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Coo, \
A
TEXT-BOOK OF PHYSICS
PROPERTIES OF MATTER
A TEXT=BOOK OF PHYSICS
BY
H. POYNTING, J. J. THOMSON.
SCO., P.B.5., AND »I.A,,F.R.S.,
t Trinttv CslJcte. Cambridge Fellow at Ttuiily Collegi, Carnl
ToCPhyilcj, BinniDctiam oF Experimental Phyiiu in ibi
Upivcniiy. or Cajbbndgc,
Volume I. Fourth Edition. Revised, Fully IllQsitaled. Trice los. 6d.
PROPESRTISS OF MAXTKR
Volume II. Foitrih Edition, RevUe.l, Fully UiunraiuJ. 85. 6d.
S O XJ N D
CONTBNTI.— The Naluit of Sound and ib chief Cbatacleriuia.— The Vcluciiy of Sound in
ii ud olhcr Media.— RiBei lion and Refmction of Sound.— Freuueun' «nd Pilch of Holes.—
Honance and Forced OraJUUoqj—Aiuxly^ of Vibraiiont-The Trauwene Vibrutioai of
relclied StrioBS oi Wiret— Pipei and other Aii Caviliej.—Rod».—Platei.— Membranes. -
Ibrationi Bulnuined by Heat.— Seniittve Flamei and Jet>.—Muiic«lS>4id.—Tfae5u|icriiotiuOD
Volume III. Sbcond Edition, Fully Illusiralcd. 159.
HEAT
_.. ._KTs.— TemHratuK.— Etpannonof Solidb.— Ltquldi.— GaieL— CirculalionandCau-
vcctiaa.- Qoantily ofHal ; Specific Heal.- Conduct itity.-Fonni of Knersyi Ceiuemlian ;
McchaDieal Equivalent of Ueai.~The Kinelk Tbeoiv.-^baiige of States Uquid Vanour, —
CHlical Fointi.- Sotldi and UquidL— Almaipbeiic Ccmditioiu — Radiation.- Tbeory o( En-
cbanlu.— RadiUionandTenpeiaturc- Tliamadynainici.— Inthermiliuid AdiabatkChuiEei.
— ThamodysaiaJcioFChaiiitiol State, and Soludoni.- ThennadynamiciafRadiatlon.—lndei.
RcmaitiinB Volumes iti PrepBrtition—
XiXO^TT; HAGIfE'XISHK A> EIIL.EIC'X'RICITY'
THE EARTH'S ATMOSPHERE
By Dr. THOMAS LAMB PHIPSON.
Just pullisheJ. In Large Ciown Bvo. Il.indsome OoUi. 45. 6d. net.
THE THERMO-DYNAMIC PRINCIPLES OF ENGINE DESIGN
By LIONEL H. HOBBS,
Enginesr-LiouUnani, R.N,; Instructor in Applied Mechanici and Marine Engine De»gn
at the Royal Naval Colli^ge, Gtcenwjch.
In Handsotne (-loth. Wilb neatly ';o Illuslralions. 3s. 6d. net.
THE ELEMENTS OF CHEMICAL ENGINEERING
By J. GROSSMANN, M.A,. Ph.D , F.I.O.
With a Ftef^^i.' by Sir WILLIAM RAMSAV, K.C.B., F.R.S.
a ctipy"— CiiaMiLAL'NEw"'' ' "'""' ""'^ '''' ' '" " ^ ' " ""^ course 1 ou
In Two Volumes, Large Svo, Strongly Bound in Half-Morocco.
PHYSICO-CHEMICAL TABLES
POR TUB USE OP ANALVSTS. PHYSICISTS, CHEMICAL MANU-
FACTURERS, AND SCIENTIFIC CHEMISTS.
VOLUME I.- Chemical Engineering, Physical CiiEMJSTBv. Price 24B. net.
VOLUME II.— Cbekical Phvsics, Pure and Analviical Chbmisiby.
By JOHN CASTELL-EVANS. F.I.C.. F.C.S.
Meliki'.gyatthe FimbuiyTechniS'l^llegr**"" ""' " "
London: CHARLES URIFFIN * CO., Ltd., Exeter Strbet, Strand.
TEXT-BOOK OF PHYSICS
BY
J. H. POYNTING, ScD., F.R.S.
HON. 8c.D. VICTOBIA UNIVERSITY
LATE FELLOW OF TRINITY COLLEGE, CAMBRIDQE ; MASON PROFESSOR
OF PHTBICB IN THE DNIYERSITY OF BIRMINGHAM
AND
J. J. THOMSON, M.A., F.RS., Hon. ScD. Dublin
HON. DX. PBINCETON; HON. Sc.D. VICTOBIA: HON. LL.D. GLASGOW
HON. Ph.D. CBACOW
FELLOW OF TRINITY COLLEGE, CAMBRIDGE; CAVENDISH PROFESSOR OF
EXPERIMENTAL PHYSICS IN THE URIYBRSITY OF CAMBRIDGE;
PROFESSOR OF NATURAL PHILOSOPHY AT THE
ROYAL INSTITUTION
PROPERTIES OF MATTER
WITH 168 ILLUSTRATIGNgV //.
^ ■^ '^ J
• J
FOURTH EDITION, CAREFULLY flEVuSEDr^
» * • . •" 4 .
LONDON
• * .. •
CHARLES GRIFFIN AND COMPANY, LIMITED
EXETER STREET, STRAND
1907
\,Ail ri^hti reserved)
9 81 48
• I •
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• • •" '
•••• •••••
••••• • ^m* *••«
.••/• ...•: rr:
• •••
'.••-
- • fc -
I » • • fc » i» '
• • •
.• » » *
* • * ••,
• •
. - •
PREFACE.
The volume now presented must be regarJed as the opening
one of a series forming a Text-Book on Physics, which the
authors are preparing. The second volume, that on Sound, has
already been issued, and the remaining volumes dealing with
Heat, Magnetism and Electricity, and Light will be published
in succession.
As already stated in the preface to the volume on Sound,
" The Text-Book is intended chiefly for the use of students who
lay most stress on the study of the experimental part of
Physics, and who have not yet reached the stage at which the
reading of advanced treatises on special subjects is desiiable.
To bring the subject within the compass thus prescribed, an
account is given only of phenomena which are of special
importance, or which appear to throw light on other branches
of Physics, and the mathematical methods adopted are very
elementary. The student who possesses a knowledge of
advanced mathematical methods, and who knows how to use
them, will, no doubt, be able to work out and remember most
easily a theory which uses such methods. But at present a
large number of earnest students of Physics are not so
equipped, and the authors aim at giving an account of the
subject which will be useful to students of this class. Even
for the reader who is mathematically trained, there is some
advantage in the study of elementary methods, compensating
for their cumbrous form. They bring before us more evidently
Ti PREFACE
the points at which the various assumptions are made, and they
render more prominent the conditions under which the theory
holds good."
In the present volume the authors deal with weight, mass,
gravitation, and those properties of matter which relate chiefly
to change of form, such as Elasticity, Fluid Viscosity, Surface
Tension, Diffusion and Solution. The molecular theory of matter
has necessarily been introduced, inasmuch as investigators have
almost always expressed their work in terms of that theory.
But the detailed account of the theory, especially as applied to
gases, will be given in the volume on Heat, in connection with
the account of the phenomena which first brought it into
prominence.
PREFACE TO FOURTH EDITION.
A FEW corrections have been made in this edition. The authors
desire to thank the readers who have kindly pointed out errors
and have enabled them to make these corrections.
J. H. P.
J. J. T.
January 1907.
CONTENTS.
CHAP. PACE.
I. WEIGirr AND MASS ....... 1
U. THE ACCELERATION OF GRAVITY. ITS VARIATION AND
THE FIGURE OF THE EARTH 7
in. GRAVITATION 28
IV. . ELASTICITY 53
V. STRAIN 62
VI. STRESSES. RELATION BETWEEN STRESSES AND STRAINS 68
Vn. TORSION 78
Vin. BENDING OF RODS 85
IX. SPIRAL SPRINGS 103
X. IMPACT 109
XI. COMPRESSIBILITY OF LIQUIDS 116
XU. THE RELATION BETWEEN THE PRESSURE AND VOLUME OF
A GAS 124
Xin. REVERSIBLE THERMAL EFFECTS ACCOBiPANYING ALTERA-
TIONS IN STRAINS 131
XIV. CAPILLARITY 135
XV. Laplace's theory of capillarity . .173
XVI. diffusion of liquids 182
XVII. diffusion of gases 196
xvm. viscosity of uquids 205
index .*••••..• auO
PROPERTIES OF MATTER.
WEIGHT AND MASS.
Introductory Remarks.— Physics is the study of the properties of
inH.tt«r, and of the action of one portion of mutter upon another, and
ultimately of the ffTfcta of these actions upon our sences. The properties
studied in the various branches, Sound, Heat, Light, and Magnetism and
Electricity, are for the cior* part easily ciafiuilied under these headings.
But there are other properties chiefly connected with changes in shape and
relative position within a system which are grouped together as " Genei-al
Properties of Matter." Among theee latter propeitioa are Eliistidty,
Surface Tension, Difl^usion and Viscosity.
The most general properties of matter are really those studied in
Btitiea and DynamicA: the relation between forces, when the matter
acted on is in eqtiilibriura and the motion of matter under the muttial
action of the various portions of a nystem. But in 8t.itii:s and Dynamics
the recourse ta expeiimcnt )k so small, and when the expenroentiil foun-
dation is once laid the mathematical structure is so great, that it is con-
venient to treat these branches of Physics sepanttety. We shall assume
in this work that the reader haa already studied them, and b familiar
both with the conditions of equilibrium and with the simpler types of
motion.
We shall, however, begin with the discussion of some ijuestions which
involve dynamical considerationn. We shall show how we pass from the
idea of v>eiijhl to that of moM. and how we establish the doctrine of the
constB-Ticy of mass. We >>hall then give some account of the measuiement
of gravity at the surfaiw of the enith, and of the gravitation wliich is a
propeity of all matter wherever situated. We shall then prot-eed to the
discussion of those properties of matter which are pei haps best described
as involving change of form.
Welgrht.— All malt«r at the surface of the earth has weight, or is
pulled towards the ground. The fact that the pull is to the earth at
all parts of its surface shows conclusively that it is due to the earth.
Apparent exceptions, such as the rising oE a balloon in air, or of a cork
in water, are of course explained, not by the levity of the rising bodies,
but by the greater gravity of their surroundings. (Common experience
2 PROPERTIES OF MATTER.
with the halanoe shows that the ratio of the weights of two bodies is
constant wherever they are weighed, so long as they are both woighed
at the same point. Common experience shows too that the ratio is the
same, however the bodies be turned about on the scale-pin of the balance.
The balance does not tell us anything as to the constancy of weight of
a given body, but only as to the constancy of ratio ; for if the weights of
different bodies varied, and the vaiiation w;us always in the same ratio, the
balance would fail to indicate it. But here experiments with pendulums
supplement our knowledge. A given pendulum at a constant temperature
and in a fixed position has, as nearly as we can observe, the same time of
gwing from day to day and from year to year. This implies that the
pull of the earth on the bob is constant — i.e., that the weight at the
same place remains the same.
This constancy of weight of a body at the same point appears to hold
whatever chemical or physical changes the matter in it may undergo.
Experiments have been made on the weight of sealed tubes containing
two substances which were at first separated, and which wei^ then
mixed and allowed to form new chemical compounds. The tubes were
weighed before and after the mixture of their contents. But though
Landolt* and Heydweillert have thought that the variations which they
observed were real and not due to en'ors of expeiiment, Sanford and
JElay:t have made similar experiments, and considered that the variations
were observational errors. Where variations have been observed they are
so minute and so irregular that we cannot as yet assume that there is any
change in weight.
Again, temperature does not appear to affect weight to any appreciable
extent. It is extremely difficult to make satisfactory weighings of a body
at two different tempei-atures. Perhaps the best evidence of constancy is
obtained from the agreement in the results of ditrerent methods of
measuring liquid expansion. In Dulong and Petit's U-tube method of
determining the expansion of mercury, two unit columns have different
heights but equal weights, and it is assumed that the cold column would
expand into the hot column without change of weight. But in the
dilatometer method nearly the whole expansion is directly measured, and
only the small expansion A the envelope, measured by assuming the expan-
sion of mercury, introduces the assumption of constancy of weight with
change of temperature. The close ngi-eement of the two methods shows
that there is no large variation of weight with tempemture.
We may probably conclude that, up to the limit of our present powers
of measurement, the weight of a body at a given |)oint is constant under
all conditions.
But when we test the weight at different points this constancy no
longer holds. The common balance used in the ordinary way fails to show
variation, since both pans are equally affected.
But very early in the history of the pendulum, as we shall show in the
next chapter, experiments proved that the seconds pendulum had different
lengths at different places, or that the siime pendulum had diffei*ent times
of swing at different places. In other words, the weight of the bob varied*
Thus a body is about 1 in 300 heavier at Loudon than at the Equator.
• Zeitf. Phyiih. Chem., xii. 1, 1894.
Ztit.f, Phytik.y August 25, 1900. p. 527.
Pky%, Rev,, V. 1897, p. 247.
\
Aa RirSy as 16C2 an experiment was made by Dr. Power* in whicb a
varialion of weight nitli ctiange of level over the same point was looked
for. A body was weighed by a fixed balance, being first placed ia the
e(^e-[uin and then hung far below the &itiie pan by a string. The
experimcut was i-opeated by Hooke, and later by others, but the variation
was quite beyond the mngo of observation possible with these early
experimenters, and the results tbey obtained were due to disturbances in
the siivroundings. The fii'st to show tfaat the balance could detect a,
TBtiation was von Jolly (chap. iti. p. 41), who in 1678 described nn
experiment in whioh he weighed a kilogramme on a balnnce 5-5 metres
above the floor and then hung the kilngramme by a wii'e so that it was
near the fiour. lie detected a pain in the lower position of V5 mgm.
Later lie rejieated the experirnent on a tower, a 5 kgm. weight gaining
more than 81 mgmG. between the top of the tower and a point 21 metres
below. More recently Richam and Krigar-Menzel found a variation
in the weight of a kilogi-amme when lowered only 2 metres (chap. iii.
p. 42.)
The evidence then is convincing that the weight of a body varies from
point to point on the earth's surface, and also varies with its distance above
the snme point.
The question now arises — Is there any measurable quality of matter
which remains the same wherever it is measui-ed? Experiment shows
that there is constancy in that which is termed the mas* ofmalta:
HaSS.— Without entering into any discussion of the moat appropriatfl
or most fundamental method of measuring force, we shall assume that we
can measure forces exerted by bent and stretched springs and similar con-
trivances independently of the motion they produce. "We shall asstime
that, when agiven strain is observed in aspring,it is acting with a delinita
force on the body to which it is attached, the force being determined by
previous experinicnts on the spring. Let us imagine an ideal experi-
ment in which a spring in attuched to a certain body, which it pulls
borlBontnlly, under consti'nint free from fiiction. Let the spring be
always stretched to a given amount as it pulls the body along, so acting
on it with constant force. Then all experiments and obsei'vations go to
show that the body will move \vith the same constant acceleration wherever
the experiment is made. This constjincy of acceleration under a given
force is expi-essed by saying that the mass of the body ia constant.
Though the experiment we have imagined is unreftlisable, actual experi-
ments on the same lines are made for us by good chronometers. The
balance-wheel of a chronometer moves to and fro against the resir-tance
of tlie haii'-spring, and its acceleration is very accurately the tame for the
fiatne strain of the ;ipring at the saine temperature in diBereut lati-
tudes. The weight of the balance-whe«l decreases by 3 in 1000 if the
chronometer is caiTied from London to the Equator. If the acceleration
under given force increased in the same ratio the rate of the chronometer
would change by 3 in 2(mO, or by two minutes per day, and the
ehrouom^tei wou'd be wselefs for determinations of longitude. Again, a
tuning-fork, mtiking, say, 2,'i(I vibrations per second at Paris at 1G° will
have ^-ery accurately the same frequency at the same temperature wherever
tRbted. The same portion of matter in (he prongs has the same acceleration
for the same strain and, presumably, for the some force all the world over,
• Mackenile, The Latci nj (Jmfitaii'on, p. 3.
4 PROPERTIES OF MATTER.
This constancy of acceleiMbion of a given Iwdy under given force hoi
true likewise wLatever the nature of the bodyexerting tlieforce may be—
i.e., whether it be a bent Kpi-kug, a tpii-ul spring, air grousing, a, string
pulling, and so on.
Further experiment showfl that the acceleration pf b given body is
proportional to the force acting on it, Thus, in a very small vibration of
ft pendulum the fraction of the weight of the bob tending to restore it to
ita central position ie proportional to the displacement, and tlie eimpls
harmoDic type of the motion with ite igochronisin ehoirs at once that the
acceleration is projwrtional to the displacement, and therefore to the force
acting. When a body vibrates up and down at the end of a spiral spring
we again have simple harmonic motion with acceleration proportional to
the distance from the position of equilibrium. The variation in the force
exerted by the spring is also pro{)ortional to this distance, or acceleration
is proportional to force acting. Indeed, elastic vibrations with their
isochi'onism go, in genera), to prove this proportionality. If, then, we
accept the view that we can think of forces acting on bodies es being
measurable independently of th« motion which they produce — measurable,
8fty, by the strain of the bodies acting — we have good experimental proof
that a given poi*tion of matter always has equal accelei-ation under equal
force, and that the accaleratioos under different forces are proportionU to
the forces acting upon it.
We can now go a step farther and use the acceteratiocs to comp
different masses.
Definition of Mass. — Tf<e ■niatsesofhodmayep-oporlionaUo Ihe/uivea
producing equal accderalione in them.
An equivalent statement ia, that the masses are inversely as the
ftcceleration produced by equal forces. It follows from our definition that,
if equal accelerations are observed in different bodies, then the masses are
proportional to the forces acting.
Observation and experiment further enable us to say that:
The Titasaes of boiliea are proportional Co their weiy/iU at the same point.
To prove thia it is only necessary to show that all bodies have equal
acceleration at the same place when acted on by their weights alone — to
show, in fact, that the quantity always denoted by ^ is constant at the same
ptes.
A very simple though rough experiment to prove this consists in
tying a piece of iron and a piece of wood to the two ends of a thread and
putting the thread across a horizontal ring so that the two weights
depend at the same height above the floor. The thread is now burnt
in tlie middle of the ring and the iron and wood begin to fall at the same
instant. They reach the floor so nearly together that only a single
bump is heai'd. If the surfaces presented to the air are verj' different the
air resistance may interfere with the success of the experiment. But the
more the aii- resi.-itance is eliminated the more nearlyis the time of fall the
same. Thus, if a penny and a eheet of paper are placed on a boai-d some
height above the floor, and if the board is suddenly withdrawn, the penny
falls stiiiight while tiie paper slowly flutters down, fiow crumple up tbi
paper into a little boll and repent the experiment, when the two reach tl
ground as nearly as we can observe together.
Newton {Pnncipia, Book III., Prop. 6) devised a much i
form of the experiment, using the pendulum, in which any difference
U to J
WEIGHT AND MASS.
acceleration woiiW be cumiilntlve. anJ Buspending i
weights of various kinds of matter. He says (Motte's tranBlation) :
" It has been, now or a long time, obEorved b; otfaers, that all soitB of heavy
bodies (allownoce being isade for the inequalitj ot retardatioii, which ihey
»B!ter from a small poner of reaisljince in tlie air) descend to the Earth from
tqfiat kfigkli 1(1 equal timoa ; and that equality of timea we maj dislingiiisb to
a great accaracv. by the help of pcndalania. I tried the thing in gold, silver,
lead, glass, sand, common salt, wood, water, and wheat. I provided two
wooden boxes, round and equal. I filled the ooe with wood, and suspended an
equal weight of gold (as exactly aa I could) in the centre of oscilUlion of the
other. The boiea hanging by equal threada of eleven feet, made a couple of
penduluma perfectly equal in weight and figure, and equally receiving the
resistiince of the air. And placing tbe one by the other, 1 observed ihem to
play together forwards and backwards, for a long lime, with equal vibrations.
And therefore the quantity of matter in the gold (by Cor. 1 and 8, prop. 24,
boob 2) was to the quantity of matter in the wood, as the action of the motive
force (or t-if nofi-iz) upon all the gold, to the action of tbe aanie upon all the
wood ; that is, as the weight of the one to the weight of tbe other. And the
like happened in the other bodies. By these eiperimcnlii, in bodies of the same
weight. I could manifestly have discovered a difTcrencc of matter leas Ihaa a
thousandth part of the whole, had any sach been."
Newton liere uses "quantity of matter" wliere we shnnld now say
"mass," Eessel (BetUn Abh., 1830, Avn. Fo'jg., xsv. IMa, or
Mhnoires relati/s A la I'hysiijue, v. p. 71) made a soiiea of most cai'eful
expeiimentfi by Newton's method, fuUy confirming the concIiiBion tliat
weight at the same plnce is proportional to mass.
Constancy of Mass.— The experimenta which have led to the con-
clusion that weight at the same place ia conatant now gain another
sigtiiGcance. They show that tlie mass of a given portion of matter is
constant, whatever changes of position, of form, or of chemical or physical
condition it may underga
When we " weigh " a body by the common twlantte, say, by the
counterpoise melhoil, we put it on the p.iu, countei-poise it, mid then
replace it by bodies from the set of " weights " having an equal weight.
But our aim is not to find the weight of the body, tlie pull of (he
earth on it. We use the equality of weight possessed by equal maiises at
the same point of the earth's surface to find its mass. In buying mutter
by weight we are not ultimately concerned with weight but with mass,
and we expect the same ma-^a in a pound of it whether we buy in London
or at the Equator. A set of weights is really a set of innsseH, and when
we use one of them we ai-c using it as a mns.s through its weij^ht.
Unit of Mass.— We can make a definite unit of mass by fixing on
some piece of matter as tbe standard and saying that it contains one unit
or BO many nnils. So long as vce are careful that no portion of the
standard piece of matter ia removed and that no addition is made to it,
such a unit is bolh definite and consistent.
In thin country the unit of mass for commei'cial purposes is the piece of
plalinnm kept at fhe Standards Office at Westminster, marked " P.S.
18ii 1 lb." and called the Imperial Avoirdupois Pound. But for scientific
purposes nil over the world tbe unit of mass is the gramme, the one-
thousandth part of the mass of the piece of platinum-iridium called the
" Kilogram me- International," which is kept at Paris. Copies of this
kilogramme, compared either with it or with previous copies of it, are now
distiihuted through the world, their vfihies being known to less, perhajts.
6 PROPERTIES OF MATTER.
than 0*01 mgm. For example, the copy in the Standards Office at West
minster is certified to be
1000000070 kgm.
with a probable error of 2 in the last place.
According to a comparison carried out in 1883, the Imperial pound
contains
453-5924277 gi-ammes,
though Parliament enacted in 1878 that the pound contained
453*59245 grammes.
Of course one piece of matter only can be the standard in one system of
measurements, and the enactment of 1878 only implies that we should use
a different value for the kilogramme in England fiom that used in Franca
The difference is, however, ^uite negligible for commercial purposes.
CHAPTEIl II.
THE ACCELERATION OF GRAVITY. ITS VARIATION AND
THE FIGURE OF THE EARTH.
Contents. — Early History— Pendulnm Clock — Picard'a Experiments — HuygenB*
Theory — Newton's Theory and Experiments — Bonguer's Experiments — Ber-
nouilli's Correction for Arc — Experiments of Borda and Cassini — Eater's C<m-
vertible Pendalum — Bessel's Experiments and his Theory of the Reversible
Pendulum— Repsold's Pendulum — Yielding of the Support — Defiforges* Pendulum
— Variation of Gravity over the Earth's Surface — Richer — Newton's Theory of
the Figure of the Earth — Measurements in Sweden and Peru — Bongner's
Correction to Sea-level — Clairant's Theorem — Kater and Sabine — Invariable
Pendulum — Airy's Hydrostatic Theory — Faye's Rule — Indian Survey — Formula
for^ in any Latitude — ^Von Sterneck's Half-second Pendulums — His Barymeter
— Gravity Balance of Threlfall and Pollock.
We shall describe in this and the followinj^ chapter the methods of
measuring two quantities ; the acceleration of falling bodies due to the
earth, at its surface (the quantity always denoted by g) ; and the accelera-
tion due to unit mass at unit distance (the quantity known as the ffi'avita'
Hon constant and denoted by G). The two may be measured quite in-
dependently, but yet they are closely related in that g is the measure of a
particular case of gravitation, while G is the expression of its general
measure. The two together enable us to find the mass and therefore the
mean density of the earth.
The Acceleration of Gravity,* — ^We shall briefly trace the history
of the methods which have been used in measuring g, for in so doing we
can set forth most clearly the diificulties to be overcome and realise the
exactitude with which the measurement can now be made. We shall
then give some account of the experiments made to determine the varia-
tions of gravity and the use of the knowledge so gained to determine the
shape of the earth.
Early History. — The first step in our knowledge of the laws of
falling bodies was taken about the end of the sixteenth century, when
Stevinus, Galileo, and their contemporaries were laying the foundations
of the modern knowledge of mechanics. Stevinus, the discoverer of the
Triangle of Forces and of the theory of the Inclined Plane, and Galileo,
* A collection of the most important original papers on the pendulum
constitates vol>«. iv. and v. of Mimoircs rdatifs d la Physique, It is prefaced by an
excellent history of the subject by M. Wolf, and contains a bibliography. The fifth
volume of The O. T. Survey of India consists of an account of the pendulum
operations of the sbrvey, with some important memoirs. In the Journal de
Phygique^ vi*. 1888, are three important articles by Commandant Defforges on the
theory of the pendulum, concluding with an account of his own pendmam. The
description given in this chapter is based on these works.
8 PROPERTIES OF MATTER.
the founder of Dynamics, were both aware that the doctrine then held that
bodies fall with rapidity proportional to their weight was quite false, and
they asserted that under the action of their weight alone all bodies would
fall at equal rates. They pointed out that the different rates actually
observed were to be ascribed to the resistance of the air, which has a
greater effect on the movement of light than of heavy bodies of equal
size. Galileo made a celebrated experiment to verify this fact by dropping
bodies of different weights from the top of the Leaning Tower of Pisa,
and showing that they reached the ground in the same time. The air-
pump was not yet invented, so that the later verification by the *' gtiinea
and feather" was not then possible. But Galileo did not stop with this
experiment. Ho made the progress of dynamics possible by introducing
the conception of equal additions of velocity in equal times — the con-
ception of uniform acceleration. His first idea was that a constant force
^ould give equal additions of velocity in equal distances traversed, but
investigation led him to see that this idea was untenable, and he then
enunciated the hypothesis of equal additions in equal times. He showed
thaty by this hypothesis, the distance traversed is proportional to the
square of the time. Not content with mere mathematical deductions,
he made experiments on bodies moving down inclined planes, and demon-
strated that the distances traversed were actually proportional to the
squares of the times — 1.0., that the acceleration was uniform. By ex-
periments with pendulums falling through the arc of a circle to the
lowest point, and then rising through another arc, he concluded that the
velocity acquired in falling down a slope depends only on the vertical
height fallen through and not upon the length of the slope, or, as we
should now put it^ that the acceleration is proportional to the cosine of
the angle of the slope with the verticaL He thus arrived at quite sound
ideas on the acceleration of falling bodies and on its uniformity, and from
his inclined plane experiments could have obtained a rough approxi-
mation to the quantity we now denote by g. But Galileo had no accurate
method of measuring small periods of time in seconds. The pendulum
clock was not as yet invented, and he made merely relative measurements
of the time intervals by determining in his experiments the quantity
of water which flowed through a small orifice of a vessel during each
interval.
To Cktlileo we also owe the foundation of the study of pendulum
vibrations. The isocbronism of the pendulum had been previously ob-
served by others, but Galileo rediscovered it for himself, and showed by
further experiment that the times of vibration of different simple pendu-
lums are proportional to the square roots of their lengths. He also used
the pendulum to determine the rate of beating of the pulse and recognised
the possibility of employing it as a clock regulator. He did not publish
his ideas on the construction of a pendulum clock, and they were only
discovered among his papers long after his death.
From Galileo, therefore, we derive the conception of the appropriate
quantity to measure in the fall of bodies, the acceleration, and to him we
owe the instrument which as a free pendulum gives us the acceleration of
fall, and, as a clock regulator, provides us with the best means of deter-
mining the time of fall.
Soon after Galileo's death, Mersenne made, in 1644, the fir&t determi.
nation of the length of a simple pendulum beating seconds, and a little
THE ACCELERATION OF GRAVITY. 9
later he suggested as a piiiblein t)ie determination ot the length of a
simple pendulum equivalent to a given compound pendulum.
Pendulum Clock. — Hut it was only with the invention of the
pendulum clock by liuygens in 1G57 that the second becnme an interval
of time measurable with consistency and ease. At nnce the new clock was
widely used. Its rate could easily be deteimined by star observationR, and
determinations of the length of tlie seconds pendulum by its ad becume
common.
