weaker parts of the field, and we have the main facts of dia
magnetism explained. There is a serious difficulty in the nearly constant
permeabihty, differing only by a very small amount for a diamagnetic soHd
Hke bismuth and a magnetic gas like oxygen, the one with its molecules
crowded together, the other with its molecules comparatively wide apart.
Perhaps we can strengthen this weak point by supposing the conducting
rings of very different diameters in the two substances, or we may think
of the electric channels as different altogether from the molecules and the
AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS 259
same in number per c.c. in bismuth and in oxygen. Another difficulty Hes in
the nonexistence of permanent magnetism except in iron, nickel, and perhaps
cobalt, but it is no greater than the difficulty with iron above 800° C.
Perhaps both will ultimately find the same explanation, and until this is
forthcoming we need only say that Ampere's hypothesis is merely silent on
the point and is not necessarily unable to explain it.
Now as to magnetic energy. Dr. Lodge points out a number of facts
which suggest that magnetic disturbance is of the nature of spin round the
Hues of force. The Amperean circuits at once present themselves as being
the seat of this spinning, and the electric fluid or fluids in the channels as the
spinning material. These whirHngs of electricity, either in themselves or in
the accompanying motion of the entangled matter (Maxwell, by the way,
thought it was the matter), possess, according to Dr. Lodge, the magnetic
energy of the system. In fact, they are themselves magnetism.
So far we have been dealing with magnetism and its relation to electricity,
the spacefilling fluid, and we have come to the conclusion that magnetism
consists of vortices in this fluid. It remains to explain the nature of the
ordinary electric current, and the way in which its accompanying magnetic
field is maintained. Dr. Lodge uses for the purpose a mechanical analogue
or mechanical model, a modification of Maxwell's wellknown model, which
is described in his Scientific Papers, vol. 1, page 451, and I believe by
Dr. Garnett in Maxwell's Life.
Let us imagine the two fluids, which are to be regarded as jointly filling
space, to have a cellular construction, each consisting of spheres or little
indiarubber bags, or what you like, in contact with each other. In dielectrics
we think of contiguous cells as gearing in some way. Let these cells, when
in a magnetic field, be spinning round the fines of force, the positive in one
direction, the negative in the opposite; and let positive and negative be
alternated so that the opposite motions may be possible without sHp of
gearing.
Fig. 37 from Dr. Lodge's book illustrates this idea when the fines of force
pass into the paper from above, the axis of spin, therefore, being perpendicular
to the paper. We may further materiafise the conception by inserting teeth
round the edges of the cells; and here the family likeness to the parent
(Maxwell's model) becomes stronger. We then have Fig. 36. If we suppose
Fig. 37 to represent the unstrained state in a dielectric, then electrostatic
strain will be presented by some such deformation as that represented in
Fig. A ; not, I think, in the manner represented by Fig. 46 in Dr. Lodge's
book.
If we take Fig. 36 as our type, it is easier to think of the wheels as being
attached to some kind of framework. We may think of all the positive
17—2
260 AN EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS
wheels in Fig. 36 as arranged flat against one series of parallel rods going
down the page, and the negative wheels on another intermediate and parallel
series. We may think of the wheels, or their teeth, as not quite rigid, so that
when the positive rods are pulled down and the negative up, there is a slight
Dr. Lodge's Fig. 36. Rows of cells alternately Dr. Lodge's Fig. 37. Section of a magnetic
positive and negative, geared together, field, perpendicular to the lines of force;
and free to turn about fixed axes. alternate cells rotating oppositely.
(Another mode of drawing Fig. 36.)
Fig. A. Electrostatic strain. The cells displaced slightly in opposite directions.
C
+ 1 +1
2
Dr. Lodge's Fig. 46.
Fig. B. Cells shpping in a wire carrying
a uniformly distributed current.
relative displacement and energy is stored, and this will represent an electro
static strain. Probably, if we arranged all the forces properly, we should
be able to do without the framework of rods. Meanwhile I find I cannot
think of the model clearly without it.
AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS 261
The distinction between a dielectric and a conductor is to be represented
by the abohtion of the teeth on the wheels in the conductor. The surfaces
of the wheels are in contact, and are only more or less rough — viscously
rough, rather than frictionally rough. It appears rather difficult to think
of the layer of wheels separating a dielectric and a conductor, for on the
dielectricside they must have toothed gearing, and on the conductorside
they must only rub ; but we may get over the difficulty by supposing these
wheels double, one toothed to gear on the dielectricside, and the other
untoothed to rub on the conducting side, and both keyed on the same axis.
Dr. Lodge regards a current as represented by a slip of one row of wheels
against the next. He says (p. 206), 'Notice that in a medium so constituted
and magnetised — that is, with all the wheelwork revolving properly — there
is nothing of the nature of an electric current proceeding in any direction
whatever. For, at every point of contact of two wheels, the positive and
negative electricities are going at the same rate in the same direction; and
this is no current at all.... A current is nevertheless easily able to be repre
sented by mechanism such as that of Fig. 36 or 37 ; for it only needs the
wheels to gear imperfectly and to work with shp. At any such shpping
place the positive is going faster than the negative, or vice versa, and so there
is current there. A line of sHp among the wheels corresponds therefore to
a linear current.... Understand: one is not here thinking of a current as
analogous to a locomotion of the wheels — their axes may be quite stationary.
The shp contemplated is that of one rim on another.'
Thus in Fig. B, let the row of wheels represent the cells across the dia
meter of a wire carrying a steady current, ABCD representing the section of
the wire. Let the speeds of rotation of the wheels be as marked on each.
The resultant positive shp is + 1 at each surface of contact, a total of 6.
It is perhaps presumptuous to quarrel with a parent such as Dr. Lodge
as to his mode of developing the faculties of his own offspring, but I must
venture here to differ from him entirely in the way in which he seeks to
make his model represent current. It appears to me that he has grafted on
to his own model the representation of current in the entirely different model
of FitzGerald, and so obtains something quite inconsistent with his previous
ideas. It is only by stopping short at the centres of the bounding wheels
that he can obtain a resultant flow in one direction. Obviously, if he took
in a whole number of wheels, each entire, he could get no resultant flow, for
the flow on the opposite sides of each wheel is equal and opposite. Thus,
in the figure, if he took into account the outer sides of the two last wheels
in the wire, he would have — 6 neutralising the previously obtained { 6.
Or, to put it in another way, if we draw a plane, say, above the line of centres,
a tangent to all the wheels, there is evidently no resultant flow across that
plane. And we can think of cases of shp when we have no reason to suppose
262 AN EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS
I
ABCDOdcba
there is current. Thus, if a bar of soft iron is placed axially in a magnetic
field, near the centre of the bar the hues of force are parallel to its length
within and without the bar. According to Lodge, the electric whirls are
very rapid within the iron, and comparatively slow in the air outside. There
must, therefore, be slip at the boundary, and yet we have no dissipation of
energy to indicate the existence of the current.
Another objection is that the model worked thus would make a diflterence
in kind between the process of displacing in a dielectric, the equal and opposite
motions of the two fluids which are leading to an electrostatic strain, and the
current in a conductor. This is rather setting back the clock. Perhaps we
have gone on too fast, and the difference may exist, but I think we should
hardly accept the evidence of the model on the point.
What Prof. Lodge calls current appears to me then
to be merely sudden change of magnetic intensity. If
this is just criticism let us see if the model can be
made to represent a true current. I shall take the
case of a steady current in which the condition of
affairs is not altering. It is always better to begin
with such a case, just as it is better to begin with
statics, hydrostatics, electrostatics, than with dynamics,
hydrodynamics, and electrodynamics. When matters,
or ethers, get into changing motion they are, like the
celebrated pig of the Irishman, difficult to count.
In the model we shall suppose that the current in
the wire is represented by two equal and opposite
processions of cells or wheels along the wire, and to
help us in thinking of these processions we shall, as
before, suppose two sets of rods parallel to the axis of
the wire, the positive wheels on one set with their
spindles perpendicular to the rods, and the negative
wheels on the other set.
In Fig. C the rods alone are represented, directions of motion of the
wheels on each being shown by arrows. The + rods are to be regarded as
moving down the page, and the — rods up. The wheels on the bounding
rods, B, b, gear with the wheels on A, a in the dielectric, but sRp on those
on C, c in the conductor. Now, if the conductor extended through all space
I do not see that the motions of the rods would involve rotation of the
wheels. For any rod would be as it were surrounded by a symmetrical
system, its wheels would not know which way to turn, and so would merely
rub against their neighbours. But here the conductor is bounded by an
insulating medium in which the rods can only move a very httle way, and
that only by straining the teeth of the wheels.
Fig. C. Rods on which
the positive and nega
tive wheels are sup
posed to be fixed in
the model of a wire
carrying a current.
AN EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS 263
The result is that as B moves up past A its wheels behave as pinions
moving on a rack in the place of A. They tend to turn the wheels of C in
the opposite direction. These in turn tend to send round the wheels of B,
and so on to 0, the middle rod. Now, starting from the other side, a is at
rest, and as h moves past it its wheels must rotate. This rotation is trans
mitted inwards, as before, to 0. The result is that the wheels on are
urged in opposite directions from the two sides, and so remain at rest, while
those of D and d rub against them. Those of C and c rotate faster than those
of D and d, the friction due to the sUp balancing that due to the sHp of D
and d on 0, and so on, the rotation increasing outwards. When we come
to B, b, their wheels have the frictional resistance against the C, c wheels on
the inside, which must be balanced — since the motion is steady — by pressure
on the outside against the A, a wheels. Hence the B, b wheels are not merely
rolHng on the A, a wheels, but are pressing against them, and so turning them
round. In other words, the rotation extends out into the dielectric. Thus,
at the middle of the wire, there is no rotation — no magnetic intensity. It
increases as we go outwards to the boundary, being in opposite directions
on the two sides, and it exists in the surrounding medium. The rubbing of
the surfaces, perhaps of the rods, perhaps of the wheels, dissipates energy
which represents the heat appearing in the wire.
The model is probably indeterminate as to the way in which the energy
finds its way into the various parts of the conductor, where it appears as
heat. We might imagine the rods moved by end thrusts along the wire,
their wheels setting the outside wheels spinning, or we might imagine a spin
propagated along the outside wheels to the wire, and there setting the rods
in longitudinal motion.
I have no doubt that the model would be able to represent the phenomena
of the currentinduction when the motion is accelerating or unsteady, for
Maxwell's model does this, and the differences of construction would hardly
affect the point.
I have now given an account of the main features of Dr. Lodge's hypothesis
as to the nature of electric charge, magnetism, and current, and I have done
this at considerable length because I propose to discuss the foundations for
the hypothesis, and to see if they are sufficiently firm to bear the superstructure.
To recapitulate the main features, we have a double ether consisting of equal
f and — portions attached in some sort of way to matter as a framework —
elastically in dielectrics — loosely, with a kind of viscous connection in con
ductors. The two ethers always move in opposite directions along the fines
of electric force, and in dielectrics they store energy by their displacement past
the material framework, while in conductors they only dissipate it during the
displacement. The ethers are cellular in construction, and the cells are capable
of rotation, this rotation constituting magnetism, and the accompanying
264 AN EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS
rotational energy being magnetic energy. The axes of rotation are along
the hnes of magnetic force. The cells roll on one another as if quite rough in
dielectrics, and as if lubricated with viscous oil in conductors, and they are
alternately + and — , and moving in opposite directions, a clockwise moving
+ cell giving the same direction to the magnetic force as a counterclockwise
— cell. Current is, according to Lodge, shp of these cells against each other,
but according to the interpretation I have given of his theory, current is
a double and continuous procession of the cells in a conductor, with the
consequent rotations.
The hypothesis is avowed by its author to be an attempt to obtain a
mechanical explanation of electric and magnetic phenomena. He uses the
main idea of Maxwell's wellknown model, but replaces Maxwell's duality
of magnetic wheels and electric 'idle' wheels by a duahty of electric wheels.
It is, perhaps, open to question whether this is really a simplification, but
the attempt was very well worth making, for it is only by variation and
natural selection that the mechanical model will be suited to its environment
in the electric world.
It is, I suppose, useless to look for any other than a mechanical hypothesis
as final. Probably because we are able to picture mechanical processes, able
to think of ourselves as seeing what goes on, seeing kinetic energy manifested
in the moving parts, able to think of ourselves as part of the connecting
machinery, feehng the stresses, and helping to make the strains, we have
come to regard mechanical explanations as the inevitable and ultimate ones.
Thus, though the old mechanical hypothesis of light is for the present
discarded in favour of an electromagnetic one, I suppose no one is content
with the present position ; but we are all looking forward to the time when,
by mechanical explanation of electromagnetism, Ught shall once more become
mechanical. Nevertheless, ifis well to bear in mind that all such explanations
are merely hypothetical, and may at any time have to be discarded, as the
solidether Hghthypothesis has been discarded. Indeed, they are solely of
value as a scaffolding enabling us to build up a permanent structure of facts,
i.e., of phenomena affecting our senses. And inasmuch as we may at any
time have to replace the old scaffolding by new, more suitable for new parts
of the building, it is a mistake to make the scaffolding too solid, and to regard
it as permanent and of equal value with the building itself. It is on this
point that I find most cause for disagreement with Prof. Lodge. He appears
to me to regard his hypothesis as inevitable and permanent, or at least as
approximating to the permanent and inevitable. The scaffolding, in fact, is
made as important as the building. It behoves us, therefore, to examine
into the security of its foundations.
We may take the hypothesis as based on two statements :
I. Kegarding electricity at rest, 'Whenever we perceive that a thing is
AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS 265
produced in precisely equal and opposite amounts, so that what one body
gains another loses, it is convenient and most simple to consider the thing not
as generated in the one body and destroyed in the other, but as simply trans
ferred' (p. 9).
II. Kegarding electricity in motion: 'Magnetism is nothing more nor
less than a whirl of electricity' (p. 171).
Let us examine some cases which should come under the first statement.
We may regard the upper and under surfaces of a vessel of liquid as
respectively positive and negative, and if we measure them by their projections
on a horizontal plane they are equal and opposite element by element. We
clearly cannot create a positive without an equal negative. Bring the two
elements together, and they neutrahse each other, for both cease to exist.
We have no idea here of something transferred.
Or, take the case of a rotating cord. Its two ends, as viewed from the
outside, have equal and opposite properties, as may be seen at once if we think
of them as brought close beside each other. If the rotations were transferred
to one and the same body the net result both to themselves and to the body
would be no motion. Or if the cord were cut away in slices, bit by bit, when
it is all cut away the two ends have come together, and have, in a sense,
neutrahsed each other. Here again we think of nothing transferred.
As a third case take a magnet. We always create equal and opposite
poles at the same time with mutually neutralising properties, and to bring
this within Dr. Lodge's statement we have only to consider the magnet as
made up of two bodies joined together in the middle. Or perhaps we might
take the case of the creation of N. and S. poles by the breaking of a magnet.
No one has yet made a displacement or transferhypothesis for magnetism
except for the purpose of illustration. This, by the way, is rather remarkable,
for magnetism with its reentrant tubes of induction is so much more ready
to lend itself to such a hypothesis. We seem to have finally made up our
minds that electric is strainenergy, and that magnetic is rotational energy,
and we do not even consider the alternative of magnetic as strain and electric
as kinetic energy. No doubt the existence of magnetic rotation of the plane
of polarisation of light strongly suggests the rotational nature of magnetism,
but we can hardly claim any explanation of the phenomenon yet put forth
as complete, and certainly none is exclusive. The other facts would possibly
be equally well explained if we thought of electric tubes of induction as
strings of wheels with their spindles along the axis of the tubes, and magnetic
tubes as fines of flow of ether or of two ethers, with a certain displacement
coefiicient, greater in diamagnetics, less in paramagnetics. I do not put this
forward as worth following out, but merely to illustrate the contention that
we need not necessarily think of f and — electrification as due to a longi
tudinal transfer of something. It appears to me quite possible then to
266 AN EXAMINATION OF PROF. LODGE'S ELECTROMAGNETIC HYPOTHESIS
symbolise the positive and negative charges facing each other across a dielec
tric in other ways than by the transfer of something from one towards the
other.
Turning now to the second fundamental idea, Ampere's hypothesis, that
a magnetic molecule is a small closed electric current, this is really closely
connected with the idea that there is something moving in the direction of
an electric current and constituting the current. It is in fact dependent on
our acceptance of the first idea. For consider what an Amperean current is
phenomenally. There is no resistance, and therefore no heat developed, and
no fall of potential; there is no junction, and therefore no Peltier effect;
there is no break, and therefore no chemical effect. The one effect left is
the magnetic one. If, then, we take away the idea of some substance whirling
round and round in a channel, all that we have left is a little permanent
magnet. In other words, we explain the constitution of a magnetic molecule
by supposing that it is a molecule having a magnetic constitution.
It is true that if we get a bundle of Hues of magnetic force and tie them
together with a perfectly conducting cord with its ends joined up we have
a permanent magnet, for no more and no less Hues can pass through the ring
of perfectly conducting stuff. But this does not really explain. It does not
show that the unknown is a case of the known. For a perfect conductor is
far more difficult to think of than a permanent magnet.
It is getting time that these socalled 'perfects' were abohshed from
Physics. We have to deal with matter as it is and not as we should have had
it if we had been consulted as to how it should be made in order to simplify
our equations. The perfect gas has nearly gone and I shall be glad to see the
perfect conductor preparing also to depart.
If, however, we grant substantiality to the electric current, we are met by
a difficulty in connection with Ampere's hypothesis which, at any rate, requires
examination.
Let us suppose that we have a circuit of selfinduction L and resistance
R, and let N be the number of outside lines passing through the circuit. If
E is the E.M.F. of the battery kind in the circuit the current equation is
li E = and i? = we have an Amperean circuit for which
01* LC + N = constant.
In words, the total number of Hues of force threading the circuit is
constant.
AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS 267
If, now, N increases, either L or C must diminish. In order that L may
diminish the circuit must contract and pinch in, as it were, the Hues of force.
But meanwhile, since the total is constant, the number of hues passing through
the same area is increasing, and we should expect the spin, therefore, to increase
and tend to make the tubes of force widen out. That is, there would be
resistance to the contraction. The contraction is contrary, too, to our
ordinary experience of a circuit with a current in it, for such a circuit tends
to expand. If, on the other hand, the circuit is rigid, L is constant and C
naust diminish. If, as I gather from the general tenor of Dr. Lodge's work,
the Amperean circuit is one of the ether cells peculiarly constructed or at least
of the same order of magnitude, a diminution of C means a diminution of spin
of this cell. But the total number of lines of force through it is constant,
which seems to imply that the spin remains the same. This seems to
force us to suppose that the Amperean circuit is not of the order of the
ether cells, but larger, so that it may include within its contour a number
of these cells. Then when N increases, C round the boundary may diminish,
and still LC + N he constant, for the internal wheels may be spinning at the
same rate as before, but the external wheels more slowly, the current being
a function of their difference. It would require further examination to see
whether the total spinning energy could thus be accounted for. At first sight
it would appear that that energy is diminished by the lessening of C just when
we want it to increase.
If we are to suspend our judgment as to the substantiality of electricity,
and are not as yet to conclude that a current is a rushing round of a thing
of some kind, we are bound also to suspend our judgment as to Ampere's
hypothesis, and if we are in this state of suspense, and, for my own part, I must
confess to being so, Dr. Lodge's hypothesis and its accompanpng models are
to be used rather as illustrations — as analogies — than as ultimate solutions.
For the purpose of illustration they are of the greatest value. Like a bank
reserve, ready in case of emergency to cash the symboUc banknote, they
are in the background in the mind ready to turn into mechanical reahty the
ordinary electromagnetic symbols, such as hues of force, which are, I think,
much more easy to deal with, but which are, doubtless, wanting in reahty.
Perhaps Dr. Lodge may consider that my criticism is somewhat carping,
in that I have nothing constructive to put in the place of the hypothesis to
which I object. But I beheve that the time has hardly come for ultimate
mechanical construction, and that, at present, progress is more Ukely to be
made if we are content with an electromagnetic explanation — if we merely
carry down to the molecules and their interspaces the electric and magnetic
relations which we find between large masses and round large circuits, and
leave the ether out of account. I beheve that we may symbolise electric
and magnetic actions by means of fines of force and their motions in a way
268 AN EXAMINATION OF PROF. LODGE's ELECTROMAGNETIC HYPOTHESIS
which allows us to think clearly of the phenomena, and though the ultimate
nature of the lines of force is unknown, we can only say the same of the ether.
In the application of these lines of force to the molecules, and their constituent
atoms, there appears to be hope for some kind of explanation of the chemical
phenomena of the circuit and of the distinction between electrolytic and
metalhc conduction. The difficulties in the way are by no means small, as
everyone knows. I may, at some future time, attempt to set forth some of
the difficulties as they occur to me, in the hope that a plain statement of them
may lead some reader to help in their solution.
18.
MOLECULAR ELECTRICITY.
[Electrician, 35, 1895, pp. 644647, 668671, 708712, 741743.]
In a Paper in The Electrician of October 13, 1893*, after some criticism
of the 'wheelwork' hypothesis of electromagnetic actions, I concluded by
saying that I believed that ' the time has hardly come for ultimate mechanical
construction, and that, at present, progress is more hkely to be made if we are
content with an electromagnetic explanation — if we merely carry down to
the molecules and their interspaces the electric and magnetic relations which
we find between large masses and round large circuits, and leave the ether
out of account.'
I propose to follow out this idea, to see where it leads us, and what
difficulties we have to face. The idea is in many minds, and we have
examples of its use in the papers of Prof. J. J. Thomson and in the articles by
Mr. Chattock in Dr. Lodge's Modern Views of Electricity and the Philosophical
Magazine. The fullest account yet published is, I think, in Prof. Thomson's
Recent Researches, Chapter i, but there are some difficulties not there brought
to the front. I believe there will be some advantage in a new statement,
starting with the very alphabet of the subject, and following it up by full
examination of the consequences. In the belief that some way may yet be
found to remove the difficulties, and that even now it gives us a valuable
picture of some electrical actions, I venture here on a full description of the
shape it has taken in my own mind.
The hypothesis with which we start is that electrical and chemical forces
are identical; that electrification is a manifestation of unsatisfied chemical
affinities, and that chemical union is a binding together of oppositelycharged
atoms or groups of atoms.
Before descending to the atoms, let us briefly consider what we observe
on the large scale coming within the range of experiment. The foundation
stone of our electrical knowledge is the experimental result that the two
kinds of charge are always found in equal amounts with opposite or neutrahsing
mechanical properties, and that they always face each other on the two sides
of the insulating matter between them. We add to this the idea that the
* [Collected Papers, Art. 17.]
270 MOLECULAR ELECTRICITY
energy of the charges accompanies some kind of strain — alteration, perhaps,
of atomic or molecular configuration — in the insulating medium. The two
charges, in fact, always join hands through the dielectric; or, putting the
same idea in another form, the charges are manifestations on the bounding
surface of a state of strain within the dielectric, somewhat as hydrostatic
pressure is a manifestation on the surface of the vessel containing a Hquid
of a strain to which the liquid is subjected. When this idea of strain was
enunciated by Faraday and rendered precise by Maxwell we had only the
variation of inductive capacity to support it. But now the double refraction
of dielectrics in the field discovered by Kerr, and the wave phenomena dis
covered by Hertz, give as near a positive proof of the existence of the electric
strain as we can ever hope to get.
In a system at rest the electric strain ends at the surface of a conductor
and the electric stress or pull on the surface 2tto'^IK per sq. cm. is resisted on
the other side of the surface by ordinary mechanical stress accompanying
ordinary mechanical strain in the matter of the conductor. Some day,
perhaps, we may be able to manufacture a hypothesis identifying mechanical
and electrical strain ; but at present we are obliged to separate them and
think of them as different in kind, since mechanical strain in moving off or
dying away does not give rise to the magnetic or chemical actions which
characterise the moving off or dying away of electric strain.
We symbolise the relative value of the strain at different points of the
field by unit tubes of induction each beginning at + 1 and ending at — 1 of
electrification. The strain varies inversely as the crosssection of a tube, and
we regard 1/^, the reciprocal of the specific inductive capacity, as a kind of
modulus of electric elasticity. Though tubes of induction give us a better
description of the imagined physical condition of the dielectric, it is easier
to use lines of induction, one for each tube, when we have to draw figures.
These ideas of duality of charge, and of strain between them, which we
owe to Faraday and Maxwell, though now accepted by everyone, have hardly
yet so saturated our minds that we instinctively use them in every case. We
still too often find descriptions of elementary phenomena which entirely leave
them out of account. Like charges are still described as repelling each other,
as when the gold leaves of an electroscope diverge, without a hint that the
apparent repulsion is really a pull on each body along the fines of induction
stretching out to the opposite charges induced on the surrounding conductors.
The earth is still looked upon as a big conductor which will hold any reasonable
amount of electricity without showing signs of it, and we speak of discharging
a body into it instead of saying that the opposite charge is usually on the
earth or the walls of the room in which we work, and that when a conducting
bridge is made the two charges can come together so that the inductive strain
is refieved, and its energy is dissipated. Or to take another example where
MOLECULAR ELECTRICITY 271
common language and thought lag behind the more precise ideas given to us
by science: the earth, as the 'return' circuit of a telegraph wire, is still
described as a reservoir from which, say, + can be pumped up at the sending
end and into which it can be emptied out at the receiving end, just as if
a telegraph wire corresponded to a pipe running above a water reservoir with
a pump at one end and a spout at the other. Sometimes even we find the
process described as if the charge going to earth at the receiving end knew
its way back through the earth to the particular battery from which it started.
Whereas the earth is really not a return circuit at all, but a parallel outgoing
circuit for the opposite charge, enabUng the two electrifications to travel in
company and to face each other — one on the wire, the other on the earth —
whether the earth be the bare ground under an aerial line or the sheath of
a cable in the bed of the ocean.
This travelling of opposite charges to meet each other is to be regarded,
I am convinced, as the essential electrical part of the ordinary current. The
lines of induction connecting the charges sweep through the air or other
surrounding insulator, and this motion of the induction, or travelling onwards
of the electric strain, is accompanied by — indeed, is rendered possible by —
the magnetic induction which surrounds the conducting wire. The only case
of current without motion of charge is the gradual cessation of charge in an
imperfectly insulating condenser. Here the charges simply die away in situ
— their strain decays and their energy is dissipated ; and as there is no motion
of induction there is no magnetic field.
To realise how ordinary current may be described in terms of motion of
lines of induction let us suppose that, instead of the more usual dynamo or
battery as source, we have a charged condenser with the — terminal to earth
and the + terminal insulated, but capable of connection by a key with a wire
earthed at the farther end. Up to the instant when the key is put down the
two charges are almost entirely in the condenser, joining hands in the dielectric
between the plates. The still prevalent mode of describing the current on
contact at the key is equivalent to supposing that, as soon as the + found
a door open to it, it freed itself from the embrace of its — , put its hands in
its pockets, ran along the wire, setting all the neighbouring magnetic machinery
spinning as it rushed past, and finally plunged into the earth at the farther
end. The — , no longer kept up or 'bound' by the +, sank into the earth at
the condenser end, and both were lost in that vast electrical abyss, the globe.
But let us see what is really implied in the process of discharge if we
keep to our fundamental principle that + and — always have inductiontubes
or lines between them. Let us try the various suppositions which seem open
to us, and let us first imagine as a possibility that the charges are able to start
on their travels by breaking their inductiontubes somewhere in the dielectric
between the condenserplates, as in Fig. 1. At the broken ends we must
272
MOLECULAR ELECTRICITY
have opposite charges (just as in a broken magnet we have opposite poles),
for a charge is the end on matter of an inductiontube. If each + unit
setting out along the wire drags the broken half of its tube after it into the
wire, we shall have on the whole equal + and — travelling in the same direction
along the wire and no external magnetic effect. Indeed, as soon as the leading
+ has drawn its piece of tube completely into the wire there does not appear
to be any reason against the union of the + and — at opposite ends of the
piece, and then nothing need occur in or around the rest of the wire. But
as we have magnetic and other efiects in and around the whole length of wire,
and everything is against the passage of + and — in the same direction, we
are bound to reject this supposition of breaking tubes. As another supposition
let us imagine the tubes as remaining in the dielectric of the condenser but
growing longer where they stand, and as it were pushing their endcharges
before them, the one through the wire, the other into the earth. But this
Fig. 1.
will leave the tubes still in full strength in the dielectric, and when the dis
turbance in the wire has died away the condenser will still be fully charged,
which is absurdly at variance with experience. The only possible supposition
still left to us is that the tubes of induction spread out sideways from the
dielectric of the condenser into the surrounding air, connecting the  and —
charges as these move respectively along the wire and the earth. The
induction— that is, the condition we call electric strain — is propagated by
the working machinery from point to point through the air, and the motion
of this machinery is manifested as magnetic induction, symbolised by lines
of magnetic force which form closed circuits round the wire. We cannot
explain this mode of transfer, but the mechanical models of Maxwell and
Lodge and FitzGerald help us to accept it as possible.
We must now introduce the idea that conductors are dielectrics — of
a special kind, no doubt — admitting of electric strain, though allowing it
to decay rapidly and dissipating the energy they receive in the form of heat.
We may take as a helpful analogue to this behaviour of a conductor the
behaviour of a liquid under ordinary elastic shearstrain. A liquid can
receive shearstrain just as much as a solid, but the strain decays, that is,
rapidly loses its energy, and the lost energy is transformed into heat. But
MOLECULAR ELECTRICITY 273
the still undecayed strain remaining at any instant gives a tangential resistance
to shear, which we recognise as viscous resistance to the motion of one layer
relative to the next. So in a conductor such as a wire, electric strain may be.
produced and electric energy may be taken in ; but the strain decays almost
as fast as it is received and the energy is changed to heat. The e.m.f. along
the wire is proportional to the strain remaining still undecayed at any instant^
and this e.m.f. corresponds to the elastic stress — the viscous resistance — of a
sheared liquid.
In the case of the condenser which we are considering, as the induction
tubes spread out from it their endportions are continually moving sideways
into the wire, and to a less extent into the earth, and there melting away
and dissipating their energy as heat. New ends are continually being formed
at the junctions of the decayed and undecayed portions, and these ends travel
further and further along the circuit until the tubes are entirely propagated
into the conductors, when the discharge is complete.
A full account of this mode of regarding current, and the relations involved
between electric and magnetic induction, with an attempt to show that it
applies to the voltaic circuit as well as to such a circuit as we have here con
sidered, was given in the Philosofhical Transactions for 1885*. Prof. J. J.
Thomson has modified and elaborated the theory in the Philosophical Magazine
for March, 1891, and in his Recent Researches in Electricity and Magnetism,
Chapter i. The full theory shows that if we assume that the magnetomotive
force round a closed curve is equal to 47r x number of electric tubes cutting
the curve per second, then the magnetic field is accounted for. If, then, we
make the electric tubes move so as to account for the right electrical quantities,
the magnetic properties will follow as a matter of course, and we may therefore
concentrate our attention on the electric tubes, as we have done in the pre
ceding account of one case of current. I have given this case at length, by
way of introduction, to show that here at least the only way in which we can
think consistently of current in a wire is in terms of motion of + and of — ,
and of tubes of electric induction connecting them and sweeping through the
surrounding medium into the wire. We shall see later how such tubes may
be supposed to be furnished by the recombinations which occur in the voltaic
circuit, and I have no doubt that in all other cases of current the same kind
of explanation may be given.
There is another reason, perhaps, for giving here the foregoing account of
a condenser discharge, in that the hypothesis which I am going to describe
may not inaptly be termed a condenserhypothesis of electricity. The
properties which we find in condensers will be carried down to the molecules,
which we shall suppose to be small condensers with equally and oppositely
charged atoms forming the two plates. Of course, this will not explain
* [Collected Papers, Art. 11.]
p. n. w. is
274 MOLECULAR ELECTRICITY
electricity. It is a purely electrical hypothesis, and it shifts all responsibility
of further explanation on to the shoulders of whatever atomic hypothesis we
adopt, just as Weber's magnetic hypothesis gives no explanation of magnetism,
but assumes the molecules to be readymade magnets, and leaves to them the
burden of accounting for themselves.
We are naturally led to make the hypothesis from the consideration of the
chemical action of current. When a voltameter is included in a circuit a
perfectly definite quantity of electrolyte is decomposed for a given number
of inductiontubes moving sideways into it. All parts of the liquid between
anode and cathode are concerned in what goes on, and there are two opposite
processions of ions. This is all experimental fact. Kepresenting the action
atomically we may obtain the net result by the old Grotthus chain method of
picturing the process, and each atom set free requires the supply of a definite
constant amount of electric induction, calculable if we know the number of
atoms per gramme of the substance.
Let us now consider how electrolysis may occur. We shall imagine
a somewhat abstract kind of electrolyte, one consisting of molecules, each
with a pair of atoms, an abstraction, a simplification, which we shall have to
discard later. To account for the exact chemical equivalence of electrolysis
in different cases we shall have to suppose each atom to be supplied with the
same amount of electric induction, to have the same minute fraction of an
inductiontube starting from it if it is of one kind, ending on it if it is of the
other kind. To bring the induction up to a thinkable size we must choose
a new unit — I suppose miUions of billions of times less than the ordinary unit,
and it will be convenient to picture the atom as supplied with two, four, or
some such small number of tubes in terms of this new unit.
Here, then, is our molecule, represented in Fig. 2. A is
the positive and B the negative atom. The molecule is, in
fact, a little condenser with fixed charges, and the distance — ' g
between A and B is usually small compared with their dis pig. 2.
tances from neighbouring molecules. But we must suppose
that in general A and B do not come in contact, perhaps through motion of
vibration or of rotation relative to their centre of gravity. There is therefore
persistence or conservation of their tubes. The electrical attractions sym
bolised by these tubes are absolutely identical with the chemical attractions
of the molecules, and the electrical energy they contain with the chemical
energy of the molecule. It is easy to see, however, that two molecules
colHding may become connected, and form a more complex group, and may
even effect an exchange of atoms.
Thus, in Fig. 3, (a) to (/) represent successive stages in the process of
exchange. Just as, when two condensers are brought near with their tubes
of induction running in opposite directions, crossconnections are formed by
MOLECULAR ELECTRICITY
275
the coalescence and neutralisation of parts of the tubes, so here we may
suppose that in a molecular collision the tubes straying out coalesce, and the
parts which run in opposite directions destroy each other. In (c) we have
a more complex molecular group. If the process is continued we may arrive
at (/), which gives the original molecular configuration but with exchange
of partners. If we are ever to attempt an electrical hypothesis of elasticity
and cohesion we shall have to suppose that straying of tubes is always going
on in solids and liquids, and that complex groups like (c) are always being
formed, so that each molecule is attached to its surroundings. In gases not
in an electric field the molecules may be regarded as much more selfcontained.
(rt)
>' '' /k j\
(L)
X
,(/)
Fig. 3.
When a line or tube of induction moves sideways into an electrolyte com
posed of such condensermolecules it finds the molecules with their axes
distributed equally in all directions. It will pick out those already facing
in the direction suitable for it — those requiring the least energy for its purpose,
and those alone need be drawn.
In Fig. 4 (a) two lines of induction, XX, YY, are represented as ready to
move into an electrolyte, in which AB, AB represent the suitable molecules.
The successive stages are represented by (6), (c), (d), (e), and the final result
is that the highest B atom is delivered up to the + electrode, and the lowest
A atom to the — electrode, while all the intermediate atoms in the chain
change partners, merely forming molecules like the original ones, but with
their axes reversed.
If more lines come in they will find new molecules directed as they require,
and soon also the collisions occurring in the liquid will distribute the axes of
18—2
276
MOLECULAR ELECTRICITY
the molecules of the first chain into various directions, so that some of these,
too, will be ready for later lines of induction.
The coalescence and destruction of the parts of two tubes which run
alongside each other in opposite directions, upon which the whole process
•f Etectrode. ..«___
1^ /s
X —Electrode
Fig. 4 (a).
Fig. 4(6).
depends, will naturally tend to take place, as less potential energy will be
needed for the coalesced tubes than for the two side by side. The energy of
the final configuration is the same as that of the initial configuration, except
at the two ends, so that if XX and Y Y bring in enough energy to provide for
the new configurations at the electrodes the process will continue. If XX
MOLECULAR ELECTRICITY
277
and 77 bring in an excess of energy we may regard the change of partners
as occurring with more or less of a rush, and the excess is converted into
internal molecular energy, partly kinetic, partly potential; in fact, the chain
y
\
\
J
>
f
It
\
f
i
i
V 1
<
Fig. 4 (c).
Y )
c
Sr
B
Sr
f
Q
Sr
B
r^]
V
Fig. 4 {d).
of recombined molecules is warmer than it was, and we have the Joule C^R
effect.
For simplicity we have considered the tubes XX, 77 as stretched
through free space till they come to the chain of molecules which they
are to rearrange. But it is easy to see how a tube may be handed on from
278
MOLECULAR ELECTRICITY
y X f Electrode
N' V
B
V X
Fig. 4 (e).
— Electrode
Fig. 5 (a).
X
Fig. 5 (&).
MOLECULAR ELECTRICITY
279
chain to chain without any free existence. In Fig. 5 (a) a tube XX already
threads a chain of AB molecules and in (6), (c), (d) are shown the successive
stages of its transfer to a neighbouring chain. The process is rendered more
capable of representation by supposing each member of the second set of
molecules lifted its own depth upwards. The derivation of each stage from
the preceding is evident. This is doubtless the process which we must
Fig. 5 (c).
Fig. 5 id).
suppose to occur in all cases, and the tubes must be regarded as threading
such chains in the dielectric, even before they enter the electrolytic cell.
Let us now consider what must happen at the electrodes. If the ions
unite chemically with the electrodes, the positive and negative atoms find
respectively negative and positive atoms ready to combine with them, and
no difficulty is introduced at this point. But if the ions are set free, as are
hydrogen and oxygen at platinum electrodes in dilute acid, we are brought
280 MOLECULAR ELECTRICITY
face to face with the great, and, it is to be confessed, the yet unsolved difficulty
in the hypothesis. Take the case of the hydrogen atoms released at the
cathode, the atoms such as the lowest A in Fig. 4 (e). We may think of them
as connected at first by tubes of induction going from them to platinum atoms
on the surface of the cathode. But soon the gas bubbles up, and there is no
manifestation of the charge all of one kind which we have ascribed to the
separate hydrogen atoms. We may attempt several explanations. The
common notion appears to be that the hydrogen gives up its charge to the
platinum, and then rises up quite deprived of electricity. This might occur
by the breaking away of the tubes from the hydrogen atoms and the with
drawal of the positive ends into the platinum, the positive charge moving
across the separating space, like a disembodied soul, from its hydrogen
habitation into the platinum. The platinum having two neutralising souls,
becomes as merely material as the deserted hydrogen. But this is strongly
against our experimental knowledge of charge, which is always, so far as we
know, on matter, whereas here charge crosses over a separating space. We
might suppose this got over by thinking of the hydrogen and platinum atoms
as coming absolutely in contact, shortening the connecting tubes to nothing,
and putting them out of existence. If the tubes are not recreated when
the hydrogen rebounds the gas rises up without electricity.
But, according to the current ideas of chemistry, the hydrogen issuing
from the cell is not atomic, but molecular, and consists of paired atoms at
least; that is to say, it is a chemical compound, differing from ordinary
compounds it is true, in that the two members of each molecule are the same
in kind, but the atoms are held together by chemical attraction, and satisfy
each other's chemical affinity just as much as if they were different elements.
If our hypothesis has any truth in it, this means that electric induction exists
between the atoms and holds them together, a view supported, I think, by
the phenomena of electric discharge in gases and the electrolysis which appears
to take place in that discharge. To be consistent, then, we must suppose
that the hydrogen, when it rises up, consists of pairs of + and — atoms, and
for us the difficulty is to explain, not what has become of the hydrogen charge,
but how half the atoms have succeeded in exactly reversing their charges,
so that they are able to combine with the other half to form neutraHsing pairs.
And, of course, we must extend our supposition of the constitution of an
element as being made up of + and — atoms to the platinum also. When
the lines of induction move into the platinum, the — atoms of the surface
pairs are freed from their + partners, and are free to form pairs with the
+ hydrogen atoms. But when half the hydrogen reverses its charge half
the platinum must do the same in order that it too may effect the neutral
combination which we find when the electrolysis has ceased. Perhaps the
best course is to say that if our main hypothesis of identity of chemical
attraction and electric induction is real, then the reversal of charge, such as
MOLECULAR ELECTRICITY
281
occurs on our supposition at the electrodes, is up to the present an unexplained,
an ultimate fact. Without attempting explanation, but rather as a crude
mode of picturing a process which would lead to the result, we might think
of a + H and a — Pt as coming absolutely in contact, so that the tubeends
can shde round on the surface of the atoms and across the bridge of contact
until + is on the Pt and — on the H. Possibly Fig. 6 will make this clearer.
Fig. 6.
In (b) two atoms have come in contact, in (c) the tubes are beginning to
slide round, in (d) the ends have changed places, and in (e) the atoms have
drawn apart with reversed charges. Having thus, or otherwise, reversed half
the pairs, combination with the other half may ensue as in Fig. 7, the result
being hydrogen molecules and platinum molecules each consisting of + and
— pairs.
At first sight it seems as though the reversal occurring in the sparkdis
charge of a Ley den jar or a Hertz vibrator would give us the key to the
reversal pictured in Fig. 6. But further examination takes away this hope.
The jarreversal is never complete and exact such as we have to suppose the
atomic reversal. Indeed, the induction may have any value, f or — , less
numerically than the initial value, as it gradually dies down to zero through
the successive vibrations. We shall see later how some account may be
given of the reversal in vibrators, an account which quite destroys any analogy
with atomic reversal.
Perhaps a better analogy is afforded by two vortex rings A and B when
they are playing at leapfrog. The first widens out and lets the second go
through it. Before the passage the liquid is streaming through A towards B.
After the passage it is streaming from B towards A, so that if we think of
induction as corresponding to direction of flow we have here a reversal.
282
MOLECULAR ELECTRICITY
Pt
Pt
(a)
(b)
Pt
But I suspect that this liquid analogy is much more appropriate to a
magnetic than to an electric hypothesis.
Were it not for the magnetic difficulties involved there would be some
temptation to attempt a theory in which the sign of charge is merely a state
ment of the kind of atom concerned, and that the tubes, Hke gravitation
tubes, have nothing but ahgnment and neither + nor — direction. But this
would, I fear, be getting over one difficulty only by introducing much greater
difficulties hereafter.
We shall assume, then, that in some way or
other reversal can take place in certain cases,
and we may apply it at once to explain the
apparently neutral condition of the ions in the
electrolytic cell. We suppose that half the ionic
atoms in Fig. 4 reverse or change charges with
the electrode atoms, as in Fig. 6, and that then
they go through the process of Fig. 7 with the
other half, and so form + and — pairs, exhibiting
no external electrification.
