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PROPERTIES OF MATTER
BY THE SAME AUTHOR.
LIGHT.
Third Edition. In crown 8vo, cloth, price 7s. 6cL
DYNAMICS.
In crown 8vo, cloth, price 7s. 6d.
NEWTON'S LAWS OF MOTION.
In crown 8vo, cloth, price Is. 6d. net
PROPERTIES OF MATTER
P. G. TAIT, M.A., SEC. R.S.E.
HONORARY FELLOW OF ST. PKTKR'S COLLEGE, CAMBRIDGE,
PROFESSOR OP NATURAL PHILOSOPHY IN THE
UNIVERSITY OF EDINBURGH
FIFTH EDITION
EDITED BY
W. PEDDIE, D.Sc., F.R.S.E.
HARRIS PROFESSOR OF PHYSIOS IN UNIVERSITY COLLEGE, DUNDER,
UNIVERSITY OF ST. ANDREWS
LONDON
ADAM AND CHARLES BLACK
1907
T3
First Edition published April, 1885 ;
Second Edition, October, 1890 ; Third Edition, June, 1894 ;
Fourth Edition, November, 1899 ; Fifth Edition, July, 1907.
The Right of Translation and Reproduction is Reserved
PREFACE.
IT is desirable that a work so characteristic as this is
should remain, in a new edition, as far as possible in the
form in which it left the hands of its distinguished
author. The rapid advance of physical science, in the
few years which have passed since the last edition
appeared, has necessitated some slight additions. In
making these, the original plan of the book has been
strictly adhered to, and all additions have been placed
within brackets and initialed.
The work of revision has been a pleasure to me, as a
service gladly rendered to the memory of my former
master and friend.
W. PEDDIE.
UNIVERSITY COLLEGE, DUNDEE,
June 3, 1907.
222568
PREFACE TO THE FIRST EDITION.
THE subject of this elementary work still forms in
accordance with tradition from the days of Eobison,
Playfair, Leslie, and Forbes the introduction to the
course of Natural Philosophy in Edinburgh University.
The work is (with the exception of a few isolated
sections) intended for the average student ; who is sup-
posed to have a sound knowledge of ordinary Geometry,
and a moderate acquaintance with the elements of
Algebra and of Trigonometry.
But he is also supposed to have what he can easily
obtain from the simpler parts of the two first chapters of
Thomson and Tait's Elements of Natural Philosophy, or
from Clerk-Maxwell's excellent little treatise on Matter
and Motion a general acquaintance with the fundamental
principles of Kinematics of a Point and of Kinetics of
a Particle. To have treated these subjects at greater
length than has here been attempted would have rendered
it imperative to omit much of the development of im-
portant parts of preliminary Physics, of which, so far as
I know, there is no modern British text-book. The
work was peremptorily limited to a small volume; so
that the parts of these auxiliary subjects which have
viii PREFACE TO THE FIRST EDITION.
been admitted are mainly of two kinds : those which
are really introductory to the books just mentioned,
because treating of matters' usually deemed too simple
for special notice ; and a few which are in a sense sup-
plementary, because giving valuable results not usually
included in elementary books.
It is my present intention to complete my series of
text-books by similar volumes on Dynamics, Sound, and
Electricity. Should I succeed in bringing out such
works, I shall thenceforth be enabled to introduce
references to one or other, instead of the digressions
which are absolutely necessary in every self-contained
elementary treatise devoted to one special branch of
Physics only.
P. G. TAIT.
COLLEGE, EDINBURGH,
March 5, 1885.
PREFACE TO THE SECOND EDITION.
LN the present edition this Treatise has been carefully
revised and considerably extended : special attention
having been paid to passages where a difficulty had
been found.
For one of the most important additions I am indebted
to M. Amagat, who has very kindly enabled me to avail
myself of some of his splendid but hitherto unpublished
results. These relate to the compression of fluids exposed
to enormous pressures ; and, when published entire, will
form a singularly interesting and practically new branch
jf the subject.
To some of the scientific critics of the first edition I
am indebted for suggestions of real value, and I have
endeavoured to profit by them. I must except, however,
those which concern my treatment of the subject of Force.
I have seen so much mischief done by this quasi-personi-
fication of a mere sense -impression that, even in an
elementary book, I am constrained to protest against it.
(See 15 of the text.) I feel assured that the difficulties
which are now everywhere felt as to the great scientific
question of the day, the nature of what we call electricity,
ire in great part due to the way in which our modes of
x PREFACE TO THE SECOND EDITION.
thinking have been, by early training and subsequent
habit, encouraged to run in this fatal groove.
To some of my other critics, more aggressive because
less scientific, I have been indebted for genuine amuse-
ment. Nothing is, however, without its use in this
world, though it may occasionally be difficult to discover
that use. It would seem, then, that the function of the
unscientific critics of a scientific book is (like that of the
writers of slipshod English) to furnish examiners with
rich material for questions of the well-known kind :
"Point out all the errors in the following passage."
Nothing is more difficult than the attempt to make such
passages : and the results are usually forced and awkward.
From the critics I allude to they come in perfection.
There is one additional remark which I must make.
The majority of the illustrations in this work (whether
given in words or by diagrams) are, when the contrary is
not stated, to the best of my knowledge original. I make
the remark lest I should be supposed to have taken them
from some of the books in which they have been re-
produced without acknowledgment of their source. It is
flattering to have one's work thus appreciated, but the
honour has its little inconveniences.
P. G. TAIT.
COLLEGE, EDINBURGH,
July 1, 1890.
PREFACE TO THE THIRD EDITION.
IN spite of what I must still consider a tolerably com-
plete statement of my position ( 11-15, 108, etc.), some
of my critics persist in accusing me of deliberate incon-
sistency in my treatment of Force. I am represented by
them as first telling my readers that there is "no such
thing as Force," and then introducing utter confusion by
the assertion that "Matter is merely the plaything of
Force" ! They appear to be unable to perceive that the
idea of Force is an essential feature of Newton's Laws of
Motion : and that, until we are provided with an effi-
cient substitute for Newton's system, we must retain it
in its integrity. To have left these statements altogether
unnoticed might have been prejudicial to the book itself,
in the eyes of the many weak ones who regard the dicta of
a critic (especially when he is anonymous) as necessarily
authoritative. To take more than this passing notice of
them would be to exaggerate their importance.
Since the last edition of this book was published, the
whole of M. Amagat's splendid experimental results have
become generally accessible : and I have made consider-
able additional use of them, especially in Chaps. IX.
andX.
Parts of Chap. VIII. have been considerably modified,
xiv PREFACE.
and VI. of this work may be referred to a little pam-
phlet on Newton' 's Laws of Motion, which has just been
issued by the same publishers. In it I have discussed,
though with studied conciseness, a good deal of important
matter which the plan of this volume prevented me from
introducing.
The present issue has been kept up to date, so far as
recent real extensions of the subjects it treats of are con-
cerned. Other engrossing work has (happily ?) rendered
impracticable for the moment a scheme which I had con-
templated for a thorough revision of the whole plan of the
volume.
P. G. TAIT.
COLLEGE, EDINBURGH,
September 19, 1899.
CONTENTS.
CHAPTER I.
PAG*
INTRODUCTORY . . . . . ' . 1
CHAPTER II.
SOME HYPOTHESES AS TO THE ULTIMATE STRUCTURE OF
MATTER ...... 18
CHAPTER III.
EXAMPLES OF TERMS IN COMMON USE AS APPLIED TO
MATTER ...... 25
CHAPTER IV.
TIME AND SPACE . . . ... 48
CHAPTER V.
IMPENETRABILITY, POROSITY, DIVISIBILITY . . 83
CHAPTER VI.
INERTIA, MOBILITY, CENTRIFUGAL FORCE . . 94
CHAPTER VII.
GRAVITATION . . . . . .113
CHAPTER VIII.
PRELIMINARY TO DEFORMABILITY AND ELASTICITY . 146
xvi CONTENTS.
CHAPTER IX.
PAGE
COMPRESSIBILITY OF GASES AND VAPOURS . . 161
CHAPTER X.
COMPRESSION OF LIQUIDS . . . . .188
CHAPTER XI.
COMPRESSIBILITY AND RIGIDITY OF SOLIDS . . 205
CHAPTER XII.
COHESION AND CAPILLARITY .... 242
CHAPTER XIII.
DIFFUSION, OSMOSE, TRANSPIRATION, VISCOSITY, ETC. . 275
CHAPTER XIV.
AGGREGATION OF PARTICLES . . . 296
CHAPTER XV.
DISINTEGRATION OF THE ATOM . . . .311
APPENDIX.
I. HYPOTHESES AS TO THE CONSTITUTION OF MATTER.
BY PROFESSOR FLINT, D.D. . . .316
II. EXTRACTS FROM CLERK-MAXWELL'S ARTICLE "ATOM" 320
III. VITRUVIUS ON ARCHIMEDES' EXPERIMENT . . 335
IV* NOTE ON A SINGULAR PASSAGE IN THE " PRINCIPIA " 336
V. EXTRACT FROM LA IIMOR'S "AETHER AND MATTER" 344
VI. EXTRACT FROM KELVIN'S "BALTIMORE LECTURES" 346
INDEX 350
PKOPEKTIES OF MATTEE.
CHAPTER I.
INTRODUCTORY.
1. WE start with certain assumptions or AXIOMS, which
are by no means of an a priori character, having been
forced upon us by the observations and experiments of
many generations :
(1) The physical universe has an objective existence.
(2) We become cognisant of it solely by the aid of our
Senses.
(3) The indications of the Senses are always imperfect,
and often misleading ; but
(4) The patient exercise of Reason enables us to control
these indications, and gradually, but surely, to
sift truth from falsehood.
2. If, for a moment, we use the word Thing to denote,
generally, whatever we are constrained to allow has
objective existence : i.e. exists altogether independently
of our senses and of our reason : we arrive at the follow-
ing conclusions :
A. In the physical universe there are but two classes
of things, MATTER and ENERGY.
A
2 ' PROPERTY "o
B. TIME and SPACE, though well known to all (in
Newton's words, omnibus not!ssima\ are not things. 1
C. NUMBER, MAGNITUDE, POSITION, VELOCITY, etc., are
likewise not things.
D. CONSCIOUSNESS, VOLITION, eetu, are not physical.
3. So says modern physical science, and to its generally
received statements we cannot but adhere.
Metaphysicians, of course, who trust entirely to so-
called " light of nature," have their own views on this,
as on all other subjects ; but the number and variety of
these views, some of which are entirely incompatible
with others, form a striking contrast to the general con-
sensus of opinion on the part of those who have at least
tried to deserve to know.
In the words of v. Helmholtz, 2 one of the chief modern
authorities in science properly so-called :
"The genuine metaphysician, in view of a presumed
necessity of thought, looks down with an air of superiority
on those who labour to investigate the facts. Has it
already been forgotten how much mischief this procedure
1 " Space is . . . regarded as a condition of the possibility of
phenpmena, not as a determination produced by them ; it is a
representation d. priori which necessarily precedes all external
phenomena : "
"Time is not an empirical concept deduced from any experience,
for neither) co-existence nor succession would enter into our per-
ception, if the representation of time were not given d priori." -
KANT, Critique of Pure Reason ; Max Miiller's Translation.
2 " Hier haben wir den achten Metaphysiker. Einer angeblichen
Denknothwendigkeit gegeniiber blickt er hochmiithig auf die,
welche sich um Erforschung der Thatsachen bemiihen, herab. 1st
es schon vergessen, wie viel Unheil dieses Verfahren in den
friiheren Entwicklungsperioden der Naturwissenschaften ange-
richtet hat ? " Preface to the German Translation of the second
part of Thomson and Tail's Natural Philosophy.
INTRODUCTORY. 3
wrought in the earlier stages of the development of the
sciences 1 "
Clerk-Maxwell develops the contrast somewhat more
elaborately :
"... In every human pursuit there are two courses
one, that which in its lowest form is called the useful, and
has for its ultimate object the extension of knowledge,
the dominion over Nature, and the welfare of mankind
The objects of the second course are entirely self-con-
tained. Theories are elaborated for theories' sake, diffi-
culties are sought out and treasured as such, and no
argument is to be considered perfect unless it lands the
reasoner at the point from which he started. . . .
" The education of man is so well provided for in the
world around him, and so hopeless in any of the worlds
which he makes for himself, that it becomes of the
utmost importance to distinguish natural truth from
artificial system, the development of a science from the
envelopment of a craft."
Newton, however, had long before expressed essentially
the same ideas. He said :
"To tell us that every species of things is endowed
with an occult specific quality, by which it acts and pro-
duces manifest effects, is to tell us nothing; but to
derive two or three general principles of motion from
phenomena, and afterwards to tell us how the proper-
ties of all corporeal things follow from those manifest
principles, would be a very great step in philosophy,
though the causes of those principles were not yet dis-
covered ; and therefore I scruple not to propose the
principles of motion above mentioned, they being of
very general extent, and leave the causes to be found
out."
4 PROPERTIES OF MATTER.
Midway between Newton's time and our own, another
very great man, Young, spoke as follows of the pernicious
effects of metaphysics in the ancient world :
" A one of the departments of human knowledge were
excluded from the pursuits ... of the Grecian sages,
until Socrates introduced, into the Ionian school, a taste
for metaphysical speculations, which excluded almost all
disposition to reason coolly and clearly on natural causes
and effects."
Quotations like these might be multiplied indefinitely.
But we have given enough to j ustify fully the statements
made in the opening section. These statements must be
our guide in all that follows.
4. A stone, a piece of lead or brass, water, air, the
ether or luminiferous medium, etc., are portions of
Matter; wound-up springs, water-power, wind, waves,
compressed air, hot bodies, electric currents, as well as the
objective phenomena corresponding to our sensations of
sound and light, are examples of Energy associated with
matter. For the present, at all events, we are in the
habit of speaking of Energy as Kinetic when it obviously,
or at least certainly, depends on motion of matter. Thus
the energy of wind, waves, heat, electric currents, etc., is
kinetic. But energy stored up, as in a charged thunder-
cloud, a wound-up spring, a "head" of water, is called
Potential. Of its intimate nature we as yet know
nothing, and it is possible that it may be ultimately
kinetic : i.e. dependent upon motions of inscrutably
minute parts of matter.
5. All trustworthy experiments, without exception,
have been found to lead to the conviction that matter is
unalterable in quantity by any process at the command
of man.
INTRODUCTORY. 5
This is one of the strongest arguments in favour of
the objective existence of matter. It was usefully
employed, at the very end of last century, by Rumford in
his memorable Inquiry concerning the Source of the Heat
excited by Friction. 1
It forms also the indispensable foundation of modern
chemistry, whose main instrument is the balance, used
to determine quantity of matter with great exactness.
We may speak of this property, for the sake of future
reference, as the Conservation of Matter. It justifies one-
half of the statement in 2, A.
It is to be remarked here that the statements just made,
being the direct result of experiment, are strictly applicable
to gross matter only. The Ether or luminiferous and
electrical medium is certainly matter, in the sense of
having Inertia ( 9), but we have at present no means of
investigating its conservation.
6. So far the reader (if he resemble at all the average
student of our acquaintance) is not likely to feel much
difficulty. His every-day experience must have long ago
impressed on him the conviction of the objectivity of
matter, though perhaps he may not have learned to express
it in such a form of words.
But it is usually otherwise when he is told that energy
has an objective existence quite as certainly as has matter.
He has been accustomed to the working of water-mills, let
us say, and he cannot but allow that a " head " of water is
something other than the water ; it is something associated
with the water in virtue of its elevation. He sees and (if
he be of an economic turn) he deplores the terrible waste
of water-power which is stupidly permitted to go on all
over the world. He allows that water-power does exist,
1 Phil. Trans., 1798.
6 PROPERTIES OF MATTER.
but the waste which he laments he looks upon as its
annihilation. Till within the last fifty years or so the
vast majority even of scientific men held precisely the
same opinion.
7. The modern doctrine of the Conservation of Energy,
securely based upon the splendid investigations of Joule
and others, completes the justification of our preliminary
statement. Energy, like matter, has been experimentally
proved to be indestructible and uncreatable by man. It
exists, therefore, altogether independently of human
senses and human reason, though it is known to man
solely by their aid.
One of the most curious passages in history is that
which describes the quest of The Perpetual Motion.
This was simply the attempt to discover a continuous
Source of fresh mechanical energy. In 1775 the Academy
of Sciences declined, for the future, to consider any
scheme which professed to furnish work without corre-
sponding and equivalent expenditure. But the race of
Perpetual Motionists is by no means even yet extinct.
The doctrine of the impossibility of the Perpetual Motion
is often valuable in modern physics (see, for instance,
139 below), as it furnishes simple ex absurdo proofs of
important fundamental theorems.
The objectivity of energy is virtually admitted in a
curious way, by its being advertised for sale. Thus in
manufacturing centres, where a mill-owner has a steam-
engine too powerful for his requirements, he issues a
notice to the effect, " Spare Power to let." But, of
course, the common phrase "price of labour" at once
acknowledges the objectivity of work.
8. There is, however,, a most important point to be
noticed. Energy is never found except in association
INTRODUCTORY. 7
with matter. Hence we might define matter as the
Vehicle or Receptacle of Energy ; and it is already more
than probable that energy will ultimately be found, in
all its varied forms, to depend upon Motion of matter.
This is advanced, for the moment, as a mere introductory
statement, instances of which will be discussed even in
the present work ; but its complete treatment would
require the introduction of branches of physics with
which we have here nothing to do. One great argument
in its favour is, that matter is found to consist of parts
which preserve their identity, while energy is manifested
to us only in the act of transformation, and (though
measurable) cannot be identified. For this is precisely
what we should expect to find if energy depends in-
variably on motion of matter.
9. Beside their common characteristic, conservation,
and in strange contrast to it, we have their characteristic
difference. Matter is simply passive (inert is the scientific
word) ; energy is perpetually undergoing transformation.
The one is, as it were, the body of the physical universe;
the other its life and activity. All terrestrial phenomena,
from winds and waves to lightning and thunder, eruptions
and earthquakes, are transformations of energy. So
are alike the brief flash of a falling star, and the fiery
glow from the mighty solar outbursts of incandescent
hydrogen.
10. From the strictly scientific point of view, the greater
part of the present work would be said to deal with energy
rather than with matter. In fact, were we to speak of
weight as a property of matter, in the sense that a stone
of itself has weight, or even in the sense that the earth
attracts the stone, we should go directly in the teeth of
Newton's distinct assertion.
8 PROPERTIES OF MATTER.
For such a statement (because confined to the attract-
ing bodies alone) implies the existence of Action at a
Distance, a very old but most pernicious heresy, of which
much more than traces still exist among certain schools,
even of physicists. (See Newton's words on this subject,
160 below.)
Gravitation, like all other mutual actions between
particles of matter, such as give rise to cohesion,
elasticity, etc., must, with our present knowledge, be set
down to the energy which particles of matter are
found to possess when separated. The intervening
mechanism by which this is to be accounted for has, as
yet, only been guessed at, and none of the guesses have
been succesoM. Clerk-Maxwell's success in explaining
electric and magnetic attractions by something analogous
to stresses and rotations in the luminiferous ether shows,
however, that we need not despair of being able to dis-
cover the ultimate mechanism of gravitation.
But there is great convenience in separating, as far
as possible, the treatment of Mass, Weight, Cohesion,
Elasticity, Viscosity, etc., which we range under the
general title, Properties of Matter, from that of Heat,
Light, Electric Energy, etc., which can all in great
measure be studied without express reference to any one
special kind of matter though, as forms of energy, they
exist only ( 8 above) in association with matter. Along
with these forms of energy must of course be treated the
allied properties of matter, such as specific heat, refractive
index, conductivity, etc. Such, therefore, are foreign to
the present work. And it must be remarked that, even
in popular language, we invariably speak of the hardness
of a body, its rigidity, its elasticity, as belonging to it in
much the same sense as does its density or its atomic
INTRODUCTORY. 9
weight and certainly in a much more intimate sense
than does its temperature or its electric potential.
It is, therefore, on the two grounds of custom and
convenience that we use the term Properties of Matter as
the title of this work. The error involved is not by any
means so monstrous as that which all agree to perpetuate
hy the use of the term Centrifugal Force.
11. The word Force must often, were it only for
brevity's sake, be used in the present work. As it does
not denote either matter or energy, it is not a term for
anything objective ( 2, A). The idea it is meant to
express is suggested to us by the " muscular sense," just
as the ideas of brightness, noise, smell, or pain are sug-
gested by other senses : though they do not correspond
directly to anything which exists outside us.
It is exceedingly difficult to realize fully the fact that
noise is a mere subjective impression, even when reason
has convinced us that outside the drum of the ear then-
is nothing to correspond to it except a periodic com
pression and dilatation of the air.
Still more difficult is it to realize that outside us all it
dark; and that the objective cause of even the most
gorgeous of optical phenomena is an excessively rapid
quivering motion of the ethereal jelly which extends-
through all space.
We need not, therefore, be surprised at the tenacity
with which the great majority, even of scientific men,
still cling to the notion of force as something objective.
But if it were objective, what an absolutely astounding
difficulty would have to be faced by one who tries to
explain the nature of hydrostatic pressure; and who
finds that by the touch of a finger on a little piston he
can produce a pressure of (say) a pound weight on every
10 PROPERTIES OF MATTER.
square inch of the surface of a vessel, however large, if
full of water, and the same amount on every square inch
of surface of every object immersed in it, even if that
object consisted of hundreds of square miles of sheets
of tinfoil far enough apart to let the water penetrate
between them. All this, moreover, is found to disappear
the moment he lifts his finger !
When we communicate energy to a body, as in pushing
or drawing a carriage, the impression produced upon our
muscular sense does not correspond to the energy com-
municated per second, but to the energy communicated
per inch of the motion. For experiment has proved
that what appears to our muscular sense as a definite
tension (in a cord, let us say) is associated with the com-
munication of energy, to any mass of matter whatever,
in direct proportion to the (linear) space through which
it is exerted, altogether independently of the speed with
which the mass may be already moving in the direction
of the tension ; so that in equal times energy is com-
municated in direct proportion to that speed. When
there is no motion, no energy is communicated ; and this
would certainly not be the case if communication of
energy corresponded to the time during which the tension
was said to act.
12. The muscular sense is far more deceptive than any
other, except, perhaps, that of touch. Conjurors, ven-
triloquists, perfumers, and cooks make their livelihood by
practising on the imperfections of our senses of sight,
hearing, smell, and taste respectively. But he who has
tried the simple experiment of rolling a pea on the table
lietween his first and second fingers, after crossing one-
over the other, will at once recognise the extreme deceit-
fulness of the sense of touch. And the muscular sense
INTRODUCTORY. 11
well deserves a place beside it. So, as we know that
there is but one pea, though the sense of touch vividly
impresses us with the notion that there are two, we must
be very wary when the muscular sense plainly suggests to
us the notion of force as an objective reality.
13. Many of the terms which are now used in a strictly
scientific sense had a humbler origin, having been devised
entirely for the popular expression of common ideas. The
term Work is a specially illustrative one. Thus, in a
draw-well, the work done in bringing water to the surface
would be reckoned at first in terms of the quantity of
water raised: two raisings of a full bucket lifting twice
as much water as one. But then it was found that, for
the same quantity of water raised, the work depended on
the depth of the well : doubled depth corresponding to
doubled work. Again, if the bucket were filled with sand
instead of water, more work was required, in proportion
as sand is heavier than water. All these statements were
soon found to be comprehended in the simple form :
the work done is directly proportional to the weight
raised and also to the height through which it is raised.
Here the indications of the muscular sense stepped in, and
work came to have a general meaning, viz. the product
of the so-called force exerted, into the distance through
which it is exerted.
Had they not possessed the muscular sense, men might
perhaps have been longer than they have been in recog-
nising the important thing potential energy ( 4) ; but
when they had come to recognise it, they would have
stated that when water is raised it gains potential energy
in proportion as it is raised, and perhaps they might have
found it convenient to use a single term for the rate at
which such energy is gained per foot of ascent. This
12 PROPERTIES OF MATTER.
would probably not have been the word " Force," but it
would have expressed precisely what the word force now
expresses.
Then they would have recognised that when energy is
transmitted by a driving-belt, the amount transmitted is
(ceteris paribus) directly proportional to the space through
which the belt has run. They might have invented a
name for the rate of transmission per foot-run of the belt ;
they might even have called it the tension of the belt ; but,
anyhow, it would be precisely what is now called force.
Let us look at the matter from another point of view.
14. A stone, if let fall, gradually gains kinetic energy
or energy of motion, and experiment shows that the
energy gained is directly proportional to the vertical space
fallen through. Hence we have come to say that the
stone is acted upon by a force (its weight, as we call it)
whose amount is practically the same at all moderate
distances from the earth's surface.
But, so far as we know the question scientifically, we
can say no more than that the stone has potential energy
(just as water in a mill-pond has head) in proportion to
its elevation above the earth's surface ; and consequently,
by the conservation of energy, it must acquire energy of
motion in proportion to the space through which it
descends. Why it has potential energy when it is raised,
and why that potential energy takes the first opportunity
of transforming itself into kinetic energy : thus requiring
that the stone shall fall unless it be supported : are
questions to be approached later. (Chap. VII.)
15. That the statement above is complete, without the
introduction of the notion of force, is seen from the fact
that a knowledge of the kinetic energy acquired, after a
given amount of descent, enables us to determine fully
INTRODUCTORY. 13
the nature of the resulting motion even when the stone is
projected, obliquely or vertically, not merely allowed to
fall. The question is easily 'reduced to one of mathe-
matics, or rather of Kinematics, and as such the non-
mathematical student must, for the present, simply accept
the statement as true.
And thus we have another of the many distinct and
independent proofs that Force is a mere phantom sugges- ,
tion of our muscular sense ; though there can be no doubt
that, in the present stage of development of science, the
use of the term enables us greatly to condense our
descriptions.
But it is a matter for serious consideration whether we
do not connive at a species of mystification by thus
employing, in the treatment of objective phenomena, a
term for a mere sensation, corresponding to nothing
objective: even although it be employed solely to shorten
our statements or our demonstrations.
Every one knows that matter (e.g. corn, gold, diamonds)
has its price ; so (as we saw in 7) has energy. We are
not aware of any case in which force has been offered for
sale. To " have its price" is not conclusive of objectivity,
for we know that Titles, Family Secrets, and even Degrees,
are occasionally sold ; but " not to have its price " is at
least all but conclusive against objectivity.
We are in the habit of speaking of fresh air, sunshine,
&c., though obviously objective, as priceless !
16. These introductory remarks have been brought
in with the view of warning the reader that we are
dealing with a subject so imperfectly known that at almost
any part of it we may pass, by a single step as it were,
from what is acquired certainty to what is still subject
for mere conjecture
14 PROPERTIES OF MATTER.
An exact and adequate conception of matter itself,
could we obtain it, would almost certainly be something
extremely unlike any conception of it which our senses
and our reason will ever enable us to form. Our object,
therefore, in what follows, is mainly to state experimental
facts, and to draw from them such conclusions as seem to
be least unwarrantable.
17. But, for the classification of the properties of
matter, whether our classification be a good one or not,
it is necessary that we should have a definition of matter.
From what was said in last section it is obvious that
no definition we can give is likely to be adequate. All
that we can attempt, then, is to select a definition which
(while not obviously erroneous) shall serve as at least a
temporary basis for the classification we adopt.
18. Numberless definitions of matter have been pro-
posed. 1 Here are a few of the more important :
(a) That which possesses Inertia ( 9).
((3) The Receptacle or Vehicle of Energy ( 8).
(y) Whatever exerts or can be acted on by Force.
(8) Whatever can be perceived by our senses, especi-
ally the sense of Touch. This is closely akin
to the well-known definition of matter as a
Permanent Possibility of Sensation.
() Whatever can occupy space.
() Whatever, in virtue of its motion, possesses Energy.
(rj) Whatever, to set it in motion, requires the ex-
penditure of Work.
(0) [Torricelli, Lezioni Accademiche, 1715, p. 25.] La
materia altro non e, che un vaso di Circe incan-
tato, il quale serve per ricettacolo della forza, e de'
1 A remarkable collection of such (now historical) speculations,
due to Professor Flint, is given in Appendix I.
INTRODUCTORY. 15
momenti dell' impeto. La forza poi, e gl' impeti,
sono astratti tanto sottili, son quintessenze tanto
spiritose, che in altre ampolle non si posson
racchiudere, fuor che nell' intima corpulenza de'
solidi natural!.
(0 [The Vortex Hypothesis of Lord Kelvin.] The
rotating parts of an inert perfect fluid ; whose
motion is absolutely continuous, which fills all
space, but which is, when not rotating, absolutely
unperceived by our senses.
|(/c) Surfaces of misfit in a granular medium. [Hypo-
thesis of Osborne Reynolds.]
(A) Loci of intrinsic rotational strain in the ether.
[Hypothesis of Larmor.] See App. V.
(/A) Fundamentally of electric constitution. [J. J.
Thomson and Kelvin.] See Chap. XV. and
App. VI. , ; It should be noted that, as Larmor
has pointed out, the vortex hypothesis gives no
account of electric action. W. P.]
19. The mutual incompatibility of certain pairs of
these definitions shows that some of them, at least, must
be of the so-called metaphysical species ( 3).
(a), (/?), (), (rj), above, have much in common, and,
with further knowledge, may perhaps be found to differ
in expression merely. At present, from want of informa-
tion, we cannot be certain that any two of them are
precisely equivalent.
Berkeley virtually asserted that all motion is produced
by the direct action of spirits on matter. Even then, the
statement (/3) that matter is the receptacle or vehicle of
energy holds good (but how then does energy exist in
the spirit ?).
But the statement that matter is whatever can exert
16 PROPERTIES OF MATTER.
force (y) is to be rejected ; though it was virtually intro-
duced by Cotes in his Preface to the second edition of
the Principia.
(8) must be rejected, if only because there is another
thing besides matter (in the physical universe) which we
know of, and of course only through our senses ( 1).
But this is not all the error ; for we get the notion of
force through our muscular sense ( 11), and force is not
matter, not even a thing.
Torricelli's language is poetical, and therefore his
statement (6) must not be taken too literally. In his
time, as in all subsequent time till well within the last
half century, energy and force were very rarely distin-
guished from one another. Even now they are too often
confounded.
(i), the most recent of these speculations, has the
curious peculiarity of making matter, as we can perceive
it, depend upon the existence of a particular kind of
motion of a medium which, under many of the defini-
tions above, would be entitled to claim the name of
matter, even when it is not set in rotation.
20. But as we do not know, and are probably incapable
of discovering, what matter is, we must content ourselves
for the present with a definition which, while not at
least obviously incorrect, shall for the time serve as a
working hypothesis.
"We therefore choose (e) above, i.e. we define, for the
moment, as follows :
Matter is whatever can occupy space.
Experience has proved that it is from this side that the
average student can most easily approach the subject, i.e.
here, as it were, the contour lines of the ascent ( 80) are
most widely separated.
INTRODUCTORY. 17
21. A But this definition involves three distinct proper-
ties: (1) the Volume, (2) the Form or Figure, of the space
occupied ; and (3) the nature or quality of the Occupation.
Hence the older classical works on our subject almost
invariably speak of matter as possessing (1) Extension,
(2) Form, and (3) Impenetrability. It is mainly for the
sake of the first of these, and the preliminary discussions
which it necessarily introduces, that we have chosen the
above definition as our starting-point.
22. Before we take these up in detail, however, it may
be useful to devote a short chapter to a digression on
some of the more notable of the hypotheses which have
been propounded as to the ultimate structure of matter.
We advisedly use the word structure instead of nature,
for it must be repeated, till it is fully accepted, that the
discovery of the ultimate nature of matter is probably
beyond the range of human intelligence.
Another chapter, of a very miscellaneous character, will
follow, devoted to the examination of some of the terms
popularly applied to pieces of matter, and a rapid glance
at the physical truths which underlie them. This is
introduced to give the reader, at the very outset of his
work, a general idea of its nature and extent.
CHAPTER II.
SOME HYPOTHESES AS TO THE ULTIMATE STRUCTURE OF
MATTER.
23. THE hard Atom, glorified in the grand poem of
Lucretius, but originally conceived of, some 2400 years ago,
by the Greek philosophers Demokritus and Leukippus,
survives (as at least an unrefuted, though a very improb-
able, hypothesis) to this day. Newton made use of the
hypothesis of finite, hard, atoms to explain why the speed
of sound in air was found to be considerably greater than
that given by his calculations ; which were accurate in
themselves, but founded on erroneous or, rather, incom-
plete data. But in this problem Laplace found the vera
causa, and in consequence Newton's apparent support of
the hypothesis of hard atoms is no longer available.
Many of the postulates of this theory are with difficulty
reconciled with our present knowledge ; some have been
contemptuously dismissed as "inconceivable." But any
one who argues on these lines becomes, ipso facto, one of
the so-called metaphysicians.
Let us briefly consider the main statements of this
theory, but without regard to the order in which
Lucretius gives them.
24. Nature works by invisible things : thus paving-
ULTIMATE STKUCTURE OF MATTEK. 19
stones and ploughshares are gradually worn down without
the loss of any visible particles.
Reproduction [i.e. agglomeration of scattered particles
so as to produce visible bodies] is slower than decay
[i.e. the breaking up of bodies into invisible particles], and
therefore there must be a limit to breakage, else the
breaking of infinite past ages would have prevented any
reproduction within finite time. Hence there exists a
least in things [i.e. unbreakable parts or Atoms, "strong
in solid singleness "].
But there is also void in things, else they would be
jammed together, and unable to move. Here Lucretius
takes the case of a fish moving in water, showing that
void is necessary in order that it may be able to move.
[Our modern knowledge of circulation, i.e. the motion of
fluids in re-entrant paths, shows that this reasoning is
baseless.]
There can be no third thing besides body and void.
For nothing but body can touch and be touched; and
what cannot be touched is void. [Here we have the germ
of the erroneous definition of matter (8) in 18 above.]
The atoms are infinite in number, and the void in
which they move [space] is unlimited.
They have different shapes ; but the number of shapes
is finite, and there is an infinite number of atoms of each
shape.
Nothing whose nature is apparent to sense consists of
one kind of atoms only.
The atoms move through void at a greater speed than
does sunlight.
Besides this, there is a great deal of curious speculation
as to bow a vertical downpour of atoms [supposed to be
a result of their weight] is, in some arbitrary way, made
20 PROPERTIES OF MATTER.
consistent with their meeting one another and agglome-
rating into visihle masses of matter.
The basis of the whole of Lucretius' reasoning in
favour of the existence of atoms lies in the gratuitous
assumption that reproduction is slower than decay. This
is by no means consistent with our modern knowledge,
for potential energy of different masses [whether gravita-
tional or chemical] is constantly tending to the agglomera-
tion of parts, and on a far grander scale than that in which
any known cause tends to decay or breaking up.
But if there be hard atoms, they must (in all known
bodies) have intervals between them ; for compressi-
bility : i.e. capability of having the component atoms
brought more closely together : is a characteristic of all
known bodies. [Contrast this mode of arriving at the
conclusion that " there must be void in things," with the
erroneous mode employed by Lucretius.]
25. A refinement of this theory, mainly due to
Boscovich, gets rid of the material atom altogether,
substituting for it a mere mathematical point, towards
or from which certain forces tend. It is supported by
the assertion that we know matter only by the effects
which it produces (or seems to produce), and therefore
that, if these effects can be otherwise explained, we need
not assume the existence of substance or body at all.
This theory was, at least in part, accepted by so great an
experimenter and reasoner as Faraday. It virtually
substitutes force for matter as an objective thing ( 2) }
and it essentially involves the heresy of distance-action
( 10). But the fatal objection to which it is exposed is
that it does not seem capable of explaining inertia, which
is certainly a distinctive (perhaps the most distinctive)
property of matter
ULTIMATE STRUCTURE OF MATTER. 21
This theory must be regarded as a mere mathematical
fiction, very similar to that which (in the hands of
Poisson and Gauss) contributed so much to the theory
of statical Electricity ; though, of course, it could in no
way aid inquiry as to what electricity is.
26. A much more plausible theory is that matter is
continuous (i.e. not made up of particles situated at a
distance from one another) and compressible, but in-
tensely heterogeneous ; like a plum-pudding, for instance,
or a mass of brick-work. The finite heterogeneousness
of the most homogeneous bodies, such as water, mercury,
or lead, is proved by many quite independent trains of
argument based on experimental facts. If such a con-
stitution of matter be assumed, it has been shown l that
gravitation alone would suffice to explain at least the
greater part of the phenomena which (for want of know-
ledge) we at present ascribe to the so-called Molecular
Forces. But it does not seem to be compatible with
experimental facts; especially some of the simpler
phenomena presented by gases. ( 55, 322.)
27. The most recent attempt at a theory of the
structure of matter, the hypothesis of Vortex Atoms, is
of a perfectly unique, self-contained character. Its postu-
lates are few and simple, but the working out of anything
beyond their immediate consequences is a task to tax to
the utmost the powers of the greatest mathematicians for
generations to come. A vortex filament, in a pefect
fluid, is a true " atom ; " but it is not hard like those of
Lucretius ; it cannot be cut, but that is because it
necessarily wriggles away from the knife.
The idea that motion is, in some sort, the basis of
what we call matter is an old one : but no distinct con-
1 "W. Thomson, Pro c . R.S.E. t 1862.
22 PROPERTIES OF MATTER.
ceptions on the subject were possible until v. Helmholtz,
in 1858, made a grand contribution to hydrokinetics in
the shape of his theory of vortex motion. 1 He proved,
among other entirely novel propositions, that the rotating
portions of a continuous incompressible fluid, in which
there is neither viscosity nor finite slipping, maintain
their identity : being thus for ever definitely differenti-
ated from the non-rotating parts. He also showed that
these rotating portions are necessarily arranged in con-
tinuous, endless filaments : forming closed curves, which
may be knotted or linked in any way : unless they
extend to the bounding surface of the fluid, in which
alone they can have ends. Thus, to give ends to a
closed vortex filament (i.e. to cut it) we must separate
the fluid mass itself, of which it is a portion : so that
on Lord Kelvin's theory we must (virtually) sever space
itself.
Such vortex filaments (though necessarily of an im-
perfect character) are produced when air is forced to
escape from a box, through a circular hole in one side,
by sharply pushing in the opposite side. If the air in
the box be filled with smoke, or with sal-ammoniac
crystals, the escaping vortex ring is visible to the eye;
and the collisions of two vortex rings, which rebound
from one another, and vibrate in consequence of the
shock, as if they had been made of solid india-rubber,
are easily exhibited. Experimental results of this kind
led Lord Kelvin 2 to propound the theory that matter,
such as we perceive it, is merely the rotating parts of a
fluid which fills all space. This fluid, whatever it be,
must have inertia : that is one of the indispensable
1 Crelle, 1858. Translated (by Tait) in Phil. Mag., 1867.
2 Proc. R.S.E., 1867.
ULTIMATE STKUCTURE OF MATTER. 23
postulates of v. Helmholtz's investigation ; and the great
primary objection to Lord Kelvin's theory is, that it
explains matter only by the help of something else which,
though it is not what we call matter, must possess what
we consider to be one of the most distinctive properties
of matter.
28. This theory is still in its infancy, and we cannot as
yet tell whether it will pass with credit the severe ordeal
which lies before it, when the properties of vortices (which
must be discovered by mathematical investigation) shall
be compared, one by one, with the experimentally ascer-
tained properties of matter. As we have already said,
this theory is self-contained ; no new hypotheses can be
introduced into it ; so that it possesses, as it were, no
adaptability, or capability of being modified, but must fall
before the very first demonstrated insufficiency, or contra-
diction, if such should ever be discovered.
29. But the really extraordinary fact, already known in
this part of our subject, is the apparently perfect similarity
and equality of any two particles of the same kind of gas,
probably of each individual species of matter when it is
reduced to the state of vapour. Of such parts, therefore,
whether they be further divisible or not, each species of
solid or liquid must be looked on as built up. This
similarity of parts, very small indeed but still of essenti-
ally finite magnitude, has been so well treated by Clerk-
Maxwell that, instead of insisting upon it here, we give a
considerable extract from one of his remarkable articles
in Appendix II. below.
30. The further treatment of the subject of structure,
involving the question of how the component parts (be
they atoms or not) of bodies are put together, must be
deferred to the end of the work. What has been said
24 PROPERTIES OF MATTER.
above must be looked on as a mere preliminary sketch,
not intended even to be fully understood until the
experimental data, on which all our reasoning must be
based, are brought before the reader as completely as our
limits permit.
CHAPTER III.
EXAMPLES OF TERMS IN COMMON USE AS APPLIED TO
MATTER.
31. BEFORE we proceed to a more rigorous treatment of
our subject, it may be well to consider what physical truth
underlies each of some of the many adjectives in common
use as applied to portions of matter, such as Massive,
Heavy, Plastic, Ductile, Viscous, Elastic, Rigid, Opaque,
Blue, Coherent, etc.
This course secures a twofold gain, so far as the
beginner is concerned ; for, first, he is introduced by it, in
a familiar way, to some of the more important terms
which are indispensable in scientific description ; and
second, he obtains a glance here and there through the
whole subject of Natural Philosophy, because the pro
gramme before us is so vague as to leave room for
innumerable digressions, each introducing some novel but
important fact or property. But we must endeavour to
be brief, for whole volumes would have to be written
before this subject could be nearly exhausted.
32. Every one who has used his senses to any purpose
knows, before he comes to the study of our science, a
great many of its phenomena, among them some of the
yet unexplained. But he knows, as it were, each by
26 PROPERTIES OF MATTER.
itself, and only in its more prominent features; the
analysis of the appearances or impressions which he has
seen and experienced, and the explanation of the physi-
cal fact or process which underlies each of them, are
absolutely necessary before he can understand the mode
in which they must be grouped, and the reasons for such
grouping.
33. Thus he knows that the moon keeps company with
the earth, never receding nor approaching by more than a
small fraction of the average distance. He also knows
that the earth keeps, within narrow limits, at a definite
distance from the sun. He has a general notion, at least,
that the state of matters on the earth would become
serious, as regards both animal and vegetable life, if we
were to approach to even half our present distance from
the sun, or recede to double that distance. But he would
require to be a Newton if, without instruction, he could
divine that these results are due to the very cause which
keeps the bob of a conical pendulum moving in a horizontal
circle.
He sees ripples running along on the surface of a pool,
but requires to be told that their motion depends upon
the cause which rounds the drops of water on a cabbage-
blade, or in a shower, and which renders it almost
impossible to keep a water-surface clean.
He sees what he calls a flash of lightning, but he
requires to be told that what he sees is mainly particles
of air heated so as to be self-luminous.
He looks at the stars and thinks he sees them as they
are, but he requires to be informed that he fees even the
nearest of them only as it was three years ago and that it
may have changed entirely in the interval.
And he will certainly require to be informed, even
TERMS APPLIED TO MATTER. 27
with patient iteration, that air is made up of separate and
independent particles : the number of which in a single
cubic inch is expressed by twenty-one places of figures, a
multitude altogether beyond human conception : a busy
jostling crowd, each member of which darts about in
all directions, impinging on its neighbours some eight
thousand million times per second.
But when he has got so far, and has been told that
this astounding information is as nothing to what we
feel convinced that science can yet reveal, he cannot help
marvelling alike at the arcana of physics, and at the
patient efforts of genius which have already penetrated
so far into the darkness shrouding its mysteries.
34. Take the terms Massive and Heavy as applied to
a piece of matter, or the corresponding substantives, the
Mass and the Weight of a body.
The terms are usually regarded as equivalents, but in
their origin they are completely distinct. The one is a
property of the body itself, and is retained by it without
increase or diminution wherever in the universe the body
may be situated. The other depends for its very exist-
ence on the presence of a second body, the Earth : and
varies in inverse proportion to the square of the distance
from the earth's centre.
The destructive effects of a cannon-ball are due entirely
to its mass and to the relative speed with which it
impinges on the target. They have nothing to do with
its weight, for they would be exactly the same (for the
same relative speed) in regions so far from the earth, or
other attracting body, that the ball had practically no
weight at all.
When an engine starts a train on a level railway, or
when a man projects a curling-stone along smooth ice,
28 PROPERTIES OF MATTER.
the resistance which either prime mover has to overcome
is due to the mass of the body to be moved. Its weight,
except indirectly through friction, has nothing to do with
it. So when we open a large iron gate properly sup-
ported on hinges, it is the mass with which we have to
deal ; if it were lying on the ground and we tried to lift
it, we should have to deal simultaneously with its weight
and with its mass.
The exact proportionality of the weights of bodies to
their masses, at any one place on the earth's surface,
was proved experimentally by Newton, and is thus no
mere truism, but an essential part of the great law of
gravitation.
Thus a pound of matter is a definite amount, or mass,
of matter, unchangeable whithersoever that matter may
be carried. But the weight of a pound of matter, or a
" pound- weight," as it is commonly called, is a variable
quantity, depending upon the position of the body with
respect to the earth ; and changes (to an easily measurable
amount) as we carry the body to different latitudes, even
without leaving the sea-level.
35. The common use of the balance as a means of
measuring out equal quantities of matter is justified by
Newton's result ; but the process is essentially an indirect
one, for the balance tells only of equality of weight. If
the earth were hollow at the core, the balance would
cease to act in the cavity ( 140). Bodies would preserve
their masses there, but would be deprived of weight, at
least so far as the earth is concerned.
To sum up for the present, the mass of a body is
estimated by its inertia, and is taken as the measure of
the amount of matter in the body ; while the weight is
an accidental property, connected with the presence of
TEEMS APPLIED TO MATTER. 29
another mass of matter. But it is a most remarkable
fact that under the same given external conditions the
weight depends upon the quantity only, and not on the
quality, of the matter in a body.
If a body, A, becomes heavy in consequence of the
presence of another body, B, so in like wise does B become
heavy in consequence of A's presence. And the weights
of the two, each as produced by the attraction of the
other, are exactly equal. Hence, if they be free to
move, the quantities of motion (i.e. the momenta) pro-
duced in a given time are equal and opposite. [Newton's
Lex iii. 128.] But as the momentum is the product of
the mass and the velocity, the parts of the velocities of
the two bodies, due to their mutual gravitation alone,
will be in amount inversely as their masses. Thus,
though the weight of the whole earth, produced by the
attraction of a stone, is exactly equal to that of the stone
produced by the attraction of the earth, the consequent
rate of fall of the earth towards the stone is less than
that of the stone towards the earth in the same ratio that
the mass of the stone is less than that of the earth, and
is therefore usually so small as to escape observation.
The moon, however, is a stone whose mass is not exces-
sively smaller than that of the earth, and the consequences
of the earth's fall towards the moon have to be taken
account of in astronomy.
36. To properties such as mass, which depends on the
size as well as on the material of a body, and weight
which, in addition, depends on a second body, there
correspond what are called specific properties, characteristic
of the substance and independent of the dimensions of the
particular specimen examined.
Thus the mass of a cubic foot of any kind of matter
30 PROPERTIES OF MATTER.
may be called its specific mass. But this quantity, i.e.
the amount of matter in unit bulk, is usually expressed
by the term Density.
The weight of a cubic foot of each particular kind of
matter in any locality may be called the specific weight.
But as this varies, though in the same proportion for all
bodies, from place to place, we use instead of it the ratio
of the weight of a cubic foot of the substance to that of
a cubic foot of some standard substance. This is called
the Specific Gravity. Pure water, at the temperature
called 4 C. (its maximum-density point), is usually taken
as the standard substance.
Newton's experimental result shows that the density
and the specific gravity of any substance are proportional
to one another, so that if the density of water at 4 C.
l)e taken as unit-density, a table of specific gravities is
identical with a table of densities. ( 166.) But we
must repeat, the coincidence is an experimental fact, not
(as yet at least) in any sense a truism.
Specific gravity is, in general, much more easily
measured with accuracy than is density, so that it is usually
the property to be directly determined, the other simply
following from it in consequence of Newton's discovery.
37. To vary the subject widely, let us now consider
the term Viscous as applied to fluids. The contrasted
adjective is usually taken as Mobile.
When a liquid partially fills a vessel, and has come to
rest, it assumes a horizontal upper surface. If the vessel
be tilted, and held for a time in its new position, the
liquid will again ultimately settle into a definite position,
with its surface again horizontal. Practically it occupies
the same bulk in each of these positions. Hence the only
change it has suffered is a chsnge of /
TERMS APPLIED TO MATTER.
31
But this change of form is much more rapidly attained
in some liquids than in others, even v/hen they are of
nearly the same density. Some (such as sulphuric ether)
attain their equilibrium position so quickly that they
retain energy enough to oscillate about it for some time
before coming to rest; others (such as treacle) attain it
only after a long time and, unless in great masses and
when violently disturbed, do not oscillate but gradually
creep to their final shape. Hence we call treacle viscous.
To analyse this result let us consider (in a very ele-
mentary case, for the general analysis of the process
requires higher mathematical methods than we can employ
in a work like this) what is involved in Shear: i.e.
change of form of a body without change of bulk.
38. When water flows, without eddies, slowly in a
rectangular channel of uniform width and depth, we
know, by observation of particles suspended in it, that
the upper parts flow faster than the lower, and (practically)
in such a way that a column of the water, originally
straight and vertical, inclines, as a whole, forwards more
and more in the direction of its motion. Hence in a
vertical section, along the middle of the channel, the
particles originally forming the line ab in the figure will,
c a' c'
6 d V d,'
. FIG.1.,
after the lapse of a certain time, be found approximately
in the line a'b'. Similarly those which were originally in
32 PROPERTIES OF MATTER.
cd parallel to ab, will be found in c'd', parallel to a'b', and
so situated that a'c' = ac, and of course also b'd' = bd. The
figures ad, a'd", are thus parallelograms on equal bases
and between the same parallels, and therefore equal in
area. This shows that the water enclosed between
vertical cross sections through ab and cd has the same
volume as that between inclined sections (perpendicular
to the sides) passing through a'b' and c'd'. There has
thus been change of form only in this mass of water, and
we see that it has been produced by the sliding of every
horizontal layer of the water over that immediately
beneath it. [The same result follows even if a'b' be not
straight, for c'd' will necessarily be equal and similar to
it.] A good illustration of the nature of this kind of
distortion will be seen in the leaves of an opened book,
especially a thick one, such as the London Directory. It
is often well exhibited by piles of copies of a pamphlet,
or of quires of note-paper curiously arranged in a shop-
window. Now when there is resistance to sliding of one
solid on another we call it Friction. Thus the viscosity
of a fluid is due to its internal friction, just as the slower
motion at the bottom than at the top of the channel is
to be ascribed to the friction of the liquid against the
solid. [Another illustration of the subject is frequently
furnished by the way in which we can judge of the
direction of the wind from the mere form of detached
clouds, as seen at a single glance. For such clouds, if
originally nearly spherical, are distorted into ellipsoids
whose longer axes overhang, as it were, the direction of
motion.]
39. We now see why it is that disturbances of liquids
gradually die away : why the waves on a lake, or even
on an ocean, last so short a time after the storm which
TERMS APPLIED TO MATTER. 33
produced them has ceased. Also why winds (for there
is friction in gaseous fluids as well as in liquids, though
the mechanical explanation of its origin may not be quite
the same) gradually die out. In either case the energy
apparently lost is, as in the case of friction of solids,
merely transformed into heat. We also see why it is
that winds have the power of raising water-waves.
The stirring of water, or oil, and the measurement of
the consequent rise of temperature when the whole had
come to rest, the work done in stirring being also deter-
mined, was one of the processes by which Joule found,
with great accuracy, the dynamical equivalent of heat.
40. It is very instructive to watch the ascent of an
air-bubble in glycerine, and to compare it with that of
an equal bubble in water. The experiment is easily
tried with long cylindrical bottles, nearly full of different
liquids, but having a small quantity of air under the
stopper. When the bottle is inverted the bubble has to
traverse the whole column.
The (apparent) suspension in water of mud, and ex-
ceedingly fine sand (to whose presence the exquisite
colours of the sea and of Alpine lakes are mainly due), is
merely another example of viscosity. So is the suspen-
sion of fine dust, and of cloud particles, in the air.