Picard's Experiment.— In 1(IC9 Picard determined this length at
Paris, using a copper ball an inch in diameter suspended by an aloe fibre
from jaws. This suspension was usual in early work, the aloe fibre being
unafiected to any appreeinble extent hy moisture. Pirard's vnlue was
3G inches 8J lines Paris measure. The Pans foot may be taken bb
}Ji or M)C5 English feet, and there are 12 lines to the inch, so that the
length found was 39'(13 English inches. Picnnl states that the value had
been found to be the same at London and at Lyonn.
Huyjrens' Theory.— In 11'3 Iluygens propounded the theory ot the
cycloidal pendulum, proving its exact isochronism, and he showed how to
construct such a pendulum by allowing the htring to vibriit« between
oycloidal cheeks. lie determined the length beating seconds nt Paris,
confirming Picard's value, and fi-nm the formula which we now put in the
form g = ir'l he found 2 the distance of free fnll in one sei^ond, the
quantity which was at first nsed, inste-nd of the full acceleration we now
employ. His value was 15 ft. I in. \^ liues, Paris measure, which would
give 3-32-16 English feet,
Huygens at the same time gave tlio theory of uniform motion in a
drcle and the theory of the cnnical pendulum, and above all in importance
he founded the study of the motion of bodies of finite Bii;e by solving
Mersenne'e problem and working out the theory of the compound
pendulum. He discovei-ed the method of determining the centre of
oscillation ajicl showed its interchangeahility with the centre of suspension.
Newton's Theory and Experiments.— Newton in the Princifia
made great use of the theory of the pon<lulum. He there for the fii-st
time made the idea of mass definite, and by his pendulum experiments
{Prineipia, sect, vi.. Book II,, Proj>, 34), he proved that mass is
proportional to weight. He used pendulums too, tn investigate the
resistance of the air to bodies moving through it, and repeated the
pendulum esperimenta of Wren and others, by which the laws ot impact
hod been discovered. But his great contribution to our present subject
was the demonstration, by means of the moon's motion, that gravity is
& particular case of gravitation and acts accoi-ding to the law of inverse
squares, the attracting body being the earth. lu Book III., Prop, 4, he
calculates the acceleration of the moon towards the earth and shows that,
starting from rest with this acceleration, it would fall towards the earlh
15ft. 1 in. ]>- lines (Paris) in the first minute. If at the surface of the
earth 60 times nearer the accelei-ation is GO* times greater the same
distance would here be fallen through in one second, a distance ainiost
exaotly that obtained by Huygens' experiments.
In a later proposition (37) he returns to this calculation, and now,
assuming the law of invei'se squares to lie cornet, he mnkes a. more exact
determination of the moon's acceleintion, and fiom it deduces the value
10
PROPERTrES OF MATTER.
of gravity at the menn radius of the enrth in lutitudo 45°. Ih(
theory of the variation of gr:ivity with latitude, of wliich we i
Bome account below, he finds tha value at Paris, lie corrects the value
thiiB found for the rentrifugal force at Paris and (in Prop. 19) for tha
air displaced, which he takes as j^omr '^^ ^^^ weight of the boh used in the
pendulum experiments, and finally arrives at lb ft. 1 in. 1 J lines (Paris),
differing from HHygens' vaUie by about 1 in 75uO.
BOUg:uer'S Experiments.— Though Newton was thus aware of tlw
need of the correction (or the buoyancy of the uir, it does not appear to
have been applied again until Bouguer made his celebrated experimenta
in the Andes in 17;17. The^e are especially interesting in regard to
the variations of gravity, but we may here mention lioma important
points to which Buuguer attended. While hh predecessors probably
altered the length of the pendulum till it swung secoDds a« exactly
as could be observed, Bouguer introduced the idea of an " invariable
pendulum," making it always of the same length and observing how long
it took to lose so many vibratious on the secon<ls clock. Fur this purpose
the thi-ead of the pendulum swung in front of a scale, and he noted the
time when the thi-ead moved pa.st the centre of the ecale at the same
instant that the beat of the clock was heard. Here we have an elementary
form of the " method of coincidences," to be described later. He need,
not the measured length from the jaw suHpension to the centre of the bol
which was a double truncatetl cone, but the length to the centre of oscillt
' tion of the thread and bob, and he allowed for change of length of )ii^
measuring-rod with temperature. He also assured himself of the coinci*
dence of the centre of figure with the centre of gravity of tbe bob bj
ahowing that the time of swing was the same wheu the bob was inverted.
Ue determined the density of the air by finding the vertical height through
which he must carry a barometer in order that it should fall one line, and
he thus estimated the density of the air on the summit of Piuhincba at
^1^^)^ that of the copper bob of his pendulum. Applying these correc-
tions to his observations he calculated the length of the seconiU pendulum
CorreetiOD for Arc. — In 1747, I>. BemouilU showed how to correct
the observed time of vibration to that for an infinitely small arc of swing.
The observed time is to a first approximation longer than that for aa
infinitely small e
1 the ratio 1 + tt, whet
< the amplitnde of thf
angle of swing. The correction has to be modified for the decrease in
amplitude occurring during an observation.
Experiments of Borda and Cassinl.— The next especially uoUf
worthy experiments are those by Itordaand Ciissini niade at Pai-is in 1793
in connection with the investigations to determine a new standai'd (a
length, when it was still doubtful whether the seconds pendulum might
not be preferable to a unit related to the dimensions of the earth. The
form of pendulum which they used is now named after Uordn, It con-
sisted of a platinum ball nearly 1 i inches in diameter, huag by a fine iron
wire about 12 Pjiris feet long. It had a halt-period of about two seconds
The wire was attached at its upper end to n knife edge — the advantages of
a knife-edge suspension having been already recognised— and the knife
edge and wire -holder were so formed that their time of swing alone was tha
same af> that of the pendulum. In calculating the moment of inertia,
I
THE ACCELERATION OP GRAVrTY. H
tfaey could thereFore be left out of account. At the lower end the wire
vma attached to a bIdUIow cup K'ith the concavity donnwarda, and the ball
exactly fitted into tliis cup, being made to adhere to it by a little greaee.
The ball could therefore be enittly and exactly reversed without altering
the pendulum length, and any non-coincidence of centi'e of gravity and
centre of figure could be e)iininate<l by taking the time of swing for each
position of the btill. The peudulum was bung in front of a eeconds clock,
with its bob a little below the clock hub, and on the lalt«r wiw fixed a.
black pnper with a whit© X-shaped cross on it. The vibrations were
watched through a telescope from a short diHtance away, and a little in
front of the pendulum was a, blsu^k screen coveiing half the field. When
the pendulums were at rest in tlie tivid the edge of this screen covered
half the ci-osa and half the wire. When the swings were in progress the
times were noted at which the peudiiliim wire just bisected the cross at the
instant of diwippearanc© behind the ecreen. This was a "coincidence,"
and, since the clock bob made two swings to one of tlie pendulum, the
interval between two succeaaive " coincidences " was the time in which the
clock gained or lost one complete vibration or two seconds on the wire
pendulum. The exact second of a coincidence could not be determiued
but only estimated, as for many secoada the wire and cras!4 appeared to
pass the edge together. But the advantige of the method of coincidences
was still preserved, for it lies in the fact that if the uncertainty is a
amall fraction of the interval between two suuce^ive coincidences the
error introduced is a very much smaller fraction of the time of vibration.
For, suppose that the wire pendulum makes n half swings while the clock
makes '2n + 2. If the clock beats exact seconds the time of vibration of
the wire pendulum la
If there is a passible error in the determination of each of two successive
coincidences of tn seconds, or at the most of 2m in the interval of 'in + 2
seconds, the observed time might be
In one case Borda and Oas^ni employed an interval of 2» = .%00 seconds,
and found an uncertainty not moi'e than 30 seconds fur the instant of
coincidence. Thiu
m^ 30 1
m' ibOlf 7!iW0
Now, OB tliey observed for about four hours, or for five intervals in succes-
■ion, the error was reduced to J. or sr^mm °^ ^^^ vaiue of (. Practically
the method of coincidences determiued the time of vibration of the
pendulum in terms of the clock time with suHicJent accuracy, and the
responsibility for error lay in the clock. The pendulum was treated as
forming a rigid system, and the length of the ei:[uivalent ideal simple pen -
dulum was calculated therefrom. Correctiooe were made for air displaced,
for arc of swing, and for variations in length with temperature.
The final value obtained wan : Seconds pendulum at Furia = 410'5593
12 PBOPERTIES OF MATTER.
lines (Paris). As the metre =• 443*290 Paris lines, this gives 993JiD mm.,
nod, corrected to sea-level, it gives 99l'>'85 toiQ.
Rater's Convertible Pendulum.— The difficuUies in measuring the
length and in calculating the moment of inertia of the wire-BUfipended or
so-called simple pendulum led Pi-ony in 1800 to propose a pendulum
employing the priuriple of interchnngeability of the centres of oscilljition
Rnd SI 1.1 pension. Tim pendulum wns to have two knife ed^es tmii'^d
inwards on opposite sides of the centre of gravity, so that it coiilil bo
swung from eitljer, and was to be so adjusted that the time of
swing was the same in both cases. The distance between the
knife edges would then be the length of the equivalent simple
pendulum, Prony's proposal was unheeded by hiscontemporai'ie",
and the paper describing it was only published eighty )'eai's later.*
In 1811, Eohnenberger made the same proposal, and again
in 1817 Captain Kater independently hit on the idea, and for
the first time carrieil it into practice, making his celebrated
determination of if at London with the form of il1^trument since
known as " Katei's convei-tible pendulum." This pendulum is
shown in Fig. 1. On the rodnre two adjustable weights, to and s.
I ■j'"' The liirger weij!ht ?« is moved about until the times of swing
from the two knife edges it-, k, are nearly equal, when it is
screwef! in po>ition. Then s is moved by means of a sci'ew to
make the final adjustment to equality, Kater determined the
time of vibration by the method of coincidences, his use of it
fs being but slightly different from that of Borda. A white circle
on black paper was fastened on the bob of the clock pendulum ;
the convertible pendulum was suspended in fi*ont of the clock,
and when the two were at rest the tail-piece t of the former just
covered the white cii-cle en the latter as viewed by a telescope a
few feet away. A slit was made in the focal plane of the
eyepiece of the telescope just the width of the images of the
white patch and of tbe pendulum tail. A coincidence was the
instant dujing an observation at which the white circle waa
quite invisible as tbe two pendulums swung past the lowest
point together. A series of swings were made, lii'St fivm cne
knife edge and then from the other, each series lasting over
four or five coincidences, the coincidence interval being about
no. i.-~ 500 seconds. The line weight was moved after each series till
„ ''"''^r-i!], the number of vibrations per twenty-four hours only dill'ered by
Peiiduliim, B small fiuction of one vibration whichever knife edge was used,
and then the diirerence "was less than errors of observation, for
the time was sometitne.s greater from the one, sometimes greater fi'om the
other. The mean time observed when this stage was reached was corrected
for amplitude, and then taken as the time of the simple pendulum of
length equal to the distance between the knife edges, this distance being
Ciirefully measured. A correction was made for the air di^plnced on tha
assumption that gravity was diminished thereby in tbe ratio of weight of
pendulum in air to weight of pondidum in vacuo. The value was then
corrected to sea-level. The final valueof the length of the seconds pendulum
at sea-level in the latitude of London was determined to be39'lSU^Uinchea.t
ler ID tie Phil. Trans, for ISIS "An
THE ACCELERATION OF GRAVITY. 13
Bessel's Experiments and his Theory of the Reversible
Pendulum. — In 18:2G Besael luude expeiimenU to tletermine the length
of the seconds pendulum &l Koenigslier^. He used a wires tiapend<3d
peoJulum, swung livst from one point nnd then from another point,
esactly a '"Tuise of Peru"" higher up, the bob being at the same level in
each wise. Assuming that the pendulums are truly Fimple, it will easily be
Feen thnt the diftoience in the squares of the times is tlie square of the
time for a simple pendulum of leogth equal to the diflereiiee in lengths,
and theiefore the actual length need not be known. But the practical
pendulum departs from the ideal simple ty|>e, and so the actual lengths
have to be known. As, however, they enter into the eiipresfiion for the
difference of the aquares of the times, with a veiy small quantity as co-
efficient, tliey need not be known with such accuracy aa their diS'ei'encea.
Beseel took et.pecial core that this diR'erence should be accurately equal to
tbetoise. At the upper end, in place of j.^ws or a knife edge, heused-a hori-
zontal cylinder on which the wire wrapped and unwRipped. He introduced
corrections for the stiflness of the wire and for the want of rigidity of
connection between bob and wire. The necessity for the latter correction
was pointed out by Laplace, who showed that the two, bob and wire, could
not move aa one piece, for the bob aciiuires nnd loses angular momentum
■round its centre of gravity, which cannot be accounted for by forces
passing through the centre, such as would alone act if the line of the wire,
produced, always passed through the centre. In reality the bub turns
through a slightly greater angle thnn the \vii-o. so that the pull of the wire
L8 now on one side and now on the other side of the centre of gravity.
The correction ia, however, small if the bob has a radius small in couipurison
with the length of the wire.
If { is the length of the wire, r the distance of the centre of gravity of
the bob from the point at which the wire is attached to it, and s the nidiua
of gyration of the bob about an axis through the centre of gravity ; then,
neglecting higher powers than k', the equivalent simple pendulum can be
shown to be
the last term being due to the correction under consideration. As an
illustration, suppose the bob ts a sphere of 1 inch radius and the wii'e
is 38 inches long; then the equivalent simple pendulum in inches ia
39 + '010i56 + -00tl|02, and the last term, 1/400000 of the whole length,
need only to be taken into account in the most accurate work.
Besael also made a very important change in the air correction. The
effect of the air on the motion may be separated into three parte —
(1) The buoyancy, the weight of the pendulum being virtually
decreased by the weight of the air which it displaces.
(2) The flow of the nir, some of the air moving with the pendulum,
and so virtually increasing its mass.
account of experimi'ntB for (tetermiziing; thn loDK^b of the pendnluin vibrating
seconds in the latitude ot London," and fn a paper In the Phil. Tram, for 1819,
" Experimenta for determiDing tbe variations in the leogcb of the pendulum
vibmting ■econds." Eater applies turlber conections and gives the above value.
• The '-ToUe ot Pern '■ was a sUmdaid bar at tbe Paris Observatory, 6 Paris feet
Ot about 1S19 millimetres long.
14 PROPERTIES OF MATTER.
(3) The air drag, a viscous resistanco which comes into play between
the different layers of air, moving at different ratas, a i^esistance trans-
mitted to the pendulum.
As far back as 1786 Du Buat had pointed out the existence of the
second effect, and had made experiments with pendulums of the same
length and form, but of different densities, to determine the extra mass for
various shapes. Bessel, not knowin<]f Du Buat's work, reinvestigated the
matter, and again by the same method determined the virtual addition to
the mass for various shapes, and among others for the pendulum he used.
The vitcous resistance was first placed in its true relation by Stokes'
investigations on Fluid Motion in 1847. In pendulum motion we may
regard it as tending to decrease the amplitude alone, for the effect on the
time of vibration is inappreciable. We may represent its effect by
introducing a term proportional to the velocity in the equation of motion,
which thus becomes
The solution of this is d^Ae" »cosi \fx-—t-a \
where A and a are constants.
The period is T • — -^L-. where y depends on the viscosity.
Approximately T - _?_(l+__y or the time is increased by the
viscosity in the ratio 1 + — : 1,
or since ^ = Tp^^ (nearly), in the ratio 1 + '. 1.
To see the order of this alteration, suppose that p^ p, represent two
succeeding amplitudes on opposite sides of the centre — i.e., values for which
^* = 0, or cos (^^ '* J ~ " jj ) = ; then p^=e^ or, taking logarithms,
iogei=x=?:I
*
Now in one of Kater*s experiments the arc of swing decreased in
about 500 seconds from 1-41° to 118°, or in the ratio 1195 : 1.
Then (f*^Y' ^ i'ld:) and 500X=Iog, 1195 = 0178
whence X = 000356 and -^ = ~ = 6 x lO"'-' about.
In Borda*8 pendulum the effect was about the sUmQ — f.tf., one that ii
practically quite negligible.
THE ACCELERATION OF GRAVITY. 15
Bessel also used the pendulum to investigate afresh the correctness of
Newton's proof that mass is proportional to weight, carrying out a series
of experiments Vhich still remain the best on the subject. But BesseVs
chief contribution to gravitational research consisted of his theory of the
" reversible pendulum." He showed that if a pendulum were made
symmetrical in external form about its middle point, but loaded at one
end, to lower the centre of gravity, and provided with two knife edges,
like Kater's pendulum, one very nearly at the centre of oscillation of the
other, the length of the seconds pendulum could be deduced from the
two times without regard to the air effects. Laplace had shown that the
knife edges must be regarded as cylinders, and not mere lines of support.
Bessel showed, however, that if the knife edges were exactly equal
cylinders their effect was eliminated by the inversion, and that if they were
different cylinders their effect was eliminated by interchanging the knife
edges, and again determining the times from each — the " erect " and
" inverted " times as we may conveniently term them.
We shall consider these various points separately.
In the first place, Bessel showed that it was unnecessary to make the
erect and inverted times exactly equal. For if T, and T, be these times,
if A, A, be the distances of the centre of gravity from the two knife edges,
and if k be the radius of gyration round an axis through the centre of
gravity, the formula for the compound pendulum gives
9 T,_ V + «c' g ma_ V + *c*
Multiply respectively by Aj, A„ subtract and divide by A, - A, and we
have
h T*-AT*
Let us put ^I'l y^ ^T*
We shall term T the computed time. We see that it is the time corresponding
to a length of simple pendulum h^ + A,. It may be expressed in a more
convenient form, thus :
LetT»=±i-±ii. anda» = ±J— -!j-,
2 2
then Tj' = r' + a-, T,^ = r^ - a', and substituting in T* we get
A,T,>-A,T,^ _^,^^,A,+A,_ T,^ + T,^ ^ T,^-T,^A,+A,
A,-A, A,-A, ^ ^ K-h,
Now A, + A, is measurable with great exactitude, but h^ and ^,, and
therefore h^ - Aj, cannot be determined with nearly such accuracy. The
method of measuiing them consists in balancing the pendulum in horizontal
position on a knife edge and measuring the distance of the balancing knife
edge from each end knife edge. But the formula shows that it is not
liecessary to know A, - A, exactly, for it only occurs in the coefficient of
T,* - T,', which is a very small fraction of T,^' + T,*. Knowing, then, Aj + A,
exactly and A, «> A, approximately, we can compute the time corresponding to
tn rROFKrrriKs of matter.
hf ^ hf trtrtft i^M* if mM tti ih^ttfrftet and inverted pomt ions and avoid the troaUe-
mmit* Mirl^A (^ tmiw whith Kater made before obtaining exact equality for
Umw Irnm tmtiU kriff#i erl^A,
Now l#*i tf«i Mmn\ihr thft air effect. Take first the erect position of the
imiiihiUiin. Wa may rf^frenent the buoyancy by an upward force applied
at ilii« M«ritfit c«f gravity of the displaced air, and equal to its weight my.
\H MiIm muirfi or gravity Ihi dintant $ from the centre of suspension.
TUtt imm of air flowing witb the pendulum will have no effective weight,
rimm It U bitoytKl up by the surrounding air. It is merely an addition to
ihn imim movf«d and mfrves to increase the moment of inertia of the
IHtiMluluini IM UN reprc«c*nt it by the addition of a term mcP when the
imitilMlum In «riNit«
4fr'' MA, - WW MAj \ MA|/
^A," + «' A,* + «' WW . nt'(^
* A, A, MA, MA,
hn^lm«ihi|t miimrm nnd prod(tct« of ^ and ^, since in practice these
ilUMiiMilivi nii« of th«» ortior tO^\
Niiw lnvt«it Aiui nwih^ fixmi an axis near the centre of oscillation.
11m yi\\\\^ of m in Mtt> miui«, but iU iHMitTe of gravity may be at a different
ilki^MlUH« t\^\\\\ tht« imw nuw^H^UHion, 8ay s\ The air moving may be different,
m\ k\\^% ^^ lUMut now put m^f* instn^d of m'iP. We have then
«'*''!* ^V ^ «Wt!! *^' + !?!^
4(r* A» A, MA, MA,
\t vn^ |\M^ A^A^ - K* A» an np|H\xxinvitioii in the coefficieiits cf the
fV^^W \^\M\^ \s^\\\iS\\\\\\^ yl tl\' \>Mii|'Ul«d liiii« T i$ j^i\>en by
V
«^ 1^1 \f m^ «aak» iK^ «xtiM«ftl focm cf tlM
^ ^ y^iMs^^^s^w^ ^>'w;«M<a«>MiI aK>«l k» Midle point*
VN>^. t>vr aa: <«il^ i$^ rfiwriinum i m ikm
v'^-^y ^>;^M^ it. ^^fci^.
>\N .^^^^'*f v>nv >»^its^. ^ tlHidf ,; )«(^
THE ACCELERATION OF GRAVITY. 17
^ the radius of the erect one heing p^ that of the inverted one p^. If 0, Fig. 2,
is the centre of curvature of the knife edge, O the point of contact, G the
centre of gravity, then CG = A|4-pi and the work done is the 8am e as if
G were moved in a circle of radius h^ 4- ^p since the horizontal travel of C
does not affect the amount of work. The instantaneous centre of motion
is the point of contact O. The kinetic energy is therefore
•
M(.« + OG0j
But OG» = OC» + CG"-20CCGcosd
■■A,' neglecting Pi\0^ and smaller quantities.
Then the kinetic energy is M(A|* + k^)-^ .
The work done from the lowest point is
^9{K + Pi) (1 - cos ^) = Mi7(A, + p^t
Hence the erect time is given hy *
4,r» A,+p, A, V V
the inverted time is given by
In the computed time we may put K^sAjAfin the coefficient of the small
quantities p, and p,, and therefore
4t»
Now interchange the knife edges. Assuming that no alteration is
made except in the interchange of p, and p„ the computed time T is
given by
adding the two last equations together and dividing by 2,
* If In simple harmonic motion the kinetic energy at any point is \aff^ and the
work from the centre of swing is ^6^, then the periodic time is easily seen to be
18
PROPERTIES OF MATTER.
RepSOld'S Pendulum. — Bessel did not himself construct a peudulum
to fulfil these conditions, hut, af ber his death, Kepsold in 18G0 devised a form
with interchangeahle knife edges and of symmetrica)
form now known as Repsold's Reversible Pendulum
(Fig. 3), in which he carried out Bessel's suggestions.
The stand for the instrument was, perhaps fortunately,
far from sufficiently firm, for as the pendulum swung
to and fro the stand swung with it. Attention was
directed to the investigation of the source of error.
Its existence was already known, but its magnitude
was not suspected till Peirce and others showed how
seriously it might affect the time.
Yielding* of the Support.— The centre of gravity
moves as if all the forces acted on the whole mass
collected there, so that if we find the mass acceleration
of the centre of gravity, and subtract the weight,
M(7, we have the force due to the support. Reversing,
we have the force on the support.
The acceleration of the centre of gravity is hfi
along the arc and hjd^ towards the point of support.
Resolving these horizontally and vertically,
horizontal acceleration
= h^6 cos 3 " A|^ sin ^ s hfi approximately ;
vertical acceleration
Fio. 8.— Repeold's Re-
The*^Rn88?an "ren^ " Aj^ sin 6 + h^O^ COS 6 = h^e& + h^il^ approximately ;
dnlura used in the » ah B
Indian Survey. but 0= --^_L-_
Then the horizontal force on the stand is Mr/-— ' — 6
= M^r— -1- - since i^ = h.h^
h^ + h^
If a is the amplitude of 0, then ^= ,-pL- (a' - &)
and the vertical force upwards, on the pendulum
Now in finding the yielding of the stand we only want the varying
part of this. Reversing it, the variation in the force on the stand
A, + *c' A, + A,
which is of the second order in 0, and it can be shown that the effect on
the time of swing is negligible in comparison with that of the horizontal
yielding.
THE ACCELERATION OF GRAVITY.
19
Let the yielding to a horizontal force he c per dyne. Let DC (Fig. 4)
be the vertical position, AG the position when displaced through ange 0.
Then tlie yielding OA =^e- j-
produce GA to 0', then OO'^OA/O =e-^'-g^d, say,
or the instantaneous centre is raised d^ above 0, and the
centre of gravity is moving in a circle of radius
^i+^i
MA.
Fio. 4.~ Yielding
of the Support
Let the instantaneous centre be raised d^ = e /'*"? -a
when the pendulum is inverted,
fienoe the erect time is given by
4^ Ai + rfj * * \ + rf,
Uie inverted time by
4ir» A, + rf, ' ' h, + d,
and the computed time by ^ - = Aj + A, + cMy, since h^d^ = A,(£,.
We see that eM.g is the horizontal displacement of the support due to
the weight of the pendulum applied horizontally.
Defforgres* Pendulums.— Starting from this point, Commandant
Defforges his intnxluced a new plan to eliminate the eflect of yielding,
using two convertible pendulums of the Repsold type, of equal weight, of
different lengths, and with a single pair of knife edges, which can be trans-
ferred from one to the other. The ratio of A, : A, is made the same fur
each.
Let the radii of curvature of the knife edges be denoted by p^ p,, let
Aj + A, = /, refer to the first pendulum, A'j + A', = /, reff^r to the second.
The effect of yielding is the Fame for each, increasing the length by i.
Let T V be their computed times,
then
and
^"h + ^ + Jp-IT'^t-Pi)
V-V
£0 PROPERTIES OF MATTER.
and X(T. - T'O = ^ - ^. + (p. - p.) (^-l--- - ^_^._) .
A A '
since t^ = r^ the co-efficient of p, - /9j disappears, and it is not necessary
to interchange the knife edges on the same pendulum. Hence the pen-
dulums are convertible, and we have
The United States Coast and Geodetic Survey have recently constructed
a pendulum in which the planes are on the pendulum and the knife edges
on the support. The one disadvantage is the difficulty of so suspending
the pendulum that the same part of the plane is always on the knife edge,
but against this is to be set the probable greater accuracy of measure-
meiit of ^i + A, and the freedom from the necessity of interchange of
knife edge. Further, should a knife edge be damaged it can be reground
without affecting the pendulum, whereas in the ordinary construction
regrinding really alters the pendulum, which practically becomes a
different instrument.
Variation of Gravity over the Surface of the Earth.
Richer. — The earliest observation showing that gravity changes with
change of place was made by Richer, at the request of the French
Academy of Sciences, in 1672. He observed the length of the seconds
pendulum at Cayenne, and returning to Paris found that the same
pendulum must there be lengthened 1^ Paris lines, 12 to the inch.
Newton's Theory. — This observation waited no long time for an
explanation. Newton took up the subject in the Principia (Book II I.,
Props. 18-20) and, regarding gravity as a terrestrial example of uni-
versal gravitation, he connected the variation with the form of the
earth. He showed first that if the earth is taken as a homogeneous
mutually gravitating fluid globe, its rotation will necessarily bring about a
bulging at the Equator, for some of the weight of the equatorial portion
will be occupied in keeping it moving in its daily circle while the polar
part has but little of such motion. A column, therefore, from the centre
to the surface must be longer at the Equator than at the Pole in order
that the two columns shall produce equal pressures at the centre. Assuming
the form to be spheroidal, the attraction will be different at equal dist-anees
along the polar and equatorial radii. Taking into account both the
variation in attraction and the centrifugal action (^\^ of gravity at the
Equator), Newton calculated the ratio of the axes of the spheroid. Though
his method is open to criticism, his result from the data used is perfectly
correct, viz., that the axes are as 230 : 229. Taking a lately measured
value of 1" of latitude, he found thence the radii, and determined their
difference at 17*1 miles. He then found how gravity should vary over
such a spheroid, taking centrifugal action into account, and prepared
a table of the lengths of 1° of latitude and of the seconds pendulum
T every 5^ of latitude from the Equator to the Pole. From Ujs table
THE ACCELERATION OF GRAVITY.
Bl
tbe pendulum lengtli at Cayenne, in
lees than at Ptiris in Utttiido -((S" W.
of this from the diminulion of Ij line
of the scale with higher temperatui
latitude 4° 55', should be 1 line
Ho asBigns part of tlio difference
observed by Richer to expansion
the EVjiiator.
The Swedish and Peruvian Expeditions.— Newton's theory of
tlie figure of the eartli as depending on gravitation and rotation led eai'ly
in the eijjhteenth century to measure ments of a degree of latitude in Peru
and in Sweden. If the earth were truly spheroidal, and if tbe plumb-
line were everywhere perpendicular to the surface, two such mensuretueiita
would BulBce to give the axes a and b, inasmuch as length of arc of 1*
MI-f + 3.
. X + X'
i3C00sinI" whei-e (=-
=the ellipticity and XX'
are the latitudes at the beginning and end of the arc.*
We know now that through local variations in gravity the plumb-line
is not perpendicular to a true spheroid, but that there ore humps and
hollows in tbe surface, and many measure men ts at different ports of the
earth are needed to eliminate the local variations and find the axes of the
spheroid most nearly coinciding with tbe real surface. But the Swedi-h
and Pei-uvian expeditions clearly proved the increase of length of a degree
in northerly regions, and so proved the flattening at the Pules. These
expeditions have another interest for us here in that pendulum observa-
tions were made. Thus MaupertuLs, in the northern expedition, found
that a certain pendulum clock gain'ed 59'1 seconds per day in Sweden on
itfl rate in Paris, while Bouguer and La OandH.raine, in the Peniviaa
expedition, found that at the Equator at sea-level the seconds pendulum
was 1*20 Paris lines shorter than at Paris. Bouguer's work, to which we
have already referred, was especially important in that he determined the
length of the Eeconds pendulum at three elevations : (I) At Quito, which
may be regai-ded as a tableland, the atatioa being HGli toises t above sea-
level ; (3) on the summit of Ficliincha, a mountain rising above Quito to
a height of 2434 toises above sea-level ; and (3) on the Island of Inca, on
the river E^<me^alda, not more than thirty or foiiy toises above sea-level.