A similar reversal will enable us to give an
electrolytic account of metallic conduction. The
great distinction between electrolytes and metals
is that in the former there are opposite atomic
processions, while in the latter there is apparently
no motion in either direction. Electrolysis with
out procession means, as we have just seen with
the electrodes, where the procession stops, re
versal of charge. Suppose, then, that such reversal is possible with every
metallic molecule. Imagine a copper wire carrying a current to be made up
of molecules, as represented in Fig. 8 {a), and let tubes of induction XX, YY
be just movmg into the wire. Before the current is established the molecular
axes are evenly distributed in all directions, but the tubes entering in select
those most suitable, as shown in {a), turning the configuration to that shown
in (6) ; if now reversal takes place we get (c). Let two more tubes come in,
and we get (d), a reversal of which gives the same arrangement as (a) as far
as the copper is concerned, but the four entering tubes of induction have
entirely disappeared.
We must now attempt to give some account of the action at the source
of a voltaic current. As we simplified the electrolyte so we shall simplify
the active liquid of the voltaic cell by imagining that we have merely sulphuric
acid, that is, we shall neglect the solvent, water. We shall take as the two
metals zinc and copper.
We know that the result of putting the copper and zinc in the acid is that
(c)
Fig. 7.
MOLECULAR ELECTRICITY
283
the air above the acid tends to become the seat of tubes of induction running
from copper to zinc. This will be neutralised by the oxidising tendency of
the air on the zinc— at any rate that is the 'chemical theory' of voltaic
action. To eliminate this action of the air we shall suppose the zinc to have
Fig. 8 (a).
Cu.
■Cu^
. Cu,
•Cu,
.Cu,
ii
Ou.
Cu.
.Cu^
Cu.
Cu,
T
Fig. 8 (6).
U
OUc
Cug
.Cu^
Cuj
Cu.
Cu,
■ Cu,
Cu,
Cu,
Cu,
Fig. 8 (c).
Fig. 8 id).
a copper terminal in the air, as it always has in practice. Experiment shows
that these tubes of induction always run from copper terminal to zinc terminal
with a fall of potential which may be as much, say, as a volt and a half. To
represent the establishment of this induction let us imagine the cell initially
to consist of such molecules as are shown in Fig. 9 (a), where we represent the
284
MOLECULAR ELECTRICITY
acid molecules as consisting of + Hg paired with  SO4, and the copper and
zinc plates as ± Cu and ± Zn.
Of course, the molecules in each substance are evenly distributed in all
directions, but only those are shown in the figure which are suitably directed
for the action which is going to occur. The first stage is represented in
Fig. 9 (6), where change of partners has occurred between metal and acid
both with copper and zinc. The HgZn molecules do not reverse, for it is an
experimental fact that the hydrogen does not rise up where it is first turned
out. The reversal may perhaps be prevented by the presence of the ZnS04
Cu
Cu
j\ /(
■Cu
• Cu
Ou
Cn
SO^
>S Ji
Zn —
H
SO4
>
1 d
k
, ,
f
—<
— <
r H
7n
*^
SO4 H2 SO4 Hg
Fig. 9 (a).
•Cu
■Cu
Cii
Cu
Zn
Zn
SO^
SO4 Hj SO4 Hj
Fig. 9 (&).
SO4
Cu
Cu
molecules, but it is more probably due to the electrical energy put into the
HgZn. For the ZnS04 contains much less energy than the H2SO4 it replaces ;
so that the HgZn contains much more than the ZnZn it replaces. The
electrical energy put in makes the atoms separate too widely, we may imagine,
to allow of the contact needed for reversal. Probably at first there is no
reversal in the HgCu molecules at the other plate, and for similar reasons.
The electrical energy in the HgZn and HgCu molecules will imply that the
acid is at a higher potential than the metals, and tubes of induction will
spread out from the Hg atoms. Since the HaZn molecules .contain the most
energy, we may represent their tubes as spreading rather than those of the
MOLECULAR ELECTRICITY
285
HgCu, and in Fig. 9 (c) we have the tubes shown as going out towards the
copper and then doubling back again. In (d) the tubes have entered into
the neighbouring molecules by coalescence with oppositelydirected tubes,
and we are left with two tubes running from the + to the — terminal with
a ZnS04 molecule and an HgCu molecule, the CUSO4 molecule having been
dissociated in the process. The molecules of acid have changed partners, but
still have the same constitution.
If the action is not continuous, but merely goes on till the terminals are
charged, we must suppose that HgZn and HaCu pairs are left against each
Zn
SO4
Cu
8O4 Hj 8O4 Hg
Fig. 9 (c).
Ou
Cu
„J_L M M M
Zn
80^
Cu
Cu
Cu
Fig. 9 (d).
plate with an average fall from H to metal of, say, 1 volts. The pairs against
the copper plate are no doubt the agents in polarisation. The reverse current
which we get on replacing the zinc by a fresh copper is to be set down to the
strayingout of the tubes of induction of these HgCu pairs. If the action is
made continuous by connecting the terminals with a wire, the hydrogen rises
up from the copper plate, and we must suppose that half the HgCu pairs have
reversed and have then changed partners with the other half. Perhaps this
reversal is rendered possible by the resolving of the CUSO4 molecules, perhaps
by the outlet provided for the energy of the HgCu pairs in the external circuit.
286
MOLECULAB ELECTRICITY
In a very similar way we may account for the contact difEerence of potential
of copper and zinc in air. We know that if the two metals are brought into
contact a fall of potential occurs from the air near the zinc to the air near the
copper ; that is, electric induction passes from the neighbourhood of one to
the neighbourhood of the other. Both metals can be oxidised, but zinc by
far the more readily. Let us suppose the oxygen molecules in the air to be
made up of pairs of opposite atoms, and that before contact we have a state of
affairs, represented by Fig. 10 {a), developing into (6) by the actions at the
metals. On each metal we have the normal oxides ZnO, CuO, where the
positive atom comes first, and the unstable molecules OZn and OCu. If the
tubes of induction of these unstable molecules stray out they have to double
Zn
Zn
U V
Zn
Zn
qO
_
o o
""^~
Q
. .
' 1
' '
O A«
/■H ■
— < —
Oo
o
Fig. 10(a).
Zn
.1 >i
r V
Cu
CU
Cu
Cu
Cu
Cu
Zn
Zn
O
<
o
*:
Cu
Cu
o o
Fig. 10 (&).
back on themselves, so that as long as there is no contact there are equal
numbers in the two directions passing through the air, and no fall of potential
except from the surfacelayer of the air to the surfacelayer of the metal close
to it. This fall will be greater at the zinc than at the copper, since there is
presumably more energy given up at the zinc surface. Probably reversal is
prevented either by the presence of the normal oxides or by the want of
outlet for the energy. If, however, it does occur in some of the unstable
molecules there will be repairing and the formation of new metal molecules
and new oxygen molecules, with the net result that each metal is left slightly
oxidised. But let us make the two metals touch as at J in Fig. 11 {a). If the
tubes of induction of the OZn molecule now move out Uke those of the HgZn
molecule in Fig. 9 (c), the upper returning part can enter into the continuous
metal bridge and there be dissipated; while the lower outgoing part will
MOLECULAR ELECTRICITY
287
thread the oxygen molecules, as shown in Fig. 11 (b), and decompose the CuO
molecule already found. We shall as a net result have ZnO on the surface
of the zinc, a fall of potential from on the zinc surface to on the copper
surface, and the unstable OCu, which possibly reverses and repairs, leaving
the copper surface unacted on. If it remains then the copper is polarised, and
if the zinc were suddenly removed and replaced by a new copper it would
appear that the old copper should show a fall of potential towards the new.
Zn
Zo
Zn
Zn
Zn
Zn Oy
Co Cu
Cu
n >i
< — — < — — < — — e
O O GO
< 1 t < — I I < 1 I ^
> 1 I — < 1 I < — I I ►
o o O O O o
> — — ^ — — < — — )>
Cii
Cu
Cu
Cu
Fig. 11 (a)
o o o o
Fig. 11 (b).
Zn
Cu
Fig. 11 (c).
But any such fall would probably be disguised by the induction ending in
the atoms next to the copper. We have probably gone too far in this
account in supposing that all the induction of the OZn molecules goes out in
this way. If, for example, we suppose that only half goes out, we get a chain
extending from Zn to Cu, as in Fig. 11 (c). Probably some such supposition
must be made in order to explain how the + atoms remain at the zinc
surface and the — atoms at the copper surface after the break of contact.
That they do remain is shown by the i electrification of the zinc and the
288
MOLECULAR ELECTRICITY
— of the copper when tested by an electrometer. But our account must be
regarded as a first attempt and not as a complete explanation.
m
L i ' M i
f .
•^
i
t.
< '
'
k , . . ^
r
Fig. 12(a).
\<
.1 •
Fig. 12(c)
■!■'
  w
1 ' y
't
— u
!■
, ,,
Fig. 12(e).
Fig. 12(6).
V ^^ >^
V V V
u >r >r
Fig. 12 (cZ).
Leaving electrolysis, let us consider how an ordinary insulator may be
affected by induction, and how we may represent the condition of affairs
leading up to sparkdischarge. We shall thus get some useful ideas which
will supplement our account of electrolysis and make it somewhat easier to
understand the true nature of the process.
MOLECULAR ELECTRICITY
289
We shall suppose that we are deaUng with a dielectric consisting of paired
+ and — atoms, and we shall suppose, further, that these molecules are
initially selfcontained — that is to say, that their tubes do not stray out to
surrounding molecules. This is, no doubt, a simplification not existing in
nature; but, as with electrolysis, so here also we must be content to begin
with an abstract case. When there is no apparent electrification in the system
the axes of the molecules will be equally distributed in all directions. But
if electrification is communicated to a pair of conductors bounding the
dielectric, tubes of induction move into the dielectric, and, selecting the
suitably arranged molecules, connect these in chains. It will be convenient
now to suppose at least four tubes of induction to pass from atom to atom
in a molecule. Let Fig. 12 (a) represent a number of molecules ready for
a tube of induction to affect them. In (6) it has moved in and formed the
molecules into a chain stretching right through the dielectric. If we suppose
another tube to move in, as in (c), we get a condition of instability, for now
we are just halfway to a change of partners all along the line. A third tube
will give us (d), a configuration with the same amount of energy as (b), since
the molecules in the two cases are similar, the axes only being reversed.
Hence in passing from (c) to (d) energy is given up, another way of saying
that (c) is unstable. A fourth line entering will change {d) to (e), where the
change of partners is complete, and where electrolysis has occurred.
We may usefully follow out the process by the aid of a diagram representing
the relation between energy put in and induction. Beginning with an ordinary
condenser, let distances along OX (Fig. 13) represent induction put in per unit
area crosssection, distances along Y difference of potential between the end
plates.
At first we have V = r^ >
p. c. w. 19
290 MOLECULAR ELECTRICITY
where V is the potential difference, d the thickness, a the surface density,
and K the specific inductive capacity.
But since the induction D = cr.
If then K is constant, the relation between F and D is represented by a straight
line, which makes with OX an angle 6 given by
4:77(Z
tan c/ = ^ .
The energy stored per unit crosssection is equal to ^, or equal to the
area from the origin up to the ordinate V, and bounded by the hne representing
the relation between F and D. We may indeed regard the diagram as showing
the relation between induction and energy stored, and this is probably a better
point of view, since we cannot attach much idea to potential in the later
parts of the process now to be considered. From this point of view the
abscissa is the induction through unit area and the ordinate is the energy
added per unit addition of induction.
All experiments hitherto made appear to show that K is practically
constant so long as the medium can continue to store up energy, though
the Kerr effect shows that K does alter slightly as D increases, apparently
sometimes increasing and sometimes diminishing, or, perhaps, always in
creasing, but sometimes most along the lines of induction, at other times
most at right angles to them.
Sooner or later, however, a point of instability is reached — sooner in gases,
later in liquids and solids ; and discharge occurs along one track and all the
energy is dissipated. We can see how the curve in Fig. 13 must run in order to
represent this instabiHty. Making the unit area small enough to represent the
crosssection of a molecule, the curve expresses the relation between induction
going right through a chain of molecules from plate to plate and energy put
in. At first, while only a small part of the induction is continuous, as in
Fig. 12 (6), we know that the energy stored is proportional to the square of
the induction, and the curve is, as we have seen, a straight line. But as more
induction becomes continuous from plate to plate, and as the atoms are
pulled in both ways, the force resisting separation does not go on increasing
so rapidly, and the curve falls below a straight line.
At some point on the way to instability and subsequent change of partners
the force will reach a maximum, and after that the curve will turn down,
successive equal additions of induction requiring diminishing additions of
energy. At last, when the fines of induction from each atom run half one
way, half the other, as in Fig. 12 (c), the energy put in is a maximum, and the
curve crosses the X axis, as at A (Fig. 13). The configuration is now unstable.
MOLECULAR ELECTRICITY
291
and will of itself pass through (d) and (e), absorbing two more positive hnes or
extruding two negative hnes till we arrive at (e), represented by the point
B in Fig. 13, when the condition is again stable, like that at the beginning.
The energy put in between and A is given out again between A and B in
part, no doubt, as the Hght, heat, and so on of the discharge. The curve
OAB may be termed, perhaps, a molecular characteristic. It should be noted
that the passage from (c) to (d) (Fig. 12), or past A (Fig. 13), may be effected
D
D
Fig. 14(a).
Fig. 14(6).
Fig. 14 (c).
M V
^' " A
V ^r
Fig. 14 id).
either by absorbing more positive tubes, or by sending out negative tubes.
If the passage through the position A occurs with a rush, then we may regard
the chain as sending out negative tubes by some such process as is illustrated
in Fig. 14, (a) to (d). Though the instability of a single chain will not be
reached till its condition is represented by A (Fig. 13), the instability for
a great number of parallel chains is reached as soon as they are all at or near
the highest point of the characteristic. In the ascending part of the curve
all the parallel chains will tend to have the same amount of induction through
19—2
292
MOLECULAR ELECTRICITY
them, for if any one has more than another, energy will be yielded up by
a redistribution between them. Thus let one chain have induction OM
(Fig. 15), another near it induction ON, the energies stored being OMP, ONQ.
If L bisect MN the second chain may give up induction LN = LM to the
first, and at the same time there will be a yield of energy equal to the difference
between the areas PL and QL. Probably the induction is really distributed
about an average, some of the chains having more, others less, for no doubt
the condition is kinetic, and when there appears to be equilibrium it is not
static but ' mobile.'
Fig. 16.
If the average condition is represented by a point far up the slope and near
the crest, and if any one of the chains gets past the crest, as far below it on
the other side as the average is below on the first slope, then this advanced
chain will at once receive induction from the others, and continue to move
down the second slope towards the point of instabihty. Thus, let there be
n chains in all. Let OH A (Fig. 16) be the characteristic of that with the
greatest induction, OKA' the sum of the other n — 1 characteristics. If the
first chain is at P while the others are at p on the same level, by a transfer
of induction mn = MN from the general body in the first chain, there is
a yield of energy, since the area of pmnq is greater than the area of PMNQ,
for the two slips are of the same breadth, but the slope of PQ is steeper than
that of fq. The transfer will therefore take place, and the chain moves from
P to Q. At Q, a fortiori, a new transfer will take place, and so on, and the
chain will move towards the position of instability. This will all probably
MOLECULAR ELECTRICITY 293
occur even when the average is far below the crest if the induction is widely
distributed about that average, for as soon as one chain gets over the crest
it will probably find others between it and the average ready to hand on their
induction and energy to it, and send it down the second slope.
As, then, the general average rises towards the crest, the most advanced
chains, as soon as they pass the crest, tend to discharge the rest, and there
will at a certain point be a rush in sideways on to these advanced tubes.
They will move down to the condition of instability represented by A on the
diagram. They can of themselves move past this point, extruding negative
tubes, till they arrive at the point B, where the change of partners is complete.
It appears at least probable that this yield of negative tubes gives the
opposite charging of the medium which takes place in the second quarter
period of the oscillation accompanying rapid discharge, now made so familiar
by the work of Hertz. We may, perhaps, think of the process somewhat
as follows: If the inrush of positive tubes during the first stage is rapid
they will concentrate on the chains in the neighbourhood of that which began
the breakdown, and carry them past the unstable point. Then will begin
the outrush of negative tubes, and when this is complete the central chains
will be discharged and in the condition of Fig. 12 (e), while the surrounding
medium will contain negative induction. There will be now an inrush of
negative induction into the locus of the first discharge, for not only is this
free from induction, but also its molecules are suitably arranged to take up
negative tubes, and during the second halfperiod of swing from negative
to positive there will be a second change of partners along the same line.
And so on with the successive alternations of charge, and there is a tendency,
evidently, to keep the same line of discharge. Each change will give atoms
at the two endplates, which will combine either with each other or with the
electrodes, or perhaps be taken again into the chains in the following changes.
In the sudden changes of partners some of the energy goes to atomic vibration,
and we have evidence of this in the atomic radiation which we call spark.
There is also energy of translation of the atoms from one partner to another,
which appears as heat in the molecules. This heat possibly produces the
sound of the spark through the sudden expansion. We have then dissipation
of energy as well as the radiation out to space in the Hertzian waves, and the
two gradually reduce the electrical energy of the system.
We can see, too, how the amplitude of charge may lessen in the process.
For if the negative tubes begin to move out at the middle of each oscillation
before the positive tubes have all moved in, or vice versa, then the first of the
issuing kind will destroy the last of the incoming kind, and the final charge
will be diminished by the amount of this overlap. In the extreme case of
slow discharge, as through a wet thread, the issuing tubes are neutralised as
they come out by the incoming ones, and the motion is deadbeat.
294
MOLECULAR ELECTRICITY
These negative tubes turned out may also supply the negative tubes
required in the theory of Prof. J. J. Thomson (Recent Researches, Chapter i).
Now let us take a conducting dielectric such as water. . As guiding us to
an account of what goes on we have the facts (1) that, however small the
E.M.F., conduction and presumably electrolysis take place, and no finite
difference of potential can be maintained between the electrodes unless we
continually supply fresh energy, and (2) that a solute such as sulphuric acid,
which appears to combine with the solvent in some way, enormously increases
the conductivity. From (1) we gather that some of the molecules must be
just ready for change of partners, and from (2) we may at least guess that
molecular groups are formed much more complex than the atomic pairs we
have hitherto dealt with. We can see how such molecular groups might
arise by the straying out and coalescence of inductiontubes of neighbouring
molecules, and the consequent formation of new connections. Thus, if two
OC
'^ (fl)
a {(')
Fig. 17.
Fi2. 18.
pairs come together, as in Fig. 17 (a), we may have them connected into
a single group, such as (6), and any number of molecules may be brought
into circuit in the same way, so that we may have a parallel arrangement
as in Fig. 18. Or if the coalescence occurs by the approach of two or more
groups, such as Fig. 17 (6), we may have a series arrangement as in Fig. 19,
the kind which we shall suppose to be effective in conduction. We must
further suppose that small quantities of acids or salts in solution enormously
increase the number of complex molecules. Some of these groups would
appear to be more energetic than the initial pairs. Probably on that account
they are continually breaking up and reforming, the energy of translation
being no doubt converted at each collision and reformation into energy of
electrical separation. There are no doubt all degrees of connection from those
of Fig. 17 (h) and Fig. 19 (where the result may be obtained practically by
adding one closed ringtube of induction to a series of pairmolecules) to the
case where the tubes from each atom go half one way and half the other.
MOLECULAR ELECTRICITY
296
But I imagine that at any given instant only a small fraction of the molecules
are thus connected into groups or circuits.
If a tube of induction running from above downwards moves sideways into
such a group as that in Fig. 19 or Fig. 20 (a), it finds the righthand side made
ready for it, and we may possibly have in succession Fig. 20 (a), (b) and (c),
where we suppose that the lefthand side splits up into pairs, while the right
hand side remains threaded on the incoming hne. When a liquid contains many
such groups, some of the chains of molecules are almost readymade, and the
chains are very easily completed from plate to plate, since the entering tubes
have only to furnish a link, as it were, here and there. The ' electric elasticity'
\t yr yr
>r V y
Fig. 19.
Fig. 20(a).
>f >r 1,
>r V ^'
;. M '<
>r \f >r
i^ A M
>f V >' ^f >k i^ ik
Fig. 20 (&).
Fig. 20 (c).
IjK will be a sort of average of the elasticities or of the difficulties of forming
chains in all the various kinds of molecules present. To give a numerical
illustration, let us have two condenserplates in a liquid in which we suppose
that in the simply paired molecules, such as those on the left in Fig. 20 (c),
the value of K is 177. Let us suppose also that these occupy 99 per cent,
of the paths from plate to plate, while the other 1 per cent, is occupied by very
muchconnected molecules with K equal to 7500. Then if the capacity of the
condenser in air were 0, it would be with this arrangement of molecular paths
99 X 177C + 7500C
100
= 7676C,
or the resultant specific inductive capacity is 7676.
296 MOLECULAR ELECTRICITY
But this is only true in the mass. If we have very minute electromagnetic
waves going through the liquid they will, for 99 per cent, of the molecules, use
K= 177. For a very large fraction of the remaining 1 per cent, they will
probably also use this K, for they are not concerned with the group as a
whole, but only with the individual members to which they add or from which
they subtract small quantities of induction ; and as long as the points repre
senting these members of groups are on the straight part of the characteristic
their K for waves is still 177. We may therefore expect the value for small
waves generally to be very little more than 177.
This appears to indicate a possible explanation of the high inductive
capacity of such substances as water and alcohol. Accompanying this high
value there is generally conductivity and no doubt electrolysis. If we
suppose that some of the groups are already so far towards decomposition
that half the tubes from each atom run one way and half the other, tubes
entering the substance will concentrate on such groups, for a further addition
of induction will yield energy instead of requiring it, and the point of instability
being passed electrolysis will occur. Perhaps even before the tubes are thus
evenly divided, tubes entering the substance may prefer to pass through the
groups rather than through the simply paired molecules, for while more
energy may be stored in the one half of such a group as that represented in
Fig. 20, when a new tube enters in it, less may be required in the discarded
half, and so, on the whole, energy may be given up. But it does not seem
possible to give any satisfactory account of the process in our ignorance of
the real constitution of the complex molecules. All we can say is that some
such process probably occurs, inasmuch as conduction does occur even with
the smallest external e.m.f.
We may perhaps suppose that in metallic conduction we have a similar
process. If the metallic molecular structure is very complicated, with many
groups having unstable, or nearly unstable, construction, then tubes of
induction entering a metal will select such groups in preference to the more
simple stable molecules and electrolyse them. To account for the absence
of transfer of atoms along the line of current we must, I think, introduce
the supposition of change of charge as already explained on p. 282, and to
account for the continuance of conduction we must suppose that there is
a continuous formation of new groups as the old ones are broken up. Perhaps
we have here some key to the rise of resistance with rise of temperature. As
the molecules become more energetic we may expect that the groups will be
more broken up by the motions of vibration and translation, and that the
number of unstable groups is diminished and their rate of formation is
decreased. Hence the entering tubes will find fewer groups ready for them,
and if the external e.m.f. remains constant, the rate at which the tubes are
dissipated will be decreased. The difficulty of such explanation consists in
MOLECULAR ELECTRICITY 297
understanding its inapplicability to electrolytes. Perhaps, too, we have here
a hint as to the superior conductivity for heat of metals. If a metal consists
largely of manyatomed groups entangled together, and continually breaking
up and reforming, energy supplied to one molecule will, we may imagine,
be more readily transferred to its neighbours than if each molecule is self
contained and permanent. This will certainly be the case if energy given to
an atom in a molecule is more rapidly transferred to its fellows in the same
molecule than to atoms at the same distance in the surrounding molecules.
To use an illustration which may put the suggestion in a clearer way, news
will be transmitted through a population dwelling in villages and towns much
more rapidly than through an agricultural population of the same average
density scattered over a country in isolated homesteads.
The theory which I have been trying to set forth may be regarded as
a theory of the conservation of inductiontubes, and of their beginning and
ending on atoms. That the atoms always have charges on them, and there
fore have tubes proceeding to or from them, appears to be generally held as
necessary if we accept the electromagnetic theory of light. If the molecules
give rise to waves of electric induction they must necessarily be electric
systems, and their parts must almost certainly be bound together by electric
forces. Whether it is possible for an inductiontube to exist without atoms
is at present merely a matter of speculation. At present we know of no such
thing. If such a tube exist, it can only be as a closed ring, like a closed
ringtube of magnetic induction round a current, for an unclosed tube would
have opposite charges at its ends in free space, and charges not on matter
are so entirely outside experience that we cannot accept their existence. The
weight of evidence appears to me rather against the view of matterfree
induction, and though at first we might be inclined to think that the passage
of lightwaves across interstellar space implied such induction, yet even in
this case we have possibly quite enough matter to supply atomic ends for
the tubes to attach themselves to. If we accept the electric discharge theory
of comets' tails we apparently assume the existence of enough matter to
carry electric induction from a cometic nucleus outwards. I suppose that
the theory implies that the nucleus is charged in one way, say positively, and
that it has become separated from the matter bearing the negative, which
remains far out in space. The sun is itself to be regarded as charged with
the same sign as the nucleus, while the corresponding solar negative is also
somewhere in space. When the induction of the comet is added to that of
the sun the strain is sufficient to break down the feeble insulation of inter
planetary space, and a discharge results straight out, or nearly straight out
'from the nucleus, and this discharge is through the interplanetary matter.
But though this appears to be the only reasonable account of the discharge
theory, after all it is bringing Httle more than a speculation to bolster up
298 MOLECULAR ELECTRICITY
another speculation, viz., the existence of matter sufficient to carry waves of
induction in the interatomic form wherever light waves travel.
At first sight this theory of molecular electricity appears to be very different
from the chemical dissociationtheory now generally held ; but if the dissociated
atoms of that theory have charges, they have also tubes of induction proceeding
from the charges. When the tubes are taken into account they must, I believe,
lead to some such hypothesis as that of which I have attempted an inadequate
and imperfect explanation. I am only too deeply conscious of the difficulties
unsurmounted. But in working at the subject I have felt all through that,
since so much is nearly but not quite explained, there must be hope of progress
on these or similar Hues if we can only supply some as yet unrecognised idea.
Perhaps the very imperfections of my account may stimulate some reader
to take up the subject afresh from some better point of view, and, with new
ideas, achieve success.
PAET III.
WAVE PROPAGATION— KADIATION— PRESSURE
OF LIGHT— AND RELATED SUBJECTS.
19.
NOTE ON AN ELEMENTARY METHOD OF CALCULATING THE
VELOCITY OF PROPAGATION OF WAVES OF LONGITU
DINAL AND TRANSVERSE DISTURBANCES BY THE RATE
OF TRANSFER OF ENERGY.
[Birmingham Phil. Soc. Proc. 4, (1885), pp. 5560.]
[Bead Nov. 8, 1883.]
Waves of Longitudinal Disturbance.
A wave of sound may be considered as energy of a particular type, partly
potential and partly kinetic, which is being passed on from point to point
through the medium, so that the energy which is at any moment occupying
a particular portion of space will have passed in a second later to a distance
equal to the velocity of sound. Of the two energies the potential is due to
the strain of the medium, and, when in this strained condition, each part of
the medium exerts force on the neighbouring parts. But it also has kinetic
energy, that is, the part considered is in general in motion and there is therefore
motion of the point of application of the force which it exerts on the contiguous
parts through the existence of its potential energy ; that is, it does work and
passes on energy to the contiguous parts. If we consider for instance a series
of plane waves of sound which move on unchanged, the work done in any
small time t at any plane perpendicular to the direction in which the sound
is moving must be equal to the soundenergy contained in the space immedi
ately behind the plane through which the sound will travel in the time t.
This gives us one relation between the various quantities, and we obtain
another from the consideration that any condition as to velocity and dis
placement which is now at a particular point will have travelled on unchanged
300 VELOCITY OF PROPAGATION OF WAVES OF LONGITUDINAL AND
in the time t to a distance Ut where TJ is the velocity of sound. From these
two relations we can at once find the velocity TJ*.
Let BA be the direction in which the sound is travelling, and let AP be
the trace of a plane perpendicular to AB. Draw a curve CPQD whose height
above each point of AB shall represent the displacement of the particle at
that point (these displacements are actually of course along AB). Thus AP
represents the displacement at A along AC, and MQ the displacement at M
along MA.
Consider a small volume V with unit area on the plane through AP as
base and height AM = V. The volume of the medium which had height F
before the disturbance reached A will now be compressed; for the end A
has moved forward a distance AP, while the end M has moved forward
a greater distance MQ. The compression is therefore the difference between
these, viz.; QR = v, say. But if the sound takes a time t to travel over MA,
p
Q,__
1 — "^'^ n
^
■^ R
^^^^^\
c
A
M
DB
after that time the displacement of A will be equal to the present displacement
of M. Or if u be the actual velocity of the particle at A,
Qn = v= ut (1)
Now, considering the energies, we have the kinetic energy of the volume
V = p ^, where p is the density of the medium. The potential energy equals
the work done in compressing. If P is the original pressure, and P + f the
pressure in the compressed state, P f ^ is the average pressure during the
compression, and the distance through which this has moved is numerically
equal to the diminution of volume v. Then the potential energy is
(p.g.
* This method of treating the subject of wave propagation is given by Lord Rayleigh in
a note in vol. 9, no. 125, of the Proceedings of the London Mathematical Society (republished at
the end of vol. 2 of his Theory of Sound). This paper is merely an application of his method to
two particular cases.
TRANSVERSE DISTURBANCES BY RATE OF TRANSFER OF ENERGY 301
The work done in t sees, across unit area at A is equal to the pressure
exerted by the medium to the right on the medium to the left, multiplied
by the distance which the particle has moved in the time t, or (P + p) ut.
We may equate this to the sum of the two energies, potential and kinetic,
contained in 7, for this energy is passed across the plane in the time t.
Then (P + ^)^,^=(p + g^ + pIJ.^ (2)
But ut = y by equation (1) ;
or ^ = P t^
2^2
Vv
But V = AM = (distance travelled by the sound in t) = Ut;
Vv V V^ V ^^„
p V
Vv V
But —  , or — , that is the ratio of a small increase of pressure to the
V
change per unit of volume thereby produced, is the elasticity of the medium.
We therefore obtain
jj^ elasticity
density '
or velocity = Velasticity f density.
Waves of Transverse Disturbance.
The velocity of propagation of plane waves in which the disturbance is
in the plane of the wave can also be easily found by this method. Since
the displacements are all perpendicular to the direction of propagation, the
force acting across a plane perpendicular to this direction will be entirely
tangential and the rigidity of the medium will alone be brought into play.
As before, let the waves be travelling in the direction BA and let the
curve CPQD represent the displacement. The curve may now represent the
actual displacements since they are perpendicular to AB.
If, as before, u is the velocity of the particle displaced from A, andt the
time the wave takes to travel over the small distance MA, then the displace
ment AP becomes equal to MQ in t, or QR = ut.
302 VELOCITY OF PROPAGATION OF WAVES OF LONGITUDII^AL AND
Now consider the energies in the volume V, with unit area on the plane
through AP and height AM numerically equal to V.
If p is the density, the kinetic energy is p —^ .
The potential energy is equal to the work done in bringing the medium
from its normal state into its present state of shear — the angle of shear at
A being QPR, and since MA is very small this angle is measured by
QR QR ut
— — or — — = —
PR V V
If G be the modulus of rigidity, the tangential force per unit area is
ut
G X angle of shear = G ^.
Now the average value of this force in bringing the shear to its present
value is half this, and the distance of displacement of the force is QR. Then
the potential energy in the volume V is
The work done in t sees, across unit area at A is equal to the tangential force
at A multiplied by the distance through which the particle at A moves in t ;
lit u^t^
or ^ V ■ ^^^ "" ^ 'v •
Equating this to the sum of the two energies, potential and kinetic, con
tained in V we have
or GjpV.
But V = AM = (distance through which the wave travels in t) = Ut;
G V^
P t'
or U ■= Vmodulus of rigidity h density.
It follows at once from the above that the energy in any part of the
medium is equally divided between the two forms kinetic and potential, for
the equation
~r ^ 2T "^ ^ 2
gives G ^y ^^'2 '
or potential energy in F =^ kinetic energy in F.
TRANSVERSE DISTURBANCES BY RATE OF TRANSFER OF ENERGY 303
This result also holds for waves of longitudinal disturbance if the initial
pressure is zero. That is if we may put P = in the equation
(P + V)v=(^P + l)v + p^^
for the potential energy is then ^, and the above equation gives us
pv Vv^
or potential energy in F = kinetic energy in V.
20.
RADIATION IN THE SOLAR SYSTEM: ITS EFFECT ON
TEMPERATURE AND ITS PRESSURE ON SMALL BODIES. •
[Phil. Trans. A, 202, 1903, pp. 525552.]
[Received June 16. Read June 18, 1903.]
PART I.
Temperatuee.
When a surface is a full radiator and absorber* its temperature can be
determined at once by the fourthpower law if we know the rate at which
it is radiating energy. If it is radiating what it receives from the sun, then
a knowledge of the solar constant enables us to find the temperature. We
can thus make estimates of the highest temperature which a surface can
reach when it is only receiving heat from the sun. We can also make more
or less approximate estimates of the temperatures of the planetary surfaces
by assuming conditions under which the radiation takes place f, and we can
determine, fairly exactly, the temperatures of very small bodies in inter
planetary space.
These determinations require a knowledge of the constant of radiation
and of either the solar constant or the effective temperature of the sun, either
of which, as is well known, can be found from the other by means of the
radiationconstant. It will be convenient to give here the values of these
quantities before proceeding to apply them to our special problems.
* A surface which absorbs, and therefore emits, every kind of radiation is usually described
as 'black,' a description which is obviously bad when the surface is luminous. It is much better
described as 'a full absorber' or 'a full radiator.'
t This was pointed out by W. Wien in his report on 'Les Lois Theoriques du Rayonnement '
{Congres International de Physique, vol. 2, p. 30). He remarks that Stefan's law enables us to
calculate the temperatures of celestial bodies which receive their light from the sun, by equating
the energy which they radiate to the energy which they receive from the sun, and states that
for the earth we obtain nearly the mean temperature, using the reflecting power of Mars, while
the temperature of Neptune should be below  200° C.
RADIATION IN THE SOLAR SYSTEM 305
The Constant of Radiation.
If R is the energy radiated per second per square centimetre by a full
radiator at temperature 6° A (where A stands for the absolute scale), the
fourthpower law states that
R = ad\
where a is the constant of radiation.
According to Kurlbaum* the constant is
CT = 532 X 105 erg/cm. 2 sec. deg.*.
The Solar Constant.
The solar constant is usually expressed as a number of calories received
per minute by a square centimetre held normal to the sun's rays at the distance
of the earth. The determinations by different observers differ so widely that
it is not necessary for our present purpose to consider whether the constant
really exists or whether there are small periodic variations from constancy.
Angstrom estimated the value as 4 calories per square centimetre per
minute, and this value is adopted by Crova as very probable I . When
converted to ergs per second this gives
Sa = 028 X 10' ergs/cm.2 sec,
where the suffix denotes that it is Angstrom's value.
LangleyJ assumed that the atmosphere transmits about 59 per cent, of
the energy from a zenith sun, and from his measurement of the heat reaching
the earth's surface he estimated the value of the constant at 3 cal./cm.^ min.
This gives
Si = 021 X 107 ergs/cm.2 sec.
Rosetti§ assumed a transmission of 78 per cent, from the zenith sun, but
Wilson and Gray consider that 71 per cent, represents Rosetti's numbers
better than 78 per cent. If in Langley's value we replace 59 per cent, by
71 per cent., we get 25 cal./cm.^ min. This gives
Sr = 0175 X 107 ergs/cm.2 sec.
* Wied. Ann. vol. 65, 1898, p. 748.
f Congres International de Physique, vol. 3, p. 453.
J Phil. Mag. vol. 15, 1883, p. 153, and Researches on Solar Heat.
§ PhU. Mag. vol. 8, 1879, p. 547.
II Phil. Trans. A, 1894, p. 383.
p. c. w. 20
306 RADIATION IN THE SOLAR SYSTEM:
The Radiation from the Sun's Surface.
If s is the radius of the sun's surface, R the radiation per square centi
metre, then the total rate of emission is 4:7ts^R. This passing through the
sphere of radius r, at the distance of the earth and with surface 4^r^, gives
where S is the solar constant.
Hence R = p= (ifx" lO^' ^ = ^^'''^^^
Corresponding to the three values of S just given we have three values of R,
viz.,
Ra = 129 X 10" ; Ri = 0945 x lO^i ; R^ = 0805 x lO^i.
The Effective Temperature of the Sun.
If we equate the sun's radiation to g6^, where g is the radiationconstant,
we get 6, the 'effective temperature' of the sun, that is the temperature of
a full radiator which is emitting energy at the same rate.
Thus 532 X 105 ^^4 _ 1.29 x lO^S
whence 0^ = 7000° A approximately.
Similarly 6', = 6500° A ; 9, = 6200° A.
Wilson* made a direct comparison of the radiation from the sun with
that from a full radiator at known temperature. Assuming a zenith trans
mission of 71 per cent., he obtained 5773° A as the effective solar temperature.
If we put
46,0006' = 532 X 10^ x 5773^,
we get S^ 0128 x 10^
This is no doubt too low a value. Either then Wilson's zenith transmission
was less than 71 per cent, or Kurlbaum's constant is too small.
The low value is probably to be accounted for chiefly by the first supposition.
Wilson points out that if x is the true value of the transmission, his value of
the temperature is to be multiplied by (lljx)^. If we take 9^ = 6200°^ as
the true value, then x will be given by
This low value is not necessarily inconsistent with the much higher value
71 per cent, used above in finding Rosetti's solar constant, for no doubt the
transmission varies widely with time and place, and we have no reason to
* Roy. Soc. Proc. vol. 69, 19012, p. 312.
ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 307
I assume that 177 calories per minute, obtained by Langley, would have been
received from the zenith at the time and in the place where Wilson was making
his determination.
The Effective Temperature of Space.
In determining the steady temperature of any body as conditioned by
the radiation received from the sun, we have to consider whether it is necessary
to take into account the radiation from the rest of the sky. If it receives S
from the sun, p from the rest of the sky, and if its own radiation is R, then in
the steady state
R = S\p or Rp = S.
It behaves therefore as if it were receiving S from the sun, but as if it
were placed in a fully radiating enclosure of such temperature that the
radiation is p. This temperature is the 'effective temperature of space.'
The temperature may perhaps be more definitely described as that of
a small full absorber placed at a distance from any planet and screened from
the sun. Various wellknown attempts have been made to estimate this
temperature, but the data are very uncertain. The fourthpower law however
shows that it is not very much above the absolute zero, if we can assume that
the quality of starlight is not very different from that of sunlight.
According to Hermite* starlight is onetenth full moonlight. Full
moonlight is variously estimated in terms of full sunlight. Langley f takes
it as 4QoVoo These two values combined give sunlight as 4 x 10^ starlight.
But starlight comes from the whole hemisphere, while the sun only occupies
a small part of it. In comparing temperatures we
have to use the brightness of sunlight as if the
whole hemisphere were paved with suns.
If B is the illumination of a surface at 0,
Fig. 1, lighted by the sun in the zenith at S, and
if TTS^ is the area of the sun's diametral plane,
then B/tts^ is the illumination at due to each
square centimetre. If the hemisphere were all
of the same brightness as the sun, the illumina
tion at due to the ring of sky between 6 and 6 \ dd would be
^27rf2sin6>cos^^^,
7TS^
where r is the distance of the sun.
Integrating from ^ = to ^ = 7r/2, we have
Total illumination = Br^js^ = 46,000 B.
* U Astronomie, vol. 5, p. 406.
t 'First Memoir on the Temperature of the Surface of the Moon.' National Academy of
Sciences i Memoirs, vol. 3, 1884.
20—2
308 RADIATION IN THE SOLAR SYSTEM I
The illumination from a hemisphere paved with suns is therefore
46,000 X 4 X 10^ = 184 X lO^^ times that from the stellar sky.
If we assume that the quality of the radiation is the same in both cases,
that is, if we assume that the energy is proportional to the lightpart of the
spectrum, we have by the fourthpower law
^ ^ . „ effective temperature of sun
Enective temperature oi space = i
(0184 X 10i2)t
effective temperature of sun
f ^ 655
As the temperature of the sun probably lies between 6000°^ and 7000^^,
this gives
Effective temperature of space = 10° A.
If, then, a body is raised by the sun to even such a small multiple of 10°
as, say, 60°, the fourthpower law of radiation implies that it is giving out,
and therefore receiving from the sun, more than a thousand times as much
energy as it is receiving from the sky.
The skyradiation may therefore be left out of the account when we are
dealing with approximate estimates and not with exact results, and bodies
in the solar system may be regarded as being situated in a zero enclosure
except in so far as they receive radiation from the sun.
Temperature of a Planet under Certain Assumed Conditions when
placed at a Distance from the Sun equal to that of the Earth.
The real earth presents a problem of complexity far too great to deal
with. I shall therefore consider an ideal earth for which certain conditions
hold, more or less approximating to reality, and determine the temperature
of its surface on the assumption that it receives heat from the sun only.
Let us suppose :
1. That the planet is rotating about an axis perpendicular to the plane
of its orbit, which is circular.
This will give us too high a temperature at the equator, and the absolute
zero, which is too low, at the poles. The mean, however, over the planet
will probably be not much affected by the supposition.
2. That the effect of the atmosphere is to keep the temperature in any
given latitude the same, day and night.
This is not a great departure from reality. On the sea, which is more than
twothirds of the earth's surface, the daily range is very small, of the order of
1° or 2° C, while even on the land it is, in extreme cases, not more than
15° C, which is not a large fraction of the absolute temperature.
ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 309
3. That the surface and the atmosphere over it at any one point have one
effective temperature as a full radiator. This is no doubt a departure from
reality. How wide a departure we have no present means of estimating.
4. That there is no convection of heat from one latitude to another.
This is a very wide departure from reality. But, as we shall see below, the
mean temperature of the planet is very little affected by convection, even if we
assume that it is so extensive as to make the surface of uniform temperature.
5. That the reflection at each point is fjj of the radiation received.
This is probably of the order of the actual reflection from the earth.
According to Langley* the moon reflects about J of the radiation received.
The earth certainly reflects less. The temperatures determined hereafter are
proportional to the 4th root of the coefficient of absorption. Even if this
coefficient is as low as 09 its 4th root is 0974. Hence if the actual value is
anywhere between 09 and 1, the assumed value of 09 will not make an error
of more than 2J per cent, in the value of the temperature.
6. That the planet ultimately radiates out all the heat received from the
sun, no more and no less.
This again is very near the condition of the real earth, which, on the whole,
radiates out rather more than it receives — perhaps on the average a calorie
per square centimetre in three days.
r cos \ d\ T"" / \r dA
Fig. 2.
Making these six suppositions, let us calculate the temperature of various
parts of this ideal planet.
Consider a band between latitudes A and A + dX. The area receiving
heat from the sun at any instant, if projected normally to the stream of
solar radiation, is (Fig. 2)
2r cos XrdXcosX= 2r^ cos^ XdX,
where r is the radius of the planet.