Stokes 1 calculates that a droplet of water, a thousandth
of an inch in diameter, cannot fall in still air at a much
greater rate than an inch and a half per second. If it be
of one-tenth of that size it will fall a hundred times
slower, i.e. not more than one inch per minute ! It is
very remarkable that the resistance in such cases varies
as the diameter of the drop. (See 316.) With larger
1 On the Effect of the internal Friction of Fluids on the Motion
of Pendulums. Camb. Phil. Trans, ix. (1851), eq n (127).
34 PROPERTIES OF MATTER
bodies, moving faster, the resistance is proportional to the
sectional area, being then in great part due to another
cause than viscosity.
41. Bodies are called Elastic or Non-elastic. Compare,
for instance, the properties of a wire of steel with those
of a lead wire ; or of a piece of india-rubber and a piece
of clay or putty. But the popular use of these terms is
generally very inaccurate. The blame rests mainly with
the ordinary text-books of science, which are (as a rule)
singularly at fault with regard to the whole of this special
subject, including even its most elementary parts.
Elasticity, in the correct use of the term, implies that
property of a body in virtue of which it recovers, or tends
to recover, from a deformation.
The phrase " tends to recover " is scarcely scientific ;
we should preferably say " requires the continued applica-
tion of deforming stress ( 128) to prevent recovery, entire
or partial, from deformation."
Kinematics shows us that any deformation, however
complex, is made up of mere changes of bulk and of form.
A distortion may therefore be wholly Compression, or
wholly Shear ( 37), or made up of these in any way.
Hence there are two distinct kinds of Elasticity, viz.
Elasticity of Bulk and Elasticity of Form. The former
is possessed in perfection by all fluids, while the second
is wholly absent. In solids both are present, but neither
in perfection (except perhaps in very special cases, and
then within very narrow limits).
Thus we see that, as a necessary preliminary to in-
vestigations on elasticity of bodies, we must study their
capabilities of being distorted : a whole series of pro-
perties, such as compressibility, extensibility, rigidity, etc.
This investigation is given in Chap. VIII., and its
applications in Chaps. IX., X., XL below.
TERMS APPLIED TO MATTER. 35
42. In popular language, bodies are said to be White,
Black, Blue, Red, etc. The investigation of the under-
lying scientific facts, on which these depend, is partly
physical (and therefore within our scope), but also partly
physiological. The subject is thus a somewhat complex
one.
What do we mean by White Light ? This is a question
much more physiological than physical ; dealing, as it
does, with phenomena which are subjective rather than
objective. Probably the true answer to it depends upon
circumstances, or conditions, which may be varied in-
definitely, and with them will necessarily vary what is
described in terms of them.
Thus, in a room lit by gas, a piece of ordinary writing-
paper, or of chalk, appears white : at least if we have
been in the room for some little time. But if, beside it,
there be another piece of the same paper or chalk on
which, through a chink, a ray of sunlight is allowed to
fall (weakened, if necessary, so as to make the two appear
of nearly the same brightness), we at once call the first
piece of paper or chalk yellow, allowing the second to be
white. Here we enter on a purely physiological question.
In fact, if we accustom ourselves, for a sufficiently long
time, to the observation of bodies in a room lit up only
by burning sodium (which gives almost homogeneous
orange light), we may ultimately come to regard bright
bodies such as chalk, etc., as being white : others, of
course, being merely of different shades, or degrees of
blackness. This, therefore, is foreign to our present
subject. For all that, it furnishes us with the means of
answering an important question somewhat different from
that proposed above, but now a physical question : viz.
What do we mean by a white body ?
36 PROPERTIES OF MATTER.
43. Suppose two sources to exist in the room, giving
different kinds of homogeneous light; one being incan-
descent sodium as before, the other incandescent lithium,
which (at moderate temperatures) gives a homogeneous red
light. Chalk and ordinary writing-paper will still appear
as white bodies to an eye which has become accustomed
to the light in the room ; other bodies appear darker, but
some are reddish, some of an orange tint.
And thus we obtain the idea that what we call a white
body is one which sends to the eye, in nearly the same
proportion as it receives them, the various constituents
of the light which falls upon it ; while a black body
sends none ; and coloured bodies send back light which,
while (in general) necessarily made up of the same con-
stituents as the incident light, contains them in different
proportions to those in which they fell upon it. [It
would only confuse the student were we here to refer to
Fluorescence.]
44. Thus white light would seem to be a mere relative
term. It is conceivable that the inhabitants of worlds
whose sun is a blue star, or a red star (and there are
many notable examples of such stars), may have their
peculiar ideas of white light, formed from their own
circumstances ; as ours is formed from the light of our
own sun, which is what, in contrast with these, we must
call a yellow star.
However this may be, the discussion above has shown
what is meant by a white body. A blue body is, by
similar reasoning, one which returns blue rays in greater
proportion than it does those of other visible light. It is
therefore said to absorb the other rays in greater proportion
than it absorbs the blue rays.
Now we are in a position to understand why blue and
TERMS APPLIED TO MATTEfc. 37
yellow pigments, mixed together, give green : while a
disc, painted with alternate sectors of the same blue and
yellow, appears of a purplish colour when made to rotate
rapidly. .For the light given out by the rotating disc is
a mixture, in the proportion of the angles of the sectors, of
the kinds of light returned by the blue and yellow separ-
ately. But that which the mixed pigments send back
has in great part penetrated far enough into the mass to
run, as it were, the gauntlet of absorption by each of the
separate components in turn, and therefore is finally made
up of those rays alone which are not freely absorbed by
either.
To this discussion we need only add, in illustration of
the conservation of energy, that a body is always found
to be heated in proportion to the amount of light-energy
which it absorbs.
45. Shifting our ground again, we next take the words
Malleable, Ductile, Plastic, and Friable, as applied to
solid bodies.
All of these refer specially to the behaviour of solids
under the action of stresses which tend to change their
form; for the change of volume of solids, even under
very great pressures, is usually very small. The first
three indicate that the body preserves its continuity while
yielding to such stresses, the fourth that it breaks into
smaller parts rather than allow its form to be changed.
And, in popular use at least, the terms imply in addition
that the body is not sensibly elastic.
46. The most perfect example of a malleable body is
metallic gold. The gold leaf employed for " gilding," as
it is called, is prepared by a somewhat tedious process,
which requires a high degree of skill in the workman.
The gold is first rolled into sheets thinner than the thin-
38 PROPERTIES OF MATTER.
nest writing-paper (thus already showing a high amount of
plasticity) ; next it is beaten out between leaves of vellum,
till its surface is increased, and therefore its thickness
diminished, some twenty-fold. A small portion of this
fine leaf is then placed between two pieces of gold-
beater's skin ; and a more skilful workman, with a lighter
hammer, again extends its surface twenty-fold. This
operation can be repeated without tearing the thin film of
metal, so great is its tenacity. ( 226.)
Here we have one dimension (thickness) diminished
in a marked manner, but the product of the other two
dimensions (the surface of the leaf) is of course pro-
portionally increased.
47. The action of the hammer may be practically
viewed as equivalent to that of an intense pressure exerted
through a very small volume, thus at every stroke apply-
ing a finite amount of energy. One portion of this is
changed into heat in the hammer, the anvil, and the
gold leaf ; the rest is employed in doing work against the
molecular forces of the gold, and thus altering its form.
To show that this is the true explanation of the
observed effect, we may vary the experiment by subject-
ing a leaden bullet to the action of a hydrostatic press.
A few strokes of the pump suffice to bruise the bullet
into a mere cake. The process is essentially the same as
that of gold-beating, but lead is by no means so malleable
as gold.
48. This leads us, in our present discursive treatment
of parts of our subject, to inquire how it is, that by
means of such a machine as the Bramah press, a man can
apply pressure sufficient to mould a piece of lead, whose
shape he could scarcely alter to a perceptible amount by
the direct pressure of the hand.
TERMS APPLIED TO MATTER. 89
Here we have a first inkling of the Function of a
Machine. A machine is merely a contrivance by which
we can apply work in the way most suitable for the
purpose we have in hand. Work (as a form of energy)
is a real thing, whose amount is conserved. But we
have seen that it can be measured as the product of
two factors the (so-called) force exerted, and the space
through which it is exerted. Hence, because even when
a machine is perfect it can give out only the energy
communicated to it, if there be but one movable part
to which energy is supplied and another by which it is
given off, the simultaneous linear motions of these two
parts must be in the inverse ratio of the forces applied to
them, or exerted by them, in the direction of these
motions respectively. Thus we are not concerned with
the interior structure, or mode of action, of a perfect
machine : all we need to know is the necessary relation
of the speeds of the two parts or places at which energy
is taken in and given out. This is a matter of kinematics,
and can be made the subject of direct measurement when
the machine is caused to move, whether it be transmitting
work or not.
The statement just made is embodied in the vernacular
phrase
What is gained in power is lost in speed.
Objections may freely be taken to this form of words,
but it is meant to imply precisely what was said above
as to the action of a perfect machine.
If the machine be imperfect, as, for instance, if there
be frictional heating during its working, the heat so
produced represents some of the energy given to the
machine, and the remainder of it is alone efficient.
40 PROPERTIES OF MATTER.
49. A substance is said to be ductile when it can be
drawn into very fine wires i.e. when it admits of great
exaggeration of one of its three dimensions (length) at
the expense of the product of the other two (cross
section). Wire -drawing is, essentially, a very coarse
operation, for it has to be effected by finite stages, the
wire being drawn in succession through a number of
holes in a hard steel plate, in which each hole is a little
smaller in diameter than the preceding one. The more
nearly continuous the operation is made, the more tedious
and therefore the more costly it becomes.
The associated tenacity and plasticity of silver render
it one of the most ductile of metals. And an ingenious
idea of Wollaston's enables us, as it were, to impart to
other metals much of the ductility of silver. His idea
may be briefly explained by analogy as follows. Suppose
a glass rod, whose core is coloured, be drawn out while
softened by heating, the diameter of the core is found to
be reduced in the same proportion as is that of the rod.
Thus, to obtain platinum wires much finer than could be
procured by direct drawing, Wollaston suggested the
boring of a hole in the axis of a cylindrical rod of silver,
plugging the hole with a platinum wire which just fitted
it, and then drawing into fine wire the compound
cylinder. When this operation has been carried to its
limit, practically determined by the ductility of the
silver, the diameter of the platinum has been reduced
nearly in the same proportion as that of the silver ; and
the silver may be at once removed from the fine platinum
core by plunging the whole in an acid which freely
attacks silver but has no effect on platinum.
50. Plasticity is shown, on the large scale, by many
substances which, in hand specimens, appear fragile in
TERMS APPLIED TO MATTER. 41
the extreme. Glacier-ice is one of these, but its behaviour
is so closely connected with its thermal properties that
we can only mention it here.
The earth as a whole, though its rock-structure appears
so rigid, has been found to be more plastic (under the
tidal attraction of the moon) than a globe of glass of the
same size would have been.
But it is specially under the action of small but
persistent forces that bodies, which are usually regarded
as brittle or friable, show themselves to be really plastic.
A good example of this is given by an experiment due to
Lord Kelvin. Cobbler's wax is usually regarded as a
very brittle body ; yet if a thick cake of it be laid upon
a few corks, and have a few bullets placed on its upper
surface (the whole being kept in a great mass of water
to prevent any but small changes of temperature), after
a few months the corks will be found to have forced
their way upwards to the top of the cake, while the
bullets will have penetrated to the bottom.
[Tresca has investigated the flow of solid metals, and
has shown that even steel flows under the action of
sufficient stress. W. P.]
51. For variety, let us next take the terms Trans-
parent, Translucent, and Opaque.
These refer, of course, to the behaviour of a substance
with regard to the passage of light through it. In
common speech, a pane of ordinary window-glass is
called transparent, while a piece of corrugated or of
ground glass is translucent : the latter transmits rays,
no doubt, but with their courses so altered that they are
no longer capable of producing distinct vision of the
source from which they come. Consistency would require
that the term translucent should also be applied to
42 PROPERTIES OF MATTER.
irregularly-heated air, or to a mixture of water and strong
brine before diffusion has rendered it uniform throughout.
Translucent is hardly a scientific word, unless we
choose to limit its application to heterogeneous bodies.
In science we speak of the degree of transparency of a
homogeneous substance ; as, for instance, water more or
less coloured, and employed in greater or less thickness.
In such cases, besides the inevitable surface-reflection,
there is more or less absorption ; and the percentage of
any definite kind of incident light which unit thickness
of the substance transmits is called its transparency for
that kind of light.
Opacity may arise from either of the two causes just
mentioned. Light may enter a body, and be unable to
proceed farther, as is the case with lamp black. Or it
may fall on a highly polished surface, such as thinly
silvered glass, and be in great part reflected without
entering.
In the former case it is said to be absorbed ; and, when
this happens, the absorbing body is raised in temperature.
The incident energy is converted from the radiant form
into that of heat.
In the latter case part only can enter the body; and, if
it meet in succession other reflecting surfaces in sufficient
number, practically the whole of it may be reflected.
This is the case with a heap of pounded glass, a cloud,
a mass of snow, or of froth or foam. All of these
materials are transparent, but they reflect some of the
incident light; and, in consequence of the multiplicity
of surfaces which the light has to encounter, the greater
part of it is reflected before it has penetrated deeply into
the mass. Hence the whiteness and brightness of snow
and clouds in full sunshine.
TERMS APPLIED TO MATTER. 43
52. We have here an excellent opportunity of calling
the student's attention to the distinction : a very pro-
found one : between Heat and Temperature.
For we have seen that energy, in the form of light,
when absorbed, becomes heat in the absorbing body, and
thus raises its temperature. But if the same quantity of
heat had been given to a body, of the same nature but
of twice the mass, the rise of temperature would have
been only half as great. The very form of words here
used shows at once how different are the meanings of
the words temperature and heat. For the quantity of
heat (so much energy, a real thing) is perfectly definite,
but the effect it produces on the temperature (a mere
state) depends on the quantity and quality of the mass to
which it is communicated.
Heat is therefore a thing, something objective; tem-
perature is a mere CONDITION of the body with which
the heat is temporarily associated ; a condition which
in certain cases determines the physical state of the body
itself, and in all cases determines its readiness to part
with heat to surrounding bodies or to receive it from
them.
Heat may, in this connection, but only for illustration,
be compared with the air compressed into the receiver of
an air-gun ; temperature would then be analogous to the
pressure of that air. Neither of two receivers would
(except by diffusion, with which we are not at present
concerned) give air to the other, when a pipe is opened
between them, if the pressure were the same in both ;
but air would certainly flow from the receiver in which
the pressure is greater to the other ; and this, altogether
independently of the relative capacities of the two receivers,
or the consequent amounts of their contents.
44 PROPERTIES OF MATTER.
53. As another example, take the terms Cohesive, In-
coherent, Repulsive.
A lump of sandstone has considerable tenacity, which,
of course, is to be ascribed to those molecular forces of
which we spoke in 26. But when, in virtue of its
friability, it has been pounded down into sand, it becomes
an incoherent powder. And we know that it must at
some time previously have been in this form, for it often
contains fossil plants or fish, and it may even have pre-
served (perhaps for a million or more of years) records of
surface-disturbance in the form of dents made by rain
or hail, or by the feet of birds or reptiles. When a
sufficiently deep layer of sand is deposited, by drifting
or otherwise, above this portion, its loose particles are
brought by the consequent pressure so close together that
their molecular forces once more come into play.
The graphite, or plumbago, which forms the material
for the finest drawing-pencils, is a somewhat rare and
valuable mineral. In cutting it up into " leads," however
carefully, a considerable portion is reduced to powder
i.e. sawdust. But if this powder be exposed, in mass, to
pressure sufficient to bring its particles once more within
the extremely short mutual distance at which the molecular
forces are sensible, these forces again come into play, and
the powder becomes a solid mass, which can in turn
be sawn into " leads " for a somewhat inferior class of
pencils.
The whole of this part of the subject, especially as
regards liquids, will be fully treated later, so that we need
not further consider it here.
54. But let us contrast, with the behaviour of the
particles of a solid or a liquid, that of the particles of a
gas or vapour. Such substances require to be subjected
TERMS APPLIED TO MATTER. 45
to external pressure in order to prevent their particles
from being widely scattered. When a small quantity of
air is allowed to enter an exhausted receiver it dilates so
as to occupy with practical uniformity the whole interior
of the receiver, however large that may be.
This result was, naturally enough, at first ascribed to
a species of repulsion between the various particles ; but
the notion was found to be an erroneous one. For the
effects of a true repulsion, capable of producing the
practically infinite dilatation already spoken of, could not
all be consistent with the corresponding observed results.
The mode of departure from them depends upon the law
according to which the repulsion may be supposed to vary
with the distance between two particles. Some assumed
laws would give as a consequence that the particles would
all be driven to the sides of the vessel, leaving the interior
void. Others would require that the pressure should
change in value if we were to take half the gas and con-
fine it in a vessel of half the content. Others would
make it different at different parts of the surface unless
the vessel were truly spherical, etc. etc.
The true explanation of the phenomenon becomes
obvious to us when we apply heat to the gas. For it
then appears that the pressure requisite to maintain the
whole at a constant volume increases as the temperature
is raised ; and thus that heat is, in some way, the cause
of the pressure.
55. Hence we are led to what is called the Kinetic
Theory of gases, whose fundamental assumption ( 33) is
that the particles dart about in all directions (with an
average speed which is greater the higher the tempera-
ture), impinging on one another, and also upon the sides
of the containing vessel. This continued series of very
46 PROPERTIES OF MATTER.
small but very numerous impacts (each, by itself, absolutely
escaping observation) is perceived by our senses as the
so-called "pressure" exerted by the gas. Experiment
shows that, when a gas is confined in a vessel of definite
size, the changes of its pressure are nearly proportioned
to the changes of temperature, as measured by a mercury
thermometer, whether these changes be in the direction
of a rise or a fall. If we assume, for a moment, that
this statement is true for all ranges of temperature, even
beyond those attainable in experiment, it leads us to the
very important question : At what temperature does the
pressure of a gas vanish ?
Calculations carried out in the above way showed that,
under the assumption just mentioned, all of the (so-called)
permanent gases cease to exert pressure at one common
temperature (about- 273 C.). Therrnodynamical theory
comes to our assistance and shows that the above guess is
not far from the truth : that a body, cooled to 274 C.,
cannot be cooled any farther; that it then is, in fact,
totally deprived of heat
We might, therefore, fancy that a gas, if it could be
brought to this temperature, would be reduced to a mere
layer of incoherent dust or powder, deposited by gravity
on the lower surface of the containing vessel. But
experiment has shown that gaseous particles, even while
in motion, if only close enough together, exert mutual
molecular forces, so that the result (on any gas) of the
conditions above specified would probably be its assuming
a liquid or even a solid form. In fact Andrews has shown
that, for each particular substance, there is a temperature
(usually called the Critical Point) such that, while the
substance is at any higher temperature it is necessarily a
gas, in the sense that it cannot be liquefied by any pressure,
TERMS APPLIED TO MATTER. 47
however great. At any lower temperature it can be
liquefied by the application of sufficient pressure, and is
therefore to be regarded as a true vapour.
56. We speak of bodies as Hard and Soft. These are
barely scientific terms; because, unless they are strictly
defined, they may bear a great variety of meanings.
Thus, for instance, we have the mineralogist's Scale of
Hardness, which is often of great practical value in field-
work. For there are numerous instances in which two
quite different minerals (sometimes a very valuable and a
very common one) are almost undistinguishable from one
another so far as colour, density, and crystalline form are
concerned. Chemical tests (even the comparatively coarse
blowpipe tests), though they would settle a question of
this kind at once, are not readily applied in the field.
Hence the use of the scale of hardness, in which minerals
are so arranged that every one can scratch the surface of
any other which is lower in the scale. By carrying a set
of twelve small specimens only, of selected minerals, the
finder of a doubtful crystal can readily determine its rank
among them as regards scratching; and can thus often
settle in a moment what would otherwise require some
time, even with the facilities of a laboratory.
In such a scale diamond, of course, stands at the top,
while native copper, one of the toughest of substances, is
far below it.
But if we were to test relative hardness by some other
method, say by blows of a hammer, we should be led to
arrange our specimens in a very different order. The scale
above spoken of is, therefore, by no means a scientific one ;
though, as we have seen, it may often give easily some
useful information.
CHAPTER IV.
TIME AND SPACE.
57. WE begin with an extract from Kant, who, as mathe-
matician and physicist, has a claim on the attention of
the physical student of a very different order from that
possessed by the mere metaphysicians.
" Time and space are two sources of knowledge, from
which various a priori synthetical cognitions can be
derived. Of this pure mathematics give a splendid
example in the case of our cognitions of space and its
various relations. As they are both pure forms of
sensuous intuition, they render synthetical propositions a
priori possible. But these sources of knowledge a priori
(being conditions of our sensibility only) fix their own
limits, in that they can refer to objects only in so far as
they are considered as phenomena, but cannot represent
things as they are by themselves. This is the only field
in which they are valid ; beyond it they admit of no
objective application. This peculiar reality of space and
time, however, leaves the truthfulness of our experience
quite untouched, because we are equally sure of it,
whether these things are inherent in things by themselves,
or by necessity in our intuition of them only. Those,
on the contrary, who maintain the absolute reality of
TIME AND SPACE. 49
space and time, whether as subsisting or only as inherent,
must come into conflict with the principles of experience
itself. For if they admit space and time as subsisting
(which is generally the view of mathematical students of
nature), they have to admit two eternal infinite and self-
subsisting nonentities (space and time), which exist with-
out their being anything real only in order to comprehend
all that is real. If they take the second view (held by
some metaphysical students of nature), and look upon
space and time as relations of phenomena, simultaneous
or successive, abstracted from experience, though repre-
sented confusedly in their abstracted form, they are
obliged to deny to mathematical propositions a priori
their validity with regard to real things (for instance in
space), or at all events their apodictic certainty, which
cannot take place a posteriori, while the a priori concep-
tions of space and time are, according to their opinions,
creatures of our imagination only." l
On matters like these it is vain to attempt to dogma-
tise. Every reader must endeavour to use his reason, as
he best can, for the separation of the truth from the
metaphysics in the above characteristic passage.
58. We must now take up, as indicated in 21, the
property Extension, which is one of those expressly in-
cluded in our provisional definition of matter.
It implies that all matter has volume, or bulk. The
thinnest gold leaf has finite thickness, ths finest wire has
a finite cross section.
In popular language this is recognised by the use of
the associated terms length, breadth, arid thickness.
In other words, the term extension recognises the
essentially Tridimensional character of space.
1 Critique of Pure Reason. Max Muller's Translation.
50 PROPERTIES OF MATTER.
Why space should have three dimensions, and not
more nor less, is a question altogether beyond the range
of human reason. Only those who fancy that they know
what space is, would venture (at least after well con-
sidering the meaning of their words) to frame such a
question.
59. The proof that our space has essentially three
dimensions is given in its most conclusive form by the
statement, based entirely upon experience, that
To assign the relative position of two points in space,
three numbers (of which one at least must be a multiple of
the unit of length) are necessary, and are sufficient.
It is an easy matter for us, accustomed to tridimen-
sional space, to imagine one or more of its dimensions
to be suppressed. In fact so-called Plane Geometry is
the geometry of one particular kind of two-dimensional
space; Spherical Trigonometry that of another. We
cannot well speak of the geometry of space of one, or of
no dimensions ; but the idea we should thus attempt to
express is a correct one, though the term is inappropriate.
When, however, we try to conceive space of four or
more dimensions, we are attempting to deal with some-
thing of which we have not had experience ; and thus,
though we may by analogy extend our analytical and
other processes to an imagined space, in which the rela-
tive position of two points depends on more than three
numerical data, we can form no precise idea of how the
additional dimensions would present themselves to our
senses or to our reason.
A few remarks on this subject will be made at the end
of the chapter.
60. Space of no dimensions is a geometrical point, of
which nothing further can be said
TIME AND SPACE. 51
61. Space of one dimension: let us call that dimen-
sion Length : is a mere geometrical line which may be
curved or straight. But to be sure of the existence of
this characteristic, and to understand its true nature, we
must have cognisance of space of two dimensions if it be
a plane curve, of three if it be tortuous. The study of
all the properties of space of one dimension, though an
excessively simple affair, is of very great intrinsic import-
ance, besides being a necessary step towards that of the
higher orders. We will, therefore, treat it so fully that a
far less amount of detail will be necessary when we come
to two and to three dimensions.
62. Every one, whether he be aware of the fact or not,
is acquainted by experience with at least the elements of
this subject. Suppose, for instance, we take as our one-
dimensioned space any one of the roads or railways lead-
ing from Edinburgh to London ; which we will, for the
moment, suppose to be straight, and to run due south.
The mile-stones, set up at equal distances along the road,
mark the positions of various points in terms of the one
dimension, length, which is alone involved, or, rather, to
which for the present we restrict our consideration. And
a Gazetteer or a Railway Guide gives us the positions of
the towns or stations along the road or line : the position
of each being fully described by a single number, under-
stood as a multiplier of a mile or of some other specified
unit of length, and with a qualification which will
presently be introduced.
But these numbers refer to the distance from some
assumed starting-point, or Origin as it is technically
called ; say, in this case, London. Thus we find in an
old Road Guide, for the particular one-dimensioned space
52 PROPERTIES OF MATTER.
called the East Coast Route, a column of data from which
we extract the following :
Miles.
London .........
York .196
Berwick 337
Edinburgh ........ 395
Fractions of miles are omitted, to avoid mere arithmetical
complication.
From this table, by ordinary subtraction, we form a
list, as below, of the lengths of what we may call the
various stages of the route. Thus
Miles.
London to York . , . . . . .196
York to Berwick . 141
Berwick to Edinburgh . . . . .58
It will be seen that, in this list, the origin from which
each number is measured is the first named of the two
corresponding places, and the number itself is found by
subtracting, in the first list, the number corresponding
to the first of the two places from that corresponding to
the second.
63. Now let us at once take the only step which
presents any difficulty. Choose York as our origin, and
boldly apply the rule just given, no matter what the
consequences may be. The result is
Miles.
London . . . -196
York
Berwick. . V 141
Edinburgh 199
Here there is no difficulty whatever in understanding the
numbers for Berwick and for Edinburgh. They are, as
before, the numbers of miles by which Berwick and
Edinburgh are separated from York. Also the number
TIME AND SPACE. 53
for London, when York is the origin, differs from that for
York, when London is the origin, only by change of
sign.
So that we at once recognise the meaning of the
negative sign as applied to a length in our one-dimen-
sional space : it measures the length in the opposite
direction to that in which a positive length is measured.
The necessity for this convention, and its extreme
usefulness, were early recognised in Cartesian geometry,
but they had long before been applied in common arith-
metic as well as in algebra.
Perhaps the simplest view we can take of the subject
is that afforded by a man's " balance " at the bank. So
long as this is on the right side (i.e. positive) he can draw
any less amount and still be on the credit side; if he
overdraws (i.e. takes more out of the bank than his
balance), the difference is negative, and he is to that
amount indebted to the bank.
64. In the first of the three little tables above, all the
places involved lay to the north of the origin (London),
and were all therefore affected by the same sign (which
we happened to take as + ). When York was taken as
origin, Berwick and Edinburgh were to the north, and
their numerical quantities were still + . But London
being to the south, had a - number.
It would be easy to give multiplied examples of this,
but they are unnecessary. The only additional com-
ments which we need make are these :
(1.) When the northern direction along a line was
called +, the southward necessarily became . Simi-
larly had we chosen southward as + , northward would
have become - .
(2.) We chose for our special example a northward-
64 PROPERTIES OF MATTER.
running line, but we might equally well have chosen an
eastward one, etc. Hence pairs such as N". and S., E.
and W., up and down, etc., must he regarded as having
their members contrasted exactly as are the + and - of
Algebra or of Analytical Geometry.
And, just as a displacement in either direction along a
line may be regarded as +, while a displacement in
the opposite direction must then be regarded as - , so it
is with rates of motion, i.e. Speeds, in space of one
dimension.
Thus the relative speed of two trains running north-
ward, A at 60 miles an hour, B at 40, is 20 miles an
hour northward as regards A seen from B, and 20 south-
ward as regards B seen from A; so if A be moving
southward at 60 miles an hour, and B northward at
40, the speed of A with regard to B is 100 miles per
hour southward, and of B with regard to A 100 miles per
hour northward.
The idea of speed, as so many units of linear space
described per unit of time, is a complex one : involving
both of the fundamental ideas. We express this by
saying that its Dimensions are
L
[*}
This implies that, in whatever proportion we increase
our unit of length, the measure of a speed is diminished
in that proportion : while it is increased in the same
proportion as that in which the unit of time is
increased.
Thus a speed of 5280 feet per second is but 1 mile
per second; while a speed of 1 foot per second is 60
feet per minute.
65. A precisely similar distinction (as to + and - ) is
TIME AND SPACE. 65
observed when our one-dimensional space is a curved
line; take for example the orbit of a planet. To
describe fully the position of the planet, when the orbit
is given, one number alone (say the angle-vector, the
angle which the radius-vector, or line joining the centres
of planet and sun, makes with some fixed line in the
plane of the orbit) is required. This, however, must
again be qualified as + or - . (In the case of angles, we
agree to call them + when they are measured in the
opposite direction to that of the motion of the hands of a
watch; that is, when they are described in the same
sense as that in which the northern regions of the earth
turn about the polar axis.) Angular velocity in one plane
(i.e. rate at which the radius-vector turns) is similarly
characterised.
In aU cases where motion is restricted to one line the
same thing holds. Thus the position of a pendulum is
at every instant completely assigned by the angle the
rod makes with the vertical, provided we are also told on
which side the displacement is.
The record kept by a self-acting tide-gauge gives at
any instant the elevation or depression (again + and )
of the water above or below the mean level. Similarly
with registering barometers, thermometers, etc. But,
for the full appreciation of the indications of these
records, they are usually made in two dimensions by the
use of an important principle which will presently be
explained. (68.)
66. In what precedes we have been dealing with a
kind of space in which the only displacements are
forward or backward ; nothing is possible (nor even con-
ceivable) sideways or upwards.
This characteristic applies to Time, as well as to space
56 PROPERTIES OF MATTER.
of one dimension, and therefore we should expect to find,
as we do find, that (with the necessary change of a word
or two) all that has just been said with reference to relative
position is true of events in time, as well as of points in
one-dimensional space. There is no such thing as motion
or displacement in time, so that this part of the analogy
is wanting. Every event has its definite epoch, for ever
unalterable. And of course there is no going sideways or up-
wards, as it were, out of the one-dimensional course of time.
Thus we find that to assign definitely the position of
an event in time, provided our origin is assigned, all
we need know is a single number (a multiplier of the
time-unit) with its sign, + or - , signifying time after or
time before that origin.
Our usually adopted origin is the Christian era, and
we speak of 1899 A. D. as the present year [I leave these
dates as written. W. P.], while the date of the battle of
Marathon is recorded as 490 B.C. The difference between
the characteristics A.D. and B.C. is of precisely the same
nature as that between north and south, or + and .
Hence, if we wish to find the interval between the
present time and the battle of Marathon, we have to
subtract +1899 (the position of the new origin) from
- 490. The result is - 2389, i.e. Marathon was fought
2389 years ago. Thus to change the origin, or epoch,
we must perform precisely the same operation as that
which gave us the table in 63, from the first table in
62. Similarly, to change our system of chronology to
the year of the world (designated by A.M.) or to the old
Roman (marked A.U.C.), all we need do is to subtract from
each date (A.D. or B.C., regarded as + and - respectively)
the assumed date of the creation of the world (4004 B.C.)
or of the foundation of the city of Rome (753 B.C.).
TIME AND SPACE. 57
We need say no more on such a matter. Every intel-
ligent reader can make new and varied examples for
himself.
67. Passing next to space of two dimensions, whether
plane or spherical, we see at once from a map, or a globe,
that the position of a place is given by two numbers, its
Latitude and Longitude. But each of these has to be
qualified for definiteness by the + or - sign, or something
equivalent. Thus we have K or S. latitude, and E. or
W. longitude.
But there are two methods, specially applicable to the
plane, which deserve closer attention in view not only of
their intrinsic usefulness, but also of their bearing on the
general question of tridimensional space. These are known
in geometry as Rectangular and Polar co-ordinates.
68. In the first we assume two reference lines at right
angles to one another, both passing through the origin,
and assign the position of a point by giving its distances
from these two lines. These distances are looked on as
drawn towards the point from either line, and each there-
fore changes sign when the point is taken on the other
side of the corresponding reference line. This is symbol-
ised in the cut. Ox, Oy are the two reference lines, the
origin. The perpendiculars PM, PN, let fall from P on
these lines, completely, and without ambiguity, define its
position. For if we know OM or NP, the # of P, i.e. its
distance from Oy, that condition alone limits our choice
for P to points lying in PM, a line drawn parallel to
Oy and everywhere at the assigned distance, x, from it.
Similarly, y being given as ON or MP, the choice
of points is limited to those on the line NP, all of which
have this property.
But two lines at right angles to each other must
58
PROPERTIES OF MATTER.
intersect, and in one point only. Thus the point P is
determined by the conditions without ambiguity.
-X
X
FIG, 2.
If P lie to the left of Oy, its x is negative ; if below
Ox, its y is negative. The lettering in the cut, at the ends
of the lines bounding the four quadrants, shows at a glance
the signs of x and y when P is situated in any one of them.
In general, any given relation, between the x and y of
a point, limits its position to a definite Curve in the plane
of the reference lines. It is often very convenient to
represent such a relation by a curve ; and, in fact, most
self-registering instruments actually trace such a curve
for us. Thus, if intervals of time (as OM) measured
from a definite instant (represented by 0) be laid off
along Ox, with the corresponding heights of the thermo-
meter, barometer, tide, etc., erected as perpendiculars
(MP) at their extremities, we have (as the Locus of P) a
curve showing the mode in which temperature, pressure
of the atmosphere, etc., change as time goes on. But
such curves can be traced by a pencil attached to the
instrument, or by photographic processes, on a long band
TIME AND SPACE. 50
of paper which is drawn horizontally past it, at a uniform
rate, by clockwork.
69. In the second method mentioned in 67 the data
are the length of OP (the radius-vector), and the magni-
tude of < MOP (the angle-vector), 65. These are usually
denoted by r and 9, respectively. Here r is always taken
as a positive (or rather a signless) quantity, while 6 is posi-
tive if it be measured round from Ox counter-clockwise.
This is the method adopted by a surveyor when, with a
chain and a theodolite, he measures a field. His reference
line, Ox, is usually given by a magnetic needle attached to
the theodolite. He measures the angle a;OP and the dis-
tance OP, P being a corner (let us say) of the field. These
two data, determined for each prominent part of the bound-
ary, enable him to plot the field ; and therefore contain all
the necessary numerical data for calculating its area, etc.
It is also the method usually employed in dealing with
orbital motion of any kind in one plane.
Comparing the two methods, we see that the directed
line OP may be resolved (as it is called) into OM and
MP, lines in directions perpendicular to one another.
Also that this resolution, in any direction, is effected by
means of the cosine of the angle involved.
<
For x = OM = OP cos zOP = r cos 6,
<
y= MP = OP cos ?/OP = r sin 6.
It is clear that, though we have hitherto spoken of
and P as the simultaneous positions of two points, we
may look on them as successive positions of one (moving)
point. If we look on the displacement as having been
produced uniformly, and in one second, it represents in
magnitude and direction the Velocity of the moving
60 PROPERTIES OF MATTER.
point ; and OM, MP represent, on the same scale, its
resolved parts or components, parallel to Ox and O?/.
These components are entirely independent of one
another, so that to compound two or more displacements
or velocities we have only to resolve each into its east-
ward and its northward components, and deal with these
respectively by the ordinary algebraic process, to obtain
the corresponding components of the resultant.
70. As examples, we give one or two results which
will be specially useful to us in later chapters.
If a point be moving, in any manner whatever, we may
consider its velocity alone, independent altogether of the
actual path pursued. Here we are introduced to a new
idea, that of Acceleration. For, as velocity is rate of
change of position, acceleration is rate of change of
velocity.
Take any fixed point, 0, and let OP represent, in
magnitude and direction, the velocity of the moving
point. After one unit of
time let the velocity be
represented by OP X ; after
two units, by OP 2 ; and
so on. It is clear that all
the points P, P v P 2 , etc.,
lie on some definite curve,
which will be the more
accurately traced the greater the number of points we
obtain in any assigned portion of it ; i.e. the smaller we
assume our unit of time. If the motion whose properties
are thus studied be that of a particle of matter, this
curve (which is called the Hodograpli) is necessarily
continuous, for the velocity cannot alter by starts ( 120)
either in magnitude or in direction. And, as OP passes
TIME AND SPACE. 61
to a proximate position, OQ, by having a velocity PQ
compounded with it, the Acceleration of OP is the
velocity with which P moves in its curve. If the path
be a plane curve, the hodograph is also plane.
This construction enables us at once to solve a number
of elementary problems in kinematics, which will be of
great use to us in the sequel.
In 64 above, we showed that the dimensions of speed
(V) are
In precisely the same way we see that those of accelera-
tion (A) are
Thus the numerical measure of acceleration is diminished
in proportion as the unit of linear space is increased :
but is increased in the duplicate ratio of that of the time
unit.
An acceleration of 1 foot per second, per second, is
obviously the same as 3600 feet per minute, per minute.
71. Suppose a point to move uniformly, with speed V,
in a circle of radius R. OP in the hodograph (Fig. 3)
has constant length V, and its direction rotates uniformly.
Hence the hodograph is another circle, also uniformly
described in the same sense (i.e. clockwise or counter-
clockwise), and in the same period of time. Hence the
speed of P must be such that it describes a circle of
radius V, in the time that a point whose speed is V takes
to go round a circle of radius R. It must, therefore, be
V 2 /R. Also the direction of this speed is perpendicular
to OP, and therefore along the radius of the first circle.
And its direction is towards the centre of that circle,
62 PROPERTIES OF MATTER.
because both circles are described clockwise, or counter-
clockwise.
Let, now, the figure repre-
sent the circle of radius R,
and draw any diameter, ACA'.
Then N moves round with
speed V, and the acceleration of
its motion is V 2 /R along NC.
Remembering that accelerations
and velocities are resolved like
lines, we see that if NM be
drawn perpendicular to AA', the speed of the point M
along MC will be
V MN
NO'
and its acceleration along MC, and towards C, will be
R CN = JR 2 ^ M *
The motion of M, thus defined, is called Simple Har-
monic. It obviously consists in a vibration back and
forward along the line AA', the speed being greatest at
C, and vanishing at A and A'. The special characteristic
is that the acceleration is always directed towards C, and
is proportional to the displacement of M from that point.
72. If we use Newton's Fluxional Notation, in which
the rate at which a quantity increases per unit of time
is expressed by putting a dot over the symbol for that
quantity, a second dot placed over it will signify the rate
at which that rate increases, and so on.
Thus, if CM above be denoted by x, the speed of M is
x, and its acceleration is x. And we see ab once from the
result of last section that
TIME AND SPACE. 68
the negative sign being prefixed because while x is directed
to the right in the figure, the acceleration is directed to
the left, and conversely. Whenever, in future chapters,
we meet with a relation of this kind, we will, therefore,
interpret it as expressing simple harmonic motion. The
multiplier of the right-hand side depends only on the ratio
of V to E : what is called ( 65) the angular velocity of
the radius-vector CN. If we denote this by w, the equa-
tion may be written
X = -u?x;
an equation which belongs to all simple harmonic motions,
whatever be their range of vibration, provided the angular
velocity in the corresponding unform circular motion be
(o, or the period of a complete revolution 27r/o>. Any such
motion is therefore fully described by
05 = a COS.
where a and a are absolutely arbitrary.
73. The result above was obtained by projecting uni-
form circular motion on a diameter of the circle, or, what
comes to the same thing, on a plane perpendicular to the
plane of the circle.
But an exceedingly interesting result is obtained by
projecting the circular motion on any other plane. In
orthogonal projection equal areas are projected into equal
areas, and a circle is projected into an ellipse whose centre
is the projection of the centre of the circle.
Hence the projection gives motion in an ellipse, the
radius-vector drawn from the centre of the ellipse tracing
out equal areas in equal times, and the acceleration being
still directed inwards along the radius-vector, and still
bearing the same proportion to it.
74. Another extremely useful result may be obtained
64
PROPERTIES OF MATTER.
by supposing the unform angular velocity in the circle
to be maintained, but with a continual shrinking of the
radius at a rate measured (per second) by K times its
length at each instant.
The velocity of the moving point is thus made up of
two components, one along the circle, the other along the
radius, each proportional to the radius. Hence the path
is a spiral which makes a constant angle with the radius,
what is called the Equiangular, or Logarithmic, Spiral.
The radius- vector still revolves uniformly. 1
Let PB be the spiral, SP any radius. Then, if PT be
FIG. 5.
the velocity of P, and a the (constant) angle between its
direction and that of PS, we see at once that
whence
PT sin a = o>SP, PT cos a = /cSP,
K = <a cot a.
If SO be equal and parallel to PT, Q is a point in the
hodograph. But as PT, and therefore SQ, is proportional
to SP, and the angle QSP is the supplement of a, the
hodograph is the same spiral rotated through a given
angle, and altered in its linear dimensions by the factor
Thus the holograph of the hodograph is another
l Proc. K.S.E., December 19, 1867.
riME AND SPACE. 65
similar spiral, again turned through the same angle, and
with its radii altered in the ratio ( -^ ) . If PU be
\sm a/
drawn, making an angle a with TP produced backwards,
and meeting QS in U, it will therefore be the direction of
acceleration at P.
But PU may be resolved into PS, SU, the first of
which is along the radius-vector, the second parallel to
the tangent at P. The parts of the acceleration in these
directions are, respectively,
The latter of these, by the first equation above, may be
written as
2(^-yPT slnaCOSa ~2c, cota.PT-2PT.
\sm a/ <
Hence the motion of P is due to an acceleration along,
and proportional to the length of, PS, and another along,
and proportional to the length of, TP.
And of course the resolved part of the motion along
any line in the plane possesses the same characteristics.
If x represent the distance between the projections of S
and of P, on such a line, we see at once that we have
r- 2
x= -2ocoto. x-
-S-
Sin 2 a
or, introducing the value of K above,
This differs from the equation for simple harmonic
motion ( 72) by the term involving K, whose physical
interpretation (when it is multiplied by the mass of the
oscillating particle) is a resistance proportional to the
speed of the motion. But the preceding investigation
E
66 PROPERTIES OF MATTER.
shows us that an equation of this form represents the
resolved part (in some definite direction) of uniform
circular motion with angular velocity to, the radius of the
circle shrinking in each second by the fraction K of its
amount. (This is the same thing as saying that its
logarithm diminishes by K in unit of time.) Or we may
call it simple harmonic motion whose scale is constantly
diminishing at a definite rate.
This special case of motion is fully described by the
equation
Compare the result of 72.
75. The step to three-dimensional space is now easy.
We will take it from a somewhat altered point of view.
Our reference system is now three planes at right angles
to one another ; say the floor, the north wall, and the
west wall of a room, the corner in which these three meet
being for the time our origin.
And the position of a point is determined without
ambiguity if we know its distances from these planes,
with the proper sign of each.
For, knowing only its distance from the floor, we limit
it to the horizontal plane which is everywhere at that
distance from the floor. Similarly the second condition
limits it to a definite plane parallel to the north wall.
These two conditions together limit it to a certain hori-
zontal line lying east and west. The third condition
limits it to a certain plane parallel to the west wall ; and
this is intersected in one point, and one only, by the east
and west line just mentioned. That one is the sole point
which satisfies all three conditions.
Thus, let represent the origin, yOz the north wall,
TIME AND SPACE.
67
zOx the west wall, and xOy the floor. The figure is
drawn as seen by an eye equidistant from these three
planes, and in the room, i.e. on the positive side of each
of them. And it will be noticed that the lettering, x, y, z
of the ends of the edges, which meet in 0, is so applied
that rotation from Ox to Oy, from Oy to Oz, and from
Oz to Ox again, will all alike appear to be counterclock-
wise.
The position of any point, P, is then found thus :
Draw PIS' perpendicular to the floor, meeting it in N ;
thence NM perpendicular to Ox. Then OM = ic is the
distance of P from the north wall ; MN = y is its distance
from the west wall ; and JSTP = z is its distance from the
floor.
If P assume a new position which requires it to pass
through any one of these planes, the corresponding co-
68 PKOPERTIES OF MATTER.
ordinate changes sign ; if it pass through an edge (i.e.
the intersection of two of these planes) two co-ordinates
change sign ; and if it pass through (where the three
planes meet) all three co-ordinates become negative.
This is illustrated by the negative lettering at the (dotted)
prolongation of each edge through 0.
76. But, in analogy with the second method of 69,
we see that the position of P will be fully specified if
we know the vertical plane through in which it lies
(i.e. the plane zON), the angle NOP in that plane, and
the length of OP. The first is determined if we know
the angle xON. Hence we determine P by its distance
from 0, and two angles which (together) enable us to
assign the direction of OP. The angle icON is called the
Azimuth of the plane zQN ; let us denote it by 0. The
angle NOP is called the Altitude of P, as seen from ;
let us denote it by <. Also let the length of OP be, as
before, called r.
Comparing, as before, the results of the two methods,
we see that ON = r cos <f>, and therefore
x = OM = ON cos 6 = r cos Q cos 6,
y = MN = ON sin = r cos <p sin 0,
z = NP =rsinp.
The elements of spherical trigonometry show that the
multipliers of r, in the values of x, y, z respectively, are
the cosines of the angles between the line OP and the
lines Ox, Oy, Oz. Hence the more symmetrical method,
in which these cosines are represented by I, m, n respec-
tively, gives
x = rl, y^rm, z = rn,
with the condition
TIME AND SPACE. 69
It is easy to see that the remark in 69, as to resolu-
tion of a velocity in two dimensions, holds with respect
to three.
Then Newton's Second Law of Motion (Chap. VI.) at
once extends these conclusions to Forces.
77. A remark of great importance must be made here.
We saw in 68 that a point was determined, in x, y co-
ordinates (i.e. plane space of two dimensions), as the
intersection of two straight lines, to one of which it was
confined by its x being given in value, to the other by
the value of its y. But any two independent conditions
connecting x and y will, also, determine their values. A
single condition connecting x and y is known as the
Equation of a Curve, and, when given, limits the position
of P to that curve. Two such conditions, therefore, give
P by the intersection of two curves, on each of which it
must lie. Such a condition applied to a physical particle
is called a Degree of Constraint. In two-dimensional
space a free particle has but two Degrees of Freedom, one
of which is removed by each degree of constraint to which
it is subjected.
78. Similarly we saw that, in three dimensions, the
point given by x, y, z is determined as the intersection of
three planes, on each of which it must lie. But any one
condition connecting the values of x, y, and z is the
Equation of a Surface, and, when it is given, a particle
at the point is subjected to one degree of constraint.
When free, it has but three degrees of freedom ; and thus
three degrees of constraint, by completely determining its
x, y, and z, fix its position.
We should arrive at the same result by considering
relations among the r, 0, <f> co-ordinates. But it suffices
to consider merely what species of constraint each of
70 PROPERTIES OF MATTER.
these imposes when its value is given. All points for
which r has a given value lie on a sphere whose centre is
at 0. When is given, the point must lie somewhere
in the vertical plane zOK When < is given, it must lie
somewhere on a right cone of which is the vertex and
Oz the axis.
[The two latter statements are easily illustrated by
means of a telescope, mounted (in the common way) on
a stand which allows it to rotate either about a horizontal,
or about a vertical, axis. Place it in any azimuth, and
vary its altitude, it turns in a vertical plane about the
horizontal axis. Place it at any altitude, and vary its
azimuth, it rotates conically about the vertical axis.
Hence, by means of these co-ordinates, or conditions,
each definite point in its axis is constrained to lie on a
sphere, a plane, and a cone, simultaneously.]
79. Two devices are in common use for enabling us to
represent, on a plane (or other space of two dimensions)
the third dimension.
Thus, in an Admiralty chart, we find the sea-area
marked over with figures denoting Soundings : i.e. the
average depth of the water at certain places is written in
in fathoms. These soundings are of course more numerous
in regions where there are shoals or intricate channels.
But it is obvious that, if they were numerous enough,
they would enable us to construct a model of the sea-
bottom. The soundings, therefore, supply, as it were,
the necessary third dimension. But this process, though
usually sufficient for purposes of navigation, is at best a
rude and incomplete one.
The other method, however, rises to a very high order
of scientific importance, not merely from the point of view
for which it was originally devised, but on account of the
TIME AND SPACE. 71
extent to which its essential principles are now applied
throughout the whole range of physics. We therefore
devote some space to its full explanation.
80. This is called the method of Contour Lines, and is
employed with great effect in the best maps, such as those
of the Ordnance Survey.
A contour line passes through all places which are at the
same height above the sea-level.
Thus the sea-margin is itself the contour line of no
elevation. Suppose the water to rise one foot (vertically).
There would be a new sea-margin, in general encroaching
more on the land than the former ; encroaching most at
places where the beach has the gentlest slope, not encroach-
ing at all on a perpendicular cliff, and thrust out (seawards)
from an overhanging cliff. This is the contour line of one
foot elevation. It is clear that by supposing a gradual
rise of the sea, or subsidence of the land, foot by foot,
we could obtain a series of curves (each in its turn a
sea-margin) gradually circumscribing the uncovered portion
of the land, and finally closing in over its highest peak.
We require no such natural convulsion as that just im-
agined. Cloud strata, or fog-banks, with definite horizontal
surfaces, constantly show us these appearances in hilly
countries. But it is a simple matter of Levelling to trace
out contour lines, and to draw them on a map of the
district. For practical purposes it is usually sufficient to
draw them for every 50 or 100 feet of additional elevation
above the sea-level.
The celebrated Parallel Roads of Glen Roy are merely
contour lines, etched on the sides of the valley by long-
continued but slight agitation of the margin of the water
which filled the glen to various depths in succession, as the
barriers which dammed it ut> were, at intervals, broken down.
72
PROPERTIES OF MATTER.
Eeferring to 78, we see that a surface can be fully
described in terms of one relation among x, y, and z. Let
the plane of Oar, Oy, be that of the sea-level, and let the
relation expressing the surface of the land be
f(x, y, s) = 0.
Then the contour lines, as traced on the (two-dimension)
map are the curves
f(x, y, 0)-0,
f(x t y, 100) = 0,
f(x, y, 200) = 0, etc.
81. To familiarise the student with the general appear-
FIG. 7.
ance of contour lines, and their relation to the form of the
corresponding surface, we give those of a right cone whos^
TIME AND SPACE. 73
axis is vertical, of a hemisphere, and of a fusiform or
spindle-shaped body.
The fusiform body, whose contour lines are drawn, is
formed by the rotation of a quadrant of a circle about a
vertical tangent, the point of contact being the apex.
And the contour lines are drawn, in each case, at suc-
cessive heights increasing by one-fifth of the whole height
of the figure. Thus the distances between successive
contours, in the two last figures, form the same series of
values, but in opposite order.
The equality of distance between the successive contour
lines of the cone indicates uniform steepness throughout.
In the hemisphere the lines are closer together near the
boundary of the figure, in the spindle they close in on one
another towards the centre ; the hemisphere being steepest
at its edges, and the spindle surface steeper towards the
point.
82. In fact, the Gradient of a surface in any direction
(i.e. the amount of rise per horizontal foot) is obviously,
at any point, inversely as the distance in that direction
between successive contour lines, for they are traced at
successive equal differences of level ; and thus the dis-
tance between them, along any line drawn on the map,
is the space by which we must advance horizontally
along that line while ascending or descending vertically
through 100 feet.
83. The line of steepest slope at any point of a surface
is, of course, perpendicular to each contour line where it
meets it. For the contour line is horizontal, i.e. has nc
slope. And in the projection on a horizontal plane this
perpendicularity remains. Thus the line of greatest slope
at any point is represented on the map by the shortest
line which can be drawn from that point to the nearest
74
PROPERTIES OF MATTER.
contour line. It is the path along which a drop of water
would trickle down. It is therefore called a Stream-line.
84. If the surface be like that of a saddle (concave up-
wards along the horse's back, convex upwards across it),
we have at the middle of the saddle what is called, in
geography, a Col or mountain-pass : the lowest point
of the ridge between two neighbouring summits. The
characteristic of the col is that, at such a point, a contour
line intersects itself. The following sketch shows the
general form of the contours near such a point.
FIG. 8.