The Equator runs between Quito and the third station, and they are only
a few miles from it. In space free from matter rising above sea-level gravity
might be expected to decrease accoiiling to the inverse square law starting
from the earth's centre, so that if A is the height above sea^level and r ia
Btatlon.
AbOTO
SeH-kvcl
SMtmdi
In'unai"
Comction
Buojauey.
Fncllon
rrvAi-xy
SqUUW
U»2A/r.
Pichincha .
Qaito . .
Isle ol Inca
2131
IIGO
43870
433-33
439 07
-05
+ ■075
+ ■04
+ ■05
+ ■08
438 '83
439^21
rii
n't.
I
22 PROPERTIES OF MATTER.
the earth's radius, the decrease should be 2A/r of the original value. In
the table on p. 21, Bouguer's ressults are given. In the last column but
one is the decrease observed at the upper stations, and in the last column
the decrease calculated by 2hjr.
It will be seen that gravity decreased more slowly than by the inverse
square law. Centrifugal force would act in a contrary way, though, aa
Bouguer showed, by a n^ligible amount. The excess of gravity, as
observed, above its value in a free space must therefore be ast^igned to the
attraction of the matter above the sea-level. Bouguer obtained for the
value of gravity gi^ on a plateau of height h, as compared with its value at
sea -level g^
/, 2A^3Aa\
where h is the density of the plateau and A the density of the earth.
This formula, now known as Bouguer s Rule, seems to have dropped
out of sight till it was again obtained by Young in 1819, but on its
revival it was generally employed to reduce the observed value at a station
to the sea- level value in the same latitude.
Putting it in the form ?iJl?* = ^/l -| 1]
3093
and using the values at Quito and sea-level, A == -^ — i
oDO
Bouguer remarked that this result sufficed to show that the density of
the earth was greater than that of the Cordilleras, and consequently that
the earth was neither hollow nor full of water, as some physicists had
maintained. We now know that the value of a fo obtained is far too great,
and shall see later what is the probable explanation.
Clairaut'S Theorem.— In 1743 Clairaut published his great treatise,
Theorie di la Figure de la Terre, which put the investigation of the tigiire of
the earth on lines which have ever since been followed. In this work he
takes the surface of the earth as a spheroid of equilibrium — i.e., such that a
layer of water would spread all over it, and assumes that the internal density
varies so that layers of equal density are concentric co-axial spheroids.
Denoting gravity at the Equator, Pole, and latitude X, by g„ g^^ g^ respec-
tively, and putting w =^ centrifugal force at Equator /</, and c = e11ipticity=a
difference of equatorial and polar radii / equatorial radius, he shows (1) that
gx=gJ^-\-nsm*\) (1)
where n is a constant : (2) that
9. 2 W
Fi-om (1) and (2) we get
^A = i/.|l + Kw*-Osin^xj,
a result known as Claii-aut's Theorem.
Laplace showed that the surfaces of equal density might have any
THE ACCELERAtlON OF GRAVITY.
S3
nearly spherical form, and Stokes {Hath. Fkya. Papert, vol. U. p. 104),
going furtlier, showed lli»t it is UBueo«ssary to assume &ny law of density
BO loDg an the estemal surface iH a suLeroid of equilibrium, for the theorem
still remains true.
Fium Clairaut's Theurem it follows that, if the earth ia mi obUta
apheroid, itd ellipticity can be detenuioed from pendulum ezperimenta on
the variution of gravity without a knowledge of it* absolute value, except
in Bo far as it is involved in m. And if the theorem were exactly
true, two relative determinations at stations in widely different latitudes
ithoiild RutHce. liut here a^ain, as witli arc measurements, local variations
interfere, and many detei-uiiriatiomj must be made at widely scattered
stations to eliminate their effect.
Kater and Sabine. Invariable Pendulums.— During the last half
of the eighteenth century much pendulum work was cai-ried on, but hardly
with GuQideut accuracy to make the lei^tilts of value now, an<l wo may con-
sider that modern research be(;Ln^ with Kat«r, who con8truct«d a number of
"invariable pendulums," neaily beating seconds, and in shape much lika
his convertible pendulum without the reverse knife edge. The principle
of " invariable pendulum " work consiEts in using the same pendulum at
different stations, determining its time of vibmtion at each, and correcting
for temperature, air eS'ect, and height above sea-level. The relative values
of gravity are thus known, or the equivalent, the relative lengths of the
secondn pendulum, without measuring the length or knowing the moment
of inertia of the pendulum. Kater himself determined the length of tJie
seccmds pendulum at stations scattered over the British Islands, and
Sabine, between 1^30 and 1535, carried out observations at stations
ranging from the West Indies to Greenland and Spitsbergen. About the
lame time Freycinet and Duperry made an extensive series ranging far
into the Southern Hemisphere, and other obacrvers contributed observa-
tions. Now, though diffoi*6nt pendulums were used, these sci'ies over-
lapped and could be connected together by the obseirations at common
stolions; and Airy in 18;(0 (Kiicijc. Met., " Figure of the Earth ") deduced a
value of the ellipticity of about t,\-^-
Breaking: down of BOUgUer'S Rule.— Subsequent work brought
into ever- increasing prominence the local divergences from Clairaut's
formula, and it gradually became evident that on continents and on high
ground the value ot gravily was always less than would be especteii from
Cliiimul's formula when corrected by Bouguer's rule, while at the sea
coast and on oceanic fclnnds it was greater,
Indian Survey.— Thus, in the splendid series of pendulum ex-
periments curried out in connection with the Indian Trigonometrical
Survey between 18G,') and 1875 (G. T. Survey of India, vol. v.) tha
variations were very marked. In these experiments, invaiiable pen-
dulums, Kater'a convertible and Repsold's reversible pendulum were all
ns6d, and observations were made by Bosevi and Heaviside from More, on
the Himalayns, at a height of 15,427 fe«t,down to the sea-level. Theseries
was connected with others by swinging the pendulums at Kew before
their transmission to India, and very great precautions were taken to
CoiTect for temperature, and the air effect was eliminated by swinging in
a vacuum. At Morii the defect of gravity was very marked.
Aipy's " Hydrostatic " Theory. Faye's Rule.— Airy {Phil. Trans.,
1865, p. 101) had already suggested that elevated masses are really
M PROPERTIES OF MATTER.
boojed np by matter at their base lighter than the aTerage ; thai in faei
thej float OD the hqaid or more probabl j Tiflooiis aolid interior Teiy much
aa ieebergs float on the sea. If the high ground is in eqnilibriam, neither
rising nor Calling, we maj perhaps regard the total qnantitj of matter
underneath a station as being equal to that at a station at sea-level
in the same latitude^ This hjdiosfcatie theoiy has led F^je to suggest
that the term - — in Bongner's rule should be veplaoed bj a term only
taking into aoooont the attraction of the excess of matter under the
station above the average level of the near neighbourhood, a 8uggesti<m
embodied in Faje s rule.
Recent woHl by the American Survey (Jaier. Jcnam, Seienet^ March
1896, G. R. Putnam) has shown that <m the American continent Faye's
rule gives results decidedly more consistent than those obtained from
Bonder's rule.
By a considoation of tiie results obtained up to 1880 by the pen-
dulum, Clarke (Geodesy^ p. 350) gives as the value of the ellipticity
< » : — ttt:. — ^-^t A value almost ooincidinff with that obtained from measure-
2V2'2± 1-5 ^^
ments of degrees of latitude. Helmert, in 1884, gave as the result of
pendulum woik , and we may now be sure that the value differs very
little frcMn -— --.
300
Helmert {Theorieen der hoheren Geoddsie^ Bd. H. p. 241) also gives
as the value of ^ in any latitude X,
^^ = 978-00(l + 0005310 8in»X)
and this may be taken as representing the best results up to the present.
Von Stemeck's Half-second Pendulums.— The kbour of the
determination of minute local variations in gravity was much lessened by
the introduction by von Stemeck, about 1880, of half -second invariable
pendulums, and his improved methods of observation have greatly in-
creased the accuracy of relative deteiminations at stations connected by
telegraph.
With half the time of swing the apparatus has only one-fonrth the
linear dimensions, and it can be made at once more steady and more
portable. The size of the pendulum being thus reduced — it is about
10 inches long — it can without much trouble be placed in a chamber which
ean ba exhausted and wliich can be maintained at any desired temperature.
Each pendulum can therefore be made to give its own temperatmre and air
correction8 by preliminary observations. The form of the pendulum is
shown in Fig. 5. The chief improvements in the mode of obeervation
introduced by von Bterneck consist, Ist, in the simultaneous comparison
with the same clock of the swinging of two pendulums at two stations at
which gravity is to be compared. For this purpose the two stations are
oonnectc'd by an electric circuit containing a half -seconds ''break circuit"
chronometer, which sends a signal through each station every half-second,
and thus clock-rates are of little importance. And, 2nd, Uie method of
observing the coincidences of the pendulum with the chronometer m gni^liy^
In the final form this consists in attaching a small mirror on the pendulom
THE ACCELERATION OF GRAVITY. «5
knife e^lge (not shown in Fig, 5, which represents an earlier form) per-
peadicuW to tlje pliine of vibmtion of the penduhiio, iind pUdng a HxeJ
mirror close to tlie other &ad piirallel to it whou the peuiiulum in ut iciit.
The chronometer aignaU work a relay, givi
ia reflected into a telescope from both mitra
rest the image of the spark in both mirrora appears on the horizontal
cross-wire, and when the pendulum m vibrating a coincideooe occurs when
the two imaged are in this position. The method admits of exceedingly
26
PROPERTIES OF MATTER,
accurate determination. We shall see later how von Sterneck used the
method in gravitation experiments. Here it is sufficient to say that he
has used it in many local determinations of gravity, and that his pendulums
have been used without the simultaneous method for determinations at
various stations in both hemispheres. The American Geodetic Survey haa
adopted very similar apparatus and methods, and it appears probable that
we ^*haIl soon have a knowledge of the variation of gravity over the surface
of the earth of a far more detailed and accurate kind than could possibly
be ohtjiined by the older methods.
Differential Gravity Meters. — Before invariable pendulums were
brought to theii* present accuracy and portability, there was some hope
that for relative determinations the pendulum might be superseded by a
statical measurer of gravity which would do away with the need for time
measurements. Such an instrument must essentially consist of a mass
supported by a spring, and the variation in gravity must be shown by the
alteration in the spring due to the alteration in the pull of the earth on
the masif. The earlier instruments devised for the purpose need not be
described, for they were quite incapable of the accuracy attained by
invariable pendulums. The first instrument which promised any real
success was devised by von Sterneck, and is termed by him the Barymeter
(Mittheilungen des K. K. Militar-Geog, Inat.^ Wien, v. 1885).
Von Sterneck*s Barymeter.
O —A brass plate P (Fig. 6), 30 cm. x
20 cm., is balanced on a knife edge, «.
Along a diagonal is a glass tube
terminating in bulbsO and TJ, 5 cm. x
6 cm., so that in the equilibrium
position O is about 25 cm. above 17.
The tube and about \ of each bulb
is filled with mercury, and above the
p. mercury is nitrogen. The apparatus
s^ \Aw is adjusted so that at 0° C, and for
^v-T a cei-tain value of gravity the edges
of the brass plate are horizontal
and vertical, a level W showing
when this position is attained. If now gravity were to increase, the
weight of the mercury would be greater, and it would tend to flow from O
and compress the gas in U. Thus the balance would tilt over to the left,
and the tilting still further increasing the pressure on XJ, the flow
downwards is increased. The instrument can thus be made of any
desired sensitiveness, and its deflections can be read by scale and measured
in the usual way. To compensate for changes of temperature, a second
tube terminating in smaller bulbs o and u, each about 6 cm. x 3 cm., is
fixed along the other diagonal. This contains some mercury, but above
the mercury in u is alcohol, and only o contains nitrogen. If the
temperature rises the mercury becomes less dense, and on this account it
is driven from U to O in the larger tube, but still more is it driven in this
direction from the fact that the increase of pressure of the gas in XT is
greater than in O. Meanwhile, the alcohol in u expanding, drives the
mercury in the smaller tube into o, and by suitable adjustments of volume
the two can be made to balance sufllciently for such small temperature
variations as will arise when the whole is placed in a box surrounded witk
Fio. 6. — Von Sterneck's Barymeter,
THE ACCELERATION OF GRAVITY.
27
melting ice, and it is thus that the instrument is used. With this
instrument von Stemeek could detect the change in gravity in going from
the cellar of a building to a height of 25 metres.
Threlfall and Pollock's Quartz-thread Gravity Balance.—
In the FhilTraiis.i A. 193, 1899, p. 215, Thi-elfail and Pollock desciibe an
instrument for m&isuring variations in gravity statically which is both
accurate and portable.
The essential features of the instrument are represented in Fig. 7.
A and B are two metal rods which can slide along their common
axis. Ois a coach-spring attached to A. H I is a quartz thread 30*5 cm.
long and '0038 cm. in diameter stretched horizontally between B
and C. D is a piece of gilded brass wire soldered to the quartz thread.
Its weight is -018 gm., its length 5*3 cm., and its centre of gravity is
a little to one side of the quartz thread. Its weight therefore tends
to pull it into the vertical position and twist the quartz. But such a twist
A
n
[r^
n
^
A
4
6^
u
FiO. 7.— Threlfall mid X^oUock's Quart z-thremd Gravity Balance.
can be put on the quartz thread by rotating the arm G, which carries a
vernier, that D is brought into the horizontal position. For this about
three whole turns are required. The end of D when in the horizontal
position is on the cross-wire of the horizontal microscope E. The hori-
zontal position of the brass wire is only just stable. If it be twisted a few
d^rees more the point of instability is reached and the wire tends to
continue moving round, and would do so but for an an^ester. The mode
of using the instrument consists in dctermiuing the tviist put on the quai-tz
thread by the arm G to bring it into the horiz(>ntal position. If gravity
increases, the moment of the weight of D increases and a greater twist is
required. To calibrate the instrument the change in reading of the vernier
on G is observed in passing from one station to another, at both cf which
g is known — the two stations selected being Sydney and Melbourne. Of
course, temperatuie corrections are necessary both on account of the change
in length of D and the change in rigidity of the quartz. Preliminary
determinations of these were made at one station. For the details of the
instrument and the mode of using it we refer the reader to the original
account. It suffices here to say that it has given very faii-ly consistent
results at stations wide apart and that it promises to rival the invariable
pehduluin.
OHAFrER m.
GRAVITATION.
OoNTBNTS.*— The Law of Gravitation— The Gravitation Constant and the Mean
Density of the Earth.
The full statement of Newton's Law of Gravitation is that any particle
of mass M| attracts any other particle of mass M, distant d from it with a
force in the line joining them proportional to M^'M.JtP. The evidence for
the law may be briefly summed up as follows :
Starting with any single planet — say the earth — and referring its
position to a system, fixed relatively to the sun and the distant stars, direct
astronomical observation shows that it may be described with a close
approximation to the truth, as moving in an ellipse with the sun in one
focus, at such speed that the line from the centi'e of the sun to the centre
of the planet sweeps out equal areas in equal times. This implies, as
Newton showed, that the acceleration of the planet is towards the sun and
inversely as the square of its distance from that body.
Now, comparing the different planets, observation shows that (length of
year)Y(mean distance)' is the same for each, and from this it follows that
the constant of acceleration is the same for all, or that at the unit distance
from the sun they would all have the same acceleration if the law holding
for each in its own orbit held for it at all distances.
So far this is mere time-geometry, or a description of position and rate
of change of position, and we might have other equally true, if less
convenient, modes of description referred to other standards, such as the
epic} clic geocentric mode of the ancients, or the practical mode in common
use iu which the co-ordinates of a planet are measured with regard to some
observatory, its meridian, and horizon.
But if we regard the accelerations as indicating forces, the different
method.^ of description are no longer equivalent. We must select that
which gives a system of forces most consistent in itself and most in accord
with our terrestrial experience. Here the heliocentric method, with the
modification described hereafter, is immensely superior to any other, and,
adopting it, we must suppose that the accelerations of the planets indicate
forces towards the sun, and since the constant of acceleration is the same
for all, that the forces on equal masses are inversely as their distances
squared from the sun, whatever planets the masses belong to. In other
*
* This chapter is larcrelj t^kcn from The Mean Dentity of the Earth, and papers
cnmmunicated to the Rojal Institation and the Birmingham Natural Uiattory and
Vhilosophical Society, by J. H. Fojnting.
GRAVITATION. i.<t
words, the sun has do favourite among ita attendantx, but pullR on each
pound of each according to the eame rule.
But the assumption that the accelerations indicate forces of the kind
we experience on I he earth, carries with it the suppuf'ition of equnlil-y of
action and reaction, and so we conclude that each plnnet reacts on the sun
with a force equal and oppoBite to that exerted by the sun on the pinnet.
Hence, each acts with a force proportional to its own miws, nnd invereely
ns tlie S(|uare of iU distauce away. If we suppose that there is nothing
special in the nttraction of the sun beyond great mHgiiitude corresponding
to great masti, we inuFt (oncluile that the suit also acta with a force propor-
tional to it« niHss. But we have just shown that the force is proportional
to the mass acted on. Hence, we have the force on any planet proportional
to mass of sun x mass of planet / (distance apart)',
Nov, turning to any of the smaller systems CDDsisting of a primary
and its saletlitcs, the shnpe of orbit and the motion of the satellites agree
wiih the supposition that the primary is acting with a force according to
the invci'se square law. It is important for our special problem to note
here that in the case of the earth we must include in the term " satellite "
any body at its sui-face which cnn be Wf'iglied or moved.
We are theref-re led to coacliiJe that the law is general, or that it we
have any two bodies, of masses M[ and M„ at d distJinco apart, the force
on either ia
GM,M,
where G is a constant — the constant of gravitation.
The acceleration of one of them, say M„ towards the other is --^-l
If this conclusion is accepted, we can at once determine the masses of
the various primaries in terms of that of the sun for —
acceleration of satellite towards primary = g_^!^'>^_ P"""* !T-
distance of satellite*
I ind acceleration of primary towaids sun =0^.- ^"^s oF^uji
distance of piimnry'
By division G is eliminated, and we obtain the ratio of the masses in terms
, of qiinntiiies wliidi may be measured by obsei'vation.
. As an illii-tiation, let us make a rough determination of the mass of
I tiie sun ill tei ms of the mass of the earth.
We may tiite the acceleration of the moon to the earth as nppi-oxi-
t mately *»„* x i^m. where w„ is the angular velocity of the moon and rf„ its
I distiince from the earth, and the acceleration of the Utter to the sun as
•hV^e where u,,, is the angular velocity of the earth, and d^_ its distance
u the sun. Lot the mass of the sun be 8 and ihat of the earth be E.
I then *''"'''^'"'**'' °" P ^ Moon ^ u„'xd„ Ex d,,'
Accel ci'aiion of Eaith ug'xiig tixd,^'
27 \-/n2(>oofvm\= „ ,
50 PROPERTIES OF MATTER.
A confirmation of the generality of the law ia ohtained from the
perturbations of the planets from the elliptic orbits which we have for
simplicity supposed them to describe.
These perturbations, in any one planet, can at least approximately be
analysed into separate disturbances, each due to one of its fellow planets,
acting with a foi-ce inversely as the square of its distance away, and if we
assume this force proportional to the mass of the disturber we obtain
another measure for this mass in terms of that of the sun.
The concordance of the two methods is as complete as we could
expect.
The determination of the masses of the different members of our system
in terms of that of the sun enables us to choose a still more satisfactory
origin for our system of reference than the centre of the sun — viz., the
centre of mass of the whole system. The change is small, but without it
we could not account for all the motions merely by a set of inverse square
forces in which action and reaction were equal and opposite.
We have for simplicity considered the sun and planets as without
appreciable dimensions as compared with their distances apart. But
measurement shows that they are all approximately spheres, and the
attraction on a sphere with density varying only with the distance from
the centre — t.^., consisting of homogeneous concentric shells, if itis considered
as the resultant of the attractions on the separate particles, all according
to the same inverse square law, is the same as that on the whole mass
collected at the centre of the sphere. Further, if the attraction is due, not
to the attracting body as a whole but to its separate pai-ts, each acting, as
it were, independently and according to the same law, then an attracting
sphere acts as if it were all concentrated at its centre. Since the planets,
with a close approximation, behave as if they were merely concentrated
masses at their centres, and since the deviations from this behaviour, such
as the earth's precession^ can all be accounted for by their departure from
sphericity, we have strong presumption that the attraction is really the
resultant of all the attractions), each element m^ of one body acting on each
element m, of the other with force Qm^mjd^.
Astronomical observation enables us, then, to compare the masses of
the various members of the solar system with each other, and, by taking
into account the sizes of the planets, to make a table of specific gravities,
choosing any one as the standard substance. Thus, if we take the earth
as standard, the mean specific gravity of the sun is about 0*25, that of
Mercury about 1*25, that of Venus and Mars about 0*9, and so on.
But this does not give us any idea of the specific gravity in terms of
known terrestrial substances or any idea of the masses in terms of the
terrestrial standards, the kilogramme or the pound. It is true that Newton,
with little more than the astronomical data at his command, made a
celebrated guess on the specific gravity of the earth in terms of water,
which runs thus in Motto's translation of the Principia (vol. ii. p. 230,
ed. 1720, Book III., Prop. 10) : ** But that our globe of earth is of greater
density than it would be if the whole consisted of water only, I thus make
out. If the whole consisted of water only, whatever was of less density
than water, because of its less specific gravity, would emerge and float
above. And upon this account, if a globe of terrestrial matter, covered on all
sides with water, was leis dense than water, it would emerge somewhere :
and the subsiding water falling back, would be gathered to the opposite
GRAVITATION. 31
side. And such is the condition of our earth, which, in great measure, is
covered with seas. The earth, if it was not for its greater density, would
emerge from the seas, and according to its degree of levity, would be raised
more or less above their surface, the water and the seas flowing backwards
to the opposite side. By the same argument, the spots of the sun which
float upon the lucid matter thereof, are lighter than that matter. And
however the Planets have been formed while they were yet in fluid masses,
all the heavier matter subsided to the centre. Since, therefore, 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 or four, or even five
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,
especially since I have before shewed that the earth is about four times
more dense than Jupiter."
It is not a little i-emarkable that Newton hit upon the limits between
which the values found by subsequent researches have nearly all lain.
In order, then, to complete the expression of the law of gravitation we
must connect the celestial with the terrestrial scale of densities. In fact,
we must do for the masses of the solar system that which we do for their
distances in the determination of the solar parallax, though we cannot
proceed quite so directly in the former case as in the latter in connecting
the celestial and terrestrial measures. If we could measure the accele-
ration, say, of the moon, due to any terrestrial body of known shape
and density— if, for instance, we knew the form and extent of our
tidal- wave and its full lunar effect— we could at once find the mass of
the earth in terms of that of the wave, or its density as compared with
sea-water.
But at present this cannot be done with any approach to accuracy, and
the only method of solving the problem consists in finding the attraction
between two bodies on the earth of known masses a known distance apart,
and comparing this with the attraction of the earth on a known mass at
its surface instead of its attraction as a heavenly body. Since the law of
attraction is by observation the same at the surface of the earth and at a
distance, we can thus find the mass of the earth in terms of either of these
known masses.
To take an illustration from an experiment hereafter described, let us
suppose that a spherical mass of 20 kilos, is attracted by another spherical
mass of 150 kilos, when the centres are 30 cm. apart with a force equal to
the weight of J mgm. or sirirojjjyTny ^^ *'^® weight of the 20 kilos, when
the latter is on the surface of the earth and 6 x 10^ cm. from its centre,
we have :
Mass of Earth 150000
(G X lOy ' SO'
— 1 • 1
l^hence mass of earth = 5x10'' grammes nearly.
The volume of the earth is about 9 x 10'-'* c.c, whence the mean density
of the earth A is about 5' 5.
Or, using the experiment to give the constant of attraction, and
expressing the masses in grammes, the weight of ^ mgm. or
•00025(7 = ^xi^QQQQx^OQOQ
S2
PROPERTIES OF MATTER.
Whence, if ^ = 98; G =
981 X -00025 X 30'
150000 X 20000 10*
(nearly).
I** station
Out SoofK of
Suniin> on Slope
♦*Tir§uR5;
ftrtf SUlioa
2
9
A determination of G completes the expression of the law of
gravitation.
This example shows that the two problems, the determination of the
gravitation constant G and the determination of the mean density of the
earth A, are practically one, inasmuch as our knowledge of the dimensions
of the earth and the acceleration of gravity g at its surface at once
enable us to determine G if we know A, or to deteimine A if we
know G.
The Methods of Experiment.
These naturally fall into two classes. In the one class some natural
mass is selected, either a mountain or part of the earth's crust, and
its mass and form are more or less accurately
determined by surveys and mineralpgiad
examination. Its attraction on a plumb-
bob at one side, or on a pendulum above or
below it, is then compared with the attrac-
tion of the whole earth on the same body.
In the other, the laboratory class of
experiment, a smaller mass, such as may
be easily handled, is placed so as to attract
some small suspended body, and this attrac-
tion is measured. Knowing the attracting
and attracted masses, the attraction gives G.
Or, comparing the attraction with the attrac-
tion of the earth on the same body, we get A.
The Experiments of Bousruer in
Peru. — The honour of making the first
experiments on the attraction of terrestrial
masses is to be accorded to Bouguer. He
attempted both by the pendulum experi-
ments described in the last chapter, and by
plumb-line experiments, to prove the exist-
ence of the attraction of mountain masses in the Andes, when engaged in
the celebrated measurement of an arc of the meridian in Peru about the
year 1740. The pendulum experiments are sufficiently described in the
last chapter.
In his plumb-line experiments he attempted to estimate the sideway
attraction of Chimboi-azo, a mountain about 20,000 feet high, on a plumln
line placed at a point on its side. Fig. 8 will show the principle of the
method. Suppose that two stations are fixed, one on the side of the
mountain due south of the summit, and the other in the same latitude,
but some distance westward, away from the influence of the mountain.
Suppose that at the second station a star is observed to pass the meridian —
we will say, for simplicity, directly overhead, then a plumb-line hung
down will be exactly parallel to the observing telescope. At the first
station, if the mountjun were away, it would also hang down parallel to
the telescope when directed to the same star. But the mountain pulls the
plumb-line towards it, and changes the overhead point so that the star
Fio. 8.— Boaguer'8 Plumb-line £x-
pArimoot on the Attraciiou of
Ubimboraso.
GRAVITATION.
39
\
I
I
appears to northward instead of in the zenith. The method simply con-
sists in determining how mucli the star appears to be shifted to the noi'th.
The angle oF appatent ebift is the ratio of the horizontal pull of tits
mountain on the plumb-bob to the pull of the earth.
To cari'y out the experiment, Bouguer fixed the first station on the
south slope of Chimborazo, just above the perpetual snow-line, and the
second nearly on the same level, several milea to the westward. Ho
describes (Figure de la Terre, 7th section) how hia expedition reached the
firet Gtation after a moat toilsome journey of t«n hours over rocks and
snow, and how, when they reached it, they had all the time to fight against
the snow, whicli threatened to bury their tent. NeverthuleuH, they
succeeded in making the necessary observations, and a few days later they
were able to move on to the second station. Here they hoped for better
things, as they were now below the snow-line. But their difiiculties were
even greater than before, as now they were exposed to the full force of the
wind, which filled their eyes with sand nnd was continually on the point
of blowing away their tent. The cold was intense, and so hindered the
working of their instruments that they had to apply fire to the levelling
screws before they could turn them. Still they made their observations,
and found that the plumb-line was drawn aitide about 8 seconds. Had
Chimbot^zo been of the density of the whole earth, Bouguer calculated,
from the dimensions and distance of the mountain, that it would Iiave
drawn aside the vertical by about twelve times this, so that the earth
appeared to be twelve times as den.se as the mountain, a result undoubtedly
very far wide of the truth. But it is little wonder that under
such circumfitancea the experiment failed to give a good result, and all
honour ia due to Bouguer for the ingenuity and perseverance which enabled
him to obtain any result at all. At least be deserves the credit of first
showing that the attraction by mountain masses actually exists, and that
the earth, as a whole, is denser than the surface strata. As he remarks,
hia experiments at any rate proved tliat tlie earth was not merely a hollow
shell, Bfi some had till then held ; nor wasit aglobe full of water, as others
had maintained. He fully recognised that his expenments were mere
trials, and hoped that they would be repeated in Europe.