* 'Third Memoir on the Temperature of the Moon.'. National Academy of Sciemes, Memoirs
vol. 4, part 2, p. 197.
310 BADIATION IN THE SOLAR SYSTEM I
If S is the solar constant, this band is absorbing, with coefficient 09,
09^ X 2r^cos^XdX.
But the band all round the globe is radiating equally, according to the second
supposition, and the radiating area is
27rr cos A . rc^A = 27Tr^ cos A^^A.
Hence the radiation emitted per square centimetre per sec. is
0'9S . 2r^ cos^ XdX _ 098 cos A
27rr^ cos XdX tt '
If the effective temperature in this latitude is 6^, we have
09iS cos A
532 X 10^ 6>A^
77
or
/ 09 X 10^S \i
[ 53277 ;
cos* A.
If we put A = 0, we get the equatorial temperature corresponding to each
of the different values of S given above, viz. :
Equatorial 6^ = 350° A approximately.
e^  325° A
„ e, ^ 312° A
The temperature in latitude A is
6^ = equatorial temperature x cos* A.
Thus, in latitude 45°, it is 0917 x equatorial temperature.
The average temperature over the globe is
J r2
f 1
277/2 cos A 6e cos* XdX,
'J
where 6j^ is the equatorial temperature.
The average temperature, then, is httle more than 1 per cent, above the
temperature in latitude 45°.
If we use the three values of 6^ just given, we have
Average 6^ = 325° A approximately.
„ 01 = 302° A
d, = 290° A
Our fourth supposition was that there is no convection by wind or water
from one latitude to another. Let us now go to the other extreme and
ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 311
suppose that the convection is so great that the temperature is practically
uniform all over the globe. We then have a receiving surface virtually irr^,
and a radiating surface iTrr^. Then we get the radiation emitted per square
centimetre
47rr2 ~ 40'
and if 6 is the temperature required for this,
532x105^4=^;
whence Uniform d^ = 330° A approximately,
01 = 307° A
Or = 293° A
values not more than 5° above those obtained for the average on the
supposition of no convection.
Comparing these results with the temperature of the real earth, it is seen
at once that they are of the same order.
The average temperature of the earth's surface is usually estimated at
about 60° F., say 289° A. The temperature of the atmosphere is on the
whole decidedly lower than that of the surface below it. We should therefore
conclude that the earth's effective temperature is somewhat below 289° A.
Again, the earth and the atmosphere, taken as one surface, do not con
stitute a full absorber, but are to some extent selective. Hence we should
expect the earth to be, if anything, of a higher temperature than a full absorber
and radiator under the same conditions.
For both these reasons, then, the ideal planet might be expected to have
a temperature below rather than above 289° A. The lowest estimate obtained
above is therefore probably nearest to the truth, and it would appear that
even that is somewhat too high. This tends to show that, if we accept
Kurlbaum's value of the radiationconstant, we cannot put the solar constant
so high as 3 or 4, but must accept a value much nearer to that which I have
called Rosetti's value, viz., 25.
In what follows I shall therefore take Rosetti's value and the resulting
value of the solar temperature, viz., 6200° A.
The calculation made above may be turned the other way round, and may
be used for a
Determination of the Effective Temperature of the Sun from the Average
Temperature of the Earth.
Assuming that the real earth may be replaced by the ideal planet already
considered, the radiation per square centimetre from the equatorial band is
. But the radiation per square centimetre from the sun's surface is
312 RADIATION IN THE SOLAR SYSTEM:
46,000/S'. If then B^ is the earth's equatorial temperature, and 6^ is the solar
temperature,
— ^ : 46,000>S = Se^ : Os^
7T
whence 0^ = ^sl^^
The average temperature of the earth is 093 of the equatorial temperature.
If this average is ^^ , then
e^ = es/215.
If we take the temperature of the real earth as 289° A, and as being equal
to that of the ideal,
d^ = 215 X 289° = 6200° A approximately.
Upper Limit to the Temperature of a Fully Radiating Surface exposed normally
to Solar Radiation at the Distance of the Earth from the Sun.
The highest temperature which a full radiator can attain is that for which
its radiation is equal to the energy received. This will only hold when no
appreciable quantity of heat is conducted inwards from the surface.
To obtain the upper limit in the case under consideration, we have to
equate the radiation to the solar constant, which we shall now take as
Sr = 0175 X 10^. Then
532 X 105^4 = 0175 X 10^
whence 6 = 426° A.
If the surface reflects some of the radiation and absorbs a fraction x of
that falling on it, then the effective temperature is
x^ X 426° A.
The Limiting Temperature of the Surface of the Moon.
We may apply this result to find an upper limit to the temperature of the
moon's surface. This upper limit can only be attained when it is sending
out radiation as rapidly as it receives it, and is therefore conducting no
appreciable quantity inwards.
117 r, n . 1 T 1 ? ^ . /7 •. X i reflected radiation 1
We shall take Langley s estimate (loc. cit.) of — ^——^ :rr~. — = —  .
^ ^ ' emitted radiation 67
This is represented nearly enough by ;z; =  .
The upper limit of temperature of the surface exposed to a zenith sun is,
therefore,
^ = 426 X (I)* = 426 X 0967  412° A.
This, then, is the upper Hmit to the temperature of the hottest part of an
airless moon.
ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 313
For a surface with normal at angle A with the line to the sun,
^x = 412 cos* A.
If we take this as the law of temperature of the side of the moon exposed
to the sun, we can find the effective temperature of the full moon as seen
from the earth, i.e., the uniform temperature of a flat disc of radius equal to
that of the moon, sending to us the same total radiation.
If Ndo) is the normal stream of radiation from 1 cm.^ of surface of
the moon immediately under the sun sent out through a cone of angle doj,
that sent out in direction A to the normal is iVcosA<?a;. But 1 cm.^ on the
moon's surface, with normal inclined at A to the sun's rays, only receives
cos A of the radiation received by the surface immediately under the sun.
It therefore sends in the direction of the earth, also at A to the normal, onlv
iVcos^A^o). Hence the total radiation to the earth, obtained by inte
grating, is
/
2 N cos^ A . 27rm2 sin XdX
^^
where m is the radius of the moon and r is its distance from the earth.
Let Nj^ be the normal stream from the equivalent flat disc, then
Trm^Nn _ 277 m ^
2
whence Nz) = ^N.
o
The effective temperature of the flat disc is therefore Vf that of the
surface immediately under the sun at the same distance from it.
Then the effective average = 412 x v^f = 412 x 09 = 371° A. The upper
limit, then, to the average effective temperature of the moon's disc is just
below that of boiling water.
This is very considerably above Langley's estimate, that the surface of
the full moon is a few degrees above the freezingpoint. There can be no
doubt that a very appreciable amount of heat is conducted inwards. The
observations during eclipses by Langley* and by Boeddicker show that some
heat is still received from the moon's surface when it has entered the full
shadow, and that it takes time after the eclipse has passed to establish a
steady temperature again. It might be possible to make some rough estimate
of the amount conducted inwards from the Fourier equation, but the problem
is not an easy one. Perhaps we get the best estimate by comparing the actual
temperature with that found above.
* 'Third Memoir,' p. 159.
314 RADIATION IN THE SOLAR SYSTEM:
If the actual temperature is taken as about^  of the upper limit, say 297°^,
then the radiation outwards is of the order ()* = 041 of that where no con
duction exists. Then nearly f of the heat is probably conducted inwards.
If the moon always turned the same face to the sun instead of to the
earth, the upper limit would be approached.
Temperature of a Spherical Absorbing Solid Body of the Order 1 cm.
in diameter at the Distance of the Earth from the Sun.
The calculation of the temperature of such a body is interesting for two
reasons. Firstly, the body will be at nearly the same temperature through
out, and secondly, as we shall show in the second part of this paper, the
mutual repulsion of two such bodies, due to the pressure of their radiation,
is of the same order as their gravitative attraction.
If the radius of the body is a, its effective receiving area is 7ra^, and it
receives
TTa^S ergs/sec.
Its radiating surface is iira'^, and therefore its average radiation per square
centimetre per sec. in the steady state is
Tra^S/iTTa^ = iS.
If we take S=2'6 cal./cm.^ min. or 004 cal./cm.^sec, and if the conductivity
is of the order of that of terrestrial rock lying, say, between 001 and 0001, it
is evident that a difference of temperature of only a few degrees between the
receiving and the dark surfaces will convey heat sufficient to supply radiation,
001 cal./cm.^ sec, equal to the average. Thus, if the conductivity is 0001
and the diameter is 1 cm., a difference of temperature of 10° suffices.
We may therefore take the temperature of the surface as approximately
uniform when the steady state is reached. Let the temperature be 6, and
let the solar temperature be Og. Then we have
and
If ^e =
This will be the temperature of fully absorbing bodies of diameter less
than 1 cm., so long as they are not too small to absorb the radiation
falling on them.
: 0s'
= 1 : 46,000;S
e
207'
6200°
A,
300°.
4 approximately
ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 315
Variation of Temperature with Distance from the Sun.
Since the radiation received varies inversely as the square of the distance
from the sun, that given out varies in the same ratio. The temperature of
the radiating surface varies therefore as the fourthroot of the inverse square,
that is inversely as the squareroot of the distance.
This enables us to deduce at once the temperatures of the various surfaces
and bodies which we have considered, if placed at the distances of different
planets as well as at the distance of the earth. We have merely to multiply
iu ^J. 1,'j.v. J. £ 1 T_ /earth's distance
the results hitherto found by </ ^ , ... .
V planet s distance
The following table contains the values of the temperatures at selected
distances, all on the absolute scale :
Table of Temperatures of Surfaces at Different Distances from the Sun.
All on the Absolute Scale.
I
II
III
IV
V
VI
VII
VIII
IX
Equatorial
Average
tem
Upper
limit of a
Average
tem
perature
of
Tem
peiature
Tem
perature
At the
Distance,
Square
tempera
surface
four
of
distance of
Earth's
root of
ture
perature
of
ideal
planet
reflecting
fifths
small
the planet
distance = 1
(distance)i
of ideal
oneeighth
that of
absorb
planet
under
zenith sun
lent disc
equiva
lent disc
ing
sphere
Mercury
03871
161
502
467
664
598
478
483
Venus
07233
118
368
342
486
438
350
358
Earth
10000
100
312
290
412
371
297
300
Mars
15237
081
253
235
337
300
240
243
Neptune
300544
018
56
52
74
67
53
54
We have omitted the larger planets except Neptune, since in all probability
they radiate heat of their own in considerable proportion. Neptune is inserted
merely to show how low temperatures would be at his distance if there were
no supply of internal heat.
The results given in the table may not be exactly applicable to any of the
planets, but they at least indicate the order of temperature which probably
prevails.
If, for instance, Mars is to be regarded as having an atmosphere with,
regulating properties like our own, his equatorial temperature (Column IV)
is probably far below the temperature of freezing water, and his average
temperature (Column V) must be not very different from that of freezing
mercury. If, on the other hand, we suppose that his atmosphere has no
regulating power, we get the upper limits not very different from those in
316 RADIATION IN THE SOLAR SYSTEM:
Columns VI and VII. These are the Hmits for the bright side, and they
imply nearly absolute zero on the dark side. If we regard Mars as resembling
our moon, and take the moon's effective average temperature as 297° A, the
corresponding temperature for Mars is 240° A, and the highest temperature
is I X 337 = 270°. But the surface of Mars has probably a higher coefficient
of absorption than the surface of the moon — it certainly has for light — so
that we may put his effective average temperature on this supposition some
few degrees above 240° A , and his equatorial temperature some degrees
higher still.
It appears exceedingly probable, then, that whether we regard Mars as
like the earth, or, going to the other extreme, as like the moon, the temperature
of his surface is everywhere below the freezingpoint of water. The only
escape from this conclusion that I can see is by way of a supposition that an
appreciable amount of heat is issuing from beneath his surface.
We cannot draw any definite conclusions as to the temperatures of Mercury
and Venus till we know whether they have atmospheres and whether they
rotate on their own axes. If we make both these suppositions and further
suppose that their conditions approximate to those (given in Columns IV
and V) of the ideal planet at their distances, then they may well be surrounded
by hot clouds, as is sometimes supposed, entirely screening their solid bodies
from us. If, on the other hand, their atmospheres are ineffective as regulators
and if they always present the same face to the sun, the hottest part of Mercury
is probably not far from 650° A, and that of Venus not far from 500° A.
If a comet consist of small solid particles of diameter of the order 1 cm.
or less, then the temperatures of these particles are given in Column IX. At
onequarter of the earth's distance, say 23 million miles from the sun, the
temperature is 600° A, about the meltingpoint of lead. At onetwentyfifth,
say 3 million miles, it will be about 1500° A, say the meltingpoint of cast
iron. Nearer than this the temperature no doubt increases rapidly, but the
law of temperature, deduced from the inversesquare law for the radiation
received, requires amendment, as that law was based on the supposition that
a hemisphere only is lighted by the sun, and that the whole of his disc is
visible from every part of that hemisphere. Both of these suppositions
cease to hold when the distance from the sun is only a small multiple of his
radius.
PART 11.
RadiationPressures.
The pressure of radiation against a surface on which it falls, first deduced
by Maxwell from the Electromagnetic Theory of Light, is now established
on an experimental basis by the work of Lebedew, confirmed by that of
Nichols and Hull.
ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 317
Though this pressure was first deduced as a consequence of the Electro
magnetic Theory, Bartoli showed, independently, that a pressure must exist
without any theory as to the nature of light beyond a supposition which
may perhaps be put in the form that a surface can move through the ether,
doing work on the radiation alone and not on the ether in which the radiation
exists. Professor Larmor* has given a proof of this pressure and has shown
that it has the value assigned to it by Maxwell, viz., that it is numerically
equal to the energy density in the incident wave, whatever may be the nature
of the waves, so long as their energydensity for given amplitude is inversely
as the square of the wavelength. We may, in fact, regard a pencil of radiation
as a stream of momentum, the direction of the momentum being the axis of
the pencil. If E is the energydensity of the pencil, U its velocity, the
momentumdensity may be regarded as EjU.
If the stream of "radiation is being emitted by a surface, the surface is
losing the momentum carried out with the issuing stream, and is so being
pressed backwards. If the stream is being absorbed by the surface, then it
is gaining the momentum and is still being pressed backwards, the forces
being in the line of propagation.
As the expressions for the radiationpressure in various cases are probably
not very well known, it may be convenient to state them here for use in what
follows.
Values of RadiationPressure in Different Cases.
If 1 cm.'^ of a full radiator is emitting energy R per second, and if N dw
is the energy it is emitting through a cone dco, with axis along the normal,
then in direction d its projection is cos d, and it is emitting iVcos ddoj through
a cone do). Putting dcxi = 2tt sin Odd, and integrating over the hemisphere,
we have
R= \ iV cos ^ . 277 sin (^(9 = ttN.
J
If we draw a hemisphere, radius r, round the source as centre, the energy
falling on area r^doj is iVcos ddoj per second, and, since the velocity is U cm.
per second, the energydensity just outside the surface on which it falls is
iVcos diUr^, and this is the rate at which the momentum is being received,
that is, it IS the normal pressure. The total force on area r^dcj is iVcos Odw/U.
This is the momentum sent out per second by the radiating square centimetre
through the pencil with angle dw, in the direction 6, and is therefore the force
on the square centimetre due to that pencil.
Resolving along the normal and in the surface we have
Normal pressure = N cos^ ddco/U,
Tangential stress = iV cos ^ sin Odco/U.
* Brit. Assoc. Report, 1900; Encyc. Brit. vol. 32, Art. 'Radiation.'
318 RADIATION IN THE SOLAR SYSTEM I
Putting d(x) = 277 sin ddd and integrating over the hemisphere, we get
IT
Total normal pressure = ( [N cos^ d . Iir sin d dOjU)  ^ttNJW = 2i?/3C7.
J
Total tangential stress = 0, since the radiation is symmetrical about the
normal.
If the surface is receiving radiation, let us suppose that the stream is
a parallel pencil S ergs per second per square centimetre held normal to the
stream, and that it is inclined afc an angle 6 to the normal to the receiving
surface. The momentum received per second is S cos djTJ. This produces
Normal pressure ^ S cos^ djU ,
Tangential stress =^ S cos d sin djU.
If the stream is entirely absorbed both these forces exist.
If the stream is entirely reflected, the reflected pencil exerts an equal
normal force and an equal and opposite tangential force, and we have only
normal pressure of amount 2S cos^ d/U.
If only a fraction /x is reflected, the incident and reflected streams will give
Normal pressure = (1 + /jl) S cos^ O/U,
Tangential stress = (1 — /jl) S cos 6 sin d/U.
To the normal pressure must be added the pressure due to the radiation
emitted from the surface.
Radiation Pressure in Full Sunlight.
If a full absorber is exposed normally to the solar radiation at the
distance of the earth the pressure on it is S/U, or  ^ .w.f.~ = 58 x 10~^
^ ' 3 X 10^^
dyne/cm.^.
The RadiationPressures between Small Bodies. Comparison ivith
their mutual Gravitation.
It is well known that the radiationforce on a small body, exposed to solar
radiation, does not decrease so rapidly as gravitative pull on the body when its
size decreases. If the body is a sphere of radius a and density p, and with
a fully absorbing surface, and if it is so small that it is practically at one
temperature all through, it is receiving a stream of momentum
• ira^SjU
directed from the sun. Its own radiation outwards being equal in all
directions has zero resultant pressure.
ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 319
The gravitative acceleration towards the sun at the distance of the earth
is about 059 cm./sec.^. Then we have
Radiationpressure. tto^S
Gravitationpull U x ^ira^p x 059 '
The two will be equal when
a 
S
Up X 059'
If we put /o=l; /S = 0175 X 107; U = ^xlO^^;
we get
a = 74 X 106.
This is the wellknown result that a body of diameter about two wave
lengths of red light would be equally attracted and repelled if we could
assume that a surface so small still continued to absorb. But, of course, when
we are getting to dimensions comparable with a wavelength that assumption
can no longer be made.
It is not, I think, equally well recognised that if the radiating body is
diminished in size, the radiationpressure due to it also decreases less rapidly
than the gravitative pull which it exerts. For the radiation decreases as
the square of the radius of the emitting body and its gravitative pull as the
cube.
We can easily compare the forces due to radiation and gravitation between
two bodies, if for simplicity we assume that their distance apart is very great
compared with the radius of either.
Fig. 3.
Let A, B, Fig. 3, be two spheres with full radiating surfaces. Let their
radii be a, b, and let their centres o, o' be d apart. If this distance is great
compared with a and 6, each may be regarded as receiving a parallel stream
from the other.
Let A send out a normal stream N dco per square centimetre through cone
do), while B sends out N' dco.
B receives the stream of crosssection nb'^ or the angle of the cone is nb^/d'^,
and it issues virtually from area na^, for at B, A will appear as a uniformly
bright flat disc.
320 RADIATION IN THE SOLAR SYSTEM:
Then the total force on B is
where R — ttN.
The force on A due to B is ira^hm' jUd^, where E = ttN\
These are not equal unless R = R' , i.e., unless the two bodies have the
same temperature, an illustration of the fact that equality of action and
reaction does not hold between the radiating and receiving bodies alone.
They no longer constitute the whole of the momentumsystem. The ether,
or whatever we term the lightbearing medium, is material, and takes its
part in the momentumrelations of the system.
If the surfaces are partially or totally reflecting, the forces are easily
obtained. Thus if one is totally reflecting, it can be shown that the force is
only half as great as when it is fully absorbing. But it will be sufficient to
confine ourselves to the case of complete absorption, followed by radiation
of the absorbed heat equally in all directions from all parts of the surface.
More general assumptions do not alter the order of the forces found.
If G is the constant of gravitation = 667 x 10"^, and if p, p' are the
densities of A and B, the gravitationpull, P, is 6^ ^ ar}2^^^ •
Radiationpush F _ 97Ta^b^R
Gravitationpuff P WGUn^a^b^pp"
F 9R
P WGUTTabpp"
lia = b; p=p'; R^ 532 x 10^^^ we have
069 X 104^2 /p ^
or
If we suppose the two bodies to have the temperature of the sun, say
6200°^, and its density, say 1375, then F = P, when a =1930 cm. or
193 metres f.
Of course two globes of this size would soon cool far below the temperature
of the sun, even if for an instant they could be raised up to it.
If we suppose 6 = 300° A — the approximate temperature of small bodies
at the distance of the earth from the sun — and if we take /o = 1, then F = P
when a = 62 cm. I
* [The original has 218, instead of 069, which is a shp due to the omission of ^l^ "^^^
density of the sun is also wrongly taken as 025 instead of 1375 (see Art. 65, p. 709). This
necessitates some corrections in the succeeding part of the paper, which have in all cases been
marked with a f.' Ed.]
t [See note above. Ed.]
ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 321
Thus two globes of water — probably nearly full absorbers at 300° A —
will at that temperature neither attract nor repel each other if their radii
are about 6 cm.f
If the density of the spheres is 11, about that so often used for masses
in the Cavendish experiment, F = P when
a = 0564 cm.t
This does not throw any doubt on the results of Cavendish experiments,
for it only holds when the radiators are in an enclosure of very low absolute
temperature. In all Cavendish experiments the greatest care is taken to
make the attracted body and its enclosure of one uniform temperature.
The really interesting case is that of two small meteorites, in interplanetary
space. To judge from the specimens which succeed in penetrating the earth's
atmosphere they are very dense. Let us suppose them to have density 55 —
that of the earth — and temperature 300° A, that which they will have at
the earth's distance. Then F = P when
a = 113 cm.f
If the radii of the bodies are less than the values found for equality of
F and P in the different cases, the net effect is repulsion.
The ratio of F to P is inversely as the square of the radius, so that, as
the radii are decreased from the values giving F = P, the radiation repulsion
soon becomes enormously greater than the gravitationpull, and the latter
may be neglected in comparison. Thus for two drops of water at 300° A in
a zero enclosure, with radii 0001 cm., the pressure is nearly 40,000,000 times
the pullf.
It is not, however, that the radiationforce is great, or even its acceleration.
The force becomes exceedingly minute, but the gravitation much more minute.
Thus consider two drops of water at 300° A placed in a zero enclosure at
a distance d = 10a apart. Our assumption of parallel radiation from one to
the other is now only a rough approximation, but the result will be of the
right order.
The radiationpush is Tra^R/Ud^, and the acceleration is
3aR/4:Ud^ =" TTv? ^ ~ approximately.
This only becomes considerable when the drops approach molecular
dimensions, and long before this they cease to absorb fully the stream of
momentum falling on them. Still, even molecules are selective absorbers,
and absorb especially each other's radiations. And we may expect that if
two gasmolecules collide and set each other radiating much more violently
than before, they will be practically in an enclosure of much lower temperature
than their own, and their mutual radiation may result in very rapid repulsion
— repulsion of the order of the fourth power of the temperature reached.
t [See note on p. 320. Ed.]
P. c. w. 21
322 RADIATION IN THE SOLAR SYSTEM:
RadiationPressure between Small Bodies at Different Distances from the Sun.
We have seen above, that if two small spheres of density 55 are at the
distance of the earth from the sun, their gravitation will be balanced by their
radiationpressure when the radius of each is 113 cm.f Now the balancing
radius is proportional to the square of the temperature, that is, inversely
proportional to the distance, since the temperature (Part I) is inversely as
the squareroot of the distance. Thus, at the distance of Mercury, the radii
would be about 3 cm.f ; a million miles from the sun's surface they would
be about 100 cm.j ; out at Neptune they would be about 04 mm.
We see then that the mutual action between small bodies of density that
of the earth, will, at different distances, change sign for different sizes of
body, ranging from something of the order of 2 metres diameter")* near the sun
to the order of 1 mm. diameter f at the distance of Neptune. A ring of
small planets, each of radius 113 cm.f, and density 55, would move round
the sun at the distance of the earth without net mutual attraction or repulsion,
and each might be regarded as moving independently of the rest. It appears
possible that if Saturn is hot enough, considerations of this kind may apply
to his rings.
The repulsion between small colliding bodies, even if not heated by the
sun, must lead to some delay in their final aggregation. This is obvious
when there are only two small bodies, and their temperature is very con
siderably raised by the collision. But there is also delay if instead of a
single pair we suppose two swarms to collide. Near the boundary of the
colliding region, a body will experience radiationpressure chiefly on one side,
and will tend to be driven out of the system. Of course, if the swarms are
so dense that a member near the outside cannot see through the rest, this
effect will be less. A body in front of another entirely screens its radiation,
but the gravitation is not screened. Hence, a body near the boundary of
a denselypacked region of collision may be repelled only by the colliding
bodies just round it, while it will be attracted by all ; or, to put the same idea
in another way, a body in a spherical swarm of uniform temperature will
only be pulled equally in all directions at the centre of the swarm, but it will
be equally repelled in all directions as soon as it is sufficiently deep to be
surrounded by its fellows wherever, so to speak, it looks.
Inequality of Action and Reaction between Two Mutually Radiating Bodies.
We have seen that two distant spheres push each other with forces
TTa^b^R/Ud^ and na^b'^R' jU d^, and that these, though opposite, are not equal
unless R = R' .
It would be easy to imagine cases in which the forces were not even
opposite or in the same directions. At first sight, then, it would appear that
t [See footnote, p. 320. Ed.]
ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 323
we have two bodies acting upon each other with unequal forces, but of course
this statement is inexact. The bodies do not act upon each other at all;
each sends out a stream of momentum into the medium surrounding it.
Some of this momentum is ultimately intercepted by the other, and in its
passage the momentum belongs neither to one body nor to the other. If we
assume that the momentum is conserved, and of course everything in the
methods of this paper depends on that assumption, the action on one of the
bodies is equal and opposite to the reaction on the lightbearing medium
contiguous to it. There is no failure of the law of action and reaction, but
an extension of our idea of matter to include the medium. There should
be no difficulty in this extension ; indeed, we have made it long ago in endowing
the medium with energycarrying properties. Whether the momentum in the
medium is in the form of mass m moving with velocity v in the direction of
propagation is perhaps open to doubt. We may, perhaps, have different
forms of momentum just as we may have different forms of energy, and
possibly we ought not to separate the momentum in radiation into the factors
m and v, but keep it for the present as one quantity M.
An interesting example of inequality of the radiationforces on two
mutually radiating bodies is afforded by two equal spheres, for which, at
a given temperature, the radiationpush F balances the gravitationpull P.
Raise* one in temperature so that the push on the other becomes F\ Lower
the other so that the push on the first becomes F", but adjust so that
r + F" = 2F=: 2P,
then PF" = F' P.
There will then be equal accelerations of the two in the same, not in opposite
directions, and a chase will begin in the line joining the centres, the hotter
chasing the colder. If the two temperatures could be maintained, the velocity
would go on increasing; but the increase would not be indefinitely great,
inasmuch as a Doppler effect would come into play. Each sphere moving
forward would crowd up against the radiation it emitted in front, and open
out from the radiation it emitted backwards. This would increase the front
and decrease the back pressure, and ultimately the excess of front pressure
would balance the accelerating force due to mutual radiation.
Let us examine the effect of motion of a radiating surface on the pressure
of its radiation against it.
Application of Doppler^s Principle to the Radiation Pressure against
a Moving Surface.
If a unit area A, Fig. 4, is moving with velocity u in any direction AB,
making angle with its normal AN, the effect on the energydensity in the
stream of radiation issuing in any direction AP is twofold. If the motion
is such as to shorten AP, the waves and their energy are crowded up into
21—2
324
RADIATION IN THE SOLAR SYSTEM:
less space, and if such as to lengthen AP, they are opened out. At the same
time, in the one case A is doing work against the radiationpressure and in
the other is having work done on it. We shall assume, as in the thermo
dynamic theory of radiation, that this work adds to or subtracts from the
energy of radiation. Both effects, (1) the crowding, and (2) the work done,
or the reverse of each, combine to alter the energy and therefore the radiation
pressure. We have no data by which we can determine whether the motion
alters the rate at which the surface is emitting radiation, but it appears worth
while to trace consequences on the assumption that the radiation goes on as
if the surface were at rest*, but that it is crowded up into less space or spread
over more, and that we can superpose on this the energy given out to, or
taken from, the stream by the work done by, or on, the moving surface by
the radiation pressure. This work can evidently be calculated to the first
order of approximation by supposing the pressure equal to its value when
the surface is at rest.
Let us draw from A as centre a sphere of
radius U, equal to the velocity of radiation.
The energy which, in a system at rest, would
be radiated into a cone with A as vertex,
length U, and solid angle do), in the direction
AP making an angle x with the direction of
motion AB, will now be crowded up into a cone
of length U — u cos x, since u cos x is the velocity
of A in the direction AP. We shall suppose that
u/U is very small. Hence the energydensity
in the cone is increased in the ratio U + u cos x
ucosx
Considering now the effect of the work done, the force on A due to the
stream in dw is N cos ddoj/U, and the work done in one second is
{N cos ddco/U) X u cos X
When A is at rest the energy in this cone is
N cos ddo).
U or by the factor
* Added August 20, 1903. Since the above was written Professor Larmor has pointed out
to me that the results obtained in the text from this assumption, along with the hypothesis of
crowding of the radiation and its increase by an amount equivalent to the work of the radiation
pressure, can be justified by an argument based on the following considerations. A perfect
reflector moving with uniform speed in an enclosure, itself also moving at that speed, and so in
a steady state, must send back as much radiation of every kind as a full radiator in its place.
Now the electrodynamics of perfect reflection are known ; hence the effect of motion of a full
radiator on the amount of its radiation can be determined. The result is equivalent to the
statement that the amplitudes of the excursions of the optical vibrators are the same at the
same temperature whether the source to which they belong is moving or not.
ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 325
When A is movijig it is increased to
T,, ^ , N cos ^c?ct>
iv cos Udoj H ^ u cos ;)^,
that is N cos 6>^a; (l + ^i^^ .
Thus the effect of the work done is equal to that of the crowding, and the
energydensity on the whole is increased in the ratio
2i^cosx..
"^ U
The pressure is increased in the ratio of the energydensity*. Then the force
on A due to the radiation through dco is increased from
Ncosddco , N cosddco /^ 2w cos v i
to == 1 + ^
U U \ ' u
If we resolve this along the normal to the surface A and integrate over
the hemisphere we obtain the total normal pressure. As we only want to
know the change in pressure P we may neglect the first term which gives
the pressure on A at rest, and we have
^ [N cos^ 9 2u cos y ,
If cf) is the angle between the normal planes through B and P we have
cos X = cos 6 cos e/f + sin ^ sin ijj cos cj).
Putting dco = sin Odddcf), we get
P =
r2 c'i^2Nu
I TtT ^^^^ ^ ^^^ ^ (^^^ ^ ^^^ 'A + si^ ^ ^^^^ 'A COS (f)) ddd(f)
ttNu cos i/r Ru cos j/f
The change in the tangential stress, T, is evidently in the direction AC,
that of the component of u in the plane of A.
We may therefore resolve each element of tangential stress in the direction
AC. Omitting the first term again, since in this case it disappears on inte
gration, the element due to dco in the direction AP will contribute
N cos 6 sin 6 cos 2u cos x
U ' U
and integrating over the hemisphere we have
dco,
T = \ jj^ cos 6 sin^ 6 cos cf) (cos 6 cosip + sin 6 sin ip cos ^) (?^ defy
ttNu sin j/f i??i sin iff
^ 2JP '~^ ~~2W~'
* [See note, p. 330. Ed.]
326 RADIATION IN THE SOLAR SYSTEM:
Force on a Sphere moving with Velocity 'u' in a Given Direction.
If a sphere, radius a, is moving with velocity u, we may from symmetry
resolve the forces on each element in the direction of motion. The resolutes
will be P cos ifj and T sin ifj. Evidently it is sufficient to integrate over the
front hemisphere and then double the result. We have the
2
r. , T T^ ^ r /^^ cos^ w , Ku sm2 e/f\ ^ . , , ,
Ketardmg _borce = ^ I fj^ \ i^jj2 j ^tt"^ sm i/jdifj
va
3 U2
It is noteworthy that one half of this is due to the normal, the other half
to the tangential stresses.
If the sphere has density p the acceleration is obtained by dividing by
iTTCi^p, then
du/dt =  IRujU^pa.
Effect on Rotation.
If the sphere radius a is rotating with angular velocity co, then any element
of the surface, A from the equator, is moving with hnear velocity aa> cos A in
its own plane. This does not affect the normal pressure, but it introduces
a tangential stress opposing the motion
Rii/2U^ = RacocosXI2U\
Taking moments round the axis and integrating over the sphere, we
obtain a couple
TT
ina^p . ^a^ , = ^_^ 27ra^ cos'^ A dX,
^ "" dt 2U^ j _^
2
whence dco/dt = — qRco/U^pa.
The rate of diminution of oj is therefore of the same order as that of v.
To obtain an idea of the magnitude of the retardation of a moving sphere,
let us suppose that one is moving through a stationary medium. Let its
radius he a = 1 cm., its density p = 55, its temperature 300° A.
Idu 2 X 532 X 105 X 300*
Then
u dt 9 X 1020 X 5.5
= 175 X 1016.
This will begin to affect the velocity by the order of 1 in 10,000 in, say,
10^2 seconds, or taking the year as 315 x 10^ seconds, in about 30,000 years.
ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 327
The effect is inversely as the radius, so that a dustparticle 0001 cm.
radius will be equally affected in 30 years.
The effect is as the fourth power of the temperature, so that with rising
temperature it becomes rapidly more serious.
Equation to the Orbit of a Small Spherical Absorbing Particle Moving in
a Stationary Medium Round the Sun.
It is evident from the above result, that the effect of motion on radiation
pressure may be very considerable in the case of a small absorbing particle
moving round the sun.
We shall take the particle as spherical, of radius a and distance r from the
sun. We shall suppose the radius so small that the particle is of one tempera
ture throughout, the temperature due to the solar radiation which it receives,
but that it is still so large as to be attracted much more than it is repelled by
the sun. Both attraction and repulsion are inversely as the square of the
distance, so that we shall have a central force which we may put as producing
acceleration A/r^, where A is constant.
We know that at the distance of the earth, putting r = b, A/b^ = 059
cm. /sec. ^ say 06 cm./sec.^. Then A = OQb^. The force acting against
the motion produces retardation — 2RulU^pa.
If S is the solar constant at the distance 6, its value at distance r is
Sb^rK
Putting iTram = ira^Sb^/r^,
R = (S/i) {b^lr%
then the acceleration in the line of motion is
_ Sb"^ u__Ts
2lPpa ' r2 ~ /•2 ^
where T = Sb^l2U^pa, and s is now written for the velocity u.
The accelerations along and perpendicular to the radiusvector give the
equations
'"•=^s <■'
ls"''>W^^ ■ ; <^>
From (2) we get  {r^O) =  ^^^
whence r^O = C  TO, (3)
where C is the constant of integration.
328 RADIATION IN THE SOLAR SYSTEM:
If ^ is when t = 0, then C is the initial value of rW. Further, as 6 increases
rW decreases and is when d = C/T. This gives a limit to the angle described.
Equation (1) may be written
rre^=i^ (4)
Putting u for r~^ we have
dr X 1 du A ,„ „^, du
^ = ^ = ^^i^ = (^^^)i f^^^^^)'
T{CTe)n^^g{C^Teru^^^ from (3).
Substituting in (4)
de^ ' (C  TOY'
This can probably only be integrated by approximation. We can see the
effect on the motion at the beginning by putting
d'^u A / 2T
4 ^^ = ^ 1  ^.
dd^ ' 02
{■¥•).
since TjC is small if we begin at the distance of the earth and with a particle
having the velocity of the earth.
An integral of this is
The complementary function will be periodic and may be omitted. To the
order of approximation adopted
Then initially rjr =  (ITjC) 6.
In applying these results, we may note that T = Sb^l2U^pa is constant
for all distances, and that b, the earth's distance, is 493 U. Inserting the
value of the solar constant, 0175 x 10'^, and taking p ^ 55, we get
T  39 X 1010 . ai.
C will depend on the initial conditions. Assuming that the body considered
is initially moving in a circle, then, at the beginning
re^^~ or ^= /^= /^,
since at r = 6 the acceleration to the centre is 06.
Then C =^ r^O = V6^¥r.
ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 329
Substituting these values in r/r we have
r__ 78 X IQio
This gives only the initial value of  and cannot be taken to hold for a time
which will make T^d^/C^ appreciable. But by (3) we see that r = if
6 = C/T, so that CI27tT is a superior limit to the number of devolutions,
even if we suppose the way clear right up to the centre.
Putting the numerical values we get
C/27rT= 465A*.
Suppose, for example, that r = b = 493 x 3 x 10^^ ; a = 1, then
r/r =  35 X 1016.
If we multiply by 315 x 10^, the seconds in a year, we obtain
(r/r) X 315 x 10^ = M x IQ^.
This implies that a sphere 1 cm. radius and density 55, starting with
the velocity of the earth, and at its distance from the sun, will move inwards
10,0 00 ^^ ^^^ distance in about 10,000 years. It cannot in all make so many
as 465 X 6^ = 179 x 10^ revolutions.
If we put a = 0001 cm., since the effects are inversely as a, then its
distance will decrease by about 1 in 10,000 in 10 years, and it cannot make
in all so many as 179 x 10^ revolutions.
If instead of starting from the distance of the earth, the particle starts
from, say, 01 the distance, the effect in the radius is 100 times as great and
the number of revolutions is v'^lO times less. Then with radius 1 cm. the
distance decreases by foTooo ^^ ^^^ years, and there are not so many as
80 X 10^ revolutions §, while with radius 0001 cm. the distance decreases
by ^Q QQQ in 01 year, and there are not so many as 80,000 revolutions §.
Small particles, therefore, even of the order of 1 cm. radius, would be
drawn into the sun, even from the distance of the earth, in times not large
compared with geological times, and dustparticles if large enough to absorb
solar radiation would be swept in in a time almost comparable with historical
times. Near the sun the effects are vastly greater. The application to
meteoric dust in the system is obvious.
There should be a similar effect with dust and small particles circulating
round the earth. If, for example, any of the Krakatoa dust was blown out
so far beyond the appreciable atmosphere, and was given such motion that
the particles became satellites to the earth, at no long time the dust will
* [The original has 6 Iria. In what follows, the necessary alterations consequent upon tliis
correction have been made. Ed.]
§ [The original has 80,000 and 80 respectively. The correction necessitates a modification in
the views expressed in the succeeding paragraphs. Ed.]
330  RADIATION IN THE SOLAR SYSTEM:
return. A ring of dustparticles moving round a planet and receiving heat
either from the sun or from the planet will tend to draw in to the planet.
[Note added October 31. Since the foregoing paper was printed I have
reexamined the theory of the pressure on a fully radiating surface when in
motion, and have come to the conclusion that the change in pressure due to
the motion is only half as great as that obtained on p. 325. In that investi
gation the pressure was assumed to be equal to the energydensity, whether
the surface was at rest or in motion, whereas it appears, if the following mode
of treatment is correct, that the pressure on a radiating surface moving
u
forward is only 1 — yy of the energydensity of the radiation emitted.
Let us suppose that a surface A, a full radiator, is moving with velocity
u towards a full absorber J5, which, with the surroundings, we will suppose
at 0° ^. Consider for simplicity a parallel pencil issuing normal from A with
velocity U towards B. Let the energydensity in the stream from AhQ E
when A is at rest, and E' when it is moving. Let the pressure onAhaf^E
when it is at rest, and f' when it is moving. When moving, A is emitting
a stream of momentum f' per second and this momentum ultimately falls
on B. Let A start radiating and moving at the same instant; let it move
a distance d towards B, and then let it stop radiating and moving. It emits
momentum f' per second for a time dju and therefore emits total momentum
f'dju. Since B is at rest, the pressure on it, the momentum which it receives
per second, is W . But since A is following up the stream sent out, B does
not receive through a period as long as dju, but for a time less by djU. If we
assume that the total momentum received by B is equal to the total sent out
by A, we have
f'dju = E' (dju  djU),
or rp' =.E' (I ujU).
To find E' in terms of E we must make some assumption as to the effect
of the motion on the radiation emitted. In the paper I have assumed that
the emitting surface converts the same amount of its internal energy per
second into radiant energy as when it is at rest, but that ^p'u of the energy
of motion of the radiating mass is also converted into radiant energy. Since
the radiation emitted in one second is contained in length TJ — ^/, we have
E' {U  u) =^EU + p'u =EU\Fy (^— ) ^,
whence E' = E  ^' ,, = ^ (1 + 2WC/).
The same result is obtained if we assume that the amplitude of the
emitted waves is the same whether the surface is moving or not, and that
ITS EFFECT ON TEMPERATURE AND ITS PRESSURE ON SMALL BODIES 331
the energydensity is inversely as the square of the wavelength for given
amplitude.
We have, therefore, if the above application of the equality of action and
reaction is justified,
In a similar way we can find the effect of motion of an absorber on the
pressure against it due to the incident radiation.
Let a stream of energydensity E be incident on a fully absorbing surface
moving towards the source with velocity u. Let the surface be at 0° ^, so
as to obtain the effect of the incident radiation only. When the surface is
at rest, we may regard the stream as bringing up momentum E per second,
or as containing momentum of density E/U brought up with velocity U to it.
If the surface is moving towards the source, it takes up in one second the
E
momentum in length U + u, or receives yj{U + u), and the pressure on it is
,'^.(i.a = ,(i.^).
It is easy to show that when a perfect reflector is moving, the pressure
2u'
upon it is altered from ^^ to ^ ( 1 + jj
In the paper, the case of a full radiator in an enclosure at zero has alone
1 IT 11 •<• . ^ u ^ u cos Y ,
been considered, so that the correctmg factor is 1 + yy or 1 H ^, when
the motion is at an angle x to the line of radiation. Hence the forces obtained
2u
in the paper when the factor was 1 + 77 are all double those obtained with
the factor now given. The process of drawing in small particles to the sun
is correspondingly lengthened out.
It is, perhaps, worth noting that the motion of a body round the sun
produces a small aberrationeffect. If the body is a sphere, the sunlight
does not fall on the hemisphere directly under the sun, but on one turned
round through an angle u/U. The pressure of the radiation, though still
straight from the sun, does not act through the centre but through a point
^jj^ X (radius of sphere) in front of the centre. Thus, in the case of the
earth, it will tend to stop the rotation. But the effect is so minute that
if present conditions as to distance and radiation were maintained, it
would take something of the order of 10^^ years to stop the whole of the
rotation. J. H. P.]
21.
NOTE ON THE TANGENTIAL STRESS DUE TO LIGHT
INCIDENT OBLIQUELY ON AN ABSORBING SURFACE.
[PM. Mag. 9, 1905, pp. 169171.]
[Read at Section A, British Association, Cambridge, August, 1904.]
The existence of pressure on a surface due to tlie incidence of a normal
beam of light, first deduced as a consequence of the electromagnetic theory
by Maxwell, has been fully confirmed by the experiments of Lebedew, and
quite independently by the exact work of Nichols and Hull. These experi
ments show that the pressure exists and that it is equal to the energy per c.c.
or to the energydensity in the incident beam.
In so far as it produces this pressure we may regard the beam as a stream
of momentum, the direction of the momentum being along the line of
propagation, and the amount of momentum passing per second through unit
area crosssection of the beam being equal to the density of the energy in it.