In the shaded regions depicted to the right and left oi
the col the ground rises, in the unshaded regions depicted
above and below it falls. [The figures on the contour
lines show their order of altitude above the sea-level.]
Other very special peculiarities might be mentioned,
but they are not necessary for the beginner; and the
more mathematical reader can easily work them out for
himself. 1
iSee Cayley, Phil. Mag., XVIII. 264 (1859); Clerk-Maxwell,
Ibid., December 1870.
TIME AND SPACE. 76
85. If we draw, by the help of the contour lines, the
stream-lines (which, 83, cut them at right angles), we
find that they have the following property. In regions
above the level of a col, they fall away on both sides from
that particular one of their number which passes from a
mountain Summit down to the col, and thence up to the
neighbouring summit. This particular line, then, is the
Watershed, separating two valleys or drainage areas. If
we follow the course of the stream-lines into regions below
the col, we find that they usually approach to the special
stream-lines drawn downwards from the col on opposite
sides. These will therefore be fed by all the little rills
in succession, and thus they become the Watercourses.
A watercourse is thus the stream-line drawn from a col so
as to pass through an Imit, or lowest point of the surface.
If we were to take a cast from a model of a surface
(with its contour lines) and treat it as a model of another
surface, contour lines would remain contour lines, and
stream-lines stream-lines ; but summits would become
imits, and imits summits, while watercourses would
become watersheds, and conversely.
86. So far, we have been dealing with contour lines in
the ordinary sense of the word. But essentially the same
sort of thing is presented by the meteorological curves
called Isobars, and by Isothermals, Lines of Equal Magnetic
Variation, of Equal Dip, etc. etc. In each case the lines
are drawn, on a two-dimension map, so as to pass through
all places where the barometer, or the thermometer, stands
at a given reading or level, where the compass deviates a
given amount from true north, etc. etc. Thus they have
a characteristic similar to that of contour lines, viz. that
all points on any one line possess some definite property
to exactly the same amount. These applications of the
76 PROPERTIES OF MATTER.
principle are of great importance, but they do not belong
so immediately to our subject as do others, of which we
will now give an example or two.
87. Just as water trickles from places at higher, to
others at lower, level, and as heat flows, in a conducting
body, from places of higher, to others of lower, tempera-
ture, so electricity is said to flow from places of higher, to
places of lower, potential. Hence, to study the flow of
electricity in a sheet of metal, we require to know the
lines of equal potential.
The first investigation of this subject, by Kirchhoff, 1
supplies an exceedingly simple and beautiful example.
Putting the wires attached to the ends of a galvanic
battery into contact with a very large sheet of uniform
tinfoil, at points A and B, (Fig. 9) we establish and main-
tain a definite difference of potential between those points
of the sheet. Hence there is a steady flow of electricity
from the one to the other; and it must take place, at
every point, in a direction perpendicular to the equi-
potential line passing through that point. Thus, to find
the lines of flow of electricity, we must have a means of,
as it were, contouring the plate electrically, and finding
its lines of equal potential. This is furnished by a
galvanometer, for that instrument indicates at once any
current passing through its coil of wire. But, if the
ends of its coil be kept at equal potentials, no current
will pass. Hence, if we put one end of the galvanometer
coil in contact with the tinfoil at any point, P, and move
the other end about on the foil until no current passes,
the point, Q, with which it is then in contact, is at the
same potential as P. By fishing about, therefore, we
can, point by point, trace out the equipotential line PQ
1 Pogg. Ann., 1845, Ixiv.
78 PROPERTIES OF MATTER.
passing through P. And the same may be done for
other points, till we have covered the tinfoil with as many
lines of this kind as we desire.
In the special case which we have taken, it was found
that when the plate of tinfoil is very large in comparison
with the length AB, these lines are circles, whose centres
lie in the line AB, and in each of which the ratio
BQ/AQ is the same throughout (see 141); though of
course its values are different for different circles of the
series. A few of these circles are given (in full lines)
in the figure. [To ensure proper contact with the battery,
little circular discs of copper (indicated in white) are
attached to the tinfoil at A and B. The edges of these
discs (on account of the superior conductivity of the
copper) are (practically) equipotential lines, but the points
A and B are not exactly at their centres.]
Now geometry tells us that the lines, which cut at
right angles all circles drawn according to the above
law, belong to another series of circles : viz. those
which are determined by the condition that each passes
through the two points A and B. These circles (some
of which are represented by the dotted lines in the
figure) are therefore the current lines along which the
electricity passes in the tinfoil.
The full circles are drawn for successive equal changes
of potential ; and the dotted circles which are drawn
are so selected that the amount of electricity which
flows in a given time through the space bounded by
portions of each contiguous pair is the same. If the
full lines be regarded as contour lines of a surface, and if
A be connected with the positive pole of the battery, the
left hand side of the figure represents a hill, and the right
hand an exactly equal and similar hollow ; so that the
TIME AND SPACE. 79
halves, as separated by the single straight contour line,
would exactly fit into one another if the whole could be
folded along that line. [This illustrates the last paragraph
of 85.]
If both A and B be connected with the positive pole
of the battery, and its negative pole be connected with
a massive ring of copper, or other good conductor, which
borders the sheet of tinfoil all round at a very great
distance from A and B, the equipotential lines are what
mathematicians call Cassini's Ovals. One of them is the
Lemniscate of Bernoulli, and its double point corresponds
to a col. The figure resembles in general form that of
84, and the current lines are a series of rectangular
hyperbolas.
88. As a final example, somewhat more pertinent to
our present work, take the relation between the pressure,
volume, and absolute temperature, of a given mass of air.
Experiment has proved that when any two of these
three quantities are given, the third is determined.
Calling them p, v, and t respectively, the relation between
them is (nearly enough for our present purpose) found
to be represented by the expression
pv=Rt. . . . . . (1)
where R is a known constant quantity. [In a later
chapter we will study the precise relation. What we seek
at present is an illustration of method, -not a specially
exact representation of fact.~\
Now we may treat p, v, and t just as we treated x, y, z
in 80 above. In this statement lies the essence of the
value of the contour-line idea as applied to questions of
general physics.
Thus the experimental relation among p, v, t, (1) above,
80
PROPERTIES OF MATTER.
may be looked on as the equation of a surface. Let us
draw its contour lines on the plane in which p and v are
measured.
Equation (1) shows that these lines (three of which are
marked in the figure with their temperatures t lt t 2 , 3 ,) are
all rectangular hyperbolas, of which the asymptotes are
the axes of volume and pressure, Ov and Op. Any line
of equal pressure Av-jV^Vf is divided by them so that
FIG. 10.
Av 2 , Av s , etc., are proportional to the absolute
temperatures. So with a line of equal volume
And one special advantage of this mode of representation
is, that the work required to. compress the gas at any con-
stant temperature, as t lt from volume OB' to volume OB,
is given by the area B'j9 / 1 ^? 1 B, which is contained between
the curve t lt the axis of volume, and the lines of equal
TIME AND SPACE. 81
volume B'p'v Bp r This follows at once from the fact
that the work done during an elementary change of
volume dv, under pressure p, is represented by pdv ; a
little element of area bounded by the curve, the axis of
v, and two contiguous ordinates.
Draw a tangent PT to one of these curves at a point
P, and draw PQ parallel to Ov. The compressibility of
a gas, at constant temperature, is the fractional change
of volume per unit increase of pressure. It is therefore
represented by
5? . J_ or -L
QT OM' QT'
or (by a property of the hyperbola) ,
i.e. it is inversely as the pressure.
The expansibility, at constant pressure, is found simi-
larly by producing QP to cut the proximate curve t 2 in
E, ; for it is expressed by the fractional change of volume
per unit rise of temperature, that is
J_ or WH>) OM J_ 1
OM ' ti-ti ZjOM ' tz-ti *!*
This is a mere portion of what is called, in Thermo-
dynamics, "Watt's Diagram of Energy, the whole of which
is an application of the contour idea.
89. We must now, as promised in 59, say a few
words as to a (possible 1) fourth dimension of space.
Let us treat this from the point of view of what we
may imagine weuld be the experience of beings, endowed
with something corresponding to human reason and
human senses, but inhabiting space of one or of two
dimensions.
In one-dimensional space the inhabitants can have
F
82 PROPERTIES OF MATTER.
length only, and have absolutely no hint from experience
of what another dimension could be. Yet we might
imagine them, if they were not mere points, to experi-
ence some perfectly novel, and to them unaccountable,
sensation in passing from part to part of their space
if its curvature were not everywhere tfa same. Suppose,
for instance, their space to be a line with knots on it.
Similarly, an inhabitant of two-dimensional space (a
bookworm, as Sylvester once called him) might, even if
his dimensions were both finite, pass from place to place
in a plane or a spherical space without feeling any new
sensation. But, if a part of his space were creased or
folded, he might be imagined to feel some strange sensa-
tion while he passed through such a part. This is a
question of surface-curvature, which would be totally
unintelligible to a being whose experience (limited to
two dimensions) had not prepared him for it.
So, if there should be a fourth dimension, our three-
dimensioned space may appear to a four-dimensional
observer to have something analogous to curvature or
creasing ; and if, in the course of our solar system's rapid
progress through space, we should come to a region of
that kind, we may fancy that some absolutely novel form
of experience would be the result.
90. Speculations of this nature, however, though based
to a certain extent on scientific facts, necessarily involve
the question of sensation or perception ; and, in so far as
they do so, they pass from the domain of physical science
into the realms of Physiology.
CHAPTER Y.
IMPENETRABILITY, POROSITY, DIVISIBILITY.
91. OUR working definition of matter ( 21) involves
another property besides those discussed in last chapter
viz. Impenetrability. The sense in which we are to
understand this term depends upon the use of the word
occupy as applied to space.
On the theory of ultimate atoms, whether the old
hard atom ( 23) or the vortex atom ( 27), the occupa-
tion of space is complete so far as each atom is concerned.
Where one atom is, it fills space to the absolute exclusion
of every other. But space is not continuously filled by
the atoms of any portion of tangible matter ( 24) ; hence
there may be mixtures of atoms of different kinds, which
will be the more perfect and intimate the smaller we
suppose the individual atoms to be. But there is no use
in discussing questions of this kind, at least until we
prove the existence of atoms. Thus the strictly scientific
use of the term impenetrability need not occupy us.
92. There is, however, a semi-scientific use of the word
which is of considerable importance. For, whether matter
be impenetrable in the strict sense or not, we may use-
fully discuss the consequences of its not being penetrated.
Thus the hollow of a mould, and only the hollow, is
84 PROPERTIES OF MATTER.
accessible to the liquid metal poured into it. Otherwise
"casting" would be impossible.
One of the most important of these consequences was
long ago given by Archimedes, viz. an easy mode of
comparing the volumes of bodies of shapes so irregular or
complex as to defy the powers of calculators working
from mere linear measurements. All that is required is
to immerse them successively in a vessel partly filled with
water, and to note the amount by which the level of the
water is disturbed, i.e. (in the usual phrase) the amount
of water displaced. Bodies which, when thus tried,
displace equal amounts of water have equal volumes,
however different may be their figures and materials.
93. Hence, to measure the volume of an irregularly^
shaped body : a lump of stone or coal, for instance :
grease it or varnish it all over, to prevent water from
entering its pores, i.e. to secure non-penetration ; and im-
merse it completely in water which partially fills a vessel.
Mark the height to which the water-level rises. With-
draw the stone, and pour in mercury until the same
disturbance of water-level is again produced. The volume
of the mercury is the same as that of the stone. The
mercury has the advantage of taking at once the form of
any vessel in which it may be placed, so that its volume
may be promptly determined by pouring it into a properly
graduated beaker.
This simple consideration forms one of the bases of
the common method of measuring specific gravity ( 36)
by weighing a body, first in air, and then when it is
suspended in water.
94. But it is not solids (such as the stone above) or
other liquids (such as mercury) alone which can thus
displace their own bulk of water. Air will do equally
IMPENETRABILITY DIVISIBILITY. 8B
well. Thus a diving bell is merely an open-mouthed
vessel, inverted and let down into water. The air it
contains is not penetrated by the water, and thus dis-
places (just as a solid or a liquid would have done) its own
volume of water. Its volume, no doubt, becomes less as
the bell descends under the water, but this is due to the
increase of hydrostatic pressure to which it is subjected.
Still, however it be compressed, as it is not penetrated
it displaces at every instant its momentary bulk of water.
95. When one liquid mixes with, or when it combines
with another, it does not usually displace its own bulk of
the other. In such cases there is something equivalent
to interpenetration.
Thus, when twenty-seven parts (by weight) of water
are mixed with twenty-three of alcohol, the volume of
the mixture is less by 3*6 per cent, than the sum of the
volumes of the constituents. [Leslie.]
When an alloy of tin and copper, such as used to be
employed for the specula of large reflecting telescopes, is
formed, the joint bulk may be as much as 7 or 8 per cent*
less than the sum of the bulks of the constituents.
And Faraday showed that, when potassium is oxidised,
the resulting potash has a less volume than either of the
constituents.
96. But, as a rule, in these cases of contraction, other
physical phenomena present themselves. Thus the mixture
of alcohol and water above described becomes more than
8 C. warmer than the components, if both were taken
at the same (ordinary) temperature. A rod of tin dipped
into melted copper (at a very dull red heat) produces vivid
incandescence as it melts and is alloyed. And the com-
bination of oxygen and potassium develops kinetic energy
at an almost explosive rate.
86 PROPERTIES OF MATTER.
There are other cases, which we need not treat of here
(especially as they belong properly to Heat and to
Chemistry), in which the volume of a mixture is greater
than the sum of the volumes of its constituents.
97. These examples show that Archimedes' notable
process might altogether have failed in its application.
For he is said to have been asked to decide whether the
votive crown made for Hiero of Syracuse really consisted
of the amount of gold furnished for its manufacture, or
whether a part of the gold had been abstracted, and its
place supplied by an equal weight of silver.
He procured lumps of silver and of gold, each equal in
weight to the crown. These he immersed successively in
a vessel filled to the brim with water, measuring in each
case the amount of overflow, which he found to be greater
for silver than for gold. The vessel being once more filled,
the crown itself was immersed, and was found to displace
more water than did the gold. Hence, by calculation,
Archimedes found how much silver had been substituted
for an equal weight of gold. 1 This calculation, of course,
must have proceeded on the supposition that the bulk of
the alloy was equal to the sum of the bulks of the com-
ponent metals.
But interpenetration, of which he had no knowledge,
might have completely baffled the great mathematician.
If a similar question were raised now, it would of course
be decided at once by the processes of the chemist, not by
those of the physicist.
98. We have seen ( 24) that, on the hypothesis of
hard atoms, there must necessarily be interstices between
them, else bodies could not be compressible.
But it is an experimental fact, independent of all
1 The original passage is given as Appendix III.
IMPENETRABILITY DIVISIBILITY. 87
hypotheses, that bodies in general are Porous. By the
term pores we do not refer to visible channels, such as
those which run in all directions through a piece of sponge,
but to microscopic channels, which pervade even the most
seemingly homogenous and continuous substances, such as
solid lead, silver, gold, etc.
The proof that such channels exist was given experi-
mentally by Bacon, who tried to compress water by
squeezing or hammering a leaden shell filled with water
and closed. The water exuded like perspiration through
the pores of the lead. The Florentine Academicians tried
the same experiment with a silver shell, but obtained the
same result. They then tried to prevent the escape of
the water by thickly gilding the shell, but again in vain.
99. When a corner of a piece of blotting-paper, or of
a lump of loaf-sugar, is dipped in water, we see (especially
if the water be coloured) the rapid entrance it effects
into the pores. Why it enters, under these conditions, is
another question. (Chap. XII.)
The porosity of wood, necessary for the circulation
of the sap, is beautifully shown by the fact that, from
microscopic examination of a thin slice of a fossil tree, a
botanist can tell at once the species to which it belonged.
The greater part of the material of the wood has dis-
appeared for it may be millions of years, but its micro-
scopic structure has been preserved by the infiltration of
silicious or calcareous materials which, hardening in the
pores, have thus preserved a perfect copy of the original.
The rapid passage of gases through unglazed pottery,
iron and (hot) steel, etc., shows the porosity of these
bodies in a very remarkable manner. So does the strange
absorption of hydrogen by a mass of palladium. (See
Chapter XIII.) The porosity of steel has recently been
88 PROPERTIES OF MATTER.
shown in a most remarkable manner by Amagat, who
forced mercury through a thickness of more than 3 inches
under a pressure of at least 4000 atmospheres. ( 196.)
The metal was quite impervious to glycerine under the
same pressure. 1
Another beautiful instance is afforded by the silicious
concretion, Tdbasheer, found in the joints of sugar-canes.
It is opaque when dry, but when immersed in water for
a short time becomes transparent. Certain agates, called
Hydroplianes, exhibit the same property.
100. No decisive proof of the porosity of vitreous
bodies, such as glass, seems yet to have been obtained.
That they form almost a solitary class of exceptions to an
otherwise general rule seems highly improbable. And
instances, such as those given below, seem to indicate that
these vitreous bodies have not yet been proved to be
porous solely because we have not discovered the proper
mode of testing them.
When polished marble is wetted with water, very little
enters the pores ; while oil, on the contrary, is rapidly
absorbed.
A bag of cambric or gauze, the holes in which are
visible to the eye, holds mercury securely, until sufficient
pressure is applied to force out the liquid. ( 288.)
Glazed pottery-ware, which is practically impervious to
hydrogen and to pure water, is rapidly penetrated by a
strong aqueous solution of bichromate of potash. This
solution, crystallising in the pores, disintegrates the whole,
just as water, freezing in the pores of a rock, gradually
breaks its surface-layers into small fragments, to be after-
wards washed down to the plain as alluvial soil.
The question of the porosity of colloidal bodies, such as
1 Comptes Hendus, March 2, 1885.
IMPENETRABILITY DIVISIBILITY. 89
gelatine, albumen, and, from some points of view, india-
rubber, is somewhat puzzling. We will refer to it in
Chapter XIII.
101. The Divisibility of matter, in the strict scientific
sense, at once raises the question of the existence of finite
atoms. For, if there be such atoms, division has them
for its limit, whatever processes may be employed. 1 We
are not prepared to face this aspect of the question, and
must, therefore, confine ourselves to examples of extremely
minute Division.
An " impalpable " powder is one which gives no gritty
sensation when we rub it between the thumb and fingers.
The process of Levigation, depending on fluid friction
( 38), is employed for the assortment of solid particles into
packets of different degrees of fineness. Thus, if ground
emery be thrown into a tall vessel full of water, we may
remove from the bottom of the vessel successive crops, as
it were, of gradually increasing fineness. Yet even the
finest of these powders can be used for grinding metallic
or glass surfaces, showing that its particles still possess
properties similar to those of the coarser powders.
Silica may be thrown down, by chemical processes,
in such an extreme state of division that when it is dried
and poured into a trough it behaves almost like a liquid.
Especially when it is heated, we observe that, like a
liquid, it is capable of propagating gravitation-waves.
Calcined magnesia and other very fine powders show
similar properties.
102. Even the rough process of scratching the polished
surface of glass with a diamond point can be carried out
by machinery to such an extent of delicacy that groups
of equidistant parallel lines may be traced, some of which
1 See again Appendix II.
90 PROPERTIES OF MATTER.
can only be " resolved " into their components by the very
best microscopes ; others, which we have every reason to
believe capable of resolution, have not yet been resolved.
These pieces of ruled glass are known to microscopists as
Noberfs Test.
Ordinary gold-leaf, though prepared by a very rough
process, has a thickness of about l/300,000th of an inch
only. But, as Faraday showed, it can be rendered very
much thinner by immersion in a solvent such as cyanide
of potassium. And, by a species of inversion of Wollaston's
process ( 49), i.e. drawing into very fine wire a silver rod
thickly gilt, we obtain a continuous film of gold, whose thick-
ness is estimated at less than l/4,000,000th of an inch.
103. The average size of the particles of water in a
light cloud is easily estimated from the diameter of the
coloured rings, or Coronce, which it produces when it
covers the sun or moon. 1 If the radius of the innermost
ring be 15, the diameter of the particles must be about
1 /13,000th of an inch. Such must have been the average
size of the dust particles from the Krakatao eruption of
1883, which produced the remarkable sunsets, as well as
the corona seen about the sun when no cloud was visible.
The length of time these particles remained in suspension
is accounted for in 40 above.
104. Leslie, in his Natural Philosophy (1823), says:
"A single grain of musk has been known to perfume a
large room for the space of twenty years. Consider how
often, during that time, the air of the apartment must
have been renewed and have become charged with fresh
odour ! At the lowest computation the musk had been
subdivided into 320 quadrillions of particles, each of them
capable of affecting the olfactory organs." Leslie does not
1 Tait's Light, 3rd ed., 180, 246.
IMPENETRABILITY DIVISIBILITY. 91
tell us how the computation was made, nor even what we
are to understand by quadrillions.
[The usual British reckoning gives a quadrillion as a
billion billions, each billion being a million millions,
while the French reckoning makes it only a thousand
million millions. This confusion is entirely removed
by the modern mode of writing large numbers, when we
know them only in rough approximation. We write the
first two or three significant figures, and indicate the
number of the remaining ones by the corresponding
power of 10.]
Thus Leslie may have meant either 320 x 10 24 , or
320 x 10 15 . If, as is most probable, he meant the former
of these numbers, the result of his computation has
been singularly verified by recent discoveries, some of
them apparently altogether unconnected with the question
before us.
105. One of the most striking instances of division is
that furnished by holding, in an otherwise slightly
luminous flame, a particle of common salt or of some
other metallic chloride. Fox Talbot, in 1826, wrote :
" A particle of muriate of lime on the wick of a spirit-
lamp will produce a quantity of red and green rays for a
whole evening without being itself sensibly diminished."
Swan traced the source of the peculiar orange ray which
appears in the light of almost every flame to the wide
diffusion of exceedingly small quantities of common salt.
These phenomena are nowadays known to all in connec-
tion with Spectrum Analysis. The quantity of common
salt which, for a considerable time, will continue to give
the orange tinge to the flame of a large Bunsen lamp is
minute in the extreme. The effect is now proved to be
due to vapour of sodium.
92 PROPERTIES OF MATTER.
106. A conviction of the practically infinite divisibility
of matter must be felt by all who would hold a warranted
belief in the "dilutions" which are at least popularly
supposed to be one of the main characteristics of homoeo-
pathic medicines. When a single grain of atropine is
dissolved in a gallon of water, and one drop of this is added
to another gallon of water, we have what is called the
first dilution. Add a drop of this to a third gallon of
water, and we have the second dilution. And so on.
Tenth dilutions are said to be sometimes administered.
If we take the diameter of a drop as about l/8th of an
inch, we find, by an easy calculation, that (as there are
277 cubic inches in a gallon) the tenth dilution should
contain about 2/1 60 of a grain of atropine per drop ! If
that drop were magnified to the size of the whole earth,
the atropine in it (magnified, of course, in proportion,)
would correspond to a particle of somewhere about one
three-billionth of an inch in diameter ! ! So that, unless
atropine can be separated into particles of this excessive
smallness, one drop of the so-called dilution will be totally
different from another : one being (according to the pre-
scriber's principles) a potent medicine, the other pure water !
107. The kinetic theory of gases informs us that, in
a cubic inch of any gas at atmospheric pressure, and at
ordinary temperatures, there are somewhere about 3 x 10 20
detached particles absolutely similar and equal to one
another. These cannot be Lucretian atoms, for they
have each many different modes of vibration, even when
they belong to a simple and not to a compound gas.
Here we reach the limit of our present knowledge as to
division of matter. What is the structure of these
gaseous particles on which their vibrations depend ( 29),
and how far further divisible each particle must be
IMPENETRABILITY DIVISIBILITY. 93
supposed in consequence, are matters beyond our know-
ledge. [Knowledge on this point has greatly advanced
since the above statement was made, see Chap. XV. W. P.]
[These results of the kinetic gas theory are confirmed by
altogether independent lines of physical reasoning with
which we are not concerned here.]
We may take, as a rough approximation, that the
grained structure ( 26) of the most nearly homogeneous
solid or liquid bodies is of the order of 5 x 10 8 to the
inch linear. To give a notion of the amount of division
which this indicates, suppose we magnify a cubic inch of
such a substance to a cube whose side is the diameter of
the earth. The earth's polar diameter is 5 x 10 8 inches,
very approximately. Thus, in the enormously magnified
cube, there is one particle in every cubic inch or so. We
say nothing, it is to be noticed, as to the size of the
particle or granulation itself. [The estimates hitherto
made of this quantity can hardly be called even rough
approximations : so widely do they differ from one
another, according to the particular physical phenomenon
on which each is based. But probably the particle does
not occupy so much as 5 per cent, of its share of the
whole content.]
All that can be said of the above estimates (of number
of particles per cubic inch) is that they are, at least
nearly, of the proper order of magnitude. And it is
curious to find that the result of Leslie's old " computa-
tion" ( 104) agrees fairly enough with our present know-
CHAPTER VI.
INERTIA, MOBILITY, CENTRIFUGAL FORCE.
108. We commence with Newton's
FIRST LAW OF MOTION.
Every body perseveres in its state, of rest or of uniform
motion in a straight line, except in so far as it is compelled
by forces to change that state.
The property, thus enuntiated as belonging to all
bodies, is usually called Inertia. And it is clear from the
statement above that it may be described as passivity, or
dogged perseverance, but in no sense whatever as
revolutionary activity. This consideration will be found
presently to be of great importance.
Matter is, in Newton's system, regarded as the play-
thing of force ; submitting to any change of state that
may be imposed on it, but rigorously persevering in the
state in which it is left, until force again acts upon it.
109. The state referred to is one of rest, or of uniform
motion in a straight line (of which rest is a mere particular
case). Here we meet with a serious difficulty
All transiatory motion (including rest, or null motion)
is, from the very nature of space, essentially relative.
INERTIA CENTRIFUGAL FORCE. 96
Eelatively to what, then, are we to say that a body not
acted on by force moves uniformly in a straight line?
The answer, so far as we can give it, is
Relatively to any set of lines drawn in a rigid body, of
finite dimensions; which is not acted on by force, and which
has no rotation.
As will be seen later, ( 131) Newton has pointed out a
physical test, by which it can be ascertained whether a
body has rotation or not.
But questions of this kind can only be alluded to,
certainly not fully discussed, in an elementary work.
110. The grand proof of the truth of the first as well as
of the other Laws of Motion is furnished by the celestial
motions. So irregular is the motion of the moon, when
considered carefully, found to be, that no amount of the
most exact observation alone (i.e. unaided by physical
investigations) could enable us to predict its place, even
twenty-four hours beforehand, with anything like the
accuracy with which it is predicted four years before-
hand in the Nautical Almanac. So convinced have
astronomers become of the truth of the laws of motion,
which are necessarily involved in all their lunar and
planetary calculations, that when a discrepancy between
prediction and observation is found to occur no one now
questions the bases of the calculation. The discrepancy
is used to correct our previous estimates of the elements
of the lunar or planetary orbit; or, as in the notable
case of Uranus, it is employed as an indication of where
to seek for some undiscovered body whose influence has
not been taken into account.
111. Familiar instances of Inertia present themselves
in all direction^. When a railway carriage is running
uniformly on a straight piece of road, we become uncon-
96 PROPERTIES OF MATTER.
scious of the motion unless we look out at external
bodies ; but we detect at once any sudden, or even rapid,
change of speed. If the motion of the train be checked
by a sudden application of the brake, their inertia (which
really maintains their motion) appears to urge the passen-
gers forwards. A sudden starting of the train produces
the opposite effect. While the steady motion continues,
a conjuror can keep a number of balls in the air just as
easily as if the carriage were at rest. But these things
need not surprise us. Our rooms are always like perfect
railway carriages in respect of their absolutely smooth,
but very rapid, motion round the earth's axis. The
whole earth itself is flying in its orbit at the rate of a
million and a half of miles per day ; yet we should have
known nothing of this motion had our globe been per-
petually clouded over like that of Jupiter. The whole
solar system is travelling with great speed among the
fixed stars, but (at least until the recent introduction of
spectroscopic methods) we knew of the fact only from
the minutely accurate observations of astronomers, aided
by all the resources of the Theory of Probabilities.
112. When a bullet is dropped from a definite point in
a uniformly running carriage, it strikes the same point of
the floor whatever be the speed of the motion ; for, by its
inertia, it preserves while falling the forward motion of the
carriage which it obviously had while it was held in the
hand. But, if the bullet be dropped from the yard of a
vessel to the deck, it will not fall always on the same spot,
however uniform be the ship's progress, if there be any
pitching. For, when the vessel pitches, the yard moves
forward alternately faster and slower than does the deck.
Now the top of a tower (unless it be at one of the
poles) is farther from the earth's axis than is the foot, or
INERTIA CENTRIFUGAL FORCE. 97
the ground on which the tower is built ; and, therefore,
as both complete their revolution in twenty-four hours,
the top of the tower moves permanently faster than does
the base. Hence even a truly spherical bullet, dropped
from the top, does not fall vertically. It deviates to the
east of the vertical, because it preserves while falling its
superior eastward speed. In this way we obtain one
physical proof of the earth's rotation.
113. The upsetting of buildings by an earthquake
furnishes a striking instance of inertia. So does the
almost perfect immunity we experience from the millions
of meteoric stones which are constantly encountering the
earth with planetary velocities. This is due to the
inertia of the air, which, in its turn, is one indispensable
cause of the destructive action of a tornado ; just as, on
a smaller scale, a cannon-ball would be harmless without
inertia, while an earthwork, without inertia, would afford
no defence against it. But we need give no more in-
stances the reader will easily supply others from his own
experience.
114. So far, we have been speaking of inertia as mani-
fested by the tendency of a body to persevere in its
motion with unaltered speed. But we must carefully note
that this is only one part of Newton's Law. The state
in which he tells us that bodies persevere by inertia is
not one of uniform motion merely, but of motion in a
straight line. The preservation of the rectilinear path
is quite as essential a part of the functions of inertia as is
the preservation of the uniform speed. Hence, just as
(following Newton's system) we attribute any change in
the speed of a body to the action of force, so, if its line of
motion be not straight, (whether the speed be unaltered or
no) its curvature also must be due to the action of force.
G
98 PROPERTIES OF MATTER.
115. How the force must be applied which causes a
body, in spite of its inertia, to move in a curve is easily
understood from some common instances, though it is
pretty obvious that it must be in a direction perpendicular
to that of the motion ; and, of course, in the plane in
which the curvature takes place. For any force in the
direction of motion must tend only to increase or to
diminish the speed.
It is found that, at a curve on a railway line, it is the
outer of the two rails which is most worn (i.e. that one
which forms the convex side of the track). And, when
a sharp curve has to be taken rapidly, the outer rail has
generally to be laid a little higher than the other. But
(except when the brake is on) the pressure is mainly
perpendicular to the rails. Hence the force which causes
the carriages to move in a curved path must be directed
inwards to the centre of curvature.
When we whirl a sling with a stone in it, we feel the
tension of the cord (which is constantly pulling the stone
from its natural straight path in towards the hand)
increased as we cause the sling to rotate faster.
A bullet, suspended by a string, forms what we call
a simple pendulum. It can, by proper initial projection,
be made to revolve uniformly in a horizontal circle. It
is then what is called a Conical Pendulum. Here the
tension of the string may be resolved into two parts ;
one vertical, which supports the weight of the bullet, the
other horizontal, which continually deflects the bullet
from its natural rectilinear path. If the string could be
made long enough, the time of revolution might be made
twenty-four hours ; and if the pendulum were then set up
at the north pole, and made to describe its circle in the
positive direction ( 65), it would appear to remain sus-
INERTIA CENTRIFUGAL FORCE. 99
pended at rest in the air, the supporting string not being
vertical ! If it were made to revolve in the negative
direction, it would appear to complete a revolution in
twelve hours.
The moon is caused to move in an (approximately)
circular path about the earth by the same attraction
which causes stones to fall vertically downwards.
116. One of Newton's remarks on the First Law of
Motion runs thus :
"A hoop (or top, Trochus) whose parts by their cohesion
perpetually draw one another aside from rectilinear motions,
does not cease to rotate, except in so far as it is retarded
by the air."
Thus the uniformity of the earth's rotation about its
axis, which is the basis of our measurement of time, is
merely an example of the First Law of Motion.
But when a fly-wheel, or a grindstone, is made to
rotate so fast that the cohesion of its parts is no longer
capable of supplying the forces requisite to keep them
moving in their circular paths, it bursts (this is the tech-
nical word), and the fragments fly off in paths which are
tangential and rectilinear, except in so far as gravity
modifies them.
If the rotating body be plastic, as must have been
the case long ago with the earth as a whole, its form
will be modified by the tendency of every particle to
preserve its rectilinear path. Thus it swells out in all
directions perpendicular to the axis of rotation. Jupiter
and Saturn, being much larger than the earth, and also
rotating more rapidly, show this effect in a much greater
degree.
A beautiful example is furnished by suspending an
endless chain by a cord, and (by very rapidly twisting
100 PROPERTIES OF MATTER.
the cord by means of multiplying gear) throwing it into
rotation. When the rotation of the whole is sufficiently
rapid it assumes almost exactly the form of a horizontal
circle, all its links being equidistant from the (vertical)
axis of rotation.
117. The old notion, probably suggested by such
instances as the pull which the stone in a sling seems to
exert on the hand, was that bodies have a tendency to
fly outwards from the centre about which they are revolv-
ing. Hence they were said to exert Centrifugal Force,
and a Centripetal Force was of course required to balance
this. The term Centrifugal Force has become rooted
in our scientific language. It is a convenient enough
expression, provided we do not split it up, thus taking
it to imply force, and flying from a centre ; but interpret
it merely as indicating that, to keep a body moving in
a curve instead of in its natural straight line, a force
directed towards the centre of curvature is always
required. But, as the third law of motion ( 128) tells
us, a force is only one-half of a stress, so that when force
is exerted to pull the body inwards from the tangent, an
equal force must be exerted at the centre tending outwards
from it.
We might quite as justly speak of the Onward Force
of a cannon-ball, which requires a resistance to check it ;
as of the centrifugal force (understood not as a single
term but as two words, each with its ordinary meaning)
which must exist because it requires centripetal force
to balance it.
118. Calculating (as in 71) from the earth's mean
equatorial radius, 3962 miles, and the number of seconds
in a sidereal day, 86164, we find that the acceleration of
a point on the equator is about (HI 16 foot per second,
INERTIA CENTRIFUGAL FOllCE. 161
per second : while the acceleration due to gravity is
about 32, in terms of the same units. Thus about -^l-g-th
of its weight is required merely to keep a body on the
earth's surface at the equator. By this amount its weight
(as indicated by a Sjr/ring-balanee, 165) would be
diminished.
If the earth had been revolving seventeen times faster
than it does, this apparent diminution of weight would
have been 17 2 (or 289) times greater than it is, i.e. bodies
at the equator would have shown no apparent weight,
provided they moved along with the same speed as
the ground below them.
119. As is pointed out in the preceding extract from
Newton ( 116), a wheel or other body, rotating about
an axis and not acted on by forces, perseveres by inertia
in its uniform rate of rotation. But it does more; it
preserves (even when acted on by small forces) the direc-
tion of its axis of rotation, provided at least that it be
rotating about its axis of greatest or of least moment
of inertia ( 132). It is for this reason that rifling of
the bore of a gun has been introduced ; and also that a
skilled player, when throwing a quoit, gives it rotation
in its own plane.
The rotation of the earth about its axis is a more
complex phenomenon, because it takes place under the
action of considerable forces which tend to make the
earth revolve about axes lying in the plane of its equator.
Yet, because the moments of inertia ( 132) about all
such axes are approximately equal, the period of the daily
rotation is not altered, though the direction (in space) of
the polar axis is affected by Precession and by Nutation.
We cannot, however, do more than allude to matters of
this order of difficulty. They are all beautifully illus-
102 PROPERTIES OF MATTER.
trated by means of Gyroscopes, Gyrostats, etc., but the
full study of the phenomena requires higher mathematics
than we can introduce here. These are properly questions
of Abstract Dynamics.
120. Newton's SECOND LAW OF MOTION is as follows:
Change of momentum is proportional to force, and takes
place in the direction in which the force acts.
Thus, according to JSTewton, a force always produces
change of momentum. Hence there is no balancing of
forces, though there may be balancing of the effects of
forces.
Every force (however small) produces its proper change
of momentum. This used to be stated, under the name
of Mobility, as a characteristic property of matter.
This change is always gradual, never abrupt. An
infinite force would be required to produce a finite change
of momentum abruptly.
As change of momentum alone is mentioned, it is clear
that Newton means that the effect of a force is independent
of the state of motion of the body to which it is applied.
Hence if a force be uniform, as for instance is practically
the case with the action of gravity upon a falling body,
the additional momentum produced by it in each and
every second will measure its amount. But if it be
variable, we must measure it by the rate at which
momentum is produced by it instead of the momentum
produced by it in one second. Thus the true measure of
a force is the rate of change of momentum ; or, to use the
kinematical term, the product of the mass of a body into
the acceleration of its velocity.
121. Two special cases, of great importance, must now
be treated : uniform acceleration in the direction of
motion, and uniform circular motion.
INERTIA CENTRIFUGAL FORCE. 103
it is found that, in vacua, all bodies acquire, per
second, an additional vertical velocity of about 3 2 '2 feet
per second. This quantity (which varies with the lati-
tude, height above sea-level, etc. 165) is usually denoted
by the letter g. Hence, if M be the mass of a body, its
weight (i.e. the force which accelerates its fall) is measured
by the product Mg.
Kinematics (as we saw in 71) shows us that when a
point moves uniformly in a circle, the acceleration is
directed inwards to the centre, and its magnitude is the
square of the speed multiplied into the curvature of the
path. Hence to keep a body, of mass M, moving with
uniform speed V in a circle of radius K, a force whose
magnitude is MV 2 /E, directed towards the centre of the
circle, must constantly act upon it. As Mg is the weight
(W) of the body, we may express this force as
If to be the Angular Velocity in the circular path, i.e.
the angle described in unit of time by the radius drawn
to the moving body, we have obviously
V-Ra.,
and the expression for the requisite force takes the form
MR 2 , or W.
122. Newton shows that, as an immediate consequence
of the Second Law, we have the Law of Composition of
Forces acting at one point ; the so-called Parallelogram,
or Triangle, of Forces. This follows from the facts that
(a) the changes of velocity produced in the same time, by
different forces acting on the same body, are proportional
104 PROPERTIES OF MATTER.
to and in the directions of the several forces, and that (b)
the effect of each force is independent of the simultaneous
action of the others. Thus the problem is reduced to the
obvious kinematical composition of velocities ( 69).
123. We are now prepared to measure both Masses and
Forces. But, for this purpose, it is necessary to have units
in terms of whicli to measure.
The British unit of mass is the Standard Pound, whose
amount was probably adopted in old times for reasons of
convenience, but is now fixed by law.
The French, or Metrical, unit of mass is the Kilo-
gramme, originally intended to be the mass of a cubic
decimetre, or litre, of water at its maximum density
point ; but, practically, defined by a platinum standard.
The Kilogramme is about 2-20462 Pounds.
1 24. . The British Unit of Force is such that, when it
acts for one second on a mass of one pound, it produces
in it a speed of one foot per second.
The C. G. S. Unit of Force (or Dyne), which is coming
into use for scientific measurements, is such that, when it
acts for one second on a mass of one gramme, it produces
in it a speed of one centimetre per second.
The British Unit of Force is about 13,825 Dynes.
Since the speed acquired by a body falling for one second
(in vacuo) is ( 120) about 32'2 feet per second, the
Weight of a Pound is about 445,165 Dynes.
125. Generally, if a definite force act upon any mass
for one second, it will generate in it a speed whose magni-
tude is inversely as the mass.
Thus the comparison of masses, i.e. their measurement
in terms of some standard unit, becomes a perfectly definite
scientific process.
It may not be easy to carry out, and in fact it is not ;
INERTIA CENTRIFUGAL FORCE. 105
at least by any very direct application of the principles
just explained. That, however, is another matter. No
one in his senses would question the perfectness of Euclid's
process for dividing a straight line into a given number of
equal parts, on the ground that it is practically inappli-
cable when we try to carry it out.
What we have sought is an accurate, and an easily
intelligible, method of comparing masses. If it be not
easily workable in practice, we must find something more
workable : just as we now use a screw dividing-engine
instead of Euclid's unrealisable straight lines, and still
more unrealisable parallel straight lines.
126. When we have measured the mass of a body, and
also its volume ( 92), its average, or mean, density follows
at once. If the body be homogeneous, this is its actual
density throughout : and whether it be homogeneous or
not, its mean density is simply the average amount of
mass per unit of volume.
On account, chiefly, of the remarkable result of New-
ton's ( 34), we postpone all considerations regarding the
densities of various kinds of matter until we are dealing
with their specific gravities also.
127. The process of weighing is, as Newton showed
( 34), essentially a comparison of masses. So that our
measurement of mass is practically carried on by means
of the Balance, which is one of the most delicate and
accurate instruments of precision yet invented.
The processes for measuring force are not yet nearly
so accurate. Numerous instruments have been devised
for the measurement of special classes of forces, the great
majority depending upon elasticity of matter. Some of
the more important of these will be mentioned when we
require them ; but the reader must be reminded that on
106 PROPERTIES OF MATTER.
Newton's system the true measure of a force is the
momentum it produces in one second.
128. The first two laws of motion (applied with suf-
ficient mathematical resources) enable us to solve any
problem whatever regarding the motion of a body, treated
as a mere particle, under the action of any given forces.
Conversely, they enable us, from the given motion of a
particle, to find the forces under which the motion has
taken place. But they do not suffice for the calculation
of the motion of two or more particles which mutually
influence one another, whether by gravitation, or cohesion,
or by any physical mode of attachment. Hence the
necessity for the
THIRD LAW OP MOTION.
To every action there is always an equal and contrary
reaction; or, the mutual actions of any two bodies are
always equal and oppositely directed.
In modern speech Newton's first explanation of the
sense in which this statement is to be understood may
be simply expressed thus :
Every action bettveen two bodies is a Stress.
In this sense it is very closely connected with the
first law. For a system of two bodies, considered as one,
cannot set itself in motion. Even when the masses are
not connected in any way, the equality of action and
reaction involves transference of momentum between
them which leaves the motion of their Centre of Inertia
unaffected. It was by floating on water a magnet and a
piece of iron, both attached to the same light board, that
Newton proved the equality of action and reaction for
magnetic force. He also proved it by showing that, if
INERTIA CENTRIFUGAL FORCE. 107
the gravitation action of one part of the earth on the rest
were not exactly reciprocated, the earth (as a whole)
would alter its existing state of motion.
129. The term centre of inertia, employed above, re-
quires a few words of explanation. We assume the
following proposition, which can be established by very
elementary mathematics.
In every group of massive particles there is one
definite point, such that, whatever plane be drawn in
space, the distance of the point from that plane, multi-
plied by the sum of the masses, is equal to the sum of
the separate products formed by multiplying each mass
into its distance from the plane. It can, therefore, be
found by executing the requisite calculations for any
three planes whatever, provided no two of them are
parallel. This is the centre of inertia of the particles.
When the particles are severally acted on by forces,
it follows from Newton's Third Law that the centre of
inertia of the group moves as if the whole mass were
there concentrated, and acted on by all the forces simul-
taneously. This consideration greatly simplifies kinetical
problems connected with a rigid system or a group of
particles ; for it enables us to commence by determining
the motion of the centre of inertia as if it were a mere
particle, and afterwards to study the motion relatively to
that centre.
130. But Newton proceeds to point out that there is a
second sense in which the terms action and reaction in the
Third Law may be interpreted, the law itself still remain-
ing true. In modern phrase it may be expressed as
The activity of an agent (or the rate at which it does
Work), is equal to the counter-activity of the resistance.
Newton's statement of this second mode of interpret-
>08 PROPERTIES OF MATTER.
ing the Third Law has been shown to require compara-
tively little addition to make it a complete enunciation
of the Conservation of Energy ( 7).
131. If two masses be connected by a spiral spring,
but be otherwise free, and if the spring remains stretched
to a constant amount, this can only be because the bodies
are revolving about one another. For, otherwise, the
stress in the spring would have caused them to approach
one another. This is Newton's test of the existence of
Absolute Rotation, 109. For, by the first law, the stress
which certainly acts on the masses must interfere with
their states, and by the third law it must do so in opposite
directions. Each, therefore, must be describing a curved
path relatively to the other, and this must of course be
circular.
Nothing is known, nor is anything conceivable even by
the most transcendental of metaphysicians, which could
give us an indication of Absolute Translation.
132. We conclude the chapter with a few additional
illustrations and explanations connected with inertia.
In 119 above we introduced, without explanation,
the important term Moment of Inertia. This quantity is
defined, for any body, with reference to any assigned
axis. It is the sum of the products obtained by multi-
plying the mass of each small portion of the body into
the square of its distance from the axis. Its use is two-
fold.
(a) If o) be the angular velocity of a rigid body about
an axis, r the distance of the particle whose mass is ra
from that axis, the speed of ra is rw, and the kinetic energy
of rotation (half the product of each part of the mass into
the square of its speed) is
INERTIA CENTRIFUGAL FORCE. 109
half the product of the moment of inertia into the square
of the angular velocity.
(ft) Again, the Moment of the momentum of a particle
about an axis is denned as the product of its momentum by
the shortest distance between the axis and the line of
motion of the particle. Hence the moment of momentum
of m about the axis is mru.r, and the whole moment of
momentum of the body is
the product of the moment of inertia into the angular
velocity.
133. It is shown in treatises on Dynamics that the
effect of a pair of equal and opposite forces, whose lines
of action are different (called by Poinsot a Couple) is to
produce moment of momentum in proportion to the time
it acts and to the moment of the couple. Hence, if Q
be the (constant) moment of the couple, o> the angular
velocity it produces in time t, when its plane is perpen-
dicular to the axis above spoken of,
Io> = 2(wr 2 ) = Q, . . . . (1)
whence
Ia, 2 =i2(mr 2 )o, 2 = Q.^ - Qa, . . . (2)
where a is the angle through which the body has turned.
For co grows uniformly, and therefore its average value
during the time t was w 2, so that the whole angle
described is <at/2.
But if a (constant) force P act on a particle, of mass M,
and produce in time t a speed v, we have
Mi = P; ...... (3)
The speed increases uniformly, so that its average value
110 PROPERTIES OF MATTER.
is 0/2, and therefore the space described is s vt/ 2.
Hence, by multiplying both sides by w/2, we get
Ps ...... (4)
It is obvious that in the former pair of equations, (1) and
(2), the quantities I and Q, to and a, play exactly the same
parts as do M and P, v and s, respectively, in the latter
pair, (3) and (4).
This analogy shows, at least in part, the great con-
venience of the idea of the moment of inertia.
For special purposes we often write I in the form M& 2 ,
k being then the common distance from the axis at which
every one of the particles must be placed, so that the
whole may have the same moment of inertia as before.
It is called the Radius of Gyration.
134. As an illustration of the application of the two
interpretations of the third law, suppose a fly-wheel whose
axle (horizontal) is carefully mounted on friction rollers,
to be set in rotation by the descent of a weight attached
to a string wound round the axle.
Let to be the angular velocity produced in the fly-
wheel when a length x of the cord has been unwound, a
the radius of the axle, M the mass of the appended
weight, I the moment of inertia of the wheel, and T the
stress in the cord.
Then the rate of increase of momentum of the mass
M is Mo; (with Newton's notation, 72). This must be
the measure of the force producing it, so that
Mx = M0-T .... (1.)
The rate of increase of moment of momentum of the
fly-wheel is Io>, which must measure the couple produc-
ing it. Hence
I=T ..... (2.)
INERTIA CENTRIFUGAL FORCE. Ill
But aa> is the amount of cord unwound per second,
i.e. the rate of descent of the weight. Thus
ao> = x ..... (3.)
(1) and (2) are dynamical equations, (3) is kinematical.
x, <o, and T are to be found from the three. They give
Ma 2 *? May , 4 .
Ma 2 + I Ma 2 + rf '
where m is the mass of the wheel, and k its radius of
gyration. ( 133.)
If the wheel had no moment of inertia this would
become
*-*,
the ordinary equation of acceleration of a free falling
body.
Hence, the only effect of the fly-wheel is to diminish
the effect of gravity on the weight in the proportion
Ma 2 : Ma 2 + m& 2 . The measure of the stress on the
cord is
m_ M.lg M
~Ma 2 +I = Ma 2
and it therefore remains the same throughout the motion.
It increases with increase of the radius of gyration of the
wheel, but not indefinitely. Its utmost value, as was to
be expected, is (M<7) the weight of the appended mass.
135. But the solution of the same problem, by the
help of Newton's second interpretation of the third law,
is far more simple.
The rate at which the agent (the weight of the falling
body) is doing work is, at any instant,
The rate at which energy is being gained by the falling
112 PROPERTIES OF MATTER.
body is M.XX. The rate at which energy is gained by the
fly-wheel is loxo.
Hence
or with the help of (3), our kinematical condition,
M#a a = Ma 2 x+IaJ, . . . . (4.)
which is the same equation as before.
136. But, instead of reckoning rates of transference of
energy, we may still more simply proceed by expressing
the conservation of the whole amount of energy in the
system ( 7).
The falling body has lost WLgx, and has gained
The fly-wheel has gained JIo> 2 . Hence
or by (3)
which is the fluent, or integral, of (4) when the terms of
that equation are multiplied by x.
137. If we consider these three solutions of the same
problem, we see that, while the stress between the
members of the system plays a prominent part in the
first, it is altogether unnoticed in the two latter.
This might, at first sight, tend to induce us to ignore
stress altogether ; and, undoubtedly, we can do so in all
cases, except when we study the condition of the intervening
medium, while energy is stored in any part of it ; or while
energy is being transferred through it from one part of the
system to another. The consideration of this view of the
subject is deferred to our chapters on Elasticity. See,
especially, 169.
CHAPTER VII.
GRAVITATION.
138. WITHOUT preface we simply give a statement, com-
pounded from various parts of the Principia (especially
the Third Book), which comprehends all the essentials of
Newton's great generalisation.
Every particle of matter in the universe attracts every
other particle with a force whose direction is that of the
line joining the two, and whose magnitude is directly as the
product of their masses, and inversely as the square of
their distance from each other.
Thus, if M, m, are the masses, D their distance, their
attraction (i.e. the weight of either as due to the other) is
Mm
*W
Here k is called the Coefficient of Gravitation. Its value
obviously depends on the units employed, and can be
found in terms of these by means of the apparatus for
the mean density of the earth, whose determination is one
of the objects of the present chapter.
This statement is made in terms of attraction : i.e.
force : a form convenient for our present purpose.
But it will be shown later ( 159) that all we know on
the subject can be expressed (and still more simply) in a
form which ignores even the very name of force.
It divides itself, for proof, into a number of separate
heads ; as follows :
(a) The Universality of Gravitation.
H
114 PROPERTIES OF MATTER.
(5) The direction of the force between two particles.
(c) The proportionality of the force to the product of
the masses.
(d) The law of the inverse square of the distance.
Besides these more immediate assertions the statement
also raises the questions
(e) What do we mean by " attraction " ?
(/) What is the cause of gravitation ?
And other matters of great importance naturally present
themselves, such as, " What is the mass of the Earth," etc. 1
These questions must be kept before us, so that we may
give to each of them (so far as our knowledge yet extends,
and so far as is consistent with the scope of this work) a
sufficient answer. (/) is still an open question, for the
attempts at answering it have not been very successful, (a)
of course can only be answered either in an approximate or in
an indirect manner, because we cannot (by our most delicate
instruments) even prove the existence of gravitation- attrac-
tion between two particles of matter. Here, however, we
tread (as will be seen) on comparatively safe ground.
And the same may be said for (b), (c), and (d), because
the reasoning and experiment which sufficiently answer
(a) will be found here even more complete, (e) will be
discussed along with (/).
139. (a) One strong argument for the universality of
gravitation is that the weight of a body is the sum of the
weights of its parts. This is, of course, a matter which
can be tested to a very great degree of accuracy by means
of the balance. Thus each particle of the body contributes
its share to the weight of the whole.
And the weight of a given quantity of matter does not
depend upon its form. A mass of gold retains exactly
the same weight when it is beaten out into the finest leaf,
GRAVITATION. 115
or dissolved in any quantity, however great, of an acid.