Thirty years later his hope whs fulfilled. Maskelyne, then the
English Astronomer Royal, brought the subject before the Boyal Society
in 1772, and obtained the appointment of a committee " to consider of a
proper hill whereon to try the experiment, and to prepare everything
necessary for carrying the design into execution." Cavendish, who wa^
himself to carry out an earth-weighing experiment some twenty-five years
later, was probably a member of the com mittee, and was certainly deeply
int«re8ted in the subject, fur among his papers liave been found calcula-
tions with regard to Skiddaw, one of several English hills at first con-
sidered. Ultimately, however, the committee decided in favour of
Scliieballion, a mountain near L. Rannocli, in Perthshire, 3547 feet high.
Here the astt'onomical part of the experiment was carried out in 1774,
and the eui'vey of the diiitrict in tliat and the two following years. The
mountain has a short east and west ridge, and slopes down steeply on the
north and south, a shape very suitable for the purpose.
Maskelyne, who himself undertook the astronomical work, decided to
work in a way very like that followed by Bouguer on Chimboraio, but
modified in a manner suggested by bim. Two stations were selected, one
Si
PaOPERTIES OF MATTER.
on the south and the otlier od the north slope. A small obaervatory wu
«rect«d first at the Houth station, nod the angular distance of some stars
from the senith, when they were due south, woe most carefully measured.
The stara selected nil p.issed nearly overhead, ao that the angles nieomired
were very small The inHtrumeiit used was the zenith sector, a t«lescope
rotating about a horizontal east and wiist axia at the object-glass end, and
provided with a plumb-line hanging from the axis over a graduated scale at
the eyepiece end. This showed how far the telescope was from the vertical.
After about a month's work at this station the observatory was moved
to the north station, and again the same stars were observed with the
zenith sector. Another month's work completed this part of the ez-
periment. Fig. 9 will show how the observations gave the attraction
due to the hill. Let us for the moment leave out of account the curvature
of the earth, and suppose it ftat. Further, let us suppose that a star ii
being observed which would be directly overhead if no mountain existed.
Then evidently at S, the plumb-line i
t ^ I pulled to the north, and the zenith i
'; / shifted to the south. The star therefore
• / appears slightly to the north. At N.
': ■ there is an opposite eB'ect, for the moun-
'; J, tain pulls the plumb-line southwards,
n I >, and shifts the zenith to the north ; and ,
U J \ i ^°^ ^^^ ^^''■^ appears slightly to th* 1
111 X \iL> south. The total shifting of the star is j
•Yt/» "XC .,. double the deflection of the plumb-linB 1
I at either station due to the pull of ths
mountain.
But the curvature of the earth n
deflects the verticals at N. and S., and
in the same way, ho that the obeerved
shift of the star is partly due to the mountain and partly due to the
curvature of the earth. A careful measure was made of the diatanos
between the twoatationa, and this gave the curvature deflection as about 48".
The observed deflection was about 55", ao that the eflisct of the mountain, I
the difference between these, was about 12".
The nest thing waa to lind the form of the mountain. This was befora J
the days of the Ordnance Survey, so that a complete survey of the district 1
waa needed. When this was complete, contour mapa were made, giving I
the volume and distance of every part of the mountain from each station, j
Button was associated with Maakelyne in this part of the work, aiid hs I
carried out ail the calculations bitsed upon it, being much assisted bj J
valuable suggestions from Cavendish. '
Now, had the mountain had the same density as the earth, it ^ _
calculated from its shape and distance that it should have deflected the J
plumb-lines towai-da each other through a total angle of 20-0", or 1 J tim^ ]
the observed amount. The earth, tlien, is IJ times as dense i
mountain. From pieces of the rock of which the moiintain is composeiL 1
ite density was estimated as 21 times that of water. The earth should I
have, therefore, density I J x 2J or 4J. An estimate of the density of thsJ
mountain, based on a survey made thirty years later, brought the neult J
up to 5. All subsequent work has shown that this number is not y^tfM
fw from the truth. ^
— M«Bke1yne'B riuroh-li:
1
I
GRAVITATION.
S5
I
I
An exactly dmilar experiment was made eighty years later, on tlie
completion of the Ordnance Survey of the kingdoui. Certain anomalies
in the direction of the vertical at Edinburgh led Colonel James, the
director, to repeat the Schiehiillion experiment, using Arthur's Seat as
the deflecting mountain. The value obtained for the mean density of the
earth was about 5^.
Repetitions have also been made of the pendulum method, tried by
Bouguer in the Andes.
The first of these wiia by Carlini, in 1 821. He observed the length of
a pendulum swinging seconds at the Hospice on Mont Cenis, about 6000
feet above sea-level, and so obtained tlie value of gravity there. The
value due to mei-e elevation above the sea-level was easily calculated, but
the observed value was greater than that calculated by about 1 in 5000.
In other words, the pull of the whole eai-th was 5000 times greater than
that of the mountain under the Hospice. Knowing approximately the
shape of the mountain, and estimating its density
from ppecimena of the rock, Cai'liiii found the
density of tiie earth to be about 4;^ times that of
Another experiment of the same kind was
made by Mendenhall, in Japan, in 18SI). Here I
he determined the value of gravity on the
summit of Fujiyama, a mountain neiirly 2^ milett
high. He found it greater tlian the viilue
calculated ii-om the increased distance from the
earth's centre by about I in 5000, as Carlini had
done on Mont Cenia. Fujiyama, though the
higher, is moi-e pointed and loss dense than
Uont Cenis. Mendenhall estimated the i
6-77.
Airy applied the pendulum to solve the problem In a somewhat different
way, using, instead of a mountain, the crust of the earth between the top
and the bottom of a mine. Him first attempts were made in 1826, at the
Dolcoatfa copper mine, in Cornwall. Hei'e he swung a pendulum first at
the surface and then at the bottom of the mine. At the point below we
may consider that the weight of the pendulum was due to the pull of the
part of the earth within the sphere with radius reaching from the enrth's
centre to the point (Fig. 10). Knowing the value of gravity below, it
was easy to calculate what it would have been at the level of the surface
had no outer shell exi&ted, and had the change in value depended merely
on the greater distance from the earth's centre. The observed value wsa
greater than this throngh the pull of the outer shell, and it was hoped
that the difference would be measured sufficiently accurately to show how
much greater is the mass of the earth than that of the crust. The first
attempt was brought to an end by a curious accident. As one of the
pendulums used was bebg raised up the shaft, the box containing it took
fire, the roje was burnt, and the pendulum fell to the bottom. Two years
later another attempt vas made, but this was brought to an end by a
fall in the mine, which stopped the pump so that the lower station was
Hooded.
Many years later, in 1854, the experiment was again undertaken by
Airy, this time in the Hartoa coal-pit, near Sunderland. The method wu
'riu,iit.le ol Alry'n
■il Experiment.
I density of the earth aa
36 PROPERTIES OF MATTER.
exactly the same, a pendulum being swung above and below the surfaoe,
and the diminution in gravity above carefully determined. The experiment
was carried out with the greatest care and in a most thorough way, two
pendulums being swung at the same time — one above and one below — the
two being interchanged from time to time. Several assistants were
occupied in taking the observations, which extended continuously night
and day for about three weeka Now gravity at the surface was greater
than it would have been, had no outer shell existed of thickness equal to
the depth of the pit, by about 1 in 14,000, so that the pull of the earth
was about 14,000 times that of the shell. The density of the shell was
determined from specimens of the rocks, and Airy found the density of
the eai*th about 6^.
Some very interesting experiments have since been made in a similar
way by Von Sterneck in silver mines in Saxony and Bohemia. Using the
invariable pendulums described in the last chapter he obtained different
results with different depths of mines, the value of the mean density
increasing with the increasing thickness of the shell used. This shows
very evidently that there were sources of disturbance vitiating the method.
Von Sterneck found, on comparing his observations at the two mines, that
the increase in gravity on descending was much more nearly proportional
to the rise of temperature than to the depth of descent. This appears to
indicate that whatever disturbs the regularity of gravity disturbs also the
slope of temperature.
All the methods so far described use natural masses to compare the
earth with, and herein lies a fatal defect as regards exactness. We do not
know accurately the density of these masses and what is the condition of
the surrounding and underlying strata. We can really only form at the
best rough guesses. Indeed, the experiments might rather be turned the
other way about, and assuming the value of the mean density of the earth,
we might measure the mean density of the mountain or strata of which
the attraction is measured.
The Cavendish Experiment.
We turn now to a different class of experiment, in which the attracting
body is altogether on a smaller scale, so that it can be handled in the
laboratory. The smallness of the attraction is compensated for by the
accuracy with which we know the size and mass of the attracting body.
The idea of such an experiment is due to the Rev. John Michell, who
completed an apparatus for the purpose but did not live to experiment
with it.
MichelFs plan consisted in suspending in a narrow wooden case a
horizontal rod G feet long, with a 2-inch sphere of lead hung at each end
by a short wire. The suspending wire for the rod was 40 inches long.
Outside the case were two lead spheres 8 inches in diameter. These were
to be brought up opposite the suspended spheres, one on one side, the
other on the other, so that their attractions on those spheres should con>
spire to turn the rod the same way round. Now moving each large sphere
on to the other side of the case so as to pull the suspended sphere with
equal force in the opposite direction, the rod should turn through twice the
angle which it would describe if the spheres were taken altogether away.
Hence half this angle would give the twist due to the attractions in one
GRAVITATION. S7
position aloDO. KDowiog the torsion couple of the suspending wire for a
given angle of twist and the length of the rod, the attracting force would
be calculable. To find the torsion couple, Michell proposed to set the rod
vibrating. From its moment of inertia and time of vibration the couple
could be found.
Neglecting all corrections, the mathematics of the method may be
reduced to the following :
Let the two suspended balls have mass m each, the two attracting balls
mass M each. Let the rod have length 2a and with the suspended balls
moment of inertia I ; let d he the distance apart of the centres of attracting
and attracted balls, and let be the angle through which the attraction
twists the rod.
If /i is the toision couple per radian twist, and G the gravitation
constant, then
^ 2GMma,
The time of vibration
whence, eliminating ft.
d'
N = 27r JYJJl,
4vrie ^ 2GMma .
N» d'
Now we may obtain another equation containing G by expressing the
acceleration of gravity in terms of the dimensions and density of the
eaith,
^ 4 r'A 2^.^
^ = G..._ = -GAO,
where r is the radius, C the circumference, and A the density of the
earth. Eliminating G between the last two equations and putting for
g/v^ the length of the seconds pendulum L — a useful abbreviation — we
find
. 8 L Mma N%
A = - X — X X — *
4. C d' id
where all the terms on the right hand are known or may be
measured.
On Michell's death the apparatus which he had collected for his
experiment came into the po&^ession of Prof. Wollaston, who gave it to
Cavendish. Cavendish determined to carry out the experiment, with
certain modifications ; but he found it advisable to make the greater part
of the apparatus afresh, though closely following Michell's plan and
dimensions.
The actual work was done in the summer of 1797 and the following
spring of 1798*
He selected for the experiment, according to Baily, an outhouse in his
garden at Clapham Common, and within this he appears to have constructed
an inner chamber to contain the apparatus, for he states that he ** resolved
to place the apparatus in a room which should remain constantly shut, and
to observe the motion of the arm from without by means of a telescope,''
in order that inequalities of temperature and consequent air currents within
the case should be avoided.
* Szperimenti to deter mine the density of the earth. PM(. Irat;s., Izxzviii..
1798.
PROPERTIES OF MATTER.
Tlie torsion rod A A (Fig, U, reduced from the figure in Cavenijish'g j
paper) was ot deal, feet long, strengthened by a silver wire tying the e
to an upright m g' in the middle. The two attracted balls x x were lead, I
2 inches in diameter, and hung by short wires from the ends of the rod. I
The torsion wire was 30| inches long, of silvered copper, and at firet of I
such croBS section aa to give a time of oscillation about IStu. This n
soon changed fur ono with a time of oscillation about 7m.
The position of the rod was determined by a fi.ted scale on ivory divided I
to s'o^h inch near the end of the arm, the arm itself carrying a vernier of I
five divisioDK. Tliit; was lighted by a lamp outride the I'oom, and )
viewed Ihrough a telescope passing thi'ough a hole in the wall.
The torsion case was supported on four levelling screws. The attracting
I
FlO. 11.— CmvBniiiali'a App»rmtnB. A A, torsion roil hung by wire / y ; x x,
kttrictod balls huDir Ironi IM euda ; W W, lUmoliug louisee inovaLls
roDnd axis F. T T, lolescopea lo view poailinn ul taraion rod.
masees, lead spheres 13 inches in diameter, WW, hung down from a c
bar, being suspended by vertical copper rods. This bar could be rotated I
by ropes passing outside the room round a pin fixed to the ceiling in
continuation of the torsion axis.
The masses were stopped when J inch from the case by pieces of wood <|
fastened to the wall of the building. When the masses wei-e against tha J
stups their centres were 8'85 inches from the central line of the case. 1
The method of espcriment was somewhat as follows: The torsion rod I
was never at rest, and the centre of swing was taken as the position
which it would be if all disturbances could be eliminated. This centre ot 1
swing was determined from three succeeding e^ttremities of vibration when
the attracting masses WW were against the stops on one side. They «
then swung round so as to come against the stops on the other side of the
attmcted maiises, and the now centre of swing was observed. In a
pailicular experiment the difference between the two centres was about
six scale divisions. The time of vibration was observed from several euo-
cesaive passages past the centre of swing, the value obtained in the a
GRAVITATION.
S9
experiment being about 427 sees,, und the masses were then movei] bock
to their first poKition, giving a secoDil value for the deflection.
In computing the results various correctious had to be inti'oduced into
the equivalents of the simple formulae which have been given above.
Taking the attraction formulit.
a correction had to be made, because the attracting masses were not quite
oppofiito those attracted, as the suspending bar was a little too c^hoi't.
Then allowance was necessary for the attrsctioD on the tondon rod, and a.
negative correction had to be applied for the attraction on the more
distant ball. The copper suspending rods were also allowed (or, n.nd a
further correction was made for the change in attraction with change of
scale reading— i.e., for change of distance between attracting and attracted
masses. This correction was pitiportionnl bo the deviation from the central
position, and may be regarded as an alteration of fi.
As to the case, it would evidently have no effect when the rod was
central, but it was necessary to examine its attraction when the rod was
deflected. Cavendish found that in no case did it exceed 1/1170 of the
attraction of the masees, and therefore neglected it.
Turning now to the vibration formula,
N = 2xv^l/^;
this was correct when the ma.sses were in the " midway" position — i.e., in
the line perpendicular to the torsion rod. But when they were in the
positive or negative position, the variation in their attraction, as the balls
approached or recedetl from them, made an appreciable alteration in the
value of the restoring couple, and thus virtually altered ft. The time had
therefore to bo reduced by (i/I85 of ita observed value where o was the
deflection in scale divisions due to the change of the masses from midway
to near position.
But it is to be observed that, if the weights were moved from one near
position to the other, and the time of vibration was taken in either
osition, then the same correction having to he applied to /i in both
posit
lite, it might be omitted from both.
In all, Cavendish obtained twenty-nine results with a i
By a
M the n
Istake
D = 5-i48±-033.
1 his addition of the results, pointed out by Baily, 1
ri 5'4S.
Repetitions by Reich, Baily and Cornu and Baitle.— His
experiment has since been repeated several times. Reich made two
experiments in Germany by Cavendish's method, obtaining in 1837 a
value 5'40, and about 1841) a value 5~fi8, In England it was repeated
by BaUyahout 1841 and 1843. Itaily'K experiment excited great attention
st the time, and the result obtained, 5'G74, was lung supposed to be very
near indeed to the truth. But certain discrepancies in the work gradually
impaired confidence in the final result, and in 1870 MU. Comu and
Bailie, in France, undertook a repetition, with various improvements and
refinemeota. In planning out their own work they succeeded in detecting
PROPERTIfi^OF MXfflER.
40
probably the chief source oE error io Baily's work. They hava as yet only
given an interim result of about 5'.">, and have shown that Baily's work,
if properly interpreted, should bring out a not very different result. Their
final conclusion is ntill to be published.
Boys'S Cavendish Experiment. — In the Philosophical TrajuaetioM
for 18'J5(vol, 186, A. p. 1) is an account of a determination of the grarita-
tion constant carried out with the greatest care by Prof. Boys. He had
discovered a method of drawing exceedingly fine quartz fibi-es and had
found them exceedingly
strong and true in their
elastic properties. They are
therefore pre-eminently ap-
plicable in torsion experi-
ments where smiill forces are
to be measured. Using a
qiiiirti fibre as the torsion
wire in a Cavendish appara-
tus, he was able to reduce
the attracted weight and
the whole apparatus and yet
reduce the diameter of the
suspending fibre so far that
the Keusitivenesa wait as great
as in earlier experiment^;.
At the same time the small-
ness of the apparatus allowed
it to be kept at a much more
uniform temperature, and
the disturbances due to con-
vection air currents were
much lessened. These dis-
turbances had much troubled
the earlier workers. In Fig.
12 is a diagrammatic repre-
sentation of the apparatus.
The Httracted masses mm
were of gold, one pair 0'2
inch, another pair O'Sfl inch
in diameter. The torsion
rod N was 0-9 inch long
and was itself a mirror in
S3 feet, and divided to 50tha
I 17 inches long.
diameter. Had
SojB'ii Appantua.
which the reflection of a Kcale distant about
of HI) inch, was viewed. The (juai'tz fibre wa
The attracting masses MM were lead halls 4^ inches
the masses all been on one level, as in the original arrangement, with such
a short torsion i-od the attracting masses would have attracted both gold
balls nearly equally. To avoid this, Boys had one attracting and one
attracted mans at one level and the other two at a level six inches below.
The balLs mm were hung from the torsion rod by quartz fibres inside a
tube about 1^ inches diameter. The attracting masses MM wera buog
irom the revolving lid of n concentric tubular case about 10 inches in
diameter. Thete lumisc^ were uminged in the position in which they
I
GRAVITATION. 41
exerted the mnjiimum couple on the gold halls first in one direction and
then in the opjiosite. The deflection varied from 351 to 577 divisions,
according to the balls used and the times of vibration from 188 to 342
seconda. The apparatus was moat exactly constructed and measured, and
the i-esulta were very concordant.
The final value, probably the best yet obtained, was ;
G = 6-657G X 10-»; whence A = 5.'i270
Braun'S ExpePiment (Deniaehrijl. der Math. Nat. Clause drr Kai».
Akad. Wien. 1»'J6. Bd. Ixiv.).— In 18110 Dr. Braun published an account
of an experiment carried out by him. He used the torsion-rod method,
and though his apparatus was considerably larger thtin that of Boyn, it
was still much smaller than that of Cavendish, Reich or Uaily. The
rod was about 2i cm. long and was sutipended from a tripod by a brass
torsion wire nearly one metre long and 0'055 mm, in diameter. The
whole torHioQ arrangement was under a glass receiver, about a metre high
and 30 cm. in diameter, resting on a flat glafS plate. The receiver could be
exhausted and in the later ex[jerimenta the pressure was about 4 mm. of
mercury and the disturbances due to air currents were very greatly
reduced. The attracted masses at the end of the I'od were gilded bniss
spheres each weighing about 54 gms. Round the upper part of the
receiver, and outside it, was a graduated metal ring which could be
revolved about the axes of the torsion wire; from this were suspended,
about 42 cm. apart, the two attracting mapses. Two pairs were used, one
a puir of bi-ass spheres about five kgms. each, the other a pair of iron
spheres filled with mercury and weighing about nine kgrns. each.
Special aiTangements had to be useil to determine the position of the
rod by means of a mirror fixed on ita centre, the beam being reflected
down through the bottom of the plate. The time of vibration was about
1275 sees. The result obtained was very near to that of Boys, vis. :
G = 6-65786x10';
.2725
I
A result very nearly the same has recently Lieen obtained by von
Eijtvos (IPted. Ann. 59, 189C, p. 354), but he has not yet completed the
Wilsingr's Experiment.— About 1886, Dr. Wilsing, of Potsdam,
devised a modified form of Cavendish's experiment, in which a sort of
double pendulum is used — i.e., one with a ball below and another at a
nearly equal distance above the suspension. The pendulum is then in a
very sensitive state, and a very small horizontal force pulls it tlirougli a
large nngle.
It is then just like a toi-sion balance, but with a vertical instead of a
horizontal rod. If weights are brought up, one to pull the upper ball to
one side and the other to pull the lower ball to the other side, the
pendulum twists round slightly. From the observe*! twist and the time
of swing the btti-action can be measured and compared with the pull ol
the earth. Wilsing found that the earth had a mean density of 5*570.
Experiments with the Common Balance-
Von Jolly's Experiment.— in 1878 and in I8S1 Professor von Jolly
described a method which he iiad devised. Ke had a balance fixed at the
42
PROPERTIES OF MATTER.
top of a tower in Munich, and from the scale-pans hong wires supporting
two other scale- pans at the bottom of the tower (21 metres below).
Imagine that two weights are balanced against each other at the top of
the tower. If one is now brought down and put in the lower scale-pan on
the same side it is nearer the centre of the earth, and, therefore, heavier.
Yon Jolly found a gain of about 32 milligrammes in 5 kilogrammes. He now
built up a large lead sphere under the lower pan, a yard in diameter, so that
its attraction was added to that of the earth. The gain on transferring
the weight from the upper to the lower pan now came oat to about half a
milligramme more, so that the attraction of the sphere was this half milli-
gramme. The earth's attraction was about 10,000,000 times that of the
sphere, and its density was calculated to be 5'G9.
r
^ %:-;>■ - '■ ■ '^;ir;; • ; . ■/,■ ^^-'^^^-^ 'Z< (^ O '
Kr ''/ ,■ . '•■, . '^ ■., '-- i' >-, ^%^ V— / '
V//^/^AV/y/»A
Fio. IS.^Bichars and Krigir-Mesiel*8 Experiment
Experiment of Richarz and Krigrar-MenzeL— An experiment
very much like that of Yon Jolly in principle has been carried out by
Drs. Richarz and Krigar-Menzel at Spandau, near Berlin (Ahha/nd, dir
Konigl. Preusa Akad. Berlin, 1898). A balance with a beam 28 cm.
long was supported at a height above the floor, and from each end
were suspended two pans, one near the beam the other near the floor,
more than two metres lower, Fig. 18. In principle the method was as
follows : Spherical gilded or platinised copper weights were used, and to
begin with thase were placed, say, one in the right-hand top pan, the other
in the left-hnnd bottom pan. Suppose that in this position they exactly
balanced. The weights were then moved, the right-hand one into the
right lower pan, when it gained weight through the increase of gravity
with a descent of over two metres ; the left-hand one into the left upper
pan, when it lost weight through the ascent of the same amount. The
result after corrections was that the right-hand pan appeared heavier by
1*2458 mgm., half this being due to the change in position of a single
kilogramme.
GRAVITATION. 4.1
A lead parallel oplped was now builb up of separate blocks, between the
upper and lower puns, 2 metres high ami 2'1 metres square, ho lizon tally,
with passages for the wire^ suspending the lower pans. The weighing
of the kilogramme.') was now repeated, btit the attraction of the lead,
which was reversed when a. weight was moved from bottom to top, was
more than enough to make up for the decrease in gravity, and the right-
hand now appeared lighter on going throtigh the same operation by
O'lSll mgm.; whence the attraction of the lead alone made n difference
of I'3664 mgm. This in four times the attraction of the lend on a single
kilogramme. Knowing thus the pull of ft block of lead of known form and
density on the kilogr.imme at a known difidince, and knowing too the ptdl
of the earth on the same kilogramme, viz., 10' mgm., the mean density of
the earth could he found.
The final result was :
G = r,CS5x io-«
I
I
Poyntingr's Experiment.— The method of using the bftknce in this
experiment will be gathered from Fig. 11. A B are two lead weights
about 50 lb. each, hanging down from the ends of a very large and strong
balance inside a protecting wood case. M is a large lead sphere, weighing
about 350 lb., on a turn-table, so that it can move round from tuider A till
it comes under B. The distance between the centres of M and A or M
and B is about one foot. When under A, M pulls A, and so increases its
weight. Wlieu moved so as to come under B the increase is taken from
A and put on to B. The btktance is free to move all the time, bo that it
tilts over to the B side an amount due to double the attraction of M
on either, m was a balance weight half the mass of M, but at double the
distance. Before this was used it was found that the movement of M
tilted the floor, and the balance, which was a very sensitive level, was
affected by the tilt.
To observe ttie deflection due to the alteration in weight, a mirror was
connected with the balance pointer by the " double sus[>ension " method,
due to Lord Kelvin, and shown in Fig. lb.
With the suspension the mirror turned through an angle 150 timei^ as
great as that turned through by the balance beam. In the room above
was a telescope, which viewed the reflection of a scale in the mirror, nud
as the mirror turned round the scale moved across the field of view. The
tilt observed meant that the l)eam turned through rather more than I",
and that the weight moved nearer to the mass by about ^u'^ir of an inch.
The weight in milligrammes producing this tilt had to be found. This was
done virtually (though not exactly in detail) by moving a centigramme
rider about 1 inch along the beam, which was equivalent to adding to one
side a weight of about -^ milligi-amme. The tdt due to the transfer was
observed, and was found to be very nearly the same as that due to the
attraction, so that the effect of moving M round from A to S was
equivalent to increasing B by y'^ milligramme, or tooond rnt ''^ its previous
weight. The pull on either is half this. In other wuids, the earth pulled
either about 100,000,000 times as much as the mass M, and the earth,
which is 20,000,000 times as far away, would at lihe same distance have
eierted 400,000,000,000,000 times 100,000,000 times the pull, and is,
therefore, bo many times heavier. Thus wo find that the earth weighs
44 PROPERTIES OF MATTER.
about ]'35 X 10" lb. In obtaining the attraction of M on A or B, the
attraction oo the beam had to be eliminnted. Thi§ was done by moving
B into the poaitiona A.' B' one foot higher, and finding
tia. 14.— PoTDtiDg*! GiporimeDt A B,walghUi, Mch kbont 60 lb., hughig from
the two knns o( Muice. M, kItracUng niBas od tum-table, moTaUa » u to
ooDW onder either A or B. m, babociDg matw. A' B'. •econd ponltiom for A
ud B. In tli<B poBition the uttiKtion of U on Ibv beam ud BnapendlnE wlr>i
Is the einie u br(oi«. ao tbat Ibe diffiireiice of Kitrai:tion on A and Bin tbe
two poeiliooa U due lo the difference ia distuce of A and B oalf, and Ibiu the
attiartion on the beam, Ac, ia EiiminaUd.
the attraction in this position. The difiereiice was due to the change
in A and B alone, for the attraction on the beam remftined the same
throughout.
The final resnlt waa —
G = eG984xlO-«
A -5-4934
GRAVITATION.
45
Experiments on the Qualities of Gravitation.
The Bangfe of Gravitation.— The first question which arises is,
whether the Jaw of gravitation holds down to the minutest masses and
distances which we can deal with. All our observations and ezpeiimeiits
go to show that it holds throughout the long range from interplanetuiy
MUroscopt stxige
H
Bracket
2n
e
L.
I
e^
Mirror
Vanes un/rking
im cLashpot
Fio. 15. — Doable Snspens'.on Minor (half t^Wb\
distances down to the distances between the attracting bodies in the
laboratory experiments described above.
The first step in the descent from celestial spaces is justified by the fact
that the acceleration of gravity at the earth's surface agrees with its value
on the moon, as attracted by the earth. The further step downward
appears to be justified by the fair agreement of the results obtained by the
various forms of Cavendish, balance, and pendulum experiments on the
mean density — experiments which have been conducted at distances vaiying
from feet down to inches. Where the law ceases to hold is yet a matter for
experiment to determine. When bodies come into what we term '* coutact,"
the adhesion may possibly still be due to gravitation, according to the inverse
iiqaare law, though the varying nature of the adhesion in difierent cases
aoems to point to a change in the law at such minute distances.
PROPERTIES OF MATTEft.
Gravitation not Selective.— It might he possible that some matter {
ia attracted more than in proportion to its mnssand some lees. Th« agree-
ment of astronomical obaervations with deductions from tlie general law ia
not perfectly decisive as to this possibility, for there mi^'ht be auch a
mixture of difl'erent kinds of ma-lter ia all the pknet« that the general
average attmclion was in aecordance with the law though not the attraction
on each individual kind. A supposition somewhat of this dencription is
required in an explanation which has been given of the formation of j
comets' tails, some matter in the comet bping niipposed to be acted on hy
the sun, not by the ordinary law but by a repulsion. This explanation is,
however, now generally abandoaed, att electrical origin of tlie tails being
regarded as more protwible.
But, with regaiil to ordinary terrestrial matter, Newton's lioUow
pendulum experiments {Prineipia, Book III., Prop. fS) repeated with n
detail and pi-ecision by Bessel (Vereuc/te iiber die Jiraji, jiiit welcher die I
Krde Kirrper von verschiedenerBeackaffenUtit arneiht, Abhand. der Berl.
Ak. 1830, p. 41; or Memoirea relaCi/s A la Physiq^te, toroe v. pp. 71-
133) prove that the earth as a whole is not selective. Still, the reeulte |
unly Stnlgbt FIvM
might just conceivably be due to an average of equal excesses and defects.
But again we may quote the various mean density experiments, and especially
those made by Baily, in which a number of different Jittracting and attracted
Bubstances hiive been used with nearly the same i-eaults.