Let E denote this energydensity. If the beam is inclined at 9 to the normal
to a surface on which it falls, the momentumstream on to unit area of the
surface is E cos 9 per second, and this is the force which the beam will exert
in its own direction. If the beam is entirely absorbed, the result is a pressure
E cos^ 9 along the normal and a tangential stress in the plane of incidence
E sin 9 cos 9 = IE sin 29. If ^ of the incident beam is reflected, the normal
pressure is {I \ ^x) E cos^ 9 , and the tangential stress is  ^sin2^*.
When there is absorption the tangential stress has a maximum value at 45°
if IX is constant. When there is no absorption the tangential stress disappears.
The tangential stress is much more easily detected than the normal
pressure. For the action of the gas surrounding the surface is normal to it
and is with difficulty disentangled from the normal lightpressure. But the
gasaction is at right angles to the tangential stress, and it is merely necessary
to arrange a surface free to move in its own plane to eliminate the action of
the normal forces and to reveal the tangential stress.
* These expressions are given in 'Radiation in the Solar System,' Phil. Trans. A, 202, p. 539.
[Collected Papers, Art. 20.]
NOTE ON THE TANGENTIAL STRESS DUE TO LIGHT, ETC. 333
With the assistance of my colleague Dr. Guy Barlow, to whom I am much
indebted for help in the work, I have made the following experiment to show
the existence of the stress.
Two circular glass discs, each 275 sq. cm. area, were fixed at the ends of
a horizontal light glass rod 5*3 cm. long, the discs being perpendicular to the
rod and fixed to it at their highest points. One of the discs was lampblacked,
and the other silvered. The rod was placed in a light wire cradle and
suspended by a fine quartz fibre about 25 cm. long in a brass case with glazed
sides. On the cradle was a mirror by which deflections could be observed
with a telescope on a millimetrescale 18 metres distant. The moment of
inertia of the system was 2*35 gm. cm.^ and the time of vibration was
146 seconds. A deflection of 1 scaledivision therefore corresponded to a
tangential force on a disc of about one twomillionth of a dyne — more exactly
0483 X 106 (jyne.
The air was pumped from the case till the pressure was less than 1 cm. of
mercury. At this pressure the irregularity of the disturbances due to the
residual gas is very greatly reduced. A parallel beam of light from a Nernst
lamp was then directed so as to be incident obliquely on the lampblacked
disc. From the arrangement of the discs it is obvious that a uniformly
distributed normal force would have no moment tending to twist the system,
while a tangential force would have a moment and would twist it. In all
cases the disc moved away from the source of light. The deflection was
a maximum when the incidence was not very far from 45°, and fell off on each
side of the maximum value.
As there are various sources of error not yet removed, we have not made
a complete series of measurements but have only made sure that the effect
is of the order to be expected from the theory, by finding the deflection for
an angle of 45°.
The beam from the Nernst lamp when incident at 45° turned the rod
through 165 scaledivisions. Assuming total absorption, the tangential
force should be ^E sin 2d x area of disc == ^E x 275.
Equating to the value of the force given by the deflection, viz.,
0483 X 106 X 165,
we have E = 58 x 10^ erg/cm.^.
The same beam was then directed on to a small lampblacked silver disc
of known heatcapacity, through a glass plate of thickness equal to that of
the side of the case. The initial rise of temperature per second was measured
by a ther mo junction of constantan wire soldered to the disc. The energy
density of the stream was thus found to be ^ = 65 x 10"^ erg/cm. 3.
334 NOTE ON THE TANGENTIAL STRESS DUE TO LIGHT, ETC.
The agreement of the two values is quite as close as could be expected
in so rough a determination*.
When the beam was directed on to the silver disc at the other end of the
torsionrod, the deflection was much less, as was to be expected.
We have also made some qualitative experiments with a blackened glass
cyhnder — a ring cut from a testtube — suspended by a quartz fibre with its
axis vertical. When a beam fell on this in any direction not along a diameter,
there was always a twist in the direction corresponding to the tangential
stress.
* [A redetermination of the various constants and a revision of the calculations gave a still
closer agreement. The observed torque was 21 x 10"^ cm. dynes, while the torque calculated from
the energy was 22 x 10"^ cm. dynes. This correction is given in The Pressure of Light, p. 55.
(Romance of Science Series, S.P.C.K. 1910.) Ed.]
Wl
[Note by G. Barlow, July 1916.
Particular care was taken to make the torsionsystem very symmetrical
th respect to the axis of suspension. The success of the experiment
depended, also, on using a uniform parallel beam of light of crosssection
slightly greater than the projected area of the disc. During the small oscilla
tions of the system the disc, therefore, remained uniformly illuminated.
It was found that when a circular patch of light was focussed centrally on the
disc the deflections were irregular. On one occasion we used a beam of
sunlight reflected from a heliostat, but owing to the very variable absorption
by the town atmosphere the observed pressure showed great fluctuations.
The experiment with the cylinder has since been repeated, and was
shown to some members of the British Association at the Birmingham meeting
of 1913. The cylinder was of aluminium and was turned very accurately
by a watchmaker. The ends were closed, but near the axis two small air
holes were drilled. The surface was blackened with a deposit of asphaltum.
The observed torques due to the lightpressure were always of the order of
magnitude expected, and variation of the gaspressure over a considerable
range did not greatly afi^ect the results. This method is suitable for lecture
demonstration but it does not appear satisfactory for exact measurements.]
22.
RADIATIONPKESSURE *.
[Phil. Mag. 9, 1905, pp. 393406.]
[Presidential Address, delivered at the Annual General Meeting of the
Physical Society, February 10, 1905.]
A hundred years ago, when the corpuscular theory held almost universal
sway, it would have been much more easy to account for and explain the
pressure of light than it is today, when we are all certain that light is a form
of wavemotion. Indeed, on the corpuscular theory it was so natural to
expect a pressure that numerous attempts were madef in the eighteenth
century to detect it. But the early experimenters had a greatly exaggerated
idea of the force they looked for. Even on their own theory it would only
have double the value which we now know it to possess, and their methods of
experiment were utterly inadequate to show so small a quantity. But had
these eighteenthcentury philosophers been able to command the more refined
methods of today, and been able to carry out the great experiments of
Lebedew and of Nichols and Hull, and had they further known of the emission
of corpuscles revealed to us by the cathodestream and by radioactive bodies,
there can be little doubt that Young and Fresnel would have had much
greater difficulty in dethroning the corpuscular theory and setting up the
wavetheory in its place.
The existence of pressure due to waves, though held by Euler and used
by him 160 years ago to explain the formation of comets' tails by repulsion,
seems to have dropped out of sight, till Maxwell, in 1872, predicted its existence
as a consequence of his Electromagnetic Theory of Light. It is remarkable
that it should have been brought to the front through the investigation of
such a special type, such an abstruse case, of wavemotion, and that it was
not seen that it must follow as a consequence of any wavemotion, whatever
the type of wave we suppose to constitute Light. I believe that the first
suggestion that it is a general property of waves is due to Mr. S. Tolver
Preston, who in 1876 { pointed out the analogy of the energycarrying power
* [This address is included here because it contains an account of some original work not
elsewhere described. Ed.]
t Some account of these methods is given by Nichols and Hull in 'The Pressure due to
Radiation,' Proc. Am. Ac. vol. .38, no. 20, p. 559. See also Priestley, On Vision, p. 385.
t Engineering, 1876, vol. 21, p. 83.
336 KADIATIONPRESSURE
of a beam of light with the mechanical carriage by belting, and calculated
the pressure on the surface of the Sun by the issuing radiation, obtaining
a value equal to the energydensity in the issuing stream, without assumption
as to the nature of the waves. But though the analogy is valuable, I confess
that Mr. Preston's reasoning does not appear to me conclusive, and I think
it still remains an analogy. There is, I suspect, some general theorem yet
to be discovered, which shall relate directly the energy and the momentum
issuing from a radiating source. It seems possible that in all cases of energy
transfer, momentum in the direction of transfer is also passed on, and there
fore there is a back pressure on the source. Such pressure certainly exists
in material transfer, as in the corpuscular theory. It exists too, as we now
know, in all wavetransfer. From the investigation below (p. 338) it appears
to exist when energy is transferred along a revolving twisted shaft. In heat
conduction in gases, the kinetic theory requires a carriage of momentum from
hotter to colder parts ; so that there is some ground for supposing the pressure
to exist in all cases.
Though we have not yet a general and direct dynamical theorem accounting
for radiationpressure. Professor Larmor* has given us a simple and most
excellent indirect mode of proving the existence of the pressure, which applies
to all waves in which the average energydensity for a given amplitude is
inversely as the square of the wavelength. Let us suppose that a train of
waves is incident normally on a perfectly reflecting surface. Then, whether
the reflecting surface is at rest, or is moving to or from the source, the perfect
reflection requires that the disturbance at its surface shall be annulled by the
superposition of the direct and reflected trains. The two trains must there
fore have equal amplitudes. Suppose now that the reflector is moving
forwards towards the source. By Doppler's principle, the waves of the
reflected train are shortened, and so contain more energy than those of the
incident train. This extra energy can only be accounted for by supposing
that there is a pressure against the reflector, that work has to be done in
pushing it forward. When the velocity of the reflector is small, the pressure
is easily found to be equal to E ll + jj), where ^ is the energydensity
just outside the reflector in the incident train, U is the wave velocity, and
a the velocity of the reflector. If u = 0, the pressure is E ; but it is altered
2u
by the fraction ^ when the reflector is moving, and the alteration changes
sign with ii. A similar train of reasoning gives us a pressure on the source,
increased when the source is moving forward, decreased when it is receding.
It is essential, I think, to Larmor's proof that we should be able to move
the reflecting surface forward without disturbing the medium except by
* Encyc. Brit. vol. 32, ' Radiation,' p. 121.
RADIATIONPRESSURE 337
reflecting the waves. In the case of lightwaves it is easy to imagine such
a reflector. We have to think of it as being, as it were, a semipermeable
membrane, freely permeable to ether, but straining back and preventing
the passage of the waves. In the case of soundwaves, or of transverse waves
in an elastic solid, it is not so easy to picture a possible reflector. But for
soundwaves I venture to suggest a reflector which shall freeze the air just in
front of it, and so remove it, the frozen surface advancing with constant
velocity u. Or perhaps we may imagine an absorbing surface which shall
remove the air quietly by solution or chemical combination. In the case
of an elastic solid, we may perhaps think of the solid as melted by the ad
vancing reflector, the products of melting being passed through pores in the
surface and coming out to solidify at the back.
Though Larmor's proof is quite convincing, it is, I think, more satisfying
if we can realise the way in which the pressure is produced in the different
types of wavemotion.
In the case of electromagnetic waves. Maxwell's original mode of treatment
is the simplest, though it is not, I believe, entirely satisfactory. According to
his theory, tubes of electric and of magnetic force alike, produce a tension
lengthways and an equal pressure sideways, equal respectively to the electric
and magnetic energydensities in the tubes. We regard a train of waves as
a system of electric and magnetic tubes transverse to the direction of propaga
tion, each kind pressmg out sideways^that is, in the direction of propagation.
They press against the source from which they issue, against each other as
they travel, and against any surface upon which they fall. Or we may take
Professor J. J. Thomson's point of view*. ' Let us suppose that the reflecting
surface is metallic; then, when the light falls on the surface, the variation
of the magnetic force induces currents in the metal, and these currents pro
duce opposite effects to the incident light, so that the inductive force is
screened off from the interior of the metal plate : thus the currents in the plate,
and therefore the intensity of the light, rapidly diminish as we recede from the
surface of the plate. The currents in the plate are accompanied by magnetic
force at right angles to them ; the corresponding mechanical force is at right
angles both to the current and the magnetic force, and therefore parallel to
the direction of propagation of the light.' In fact, we have in the surface of
the reflector a thin* currentsheet in a transverse magnetic field, and the
ordinary electrodynamic force on the conductor accounts for the pressure.
In soundwaves there is at a reflecting surface a node— a point of no
motion, but of varying pressure. If the variation of pressure from the
undisturbed value were exactly proportional to the displacement of a parallel
layer near the surface, and if the displacement were exactly harmonic, then
the average pressure would be equal to the normal undisturbed value. But
* Maxwell's Electricity and Magnetism, 3rd edition, vol. 2, p. 441, footnote.
P. c.w. 22
RADIATIONPHESSURE
consider a layer of air quite close to the surface. If it moves up a distance y
towards the surface, the pressure is increased. If it moves an equal distance
y away from the surface, the pressure is decreased, but by a slightly smaller
quantity. To illustrate this, take an extreme case, and for simplicity suppose
that Boyle's law holds. If the layer advances halfway towards the reflecting
surface, the pressure is doubled. If it moves an equal distance outwards from
its original position, the pressure falls, but only by onethird of its original
value ; and if we could suppose the layer to be moving harmonically, it is
obvious that the mean of the increased and diminished pressures would be
largely in excess of the normal value. Though we are not entitled to assume
the existence of harmonic vibrations when we take into account the second
order of small quantities, yet this illustration gives the right idea. The
excess of pressure in the compressionhalf is greater than its defect during
the extensionhalf, and the net result is an average excess of pressure — a
quantity itself of the second order — on the reflecting surface. This excess in
the compressionhalf of a wavetrain is connected with the extra speed which
exists in that half, and makes the crests of intense soundwaves gain on the
troughs.
Lord Rayleigh*, using Boyle's Law, has shown that the average excess on
a surface reflecting soundwaves should be equal to the average density of
the energy just outside ; and I think the same result can be obtained by his
method if we use the adiabatic law. But the subject is full of pitfalls, and
I am by no means sure that the result is to be obtained so easily as it appears
to be. It is perhaps worth while to note one of these pitfalls, of which I have
been a victim. It is quite easy to obtain the pressure against a reflecting
surface by supposing that the motion just outside it is harmonic. But the
result comes out to (y + 1) x energydensity, where y is the ratio of the specific
heats. Lord Rayleigh kindly pulled me out of the pit into which I fell, pointing
out that when we take into account secondorder quantities the ordinary
soundequation does not hold. In fact we cannot take the disturbance as
harmonic, and the simple mode of treatment is illusory.
The pressure in transverse waves in an elastic solid is, I think, to be
accounted for by the fact that when a square, ABCD, is sheared into the
position aBCd (Fig. 1) through an angle e, the axes of the shear, aC and Bd,
6
no longer make 45° with the planes of shear AD, BC. Since ACa = , the
6 . .
pressureline aC is inclined at 45° —  to the direction of propagation, and
the tensionline at 45° + ^ to that line. The result is a small pressure
perpendicular to the planes of shear, that is, in the direction of propagation ;
.and this small pressure is just equal to the energydensity of the waves.
* Phil. Mag. vol. 3, 1902, p. 338, 'On the Pressure of Vibrations.'
RADIATIONPRESSURE
339
For let PQR (Fig. 2) be a small triangular wedge of the solid, PQ being
a plane of shear perpendicular to the direction of propagation. Let this
wedge have unit thickness perpendicular to the plane of the figure. Let
FR be along a pressureline and QR along a tensionline, and let pressure
and tension each be P. Resolve the forces on PR and QR perpendicular
to PQ. Then we have a force from right to left,
P . QR cos PQR P .PR cos QPR
= P.PQ cos2 ^45°   cos^ ^45° + ^\i = P . PQ .e.
oz/Tccr/OAA^
Of^ f/fOPAOATWA/
Fig. 1.
Thus, to prevent motion in the direction of propagation there must be
a pressure on PQ equal to Pe = ne^, where n is the rigidity modulus. But
the strainenergy per unit volume is ^ , and the kinetic energy is equal to
it. The total energydensity is therefore ne^, and the pressure is equal to this.
The pressure of elasticsolid waves appears to be beyond experimental
verification at present. But that of soundwaves has been demonstrated
most successfully by Altberg*, working in Lebedew's laboratory at Moscow.
A small wooden cylinder, 21 mm. diameter, was suspended at one end of
a torsionarm, with its axis horizontal and transverse to the arm. One end
of the cylinder occupied a circular hole in the middle of a board, there being
just sufficient clearance to allow it to move, and the plane end was flush
with the outer surface of the board. When very intense soundwaves 10 cm.
in length, from a source 50 cm. distant, impinged on the board, the cyHnder
was pushed back, the pressure sometimes rising to as much as 024 dyne/cm. 2.
The intensity of the sound was measured independently by the vibrations
of a telephoneplate, in a manner devised by M. Wien, and through a large
range it was found that the pressure on the cylinder was proportional to the
intensity indicated by the telephone manometer.
* Ann. der Physik, vol. 11, 1903, p. 405.
22—2
340 RADIATIONPRESSURE
Just lately Professor Wood* has devised a strikingly simple experiment
to illustrate soundpressure. The soundwaves from strong inductionsparks
are focussed by a concave mirror on a set of vanes like those of a radiometer,
and when the focus is on the vanes as they face the waves the mill spins
round.
Theory and experiment, then, justify the conclusion that when a source
is pouring out waves, it is pouring out with them forward momentum as well
as energy, the momentum being manifested in the reaction, the backpressure
against the source, and in the forward pressure when the waves reach an
opposing surface. The wavetrain may be regarded as a stream of momentum
travelling through space. This view is most clearly brought home, perhaps,
by considering a parallel train of waves which issues normally from a source
for one second, travels for any length of time through space, and then falls
normally on an absorbing surface for one second. During this last second,
momentum is given up to the absorbing surface. During the first second,
the same amount was given out by the source. If it is conserved in the
meanwhile, we must regard it as travelling with the train.
Since the pressure is the momentum given out or received per second, and
the pressure is equal to the energydensity in the train, the momentumdensity
is equal to the energydensity H wavevelocity.
This idea of momentum in a wavetrain enables us to see at once what is
the nature of the action of a beam of light on a surface where it is reflected,
absorbed, or refracted, without any further appeal to the theory of the wave
motion of which we suppose the light to consist f.
It is convenient to consider the energy per linear centimetre in the beam,
and the total pressureforce, equal to this linear energydensity, so as to avoid
any necessity for taking into account the crosssection of the beam.
Thus, in total reflection, let a beam AB (Fig. 3) be reflected along BC, and
let AB = BC represent the momentum in each in length V equal to the
velocity of light.
Produce AB to D, making BD = AB.
Then DC represents the change in the momentum per second due to the
reflection — the force on the beam, if such language is permissible ; and CD is
the reaction, the total lightforce on the surface.
If there is total absorption, let AB (Fig. 4) represent the momentum of
the incident beam. Eesolve AB into AE parallel and EB normal to the
surface. Then, since the momentum AB disappears as lightmomentum,
there must be a normal force EB on the surface and a tangential force AE
* Phys. Zeitschrift, 1 Jan. 1905, p. 22.
•j A discussion, on the electromagnetic theory, of the forces exerted by light is given by
Goldhammer, A7in. der Phys. vol. 4, 1901, p. 483.
RADIATIONPRESSUKE
341
parallel to the surface. I have lately* described an experiment which shows
the existence of the tangential force AE.
If there is total refraction, let AB (Fig. 5) be refracted along BC with
velocity V. HE is the energy in unit length of AB, and if E' is the energy
in unit length of BC, the equahty of energy in the two beams is expressed by
VE = V'E'.
Fig. 3.
Fig. 4.
But if M is the stream of momentum passing per second along AB, and
if M' is that along BC, then
M=E and M' = E'.
Whence
and
VM = F'M'
M' = y, M = fxM.
Let AB = M, and BC along the refracted beam = M' = fjuM = fiAB.
Draw CD parallel to BA, meeting the normal BN in D. Then
CD = CB sin r/sin
CB
= AB=M.
Phil. Mag. Jan. 1905, p. 169. [Collected Papers, Art. 21.]
342
RADIATIONPRESSURE
Hence, by the refraction, momentum DC has been changed to momentum
BC, or momentum BD has been imparted to the light. There is therefore
a reaction BB on the surface. The force DB may be regarded as a pullout
or a pressure from within, and it is along the normal*.
If the refraction is from a denser to a rarer medium, CB will now represent
the incident stream and BA or CD the refracted stream. BD is the stream
added to CB to change it to CD, and DB is the force on the surface, again
a force outwards along the normal.
In any real refraction with ordinary light, there will be reflection as well
as refraction. The reflection always produces a normal pressure, and the
refraction a normal pull. But with unpolarised light, a calculation shows
that the refractionpull, for glass at any rate, is always greater than the
reflectionpush, even at grazing incidence.
The following table has been calculated from Fresnel's formula for
unpolarised light by Dr. Barlow :
P = total pull on surface.
M = momentum per second in incident beam.
R = reflectioncoefflcient for angle i.
/x = 15.
i
R
PjM
•0400
•4000
20
•0402
•4240
40
•0458
•4925
50
•0572
•5310
60
•0893
•5720
65
•1205
•5771 Maximum
70
•1710
•5683
75
•2531
•5329
80
•3878
•4521
89
•9044
•0738
90 f/^
2^,^de
90
i"ooo6
•0000
If a ray of light passes obliquely through a parallel plate, there is a normal
pull outwards at incidence and a normal pull outwards at emergence : and
if the refraction were total, this would result in a couple. But since some of
the light returns into the first medium, it is easy to see that the net result is
a normal repulsion and a couple.
An experiment which I have lately made in conjunction with Dr. Barlow
will serve as an illustration of the idea of a beam of light regarded as a stream
* It has been pointed out by J. J, Thomson, Electricity and Matter, p. 67, 'that even when
the incidence of the light is oblique, the momentum communicated to the substance is normal
to the refracting surface ' The change of momentum of a beam of light is, it may be noted,
the same on the wave and on the corpuscular theory.
RADIATIONPRESSURE 343
of momentum. A rectangular block of glass, 3 cm. x 1 cm. x 1 cm., was
suspended by a quartz fibre so that the long axis of the block was horizontal.
It hung in a case with glass windows, which was exhausted to about 15 mm.
of mercury. A horizontal beam of light, from either a Nernst lamp or an
arc, was directed on to one end of the block so that it entered centrally as AB
in Fig. 6, and at an angle of incidence about 55°. After two internal reflections
it emerged centrally as EF from the other end. Thus a stream of momentum
AB was shifted parallel to itself into the line EF, or a counterclockwise
couple acted on the beam. The reaction was a clockwise couple on the block.
Using mirror, telescope, and a millimetrescale about 184 cm. distant, with
the strongest light a very small deflection in the right direction could just
be detected. But the quartz fibre was rather coarse, indeed needlessly
strong; and as the time of vibration was only 39 seconds, the deflection was
very minute. To render the effect more evident we used intermittent passage
of the beam, sending it in during the halfperiod of vibration while B was
F
Fig. 6. Plan.
moving from A, and shutting it off while B was moving towards A. The
swings then always increased. When the beam was sent in during the
approaching half and shut off during the receding half, the swings always
decreased, and always rather more rapidly than they increased during the
first half. For in the first case the natural damping acted against the light
couple, and in the second with it. In one experiment the average increase
was 55 scaledivision and the average decrease 61 per period, and was fairly
regular in each case. The mean was 58. The steady deflection is half this,
or 029 division, giving a couple 11 x 10"^ cm. dyne. We made a measure
ment of the energy in the beam by means of the rate of rise of a blackened
silver disc; but it was necessarily very inexact, as we had no means of
securing constancy in the arc used in this experiment. This energy measure
ment gave as the value of the couple 6 x 10"^, and the agreement is sufficient
to show that the order of the result is right.
An analysis of this experiment shows that the couple was really due to
the pressures at the two internal reflections ; for, as we have seen, the forces
at incidence at B and emergence at E are normal and produce no twist.
344 RADIATIONPRESSURE
Another experiment which we have made is, I think, more interesting,
in that it brings into prominence the pull outwards or push from within
occurring on refraction. Two glass prisms, each with refracting angle 34°,
another angle being a right angle, and with refracting edge 16 cm. long, were
arranged as in Fig. 7 (which shows the plan) at the ends of a thin brass torsion
arm suspended at its middle point from a quartz fibre in the same case as
that used in the last experiment. The two inner faces were 3 cm. apart,
and their width was 185 cm. A mirror gave the reflection of a millimetre
scale 1714 cm. distant. The moment of inertia of the system was 48 gm. cm.^,
and the time of vibration was 317 seconds. The airpressure was reduced as
before. When a beam of light from a Nernst lamp was sent through the
system, as shown in the figure, it was shifted parallel to itself through a
distance about 164 cm. The torsionarm moved round clockwise by an
easily measurable amount. In one experiment the deflection was 33 scale
divisions, indicating a couple 184 x 10~^ cm. dyne. The same beam directed
Fig. 7. Plan.
on to the blackened silver disc gave the linear energydensity as 98 x 10"^,
which should have given a couple 16 x 10~^. Though the agreement is
perhaps accidentally close, yet, as we could use a Nernst lamp, the measure
ments were much more trustworthy than in the last experiment*.
The interesting point here is that the effect could only be produced by
a force outwards at B and E. Whatever forces exist at C and D would be
normal to the surfaces and would give no twist.
A very short experience in attempting to measure these lightforces is
sufficient to make one realise their extreme minuteness — a minuteness which
appears to put them beyond consideration in terrestrial affairs, though I have
tried to showf that they may just come into comparison with radiometer
action on very small dustparticles.
In the Solar system, however, where they have freer play and vast times
to work in, their effects may mount up into importance. Yet not on the
larger bodies ; for on the earth, assumed to be absorbing, the whole force of
* [See The Pressure of Light, p. 61 (S.P.C.K. 1910), where these values are slightly corrected,
and results are given for a similar experiment with smaller prisms. Ed.]
t Nature, Dec. 29, 1904, p. 200. ^Collected Papers, Art. 65.]
RADIATIONPRESSURE 345
the light of the sun is only about a 50 millionmillionth of his gravitation
pull. But since the ratio of radiationpressure to gravitationpull increases
in the same proportion as the radius diminishes if the density is constant,
the pressure will balance the pull on a spherical absorbing particle of the
density of the earth if its radius is a 50 billionth that of the earth — a little
over a hundredthousandth of a centimetre, say, if its diameter is a hundred
thousandth of an inch.
We may illustrate the possible effects of radiationpressure without
proceeding to such fineness as this. Let us imagine a particle of the density
of the earth, and a thousandth of an inch in diameter, going round the sun
at the earth's distance. There are two effects due to the sun's radiation.
In the first place, the radiationpush is j^^y of the gravitationpull ; and the
result is the same as if the sun's mass were only 99/100 of the value which it
has for larger bodies like the earth. Hence the year for such a particle would
be longer by ^^^, or about 367 instead of 365 J days. In the second place,
the radiation absorbed from the sun and given out again on all sides is crushed
up in front as the particle moves forward and is opened out behind. There
is thus a slightly greater pressure due to its own radiation on the advancing
hemisphere than on the receding one, and this appears as a small resisting
force in the direction of motion. Through this the particle tends to move
in a decreasing orbit spiralling in towards the sun, and at first at the rate of
about 800 miles per annum.
Further, if there be any variation in the sun's rate of emitting energy,
there will be a corresponding variation in the increase of the year and the
decrease of the solar distance, and the particle, if we could only observe it^
would form a perfect actinometer.
Though, unfortunately, we cannot observe the motion of independent
small particles circling round the sun at the distance of the earth, there is
good reason to suppose that some comets at least are mere clouds of dust.
If we are right in this supposition, they should show some of these effects.
Encke's comet at once suggests itself as of this class ; for, as everyone knows,
it shortens its journey of 3 J years round the sun on every successive return,
and on the average by about 2i hours each revolution. Mr. H. C. Plummer*
has lately been investigating this comet's motion; and he finds that if it
were composed of dust particles, each of the earth's density and about J^j mm.
or rather less than a thousandth of an inch in diameter, the resisting force
due to radiationpressure would account for its accelerating return. But the
sun's effective mass would be reduced by about 1/80; and on certain sup
positions he finds that the assumed mean distance as calculated from Kepler's
law, without reference to radiation, is greater than the true mean distance
* Monthly Notices B.A.S. Jan. 1905, 'On the Possible Effects of Radiation on the Motion
of Comets, with special reference to Encke's Comet.'
346 RADIATIONPRESSURE
by something of the order of 1 in 400, and he thinks such a large error is hardly
possible. So that radiationpressure has not yet succeeded in fully explaining
the eccentricities of this comet. But comets are vague creatures. As
Mr. Plummer suggests, we hardly know that we are looking at the same matter
in the comet at its successive returns ; and I still have some hope that the
want of success is due to the uncertainty of the data.
There is one more effect of this radiationpressure which is worthy of
note : its sorting action on dustparticles. If the particles in a dustcloud
circling round the sun are of different sizes or densities, the radiationaccelera
tions on them will differ. The larger particles will be less affected than the
smaller, will travel faster round a given orbit, and will draw more slowly in
towards the sun. Thus a comet of particles of mixed sizes will gradually
be degraded from a compact cloud into a diffused trail lengthening and
broadening, the finer dust on the inner and the coarser on the outer edge.
Let us imagine, as an illustration of this sorting action, that a planet,
while still radiating much energy on its own account, while still in fact a small
sun, has somehow captured and attached to itself as satellite a cometary cloud
of dust. Then, if the cloud consists of particles of different sizes, while all
will tend to draw in to the primary, the larger particles will draw in more
slowly. But if the larger particles are of different sizes among themselves,
they will have different periods of revolution, and will gradually form a ring
all round the planet on the outside. Meanwhile the finer particles will drift
in, and again difference in size will correspond to difference in period and
they too will spread all round, forming an inner fringe to the ring. If there
are several grades of dust with gaps in the scale of size, the different grades
will form different rings in course of time. Is it possible that here we have
the origin of the rings of Saturn ?
The Radiation Theory is only just starting on its journey. Its feet are
not yet clogged by any certain data, and all directions are yet open to it.
Any suggestion for its future course appears to be permissible, and it is only
by trial that we shall find what ways are barred. At least we may be sure
that it deals with real effects and that it must be taken into account.
[Compare Rayleigh, 'On the Momentum and Pressure of Gaseous
Vibrations,' Phil. Mag. vol. 10, 1905, p. 364. Ed.]
23.
ON PKOF. LOWELL'S METHOD FOR EVALUATING THE SURFACE
TEMPERATURES OF THE PLANETS ; WITH AN ATTEMPT
TO REPRESENT THE EFFECT OF DAY AND NIGHT ON THE
TEMPERATURE OF THE EARTH*.
[Phil. Mag. 14, 1907, pp. 749760.]
Prof. Lowell's paper in the July number of the Philosophical Magazine
marks an important advance in the evaluation of planetary temperatures,
inasmuch as he takes into account the effect of planetary atmospheres in
a much more detailed way than any previous writer. But he pays hardly
any attention to the 'blanketing effect,' or, as I prefer to call it, the 'green
houseeffect' of the atmosphere. He assumes in fact that the fourth power
of the temperature is proportional to the fraction of solar radiation reaching
the surface, and he neglects both the surfaceradiation reflected down again
and the radiation downwards of the energy absorbed by the atmosphere.
This is brought out clearly in the footnote on p. 172, where he uses a
formula of Arrhenius, to which I am unable to refer, but wliich I think he
must misinterpret in making it give his result. The inadequacy of his method
is well shown by its application to the cloudcovered half of the earth's surface.
He finds that this half only receives 02 of the radiation which the clear sky
half receives. The surfacetemperature under cloud should therefore be only
V 02 = 067 of that under clear sky. If the latter is 300° A. the former is
only about 200° A. Common observation contradicts this flatly, for the
difference is at most but a few degrees.
On another point common observation appears, at any rate at first sight,
to contradict Professor Lowell. He assumes that the loss in the radiation of
the visible spectrum in its passage through the atmosphere is practically all
due to reflection, and he puts it down as about 07 of the whole in clear
sky. If this were true the reflection from the sky opposite to the sun would
I think be vastly greater than it is. White cardboard reflects diffusely
about 07 of sunlight. But when a piece of white cardboard is exposed
normally to the sun's rays it is several times brighter than the cloudless sky.
* In Phil. Trans. A, vol. 202, p. 525 f I attempted an evaluation, in which the atmosphere
was taken into account as keeping the temperature at a given point practically the same day
and night. I did not then know that Christiansen {Beibldtter zu den Ann. der Physik und Chemie,
vol, 10, 1886, p. 532) had nearly twenty years earlier applied the fourthpower law to calculate
planetary temperatures. His work deserves recognition as the first in which this law was applied.
t [Collected Papers, Art. 20.]
348
The * greenhouse effect' of the atmosphere may perhaps be understood
more easily if we first consider the case of a greenhouse with horizontal roof
of extent so large compared with its height above the ground that the effect
of the edges may be neglected. Let us suppose that it is exposed to a vertical
sun, and that the ground under the glass is ' black ' or a full absorber. We
shall neglect the conduction and convection by the air in the greenhouse.
Let S be the stream of solar radiation incident per sq. cm. per sec. on the
glass. Of this let rS be reflected, aS be absorbed, and tS be transmitted by
the glass. Then r + a+ t = 1. Let the ground send out radiation R per sq.
cm. per sec. and of this let r^R be reflected, a^R be absorbed, and t^R be
transmitted by the glass. Here also rj + a^ + ^^ = L It is to be noted that
since the edges are far distant R is incident on each sq. cm. of glass. The
glass, then, absorbs aS + a^R, and as it is thin it may be taken as having
the same temperature on each side, so that it sends down to the ground
J {aS + a^R), the other half going upwards into space. Equating receipt
and expenditure of radiation by the ground,
R=.tS+riR+ i {aS + a^^R),
whence on putting 7\= 1 — n^ — t^ we obtain
a
i2 = S.
The values of t and a depend upon the glass. By way of illustration
let us take t = 06, a = 03. For radiation from a surface under 100° C.
Melloni found that even thin glass is quite opaque. We have then t^ = 0,
and if we neglect reflection, probably small, Oi == L
Then R = ^S=l'bS.
If the glass were removed we should have
R^S.
The temperature of the ground is therefore \/'iD = 11 times as high
under the glass as it is in the open. If, for instance, it is 27° C. or 300° A.
in the open, it is 330° A. or 57° C. under the glass.
If the glass reflects some of the radiation R then a^ is less and the ground
temperature is still higher.
If the ground, instead of being black, reflects a fraction p of the incident
sunlight, or has total albedo p, the formula must be modified. If we take
into account merely the first reflection from the ground and assume that the
glass has absorption a for it, then we easily find
SURFACETEMPERATURES OF THE PLANETS 349
t +
n^sJl^
If we take p = 01 the numerator is 078 instead of 075, and if we assume
the fourthpower law for the lowtemperature radiation emitted by the
surface, the temperature is about 1 per cent, higher*. But the ground will
probably reflect a much smaller fraction of the whole spectrum, and the
correction for total albedo becomes inconsiderable.
If we replace the sun by cloud the radiation is, on the average, of much
lower temperature, and t and a are much nearer to t^ and a^. The value of
R/S is then much nearer to 1, and the covered ground has a temperature
much ]ess raised above that of the open ground. This agrees of course with
common experience.
A planetary atmosphere no doubt acts in some such way as the green
house glass. Let us, for the sake of comparison with Prof. Lowell's results,
assume, as he has done, that we have a steady state, with the incident radiation
normal to the surface. I do not see how to estimate the distribution of the
radiation from the air between the upward stream into space and the down
ward stream to the surface. Since the lower layers of air are warmer than the
upper probably more than half comes down, and the truth probably lies
between the assumptions that the atmospheric radiation is J {aS + a^R) as
it is with the greenhouse, and that it is aS + a^R when all the radiation would
be downwards. Let us suppose that  (aS + a^R) comes downward.
The albedos of the surfaces of both the Earth and Mars average, according
to Lowell, 01 for visible radiation. They must be much less for the whole
spectrum. Where all the data are uncertain the effect of small albedo may
be neglected, and indeed in our ignorance of the dependence of temperature on
radiation in the case of a partially reflecting surface, it is safer to neglect it. If
dg is the actual surfacetemperature under a vertical sun, and 6 is the tempera
ture which the surface would have without atmosphere, it is easily found that
t\ a/n
^1 ^ {n— l)a^/n
Earth. If we use Lowell's figures for the Earth under a clear sky,
t = 042, a = 05 X 065 = 0325,
t^ = 05, since of the invisible radiation half is transmitted,
a^ = 05, very httle is reflected.
* [It would appear that in the preceding equation for R the term ( 9 ~ P ) ^ should he(^l]pt.
This would give the numerator as 070, and the temperature about 2 per cent, lower. Ed.]
350 PROF. Lowell's method for evaluating the
We shall suppose in succession that
(a) half of the radiation is downwards or that n = 2,
(b) twothirds ^ = f ,
(c) all n=l.
We then find
{a) 1 = 094; (6) J=099; (c) f=M2.
For the case of a cloudcovered earth the data are very uncertain. Lowell
takes t = 02 of 042 = 0084, assuming that the atmosphere has already
reflected and absorbed 058 before the cloud is reached, surely an overestimate,
since the cloudsurface is in the higher air. Let us guess that ^ = 0L The
absorption without cloud is according to Lowell about 03. With cloud
much is reflected back without reaching the lower and more absorbing regions.
Let us guess that a = 02. Of the radiation from the surface we may suppose
perhaps that 02 passes through, that 07 is reflected, and that 01 is absorbed.
Of the 02 passing we may suppose that 01 is absorbed and 01 goes into
space. Then ^^^ = 01 and a^ = 02.
With these values we get for the different values of n
. («) J=l; (b) J =108; (c) 1=131.
These guesses, then, make the temperature under a cloudy sky at least
as great as under a clear sky. But this is certainly not true in common
experience, where, however, we may have clouds accompanied by cold winds
and no approach to the steady state here assumed. The results merely serve
to show that with certain absorptions and transmissions clouds might actually
raise the surfacetemperature, and that for the present it is better to neglect
them.
Mars. If we apply Lowell's data for Mars we have
t = 064, and a = 040 x 065 = 026,
ti = 06, and a^ = 04, since R is dark radiation.
With these values we get for the different values of n
(a) 1 = 099; (b) =]02; (e) =M0.
Comparison of the Earth and Mars. Let us take the temperature of the
Earth as 17° C. or 290° A. If it were removed to the distance of Mars
its temperature would be inversely as the squareroot of the distance, which
is 1524 that of the Earth, or 290/1235 = 235° A.
StlRF ACETEMPERATURES OF THE PLANETS 351
With the different values of n the temperature of Mars should be
(a) 235 X If =247° A. or 26°C.,
(b) 235 X ^2. _ 242° A. or 31°C.,
(c) 235 X j§ = 231° A. or  42° C.
Of course the data are very uncertain and the formula used is only an
approximation. But with these data it is hard to see how the temperature
of Mars can be raised to anything like the value obtained by Professor Lowell.
Perhaps the data are quite wrong. It is conceivable that Mars has a quite
peculiar atmosphere practically opaque to radiations from the cold surface.
Those who believe that there is good evidence for the existence of intelligent
beings on that planet, should find no difficulty in supposing that they have
been sufficiently intelligent to cover the planet with a glass roof or its equiva
lent. Then we might easily have ^+ = 077 and ^^+^^ = 05, and then
the temperature might be raised to 281° A. or 8° C. Indeed, if the glass were
of such kind as to transmit solar radiation, and if it were quite opaque to
dark radiation while still reflecting a considerable proportion, the temperature
might easily be raised far above this.
An Attempt to represent the Effect of Day and Night
on the Temperature of the Earth.
The 'greenhouse' formula, which has been used in the foregoing dis
cussion, would hold only if all the conditions were steady. But in reality
the alternations of day and night prevent a steady state, and we can only
hope that the neglect of these alternations does not greatly affect the ratios
of the temperatures found for different planets or for different elevations on
the same planet.
I shall now attempt to represent the effect of the diurnal variation in the
supply of solar heat to the Earth, or rather to an abstract Earth. For even
if we could represent the actual conditions we should obtain differential
equations so complicated that they would be useless for practical purposes.
To simplify matters, let us suppose that we are dealing with the equatorial
region of the earth at the equinox, that the air is still, that the surface is solid
and black, and that the sky is clear.
The temperature of the air except near the surface can change but little
during 24 hours. For over each square centimetre at sealevel we have
1000 gms. of air with specific heat 02375, and therefore with heat capacity
2375. Consider a band of the atmosphere 1 cm. wide round the equator.
A stream of solar radiation of length equal to the diameter 2r of the earth
352 PROF. Lowell's method for evaluating the
enters a band of air of length equal to half the circumference. If the solar
constant is 3 the average energy entering a sq. cm. column is
2rS 2S 6 , . .
= — =  cal./mm.
Then in 12 hours 1375 cal. enter on the average, and if this heat were all
absorbed and retained it would raise the temperature on the average about
1375/2375 = 5°8 C.
As the absorption is only partial and as radiation takes place from the
air, the rise cannot really average nearly as much as this.
Again, consider the radiation during the twelve hours of night. If the
air ^vere a black body and of temperature 300° A., and these are absurdly
exaggerated estimates of its radiating power and of its average temperature,
it would only radiate about 12 cal./min. per sq. cm. column from its two
surfaces, or 864 calories in the twelve hours, and neglecting the radiation from
the ground the temperature would only fall about 864/2375 or 3°6 C.
Obviously, then, the air as a whole cannot undergo much variation in tempera
ture as day alternates with night. It is indeed a flywheel storing the energy
of many diurnal revolutions. We may, then, in a rough estimate consider
that its temperature, and therefore its radiation, remains constant during the
24 hours.
If the total radiation from a sq. cm. column per second is A, there will
be a stream D downwards and U upwards where Z) + U = A. We can find
an expression for A by equating it to the average absorption. Considering
an equatorial band 1 cm. wide, the average energy entering it per sq. cm. in
the 24 hours is  . Let the average amount absorbed be  . The value of
77 77
a at sealevel varies for clear sky from perhaps 03 with the zenith sun to very
nearly 1 with the setting sun. Let the average radiation from the surface
during the 24 hours be /?, of which a^ R is absorbed by the atmosphere. Then
neglecting conduction through the air, the constanttemperature assumption
gives us
A=^ V a^R.
77
If a fraction is radiated downwards
n _
j^ dS a^R
rnr n
The actual surfacetemperature depends not only on radiation but also
on conduction both by ground and air. But we shall neglect this conduction
and shall suppose that the surface has reached an equilibrium between receipt
and expenditure of radiation. This is a condition to which the surface tends
at or soon after noon by day, and before dawn at night. We shall suppose
SURFACETEMPERATURES OF THE PLANETS 353
that the lowtemperature radiation from the surface is either transmitted
or absorbed, so that, using the previous notation,
^j + ^1 = 1 and r^ = 0.
If R^ is the equilibrium surfaceradiation reached, we suppose about noon,
yiTT n
If Rn is the equilibrium surfaceradiation in the later part of the night, we
have to omit tS, and _
fiTT n
To proceed further, we must express R in terms of S. We can only do
this by some assumption. Probably it is not very far from the truth to
assume that R = i (Ra + Rn), and we shall take this value. It gives us
t a
n
and substituting in the values of day and night radiations we get
t
a
s
a
+ <Li
2 +
mr
= t +
rnr n
1
"^
n
t
d
i>
ai2'^
Rn
a
+
mr
s
rnr
^1
~'
n
Though these formulae are only obtained by making large assumptions,
and by neglecting important considerations, they nevertheless show the
tendency of the day and night effect, and it is worth while to apply them to
the Earth, taking the best data at our command.