Thus terrestrial gravity acts as freely upon the particles
when they are surrounded on all sides by the solid mass, as
when they are directly exposed by the beating, or solution.
In fact, it is quite easy to see that, were this not the
case, were it, in fact, possible to find a screen through
which gravity could not act, i.e. were it possible to inter-
fere with the universality of gravitation, we should also
be able to produce The Perpetual Motion : an inexhaust-
ible source of new energy. This we know ( 7) cannot be.
To show, however, that the above hypothesis would
lead to this result, we have only to think of a fly-wheel,
one part of which shall be screened from the earth's
attraction, the rest unscreened. Every part loses weight
as soon as it enters the shadow, as it were, of the screen,
and gains it again when it emerges. Thus the wheel,
being constantly heavy on one side and weightless on the
other, constantly gains energy from nothing.
The wheel would become as it were a tread-mill :
working of itself, instead of by the hard labour of a gang
of convicts climbing, without mounting, up one side.
140. (a) continued. Newton attacked the question by
assuming the law of gravitation for the separate particles
of a body, and thence finding what should be the law of
attraction towards the body as a whole. He thus arrived
at two exceedingly simple and beautiful theorems. The
first is as follows :
A spherical shell of uniform gravitating matter exerts no
attraction on a particle within it.
[For the proof of this, and of the succeeding proposition,
we assume the following results of pure mathematics :
The area of a transverse section of a cone of small angle
is proportional to the square of its distance from the vertex.
116 PROPERTIES OF MATTER.
The measure of the spherical opening of such a cone is
the area it cuts off from the unit sphere whose centre is
its vertex ; which is the same as the area of the transverse
section at unit distance from the vertex.
An oblique section has greater area than the transverse
section, at the same distance from the vertex, in propor-
tion to the secant of their inclination to one another.]
Take any point B, within the spherical shell, and let it
be the vertex of a double cone of exceedingly small angle.
This cuts out two minute areas on the spherical surface,
obviously at equal inclinations to
the axis of the cone. Hence their
areas, and therefore their masses,
are as the squares of BP, BQ.
But their attractions on B are
inversely as the squares of BP,
BQ. Thus these attractions balance
one another. And the whole shell
may thus be divided into pairs of
parts, whose attractions exactly
balance one another on B. Hence the proposition, which
is obviously true of any uniform shell, however thick, if
only bounded by concentric spheres. And it is true, if
the shell be made up of concentric layers of different
densities, provided the density of each layer be uniform.
No other law than that of the inverse square of the
distance is capable of giving this result.
141. The second of Newton's theorems is :
A spherical sJiell of uniform gravitating matter attracts
an external particle as if its whole mass were condensed at
its centre.
Let A be the external particle, C the centre of the
shell. Cut off CB, a third proportional to CA, CD ; and
GRAVITATION. 117
divide the shell by small double cones whose vertices
are at B. Let PBQ be such a cone. Then if w be its
spherical opening, the areas of the sections at P and Q are
BP 2 ft> sec CPB, BQ2o, sec CPB,
and their attractions on unit mass at A are
BI"...eeCPB ft and Eg., sec CPB ft
Ax AQ
where p is the surface-density, i.e. the mass per unit area.
FIG. 12.
But the geometry of the figure shows us at once that
<CPB= < PAD= < QAD, and BP : AP : : CP : AC : :
BQ : AQ. Hence the elements at P and Q attract A
equally, and the resultant of their attractions is therefore
along AC. Its value is
2CP 2 .c0p
AC 2
in which the multiplier of <o is constant ; i.e. each portion
of the shell produces a share, of the whole attraction
along AC, proportional to the angular opening it subtends
at B.
The sum of all possible values of w is the area of the
118 PROPERTIES OF MATTER.
surface of the unit hemisphere, i.e. '2ir. Hence the whole
attraction is
47rCP 2 p
AC 2
Now 47rCP 2 is the surface-area of the shell, so that the
above expression is merely
Mass of shell
Square of distance from centre '
and the proposition is proved.
It can at once be extended, as the former was, to a mass
made up of concentric shells of different densities, pro-
vided each have the same density throughout.
No other law of force, except the law of the direct
distance, gives this result.
142. Hence a uniform spherical shell, or a mass made
up of uniform concentric shells, has a true Centre of
Gravity, so far as bodies external to it are concerned ;
for it attracts, and therefore is attracted by, all external
bodies, as if it were condensed in its centre.
It is only a very special class of bodies (though of
course its members are infinite in number) which have a
true centre of gravity in the sense just explained. When
such a point exists, it always coincides with the centre of
inertia, as we see at once by supposing the attracting
body to be so distant that its action on different parts of
the attracted body is in parallel lines, and proportional
simply to the relative masses : and, for many purposes,
it is sufficiently accurate to assume that the centre of
inertia of a body may be treated as a centre of gravity.
But we must beware of making too free a use of this
hypothesis. If, for instance, the earth had a true centre
of gravity, and were rotating about its axis of greatest
moment of inertia (through that point), there could be
neither Precession nor Nutation.
GRAVITATION. 119
143. (a) continued. Armed with these results, Newton
was justified in dealing with masses approximately
spherical, such as those of the sun and planets, as if each
had been a mere particle, condensed at its centre. And
here he had the benefit of the altogether extraordinary
labours of Kepler ; who, by sheer guessing, often of the
wildest kind but followed up by persevering calculation,
had reduced to a few simple statements the chief kine-
matical results deducible from the observations of Tycho
Brahe. These were given in Kepler's work, De Motibus
StellcB Martis, Prague, 1609, and are now universally
Kepler's Laws.
I. Each planet describes an Ellipse (with comets this
may be any Conic Section) of which the Sun's centre
occupies one focus.
II. The radius-vector of each planet describes equal
areas in equal times.
III. The square of the periodic time (in an elliptic
orbit) is proportional to the cube of the major axis.
144. (b) Newton showed that, as an immediate conse-
quence of Kepler's Law II. above, the direction of the
attraction of the sun for a planet must be that of the line
joining their centres.
In fact, double the area described by the radius-vector
of a planet in one second is the moment of its velocity
about the sun's centre. But the moment of the resultant
of two velocities is the sum of their separate moments.
Hence, as the moment of the planet's velocity remains the
same, the moment of each successive increment which it
receives must be nil, i.e. these increments (i.e. the
accelerations) must be directed towards the sun's centre.
120 PROPERTIES OF MATTER.
We may prove this part also of the law of gravitation
by showing that, were it not true, The Perpetual Motion
would be attainable. But the reader may easily make out
this proof for himself.
145. (c) That the attraction varies directly as the
product of the masses will be proved at once if it be
shown to be proportional to one of the masses while the
other remains constant. For it must be remembered
that, by the third law of motion (see 128), gravitation-
attraction is mutual ; each of the two attracting bodies has
as much of a share in producing it as has the other. It is
clear, then, that the proof of this part of the law will be
obtained at once if we can show that the weights of bodies
are, in any and every one locality, proportional to their
masses ( 34).
We have seen that the measure of a force is the
momentum it produces in one second. Submit a number
of bodies to the action of their own weights alone, each
will acquire in one second a momentum proportional to
its weight. But if the weight be proportional to the
mass, the momentum must also be proportional to the
mass, and thus the speed acquired must be the same for
all. That is, if they be under the action, each of its own
weight alone, they will fall side by side through any space
whatever. Now this is known to be very nearly the case
when we let stones or bullets, or even lumps of wood,
fall ; while it is obviously not so with feathers, paper, or
gold leaf. But these exceptions show at once why the
trial is not a fair one. The falling bodies are all resisted
by the air, some only slightly, others with forces not much
less than their whole weights. Hence, to make the
experiment as nearly as possible free from such interfering
causes, Newton made the fall extremely slow, but in such
GRAVITATION. 121
a way that it could be repeated over and over again under
precisely similar circumstances, and therefore its period
could be measured very exactly. He used, as the bob of
a simple pendulum, a light hollow shell which could be
filled successively with different kinds of matter.
In Book II. sec. vi. prop. xxiv. of the Principia, he
proves that the mass of the bob of a simple pendulum of
given length is directly as its weight and as the square of
its time of oscillation in vacuo. And, in the 7th Corollary
to this proposition, we read :
" Hence appears a method both of comparing bodies one
among another, as to the quantity of matter in each, anc
of comparing the weights of the same body in different
places, to know the variation of its gravity. And, by
experiments made with the greatest accuracy, I have
always found the quantity of matter in bodies to be
proportional to their weight."
Thus gravity depends on the quantity, but in no way
on the quality, of the matter in a body ; and it is in all
cases attractive. In these respects it stands in marked
contrast to magnetic forces.
146. (d) An immediate deduction, from the first two
of Kepler's Laws, is that the Hodograph ( 70) of a
planet's orbit is a circle. For (see Fig. 13) the moment
of the velocity, V, of P, about the sun, S, is constant
( 144). And by Kepler's Law I., the orbit ABA' is an
ellipse of which S is one focus. Let fall the perpendicular
SQ on the tangent at P, then Q lies on the circle whose
diameter is the major axis AA' of the orbit. Thus V.SQ
is constant. But if QS cut the circle again in R, SR.SQ
is constant. Thus SR is proportional to V. Hence SR
is drawn from a fixed point S, in a direction perpendicular
to that of the motion of P, and its length is proportional
122
PROPERTIES OF MATTER.
to the speed of P. The locus of R, the auxiliary circle,
is therefore a curve similar to the hodograph, but turned
through a right angle.
The tangent at R, which is the direction of the accelera-
tion of the velocity SR, is therefore perpendicular to SP.
[In fact CR is parallel to SP, by a property of the ellipse.]
The magnitude of the acceleration of P is proportional to
the speed of R, i.e. proportional to the angular velocity of
CR ; i.e. to the angular velocity of SP. But the moment
of P's velocity, about S, which is constant, can also be
expressed as the product of SP 2 into the angular velocity
- R
FIG. 13.
of SP. Hence the angular velocity of SP, and therefore
also the acceleration of P, must be inversely proportional
to SP 2 . Thus we have the law of change of attraction
with distance.
147. The detailed investigation is easily given : thus,
GKAVITATION. 123
if SP = r, <A'SP = 0, and if h be twice the area
described by SP in unit of time,
r*6 = h.
But SQ.V-A,
while SQ.SR=AS.SA' = BC 2 ,
where BC is the semi-axis minor of the ellipse.
Thus SR-^-V.
But, on the same scale, the acceleration of P is measured
by the velocity of R, which is CR.0, or CA.0.
Hence the actual acceleration of P is
h PA ,,_ fr 2 .CA 3 1
"BC 2 - BC^CA'r 3 "
Now twice the area of the ellipse is 27rBC.CA ; and, if
T be the periodic time, it must also be h. T. Hence
Acceleration of P = ' JL
Kepler's Third Law tells us that CA 3 /T 2 is the same
for all the planets. Hence we conclude that it is the
same gravitation, diminishing as the square of the distance
increases, which acts on each one of the planets.
148. The result of 146 might at once have been
obtained from Kepler's third law. For if we suppose the
orbits of the planets to be circles (which they are approxi-
mately), that law gives
T 2 <x R 3 ,
where T is the periodic time, R the radius of the circle.
But, if V be the planet's speed in its circular orbit, we
have the kinematical result
2 <x R 2 .
124 PROPERTIES OF MATTER.
From the two we obtain
z'.e. (see 121) the accelerations are inversely as the
squares of the distances.
But it is better to derive, as Newton did, the law of
inverse square from the two first of Kepler's laws ; and
then the third gives us the further information that every
planet behaves exactly as any other would do if substituted
for it, i.e. that the sun's gravity pays no attention to the
quality of matter.
149. Having found that, in these general matters at
least, the assumed law of gravitation is in agreement with
the planetary motions, Newton turned to particulars, and
the special one which he took as a test was the moon's
revolution about the earth. He says :
" That the circum terrestrial force likewise decreases in
the duplicate proportion of the distances, I infer thus.
" Let us then assume the mean distance. of the moon
60 semi-diameters of the earth, and its periodic time in
respect of the fixed stars 27 d 7 h 43 m as astronomers have
determined it. And a body revolved in our air, near the
surface of the earth supposed at rest, by means of a
centripetal force which should be to the same force at the
distance of the moon in the reciprocal duplicate propor-
tion of the distances from the centre of the earth, that is,
as 3600 : 1, would (secluding the resistance of the air)
complete a revolution in l h 24 m 27 s .
" Suppose the circumference of the earth to be
123,249,600 Paris feet, as has been determined by the
late mensuration of the French, then the same body,
deprived of its circular motion, and falling freely by the
GRAVITATION. 125
impulse of the same centripetal force as before, would, in
one second of time, describe 15^- Paris feet.
" This agrees with what we observe in all bodies about
the earth. For by the experiments of pendulums, and a
computation raised thereon, Mr Huygens has demonstrated
that bodies falling by all that centripetal force with which
(of whatever nature it is) they are impelled near the
surface of the earth, do, in one second of time, describe
15_^. Paris feet."
The comparatively accurate measurement, of the length
of a degree of latitude on the earth, by Picard was un-
doubtedly the cause which ultimately led to the publica-
tion of the Principles, of which the fundamental proposi-
tions had been obtained nearly twenty years before.
For Newton, using the rough estimate of 60 miles to
a degree, had found that the moon's deflection by gravity,
in one second, from a rectilinear path, was not -g-gVffth of
the space through which a stone falls in one second at
the surface of the earth, and had in consequence put his
investigations aside, until he was led to resume them by
hearing the result of Picard's measures.
150. Having thus established the law of gravitation by
calculations founded mainly on Kepler's laws, Newton
proceeded to show that these laws could not themselves
be accurate. For a single spherical planet, revolving
about a spherical sun, the first two laws would still be
true, but a second planet would at once interfere with
this state of matters : the orbits would no longer be
ellipses, and equal areas would no longer be described
in equal times. Again, the third law could never be
exactly true, even if the planets did not attract one
another, unless they contained each the same fraction of
the sun's mass. But the consideration of questions like
126 PROPERTIES OF MATTER.
these belongs to Physical Astronomy, with which we have
nothing to do here. Suffice it to say that Newton's own
magnificently- extended deductions, supplemented as they
have been by those of successive generations of illustrious
mathematicians, have verified already to a very high
degree of nicety the competence of the law of gravitation
to account for the excessively complex motions and
perturbations observed in the solar system.
151. We have already ( 118) adverted to the apparent
loss of weight by bodies at the equator. This loss, due
(in part, at least, for there is another part ( 165) due to
the figure of the earth) to the so-called Centrifugal Force,
is, of course, directly proportional to the mass of each
body. But experiment with the most delicate balances
has shown that bodies of any kind which equilibrate
in one latitude equilibrate in all. Hence their weights
remain equal when, from that of each, is subtracted an
amount proportional to the mass. This can only be if
the weights are themselves proportional to the masses.
Thus we have an independent experimental proof of the
truth of clause (c) of Newton's statement.
152. We can scarcely yet be said to have proof that
gravitation exists, as we know it, in stellar systems. For
the data, from which to calculate orbits of double stars,
have to be obtained under circumstances which do not
admit of more than rude attempts at approximation. We
know that there are hundreds of systems in which two or
more stars revolve about one another in a way which
leaves no doubt that they are physically connected. But
the observations which have as yet been made have been
applied, not to prove that the relative orbits are consistent
with Kepler's laws but, to find the approximate dimen-
sions of the orbits, and thence the amounts of matter in
GRAVITATION. 127
the mutually influencing bodies, on the supposition that
Kepler's laws hold even in these remote systems.
Thus we cannot, at least for the present, look for proof
of the universality of gravitation in this direction. But
we have ample direct proofs that parts of the earth, and
not merely the earth as a whole, exert gravitating force.
Some of these will be considered in the immediately
succeeding sections.
153. The most direct of these (albeit not the earliest)
is what (though devised by Michell) goes by the name of
The Cavendish Experiment.
In this, by means of the elasticity of a wire or fibre, the
attraction between two spheres of manageable size is not
only demonstrated, but measured. The following sketch
shows a horizontal section through the main parts of the
arrangement.
Two small balls, A and B, an inch or two in diameter,
are connected by a stiff, but very light, horizontal girder
or tube, which is suspended at its middle point (E) by a
long fine wire. The whole of this part of the apparatus
is enclosed in a case, carefully coated with tinfoil or gold-
leaf, to prevent (as far as possible) irregular heating and
consequent currents of air ; perhaps, also, slight electrifi-
cation. To the girder is attached a small mirror, whose
plane is vertical. A little glazed window in the case
allows any motion of the mirror to be measured by the
consequent deviations of a ray of light reflected by it.
Outside the case are placed two equal, but much more
massive, spheres, usually balls of lead a foot or more in
diameter, so mounted that they can be made to move
(without jerk of any kind) from the positions C p D v to
128
PROPERTIES OF MATTER.
the positions C 2 , D 2 , and C 3 , D 3 , or back again, at will.
[In Cornu's recently-constructed apparatus there are four
spherical iron vessels, of equal size, placed once for all at
FIG. 14.
C 15 C 3 , Dj, D 3 , and so connected, two and two, that Cj
or D 3 , and simultaneously D x or C 3 , may be filled with
mercury, the other of each pair being left empty. All
four can be left empty when required.]
GRAVITATION. 129
Cavendish, and all who have since made the experi-
ment, found that the apparatus was never at rest. In
order to determine the equilibrium position it was
necessary, therefore, in all cases to measure the limits of
successive oscillations, and to compare the mean of two
successive deflections to one side, with the intervening
deflection to the other side. The time of each oscillation
was also carefully measured.
When the large masses were placed at Cg, Dg, in a line
perpendicular to the girder (i.e. each half-way between
its extreme positions), the oscillations were due practically
to torsion alone, and the couple required to twist the
suspending filament through a given angle could be
determined from the period of free oscillation, taken
along with the length of the girder and the masses of
the two small balls.
When the masses were placed at C 15 D 15 within a
couple of inches of the small balls, the range of the
oscillation was completely altered. From the observa-
tions (made as before) the new position of equilibrium
could be calculated. A fresh set of observations was
then made with the balls at C 2 , D 2 , and then they were
shifted to C 3 , D 3 . Thus is determined the deflection
ivhich would have been produced if the sensitive part of
the apparatus could have been reduced to rest.
But from this deflection, and the ascertained coefficient
of torsion of the wire, the force acting on each of the
small balls can be calculated. This is to be compared
with the weight of one of the small balls, and then the
question is, " What must be the mass of the earth when
it attracts a mass at its surface (i.e. 4000 miles from its
centre) with a force greater in a known ratio than that
with which the same mass is attracted by a given
I
130 PROPERTIES OF MATTER.
spherical mass of lead, whose centre is placed at a given
distance ? " The law of gravitation at once enables us to
write the requisite condition. The mass of the earth,
thus found, has only to be divided by its volume ( 126)
to give the mean density.
The quantities compared in such a case, i.e. the attrac-
tions, may be taken as approximately in proportion to
the radius and the mean density of the earth, and of the
leaden sphere, respectively. They are, therefore (as the
density of lead is double that of the earth), in the ratio
4000 x 5280 : 2 ; or 10 7 : 1 roughly : taking 1 foot as the
radius of the leaden sphere. Hence, to estimate correctly,
to two significant figures only, the earth's mean density,
we require to measure a force of the order of the hundred-
millionth part of the weight of the small ball. This
rough calculation gives some idea of the delicacy of the
experiment.
154. The details of the necessary precautions, as well
as of the results of various repetitions of this experiment,
do not suit a work like this, and must be sought in the
original descriptions. 1
Cavendish's result for the mean density of the earth was
5 '48 (the density of water being taken as unit); Reich
obtained 5'49 ; Baily 5'67, since reduced (by the recal-
culations of Cornu) to 5*55. Cornu's own result is 5 '50.
It is very remarkable that Newton, in Book III. of
the Principia, prop, x., made the following guess :
" Since the common matter of our earth, on the surface
thereof, is about twice as heavy as water, and a little
lower, in mines, is found about three, four, or even fi ve
1 Cavendish, Phil. Trans., 1798. Baily, Mem. Ast. Soc., 1843.
Reich, Abhand. d. K. Sachs, Ges., 1852, Cornu, Comptes Rendus,
1870-78,
GRAVITATION. 131
times more heavy, it is probable that the quantity of the
whole matter of the earth may be five or six times greater
than if it consisted all of water."
Every one of the experimental results, above given,
lies almost exactly half-way between the limits thus
assigned, and published, more than a century before even
the earliest attempt at direct determination was made.
155. Good results have been obtained by a modifica-
tion of this experiment, which enables the experimenter
to employ an ordinary balance ; an attracting sphere of
considerable mass being applied beneath a sphere attached
to one arm of the balance, and already counterpoised
(at a different level) by weights in a scale-pan. Thus the
uncertainties of torsion are avoided. Of late, however;
fibres of quartz have been drawn, which are found to be
singularly certain in their working, so that the original form
of the Cavendish apparatus has been reverted to by several
experimenters, with its scale very considerably reduced. 1
156. Other methods, which have been employed for
the determination of the mean density of the earth,
depend upon the comparison of the attraction exercised
by a mountain, or by some other part of the earth, with
that of the whole earth, when these act simultaneously,
but in different directions, on the same body. The earliest
recorded trial of this method was made by De la Conda-
mine and others, among the Andes. It was first carefully
worked out by Maskelyne on a prominent Perthshire
mountain, and has consequently been called
The Schehallien Experiment.
By geodetic measures, altogether uninfluenced by
gravitation, the actual distance between two stations,
1 Boys, Nature, xxxix., 65, 1889.
132 PROPERTIES OF MATTER.
one north the other south of the mountain, can be found,
and from it can be calculated the difference of their
(geographical) latitudes. But the true latitude of each
station separately can be determined by the usual astro-
nomical methods, depending on the observed meridian
altitude of a star. The difference between the geographi-
cal and the true latitude of each station depends upon
the attraction of the mountain for the plumb-line, or
the trough of mercury, which is used to determine the
vertical. The station south of the mountain (in the
northern hemisphere) has its latitude made less than
the geographical, that to the north made greater by this
action. Hence, if everything were symmetrical on the
two sides of the mountain, the difference of the astro-
nomically determined true latitudes at the two stations
would be greater than that of their geographical latitudes
by double the deviation produced in the plumb-line by
the mountain.
The mountain must now be contoured ; then studied
by a geologist, so as to enable him to decide on the most
probable distribution of matter in it; then the specific
gravities of samples of these kinds of matter must be
determined. Next a laborious process of Quadrature
must be gone through to calculate the action on the
plummet, taking account of the form and density of the
mass. Finally, the deflection of the plumb-line is de-
rived from this result, in terms of the (unknown) mean
density of the earth, and compared with the measured
deflection.
Maskelyne's 1 observations, developed successively by
Hutton 2 and by Playfair, 3 gave as result for the earth's
mean density 4*48 and 4'86. The great objection to this
i Phil. Trans., 1775. 2 Ibid., 1778. 3 Ibid., 1811.
GRAVITATION. 133
method is the uncertainty under which we must remain
as to the internal structure, not only of the mountain
itself but of the whole crust of the earth in its neighbour-
hood. This cannot be got over completely, so that the
result is liable to considerable error.
157. The Harton Experiment was made by Airy in the
Harton pits. It consists in comparing the intensity of
gravity at the earth's surface with that at the bottom of
a mine : the same pendulum being used successively at
the two stations; or, still better, two pendulums being
made to vibrate simultaneously, one at each station, but
now and again interchanged. This method, with the
help of modern electrical processes for comparing the
behaviour of the pendulums, is probably (so far as exact-
ness of measurement is concerned) a really good one,
The intensity of gravity at the bottom of the mine differs
from that at the surface on two accounts. Suppose a
surface drawn inside the earth, but everywhere at a depth
equal to that of the mine ; so as to divide the earth into a
core and a uniformly thick skin, as it were, enclosing it.
Gravity at the top of the pit depends on the combined
attractions of these parts. At the bottom of the pit the
skin (being, at least very approximately, spherical) ceases to
attract (by Newton's proposition, 140), but we have come
nearer to the core. Hence the observations enable us to com-
pare the attraction of the core with that of the skin. Now
we know the volume of the skin, but it has to be assumed
(and this is the fatal defect of the method) that the skin is
everyivhere of the mean density determined from examina-
tion of the various strata passed through in sinking the pit.
It is not, therefore, surprising that the result of this
experiment, 1 viz. 6*56, should differ very materially from
1 Phil. Trans., 1856.
184 PROPERTIES OF MATTER.
the consistent results obtained by the various workers at
the Cavendish experiment.
158. It was suggested byRobison 1 that the alternate
filling and emptying of an estuary or bay, at different
states of the tide, might supply an excellent mode of
measuring the earth's mean density by means of observa-
tions of the consequent twelve-hourly periodic changes of
latitude. The contouring required would be very easy,
in fact the two chief contours required are given at once
by the sea-margins at high and low water ; the density of
sea-water is practically uniform, and there are places
where the whole rise of the tide sometimes amounts to
1 20 feet or so. But this promising method seems not to
have got beyond the stage of suggestion. Still, it is the
only one yet proposed, besides the Cavendish method and
its mere modifications, which has not some inherent and
fatal weakness.
159. (e) and (/) of 138 above. That two pieces of
matter behave as if they attracted one another according
to Newton's law, is certain. But it by no means follows
that they do so attract. All that we are entitled to say,
from the facts given above, is as follows :
The part of the energy of a system of two particles of
matter, of masses m and m', which depends upon their
distance, r, from one another, is less than if they were in-
finitely far apart by
and this is not altered by the presence of other particles.
This, taken along with the conservation of energy,
enables us fully to investigate the motions of any system
1 Elements of Mechanical Philosophy, 1804, p. 339. See also
Forbes, Proc. R.S.E., II. p. 244.
GRAVITATION. 135
of gravitating masses. It represents, in fact, our whole
knowledge on the subject. And it is important to
observe that the statement is altogether free from even
the mention of the word attraction or force. [See, again,
15, 137.]
160. We may, however, briefly notice some hypotheses
which have been framed as to the mechanism on which
gravitation depends. For Newton, in his celebrated
Letters to Bentley, expressly says :
"You sometimes speak of gravity as essential and
inherent to matter. Pray do not ascribe that notion to
me ; for the cause of gravity is what I do not pretend to
know, and therefore would take more time to consider of
it."
" It is inconceivable that inanimate brute matter should,
without the mediation of something else which is not
material, operate on and affect other matter without
mutual contact, as it must do if gravitation in the sense
of Epicurus be essential and inherent in it. ... That
gravity should be innate, inherent, and essential to matter,
so that one body may act upon another at a distance
through a vacuum^ without the mediation of anything
else, by and through which their action and force may be
conveyed from one to another, is to me so great an
absurdity, that I believe no man who has in philosophical
matters a competent faculty of thinking, can ever fall into
it. Gravity must be caused by an agent acting constantly
according to certain laws; but whether this agent be
material or immaterial, I have left to the consideration of
my readers."
161. When we come to deal with molecular forces we
shall find that small bodies, such as sticks, straws, air-
bubbles, etc., floating on water, are made to aggregate
136 PROPERTIES OF MATTER.
themselves into groups by molecular tension in the water-
surface ( 288). Hence the idea that stress, in a medium
filling all space, might account for the apparent mutual
attraction between bodies entirely surrounded by this
medium.
Newton, in the Queries at the end of his Optics, speaks
of a possible explanation to be obtained by assuming that
dense bodies rarefy the ether surrounding them, to an
amount which is less as the distance is greater.
Clerk-Maxwell says on this point : l
" To account for such a force by means of stress in an
intervening medium, on the plan adopted for electric and
magnetic forces, ... we must suppose that there is a
pressure in the direction of the lines of force, combined
with a tension in all directions at right angles to the lines
of force. Such a stress would, no doubt, account for the
observed effects of gravitation. We have not, however,
been able hitherto to imagine any physical cause for such
a state of stress. It is easy to calculate the amount of
this stress which would be required to account for the
actual effects of gravity at the surface of the earth. It
would require a pressure of 37,000 tons' weight on the
square inch in a vertical direction, combined with a
tension of the same numerical value in all horizontal
directions. The state of stress, therefore, which we must
suppose to exist in the invisible medium is 3000 times
greater than that which the strongest steel could support.''
162. Other attempts have been made, with the view
of showing that waves, or pulsating motion, in a medium,
would have the effect of drawing immersed bodies
together. Again, Lord Kelvin has shown that if space
be filled with an incompressible fluid, which comes into
1 Eney. Brit., ninth ed., Art. "Attraction."
GRAVITATION. 137
existence in fresh quantities at the surface of every
particle of matter, at a rate proportional to its mass, and
is swallowed up at an infinite distance, or, if each
particle of matter constantly swallows up an amount
proportional to its mass, a constant supply being kept
up from an infinite distance, in either case gravitation
would be accounted for. This is, however, virtually
a suggestion of a dynamical mode of producing the
diminution of pressure required in Newton's attempt at
explanation.
163. An attempt at explanation, from a totally different
point of view, was made by Le Sage in 1818. The fol-
lowing account of it is taken from Clerk-Maxwell's article,
" Atom," already referred to :
"The theory of Le Sage is that the gravitation of
bodies towards each other is caused by the impact of
streams of atoms flying in all directions through space.
These atoms he calls ultramundane corpuscules, because
he conceives them to come in all directions from regions
far beyond that part of the system of the world which
is in any way known to us. He supposes each of them
to be so small that a collision with another ultramundane
corpuscule is an event of very rare occurrence. It is by
striking against the molecules of gross matter that they
discharge their function of drawing bodies towards each
other. A body placed by itself in free space and exposed
to the impacts of these corpuscules would be bandied about
by them in all directions, but because, on the whole, it
receives as many blows on one side as on another, it cannot
thereby acquire any sensible velocity. But if there are
two bodies in space, each of them will screen the other
from a certain proportion of the corpuscular bombard-
ment, so that a smaller number of corpuscules will strike
138 PROPERTIES OF MATTER.
either body on that side which is next the other body,
while the number of corpuscules which strike it in other
directions remains the same.
" Each body will therefore be urged towards the other
by the effect of the excess of the impacts it receives on
the side farthest from the other. If we take account of
the impacts of those corpuscules only which come directly
from infinite space, and leave out of consideration those
which have already struck mundane bodies, it is easy to
calculate the result on the two bodies, supposing their
dimensions small compared with the distance between
them.
"The force of attraction would vary directly as the
product of the areas of the sections of the bodies taken
normal to the distance and inversely as the square of the
distance between them.
" Now, the attraction of gravitation varies as the pro-
duct of the masses of the bodies between which it acts,
and inversely as the square of the distance between them.
If, then, we can imagine a constitution of bodies such
that the effective areas of the bodies are proportional to
their masses, we shall make the two laws coincide. Here,
then, seems to be a path leading towards an. explanation
of the law of gravitation, which, if it can be shown to be
in other respects consistent with facts, may turn out to be
a royal road into the very arcana of science.
"Le Sage himself shows that, in order to make the
effective area of a body, in virtue of which it acts as a
screen to the streams of ultramundane corpuscules, propor-
tional to the mass of the body, whether the body be large
or small, we must admit that the size of the solid atoms
of the body is exceedingly small compared with the
distances between them, so that a very small proportion
GRAVITATION. 139
of the corpuscules are stopped even by the densest and
largest bodies. We may picture to ourselves the streams
of corpuscules coming in every direction, like light from a
uniformly illuminated sky. We may imagine a material
body to consist of a congeries of atoms at considerable
distances from each other, and we may represent this by
a swarm of insects flying in the air. To an observer at a
distance this swarm will be visible as a slight darkening
of the sky in a certain quarter. This darkening will
represent the action of the material body in stopping the
flight of the corpuscules. Now, if the proportion of light
stopped by the swarm is very small, two such swarms will
stop nearly the same amount of light, whether they are in
a line with the eye or not, but if one of them stops an
appreciable proportion of light, there will not be so much
left to be stopped by the other, and the effect of two
swarms in a line with the eye will be less than the sum
of the two effects separately.
" Now, we know that the effect of the attraction of the
sun and earth on the moon is not appreciably different
when the moon is eclipsed than on other occasions when
full moon occurs without an eclipse. This shows that
the number of the corpuscules which are stopped by bodies
of the size and mass of the earth, and even the sun, is
very small compared with the number which pass straight
through the earth or the sun without striking a single
molecule. To the streams of corpuscules the earth and
the sun are mere systems of atoms scattered in space,
which present far more openings than obstacles to their
rectilinear flight.
" Such is the ingenious doctrine of Le Sage, by which
he endeavours to explain universal gravitation. Let us
try to form some estimate of this continual bombardment
140 PROPERTIES OF MATTEtt.
of ultramundane corpuscules which is being kept up on
all sides of us.
" We have seen that the sun stops but a very small
fraction of the corpuscules which enter it. The earth,
being a smaller body, stops a still smaller proportion of
them. The proportion of those which are stopped by a
small body, say a 1 Ib. shot, must be smaller still in an
enormous degree, because its thickness is exceedingly
small compared with that of the earth.
" Now, the weight of the ball, or its tendency towards
the earth, is produced, according to this theory, by the
excess of the impacts of the corpuscules which come
from above over the impacts of those which come from
below, and have passed through the earth. Either of
these quantities is an exceedingly small fraction of the
momentum of the whole number of corpuscules which
pass through the ball in a second, and their difference
is a small fraction of either, and yet it is equivalent to
the weight of a pound. The velocity of the corpuscules
must be enormously greater than that of any of the
heavenly bodies, otherwise, as may easily be shown, they
would act as a resisting medium opposing the motion of
the planets. Now, the energy of a moving system is half
the product of its momentum into its velocity. Hence
the energy of the corpuscules, which by their impacts on
the ball during one second urge it towards the earth,
must be a number of foot-pounds equal to the number of
feet over which a corpuscule travels in a second, that is to
say, not less than thousands of millions. But this is only
a small fraction of the energy of all the impacts which
the atoms of the ball receive from the innumerable
streams of corpuscules which fall upon it in all directions.
" Hence the rate at which the energy of the corpuscules
GRAVITATION. 141
is spent in order to maintain the gravitating property of a
single pound, is at least millions of millions of foot-pounds
per second."
1 64. One common defect of these attempts is, as Clerk-
Maxwell points out, that they all demand some prime-
mover, working beyond the limits of the visible universe
or inside each atom : creating or annihilating matter,
giving additional speed to spent corpuscles, or in some
other way supplying the exhaustion suffered in the pro-
duction .of gravitation. Another defect is that they all
make gravitation a mere difference-effect as it were ; there-
by implying the presence of stores of energy absolutely
gigantic in comparison with anything hitherto observed or
even suspected to exist, in the universe; and therefore
demanding the most delicate adjustments, not merely to
maintain the conservation of energy which we observe, but
to prevent the whole solar and stellar systems from being
instantaneously scattered in fragments through space.
In fact, the cause of gravitation remains undiscovered.
165. The ordinary balance, as we have already seen,
merely tests equality of masses. To find the weight of a
body we must measure directly the earth's attraction for it.
This can be done, perfectly in principle but only with
a rude approximation to accuracy in practice, by means
of a Spring-Balance, or by some other contrivance which
depends on the elastic resilience of a special kind of matter.
By far the most accurate instrument for measuring the
intensity of gravity, from which, of course, the weight of
any body (whose mass is known) may be immediately
calculated, is the pendulum.
A simple pendulum ( 115) exists, of course, only in
theory ; but by means of a theorem of abstract dynamics
we can calculate the length of the simple pendulum
142 PROPERTIES OF MATTER.
which will vibrate in the same period as does a mass,
of any form and dimensions, freely supported in any
assigned way on a horizontal axis. This the reader must
take for granted. 1 Hence we can reduce observations
made with any pendulum to those with the corresponding
simple pendulum.
The following expression, whose form is suggested by
the theory of the Figure of the Earth, and whose constants
have been determined and verified by pendulum observa-
tions made all over the world, gives approximately the
value of g ( 120) at sea-level in any latitude X.
32-088(1 + 0-00513 sin 2 .x).
166. We conclude the chapter with a small table of
(approximate) Specific Gravities, or what is the same thing
( 36), Densities, and a few remarks suggested by it.
None of the numbers for solids can be given wich any
great accuracy (except perhaps those for natural crystals) :
for, even if the substance be pure, its density may be
altered to a considerable amount by the processes through
which it has passed in assuming the state in which it is
tested. Such a table as the present must be looked on as
affording materials for rough calculations only. When
better results are required, special determinations must be
made for each substance dealt with.
Hydrogen 0*000089
Helium . .'"..., . 0-000177
Steam . . . . > . 0-0006
Nitrogen . . . . . '001 25
Air . . . . . - . 0-00129
Oxygen . . , . . 0*00143
Argon . . . . ' . . 0*00178
(The above are at 1 atmosphere; steam (of course) at 100 C.,
the others at C.)
1 Thomson and Tait's Elements of Nat. Phil., Appendix, g.
GRAVITATION. 143
Cork 0-24
Lithium 0'59
Potassium 0'86
Gutta Percha . . . . 0'98
Water I'OO
Magnesium . . . . .175
Quartz 2 '65
Aluminum ..... 2'67
Granite, Marble, Slate . .27
Glass 27 to 4 '5
Basalt. .... 2-9
Bromine . . . . .3*0
Zinc 7'2
Tin 7'3
Iron 7'8
Nickel 87
Copper 8 '9
Silver 10'6
Lead 11-8
Mercury 13 '6
Gold 19-4
Platinum 21 '5
Indium 22 '4
[It is well to note that, as the mass of a cubic foot of water is
about 62'5lbs., these numbers, each multiplied by 62'5, give in
pounds very nearly the mass per cubic foot of the corresponding
substance.]
The chief additional remark suggested by the table is
that, not only are there bodies which, though liquid at
ordinary temperatures, are denser than the great majority
of solids, but that a comparatively moderate pressure,
such as a few hundred atmospheres, would (without pro-
ducing liquefaction) make the density of air or oxygen
greater than that of some solids ; so that, for instance, if
chemical action could be prevented, we might easily have
solid lithium floating upwards in compressed oxygen, as a
cork rises in water.
The ratio of the densities of iridium and of hydrogen,
144 PROPERTIES OF MATTER.
as given in the table, is about 250,000:1. But, by means
of a Sprengel pump, the density of the hydrogen might
easily be reduced to a four-thousandth of its former value.
Thus we can place beside one another specimens of
matter, one of which has one thousand million-fold the
density of the other. Such a comparison may help us to
understand the possibility of the existence of the lumini-
ferous medium ; which is certainly matter, yet of a density
perhaps smaller in comparison with that of attenuated
hydrogen, than is the latter in comparison with the
density of iridium. In the present work the ether does
not come in for treatment. We know it only in so far
as it is the vehicle of radiation and electrical energy :
so that it is to works on Light and Electricity the student
must be referred.
167. By considering the earth, for a moment, as a
liquid mass, it is easy (on hydrostatical principles) to
calculate the whole pressure across any plane section of
it. 1 This is, of course, the resultant gravitation attraction
between the parts separated by the plane of section.
Assuming the result of 154 for the mean density, we
find that the average attraction, per square foot, across a
diametral plane is about 18 x 10 8 Ibs. weight. The tenacity
of sandstone is about 72 x 10 3 Ibs. weight per square foot.
Thus gravitation is 25,000 times as effectual in keeping
the earth together, as would be its cohesion if it were solid
sandstone. Even if the earth were as tenacious as steel,
its cohesion across a diametral plane would be only about
1 per cent, of the attraction across it.
Since the cohesion between two halves of a globe is,
ceteris paribus, as the area of a diametral plane, i.e. as the
square of the radius, while the gravitation attraction is
1 Tait, Proc. M.S.R, 1875.
GRAVITATION. 145
as the sixth power of the radius directly, and as the
square of the radius inversely, a sphere of the earth's
mean density and of the tenacity of sandstone would
require to be of about 25 miles radius only, in order that
cohesion may be as effective as gravity in keeping two
hemispheres together. If the tenacity were that of steel,
the radius would be about 400 miles.
Hence the earth's strength depends almost wholly on
gravitation, while that of a stone, less than a mile or so
in diameter, depends almost wholly on cohesion, and the
more completely the smaller it is.
CHAPTER VIII.
PRELIMINARY TO DEFORMABILITY AND ELASTICITY.
168. A SUBSTANCE is said to be elastic when, on being
left free, it recovers wholly or partially from a deforma-
tion ( 41).
This definition is sometimes given in another form :
a substance is said to be elastic when it requires the
continued application of stress to keep it deformed. But
this is by no means an equivalent of the former state-
ment ; and, besides, it usually introduces complications ;
for in many substances the force requisite to maintain a
distortion becomes less and less with the lapse of time ;
and the continued application of a given distorting force
often produces a constantly increasing distortion. To
this, and to another curious property called the Fatigue
of Elasticity, we will recur, but we will for the present
adhere to the first definition given above.
Hence, as an introduction to this part of the subject,
we must inquire into the nature and mechanism of the
simpler kinds of deformation.
169. The term usually employed for deformation of
any kind is Strain. The treatment of strains is an
entirely geometrical, or (more properly) kinematical,
question. But when we inquire how a strain is produced
DEFOKMABILITY AND ELASTICITY. 147
in a given piece of matter, the question becomes a
dynamical one, and we are led to the notion of a system
of equilibrating forces, called a Stress. (See, again,
137.) And we figure to ourselves that every stress
produces a corresponding strain, which will be of greater
or less amount as the specimen of matter operated on is
of more or less yielding quality.
It is sometimes convenient to speak of the property of
yielding to a particular stress, as when we speak of the
Compressibility of a substance ; sometimes it is more con-
venient to speak of the property of resistance to a stress,
as when we speak of a body's Rigidity. But the resist-
ance to a stress is measured by the reciprocal of the
amount of yielding (just as the electric resistance of a
wire is the reciprocal of its conducting power), so that
either of these numerical quantities is immediately
deducible from the other.
It will be seen shortly that if P be the measure of
any one kind of stress, and p that of the corresponding
strain (supposed small), experiment points to a general
relation of the form
where C is a constant depending on the special substance,
and the special form of stress. C is obviously greater,
the smaller is the strain for a given stress ; and it there-
fore measures the resistance of the substance to the
particular kind of stress denoted by P.
As stress is force per unit of surface, while strain has
no dimensions, the dimensions of C in the above expres-
sion are
M
Hence the numerical value of C changes, in passing
148 PROPERTIES OF MATTER.
from one system of units to another, directly as the unit
of length and the square of the unit of time are increased,
and inversely as the unit of mass is increased.
170. We shall not require for our elementary treat-
ment of the question more than the simplest portions of
the subject of strain, and shall therefore be concerned
with Homogeneous Strain only.
By this term it is implied that all originally similar,
equal, and similarly situated portions of a substance
remain after the strain similar, equal, and similarly
situated, however their forms and dimensions may be
changed. Hence points originally in a straight line, or
in a plane, remain in a straight line, or in a plane. Also
equal parallel lines remain equal parallel lines. There-
fore a parallelogram remains a parallelogram, an ellipse
remains an ellipse, a parallelepiped remains a parallel-
epiped, and an ellipsoid remains an ellipsoid.
A most important case is that of a sphere inscribed in
a cube. The diameters which pass through the points of
contact form a conjugate system : i.e. the tangent plane
at the extremity of any one of them is parallel to the
other two. This parallelism is not affected by the strain ;
so that when the sphere becomes an ellipsoid, the cube
becomes a parallelepiped whose faces are parallel to a set
of conjugate planes. Conversely, every set of three con-
jugate planes of the ellipsoid was originally a set of three
mutually perpendicular diametral planes of the sphere.
An immediate consequence of this is that the principal
axes of the ellipsoid, which are at right angles to one
another, were originally at right angles to one another in
the sphere.
171. Now suppose small, equal, and similarly situated
cubes to be traced in the unstrained body. This will be
DEFOKM ABILITY AND ELASTICITY. 149
effected by three imagined series of equidistant parallel
planes, those of each series being perpendicular to those
of the other two. After the strain the cubes become
equal, similar, and similarly situated parallelepipeds.
And it is clear that if one of the cubes, and the corre-
sponding parallelepiped, be given, everything else can be
determined.
But there is one special set, of three series of rect-
angular planes, with which it is best to commence : viz.
those which, as shown in last section, become the princi-
pal planes of the ellipsoid which is formed from the
sphere inscribed in the cube. This elementary consideration
produces a marvellous simplification of our investigation.
172. For we now see that every homogeneous strain
may be looked on as having been produced by uniform
extensions, or compressions, parallel to three mutually
perpendicular lines (the amounts parallel to these being
generally different), and a subsequent rotation of the
whole as if it were rigid. We shall not require to
consider the rotation, for we are concerned only with the
deformation which each small part suffers.
Thus, taking account of these permissible simplifica-
tions, we need only inquire into the circumstances under
which an originally cubical portion of the substance
becomes in general brick-shaped, without change of the
directions of its edges. The investigation presents no
grave difficulties when the strains are of finite magni-
tude, but we will, for simplicity as well as convenience
( 174, 177), consider them as small.
173. We will first consider the particular cases which
are of greatest importance.
Nothing need be said of the case where a cube remains
a cube, though with altered edges, except that it involves
150
PROPERTIES OF MATTER.
uniform dilatation, or a condensation such as is due to hydro-
static pressure. This is the only strain (except, of course,
pure rotation) which does not alter the figure of a body.
But, of the strains which alter the figure of a body,
without altering its volume, the most important is that
which converts a cube into a brick shape by lengthening
in a given ratio one set of parallel edges, shortening a
second set in the same ratio, and leaving the third set
unaltered. Here it is obvious that the volume also
remains unaltered. Let the ratio of extension be 1 + 1 : 1,
that of contraction will be 1 1 : 1, on account of the
smallness of the fraction Z in all the cases which we have
FIG. 15.
to consider. Let the figure represent (in its successive
states) one of those faces of the cube, of which all the
edges have been altered. The square inscribed in that
face is obviously distorted into a rhombus, of which two
of the angles are greater, and two less, than right angles,
by the same amount, 6 suppose.
Then it is clear that the ratio of the diagonals of the
rhombus may be expressed in either of the following
forms :
DEFORMABILITY AND ELASTICITY. 151
and thus, as is very small, so that the arc may be
written in place of its tangent,
0=2*.
This shows the relation between the difference of the
angles of the rhombus from right angles, and the fractional
alteration of the edges of the original cube.
174. So far for the strain, let us now consider its
cause. Every equilibrating system of forces (i.e. every
stress) can be reduced to simple stresses, each consisting
of equal and opposite forces in the same line, that is,
thrusts, or tensions. Thus we have now to inquire what
thrusts or tensions will convert a cube of deformable
matter into an assigned brick shape : in which, of course,
is included the simple case of its remaining a cube, though
with altered edges. These must evidently be spread
uniformly over each of its surfaces, for every one of any
number of smaller equal cubes, into which it may be
supposed to be divided, suffers precisely the same propor-
tionate deformation.
And as ( 172) we confine ourselves to very small
deformations, any number of them may be superposed,
without interfering with one another i.e. they may be
successively inflicted in any order, with the same final
result. It is mainly for this reason that we restrict
ourselves to small strains.
175. The problem is too difficult for an elementary
work, unless the portion of matter dealt with be not only
homogeneous, but isotropic, i.e. unless it possess exactly
the same properties at all parts and in all directions, so
that the effect of a given stress on a unit cube of it is
exactly the same however and wherever the cube be cut
out of the original material.
152 PROPERTIES OF MATTER.
Hence we see that, for cubes which become brick-shaped,
without change of direction of the edges, the thrusts or
tensions must each be perpendicular to the face on which
it acts. And ( 169) we measure each by its amount per
unit area.
[It is most particularly to be remarked that, in all that
follows on this subject, it is understood that the body
operated on is kept at a definite temperature, alike through-
out its substance and throughout the whole period of the
operation.
The study of the heat developed by sudden applica-
tions of stress belongs entirely to Thermodynamics, upon
which we do not enter in this work. In fact, we here
confine ourselves to Isothermals, and have nothing to do
with AdiabaticsJ]
176. The simplest case of all, and that which alone
we require when we deal with fluids, is when the stress is
pressure or tension, the same on each face of the cube.
Here the cube obviously remains a cube, but its edges
are diminished or increased in length. Let unit of edge
become 1 -/ (where / is very small) under pressure P
per square unit of each face ; what is called hydrostatic
pressure, pressure the same in all directions, and always
normal to the surface. Then the volume of unit cube
becomes 1 3/.
The compressibility of an isotropic body is measured by
the ratio of the compression per unit volume to the hydro-
static pressure applied.
Hence the compressibility is 3//P, and the Resistance
to compression ( 169), usually called k, is P/3/, so that
177. When we deal with solids, in which the stress is
DEFORMABILITY AND ELASTICITY. 151
not necessarily of the nature of hydrostatic pressure, some
further considerations must be attended to.
We now assume, consistently with experiment (as will
afterwards be shown), that, if the strain produced by any
stress be small, the reversed stress will produce exactly
the reversed strain. This is another reason ( 172) for
confining our work to small strains.
Suppose the pairs of opposite faces of a cube be called A,
B, and C ; the edges joining the corners of each pair a, b,
c, respectively. Then a tension P, per unit of area, on the
A faces will increase a in some definite ratio 1 +p : 1, and
diminish b and c in some common ratio 1 q:l. Now
superpose a pressure P, per unit area, on the B faces.
Since the body is isotropic this will compress b in the
ratio 1 - p : 1, and extend a and c in the ratio 1 + q : 1.
Hence the result of tension P on the A faces and pres-
sure P on the B faces is that a is extended in the ratio
1 +p + q : 1, b is compressed in the ratio I p-q:l t
while the length of c is unaltered.
The effect is, therefore, (as in 173) to change the
form of each section of the cube parallel to the C faces,
but to leave the area of that section and the volume of
the cube unaltered. This strain is called a Simple
Shear, and the corresponding stress is called Shearing
Stress.
There is, however, another mode of looking at this
matter, to which we must devote a little space. It is
usual, in defining Rigidity, to consider the deformation
produced in the unit cube by equal tangential forces,
applied to two pairs of its sides, in directions parallel
to the third pair of sides, as indicated in the diagram
below. These forces, as shown in the figure, obviously
constitute a balancing system, or Stress. But it may be
154 PROPERTIES OF MATTER.
analysed into a much simpler one. For, if we draw
either diagonal in the figure, the resultant of the forces
applied to either pair of faces on
one side of it is easily seen to
be P v /2, in a direction perpen-
dicular to the diagonal. But the
length of the diagonal is J2.
Hence the stress perpendicular to
either diagonal plane is P per
square unit. And it is clearly a
pressure perpendicular to one dia-
gonal plane, and a tension perpen-
dicular to the other. It is therefore the system already
studied in 177, and the effect on the cube above is that
studied in 173.
178. We now define as follows :
The rigidity of an isotropic solid (i.e. the resistance to
change of form under a stress such as that in the above
figure) is directly proportional to the tangential force per
unit area, and inversely as the change of one of the angles
of the figure.
Hence, using the common designation, it, and calling 6
(as before) the change of each of the angles of the figure,
we have
Rigidity -n-P/0,
or, by 173, 177,
P+l-1 ..,.. (1.)
179. But, by 177, the effect of pressure P, applied
simultaneously to all the sides of the cube, would be to
reduce the lengths of the edges in the common ratio
or (approximately)
DEFOKMABILITY AND ELASTICITY. 155
Hence by 176, where it is obvious that / stands for
what we now call p - 2q ;
P-2<1 = ^ .... (2.)
180. From (1) and (2) we have at once
These represent respectively the extension of one set of
edges of the unit cube, and the common contraction of
the other two, when it is subjected to tension P parallel
to the former set.
[These results might have been obtained, perhaps even
more simply, by assuming the existence of compressibility
with absolute rigidity, then assuming pliability with
absolute incompressibility, and superposing the effects.
But the logic of this process is more likely to puzzle the
beginner.]
181. Hence the extension, per unit of length, of a rod
or bar, under longitudinal tension P per square inch of its
cross-section, is
p 3k+n
~Wcn'
The applications of this formula are very numerous
and important, as will be seen in 224, &c., below.
The corresponding diminution, per unit area, of cross-
section is
And thus the increase per unit volume is P/3&, a result
156 PROPERTIES OF MATTER.
which we might have obtained directly in many other
ways.