Gravitation not Affected by the Medium.— When we comparo
gravitation with other knuwu forces (and those which have been most
closely studied are electric and magnetic forces) we are at once led to
inquire whether tlie lines of gravitntive force are always straight linea
radiating from or to the mass round which tbey centre, or whether, like
electric and magnetic lines of force, they iiave a pi-efei-ence for some media
and a di.staste for otliers. We know, for example, tluit if a mjignetio
sphere of iron, cobalt or manganese \a placed in a previously straight field,
its perme;ibiHty is greater than thr, air it replaces, and the lines of force
crowd into it, as in Fig. II!. The magnetic jiction ia then utronger in the
presence of the sphere neai' the ends of a diameter parallel to the original
course of the lines of force, aud the lines are deflected. If the sphere be
diamagnetic, of water, copper, or bismuth, the permeability being lesa
than that of air, thew ia Pa oppiwite < , as in Fig. 1 7, and the (ield is
weakened at the ends of a diameter p- '* f'"^«. fl""", ftJ?"'."
the linea aie deBectwI. Similarly. " ^ '" »" ^i«'''"<'
field gathei s in the lines ^ "» ">« ■">«' enter
and leave stronger than
I
GRAVITATION.
♦7
If we enclose a magnet in a lioltow box of Boft iron placed in a
nagnetic field, the lines of force ai-e jriitliered into tlie iron and krgely
'Tle&i«d away from the inside cavity, so t!iut iLe magnet 18 screened from
external action.
Afltronoinical ohaervations are not conclusive agninst any euch efl'ect of
the medium on gravitation, for the medium intervening between the sun
and planets approachea a vacuum, where no far we have no evidence for
variation in quality, even for electric and magnetic induction. In the caae
of the enrth, too, ila apherical form might render ol«ervation inconclusive,
for just as a upliere composed of concentric dielectric ahells, each with its
Kurface uniformly electrified, would have the same external field in air,
whatever the dielectric constant, if the quantity of electrification within
were the snme, bo the earth might have the same field in air whatever the
varying quality of the underlying strata, as regards the tranamisajon of the
action acrosa them, if they were only suitably a
But common ezpeiience
might lead ua at once to
say that there is no very
considerable effect of the
kind with gi-avitation. The
evidence of oivilinary weigh-
ings may, perhaps, be re-
jected, inasmuch as both
sides will be equally af-
fected as the balance ia
commonly used. But a
spring balance should show
if there is any large efloct
when used in diflerent positions above different media, or in different
enclosures. And the ordinary balance is used in certain experiments in
which one weight is suspended beneath the balance case, and surrounded,
perhaps, by a metal case, or, perhaps, by a WHter-bath, Yet no appreiiable
variation of weight on that account has yet been noted. Nor does the
direction of the vertical change rapidly from place to place, as it would
withvarying permeability of the ground below. But perhaps the agreement
of pendulum results, whatever the blocb on which the pendulum is placed,
and whatever the case in which it is contiiined, gives the be.st evidence
that there is no great gathering in, or opening out of the lines of the
earth's force by different media.
Ktill, a dii-ect experiment on the nttraclion between two masses with
different media interposed was well worthy of trial, and such an experiment
has been carried out by Messrs. Austin and Thwing.* The effect to be
looked for will be understood from Fig. 18. If a medium more peimeable
to gravitation ia interposed between two bodies, the lines of foree will
move into it from each side, ami the gravitative pull on a body, near the
interposed medium on the sido away from the attracting body, will be
increased.
The apparatus they used was a modified kind of lioys's apparatus
(Fig. 19). Two .small gold masses in tlie form of short vertical wires, each
■4 gm. in weight, were arranged at different levels at the ends virtually of
a torsion rod $ mm. long. They are represented in the figure by the two
• Phxiieai RttiriB, v. 1897, p. 29*.
thickenings on the HufipeiiJing fibre. The attracting masses M|M, were lead,
each about 1 kgin. These were first in the potiitioiis shown bj black lines in
the figure, and were then moved into the positions shown by dotted lines.
The attraction was measured first when merely the air and the case of the
instrument intervened, and then when various slabs, each 3 cm. thick, 10
cm. wide and 29 cm. high, were interposed. With screens of lead, zinc,
mercury, water, alcohol or glycerine, the change in attraction was at the
moat about 1 in 500, and this did not exceed the errors of experiment.
That iH, they found no evidence of a change in pull with change of medium.
If such chnnge exisUs, it in not of the order of the change ol' electric pull
with change of medium,
but something far smaller.
It still remains just pos-
sible, however, that there
are variations of gravita-
tional permeability compar-
able with the variations of
magnetic permeability in
media such as wat«i' and
alcohol.
Gravitation not Di-
rective. — Vet another
kind of effect might be sus-
liected. In most crystalline
substances the physical pro-
perties are different along
difi'erent directions in a
crystal. They expand dif-
ferently, they conduct heat
differently, and they trans-
mit light at different speeds
in different directions. We
might then imagine that
J thelinesofgravitativeforce
spread out from, say, a crys-
tal sphere unequally in Af-
ferent directiona Soma
years ago Dr. Mackenzie* made an experiment in America, in which h6
sought for direct evidence of such unequal distribution of the lines of
force. He used a form of apparatiis like that of Professor Boys (Fig. 12),
the atti'octing masses being calc spar spheres about 2 inches in diameter.
The attracted masses in one experiment were small lead spheres about
^ gm. eacJi, and he measured the attraction between tlie crystals and the
lead when the axes of the crystals were set in various positions. But the
variation in the attraction was merely of the order of error of experiment.
In another experiment the attracted masses were small calc spar crystal
cylinders weighing a little more than J gm. each. But again there was no
evidence of variation in the attraction with variation of axial direction.
Practically the same problem was attacked in a different way by
Poyuting and Gray.t They tried to find whether a quartz crystal sphere
• Phyiicnl Remev,, il. 1895, p. 321.
t PhU. Trant., 102, 1899, A. p. 215.
4
4
GRAVITATION.
*9
I
had any directive action on another quarts crjiitBl i^phere clothe to it, whelbirr
they tended to set nitli their axes [larullel or crossed.
It nitty easily be seen that this is the same problem by consideritig
what mufit hap|.>eii if thei'e is any difference in the attraction between two
fiucb spheres when their axes are parallel and when they are crofised-
Suppobe, for example, that the attraction is always greater when their axeH
are parallel, and tliia seems a reasonable suppotdtion, inasmuch as in
etraightforwai-d orj stall isation siiccessive parts of the crystal are a<Ided to the
existing crystal, all with their axes pamllel. Begin, then, with two quartz
crystal spheres near each other with their axes in the same plane, but
perpendicular to each other. Hemove one to a very great distance, doing
work against their mutual attractions. Then, when ii is quite out of range of
appreciable action, turn it round till its axis is parallel to that of the tixed
crystal. This absorbs no work if done slowly. Then let it retui'n. The
force on the return journey at every point id greater than the force on the
outgoing journey, and more work will be got out than was put in. When
the sphere is in its first position, turn it round till the axes are again at
right angles. Then work must be dune on turning it through this right
augle to supply the diiference between the outgoing and incoming works.
Fur if no work were done in the turning, we could go through cycle after
cycle, always getting a balance of energy over, and this would appear to
imply either a cooling of the crystals or a diminution in their weight, neither
supposition being admissible. We are led then to say that if the attracLiun
with pandlel axes exceeds that with ciossed axes, there must be a directive
action resisting the tm-n from the crossed to the parallel positions. And
conversely, a directive action implies axial variation in gravitation.
The straightforward mode of testing the existence of this directive
action would consist in hanging up one sphere by a wire or thread, and
turning the other round into various positions, and observing whether the
hanging sphere tended to twist out of position. But the action, if it exists,
is so minute, and the dLstui-bances due to air currents are bo great, that it
would be extremely dilficult to observe ite effect directly. But the prin-
ciple of farced oscillations raay be used to magnify the action by turning
one sphere round and round at a constant rate, so that the couple would
act first in one diiection and then in the other alternately, and so set the
hanging sphere vibi'ating to and fro. The nearer the complete time of
vibration of the applied couple to the natural time of \ibration of the
hanging sphere, the greater would be the vibration set up. This is well
illutitrated by moving the point of suspension of a pendulum to and fro in
gradually decreasing periods, when the swing gets longer and longer till
the penod is that of the pendulum, and then decreases again. Or by the
flxperimont of varying the length of a jar resounding to a given fork, when
the sound suddenly swells out as the length becomes that which would
uatarally give the same note as tlie fork. Now, in looking for the couple
between the crystals, there are two possible cases. The most likely is that
in which the couple acts in one way while the turning sphere is moving
from parallel to crossed, and in the opposite way during the next quarter
turn from crossed to pai-allol. That is, the couple vanishes four times
during the revolution, and this we may term a quadrantal couple. But it
ia just possible that a quartz crystal has two ends like a magnet, and that
lika poles tend to like dii-ections. Then the couple will \-anish only twice
I revolution, and may be termed a scmiciiculai- couple. Both wero
so PROPERTIES OF MATTER.
looked for, but it is enough now to consider the possibility of the quad mntnl
couple only.
The mode of working will be Been from Fig. 20. The hanging ephei-e,
■9 cm. in diameter and 1 gm. in weight, was placed in a light aluminium
wire cage with a mirror ou it, and suapended by a long quartz fibre in a
brasa caae with a window in it opposite the mirror, and surrounded by a
double-walled tinfoiled wood case. The position of the sphere was read in
Fio. SO. —Experiment on directive Action oF ods QoitIz Cryatal on anathar.
the usual way by scale and telescope. The time of swing of this little
sphere was 130 seconds.
A larger quartz sphere, 6'6 cm. diameter and weighing 400 gras., was
fixed at the lower end of an axis which could be turned at any desired rate
by a regulated motor. The centres of the spheres were on the same level
and D'9 cm. apart. On the top of the axis was a wheel with 20 equidistant
marks on ils rim, one passing a fixed point every II '3 seconds.
It might be expected that the couple, iF it existed, would have the
greatest eSect if its potiod exactly coincided with the 120-second period of
the hanging sphere — i.e., it the Itu-ger sphere revolved in 2-10 seconds. But
in the conditions of the experiment the vibrations of the small sphere were
very much damped, and the forced oscillations did not mount up as they
wnnld in a freer sivipg. The disturlinnces, which were mostly of au iu-
puleive kind, contimioKy Bet the ]mnging sphere into large vibration, and
these might eAsily be taken aa dtte to the revolving sphere. In fact,
looking for the couple with exactly coincident periods would be something
} i a \
PkrvoiL 12S
Fm, 21.— Upjwr riirvo > rcRoIar Vibimlioi
I
I
like trying to find if a fork Fet the air in a resonating jar vibrating when
a brass band was playing nil round it. It was necessary to make the
couple period, then, a little difiereut from the natural 12t)-second period,
and accordingly the large sphere was revolved once i
the supposed quadrantal couple would have a
period of 1 1 5 seconds,
Figs. 21 and 22 may help to show how
this tended to eliminate the disturbonc-es.
Let the ordinates of the curves in Fig. 21
represent vibrations set out to a horisontal
time scale. The upper curve is a regular
vibration of range + 3, the lower a disturbance
beginning with range ±li'- The first has
peiiod 1, the second period 1-2-'). Now, cutting ___
the curves into lengths equal to the period of ^ _
the shorter time of vibration, and arrnriging
the lengths one under the other, aa in ¥ig. 2'2,
it will be seen that the maxima and the
minima of the regular vibration always Fall at
the same points, so that, taking 7 periods, and
adding up the ordinates, we get 7 times the
range, viz., ± 21. But in the disturbance the
maxima and minima fall at different points,
and even with 7 periods only the range is
from + IR to - 13, or less than the range due F;o. 22.-ne.xilii
to the addition of the much smaller regular p"° j, ,.„u^f' ^ Hg Pjnod of
vibration. the regulu- one.
In the experiment the couple, if itexisted,
would very soon estjvhlish it.s vibration, whiirh would always be there, and
would go through ail its valuc.s in llf> seconds. An obsorvei', watching
PROPERTIES OF MAITER.
the wheel at the top oE the revolving axis, gave the time signals erery ll'S 1
Heconds, roguluting tlie speed if aece^mry, and an observer nt the telescope n
gave the scale reading ut eveiy eignai, that is, 10 times during the pei iod.
The values were aii-ituged in 10 columns, each horizoutol line giving the
readings of a period. The experimeut was carried on for about 2^ hourd
at a time, coveiing, say, 80 periods. On adding up the columos, the
maxima and minima of the couple effect would always fall in the same two
columnfi, and ho the addition would give 80 times the swing, while the
masima and miuinin of the natural swings due to disturbances would fitll
in different columns, and so, in the long run, neutrulise each other. The
resulta of different days' work miyht, of course, be added together.
There always was a small outstanding effect such as would be produced 1
by a quadrantaJ couple, but it«i efi'ect was not always in the same columns, 1
and the net result of observations over about ,^&Operiods was that there waa i
DO ll&aecond vibi'ationof more than 1 eecond of arc, while the diaturhnnces
were sometimes 50 times aa great. The semicircular couple required the
turning sphere to revolve in 1 15 secumls, Here, want of symmetry in the
apparatus would come in with the same elfect aa the couple sought, and
the outstanding result was, accordingly, a Uttle liirger. But in neithercase
could the ex j>erinients be takenasbhowingareal couple. They only showed
that, if it existed, it was incapable uf piuduciiig an effect greuier than that
observed, Ferhapsthebestwayloput thereiiuliof the uurkisthis: Imagine
the small sphere set with its axis at 45° to that of the other. Then the
couple is not greater than one which would take 5^ hours to turn it
through that 45° to the pai'allel position, and it would oscillate about that
position in not less than SI hours.
The Bemicireulai' couple is not greater than one which would turn from I
crossed to parallel position in 4^ houi-s, and it would oscillate about that ]
position in not less than 17 hours.
Or, if the gravitation is less in the crossed thau in the parallel position, M
and in a constant ratio, the dtfftrence is less than 1 in 16,000 in the ona |
case and less than 1 in 2800 iu the othei'.
We may compare with these numbers the difference of rate of travel I
of yellow light through a quartz crystal along the asia and perpendicular ■
to it. That difference is of quite another order, being about 1 in 170, f
Other possible Qualities of Gravitation.—Quite indecisive ex- ]
periments have been made to discover u possible alteration of mass oak
chemical combinutiou.* Alterations have appeared, but they are too smaltl
and too irregular to enable any conclusion to be di^awn as yet,
So far, too, there is no reason to suppose that temperature offectfl'l
gravitation. Indeed, as to temperature effect , the agreement of weighll
methods and volume methods of measui'ing expansion is good, as far bj '
goc.-, in showing that weight is independent of temperature,
Ko research yet made has succeeded in showing that gravitation
related to anything but the masses of the attracting and the attracted
bodies and their distance apart. It appears to have no relation to physical
or chemical conditions of the acting masses or to the intervening medium.
• Landolt, PrrKm. At. Wim. mrlln, Sil:. Bei-., vili. IBOO, p. 206, or Clitmiatl
.VriDj, xcili. lUOiJ, p. 271, hail given an account of eiperimcots wbicli perhape give
eriijence of loixt of ntasB In the vessel oootalning' tlie combining EUlMlancaa in
oenain cueH. The resuUs are very incousiateni. The loss, if ptovcd to exisi, ma;
be due to escape Ihioii^h Che h'^ssx. aiid not to Hlturution of niau or torubini '
Lniiiloh'a iiajii't uuiiliiiiis ii!liMi:iiiiiis lo ulLor work.
CiiAritu IV.
ELASTICITY.
Contents. — Limits of Elasticity — Elastic after efFect — Viscosity of Metals and
Elastic Fatigae — Anomalous Effects of fiist Loading a Wire — Breakitg t:>tress.
Ik this chapter we shall consider changes in the cori formation of solid
bodies and the connection between these changes and the forces which
produce them.
Many of the points with which we shall have to deil are well
illustrated by the simple case of a vei-tical metal wire the upper end of
which is fixed while the lower end carries a scale-pan. If we measure
the increments of elongation of the wire when different weights are
placed in the scale-pan and piot our results as a curve in which the
abscissie are the elongations of the wire — ie.y the extension of the wire
divided by its unstretched length, and the ordinates the stretching weight
(inclusive of the weight of the scale-pan) divided by the area of cross
section of the unstretched wire, we obtain results similar to those shown
in Fig. 23 (from A History of the Thtory of Elasticity aiid of the Strength
of Materials)^ which represents the results of experiments made by Professor
Kennedy on a bar of soft steel.
The first part of the curve — when the stretching force per unit area is
less than a certain value, is a straight line — 1.«., up to a certain point the
elongation is proportional to the load per unit area of cross section,* and
up to this point we find that when we remove the weight from the scale-
pan the stretched wire shortens until its length is the same as it was
before the weights were put on (the elongations in this stage are so
small that on the scale of Fig. 23 this part of the curve is hardly distinguish-
able from the axis AB). When, however, we get beyond a certain
point B on the curve — i.«., when the stretching foi-ce per unit area is
greater than the value represented by AB, the curve becomes bent, and
we find on removing the weights that the wire does not return to its
original length, but is permanently lengthened, and is said to have
acquired permanent set.
The range of elongations over which the wire, when unloaded, recovers
its original length, is called the range of perfect elasticity; when we
go beyond this range we are said to exceed the elastic limit.
• Thia seems to be only approximately true for cert.iin kinds of iron. (A HUtory
of the Theory of ELatticiiy and of the Strength of Materials. Todhunter and Pearson,
Vol. i. p. 893.
54
PROPERTIES OF M
After pauwng the p(»nt represented b j B a sUge b readied where the
ea rt eoiion beoomes rerj krge. The scale-paa runs rmpidlj dovn mud the
wire looks m if it were mbcmt to break. Bj far the greater part of this
ezteocioo is permaiieDt, and the wire, after passing Uie state repres«nted
bj Cy is not able to sustain as great a pull as before without snfiering
further elongation ; this is shown bj the bending back of the corre. The
plaoe C where this great extendon begins is called the jidd-pomt; it
■eems to be alwajs further along the curve than the elastic limit B.
H Exlension/.
Fio. 23.~Elongation of a Stietched Wire.
The part of the increment of elongation which disappears on the
removal of the stretching weight, between the elastic limit and the yield-
point, is proportional to the stretching weight, and the ratio of this
ELASTlCrrV. 55
movement to tlie stretching weight per unit nreii i-s itccording to the
vxperimenbi; of Professor Kennedy, the same as that ivithio the limits of
perfect elasticity (see Tudhunter aud Pearson's History of Elastkiiij,
p. 88!t).
After piiswng tiie yield-point the elongation increases vei-y rapidly
with the load, and at this stage the wire is plastic, the eloogitiiuu
depending upon the time the sti-etching force acta. The extension rapidly
increases and the urea rapidly contracts until the breaking- point E is
reached. The apparent maximum for the load per unit area shown in
Fig. 23 is dae to the contraction of the area, so that the pull per unit ai-ea
of the stretched wire is no longer represented even approximately by the
ordinates. About the point D the wire begins to thin down or flow
locally, so that its cross section is no longer iinifoim, some parts being now
smaller than the rest.
The portion <jHG' of the curve represents the ofTect of unloading
and reloading at a point O past the yield-point. We see, from the shape
of this portion of the curve, that the limit of perfect elasticity for this
permanently stretched wire has been extended beyond the yield'point of
the wire before it was permanently stretched. The range between the
limit of perfect elasticity and the breaking-point is veiy different for
different substances ; for ductile substances, such as lead, it is conf.idersl>Ie,
while for brittle ones, such as glass, it is evanescent.
We are thus from our study of the loaded wire led to divide the
phenomena shown by substances acted upon by forces into two divi.sions^
one division in which the solid recovers its original form after the
removal of the forces which deformed it, the other division in which a
permanent change ia produced by the application of the force. Even
within the limits of perfect elastifity different bodies show distinct
differences in their behaviour. Some recover their form immediately
after the removal of the force, while others, though they recover it
ultimately, take considerable time to do so. Thus a thread of quarti fibre
will recover ita shape imujediately after the removal of the tensional
and torsional forces acting upon it, while a glass fibre may, if the forces
have been applied for a considerable time, be several hours before it
regains its onginal condition. This delay in recovering the original
condition of the substance is calleil the elastic after-effect ; it may be
conveniently studied in the case of the torsion of glass fibres.
Take a long glass fibre and fasten to it a mirror from which a spot of
light is reflected on to a scale, twist the fibre about its axis and keep it
twisted for a considerable time. Then remove the twisting couple : the
spot of light will at once come back a considerable distance towards its old
position, but will not reach it, and the rest of the journey will be a slow
creep towards the old position, and several hours may elapse before the
journey is completed. The larger the initial twibt and the longer the
time for which it was applied the greater is the temporary deflection of
the spot of light from its original position.
The general shape of the curve which represents the relation between
the displacement of the zero — i.e., the displacement of the position of the
Fpot of light—and tlio time which hcis elapsed since the removal of the
twixt, is shown in Fig. 24. In this curve the ordinates represent the
displacement and the a1>si isMe ihe time since the removal of the twist.
ITie altitude PN, when the abscisea ON is given, depends upon the
56
PROPERTIES OF MATTER.
magnitude of the initial twist and tbe time for which it was applied ; the
curve 18 steep at first but gets flatter and flatter as the time increases.
The longer the initial twist is applied the more slowly does the zero
approach its original position. Very complicated movements of the zero
may occur if the fibre has been twisted first in one direction and then
in the opposite for a considerable number of times. The general features
of this phenomenon will be illustrated by the following simple case. Suppose
that immediately after the removal of the first twist, whose after-effect,
if it were alone, would be represented by the curve (I), Fig. 24, a second
twist in the opposite direction is applied for a time represented by ON and
then removed. Suppose that the deflection of the zero due to this twist
alone is represented by the dotted curve (II) (as the twist is in the opposite
7^ K
Fio. 24.— Carre showiDg tbe Elastic After-effect in a Twisted Glass Thread.
direction, the ordinates represent negative deflections). Then if we can
superpose the efiects, the displacement of the zero at a time NK after the
removal of the second twist will be represented by the differences between
the ordinates KR, KS of the two curves. The ordinate of the second curve
may be above that of the first at the time the second twist is removed, and
yet, as the curve is very steep just after the removal of the twist,
curve (II) may drop down so quickly as to cut the first, as shown in the
fi;^re. Thus in this case we should have the following effects: immediately
after the removal of the second twist there would be a displacement of
the zero in the direction of the last applied twist, the spot of light would
then creep back to the zero but would not stay there, but pass thron<yh
the zero and attain a maximum deflection on the other side ; it would then
creep back to the zero and would not again pass through it. In this
way, by superposing twists of different signs, we can get very complicated
movements of the zero, which are a source of trouble in many instruments
which depend upon the torsion of fibres. With quartz fibres the residual
EI.ASriCITY.
ill, niul thi.>
S7
greatly
efTect is exceedingly small, niul thi.> is one of the cliiet cm
tlieii' use so valuaUe. The re^tdiml after-effect in gldss is n cause ot
tfoiible in tbermoinetry, eocli cliaDge of temperature causing a temporary
change in the zero.
The magnitude of the elastic after-effect seems t
when there is a want of homogeneity in the
constitution of the body. In the most homO'
geneuus bodies we know, crystnt^, it is exceedingly
small, if it exists at all, nhile it is veiy large m
glass which is of composite character, being a
mixture of diflerent silicates ; it exists in metals,
although not nearly to tbe flame esteut as in
glass. A similflr dependence ujion want of
uniformity seems to characterise another similar
effect — tbe residual charge of dielectrics [tea
volume on Eleclritiity and Mo^netit^ni), the laws
of which are closely analogous to those of the
elastic after-effect.
The phenomenon of elaaltc after-effect may
be illustrated by a mechanical mode! similar to
that shown in Fig. 25.
A 13 a Efiring, from tho end, U, of which
Knotber spiing C is suspended, c.irryiug a
dumper IJ, which moves in a very viscous
liijuid. If B is moved to a position B' ai.d kept
there for on!y a short time, so short thikt D has
not time to move appteciably from its original
position, then when B is let go it will return at
once to its original zero, for 1) has not moved, to
that the conditions are the same as they were
before B was displaced. If, however, B is kept
in tbe position B' for a long time, 1) will slowly roi
tuch that D' is as mucli below B' as i) was below B.
it will not at once return 10 B, for in this position the spring between B
and D is extended, B will slowly move back towards its old zero, and will
only reach it when the slow moving D' has returned to D.
Viscosity of Metals and Elastic Fatigue.— If two vertical wires,
one made of steel and the other of
sine, are of the same length and ^ O o ^
diamet«r, and carry vibration bars Fio. 26.
of the same diameter, then if
these bars are set vibrating tbe vibrations die away, but at very difierent
rates: the steel wire will go on vibrating for a long time, but the zinc
wire will come to rest after making only a small number of vibrations.
Thia decay in the vibrations of the wire is not wholly nor even mainly
due to the resistance of the aii; for this is the sauie for both wires; it is
due to a dissioation of energy taking place when the parts of a metal wire
ore in relative motion, and may, from analogy with the cose of liquids
and gases, be said to bo due to the nUeosit^ of tho metal. We can
see Lhnt eln-sttc after-elfect would causa a decay in the vibrations of
wire. For suppose O, Fig. 2ii, represents the original zero — i.e., the
place where the force acting on the system vanishes, then if the wiie is
e off to a
58 PROPEirriES or matter.
diftfilnc«d to A and then let go the new zvro will be at O*, a point between
& aad O; thus the force will tend to stop tlie vibratMn as soon as the
wire passes 0' — scouer, that if, tlinii it would do if iLei-e were no aftw-
eflijct. Again, when the wire is on Ibe oilier side of O, the lero will be
di.-<|>!ii(.-(.xl by the einstic after-effect to 0", a point between and B, aod
thus Bgiiin the force tending to stop ihe vihratiou will begin to act sooner
tbiin it would if there were no
elastic ufter-eflect. We can see the
Slime ihitif; from the study of the
model in Fig, 25, for some of the
kinetic energy will be converted into
beat by the friction between the
Tiscous diitd and the damper D.
Lord Kelvin discovered a remark-
able property of the viscosity of
metnia which he called elattie faligua.
He found that if a wire were kept
vibrating ulnioRt continuously the
rate at which the Wbralions died
away got gi«ater and greater; in
fact, the wire behnvtd ns if it got
tii«d and could only with diSicuIty
kcop on vibrating. If the wire
were given a rest for a time it
rccovereil itself, and the vibrations
for a short time after the rest did
not die away ue&i'ly so rapidly as
they hud gone just before ibt rent
1 n^ „, began. Muir (Proe. Roy. Soe., Isir.
p. BS7) found that a metal wire
reciivereil from its fiitigue if it were wiirmed up to a temperature above '
I 00° C.
Anomalous Effects on first Loading a Wire.— The extension pro-
duced by n given Irmd placed on a wire for the fii-st time is not in general
quite the Siime as that produced by subsequent loadiog ; the wire require^
I
ELASTICITY.
S9
i and tlie other effects w«
to be loaded and unloaded several limes boFore it gets into a steady ttsXe.
The first load after a rest also gives, in geneml, an irregular result. It
eeema as if slraining a wire produced it cliange iu its structure from whicb
it did Dot recover for some time.
Great light will pi'ob.tbly be tliio'
hiive been considering by tbu es:imina-
lioQ by the microsc-ope of sections of
the metals. \Vh«u exiiniiued in tbis
way it ia found that metals pos£e.-.s a
structure coarse enough to be easily
rendered visible. Fi^s. 27, if
ehow the appearance under the n
scope of cei-tain metal.-*. It will be |
seen from these figures that in tbi
metals we have aggiegates of cryat
of very great complexity — the liu(
dimension of the.'e aggregateti is son
times a considerable fraction of a
millimetre. These lai-ge aggie^ntes
aro certainly altered by large Btraii.s.
Thus Ewing and Rosenhain {Proc.
Eoij. Soe., ilv. p. 85) have made the
very interesting discovery that when a, m
point there is a slipping of the cryafale, n
along their ptanee of cleavage. The appearance of a piece of iron after
straining fast the yield-point is shown in Fig. 3U; the markin
the ligure ui'6 due i
' stcplike sU'Ucture of the aggregates caused
''■''''^i?'''i'i/^//!('i'':^y^^
by the slipping past each other during the strain of the crystals in
the aggi'egates, as in Fig. 31. Plasticity may thus be regarded as the
yiddiag, or rather slipping post each other of the crystals of the Luge
•ggregates which the microscope shows exist in metiila.
In harmony with this view is the obsenatioa of McOonnel and Kidd
{Proc. Uoy. Soc, xliv. p. .i:!l} that ice in mass ia plastic when consisting vS
crystals irregularly arranged. In later esjiciiments {/'roc. Itvy. Soc, slir.
p. 3S3), McUonucl found that a tingle ciystal of ice is not plastic under
pressure applied along the optic asia, but that it does yield under pressure
60 PROPERTIES OF HATTEB
inclined to tlie axis, ns if there were slipping of the planes perpemliciibr
If tliere is a ganeral change in these aggregates under targe strains it
ia poesible that there are some aggregates wbicb are unstable enough to
be broken up bj* Htnaller strains, and that the first appliontion is aocom-
oanied bj a breaking up of some of the mora unstable groups, bo that the
structure of the metal is slightly changed ; ne can then understand the
irregularities observed when a wire is first loaded and also the existence
of the elastic after-effect. Indeed, it would seem almo-t JneWtable that
any strain among such irregular shaped bodies as those shown in Pig. 28
would result in some of thero getting jatomed, and thus beeoming exposed
to very great pressures, pressures which might be sulSiient to break up
some of the weaker Aggregates, and thus give relief to the system. The
existence of such a structure as that shown in Fip. 28 cnuses us to
wonder whether, if a succession of very accurate observations of tlie
elastic properties of a metal were made, the results would not differ
fi-om each other by more than could be accounted
* r by the errors of experiraeut.