At the surface let us take t = 04:2 and aj = 05 as before. For d we have
no trustworthy observations, and I doubt whether a calculation from Langley's
observations is of any more value than an estimate. Since a varies from
perhaps about 03 to 1, let us take a = 0628 or 27r/10, a value simpHf5H[ng
arithmetic.
At the level of Camp Whitney 3550 metres above sealevel, with barometer
about 500 mm., and therefore with about  of the atmosphere below it, we
may take ^ = 06 and a^ = 04. For a we must take a value much smaller
than that at sealevel. Since the most absorbing third of the atmosphere is
below, I do not think it is far wrong to take d as having half the value at the
p. c. w. 23
354 PROF. Lowell's method for evaluating the
lower level, and I therefore put a = 0314. But I have also examined the
consequences of putting it equal to 0419, i.e. f of its value at the lower level,
and the results are given below to show how much the figures are affected by
the variation in the value taken.
We have no data for n. I have therefore calculated the values of R^ and
Rn in terms of S for successive values of n equal to 1, , , f , 2 ; corresponding
to D equal to A, ^A, ^A, %A, and \A respectively.
In the following tables the values of RajS and RJS are given, and also
the mean RjS = J (R^ + Rn) S Then follow the ratios of the day and night
temperatures, 6^ and 6^, to the temperature ^ of a black surface radiating S,
and then the mean value djd. The last column gives the range 6^ — 9^ on
the supposition that = 300° A.
The third table is only given to show that the change in the value of a
does not greatly affect the results. The value of a of Table II is much more
reasonable if that of Table I is near the truth. We need, therefore, only
compare the results given in the first two tables.
If we take the same values of n in each table, the value of 7^ is less at the
higher level than at the lower in every case except that in which n has the
extreme and probably inadmissible value of 2. The value of 6 is less at the
higher level in every case. But it appears most probable that 1/n or DjA
is greater at the lower level than at the higher. For consider a thin layer
of air at sealevel. It is radiating equally up and down, but of the half
going upwards a considerable fraction will be intercepted by the superin
cumbent and strongly absorbing layers. Now consider a thin layer close
to the surface at the higher level. It, too, radiates half up and half down.
But of the half going upwards a less fraction will be intercepted since the
superincumbent layers are now less absorbing. Thus DjA will be greater
at the lower than at the higher level*. We should, therefore, compare the
results for any value of DjA in Table I with the results in Table II for a some
what lower value.
We may exclude the extreme cases of 9^ = 2 and n= 1, as the true value
is certainly between these, and confine our examination to intermediate values.
Suppose, for example, that DjA = 4/5 at the lower level, while it is 3/4
at the upper level. Then djd = 088 from Table I at the lower level, while
O/e = 083 from Table II at the upper level. Or if D/A = 3/4 at the lower
level, while it is 2/3 at the upper level, O/O = 086 below, while ejO  081
above. Or in each case the mean temperature is higher at sealevel by about
5 in 87 or by about 17° in 300°.
* Another consideration leading to the same conclusion is that the atmosphere acts like
a plate with its lower surface much warmer than its upper. When we only have the part above
an elevated region the difiference of temperature between the surfaces is much less than for the
whole air, and the radiations up and down are more nearly equal.
SURFACETEMPERATURES OF THE PLANETS
356
Table I.
At sealevel, t = 042, a^ = 05, a = 06^8.
Range
n
DIA
^d/S
iiJS
R/S
eje
^nl^
ejd
about
300° A.
1
1
103
061
083
101
088
095
41°
5/4
4/5
083
041
062
095
080
088
51°
4/3
3/4
079
037
058
094
078
086
56°
3/2
2/3
072
030
051
092
074
083
65°
2
1/2
062
020
041
089
067
078
85°
Table II.
At 3550 m. above sealevel. Barometer 500 mm.
t = 06, ai = 04, a = 0314.
Range
n
D/A
Rd/S
njs
R/S
Sdie
Kid
dfd
about
300° A.
1
I
097
037
067
099
078
089
71°
5/4
4:15
086
026
056
096
071
084
89°
4/3
3/4
084
024
054
096
070
083
94°
3/2
2/3
080
020
050
095
067
081
100°
2
1/2
074
014
044
093
061
077
125°
Table III.
At 3550 m. above sealevel and with. ^ = 06, ai = 04,
but with a = 0419 = 2/3 of 0628.
Range
n
D/A
RdlS
Rjs
R/S
dale
eje
did
about
300° A.
1
1
102
042
072
101
081
091
66°
5/4
4/5
090
030
060
097
074
086
80°
4/3
3/4
087
027
057
097
072
085
88°
3/2
2/3
083
023
053
095
069
082
95°
2
1/2
076
016
046
093
063
075
120°
23—2
356 METHOD FOR EVALUATING SURFACETEMPERATUBES OF THE PLANETS
It is to be observed that the lower mean temperature at a higher level
must hold good if the higher level is so much higher that there is practically
no atmosphere above. For then t = 1 and a^ = 0, so that R^^ S and i?„ = 0.
Therefore djd = 1 and 6 JO = and 0/6 = 1/2.
The lower mean temperature of elevated parts of the earth's surface is
a wellestablished fact. Perhaps if it were only observed in the case of
mountain peaks it might be ascribed to the cold air blowing against them.
The fall of temperature in free air as we go upwards tends towards that
given by convective equilibrium, though recent observations show that it
is not so great as that given by the adiabatic law. Thus for a rise of 3500
metres the adiabatic law would give a fall of about 32° C. if the sealevel
temperature were 300° A. ; whereas the observations of Teisserenc de Bort
at Trappes show a mean annual fall of about 16° C. for this rise (Encyc. Brit.
vol. 30, Meteorology, p. 695). A continual blast of air thus cooled might of
course reduce the temperature on the mountain peaks, even if radiation did
not tend to any such reduction. But we can hardly account in this way for
the equally wellestablished lower temperature of elevated continental plateaus.
According to Abbe {loc. cit. p. 694) 0°5 C. must be subtracted from sealevel
temperature for every 100 metres general elevation of the landsurface or
about 18° for an elevation of 3500 metres, and this fall may be ascribed to
radiation in some such way as that here set forth.
If the atmosphere of Mars is comparable with our own atmosphere at
high levels, and if the effect is of the same general character in the two cases,
it appears probable that the surfacetemperature of Mars is actually lower
by many degrees than that which the surface of the Earth would have at
the same distance from the Sun.
24.
THE MOMENTUM OF A BEAM OF LIGHT.
[Atti del IV Congresso internazionale dei Matematici
(Rome), 3, 1909, pp. 169174.]
[The substance of this paper is contained in the Address to the French
Physical Society, March 1910. Collected Papers, Art. 70. Ed.]
25.
ON PKESSURE PERPENDICULAR TO THE SHEARPLANES IN
FINITE PURE SHEARS, AND ON THE LENGTHENING OF
LOADED WIRES WHEN TWISTED.
[Roy. Soc, Proc. A, 82, 1909, pp. 546559.]
[Read June 24, 1909.]
In the Philosophical Magazine, vol. 9, 1905, p. 397*, I gave an analysis of
the stresses in a pure shear which appeared to show that if e is the angle of
shear and if n is the rigidity, then a pressure ne^ exists perpendicular to the
planes of shear. That analysis is, I believe, faulty in that the diagonals
of the rhombus into which a square is sheared are not the lines of greatest
elongation and contraction, and are not at right angles after the shear, when
secondorder quantities are taken into account, i.e., quantities of the order
of 6^ ; I think the following analysis is more correct, and though it does not
give a definite result, it leaves the existence of a longitudinal pressure an
open question. The question appears to be answered in the affirmative by
some experiments, described in the second part of the paper, in which loaded
wires when twisted were found to lengthen by a small amount proportional
to the square of the twist.
I. Stresses in a Pure Shear.
Let a square ABCD (Fig. 1) of side a be sheared into EFCD by motion
through AE = d, the volume being constant. The angle of shear is ADE = e,
and tan e = dja exactly ; neglecting e^, we may put e = dja.
To find which line is stretched most by the shear, consider the Kne r
drawn from B to P and making an angle 6 with DC before stretching.
Let it stretch to p, making an angle 6' with DC ; we have r = a /sin 6 and
p = a/sin 6' ;
also p2 ^ ^2 __ 2rd cos 6 { d^;
thus /)2/r2 = 1 + 2dlr . cos ^ h d^/r^
= 14 2dla . sin ^ cos ^ 1 d^/a^ . sin^ 6.
* [Collected Papers, Art. 22, p. 338.]
ON PRESSURE PERPENDICULAR, TO THE SHEARPLANES
359
Differentiating p^jr^ with respect to 6, it is a maximum when
Mja . cos 2d + d^la^ . sin 26 = 0,
or tan 29 = — 2a/d = — 2 cot e.
Put 6 = 45° + 8, then tan 28 = J tan e,
or 8 = Je to the second order, so that r makes an angle J e with the diagonal
DB of the square through D, and on the upper side.
If the same shear is now made in the opposite direction p contracts to r,
and the same directions of p before, and r after shear, give the maximum
contraction. It is almost obvious that p makes an angle Je with DB on the
lower side, but it may be verified by putting
r^ = p^ 2pd cos d' + d^,
and finding the maximum value of r^/p^ after putting p = a/ sin 6' on the
right.
J) C A J2a. ^
Fig. 1.
Hence the lines of maximum elongation and contraction are at Je with
the diagonals of the square, and are at right angles before and after the strain,
to the order of e^. It is noteworthy that as the shear increases the fibres
which undergo maximum elongation and contraction change.
To find r and p put
r = a/sin 6 = a/sin (45° + \e),
then r  y/2a (1  Jc + A^') ;
and changing the sign of e we get
p = V2a (1 + ie + i^e^).
It is easily seen that the elongation and contraction are respectively
e=(p r)lr = ^c (1 + Jc) ; c = {r  p)/p = Jc (1  Je).
We shall now consider the stresses. We shall assume that a pressure P
is put on in the direction of maximum contraction and a tension Q in the
direction of maximum elongation, these being, as we have seen, at right
angles ; and we shall consider the equilibrium of the wedge ABC (Fig. 2) when
360 ON PRESSURE PERPENDICULAR TO THE SHEARPLANES
sheared, assuming that P and Q are the only forces on AC and BC, Let
AB = 2a ; AC = p\ BC = r; these having the values just found.
Kesolve in a direction perpendicular to the base, and let R be the pressure
against the base, then
R.2a = Pp cos (45°  Je)  Qr cos (45° + Je)
= Pa cot (45° i€)Qa cot (45° + Je)
= Pa (1 + Je + le2)  Qa (1  Jc + Je^)
= (P  g) a + (P + Q) . ia€ + (P  g) . lac^,
where P and Q can only be taken as equal to the first order. Proceeding to
the second order, we must put
P ^ 7l€ + fe^ ;
then Q = ne  pe^,
where p is a constant to the second order.
Thus PQ = 2pe^ and P + Q = 2ne,
and P = (Jn + p) e^,
the third term being negligible. If we resolve parallel to the base, it is easily
found that the tangential stress is
T = 1 (P + Q) = nc.
If the shear is produced by a tangential stress T, then it requires the
system P, Q, and R to maintain equilibrium with it.
It is possible that a stress exists perpendicular to the plane of the figure
in Fig. 1. It can only be assumed that the changes of dimension in that
direction neutralise each other to the first order when equal pushes and
pulls are put on in the plane of the figure ; when the dimensions perpendicular
to the figure are constrained to remain the same to the second order — and
this is our supposition — it may require a tension or pressure to effect this.
Let us suppose that a pressure S = qe^ is introduced, a tension if q is
negative. To make P = we should require to have f = — ^w, also P would
then be less than Q. If pressure perpendicular to AC is exerted alone, and
then tension perpendicular to BC is exerted alone, it appears probable that
for very large equal compressions and extensions P is greater than Q. If we
suppose that when they are simultaneous the tendency is in the same direction,
then R should have a positive value, or the longitudinal pressure perpendicular
to AB should exist.
Let us examine the consequences of the supposition that both R and S
exist. Let a thin tube of length I and of radius a be fixed at one end, and let
IN FINITE PURE SHEARS, ETC. 361
the other end be twisted through an angle 6 so that the angle of shear is
€ = ad/l. Let an endpressure R = (^n + p) e^ be put on, and also a side
pressure S = qe^ so as to maintain constant dimensions.
The sidepressure S may be replaced by a uniform pressure S over the
whole surface, and a tension S over the ends. We have then an endpressure
R — S and a pressure S all over.
Now suppose that these forces are removed. Through the removal of
R — S we shall have a lengthening dl^ given by
(in + p q) £2 = Ydljh
and a contraction S^ of the diameter given by
Sj2a = adljl,
where Y is Young's modulus and o is Poisson's ratio.
Through the removal of the pressure S we shall have a lengthening dlz
given by qe^ = ^Kdljl, where K is the bulkmodulus, and an expansion §2
of the diameter given by qe^ = ZKh.J2a.
The endlengthening is therefore
dl = dl^ + dl^ = {{in + pq)IY+ q/SK} k^ ;
or putting 1/3Z = 3/7  1/n,
dl = {(Jn + p)IY+ (2/F  l/n) q} k^ = sU^ = sa^d%
where s is put for (Jn + p)/7 + {2/Y — l/n) 5'.
The diameter decreases by
S = Sj _ §2 = {{in +pq)alY q/SK} 2a€^
= {{in + p)cjIY [(3 + c7)/r  l/n] q} 2a^6^ll\
It would not be easy to test this result with a thin tube. But if we suppose
that a wire extends by the amount equal to the average extension of the tubes
into which it may be resolved, we get
dl=—J ^"""^dr = isa^e%
I now proceed to describe some experiments which show that such an
extension exists.
II. The Lengthening of Loaded Wires when twisted.
Experiments were made on several wires hung vertically from a fixed
support, and loaded in order that kinks or remnants of the spiral due to the
coihng to which they had been subjected might be taken out. This was
considered to be effected when the stretch for a given addition of load was
362
ON PRESSURE PERPENDICULAR TO THE SHEARPLANES
sensibly the same whether the wire was twisted or not. An account of the
twisting of a steel wire before this stage was reached will be given later.
Fig. 3 represents the arrangement more or less diagrammatically. The upper
end of the wire to be twisted was fixed to a
stout bracket B near the ceiUng. The wire
was always about 231 cm. long ; its lower end
was clamped in jaws in the upper end of a
turned steel rod rr, 51 cm. long, which passed
through a hole in the table T on which was
the observing microscope M, and a parallel
plate micrometer m. One division of this
micrometer was equal to 000974 mm. At
the lower end of the rod was a horizontal
iron crosspiece cc, 19 cm. long and 16 cm.
square. From the lower end was suspended
a carrier for the weights, or for the two stouter
wires the weight itself, connected to the rod
by a flat steel strip twisted in its middle, so
that the upper and lower halves were in two
vertical planes at right angles. Below the
weights was a set of vanes immersed in a
shallow bath of oil to damp vibrations. This
bath rested on a circular turntable tt, on
which were two uprights ii, ii at opposite ends
of a diameter, with horizontal screws at their
upper ends which could be brought to bear
against the ends of the crosspiece as shown
in the plan, Fig. 4. The screws ended in small
steel balls and the sides of the crosspiece were
polished. On rotating the turntable the screws
came against the crosspiece and turned it
round ; and so the wire was twisted by a couple
with vertical axis. The axis of the turntable
was made vertical by means of the levelling
screws I, I. To adjust this axis in the axis of
the wire prolonged, the turntable could be
moved over the baseplate by means of the
horizontal screws s, of which only one is
represented in Fig. 3. All are shown in Fig. 4.
A horizontal microscope, not represented in the figure, was attached to one
of the uprights and focussed on the edge of the rod rr. The adjustment by
the screws s was continued until the microscope always saw the edge of the
rod in the middle of the field, however the turntable might be turned.
Fig. 3. Elevation of Arrange
ment for Twisting the Wires.
IN FINITE PUEE SHEARS, ETC.
363
To give a definite point of view in the microscope M, in the earlier experi
ments starchgrains were put on the wire about 1 cm. from the lower end.
These were illuminated, and a suitable one was selected.
In the later experiments a needle, about 1 cm. long, was fixed, point
upwards, on the upper end of the rod close alongside the wire, and the needle
point was viewed. This was better than the starchgrains.
In the earlier work the temperature of the room was fairly steady, and
the changes in length due to temperature variations were too slow to give
trouble. But in some gusty weather occurring later there were such rapid
and considerable variations in the temperature of the room that it was
necessary to enclose the wire in a wooden tube. After this was done tempera
ture gave no further trouble, whatever the weather.
In order to observe the effect of a twist the
turntable was levelled and adjusted axially when
the wire and crosspiece were free. The turntable
was rotated till the screws on the uprights just
touched the crosspiece. Then chalkmarks were
made on the turntable and on the plate below, one
just over the other. The microscope was adjusted
exactly to sight the upper or lower edge of a starch
grain on its horizontal cross wire, and the micrometer
was read. Then the turntable was rotated so many
whole turns, and the micrometerplate was moved till the edge of the grain
was again on the crosswire and the micrometer was read again.
Except in the case of a wire stretched only by the weight of the rod and
crosspiece, in some experiments described later, there was always a lengthening
on twisting, of the same order whether the twist was clockwise or counter
clockwise. The lengthening was nearly proportional to the square of the
twist put on. It was necessary to limit the twist to a few turns to avoid
permanent set, and when such a small twist had been given and the wire
was untwisted it returned sensibly to its original length.
The lowering was entirely due to twisting and not to any giving of the
support, for when a microscope was sighted on a point on the wire close to
the upper end, no change in level could be detected, when the wire was
twisted through 5. turns at its lower end. This was further verified by an
experiment on a steel wire from the same piece as No. 3 below, which
showed that the extension halfway down the wire was, within the limits of
experimental error, half that at the lower end. A microscope and micro
meter were fixed on a table halfway up the wire, and a needlepoint was
fixed here as well as at the lower end. At each twist and untwist both
micrometers were read. I give the observations in this experiment in full,
as they will show the sort of accuracy attained.
Fig. 4. Plan of the Cross
piece and Turntable.
364 ON PRESSURE PERPENDICULAR TO THE SHEARPIANES
The lower end was twisted from a starting twist of J turn to 4 J turns.
MicrometerReadings at Lower End.
J turn
4J turns
Lowering
223
186
37
225
190
35
230
192
38
226
194
32
229
196
33
226
195
31
230
195
35
230
189
41
227
192
35
229
196
33
Mean lowering, 350 divisions.
One division of micrometer = 000974 mm.
The lowering is 00341 mm.
MicrometerReadings Halfway up the Wire.
^ turn
4J turns
Lowering
304
283
21
305
284
21
302
28 1
21
305
285
20
305
280
25
304
287
17
310
284
26
306
285
21
318*
299
19
316*
29lt
25
Mean lowering, 216 divisions.
One division of micrometer = 000751 mm.
The lowering is 00162 mm.
If the lowering at the end is accurate, that halfway up should be
00171 mm. The observed lowering is as nearly equal to this as could be
expected.
With the first wire, determinations of extension due to an addition of
520 grammes were made both in the untwisted and twisted conditions, as
it was only when these became sensibly equal that the lowering on twist
became equal for different loads. The extra load could be put on or taken
ofi by lowering or raising a lever, not represented in Fig. 3. It is unnecessary
* Another point on the needle sighted,
■f [This number has been corrected from 299 in accordance with the original MS.
other small shps in the results which follow have been similarly corrected. Ed.]
A few
IN FINITE PURE SHEARS, ETC. 365
to describe the details of this arrangement. The experiments with the other
wires were made with such loads that it was not considered necessary to
observe the stretch due to addition of load.
Results.
la. Steel piano wire, diameter 0720 mm. (mean of 10 measurements
at different points), length to observing point in this and all cases 230 cm.
Permanent set, after putting on eight turns twist and then untwisting, only
a very few degrees. Total load, 7081 grammes.
The twist is termed clockwise when the turntable as viewed from above is
moved clockwise.
Clockwise twist, 04 turns ; lowering 00181 mm., mean of 10 observations.
„ 08 „ „ 00732
The ratio of these is 404 : 1.
The extension due to an addition of 520 grammes was :
No twist on the wire 0143 mm., mean of 10 observations.
4 turns „ 0141 „ „ „
8 ,, ,, u14o „ ,, ,,
lb. Same wire.
Total load, 9081 grammes.
Clockwise twist, 04 turns; lowering 00180 mm., mean of 20 observations.
„ 08 „ „ 00749
The ratio of these is 415 : 1.
The extension due to an addition of 520 grammes was :
No twist on the wire 0142 mm., mean of 10 observations.
8 turns „ 0144 „ „ „
Taking the mean lowering for the two loads of 7081 and 9081 granmies
for eight turns twist, viz., 0074 mm., and taking it as proportional to the
square of the twist, the lowering for one turn is 000116 mm., and
s = 2ldllaW^ = 1043.
The moduli of elasticity of this wire were found to be
n = 0769 X 1012, y _ 2013 x 10^2,
whence n/Y = 0382. The value of n, found for loads of 1081 grammes and
9081 grammes respectively, was identical.
2. The same wire was raised to a red heat, by an electric current, with
the load of 9081 grammes on it. It lengthened about 3 cm., and this length
was cut off. The surface oxidised, and when the oxide was rubbed off the
diameter was 0696 mm. (mean of 10 measurements).
366 ON PRESSURE PERPENDICULAR TO THE SHEARPLANES
The permanent set after twisting and untwisting was greater, and so only
three turns were given.
Total load, 9081 grammes.
Clockwise twist, 03 turns ; lowering 00129 mm., mean of five observations.
The extension due to an addition of 520 grammes was :
No twist on the wire 0155 mm., mean of 10 observations.
3 turns „ 0154 „ „ „
The lowering for one turn according to the squarelaw is : 000143 mm.,
whence 5 = 1376 mm.
The five values of the lowering were : 15, 12, 12, 12, 15 divisions, mean
132 divisions. With such small lowering no accuracy could be expected,
and it would be difficult to verify the squarelaw.
The moduli of elasticity for the softened wire were :
n = 0809 X 1012 and Y = 206 x IO12,
whence n/ 7 = 0393.
3. Steel piano wire, diameter 0970 mm. (mean of 10 measurements).
A needlepoint fixed at the side of the wire was viewed in the microscope.
After twisting and untwisting, a slight permanent set threw the point out of
focus if the start was from no twist. A quarter turn was therefore put on
initially, and the twisting was from this, and the untwisting was back to it.
Total load, 19,504 grammes.
Clockwise twist, J2J turns; lowering 00088 mm., mean of 10 observations.
55 55 4~^4 '5 " 00343 ,, ,, ,,
Counterclockwise twist, ^2^ turns; lowering 00090 mm., mean of 10 obs.
141 00844
Mean lowering, J2J turns, 00089 mm.
141 turns, 00344 „
By the squarelaw the lowerings for 4, 2J, and J should be as 289 : 81 : 1,
and the difference should be as 288 : 80 = 18 : 5.
The observed differences are as 194 : 5.
The lowering for one turn deduced from the difference between J and 4J is
000191 mm., whence s = 0946.
Comparing the lowerings for one turn of this wire with the hard wire
No. 1, if the lowering is proportional to the square of the diameter, we ought
to have for No. 1 a lowering of 000191 x (72/97)2 = 000105 mm. The
observed lowering was 000116 mm., which is as near the calculated value as
could be expected.
EST FINITE PURE SHEARS, ETC. 367
4. The same wire was then raised to a red heat by an electric current with
the load on. After being rubbed down its diameter was 0947 mm. (mean of
10 measurements). Same load as in experiment 3.
Clockwise twist, J3J turns, lowering 00207 mm., mean of 10 observations.
The deduced lowering for one turn is 000197 mm.
The value of s is 1025 mm.
Comparing the lowerings for one turn of this wire with the softened wire
No. 2, the squarelaw for the diameter should give for No. 2 a lowering
000197 X (696/947)2 = 000106 mm. The observed lowering was 000143 mm.,
a considerable divergence.
6. Copper wire, diameter 0655 mm. (mean of 10 measurements). Load,
7081 grammes.
Clockwise twist, J2J turns; lowering 00066 mm., mean of 10 observations.
Counterclockwise twist, ^2^ turns ; lowering 00083 mm., mean of 10 obs.
It was not safe to give a greater twist owing to the largeness of the
permanent set. With 2^ turns the set was still small.
The larger value of the lowering for the counterclockwise twist is almost
certainly real, and not merely error of observation. Some other observations
showed an even greater excess, though they were very irregular owing to
temperature variations, and are not worth recording. The extension due to
an addition of 520 grammes was :
No twist on the wire 0268 mm., mean of 20 observations.
3 turns „ 0269 ,, „ ,,
Taking the mean for clockwise and counterclockwise twist, the lowering
for one turn is 000149 mm., and s = 162 mm.
6. Brass wire, diameter 0928 mm. (mean of 10 measurements). Load
19,504 grammes.
Clockwise twist, i2J turns; lowering 00169 mm., mean of 10 observations.
141 00540
Counterclockwise twist, j2J turns; lowering 00135 mm., mean of 10 obs.
„ i41 „ „ 00479
The difference between clockwise and counterclockwise twisting is too
large for errors of observation.
For the squarelaw the lowerings for J4J and for J2 J turns should be in the
ratio 18 : 5. They are in the ratios 16 : 5 for clockwise, and 177 : 5 for counter
clockwise twisting. The lowering for one turn clockwise, as deduced from
i4J turns, is 000300 mm., and for one turn counterclockwise is 000265 mm.
The mean value of s = 1537 mm.
368 ON PRESSURE PERPENDICULAR TO THE SHEARPLANES
Experiments with Smaller Loads.
When the piano wire diameter 072 mm. was loaded only with the rod and
crossbar weighing 1081 grammes, there was a rise on twisting.
Clockwise twist, 04 turns, rise 0041 mm.
08 „ 0139 „
Counterclockwise twist, 04 „ 0023 „
„ 08 „ 0108 „
The extension for an addition of 520 grammes was :
No twist on wire 0137
mm..
mean
of 6 observations.
4 turns clockwise 0170
jj
55
6
55
8 „ „ 0237
?j
55
3
55
4 turns counterclockwise 0156
5J
55
6
55
8 „ .. ,. 0238
55
55
3
55
If by means of the observed extensions we calculate the positions of the
point viewed, when the load of 1081 grammes is taken off, we find that the
total rise for clockwise twist would be: for four turns 0110 mm., and for
eight turns 0347 mm.
The rise appears to be due to coiling up of the wire on twisting, through
some remnant of the spiral condition in which it existed before suspension.
This is confirmed by the very large increase in extension, due to addition of
load as the twist on the wire is increased. It may be a coincidence that the
rise on twisting and the increase of stretch are both nearly proportional to
the square of the number of turns.
Experiments were then made with greater loads to find how the lowering
and extension changed. Only clockwise twist was observed.
Load 3081 grammes, the rise changed to lowering.
Twist, 04 turns; lowering 00131 mm., mean of 20 observations.
55 08 „ „ 00498
The extension due to an addition of 520 grammes was :
No twist on the wire 0144 mm., mean of 10 observations.
4 turns „ 0143 „ „ „
^ 55 55 Uliy ,, ,, ,,
Showing a still slight excess of extension in the most twisted condition.
Load 5081 grammes.
Twist, 04 turns; lowering 00164 mm., mean of 20 observations.
„ 08 „ „ 00660
m FINITE PURE SHEARS, ETC. 369
The extension due to an addition of 520 grammes was :
No twist on the wire 0141 mm., mean of 10 observations.
4 turns „ 0142 „ „ 15
8 „ „ 0144 „ „ 15
The results for loads of 7081 and 9081 grammes are already recorded
under 1 a and 1 h. There is obviously a tendency for the lowering to increase
with load until the extensions under different twists become more nearly
equal with equal added load.
When the same wire was softened and loaded with 3081 grammes the
lowering for three turns was 00093 mm. (mean of 10 observations).
The extension due to an addition of 520 grammes was :
No twist on the wire 0149 mm., mean of 10 observations.
3 turns „ 0147
With load 9081 grammes the same wire gave the results recorded under 2,
which show a greater lowering for an equal twist but the same extension with
added load. "
The copper wire diameter 0655 mm. (No. 5 above) with load 4081 grammes
gave:
Clockwise twist, 03 turns, 000965 mm., mean of 10 observations.
Counterclockwise twist, 03 turns, 00156 mm., mean of 10 observations.
Taking the mean of these, the lowering for one turn is 00014 mm.
The extension due to an addition of 520 grammes was :
No twist on the wire 0268 mm., mean of 5 observations;
3 turns „ 0270
extensions agreeing very nearly with those recorded above for a load of
7081 grammes on the same wire.
Remarks on the Results of Measurements.
The lowering was never so much as 01 mm. and was usually much less.
The accuracy attained could hardly be expected to be great. The measure
ments, however, appear to show that when a wire is sufficiently loaded to be
straightened, it is lengthened by twisting by an amount proportional to the
square of the twist and, with a given number of turns, inversely as the length.
It might be thought possible that the effect observed was due to rise of
temperature, either through adiabatic strain or through dissipation of strain
energy as heat. But the observations give no support to this explanation.
When the wire was extended by twisting, it remained extended, and when
untwisted it returned. Temperatureeffects would be a maximum the instant
after twisting, and would then gradually subside. It may be noted that
p. c.w. 24
370 ON PRESSURE PERPENDICULAR TO THE SHEARPLANES
the adiabatic change of temperature is proportional to a^B^jl, but it is a
cooling, and its amount is such as to shorten the wire, in the case of steel, by
something of the order of 1/100 of the observed extension. If we suppose
that some definite fraction of the strainenergy put in is dissipated, again the
change, now a warming, is proportional to a^O^jl. The whole strainenergy,
in the case of steel, would only raise the temperature by an amount accounting
for something of the order of 1/10 the observed extension, and, in fact, only
an exceedingly minute fraction of the strainenergy is dissipated.
A comparison of the wires (1) and (3) appears to show that the lengthening
for a given twist is proportional to the square of the radius.
If we put the lengthening
s for steel is in the neighbourhood of 1. For copper and brass s is in the
neighbourhood of 15. The lowering for the copper and brass wires tested for
twists in opposite directions is not the same.
With a hard steel wire with small load the end of the wire rises on twisting,
probably through coiling.
The value of s = (\n + p)/Y — (21 Y — Ijn) q appears to be measurable,
but its value gives us no clue to the values of f and q.
If we could assume q = 0, then for steel we should have f about 2n, but
I see no justification for the assumption.
If we could measure the decrease in diameter, we should obtain the value
of (\n + f — q) (jjY — {(?> + g)IY — Ijn) q, and knowing n, Y and a we should
be able to find p and q. But a thin wire is quite unsuitable for this
measurement. The decrease is probably of the order of 2a/? x lengthening.
With the wires I have used this is of the order 1/1000 x lengthening, and an
accuracy of measurement of 10"^ mm. would be required at least. With a
shaft of considerable diameter it might be possible to measure the quantity,
though the experimental difficulties are obviously very great*.
The Effect of the Lengthening of a Wire on its Torsional Vibration.
If a wire is loaded with mass M having moment of inertia /, when M is
set vibrating torsionally it falls and rises as it swings, its distance below the
highest point being given by
X = isaW/l.
The kinetic energy is T = ^16^ + iMx\
The last term is easily found to be negligible.
The potential energy is
F = Inira^e'^ll  Mgx = Inira'^d^ll  iMgsaW%
* [For the experimental carrying out of these measurements see Collected Papers, Art. 30. Ed. ]
IN FINITE PURE SHEARS, ETC. 371
The equation of motion is
• 10 + (JwTraV^  Mgsa^/l) d = 0.
Whence T^  ^^^^^
~n7Ta^{l2Mgsln7Ta^)'
and T is greater than it would be if s were 0, by the factor
1 + Mgslmra^.
If Y is Young's modulus and if e is the elongation of the wire due to the
load Mg,
Mg/ira^ = Ye,
so that the factor may be conveniently written as 1 + seY/n.
If the vibrations are used to determine the modulus of rigidity n, then the
value of n will be greater than that deduced by neglect of s, and by the factor
1 + 2seY/n,
To give an idea of the effect on the determination of the modulus of
rigidity, let us suppose that a quite straight steel wire, diameter 07 mm., has
a load of 2000 grammes. For steel Y/n is about 26. For the given diameter
e is about 2 x 10"*. We have found that s is about 1. The correcting
factor is then about 1001, or the true rigidity exceeds the value calculated
in the ordinary way by about 1 in 1000. If the wire is not sufficiently loaded
to be straight the value of s is less. If very lightly loaded the sign of s may
be changed and the true rigidity may be less than the value as ordinarily
calculated. The correction is hardly needful in practice, as the modulus of
rigidity is probably not measurable to three figures.
Distortional Waves.
In purely distortional waves in a medium of great extent it is evident
that the pressure S perpendicular to the axes of shear, if it exists, will not
produce any motion. To keep the waves purely distortional, i.e. with
motion perpendicular to the direction of propagation only, a force must
be applied from outside dRjdx per cubic centimetre in the direction of
propagation. If this force is not applied then longitudinal motion must
result, obviously of the second order, unless Jn + p = 0. This is probably
the condition for an incompressible medium. If \n + p is not zero it
appears possible that dispersion may exist. If the longitudinal motion is
neglected the pressure in the direction of propagation is (n + f) e^ and all
that we can say, at present, is that it is probably of the order of ne^.
24—2
2t).
THE WAVEMOTION OF A EEVOLVING SHAFT, AND A SUG
GESTION AS TO THE ANGULAR MOMENTUM IN A BEAM
OF CmCULAELY POLARISED LIGHT.
[Roy. Soc. Proc. A, 82, 1909, pp. 560567.]
[Read June 24, 1909.]
When a shaft of circular section is revolving uniformly, and is transmitting
power uniformly, a row of particles originally in a line parallel to the axis
will lie in a spiral of constant pitch, and the position of the shaft at any
instant may be described by the position of this spiral.
Let us suppose that the power is transmitted from left to right, and that
as viewed from the left the revolution is clockwise. Then the spiral is a
lefthanded screw. Let it be on the surface, and there make an angle e with
the axis. Let the radius of the shaft be a, and let one turn of the spiral have
length A along the axis. We may term A the wavelength of the spiral.
We have tan e = IrrajX. If the orientation of the section at the origin at
time t is given by ^ = 27rNt, where N is the number of revolutions per second,
the orientation of the section at x is given by
e = 'lirNt   tan e = ^ (iVA^  x), (1)
a A
which means movement of orientation from left to right with velocity iVA.
The equation of motion for twistwaves on a shaft of circular section is
^^^U^^ (2)
where JJ^^ = modulus of rigidity/density = njp.
Though (1) satisfies (2), it can hardly be termed a solution, for d^6/dt^, and
dW/dx^ in (2) are both zero. But we may adapt a solution of (2) to fit (1) if
we assume certain conditions in (1).
The periodic value
e = esm^{Ujx)
satisfies (2), and is a wavemotion with velocity Un and wavelength I. Make
WAVEMOTION OF A REVOLVING SHAFT 373
t SO great that for any time or for any distance under observation Unt/l and
x/l are so small that the angle may be put for the sine. Then
e=9^{Ujx) (3)
This is uniform rotation. It means that we only deal with the part of
the wave near a node, and that we make the wavelength I so great that
for a long distance the 'displacementcurve' obtained by plotting 6 against
t coincides with the tangent at the node. We must distinguish, of course,
between the wavelength I of the periodic motion and the wavelength A of
the spiral.
We can only make (1) coincide with (3) by putting
0/?=l/A and NX= C/„.
Then it follows that for a given value of N, the impressed speed of uniform
rotation, there is only one value of A or one value of € for which the motion
may be regarded as part of a natural wavesystem, transmitted by the elastic
forces of the material with velocity = \/(nlp). There is therefore only one
'natural' rate of transmission of energy.
The value of e is given by
tan e = 27Ta/X = 27TaNINX = 27raiV/t/'„ = 27TaN^{pln).
Suppose, for instance, that a steel shaft with radius a = 2 cm., density
/) = 78, and rigidity n = 10^^ ig making iV = 10 revs, per sec. We may put
tan € = €, since it is very small. The shaft is twisted through 2?? in length
A or through 27r/A per centimetre, and the torque across a section is
smce
A
U^ _ 1 In
The energy transmitted per second is
27rNG ^ 27T^a^NWM
Putting 1 H.p. = 746 x 10^ ergs per second, this gives about 38 h.p.
But a shaft revolving with given speed A^ can transmit any power, subject
to the limitation that the strain is not too great for the material. When the
power is not that 'naturally' transmitted, we must regard the waves as
'forced.' The velocity of transmission is no longer j[7„, and forces will have
to be applied from outside in addition to the internal elastic forces to give
the new velocity.
Let H be the couple applied per unit length from outside. Then the
equation of motion becomes
^ _ p 2 ^ ^ 2^
dt^ ' "" dx^ TraV
374 WAVEMOTION OF A REVOLVING SHAFT, AND A SUGGESTION AS TO
where ^Tra* is the moment of inertia of the crosssection. Assuming that the
condition travels on with velocity TJ unchanged in form,
or H has only to be applied where dW/dx^ has value, that is where the twist
is changing.
The following adaptation of Rankine's tubemethod of obtaining wave
velocities* gives these results in a more direct manner. Suppose that the
shaft is indefinitely extended both ways. Any twist disturbance may be
propagated unchanged in form with any velocity we choose to assign, if we
apply from outside the distribution of torque which, added to the torque due
to strain, will make the change in twist required by the given wavemotion
travelling at the assigned speed.
Let the velocity of propagation be U from left to right, and let the dis
placement at any section be 6, positive if clockwise when seen from the left.
The twist per unit length is
dd_ I d_d^ I
dx" Udt~' U'
The torque across a section from left to right in clockwise direction is
1 ^ dd mra'^ X
Let the shaft be moved from right to left with velocity U ; then the dis
turbance is fixed in space, and if we imagine two fixed planes drawn
perpendicular to the axis, one, ^, at a point where the disturbance is 6 and
the other, B, outside the wavesystem, where there is no disturbance, the con
dition between A and B remains constant, except that the matter undergoing
that condition is changing. Hence the total angular momentum between A
and B is constant. But no angular momentum enters at B, since the shaft
is there untwisted and has merely linear motion. At A, then, there must be
on the whole no transfer of angular momentum from right to left. Now,
angular momentum is transferred in three ways :
1. By the carriage by rotating matter. The angular momentum per unit
length is ^pna'^O, and since length V per second passes out at A, it carries
out IpTra^dU.
2. By the torque exerted by matter on the right of A on matter on the
left of A. This takes out  nTra^d/^U.
3. By the stream of angular momentum by which we may represent the
forces applied from outside to make the velocity U instead of Un .
* Phil. Trans. 1870, p 277.
ANGULAR MOMENTUM IN A BEAM OF CIRCULARLY POLARISED LIGHT 375
If H is the couple applied per unit length, we may regard it as due to the
flow of angular momentum L along the shaft from left to right, such that
H = — dL/dx. There is then angular momentum L flowing out per second
from right to left. Since the total flow due to (1), (2), and (3) is zero,
ifma*dU  n7ra^e/2U  L = 0,
and X = '?(.^^) = '^(^"^„^)=^(^.A
li H = 0, either U^ = C7„2 when the velocity has its 'natural value,' or
d^d/dx^ = 0, and the shaft is revolving with uniform twist in the part con
sidered.
Now put on to the system a velocity U from left to right. The motion
of the shaft parallel to its axis is reduced to zero, and the disturbance and
the system H will travel on from left to right with velocity U. A 'forced'
velocity does not imply transfer of physical conditions by the material with
that velocity. We can only regard the conditions as reproduced at successive
points by the aid of external forces. We may illustrate this point by con
sidering the incidence of a wave against a surface. If the angle of incidence
is i and the velocity of the wave is F, the line of contact moves over the
surface with velocity v = 7/sin i, which may have any value from F to
infinity. The velocity v is not that of transmission by the material of the
surface, but merely the velocity of a condition impressed on the surface from
outside.
Probably in all cases of transmission with forced velocity, and certainly
in the case here considered, the velocity depends upon the wavelength, and
there is dispersion.
With a shaft revolving N times per second U = NX, and it is interesting
to note that the group velocity, U — XdU/dX, is zero. It is not at once evident
what the groupvelocity signifies in the case of uniform rotation. In ordinary
cases it is the velocity of travel of the 'beat' pattern, formed by two trains
of slightly different frequencies. The complete 'beat' pattern is contained
between two successive points of agreement of phase of the two trains. In
our case of superposition of two strainspirals with constant speed of rotation,
points of agreement of phase are pomts oi intersection of the two spirals.
At such points the phases are the same, or one has gained on the other by
27T. Evidently as the shaft revolves these points remain in the same cross
section, and the group velocity is zero.
With deepwater waves the group velocity is half the wave velocity, and
the energyflow is half that required for the onward march of the waves*.
* 0. Reynolds, Nature, August 23, 1877 ; Lord Rayleigh, Theory of Sound, vol. 1, p. 477.
376 WAVEMOTION OF A REVOLVING SHAFT, AND A SUGGESTION AS TO
The energyflow thus suffices for the onward march of the group, and the
case suggests a simple relation between energyflow and group velocity.
But the simpHcity is special to unforced trains of waves. Obviously,
it does not hold when there are auxiliary working forces adding or subtracting
energy along the waves. For the revolving shaft the simple relation would
give us no energyflow, whereas the strain existing in the shaft implies trans
mission of energy at a rate given as follows.
The twist per unit length is dd/dx, and therefore the torque across a
section is — ^rnra^ dd/dx, or ^nTra^d/U, since dO/dx = — 6/U. The rate of
working or of energyflow across the section is ^mra^d^lU.
The relation of this to the strain and kinetic energy in the shaft is easily
found. The strain energy per unit length being J (couple x twist per unit
length) is ^rnra^ (dO/dx)'^, which is ^mra'^b^lU^. The kinetic energy per unit
length is Ipna^O^, or, putting p = n/Un^, is l^iTra^d^/UJ.
In the case of natural velocity, for which no working forces along the
shaft are needed, when U = Un = s/in/p), the kinetic energy is equal to the
strain energy at every point and the energy transmitted across a section per
second is that contained in length [/"„ .
But if the velocity is forced this is no longer true*, and it is easily shown
2U
that the energy transferred is that in length fj^Tfj—2> which is less than
U a U > Un, and is greater than U ii U < 17^.
It appears possible that always the energy is transmitted along the shaft
at the speed Un • If the forced velocity U > 11^, we may, perhaps, regard the
system in a special sense as a natural system with a uniform rotation super
posed on it.