Thus, in pulling out an india-rubber band with a given
tension, we increase its volume by one-third of the amount
by which it would be diminished by hydrostatic pressure
of the same value.
Also by pulling out a truly cylindrical and uniform
tube, filled to a definite mark with a liquid, we may
measure directly the value of k for the matter of the
tube.
182. From the foregoing formulae the result of the
application of any (moderate) stress to an isotropic body
can be calculated.
As an example, suppose we desire to find what stress
will produce extension of an isotropic bar or cylinder
unaccompanied by lateral change of any kind.
If we have tensions, P along, and P' in all directions
perpendicular to, the axis of the bar, we have for the
longitudinal extension ( 177)
, 2F
P-^*i
and for the extension in any radial direction
P>_P + P'
P P ~ q '
The latter must vanish, by our assumed condition, so that
which gives the required relation between P' and P ; and
thus the extension is
P
183. In the chapters which immediately follow, it
DEFOEMABILITY AND ELASTICITY. 157
will be seen that to determine the compressibility of a
fluid we require (at least in all the ordinary modes of
experimenting) to know the distortion produced in the
vessel which contains it.
When the same hydrostatic pressure is applied simul-
taneously to the outside of the vessel and to its contents,
the correction for diminution of the interior volume is of
course, 176, 212, PV/&: where P is the pressure per
unit surface, V the interior volume, and k the reciprocal
of the compressibility of the material of the vessel. This
is to be added to the apparent compression of the fluid.
But when the pressure on the vessel is mainly internal
(as in Andrews' experiments on carbonic acid, 205),
or wholly external (as in glass manometers, 233), the
correction is not so simple. It can, in every case, be
determined by means of the equations of 180 ; but the
investigation even of symmetrical cases is beyond the
limits here imposed on us. We therefore merely state
the results for the forms of vessel most commonly used,
viz. tubes and bulbs. For simplicity we assume the
tubes to be cylindrical, and the bulbs to be spherical,
each being of uniform material and of uniform thickness
throughout. The internal and external radii are, in
both cases, denoted by and a^ respectively ; and the
cylinders are supposed free to alter in length as well as
in cross-section.
Then the diminution per unit of content, by external
hydrostatic pressure P, is
In cylinders
In spheres
158 PROPERTIES OF MATTER.
The increase per unit of content, by internal hydro-
static pressure F, is
In cylinders F
In spheres F V 3 fU^|Y
a i -o \* V*/
When there are simultaneous hydrostatic pressures out-
side and inside, the corresponding results, calculated
from these expressions, are to be simply superposed
( 174).
Thus, if P and P' be simultaneous and equal, we have,
alike in cylinders and spheres, for the diminution of
unit internal content, P/& as above.
When an exceedingly thick vessel (at least a vessel in
which al is very small in comparison with a-f) is exposed
to internal pressure only, the effect on unit of its content
practically depends on its rigidity alone, and is P'/n for
a cylinder, and 3P'/4w< for a sphere. This is a very
striking result.
When such a vessel is exposed to external pressure
the result is
For cylinders P (1 + ~\
For spheres P (- + ^
\k 4%
This shows the fallacy of the too common notion that,
by making the bulb of a thermometer thick enough, we
enable it to " defy pressure" ', as, for instance, when it is
to be employed to measure temperatures in a sounding
of 3000 or 4000 fathoms.
184. It is very interesting to study the cases of
heterogeneous strain presented by the walls of cylinders
DEFORMABILITY AND ELASTICITY. 159
and bulbs when the internal and external hydrostatic
pressures are different. The following data will show
the student the form and volume of the strain-ellipsoid,
i.e. the ellipsoid into which a very small part of the wall,
originally spherical, is distorted. "We give the formulae
for a cylinder under external pressure. Let the original
position of the centre of the little sphere be at a distance,
r (intermediate, of course, between and a^, from the
axis. Then it is deformed into an ellipsoid, whose axes
are (a) radial, (/?) parallel to the axis of the cylinder,
(y) at right angles to these two. If we denote by 1 the
original radius of the little sphere, the semi-axes of the
ellipsoid are
These are, in order of increasing magnitude, (/?), (y), (a).
The axes (ft) and (y) are always reduced in length, but
the radial axis (a) will be increased in length by the
07.
strain provided r 2 < a 2 .
In ordinary flint glass this condition becomes, approxi-
mately
So that the interior layers of a glass tube, exposed to
external pressure only, are always extended in the radial
direction. This extension is greatest at the interior
surface, and vanishes in the layer whose radius is about
160 PKOPE.RTIES OF MATTER.
1 '6a . If the external radius be greater than this, the
outer layers are radially compressed, and the more the
farther they lie beyond the limit of no extension.
185. The theory of the propagation of Waves, whether
of compression or of distortion, in an elastic body, is
beyond our limits ; but we may make the statement
that, if we could set aside the effects of sudden stress
in producing changes of temperature, and thus altering
the coefficients of compressibility and rigidity (for this
question belongs properly to Thermodynamics), the rates
of propagation of waves of different kinds depend only
upon one or both of the elastic constants (k and n), and
upon the density of the body. When the coefficients
are measured in terms of the weight of unit bulk of
the body, they are called Moduli. Hitherto we have
measured them in terms of pressure or tension, i.e. force
per unit area. But, if we measure the force by the
length of the column of the substance, of unit section,
whose weight it can just support, we obviously take
account of the weight of unit bulk. Now the theoretical
result (under the conditions above specified) is that the
speed of a wave is that which would be acquired by
a free body falling, under uniform gravity, through a
height equal to half the length of the modulus corre-
sponding to the particular kind of distortion which is
propagated. Thus the speed of sound in air or water
depends upon the value of k alone ; that of a shearing
wave, such as light and some forms of earthquake, on n
alone. When a wave of extension is sent along a wire,
as (for instance) to set a distant railway signal, Young's
modulus ( 224) comes in; and, when we deal with
plane sound-waves in a solid, we must take the corre-
sponding modulus as given in 182.
CHAPTER IX.
COMPRESSIBILITY OF GASES AND VAPOURS.
186. A VERY general proof of compressibility and of
elasticity of bulk is afforded at once by the fact that the
great majority of bodies are capable of transmitting
sound-waves. For the propagation of sound consists
essentially in the handing on by resilience, from layer to
layer of the medium, of a state of compression or dilata-
tion ; the (small) disturbance of each particle taking
place to and fro in the direction in which the sound is
travelling. All ordinary sounds are propagated in air.
But the rate of passage of sound has been measured in
the water of the Lake of Geneva and elsewhere ; and
miners are in the habit of signalling to one another by
the sounds (of taps with a pick) conveyed through solid
rock.
187. Compressibility, elasticity, and inertia of air
are all demonstrated by the action of an air-gun. Its
reservoir is charged, by means of a pump, with some
forty or sixty times the quantity of air which it would
contain at the normal pressure and temperature ; the
moment the valve is thrust down, by the fall of the
hammer, a portion of the air is forced out by its elas-
ticity ; and this rapid stream, by its inertia, communi-
L
162 PROPERTIES OF MATTER.
cates motion to the bullet. The same thing is shown,
in a very beautiful form, by allowing the compressed air
to escape in a fine jet; for a ball of cork can be sus-
pended in the jet, as a metal shell is suspended in a
fountain-jet of water, but in this case without any visible
support.
188. In 1662 Robert Boyle published his Defence of
the Doctrine touching the Spring and Weight of the Air.
The following extract, especially, is still of great interest.
It occurs in Part II. chap. v.
" We took then a long Glass - Tube, which by a
dexterous hand and the help of Lamp was in such a
manner crooked at the bottom, that the part turned up
was almost parallel to the rest of the Tube, and the
Orifice of this shorter leg of the Siphon (if I may so call
the whole Instrument) being Hermetically seal'd, the
length of it was divided into Inches (each of which was
subdivided into eight parts) by a straight list of paper,
which containing those Divisions was carefully pasted all
along it : then putting in as much Quicksilver as served
to fill the Arch or bended part of the Siphon, that the
Mercury standing in a level might reach in the one leg
to the bottom of the divided paper, and just to the same
height or Horizontal line in the other ; we took care, by
frequently inclining the Tube, so that the Air might
freely pass from one leg into the other by the sides of
the Mercury, (we took (I say) care) that the Air at last
included in the shorter Cylinder should be of the same
laxity with the rest of the Air about it. This done, we
began to pour Quicksilver into the longer leg of the
Siphon, which by its weight pressing up that in the
shorter leg, did by degrees streighten the included Air :
and continuing this pouring in of Quicksilver till the Air
COMPRESSIBILITY OF GASES AND VAPOURS. 163
in the shorter leg was by condensation reduced to take
up but half the space it possess'd (I say, possess' d not
fill'd) before ; we cast our eyes upon the longer leg of the
Glass, on which was likewise pasted a list of Paper care-
fully divided into Inches and parts, and we observed, not
without delight and satisfaction, that the quicksilver in
that longer part of the Tube was 29. Inches higher than
the other. Now that this Observation does both very
well agree with and confirm our Hypothesis, will be easily
discerned by him that takes notice that we teach, and
Monsieur Paschall and our English friends Experiments
prove, that the greater the weight is that leans upon the
Air, the more forcible is its endeavour of Dilatation, and
consequently its power of resistance, (as other Springs
are stronger when bent by greater weights.) For this
being considered it wil appear to agree rarely- well with
the Hypothesis, that as according to it the Air in that
degree of density and correspondent measure of resistance
to which the weight of the incumbent Atmosphere had
brought it, was able to counterbalance and resist the
pressure of a Mercurial Cylinder of about 29. Inches, as
we are taught by the Torricellian Experiment ; so here
the same Air being brought to a degree of density about
twice as great as that it had before, obtains a Spring
twice as strong as formerly. As may appear by its being
able to sustain or resist a Cylinder of 29. Inches in the
longer Tube, together with the weight of the Atmo-
spherical Cylinder, that lean'd upon those 29. Inches of
Mercury ; and, as we just now inferr'd from the Torri-
cellian Experiment, was equivalent to them.
" We were hindered from prosecuting the tryal at that
time by the casual breaking of the Tube. But because
an accurate Experiment of this nature would be of great
184 PROPERTIES OF MATTER.
importance to the Doctrine of the Spring of the Air,
and has not yet been made (that I know) by any man ;
and because also it is more uneasie to be made , >
then one would think, in regard of the diffi-
culty as well of procuring crooked Tubes fit for
the purpose, as of making a just estimate of
the true place of the Protuberant Mercury's
surface; I suppose it will not be unwelcome
to the Reader, to be informed that after some
other tryals, one of which we made in a Tube
whose longer leg was perpendicular, and the
other, that contained the Air, parallel to the
Horizon, we at last procured a Tube of the
Figure exprest in the Scheme ; which Tube,
though of a pretty bigness, was so long, that
the Cylinder whereof the shorter leg of it
consisted admitted a list of Paper, which had
before been divided into 12. Inches and their
quarters, and the longer leg admitted another
list of Paper of divers foot in length, and
divided after the same manner : then Quick-
silver being poured in to fill up the bended
part of the Glass, that the surface of it in
either leg might rest in the same Horizontal
line, as we lately taught, there was more and "
more Quicksilver poured into the longer Tube ;
and notice being watchfully taken how far
the Mercury was risen in that longer Tube
when it appeared to have ascended to any FlG - 17 -
of the divisions in the shorter Tube, the several
Observations that were thus successively made, and as
they were made set down, afforded us the ensuing
Table.
COMPKESSIBILITY OF GASES AND VAPOURS. 165
A TABLE OF THE CONDENSATION OF THE Am.
A.
A.
12
11
10
4
3|
3i
3
B.
C.
00
02H
04*
06*
10*
12*
15*
17**
21*
25*
29H
32*
34**
37H
41*
45...
58*
71*
78H
88*
nto
<M
3
D.
29*
30*
31**
33*
35*
37...
41**
47*
50*
61*
64*
67*
70H
74*
82**
93*
100*
107H
117*
E.
29*
30*
31**
33f
35...
361|
41*
46|
50...
63*
66f
70...
77*
82*
87f
107*
116*
A. A. The number of
equal spaces in the
shorter leg, that con-
tained the same par-
cel of Air diversely
extended.
B. The height of the
Mercurial Cylinder
in the longer leg,
that compress'd the
Air into those
dimensions.
C. The height of a Mer-
curial Cylinder that
counterbalanc'd the
pressure of the At-
mosphere.
D. The Aggregate
of the two last
Columns, B and C,
exhibiting the pres-
sure sustained by
the included Air.
E. What that pressure
should be according
to the Hypothesis,
that supposes the
pressures and ex-
pansions to be in
reciprocal propor-
tion."
189. The form of apparatus employed by Boyle is still
recognised as by far the best for the purpose. With a
few necessary modifications, to adapt it to difference of
circumstances, it was employed by Amagat l in the most
important recent experimental determinations of the
effects of great pressures on the volume of a gas.
Its action depends on the two hydrostatical principles
1 Annales de Chimie, 1880.
166 PROPERTIES OF MATTER.
stated below, the truth of which we are here content to
assume. 1
In a mass of fluid, at rest, the pressure (per square inch)
is the same at all points in any horizontal plane.
The change of pressure from one Jiorizontal plane to
anotlwr is equal to the weight of a column of the fluid, one
square inch in section, extending vertically between these
planes.
From these it follows that the pressure of the gas
operated on, i.e. the pressure on the mercury surface at
A (Fig. 17) is the same as that at the same level, B, in
the other branch of the tube : and this, again, exceeds
the pressure at C (the atmospheric pressure), by the
weight of a column of mercury of square inch section and
of height BC.
190. In his comments on this experiment Boyle
says :
"For the better understanding of this Experiment it
may not be amiss to take notice of the following particu-
lars :
" 3. That we were two to make the observation to-
gether, the one to take notice at the bottom how the
Quicksilver rose in the shorter cylinder, and the other
to pour it in at the top of the longer, it being very hard
and troublesome for one man alone to do both accurately.
" 6. That when the Air was so compress'd, as to be
crouded into less than a quarter of the space it possess'd
before, we tryed whether the cold of a Linen Cloth dipp'd
in water would then condense it. And it sometimes
1 See, for instance, Thomson and Tait, Elements of Natural
Philosophy, 692, 694.
COMPRESSIBILITY OF GASES AND VAPOURS. 167
seemed a little to shrink, but not so manifestly as that
we dare build anything upon it. We then tryed likewise
whether heat would notwithstanding so forcible a corn-
pressure dilate it, and approaching the flame of a Candle
to that part where the Air was pent up, the heat had a
more sensible operation than the cold had before ; so
that we scarce doubted but that the expansion of the Air
would notwithstanding the weight that opprest it have
been made conspicuous, if the fear of unseasonably
breaking the Glass had not kept us from increasing the
heat.
" And there is no cause to doubt, that if we had been
here furnished with a greater quantity of Quicksilver and
a very strong Tube, we might by a further compression of
the included Air have made it counterbalance the pres-
sure of a far taller and heavier Cylinder of Mercury.
For no man perhaps yet knows how near to an infinite
compression the Air may be capable of, if the compressing
force be competently increast.
" And to let you see that we did not (a little above)
inconsiderately mention the weight of the incumbent
Atmospherical Cylinder as a part of the weight resisted
by the imprisoned Air, we will here annex, that we took
care, when the Mercurial Cylinder in the longer leg of
the Pipe was about an hundred Inches high, to cause
one to suck at the open Orifice ; whereupon (as we ex-
pected) the Mercury in the Tube did notably ascend. . . .
And therefore we shall render this reason of it. That
the pressure of the Incumbent Air being in part taken
off by its expanding it self into the Sucker's dilated chest ;
the imprison'd Air was thereby enabled to dilate it self
168 PROPERTIES OF MATTER.
manifestly, and repel the Mercury that comprest it, till
there was an equality of force betwixt the strong Spring
of that comprest Air on the one part, and the tall Mer-
curial Cylinder, together with the contiguous dilated Air,
on the other part."
It is scarcely necessary to call attention to the truly
scientific caution with which Boyle thus gives his con
elusions from this notable experiment.
191. BOYLE'S LAW (as it is called in Britain) is now
stated in the extended form :
TJie volume of a given mass of gas, kept at a given
temperature, is inversely as the pressure. 1
In symbols this is merely
jw-C . . . . . (1.)
where C is a quantity depending upon the mass of gas,
and on its temperature. [This law is only approximately
true. In 196-207 below the relation between pressure
and volume will be more exactly stated.]
From the definition of density as the quantity of
matter per unit of volume, we see at once that Boyle's
Law may be stated in the form
The density of a gas, at constant temperature, is propor-
tional to the pressure.
192. The compressibility follows at once. For a
small increase, &, in the pressure, corresponds to a small
diminution, to, in the volume, such that we still have
(p + za}(v - ) = C = pv.
Neglecting the product of the two small quantities we
have
1 This Law usually goes by the name of Mariotte in foreign
books. See Appendix IV.
COMPRESSIBILITY OF GASES AND VAPOURS. 169
Here the change, per unit of volume, is at/v, so that
the compressibility ( 176) is
=1.
arv p
The resistance to compression is therefore proportional to
the pressure; i.e. inversely as the volume. This result
was obtained by a graphic process in 176 above.
193. So closely does air follow Boyle's Law through
all ordinary ranges of pressure, that it is constantly used
in Manometers for the direct measurement of pressure.
The manometer is, in its elements, merely a carefully
calibrated tube containing dry air, from whose volume
(when it is kept at constant temperature) the pressure is
at once calculated.
The chief defect of such manometers is that successive
equal increments of pressure produce gradually diminish-
ing effects on the volume of the gas; and thus the
inevitable errors of observation become more serious, in
proportion to the quantity to be measured, as higher
pressures are attained. Various ingenious devices, such
as tubes of tapering bore, have been devised to remedy
this defect. In all such modifications most careful
calibration is essential.
194. All gases, at temperatures considerably above
what is called their critical point ( 55, 206), follow
Boyle's Law fairly through a somewhat extensive range of
pressures. But a gas, at a temperature under its critical
point, is really a vapour, and can be reduced (without
change of temperature) to the liquid state by the appli-
cation of sufficient pressure, at least if nuclei be present.
The compression of vapours will be treated farther on.
195. So far, we have been dealing with the effects of
increased pressure. But Boyle carried his inquiry into
170
PROPERTIES OF MATTER.
the effects of diminution of pressure also. His apparatus
was of a very simple kind, though still useful, at least
for class illustration. The following extract, while highly
interesting, sufficiently describes his results and method :
"A TABLE OF THE KAREFACTION OF THE Am.
A. B.
2
3
4
5
6
7
8
9
10
12
14
16
18
20
24
28
32
oo
lOf
2of
22f
25|
26f
26f
27i
27|
27f
27J
28 +
28|
28f
28!
C.
D.
E.
29|
2|t
Iff
ll!
OHI
A. The number of equal spaces
at the top of the Tube, that
contained the same parcel of
Air.
B. The height of the Mercurial
Cylinder, that together with
the spring of the included
Air, counterbalanced the
pressure of the Atmosphere.
C. The pressure of the Atmos-
phere.
D. The Complement of B to C,
exhibiting the pressure sus-
tained by the included Air.
E. What that pressure should
be according to the Hypo-
thesis.
" To make the Experiment of the debilitated force of
expanded Air the plainer, 'twill not be amiss to note some
particulars, especially touching the manner of making the
Tryal ; which (for the reasons, lately mention'd) we made
on a lightsome pair of stairs, and with a Box also lin'd
with Paper to receive the Mercury that might be spilt.
And in regard it would require a vast and in few places
procurable quantity of Quicksilver, to employ Vessels of
such kind as are ordinary in the Torricellian Experiment,
we made use of a Glass-Tube of about six foot long, for
COMPKESSIBILITY OF GASES AND VAPOURS. 171
that being Hermetically seal'd at one end, serv'd our
turn as well as if we could have made the Experiment in
a Tub or Pond of Seventy Inches deep.
" Secondly, We also provided a slender Glass-Pipe of
about the bigness of a Swan's Quill, and open at both
ends ; all along which was pasted a narrow list of Paper
divided into Inches and half quarters.
" Fourthly, There being, as near as we could guess,
little more than an Inch of the slender Pipe left above
the surface of the restagnant Mercury, and consequently
unfill'd therewith, the prominent orifice was carefully
clos'd with sealing Wax melted ; after which the Pipe
was let alone for a while, that the Air dilated a little by
the heat of the Wax, might upon refrigeration be reduc'd
to its wonted density. . . .
" Sixthly, The Observations being ended, we presently
made the Torricellian Experiment with the above mention'd
great Tube of six foot long, that we might know the
height of the Mercurial Cylinder, for that particular day
and hour ; which height we found to be 29f Inches.
"Seventhly, Our Observations made after this manner
furnish'd us with the preceding Table, in which there
would not probably have been found the difference here
set down betwixt the force of the Air when expanded to
double its former dimensions, and what that force should
have been precisely according to the Theory, but that the
included Inch of Air receiv'd some little accession during
the Tryal ; which this newly- mention'd difference making
us suspect, we found by replunging the Pipe into the
Quicksilver, that the included Air had gain'd about half
an eighth, which we guest to have come from some little
172 PROPERTIES OF MATTER.
aerial bubbles in the Quicksilver, contained in the Pipe
(so easie is it in such nice Experiments to miss of
exactness)."
196. We must now state how far these results of
Boyle have been verified by modem experimenters, and in
what direction they are found to deviate from the truth.
But before we do so we must introduce a definition.
The unit usually adopted for the measurement of
pressure is called an Atmosphere, roughly 147 Ibs. weight
per square inch.
Its definition is, in this country, the weight of a column
of mercury at C., of a square inch in section, and
29*905 inches high; the weighing to be reduced to the
value of gravity at the sea-level in the latitude of London.
(See 165.)
The value of an atmosphere, in C.G.S. units, is about
1,014,000 dynes per square centimetre.
197. It is to Eegnault that we owe the first really
adequate treatment of the subject, but the range of
pressures he employed was not very extensive.
Regnault showed that air and nitrogen are, for at least
the first twenty atmospheres, more compressed than if
Boyle's Law were true, but that hydrogen is less
compressed.
Then Natterer made an extensive but rough series of
experiments at very high pressures (sometimes nearly
3000 atmospheres), whose result showed that air and
nitrogen, as well as hydrogen, are less compressible than
Boyle's Law requires, and deviate the more from it the
higher the pressure.
198. Andrews, 1 in his classical researches which
established the existence of the critical point ( 55), first
1 Phil. Trans., 1869.
COMPKESSIBILITY OF GASES AND VAPOURS. 173
gave the means of explaining this very singular fact.
We will recur to it when we are dealing with vapours,
but we give a few of Andrews' data here. The way in
which the compressibility varies with pressure is obvious
from the curves in the diagram ( 205), when interpreted
as in 176. But from Andrews' tables of corresponding
volumes of air at 13-1, and carbonic acid at 35'5,
subjected simultaneously to each of a series of increasing
pressures, we extract the numbers in the two first
columns :
CARBONIC ACID (GAS) AT 35 '5 C.
Recip. of Vol. Recip. of Vol. of
of Air. Carbonic Acid.
81-28 228-0
86-60 351-9
89-52 373-7
92-64 387-9
99-57 411-0
107-6 430-2
pv for Garb. Acid.
356
246
239
239
242
250
Andrews points out that the deviation of air from
Boyle's Law is, even at the highest of these pressures,
inconsiderable. Taking the reciprocals of the volumes
of air, therefore, as measuring pressures with sufficient
accuracy, we form the third column of the table. This
shows that in carbonic acid, a few degrees above its
critical point, the deviation from Boyle's Law is like
that in air and nitrogen for the first 90 atmospheres,
and, after that, resembles that in hydrogen. Unfor-
tunately the bursting of the tubes prevented Andrews
from carrying the pressure beyond 108 atmospheres.
199. The remarkable researches of Amagat already
alluded to ( 189) were carried out in a gallery of a deep
coal-pit, where the temperature remained steady for long
174 PROPERTIES OF MATTER.
periods. The shorter branch of his apparatus, that which
contained the gas whose compression was to be measured,
terminated in a very strong glass tube of small bore, care-
fully calibrated. The longer branch was made of steel,
and extended to a height of 330 metres (about 1000 feet)
up the shaft of the pit. A small but powerful pump was
employed to force mercury into the lower part of the
apparatus until it began to run out at one of a set of stop-
cocks which were inserted at measured intervals along the
tall tube. Then a measurement of the volume of the
compressed gas was made, the stopcock closed, and that
next above it opened in turn for a measurement at a
higher pressure.
200. The following short table gives an idea of Ama-
gat's earlier results 1 for air at ordinary temperature :
Pressure in Atmospheres. pv.
1-00 1-0000
31-67 '9880
45-92 -9832
59-53 -9815
73-03 -9804
84-21 -9806
94-94 -98] 4
110-82 -9830
133-51 -9905
176-17 1-0113
233-68 1-0454
282-29 1-0837
329-18 1-1197
400-05 1-1897
[As Amagat's pressure data were obtained direct from a
column of mercury, they supply by far the most accurate
1 Ann. de Chimie, 1880 ; supplemented from Comptes fiendus,
1884.
COMPRESSIBILITY OF GASES AND VAPOURS. 175
means of finding the unit for pressure gauges. Hence it
may be well to note that, at ordinary temperatures, for a
pressure of 152*3 atmospheres, or one ton- weight per
square inch, dry air almost exactly follows Boyle's Law,
i.e. it is reduced to 1/152-3 of its volume at one atmosphere.
Hence, practically, when dry air is compressed to anything
from 1/140 to 1/160 of its bulk under one atmosphere,
Boyle's Law may be used to calculate the pressure.]
It is very difficult to assign with exactness the position
of the minimum value of pv, as inevitable errors of
observation may rise to considerable importance when a
quantity varies very slowly ; but it may be put down as
corresponding to about 78 atmospheres.
201. Amagat's direct measures with the mercury
column were made on the volume of nitrogen. But
when these had been carefully made, once for all, the
nitrogen manometer was used in connection with a similar
instrument filled with some other gas. Thus the relation
of pv to p was determined with accuracy for hydrogen,
oxygen, air (as above), carbonic oxide, carbonic acid,
ethylene, etc. In later papers 1 Amagat has extended
these results through a considerable range of temperatures.
For the numerical data we must refer to the papers them-
selves ; but we reproduce three of the most important of
his graphic representations of the earlier results.
The diagram following consists of two parts. The
upper part shows the relation of pv to p, through a range
of about 80 C., for nitrogen, whose behaviour is typical of
that of a large number of gases. The minimum value of
pv is distinctly shown at every temperature. The lower
diagram exhibits the exceptional case of hydrogen, where
all the curves are, practically, straight lines. The
1 Annales de Chimie, xxii. 1881, xxix. 1893.
FIG. 18.
COMPRESSIBILITY OF GASES AND VAPOURS. 177
pressure unit is a metre of mercury, i.e. 100/76 atmos-
pheres.
The diagram on the next page shows the corresponding
relations for carbonic acid, at temperatures above its
critical point ; as well as for liquid carbonic acid at 18*2
C. In this last case the curve is given only for pressures
from 80 to 260 metres of mercury. This diagram gives
very valuable information. Especially it shows the
marked influence of change of temperature on the pressure
corresponding to the minimum value of pv. Ethylene
gives a diagram somewhat resembling this, but the
changes in the value of pv are so disproportionately
greater that its behaviour could not be satisfactorily
exhibited on a scale so restricted as a page of this book.
The reader should be reminded that, had the law of
Boyle been accurate, all of these curves would have been
simply horizontal straight lines.
The more recent researches of Amagat, 1 above referred
to, have extended this enquiry to the results of very much
higher pressures, such as 3000 atmospheres, under which
the density of gaseous oxygen becomes greater than that
of water. The exact measurement of these great pressures
was effected by means of an exceedingly ingenious instru-
ment, the Manometre d pistons Ubres, which Amagat con-
structed for the purpose. In this instrument there are
two pistons, of very different sectional area, subjected to
the same total thrust. Thus the pressure (per square inch)
on each is inversely as its section. The pressure on the
smaller piston is that of the substance compressed, that
on the larger is measured directly by means of a column
of mercury. The unit for graduation (which of course
depends on the ratio of the effective sections of the
1 Comptes Rendus, Sept. 1888.
M
m
\\\\
2O W 60 80 WO
FIG. 19.
COMPRESSIBILITY OF GASES AND VAPOURS. 179
pistons) was determined, once for all, by comparison with
the nitrogen gauge. The special feature of this instru-
ment, on which its precision depends, is that the pistons
fit all but tightly in their cylinders ; a very thin layer of
viscous fluid passing with extreme slowness between each
piston and its cylinder. Exact adjustment is secured by
giving slight rotation to each piston in its bearings. For
the larger piston castor-oil is used, for the smaller treacle.
But each piston, before being inserted, is most carefully
lubricated with neats-foot oil. We have been thus
particular in describing the main characteristics of this
instrument, because it supplies efficiently what has long
been felt as an extremely serious want in the physical
laboratory.
The pressure which reduced the gas to a given volume
was determined by an electrical method which will
presently be described (211). In the table below, 1 the
volume of each gas at C. and one atmosphere is taken
as unit ; and the temperature throughout was about 1 5 C.
TRUE VOLUMES OF VARIOUS GASES ABOUT 15 C. UNDER
VERY GREAT PRESSURES.
Volumes.
Pressure in
Atmospheres.
Air.
157.
Nitrogen.
16-0.
Oxygen.
15 '6.
Hydrogen.
15 -4.
1000
0020615
0021340
0018000
0017780
1500
17935
18540
15710
14180
2000
16430
16990
14440
12225
2500
15420
15960
13600
11010
3000
14660
15225
12960
10125
Amagat found the compressibility of the glass employed
to be about 0*0000023 per atmosphere. As the pres-
sures were equal inside and outside his piezometer, this
was the only elastic coefficient required. ( 183.)
1 Ann. de Chimie, xxix., 1893.
180 PROPERTIES OF MATTER.
Thus, referring to 166, we see that oxygen at 15 has
almost reached the density of water at a pressure of 2500
atmospheres. Even nt 1000 atmospheres it has about
three-fourths of the density of water.
202. There is, unfortunately, a considerable variety of
statement as to the relation between pressure and volume
in air and other gases, when they are considerably rare-
fied. This is not to be wondered at, for the experimental
difficulties are extremely great.
The experiments of Mendeleeff gave a gradual descent
of value of pv, in air, from
1-0000 at -85 atm.
to
0-9655 at 0'019 atm.
These would tend to show that, at pressures lower than
an atmosphere, air behaves as hydrogen does for pressures
above an atmosphere.
The experiments of Amagat do not show this result.
They rather seem to indicate that pv remains practically
constant for air, from one atmosphere down to at least
g^th of an atmosphere.
203. But the real difficulty in all such experiments
arises from the shortness of the column of mercury by
which the pressure must be measured. It is not easy to
see how this difficulty can be obviated without introduc-
ing a chance of graver errors of another kind, due for in-
stance to vapour-pressure or to capillary forces.
We shall find, later, that a fair presumption from
Andrews' investigations would be that, in air and the
majority of gases, pv should increase (of course very
slightly) with diminution of pressure from one atmos-
phere downwards; while (possibly) hydrogen may give
COMPRESSIBILITY OF GASES AND VAPOURS. 181
values of pv diminishing to a minimum, and then in-
creasing as the pressure is still further reduced.
204. Passing next to the compressibility of vapours, it
would appear natural that we should specially consider
aqueous vapour, which is constantly present in the atmos-
phere as superheated, sometimes even as saturated, steam.
And we have for it the splendid collection of experimental
results obtained by Regnault. But the critical point of
water-substance is considerably higher than the range of
temperature in Eegnault's work ; so that we will deal
chiefly with carbonic acid, for which we have Andrews'
data both above and below its critical point, and which
may be taken as affording a fair example of the chief
features of the subject.
205. Without further preface we give Andrews' dia-
gram, which will be easily intelligible after what has been
said in 88. It shows, in fact, how the figure in that
section, which is drawn from Boyle's Law, is modified in
the case of a true gas, and of a true vapour, each within
a few degrees of the critical temperature.
[To save space, a portion of the lower part of the dia-
gram (containing the axis of volumes) is cut away, so that
pressures, as shown, begin from about 47 atmospheres.
The dotted air-curves are rectangular hyperbolas, as in
88, but the (unexhibited) axis of volumes is their hori-
zontal asymptote.]
The critical temperature of carbonic acid was given by
Andrews as about 30 '9 C., so that the isothermals indi-
cated by full lines in the figure, and marked 13 '1 and 21*5
respectively, belong to vapour, or liquid, or vapour in pres-
ence of liquid, the others to gas. [The true critical point is
a little higher (about 31 *3). There was a trace of air, some
l/500th of the whole, in the gas operated on by Andrews.]
FIG. 20.
COMPRESSIBILITY OF GASES AND VAPOURS. 183
Let us study, with Andrews' data, the values of the
product j)v for the isothermal of 13'l C. The following
table is formed precisely on the same principle as that of
198 for the isothermal of 35 '5 C.
CARBONIC ACID (VAPOUR AND LIQUID) AT 13-1 C.
Recip. of Vol. Recip. of Vol. of
of Air.
47'5
4876
48-89
49-0
49-08
50-15
50-38
54-56
75-61
90-43
Carbonic Acid.
76-16
80-43
80-90
105-9
142-0
462-9
471-5
480-4
500-7
510-7
pv. for Carb. Acid.
653
606
600
462
345
108
106
113
151
196
206. Near to 49 atmospheres liquefaction commences,
the vapour being condensed to -^-st of its volume at one
atmosphere, and we see that an exceedingly small increase
of pressure produces a marked change of volume. Had
it been possible to free the carbonic acid perfectly from
air, no additional pressure would have been required till
the whole was liquid, at about ^^d of its original volume.
The numbers pv diminish, as in the case of air (but much
more rapidly), till the liquefaction begins : then they
ought to diminish exactly as the volume diminishes (the
pressure being constant) till complete liquefaction : after
which, of course, they begin to rise rapidly, as it is now a
liquid which is being compressed.
We need not give the experimental numbers for the
isothermal of 21 -5 C. ; but the cut shows that the stages
of the operation were much the same, only that the pres-
184 PROPERTIES OF MATTER.
sure had to be raised over 60 atmospheres before liquefac-
tion began, and liquefaction was complete before the
volume had been reduced so far as at the lower tempera-
ture. Thus the range of volume in which the tube was
visibly occupied partly by liquid, partly by saturated
vapour, and therefore (but for the trace of air) necessarily
at constant pressure, was shortened at each end. The
dotted line in the lower part of the figure, introduced by
Clerk-Maxwell, bounds the region in which we can have
the liquid in equilibrium with its vapour. This region
terminates at the critical isothermal, for above that there
can be neither vapour nor liquid.
But the properties of the gas, above the critical point,
maintain at any particular volume a certain analogy to
those of the vapour or liquid below it. For volumes
somewhat above the critical volume (0*0042) the gas has
properties analogous to the superheated vapour, i.e. pv
diminishes with increase of pressure. For volumes some-
what under the critical volume its properties are analogous
rather to those of the liquid, and pv increases with increase
of pressure. Thus there is in each isothermal of the gas
a particular pressure, for which pv is a minimum. This
feature of the isothermal becomes less marked as the
temperature is raised. [This, however, has been already
exhibited more fully on Amagat's diagram, p. 1 77.] We
might introduce a continuation, beyond the critical point,
of the left-hand portion of the dotted curve, which should
pass through the points on each isothermal at which pv is
a minimum. This line would divide the wholly gaseous
region into two parts ; that to its right, in which the gas
has properties somewhat resembling those of superheated
vapour ; to the left, that in which its properties resemble
rather those of a liquid.
COMPRESSIBILITY OF GASES AND VAPOURS. 185
An ingenious suggestion of J. Thomson substitutes for
the horizontal part (liquid in presence of vapour) of
Andrews' curves (p. 181) the continuous curve shown (by
dashes) on the isothermal of 21'5 C. The middle portion
of this curve (where pressure and volume increase to-
gether) is physically unstable, but the other parts can be,
to some extent, realised. The subject properly belongs
to Heat. It is known that liquids may, in certain cases,
be raised considerably above their boiling points without
boiling; and Aitken ( 291) has proved that a nucleus of
some kind is necessary for the condensation even of super-
saturated vapour. The first of these phenomena may
account for a portion of the new part of the curve near
the liquid region, the second for that near the vapour
region. The rest, belonging to an essentially unstable
condition, cannot be realised experimentally.
The apparently anomalous behaviour of hydrogen may
perhaps be traced to the fact that, at ordinary temper-
atures and pressures, it is in that region of its gaseous
state which has more analogy with the liquid than with
the vaporous state. Thus it is possible that if hydrogen
be examined at sufficiently low pressure, and temperature
not far above its critical point, it also will show a mini-
muni value of pv ( 203).
207. The reduction of various gaseous bodies to the
liquid form was one of the earliest pieces of original work
done by Faraday. Some of them he liquefied by cooling
alone, many others by pressure alone ; and he pointed out
that, in all probability, every gas could be liquefied by
the combined influences of cooling and pressure, provided
these could be carried far enough.
Thilorier prepared large quantities of liquid carbonic
acid, and took advantage of the cooling produced by its
186 PROPERTIES OK MATTER.
rapid evaporation, at ordinary pressures, to reduce it to
the solid state.
Cagniard de la Tour succeeded in completely evaporating
various liquids (including ether, and even water) in closed
tubes, which they half-filled while in the liquid state.
It was Andrews' work, however, which first cleared up
the subject, and, as an early consequence of it, several of
those gases which had resisted all attempts to liquefy
them were, at the end of 1877, shown to be capable of
liquefaction by the production of a momentary mist on
sudden adiabatic expansion from a pressure of several
hundred atmospheres to one. These important results
were obtained by Pictet; and in the year 1883 Wrob-
lewski and Olzevvski succeeded in getting oxygen, nitrogen,
&c., in the static condition.
More recently, Dewar has made a careful study of the
remarkable properties of air, oxygen, and nitrogen :
obtained in large quantities in the liquid state.
In 1898 he succeeded in getting liquid hydrogen in
the static condition, and by evaporation in vacua has got
it recently in the solid condition.
Van der Waals, Clausius, Sarrau, and others, working
from various assumptions, have given formulas which
accord somewhat closely with the observed phenomena,
and with J. Thomson's suggested modification of the
diagram. One of the simplest expressions of the kind
(which takes the place of (1) of 191) is of the form
Here C is as before, and A, a, /3 are parameters depending
on the properties of the substance as well as on its
temperature. The " critical point " is determined by the
COMPRESSIBILITY OF GASES AND VAPOURS. 187
condition that the three values of v, given by this equation,
shall be equal. These formulae are based upon what is
called the Virial equation, a result of pure dynamics,
given by Clausius in 1870. This represents a relation of
equality between the average values of the kinetic energy
of a system, and of a certain function of the forces exerted
between its parts ; which obtains whenever the motion is
" stationary ". The part of this function which depends
on the retaining walls is proportional to pv. Hence,
when mutual action between the particles is left out of
account, we have at once the equation of 191, C being
proportional to the whole kinetic energy of the gas.
But the full treatment of such matters belongs to
Thermodynamics, and is not for a work like this. Nor
have we anything here to do with the employment of
these liquefied gases for the production of exceedingly low
temperatures ; though, from the experimental point of
view, this application promises to be (for the present at
least) their most valuable property.
CHAPTER X.
COMPRESSION OF LIQUIDS.
208. A GLIMPSE at the negative results of the early
attempts to compress water was given in 98. The
problem is a complicated one, because (at least in the best
methods hitherto employed) the quantity really measured
is the difference of compressibility of the liquid and the
containing vessel. Hence it involves the compressibility
of solids also : and this, as we shall find (231) is a very
difficult problem indeed. Difficult as it is, it has to be
encountered, for the compressibility of glass is by no means
small in comparison with that of the greater number of
liquids. The first to succeed in proving the com pressibility
of water was Canton, 1 the value of whose work seems not
to have been fully appreciated. His second paper, in fact,
has dropped entirely out of notice.
Noting the height at which mercury stood in the narrow
tube of an apparatus like a large thermometer, immersed
in water at 50 F., the end of the tube being drawn out to
a fine point and open, he heated the bulb till the mercury
filled the whole, and then hermetically sealed the tip of
the tube. When the mercury was cooled down to 50 F.
it was found to have risen in the capillary tube. This
was due partly to expansion of mercury, released from the
1 Phil. Trans., 1762,
COMPRESSION OF LIQUIDS. 189
pressure of the atmosphere, partly to the compression of
the bulb, due to one atmosphere of external pressure.
Then he filled the same apparatus with water, performed
exactly the same operations, and obtained a notably larger
result. This, of course, proves that water (if not also
mercury) expands when the pressure of the atmosphere is
removed from it.
To get rid of the effect of unbalanced external pressure,
and thus (as he thought) to measure the full amount of
expansion, he placed his apparatus (with its end open) in
the receiver of an air-pump. He could also place it in a
glass vessel, in which the air was compressed to two
atmospheres. He observed that, on the relief of pressure,
the water rose in the stem, while on increase of pressure
it fell. He gives the fractional change of volume per
atmosphere, at 50 F. (10 C.), as 1/21740 or 0-000046.
He applied no correction for the compressibility of glass,
giving the completely fallacious (212) reason that he had
obtained exactly the same results from a thick bulb and
from a thin one. [This, however, proves the accuracy of
his experiments.] His result, considering its date, is
wonderfully near the truth.
209. In a second paper, 1 published a couple of years
later, Canton made some specially notable additions to our
knowledge. For he says, referring to his first paper :
" By similar experiments made since, it appears that water
has the remarkable property of being more compressible in
winter than in summer, which is contrary to what I have
observed both in spirits of wine and in oil of olives ; these
fluids are (as one would expect water to be) more com-
pressible when expanded by heat, and less so when con-
tracted by cold."
1 Phil Trans., 1764, vol. liv. 261.
190 PROPERTIES OF MATTER.
By repeated observations, at " opposite " seasons of the
year, he found that the effect of the " mean weight of the
atmosphere " was, in millionths of the whole volume
At 34 F. At 64 F.
Water 49 44
Spirit of Wine 60 71
He also gives a table of compressibilities in millionths of
the volume, per atmosphere of 2 9 '5 inches, and of specific
gravities ; for different liquids, at 50 F. ; as follows :
Compressibility. Spec. Gravity.
Spirit of Wine ... 66 846
Oil of Olives . . 48 918
Rain Water . . 46 1000
Sea Water . 40 1028
Mercury . 3 13595
and he observes that the compressions are not " in the
inverse ratio of the densities, as might be supposed."
He calculates from the result for sea water that two
miles of such water are reduced in depth by 69 feet 2
inches ; the actual compression at that depth being 1 3 in
1000. This, of course, assumes that the compressibility
is the same at all pressures, which, as we shall see
immediately, is by no means the case.
210. Perkins, in 1820, made a set of experiments on
the apparent compressibility of water in glass, of a some-
what rude kind; but in 1826 1 he gave some valuable
determinations, unfortunately defective because of the
inadequate measure of the pressure unit. Thus he did
not give accurate values of the compression, but he intro-
duced us to a higher problem : how the compressibility
depends upon the amount of pressure. Perkins' results
1 " On the Progressive Compression of Water by high Degrees of
Force." Phil. Trans.
COMPKESSION OF LIQUIDS. 191
are all for 50 F. (10 C.), and are given in figures, as well
as in a carefully-executed diagram plotted by the graphic
method. His measurement of pressures depended upon an
accurate knowledge of the section of a plunger : an
exceedingly precarious method : and he estimated an
atmosphere at 14 Ibs. weight only per square inch. It is
not easy to make out his real unit, especially as we know
nothing about the glass he used, but it seems to have been
about 1*5 times too great ; i.e. when he speaks of the effect
of 1000 atmospheres he was probably applying somewhere
about 1500. Hence it is not easy to deduce from his data
anything of value as to the amount of compression. But
the novel point, which he made out clearly, is that (at 10
C.) the compressibility of water decreases, quickly at first,
afterwards more slowly, as the pressure is raised. We
obtain from Perkins' diagram the following roughly
approximate results, in which we have made no attempt
to rectify his pressure unit :
Pressure Compression of Water Average Com- True Com -
in in Millionths of pressibility per pressibility per
Atmospheres. Orig. Vol. Atmosphere. Atmosphere.
150 10,000 66 51
300 17,500 58 48
900 43,400 48 39
and from a further isolated statement we obtain
2,000 83,300 42
In this paper Perkins mentions a remarkable experi-
mental result he had obtained : viz. the solidification of
acetic acid by pressure. Amagat has recently succeeded
in solidifying tetrachloride of carbon by pressure. 1
211. Orsted's improvement in the experimental method
(1822) consisted chiefly in applying pressure, as in Canton's
1 Comptes Rendus, 1887.
192 PROPERTIES OF MATTER.
process, in such a way that the effects of pressures up to
40 or 50 atmospheres can be read off at every stage of the
experiment.
The liquid operated on fills the bulb and the greater
part of the stem of the apparatus (called a Piezometer},
and is separated by mercury contained
in a U tube from the water-contents of
a strong glass cylinder, in which the
pressure is produced by forcibly screw-
ing in a piston or plug. As in Canton's
apparatus, the stem of the piezometer is
carefully calibrated and divided into parts
corresponding to equal volumes, and the
cubic content of the bulb is determined.
Hence the ratio of the content of one
division of the tube to the whole content
of bulb and stem is found.
When pressure is applied, the mercury
is seen to ascend in the stem to an
amount nearly in proportion to the
pressure. The pressure is roughly cal-
culated (by Boyle's law) from the ob-
served change of volume of air contained
in a very uniform tube, closed at the
FIG. 21. t^ anc j immersed along with the
piezometer, in the water of the compression vessel.
The only serious defect of this apparatus, besides the
inadequate measurement of pressure, is the limitation of
the pressure to what the exterior vessel can resist, some
50 or 60 atmospheres only. When higher pressures are
to be applied, iron or steel must be used for the compres-
sion vessel ; and then the piezometer must be made, in
some way, to record the change of volume of its contents.
COMPKESSION OF LIQUIDS. 193
The most common device is to have (as in a maximum
thermometer) a little index resting on the mercury and
prevented, by attached hairs, from moving too freely.
It contains a small piece of iron, so that it may be adjusted
from without by a magnet. This method is liable to the
objection that, unless the pressure is relieved very
cautiously, the index may be displaced by the current of
liquid : so that the results are sometimes a little too small.
Cailletet gilt the inside of the stem, and the eating away
of the film of gold showed the height to which the mercury
had risen. An exceedingly thin film of silver, deposited
by sugar of milk, has also been employed. But all such
devices are very troublesome, for the compression vessel
has to be opened after every experiment. Hence Tait 1
suggested the sealing of a number of fine platinum wires
into the stem of the piezometer, and by an obvious
electrical method detecting the instant at which the
mercury reaches one of them. Thus, instead of measuring
the compression produced by a given pressure, we measure
the pressure necessary to produce an assigned compression.
This method was employed by Amagat in his later experi-
ments ( 201, 217), and he says of it elle ne laisse reelle-
mentpresque rien a desirer.
212. Orsted verified Canton's result that the compressi-
bility of water diminishes with rise of temperature, and
suspected that the rate of diminution becomes less as the
temperature is further raised; but he did not obtain
Perkins* result. In fact he states that at any one tempera-
ture the compression is the same, per atmosphere, up to
70 atmospheress.
Orsted, and too many who have followed him, held the
opinion that, if the walls of the piezometer were very
'* Proc. R.S.E., 1884.
N
194 PROPERTIES OF MATTER.
thin, its internal volume would suffer no perceptible
change under equal interior and exterior pressures. That
this (like the somewhat similar notion of Canton) is a
fallacy, we see at once from the consideration of the effect
of hydrostatic pressure on a solid ( 176). If we suppose
the solid to be divided into an infinite number of equal
cubes, these would be changed into equal but smaller
cubes, in consequence of compression. The strained and
the unstrained vessel may therefore be compared to two
vaults of brickwork, similar in every respect as to number
and position of bricks, but such that the bricks in the one
are all less in the same ratio than those in the other.
From this point of view it is clear that the interior content
of the bulb is diminished just as if it had, itself, been a
solid sphere of glass.
Thus the numbers obtained from the piezometer must
all be corrected by adding the compression of glass under
the same pressure.
Another fallacy much akin to this, one which is still
to be found in many books, is the notion that by filling
the bulb of the piezometer partly with glass, partly with
water, and making a second set of experiments, we shall
be able to obtain a second relation between the compressi-
bilities of glass and of water; and that, therefore, we
shall be able to calculate the value of each by piezometer
experiments alone. What we have said above shows
that this process comes merely to using a piezometer
with a smaller internal capacity ; and therefore gives no
new information.
If we had a substance of known compressibility, say for
instance it were incompressible, and were partly to fill the
cavity of the piezometer with this, we should be able to
get the second relation above spoken of.
COMPRESSION OF LIQUIDS. 195
In fact the piezometer gives differences of compressibility
only ; so that, for absolute determinations with it, we
must have one substance whose compressibility is known
by some other method.
When very great pressures are applied, the correction
of the apparent compressibility is not quite so simple. If
e be the true compressibility of the liquid, that of the
piezometer, the ordinary formula is
e = e + m/p
where ra is the fractional diminution of volume. It is
easy to see, however, that the exact relation is
213. Regnault's l apparatus, though managed by a
master-hand, was by no means faultless in principle. For
pressure was applied alternately to the outside and to the
inside of his piezometer, and then simultaneously to both.
There are great objections to the employment of
external or internal pressure alone, at least in such deli-
cate inquiries as these. For, unless a number of almost
unrealisable conditions are satisfied by the apparatus, the
theoretical methods (which must be employed in deducing
the results) are not strictly applicable. They are all
necessarily founded on some such suppositions as that the
bulbs are perfectly cylindrical, or spherical, and that the
thickness of the walls and the elastic coefficients of the
material are exactly the same throughout. These require-
ments can, at best, be only approximately fulfilled ; and
their non-fulfilment may (in consequence of the largeness
of the effects on the apparatus, compared with that on its
contents) entail errors of the same order as the whole com-
pression to be measured. Jamin has tried to avoid this
$e I'Acad. des Sciences, 1847.
196 PROPERTIES OF MATTER.
difficulty by measuring directly the increase of (external)
volume, when a bulb is subjected to internal pressure ;
but, even with this addition to the apparatus, we have still
to trust too much to the accuracy of the assumptions on
which the theoretical calculations are based.
Finding that he could not obtain good results with glass
vessels, Eegnault used spherical bulbs of brass and of
copper. With these he obtained, for the compressibility
of water, the value
0*000048, per atmosphere
for pressures from one to ten atmospheres. The tempera-
ture is, unfortunately, not specially stated.
214. Grassi, 1 working with Regnault's apparatus, made
a number of determinations of compressibility of different
liquids, all for small ranges of pressure.
He verified Canton's specially interesting result, viz.
that water, instead of being (like the other substances,
ether, alcohol, chloroform, etc., on which he experimented)
more compressible at higher temperatures, becomes less
compressible. Here are a few of his numbers.
Texture 0.
0*0 0-0000503
1'5 515
4'0 499
10'8 480
18'0 462
25'0 455
34'5 453
53'0 441
These numbers, when exhibited graphically, show
irregularities too great to be represented by any simple
formula. We now know them to be in many cases very
1 Ann. de Chimie, xxxi., 1851.
COMPRESSION OF LIQUIDS. 197
far from the truth, but we give them, not at all because
they are (even yet) frequently referred to as authoritative,
but because some important reductions of physical data
have been based upon them ; so that it is well to furnish
the means of correcting such reductions.
Grassi assigns, for sea-water at 17'5 C., 0'94 of the
compressibility of pure water, and gives 0*00000295 per
atmosphere as the compressibility of mercury. But he
asserts that alcohol, chloroform, and ether have their
average compressibility, from one to eight or nine atmos-
pheres, at ordinary temperatures, considerably greater
than the compressibility for one atmosphere. As this
result was shown by Amagat to be erroneous, little con-
fidence can be placed in any of Grassi's determinations.
Amagat 1 gave, among others, the following numbers
for ether :
Pressure in Average Compression
Temperature C. Atmospheres. per Atmosphere.