The t«rm Tiscoaity is often u^ed in another
nse besides that on p. 57. We call a substance
SC01IS if it mnnot resist the applicntion of a
small force acting for a long time. Thus we call
pitch viscous becnuse, if given a sufficiently long
time, it will flow like water; and yet pitch can
sustain and recover fi-om a cimsidei'able force if
this acts only for a short time. Fig. 32 shows
the way in which some very htkrd pitch has
flowed through a vertical funnel in whicli it has
been kept in the Cavendish Lalwratory for nine
years. In an espeiiment, due to Lord Kelvin,
pieces of lead plnoed upon a plate of pitch found
in course of time their way through the plate.
Many substances, however, show no trace of
viscosity of this kind, for the existence of sharp
impresGioDS on old coins, the preservation of
bronze statues and the like, show that metals can
sustain indefinitely (or at any rate so nearly
indefinitely that no appreciiible change can be detected after thousands ot
years) their shnpe even under the application of small forces.
Breaking- of Wires and Bars by Tension.— The following tablo,
due to Wertlieini, gives the load in kilogrammes per square milliinetra
necessary tobrPak wires of different sub.stnnces I
Lead . . .21 Copper . , 40'3
Tin . . . 25 Platinum . . 81-1
Gold ... 27 Iron . . G\
Silver ... 2D Steel Wirt . . 70
Zinc . . . 128
The process of drawing into wire seems to strengthen the material,
and the finer the wire the greater is the pull, estimated per unit area of
cross section, rer|uired to break it. This is shown in the following table
given by Baumeister (Wiedemann, Anjuilm, xviii. p. 607) :
Fto. 82.
ELASTICriT.
6l
Material.
Diameter of wire
in mm.
Swedish Iron
. -72
» » '
. -50
II i>
■30
i» » '
. -25
»> »
. 15
>f >»
. -10
Brass .
•75
II
. -25
II • • •
. -10
Pull in kilogrammes
per si\. mm. required
to break the wire.
64
83
9G
94
98
123
70
98
98
The effect of temperature on the pull required to break a wire is com*
plex. Iron wire shows several maxima and minima between 15° C.
and 400° C. (Pisati, Bend, Ace, LinceL 1876, 76) ; the strength of copper,
on the other hand, steadily diminishes as the temperature increases.
The strength of a material is sometimes very seriously affected by the
addition of only a small quantity of another substance. Thus Sir William
Roberts- A listen found that gold, to which 2 per cent, of potassium had
been added, could only sustain 1/12 of the weight required to break
pure gold. In the case of steel, the addition of small quantities of carbon
to the iron increases the strength. The microscopical examination of the
structure of metals, such as is shown in Figs. 27-30, may be expected to
throw a good deal of light on effects of this kind. In this way it has been
shown that the foreign substance is sometimes collected between the
aggregates of the crystals of the original metals forming a weak kind of
mortar, and thus greatly reducing the strength of the metal. In other
cases, such as steel, a carbide is formed, and the appearance of a section
of the steel under the microscope shows that the structure is much
finer than in pure iron. It would seem from Sir William Roberts- Austen's
ezpeiiments that the addition to gold of a metal of greater atomic volume
than the gold diminishes, while \%. metal of smaller atomic volume increases
the strength.
CirAPTER V.
STRAIN.
Contents. — Homogeneons Strain — Principal Axea of Strain — Pure Strain—
Elongation — Dilation or Compression — Contraction — Shear — ^Angle of Shear.
When a body changes in shape or size it is said to be strained, and the
deformation of the body is called strain.
HomOgfeneoUS Strain. — We shall restrict ourselves to the most lim pie
class of strain to which bodies can be subjected ; this is when any t^ lines
which are equal and parallel before straining remain equal and parallel
after straining. This kind of strain is called homogeneons strain.
Thus by a homogeneou» strain a parallelogram is strained into another
parallelogram, though its area and the angle between its sides may be
altered by straining; parallel planes strain into parallel planes, and
Fio. 83.
parallelopipeds into parallelepipeds. Figures which are similar before
straining remain similar after the strain.
It follows from the definition of homogeneous strain that the ratio of
the length of two pai'allel lines will be unaltered by the strain. Let AB
and CD (Fig. 80) be two parallel lines. Let the ratio of AB to CD be tn : n.
Then, if m and 11 be commensurable, we can divide AB and CD respectively
into N?>i and N??, equal parts each equal to a. Then, as before straining all
these parts are equal and parallel, they will remain so after a homogeneous
strain. Thus AB, after straining, will consist of Nm and CD of Nn parts,
each equal to u ; and the ratio of the strained lengths is m : n, the same
as that of the unstrained lengths. If in and n are not commensurable we
can deduce the same result in the usual way by the method of limits.
From this result we can at once prove that a sphere is strained into an
ellipf^oid, and that three mutually perpendicular diameters of the sphere
STRAIN.
65
strain into three conjugate diameters of the eHipsoid. As some of our
readers may not be familiar with solid geometiy, we shall confine our
attention to strains in one plane and prove that a circle is strained into
an ellipse; the reader who is acquainted with solid geometry will not
have any difficulty in extending the method to the case of the sphere.
Let ABA'B' (Fig. 34) be a cii'cle, centre C, which strains into aha!h\
corresponding points on the two figures being denoted by corresponding
letters. Let P be a point on the circle, PL and PM parallel to CA
Fio. 84.
and CB respectively; let these lines on the strained figure be denoted
by pi, pm.
Thus, since the ratio of parallel lines is not altered by the strain
TL^pl
CA ca
PM _ p7n
But since P, A, B are on a circle whose centre is O
PL- PM^ _ .
hence
or p is on an ellipse of which ca and cb are conjugate diameters. Thus
a circle is strained into an ellipse, and two diameters at right angles to
each other in the circle strain into two conjugate diameters of the ellipse.
Now there are two, and only two, conjugate diameters of an ellipse
(unless the ellipse degenerates into a circle) which are at right angles to
each other. Hence there are two, and only two, diameters at right angles
to each other before straining which remain at right angles after the strain.
Now, though in geneml these diameters will not have the same direction
64
PROPERTIES OF MATTER.
after straining as they had before, yet we shall not be introducing any
real limitation on the strain in so far as it affects the forces called into play
by elasticity if we suppose they retain the same direction after straining
as before. For, suppose OA, OB (Fig. 35), are the unstrained directions,
Oa, 06, the strained ones, we can make Oa, 06 coincide with OA, OB by
rotating the strained system as a rigid body through the angle AOa.
This rotation as a rigid body will not involve any relative motion of the
parts of the system, and so will not call into play any forces depending
upon the elasticity of the system ; if, then, as is at present the case, our
object is to investigate the connection between these forces and the strains,
we may leave the rotation out of account.
The three directions at right angles to each other which remain at right
angles to each other after straining are called the principal axes of strain.
If these axes have tJie same dii*ection after straining as before, the strain
is said to be a pure strain ;
if it requires a rotation to
make the principal axes
after straining coincide
with their position before
the strain, the strain is
said to consist of a pure
strain and a rotation.
Thus the most general
homogeneous strain may
be resolved into extensions
(regarding a compression
as a negative extension)
along three directions at
right angles to each other.
fake these directions as the axes of aj, i/, « respectively, then if a line of
unit length parallel to the axis of x has, after the strain, a length 1 + e;
one parallel to the axis of y a length 1 +// and one parallel to the axis of z
a length l-¥gy e, /, g are called the principal elongations. If e^f-g^
then a sphere strains into a sphere, or any figure into a similar figiire,
the strained figure being an enlarged or diminished copy of the unstrained
one. These cases, which are called uniform dilatation or compression,
involve changes in size but not in shape.
A cube whose sides were parallel to the axes before straining and one
unit in length becomes after straining a rectangular parallelopipfd, whose
edges are 1 +c, 1 +/, 1+g respectively, and whose volume is (1+e) (1 +/)
(1 +y). If, as we shall suppose all through this chapter, the elongations
Bf/f g are such small fractions that the products of two of them can be
neglected in comparison with c,/, or^, the volume of the parallelepiped
isl+e+f+g.
Hence the increase of unit volume due to the strain is e-\-f+g. This
is called the cubical dilatation. We shall denote it by 3, and we have
If the strain is a uniform dilatation e=f=gy and therefore
«o that in this case the cubical expansion is three times thc^ Huear elongation.
Fio. 35.
STRAIN.
65
A
B
Resolution of a Homogreneous Strain Into Two Strains, one of
which changres the Size but not the Shape, while the other
changres the Shape but not the Size.
Let lis consider the ca«e of a Rtrain in one plane. Let A, OB ( Fig. 36)
be the principal axes of strain. Let P be the initial position of a point, P' its
position after the strain. Then if e, f are the elongations parallel to OA and
OB, i ftnd jy the displacements of P pai-allel to OA and OB respectively,
£ = eON = i(e +/)0N + \{e -/)0N,
,, =yOM = i(6 +/)0M - i(i. -/)0M.
From these expressions we see that we may regard the strain 0, /
aa made up of a uniform a
dilatation equal to i(«+/),
together with an elongation
J(tf -/) along OA, and a con-
traction i(«-/) along OB.
Thus the strain superposed
on the uniform dilatation con- If
sists of an expansion along
one of the principal axes and
an equal contraction along
the other. This kind of sti*ain
does not alter the size of the
body ; for if <r is the elonga- ^
tion along OA and the con-
traction along OB, then a
square whose sides are one unit in length and parallel to the principal
axes becomes a rectangle whose sides are 1 + o-, and 1 - o- respectively ; the
area of this rectangle is 1 - ^, or since we neglect the square of o- the aiea
is unity, and thus is not altered by the strain. A strain which does not
alter the size is called a shear. Thus any strain in one plane can be
resolved into a uniform dilatation and a shear.
We have considered a shear as an extension in one direction and an
equal compression in a direction at right angles to this ; there is, however,
another and more usual way of considering a shear, which may be deduced
as follows :
Let OA, OB (Fig. 37) be tho axes along which the extension and
contraction take place. Let OA =!OB==OA' = OB'= 1, so that before
straining AB A'B' is a square ; let this square after straining be represented
by ahah\ which will be a parallelogram.
Since Oa = 1 + ^
06=-l-(r
= 2
as we suppose that tr is so small that its square may be neglected. Thus
06 = A B. HeJnce we can move ahah' as a rigid body and place it so that ab
coincides with AB, as in Fig. 38. Then, since the area of ahah' is equal to
that of AB A'B', when the figures are placed so as to have one side in common
66
PROPERTIES OF MATTER,
they will lie between the same parallels. Thus, if a'b" be the position of aV
when ab is made to coincide with AB, a'b" (Fig. 38) will lie along A'B';
hence, except with regard to the rotation, the expansion along AG and the
Fig. 37.
contraction along OB is equivalent to the strain which would bring ABA'B'
into the position ABa'b", But we see that this could be done by
keeping AB fixed and sliding
every point in the body par-
allel to AB through a distance
proportional to its distance
from AB. We can illustrate
this kind of strain by a pack
of cards lying on the table,
with their ends in vertical
"^ planes; now slide the cards
forward, keeping the lowest
one at rest in such a way
that the ends are still flat
although the planes are no
longer vertical; each card
will have been moved forwards
through a distance propor-
tional to its, distance from the
lowest card. The angle A'Ba"
through wliich a line is dis-
placed which to begin with is perpendicular to AB is called the angle of
shear. The plane of the shear is a plane parallel to the direction of
motion and at right angles to the fixed plane.
The relation between — the circular measure of the angle of shear — and
the elongation o- along GA, and the contraction a along OB can bo found as
follows. Before the rotation making ab coincide with AB, ba' make.^
with BA' the angle Bqb ; to make ab coincide with AB (Fig. 87) the system
has to be rotated through the angle Bpb, so that after the rotation haf will
Fig. 38.
STRAIN. 67
make with BA' the angle Bqh + Bpb. Now by the figure, Bqh = Bpb, hence
the angle of shear is 2 ^ Bqb = 2 ^apA. If Am is perpendicular to ap (Fig. 37),
then, since the angle apA is by hypothesis small, its circular measure
Aw»_ Aasin45 Aa __
^Tp~ \AOjf ^AO"'''
hence 6, the circular measure of the angle of shear, = 2ff.
If e and f are the extensions along two principal axes in the general
case of homogeneous strain in two dimensions, we see from p. 65 that this
strain is equivalent to a uniform dilatation ^ (e +/) and to a shear the
circular measure of whose angle is e -f.
CHAPTER VL
STRESSES. BELATIOH BETWEEN STBESSES AID STBAIIISL
C09TK9T9, — General CoiMideratioos — Hooke's Law — Work reqaired to prodaoe uuj
8traio — Bectangalar Bar acted npoa at Right Alices to its Fi
In fftAtir that a body may be strained forces most act upon it. Consider a
fiinall cijl>e in the middle of a strained solid, and soppofae for a moment that
the ext^m^l forces are confined to the surface of this solid. Then the forces
which Mtrain this cabe must be dae to the action exerted opon it by the
siirrfM I riding matter. These forces, which are dae to the action o^ the
molec;ijles outside the cube on those inside, will only be appreciable at
molecular distai;ces from the surface of the cube, and may therefore
without appreciable error be supposed to be confined to the surface. The
most general force whii^ can
act on a face ABOD of the
cube may be resolved into
three components, one at right
angles to ABOD, the other
two components in the plane
of ABOD, one parallel to AB,
the other to BO: similarly
over the other faces of the
cube we may suppose similar
forces to act. These forces
are called stresses; the com-
ponent at right angles to a
face is called a normal stress^
the component parallel to the
face a tangential stress. The
intensity oi' any component of
tlio HtTPHH is the amount of the component over the f«ice divided by the
arm of tlid faco. We Hhall for brevity leave out the word "intensity"
anil H|MMik of it. win ply as the stress. The dimensions of a stress are those
of a forco divi<l(Ml hy an area or ^j^JV^, It is measured in dynes per
Hi|uarn <M»nt iin«*t.r(S on tho COS. system of units the pressure of the
nt.ffKHplMMo in about 10" units of stress.
WluMi we know the stresses over three planes meeting at a point O
{V\\l. no wo<'n'n detormino the stresses on any other plane through O. For
h»t OAlUy bo a very small t/ctrahedron, AOB, BOG, COA being the plani3S
ov(»r which wo know tho stresses, and ABO being parallel to the plane across
whi(*.h wo wish to determine tho stress. Then as this tetrahedron is in
A<|uilibriuni under the action of forces acting on its four faces, and as we
rio. 3u.
STRESSES. 6<J
know the forces over tliree of the facea, OAB, OBC, OCA, we can
determine the force, and hence the Btrecui, on the fourth. Wu need not
take into account any exteriial forces which are proportional to the volume
on wtiich they act, for the forces due to the stre^en are proportional to the
area of the faces, that is, to the Bqiin.re of the linear dimensions of the
tetrahedron, while the external forces are proportional to the cube of the
linear dimensions, and by making the linear dimeuMons of the tetrahedron
exceedingly small we can make the efiect of the vohime forces vanish iu
comparison with that of the hutface forces.
The stresses in a strained solid constitute a system of forces which are
in equilibrium at each part of the solid with the external forces acting ou
the solid. If we call the external forces the load, then if a load W pro-
duces a system of etreaies F,
and a load W a system of
Btresses F, then when W and
W act together the stresses
will be P + F if the deforma-
tion produced by either load
Hooke'S Law.— The fun-
damental law on which all
applications of matliemntics
to elasticity are based is due
to Kooke, and was stiited by
him in the form iU letisio sic
vit, or, in modern phraseology,
that the strains are propor-
tional to the loads. The truth
of this law, when the strains
do not esceed the elSiStic limit
(m< p. bS), has been verified
by very careful experiments
on most mateiials in common ,
use. Another way of stating Fio, 40.
Hooke's I^w is that if a load
W produces a strain S, and a load W a strain S', then a load W + W will
produce a strain S + S'. Hence, it follows from the last article that if a
system of stresses P correspond to a f>yst«m of Ktrains B, uud a system of
stresses F to a system of strains S', then a system of stresses P+ P' will
correspond to a system of strains S-i-S'. Hence, if we know the stress
corresponding to unit strain, we can find the sti-ess coirespondiog to a
strain of any magnitude of the same type. Thus, as long an Hooke's law
holds good, the stress and strain will be connected by a relation of the
Stress = e X strain
where e is a quantity which does not depend either upon the stress or the
strain. It is called a modulus of elasticitf. Thus, if the strain corresponds
to a change in size but not in shape, then the stress is a uniform pressure,
and the strain the diminution in volume of unit volume of the unstrained
substance; in tbid case e h called tha moduloB of elasticity of bulk, or
more frequently the bulk modulus. A^uin, if the strtiin is a shear which
70
PROPERTIES OF MATTER,
mhcn the shape bat not the size, the strain is measui^ by the angle of
sh€*r and the stress by the tangential force per unit area, which most be
appU^to produce this shear. In this case e is called the modnhig of
rifidttj. If we stretch a wire by a weight, the stress is the weight divided
yf the area of cross section of the wire, the strain is the increase of length
HI unit length of the wire, and in this case c is called Youngs modnliu.
feinoe we can reduce the most general 8y>tem of homogeneous strain to
a uniform expam»on or contraction and a s>^tem of shears (see p. 65) it
follows that if we know the behaviour of the body (1) when its size but not
it» Oiape is changed, and (2) when its shape but not its size is changed, we
can determine its behaviour under any homogeneous strain. This is true
when, and only when, the properties of the substance are the same in all
directions, so that a uniform hydre8tAtic pressure produces no change in
A L M N «'
Fio. 41.
shape, and the tangential stress required to produce a givdn angle of shear
is independent of the plane of the shear. This statement is equivalent to
saying that it only requires two moduli — i.e., the bulk modulus and the
modulus of rigidity, to fix the elastic behaviour of the substauce, so that all
other moduli, such as Young's modulus, must be expressible in terms of
these two.
Work required to produce any Strain.— The result for the most
general case, an<l the method by which it can be obtained, may be illus-
trated by coHHidering the work required to stretch a wire. Let us suppose
that the load is added so gradually that the scale-pan in which the weights
are placed never acquires an appreciable velocity, so that none of the work
done is converted into kinetic energy, but all is spent in stretching the
wire. When this is the case, the weight in the scale- pan when in any
position never exceeds by moi*e than an infinitesimal amount the weight
required to stretch the wire to that position.
Let the straight line AB, Fig. 41, represent the relation between
the weight in the scale-pan and the extension of the wire, the
weight being the ordinate and the extension the abscissa ; let OA repre-
STRESSES.
T!
sent the unstretched length of tbe wire. Coneider the work done
in stretching the wire from L to M, where L and M are two points very
near together. T)te force will be approKimatel/ equnl to PL ; thus
the work done in stretching from L to M will be PL x LM — i.e., the
area PLMQ' ; similarly, the work done iu stretching tbe wire from M to N
will be represented by the ai'ea QHNR', and thus the work spent la
stretcliing the wire from OA to OC will be represented by the sum of the
little rectangular areas ; but when these rectangular areas are very email,
their sum is equal t« the area ABO, and this equals JBO x AG — i.e., one-
half the final weight in the scale-pan x extension of the wire. Let a be
tbe area of crosa section of the wii-e and I the length, then BC = ax stress
and AC— f xstrain. Thus the work done in stretching the wire is equal
to o^ X ^ strain x stress. Now al is the volume of the wire, hence the
energy in each unit volume of tbe wire is J strain x stress. Though we
have considered a special case, it will be seen that the method is of general
application, and that the result will hold whenever Kooke's law is true.
We have com^idered two ways of regarding a shear : one where the
particles of the body were pushed forward by a tangential force as is
represented in Fig. 38. In this tasetlie work done on unit volume, which
is the energy possessed by the sheared body, is
where T is the tangential force per unit area and 6 the angle of shear.
The other way of regarding a shear is to consider it as an extension in
one direction combined with an equnl contraction in a direction at right
angles to the extension. Lot e be the magnitude of the extension or
contraction, P the pull per unit area producing the extension ; this is equal
to the push per unit area producing the contraction. Considering unit
volume of the strained body, the work done by the pull is J Ve, and that
by the push is also J Pe; henc« the energy per unit volume is J Pe + J Pe — Pfl,
Vut this energy la also equal to ^ TO, hence
Pe^^JTB.
But we know (p. 67) that fl-2e, henca
P = T.
Hence the pull or push per unit area in the one way of considering a
shear is equal to the tangential stress per unit area which occurs in the
other way.
If n is the coefficient of rigidity, then by the definition of n given on
p. 70,
hence
Rectangular Bar acted on by Forces at Bight Angles to Its
Faeeg._l,et ABCDEFGH, Fig. 4^. be a rectangular bar. Let the
faces CDEF, AHGH be acted on by normal pulls equal to P per unit area,
the faces A BCD, EFHG by normal pulls equal to Q per unit area, and the
faces DEGB, CFHA by normal pulls e<jual to B per unit area. We shall
72
PROPERTIES OF MATTER.
H
proceed to find the deformation of the \mr. GoDsidering the har ns made
up of rectangular parallelopipeds, wiUi their faces parallel to the bar, we see
that these will all be in equilibrium, whether they are in the interior of the
bar or whether some of their faces are on the surface of the bar, if the
normal stresses parallel to AC, CD, DE are respectively equal to P, Q, R,
and if there are no tangential stre&>es. Each of these parallelopipeds will be
subject to the same stresses, and wiU therefore be strained in the same
way. Let e^J^g he the extensions parallel to P, Q, R respectively. Con-
sider for a moment wliat the strains
would be if the stress P acted alone : P
would produce an extension proportional
to P in the direction of P; let us call
this XP; it would also produce contraction
proportional to P in any direction at
right angles to P ; and if the properties
of the strained substances were the same
in all directions, then the contractions
would be the same in all directions at
right angles to P; let these contractions
'^^be /iP. Then when P acts alone the
exiennans parallel to P, Q, R respectively
are XP, -/iP| -fP; similarly when Q
acts alone the extensions in these directions
are -/iQ. XQ, -/iQ, and when R acts
"'-^ alone the extensions are - /iR, — ^R, XR ;
consequently when these stresses act aimul*
taneously we have
D
e= XP-;iQ-^R
/=-,iP + AQ-^R
g^ -/1P-/1Q + XK
}
0)
Now we have seen (p. 70) that the
' elastic properties of the substance are
completely defined if we know the bulk
Fio. 42. modulus, which we shall denote by k, and
the modulus of rigidity which we shall
denote by n. Hence we must be able to express X and /i in terms of n
and k. We proceed to do this. If we apply a uniform tension to each
side of the bar equal to P the dilatation- of unit volume is equal to P/>t,
by the definition of k ; but in this case the dilatation is uniform in all
directions, and the linear dilatation is one-third of the volume dilatation
— i.e., it is equal to P/3A;.
p
Hence, when P = Q = R, c =/= 5^ = ^>
hence, from equations (1 ) -— = X - 2/i.
OrC
Let us now shear the body in the plane of PQ — i.e., put Q=» - P an4
R = 0. In this tase e= -/=F/27i (see p. 71); hence by equations (1)
2;i=^+M.
STRESSES. 7S
^ ^\2n U; 18«^
3\n U) dnk
Youngr's Modulus. — A very important caso is that of a bar acted on
by a pull parallel to its length, while no forces act at ri^ht angles to the
length. In this case Q = R = 0, and we have
But in this case the stress, divided by the longitudinal strain, is called
Yoimg's modulus ; hence, if we denote Young's modulus by q, we have,
F = 5^6, or g' = - =
9vk
X ^k + n
This equation gives Young's modulus in terms of the bulk modulus and
the rigidity.
PoiSSOn's Batio* — Poisson's ratio is defined to be the ratio of the
lateral contraction to the longitudinal extension for a bar acted on by a
stress parallel to its length. If we denote it by o-, then by this definition
9- -/, when Q = R«0.
6
Thusir = e= 3^-2^
X 2{3k + 7i)
Since n is a positive quantity, we see from this expression that v must
be less than 1/2. According to a molecular theory worked out by Cauchy
and Poisson, o-, for all non-crystalline substances, is equal to 1/4. The
determinations of tr given in the table of elastic constants on p. 102 do
not lend much support to this view.
Bar stretched longritudlnally, with Its Sides fixed.— The
equations (1) may be written
«=-(p-a(Q + R))
/„i/Q-.^P + R)\
^ = 1(r-^P + Q)).
If the bar is prevented from contracting laterally,
hence Q = R = ^ — ,
1 —a
. . 2»>
80 thai
f('-^.)
74
PROPERTIES OF MATTER.
HeDce the elongation is less than if the sides of the bar were free m
the ratio of 1 - , to 1. In the case of a steel bar for which a = '268
1 — tr
the elongation if the sides were fixed would be about 4/5 of the elongar
tion when the sides are free.
Determination of Young's Modulus.— A simple way of measuring
Young's modulu-* for a wire of which a considerable length is
available is the following : Fix as long a length of the wire
AB, Fig. 43, as is available firmly to a support. Another
wire, CD, which need not be of the same material, hangs from
the same support down by the side of the first wire. CD
carries a millimetre scale, the length of the scale being parallel
to the wire ; a weight is attached to the end of this wire to
keep it straight. A vernier is attached to the wire AB and
moves against the scale fixed to the wire CD. The wire AB
carries a scale-pan into which various weights can be placed.
By reading the vernier when different weights are on the
scale-pin we get the vertical depression of a fixed point on the
vernier, that is of a known point on the wire, produced by a
given weight. Let this depression be c, when the weight in
the scale-pan is increased by W. Measure the length of the
wire between the fixed support and the point of attachment to
the vernier ; let this be i, then the elongation per unit length
is e/L If w is the cross section of wire, then the stress which
produces this elongation is W/w, so that, as Young's modulus
is stress divided by strain, it is equal to ^
To determine the cross section, the most accurate way is to
weigh a known length of the wire, first in air and then in
water. The difference of the weighings in grammes will be
the volume of the wire in cubic centimetres, and if we divide
the volume by the length we get the cross section. Preliminary
measurements should have been taken with a screw gauge to
see that the wire was uniform in section. It is advisable to
load and unload the wire several times before making the final
measurements. This serves to straighten the wire, and avoids
the anomalous results which, apart from straightening, are
obtained when a wire is loaded for the first time after a rest.
We owe the following improvements of this method to Mr.
G. F. C. Searle. Two brass framas, CD, 0'D\ Fig. 44, hang from
the lower ends of the wires and support the two ends of a sensitive level L.
One end of the level is pivoted to the fnime CD by the pivots H, the
other end of the level rasts upon the end of a vertical screw S working in
a nut attached to the frame CD'. The two links, K, K', prevent the
frames from twisting relatively to each other about a vertical axis, but freely
allow vertical relative motion. When these links are horizontal the two
wires are parallel to each other. A mass M and a pan P hang from the
lower ends of the fi-araes, and the weights M and P are sufficient to
straighten the wires. The Qoi^nections l^twe^^ the wir^ f^d the frames
Fio. 43.
T5
are nude b; the swivels F, into which the ends of the wiree are soldered.
The swivels prevent the torsion of the wire. The head of the screw is
divided, say, into 100 parts, while the piteh of the screw may be -5 mm.;
thus each division on the bend corresponds to 1/200 mm. The
uneasurempnts are made in the following way : Adjust the screw so that
one end of the bubble is at zero ; if a weight be [Jaced in the pan P the
wire A' is stretched, and the bubble moves towards H ; bring the bubble
back to zero by turning t)ie screw ; the distance through which the screw
is moved is equal to the eslension of the wti*.
When the substance for which Young's modulus is to be detormined
■sa bar and not a wire, the extensions obtained by any practicable weight
would be too small to be measured in the w^y just described. In this case
Ewing's extonsometor may be used. This instrument is represented in
Fig. 45. A is the rod whose extension is to be measured, B and C
are pieces attached to A by set screws about the axes of which they
revolve; the arm B' fixed to B ends in a rounded point F, which fits
into ft V-sbaped dot cut tranaveisely acroaa the end of the piece O.
76
PROPERTIES OF MATTER.
Thus, when the rod A is stretched, the point P cicts as a fulcrum, and
Q^ the opposite end of C, moves down through a distance proportional
to the extension between the axes of the set screws. The displacement of
Q is PQ/OP times the extension of the bar. This displacement is observed
by a microscope which is attached to the bar B, and sights an object
at Q. The displacement is measured by means of a micrometer scale en-
graved on glass in the eye-piece of the microscope ; extensions of 1/20,000
of a centimetre are readily measured in this way. There is a fine screw,
with a divided head between B' and the point P. This serves to bring Q
into a convenient position for sighting, and also to determine what is
®
m
Fio. 45.
the absolute amount of extension corresponding to a division of the
eye-piece scale ; for if we know the pitch of the screw we know the dis-
placement of Q when the Fcrew-head is turned through one revolution ;
if we find how many divisions of the micrometer scale this corresponds
to we can at once standardise the scale. The pull is applied to Uie bar
by means of a small testing machine.