Let us suppose that the whole of the strain energy in length U^ is trans
ferred per second while only the fraction ijl of the kinetic energy is transferred,
the fraction 1 — /x being stationary.
The energy transferred : strainenergy in U„ : kinetic energy in 11^
= IIU:VJW^:UJ2U„\
Put U = fU.n , and our supposition gives
pu: 2im,jiu„ " /" /* p^ V p) ■
If the forced velocity U < 11^, we may regard the system as a natural one,
with a uniform stationary strain superposed on it.
* In the Sellmeier model illustrating the dispersion of light, the particles may be regarded
as outside the material transmitting the waves and as applying forces to the material which
make the velocity forced. The simple relation between energyflow and group velocity probably
does not hold for this model.
ANGULAR MOMENTUM IN A BEAM OF CIRCULARLY POLARISED LIGHT 377
We now suppose that the whole of the kinetic energy is transferred, but
only a fraction v of the strainenergy, and we obtain
It is perhaps worthy of note that a uniform longitudinal flow of fluid may
be conceived as a case of wavemotion in a manner similar to that of the
uniform rotation of a shaft.
A Suggestion as to the Angular Momentum in a Beam of Circularly
Polarised Light.
A uniformly revolving shaft serves as a mechanical model of a beam of
circularly polarised light. The expression for the orientation 6 of any section
of the shaft distant x from the origin, d = 27t\'^ (Ut — x), serves also as
an expression for the orientation of the disturbance, whatever its nature,
constituting circularly polarised light.
For simpUcity, take a shaft consisting of a thin cylindrical tube. Let the
radius be a, the crosssection of the material s, the rigidity n, and the density
p. Let the tube make N revolutions per second, and let it have such twist
on it that the velocity of transmission of the spiral indicating the twist is the
natural velocity 11^ = \^{n/p).
Repeating for this special case what we have found above, the strain
energy per unit length is ^neh, or, since e = adO/dx = — aO/Un, the strain
energy is Inahe^jTIr? = Ipahd^.
But the kinetic energy per unit length is also ^pahd^, so that the total
energy in length Z7„ is pahO^Un. The rate of working across a section is
nesaO = na^sO^/Ur, = pa^sd^U^,
or the energy transferred across a section is the energy contained in length ?/„ .
If we put E for the energy in unit volume and G for the torque per unit
area, we have
GsO=EsUn,
whence G = EUJO = ENXjIirN = EXj^tt.
The analogy between circularly polarised light and the mechanical model
suggests that a similar relation between torque and energy may hold in
a beam of such light incident normally on an absorbing surface. If so, a
beam of wavelength A containing energy E per unit volume will give up
angular momentum ^A/27r per second per unit area. But in the case of
light waves E = P, where P is the pressure exerted. We may therefore
put the angular momentum delivered to unit area per second as
PA/27r.
378 WAVEMOTION OF A REVOLVING SHAFT, AND A SUGGESTION AS TO
In the Philosophical Magazine, 1905, vol. 9, p. 397*, I attempted to show
that the analogy between distortional waves and lightwaves is still closer,
in that distortional waves also exert a pressure equal to the energy per unit
volume. But as I have shown in a paper on ' Pressure Perpendicular to the
ShearPlanes in Finite Pure Shears, etc.f,' the attempt was faulty, and a
more correct treatment of the subject only shows that there is probably a
pressure. We cannot say more as to its magnitude than that if it exists it
is of the order of the energy per unit volume.
When a beam is travelling through a material medium we may, perhaps,
account for the angular momentum in it by the following considerations. On
the electromagnetic theory the disturbance at any given point in a circularly
polarised beam is a constant electric strain or displacement / uniformly
revolving with angular velocity 6. In time dt it changes its direction by dd.
This may be effected by the addition of a tangential strain /c?^; or the
rotation is produced by the addition of tangential strain fO per second, or by
a current fO along the circle described by the end of /. We may imagine
that this is due to electrons drawn out from their position of equilibrium so
as to give/, and then whirled round in a circle so as to give a circular con
vectioncurrent/^. Such a circular current of electrons should possess angular
momentum.
Let us digress for a moment to consider an ordinary conductioncircuit
as illustrating the possession of angular momentum on this theory. Let the
circuit have radius a and crosssection s, and let there be N negative electrons
per unit volume, each with charge e and mass m, and let these be moving
round the circuit with velocity v. If i is the total current, i = Nsve. The
angular momentum will be
Ns27ra . mva = ^iraHmje = 2Aim/e = 2Mm/e.
where A is the area of the circuit and M is the magnetic moment. This is
of the order of 2M/10'.
It is easily seen that this result will hold for any circuit, whatever its
form, if A is the projection of the circuit on a plane perpendicular to the
axis round which the moment is taken, and if M = Ai. If we suppose that
a current of negative electrons flows round the circuit in this way and that
the reaction while their momentum is being established is on the material
of the conductor, then at make of current there should be an impulse on
the conductor of moment 2M/W. If the circuit could be suspended so that
it lay in a horizontal plane and was able to turn about a vertical axis in
a space free from any magnetic field, we might be able to detect such impulse
if it exists. But it is practically impossible to get a space free from magnetic
intensity. If the field is H, the couple on the circuit due to it is proportional
to HM. It would require exceedingly careful construction and adjustment
* [Collected Papers, Art. 22, p. 338.] j [Collected Papers, Art. 25.]
ANGULAR MOMENTUM IN A BEAM OF CIRCULARLY POLARISED LIGHT 379
•Pi
■Pi
[of the circuit to ensure that about the vertical axis the component of the
jouple due to the j&eld was so small that its effect should not mask the effect
Fof the impulsive couple. The electrostatic forces, too, might have to be
[considered as serious disturbers.
Returning to a beam of circularly polarised light, supposed to contain
electrons revolving in circular orbits in fixed periodic times, the relations
between energy and angular momentum are exactly the same as those in
a revolving shaft or tube, and the angular momentum transmitted per second
per square centimetre is EXJ^tt = PXj2tt, where P is the pressure of the
light per square centimetre on an absorbing surface.
The value of this in any practical case is very small. In
lightpressure experiments, P is detected by the couple on a
small disc, of area A say, at an arm h and suspended by a
fibre. What we observe is the moment APh. If the same
disc is suspended by a vertical fibre attached at its centre and
the same beam circularly polarised in both cases is incident
normally upon it. according to the value suggested the torque
is APXI27T.
The ratio of the two is A/27r6. Now b is usually of the
order of 1 cm. Put A = 6 x 10~^ cm., or, say, 27r/10^, and
the ratio becomes 10~^. .
It is by no means easy to measure the torque APb accurately,
and it appears almost hopeless to detect one of a hundred
thousandth of the amount. The effect of the smaller torque
might be multiplied to some extent, as shown in accompanying
diagram.
Let a series of quarter wave plates, Pi, ^2' Vs^ ..., be suspended by a fibre
above a Nicol prism N, through which a beam of light is transmitted upwards,
and intermediate between these let a series of quarter wave plates, g'l, g'2^ ?3> •••>
be fixed, each with a central hole for the free passage of the fibre. The beam
emerges from N plane polarised. If N is placed so that the beam after passing
through 2?i is circularly polarised, it has gained angular momentum, and there
fore tends to twist p^ round. The next plate q^^ is to be arranged so that the
beam emerges from it plane polarised and in the original plane. It then
passes through ^2 > which is similar to pi , and again it is circularly polarised
and so exercises another torque. The process is repeated with q^ and f^, and
so on till the beam is exhausted. By revolving iV through a right angle round
the beam, the effect is reversed. But, even with such multiplications, my
present experience of lightforces does not give me much hope that the effect
could be detected, if it has the value suggested by the mechanical model.
27.
PKELIMINARY NOTE ON THE PRESSURE OF RADIATION AGAINST
THE SOURCE: THE RECOIL FROM LIGHT.
By J. H. PoYNTiNG and Guy Barlow, D.Sc.
[British Association Report, 1909.]
[This is merely a preliminary account of the following paper Art. 28.]
• 28.
THE PEESSUKE OF LIGHT AGAINST THE SOURCE:
THE RECOIL FROM LIGHT.
BAKERIAN LECTURE.
By J. H. PoYNTiNG and Guy Barlow, D.Sc.
[Roy. Soc. Proc. A, 83, 1910, pp. 534546.]
[Read March 17, 1910.]
All experiments on the pressure of radiation have hitherto been made on
the force exerted by light or radiation on a receiving surface. The experiment
now to be described shows the pressure of radiation against the source from
which it starts, and from analogy with a gun we may term this the recoil
from hght. It does not appear practicable to show this effect by using a
source in which heat is developed intrinsically. But if radiation falls on an
absorbing body it heats the body and the heat so developed issues again as
radiation, and it is possible to detect the efEect of this issuing radiation.
Theory.
We may see the nature of the action to be looked for by considering an
ideal case in which we allow a beam of light with energy P per cubic
centimetre to faU normally in a perfect vacuum in turn on each of four discs,
the front and back surfaces of these discs being respectively as in Fig. 1,
BB BS SS SB
' BACK
Fig. 1.
where B represents a fully absorbing or * black' surface, and S a fully reflecting
or nonradiating surface.
When the radiation falls on an absorbing face, as in the case of either of
the discs (1) and (2), the temperature of the disc rises till a steady state is
reached in which emission equals absorption. We may suppose that the
discs are so thin that the two faces are sensibly at the same temperature. If
we did not take into account the pressure due to issuing radiation, or if we
382 THE PRESSURE OF LIGHT AGAINST THE SOURCE:
only considered the initial effects before heating took place, the pressures on
the first two discs would be P in each case, due to the incident beam alone,
and on the last two would be 2P, due to the sum of the incident and reflected
beams. We should have, therefore, pressures respectively
(1) (2) (3) (4)
P P 2P 2P
But when a steady state is reached, the discs (1) and (2) must be giving
out as much radiant energy as they receive. The first disc gives out equal
amounts on the two sides, producing equal and opposite pressures. All the
radiation from the second disc is given out at the front side and is equal in
energy to that of the incident beam. Assuming this emitted radiation is
distributed according to the cosinelaw, the pressure resulting from it is
easily shown to be P, so that the total pressure on this disc is fP.
Since there is no absorption by discs (3) and (4), we still have the pressures
2P ; hence we have now
(1) (2) (3) (4)
P fP 2P 2P
In a real case these results are modified in two ways :
(i) By the possession of some small reflecting power by surface B, and of
some small absorbing and radiating power by surface S.
(ii) By an inequality of temperature between front and back surfaces
conditioned by the energy which is carried through from front to back to be
radiated thence. The vacuum is not perfect, and there is radiometeraction
due to the residual gas, which, owing to the inequality of temperature, is not
the same on the two sides. This is probably the only way in which gas
action is sensible, for the effects due to ordinary convection and conduction
in the residual gas are negligible. The temperaturediflerence, though
sufficient to produce a differential radiometeraction, is so small that in
estimating the radiation from the two sides of a disc we may take them as
being at the same temperature.
In the experiment to be described the diffusion is so sHght that we dis
regard it. Considering, then, only the reflection and absorption, let r be the
coefficient of reflection of the surface B for the incident radiation, p that of S,
a the coefficient of emission of B for the emitted radiation, a that of S.
It is then easy to show that the total radiationpressures on the four discs
are respectively
(1) (2) (3) (4)
The emitted radiation is not of the same quality as the incident radiation ;
and strictly we are not justified in assuming that the emissive powers for the
THE RECOIL FROM LIGHT
383
two surfaces for the one quality of radiation are in the same ratio as the
absorbing powers for the other. But we shall for simpUcity suppose that
the ratios are the same, a supposition which enables us to proceed, and which
is probably not very far from the truth. We have, therefore,
a=l— r, a=l— /).
On this assumption Table I has been constructed, giving the four pressures
for different values of r and p. The pressureratios in the last two columns
are of interest in connection with the experimental results, and will be referred
to later.
Table I. Pressures on Discs, calculated for Different Values of the
Reflection Coefficients .
Reflecting
power of B
Reflecting
power of 8
Pressures taking P =
1
Pressureratios
r
P
BB
BS
88
8B
B8
BB
BS
\{88 + 8B)
100
100
167
200
200
167
167
095
100
160
195
192
160
166
090
100
154
190
185
154
165
005
100
105
168
200
200
160
168
005
095
105
162
195
192
154
167
005
090
105
156
190
185
149
166
010
100
110
170
200
200
155
170
010
095
MO
164
195
192
149
170
010
090
MO
158
190
185
144
169
The modification of the values in the above table by radiometer action
will be greater for the BB disc than for the others. In that disc energy
proportional to JP has to be carried through the disc, and therefore the
temperaturedifference is the greatest. It is only possible to guess at the
relative magnitudes of the radiometeractions for the different discs. We
have therefore sought to make the vacuum so high that the action nearly
disappeared.
The Experiment.
In the final form of the experiment each disc consisted of a pair of
circular coverglasses, 1*2 cm. in diameter and about 01 mm. thick, between
which was squeezed a layer of asphaltum also about 0*1 mm. thick, the
temperature being first raised sufficiently to render the asphaltum molten.
It is difficult to make the discs of uniform thickness and free from bubbles
of gas ; but a great number were made and the four best were selected for
use. Such a compound disc appears to be perfectly opaque, and its surface
is the blackest black and the least diffusing that we have yet been able to
obtain.
384
THE PRESSURE OF LIGHT AGAINST THE SOURCE
The reflecting surface was made by depositing silver on the outside of the
compound disc by means of the discharge from a silver cathode in an
exhausted receiver. A similar deposit on clear glass just allowed an arc
light to be seen through it.
Four holes the size of the discs were cut in a stout plate of mica ABCD,
the centres of the holes being at the corners of a 2 cm. square (Fig. 2).
The discs were then fixed in these holes by a minute amount of celluloid
varnish. The suspensionrod E and the mirrorholder F were attached to
H
:i
<}.
the mica plate, at the middle points of its top and bottom edges respectively,
by very small copper clips without any cement. A platinised mirror was
cut in half, and the two portions M^, M^ were mounted back to back in
a suitable clip of copper foil at the extremity of the rod F, the plane of the
mirrors being perpendicular to the mica plate. This system was suspended
by a quartz fibre G, 9 cm. long, in the centre of a glass flask of 16 cm.
diameter. The upper end of the quartz fibre was fixed to a brass collar H
held by friction in the neck of the flask. Both ends of the fibre were silvered
and coppered, so that they could be soldered to E and to the support. After
THE RECOIL FROM LIGHT
385
suspension the mouth of the flask was sealed off, a lateral tube in the neck
being still available for connection with the exhausting apparatus.
To carry out the exhaustion of the flask to a very high degree, the general
arrangement shown in Fig. 3 was adopted, and the successive stages in the
process were as follows :
(1) Prehminary exhaustion by aid of a Gaede pump. The experimental
flask A was kept hot by gasburners below it, and the charcoal bulbs Cj and
Cg were strongly heated electrically by enclosing them in asbestos tubes con
taining coils of platinum wire. From time to time the whole apparatus was
washed out with dry oxygen generated from the manganese dioxide in the
bulb E. A small discharge tube G indicated the state of the vacuum, and
examination of the spectrum gave useful information as to the gas given off
from the bulbs C^ and C2. At the end of three days these bulbs appeared to
Fig. 3.
have ceased to give off any gas, and the Gaede pump would bring the vacuum
down to a hard Xray stage in about 10 minutes after admitting oxygen.
The temperatures of C^ and C2 were then somewhat reduced, and the apparatus
was sealed off at Si. B and F in the figure represent phosphorus pentoxide
drying tubes, and H is a manometer.
(2) The charcoal bulb Cj was put in liquid air, and the temperature of the
second bulb Cg and of the flask A was then slowly reduced to the room
temperature. After 30 hours the apparatus was finally sealed off at 82
During both these stages the Utube D was kept always immersed in liquid
air. This arrangement formed a perfect trap for mercury vapour, which
otherwise diffused from the pump into the flask and attacked the silver
mirrors. A roll of silver foil was placed in one limb of the Utube, with the
object of making the trap more effective, but this precaution was probably
unnecessary.
p. c. w.
386 THE PRESSURE OF LIGHT AGAINST THE SOURCE:
(3) In the final stage of the exhaustion, the charcoal bulb Cg was
surrounded by liquid air, which was boiled off continuously at the reduced
pressure of about 2 cm, of mercury for several hours before and during the
whole of the measurements. Experience showed that the highest vacuum
was obtained probably two hours after the application of fresh liquid air, and
that it was necessary to renew the liquid air after every four or five hours.
The source of light S (Fig. 4) was an Ediswan 50 volt ' Focus lamp,' which
was fed from accumulators. By means of an adjustable resistance in series
with the lamp, the voltage was maintained exactly at 60 volts, the lamp then
taking a current of 537 amperes. The light was so steady for hours at a
B
o
s
Li.
La.
; ,'
G :
::©
::::::::
.■:::;:;i::
'€
o
so
I 1 :.
IOC
A
Cms
Fig. 4.
time that adjustment of the resistance was seldom necessary. A photo
graphic lens Li, of 15 cm. focal length, provided with an iris diaphragm, was
arranged to throw an image of the lamp filament on an achromatic lens Zg
of 19 cm. focal length. This seeond lens then formed a uniformly illuminated
image of the iris diaphragm on the disc to be worked with. By adjustment
of the diaphragm this image was made rather smaller than the disc, so that
when the beam was centred on the disc an unilluminated margin about  mm.
wide was left all round it. The lamp and lenses were fixed to a board which
could be moved parallel to itself vertically and horizontally between guides,
so that the beam could be easily directed on to the four discs in succession.
The centring of the image on the disc was made by eye.
The flask was mounted on an iron turntable with the quartz fibre
accurately in the axis of rotation. By rotation of the flask through 180°
it was possible to experiment on the reverse sides of the four discs. We
9 THE RECOIL FROM LIGHT 387
shall refer to the observations raade in the two positions as 'direct' and
'reverse.' Thus the 'direct' BS disc becomes the 'reverse' SB disc. The
flask was shielded from electrification and from extraneous radiation by
enclosure in a cylindrical case of tinned iron, blackened inside and provided
with windows to admit the beam of light and to allow the deflections to be
observed.
For reading the deflections the image of an electric lamp B (Fig. 4) on a
millimetre scale C, at a distance of 113 cm., was used. The definition of the
image was sufficiently good to allow deflections to be read accurately to
02 mm., although the optical irregularities of the glass flask rendered a
telescope useless.
In finding the centre of swing from the deflections, a curious periodic
motion was observed in its position, the centre moving to and fro with simple
harmonic motion. The complete period of the torsional vibrations of the
suspended system was 746 seconds with no appreciable damping; and the
period of the motion of the centre of swing was found to be about seven
minutes, that is nearly 11 torsional halfperiods. The periodic motion was
ultimately traced to pendulummotions of the system set up by external
disturbances, for the amplitude of the displaced centre of swing was found
to increase with such pendulummotion. The pendulumperiod of the system
was slightly longer for motion in the plane of the mica plate than for motion
in a perpendicular plane. Hence a pendulummotion once set up changed
periodically from motion in a straight line to motion in an ellipse, and the
complete cycle of these changes was actually gone through in seven minutes.
This meant a periodic change in the angular momentum of the system about
the axis of suspension, and to neutralise this the mica plate tended to turn
with equal and opposite angular momentum about the vertical axis, and so to
give a twist to the fibre. Accordingly, we should expect the maximum twist
to take place when the vibrations are lineat, and this was observed to be the
case. To eliminate this effect, 12 consecutive turning points were always
taken (i.e. observations over a period of seven minutes), and the mean centre
of swing calculated from these.
When the beam was allowed to fall on a disc, some initial effects were
observed, of which we shall give an account below, and shall then suggest a
tentative explanation.
The Results.
The observed deflections for the four discs are given in Table II.
These results are divided into two series, A and B. In Series A the beam
of Hght was kept on the disc under experiment until the deflection seemed
nearly constant, the time of exposure being generally about 20 or 30 minutes
before the deflection was read. Attention was chiefly given to the BB and
25—2
388
THE PRESSURE OF LIGHT AGAINST THE SOURCE *.
BS discs ; the deflections for the SS and SB discs are not so reUable, as it
became afterwards evident that these discs require even longer time to recover
from the initial disturbances (see below).
Table II.
i
BB
BS
SS
SB
Series A
Light on for about 20 or
30 mins. ; zero observed be
tween each exposure to light
D
1491
1556
1503
2045
2329
2077
3232
2722
Mean D
1517
2150
3232
2722
R
1715
1701
2289
2394
2736
2619
Mean R
1708
2342
2736
2619
Mean D and i?
1613
2246
2984
2671
1
Series B
Light on for 1 hour; zero
observed only at beginning
and end of observations on
several discs
D
(1378)*
1520
1471
2107
2044
3071
2851
2915
2795
Mean I)
1496
2076
2961
2855
R
1711
2337
2772
2742
Mean D and R
1604
2207
2867
2799
Final Values
BB and BS taken as mean for
Series A and B
! SS and SB from Series B alone
161
223
287
280
Deflections in scaledivisions for the four discs. Observations on the 'direct'
and 'reverse' sides are denoted by D and R respectively.
* Rejected in taking the mean as, owing to a breakdown of the pump for exhausting the
liquid air, the vacuum had doubtless deteriorated.
In the Series B, made a month later, the beam was kept on each disc for
one hour before reading the deflection, with the object of obtaining a still
closer approach to a steady state. One hour was also allowed before taking
the zero after cutting off the light. The much greater time now required
made it impracticable to observe the zero more often than twice while making
observations on all four discs. Experience showed that, provided the vacuum
was well maintained, the zero seldom changed more than one or two tenth
THE RECOIL FROM LIGHT
389
divisions during five hours. There appeared, therefore, no objection to this
course.
Examination of the table shows that in most cases rather different values
are given by the * direct' and 'reverse' observations. This is particularly
evident in the case of the BB disc, where the values, roughly 15 and 17
divisions respectively, differ by about 13 per cent. The explanation of
this appears to be that the radiometeraction was still sensible and acted
differently in the ' direct ' and ' reverse ' positions of the disc. The difference
was probably due to inequality in the thickness of the two coverglasses
enclosing the asphaltum layer.
Some observations were made without using liquid air, i.e. with the charcoal
bulb at roomtemperature. The deflections were then always very unsteady,
and the zero showed erratic behaviour. A few deflections obtained under
these conditions are given in Table III. They merely serve to indicate the
general effects of the gasaction. On the BB disc the want of symmetry
referred to above is now greatly exaggerated, and points to the gasaction
being a suction on the 'direct' side and a pressure on the 'reverse.' It is
also to be noted that the action is a suction on the BS disc. Any tendency
towards this action in the highvacuum experiments would, therefore, tend
to mask the recoilpressure sought for.
Table III.
BB
BS
SS
SB
Direct
5
37
 19
 77
no obs.
45
93
58
Reverse
Deflections for the four discs with charcoal bulb at roomtemperature. (The
minus sign indicates suction.)
On account of the uncertain values of the various corrections required, and
on account of the existence of outstanding disturbances, it was felt that mere
multiplication of observations would not lead to more exact results. As the
final values, in scaledivisions, for the pressures on the four discs given by the
experiment we therefore take
BB BS ss SB
161 223 287 280
The values for BB and BS are the means given by both Series A and B,
the values for SS and SB are from Series B alone, since for these discs the
steady state was not attained properly in the observations of Series A.
Hence we have
BS
BB
= 139,
BS
J (SS + SB)
= 158.
390 THE PRESSURE OF LIGHT AGAINST THE SOURCE:
We select for comparison with these ratios those calculated for r = 005,
p = 095 in the last columns of Table I, i.e. the values
m~^'^^' UssTsB) ''"''•
The latter ratio agrees better with the experiment than the former. This
is what we might expect, since the latter ratio does not involve the pressure on
the BB disc, and that is the pressure which is most affected by the radiometer
action. As to the actual reflectioncoefficients for the surfaces of the discs,
we can do but httle more than make a guess. For the black surface we may
take the glass surface as reflecting 4 per cent, of the incident Ught, and
allowing another 1 per cent, for the asphaltum — probably a reasonable
estimate — we have, finally, r = 005. In the case of the silver surface the
reflection was tested by means of a thermopile, and it was concluded that at
least 96 per cent, of the beam used was reflected.
The Energy of the Beam.
A determination of the energy of the beam used was made and afforded a
means of calculating the absolute values of the pressures to be expected.
As in the experiments of Nichols and Hull, and in other experiments on
lightpressure which we have made, the energy was measured by allowing the
beam to fall on a blackened disc of pure silver (2 cm. diameter and 028 cm.
thick), and by observing the initial rate of rise of temperature by means of
a constantansilver thermoelectric junction soldered to the disc. A Rubens
panzergalvanometer was used and was adjusted to have a period of about
two seconds and was then made deadbeat. The transit of each centimetre
division of the scale across the field of view of the observing telescope was
recorded on the drum of an electrically driven chronograph. Just before and
just after each series of transits, a set of 5second intervals were also recorded
in order to give the peripheral speed of the drum. The number of micro
volts per scaledivision was then determined, and the thermoelectric power
of the couple being known from a separate experiment, it was possible, by
means of a graphical representation of the chronographrecord, to calculate
the rate of rise of temperature, and hence the energy of the beam was
calculated to be 33 x 10~^ erg per centimetre length. This would be the
force in dynes on a fully absorbing surface. The moment of inertia of the
suspended system was 0770 grammecm. 2, and its period was 746 seconds.
The arm was 100 cm. Hence the beam should give a deflection of 136
divisions of the scale used when falling on a disc fully absorbing on both
sides. Assuming that the BB disc reflects 5 per cent., the deflection should
be 143 divisions. This is in close agreement with half the value obtained
with SS and SB, and the excess over 143 of the observed value 161 obtained
THE RECOIL FROM LIGHT 391
with BB is probably to be ascribed to residual radiometeraction. The
smallness of the excess shows that the radiometeraction was reduced to
a very small amount.
Initial Effects.
We have already referred to the necessity for exposing the discs in general
to the beam for a long time before taking the observations of the deflections.
On the BB disc, however, there was no marked initial effect, and after a few
periods the deflection was nearly constant. When the beam was first allowed
to fall on one of the other discs an initial disturbing effect was evident, and
in some cases even half an hour seemed insufficient to produce a steady state.
On the BS disc there were sometimes indications of a pressure for the first
few seconds, but a strong suction always set in, and this suction, after reaching
a maximum, rapidly subsided, giving place finally to a pressure increasing to
the Umiting value corresponding to the steady state.
The SS disc showed want of symmetry. On the direct side there was
suction followed by pressure, as in the last case, but on the reverse side there
was at first excessive pressure, reaching a maximum and then slowly falling
to the value given in the steady state.
The SB disc showed an initial excess pressure very similar to that on the
reverse side of the SS disc.
For both the SS and SB discs the duration of these initial effects was so
drawn out that an appreciably steady state was not attained in much less
than an hour. In Fig. 5 the effects on all four discs are represented by curves
in which the time is taken as abscissa and the deflection as ordinate. The
residual effects which follow the cutting off of the Hght are also indicated.
As an explanation we suggest that these effects are due to the heat of the
beam driving off occluded gas from the silver films, the expulsion of the gas
causing a back pressure on the film. In the case of the SS disc it was known
that the silver film on one side was decidedly thicker than that on the other,
so that there is no difficulty in accounting for the want of symmetry observed
in that case.
This explanation is supported by two observations : first, that with a
stronger beam the steady state is sooner reached ; secondly, that in the case
of the BS disc it was noticed that if after the estabhshment of the steady
state one cut off the light for a minute or two, and then put it on again, an
initial suction took place but was much less than originally.
Assuming that the effects observed were really due to the expulsion of
occluded gas as suggested above, it is possible to form an estimate of the total
mass of gas given off by calculating from the curves (Fig. 5) the total impulse
given to the disc. Thus, if we assume the gas to be oxygen and suppose,
392
THE PEESSURE OF LIGHT AGAINST THE SOURCE:
further, that the molecules leave the film normally with the ordinary molecular
velocity, we find that for the BS disc the total mass given off was about
1*7 X 10' gramme. Taking the volume of the experimental flask as 2 litres.
BB
fi
a
SB
mr/7S
Fig. 5. Initial Effects. The ordinate represents the deflection, plotted against the time as
abscissa. The direction of the deflection for pressure is shown by the arrow. For the
SS disc curves D and R correspond to the ' direct ' and ' reverse ' sides respectively.
The light is put on at a and cut off at ^.
this quantity of gas would give a partial pressure of about ^^ dyne/cm. 2.
So that unless the gas is rapidly absorbed by the charcoal it would appear
that the vacuum might be sensibly affected. Unfortunately we had no
means of forming even a rough idea of the actual pressure of the residual
gas in the apparatus.
THE RECOIL PROM LIGHT 393
The Temperature of the Discs.
In the steady state all the energy of the beam absorbed by a disc must be
! radiated from the faces. By assuming the fourthpower law of radiation,
we may therefore estimate roughly the rise in temperature of the disc. The
results are 55° C. and 90° C. for the rise in temperature of the BB and BS
discs respectively.
For the BB disc it was also estimated that the temperaturedifference
between the two faces was probably less than J^° C. The temperature
differences for the other discs should be still less. It should be noticed that
the glass is black for the issuing radiation, hence the asphaltum layer alone
counts in the case of the BB disc.
Early Experiments with Platinum Discs.
In some early experiments we used two discs, BB and BS, oi platinum foil
J mm. thick, the black surfaces being formed by depositing platinum black.
Results were obtained somewhat like what we expected as to the ratios, but
the deflections depended very greatly on the state of the vacuum, and under
the best conditions were about 50 to 100 per cent, greater than the values
calculated from the energy of the beam. This disagreement was doubtless
chiefly due to radiometeraction. The black surface, being flocculent, is
obviously badly conducting, the temperatureslope is therefore increased, and
we have in consequence a big differential radiometeraction on the faces of
the discs. Moreover, the polished platinum is a poor reflector, so that such
discs quite fail to approach the ideal conditions. These considerations led to
the use of the asphaltum discs.
29
ON SMALL LONGITUDINAL MATERIAL WAVES ACCOMPANYING
LIGHTWAVES.
[Roy. Soc. Proc. A, 85, 1911, pp. 4'74476.]
All experiments on the pressure of light agree in showing that there is a
flow of momentum along the beam. This flow is manifested as a force on
matter wherever there is a change of medium. When the light is absorbed,
the momentum is absorbed by matter. When the beam is shifted parallel to
itself there is a torque on the matter effecting the shift. The momentum
would therefore appear to be carried by the matter and not merely by the
ether. Though there is an obvious difficulty in accepting this view when the
density of the matter is so small as it is in interplanetary space, it appears to
be worth while to follow out the consequences of the supposition that the
force equivalent to the rate of flow of momentum across a plane perpendicular
to a beam of light acts upon the matter bounded by the plane. This rate
of flow per square centimetre is equal to the energydensity or energy per
cubic centimetre in the beam. Of course, in experiments, only the average
of the rate of flow during many seconds and the average energy per cubic
centimetre in a length of beam of millions of miles is actually measured.
But on the electromagnetic theory of light, which suggested the experiments
and which gives the right value for the pressure, this pressure is equal to the
energydensity at every point of a single wave.
Let us suppose that we have a train of plane polarised electromagnetic
waves of sineform, the magnetic intensity being given by
H = H^ sin ^ (x — vt),
A
where H^ is the amplitude of H. The magnetic energy per cubic centimetre
at any point is hH^/Stt, and as the electric energy is equal at each point to
the magnetic energy, the total energy is jjlH^I^^tt.
The energy per unit volume is ~ — dx = ijuH^j^tt.
SMALL LONGITUDINAL MATERIAL WAVES ACCOMPANYING LIGHT WAVES 395
The pressure p across a transverse surface is
AfIT
p = fiH^I4^ = ^ sin2 ^(x vt)
&7T
1 — COS y (iC — Vt)
The force on an element of length dx is
^ dx =
dx
'^ J sm jr [x — vt) dx
fJiH^^ .477. ..
= — ^^ sm Y (x — vt) dx.
If ^ is the linear longitudinal displacement of the element there will be
a force due to elastic change of volume
d^$.
where q is the elastic constant for compression or extension.
If p is the density of the material, the equation of motion is
d^ M^l^,.47^
^' dx^
d^i
Pd^
2A smy (a;i;0.
477
A sin Y (x — vt — e).
Assume
Then, substituting,
( p^ . ^ v^  qA ^ j sm j(xvte)= ^^^ sm y (^  ^'0
Putting X =^ and ^ = 0, we see that e = 0. Putting q = pu^, where u
is the velocity of free elastic waves of the q type, and assuming that the
longitudinal waves are forced waves, keeping exact time with the waves of
light, we have
Xp^H ,^
32tt^P (v^  u^) '
As u/v is negligible for all ordinary matter,
. XuHi' .477. ,,
. uHi^ 477, .
The potential energy in these waves is neghgible in comparison with the
kinetic. We have then
Energy per unit volume = J p^hlx
•'
25677^^^ '
396 SMALL LONGITUDINAL MATERIAL WAVES ACCOMPANYING LIGHT WAVES
As the electromagnetic energy per unit volume is ^xH^j^,
Energy in longitudinal waves _ ^E^ _ 1 ij^H^ jpv^
Electromagnetic energy S27Tpv^ 8 Stt / 2 '
which is oneeighth of the electromagnetic energy divided by the energy
which the matter would have if it were moving with the velocity of light
in that matter.
This shows how infinitesimal is the fraction of the energy of the beam
which is located in these waves of compression of the material.
The fraction is proportional to the intensity of the beam.
As an example, take a beam of the intensity of full sunlight just outside
the earth's atmosphere, in which the energyflow is about 14 x 10^ ergs/sec.
The energydensity ixH^^j^rr is therefore 14 x 10^ ^ v. Put v = 2> x lO^^jn,
where n is the refractive index. The fraction is
1 1'4 y 10^ ^3
At the surface of the sun it would be about 40,000 times as much, say,
5 X 1022^7^.
It is interesting to note that if a beam of light is incident on any reflecting
or absorbing surface and if the pressure of light is periodic with the waves it
must give rise to ordinary elastic waves in the material of frequency double
that of the lightwaves.
30.
ON THE CHANGES IN THE DIMENSIONS OF A STEEL WIKE WHEN
TWISTED, AND ON THE PRESSUKE OF DISTORTIONAL WAVES
IN STEEL.
[Roy. Soc. Proc. A, 86, 1912, pp. 534561.]
[Read March 21, 1912.]
In the Proceedings of the Royal Society* there is an account of some
experiments which I made to show that wires when twisted lengthen by an
amount proportional to the square of the angle of twist, a result expected
from an analysis of the strains in a finite pure shear. In those experiments
it was necessary to put considerable loads on the wires.
I have now succeeded in measuring the change in the diameter of a wire
when twisted, as well as the longitudinal extension, and have found that the
change, a contraction, is also proportional to the square of the angle of twist.
It has been now found that the change is sensibly the same for large loads
and for the smallest load which could be used, when the wire was sufficiently
straightened before being twisted, so that apparently the only function of the
load is to straighten the wire.
To measure change in the diameter the wire was fastened at the bottom
of a long narrow tube, the 'wire tube,' filled with water. It passed out from
the top of the wire tube through a watertight leather washer. A capillary
glass tube rose vertically from an orifice in the side of the tube, into which
it was cemented, and the change of the water level in the capillary when
the wire was twisted indicated the change in the volume of the wire within
the wiretube.
Description of the Apparatus.
The apparatus used for the measurement of the effects is shown in Fig. 1,
where, for convenience of representation, various parts are put into the plane
of the figure, though actually they were in different planes.
An iron bracket B projected from the wall of the laboratory, and a tripod
rested on it, on three levelHngscrews. The tripod carried a conical bearing
* Series A, 1909, vol. 82, p. 546. [Collected Papers^ Art. 25.]
398 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED,
for the twisting headpiece T. When the
axis of this was exactly vertical, the tri
pod was fixed to the bracket by clamping
screws s, s. The twistinghead was pro
vided with a circular plate P, with marks
at 90° intervals, which could be set against
a fixed index i. In practice only whole
turns were given, so that only one mark
was used, except in one experiment de
scribed later.
At the lower end of T, there was a
chuck into which the upper end of the
wire was inserted, and a tighteningscrew
made a firm grip. The wire in all cases
was very nearly 1605 cm. long. At its
lower end it was gripped by a similar
chuck attached to a steel crosspiece C,
about 29 cm. long, seen endwise in the
figure.
Polished steel plates were screwed on
to the vertical sides of this crosspiece
near its ends. Four horizontal screws,
working in brackets projecting from the
wall, and with small steel balls at their
ends, were screwed up so as just not to
touch the steel plates when there was no
twist on the wire. But when a twist was
put on, the crosspiece moved up against
two of the screws, and was thus fixed in
position. Below the crosspiece there was
a rod to which was attached another rod
carrying a platform p, and on this weights
could be placed. Each weight was in two
semicircular halves. Below the platform
was a lead weight S, which I call the
sinker, with a volume of 1020 c.c. This
hung in a can, and near the can was a
water cistern, not shown, connected to it
by a rubber tube. When the cistern was
pulled up water flowed into the can so as
just to cover the sinker and lessen the
load by 1020 grm. When the cistern was
Fig. 1.
AND ON THE PRESSURE OF DISTORTIONAL WAVES IN STEEL 399
let down the water flowed back into it and the load increased to its full value.
This was used to determine Young's modulus and Poisson's ratio.
A table t was fixed to the wall independently of the bracket B, to carry
the observing microscope, and the observer sat on a platform built up about
15 metres from the floor. The brass wire tube had an internal diameter
about 2 mm. The top of the tube was fixed in a horizontal brass plate, in
which was a hole about 025 mm. wider than the wire. On this plate rested
a wellvaseHned leather washer about 15 mm. thick, drilled so that it was
fairly tight round the wire. On the washer was another brass plate, with
a hole in it about 035 mm. wider than the wire. Four screws passed freely
through holes in this upper plate, and were screwed into the lower plate.
Springs^ between the heads of the screws and the upper surface of the upper
plate gave sufficient pressure on the washer. It was found necessary to have
the holes in the plates somewhat larger than the wire, in order to adjust the
wire and the wire tube both vertical. An arm, not shown in Fig. 1, projecting
from the lower part of the apparatus, with a sliding weight on it in the plane
of the upper sidetubes, sufficed to make this vertical adjustment. At first
I tried indiarubber washers. They were quite good when first put in, but
they deteriorated rather rapidly, and, when they began to perish, they let
a small quantity of water out of the tube when the wire moved. The leather
washer only required renewal once, when it began to let water escape, and
then, on examination, it appeared to be due to action on the wire, which
was perceptibly rough on the surface where it emerged from the tube. A short
length was cut off the lower end of the wire, and an equal length was let
down through the upper chuck, so that once more a smooth part of the wire
passed through the washer. There was no further difficulty, and no evidence
again of any escape of water.
At the upper end of the wiretube there were two sidetubes. A glass
capillary tube was cemented into one, and bent as shown on the right in the
figure, the vertical branch being about 10 cm. long. When the tube was filled
with water, the level in the capillary was adjusted at the level of the micro
scope about 7 cm. above the level of the washer, as this was about the rise of
water in the capillary due to surfacetension, and there would therefore be no
hydrostatic pressure on the water at the level of the washer. The tendency
to leak would thereby be lessened, but the precaution was probably needless.
Into the tube on the left a plunger passed through a leather stuffingbox.
The plunger had a diameter of 02060 cm., and it was driven to or fro by
a micrometerscrew of J mm. pitch. On the head of this screw was a 10 cm.
plate, with 500 divisions on its circumference. This plunger was used
ordinarily to adjust the level of the water in the capillary. But it was also
used to calibrate the capillary. For this caHbration, the usual observing
microscope was replaced by a microscopecathetometer, and the change of
400 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED,
waterlevel in the capillary was measured for one turn in or one turn out of
the screw. Turning always in or always out, it was hardly possible to hit
exactly on a whole turn, but a correction could be made, of course, for the
fraction of a division in the micrometer head in excess or defect of a whole
turn. The mean of 10 measurements when the micrometer was driven
inwards 05 mm. gave a rise of 11135 mm., with a range of 0155 mm. between
greatest and least. The mean of 10 measurements when the micrometer
was drawn out 05 mm. gave a fall of 11190 mm., with a range of 0045 mm.
The value was taken as 1 1  1 6 mm. This gives the crosssection of the capillary
as 0001493 sq. cm., and its diameter as 00436 cm.
At the lower end the wiretube was soldered on to a screwcap which could
be screwed over the chuck gripping the lower end of the wire.
Below the chuck was a sidetube used to fill the wiretube with water.
For this purpose the sidetube was connected with a flask in which water
Avas boiled. The steam passed up through the crevices in the chuck and out
at the plungertube, from which the plunger was removed. A funnel con
taining water was connected on to the plungertube, and when the water
in this was boiling freely, through the passage of the steam, the flask was
allowed to cool and water was sucked back into the wiretube. When it
was full the flask was detached and a cap was screwed on to the lower tube.
The plunger was replaced and the capillary, which had been closed meanwhile,
was opened. By driving in the plunger the water was raised up to the level
of the washer and to any desired point in the capillary.
When the apparatus was not being used the open end of the capillary was
under water in a beaker, the plunger being driven in so that the capillary
was entirely filled with water. The apparatus thus remained full of water
whatever change of temperature might occur. When required for work the
beaker was withdrawn and the plunger was screwed out till the meniscus
was in the field of view of the microscope.
The wiretube was surrounded by an outer tube about 25 cm. diameter,
filled with water. This merely served as a means of reducing the effect of
outside or inside temperaturechanges. A wooden casing covered with tin foil
surrounded the whole from the floor up to the table to lengthen out still
further any effects due to temperaturechange.
To observe the changes of level due to twisting, a microscope with a
1inch objective and provided with a parallelplate micrometer was used.
The micrometerscale was calibrated by means of a millimetre divided to
tenths on a standard invar bar. Twelve determinations of 04 mm, gave
107*2 micrometerdivisions equal to 1 mm., the determinations falling within
about 1 per cent, range. Then 1 micrometerdivision = 000933 mm. Since
the crosssection of the capillary is 0001493 sq. cm., one division of the
AND ON THE PRESSURE OF DISTORTION AL WAVES IN STEEL 401
micrometer signifies a change of volume of the water in the wiretube of
1393 X 106 c.c.
When it was desired to read the height of the water in the tube the
micrometer plate was moved till the cross wire in the microscope just
touched the image of the lowest point of the capillary meniscus. The field
was well illuminated by a small lamp behind the capillary, but the image
was not always very distinct, and settings of the micrometer could not be
trusted to, I think, two or three tenths of a division in some cases, though
usually they were more exact. Close to the tube and between it and the
micrometerplate a small vertical plate of glass was fixed to the tube at 45°
to the line of sight, and this reflected the point of a needle which was also
fixed to the tube, so that its image was in the same plane as, and close to,
the image of the meniscus. This enabled the observer to note the position
of either the meniscus or the needlepoint without moving the microscope.
When the wire lengthened the wiretube was let down by an equal amount,
and the needlepoint fell. Let us call this fall NP.