13-7 11 0-000168
13'7 33 0-000152
100 11 0-000560
100 33 0-000474
Thus the diminution of compressibility with increase of
pressure is always considerable, and it is more marked
the higher the temperature.
215. A very complete series of determinations of the
compressibility of water (for a few atmospheres of
pressure only), through the whole range of temperature
from C. to 100 C., has recently been made by Pag-
liani and Yincentini. 2 Unfortunately, in their experi-
ments pressure was applied to the inside only of the
l Ann. de Chimie, 1877.
2 Sulla Compressibititd dei Liquidi, Torino, 1884.
198 PROPERTIES OF MATTER.
piezometer, so that their indicated results have to be
diminished by from 40 to 50 per cent. The effects of
heat on the elasticity of glass are, however, carefully
determined, a matter of absolute necessity when so large
a range of temperature is involved. But in these experi-
ments one datum (the compressibility of water at C.)
has been assumed from Grassi. The results show that
the maximum of compressibility, indicated by Grassi as
lying between C. and 4 C., does not exist. The
following are a few of the numbers, which show a tempera-
ture effect much larger than that obtained by Grassi :
Compressibility
Temperature C. of water.
0'0 0-0000503
2'4 496
15'9 450
49'3 403
61'0 389
66'2 389
77 '4 398
99'2 409
Thus, about 63 C. water appears to have its minimum
compressibility. The existence of a minimum does seem
to be proved, but the remarks above show that its position
on the temperature scale is somewhat uncertain.
216. Tait 1 has given the following determinations of
the average compressibility of cistern water, for pressures
up to 450 atmospheres, and temperature from to 15 C.
The compressibility of the glass of the piezometer was
found by direct experiment ( 232) to be 0'0000026.
The hair-index (, 211) was employed in the piezometer,
so that the results were given as probably somewhat too
small.
1 Phys. Chem. Chall. Exp., vol. ii. part iv., 1888.
COMPRESSION OF LIQUIDS. 199
COMPRESSIBILITY OP CISTERN WATER.
Pressure in
Atmospheres. Avera * e Compressibility
1 to 2 10 ~ 7 (520 -3 '55* + 03 2 )
1 to 153 504 3-60 0'04
1 to 306 490 3'65 0'05
1 to 458 478 370 06
where t is temperature Centigrade.
The experiments were confined to the three last ranges,
so that the data in the first line were obtained by extra-
polation. They agree, however, fairly well with two
isolated results given by Buchanan, 1 viz. :
0-0000516 at 2 '5, and 0'0000483 at 12'5 C.,
and they would have agreed almost precisely with the
results of Pagliani and Vincentini (215) had these ex-
perimenters taken, as their sole datum from Grassi, the
compressibility at 1'5 instead of that at C.
The temperature of minimum compressibility for 1
atmosphere appears to be about 60 C., and is lowered by
increase of pressure.
All the numbers in the above table are fairly repre-
sented by the approximate formula.
Q.Q01863f U P
3Q + P \ 400 10,000,
Here the unit for P is 152 -3 atmospheres, or one ton-
weight per square inch.
[If we take an atmosphere, instead of a ton-weight per
square inch, as our unit ; this gives for the average com-
pressibility of water at C., for the first p atmospheres,
the expression
0-284
5483 +p '
1 Trans. R.S.E., 1880.
200 PROPERTIES OF MATTER.
The constants in this formula were given as probably
too small. Amagat's recent results ( 217) show that
they should each be increased by a little more than 4 per
cent, only.]
The corresponding formula for sea-water is
0-OQ179/ t t* \
38 + PV 150 10, 000 /
The results have been put in the above form for the
sake of comparison with the following expression for the
compressibility, at C., of solutions of common salt,
viz.:
0-00186
36 + s + P
In this formula s represents the mass of salt dissolved
in 100 of water. It appears, from recent experiments,
that a somewhat similar result is given by aqueous solu-
tions of other salts, the addition to the denominator of
the fraction being proportional (not equal) to the mass of
salt in 100 of water. 1
Tait gives the average compressibility of mercury for
pressures up to 450 atmospheres as about 0*0000036.
This is probably a little too small, as Amagat 2 makes it
0-0000039 for the first 50 atmospheres.
217. The final results of Amagat's splendid researches
on the compressibility of liquids at enormous pressures,
which exceed in accuracy as well as in extent all previous
work, have recently been published. 3 The extremely
interesting figure opposite gives some idea of their nature
and importance. It represents the isothermals of water
1 Proc. E.S.E., June 5, 1893. 2 Comptes JRendits, 1889.
8 Ann. de Chimie, 1893.
COMPRESSION OF LIQUIDS. 201
and of sulphuric ether, up to pressures of 3000 atmos-
pheres, and for temperatures from to 50 C.
FIG. 22.
From a figure on so small a scale general notions only
can be derived. But we see clearly through how small
a range of pressures and temperatures the peculiarities
connected with the maximum density point of water re-
202 PROPERTIES OF MATTER.
main sensible. The quasi-hyperbolic form of the iso-
thermals enables us to make approximate estimates of
the utmost compression which these two liquids would
suffer under unlimited pressure. More precise informa-
tion is contained in the following numerical data.
VOLUMES OF WATER AND OF SULPHURIC ETHER UNDER
GREAT PRESSURES.
Water. Sulphuric Ether.
Atmospheres.
0C.
10'l
20 '2
1
1-00000
1 00015
1-0000
1-0320
500
97630
97780
9465
9674
1000
95600
95850
9130
9295
1500
93890
94185
8885
9020
2000
92370
92710
8684
8805
2500
91020
91370
8522
8631
3000
89830
90185
8387
8485
A convenient and fairly close approximation, deduced
from these numbers, gives the following expressions for
the average compressibility of water from 1 to p atmos
pheres :
0-295
At C.
10 C.
5725+2?
0-313
6550+^
This would indicate that water at C. cannot be
reduced to less than about 0'7 of its original volume by
any pressure, however great.
Amagat's data for ether give the corresponding for-
mulae.
20 --
2086 +p
COMPRESSION OF LIQUIDS. 203
It is observed that formulae like these, if they agree
exactly with the data for extreme and mid-range of
pressures, err slightly in defect for pressures lower than
the average, and in excess for those higher ( 260).
The apparatus which gave the magnificent results tabu-
lated above was, of course, specially adapted to the effects
of extreme pressures, and was therefore not qualified to
give very precise values for moderate pressures.
218. From the results of Andrews already given ( 205)
we find the following roughly approximate values of the
COMPRESSIBILITY OF LIQUID CARBONIC ACID AT 13-1 C.
Pressure in True Compressibility
Atmospheres. per Atmosphere.
50 0-0059
60 0-00174
70 0-00096
80 0-00066
90 0-00044
showing very great, but very rapidly decreasing, com-
pressibility. As already explained, Andrews has pointed
out that part of this, especially for the lower pressures
in the table, is due to the trace of air which, in spite of
every precaution, was associated with the carbonic acid.
219. It has long been known that, when the Torricellian
experiment is performed, the mercury will sometimes not
descend until the tube is sharply tapped, even if the
column be 4 or 5 feet in length, instead of being little
more than the 30 inches which are supported by the
atmospheric pressure. In such a case the portion of the
column which stands above the barometric height must
be in a state of hydrostatic tension. And, as in the case
of solids, ( 177) we conclude that its volume is increased
204 PROPERTIES OF MATTER.
to the same extent as it would have been diminished by
an equal hydrostatic pressure.
A very interesting experiment bearing on this subject
was made by Berthelot. 1 A strong glass tube, sealed at
one end and drawn out very fine at the other, was filled
to a definite mark with water. By immersing the whole
in warm water the contents were made to expand nearly
to the point, which was then hermetically sealed. A
very slight additional heating, slowly and cautiously
applied, caused the water in time to dissolve the small
remaining bubble of air, so that the tube was absolutely
full of liquid. When slowly cooled to its original temper-
ature it remained full of water. By the help of the mark
(checked if necessary by calculation from the temperature
of the warm water) the increase of volume could be
estimated, and thence the tension to which the water
was exposed. In this way pure water was found capable
of bearing some fifty atmospheres of tension, while eau
sucree bore nearly one hundred. It is clear that the
adhesion of the water to the glass is an indispensable
circumstance in this experiment. And as the equilibrium
is essentially unstable, throughout the whole contents, it
is remarkable that so large an effect can be obtained :
though, of course, it is far below what might (theoretically
at least) be supposed possible.
1 Ann. de Chimie, xxx. 232 ; 1850.
CHAPTER XI.
COMPRESSIBILITY AND RIGIDITY OP SOLIDS.
220. IN the two preceding chapters we had to deal with
bodies practically homogeneous (except in the special case
of vapour in presence of liquid) and perfectly isotropic ;
bodies, moreover, which are devoid of elasticity of form,
while possessing perfect elasticity of volume. Hence the
determination of (apparent) compressibility for any definite
substance of these kinds depended for its accuracy solely
on the care and skill of the experimenter, and on the
adequacy of the process and the apparatus employed.
When we deal with solids the circumstances are very
different. It is rarely the case that we meet with a solid
which is more than approximately homogeneous. Some
natural crystals, such as fluor spar, Iceland spar, etc., are
probably very nearly homogeneous ; so are metals such as
gold, silver, lead, etc., when melted and allowed to cool
very slowly. To produce homogeneous glass (especially
in large discs, for the object-glasses of achromatic tele-
scopes) is one of the most difficult of practical problems.
On the other hand, crystalline bodies are essentially non-
isotropic ; so is every substance, crystalline or not, which
shows " cleavage."
And further, very small traces of admixture or impurity
often produce large effects on the elastic, as well as on
the thermal and electric, qualities of a solid body. Think,
for instance, of the differences between various kinds of
iron and steel, or of the purposely added impurities in
206 PROPERTIES OF MATTER.
the gold and silver used for coinage. Very slight changes,
in the manipulation by which wires or rods are drawn
from the same material, may make large differences in
their final state : differences by no means entirely to be
.got rid of by heating and annealing, etc. The whole
question of " temper " is still in a purely empirical state.
Besides, we must remember that every solid has its limits
of elasticity, to which attention must be carefully paid.
Thus we can give only general or average statements as
to the amount of compressibility or rigidity of any solid,
in spite of the labour which Wertheim and many others
have bestowed on the subject.
221. In an elementary work we cannot deal, even
partially, with the elastic properties of non-isotropic
bodies. The necessary mathematical basis of the investi-
gation, though it has been marvellously simplified, is
quite beyond any but advanced students. And the
experimental study of the problem has been carried out
for isolated cases only. Hence we limit ourselves, except
in a few special instances, to the consideration of homo-
geneous, isotropic, solids.
On the other hand, the compression or distortion
produced in a solid by any ordinary stress is usually very
small. This consideration tends to simplify our work ; for,
as a rule, small distortions maybe regarded as strictly super-
posable. Thus we may calculate, independently, the effects
of each of the simple stresses to which a solid is subjected.
Our warrant for this must of course be obtained
experimentally. It was first given by Hooke.
In 1676 1 he published the following as one of "a
decimate of the centesme of the Inventions, etc."
1 A Description of Helioscopes, &e. t made by Robert HooTce,
Postscript, p. 31.
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 207
" 3. The trite TJieory of Elasticity or Springiness, and a
particular Explication thereof in several Subjects in which
it is to be found : And the way of computing the velocity
of Bodies moved by them, ceiiinossstttm."
The key to this anagram was given by Hooke himself
in 1678, 1 in the words :
" About two years since I printed this Theory in an
Anagram at the end of my Book of the descriptions of
Helioscopes, viz. ceiiinosssttuu, id est, Ut tensio sic vis ;
That is, The Power of any Spring is in the same propor-
tion with the tension thereof .- That is, if one power
stretch or bend it one space, two will bend it two, and three
will bend it three, and so forward. Now as the Theory
is very short, so the way of trying it is very easie."
He then shows how to prove the law in various ways :
with a spiral spring drawn out ; a watch spring made
to coil or uncoil ; a long wire suspended vertically and
stretched ; and a wooden beam fixed (at one end) in a
horizontal position, and loaded.
The above extracts sufficiently show in what sense
Hooke intended the words Tensio and Vis to be under-
stood : and his law is now usually stated in the (some-
what amplified) form, called HOOKE'S LAW,
Distortion is proportional to the distorting Fon-'
or, still more definitely,
Strain is proportional to Stress.
In the latter form we have made anticipatory use of it in
Chap. VIII. and elsewhere.
1 Lectures de Potentia Restitutiva, or of Spring, p. [1], This is
a very curious pamphlet, containing some remarkably close antici-
pations of modern theories, especially Synchronism and its results,
and the Kinetic Theory of Gases. The first is foreign to our
present subject, the second will be considered later ( 322).
208 PROPERTIES OF MATTER.
222. A very general proof of the accuracy of this law
is easily to be obtained in the case of bodies which can
be made to produce a musical sound : a tuning-fork, for
instance. For, if the pitch of the note (i.e. the number
of vibrations per second) do not alter as the sound grows
fainter, the vibrations must be isochronous, and the
elastic resilience therefore proportional to the distortion.
(See 72.)
223. The ordinary experimental illustrations of Hooke's
Law are given, very much as he originally gave them,
by:-
1. A rod or wire, fixed vertically and stretched by
appended weights ; or a rod or column compressed by
weights laid on its upper end.
2. A wire stretched horizontally and extended by
weights suspended at its middle point.
3. A bar or plank fixed horizontally at one end and
loaded with weights at the other.
4. A plank with its ends resting on trestles and loaded
at the middle.
5. A spiral spring, forming a helix of small step,
compressed or extended by weights.
6. A wire or rod, fixed at one end and twisted at the
other.
The mere mention of these methods is sufficient, with-
out further illustration, to suggest the means by which
the requisite measurements can be carried out. They
will be considered in detail, but not in the above order.
In all these cases experiment shows that (within
certain limits, which will be afterwards discussed) the
distortion is proportional to the distorting force.
1 and 2 are mere varieties of one experiment. The
same may be said of 3 and 4, which are examples of a
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 209
somewhat more complex form. And 5 and 6, though at
first sight very unlike, are practically one problem.
Besides, they are of a simpler character than either of
the other pairs, for they involve the coefficient of rigidity
alone ; the others involve both coefficients. But 1 and
2, on the other hand, are simpler than the rest, on a
different account, viz. that they involve homogeneous strain.
224. Young's Modulus, as it is called, is determined
from the stretching of a rod or wire by appended weights.
As defined by Young, its measure is the ratio, of the
simple stress required to produce a small shortening or
elongation of a rod of unit section, to the fractional
change of length produced. Its value is expressed, as
we see by 181, in terms of the rigidity and the resist-
ance to compression, by the formula
Qkn
For bodies like india-rubber, in which 7c is large in
comparison with n, its value is nearly 3n. Hence in
pulling out an india-rubber band the change is almost
entirely one of form (see again, 181) and therefore
the area of a cross section is diminished in nearly the
same proportion as that in which the band is lengthened.
A piece of good cork suggests, though it does not
realise, the conception of a solid in which n shall be very
large in comparison with k ; and for such a body Young's
modulus would be nearly 9/c. Since n is very large,
there is little change of form ; so that traction or pressure,
in any direction, would expand or contract a body of this
kind nearly equally in all directions. In cork the effect
is confined mainly to the dimension operated on.
From such considerations we see that Young's modulus,
though comparatively easy of measurement, is not the
210 PROPERTIES OF MATTER.
simple quantity which it at first appears to be ; and that,
in fact, it may have the same numerical value in each of
two hodies which differ widely from one another, alike
in rigidity and in compressibility.
225. The following table gives approximate values
( 166) of Young's modulus for some common materials;
the unit being 10 7 grammes' weight per square centi-
met-ve :
Qkn
Young's modulus, ^ . Tenacity.
Gold 86 0-27
Silver . . 76 . . 0'3
Copper (hard) .125 . . 0'4
Copper (annealed) 110 . . 0'8
Iron . . .180 . . 0'6
Steel . .240 . . 0'8
Oak 10 0-1
Teak . . 17 . . 0-1
Fir ... 12 .. 0-07
Glass . 40 to 60 . . 0'06
To convert these numbers (as they stand in the table)
into the common reckoning of pounds' weight per square
inch, it suffices to multiply them by about 142,000 instead
of by 10 7 . To convert to C. G. S. units, i.e. dynes per
square centimetre, multiply by 9 '81 x 10.
226. A second column (in terms of the same units)
has been added to the above table, to give an indication
of the Tenacity of each of the materials specified. This
means the utmost longitudinal stress which (when
cautiously applied) a rod or wire can endure without
rupture. It has no direct connection with Young's
modulus, nor with either of the coefficients of elasticity,
for a substance has usually to be strained far beyond its
limits of elasticity before rupture takes place, and the
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 211
dimensions of the cross section are also much reduced.
The uncertainty of the amount of this quantity, even in
different specimens taken from the same piece of matter,
leads to our giving it usually to one significant figure
only.
227. Young's treatment of the subject of elasticity is
one of the few really imperfect portions of his great
work. 1 He gives the values of his modulus for water,
mercury, air, etc. ! It is not easy to understand what
he really meant by speaking of " the " modulus of
elasticity : unless, as Lord Rayleigh suggests, he meant
that which (whatever be, in each case, its real nature)
is involved in ordinary sound waves, whether in air or
along wires. Young's modulus is, no doubt, a quantity
of great value in practical engineering : in many cases
the only elastic datum required. Yet he speaks of
rigidity, etc., in a way which is scarcely compatible with
the idea of one modulus only. But the subject was in a
state of great confusion till long after his time, mainly
in consequence of an unwarranted conclusion (deduced
by Navier and Poisson from a species of molecular
theory) that there is a necessary numerical ratio between
rigidity and resistance to compression. In fact, what
was called Poisson's ratio, that of the lateral shrinking,
to the longitudinal extension, of a bar or rod under
tension, was supposed to be necessarily equal to 1/4.
This gives ( 180)p = 4g, or 3& = 5ra.
The erroneousness of this conclusion was first pointed
out by Sir G. G. Stokes, 2 and his paper has put the whole
subject in a new and clear light. We have already given,
in 224 above, some of his illustrations, which show
- 1 Lectures on Natural Philosophy, 1807.
2 Camb. Phil. Trans., 1845.
212 PROPERTIES OF MATTER.
that there is no necessary ratio, or even relation, between
n and k.
De St Venant 1 has given complete solutions of a
number of interesting cases, such as the torsion of prisms
of different forms of cross-section, many of which are
very valuable in practical applications. Lord Kelvin, 2
besides giving the theory with extreme generality, has
also specially developed the application of Thermo-
dynamics 3 to the subject.
In spite of Stokes' exposure of the inaccuracy of the
so-called Vni-constant Theory, it has still determined
partisans. These may profitably consult the following
data, given by Amagat ; 4 though we quote them for their
intrinsic value, not for the purpose of further " slaying
the slain."
ELASTIC CONSTANTS (MEAN VALUES) AT 12 C.
Poisson's
Compressibility
Young's
Ratio.
per Atmosphere.
Modulus.
Glass 0-245
0-00000220
6,775
Steel 0-268
68
20,395
Copper 0-327
86
12,145
Brass 0'327
95
10,851
Lead 0'428
276
1,556
The unit for Young's modulus, which was determined
directly, is a kilogramme weight per square millimetre,
so that the numbers in the last column must be divided
by 100, to reduce them to the unit employed in the
table of 225.
The numbers in the first two columns are the means
of closely accordant results derived, one set from the
1 Mim. des Savans Strangers, 1855. See also Thomson and
Tait's Nat. Phil., 699, etc. 2 Phil. Trans., 1854.
3 Quarterly Math. Journal, 1855. 4 Comptes Rendus, 1889.
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 213
change of contents of a cylinder under longitudinal
traction ( 181), the other from the similar change under
external pressure alone ( 183). Along with each of
these data the value of Young's modulus, as given in the
last column, was employed.
228. We will now consider the pure Torsion of a
cylindrical rod or wire, as employed, for instance, in the
Cavendish experiment ( 153).
This is a very simple problem if the cylinder be truly
circular, and of perfectly homogeneous isotropic material.
For it is clear from what follows that equal and opposite
twisting couples, applied at its ends, will simply make
successive transverse slices, of equal thickness, rotate
about the axis each by the same amount less than the one
before it.
The length of the cylinder cannot increase under
torsion, for a reversal of the couples (which is practically
the same arrangement) would shorten it ( 177), and
vice versa. Neither can its radius change, for exactly
the same reason. Nor can a transverse section become
curved at any part. Thus the volume remains unchanged,
and therefore the coefficient of rigidity alone is involved.
Consider a thin annular portion of the cylinder bounded
by transverse sections at a very small distance, t, from
one another, and by concentric
cylinders of radii r, and r + 1. We
may subdivide this into cubes, of
side t, by planes through the axis,
making angles tjr with one another.
Let <p be the twist per unit
length of the cylinder, t(p is the
angle by which one of the parallel
sections has rotated relatively to the other, and r.t^/t, or
214 PROPERTIES OF MATTER.
r<p, is the change of angle in each of the little cubes.
Hence, if P be the tangential force per unit of area( 178)
P=nr<p.
The moment, about the axis, of the tangential force on
the cube is therefore
[Note here, for an ulterior purpose ( 235), that r 2 ^ 2 is
the moment of inertia ( 132) of the area of the face of
the cube about the axis.]
But the number of cubes is 27rr/t, so that the whole
moment is SJuwtyp.
This is the couple required to twist a circular cylinder
of radius r, and very small thickness t, through the angle
(p per unit of length.
To find the result for a solid cylinder of radius K, we
must put dr for t, and integrate. The result is
Hence the twist produced, per unit of length in a
cylinder, is directly as the twisting couple ; inversely as
the rigidity and as the fourth power of the radius.
229. This suggests an obvious and direct experimental
process for determining the rigidity of homogeneous
isotropic substances. There are two difficulties, of a
formidable character, in the way of its application : first,
the obtaining a homogeneous isotropic material, and
secondly, the making it into a circular cylinder. It is
clear that very small irregularities of form, or errors in
the estimate of the radius, may give rise to large errors
in the calculation of the rigidity, since the fourth power
of the radius is directly involved in the calculations.
And it is probable that the mode of manufacture of the
COMPRESSIBILITY AND RIGIDITY Ott SOLIDS. 215
cylinder (especially if it be drawn, 49) may render its
otherwise isotropic material markedly non - isotropic.
Hence the following numbers are given as mere approxi-
mations. The unit is, again, 10 7 grammes' weight per
square centimetre ( 225).
APPROXIMATE RIGIDITY (n).
Glass
Brass
Iron (wrought)
Iron (cast)
Steel
Copper
15 to 25
35
79
55
85
45 to 50
These values are for ordinary temperatures. As the tem-
perature is raised, the rigidity is found steadily to diminish.
230. When a spiral spring is drawn out, it is pretty
clear to every one that there is unbending, for the curv-
ature becomes less as the helix is lengthened. And
Fio/25.
the following simple experiment shows that this flexure
is accompanied by torsion. Coil up a strip of sheet
216 PROPERTIES OF MATTER.
india-rubber, as in the cut, and pull out the inner end.
It assumes the form sketched. The portion pulled out
straight is twisted merely ; the coiled part is merely
bent ; the intermediate portion is partly bent and partly
twisted. Every coil pulled out gives one complete turn
of twist. If we make kinks on the strip, as in fig. 25,
then, on pulling out the first, we find two complete turns
of twist, but on pulling out the second there is no twist,
one of the kinks giving a right-handed, the other a left-
handed, complete turn of twist. 1
When the spring is very flat, i.e. has a very small
step, the principal effect of a moderate extension is mere
torsion ; and the investigation is of a character precisely
the same as that in the preceding section. The some-
what more complex combination, of torsion and flexure
simultaneously, will be adverted to later. ( 237.)
231. Theoretically speaking, we can of course deduce
the resistance to compression from the (known) values of
the rigidity and of Young's modulus ; and it is in this way
that most of the data connected with the subject have
been obtained. But especially in cases where Young's
modulus is not very far from threefold the rigidity (as,
for instance, in india-rubber), the inevitable errors in the
determination of these might lead to enormously greater
errors in the calculated value of k.
The method which was incidentally employed by
Regnault, in his measurements of the compressibility
of liquids, consisted in applying pressure externally,
internally, and externally and internally, to a species
of piezometer containing water. The results of 183
show that (supposing it cylindrical, and unit pressure
applied) its internal volume must have been altered, in
1 Knots. Trans. R.S.E., 1877.
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 217
these three cases respectively, by the following fractions
of its whole amount:
The algebraic sum of the two first is of course equal
to the third. But the quantities measured in the two
latter cases were both less than those stated above by the
fractional change of volume of the water. The relation,
therefore, still holds, and furnishes a test of the accuracy
of the experiments. But it reduces the number of inde-
pendent equations to two, from which there are three
coefficients of elasticity to be determined. Hence
Regnault also had to fall back on the employment of
Young's modulus.
An interesting illustration of the above statements is
furnished by an experiment of Forbes. He replaced, by
an india-rubber bottle, the bulb of a piezometer. In such
an instrument the apparent compressibility of water was
found to be barely sensible.
232. Probably the best, certainly the most direct,
method is that adopted by Buchanan, 1 in which the
length of a rod is very carefully measured while it is
under hydrostatic pressure, and also while free. The
linear contraction so determined is numerically (if the
material be homogeneous and isotropic) one-third of the
compression ( 176). Unfortunately Buchanan's pub-
lished measures are confined to one particular kind of
glass. The special merit of his method is that, provided
1 Trans. E.S.K, 1880.
218 PROPERTIES OF MATTER.
the rod be of isotropic material, the regularity of its cross
section is of no consequence.
Thus we can give for this property also only a few
roughly approximate numbers. They are given in the
same units as the preceding.
APPROXIMATE RESISTANCE TO COMPRESSION (k).
Glass 20 to 40
Copper v 160
Iron (wrought) . . . , 150
Steel 185 to 200
It is greatly to be desired that more, and more accurate,
data should be obtained in this matter : though, as is
evident from 219, the problem is one of very great
uncertainty as well as difficulty. Difficulty incites rather
than repels a true experimenter, but uncertainty is
paralysing.
233. Though, as we have seen, we can give only
general and somewhat vague numerical data, there is
practical unanimity on the part of experimenters that,
within the limits of elasticity, Hooke's law is very closely
followed. Hence, although it is necessary to measure
the elastic coefficients for each specimen of each sub-
stance we employ, once that measurement is effected we
can trust to it as giving the special qualities of the
material through a range of stress which, in glass, steel,
etc., is often fairly wide. One excellent example is to
be found in the substitution of glass or steel for air or
nitrogen in the construction of instruments for measuring
hydrostatic pressure.
The first to introduce this principle seems to have been
Parrot, 1 whose ]<!laterometre was merely an ordinary
1 "Experiences de forte Compression sur Divers Corps," M6m.
ie I' Academic Imperiale des Sciences de St Petersbourg, 1833.
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 219
thermometer, with a bulb thick enough to stand great
pressure. Keeping it immersed in water at a constant
temperature, and applying great pressures, he found that
the diminution of capacity of the bulb was almost exactly
proportional to the pressure.
Instruments working on the same general principle have
since been introduced, in ignorance of Parrot's work, by
many investigators. Bourdon gauges, aneroid baro-
meters, etc., are merely special though rather complex
instances.
Sudden application of pressure produces temperature-
changes which affect especially the volume of the liquid
contents by means of which the distortion is usually
measured. But these instruments (in Parrot's form at
least) may be made practically insensible to such changes
by the simple expedient of nearly filling the bulb (which,
for this purpose, should be cylindrical) with a piece of
glass tube closed at each end. 1 The mercury in the bulb
is thus greatly reduced in quantity, and therefore the
temperature effects in the stem are very small, while the
instrument is still as sensitive as ever in indicating
changes of volume.
The dimensions and thickness of such an instrument,
for any special purpose, can be easily calculated from the
formulas of 183 ; and the unit of pressure can be deter-
mined for it, by a single comparative experiment, with
the aid of Amagat's table of compression of air ( 200).
There is great advantage in using simultaneously two
instruments of this kind, in one of which the thickness is
considerably greater (in comparison with the diameter)
than in the other. For, so long as their indications
1 Tait, Report oil the Pressure Errors of the Challenger Ther-
mometers, 1881.
220 PROPERTIES OF MATTER.
agree, loth may be trusted as following Hooke's law
very accurately.
234. The limit of pressure measurable by means of
these instruments depends upon the resistance of a glass
or steel tube to crushing by external pressure. From a
series of experiments, made for the purpose, 1 Tait has
calculated that ordinary lead glass (in the form of a tube
closed at each end) gives way when the distortion of
the interior layer amounts to a shear of about 1 -g-^,
coupled with a compression of about -^. Hence even
a very thick tube of such glass cannot resist more than
about 14 tons' weight per square inch (2130 atmospheres)
of external pressure. No corresponding experiments
seem yet to have been made for steel.
235. We now come to the case of bending of a rod or
bar. Here we have no such simple problem as in the
case of the torsion of a cylinder, and must consequently
assume the solution as given by mathematical investiga-
tion ; based, of course, on the principles already explained.
This shows us that, so long as the radius of curvature is
large in comparison with the thickness of the bar in the
plane of bending, the line passing through the centre of
inertia of each transverse section, the elastic central line
as it is called, is bent merely, and not extended nor
shortened.
The flexural rigidity of the bar, in any plane through
the central line, is directly as the couple, in that plane,
which is required to produce a given amount of curvature
in the central line. Its amount may easily be calculated
by means of the following considerations. Let the figure
represent a transverse section of the cylinder, C its centre
of inertia, CD a line in it perpendicular to the plane of
1 Proc. K.S.E., April 18, 1881.
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 221
bending, and let the centre of curvature of the bending
lie towards E Then obviously all lines parallel to the
axis of the bar on the
E-ward side of CD are com-
pressed, all towards the
other side extended ; each
in proportion to its distance
from CD and to the curva-
ture. If we contemplate a
transverse slice, of small
thickness t, we see that its
thickness remains unchanged FIG. 26.
along CD, is diminished on the E-ward side of that line,
and increased on the other. The thickness at the small
area A becomes t (1 + \ where r is the radius of
bending. This requires a tension A m, where m is
Young's modulus. The moment of this about CD is
A.AB 2 . .
r
Hence the sum of all such, i.e. the moment of the
bending couple, is multiplied by the moment of inertia
of the area of the section about CD. (The reversed
strain, in the compressed parts, requires the reversed
stress. 177.) Now through C in the plane of the
section, there are two principal axes of inertia of the
area, in directions at right angles to one another.
Hence, except in the cases of "Kinetic Symmetry" of
the section (as when it is circular, square, equilateral-
triangular, etc.), there are two principal flexural rigidities,
a maximum and a minimum, in planes (through the axis)
222 PROPERTIES OF MATTER.
perpendicular to one another. If the rigidities in these
planes be called Rj and R 2 , the flexural rigidity in a
plane (through the central line) inclined at an angle
to that of R x is
[Compare 228, in which the corresponding case of
torsion-rigidity was shown to depend upon the moment
of inertia of the area of the section about the elastic
central line. This is the third principal axis of the trans-
verse sectional area at its centre of inertia.]
236. It appears from last section that flexure (within
moderate limits) is, practically, as regards any very small
portion of the substance, the same thing as longitudinal
extension or compression, and thus cannot give us any
simple information as to the elastic coefficients of the
substance. But it has very important practical appli-
cations, and therefore we devote some sections to the more
common cases.
The principal moments of inertia of the area of a rect-
angle, sides 2a and 2fr, about axes through its centre and
parallel to pairs of sides, are 4a 3 &/3 and 4a& 3 /3. Multiplied
by m, they represent the flexural rigidities of a plank in
planes parallel to its broader, and to its narrower faces
respectively. These rigidities, multiplied by the bending
curvature, give the couple required to produce and to
maintain the flexure.
237. The Elastic Curve of James Bernouli, celebrated
in the early days of the differential calculus, is a parti-
cular case of the bending of a wire or plank, in which
the flexural rigidity in the plane of bending is the same
throughout, and a simple stress ( 128) alone is applied.
The obvious condition is that the curvature at each
COMPKESSIBILTTY AND RIGIDITY OF SOLIDS. 223
point is directly proportional to the distance from the
line in which the stress acts. For the investigation of
the equation of the curve from this condition, and for
drawings of its various forms, the reader must be referred
Fro. 27.
to works on Abstract Dynamics ; l but we figure here the
special case which corresponds to a stretched uniform
wire, of infinite length, with a single kink upon it. This
will be referred to in 289 below.
The investigation of the bending of planks, variously
supported, and under various loads, is a somewhat
generalised form of the question of the elastic curve.
The principles involved in its solution are simple, and
almost obvious ; but the mathematical treatment of it
would lead us too much out of our course. So would
that of the problem of the effect of a couple applied
anyhow to one end of a cylindrical or prismatic wire, of
any form of section, the other end being fixed. The
wire, in such a case, takes generally the form of a circular
helix : involving both flexure and twist. The extreme
particular cases are (1) when the wire is in the plane of
the couple, and there is bending only; (2) when the wire
is perpendicular to the plane of the couple, and there is
twist only.
1 See, for instance, Thomson and Tail's Nat. Phil. , vol. i. part
ii. p. 148.
224 PROPERTIES OF MATTER.
238. The results hitherto given are all approximate
only, and depend upon the radius of bending being large
compared with the thickness of the wire or bar in the
plane of flexure. Those given in 228, for torsion, may
be applied, under a similar restriction, to cases in which
the section of the wire or bar is not circular. The
mathematical treatment of the exact solution of such
problems is of too high an order of difficulty for the
present work ; but some of its results, alike interesting
and important, may be easily understood. A few of
them will now be given, but the reader must be referred
to the works already cited ( 227) for a more complete
account.
239. Thus in the flexure of a uniform bar into a
circular arc, we saw ( 235) that each fibre is extended
or compressed to an amount depending on its original
distance from the plane which passed through the centres
of inertia of its transverse sections (while it was straight),
and perpendicular to the plane of bending. But this
involves ( 177) compression or extension of the trans-
verse section of the fibre, uniform in all directions, and
to an amount proportional to the extension or shortening
;>f its length. Hence, if the section of the unbent bar
be divided into equal indefinitely small squares, each of
these will remain a square after bending. From this we
can obtain an approximate idea of the change of shape of
the transverse section.
Consider the annexed figure, which represents parts of
a series of concentric circles, whose radii increase in a
slow geometrical ratio, intersected by radii making with
one another equal angles such that the arcs into which
any one circle is divided are equal to the difference
between its radius and that of the succeeding circle.
COMPRESSIBILITY AND KIGIDITY OF SOLIDS. 226
When the circles and radii are infinitely numerous, all
the little intercepted areas are squares. The sides of
the squares along CD are obviously
greater than those of the squares
along AB by quantities proportional
to AC. Those of the squares along
EF are less than those of the squares
along AB by quantities proporcional
to AE. The figure CDFE must
therefore represent the distorted form
of the cross section of a beam, origin-
ally rectangular, and bent in a plane
through OG, and perpendicular to
the plane of the figure. The side of
the beam which is concave in the
plane of flexure is convex in a
direction perpendicular to the plane
of flexure ; that which is convex in
the former plane is concave in the
latter. The cause is, of course, the transverse swelling
of the fibres on the side towards G, the centre of bending,
and the diminution of section of those on the other side
of the bar. It is sufficiently accurate to assume that AB,
which is unchanged in length, was originally midway
between the faces of the bar.
If OG be the radius of flexure, the ratio of the exten-
sion of one of the fibres which pass through a point of
EF to its original length is AE/OG. Its lateral con-
traction in all directions must therefore be ( 180)
But it is obviously AE/OH. Hence
2 ' 3& + n)OG = (3 A - 2n)OE.
226
PROPERTIES OF MATTER.
Thus the point H is determined, and the approximate
solution is complete. A square bar of vulcanised india-
rubber shows very clearly the characteristics of this
strain.
240. In the case of torsion of a cylinder whose section
is not circular, plane transverse sections do not remain
plane. The following figure gives de St Yenant's result
for an elliptic cylinder. It represents the contour lines
of the distorted section made by planes perpendicular
to the axis. They are equilateral hyperbolas (as in 88),
the common asymptotes being the axes of the section
FIG. 29.
The torsion is applied in the positive direction to the end
of the cylinder above the paper ; and the full lines
represent distortion upwards ; the dotted, downwards.
241. Coulomb, who first attacked the torsion problem,
was led (by an indirect and unsatisfactory process) to the
result above ( 228), viz. that the torsional rigidity is pro-
portional to the moment of inertia of the area of the
transverse section about the elastic central line. This is
true only in circular cylinders or wires. It gives too
large a value for all other forms of section. From de St
Venant's paper we extract the following data. The first
numbers express the ratio of the true torsional rigidity to
the estimate by Coulomb's rule. The second numbers
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 227
show the ratio of the torsional rigidity to that of a
cylinder, of the same sectional area, but circular.
Equilateral Triangle. Square.
0-600 0-843
0725 0-883
The torsional rigidity of an elliptic cylinder, a and b
being the semi-axes of the transverse section, is
When b = a we have, of course (as in 228),
242. From these and like results we are led to see that
projecting flanges, which add greatly to the flexural
rigidity of a rail or girder, are practically of no use as
regards resistance to torsion.
Another of de St Tenant's important results is that the
places of greatest distortion in twisted prisms are the
parts of the boundary nearest to the axis.
Near a re-entrant angle in the boundary of the section
the stress and strain are usually very much greater than
at any other part, whether the stress be such as to pro-
duce torsion or bending. Hence the reason for the very
important practical rule of always rounding off such angles,
when they cannot be entirely dispensed with.
243. Still keeping to statical experiments, we have to
consider briefly the limits of elasticity.
When a solid is strained beyond a certain amount,
which depends not merely on its material but upon its
state and the mode of its preparation, one of two things
228 PROPERTIES OF MATTER.
occurs. Either, it breaks, and is said to be brittle, or it
becomes permanently distorted, and is said to ~b& plastic.
Different kinds of steel, or the same steel differently
tempered, give excellent instances. Some have qualities
superior to those of the best iron, others are more brittle
than glass. Even in the case of one definite material,
the results differ widely from one another according to
the duration of the stress, and also according to the way
in which it is applied : i.e. by successive large or small
instalments, and with longer or shorter intervals between
the applications.
244. When a body has been permanently distorted,
as, for instance, a copper wire which has received a few
hundred twists per foot, it has new limits of elasticity
(within which Hooke's law again holds, though with altered
coefficients) ; but the elasticity, at all events for distortions
of the same kind, is usually of a very curious character,
inasmuch as the strain produced by a stress will, in gen-
eral, no longer be exactly reversed by reversal of the
stress. In fact the body has been rendered non-isotropic ;
and, so far as this problem has yet been treated (though
that does not amount to much), it is of the order of
questions which we cannot enter on in this volume.
The limits of elasticity vary so much, even in different
specimens of the same material, that no numbers need
here be given. Every one who has occasion to take
account of these limits must determine them for himself
on the materials he is about to employ.
245. A curious fact, showing that elasticity may re-
main dormant, as it were, is exhibited by sheet india-
rubber. When it has been wound in strips, under great
tension, on a stout copper wire, and has been left in that
condition for years, it appears to harden in its state of
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 229
strain, and can be peeled off like a piece of unstretched
gutta-percha. But, if it be placed in hot water, it almost
instantly springs back to its original dimensions. The
experiment may be made, but with less perfect results,
in a few minutes, by merely putting the strained india-
rubber into a mixture of snow and salt.
246. Excellent instances, illustrative of the possibility
FIG. 36.
FIG. 31.
of arrangements giving peculiar kinds of non-isotropy, are
furnished by many manufactured articles, such as woollen
or linen cloth, wire-gauze, etc., in which Young's modulus
is large for strips cut parallel to the warp or woof, but
small for strips cut diagonally. Still more curious is a
special kind of wire-gauze in which the meshes are rhombic.
Another suggestive instance is a strip formed of wire
knotted as in Fig. 30, in which the flexure and torsion
rigidities for any bending or twist, and its reverse, are in
general markedly different. Similarly a shirt-of-mail
made of rings, each three joined as in the first figure (31
above), is perfectly flexible ; as in the second figure,
nearly rigid. 1
1 On Knots. Trans. E.S.E., 1877.
230 PROPERTIES OF MATTER.
247. Kinetic processes for determining coefficients of
elasticity are often based upon the pitch of the note given
out by a vibrating body. We do not give any of these,
as they belong properly to the subject of SOUND. All
require an exact determination of pitch, and (except in
the very simplest case, that of stretched wires, as those
of a pianoforte) require, for their comparison with the
other experimental data, higher mathematics than we can
introduce here.
248. There is, however, one kinetic process of a very
simple character (we have already adverted to it while
describing the Cavendish experiment, 153) by which the
rigidity of a substance is determined from torsional
vibrations.
The wire to be experimented on is firmly fixed at its
upper end, and supports a mass whose weight is sufficient
to render it straight, but not so great as to produce any
sensible effect on its rigidity. The moment of inertia of
this mass may be caused to have any desired value by
making the whole into a transverse slice of a hollow
circular cylinder of sufficient radius, which can be very
accurately turned and centred on a lathe. The wire
must be attached to the middle of a light cross bar, so as
to lie in the axis of this cylindrical vibrator.
If JS" represent the torsional rigidity of the wire, I its
length, and <p the angle through which the vibrator has
been turned, the elastic couple is
The rate at which work is done against the elastic forces
is
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 231
But this must be equal to the rate at which the appended
mass loses energy of rotation, i.e. ( 135)
-Ip
if I be its moment of inertia. Hence
This shows ( 72) that the oscillations are of the simple
harmonic character, and that the period is
27T
/B
V IT*
or, if the wire be of circular section ( 228),
'Sirll
/Sirll
VnB*'
In this expression all the factors are known, with the
exception of n, which can therefore be determined.
The chief difficulties in the application of this process
are the finding exactly the radius of the wire, and the
ensuring that its substance is really isotropic.
249. The solution just given is accurate only if all the
circumstances have been taken into account. But a very
few trials, with wires of different metals, show that the
range of vibration diminishes at every oscillation, and
with some metals much more rapidly than with others.
This cannot, therefore, be wholly due to the resistance of
the air. Part of it, at least, is undoubtedly due to the
dissipation of energy, by thermal effects of change of form,
which occur even when the elasticity is perfect. This,
however, is beyond our province, as defined in 175.
But a large part, with metals like zinc much the greater
part, is due to internal viscosity. [The manifestation of
viscosity may be regarded, as in Kelvin's view, not
necessarily as implying that resistance is proportional to
232 PROPERTIES OF MATTER.
speed, but as implying that resistance is some function of
the speed. On the Maxwellian view ( 253), energy must
be lost when molecular groups break down under stress
and rearrange themselves, the energy being dissipated
as heat. W. P.]
250. So long as we deal with steel, iron, silver, etc.,
and keep to torsions well within the limits of elasticity,
the arc of oscillation is found to diminish in simple
geometrical progression. This points to a resistance to
the motion, partly due to air acting on the suspended
mass, partly to thermal effects and to viscosity in the wire
itself, but, on the whole, proportional to the rate of
motion, i.e. the rate of distortion.
Thus the equation of 248 takes the form
The solution of the problem in this case is, therefore,
of the nature of that given in 74 above ; and we see
that, if the diminution of the arc of oscillation (per
vibration) is large, the periodic time will be perceptibly
increased. Thus the direct determination of n, by the
mode of calculation given in 248, would necessarily lead
to underestimation of its value.
The logarithmic decrement of the arc of vibration gives
us K, the time of vibration gives us w, and then we have
whence N, and therefrom n, can be found.
251. All this part of our subject is still very imperfectly
worked out. We have already seen ( 50) that even
brittle bodies may be completely changed in form by
small but persistent forces. And there is no doubt that
all elastic recovery in solids is gradual, so that, for
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 233
instance, in the torsion vibrations which we have just
considered, even when there is no sensible viscous
resistance, the middle point of the range does not coin-
cide with the original untwisted position of the wire.
It is always shifted towards the side to which torsion
was first directed, and to a greater extent the longer the
wire has been kept twisted before being allowed to
vibrate. With every vibration, however, the middle
point of the range creeps slowly back towards the original
undisturbed position, but the oscillation usually ceases
before this is reached. Still, even after the oscillation
has ceased, the wire continues to untwist, more and more
slowly, sometimes not even approximately reaching its
undisturbed position till hours or even days have passed.
When viscous resistance is considerable these results
are usually still more marked ; and Lord Kelvin l has
discovered the very curious additional fact that this
molecular friction becomes greatly increased by keeping
the wire oscillating for days together. He has pushed
this process so far with one of two similar wires that,
whereas, in that which had been made to vibrate only a
few times, the arc of oscillation became reduced to half
in 100 vibrations, the (equal) arc of that whose elasticity
had been " fatigued " fell to half after 44 or 45 vibrations
only.
252. Some of these phenomena are seen in a still more
striking form when we dispense with oscillation. Thus,
for example, suppose the wire to be kept twisted through
90 to the right for six hours, then for half an hour 90
to the left, and be then so gradually released that there
is no oscillation. When it is left to itself it turns slowly
towards the right, gradually undoing part of the effect of
1 Proc. U.S., 1865.
234 PROPERTIES OF MATTER.
the more recent twist, then stops, and twists still more
slowly to the left, thus undoing part of the quasi-permanent
effect of the earlier twist. Thus the behaviour of such
a wire, strictly speaking, is an excessively complex one,
depending, as it were, upon its whole previous history ;
though, of course, the trace left by each stage of its
treatment is less marked as the date of that stage is more
remote. This subject has of late attracted great
attention in Germany, and, under the name Elastische
Nachwirkung, has been the object of numerous researches
by Wiedemann, Kohlrausch, Boltzmann, etc.
253. Clerk-Maxwell 1 has given a sketch of a theory
of this peculiar action, from which we quote the
following :
"We know that the molecules of all bodies are in
motion. In gases and liquids the motion is such that
there is nothing to prevent any molecule from passing
from any part of the mass to any other part ; but in solids
we must suppose that some, at least, of the molecules
merely oscillate about a certain mean position, so that, if
we consider a certain group of molecules, its configuration
is never very different from a certain stable configuration,
about which it oscillates.
" This will be the case even when the solid is in a
state of strain, provided the amplitude of the oscillations
does not exceed a certain limit, but if it exceeds this limit
the group does not tend to return to its former configura-
tion, but begins to oscillate about a new configuration of
stability, the strain in which is either zero, or at least less
than in the original configuration.
" The condition of this breaking up of a configuration
must depend partly on the amplitude of the oscillations,
1 "Constitution of Bodies," Ency. Brit., ninth edition.
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 235
and partly on the amount of strain in the original
configuration ; and we may suppose that different groups
of molecules, even in a homogeneous solid, are not in
similar circumstances in this respect.
" Thus we may suppose that in a certain number of
groups the ordinary agitation of the molecules is liable to
accumulate so much that every now and then the con-
figuration of one of the groups breaks up, and this whether
it is in a state of strain or not. We may in this case
assume that in every second a certain proportion of these
groups break up, and assume configurations corresponding
to a strain uniform in all directions.
"If all the groups were of this kind, the medium
would be a viscous fluid.
" But we may suppose that there are other groups, the
configuration of which is so stable that they will not
break up under the ordinary agitation of the molecules
unless the average strain exceeds a certain limit, and this
limit may be different for different systems of these groups.
"Now if such groups of greater stability are dis-
seminated through the substance in such abundance as to
build up a solid framework, the substance will be a solid,
which will not be permanently deformed except by a
stress greater than a certain given stress. [If the tissue
of less solid material, which permeates the more solid
framework, extends continuously throughout the body,
the possibility of diffusion of the molecules is at once
ensured (see note, 299). We do riot require to assume
that the more solid framework has absolute permanency
in any part. Such a framework may always exist as a
sufficiently strong basis for rigidity, and yet be slowly
variable. This is entirely in accordance with Maxwell's
ideas. W. P.]
236 PROPERTIES OF MATTER.
" But if the solid also contains groups of smaller
stability and also groups of the first kind which break up
of themselves, then when a strain is applied the resistance
to it will gradually diminish as the groups of the first kind
break up, and this will go on till the stress is reduced to
that due to the more permanent groups. If the body is
now left to itself, it will not at once return to its original
form, but will only do so when the groups of the first
kind have broken up so often as to get back to their
original state of strain.
" This view of the constitution of a solid, as consisting
of groups of molecules some of which are in different
circumstances from others, also helps to explain the state
of the solid after a permanent deformation has been given
to it. In this case some of the less stable groups have
broken up and assumed new configurations, but it is quite
possible that others, more stable, may still retain their
original configurations, so that the form of the body is
determined by the equilibrium between these two sets of
groups ; but if, on account of rise of temperature, increase
of moisture, violent vibration, or any other cause, the
breaking up of the less stable groups is facilitated, the
more stable groups may again assert their sway, and tend
to restore the body to the shape it had before its
deformation."
254. There remains one specially complex kinetical
case of elastic reaction, i.e. the effects of Collision.
According to Newton, the " rules of the congress and
reflection of hard bodies" were discovered about the
same time by Wren, Wallis, and Huygens. Wallis had
the priority, then followed Wren. But Wren "confirmed
the truth of the thing " by a pendulum experiment (see
Appendix IV.). By " hard bodies " are meant such as
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 237
rebound from one another with the same relative velocity
as they had before collision. Newton goes on to describe
his own mode of experimenting on the subject, how he
allowed for the resistance of the air, etc., and proceeds as
follows :
" By the theory of Wren and Huygens, bodies
absolutely hard return one from another with the same
velocity with which they meet. But this may be
affirmed with more certainty of bodies perfectly elastic.
In bodies imperfectly elastic the velocity of the return
is to be diminished together with the elastic force ;
because that force (except when the parts of bodies are
bruised by their congress, or suffer some such extension
as happens under the strokes of a hammer) is (as far as
I can perceive) certain and determined, and makes the
bodies to return one from the other with a relative
velocity, which is in a given ratio to that relative
velocity with which they met. This I tried in balls of
wool, made up tightly and strongly compressed . . . the
balls always receding one from the other with a relative
velocity, which was to the relative velocity with which
they met as about 5 to 9. Balls of steel returned with
almost the same velocity : those of cork with a velocity
something less ; but in balls of glass the proportion was
about 15 to 16."
Of course results of this kind are confined to moderate
relative speeds. The question becomes a very different
and vastly more difficult one when very high relative
speeds are contemplated. When the relative speed is
such as to lead to the breaking of one of the two bodies,
we have a problem of at least as high an order of diffi-
culty as that presented by Tenacity. (226.)
255. So far as spherical bodies of small size, and im-
238 PROPERTIES OF MATTER.
pinging on one another with a moderate relative speed,
are concerned, there is yet but little to add to Newton's
results. These results, however, were fully confirmed by
the careful and instructive experiments of Hodgkinson. 1
Though the difficult mathematical problem of the defor-
mation and elastic rebound of two impinging spheres has
been magnificently treated by Hertz, 2 his results are
applicable only to very slight distortions.
But recent inquiries have shown that Newton's use of
the term " perfectly elastic " is not correct, for two bodies
may be perfectly elastic, and yet not rebound from one
another with the relative velocity of their approach. This
happens, in an easily intelligible manner, when a bell or
other body capable of vibrations is struck by a hammer.
And it is clear that the problem of impact of large masses,
where the propagation of distorting stress in each has to be
taken account of, will prove a very difficult one.
In modern phraseology the ratio of the relative velocity
of recoil to that of collision is called the Coefficient of
Restitution. It is not directly a coefficient of elasticity,
for it depends to some extent upon the sizes and shapes
of the impinging bodies, as well as upon the materials of
which they consist.
256. It is clear that, so far as direct impact of spheres
is concerned (where the whole motion of each of the
masses is in one common line), the third law of motion,
along with the value of the coefficient of restitution,
suffices for the calculation of the entire circumstances of
the motion after impact.
For if M, M', be the masses of the spheres, v, v and
v v Vi their respective speeds before and after impact,
1 Brit. Ass. Repvrt, 1834.
2 Crelle, xcii, 1882. See also Schneebeli, Arch, de Genlve, 1884.
COMPKESSIBILITY AND RIGIDITY OF SOLIDS. 289
and e the coefficient of restitution, the third law gives
Mi?! + M'< = Mv + MV
while the elastic property gives
V - *>i - ( - tf) ;
so that Vj and 1 ' are definitely determined.