Optical Measurement of Youngr*s Modulus.— Michelson's method
of interference fringes, produced by the aid of semi-transparent mirrors,
gives a very delicate way of measuring small extensions.
The principle of the method is shown in Fig. 46. A and B are plane
plates of very carefully worked glass of the same thickness. One surface of
A is coated with a thin film of metal, preferably platinum. The platinum
may be deposited on the glass by placing the glass near a platinum
cathode in an exhausted tube, and sending a current from an inductioD
STRESSES.
77
coil through the tube. The platinum sputters from the terminal and ia
deposited on the glass. This film is so thin as to be semi-transparent ; it
aUowB part of the light to pass through it. Suppose a beam of light,
starting from S, falls on the plate A, some of it is reflected from the
upper surface of the plate, and after being reflected from the mirror
returns and passes out of the plate A and enters the eye at E ; another
part of the beam passes through the plate A, is reflected at D, returns to
the plate A, where it is reflected to E. Even when the difference of path
is great, if A and B are very truly plane and of the same thickness the first
part of the beam from Swill interfere with the second part and produce inter-
ference bands. If the distance between one of the min-ors and the plate A is
Fio. 4«.
altered, the bands are shifted ; an alteration of the distance through 1/4 of
a wave-length will make the dark bands and light bands interchange
their position ; by observing the position of the bands we can measure
movements of the mirror amounting to 1/50 of the wave-length of sodium
light, or say a millionth of a centimetre. To apply this method to the
determination of Young's modulus we keep one of the mirrors fixed while
the other is c:irried by the wire whose extension we wish to measure.
Since we can measure accurately in this way very small extensions we are
able to use comparatively short wires, and so have all the conditions of
the experiment under much better control than when a long wire is
used. This method has been used by Mr. Shakespear at the Cavendish
Laboratory. He has also used the method described on p» 43 for multi-
plying the small movements of the pointer of a balance, to multiply the
movement due to the extension of a wire.
Other methods of detei^mining q will be given in the chapter on the
Bending of Bods.
CHAPTER VII.
TORSION.
Contents.— Torsion of Circular Tubes and Rods— De St. Venant's Researches-
Statical and Dynamical Methods of Measuring Rigidity.
Torsion of a thin Cylindrical Tube of Circular Section.— The
case of a thin cylindrical tube of circular section fixed at one end and
twisted by a couple whose axis is the axis of the tube, admits of a very
simple solution. We can prove that each cross-section of the tube made
by a plane at right angles to the axis is twisted as a rigid body in its own
plane through an angle proportioned to its distance from the fixed end,
and that there is no displacement of any point in the tube either radially
or longitudinally. The last result follows at once from the symmetry of
the tube about its axis ; for from the symmetry, if the radud displace-
ment is outwards at one part of the section it will be outwards at every
point, so that there would
be a swelling of the tube ;
reversing the couple ap-
plied to the tube would,
however, reverse the dis-
placement (since we sup-
pose Hooke's law to
hold) ; hence a couple in
one direction would cause
the tube to swell, while
one in the opposite direc-
tion would cause it to
contract; it is evident,
however, that whether
the tube swells or con-
tracts under a twist about its axis cannot depend upon the dii-ection of the
twist, hence we conclude that there is no radial displacement. Similar
reasoning will show that the longitudinal displacement must also vanish.
We shall now show that the tube will be in equilibrium when each
cross section is twisted as a rigid body through an angle proportional to
the distance of the section from the fixed end.
For suppose ABODEFGH is a rectangular parallelepiped cut out
of the tube before the twist was applied^ suppose the distance between
the planes ABCD, EFGH is d, and let k be the distance of the plane
EFGH from the fixed end of the tube. Then, since the angle through
which each section is twisted is proportional to its di&tance from the fixed
end, if <^ be the angle through which the section at unit distance from the
fixed end is twisted, the rotation of EFGH is k<f>, and that of ABCD
IB (k + d) (ft. If a is the radius of the tube, and if t, its thickness, is small
compared with a, each point in EFGH will be moved through a distanod
Pio. 47.
TORSION.
79
aJki^, &□<! each point of ABCD through a. distance a (k + d) ^, hence
ifter the twist the shape of the paraUelopiped ABCDEFGH will be
dmilar to EFQHA'B'CD', where AA' = BB' = CO-= DD' = (W^. Hence
the deformation of the elements will be a shear of which the angle
of 3hear = AA'/AE =a^. The tangential atress T will therefore be naifi.
Hence the etressoa on the elements will be as shown in Fig. 47,
horizontal tangential stresses equal to T on the faces ABCD, EFGH, and
vertical tangential stresses equal to T on the faces ABEF, CDHO. As f
is aniform for all parts of the tube these streaEes are constant throughout
the tube, and therefore each portion of the interior will be in
equilibrium under these stresses. To find the condition for equilibrium
under the external couple, consider a portion ABCD, Fig. 48, cut from
the tube; this portion is in ecjuilibrium under the action of the tangential
stress T on its cross section, and the external
couple whose moment we shall suppose is 0. For
equilibrium the moment of the tangential stre-sses
round the axis must equal C. The moment of the
tangential stresses is, however, T x area of cross-
aeotion of tube x radius of tube, which i» equal to
henca we have
0)
which gives the rate of twiftt ^ when the external
couple is known.
Case of a Solid Rod of Circular Section.—
We can regard the rod as made up of a series of
tubes, and hence from the preceding investigation
we see that each cross-section of the rod will be
twisted as a rigid body through an angle proportional Fio. 48.
to its distance from the fixed extremity.* The
couple required to twist the rod will be the sum of the couples required
to twist the tubes of which it is built up, or in the notation of the
integral calculus,
if a is the radius of the solid cylinder. If « is the angle through which
the lower extremity of the rod is twisted and I the length of the rod, then
Thus the couple required to twist th^ lower end of the bar thi'ougfa a
given angle varies directly as the fourth power of the radius and inversely
as the length of the bar. If instead of a bar we have a thick tube whose
* For if the croui -Beet ions oF the different tnbei were twitted Ihrough different
angleB, m> as to sbear one tuba past the next, there wuiild be tnistins couples actiog
on the inner parts at the tube, and, since tha outside o£ Iha rod is free, nolhing to
balance these on the outBide.
80
PROPERTIES OF MATTER.
inner radius is b and outer radius a, the couple O required to twist its
lower extremity through an angle ^ is given by the equation
Va
The work required to twist the cylinder through an angle ♦ can be shown
by a method exactly similar to
tuat given on p. 71 to be equal
to ^C<^; hence in the case of a
sol ill rod the energy is
The volume of the rod is /ira',
hence the mean energy stored up
in unit volume of the rod is J waV.
When the cross-section of the
bar is not a circle the problem
becomes much more diflScult. It
has, however, been solved by St
Venant for a considerable number
of sections of different shapes,
including the ellipse, the equilateral triangle and the square with rounded
comers. In every Gise except the circle a cross section made by a plane
at right angles to the axis does not remain a plane after twi&ting but is
buckled, part of the section being convex and part concave. In these
cases there is a longitudinal displacement of the particles,
some moving up and others down. The longitudinal
movement is the same for all particles that were originally
in a straight line parallel to the axis of the cylinder. We
Pf — L ^^ can see in the following way that there must he longitudinal
displacements of the particles and find the direction of the
displacement. Let us take the case when the section is
an ellipse ; then, if each section were rotated round the
axis without any longitudinal displacement, the stress in
each section at any point P would be at right angles to
the line joining O to that point. Thus, if Fig. 49
represent the section of an elliptic cylinder, twisted in the
direction represented by the arrow, the fixed end of the cylinder being
below the plane of the paper and the twist applied to the end above the
paper the stress in the section, if there were only rotation, would be at
right angles to OP *. no^> if P is a point on the ellipse, the tangent to the
ellipse will not be at right angles to OP except at the extremities of the
axes- hence in general the stress would have a component along the
normal to the cyjinder. Since, however, the sides of the cylinder are
supposed to be free and not acted upon by forces, there cannot be
equilibrium unless the stress along the normal to the cylinder vanishes ;
hence there must be some other displacements which will produce a sti-ess
to balance the normal component of the stress at right angles to OP.
This component is directed outwards in the quadrants AB, A B', inwards
in the quadrants BA', B'A ; hence the additional stress must be directed
TORSION. 81
iuwards in the quadrants AB, A'B', nml outn-nrds in the qKadranta BA',
B'A. Now suppose FQRSTUVW, Fig. 50, represents it pnrallelo piped
cut from the quadrant AB, the facen PQRS, TUVW being at right angles
to the axis of the cylinder and the latter nearer to the fixed end, the fac6.i
PQTU, RSVW being at right angles to OP ; then there must be a stress
in the plane PQRS directed from R to Q but if there is a stress in this
direction there must be a stress in RSVW pamllel to RV, otherwise the
parol] elopiped would be set in rotilion. and coi Id not be in equilibrium.
Now the stress in RW parallel to RV impl es either that the longitudinal
displacement in the direction R\ is greater than that in the same
direction in the face PQTU— i.e., that
the longitudinal displacement incrennes
Bs we recede from the axis or else that
the longitudinal displacement in the
opposite direction VR is less than that
in the face TPQU— i.e., that the longi
tudinal displaceoient diminishes as we
recede from the axis. Bat as the
longitudinal displacement vanishes at -
the axis itself, it seems clear that it
must increa.se as we recede from the
axis ', bence we conclude that the
longitudinal displacement is in the
direction RV — i.e., towards the fixed
end of the cylinder. In the quadrant
B'A' the tangential stress at right
angles to OP has a component along Fia 53,
the outward normal, hence the longi-
tudinal displacement is again towards tLe fixed end of the cylinder. In
the other quadrants BA', B'A the tangential stress baa a. component along
the inward normal, and in this case the longitudinal displacement will be
in the opposite direction — i.e., au-ay froxa the fixed end of the cylinder.
Along the axis of the ellipse there is no longitudinal displacement. In
Figs. 51, 52, 53, taken from De St. Tenant's paper, the lines of equal
longitudinal displacement are given in Fig. 51, when the cross section of
the cylinder is an ellipse, in Fig. &3, when it is an equilateral triangle,
and in Fig, 53, when it is a square. The dotted lines represent
displacements towards the fixed end of the cylinder, the full lines
displacenients away from it. The direction of twist is indicate<l by the
arrows. It will be seen that in all cases the displacement is towards the
fixed end or away from it, acconliiig as the coniponerit of the tangenli^il
stress at right nngles to OP nloDg llie noiniul to the boumiary is ilii-pcted
to the oiiteide or inside of the tyliDder. The reason for tiiis we saw
when we conside.-ed the elliptic eyUudi-r.
The appearance of cylinders under considerable twist is shown in
Fig. i)4; thiscasecanbereiiiisedby twisting a rubber spring of elliptic orrect-
angular section and observing the distortion of lines drawn on the spring.
In the case of the elliptic cylinder, De St. Yenant ahowed that the
longitudinal ditiplncement vj reckoned positive when towards the fixed end
of the cylinder at a point whose co-ordinates referred to the principal
axas of tiie ellipse are x, y is given by the equation
and 9 the rute of twiiiit.
Fio. 55.
Thus the lines of equal longitudinal displnccmeut are rectiingular
bobfl with llie axes of the ellipse for asymptotes.
The couple reijuired to produce a rate of twist f was shown by
Do St, Venant to be given by the equation
r liyper-
Tn the case of a thin strip of e1lj|jtic section where h is small compared
with a this equation is approximately
Let us compare this with the couple C required to produce the same
rate of twist in a ivire of circular section, the area of the ci-oss-Bection
being the Funie ns that of the strip. If r is the radius of ihe croEs-sectioh,
then (see p. 79)
a the areas of the cross-sections a
henca
.11 compared with C,
thus, as 6 is very small compared with a, C
Thus, if we use the torsion to measure small
couples, the strip will be very much more
Bensitive than the circular wire. Stn'ps of
thin metal are employed ia some delicate
torsion balances.
The greatest strain was shown by De St,
Tenant to be in the parta of the boundary
nearest the nsta — i.e., the extremities of the
minor axis in the case of the elliptic cylinder
and the middle points of the sides in the case
of the triangular cylinder.
The stress vanishes at a projecting corner,
as, for example, at angles of the triangle and
square. On the other hand, it becomen
infinite nt an internal angle, sucIj as in shown
in Fig. 50. These should, therefore, he
avoided in shafts subject to torsion, or if they
have to be used the angle should be rounded
off.
Determination of the Rig-idily by
Twisting".— The coefficient of rigidity n is
fre<}uently deteiinined by means of equation,
(see p. 79) which gives the relation hetweeii
the couple C required to twitt a circular rod
of mdiiis n and length t and the angle 4>
through which the rod is twisted by the
couple. The ratio of the couple to tbe angle
may be determined (I) atntically; (2) dyna-
mically.
In the statical methoil a knowu couple is
applied to the wire or rod by au ai-mugement
such as that shown in Fig. fifi,
through which a pointer or mi
measured. This gives C and 4,
equation gives n.
In the dynamical method for determining the rigidity, the wire whose
ri^dity is to be determined bangs vertically, and carries a vibration bar
of known moment of inertia. If this bar is displaced from its positi
of equilibrium it vibrates isochronously, and the time of its vibrati
C&n be determined with great accui-acy. The torsional couple tendi
d the a ^
or attached to the wire is deflected is
id if we measure n and I, the preceding
84. PROPERTIES OF MATTER.
to bring tho bar back to its position of equilibrium when it is displaced
through an angle 4» is equal to
1 4*.
hence, if MK' is the moment of inertia of the bar, the time T of a complete
vibration is given by
^ piirar/l
BttMK'/
henoe w= ,|^ . —
This experiment is easily made and T can be measured veryaamrately.
The values of n found by this method are, as a rule, higher than those
found by the statical method. Both methods are open to the objection
that, as a occurs to the fourth power, if we make an error of 1 per cent,
in the determination of a the use of the formula will lead to an error of
4 per cent, in the determination of n. Again, the use of wire in the
determination of elastic constants is objectionable, as the process of wire-
drawing seems to destroy the homogeneity of the metal, the outer layers
differing from the inner. Unless the material is homogeneous it is not
justifiable to use the equation of page 79, and any abnormality in the
outer layers would seriously affect the torsion, as it is in these layers that
the strain is greatest. Tho values of n for all metals are found to decrease
as the temperature increases. (Horton, Froc, Roy. Soc, 73, p. 834.)
CHAPTEB VIII,
BENDING OF RODS.
CosTEHTS.— Bar bent into a Circular Arc — Eoerey in Bar— Bar Loaded atone End —
Depression of End— Bar Loaded in Middle. Ends fiie- Bar Loaded in Miditle.
Ends clamped— Vibration of Loaded Bata— Eliietii; Curves— S lability ot Luailed
Fillar — Yonng's Modulus delermiaed bj Flexure— 'J' able of Uoduli of Elasticity.
Br a rod in this chapter wo mean a bar of unifonn m&terial and cross-
section whose length is great compared with its traosverse dimenGiorifl.
We shall suppotte that such a bar is. acted on by two (Hiuples, equal and
opposite, applied at the two ends of the rod, the plane of the couples
passing through the centres of gravity of all the cross-sections of the rod,
and intersecting the cros'-^-aections in a line which is an asis of symmetry
of the cross-section. Let the couples net so that the upper part of the bar
is extended while the lower part is compressed. There will, therefore, be
a part of the bar between the top and the bottom which is neither
extended nor compressed. This part of the bar is cnlled the neiib-al
surface, and the section of it by the plane of the couple is called the
ttmtiral axia. Let ua suppose the bar divided into thin filaments parallel
to its length. We shall now proceed to show that the bar will Iw in
equilibrium if each filament above the neutral surface is extended, each
filament below that surface compresned, the extension or compression
being proportional to the distnnce of the filament from the neutral
surface, the filaments being extended or compressed as tbey would be ii
the sides of the filament were free from stress ; so that if F is the tension
and e the elongation, T = qe where q is Young's modulus.
First consider the e<|uilibrium of any filament; the strain is a uni-
form extension or contraction, according ua the filament is above or below
the neutral surface. The atiiiin will, therefore, be a uniform longitudinal
tension or compression, there will be no shearing stresses and no stresi^es
at right angles to the length of the bar ; all these statements hold whether
the filament abuts on the surface or not. As the only forces attirigon
the filament are at right angles to its ends, atid are equal and opposite,
the filament will be in equilibrium. Thus each internal portion of the
bar is in equilibrium, and the bur as n whole will be in equihbrium if the
stresses due lo the strain are in equilibrium with the external forces.
Suppose that the bar is cut at C, and that EFGH (Fig. 5S) represents a
cross-section of the bai-, being the centre of gravity of the section ; then the
furces acting on the portion CA(Fig. 57)ot the bar are the osternal couple,
86
PROPERTIES OF MATTER.
whose moment we shall take to be C and the stresses acting across the
cross section. Thus the condition for equilibrium is that the stresses across
this section should be equivalent to a couple in the plane of bending whose
moment is C. Now the tension acting on the cross-section of a filament
at P is equal per unit area to qe where e is the elongation of the filament.
Now e is proportional to PN if ON is perpendicular to the plane of
bending and PN perpendicular to ON ; let e = aPN. Thus the force acting
on the filament parallel to the length of the rod is ^.a.PNa> where ut is
the cross-section of the filament, and the forces on all the filaments into
which the bar may be supposed to be divided must be together equivalent
to a couple of moment C in the plane of bending. The conditions for this
are (1) that the resultant force should vanish ; (2) that the moment of the
forces about OM, which is perpendicular to ON, should be zero ; and (3)
that the moment of the forces about
0N = 0. All these conditions can be
fulfilled if OM, ON are the principal
axes of the cross-section.
For the resultant force is S^'a.PN.w
where 2^aPN.u> denotes the sum of
the product ga.PN.w for all the fila-
ments ; this vanishes since SPNoi = 0,
O being the centre of gravity of the crosa-
section. The moment of these forces
about OM is equal to S^^aPN.PMw,
this vanishes since SPN.PM^O, as
OM, ON are principal axes. The mo-
ment of the tension about ON is
2(^aPN'u> ; this is equal to qakJ^ if A^*
is the moment of inertia of the cross-
section about ON. Hence the tensions
will be in equilibrium with the external
forces if qaAI^ = C.
To find a, let us consider the deformation of a rectangle A BCD (Fig. 59)
in the plane of bending, AB being a portion of the neutral axis. Let
A'B'O'D' be the strained configuration of this rectangle ; then, since there
is no shear, the angles at A' and B' will be right angles, and C'A', D'B'
will be normals to the curve into which the neutral axis is bent ; if these
normals intersect in O, ihen O is the centre of curvature of the neutral
axis. We have from the figure
AB' A'O
But A'B' = AB, since the neutral axis is not altered in length by the
bending, and AB = CD ;
hence 9'i>'-^I> ^'^
Fio.68.
CD
A'O
But if is the elongation along CD, e =
CD' - CD
CD
hence
«=•
AV _ AV _ AC
A'O p p
approximately.
BENDING OF RODS.
87
where p is the radius of curvature of the neutral axis at A. But with
the previous notation esa.AC, so that a = - •
P
Since qaA^k-m^C, we have q = C; or, p = q.-Z.
Thus the radius of curvature of the neutral axis is constant, so tbit the
neutral axis is a circle.
The fact thata thin bat c'r
or lath is bent into a circle
by the application of two
couples is often utilised
for the purpose of drawing
circles of large radius.
The bending of the ^
bar will be accompanied
by a change in the shape ^
of the cross section. The
elongation of the upper
filaments will be accom-
panied by a lateral con-
traction equal to a times
the elongation where a is
Poisson's ratio (see p. 73),
while the shortening of
the lower filaments will
be acoompanied.by a lateral Fio. 59.
expansion. Thus the
shape of the cross-section supposed to be originally a rectangle will after
the bending be as represented in PQLM (Fig. 60).
Suppose LM is the line where the neutral surface cuts the cross
section, then the lateral contraction of PQ is equal to
LM - PQ
LM
and the longitudinal extension is equal to
QM
, LM - PQ QM
hence — ^ ^, = a-^ —
LM p
but if LP, MQ intersect in O', then ^^il^^ = ??i
LM LO
•»"°'* -p-ny
But LO' is equal to the radius of curvature of the neutral surface in
the plane at right angles to the length of the rod. If this is denoted by
p' we have
trp'^p
Thus the ratio of the two curvatures is equal to Poisson's ratio.
88
PROPERTIES OF MATTER.
Energry in the Bar. — Consider one of the filaments into which the
bar was supposed (p. 85) to be divided. Thus, if e is the elongation in
this filament, / the length of the filament (which is equal to the length of
the bar), u> the area of its cross- section, the energy in the filament is by
p. 71,
But e = a.PN ;
hence the energy in the filament ia j^a'PNW.
The energy in the bar is the sum of the
energies in the filaments, and is thus
^qaHJlFWw ; but SPN*a> = AJfc^,
and a = l/p where p is the radius of curva-
ture of the natuial axis, and thus the
energy is equal to ^qAk^l/p^,
Again, gaA^^ = C, where G is the couple
applied to the bar,
hence the energy = J C-
P
= half the product of the couple and
the angle between the tangents at the
extremity of the bar. This result could
be deduced at once by the method already
given.
Rod bent by a Weight applied at one End.— In the case just
Fio. CO.
mi^m*
1
PiO. 61.
considered the stresses in the bar were entirely normal ; in this case, how-
ever, we see that for equilibrium the normal stresses must be accompanied
by tangential ones. 'For, suppose ACB, Fig. 61, represents the bar, the
Aveight being applied at B while A is fixed ; consider a section through
made by a plane at right angles to the length of the bar. Then the
portion CB of the bar must be in equilibrium under the action of the
stresses across the section at C and the weight W at the end of the bar ;
tiius the stresses across C must he equivalent to a vertically upward fore^
BENDING OF RODS.
8P
\V and a couple whoBe moment is W.BC : there must be, therefore, tangential
8tref8i!S acting across tlie section whose resultant is a force "W acting
upwards. We stiaO ehow, however, that if the lateral dimenpions of the
bar are very small, then, except quite close to the end B, the tangential
stresses will be verj- small compared with the normal etreaaes. For let
EFGH represent a section of the bar, O the centre of the section, and ON
aa axis at right angles to the plane of bending. Then, if A is the area of
the Ci*oas- section, T the average tangential stress over the area
Let N represent the normal stress at a point F, dui e. small ai'ea round P,
then since these normal
stresses ai-e equivalent to a
couple whose moment round
ON is W.BO, we have
/k.pn<z„-w.bc.
Thus the average normal <
stress must be of the order
of magnitude
W.BO
Ad Pm. 6!.
where <2 is a quantity comparable with the depth of the bar. Ilence,
W
since — =T, the magnitude of N is comparable with Tx BO/d, so that if the
distance of the section from the end is large compared with the lateral
dimensions of the bar, the normal stresses will be veiy large compared
with the tangential ones. In the subsequent work we shall confine our
attention to the efiect of the normal stresses, but this must be regarded as
Hn approximation only applicable to very thin rods. Let Fig. C2
reprefient a email recttingulor parallelopiped cut out of the bar, the faces
EFGS, E'FG'U' being at right angles to the length of the bar, while the
faces FFH'U, EE'GO' are parallel to the plane of bonding, then the
actual state of stress may be tliua described. The normal stresses are
confined to the faces EFGH, E'F'G'H', there being no normal stresses
over the other faces ; there are tangential stresses on the faces FfilH.',
EE'GQ', and also on the faces GG'UH' and EE'EF*, but there are no
tangential stresses over the faces EFGH, E'FQ'H'.
We may proceed to find the bending of the rod produced by the
weight at ite end in the following way. Suppose PQRS (Fig. 62a) represents
a portion of a rod bent as on p. H5, by -couplas of moment acting at its
ends, then the streaiies in the bar are sucb as to cause a couple with
moment to act across PQ and a couple whose moment is to act across
the section JtS. The stre»^ses which produce these couples, as we have
seen on p. 87, correspond to a state of strain such that the central axis of
the portion of the bar is bent into a circle whose radius p ia given by the
e>iiiatioa
At*
O.
90
PROPERTIES OF MATfER.
R
-h — M^
Now suppose that PQRS, instead of being a poi-tion of a bar acted on
by a couple, is a portion of one acted on by a force at the end A : then
neglecting, for the reasons given above, the tangential stresses across the
section, the stresses are equivalent to a couple W.AN across the section PQ
and a couple W.AM across the section RS, and as AN and AM differ but
little from AL where L is
the middle point of MN,
^4 we may regard the ends
Q S of PQRS as being acted
Fio. G2A. on by equal and opposite
couples whose moment is
W. AL. Hence, by what we have j ust seen, the central axis of PQRS will be
bent into the arc of a circle whose radius p is given by the equation
^^ = W.AL;
P
hence, when the bar is acted on by a weight applied at one end, the neutral
axis of the bar is bent into a curve such that the i-adius of curvature at a
point varies invei-soly as the distance of the point from the end to which
the weight is applied.
Depression of the Bar; Angle between Tangents at two
Points on the neutral Axis.— Suppose Fig. 63 represents the cuived
I
I
I
I
I
I
1
I
I
Fig. 63.
position of the neutral axis.^ Suppose RS are two points near together
on the neutral axLs, then the angle between the tangents at R and S is
equal to RS > where p i« the i-adius of curvature of RS ; but 1/p is equal
to W.AR///. AXr*, hence Ac the angle between the tangents at R and S is
equal to
-P---AR.RS
♦ Though this figure shows for clearness* sake considerable curvature, yet it mast
i,-^i™^J°^®''^*^ ***^^ "^ *" ^^^^^^ investigations we are only dealing with cases in
wbich the bending is very slight and the neutral axis consequently nearly straight
BENDING OF RODS. 9I
or, in the notation of the differential calculus, if 5» AH, we have
hence ^, the angle between the tangents at A and P, is given by the
equation
AP
= f ^^
J g.Ak'
o
d8 (2)
w
Suppose the tangent at P cuts the vertical through A in the point T,
we shall proceed to find an expression for AT. Let the tangents at B,S
cut the vertical line through A in the points M,N, then, remembering that
these tangents are very nearly horizontal, we have approximately, if a3 is
the angle between the tangents at 11 and S,
MN = All .Ab=^^-f.d^ by (1)
(j.AL'
AP
Now AT = 2MN = f^^' il8 = -^^'_ (3)
o **
If the end B of the bur is clamped so that the tangent is horizontal,
then the distance between A and the point where the vertical through A
cuts this tangent will be the vertical depression of A produced by the
weight W; hence, if d be this depression, we have by (3)
d= ""l kW (4)
Thus the vertical depression of the end is proportional to the weight,
to the cube of the length, and iiivertely proportional to the moment of
inertia of the cross section about an axis through its centre at right angles
to the plane of bending ; it is also inversely proportional to the value of
Young's modulus for the material of which the bar is made.
Since the depression is j;roportional to the weight, the energy stored
in the bar is equal ^WfZ, and this by equation (4) is equal to
We shall now proceed to find the depression PM (Fig. 64) of any point
P on the bai' below the horizontal tangent at B. Let the tangent to the
centitil axis at P cut the vertical line through A in the point T, and let the
honzontal line through P cut this Hue at ; then the veiiical depression
of Pis
PM = AN-AT-TO
Now TO = PC X angle the tangent at P makes with the tangent at
B, and since PO is approximately equal to AP, and the tangent at A
makes with the tangents at P and B angles whose circular measures are
92
PROPERTIES OF MATTER.
respectively W,AF^J2qAIr and W,ABy2qAk' (by equation (2)),
have
AP W
TO = ^^;-!l_( AB* - AP=)
we
2qAk'
By equation (3) we have
Thus
W
AN = — l!l-AB»
W
AT = -^, AP^
dqAk"
Fio. 64.
Hence
PM = W r AB»-AP^ _ AP(AB»-APO \
qAk'[ 3 2 /
. W BP7 3AP + 2BP )
'^qAk' 16/
(5)
Fio. 66.
Let us now find what would be the depression of A if the weight W
were applied at P. lu this case AP would be straight, and if AN.
Fig. 65, is the depression of A,
AN = PM + AP X angle which tangent at P makes with the horizontal.
Now by (4)
^""'^^^
and by (2) the angle the tangent at P makes with the horizontal it
equal to
W
29A*'
BP'
hence
BENDING OF RODS.
qAk' \ G J
93
(6)
Comparing equations (5) and (6) we see that the depression at P when the
load is applied at A is the same as the depression at A when the load is
applied at P. In the case we have just been considering one of the points
is at the end of the rod. The theorem, however, is a general one and
holds wherever the points A and P may be. '
The relation between the depression and the weight given by equa-
tion (4) gives us a means of determining q by measuring the flexure of
a beam. In experiments made with this object, however, it has been
more usual to use the system considered in the next paragraph, that of a
beam supported at the ends and loaded in the middle.