At the same time the wire contracted laterally, and the meniscus fell in
the tube, and the fall relative to the tube gave the change in volume. The
fall observed was that relative to the tube plus that of the tube or NP.
Hence, if the fall observed in the microscope is T the fall relative to the
tube is T  NP.
The Wires and their Preparation.
Two pianosteel wires were used in the experiments here described. No. I
with a mean diameter 00986 cm., the diameters in two planes at right angles
being measured with a micrometer every decimetre of its length. The
measurements ranged from 00980 to 00989 at different points. No. II had
a mean diameter of 01210 cm., measured in the same way, with a range at
different points from 01207 to 01212.
It was found necessary to straighten these wires, for, unstraightened, they
showed the effect with light loads noticed in the previous paper *, an apparent
shortening on twisting, due, I think, to coiling. To straighten them they
were loaded and an electric current was passed through them. No. I was
loaded with 50 kgrm., and received a current of 10 amperes. No. II was
loaded with 60 kgrm., and received a current of 16 amperes. In each case
the wire drew out slightly and then stopped, acquiring a blue temper without
rising to a red heat.
Of course the wires became circularly magnetised, but the magnetisation
can hardly have contributed to the results here to be described, as these
results are of the same character and order as results obtained with heavily
* [Collected Papers, Art. 26.]
p. c. w. 26
402 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED,
loaded unannealed wires in a number of preliminary experiments made
before the experiments took their final shape.
A few experiments were made on a harddrawn copper wire, mean diameter
01219 cm. (with a range from 01216 to 01224 cm.).
The Method of Measuring the Lowering on Twisting.
In making any determination of NP or T the following plan was adopted.
Suppose, for instance, that the value of T was to be found for four turns
of the wire clockwise as seen from above. The position of the meniscus
was read for no twist at a given minute, then my assistant put on four turns
clockwise — denoted by C^ — then he gave a signal just before, and again
exactly at the next half minute, and I set the cross wire on the meniscus at
the half minute. The micrometer was read, and the twist was taken off.
At the next half minute the micrometer was set as before. Again C^ was
put on, and so on, usually for 32 observations. The first two or three
readings were not taken into account, as initially there was usually some
irregularity, due probably to settling down in the bearing. The readings
were combined in threes in the usual way to give T = \ {a \ c) — h to
eliminate as far as possible any march of the zero reading. With the
meniscus there was almost always a march, due chiefly to temperature
change, for, of course, the arrangement was a very sensitive thermometer.
The mean result of the 32 observations was equivalent to 15 or 16 inde
pendent determinations. To determine NP the same course was followed,
except that the time was not noted. For, though there was often a march
in the zero, it was very much smaller, and the observations were made at
suflQciently nearly equal intervals of time without noting exact times. This
small march was doubtless partly due to temperaturechange, but also partly
due, I believe, to further settling down of the cone of the twistinghead into
its bearing through slow squeezing out of the oil. In reconstruction I should
try the effect of replacing the conical bearing by a ballbearing.
Before the reading was made it was found to be absolutely necessary to
move the headpiece some ten or fifteen times to and fro through a small
angle — perhaps diminishing from 20° — on each side of the final position.
If this was not done the wire did not sink down or rise up to its final position,
probably owing to some small friction in the leather washer. After the
alternating motion of the headpiece had been given ten or fifteen times no
further alternation made any difference in the reading. It was only after
finding the necessity of this that I obtained consistent readings.
The maximum and minimum values of NP in a set of 30 determinations
usually differed by about 04 division, only once rising to 08 division.
The maximum and minimum values of T in a set of 30 differed more, as
might be expected; the difference averaging 17 divisions, and once rising to
AND ON THE PRESSURE OF DISTORTIONAL WAVES IN STEEL
403
nearly 5 divisions. This, however, was in the case of the least load, when the
weight was probably insufficient to keep the wiretube quite vertical.
Subsidiary Experiments,
The three following subsidiary experiments were made in order to justify
the methods of measuring the changes of dimensions on twisting :
1. To show that the lengthening on twisting is not due to a change in
Young's modulus, Y.
This is satisfactorily proved by the experiments on twisting described
below. For, suppose that we have a stress P applied to the end of the wire
by a load stretching length I by dl when the wire is not twisted, we have
dl = Pl/Y. Now let the wire thus loaded be twisted through, say, four turns
and let the lowering through twisting be 8. If this is due to a change in
Young's modulus to F, dl + S = Pl/Y'. Whence S = PI (Fi  Fi), and
S should be proportional to P, whereas it is found to be very nearly the same
for loads varying from 5 to 50 kgrm. (approximately).
It appeared worth while, however, to test the question directly, by finding
the extension of wire I for very different loads when 102 kgrm. was added,
first with the wire untwisted, then with the wire twisted through four turns
clockwise. The following results, in micrometer divisions, were obtained,
each the mean of a number of measurements :
Table I.
Load
No twist
Lowering for 102 kgrm.
C4 twist
Lowering for 102 kgrm.
185
385
485
1055
1068
1050
1
1052
1080
1039*
Means
1058
1057
Thus a twist of four turns produced no measurable change in Young's
modulus.
2. To show that the rise and fall of the liquid meniscus were due to, and
measured, the change in volume of the wire in the wiretube.
The most satisfactory way of showing this appeared to consist in using the
apparatus to measure Poisson's ratio a. The load was altered by 102 kgrm.
by alternately immersing the sinker in water and letting the water run out.
* [This corrected value is that given in the original manuscript. Ed.]
26—2
404 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED,
The rise and fall of the needlepoint gave the endextension, and Young's
modulus Y could then be calculated ; while the rise and fall of the meniscus
gave the volumechange, and the sidecontraction and Poisson's ratio a could
be calculated. The rigiditymodulus n could be obtained from n = ^ Y/(l + a).
When the load was increased there was some yield of the supporting
bracket. To determine its amount, a needlepoint was fixed on the bracket
close to the upper chuck and sighted by the microscope. A series of loads
up to 50 kgrm. showed that the lowering per kilogramme was 006 division.
An addition of 102 kgrm., therefore, lowered the bracket by 0061 division,
and this had to be subtracted from the NP reading when used to find Y. In
the lowering of the meniscus T — NP, it obviously did not come into con
sideration. The observed change of volume given by T — NP had to be
corrected by a factor about 1605/156, since only 156 cm. of wire were within
the wiretube and 45 cm. were outside. The actual length outside varied
from 42 to 46, and the factor was varied accordingly.
No doubt better values of Y and a might have been obtained with a larger
change of load, but, to test the apparatus, it was important to observe lower
ings of the same order as those observed in the twisting. In the following
Table II the values of T — NP and NP, due to an addition of 102 kgrm., are
given in micrometerdivisions corrected as above described. Each value is
the mean of 30. The range between maximum and minimum in a set averaged
17 divisions for T and 057 division for NP :
Table II. Elastic Moduli and Poisson's Ratio.
Load in kgrm.
NP
TNP a
1012 y
1012 n
Steel Wire I, Diameter 00986 cm.
185
285
385
485
Mean values
1065
1060
1074
1044
2986
2957
2937
2905
0272
0271
0265
0270
211
212
209
216
0830
0835
0828
0849
—
0270
212
0835
Same Wire with C^ Twist on it
485
1027 1 2804 1 0265 j 218
Steel Wire II, Diameter 01210 cm.
0861
485
705 1 3122 1 0287  212 [
Harddrawn Copper Wire, Diameter 01219 cm
0825
185
1120 6152 0331 131
0493
AND ON THE PEESSURE OF DISTORTIONAL WAVES IN STEEL 405
The values found for a steel wire after annealing, given in the paper
already referred to*, were Y = 206 x lO^^, n = 0809 x 10^2 (by vibration),
whence or = 0273.
The values for the steel wires are sufficiently near to each other and to the
values previously found to show that the tubereadings gave, at any rate, very
nearly the true changes in volume.
3. To show that the changes were very nearly isothermal.
The change in temperature of a solid sheared adiabatically through e is
de= Xnd€y2JC^p,
where An is the decrease in rigidity per degree rise, 6 is the absolute tem
perature, Cp is the specific heat, and p is the density.
Let us suppose that a steel wire 156 cm. long and 005 cm. radius — nearly
wire I — is twisted through one turn. It is sufficient to investigate the effect
for one turn, for both the adiabatic temperaturechange and the twisting
effects are proportional to the square of the shear, and therefore in a ratio
independent of the shear. Then e = 27rr/156, where r is the distance of an
element from the axis. The mean change in temperature of such a wire is
[^ 27Trdedr _ Tr'^a^Xnd
where a = 005.
For steel we may put n = IO12, A = 2 x lO"*, p = 78, C^ = 0112. Taking
6 as 300° A. we find the heat in calories developed by the twist to be about
— 18 X 10* calories. Or on untwisting + 18 x 10"* calories.
If this heat were confined to the steel it would alter its temperature by
about 1/600° C. and its linear dimensions by about 18 x 10"^ in 1. The
twisting through one turn, as will be seen below, alters the radius by about
319 X 10*^ in 1 and the length by about 172 x 10"^. The effects of an
adiabatic change of temperature would, therefore, be appreciable compared
with the effects of twisting, especially on the radius. But the wire shares its
heat, positive or negative, with the water in the wiretube, and here it may
produce a serious effect, if it gets no farther than the wiretube, owing to the
considerable coefficient of expansion of the water. Let us suppose that the
heat or cold is shared with the water in the wiretube so rapidly that both are
at one temperature. The water, having approximately three times the volume
of the steel, has about 7/9 the heatcapacity of steel plus water. So that the
water would receive about 14 x 10"* calories. If a mass of water at a tempera
ture at which its cubical expansion is a receives H calories its change of volume
is Ha, whatever the total volume. In our case the temperature was usually
about 12° C, at which a is about 10"*. Then the volumechange would be
about 14 X 10~^ c.c. and since one micrometerdivision is about 14 x 10~' c.c.
* Loc. cit. p. 554. [Collected Paper s^ p. 366.]
406 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED,
along the capillary, one turn, through thermal effect alone, would produce a
fall on twisting and a rise on untwisting of about 01 division. The actual
change observed on twisting through one turn was about 056 division. If
then the heat or cold only slowly spread from the wire, or again if it were
rapidly shared with the water in the wiretube but only slowly spread thence,
the measurements would be seriously affected. But it is obvious that there
must, in reality, be a rapid adjustment of temperature between the wiretube
and the outer waterjacket, and it was important to find out how rapidly
the adjustment progressed. Fortunately the wire was insulated from the
tube where it passed through the washer, so that it was easy to pass an electric
current through it by connecting the terminals of a battery, one to the bracket,
the other to the wiretube. Heatingcurrents of the order of 1 to 2 amperes
were thus passed along the wire. The current was put on for 2 seconds,
the meniscus rushing up meanwhile fairly uniformly, and the point to which
it rose was read on the micrometer. Then, 15 seconds after the cut off,
the position of the meniscus was read again and the mean of a number of
determinations showed that after 15 seconds only 0032 of the original rise
remained. The original rise varied from 7 to 18 divisions with different
currents. If the twisting were made instantaneously and the reading of the
fall in the tube were made 15 seconds later, about 0032 x 01 ^056 = 0006
of the fall would be due to the cooling on twisting. But this is a very con
siderable overestimate. The twisting was usually begun 25 seconds before
reading and ended more than 15 seconds before. The effect of temperature
change may, I think, be estimated at less than 1/300 of the whole. It was
impossible to assign even an approximate value to it and as it proved to be
so small it was neglected. The effect would have been reduced altogether
beyond consideration if the readings had been taken at intervals of one minute,
but this would have introduced errors, probably much worse, through
irregularities in the march of temperature.
In the experiments on Poisson's ratio the adiabatic change of temperature
on adding a load which stretches length I by dl is
dd =  aYddl/JC^pl,
where a 7 is the change in Young's modulus per degree. The value of a for
steel is about 1/4000. This gives the heat for a stretch of 10 divisions as
about — 9 X 10^ calories. With uniform temperature of steel and water in
the wire tube the water would have about 7 x 10~^ calories, and its effect
would be about 05 division. After 15 seconds it would be about 0016 division.
As the change of level observed was about 30 divisions the effect is negligible.
The direct effect in lengthening or shortening the wire is easily shown to be
very much smaller.
AND ON THE PRESSURE OF DISTORTIONAL WAVES IN STEEL 407
Measurement of the Changes of Dimensions on Twisting,
The method of making the measurements has been described abeady. In
the case of wire I, NP, or the lengthening of the wire, was observed for various
loads for four turns and for two turns clockwise twist, denoted by C4 and Cg,
and for four turns and for two turns counterclockwise twist, denoted by CC^
and CC 2i each value being the mean of 30 determinations — once or twice of
40 — made as described above. As permanent set came in with five turns,
four turns was the maximum twist employed. The mean values of T were
also determined for the same four twists and T — NP was corrected for the
length of wire outside the wiretube. For one load on wire I the lowerings
for Og and COg were also observed.
In all cases the lowering w could be represented very nearly by the parabola
L (n \ cY = w \ h,
where n is the number of turns put on and L, c, and b are constants, not, of
course, the same for NP and T — NP. The constant c represents the fraction
of a turn, always on the counterclockwise side of the point of no twist, about
which the lowering is symmetrical. Putting n = — c, b = — w is a small
shortening, or for a counterclockwise twist c the wire has a minimum length.
The existence of c and b is due to want of homogeneity in the wire. They
may be explained by supposing that the wire in the apparently neutral
condition consists of a core and a sheath twisted against each other, as will
be shown in the theory given later. Owing to want of exact centering the
image * wobbled' somewhat in the field during twisting, and only returned
to the same vertical line after a whole number of turns, so that it was futile
to attempt to measure b. But there was fairly conclusive evidence that it
had a real existence.
According to the theory given, L is the allimportant quantity. The
internal strain only shifts the vertex of the parabola without altering its size.
To find the constants of a parabola which should fairly represent the
results, it was assumed that the curve went through the point w = 0,n = 0,
so that Lc^ = b.
The equation then becomes
L (n^ \ 2nc) = w.
Let w be the lowering for C„, and w' that for CCn, then
L (n2  2nc) = w',
whence
L== {w + w')/2n^ and c = {w  w')linL = ^n (w  w')l(w + w').
The errors are given by
hL = h{w\ w')l2n% and Sc =^ S (w  w')linL,
assuming that L is without error in c.
408 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED,
If then we find the value of L, say L^, from the lowerings at C^ and OC4,
and the value of L, say L^ , from those at G^ and CG^ , the value of the former
should have four times the weight of the latter and we may take the best
value of X as 1^ (4X4 + L^.
The value of c determined from C4 and CC4 should have twice the weight
of that determined from C^ and CC2, and we may take the best value as
J (2C4 + C2).
In the following tables the results are set out. In Table III the lengthening
of the wire I is given in micrometerdivisions, and below each lengthening the
difference, calculated — observed, is put in italics, the calculated values being
those given by the parabolas of which the constants are given in Table IV.
Similar tables are given for wire II, and for the harddrawn copper wire, but
for a single load only, sufficient to secure good centering. After the experience
with various loads with wire I, it appeared unnecessary to vary the load in the
other cases. With wire II it was not thought advisable to go beyond three
turns, and with the copper wire beyond one turn owing to permanent set,
which began to be very considerable beyond those limits. The mode of
calculating the best parabola was modified accordingly.
Table III. Lowering
NP for Steel Wire I, Dimneter 00986 cm.
Load
c.
C2
CC.^ \ CC,
i
kgrm.
485
5095
1452
0768
4233
1
+ 006
 003
1 + 013
 013
385
5353
1422
0818
4152
 005
+ 006
+ 006
 005
285
5265
1500
0858
4259
 001
+ 003
+ 004
 O'lO
185
5382
1680
0905
4091
+ O'U
 013
 004
+ 006
47
5407
1742
0797
3865
1
1
+ 013
 015
 003
+ 004
For load 285, C3 was 3048, and CC3 was 2187, and these were taken into
account in calculating the parabola, the difference in each case being + 005.
Table IV. Constants of Parabolas for Table III.
Load
485
0289
c
0226
b
0015
385
0294
0256
0019
285
0295
0237
0017
185
0302
0281
0024
47
0295
0345
0035
AND ON THE PRESSURE OF DISTORTIONAL WAVES IN STEEL
409
There was probably some permanent set given in the last two owing to
accidental overtwisting of the wire.
The mean value of L is 0295, and within errors of observation it is inde
pendent of the load. It is hardly likely that this is strictly true.
Table V. Lowering of the Meniscus T — NP for Wire /, corrected for
length Outside the Tube.
Load
c.
c^
CO,
CC^
485
1169
394
007
665
■\033
 017
+ 063
 078
385
1163
394
084
636
+ 022
 028
+ 006
 022
285
1182
400
113
638
 025
 02H
 023
+ 004
185
1099
284
100
617
+ 036
+ 037
 002
 002
47
1017
254
099
824
 014
+ 030
+ 057
 072
For load 285, Cg was 733 and CCg was 286. These were used in calcu
lating the parabola, and the differences were respectively — 001 and + 023.
Table VI. Constants of Parabolas for Table V.
Load
L
c
b
485
0559
069
026
385
0569
061
021
285
0578
061
021
185
0=525
054
015
47
0548
029
005
As the errors of observation with the last two loads were about double
those for the earlier loads, they are only given half the weight in finding L.
The value of L is taken as 0561.
Table VII. Lowering NP for Steel Wire II, Diameter 01210 cm.
Two Independent Sets.
Load
Cs
c.
ca.
cc.
I
485
4922
2368
1740
4000
+ 005
 006
 004
+ 005
II
485
4983
2243
1708
4049
 006
+ 005
 001
000
1
410 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED,
Table VIII. Constants of Parabolas for Table YII.
L
c
6
I
II
0501
0499
0154
0147
0012
0011
The mean value of L is 0500.
Table IX. Lowering of the Meniscus T — NP for Wire II, corrected
for length Outside the Tube.
Load
c^
G,
CO.,
cc^ ■
I
485
1516
773
390
922
+ 020
 025
 031
+ 033
II
485
1548
761
371
955
+ 004
 006
 007
+ O'lO
Table X. Constants of Parabolas for Table IX.
L
c
b
I
II
1
1385
1398
035
035
017
017
The mean value of L is 1392.
Copper Wire. Diameter, 01219 cm.
I was only able to use C^ and CC^ owing to permanent set.
The values of NP were 1043 and 0415, and the parabola going through
these points and the origin is 073 {n + 022)^ = iv + 003.
The values of T — NP, corrected for 42 cm. outside the tube, were 8007
and 1433, and the parabola is 472 {n + 035)^ ^w+ 058.
The work given in the former paper appears to justify the assumption of
the parabolic law for copper.
The End Elongation, SideContraction, and VolumeIncrease.
Steel Wire I. Diameter, 00986 cm.
If w is the endlowering for one turn from the position of minimum length
assumed to be L divisions,
w = L X length of one micrometerdivision
= 0295 X 933 x 10^ = 275 x lO"* cm.
AND ON THE PRESSURE OF DISTORTIONAIj WAVES IN STEEL 411
The length is I = 1605. Then
w/l = 171 X 108.
If u is the decrease in the radius a for one turn,
27raul = L X volume of one micrometerdivision of capillary,
u = (0561 X 1393 X 10«) r (2tt x 00493 x 1605)
= 157 X 108 cm.
The radius is 00493 cm. Then
u/a = 319 X 107.
The ratio sidecontraction/endelongation, namely,
u/a ^ w/l = 0187.
If dv is the volumeincrease in total volume v,
dv/v = (irahv — '2malu)/iTaH
= w/l  2u/a = 107 X 106.
All the quantities w/l, u/a, dv/v are proportional to the square of the twist
from the point of minimum length. The ratio u/a ^ w/l is the same for all
twists.
Steel Wire II. Diameter, 01210 cm.
Using the values given in the tables, we have for one turn
w = 466 X 104 cm., u/a = 524 x 10',
w/l = 290 X 106, u/a ^ w/l = 0181,
u = 317 X 108 cm., dv/v = 185 x lO^.
Copper Wire. Diameter, 01219 cm.
The corresponding quantities given by the single set of observations are
not of such weight as those for wires I and II, but I add them here :
w = 681 X 104 cm. u/a = 175 x 10^,
w/l = 425 X 106, .j^ja ^ w/l = 041,
u = 107 X 108 cm., dv/v = 075 x 106.
On comparing the results for wires I and II we see that sidecontraction
~ endelongation is very nearly the same for both. The theory given below
makes both w/l and u/a proportional to the square of the radius for wires of
the same material undergoing the same twist. But as far as these two wires
are concerned they are very nearly proportional to (radius)^'^. I do not
think the discrepancy is to be ascribed to experimental error. Perhaps the
theory is inadequate, but I think that it is more probable that slight differences
in the material, not greatly affecting the ordinary elastic moduli, may produce
412 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED,
very considerable changes in what we may term the secondary moduli, which,
in the theory below, are denoted by f and q.
I should like to have taken observations on several more steel wires with
a wider range of diameters, but I am not able to continue the work at present.
Experimental Verification of a Reciprocal Relation.
In the Philosophical Magazine for November, 1911, vol. 22, p. 740,
Dr. R. A. Houston has expressed the reciprocal relation between the stretching
and twisting of a wire (confined within limits of reversibility) in the form
dd\ fdw\
(S) =() . (1)
\aj^ / Q const. \aix/ p const.
where F is the end pull and w the increase in length, G the torque and 6
the twist on the wire (I use letters for length and torque differing from
Dr. Houston's).
As the apparatus only needed small modification it appeared to be worth
while to see how nearly this relation was verified, and wire II was used for
the purpose. Incidentally, the value of the rigidity was obtained by the
method of statical torque.
When the observations needed are worked out it is found that they are
identical, as of course was to be expected, with those needed to verify the
relation
m ^C^l) , (2)
\du J ,^, const. \^^^/0 const.
which is the more direct expression of the Conservation of Energy in these
phenomena.
Taking equation (1) we require to know on the left the extra twist dd
which must be put upon the wire to keep G the same when a load dF is
added. For this purpose the wire was initially loaded with 185 kgrm., and
the head was turned through a right angle. The bar at the bottom was
also turned through a right angle from its usual position. On the crossbar
a mirror was fixed reflecting into a telescope a millimetrescale 1565 cm.
away. The ends of the crossbar were rounded into arcs of a circle with
centre in the axis of the wire and radius 1470 cm. Horizontal threads
passed ofi these arcs to two very light horizontal spiral springs which stretched
very uniformly in proportion to the pull up to 40 or 50 grm. These springs
were attached to the bases of two travelling microscopes, of which the
horizontal scales merely were used to measure any change in stretch. Initially,
the wire was without twist, and the position of the crossbar on the scale was
read. It would have been at least very difficult to determine directly the
total stretch of the springs required to keep the crossbar in position when the
head was twisted, so the following plan was adopted :
AND ON THE PRESSURE OF DISTORTIONAL WAVES IN STEEL 413
A half turn CC was put on by the headpiece, and the springs were stretched
so as to bring the crossbar to its original position. Then a further turn and
a half CC was put on, and the additional stretch of each spring needed to keep
the crossbar in position was read. This additional stretch multiplied by
4/3 gave the total stretch of the springs, and thus the pull exerted at each
end of the crossarm to maintain two turns twist on the wire. The full
stretches thus computed were 12340 cm. on the left and 12384 cm. on the
right, corresponding to pulls, according to previous calibration, of 49*24 grm.
and 4687 grm., mean 4806 grm. The torque was therefore
G = 4806 X 981 x 294 = 138 x 10^ dynecm.
From this the rigidity is
n = 0838 X 1012.
The tubemethod gave 0825 x 10^^^ ^^d the nearness of the two values
appears to show that the springs could be trusted fairly well.
A load dF = 30 kgrm. was then added, and the torque for two turns was
thereby diminished. The springs therefore contracted, and it was observed
that they pulled the crossbar round through 1585 mm. on the scale — the
mean of five different observations ranging from 1545 to 1655, or through
an angle 000507 radian.
Denoting this angle by SO, and the radius of the crossbar arm by k, and
the decrease of torque by SG,
f=^>:^ (3)
where s is the whole mean stretch of the springs for two turns.
But we require the twist dO, which must be put on the wire from its
initial two turns, and in the opposite direction to Sd, to restore the torque
to G. This is given by
G + dd
GhG dW
here = i^r.
whence, on substituting for SG/G from (3), we get
dd^[^l)W, and jp=[~l)j^ W
Taking the right hand of equation (1), we require to know the lowering dw
for a change dG in the torque under constant load. We get the lowering
from the equation
L{ncY = w\h,
giving dw = 2 L {n — c) dn = L [n  c) —.
Also dir = —K.
dw L(nc)e , .
Ti^^^ m^^^rG' *^^
414 ON CHANGES IN DIMENSIONS OV A STEEL WIRE WHEN TWISTED,
Equating (4) and (5), we ought to find
ttG fkd \ SO
Substituting the known values on the right, viz., G = 138 x 10^, n = 2,
c = 015, h = 147, e = 4^,s= 1237, 86 = 000507, dF = 30 x 981000, we get
L = 448 X 10*.
The observed value of L is given as w on p. 411, viz.,
L = 466 X 104,
showing as close an agreement as could be expected, considering the errors of
observation.
Taking the second reciprocal relation (2), to find [jn] we must twist
\aC7/^ const.
through dd and observe dw, and then calculate what load dF must be removed
to restore the original length.
We have dw = L [n — c) — , and dF = 7ra^Ydw/l.
dF L(nc)a^Y
To find If) , we put on a load W and observe W. If hG is the
\dwJ0const.
diminution in torque at this point and dG the diminution in torque with the
original twist 6,
GhG _dW
GdG" 6 '
Substituting for SG/G from (3), this gives
the change in torque for addition W when 6 is constant.
The value of dw for this load is
dw=lWl7ra^Y,
dG\ fhO ^\iTa^YG
and ^ ^^ ^
I const
Vc^Weconst. V5 / IWd
dF for ^
\ s J
Equating (7) and (8) and putting dF for W, we get
ttG fkd ^\ SI
. OdF
the same equation as before.
If we could use a wire without any internal strain when untwisted, c would
be zero, and we could calculate L, the lowering for one turn, from observations
on the torque and load alone.
AND ON THE PRESSURE OF DISTORTION AL WAVES IN STEEL 415
Fig. 2.
A Theory of the Changes of Dimension on Twisting: The Stresses in
a Finite Pure Shear.
In the paper already referred to* I showed that in a finite pure shear
€ such as is represented in Fig. 2, in which a cube of section ABCD is sheared
into a figure of section ABKL through an angle CBK = €, the thicknesses
perpendicular to AB and to the plane of the figure remaining constant, the
lines of maximum elongation and contraction are, to the order of c^, at right
angles before the shear, making an angle e/4 with the diagonals of the square, as
AE and BG. After the shear they are
again at right angles to the order of
€^, and make an angle c/4 with the
diagonals on the other side as AF and
BH. Since we have elongation in one
direction AF, and contraction in a
direction BH at right angles, the shear
may be maintained by a pressure P
along BH and a tension Q along AF
as far as forces in the plane of the
figure are concerned.
If we go to the first order of € only,
If we go to the second order we must put
P = ne + pe^,
where ^ is a constant to that order.
If we reverse e, P becomes equal to — Q, so that we have
— Q = — ne \ fe^ or Q = ne — pe^
We can only assume that there is no pressure or tension perpendicular to
the plane of the figure, if we neglect e^. Going to the second order, we have
to allow the possibility of a pressure of that order, which we may put as
S = qe%
where if q is negative the force is a tension.
Considering the equilibrium of the wedge ABC, Fig. 3, with AC in the
direction of greatest elongation and BC in that of greatest contraction,
I showed that the tangential stress along AB is, to the second order,
T = n€,
and that a pressure is required perpendicular to AB given by
R=(in + p) e\
* Loc. cit. p. 546. ICollected Papers, p. 368.]
416 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED,
The analysis stopped here and was incomplete, as no account was taken of
the stresses on the plane CD, Fig. 3, perpendicular to AB. It requires to be
supplemented as follows :
Considering the equilibrium of the wedge CDB, let us suppose that on
CD there is a tangential stress T' along CD, and a pressure R' perpendicular
to it.
C
P
R = (^n+p)e2 T=ii£
Ii(trL+p)&2
*T=Tl&
R=(Jn+p)£^
T=ne t
E=(n+p)&^
Fig. 4.
Resolving all the forces on CDB in a direction parallel to DB,
R' . CB sin (45 + Je)  T . C5 cos (45 ^le) + Q.CB sin (45 + Je) = 0,
whence R' = T cot (45 + Je)  Q ;
or, since T = ne and cot (45 + Je) = (1 — Je), neglecting e^, as it is multi
plied by e,
R' = n€(l ic) n€ + pe^ = { ^n + f) eK
AND ON THE PRESSURE OF DISTORTION AL WAVES IN STEEL 417
Kesolving in a direction parallel to CD,
T . CB sin (45 + ^e)  R . CB cos (45 \ ie)  Q . CB cos (45 + Jc) = 0,
whence T = (R + Q) cot (45 + Jc) = {^n + p) c^ + w€  j>€^} (1  Jc)
= n€ to the second order.
On a unit cube of the material in the sheared condition then, we have, a&
in Fig. 4,
Tangential stresses along AB and CD each • ne.
Tangential stresses along ^Z) and 50 each n€.
Pressures perpendicular to ^5 and CD each (\n \ f) €^.
Pressures perpendicular to AD and BC each (— \n + p) e^.
And pressures perpendicular to the plane of the
figure each ^e^ or, in more convenient form... (q — f) e^ { peK
The Strains in a Finite ShearStress consisting of Tangential Stress T, T' only.
If an element is subjected to the system of stresses just investigated, when
we put on to it a system of tensions equal and opposite to the second order
pressures we have just found, we leave only the tangential stresses T = T' = ne.
The strains due to these tensions must be superposed on the shear €, and we
shall then have the strains due to the tangential stresses only.
We have then to examine the strains due to tensions
iin + p) e2 on AB and CD (Fig. 4).
{ in + p) e2 on AD and BC.
{q ~ p) €^+ pe^ perpendicular to the plajie of the figure.
Through the tension pe^ on every face we get an extension in all directions
pe^/SK, where K is the bulkmodulus.
The tensions ^ne^ on AB and CD and the pressures Jwe^ on AD and BC
constitute a shearstress giving an elongation parallel to BC of ^ne^/n = Je^,
and a contraction parallel to AB also Je^.
The tensions {q — p) e^ perpendicular to the plane of the figure give an
elongation perpendicular to that plane y (q — p) e^, and contractions at right
angles, viz., along AB and AD, y {q — p) e% where Y is Young's modulus and
a is Poisson's ratio.
Collecting the results, we have secondary strains accompanying the shear
€ as follows :
1 pe^ o
An elongation parallel to BC = 7 ^^ + o^ — y (9' — /^) ^^ •
i>e2 1
,, perpendicular to the plane = f^ + y (5' — V) ^^•
p. c. w.
27
418 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED,
As in the experiments, described abovp, Y and a were determined directly,
it will be convenient to replace K from the equation ^ = — y — , and the
secondary strains become
/I 1(T C7\, f 1 1G (^ \ 2 /2(7 I \ .
Equations Representing the Changes in the Dimensions of a Wire
Subject to a Torque.
I am indebted to Sir Joseph Larmor for his kindness in indicating how the
following equations should be formed and solved. Let us assume that we
put on to a wire of length I and radius a a pure shearstress proportional to
the distance r from the axis, and twisting the length I through angle 6. Then
in addition to the shear € = rd/l, this stress would produce in an element un
constrained by neighbouring material what we may term 'free strains' with
the values just found, which we may write as ar^ radial, ^r^ transverse to
the radius, and yr^ longitudinal ; where
/ 2g 1 \
=[Y^^Y^)
^=(4+ F^^T^j
/I 1  cr o \
r=(+4 + ^^Py^^j
(1)
If u is the actual radial displacement, and if w is the actual longitudinal
displacement, the strains in addition to the shear e are, in cylindrical
coordinates,
dujdr, u/r, and dwjdz.
The differences between these actual strains and the 'free strains,' viz.,
du 2 ^ w o 2 ^^ 2 /o\
^^~dr~ ' f = r~^ ' ^^cfo~^ ' *^
imply 'secondary stresses' in the wire due to adjustment of strain in neigh
bouring elements. Let these be denoted hj R, 0, W.
To find R, 0, and W, we treat e, /, g as if they were strains in an inde
pendent system. Putting A = e \f + g, the equations are
R=XA + 2fjLe, = AA + 2/x/, W = XA + 2fig, (3)
where X = K — %n = , — , — ^^^ ^^^ and jx = n
(1 + 0) (1  2(7) ^ 2(1+ C7)
AND ON THE PRESSURE OF DISTORTION AIj WAVES IN STEEL 419
The forces R, 0, and W must form a system in equilibrium, there being
no external forces to balance. Considering the equilibrium of the element
ABCD, Fig. 5,
d{RrSd) = eSddr, whence r^ + R=e. ...(4)
We obtain another equation by assuming that the wire is
so gripped at each end that sections perpendicular to the axis \ ^dr
remain perpendicular to the axis after twisting. Indeed, we ^ 1^
have already assumed this in omitting equations for shear
stress in (3). Hence w is independent of r and dw/dz is con \^d/^
stant over a section for a given wire with a given twist. Let
us put
dw/dz = h.
' Fig. 6.
Further, the load is constant, so that
r Wrdr  (5)
•0
Substituting in (4) from (3) we obtain
^,du_u_ 2A (g + i8 + y) + 6/xa  2ju^ ^
^ dr^^ dr r~ X + 2fji ^ ^^^
By putting u/r = v, we easily find the solution
u = Ar^\Bri Gr\ (7)
where ^ ^ 2A (a 4 ^ + y) 4 6/.a  2^^
8 (A + 2/x)
and B and C are arbitrary constants to be determined by the boundary
conditions.
If the wire is unstrained in all parts before twisting, the solution applies
with the same constants for all parts.
In order that u = when r = 0, we must have = 0, so that
u = Ar^ + Br (8)
When T = a,R = 0,
Substituting from (8) in the value of R in (3), and putting J? = when
r = a, we get
2 (A + /x) 5 + AA = {A (a + )8 + y) + 2/xa  (4A + 6/x) A} a\ ...(9)
From equation (5) we obtain another relation between B and h, when we
substitute for u from (8) in W from (3) and integrate from r = to r = a, viz.,
A5 + (iA + /^) /^ = ttA (a + i8 + y) + J/xy  A^} a\ (10)
and from (9) and (10) we can find B and h,
27—2
420 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED,
Since ^ is a linear function of a, ^, and y, and each of these is proportional
to 6^, h and B are proportional to 6^. Substituting for B in (8), u is also
proportional to 6^. The theory, then, gives the paraboHc law for the twisting
of a wire initially unstrained both for lengthening and for sidecontraction.
It also gives the lengthening and sidecontraction w/l and u/a for different
wires of the same material as proportional to a^.
So far the theory does not, of course, give any account of the fact that the
wires examined are always unsymmetrical, that the effects always date from
a point c, on the counterclockwise side in the wires examined, c being different
for w and u. This want of symmetry imphes initial internal strain, probably,
in reahty, very comphcated. Let us examine a simple case in which there is
a core, radius a, twisted initially against a sheath, outer radius b, and let the
opposing twists be respectively 6^ and 6^. When we put a twist 6 from
outside on to the core as a whole the core is twisted through 6 + d^., and the
sheath through 6 — Og For the core and sheath respectively we have
Uc = Ar^ + Br,
u, = A'r^ + B'r + C'r\
where ^ is a linear function of a, ^, y, and therefore proportional to {6 + 9^)^
and A' is the same function of a, ^' , y, say, and therefore proportional to
To find the constants we have
Uc = Ug when r = a,
R. = „ r = L
and 1 W^rdr^l Wgrdr=0.
These give us four equations to find B, B', C , h of the form (it appears
needless to give the detailed work)
B' = p2 (0 + e,Y +Q,{e e,Y, h = p,(e + e,Y+ q, (b e,f;
and, substituting for A' , B', C in u^, and putting r = 6, we get
Both h and Wj, are of the form
Dd^ + Ee + F,
where D does not contain 6^ or 6^. As the parabolas depend only on D,
E and F merely giving the position of the vertex, Oc and Og only affect that
position.
AND ON THE PRESSURE OF DISTORTIONAL WAVES IN STEEL 421
To find that position we may put dhjdd = for the one, dujdJd = for the
other, and since h and u are different functions of dc and dg the vertices will
be at different points for the two quantities h and n.
Taking this simple case as a guide we shall assume that internal strain
only affects the position and not the size of the parabola representing the
change of linear dimensions on twisting.
Hence if we could obtain a wire without internal strain we should have
Ln^ = w,
where L has the value found in the experiments on the actual, initially
strained wire, and we may regard the values uja and wjl for one turn as
the values for a wire initially without internal strain.
The Values of p and q in the Secondary Stresses.
We are now able to find the values of p and q. For the known values of
Y and a enable us to find a, ^ and y in equations (1) in terms of p and q
for a known twist, which we shall take as one turn, or as 2?? in length
I = 1605 cm. We also know A and /x, since
A = (7 7/(1 + 0) (1  2(7) and ^i = n = 7/2 (1 + a).
Substituting for A, fx, a, p, and y we can determine A in terms of p and q.
Then from equations (9) and (10) we can find B and h in terms of p and q.
Equating Aa^ + B to the observed value of u/a (which is negative), and h to
the observed value of w/l, we have two linear equations in p and q.
The arithmetic is straightforward, though very lengthy, and may be
omitted. I have used a slide rule in the calculations.
Using the values of the elastic constants Y and a from Table II, and the
values of u/a and w/l, on p. 411, I find for wire I
p = 167 X 1012, q= _ 070 x lO^^,
so that the force perpendicular to the plane of the figure in Fig. 4 is a tension
and not a pressure.
The Pressure in the Direction of Propagation in Distortional Waves and
the Longitudinal Waves Produced by the Pressure.
If we had a train of waves purely distortional, that is, a train in which
the strain could be represented by a pure shear e, there would be a pressure
in the direction of propagation (Jw + p) e^. But as e varies from point to
point in the train, the pressure due to the shear strain varies, and there must
be longitudinal disturbance, longitudinal waves, accompanying the distortional
waves. The longitudinal strain implies that the material yields under the
422 ON CHANGES IN DIMENSIONS OF A STEEL WIRE WHEN TWISTED,
pressure, and the pressure will, in general, have a different value from that
in a pure shear.
Let us represent the distortional train by
€ = 7; sm ^ (a:; — vt),
where v^ = n/p and tj is the ampHtude of the shear.
If f is the longitudinal displacement at the point where the shear is €,
d^/dx is the elongation of the element about the point.
Now if we shear a cube, and remove the pressure (Jw + p) e^, the cube
elongates in that direction, and if the dimensions in the two directions at
right angles are maintained the same, the removal of the pressure produces
elongation
y^ {^n + f) e^, where v = X + 2fjL = K + ^n.
This we may term the 'free elongation' in the direction of propagation
on the supposition that there is no change of length at right angles to it.
The pressure due to the shear falls from its full value {^n + p) e^ to
while the elongation increases from to its full value v~^ (Jn + f) e^. When
the elongation is d^jdx the pressure remaining is
= (\n + f) rf sin^ X ^^ ~ ^^^ "" ^^^1^^
The equation of motion for the longitudinal waves is
^ d^ ^~dx^~ '^^ + ?^) "?' X ^^^ T ^^ ~ "^'^^ "^ vd^i/dx^
an equation similar in form to that for the longitudinal waves which I have
attempted to show must accompany light waves *.
4:77
If we put f = ^ sin y {x — vt — a),
and substitute in the above equation, we find on putting a; = 0, i = 0, that
a = 0, and
_ (jn + v) rj^X
Stt {pv^  v) '
or if v' is the velocity of free longitudinal waves, since pv'^ = v and v'^ > v^y
If we substitute for d^/dx in P, we get
P = J (^n + p)7]^ h — cos Y {^ — vtn — vA ^ cos y (x — vt),
* Roy. Soc. Proc, A, vol. 85, p. 474. [Collected Papers, Art. 29.]
AND ON THE PRESSURE OF DISTORTIONAL WAVES IN STEEL 423
We may regard tkis as made up by a steady pressure J (Jw + p) r\^ and
a purely periodic pressure, of which the average is zero.
If B is the energy per cubic centimetre at any point in the distortional
waves, it is half kinetic energy, half strainenergy. The latter is Jw€^, so
that the total is wc^ or
E = ntf sin^ y (ic — vii) = \n'rf y — cos ^ (a; — v^) k
Then the average value is
E = InqK
If we denote the average pressure by P,
p^jro + y^
n
If we use the values of n and p found for wire I, we find
P = 250^.
If we put the energy per cubic centimetre in the longitudinal waves as
Average energy in longitudinal waves 1 v'^ + v^ {^n + f)
2
2 . n'
Average energy in distortional wav^s 8/o [v"^ — v^)''
so that the ratio is proportional to tf and therefore in any actual waves it is
very small.
The pressures at right angles to the line of propagation will not produce
any disturbance in a wavefront where t] is constant. Round the edges of the
wavefront, however, where t] is diminishing as we go outwards, they may
have effects, and it appears likely that they may give rise to disturbances
propagated sideways.
I have much pleasure in recording my hearty thanks to Mr. 6. 0. Harrison,
mechanic in the laboratory workshop, for his great help in planning the
apparatus used in the experiments described in this paper, for his skill in
constructing it, and for his assistance in making the observations.
31.
THE CHANGES IN THE LENGTH AND VOLUME OF
AN INDIARUBBEK COED WHEN TWISTED.
[The IndiaRvhber Jomnal, October 4, 1913, p. 6.]
In some investigations on the way in which pressure might be produced
by transverse waves in a solid, analogous to the minute pressure produced
by lightwaves, the author was led to expect that a wire with a constant
load on it would lengthen, when twisted, by an amount proportional to the
square of the twist, and he gave at the Winnipeg meeting of the British
Association an account of experiments which fully verified the expectation.
According to the theor} used, there should also be accompanying the
lengthening a diminution in the radijus, and a description has been published
in the Proceedings of the Royal Society* of experiments which show that the
diminution exists and follows the same law. The changes are very minute, of
the order of a millionth in the length and in the diameter when a steel wire
160 cm. long and 1 mm. diameter is twisted through one turn. The volume is
also slightly increased. If instead of allowing the length to increase it had
been kept constant by reducing the load, there would with steel have been a
slight outstanding increase in the volume.
The author thought it might be interesting to look for similar effects in
indiarubber. To investigate the lengthening, he used a rubber cord 118 cm.
long and 12 cm. diameter, of which the upper end was attached to a vertical
axis which could be rotated in a bearing. The lower end was attached to
a horizontal crosspiece between four stops, which allowed the crosspiece,
and therefore the end of the cord, to rise or fall, but prevented rotation.