This part of the subject will be found fully discussed in
most treatises on Dynamics. For illustrations of cases in
which, even with perfect elasticity, the coefficient of
restitution is necessarily less than unity, the reader
may consult Thomson and Tait's Natural Philosophy,
304.
257. In the case of violent impact between bodies of
small dimensions, as in golf or cricket, the mutual action
(in the first sense of Newton's Third Law, 128) usually
increases faster than in direct proportion to the deforma-
tion, measured by the approach towards one another, after
contact, of the undistorted parts of the two bodies. This
will be the case even while Hooke's Law holds, because
the greater the deformation the more extensive usually
are the parts deformed. Thus ( 71) the duration of
impact is less the greater is the relative speed ; so long at
least as no permanent distortion is produced.
The duration of compression is obviously greater than
the time in which a point, moving with the initial
relative speed, would pass over a space equal to the
deformation (measured as above). But it is less than
twice as great. For, since the mutual action increases
faster than does the deformation, and is such as to retard
the relative speed, its time-average value during the
compression must be greater than its space-average.
That is
240 PROPERTIES OF MATTER
MV MV 2
T ^ 2D"
where T is the time of compression, and D the normal
deformation. This gives
which is the statement above. The period of recovery
is longer than that of compression in the ratio 1 : e,
approximately.
Some notion of the duration of impact may be obtained
from the following experimental results. 1 A block of
hard wood, weighing 5J Ibs., fell from a height of 4 feet
on a cylinder 1J in. in diameter and 1J in. long, the
lower half of which was imbedded in a large mass of lead
resting on the ground.
Material of Cylinder. Distortion. Time of Impact.
mm. s.
Vulcanised India-Rubber 11 '5 0'0077
Vulcanite 2 '3 O'OOH
Cork 19-0 0-0166
Plane-tree 1'9 O'OOIS
It appears from the experiments that the elastic force,
called into play by the distortion, varies nearly in the
sesquiplicate ratio of the amount by which the cylinder
is shortened. It is very curious to find that this result,
deduced from experiments in which the elastic body is
violently strained throughout (being virtually struck at
both ends simultaneously) is in exact accordance with the
theoretical result of Hertz, in which the impinging bodies
were supposed to be perfectly free except in the region
1 Proc. R.S.R, 1889-90, p. 192 ; Trans. RS.K, 1891, 1893.
COMPRESSIBILITY AND RIGIDITY OF SOLIDS. 241
of their mutual contact, and the whole distortion was
assumed to be very small.
A needle-point, attached to the block, recorded its
motion on a rapidly revolving disc of glass, thinly covered
with fine printer's ink ; and time was measured by a
simultaneous record made by a tuning fork maintained in
vibration by a periodic electric current. When a golf-
ball was substituted for the cylinder, the time of impact
was about 8 '004.
Thus a golf-ball (since its mass is about 0*1 Ib. only)
has rebounded from the club before it has described a
space equal to half its radius : so that the whole time of
impact is of the order of one ten-thousandth of a second,
and the average force exerted is
01x240x10,000;
about 75,000 times the weight of the ball, or considerably
more than 3 tons' weight !
CHAPTER XII.
COHESION AND CAPILLARITY.
258. A SOMEWHAT pedantic nomenclature has introduced
the terms Cohesion and Adhesion in senses distinct from
one another. Thus contiguous parts of a piece of glass,
or of a drop of water hanging from it, are said to cohere,
while the water is said to adhere to the glass. Such
pedantry usually tends to produce confusion, as will be
seen at once if we try to state in its language how the
parts of a lump of granite, or of a drop of mixed alcohol
and water, are kept together. We will therefore use,
indiscriminately, whichever of the words happens to
present itself when we require one of them.
"We have already referred to the molecular forces which
are practically alone efficient in keeping together the
particles of a solid of moderate dimensions ( 167), and
to the resolidification (by pressure) of powdered graphite
( 53). The one characteristic of these forces, and that
which specially contrasts them with gravitation, is that
they are insensible at sensible distances. We have studied
the elasticity of fluids and of solids, and have also made
some remarks on the tenacity of solids ( 226). A few
other instances of cohesion between the particles of solids
may now be noticed, but the subject is one on which no
COHESION AND CAPILLARITY. 243
exact information can be expected in the present state
of science.
Two masses of marble, on each of which a true plane
surface has been worked, will, when these surfaces are
brought firmly together, even in vacuo, adhere so as to
overcome the weight of either (unless it be great in com-
parison with the area of contact), so that one remains
suspended from the other. Barton, early in the century,
made a set of cubes of copper whose sides were so very
true that when a dozen of them were piled on one
another the whole series adhered together when the
upper one was lifted. If a small plane surface be scraped
bright on each of two pieces of lead, and these be pressed
together (with a slight screwing motion) they adhere
almost as if they formed one mass. The processes of
gilding, silvering, nickelising, etc., and their results, are
known to all. So are the properties of lime, glue, and
other cements, all depending on the molecular forces in
and between solids.
259. Nor are we in a better position when we seek,
by what used to be considered a direct mode of measure-
ment, the force of adhesion between a solid and a liquid.
In the great majority of cases, the liquid wets the solid :
so that when we suspend a plate of the solid horizontally
from one scale-pan of a balance, and try what amount of
weights we must put into the opposite pan so as just to
detach the plate from a liquid surface, the liquid itself
is usually divided, not directly separated from the solid.
Such experiments, besides being very tedious and difficult,
lead to no results of the kind sought (see 287). We
have seen ( 219) that the adhesion of water to glass is,
at least, 800 Ibs. weight per square inch. But a force of
about 60 grains' weight only is required to draw a square
244 PROPERTIES OF MATTER.
inch of glass (wetted) from the surface of water; while,
if the plate be carefully cleaned and dried, only about
three times as great a force is required to separate it from
clean mercury. When a square inch of amalgamated zinc
is used it requires more than 500 grains' weight to remove
it from mercury. Here, however, as in the case of glass
and water, it is the liquid which is divided ( 275, 294).
260. The Internal (or molecular) Cohesion of a liquid is
suggested at once from analogy by the tenacity of a solid.
Its amount is large ( 219), but in consequence of the
mobility of liquids its direct measurement presents experi-
mental difficulties apparently insuperable. It will be seen
later than it can be calculated from the surface-tension,
though the value thus obtained will be too small. It must
be carefully distinguished from what has been called the
Internal Pressure of a liquid; a conception not very easy
to define, because somewhat difficult to grasp.
In some statical theories of molecular action, especially
that of Laplace, one of the most striking deductions is
that there must be a very great internal pressure in every
liquid mass : a pressure wholly independent of the form
and size of the bounding surface. This is usually known
as " Laplace's K." Laplace's own estimate of its value in
water is given (with the caution "une aussi prodigieuse
valeur ne peut pas tre admise avec vraisemblance ") as
the weight of the water which would fill a tube of unit
section whose length is 10,000 times the distance of the
earth from the sun: i.e. something like 10 12 tons weight
per square inch. 1 This was based on the corpuscular
theory of light, the numerical data being the refractive
index of water and the speed of light.
1 M6c. Celeste, livre x., Supplement a la The'orie de 1' Action
Capillaire.
COHESION AND CAPILLARITY. 245
No doubt the idea of molecular force was introduced to
explain, among other things, the crowding of the particles
together. But, as the particles are not supposed to be
jammed together, it is clear that they must be regarded as
acting repulsively when they are very near one another.
Thus the first layers of particles, on either side of any
imagined plane drawn in the liquid, may be considered
as repelling one another, while the constituents of the
second, third, &c., layers on either side of the plane attract
these, and one another, across it. If this attraction be
regarded as producing a pressure-stress + P, we must
recognise a simultaneous pressure-stress, - Q, due to the
repulsion. Thus, from the statical point of view, the
hydrostatic pressure is Q - P ; and depends, at any one
temperature, on the density only ; increasing steadily with
compression. When there is a free surface, the density
must obviously (262) increase, from the surface inwards,
but only through a depth of the order of the molecular
range. Thus there is a surface film in which the state of
the liquid is different from that throughout the mass. For
the rest the density is constant and corresponds to the
pressure called K; a mere difference-effect, due to the
otherwise uncompensated attraction on the skin.
The case presents a somewhat different aspect when
we look at it from the kinetic point of view : for then
we have, in place of statical molecular repulsion, the
impulse-energy of moving particles, which produces the
stress which above we called - Q.
When external pressure is applied, the values of the
opposed stresses will (on either of the above theories) be
increased, because of the further compression produced
But the excess of the repulsive over the attractive stress
will be augmented by the amount of the external pressure.
246 PROPERTIES OF MATTER.
[It is possible that the quantity II, in the empirical
formula (216, 217) for the average compressibility of a
liquid for the first p atmospheres : viz.
^
pv H +p
may represent the pressure-stress K ; for the formula may
be written as
where
a = v Q (l -e)
represents the (theoretical) volume under infinite pres-
sure. By the remark above, II increases with the com-
pression. This meets exactly the slight defects of the
formula, pointed out in 217.]
261. We now come to phenomena in which accurate
measurements are in general possible. These are the
phenomena due to the Surf ace- Tension of liquids. We
owe the idea to Segner (1751), but its development and
application are due mainly to Young. The more recent
theoretical advances in the subject were made chiefly by
Laplace and Gauss. [For a sketch of the history of this
subject the reader is referred to Clerk-Maxwell's article,
" Capillary Action," in the ninth edition of the Ency. Brit.]
262. As soon as we recognise, as a fact, the extremely
short distance at which these powerful molecular forces
are sensible, we see that there must be an essential
difference in state between parts of a liquid close to the
surface and others in the interior of the mass. For if
we describe, round any particle of the liquid as centre, a
sphere whose radius is the utmost range at which the
molecular forces are sensible, the only parts of the liquid
which act directly on that particle are those contained
COHESION AND CAPILLARITY. 247
within the sphere. So long as the sphere lies wholly
within the liquid the forces on the particle must obviously
balance one another. [Of course it must be approximately
so, unless the least distance from particle to particle is com-
parable with the radius of the sphere. We know of no
liquid for which this is the case.] But when part of the
sphere lies outside the liquid surface, i.e. when the
distance of the particle from the surface is less than the
range of the molecular forces, we can no longer make
this assertion.
Hence we should expect to find peculiarities in the
surface-film whose thickness is approximately equal to
this molecular range. [The compressibility is, happily,
small in liquids, otherwise the reasoning above would
lead to the result that the " peculiarities " should extend
to a distance from the surface somewhat greater than the
radius of the sphere of action of the molecular forces.]
We must appeal to experiment or observation to find
their nature in each particular case. And here, as we
shall soon see, a multitude of well-known facts comes at
once to our assistance. But we must first examine, after
Gauss, the theoretical conditions a little more closely.
263. An important theorem of Dynamics is that, for
stable equilibrium of a system, the potential energy of
the whole must be a minimum. It is easy to see from
the considerations given in last section that, so long as
we consider molecular forces alone, the amount of energy
of the liquid mass can vary only with the extent of the
surface, but we may formally prove it as follows.
Let the energy be e Q per unit mass of the interior
liquid, and e per unit mass for a layer of the skin, of
surface S, thickness t, and density p. Then, if M be the
whole mass of the liquid, and E its whole potential
248 PROPERTIES OF MATTER.
energy, we have, by summing the energy of the interior
mass and of the successive layers of skin (which may be
treated as having all practically the same superficies so
long as their curvature is finite, in consequence of the
shortness of range of the molecular forces),
E=(M-S. Sp)e + S. 2tpe
= M<? +S. 2tp(e-e }.
Thus, in consequence of this property of the skin, the
whole energy is increased or diminished by a quantity
which is directly proportional to S. The multiplier of
S depends upon the nature and the temperature of the
liquid, and on the nature of the substance which is in
contact with its free surface, only.
Hence, when e is greater than e everywhere through-
out the skin, as we find happens when a water surface
is exposed to air, S tends to the smallest value compatible
with the conditions. But when e is less than e , as when
water is in contact with glass, S tends to take as large a
value as possible. If part be exposed to air, and part be
in contact with glass or other substance, the final result
is more complex, but involves the same principles. [In
the case of two liquids, a tendency of S to increase leads
to puckering of the common boundary on an invisible
scale. Kelvin regards this as the commencement of the
process of diffusion. W. P.]
It is obvious that precisely similar reasoning may be
applied to the case of two liquids in contact, even after
diffusion has gone on for a little, so long in fact as there
remains a sensible difference between the energy per
unit mass in the common skin and in either of the
liquids. (See 292, 300.) In such a case, however, the
resulting changes of form must necessarily take place more
COHESION AND CAPILLARITY. 249
slowly than when a single liquid is exposed to air, as the
inertia of the whole system has to be overcome.
264. Still keeping to the theoretical view of the
subject, let us consider what is implied in the tendency
of the liquid surface to become as small as possible. It
must behave, only in an incomparably more perfect
manner, like an elastic membrane (such as a sheet of
india-rubber), which has been stretched by equal tensions
in all directions. But, while the tension of such a mem-
brane becomes less as it is permitted to shrink more and
more, the liquid film has still as great a tendency to
shrink, however small its surface may have become.
Thus it must be under a definite Surface-Tension.
If T be this tension across a line of unit-length on
such a liquid surface, it is easy to see that the work
required to stretch a rectangle, whose sides are a and b,
into another whose sides are a and b + Z, is
i.e. T multiplied by the increase of area. This quantity T
must, therefore, be the multiplier of S in the expression
(of last section) for the whole energy of a mass of liquid.
265. When the liquid is drawn out into a film, as in
blowing a soap-bubble, the tension of this film is practi-
cally 2T ; so long, at least, as the whole thickness of the
film is greater than twice the molecular range. For it
may be regarded as consisting of a layer of interior water,
with two surface-skins.
266. We now come to the observed facts which are to
be compared with these indications of theory. But, first,
we assume the mathematical theorem that the sphere is,
of all surfaces, that which, with a given content, has the
smallest superficies.
Whenever a drop of liquid is left free from all but
250 PROPERTIES OF MATTER.
its own molecular forces, we find it assumes a perfectly
spherical form. By far the most rigorous proof of this
is afforded by the rainbow. Exceedingly slight deviations
from perfect sphericity of the falling drops would suffice
entirely to alter the character of this phenomenon. Greater
deviations would altogether prevent its occurrence.
The manufacture of small-shot, in which it is im-
portant that each particle should be truly spherical, is
another good example. A shot-tower, as it is called, is
merely a gigantic shower-bath, where the liquid employed
is melted lead, which is slightly alloyed, mainly for the
purpose of making it less viscous while liquid. The
majority of the falling drops solidify in forms very nearly
spherical before they reach the water-bath which is
employed to break their fall.
The rounding off of the sharp edges of a broken piece
of sealing wax, as soon as it melts in a flame, is another
example ; and it was in consequence of the almost perfect
sphericity of the little bead, formed on the end of a glass
fibre which is held in a flame for a short time, that such
beads were employed by the early microscopists as single
magnifiers. By the use of these, Leuwenhoeck and others
anticipated many of the results which are now obtained
by means of the splendid achromatic object-glasses of
modern compound microscopes.
267. An ingenious method of guarding a liquid from
the action of any but molecular forces was devised by
Plateau. He simply placed a mass of oil in a mixture
of alcohol and water of the same density as the oil.
Here the mass assumes a perfectly spherical form.
When this arrangement is altered by evaporation, the
mixture becomes denser from the top downwards. The
globe of oil becomes flattened, because its lower parts tend
COHESION AND CAPILLARITY. 251
to rise and its upper portions to sink, being immersed re-
spectively in parts of the liquid denser and less dense than
the oil. This mass of oil can be made to fulfil definite
boundary conditions, by bringing into contact with it
various frames of wire, etc., all thoroughly oiled before-
hand.
268. If a large drop of water be laid on a clean glass
plate, it spreads itself over a considerable surface.
[Theoretically, it should wet the whole surface.] Now
let ether vapour, which is heavier than air, be poured
upon the middle of the water surface ; or let it be
touched there by a glass rod moistened with alcohol ; or
even hold the point of a red-hot poker close to it. In all
these cases the effect is to reduce, locally, the surface-
tension, so that, as it were, a weak part of the surface-
film is produced, and this is pulled out over the surface
by the greater contractile tension of the unaffected parts
of the skin. The water, in fact, often retreats on all
sides from the affected part, leaving the central portion
of the glass uncovered. The effect is precisely similar
to that which is produced in a stretched sheet of india-
rubber where one part is either thinner than the rest,
or has been slightly heated by a flame.
The surface-tension of a drop of mercury is greatly
altered when it is (electrically) " polarised." Kemarkable
phenomena of this kind were described by Strethill Wright
in I860. 1 Lippmanri has employed this surface effect in
the construction of a very sensitive electrometer.
269. The phenomenon called the "tears of strong
wine," first explained by J. Thomson, is another example.
When the sides of a drinking glass have been moistened
with strong wine, we observe that the liquid film soon
1 Phil. Mag., xix. p. 129.
252 PROPERTIES OF MATTER.
becomes corrugated. The ridges are formed of the
portions from which the greatest amount of alcohol has
evaporated, and which, therefore, have the greatest
surface-tension. As these slide, by gravity, down the
sides, we see them now and then stop, and even retract,
when they come to a part where there is more alcohol,
and therefore less surface-tension.
By far the best example, however, is furnished by
some of the less viscid oils. A few drops, let fall on the
surface of a quiet pool, seem almost to flash out over the
surface, showing in the most brilliant manner the inter-
ference colours of thin plates.
Another striking instance of the effects of surface-
tension is furnished by a piece of camphor when placed
on water. It usually dissolves more rapidly at some one
place than at others, thus relatively weakening the surface-
tension of the water in that place, and is consequently
dragged about with considerable rapidity, and in the most
capricious manner. Similar results, but of a more complex
and often much more violent nature, are obtained with a
pellet of potassium or of sodium.
270. It is the same when we deal with a double skin,
as in a soap bubble. When a soap-film is lifted by the
mouth of a glass funnel previously wetted throughout
its interior with the solution, it rapidly runs up the cone,
and fixes itself finally at the place where the area of the
cross section is least.
Van der Mensbrugghe has devised a very beautiful
and instructive experiment of this kind, which it is easy
to repeat. He lays on a soap-film, lifted by a large wire
ring, a short endless silk thread, thoroughly wetted with
the soap solution. As soon as the film is broken inside the
coil of the thread, the thread is stretched out into an exact
COHESION AND CAPILLARITY. 253
circle which bounds the hole in the film. The rupture
is easily effected by touching the film with a hot wire, or
a lucifer match just blown out. [The circle is the figure
which, with a given perimeter, has the greatest area.
Thus the film has the smallest area, subject to the
conditions.]
When a soap bubble is blown at the mouth of a
funnel, and the neck is left open, we see it shrink faster
and faster, expelling the contained air, which is thus proved
to be at greater than atmospheric pressure. Faraday suc-
ceeded in blowing out a candle by the air thus expelled.
The soap-glycerine solution, invented by Plateau, gives
films which last for hours together, enabling us to study
the phenomena of surface-tension even more simply and
accurately than can be done by the help of his earlier
method ( 267). His great work 1 on these subjects forms
a storehouse of interesting and important experiments,
all the more remarkable that they were devised by one
who had the misfortune of being permanently disabled
from seeing them. The reader must be referred to these
volumes for other examples than the few for which we
can find room.
271. It is very instructive to observe the mode in
which many problems, some of extreme mathematical
difficulty, are at once solved practically by experiments
with these soap-films.
The whole physical part of the phenomena depends
upon the fact that the film takes the form of smallest
superficies consistent with the conditions.
When the film is exposed to equal pressures on its sides,
i.e. when no air is anywhere enclosed by it, it may be
1 Statique experimental^ et theoriqiu des Liquides soumis aux
Seules Forces MoUculaires. Paris, 1873.
254
PROPERTIES OF MATTER.
made up of portions which are individually plane; in
which case the problem, though possibly complex, is
comparatively easy. But this is an exceptional case ; for,
even with equal pressures, the film is usually curved,
and it must always be curved when the pressures on its
sides are different.
272. Here it becomes necessary to consider the curva-
ture of the film, and the way in which it depends upon
the pressures to which the sides
are exposed. A very simple in-
vestigation gives us all we require
in this matter. Suppose a tape or
band, of unit breadth, while under
tension T, be wrapped transversely
round a cylinder of radius E, and
let p be the pressure which it pro-
duces on unit surface of the cylin-
der. Consider a very small portion
of its length, AB, subtending an
angle at the centre, 0, of the
cylinder. This portion of the band is kept in equili-
brium by the tension, T, at its ends, and the reaction,
p. AB, of the cylindrical surface. Resolving these forces
in the direction 00, bisecting the angle AOB, we have
2Tsin4=2>. AB = p. R0 ;
which, when is very small, becomes
T = pR, orp-.
Thus the band requires, for its support in the cylindrical
form, an excess of pressure on the concave, over that on
the convex side, amounting to -rj per unit of surface.
COHESION AND CAPILLARITY. 255
273. In the case of a soap-film, or of the surface-film of
a liquid generally, there may be simultaneous curvatures in
two planes at right angles to one another and to the tangent
plane. The effects of these are to be simply superposed, as
they are independent. Let R x be the radius of the second
curvature, then, as the film exerts equal tension in all
directions, the difference of pressures on its sides is, per
square unit,
XX)
This expression must be doubled in the case of a soap-
bubble, for ( 265) it has two surface-films.
This formula may easily be obtained in another way,
viz., by expressing the work done during an infinitesimal
normal displacement of each point of the film : first, as
the product of the difference of external and internal
pressure into the increase of contained volume, and second
as the product of the surface-tension into the increase of
surface of the film. This, however, we leave to the reader.
He will easily find that if t be the normal displacement of
the element, dS, of surface, and p the difference of pres-
sures, we have
wliatever be the value of t, the integral being extended
over the whole surface.
By a well-known geometrical theorem, due to Euler,
the quantity multiplied by T, i.e. the sum of the curvatures
in two planes at right angles to each other, and both
passing through the normal to a surface at a particular
point, is independent of the aspects of these planes.
Hence it is convenient to choose K and Rj as the principal
radii of curvature of the film.
When, as a purely mathematical problem, we seek the
256 PROPERTIES OF MATTER.
characteristic of the surfaces of least area which satisfy
given boundary conditions, we are led to the condition
that the sum of the curvatures at any point is constant.
This agrees with the physical result.
274. Thus, when a soap-film is exposed to equal
pressures on its two sides, it must satisfy the given
boundary conditions, and possess the further property
that, at every point of its surface,
i.e. whatever be its curvature in any normal section, it
must have an equal and opposite curvature in the normal
section perpendicular to the first.
Such must, therefore, if we neglect the (very slight)
disturbing effects of gravity, be the form of a soap-film
exposed on both sides to the air. Thus if we lift such a
film on a flat loop of wire it assumes a plane surface ; but,
by bending the boundary, we can make it assume forms
of marked curvature. In all its forms, however, the sum
of the curvatures at each point is nil. And the same is
the case, however ramified, linked, or knotted the wire
frame may be, provided only that there is no air imprisoned
at any place.
275. If we imprison a quantity of air by the film, as,
for instance, by forming it between the rims of two equal
funnels, and closing the neck of each with a finger, we
have in general different pressures outside and inside ;
and then we have ( 265)
ili+i
2T R R!
where p is the constant difference of pressures. By
altering the relative position of the funnels, as by shift-
ing one sidewise out of the line of symmetry, or by
COHESION AND CAPILLARITY. 257
making it rotate (otherwise than about its axis of sym-
metry), we can throw the film into extraordinary shapes ;
all of them, however, possessing the fundamental property
of constant sum of the curvatures
pv^X. at each point. But we content
\ / /^\ ~~J\ ourselves with a brief notice of
(/ ^^-^^ \ the results of gradually withdraw-
^ ng the funnels from one another,
FlG 33 while keeping their axes of sym-
metry in one line.
Thus we may begin with the film as a quasi-spherical,
or even spherical surface, having both its curvatures
moderate (Fig. 1). As we withdraw the funnels fiom
one another the longitudinal curvature diminishes, and
the transverse increases to the same amount, till at last
the longitudinal curvature vanishes altogether, and the
2
Fo. 34.
film becomes cylindrical (Fig. 2). Still further separat-
ing them, the film takes an hour-glass form as in Fig. 3,
where the increasing curvature of the transverse section
is now balanced by a gradually increasing negative curv-
ature in the longitudinal section. At a certain limit
this state of the film becomes unstable, and the positive
R
258 PROPERTIES OF MATTER.
and negative curvatures near the middle both rapidly
increase, till the walls at that part collapse into a mere
neck of water, which is ruptured, and leaves a pro-
tuberant film on each of the funnels. By a little
dexterous manipulation these may easily be made to
reunite into the original form.
276. The facts we have just described show us the
nature of the process by which a complete soap-bubble
is detached from a funnel, always leaving a film on the
funnel ready to produce a second bubble. This process
can easily be studied by completing the blowing of the
bubble with coal-gas, after it has been commenced with
air, and watching it detach itself in virtue of the light-
ness of its contents.
Even so dense a liquid as mercury can be formed into
a bubble. We have merely to shake a glass bottle filled
with water and clean mercury. The bubbles which form
on the mercury (often detached) are full of water. Some-
times we see others coming up from the interior of the
mercury. These are water-skins full of mercury.
277. When two complete soap-bubbles are made to
unite, the tendency of the liquid film is to contract, that
of the (compressed) air inside is to expand. It becomes
a curious question to find which of these actually occurs.
Let their radii, when separate, be R and R 15 and let
them form, when united, a bubble of radius r. Then, if
II be the atmospheric pressure, the original pressures in
the bubbles were
while that in the joint bubble is
in-".
r
COHESION AND CAPILLARITY. 259
By Boyle's Law the densities are as the pressures.
Hence, expressing that no air is lost, we have
or
If V be the diminution of the whole volume occupied by
the air, S that of the whole surface of the liquid film,
this condition gives at once
3nV + 4TS = 0.
As II and T are both essentially positive, this condi-
tion shows that V and S must have opposite signs.
Hence both tendencies are gratified, the surface, as a
whole, shrinks, and the contained air, as a whole, increases
in volume, simultaneously. But the work done by the
expanding gas is only about two-thirds of that done by
the contracting film.
It is worthy of notice that, as is easily proved, the air
in a soap-bubble of any finite radius would, at atmo-
spheric pressure, fill a sphere of radius greater than
before by the constant quantity 4T/3II.
278. As a practical illustration of the use of these
formulae, let us apply them to a stationary steam-boiler
of the usual cylindrical form, with the ends portions of
spheres. If R be the radius of the cylinder, Rj that
of each end, and P the excess of internal over external
pressure, the tension is
Across a generating line, RP,
Parallel to a generating line, ffR ^ = iRP,
Across any line on the end, ^RjP.
Thus, if the boiler-plate be equally tenacious in all
260 PROPERTIES OF MATTER.
directions, there is no danger of the ends being blown
off, for the boiler will rather tear along a generating line.
And, to make the ends as strong as the sides, they
require only half the curvature.
Thus, also, we see why stout glass tubes, if of small
enough bore, are capable of resisting very great internal
pressure, when, as in Andrews' experiments ( 198) on
carbonic acid, they are exposed only to atmospheric
pressure outside.
In what precedes we have neglected the weight of the
soap-film, and have consequently taken its tension as
being constant throughout. But a moment's considera-
tion of the equilibrium of a plane vertical film shows
that the tension must increase from below upwards.
This gives an immediate explanation of the difficulty
presented by the fact that bubbles cannot be blown with
pure water, though its surface-tension is much greater
than that of a soap-solution. The soap-solution is, as
Marangoni has pointed out, an excessively heterogeneous
liquid, and (within limits) can and does adjust its surface-
tension to the value required at each point. The slowness
with which the film becomes gradually thinner, so as to
display in succession the various interference colours of
thin plates, is to be ascribed to the viscosity of the liquid. 1
279. "We are now prepared to consider the phenomena
properly called Capillary, as having been detected in
tubes of very fine bore.
"When clean glass tubes, each open at both ends, are
partially immersed in a dish of water, we observe that
(in apparent deviation from the hydrostatic laws, 189)
the water rises in each to a higher level than that at
1 Lord Rayleigh "On the Superficial Viscosity of Water," Proc.
R.S., 1890.
COHESION AND CAPILLARITY.
261
which it stands outside. Also we notice that this rise is
greater the finer the bore of the tube. The cut shows
the phenomenon in section.
Perform the same experiments with mercury instead
of water, and we find that the liquid stands at lower
FIG. 35.
levels inside than outside each tube, and that this
depression is greater the finer the bore of the tube.
Turn the above cut upside down, and it will correspond
to this effect.
280. But a closer inspection at once shows the
immediate cause of the phenomena. The water surface
inside each tube is always concave outwards, that of the
mercury convex ; and the curvature of either is greater
the finer is the bore of the tube.
Remember the surface-tension of the liquid, and the
consequent excess of pressure on the concave side, over
that on the convex side, which is necessary ( 272) for its
equilibrium, and we see at once that the water immedi-
ately under the surface-film must have a less pressure
than that of the atmosphere to which its concave side is
exposed. Thus, hydrostatically ( 189), it belongs to a
higher level than the undisturbed water, whose surface
262
PfcOfEKTIES OF MATTER.
is plane, and the pressure in which (immediately under
the surface) is equal to the atmospheric pressure.
[Laplace's K ( 260), or whatever may take its place,
does not affect this conclusion ; for its value is the same
at all points of the liquid which are not in the surface-
skin].
As the surface curvature is greater in the finer tubes,
so the higher rise of water in these is a direct hydrostatic
consequence of the greater relief of pressure.
The convexity of the mercury surface, on the other
hand, requires immediately under the film a pressure
exceeding that of the atmosphere by an amount propor-
tional to the sum of its curvatures. Thus we see why
the mercury stands at a lower level in the tube than
outside it.
281. It only remains that we should account for the
concavity of the water surface, and the convexity of that
of the mercury.
In the annexed sections of a concave and of a convex
surface, in which a tangent, BA, is drawn to the liquid
film, where it meets the side of
the tube at B, the angle ABC of i D
the wedge of liquid is obviously
less than a right angle for the
concave surface, and greater than
a right angle for the convex.
Hence the problem is reduced to
the determination of this angle,
called the Angle of Contact.
That this angle must have a definite value for each
liquid, in contact with each particular solid, appears at
once from the consideration that, in the immediate neigh-
bourhood of B, the gravitational or other external forces,
B
i \
A
c
FIG. 36.
COHESION AND CAPILLARITY. 263
acting on a very small portion of the liquid, are incom-
parably less intense than the molecular tensions. Hence
the equilibrium of that portion (tangentially to the solid)
will depend upon the surface-tensions along BA, BC, BD
alone. The directions of two of these, and the magnitudes
of all three, are determinate, whatever two fluids (even
when one is gaseous) are in contact with each other and
with the solid ( 263). BA, therefore, will ultimately
assume such a direction that the surface-tension along it
will, when resolved in CD, just balance the difference
between the tensions in BD and BC. Hence, if that in
BD is the greater, the angle of contingence will be acute;
if that in BC be the greater, it will be obtuse.
282. In the case of mercury and clean glass, exposed
to air, the angle of contact is
140 (Young), 135 (Gay-Lussac), 128 52' (Quincke), 132 2'
(Bashforth).
With water and clean glass in air the angle vanishes
entirely : in fact, of the three tensions, that in BD
exceeds the sum of the other two ; but when the glass is
not clean it may reach (and even surpass) 90. When it
is exactly 90 there is no curvature of the water surface
inside the capillary tube, and it therefore stands at the
level of the undisturbed water outside. 1
283. We may now complete the explanation of the
1 One of Gay-Lussac's ingenious methods for determining the
angle of contact when it is finite must be at least indicated here.
If the liquid be introduced gradually into a small glass sphere
(from below) there will be one position in which its surface is
throughout plane. By determining this position the angle can
be at once calculated. [A plane glass plate dipped into mercury,
and inclined until the liquid under it is flat throughout, would give
an excellent determination.]
264 tROPERTlES OF MATTER.
behaviour of a liquid in a capillary tube as follows :
When the rise (or depression) exceeds several diameters
of the tube, the curvature is practically the same over
the whole free surface, which is therefore approximately
spherical. In mercury, because of the finite angle of
contact, it forms a segment less than a hemisphere ; in
water it is a complete hemisphere.
In the former case the radius is directly proportional
to that of the tube, in the latter it is equal to it. In
both cases, therefore, the relief or the increase of pressure,
and consequently the rise or depression of the liquid, is
inversely as the radius of the tube. This agrees with
the (long-known) results of experiment.
284. We may make, in a very simple manner, due to
Dr. Jurin, a calculation of the capillary elevation, which
is applicable to wider tubes than those spoken of in last
section. Suppose the radius of the tube to be r, p the
density of the liquid, a its angle of contact, T the tension
of the surface-film, and h the mean height to which it is
elevated. [This mean height is taken such that the
volume of the liquid actually raised would, if the surface
were not curved, fill the length h of the tube.] Then
the vertical component of the whole tension round the
edge of the film is obviously
27rrT cos a.
But this supports the weight
of liquid, (virtually) filling a length h of the tube.
Equating these quantities we obtain, after reduction,
A _2Tcoa a
rgp
When a>^, h is negative, and the liquid is depressed.
COHESION AND CAPILLARITY. 265
All the quantities here are easily measured except T
and a. Hence, if a can be found by a separate process,
T is at once determined. In the case of water in clean
glass we have cos a = 1, so that the above relation gives
T directly.
285. The following values of T are given by Quincke.
Each datum in the table belongs to the film at the
common surface of the substances whose names are in
the same line and column with it.
Air. Water. Mercury.
Water . 81 . - . 418
Mercury . 540 . 418
Alcohol . 25-5 . * 399
The unit here is one dyne per (linear) centimetre. To
reduce to grains' weight per inch divide by 25. Thus we
may easily calculate, from the formula of last section,
that water rises a little more than half an inch in a glass
tube whose bore is y^th inch in diameter.
286. In the Atmometer, which is merely a ball of un-
glazed clay luted to a glass tube, the whole filled with
water and inverted in a vessel of mercury, not only is
the reduction of pressure by the fine concave surfaces of
water in the pores sufficient to keep a column of 3 or 4
feet of water supported, but, as evaporation proceeds,
mercury rises to take the place of the water, sometimes
to 23 inches or more. The process has not, so far as we
know, been pushed to its limits. Thus the widest of
these pores can sustain (virtually) a column of some 26
feet of water. It is easy to put the Atmometer directly
into this condition, and the consequent great concavity of
the surface of the water in each pore renders it eminently
fit ( 291) as a nucleus for the deposition of vapour. 1
1 Proc. R.S.E., February 16, 1885.
266 PROPERTIES OF MATTER.
287. The data of 285 enable us easily to calculate the
force with which a boy's " sucker " is pressed against a
stone. Suppose we have two plates of glass, 6 inches
square, with a film of water between them whose thick-
ness is alhj-th of an inch. The force required to pull one
perpendicularly from the other, in which case the free
water surface round the edges will take a (cylindrical)
curvature of radius ^y^th of an inch, would be the
weight of a six-inch square prism of water about 5
inches high, i.e. between 6 and 7 pounds' weight. If
the film were of half that thickness (at the edges) the
force required would be double. Thus, as J. Thomson
has pointed out, two flat slabs of ice, hanging side by
side on a horizontal wire, with a film of water between
them, are pressed together with a force which may much
exceed the weight of either : and may therefore freeze
together even in a warm room. When a mere drop of
water is placed between two very true glass planes the
relief of pressure produced enables the atmospheric
pressure to force them closer together, and this effect
increases, not only by the enlargement of the wetted
surfaces, but by the increase of curvature round the
edges of the flattened drop. The pressure producible in
this way is very great, and may crack large sheets of plate
glass (if there be portions not very true) when they are
laid on one another with a drop of water between them.
On the other hand, a few small drops of mercury, in-
terposed here and there between the plates, form an
exceedingly perfect elastic cushion.
288. There are many common phenomena whose
explanation is easily traced to the action of capillary
forces. Thus air-bubbles, sticks, and straws floating on
still water, appear to attract one another ; and gather
COHESION AND CAPILLARITY. 267
into groups, or run to the edge of the containing vessel.
This is always the case with any two bodies, each of
which is wetted by the water, and it is also true when
neither is wetted. But when one of the bodies is wetted,
and the other is not, they behave as if they repelled one
another. The explanation is easily given : either from
the point of view of the various forces called into play
by the displacement of the water, or (more simply) by
the consideration of the whole energy of the liquid as
depending on the relative position of the floating bodies
( 263) and the consequent displacement of the surface.
A needle, or even a (very small) pellet of mercury,
may easily be made to float on water. The hydrostatic
condition requires merely a depression of the surface,
so that the water displaced may be equal in weight to
the floating body ; but, that this displacement may take
place, the angle of contact must be made greater than
90, which is at once ensured if the needle be very
slightly greased. Thus we explain how water flies run
on the surface of a pool.
In the same way we can explain why a piece of wood
is not wetted when it is dipped into water whose surface
is covered with lycopodium seed ; and why mercury can
be poured in considerable quantity into a bag of gauze or
cambric without escaping through the meshes. ( 100.)
An air-bubble in water assumes a spherical form, even
when it is in contact with the side of a glass vessel, and
a very small globule of mercury laid on glass becomes
almost spherical. But an air-bubble on the side of a
glass vessel containing mercury is flattened out, while a
drop of water on clean glass spreads itself out indefinitely.
In all these cases the angle of contact at once explains
the result.
268 PROPERTIES OF MATTER.
The difficulty of obtaining a clean surface of water or
mercury depends upon the great surface-tension of these
liquids relatively to that of the majority of other sub-
stances. From the reasoning of 281, and the data
of 285, we see that water ought to spread indefinitely
over a clean surface of mercury.
289. The form of section of a (cylindrical) liquid
surface, in contact with a plane solid surface, is easily
deduced from the hydrostatic principle that the elevation
(or depression) at any point is proportional to the relief
(or increase) of pressure, i.e. to the one curvature. Hence
it must be the curve of flexure (237) of a very long
uniform elastic wire, with a kink in it, under the action
of tensions at its ends ; for at every point of that curve
the curvature is proportional to the distance from the
line in which the stress acts. Hence we can at once
find the form in which the liquid surface meets a plane
solid face, whether it be vertical or not, by drawing
the corresponding elastic curve and taking account of the
inclination of the plate and of the angle of contact.
When the liquid surface is between two glass plates,
inclined at any angles to the vertical, but having their
line of intersection horizontal, the form of the cylindrical
surface is given by one of the more complex forms of
the elastic curve.
290. The surface-tension of liquids diminishes with
rise of temperature. And Andrews showed that, as
liquid carbonic acid is gradually raised to its critical
temperature, the curvature of its surface in a capillary
tube gradually diminishes.
291. Lord Kelvin 1 showed that there is a definite
vapour-pressure, for each amount of curvature of a liquid
1 Proc. R.S.E., 1870.
COHESION AND CAPILLARITY. 269
surface, necessary to equilibrium. It is less as the sur-
face is more concave, greater as it is less concave or more
convex. Hence precipitation of water-vapour will, ceteris
paribus, take place more rapidly the more concave (or
the less convex) is the surface of that already deposited.
Thus, as Clerk-Maxwell pointed out, the larger drops
in a cloud must grow at the expense of the smaller ones.
The explanation of these curious facts is given by the
kinetic theory much in the same way as is that of the
effect of the curvature of the discs of a Radiometer.
So great a pressure of vapour would be necessary for
the existence of very small globules of water (in the
nascent state of cloud, as it were), that, as Aitken has
shown, condensation cannot commence in free air without
the presence of dust-nuclei. The more numerous these
are, the smaller is the share of each, and thus we have
various kinds of fog, mist, and cloud.
292. Many extremely curious phenomena, due in great
part to surface-tension, have been investigated by various
experimenters, especially Tomlinson. Thus different
kinds of oils can be distinguished from one another,
or the purity of a specimen of a particular oil may
be ascertained, by the form which a drop takes when
let fall on a large, clean, water surface. In some cases
a drop of oil does not spread entirely over a liquid
surface, but forms a sort of lens. The angles at which
its faces meet one another, and the surface of the liquid,
are then to be determined from the respective surface-
tensions by the triangle of forces, as in 281.
Again, when a drop of an aqueous solution of a salt,
say permanganate of potash or some other highly-
coloured substance, is allowed slowly to descend in
water, it at first takes the form of a vortex-ring, bounded,
270 PROPERTIES OF MATTER.
of course, by a film of definite surface-tension. But, as
diffusion proceeds, it would appear that this film becomes
weaker at certain places (just as in the case of wine,
269), and consequently unstable. Be this as it may, the
ring breaks into segments, each of which is (as it were)
a new drop, which behaves as the original drop did,
though somewhat less vigorously. Thus we have a very
curious appearance, almost resembling the development
of a polyp ; the number of distinct individuals being
markedly greater in each successive generation. With
a drop of ink these developments take place so fast that
the eye can scarcely follow them.
The phenomena of surface-tension were found by
Bosscha to be exhibited, in some forms, by smoke. And
Van der Waals showed that smoke stands lower in the
moistened branch of a U-tube than in the dry one, ex-
hibiting a convex surface like that of mercury.
293. We may now say a word or two as to the ex-
treme limits at which the molecular forces are sensible.
It is not at all remarkable that the various estimates
differ widely from one another, for they are all obtained
by processes more or less indirect. They all agree, how-
ever, in giving small values. Experiments of Plateau
on soap-films, and of Quincke on the behaviour of water
and thinly-silvered glass, give only about 1/500,000 of
an inch. It is probable that the limits vary somewhat
with the nature of the substance experimented on ; and
the question is certainly connected, in no remote manner,
with the differences in the critical temperatures ( 194)
of various substances.
294. The separation into drops, of a liquid column
slowly escaping into air from a small hole in the bottom
of a vessel, can be studied by examining it by the light
COHESION AND CAPILLARITY. 271
of electric sparks rapidly succeeding one another. It is
a phenomenon similar to that which we have described
in 275, when a cylindrical film is drawn out between
two funnels. When the liquid is a very viscous one, as
treacle or Canada balsam, especially if its surface-tension
be small, the viscosity greatly retards the development of
this effect of instability; and such liquids can, like melted
glass, be drawn into fine continuous threads. This pro-
perty sometimes gets the special name of Viscidity.
295. The propagation of ripples, as Lord Kelvin
showed, 1 is also due mainly to surface-tension. The ex-
perimental proof is given by the fact that the shorter
are the ripples the faster they ' run, while ordinary
oscillatory waves in deep water, propagated by gravita-
tion, run faster the longer they are.
[This affords a good example of the application of what
is called the principle of Dynamical Similarity ; i.e. the
effect of scale upon physical phenomena, It is, of course,
merely a question of Dimensions as in 64. Various
instances of the application of this principle have already
been given, e.g. 40, 167, 228, 284, etc.]
In two similar ripples, of different wave-lengths, the
forces are independent of the lengths, the ranges are
directly as the lengths, and the masses of water are as the
-squares of the lengths, of the ripples. Hence the rates
of propagation are inversely as the square root of the
wave-length. In similar oscillatory gravitation waves, on
the contrary, the forces are as the squares of the lengths,
the ranges as the lengths, and the masses as the squares
of the lengths, and the rate is directly proportional to the
square root of the wave-length.
Thus very short ripples ran almost entirely by surface-
1 Phil. Mag., 1871, ii. p. 375.
272 PROPERTIES OF MATTER.
tension, while long ripples and short waves run partly
by gravity partly by surface-tension. Lord Kelvin has
shown that the limit between waves and ripples in water,
vhich is the slowest-moving surface disturbance, has
about 2/3 inch as its wave-length, and runs at a speed
of 9 inches per second. Every shorter disturbance runs
mainly by surface-tension, and may be called a ripple ;
every longer one runs mainly by gravitation, and may be
called a wave. Fairly accurate determinations of surface-
tension have been obtained by measurement of the
lengths of ripples produced when a vibrating tuning-fork
(of known pitch) rests against a trough containing a
liquid. 1
296. When a solid is exposed to a gas or vapour, a
film is deposited on its surface which, in many cases,
introduces confusion in weighings, etc. Thus, if a dry
platinum capsule be carefully weighed, then heated to
redness and weighed again immediately after it has
cooled, it is found to be lighter. If left exposed to the
air it gradually recovers its former weight. In so far as
this effect is a purely surface one, it is increased in pro-
portion as the surface of a given mass of the solid is
increased. Thus "spongy" platinum, as it is called i.e.
platinum in a state of very minute division (obtained by
reducing it by heat from some of its salts) exhibits the
phenomenon to a notable extent. Dobereiner showed
that a jet of hydrogen can be set on fire, by the heat
developed when it is blown against spongy platinum
which has been exposed to the air. The platinum is
heated to redness by the combination of the oxygen film,
already condensed on its surface, with the hydrogen
which suffers condensation in its turn.
1 C. M. Smith, Proc. R.S.K, 1890.
COHESION AND CAPILLARITY. 273
Another remarkable form of experiment, analogous to
this, consists in heating a helix of platinum wire to
incandescence in the flame of a Bunsen lamp, turning off
and then immediately turning on again the supply of
gas ; for the wire remains permanently red-hot in the
explosive mixture of air and coal-gas ; without, how-
ever, reaching a high enough temperature to inflame it
again.
The amount of surface really exposed by finely porous
bodies, especially (as Hunter * showed) cocoa-nut charcoal,
is enormous in comparison with their apparent surface ;
and in consequence they are able to absorb (as it is called)
quantities of gas altogether disproportionate to their
volume. Even ordinary charcoal, when heated red-hot
(to drive out the gases already condensed in its pores)
and allowed to cool in an atmosphere of carbonic acid
gas, absorbs from sixty to eighty times its volume of the
gas. If it be then introduced into a tube full of mercury
it can be made, by heating, to disgorge this gas, which
it reabsorbs as it cools. This property has been utilized
for the production of very high vacua; as much as
possible of the gas being removed by an air-pump while
the charcoal is hot, and the greater part of the remainder
being absorbed when it cools. 2
The student may easily understand the immense addi-
tion to the surface of a body, which is caused either by
pores or by fine division, if he reflect that a cube, when
sliced once parallel to each of its pairs of faces, obviously
has its whole surface doubled.
297. There is another form of action, analogous to
this, produced by certain substances, such as peroxide of
1 Chem. Soc. Journal., 1865-72.
2 Dewar and Tait, Proc. R.S.E., 1874.
274 PROPERTIES OF MATTER.
manganese, when in a state of fine division. When a
stream of oxygen, containing ozone, is passed through the
powder it emerges as oxygen alone. The ozone has been
reduced to the form of oxygen by what is called Catalysis;
the oxide of manganese is practically unaltered.
298. What is called the solution of a gas in a liquid is,
in many respects, analogous to the condensation on the
surface (or in the pores) of a solid.
The empirical laws of this subject, originally given by
Henry and by Dalton, have been verified for moderate
ranges of pressure by Bunsen.
According to Henry, when a solution of a gas is in
equilibrium with the gas itself, the amount dissolved in
unit volume of the liquid is proportional to the pressure
of the gas. The coefficient of proportionality diminishes
rapidly with rise of temperature.
To this Dalton added that each constituent of a gaseous
mixture is dissolved exactly as it would have been had
the others not been present.
It appears that the heat disengaged in solution is
always greater than that due to the mere liquefaction of
the gas. Hence the phenomenon is, to a considerable
extent, of a chemical character; and thus we are prepared
to find great differences in the absorption of the same gas
by different liquids. Thus carbonic acid is 2 '5 times
more soluble in alcohol than in water; while it is 1'8
times more soluble in water at C. than in water at
15 C.
CHAPTER XIII.
DIFFUSION, OSMOSE, TRANSPIRATION, VISCOSITY, ETC.
299. THOUGH we cannot mark a special group of the
particles of any one liquid or gas, so as to enable us to
see how they gradually mix themselves with the others,
we have almost perfect assurance that they do so. This
assurance is based partly upon the relative behaviour of
two miscible liquids, or two gases, put in contact with
one another; partly upon the results of the kinetic
theory, which have been found fully to explain at least
the greater number of the phenomena ordinarily exhibited
by gases. Thus, altogether independent of the convec-
tion currents due to differences of temperature, there
goes on, in every homogeneous liquid or gas, a constant
transference of each individual particle from place to
place throughout the mass. In homogeneous solids, at
least, it seems probable that there is no such transference,
but that each particle has a mean, or average, position
relatively to its immediate neighbours, from which it
suffers only exceedingly small displacements. [The fact
that diffusion can take place in solids has been shown by
Roberts- Austin. If gold and lead, at ordinary tempera-
tures, be kept in intimate contact for a few years,
penetration of gold into the lead can take place to such
an extent that chemical analysis can show the presence of
276 PROPERTIES OF MATTER.
traces of gold at distances of more than one millimetre
from the original surface of contact. At temperatures
somewhat less than 500, the rate of diffusion of gold into
lead is of the same order as that of the diffusion of soluble
salts into water. Similar results were obtained with
other metals. Henry earlier proved the diffusion of silver
into copper at a high temperature. See note, 253.
W. P.]
300. True diffusion, which is much more rapid in
gases than in liquids, is essentially a very slow process
compared with those convection processes which are
mainly instrumental in securing the thorough inter-mix-
ture of the various constituents of the atmosphere or of
dissolved salts with the ocean water. For its careful
study, therefore, great precautions are required, with a
view to the preservation of uniformity of temperature, as
the only mode of preventing convection currents. We
will suppose that these precautions have been taken.
If, by means of a tube (fitted with a stop-cock) which
is adjusted at the bottom of a tall glass cylinder nearly
full of water, we cautiously introduce by gravity a strong
solution of some highly-coloured salt (such as bichromate
of potash), the solution, being denser than the water,
forms a layer at the bottom of the vessel. If we watch
it from day to day we find that, in spite of gravity,
the salt gradually rises into the water column, which
now shows an apparently perfectly continuous gradation
of tint from the still undiluted part of the solution up
to the as yet uncontaminated water above. The result
irresistibly suggests an analogy with the state of
temperature of a bar of metal which is exposed to a
source of heat at one end. The analogy would be almost
complete if we could prevent loss of heat by the sides
DIFFUSION, OSMOSE, TRANSPIRATION, ETC. 277
of the bar ; for experiment has shown that, just as the
flux of heat is from warmer to colder parts, and (ceteris
parlbus) proportional to the gradient of temperature,
so the diffusion of the salt takes place from more to less
concentrated solution, and at a rate at least approximately
proportional to the gradient of concentration. This is,
possibly, not quite the case at first, when there are
exceedingly steep gradients of concentration, for then
(see 292) there is probably something akin to a surface-
film which for a time behaves somewhat like that between
two liquids which do not mix. This is forcibly suggested
by the result of rough stirring of the contents of a vessel
with parallel glass sides, in which there is a layer of strong
brine with clear water above it ; especially if a horizontal
beam of sunlight, from a distant aperture in the shutter
of a dark room, be made to pass through the vessel,
and be received on a sheet of paper placed a few inches
behind. However rough the stirring, if it be not too
long continued, the mixture is soon seen to settle into
layers of different densities ; and time is required before
diffusion does away with the steep gradients of concen-
tration which have been produced between the layers.
These effects can be produced again and again in the
same mixture, and show how very much more rapid is
the mixing when aided by rough mechanical processes
than when left entirely to the slow but sure effects of
diffusion. The effect of the stirring is to produce im-
mensely extended surfaces of steep gradient of concen-
tration all through the mixture, and thus greatly to
accelerate the natural action of diffusion, to which the
final result of uniform concentration is really due.
301. The first accurate experiments on this subject
are due to Graham, who employed various very simple
278 PROPERTIES OF MATTER.
but effective processes. 1 He showed that while the rate
of diffusion varied considerably with the substances
employed, these could be ranged in two great classes,
Colloids and Crystalloids, the members of the first class
having very small diffusivity compared with those of the
second. Thus he found that the times employed for equal
amounts of diffusion in water were relatively as follows :
Hydrochloric Acid ..... 1
Common Salt 2 '33
Sugar 7
Albumen ...... 49
Caramel 98
He also verified that the rate of diffusion of any one sub-
stance is proportional to the gradient of concentration,
and added the important fact that rise of temperature
has a marked effect in accelerating the process.