To the crosspiece there was attached the ordinary wheelbarometer device
for magnifying up and down motion. There was a lengthening on twisting,
somewhat irregular, not proportional to the square of the twist, but increasing
rather less rapidly. Two turns of twist gave an average lengthening of
0088 cm., or about 750 in a million, vastly greater than the lengthening
of the steel wire with a similar twist. But, as with steel wire, the lengthening
is rather more than proportional to the square of the diameter. A rubber
* [Collected Papers, Art. 30.]
CHANGES IN AN INDIARFBBER CORD WHEN TWISTED 425
cord 1 mm. in diameter, and with the length and twist of the steel, would
probably have increased in length by an amount of the same order as that
observed with steel.
To find whether there was a change in diameter, a cord of the same length
and diameter as that used for the lengthening was enclosed in a vertical glass
tube with brass ends, the lower end of the cord being attached to the lower
brass end, and the upper end to a vertical axis coming into the tube through
as close fitting a bearing in the upper brass end as could be made. This axis
could be rotated, and so any twist could be put on the cord while it remained
of constant length. The tube was filled with water, and as it was provided
at one side with a capillary tube which issued through a hole near the top
and rose above the upper end, any change in the volume of the rubber on
twisting would have been indicated by a rise or fall of the water surface in
the capillary, and this was viewed by a measuring microscope. When two
turns were put on the rubber, small changes of volume were observed, now
one way, now the other, probably due to errors of experiment. But the
changes were very small, and the mean change so minute that it appears
safe to say that the real change in volume was not so much as one in two
millions. It was therefore, if it existed at all, of an order not greater than for
the steel wire above described. If the cord had been only 1 mm. in diameter
like the steel, and had been of the same length, and had been subjected to
the same twist, the change in volume would have been vastly less than in
the case of steel.
[This appears to be the only published notice of an account of this work which was given at
the meeting of the British Association at Birmingham in 1913. Ed.]
APPENDIX BY SIE J. LAKMOR ON THE MOMENTUM OF
RADIATION.
[The following extract from a lecture 'On the Dynamics of Radiation' by Sir Joseph
Larmor, read before the Fifth International Congress of Mathematicians at Cambridge
in x4.ugust 1912, is inserted here, after consultation with the Author (whose permission
was requested), in further elucidation and illustration, chiefly from the side of the electric
theory, of Poynting's experiments resting on the momentum of radiation. Ed.]
General theory of pressure exerted hy waves.
If a perfectly reflecting structure has the property of being able to advance
through an elastic medium, the seat of free undulations, without producing
disturbance of structure in that medium, then it follows from the principle
of energy alone that these waves must exert forces against such a reflector,
constituting a pressure equal in intensity at each point to the energy of the
waves per unit volume. Of. p. 432, infra. The only hypothesis, required in
order to justify this general result, is that the velocity of the undulations in
the medium must be independent of their wavelength; viz., the medium is
to be nondispersive, as is the free aether of space.
This proposition, being derived solely from consideration of conservation
of the energy, must hold good whatever be the character of the mechanism
of propagation that is concerned in the waves. But the elucidation of the
nature of the pressure of the waves, of its mode of operation, is of course
concerned with the constitution of the medium. The way to enlarge ideas
on such matters is by study of special cases : and the simplest cases will be
the most instructive.
Let us consider then transverse undulations travelling on a cord of linear
density pQ , which is stretched to tension Tq . Waves of all lengths will travel
with the same velocity, namely c = (Tq/pq)^, so that the condition of absence
of dispersion is satisfied. A solitary wave of limited length, in its transmission
along the cord, deflects each straight portion of it in succession into a curved
arc. This process implies increase in length, and therefore increased tension,
at first locally. But we adhere for the present to the simplest case, where
the cord is inextensible or rather the elastic modulus of extension is indefinitely
great. The very beginnings of a local disturbance of tension will then be
equalised along the cord with speed practically infinite ; and we may therefore
take it that at each instant the tension stands adjusted to be the same (Tq)
all along it. The pressure or pull of the undulations at any point is concerned
onlv with the component of this tension in the direction of the cord ; this is
where 77 is the transverse displacement of the part of the cord at distance x
ON THE MOMENTUM OF RADIATION 427
measured along it; thus, up to the second order of approximation, the pull
of the cord is
T
*^o(S
The tension of the cord therefore gives rise statically to an undulation
pressure
The first of these three equivalent expressions can be interpreted as the
potential energy per unit length arising from the gathering up of the extra
length in the curved arc of the cord, against the operation of the tension Tq ;
the last of them represents the kinetic energy per unit length of the undulations.
Thus there is a pressure in the wave, arising from this statical cause, which is
at each point equal to half its total energy per unit length.
There is the other half of the total pressure still to be accounted for. That
part has a very different origin. As the tension is instantaneously adjusted to
the same value all along, because the cord is taken to be inextensible, there
must be extra mass gathered up into the curved segment which travels along
it as the undulation. The mass in this arc is
or to the second order is approximately
In the element Sx there is extra mass of amount
which is carried along with the velocity G of the undulatory propagation.
This implies momentum associated with the undulation, and of amount at
each point equal to \pqgI^\ per unit length. Another portion of the un
dulation pressure is here revealed, equal to the rate at which the momentum
is transmitted past a given point of the cord; this part is represented by
IPqG"^ i^A or \pq (^ j , and so is equal to the component previously
determined*
In our case of undulations travelling on a stretched cord, the pressure
exerted by the waves arises therefore as to one half from transmitted intrinsic
stress and as to the other half from transmitted momentum.
The kinetic energy of the cord can be considered either to be energy
belonging to the transverse vibration, viz., \\p {^\ ds, or to be the energy
4?8 APPENDIX BY SIR J. LARMOR
of the convected excess of mass moving with the velocity of propagation G*,
viz., I 2/° (t^ ) <^^^^ ; for these quantities are equal by virtue of the condition
of steady propagation jl = <^ ^ •
On the other hand the momentum that propagates the waves is transverse,
of amount p ~ per unit length ; it is the rate of change of this momentum
that appears in the equation of propagation
dtV'dt) dx\ dx)'
But the longitudinal momentum with which we have been here specially
concerned is J/jf—] c per unit length, which is p.p7^. Its ratio to
the transverse momentum is very small, being \j^', it is a secondorder
phenomenon and is not essential to the propagation of the waves. It is in
fact a special feature, and there are types of wavemotion in which it does
not occur. The criterion for its presence is that the medium must be such
that the reflector on which the pressure is exerted can advance through it,
sweeping the radiation along in front of it, but not disturbing the structure;
possibly intrinsic strain, typified by the tension of the cord, may be an essential
feature in the structure of such a medium.
If we derive the dynamical equation of propagation along the cord from
the Principle of Action 8  (T  110 dt = 0, where
[i.(5/..a„d>F=ir„(g)
2
dx.
the existence of the pressure of the undulations escapes our analysis. A corre
sponding remark applies to the deduction of the equations of the electro
dynamic field from the Principle of Action f. In that mode of analysis the
forces constituting the pressure of radiation are not in evidence throughout
the medium ; they are revealed only at the place where the field of the waves
affects the electrons belonging to the reflector. Problems connected with
the FaradayMaxwell stress lie deeper; they involve the structure of the
medium to a degree which the propagation of disturbance by radiation does
not by itself give us means to determine.
We therefore proceed to look into that problem more closely. We now
postulate Maxwell's statical stress system; also Maxwell's magnetic stress
system, which is, presumably, to be taken as of the nature of a kinetic
reaction. But when we assert the existence of these stresses, there remain
over uncompensated terms in the mechanical forcive on the electrons which
* [This specification is fictitious; indeed a factor J has been dropped in its expression just
following. There is however actual energy of longitudinal motion; as it belongs to the whole
mass of the cord, which moves together, it is very small in amount, its ratio to the energy of
transverse vibration being \ {drjldc^^. J. L.]
t Cf. Larmor, Trans. Camb. Phil. Soc. vol. 18 (1900), p. 318; or Aether and Matter, Chapter vi.
ON THE MOMENTUM OF RADIATION 429
may be interpreted as due to a distribution of momentum in the medium*.
The pressure of a train of radiation is, on this hypothetical synthesis of stress
and momentum, due entirely (p. 431) to the advancing momentum that is
absorbed by the surface pressed, for here also the momentum travels with
the waves. This is in contrast with the case of the cord analysed above, in
which only half of the pressure is due to that momentum.
The pressure of radiation against a material body, of amount given by
the law specified by Maxwell for free space, is demonstrably included in the
Maxwellian scheme of electrodynamics, when that scheme is expanded so as
to recognise the electrons with their fields of force as the link of communica
tion between aether and matter. But the illustration of the stretched cord
may be held to indicate that it is not yet secure to travel further along with
Maxwell, and accept as realities the FaradayMaxwell stress in the electric
field, and the momentum which necessarily accompanies it; it shows that
other dynamical possibilities of explanation are not yet excluded. And,
viewing the subject from the other side, we recognise how important have
been the experimental verifications of the law of pressure of radiation which
we owe to Lebedew, too early lost to science, to Nichols and Hull, and to
Poynting and Barlows The law of radiationpressure in free space is not
a necessary one for all types of wavemotion ; on the other hand if it had not
been verified in fact, the theory of electrons could not have stood without
modification.
The pressure of radiation, according to Maxwell's law, enters fundamentallv
in the BartoliBoltzmann deduction of the fourthpower law of connection
between total radiation in an enclosure and temperature. Thus in this
domain also, when we pass beyond the generalities of thermodynamics, we
may expect to find that the kAvs of distribution of natural radiant energy
depend on structure which is deeper seated than anything expressed in the
Maxwellian equations of propagation. The other definitely secure relation
in this field, the displacementtheorem of Wien, involves nothing additional
as regards structure, except the principle that operations of compression of
a field of natural radiation in free space are reversible. The most pressing
present problem of mathematical physics is to ascertain whether we can
evade this further investigation into aethereal structure, for purposes of
determination of average distribution of radiant energy, by help of the
BoltzmannPlanck expansion of thermodynamic principles, which proceeds
by comparison of the probabilities of the various distributions of energy
that are formally conceivable among the parts of the material system which
is its receptacle.
Momentum intrinsically associated with Radiation.
We will now follow up, after Poynting f, the hypothesis thus implied in
modern statements of the Maxwellian formula for electric stress, namely that
the pressure of radiation arises wholly from momentum carried along by the
waves. Consider an isolated beam of definite length emitted obliquely from
a definite area of surface A and absorbed completely by another area B. The
* For the extension to the most general case of material media cf. Phil. Trans, vol. 190 (1897)..
p. 253.
t Cf. Phil. Trans, vol. 202, A (1903). [Collected Papers, Art. 20.]
430 APPENDIX BY SIR J. LARMOR
automatic arrangements that are necessary to ensure this operation are easily
specified, and need not detain us. In fact by drawing aside an impervious
screen from A we can fill a chamber AA' with radiation ; and then closing
A and opening ^', it can emerge and travel along to B, where it can be
absorbed without other disturbance, by aid of a pair of screens B and B' in
like manner. Let the emitting surface ^ be travelling in any direction while
the absorber B is at rest. What is emitted by A is wholly gained by B, for
the surrounding aether is quiescent both before and after the operation.
Also, the system is not subject to external influences; therefore its total
momentum must be conserved, what is lost by A being transferred ultimately
to B, but by the special hypothesis now under consideration, existing mean
time as momentum in the beam of radiation as it travels across. If v be the
component of the velocity of A in the direction of the beam, the duration
of emission of the beam from A is (1 — v/o)^ times the duration of its
absorption by the fixed absorber B. Hence the intensity of pressure of
a beam of issuing radiation on the moving radiator must be affected by
a factor (1 — vJG) multiplying its density of energy; for pressure multiplied
by time is the momentum which is transferred unchanged by the beam to
the absorber for which v is null. We can verify readily that the pressure of
a beam against a moving absorber involves the same factor (1 — vjc). If the
aA^
/
B'
emitter were advancing with the velocity of light this factor would make the
pressure vanish, because the emitter would keep permanently in touch with
the beam : if the absorber were receding with the velocity of light there would
be no pressure on it, because it would just keep ahead of the beam.
There seems to be no manner other than these two, by altered intrinsic
stress or by convected momentum, in which a beam of limited length can
exert pressure while it remains in contact with the obstacle and no longer.
In the illustration of the stretched cord the intrinsic stress is transmitted and
adjusted by tensional waves which travel with velocity assumed to be prac
tically infinite. If we look closer into the mode of this adjustment of tension,
it proves to be by the transmission of longitudinal momentum ; though in
order that the pressure may keep in step, the momentum must travel with
a much greater velocity, proper to tensional waves. In fact longitudinal
stress cannot be altered except by fulfilling itself through the transfer of
momentum, and it is merely a question of what speeds of transference come
into operation.
In the general problem of aethereal propagation, the analogy of the cord
suggests that we must be careful to avoid undue restriction of ideas, so as,
for example, not to exclude the operation, in a way similar to this adjustriient
of tension by longitudinal propagation, of the immense but unknown speed
of propagation of gravitation. We shall find presently that the phenomena
of absorption lead to another complication.
ON THE MOMENTUM OF RADIATION 431
So long, however, as we hold to the theory of Maxwellian electric stress
with associated momentum, there can be no doubt
as to the validity of Poynting's modification of the
pressure formula for a moving reflector, from which
he has derived such interesting consequences in
cosmical astronomy. To confirm this, we have only
to contemplate a beam of radiation of finite length I
advancing upon an obstacle A in which it is
absorbed. The rear of it moves on with velocity c;
hence if the body A is in motion with velocity
whose component along the beam is v, the beam will be absorbed or passed
on, at any rate removed, in a time l/io — v). But by electron theory the
beam possesses a distribution of at any rate quasimoTonentum. identical with
the distribution of its energy, and this has disappeared or has passed on in
this time. There must therefore be a thrust on the obstructing body, directed
along the beam and equal to e (1 — n/o), where € is the energy of the beam
per unit length which is also the distribution of the quasimoraentum. along
the free beam.
The back pressure on a radiating body travelling through free space,
which is exerted by a given stream of radiation, is by this formula smaller
on its front than on its rear; so that if its radiation were unaffected by its
motion, the body would be subject to acceleration at the expense of its internal
thermal energy. This of course could not be the actual case.
The modifying feature is that the intensity of radiation, which corresponds
to a given temperature, is greater in front than in rear. The temperature
determines the amplitude and velocity of the ionic motions in the radiator,
which are the same whether it be at rest or in uniform motion: thus it
determines the amplitude of the oscillation in the waves of aethereal radiation
that are excited by them and travel out from them. Of this oscillation the
intensity of the magnetic field represents the velocity. If the radiator is
advancing with velocity v in a direction inclined at an angle d to an emitted
ray, the wavelength in free aether is shortened in the ratio 1 cos 9 ; thus
the period of the radiation is shortened in the same ratio ; thus the velocity
of vibration, which represents the magnetic field, is altered in the inverse
ratio, and the energy per unit volume in the square of that ratio, viz., that
energy is now € (l —  cos ^) ; and the back pressure it exerts involves
a further factor 1 cos 6 owing to the convection ; so that that pressure
is e(l — cos^) , where € is the energy per unit volume of the natural
radiation emitted from the body when at rest. The pressural reaction on
the source is in fact E'/o, where E' is the actual energy emitted in the ray
per unit time.
432 '''■ APPENDIX BY SIR J. LARMOR
Limitation of the analogy of a stretched cord.
In the case of the inextensible stretched cord, the extra length due to the
curved arc in the undulation is proportional to the energy of the motion.
The loss of energy by absorption would imply slackening of the tension ; and
the propositions as to pressure of the waves, including Poynting's modification
for a moving source, would not hold good unless there were some device at
the fixed ends of the cord for restoring the tension. The hypothesis of
convected momentum would imply something of the same kind in electron
structure.
It is therefore worth while to verify directly that the modified formula
for pressure against a moving total reflector holds
good in the case of the cord, when there is no >v
absorption so that the reflection is total. This
analysis will also contain the proof of the generalisa
tion of the formula for radiant pressure that was
enunciated on p. 426 sw^m*.
Let the wavetrain advancing to the reflector and the reflected wavetrain
be represented respectively by
7^1 = Ai cos Ml {x f ct),
7]2 = A^ cos mo {x — ct).
At the reflector, where x = vt, we must have
jvidt^ ji^.,dt;
this involves two conditions,
— = —  and m^ (c + u) = m.. (c — v).
Now the energies per unit length in these two simple wavetrains are
ipAj^ and ipA^^;
thus the gain of energy per unit time due to the reflection is
8E=(c v) Jp^2'  {c + ^) \p^i
= iM.^{,c.)(^^J
1 J 2 O ^ + ^
(c + v)
C — V
* See Larmor, Brit. Assoc. Report, 1900. [The statement that follows here is too brief, unless
reference is made back to the original, especially as a minus sign has fallen out on the right of the
third formula below. The reflector consists of a disc with a small hole in it through which the
cord passes ; this disc can move along the cord sweeping the waves in front of it while the cord
and its tension remain continuous through the hole — the condition of reflection being thus
T7j + 7/2=0 when x = vt. In like manner a material perfect reflector sweeps the radiation in
front of it, but its molecular constitution is to be such that it allows the aether and its structure
to penetrate across it unchanged. For a fuller statement, see Encyclopaedia Britannica, ed. 9 or
10, article 'Radiation.' J. L.]
ON THE MOMENTUM OF RADIATION 433
This change of energy must arise as the work of a pressure P exerted by
the moving reflector, namely it is Pv ; hence
c + V
P = lpA,K2
C — V
The total energy per unit length, incident and reflected, existing in front of
the reflector is
E^ + E^ = yA,^ + yA^^
C^ 4 7j2
Hence finally P  {E^ + E
(c  vf
2^ n2 ' ,.2^
becoming equal to the total density of energy E^ \ E2, in accordance with
Maxwell's law, when v is small.
If we assume Poynting's modified formula for the pressure of a wavetrain
against a travelling obstacle, the value ought to be
and the truth of this is readily verified.
It may be remarked that, if the relation connecting strain with stress
contained quadratic terms, pressural forces such as we are examining would
arise in a simple wavetrain*. But such a medium would be dispersive, so
that a simple train of waves would not travel without change, in contrast
to what we know of transmission by the aether of space.
The question is then suggested how far a cognate momentum can be
regarded as arising from change of aethereal inertia produced by travelling
electric strain. It will be represented by inertia attached to moving tubes
of electric force. The conclusion is reached that such a scheme can be con
sistently constructed for any steady electric system convected with uniform
speed ; also that it holds for any field of pure radiation, that is any field in
which the electric and magnetic forces are everywhere at right angles : but
that in other cases it is not possible. On the other hand any changing electro
dynamic field whatever is constituted by the superposition of pure radiations
from all the electrons belonging to its source.
[A discussion follows of the frictional resistance to the motion through
space of a radiating body, whose mere existence, as is pointed out, had been
predicted by Balfour Stewart as early as 1871. Estimates are made for
bodies of various forms, including one for the sphere which verifies Poynting's
formula in Phil. Trans. 1893 (p. 3.30 supra). The important applications
to cosmical astronomy which Poynting has there developed do not seem to
have yet received the attention they deserve.
Then it is recalled that if we assume the real existence of the Maxwell
stress in the aether, suitably modified for modern ideas, as the source of all
* Cf. Poynting, Boy. Soc. Proc. vol. 86, A (1912), pp. 53^561, where the pressure exerted by
torsional waves in an elastic medium, such as steel, is exhaustively investigated on both the
experimental and the mathematical side. [Collected Papers, Art. 30.]
p. c w. 8
434 APPENDIX BY SIR J. LABMOR
mechanical interactions between electric systems, and we retain the ascer
tained mechanical electrodynamic forces as part of its effect, then another
phenomenon is required to make up the complete result, and this can be
represented as a distribution of momentum in the aether of density equal to
the vectorproduct of aethereal displacement and magnetic induction. In
the case of trains of waves, the latter agrees with Poynting's momentum
of radiation. As regards the resultant momentum and forces for any
selfcontained svstem, th^ Maxwell stress is eliminated, and no hypothesis
as to its reality is involved.
But for such a complete system, free from external disturbance, we
require to compare this outstanding force, visualised as rate of change of
some kind of latent momentum, when the system is convected with uniform
velocity v, with what it would be for the same system at rest in the aether ;
for although the system remains the same the convection modifies the electro
dynamic field around each electron which it contains, and thus may modify
the effective electromagnetic mass of that electron as well as the distribution
of latent momentum. When this comparison is made by aid of the classical
correlation first employed by H. A. Lorentz, it turns out* that the
forcive acting on the convected system exceeds that acting on the same
system when stationary in the aether, by the effect of convection of latent
momentum specified exactly as before, together with a force equal to v _. I 2 ) '
where E is the energy in the system. On the principle that force is expressed
as 7 {mv) we can infer that an increase hE of the electrodynamic energy of
a system increases tlie effective mass of the system by SEjc^. This additional
result, as well as the momentum result, is necessitated beyond cavil by the
ascertained laws of electrodynamics, which however are themselves established
only when (v/c)^ is negligible! : extension of its validity beyond that limit
requires new postulates of 'relativity.' In astronomical applications such
as Poynting's, the effect of any change of mass due to cooling is totally
insignificant compared with the results which he derives from the latent
inomentum. J. L.j
* [Larmor : 'On the Dynamics of Radiation,' Fifth International Congress of Mathe
maticians, Aug. 1912, vol. 1.]
t [Cf. Larmor. Aether and Matter. 1900.]
PART IV.
LIGHT.
32.
ON A SIMPLE FORM OF SACCHARIMETER.
[Phil. May. 10, 1880, pp. 1821.]
The general principle of the modification of the saccharimeter which
I shall describe in this paper is well known, and has already been applied
in the construction of several standard instruments, such as Jellett's and
Laurent's. This principle consists in altering the pencil of rays proceeding
from the polariser in such a way that, instead of the whole pencil having the
same plane of polarisation, the planes of the two halves are slightly inclined
to each other. The analyser is therefore not able to darken the whole field
of view at once. In one position of the analyser the one half of the field is
quite dark; in another position, slightly different, the other half is dark;
while when the analyser is halfway between these two positions, the two
halves of the field are equally illuminated. This will be seen from the
accompanying figure.
Let CA be the trace of the plane of polarisation of the right half of the
pencil, and CB that of the other half. Let CD bisect
ACB. Then, if CE represent the plane of polarisation
of the light which alone the analyser will allow to
pass, when the analyser is turned so that CE is
perpendicular to CA the righthand side of the field
is dark. When CE is perpendicular to CB the right
hand is partially illuminated (as CA has a component
along CE), while the lefthand is dark. Halfway
between these positions, when CE is perpendicular
to (7D, both sides appear equally illuminated. The
analyser being turned round till this equality of illumination is obtained, its
position is noted on the attached circle. When an active substance is now
28—2
436 ON A SIMPLE FORM OF SACCHARIMETER
inserted in the path of the rays, the planes CA, CB are both rotated through
the same angle, and the analyser has to be rotated through this angle to
give the equal illumination once more. The circle again being read, the
difference of readings gives the rotation due to the interposed substance.
In Jellett's saccharimeter the inclination of the planes of polarisation of
the two halves of the field is obtained by interposing a prism of Iceland spar.
This is formed by cutting a rhomb nearly parallel to its optic axis, reversing
one of the pieces, and then cementing the two together again with the plane
of separation bisecting the pencil of rays.
In Laurent's instrument, for which homogeneous light is used, half the
pencil is passed through a plate of quartz cut with its axis in the surface
and parallel to its edge, the thickness being such that the extraordinary ray is
retarded half a wavelength behind the ordinary. On emergence the direc
tions of vibration in the two parts of the pencil, one of which has traversed
the quartz, are equally inclined to the edge of the crystal. The inclination
of the two to each other can be very easily altered by simply turning the
polariser.
The following arrangement is in place of the Iceland spar in Jellett's
instrument, and of the quartz plate in Laurent's. It seems to be somewhat
simpler, and gives fairly good results.
A circular plate of quartz cut perpendicular to the axis is divided along
a diameter, and one half slightly reduced in thickness. The two halves are
then reunited and interposed in the path of the pencil and at right angles
to its direction. Since one half of the pencil passes through a slightly greater
thickness of quartz, its plane of polarisation is slightly more rotated than that
of the other half ; and the pencil therefore emerges with the planes of polar
isation of its two halves slightly inclined to each other. It is of course
always necessary to use homogeneous light to avoid dispersion.
Mr. Glazebrook has very kindly given me the following numbers, which
are taken at random from a large number of sets of readings he has obtained
for the electromagnetic rotation of certain solutions of NaCl in water; the
difference of thickness of the two plates being 1 mm., and the inclination
of the planes of polarisation being therefore about 2° for the sodium light
used. The circle to which the analyser was attached reads to 3' ; but the
vernierdivisions can easily be further subdivided by eye.
In order to vary the inclination of the two planes of polarisation
to each other, one of the halves of the quartz plate might be arranged like
a Babinet's compensator, so that the difference of the two might be varied
at will. The chief objection to the method seems to be that the quartz
plate has to be adjusted very exactly perpendicular to the axis of the pencil.
%
ON A SIMPLE FORM OF SACCHARIMETER 437
Circlereadings
I. Current direct 23° 45'
23 46
23 45
23 45
Current reversed
II. Current direct ..
Current reversed
III. Current direct ...
Current reversed ... 21 45
21 47
21 48
21
36
21
34
21
39
23
15
23
16
23
18
22
19
22
19
22
20
23
30
23
30
23
28
A still simpler arrangement, which has as yet only been tried in a somewhat
rough form, consists in a cell containing some active liquid, say sugar solution.
This cell is interposed in the path of the pencil ; and in it is inserted a piece
of plateglass several millimetres thick, arranged so that one half the pencil
passes through it. This half therefore passes through a less thickness of the
active substance than the other half, and is less rotated. The two then
emerge as before, having their planes of polarisation slightly inclined to each
other. This inclination, and consequently the sensitiveness of the instrument
can be varied either by varying the strength of the active solution, or the
thickness of the plate of glass inserted in the cell.
This arrangement, as far as it has been tested, gives as good results as the
previous one, while it is much more easily constructed and adjusted.
33.
ON THE LAW OF THE PROPAGATION OF LIGHT*.
By J. H. PoYNTiNG and E. F. J. Love, B.A.
[Birmingham Phil Soc. Proc. 5 (1887), pp. 354363.]
[Read March 31, 1887.]
The general law for the propagation of light, applying both to transparent
and absorbing media, is that the intensity of illumination at a distance d
from a given source is proportional to e'^^/d^. For transparent media c = 0.
We shall describe in this paper a new experimental method of showing
that this is the law of propagation. While the method gives, we believe,
the first exact experimental verification of the law for absorbing media, a
combination of the result with those of ordinary photometric observations
shows that for transparent media c = 0, and so for these our method proves
the inversesquare law.
Since various proofs of the inversesquare law already exist it may perhaps
be necessary to justify an addition to the list by pointing out the weak points
in its predecessors.
In what may be called the a ^priori proof a cone is drawn with the source
as vertex, and crosssections of the cone are taken at different distances.
Since the areas of these sections vary as the squares of their distances from
the vertex, the amount of light falling on unit area of each is stated to vary
inversely as the square of the distance. This assumes (1) that there is some
thing, constant in amount, travelling out with a constant velocity; and
(2) that the illumination is proportional to the amount of this incident per
second on unit area. In fact, it assumes the conservation of lightenergy,
and it identifies intensity of illumination with the amount of lightenergjy
received.
It appears to be sometimes supposed that the law is proved by the con
sistency of the results obtained in photometry. Thus let I^ , I^ be the illumina
tions on a screen at unit distance from two sources ; assuming the law, when
the illuminations are equal at distances d^ and (igrespectivelyjWe have t\=^ i\'
* The substance of this paper was communicated to Section A at the 1886 meeting of the
British Association. The experiments with sodium light have been carried out since then.
ON THE LAW OF THE PROPAGATION OF LIGHT 439
This equation, holding for one pair of distances, will hold for any pair in which
the ratio d^ : dz is constant. Carstaedt {Pogg. Ann. 150, p. 551), using the
Bunsen photometer, has shown experimentally that the ratio is constant
within the limits of experimental error. But with any power of the distance
^the ratio should be constant. For the equation j^ = t^ will be satisfied by
1
any pair of values oi d^: d^ii r = (y) .
The simplest direct proof is that which shows that 1 candle at a distance
1 produces an illumination equal to that of 4 candles at distance 2, of 9
candles at distance 3, and so on. But this method is wanting in exactness,
and must be considered rather as a lectureroom illustration than as an
accurate proof.
Crookes (Phil. Trans., 1876, p. 325) has given another proof depending
on the radiometereffect. He shows by subsidiary experiments that this
effect is proportional to intensity of illumination, and then he determines
by the effect the intensity of the light received from a standard candle at
varying distances. The proof is interesting, but does not appear to be very
.exact.
Another proof is based on the observation that a uniform illuminating
surface looked at through a narrow blackened tube appears equally bright
at all distances so long as the illuminating surface entirely fills up the aperture.
The area illuminated on the retina is constant, while the area illuminating
it varies as the square of the distance. Hence the illuminating power per
unit area of the surface varies inversely as the square of the distance. This
is probably more accurate than the previous methods, but it requires the
observer to be assured that there is no gradual change in the illumination
as the distance of the source changes. The eye, however, is not very sensitive
to gradual changes of illumination, seeking always to counteract them by
altering the aperture of the pupil. But the eye is very sensitive to difference
of illumination of two surfaces presented to it at the same time.
The proof which we now give has therefore been devised to depend on
equality of illumination of two surfaces seen together. It may be regarded
as a development of the last proof, but instead of employing a single illumin
ating surface two illuminating surfaces at different distances are viewed
through a narrow blackened tube, each surface occupying half the field of
view. The illuminating powers of the two surfaces are adjusted till for a
given distance of the tube they appear equally bright. They then appear
equally bright for any other distance of the tube.
We shall first, assuming the truth of this statement, deduce from it the
law of propagation, and then give an account of the experimental verification.
440 ON THE LAW OF THE PROPAGATION OF LIGHT
I. Theoretical.
Let two uniform illuminating surfaces be arranged so that when viewed
through a narrow tube each occupies half the field, and let them be placed
at distances from the eyeend of the tube d^ and d^ respectively. Then the
surfaces sending light to a point at the eyeend of the tube may be put equal
to Mj^, hd^, respectively.
Let 7i, I2 be the illuminations produced per unit area of the illuminating
surfaces on a screen held at unit distance from an element of the illuminating
surfaces.
f ix)
Let ~ be the law of propagation, so that if I be the intensity of
X
If ix)
illumination of a screen at unit distance from a source — — is that at
x^
distance x.
If the two surfaces appear equally bright we have
j^a.fj^^jj,^.m (1)
or hf(d,) = IJ(d,) (2)
But this equation is still true by experiment if both d^^ and (^2 b^ increased
by any the same quantity £,
.. IJ(d, + e) = IJ(d, + €) (3)
Put t?i + € = X and d^ — di = y,
••• hf{^)hf(x + y) (4)
For given values of /j, I^ and y this is true for all values of x. Hence
differentiating (4) we have
Iif{^)l2fix + y) (5)
Dividing (5) by (4)
f(x) ' f{x\y) ^^^
^^' 7M = ^^^'^ (')
.. xW = x(^ + ^) (8)
Now this is true for all values of y, and can therefore only be satisfied by
X (x) = constant = — c, (9)
or / (x) = Ae"".
ON THE LAW OF THE PROPAGATION OP LIGHT 441
But at distance 1, /(I) = 1,
.. 1 = Ae^,
)r A^e".
Hence the intensity of illumination at distance x from the source is
/gc(a5l)
x^
Now ordinary photometric measures in transparent media such as air
show that if two sources give equal illuminations at distances d^ and d^, the
equality of illumination is maintained for all distances so long as d^^ : d^ is
constant.
The equality of illumination requires that
/^gc(dil) /2^c(d2l)
d^^ d
2
Putting d^ = kd^, we obtain
d^^ ^ ^Wd^"
. pcdx(lTc) ^ ^2
which can only be true for all values of d^^ when c = 0.
Hence for transparent media we have the ordinary inversesquare law.
When c has a value differing from zero e'^ is the 'coefficient of absorption.'
Of course c varies in general for different colours, so that we get equality
of tint only when we are using monochromatic light.
In this case equation (2) becomes
QY — =: g—cid^—di)
^2
1
If, then, we arrange the apparatus in such a way that Zj and /g can be
determined e^ can be found from the above equation. We have not attempted
to adapt the apparatus for this purpose, but content ourselves with pointing
it out as a possible method of finding the coefficient of absorption.
442 ON THE LAW OF THE PROPAGATION OF LIGHT
II. Experimental.
The object of the experiments was to prove the statement already made—
that *if two luminous surfaces at different distances are viewed through a
narrow blackened tube, each surface occupying half the field of view, and
the illuminating powers of the two surfaces are adjusted so that for a given
distance of the tube they appear equally bright, they will then appear equally
bright for any other distance of the tube.'
The statement requires to be demonstrated for transparent and absorbing
media. It was accordingly resolved to make observations in air with both
white and monochromatic light ; but as liquids exerting a perceptible general
absorption of white light are not to our knowledge obtainable, the experiments
with an absorbing medium were carried out with monochromatic light only.
E O
p
o
'p^xrJ
., ._
1 2 ' 1 1 'k Sl
:
H'
O
P^
Fig. 1.
The apparatus employed {vide Fig. 1) consisted of a trough, ABCD,
12" X 4" X 4" internally ; the bottom, sides, and one end are of paraffined
wood blackened, the other end being closed by a piece of plane parallel
glass, PP. A hole was pierced in the wooden end through which passed
a brass tube, HL, f" in diameter, arranged to slide in a stuffingbox, M,
fixed to the end of the trough. The end, L, of the tube was closed by a
glass plate. The bright surfaces were obtained as follows : A large plate of
opal glass was carefully examined, and a part which appeared homogeneous
was cut out and divided into two. Each of the pieces was then placed in
the trough (at 0, 0) so as to fill up half the field of view of the brass tube,
but at different distances from it, care being taken that the adjacent edges
should be those along which the plate was cut. The plates were illuminated
by lights, E, E, at some distance from the trough. In front of the lights were
placed convex lenses, G, G, somewhat nearer than their own focal length, so
as to produce the effect of a brighter beam coming from a greater distance.
On the side of each lens next the trough was placed a blackened screen, F, V,
with a rectangular slit cut in it, so as to produce a welldefined beam. One
lamp and lens was fixed, the other placed on a wooden tray moving parallel
to itself in guides, so that its distance from the trough could be varied at
pleasure, changes in its position being read off on a millimetrescale, NN,
attached to the tray, with the aid of a fiducial mark affixed to the experiment
table. To prevent light from either lamp reaching the opal on the other
side, a blackened strip of zinc, Z, was fixed in the middle line.
ON THE LAW OF THE PROPAGATION OF LIGHT
443
To get rid of internal reflection in the apparatus, the tube and trough were
"well blackened. The tube was provided in addition with four stops, H, Z,
K, L, placed respectively at the two ends, at the middle, and at onefourth
of the length from the eyeend. The diameters of the stops were H ", I ^%",
K ^}", L f^". To prevent stray light from reaching the eyes, a cardboard
screen about 12" in diameter was hung on the tube. The experiments were
carried on in the dark chamber of the Physics Laboratory at Mason College.
As sources of white light two galvanometerlamps, which burn with a very
steady hght, were employed. With monochromatic light the difficulties were
greater ; for the ordinary form of sodium flame was quite unsuitable, owing
to the different quantities of light emitted from different parts of the surface,
and the altogether uncontrollable fluctuations in brightness. The following
arrangement, represented in Fig. 2, was, however, found successful :
A stream of coalgas, whose pressure was rendered pretty uniform by
passing through a 'Stott' governor, was sent into a large glass flask, F, by
a tube, QQ, which just passed through the cork. Some granulated zinc was
Fig. 2.
placed at the bottom of the flask, together with a small quantity of con
centrated saltsolution, with a large excess of solid salt; hydrochloric acid
was added by the thistle funnel, S. The hydrogen slowly evolved from the
zinc and acid kept up a constant spirting of saltsolution into the gas ; and
as the lower end of the exit tube, RR, was near the surface of the liquid,
the gas came up charged with a quantity of finely divided saltsolution*.
This mixture was led to a large Bunsen burner, E, and a light of uniform
intensity in all parts obtained, which was freed from flickering by surrounding
the flame with an Argand lamp chimney. One of these arrangements was
used for each opal. As absorbing medium, a dilute solution of cobalt car
bonate in ammonium carbonate, which energetically absorbs yellow hght,
was employed.
* This arrangement is similar to that employed in Bunsen' s wellknown apparatus for
exhibiting the sodium absorption.
444 ON THE LAW OF THE PROPAGATION OF LIGHT
The experiments were conducted as follows : The point at which the
intensities of the lights were equal having been ascertained, the tube was
withdrawn, or pushed in, so as to alter its distance from the opals, and in
every case the two halves of the field remained equally bright. The distance
through which it was necessary to move the tray, NN, carrying one of the
lamps and its lens, in order to produce a sensible difference in the intensity
of the two halves of the field, was then noted. The distance of a source of
light which would have produced a pencil of the same dimensions and angle
as that actually employed was then found by measuring the width of the
bright image of the rectangular slit formed on a screen interposed for the
purpose at a known distance. In all the sets of experiments, the relative
intensities of the lights, their distances, and the distance apart of the opals,
were varied.
The accuracy obtained was rather remarkable. It appeared that a
difference of about 1 part in 500 in the intensities of the two halves of the
field was perceptible for white light; and of about 1 in 270 in the case of
sodium light. This could, however, only be obtained by giving the eyes
a period of rest in darkness before commencing the experiments ; accordingly
the shifting of the tray and registering of its position had to be done by an
assistant.
We append the details of one complete experiment as an example :
Light employed : Sodium Flame. Medium : Air.
Distance from image to slit 225 cm.
Width of image 44 cm.
„ „ slit 18 cm.
Effective distance of source from slit 156 cm.
The tube was at first drawn out.
Distance of slit from opal 4030 cm. Left side darker.
4023 cm. Equal.
,, ,, ,, 4018 cm. Right side darker.
The tube was then pushed in, and with the last distance, 4018, the right
side was still the darker. When the distance from the slit to the opal was
4023 the equality was restored.
The eye was therefore sensitive to a change in the distance of the moveable
source of 05 to 07 cm. — say 07 cm.
Since the effective distance of source from sht is 156 cm., and of slit from
opal 4023 cm., the total distance from source to opal is 5583 cm.
Smallest perceptible change of distance = j^g;
whence „ ,, ,, brightness = o^.
ON THE LAW OF THE PROPAGATION OF LIGHT
The results of the experiments are collected in the following table
445
Kind of light
Effective dis
tance
Smallest per
ceptible change
of distance
Change
of
intensity
White in air ... <
Sodium in air ... i
Sodium in absorb /
ing medium (
11280 cm.
10325 „
5583 „
5560 „
5430 „
6196 „
010 cm.
010 „
007 „
010 „
10 „
005 „
1 in 564
1 „ 516
1 „ 399
1 „ 278
1 „ 272
1 „ 620
Since in no case was there any difference between the positions for equality
when the tube was drawn out and pushed in, we may say that, within the
hmits of error above mentioned, the law of propagation holds good for both
transparent and absorbing media.
NOTE IN CORRECTION TO ABOVE PAPER.
[Birmingham Phil. Soc. Proc. 6 (1888), p. 168.]
In a paper read by the authors on March 31st, 1887, an account was given
of a method of experimentally verifying the law of propagation of light both
for transparent and absorbing media. In the paper it was stated that the
method used seemed to lead to much more than the usual accuracy in the
estimation of small differences of intensity of illumination. At the suggestion
of Lord Rayleigh we have reexamined the method, and have come to the
conclusion that the experiments were affected by some source of error — which
we cannot now detect — leading us to underestimate the differences noted, as
we now find that we only attain the ordinary accuracy, detecting a difference
not far from 1 % . This does not, of course, affect the method as a proof of
the law, but only gives a wider range for the possible errors of experiment.
We much regret, however, that the pubhshed figures should have given to
the experiments an appearance of accuracy to which they have no real claim.
34.
HAZE.
[Nature, 39, 1889, pp. 323324.]
I have for some time given in my lectures an explanation of the common
summer haze which appears to me to be very probable. I do not know
whether it is new, but it has not been referred to in the discussion raised by
Prof. Tyndall's letter on Alpine haze*. Some time since I mentioned it to
Prof. Lodge, and at his suggestion I send it to you, though its extension to
other kinds of haze is somewhat speculative.
It is that haze is often due to local convectioncurrents in the air, which
render it optically heterogeneous. The light received from any object is,
therefore, more or less irregularly refracted, and, through the motion of the
currents, its path is continually varying. The outline of the object, instead
of appearing fixed, has a tremulous motion, and so becomes illdefined. At
the same time, reflection occurs where there is refraction at the surfaces of
separation of heterogeneous portions. Much of the light which, in a homo
geneous medium, would come straight from the object, is thus lost for direct
vision, and the contrast between neighbouring objects is lessened. The
reflected light is diffused as a general glare. The combination of the quivering
of outline, and the loss of direct light, with the superposition of the reflected
light as a diffused glare, gives the appearance we call haze.
This explanation appears to me to accord well with the obvious facts of
summer haze — the haze which is seen in the middle of a hot, cloudless, summer
day. The lower layers of air, being heated by contact with the earth, rise
in temperature till equilibrium is no longer possible, and convection begins,
streams of the heated air rising, and streams of colder air falling to take its
place. The variation of temperature and density gives optical heterogeneity.
The existence of these streams is sometimes shown by the quivering of distant
objects, looked at through the air close to the ground, but a telescope will
often show the quivering of outline at higher levels, and when quite invisible
to the naked eye. Accompanying this refraction, reflection must occur. We
have a direct proof of its occurrence in the fact that the glare is greatest
under the sun, where reflection occurs at angles approaching grazing incidence,
* [Nature, vol. 39, 1888, p. 7.]
I
HAZE 447
for which it is a maximum ; while it is least opposite the sun, where reflection
occurs at angles approaching normal incidence, for which it is a minimum.
The opening lines of The Excursion perfectly describe the resulting appear
ance:
"'Twas summer, and the sun had mounted high;
Southward the landscape indistinctly glared
Through a pale steam; but all the northern downs,
In clearest air ascending, showed far ofif
A surface dappled o'er with shadows flung
From brooding clouds."
During the night the lower strata become colder than the upper ones, and
the atmosphere passes into a state of stable equilibrium. We should there
fore expect that, if the foregoing explanation is true, there would be complete
absence of haze, and it is well known that the air is peculiarly clear in early
morning, when we get above the foglevel.
According to this account of heathaze, it stands in sharp contrast to fog,
of which it is so often supposed to be a relative in reduced circumstances.
While the one requires convection, the other usually occurs when the air is
in stable equilibrium, the lowest strata being the coldest. In the fog, for
example, which so frequently heralds or accompanies the breakup of a frost,
the lower strata are still cold, while above the wind has changed, and the air
comes up warm and vapourladen. The vapour diffuses downwards into the
lower, cold stra