302. The subject has since been elaborately investi-
gated by various experimenters, and absolute values of
diffusivity have been calculated from their experiments
as well as from those of Graham.
Following the analogy with heat-conduction, we may
define, after Fourier's method, as follows :
The diffusivity of one substance in another is the
number of units of the substance which pass in unit of
time through unit of surface, when the gradient of con-
centration perpendicular to the surface is unit of substance
per unit of volume per unit of length.
If we use the C. G. S. system, in which unit of length
is a centimetre, unit of mass a gramme, and unit of time
a second, the numbers obtained would be exceedingly
small, so that the C. G. S. system is departed from in
practice to the extent of making a day the unit of time.
1 Chemical and Physical Researches, collected and reprinted
1876.
DIFFUSION, OSMOSE, TRANSPIRATION, ETC. 279
With this we have, according to Stefan's calculations
from Graham's results :
Temperature C.
Hydrochloric Acid . . . 5 174
Common Salt .... 5 076
.... 10 0'91
Sugar 9 0'31
Albumen .... 13 0'06
Caramel . . . .10 0'05
The meaning of this is that, for instance at 10 C., in
water which so holds common salt in solution that there
is one gramme per cubic centimetre more in each hori-
zontal stratum, than in the stratum one centimetre above
it, the upward progress of the salt is at the rate of 0'9 1
gramme through each square centimetre per day. [Solu-
tions of common salt differing by whole grammes of salt
per cubic centimetre are, of course, only a pleasant fiction
of the C. G. S. system.]
303. Fick, Voigt, Hoppe-Seyler, H. Weber, and many
others, have greatly extended Graham's work ; some
using his process (with slight variations), others employ-
ing processes depending upon special physical results
(such as rotatory polarisation, or electromotive force) due
to the salt which is diffusing. It is probable that very
good measures may be obtained, though the method
would be laborious, by using a narrow tank with parallel
glass sides (as in 300), and observing, from time to
time, the greatest refraction suffered by any part of a
horizontal beam of sunlight transmitted through the
heterogeneous liquid, the tank having been originally
half filled (as in 300) with a strong solution of a salt,
under pure water. 1 Lord Kelvin introduced a rough -
1 Tait, "On Mirage," Trans. H.S.E., 1881.
280 PROPERTIES OF MATTER.
and-ready method by letting down into the diffusion
column a number of glass beads, containing more or less
of air, and therefore having, each as a whole, different
mean densities, and observing from day to day the
position of the stratum in which each floated in equili-
brium. This method would probably be the best of all,
could we only make the beads small enough, so as not to
trench upon too many strata at once, and could we also
make certain that neither air-bubbles nor crystals should
develop upon them. The latter condition, however, is
practically unattainable.
304. It seems that the idea of comparing diffusion with
heat-conduction was originally propounded by Berthollet
before Fourier published his great investigations on the
latter subject; but Fick was the first to revive and
develop it in more recent times.
The physical explanation of the cause of diffusion of
liquids in one another, or of solids in a liquid, is vastly
more complex and difficult than that of the diffusion of
gases, though, in some of their coarser features, the
first two of these are closely analogous to the last. In
the words of Clerk-Maxwell: "It is easy to see that if
there is any irregular displacement among the molecules
of a mixed liquid, it must, on the whole, tend to cause
each component to pass from places where it forms a
large proportion of the mixture to places where it is
less abundant. It is also manifest that any relative
motion of two constituents of the mixture will be
opposed by a resistance arising from the encounters
between the molecules of these components. The value
of this resistance, however, depends, in liquids, on more
complicated conditions than in gases, and for the present
we must regard it as a function of all the properties of
DIFFUSION, OSMOSE, TRANSPIRATION, ETC. 281
the mixture at the given place that is to say, its tem-
perature and pressure, and the proportions of the
different components of the mixture."
305. The interdiffusion of gases is thus, theoretically,
a simpler question than that of liquids ; and has been
developed, from the basis of the kinetic theory of gases,
into an almost complete explanation of the observed
phenomena. We cannot here introduce the mathematical
part of the investigation, as it involves analysis of a kind
foreign to the range of an elementary book ; but we
simply state that the equations ultimately arrived at are,
in their simplest form, closely analogous to those obtained
by Fourier for heat-conduction in a homogeneous isotropic
solid. This part of the theory we owe mainly to Clerk-
Maxwell. The experimental part has been well supplied
by Loschmidt. 1 The following numbers give an idea of
his values of interdiffusivity of pairs of gases, in a
mixture at a pressure of one atmosphere. We have
preserved only two significant figures, though the
measures (which are in C. G. S. units) were given to
four.
Carbouic Acid and Air . . . . . 0'14
Oxygen and Hydrogen . . . . .0*72
Carbonic Acid and Hydrogen . . .0*55
Carbonic Acid and Carbonic Oxide . . .0*14
Carbonic Oxide and Hydrogen . . .0*64
According to the theory, as given by Maxwell, these
quantities should be nearly in inverse proportion to the
geometrical mean of the densities (at one atmosphere)
of the two gases. The higher parts of the theory of this
subject are very complex and difficult, and cannot yet be
considered as at all satisfactorily developed.
1 Sitzungs-Bcrichte d. Kais. Ac. zu Wien, 1870.
282 PROPERTIES OF MATTER.
306. Returning to the consideration of liquids, we
meet with certain very curious phenomena when two
miscible liquids are separated by a membrane such as a
diaphragm or septum of bladder, or of parchment paper.
These are usually arranged under the general title of
Osmose, sometimes pedantically divided into endosmose
and exosmose. The first careful examination of them
we owe to Dutrochet, but the earliest observation
recorded is due to Nollet in the first half of last
century.
The main phenomenon, of which all the others are
more or less complicated varieties, is simply that different
liquids pass at different rates through a porous membrane.
Thus Nollet immersed in water a vessel full of alcohol,
tightly closed by a piece of bladder, and was surprised to
find that the contents soon increased to such an extent
as almost to burst the bladder. He then filled the
vessel with water, tied on the bladder, and immersed the
whole in alcohol, when the reverse effect was obtained ;
the contents of the vessel diminished and the bladder
was forced inwards. Strange to say, after so good a
commencement, he contented himself with recording the
two observations.
307. The phenomenon is so obviously connected with
many processes which go on in living bodies, whether
vegetable or animal, that it has attracted the attention
of physiologists as well as of physicists, and an immense
mass of observations on various forms of it has already
been accumulated.
Its theoretical explanation is much more complex than
that of ordinary liquid diffusion, because it is found that
the material of the septum plays an important, often a
paramount, part in determining the rate, and sometimes
DIFFUSION, OSMOSE, TRANSPIRATION, ETC. 283
even the direction, of the osmose in a trial with two given
liquids.
Osmose is undoubtedly a case of ordinary diffusion,
complicated by the molecular actions between the material
of the septum and the various liquids employed. Thus
there need be no more reason for surprise that a
liquid, such as the sap in plants, should be osmotically
raised to great heights against gravity, than that water
should rise in a capillary tube, or that bichromate of
potash should ( 300) diffuse upwards in a column of
water.
308. Something very similar to osmose can be obtained
by ordinary diffusion, when horizontal strata of two
liquids are separated by a stratum of a third liquid .of
intermediate density. Sometimes one or other of the
extremes alone passes through the intermediate layer,
sometimes both diffuse into it. A beautiful method
of gradually developing chemical actions which, on
the large scale, would produce dangerous explosions,
is thus suggested. When nitric acid, water, and alco-
hol are the three liquids, the chemical action takes
place slowly where the two extreme liquids meet,
as they diffuse towards one another through the water-
septum.
309. Though the theory is but imperfectly understood,
the practical applications of osmose have been developed
to an important extent. Of these we need here mention
only the process of Dialysis, due to Graham. The dis-
tinction between Colloids and Crystalloids, in their
behaviour as regards a porous septum, is even more
marked than in direct liquid diffusion. Hence, when a
mixture of colloids and crystalloids, in solution, is placed
on one side of a bladder or a piece of parchment paper,
284 PROPERTIES OF MATTER.
and pure water on the other side, it is practically the
crystalloids alone which pass through the septum into
the water. Even if the colloids be originally in enor-
mous excess, one repetition of the process on the
mixture which has passed through the septum is suffi-
cient to separate the crystalloids almost entirely from
the colloids.
This process is of very great importance as an auxiliary
to chemical analysis in medico-legal questions : for the
more common of the violent poisons are with few excep-
tions crystalloids, and can be easily and almost com-
pletely separated, by dialysis, from the large admix-
ture of colloids in which they are usually found in the
viscera.
310. Graham, in his extensive series of experiments
on the passage of gases through various solids with holes
or pores, recognised several quite distinct processes, each
with its own laws.
When a gas is maintained at constant pressure on one
side of a very thin non-porous plate, which has a small
hole in it, there being vacuum at the other side, the
process of passage is called Effusion. This may be
looked on as at least roughly analogous to the passage of
a liquid through the orifice. The closer consideration of
it belongs to Thermodynamics. The work done on unit
volume as it passes out is directly as the pressure, the
kinetic energy acquired is measured by the density and
the square of the speed of effusion conjointly. Hence,
under the same pressure, the speed of effusion is inversely
as the square root of the density. This result was very
nearly realised in Graham's experiments ; witness the
following :
DIFFUSION, OSMOSE, TRANSPIRATION, ETC. 285
Time of Effusion. Theoretical Time.
Air .... I'O I'O
Nitrogen . . . 0'984 0'986
Oxygen . . . 1D50 1-051
Hydrogen . . . 0'276 0'263
Carbonic Acid . . 1'197 1'236
The only discrepancies which call for notice are with
hydrogen and carbonic acid. But Graham was able to
show from the results of another of his series of experi-
ments, that these discrepancies are due to the fact that
the perforated plate was not infinitely thin, and that the
aperture therefore behaved like a very short capillary
tube. This explanation is fully borne out by the fact
that the discrepancies are in opposite directions for these
two gases, and that this characteristic difference is required
by the mode of explanation.
From these experiments Graham concluded that the
law of this process is analogous to that of diffusion with-
out a septum. Bunsen has applied the result to the
construction of a very excellent instrument for measuring
the density of a gas.
311. Transpiration is the name given by Graham to
the passage of a gas under pressure through a capillary
tube. The results obtained were of a much more complex
character than in the case of effusion ; and the law of
the process, so far as it could be ascertained by experi-
ment alone, was of a different form. Capillary tubes,
varying in length from 20 feet down to a few inches,
were employed. It was found that the material of the
tube had no influence ; hence it has been suggested that
the tube becomes lined with a film of the gas, and that
the key to the difficulties of the problem is to be sought
mainly in connection with viscosity. The rate of trans-
piration of hydrogen is only double of that of nitrogen,
286 PROPERTIES OF MATTER.
while that of carbonic acid is much greater than that of
oxygen :
Limiting Transpiration Times
in vefy fine Capillaries.
Oxygen I'OOO
Air 0-901
Nitrogen and Carbonic Oxide . 0'875
Hydrogen 0'437
Carbonic Acid .... 0727
The two last results show the foundation of the explana-
tory remark towards the end of last section.
The following are some of Graham's comments on
this very curious subject :
"The times of oxygen, nitrogen, carbonic oxide, and
air, are directly as their densities, or equal weights of
these gases pass in equal times. Hydrogen passes in
half the time of nitrogen, or twice as rapidly for equal
volumes. The result for carbonic acid appears at first
anomalous. It is that the transpiration time of this gas
is inversely proportional to its density, when compared
with oxygen. It is to be remembered, however, that
carbonic acid is a compound gas, containing an equal
volume of oxygen. The second constituent carbon
which increases the weight of the gas, appears to give
additional velocity to the oxygen in the same manner
and to the same extent as increased density from pressure,
or from cold (as I believe I shall be able to show), in-
creases the transpiration velocity of pure oxygen itself.
A result of this kind shows at once the important bear-
ing of gaseous transpirability, and that it emulates a
place in science with the doctrines of gaseous densities
and combining volumes.
"The circumstance that the transpiration time of
hydrogen is one-half of that of nitrogen, indicates that the
DIFFUSION, OSMOSE, TRANSPIRATION, ETC. 287
relations of transpirability are even more simple in their
expression than the relations of density among gases.
In support of the same assertion may be adduced the
additional fact; that binoxide of nitrogen, although
differing in density, appears to have the same transpira-
tion time as nitrogen. Protoxide of nitrogen and
carbonic acid have one transpiration time, so have
nitrogen and carbonic oxide, as each pair has a common
density."
312. When one gas is separated from another, or from
a vacuum, by a septum of compressed graphite ( 53), the
law, and even the rate, of passage come to be very nearly
the same as those of ordinary gaseous diffusion. Thus
gases pass through such a septum at rates inversely as
the square roots of their densities, as in effusion. If the
septum is made of plaster of Paris, the results become
partially complicated by transpiration. This source of
confusion is practically non-existent when the septum is
made of " biscuit-ware," as it is technically called ; and
the same may be said of all the finer kinds of unglazed
earthenware. Here the pores are so tine that, as Graham
says, the action ceases to be molar and becomes molecular.
Each particle acts, as it were, on its own account. Hence,
when a mixture of two gases of different densities is
placed on one side of such a septum, the less dense gas
passes in greater percentage than the denser, and we
have Atmolysis : a mode of separating different gases
somewhat akin to dialysis ( 309). There are few physical
experiments more striking and suggestive than the simple
one of surrounding, with an atmosphere of coal gas, the
bulb (made of unglazed clay) of an arrangement like a
large ordinary air-thermometer. The rapidity with which
the gases pass through the bulb is extraordinary.
288 PROPERTIES OF MATTER.
313. But when the septum is made of caoutchouc the
process of penetration is quite different. The septum
now acts as a colloidal body, not as a porous one ; and
the gas combines in an imperfect chemical manner with
the matter of the septum, in which it diffuses (in the
ordinary sense of the term), until it reaches the other
side and is set free. Thus the small toy-balloons of thin
india-rubber, when originally filled with hydrogen, soon
collapse. On the other hand, when they are blown with
air and then immersed in an atmosphere of hydrogen,
they rapidly swell and burst.
The same phenomenon is beautifully shown by blow-
ing a soap-bubble with carbonic acid gas., For the gas
dissolves in the liquid film, diffuses through it, and
escapes into the air, so that the bubble soon collapses.
Similarly, an ordinary soap-bubble made to float on car-
bonic acid gas expands, gradually sinks, and finally bursts.
A good instance of gaseous diffusion is afforded by
evaporation of water, or other liquid, at temperatures
below the boiling point, when air is present. For the
process goes on until the vapour in contact with the
liquid has a pressure determined solely by the tempera-
ture, and by the curvature of the liquid surface. When
a layer of vapour of the proper pressure has once formed
at the surface, the resistance to its diffusion is so con-
siderable that, unless there be wind or convection-currents,
the rate of evaporation is reduced to that of diffusion ;
and vapour is formed (at the liquid surface) only as fast
as that which is already formed is able to get away. By
weighing, from time to time, a test-tube of known length,
which has a layer of liquid at the bottom and is open at
the top, fair measures of the rate of diffusion in air, of
vapours heavier than air, can be obtained,
DIFFUSION, OSMOSE, TRANSPIRATION, ETC. 289
314. Very curious results have been obtained by
Deville and Troost with reference to the rapid passage
of various gases through heated cast-iron. Carbonic
oxide is one of these, and as this is a highly poisonous
gas, the matter is one of great importance in relation to
the heating of rooms by stoves. They also showed that
highly heated platinum is freely pervious to hydrogen.
Graham's researches on the behaviour of palladium with
respect to hydrogen have afforded the means of obtaining
similar effects even at temperatures far below red-heat ;
and, quite recently, v. Helmholtz and Root have proved
that platinum is pervious to hydrogen even at ordinary
temperatures. Thus the question is one of importance,
not alone from the sanitary point of view, nor from the
point of view of its purely scientific explanation, but
also from the very important point of view of the con-
struction of gas-thermometers for the measurement of
high temperatures, in which the recipient must necessarily
be made of some practically infusible metal. The whole
of this part of the subject, however, has a specially
chemical interest, so that we are not called on to discuss
it further.
315. We have already employed the word Viscosity in
two somewhat different applications. In our general
discussion of common terms ( 37) we spoke of it as
applied to liquids, and also, by parity of results, to gases.
But in 249 we used it as denoting a property possessed
even by the most elastic of solids.
We must now consider, more carefully, its application
to fluid motion.
And, first, as regards liquids. Questions such as were
briefly touched upon in 37 belong, in their full develop-
ment, where eddies present almost insuperable difficulties,
T
290 PROPERTIES OF MATTER.
to Hydrokinetics, and are therefore not to be treated
farther in this work. But the passage of a liquid, under
pressure, through a capillary tube, is (so far as it is
amenable to elementary mathematical treatment) part of
our subject. So is the torsional vibration of a disc, in
its own plane, when it is suspended by a wire and im-
mersed in a fluid, especially when, as in Clerk-Maxwell's
experiments on gases, other two discs, are fixed near and
parallel to it, on opposite sides. So far as liquids are
concerned, these forms of experiment were carefully
worked out by Poiseuille and by Coulomb respectively,
and have since been extended, with various modifications,
by v. Helmholtz, O. E. Meyer, etc. Later, Graham (as
we have just seen), Clerk-Maxwell, and many others,
have applied one or other of these forms of experiment,
more or less modified, to the determination of the
viscosity of gaseous bodies.
316. Before going farther, we must define precisely
what we mean by viscosity ; and the definition will, of
course, show how it is to be measured.
Suppose a layer of fluid, of unit thickness, to fill the
interval between two plane surfaces, of indefinite extent, to
which the fluid adheres. When one of these surfaces is
made to move in a given direction parallel to the other,
with unit speed, the tangential force on either per unit of
surface is the measure of the viscosity.
Hence, if, in Fig. 1, 38, v be the speed at depth y,
the tangential stress per unit surface of the layer at that
depth is
dv
where K is the viscosity. The dimensions of Viscosity,
therefore, differ from those of Rigidity ( 178) simply by
DIFFUSION, OSMOSE, TRANSPIRATION, ETC. 291
the time unit; i.e. as the dimensions of velocity differ
from those of acceleration. This may be seen at a glance
from the equation of 250.
The establishment of a simple working definition, such
as that above, leads at once to the formation of the proper
equations of motion in all problems of this kind. The
process is precisely the same as that adopted by Fourier in
his definition of Heat Conductivity ; and it is curious to
see how all who have, in modern times, treated viscosity
without using Fourier's method, have fallen into the vague
and misleading methods of Fourier's predecessors.
317. If, with this definition, we investigate the motion
of a liquid in a capillary tube, when it has become steady,
we are led to the result (fully borne out by the experi-
ments of Poiseuille 1 ) that, ceteris paribus, the discharge
in a given time is proportional to the fourth power of the
radius of the bore. (Compare 228.)
For the solution of the problem we assume that the
motion is everywhere parallel to the axis of the tube, and
with speed dependent only on the distance from it. Let
K be the viscosity, and consider the tangential stress (per
unit length of the tube) on the surface of a cylindrical
layer of liquid of radius r, concentric with the tube. If v
be the speed of that layer, the amount of the stress is
(by the definition above)
n dv
2-imc , .
dr
Hence the difference of the tangential forces on the surfaces
of a cylinder of liquid of thickness 8r is
>KK dr\
1 Mem. des Savans Strangers, IX., 1846.
292 PROPERTIES OF MATTER.
But, as the motion is not accelerated, this must be equal
and opposite to the difference of the pressures on the ends
of the cylinder, which is (per unit length)
where x is measured parallel to the axis. This must
obviously be independent of x, and, as the motion is
always very slow under the conditions, p is approximately
independent of r. Hence
a constant, whose obvious value is the difference of
pressures at the ends, divided by the length, of the
tube.
Thus we have the very simple equation
dr
This gives
r dv == _ _ r2
dr 2 K '
the integration constant being zero, because otherwise we
should have finite tangential stress on an infinitely small
filament along the axis.
Thus, if r Q be the radius of the bore of the tube, and if
we assume that v is nil when r = r , (i.e. that there is no
finite slipping of the liquid along the walls of the tube)
The whole volume of liquid which passes in unit of
time through each cross section is
DIFFUSION, OSMOSE, TRANSPIRATION, ETC. 203
318. This expression enables us at once to calculate the
values of K from experiments such as those of Poiseuille.
Their agreement with the formula above is very close
throughout, the bores, lengths, and pressures being varied
within wide limits. The most remarkable additional
feature which Poiseuille recognised is the rapid diminution
of viscosity of water with rise of temperature. In C. G. S.
units his experiments give, approximately,
Temperature C. Coefft. of Viscosity.
0-018
10 0-013
20 O'OIO
This means that there is a tangential stress of 0*018
dynes per square centimetre on each of two parallel planes,
one centimetre apart, when one is moving relatively to
the other at the rate of one centimetre per second, and
when the interspace is filled with water at C.
It is well to note that from to 10 C. the viscosity of
water falls off at the rate of 2'8 per cent, per degree. Com-
pare this with the corresponding increase of the rate of diffu-
sion of common salt in water ( 302), which, by Graham's
results, is about 4 per cent, per degree of temperature.
319. The oscillation method of Coulomb is trouble-
some in the complex mathematical details to which it
leads, and even more so in the experimental precautions
which it requires. It has been carefully worked out, from
both points of view, by many different physicists, including
especially O. E. Meyer. Modifications of it have been
employed by v. Helmholtz and others. The theoretical
part of the investigation, which is very complex, was first
developed by Stokes, who applied it to the reduction of
pendulum experiments.
294 PROPERTIES OF MATTER.
The results of Meyer are about a sixth greater than
those of Poiseuille. Those of v. Helmholtz and v.
Pietrowski, 1 in which liquids were contained in an
oscillating sphere, were complicated by finite slipping,
which led to a new problem. Their result for water is
about one-fourth greater than that of Poiseuille.
According to Schbttner, 2 the coefficient of viscosity of
glycerine (in C. G. S. units) sinks, from about 42 at 3 C.,
to little more than 8 at 20 C.
320. The Viscosity of a gas can be calculated, at least
approximately, from the Kinetic theory, for it is easy to see
that it must depend upon the transference of momentum
by the interchange of particles between two contiguous
layers of the gas which have relative velocity.
Clerk-Maxwell, who first gave the theory of this subject,
found that the viscosity is independent of the density in
each particular gas, and increases with rise of temperature,
being directly proportional to the square root of the
absolute temperature.
But his experiments on air, made ( 315) by the oscil-
lation method, gave (in C. G. S. units) the formula
0-000,000,685(274 +
where t is the temperature centigrade. Here a different
temperature-law appears. 3
Maxwell also showed that in oxygen the viscosity is
greater than in air, and in carbonic acid less. In hydrogen
it is about half as great as in air. Theoretically it is as
the density of the gas, and the mean free path of a
particle, conjointly. The mean free path depends upon
1 Siteungsber. der K. Ac. in Wien, 1860.
2 Ibid., 1878, p. 686.
3 Stokes (Phil. Trans., 1886) attributes this discrepancy to non-
parallelism of the fixed and oscillating plates.
DIFFUSION, OSMOSE, TRANSPIRATION, ETC. 295
the size of the particles, being (ceteris paribus) inversely
as the squares of their diameters. [Compare with Graham's
results above, 310, 311.] This subject has since been
elaborately investigated by Meyer, Kundt and Warburg,
and many others, but the exact law of the temperature-
variation is still uncertain.
CHAPTER XIV.
AGGREGATION OF PARTICLES.
321. THIS chapter must be a very short one, oecause,
though experimental facts are to be had in profusion, the
subject, as a whole, has not yet been raised to the higher
level of Science from that of the mere preliminary " beetle-
hunting or crab-catching stage." Parts of it are already
much further advanced. The geometry of crystalline
forms has been very fully developed and systematised.
The physical properties of the aggregate have been
scientifically developed, as we have seen, mainly from
the basis of Hooke's Law for solids and liquids, and of
Boyle's Law for gases ; and the formulation of these laws
has enabled us to discuss, with something like a secure
foothold, the deviations from them.
But the mode of formation of the aggregate of particles,
at all events in solids and liquids, is a question of much
greater difficulty. We still require, in fact, a Kepler to
co-ordinate the facts, before there can be a chance for a
Newton to group them under some simple but all-embracing
statement.
All that our plan permits us to do is to point out
briefly how far the Ptolemy and the Copernicus, ab well
as the Tycho Brahe, of this subject have marshalled the
AGGREGATION OF PARTICLES. 297
materials for the coming Kepler. The Newton will be
later in appearing.
322. In the case of gases a real step to explanation
has been taken, but a great part of the very elements of
the Kinetic Theory ( 33, 107) is still obscure and
difficult. The earliest suggestion of it is commonly
attributed to D. Bernoulli (1738), but the following
passage from Hooke's Pamphlet De Potentia Restitutiva, 1
shows that essentially the same ideas had been published
long before.
" In the next place for fluid bodies, amongst which the
greatest instance we have is air, though the same be in
some proportion in all other fluid bodies.
" The air then is a body consisting of particles so small
as to be almost equal to the particles of the Hetero-
geneous fluid medium incompassing the earth. It is
bounded but on one side, namely, towards the earth, and
is indefinitely extended upward being only hindered from
flying away that way by its own gravity (the cause of
which I shall some other time explain). It consists of
the same particles single and separated, of which water
and other fluids do, conjoyned and compounded, and
being made of particles exceeding small, its motion (to
make its balance with the rest of the earthy bodies) is
exceeding swift, and its Vibrative Spaces exceeding large,
comparative to the Yibrative Spaces of other terrestrial
bodies. I suppose that of the Air next the Earth in its
natural state may be 8000 times greater than that of
Steel, and above a thousand times greater than that of
common water, and proportionably I suppose that its
motion must be eight thousand times swifter than the
1 See footnote to 221. Also the quotation from Boyle ( 188)
where he speaks of the space the air " possess' d notfill'd."
298 PROPERTIES OF MATTER.
former, and above a thousand times swifter than the
latter. If therefore a quantity of this body be inclosed
by a solid body, and that be so contrived as to compress
it into less room, the motion thereof (supposing the heat
the same) will continue the same, and consequently the
Vibrations and Occursions will be increased in reciprocal
proportion, that is, if it be Condensed into half the space
the Vibrations and Occursions will be double in number :
If into a quarter the Vibrations and Occursions will be
quadruple, etc.
" Again, if the containing Vessel be so contrived as
to leave it more space, the length of the Vibrations will
be proportionably enlarged, and the number of Vibrations
and Occursions will be reciprocally diminished, that is, if
it be suffered to extend to twice its former dimensions,
its Vibrations will be twice as long, and the number of
its Vibrations and Occursions will be fewer by half, and
consequently its endeavours outward will be also weaker
by half.
" These Explanations will serve mutatis mutandis for
explaining the Spring of any other Body whatsoever."
The modern revival of the theory is due to Herapath ;
and, a little later, to Joule, who was the first to make
definite calculations as to the speed of the particles of a
gas necessary for the production of the observed pressure
at different temperatures. He stated distinctly that his
results were independent of the size of the particles, and
of the number of collisions. Krb'nig is constantly cited,
especially in German works, as having advanced the
theory ; but the only novelty which his paper * seems to
contain is the somewhat startling, and certainly as yet
unverified, result that the weight of a gas when in motion
1 Pogg. Ann., 1856, xcix. p. 319.
AGGREGATION OF PARTICLES. 299
is only half what it is when the gas is at rest. [He for-
gets to take account of the additional impulse due to
restitution, 255.]
The first approach to a thorough treatment of the
theory was made by Clausius, who took account of the
mutual impacts of the particles, with the consequent con-
ception of the mean free path ; and also introduced the
statistical method of treatment. He was followed by
Clerk-Maxwell, and by Boltzmarin, and among the three
the theory was rapidly developed.
It is already competent, as is shown in works on Heat
(to which the theory now properly belongs) to explain
fully many of the properties of gases ; but it still labours
under an unsurmounted difficulty, viz. the explanation
of the diversity of values of the ratio of the two specific
heats (at constant pressure and at constant volume) in
various groups of gases. The difficulty is probably due
to our ignorance of the interior mechanism of the
particles of gases ; and it has been greatly enhanced by
an apparently unwarranted application of the Theory of
Probabilities, on which the statistical method is based. But
the examination of such questions is foreign to our work.
[Kecent views, expressed by Jeans, regarding the partition
of energy amongst the different degrees of freedom of the
particles, at least lessen the difficulty referred to. W. P.]
323. The key to the explanation of the liquid state is
undoubtedly to be sought in connection with Andrews'
grand discovery of the Critical Temperature ( 194).
Whether something akin to this does, or does not, hold
with reference to the relation between the solid and
liquid states, is a problem which does not appear to have
been attacked. It presents, undoubtedly, most formid-
able difficulties, experimental as well as theoretical,
300 PROPERTIES OF MATTER.
which are heightened by the well-known fact that there
are solids whose melting point is lowered by pressure.
But here, again, we trench on the domain of Heat.
324. The essential difference between the solid and
the liquid states of any kind of matter lies in the fact
that any distorting stress, however small, if only persist-
ently applied, produces finite change of arrangement
among the particles of a liquid, whereas it can in general
but innnitesimally alter the relative position of contiguous
particles of a solid (unless, of course, it be sufficient to
produce rupture of the mass). It is barely conceivable
that there can be in any case an abrupt transition from
one of these states to the other ; and, in fact, in the
great majority of cases at least, there is known to be a
gradual transition from the solid to the liquid states, as
the temperature is raised in the vicinity of the melting
point, so that there is continuous passage from the state
of very plastic solid to that of very viscous liquid. The
same thing is observed in various colloidal bodies such as
isinglass and other jellies, when made up with different
amounts of water. In this connection the student
should again read the remarkable statement of Clerk-
Maxwell in 253 above.
Especially on the subject of crystallisation do we
appear to get some light from such a view of inter-
molecular action. For it would seem that an essential
requisite to the formation of a homogeneous crystal must
be the comparative freedom of each particle from the in-
fluence, direct or not, of all besides those in its immediate
vicinity. That there is, even in the most homogeneous
crystals, still at least a trace of what Maxwell calls the
more stable groups, is probably indicated by the exist-
ence of Cleavage Planes, which are not in general parallel
AGGREGATION OF PARTICLES. 301
to the more prominent faces of the crystal. In fact, as
Sorby has shown, cleavage planes analogous to those of
slate-rocks can be developed in the majority of plastic
solids by the application of pressure-stress in a direction
perpendicular to them. It is thus that slates themselves
are formed from a deposit of mud. The proof is obvious
from the fact that the planes of cleavage are often in-
clined at large angles to those of stratification. But we
offer these remarks merely as a suggestion, which our
limits prevent us from developing.
In this connection it may be well to mention an acute
remark of Le Roux * to the effect that annealing appears
to preserve the state of isotropy characteristic of fusion.
He was led to this by remarking that a glass of borate
of magnesia, which was ordinarily transparent, became
like porcelain or, rather, like white marble when very
quickly cooled.
[A great amount of light has recently been thrown upon
the question of the structure of solid matter, such as solid
metals. Quincke has shown that, even in the purest
attainable molten metal, there is a sufficient amount of
impurity present to give rise to effects which are char-
acteristically exhibited when the substance solidifies.
Surface tension exists at the boundary between the molten
metal and the molten impurity, and, in consequence, the
impurity spreads out in a thin film which forms a closed
surface, or cell, containing some of the molten metal.
These cell-walls may join together so as to form a con-
tinuous "foam-structure." When the walls and their
contents are both liquid, the complex forms a liquid
"jelly." If the walls, or the contents, or both, are solid,
the jelly is stiff. If the wall material is very viscous, a
1 Comptes Rendus, 1867, p. 126.
302 PROPERTIES OF MATTER.
liquid jelly may flow with extreme slowness. Character-
istic features of ice-structure are explainable on these
views. The existence of the cells, on a microscopical
scale, in metals can readily be made evident by etching a
polished surface by an acid, provided that the acid acts at
different rates on the walls and their contents.
Under slow cooling, neighbouring cells may coalesce, so
that ultimately large cells are formed. Under rapid
cooling the cells are small. Thus arises the characteristic
difference between the annealed and the unannealed
states of a metal. In ice formed slowly, hexagonal cells
which measure a number of feet across may appear.
The cell- walls or cell-contents of a liquid jelly become
doubly refracting when strained. This condition becomes
permanent if the walls or contents solidify while strained.
The so-called "liquid crystals" recently studied by
Lehmann and others may possibly be only strained foam
cells in such a liquid jelly. On the other hand, it may
be possible that, at suitable temperatures, stable crystalline
arrangements may exist, when the fluid is at rest, although
the resistance to shear is so slight that in consequence of
surface tension crystalline form is not exhibited.
The slight rigidity and torsional set exhibited in some
very weak solutions of a colloid may perhaps be due to
foam-structure. W. P.]
325. Whether all solids tend to acquire ultimately a
crystalline structure or riot, we can at least seek to find
the forms which they are likely to assume in cases (such
as that of slow deposition from a state of solution) where
each particle is free to choose its position of least poten-
tial energy relative to those already deposited. This
was long ago very well worked out by Hatty, though (of
course) without any reference to potential energy. And
AGGREGATION OF PARTICLES. 303
the agreement, of the results of such hypothetical calcula-
tions, with the observed forms of natural and artificial
crystals, shows that we have really made a step in the
right direction (though a very short one) to the explana-
tion of these singular and beautiful results of the action
of molecular forces.
326. The simplest case is that in which the separate
particles behave as if they were spherical, i.e. as if they
exerted equal resultants of molecular arid of thermal
action (at the same distance) in all directions. Here we
can call to our assistance the analogy of well-known
results as to the piling of shot, etc.
There are two ways in which we may suppose the
first plane layer to be laid down ; i.e. in square order,
or in equilateral triangular order. Again there are two
ways in which the next layer may be (symmetrically)
deposited, i.e. particle above particle, or particle above
the middle point of each square, or of each equilateral
triangle. The two latter cases, however, are really the
same arrangement, so far as the relative positions of
contiguous particles are concerned ; as we see at once by
looking at one of the faces of a four-sided pyramid
built up on a square base. For in the face the order is
triangular. Or, if we remove one edge from the three-
sided pyramid built on an equilateral triangular base,
we find the particles in the plane of replacement to be
in square order.
We find these two forms, apparently so different, in
many species both of natural and of artificial crystals.
Thus the regular oktahedron, which is merely two four-
sided pyramids built on opposite sides of the same square
base, and its hemihedral form, the regular tetrahedron,
which is the three-sided pyramid built on an equilateral
304 PROPERTIES OF MATTER.
triangular base, are both met with in various substances
belonging to the Cubic System. The measured angles
between the several pairs of faces are found to agree
exactly with those of the geometrical solids.
327. An imperfectly developed oktahedron, when the
imperfection is symmetrical, may assume various forms,
all of which are met with in actual crystals. Thus,
suppose we remove layer after layer, symmetrically,
from the summits of the oktahedron. The new faces
thus produced form parts of the faces of a cube ; and
we may obtain the cube complete by continuing the
process till the last trace of the oktahedral faces has been
removed.
Replace, symmetrically, the edges either of the oktahe-
dron or of the cube, and we produce a set of parts of the
faces of what is called a Rhombic Dodekahedron (Fig. 41).
Many natural crystals assume symmetrical forms con-
taining faces belonging respectively to the oktahedron,
the cube truncating its summits, and the rhombic dode-
kahedron replacing its edges.
Instead of replacing each edge by a single plane
equally inclined to the faces which meet in that edge,
suppose we bevel it symmetrically by two planes. We
shall now no longer have a form with fixed angles (i.e.
invariable shape) as in the cases previously mentioned,
but one which (though its general character is determined)
will have its form dependent on the inclination of the
bevelling planes to the faces which meet in the edge
bevelled. Here, again, the geometrical results are found
to fit the forms actually presented in nature.
It is an interesting and instructive work for the
student to verify these statements by operating on a
lump of soft chalk, or (still better) stiff putty or
AGGREGATION OF PARTICLES. 305
modeller's clay, by means of a broad, but thin, bladed
knife.
328. This is not the place to enter into crystallo-
graphic details, but we may introduce the following
statement, which includes all the above results, and
is found, at all events, to accord fully with the general
appearance of any of the very numerous crystals belong-
ing to what is called the cubic system.
Any three axes which meet, but which do not lie in
one plane, being chosen, the equation of any plane
whatever, which does not pass through the origin, can
be written in the form
Here h, &, and I are finite quantities, the reciprocals of
the distances from the origin at which the plane meets
the axes respectively.
To obtain all the plane faces of any one simple
form of crystal, all we have to do is to give to
h, Jc, I in the above expression all admissible values
in succession.
329. When we take the cubic system, of which alone
we have yet spoken, symmetry shows that if the three
axes be taken as the lines joining pairs of opposite
summits of the oktahedron, obtained as in 326 above,
these will possess properties absolutely alike. Thus
symmetry further shows that the numbers h, Jc, I may be
arranged in any order, for what is true of any one of the
axes is true of each of the others. Similar considerations
show that each of h, k, Z, independent of the others, may
be either positive or negative. These quantities are
always found, in actual crystals, to have their ratios
rigorously expressible in small Whole Numbers. [This,
of itself, is a very strong argument in favour of the
U
306 PROPERTIES OF MATTER.
notion that the crystal is built up of little parts, all equal
to one another.]
The numbers h, It, I can be arranged in six different
orders ; and, as any one of these orders has eight possible
arrangements of signs, there are forty-eight symmetrically
arranged plane triangular faces on the most general simple
form of this system. This form is found in many natural
crystals, and is called a hexakis-oktahedron (Fig. 37).
When two of h, k, I are equal, it is usual to divide the
forms into two groups according as the third is less or
greater than the others. JBoth classes have twenty-four
faces only, but in the first group they are triangular, in
the second quadrilateral. These are, respectively, the
triakis-oktahedron (Fig. 38) and the eikositetrahedron
(Fig. 39) which are derived respectively from the
oktahedron by symmetrical bevelling of its edges, and
by symmetrical blunting of its summits.
Intermediate to these there is the case when all three
of h, k, I are equal, and we have the oktahedron itself.
There is also a limiting case when the third vanishes.
This is the rhombic dodekahedron (Fig. 41) discussed in
327 ; and of course it also forms the limiting case of the
series when one of h, k, I vanishes and the others are un-
equal, called the tetrakis-hexahedron (Fig. 40), obtained
by symmetrical bevelling of the edges of the cube.
When two vanish we have the
cube.
The following figure shows the
Hexakis-Oktahedron, an octant (lying
among the positive axes of x. y, 2),
being specially lettered for reference,
and we can easily see how it degrades
into the other forms. FIG. 37.
AGGREGATION OF PARTICLES.
307
Thus if xCy, and therefore yAz and zBx, become
straight lines (i.e. if the planes marked a and b t c and d,
e and/, coincide in pairs), the figure
becomes the triakis-oktahedron (Fig.
38). This degrades into the okta-
hedron if D be in the plane xyz (i.e.
if a, b, c, d, e, f, are all one plane) :
and into the rhombic dodekahedron
if DC be perpendicular to the plane
of xy (i.e. if a, b, a', b', are all parts
of one plane). FIG - 38 -
Again, if the planes/ and a, b and c, d and e, coincide
in pairs, we have the eikositetrahedron (Fig 39).
FIG. 39.
FIG. 40.
FIG. 41.
If ADB be parallel to the plane of xy, etc., the angle
ADB is a right angle, the planes / and /', a and a', b and
b', etc., coincide in pairs, and the figure is the tetrakis-
308 PROPERTIES OF MATTER.
hexahedron (Fig. 40), which becomes the cube when/',/,
a, a', etc., are all in one plane ; and the rhombic dodeka-
hedron when a, a, b and //, are all in one plane.
330. Six faces of the rhombic dodekahedron are
parallel to the line joining D with the origin, and are
situated symmetrically round it. If, then, these faces
be extended in their own planes without any alteration
of the two groups of three forming the other faces, the
whole will become a regular hexagonal prism with ends
consisting each of three equal rhombic faces with their
greater angles in contact at the summit. This is found,
by measurement, to be the form of a bee's cell. And a
series of these dodekahedra (all equally and similarly
distorted from equal rhombic dodekahedra) can (like
them) be so packed together as to fill space without
leaving interstices, as in a honeycomb.
331. If, instead of building up a mass of equal spheres,
we use similar, equal, and similarly situated ellipsoids of
revolution, we must make corresponding alterations in
our rules for h, k, I above. If the chief axes of the ellip-
soids be perpendicular to the layers of particles, the
x, y, z axes are still a rectangular system, but h (say) is
no longer interchangeable with k or with I. For h is
now a small integral multiple of a parameter which
depends on the chief axis of the ellipsoid, while k and I
are similar multiples of the (equal) parameters of the
other two axes. This greatly reduces the number of
possible faces in the simple forms.
If the particles are similar, equal, and similarly situated
ellipsoids, touching one another in a layer by the ends of
two of their axes, the x, y, z system is still rectangular,
but no two of h, k, I are interchangeable.'
In any other arrangement, which is capable of giving
AGGREGATION OF PARTICLES. 309
a homogeneous whole, the simplest x, y, z system is no
longer rectangular ; and the ellipsoids, though still
similar, equal, and similarly situated, may come in
contact (in threes) in an infinity of different ways.
There is no known form of crystallised matter whose
separate faces cannot be exactly accounted for by results
based on these premises, though there are many cases
of hemihedry, etc., in which the faces geometrically
determined for a simple form present themselves only in
selected groups.
332. In what we have said above, the only assumptions
made (for the purpose of explaining homogeneity) were
that the particles grouped were themselves equal, similar,
and similarly situated ; and that the arrangement of its
neighbours round it was exactly the same for each
particle.
But it is easy to conceive that very different results
may be obtained, even with identical materials, according
to circumstances. Every one who has seen water which
is full of excessively small ice crystals, and is pre-
vented only by currents from becoming solid ice, may
easily imagine a state of things in which the particles
(which, otherwise, would have been deposited one by
one to form a crystal) may arrange themselves in very
small, but similar and equal, groups before they are
deposited. Thus the aggregates, above contemplated,
may be built up, not directly of individual particles,
but of other less complex though (among themselves)
similar and equal sets of aggregated particles. Here
again we come back to the same idea as that in the
quotation from Clerk-Maxwell ( 253), and may employ
it for the purpose of explaining the existence of cleavage
planes, etc.
310 PROPERTIES OF MATTER.
333. The aggregations we have considered have been
such as take place freely; but if we consider what is
likely to happen under circumstances of temporary or
permanent stress, as, for instance, in a Rupert's drop,
or any other melted mass of which a portion is cooled
and solidified more suddenly than the rest, we see
that we cannot expect a result in which the potential
energy of the whole shall be as small as possible ;
and are, therefore, prepared to find that such bodies,
unless carefully annealed, are essentially in unstable
equilibrium.
[CHAPTER XV.
DISINTEGRATION OF THE ATOM.
334. A CONSIDERABLE number of the so-called elements
possess atomic weights which are exactly or very nearly
multiples of the atomic weight of hydrogen. This fact
suggests a common origin, and Prout long ago put forward
the hypothesis that the atoms of the other elements were
compounded of atoms of hydrogen. The hypothesis of a
common origin cannot now be maintained in this crude
form ; for further investigation has shown that small
divergences from integral multiples of the atomic weight
of hydrogen are real, and has shown also that large
divergences exist.
But the discovery by Mendeleeff of a periodic relation
connecting chemical and physical properties of the
elements with their atomic weights, has placed the idea
of a common origin on a more certain basis. So definite
is this relation that, by its means, the atomic weights of
unknown elements have been predicted; and the pre-
dictions have been subsequently verified. At present
the relation has a purely empirical foundation, but its
theoretical establishment in the future may be confidently
anticipated.
335. If, then, the atoms have been built up as a
congeries of smaller components, it might be expected
312 PROPERTIES OF MATTER.
that, under suitable circumstances, their disintegration
would occur. Until recently, such disintegration had
never been observed.
Phenomena which are evident when an electric dis-
charge passes through a rarefied gas in a vacuum tube
were explained by J. J. Thomson on the assumption
that they were due to the rapid propulsion of small
electrified particles from the negative electrode in the tube.
It is possible by suitable methods to measure the mass
of these particles. Whatever be the chemical nature of
the electrodes, or of the gas in the tube, the result is the
same, and the mass indicated is of the order of one
thousandth part of the mass of a hydrogen atom. These
facts are practically conclusive in favour of the idea that
we are here dealing with the elemental unit out of which
material substances are formed.
336. These corpuscles always carry individually a
definite charge of negative electricity ; and the possession
of this charge contributes, in accordance with the known
laws of electricity in motion, to their inertia. Whether,
or not, all inertia is of this type is at present an open
question. Any such " explanation " of inertia, however,
merely transfers the difficulty to another stage. Just as,
in the vortex atom hypothesis, inertia must be ascribed to
the medium whose rotating parts form the atoms ; so the
inertia which is specially manifested when the motion of
electric charges is accelerated must be ascribed to the
medium through which electrical action is propagated.
In Larmor's view (see 18 and App. V.) these
negatively charged corpuscles otherwise called electrons,
or electrions are essentially centres of rotational strain
in the ether. Without considering in detail the nature
of their relation to ether, J. J. Thomson, Kelvin (see
DISINTEGEATION OF THE ATOM. 313
App. VI.), and others, have developed to some extent the
properties of an atom regarded as constituted of electrions
at rest relatively to, or revolving round, the centre of
a uniform sphere of positive electricity. Thomson has
shown that a series of such atoms, differing in the number
of electrions present in each, exhibit periodic properties
which are analogous to those exhibited in MendeleefFs
periodic law. The vibrations of such atoms would give
rise to radiation whose spectrum would present strong
analogies to the line and band spectra of known elements.
337. A very simple case will suffice to show that the
occasional disintegration of such an atom is to be expected.
Symmetry and the laws of electrical action ensure that
four electrions will be in stable equilibrium in the interior
of a spherical space, which is uniformly charged with
positive electricity throughout its volume, provided that
they are situated at the corners of an equilateral tetra-
hedron having the same centre as the sphere has. If this
system be made to revolve round a diameter of the sphere,
centrifugal force tends to make the electrions pass to
positions remote from the axis of revolution ; and,
ultimately, at sufficient rotational speed, a new position
of equilibrium will evidently be attained with the four
electrions situated at the corners of a square in the
equatorial plane. Conversely, if the speed of rotation be
now slowed down, this square arrangement will become
unstable when the critical speed is again reached, and a
re-arrangement will again take place.
In such re-arrangements of electrions in an atom, it is
quite possible that one electrion may acquire so great
linear speed that it will be projected out of the system.
Thus disintegration is to be expected in such a system,
provided that a cause for diminution of rotational speed
314 PROPERTIES OF MATTER.
exists. Now, such a system is constantly radiating
energy. Consequently, if it receives no equivalent supply,
its energy of rotational motion must diminish, and dis-
integration will follow.
338. On the preceding view, the energy manifested on
disintegration comes from an internal store of potential
energy. Another possible source, insisted upon by Lord
Kelvin, cannot be overlooked ; that is, radiational energy
absorbed from the ether. Out of a given number of
atoms, the fraction disintegrating per second in consequence
of sudden exhaustion of internal potential energy, on the
one view, might be the same as the fraction disintegrating
per second in consequence of sudden absorption of energy
from the ether, on the other view. Either end might be
brought about as a result of slow loss of internal energy
because of radiation into space.
339. Rontgen rays, which are produced under suitable
conditions in the vacuum tube discharge, give rise to
strong fluorescence when they fall upon fluorescent bodies,
and also produce certain characteristic electrical effects
in connection therewith. H. Becquerel sought for the
converse effect an emission of Rontgen rays by fluorescent
bodies and he found that there was an emission from
uranium oxide which presented some analogies in its
action to certain of the actions produced by Rontgen
rays. Some specimens of pitch blende (mainly uranium
oxide) were found to be specially active ; and this powerful
activity was shown by the Curies to be due to the presence,
in small quantity, of a new element now well known as
radium.
The peculiar quality of radium, called its radio-activity,
is associated with its ready emission of negative corpuscles
or electrions. Indeed, as a matter of historical fact, the
DISINTEGRATION OF THE ATOM. 316
mass of an electrion was first determined by means of
observations made upon the electrions emitted by radium,
and not upon those observed in the vacuum tube discharge,
as described in 335.
The phenomenon of radio-activity is now known to be
manifested to a greater or less extent by many substances,
and it is probable that all matter exhibits it. It seems
certain that helium is produced in the disintegration
of radium; and there are reasons, although somewhat
hypothetical, for the supposition that radium results from
the disintegration of uranium, and that lead may be the
final product of the disintegration of radium. W. P.]
APPENDIX I. ( 18).
HYPOTHESES AS TO THE CONSTITUTION OF MATTER.
By Professor Flint, D.D.
1. ALL material substances are infinitely divisible into
parts of the same nature as themselves and as complex,
even qualitatively, as themselves.
2. All material substances are divisible into ultimate
indivisible homogeneous parts as complex as the wholes.
One or other of these two hypotheses (it is, perhaps,
impossible to determine which) is attributed by Lucretius
to Anaxagoras, whose real opinion, however, was probably
the one which follows.
3. All material substances are formed from a primitive
matter, "in which all things were together, infinitely
numerous, infinitely little," and of which each infinitely
little part was infinitely complex.
4. All material substances result from the combination
of a few kinds of material elements, each of which is
composed of particles like to itself, e.g., earth of earthy
particles, water of aqueous particles, air of aerial particles.
This was the hypothesis of the Hindu Kanada, the
Greek Empedocles, and a host of medieval physicists.
5. All material substances are states or stages of one
APPENDIX I. 317
primitive matter or element, e.g. of water or air. The
hypothesis of Thales, Aj^aximenes, etc.
6. All material substances are divisible into ultimate
indivisible parts, " strong in solid singleness," which have
no qualitative but only quantitative differences, and
which variously aggregate through motion in a void.
This is the atomic hypothesis as taught by Democritus,
Epicurus, etc.
7. All material substances are divisible into elementary
substances which are subdivisible into molecules, and,
ultimately, into atoms possessed of distinctive qualita-
tive as well as quantitative differences. Recently, and
probably still, the ordinary chemical hypothesis.
8. All material substances are divisible into so-called
elementary substances composed of molecular particles
of the same nature as themselves, but these molecular
particles are complicated structures consisting of congre-
gations of truly elementary atoms, identical in nature
and differing only in position, arrangement, motion, etc.,
and the molecules or chemical atoms are produced from
the true or physical atoms by processes of evolution
under conditions which Chemistry has not yet been able
to reproduce. Hypothesis of H. Spencer, etc.
9. All material substances are composed of atoms, not
hard and solid and on that account indivisible, but the
rotatory rings or infinitesimal whirls of an incompressible
frictionless fluid, supposed to be homogeneous and perfect,
but the nature of which is not otherwise described ; and
all the differences of material substances are due to the
characters and behaviour of their component rings or
whirls. The hypothesis of Sir William Thomson.
10. The matter which is the object of the senses is
the product of a world-building power moulding in accord-
318 PROPERTIES OF MATTER.
ance with eternal ideas an uncreated substratum, the
" receptacle " and " nurse " of " forms," hut itself devoid
of form and definite attributes. Plato's hypothesis.
11. The matter which is an object of sense is a
synthesis of form with a primary matter which is merely
capacity and passivity a synthesis produced by a for-
mative cause, which must be both an efficient and final
cause. Aristotle's hypothesis.
12. Impenetrability is the essence of matter. Hypo-
thesis of various physicists.
13. Extension, not impenetrability, is the essence of
matter. "Give me extension and motion and I will
construct the world." Descartes.
14. Material things are "modes" of extension, which
is one of the only two discoverable " attributes " of the
one " Substance." Spinoza.
15. Matter in its ultimate constitution consists of
metaphysical points which give rise to sensible matter by
states of effort (conatus) transitional from rest to motion.
The nypothesis of Yico. See my " Vico."
16. The ultimate elements of matter are indivisible
points without extension, but surrounded by spheres of
attractive and repulsive force which alternate according
to the distance of these points up to a certain degree of
remoteness. Hypothesis of Boscovich.
17. The physical universe is constituted by the un-
conscious perceptions of a vast collection of unextended
spiritual forces or monads, endowed with a power of
spontaneous development and with something of the
nature of desire and sentime