Fio. 66,
Beam Supported at the Ends and Loaded in the Middle.— The
ends of the beam (Fig. Gt)) are supposed to rest on knife edges in the same
horissontal line. The tangent at C, the middle point, is evidently hori-
zontal, and the pressure on each of the supports is W/2. Considering now
the portion AC of the rod, it has the tangent at C horizontal, and it is acted
upon by a vertical force equal to W/2 at A. The conditions are the same
aB for a rod of length AC clamped at C and acted on by a vertical force
'W/2, the case just treated ; hence by equation (4) d^ the vertical distance
between A and C, is given by the equation
d^
W AC»
tqKk? 3
W
AB»
48^Ayt^
Rod Clamped at both Ends and Loaded in the Middle.—
Suppose AB is a rod loaded at C, its middle point, and clamped at the
ends A and B, which are supposed to be in the same horizontal line.
I
Fio. 67,
94 PROPERTIES OF MATl'ER.
The action of the supports A, B on the rod will be ecjuivalent to a vertical
force and a couple. The magnitude of the vertical force is evidently W/2 if
\V is the weight at C. We can find the value of the couple r as follows .
By the action of the force Wj2 alone the tangent to the neutral
axis at A would make, with the tangent at C, an angle whose circular
measure is
W AC*
But since the tangent at A is parallel to the tangent at C, the couple
must bend the bar si that if it acted alone the tangent at A would make with
that at C an angle equal and opposite to that just found. Through a couple
r applied to the bar the tangents at A and C would make with each other
an angle whose circular measure is
-^p AO
qAfC-
, W AC- r An
hence — — ■ = -4— -AO
'2fjAk' 2 7AP
or r=iW.AC.
To find the depression of the middle point, we consider the effect of the
force W/2, an(l the couple r separately. In consequence of the action
of the force W/2, the middle point, C would by equation (4) be depressed
below the line AB by
_W AC^
2fiAk- 8
The couple r would bend the bar into a circle whose radius p is qAk^j r .
This would raise the point C above A by
AO-
The depression of C when both the force and the couple act is therefore
W AC^ W AC^
2ryA^■ i ~ 2qAk' 4
W .p, WAB'
24qAkr VrJqAk-
The depression of the middle point of the bar when the ends are fixed is
thus only 1/1 of tlie depression of the same bar when the ends are free.
Vibration of Loaded Bars.— Since the deflection of the bar is in all
cases proportional to the deflecting weight, a bar when loaded will execute
isochronous vibrations, the time of a complete vibration being equal to
-Vm/";:,
where M is the mass of the load and /u the force required to produce unit
BENDING OF RODS.
95
depra«?sion. From the preceding investigations we see that ft=p,qAk^/P
where / is the length of the bar and p a numerical factor, which is equal
to 3 when the weight is applied at the end of the bar, to 48 when the
weight is applied at the middle point of a bar with its ends free, and to
192 when the load is applied to the middle point of a bar with its ends
clamped.
To take a numerical example. Let us suppose we have a steel bar
80 cm. long, 2 cm. broad, and '2 cm. deep, loaded at the end with a mass
of 100 grammes. Then since for steel g = 2-139 x lO'^. and in this case
M=:100,p = 3, /=30, A = -4, Ar = J (-1)2= 0033, we find by substituting
in the formula that the time of vibration is about ^ of a second.
To take another case, suppose a man weighing 70 kilogrammes stands
on the middle of a wooden plank 4 metres long, 30 cm. wide, and 4 deep,
supported at its ends, what will be the time of swing ? For wood we may
take g = 10"; putting p = 48, M-7xlO\ ^ = 4x10-, A = 120, A:« = J(2)«
=> 1*33, we find that the time of swing is about *5 seconds.
Fio. 6a
Elastic Curve. — Let us now consider a case like that of a bow
where the force is parallel to the line joining the ends of the bar. Con-
sider the equilibrium of the portion CB (Fig. G8) under the stresses at C,
and the tension T in the string at B.
Thus the stresses across C must be equivalent to a couple T.CN and a
force T, CN being the perpendicular from C on the line of action of the
force. Confining our attention to the couple, we see that if p is the radius
of curvatures at C of the neutral axis of the rod.
Ak^
T.CN
where q is Young's modulus for the rod, A^*, the moment of inertia of
the cross-section of the rod about an axis through its centre at right
angles to the plane of bending. From equation (7) we see that l/p fs propor-
tional to CN; hence the curve into which the central axis is bent is such
Fig. 69.
96
PROPERTIES OF MATTER,
that th« reciprocal of the radius of curvature at any poiL.t vs proportiotal
to the distance of the point from a straight line. Curves having this
property are called elastic curves or eUisticas ; curves such as those shown
in Fig. 69 are included in this family ; they may be produced by taking
a flexible metal ribbon, such as a watch-spring, and pushing the ends
together. One of these curves is of especial importance — ^viz., the one
where the distance of any point on the bent rod from the line of action of
the force is very small. We shall show that this curve is the path of a
point near the centre of a circle when the circle rolls on a straight line.
To prove this it is only necessary to show that the reciprocal of the radius
of curvature of this path is proportional to the distance from the straight
line which is the path of the centre of the circle. Let us suppose that the
circle rolls with uniform angular velocity oi along the straight line. Let
C be the centre of the circle, P any position of the moving point, G the
point of contact of the circle with the line along which it rolls, PN the
perpendicular on GO. Then if i; be the velocity of the point, p the radius
of curvature of the path,
— = acceleration of P along the normal to its path
(8)
Now since the circle rolls on the line without slipping the velocity of G is
zero. Hence the system is turning about
G, so that the velocity at P is at right
angles to PG and equal to wPG;
hence PG is the normal to the path
and
V = w.PG.
Now the acceleration of P is equal to
the acceleration of plus the accelera-
tion of P relative to ; since moves
uniformly along a straight line the
acceleration of is zero, and since P
describes a circle round 0, the accelera-
tion of P relative to C is equal to wKjP
and is along PO. Thus the acceleration of P along the normal to its path
is equal to
and we have therefore by (8)
w'OPcosOPG
a;
ipQ!
= w=OP COS CPG
or
1_ OP COS CPG
o PG"2
Since the angle PGO is very small, the angle CPG is very nearly equal to
the angle PON, and PG is very nearly equal to a, the radius of the rolling
circle ; hence approximately
1 OP cos PCN _ ON
BENDING OF RODS.
97
Thus 1//D is proportional to the distance of P from the straight line
described by C.
From the equation
AX:'
p
we see that
T
The shape of the curve is shown in Fig. 71. The disttmce between
two points of inflection, that is, between two ][)oints, such as A and B,
where 1/p vanishes, is equal to wa.
Stability of a loaded Pillar.— The preceding result at once gives
us the condition that a vertical pillar with one end fixed vertically in the
ground should not bend when loaded with a weight W — t.e., the condition
that the pillar should be stable. For, suppose the pillar bends slightly,
assuming the position AB, Fig. 72, then AB is an elasticaand B must be a
point of inflection, while, since A is fixed vertically in the ground, the tangent
at A is parallel to the line of action of the force. The distance — measured
parallel to the base-lines — between a point of inflec-
tion and the point where the tangent is parallel to If
the base-line is IvJf the distance between two points
of inflection, and k| therefore, equal to ^ira, or, sub-
stituting the value of a, to
where W is the weight ; hence, in order that the
pillar should be able to bend, /, the length of the
pillar, must not be less than
2 V ^
or, in order to avoid bending.
W
irVAP
4/»
(0)
FlQ 72.
If the cross-section of the pillar is a circle of
radius 6, then Ak'' = \Trb\ Thus the weight which a
vertical pillar can support without becoming unstable
is proportional to the fourth power of the radius and
inversely proportional to the square of the length of the pillar. To take
a special case, let us consider a steel knitting-needle, 20 cm. long and
•1 cm. in radius and take ^ = 214 x 10". We find W less than 104 x 10«
I.e., less than about 1056 grammes.
Jf the rod, instead of being 6xe4 at one end, is pressed between Wo
Q
98
PROPERTIES OF MATTER.
supports so that the ends are free to bend in any direction, Fig. 78, the
ends must be points of inflection, the distance between which is «ti or
V^Aifc'/W;
hence
I
-V
W
in the limiting case when the pillar can bend. Hence for stability
W<
it^qAk"
(10)
In the case where both ends are fixed (as in Fig. 74), the tangents at
B B
Fio. '. 3.
Fio. 74.
the ends must be parallel to the line of action of the force, and there muet
be two points of inflection at, 6andc; hence the distance between the ends
is twice the distance between two points of inflection, so that
«27r
v/
w
Hence for stability
p
(")
Comparing (9) and (11), we see that a rod with both ends fixed will,
v«^ithout buckling, support a weight sixteen times greater than if ene end
were free.
Since a pillar can only support without buckling a finite wei^^t, and ee
this weight diminishes as the length of the pillar increases, it foilowa tiMit
a pole of given cross-section would, if high enough, begin to bend imder its
own weight, so that there is a limit to the height of i^ viirjbical fcUar or
BENDING OF HODS. 99
tree of given cross -Bectioii. Suppose W is the weight of the pillar, and
suppose aa an approximation tbat the problem is the »ime as if the weight
were applied at the midille point of the pillar, then if / is the length of
the piUar we see from (9) that
<V^
<-'v/^
Let OS take tlie case of a pine tree of uniform c
bottom, let the diameter of the tree be 15 en
taking the specific gravity of deal as -G, we have
■n]iir section from top to
For deal 5 = 10", aiid
" -SxaftlxlG
Z<27x IG'cm.
Ih IS the heii^ht of the tree cannot exceed about 27 metres.
Determination of Young's Modulus by Flexure.— Young's
modulus is often determined by measuring the defleciion of a beam supported
at both ends and loaded in ttie middle. If (2 is the depi-ession of the middle
of tbe bai', then (see p. 'J'3)
AB'
where W is the load, AB the length of the bar, q Young's modrilus, Ai'
the moment of inertia of the cross-section of the bar about an uxis through
the centre of gmvity of the section at right angles to the plane of bending.
The value of d can be determined by lixiog a needle point to the middle
of the bai-, and observing through a m icroscope provided with a micrometer
eyepiece the depression of the beaui when loaded in the middle with various
weights. Another method of measuring d is by means of a very carefully
made screw, the end of which is brought into contact with the bar; by
measuring the fraction of a turn through which the head of the screw
must be turned to renew the contact afttir the bar has been loaded we can
determine the value of d corresponding to given loads. The most accurate
method, however, would be an optical one, in which, by Michelson't^ method,
interfei-ence fringes are produced by the interference of light ledected
from two min-ors, one of which is fised while the other is attached to the
middle point of the b.ir. Ey measuring the displucement of the fringes
when the load is put on we could determine d, and the method is so
delicate that the displacements cotTeiiipoEiding to very small loads could be
100
PROPERTIES OF MATTER.
Another method, due to Konig, consists in measuring the angle through
which the free ends of the bar are bent. The method is represented in
Fig. 75. AB is the rod restin<r on two steel knife edges S,, S,. The mirrors
Pp P,, which are almost at right angles to the rods, are rigidly attached to
it. The vertical scale S is reflected first from the mirror P,. then from
the mirror P^ and then read through the t'Olescope F. The weight is
applied to the knife edge r, which is exactly midway between the knife
edges Sp S,. On looking through the telescope we find one of the divisions
of the scale coinciding with the cross wires ; on loading the beam another
division of the scale will come on the cross wire, and by measuring the
distance between these divisions we can determine the angle ^ through
which each free extremity of the bar has been bent. For, kt us follow
=1S
Fig. 75.
the ray backward from the telescope; when the mirror P^ is twisted
through an angle ^, the point where the reflected ray strikes the mirror
P, is shifted through a distance 2d<l>, where d is the distance between the
mirrors ; thus, if the light reflected from P, were parallel to its original
direction, the scale reading would be altered by 2c^, but the light reflected
from P, is turned through an angle 4^ ; this alters the scale reading by
4D^ where D is the distance of the scale S from the mirror P,, hence v^ the
total alteration in the scale reading, is given by
V « {2d + 4D)^
9 =
V
9 =
2d +ijy
w ?
Thus
but (see p. 91)
' 2^AJ^ 8
where I is the distance between the knife-edges.
Thus knowing v we can determine q. The advantage of this method is
that Vy the alteration in the scale reading, may be made very maoh greater
Ihan the depression of the middle of the bar.
BENDING OF RODS.
101
The following convenient method for determioing both n and q for a
wire was given by G. F. O.Searle in the rhilosophical Magatine,¥eh. J90(t.
AB, CD (Fig. 7(i) are two equal braes bars of square section, the wire
under obfler\'Btion is firmly secured by passing through horizoiitil holes
drilled through the centres O, G' of the bars. The system can he >itispe[)ded
by two parallel torsion!e.sa strings
by means of hooka attached to the
bars. If now the ends B and D
are made to approach each other
through equal distances and are
then set free the bars will yibrate
in a horizontal plane. To a lir&t
approximation the centres G and O'
remain at rettt, so that the action
of the wire on the bar, and therefore
of the bar on the wire, is a pure
couple; the wire will, therefore, be
bent into a horizontal circle and
the couple will he qAi'jp Here
q IB Youngs modulus AA* the
moment of inertia of the
Bection of the wire about a
through tne centre of gravity at
rjght angles to the plane of bending,
p the radius of curvature of the
wire which u, equal to Ij^i if / is the length of the wire and ^ the angle
through which each bans twifatetl Ilfciico, ih It if the moment of ineitia
of CD about a vertical axis through U, we ha\e
.-(Pd //AW 27Af
^di- — ^ j—f'
hence, if T, is the time of vibration,
The bars are now unhooked from the strings and one clamped to a shelf,
so that the wire is vertical ; if we make the wire execute toraional vibra
tions, and T, is the time of vibration,
hence by (12) and (13) we have
i» T*
102
PROPERTIES OF MATTER.
TABLE OF MODUU OF SLASTICITT.
The Talaes of the modali of elasticitj tbtj so mach with the trefttmeDt a metal
has received in wire-drawing, rolling, anneiding, and so on, that wbeneTer thejr
are required for a given specimen it is necessary to determine them, if any degree
of accaracj is required. The following table contains the limits within which
determinatiocs of the moduli of different metals lie. Thej are taken from the
results of experimeots bj Wertheim, Kiewiet, Lord Kelvin, Pisati, Baameister,
Mallock, Oorou, Everett, and Katsenelsohn. The values are g^ven in C.G.8. units,
n is the rigidity, q Young's modulus, k the bulk modulus, and «- Poisson's ratia
n,10ii
g/lOii
I/10U
r
Aluminium .
2-38 3-36
7-4
•13
Brass .
3-44— 4 03
9 48_lO-75
10-2— 10-85
^6— •4<;9
Copper .
3-5— 4-5
10-3— 12-8
17
•25— -35
Delta-Metal .
3-6
9 1
10
—
Glass .
1-2— 2-4
5-4— 7-8
8-4— 4-2
•20— ^26
Gold .
3-9— 4-2
r5-48(drawn)\
\ 8 (rolled) j
•
•17
Iron (cast) .
3-5 5-3
9-8 16
9*7— 14-7
•28— 31
Iron (wrought)
6-6 7-7
17—20
—
—
Lead .
•18
•5 1-8
8-7
•875
Phosphor Bronze .
3-6
9-8
—
—
Platinum
C-6-7-4
15 17
—
•16
Silver .
2-5— 2-6
7-0— 7-6
— .
•37
Steel .
7-7— 9-8
18—29
147—19
•25— -38
Tin . . .
1-6
4-2
—
—
Zinc
8-8
8-7
•20
CHAPTER JX.
SPIRAL SPRINGS.
COSTKSTS.— Flat
Tbb theories of bending itnd twisting have very important applications
to the case of spiral springs. By a spiral spring we mean a UQiform wire
or ribbon wound round a circular cylinder in such a way that the axis oi
the wire makes a constant angle with the generating lines of the cylinder.
The first cme we shall consider is that of n spiral spring
made of uniform wire of circular cross-section, and wound
round the cylinder so that the plane of the wire ia everywhere
approximately perpendicular to the axis of the cylindei' — i.e., a
" fiat " spring. Let as suppose that such a spring iu hung
with its axis vertical, and that a weight W, acting along tiie
RxiB of the cylinder, is applied to an arm attached to the
lower end of the cpring.
Considering tho equilibrium of the portion CP of the
spring, the sti-essps over the crons section F must be in equili-
brium with the force W at C, and hence these stresses must
oe equivalent to a tangential foi'ce W acting upwards, and a
couple whose moment is Wa and whose axis coincides with the
axis of the wire at F, a being the radius of the cylinder on
which the axis of the wire lies. If the diameter of the wire is
very Email compared with a we may, by the principles ex-
plained on p. 8;i, neglect the effects of the tangential force in
comparison with that of th« couple and consider the couple
alone. T'his couple is a tumiootil couple and is constant all ,
along the wire ; it will produce, therefore, a uniform mte of ;
twist ; if ^ is the i-ate of twist, i the radius of the wire, and ;
n its coefficient of rigidity, then we have (see p. 7D),
Now suppose that we have a series of arms of length a
attached to the wire at right angles, the free ends of these
arms all being in the asiH of the cyHnder. Then, if P, Q are
two points near together, the effect of the twitting is to Fia. 77.
inorease the vertical distance between the ends of the arms
attached to P, Q respectively by PQ x a^, and since a and ^ are constantfl
this result will hold whatever the distance between P and Q. Suppose Q is at
the fixed and F at the free end of tbe spring, then the increase in the
vertical distance between the arm attached to P and Q will be the vertical
depression of the weight W ; in this case PQ = /, the length of wiie in the
spring ; hence, it die the depression of W,
104 PROPERTIES OF MATTEH
Thus d varies directly as the area of the cross-section of the cylinder
and inversely as the square of the area of the cross-section of the wire. We
see that the depression of the weight is the same as the displacement of
the extremity of a horizontal arm of length a attached to the end of the
same length of wire when pulled out straight and hung vertically, the end
of the horizontal arm being acted on by a horizontal force equal to W at
right angles to the arm.
To take a numerical example : suppose we have a steel spring 300 cm.
long wound on a cylinder 3 cm. in diameter, the diameter of the wire
being *2 cm.
n = 8xl0", a = l'5, 6 = -l.
If this spring is loaded with a kilogramme so that Ws=981 x 10*, the
depression d will be given by
,__ 600x981xl0"x(l'5)«
IT X 8 X 10»^ X 10-*
B 5 cm. approximately.
Energy in the Springf.— Q, the energy stored in the spring, is
(see p. 80) given by the equation
But 0-—-
thus Q»
W'fa«
TTflb*
This result illustrates the theorem proved on p. 71.
Springrs inclined at a finite Angfle to the horizontal Plane.—
The flat spring, as we have just seen, acts entirely by torsion ; in inclined
springs however, bending as well as torsion comes into play. Let the axis of
the spring make a constant angle a with the horizontal. Let the spring
(Fig. 78) be stretched by a weight W acting along the axis of the cylinder
on which the spring is wound. Then, considering the equilibrium of the
portion AP of the spring, and neglecting as before the tangential stresses
at P, we see that the stresses at P must be equivalent to a couple whose
moment is Wa, and whose axis is PT, the horizontal tangent to the
cylinder at P. This couple may be resolved into two: — one with the
moment Wacosa and axis along the wire PQ, tending to twist the spring ;
the second, having the moment Wosina and its axis PN at right angles to
the plane of the spring at P tending only to bend the spring. Now the
twisting couple Wocosa will produce a rate of twist ^ given by
^^ Wocosq
SPIRAL SPRINGS.
105
where is a quantity depending on the shape and size of the cross-
section of the spring. When the spring is a circular wire of radius 5, we
have seen that C = vh*l2. The couple Wasina will hend the spring and
will alter the inclination of the tangents at two neighbouring points PQ by
Wosing . PQ
where D^AP, the moment of inertia of the area of the cross-section
of the wire of the spring about an axis through
its centre of gravity at right angles to the plane
of bending.
Let us now consider the effect of these
changes on the radial arms which we imagine
fixed to the spring. Let us first consider the
vertical displacements of the ends of the arms at
two neighbouring points PQ. Taking first the
torsion, the relative motion of the ends is PQ^a,
but in consequence of the inclination of the
spring this relative motion is inclined at an
angle a with the vertical so that the relative
vertical motion is
PQ.Wa'cos'a
PQo^sa = ^
Thus, if Z be the length of the wire in the
spring, the vertical displacement of the end of
the spring due to torsion is
/Wa'cos'a
Now consider the efiect of the bending on the
vertical motion of the ends of the rods at PQ.
In consequence of the bending, the relative
motion is in a plane making an angle u with the
horizontal plane and is equal to
Wasina
PQa
To get the vertical component of this we must multiply by sina, and
▼e see that the vertical displacement due to bending is
PQ
Wa'sin'a
or for the whole spring
^Wa*Fin*a
Thus the total vertical displacement is
\ nO «D J
106 PROPERTIES OF MATTER.
In addition to the vertical displacement there will be an angular dis-
f)lacement of the pointer at the end of the bar which we may calculate aa
oUows. First take the torsion. The arm at P is twisted relatively to the
arm at Q through an angle in a plane making an angle ^ - a with the
it
horizontal plane equal to PQ x ; the angular motion in the horizontal
plane is, therefore.
PQx0xcos(^-o)
or pQWasinacoFa
And the direction is such that as we proceed along the spring the arms are
rotated in the direction in which the spring is wound, so that this angular
movement due to the toi*sion is such as to tend to coil up the spring.
The angular deflection due to torsion for the whole spring is, therefore,
JLWosinacosa
Let us now consider the angular deflection due to bonding. The arm at
P is bent relatively to that at Q through an angle
pgWosina
in a plane making an angle a with the horizontal plane ; projecting this
angle on the horizontal plane the relative angular motion in this plane of
the two arms is
pQWasinacosa .
thus the angular deflection due to bending for the whole length of the
fipring is
ZWasinacosa
The deflection in this case is in the opposite direction to that due to the
torsion^ and is such as to tend to uncoil the spring. The total angular
deflection is thus
ZWasinaCOSa-f —^ — =- )•
\n\j q\j)
in the direction tending to coil up the spring. The angular deflection is
thus proportional to sin a cos a and is greatest when a = tt/^:. The deflection
tends to coil up the spiing or uncoil it according as
JL>i..
if the spring is very stiff to resist bending in its own plane, it will coil tip
under the action of the weight ; if, on the other hand, it is very stiff to
resist torsion, it will uncoil. This is exemplified by the two springs
shown in Figs. 79, 80. The first, which is made of strip metal, with the shov!^
SPIRAL SPRINGS.
107
dimension in the plane of bending, is very weak to resist bending, and so
t«nds to uncoil when stretched, while the second, which is also made of a
strip of metal, but with the long side in the plane of bending, is very stiff
to resist bending, and so tends to
coil up when stretched. In the
case of a circular wire of radius b
so that
C = ^Kb'
V
\
1__ I _ 2 /1_2y
nO qD Kb'U q]
For metals q is greater than 2n, so
that
n(} ql>
is positive, and thus a spring made
of circular wire tends to coil up
when extended.
Vibrations of a Loaded
Springf. — We can use the up and
down oscillations of a flat spiral
spring to determine the coeflicieut
of rigidity of the substance of which
the spring is made. Let us take
the case of a flat spiral spring made
of wire of circular cross-section ;
then, if the spring is extended a
distance x from its position of equi-
librium, the potential energy in the
spring is (see p. 104) equal to
where n is the coeflScient of rigidity,
b the radius of cross-section of the
wire, a the radius of the cylinder
on which the spring is wound, and
I the length of the spring. If the end of the spring is loaded with a masB
M, the kinetic energy of this mass is equal to
p«(i)'
The spring itself is moving up and down, so that there will be some kinetic
energy due to the motion of the spring. To a first approximation the
vertical motion of a point on the spring is proportional to its distance from
the fixed end, so that the velocity at a distance 8 from the fixed end will be
a dx
Fig. 79.
Fig. sa
108 PROPERTIES OF MATTER.
If p is the mass of unit length of the spring, the mass of an element of
length ds is pds and its kinetic energy is
K^ ) z^
fntegrating this expression from 8 = to 8 = /, wo find that the kinetic
energy of the spring is
or if w be the mass of the spring
iin/dxY
hence the total kinetic energy is equal to
i{
''*'W
Since the sum of the kinetic and potential energy is constant
i/tic . ni\/dxy . imh^Qi?
is constant, hence differentiating with respect to < we have
This equation represents a periodic motion, the time T of a complete
vibration being given by the equation
T = 2^a/M+W^
When T has been determined n can be found by this equation.
Angular Oscillations.* — We can prove in a similar way that T,
the time of vibration of a suspended bar about the vertical axis, is given
by the equation
^ ^674
where MA;* is the moment of inertia of the bar about the vertical axis and
q Young's modulus for the wire, by measuring T, we can determine q,
* Ayrton and Perry. Proc. JLS,^ vol. zzzvi,, p. 311 ; Wilberforce, PhiL Mag.,
Oct. 1894.
CHAPTER X.
CuNTEHTS. — Co-efficient o! R«stitiitjoii~Flcwlon'& EiperimeDts— Hodgkinson's
Eiperimeiits — Einmple of Collision of Kail way Cnrriagea — HerU's Investiga-
tiona— Table of Co-etticlecls.
Co-eflicient of fieStitutioo- — An interesting class of pho
depending on the cinsticity of matter is that of collision betiveen elitetic
bodies. The laws governing tliess colli^ona were investigated by Mewton
and his contemporaries, who used the following method. The collidipg
bodies were spherical balls suspended by strings in the way shown in
Fig. 81 ; the bails, after falling from given heights, struck againfit
e«ch other at the lowest point, and after rebounding again reached a
certain height. By measuring these heights (and allowing, as Newton
did, for the resistance of the air) the velocities of the balls before and
after collision can bo determined. New-
ton in this way showed that when the
collision was direct — ■i.e., when the rela-
tive velocities of the two bodies at the
instant of collision was along the common
normal at the point of impact — the
relative velocity after impact bore a
constant ratio to the relative velocity
before impiict — the relative velocity
being, of course, reversed in direction.
Thus, if w, V are the velocities of the
bodies before impact, u being the velocity
-f
of the I
U, V £
3 slowly moving body, while
i the velocities after impact, (
C-V. .(.-.)
(1)
where e ie a. quantity called the co-
efficient of restitution, and Newton's experiments showed that e depended
only on the materials of which the balls were made, and not on the masses
or relative velocities. A series of esperiments were made by Hodgkinson,
the results of which were in general agreement with Newton's. Hodgkinson
foond, however {Report of British Aanoc^ntion, 183i), that when the initial
relative velocity was very large e was smaller than it was with moderate
velocity.
Vincent* has shown that the coefficient of restitution is given by the
equation e = «o — 6m, where m is the velocity of approach and e„ and h are
constants.
Equation (1) and the equation
nt, Proctcdiniii Camhridfji rjiiloiophicai Soricly, vol. x p, 332.
(2)
no PROPERTIES OF MATTER.
which expresses that the momentum of the system of two bodies is not
altered by the impact, m and M being the masses of the bodieSy are suiHcient
to determine U, Y ; solving equations (1) and (2) we find
U = -— 4- e =-^{v - u)
w + M w + M '
Hence we have
JmU' + ^M V = Imu? + JM** - ^(1 - e')^^(* - «)' (3)
Thus the kinetic energy after impact is less than the kinetic energy
before impact by
Thus, if 6 is unity there is no loss of kinetic energy. In all other c/i'^ea
there is a finite loss of kinetic energy, some of it being transformed during
the collision into heat ; a small part only of it may in some cases be spent
in throwing the balls into vibration about their figures of equilibrium.
Collision of Railway Carriages.— To get a clearer idea of what
goes on when two elastic balls impinge against each other, let U8 take tho
case of a collision between two railway carriages running on frictionless
rails, each carriage being provided with a buffer spring. When the
carriages come into collision, the first effect is to compress the springs, the
pressure which one spring exerts on another is transmitted to the oarriageg,
and the momentum of the carriage that was overtaken increases, while
that of the othei* diminishes; this goes on until the two carriages are
moving with the same velocity, wiien tho springs have their maximum
compression and the pressure between them is a maximum. The kinetic
energy of the carriages is now less than it was before impact by
1 Mwi / V,
and this energy is stored in the springs. The springs having reacbf^l
their maximum compression begin to expand, iiicrea^ing still further
the momentum of the front carriage and diminishing that of the carriage
in the rear. This goes on until the springs have regained their origini 1
length, when the pressure between them vanishes and the carriages
separate. There is now no strain energy in the springs, and the kinetic
energy in the carriages after the collision has ceased id tho same as it
was before it began.
The reader who is acquainted with the elements of the differential
calculus will find it advant^i^^eous to con^ide^ the analytical solution of
the problem, which is very simple. Let Xy y be the co ordinates of the
centres of gravity of the first and second carriages respectively, /i, fi the
strength of the springs attached to these carriages (by the strength of
a spring we mean the force required to produce unit extepsiop of t})^
IMPACT. Ill
spring), C, Tf the compressions of these springs, and F the pressure between
them; then we have
m— r = P, — r— = - P
dt' ' dC
« - y =* constant - ({ + ij)
The solution of these equations is
= (t> - u)uf—- sin(i><
' M + w
where w =» 4, / - ^^^ -^i^^ — : u and « are the initial velocities of the
V ^ + /i Mm
carriages, and t is measured from the instant when the collision began,
dx me , -Mr \ Mm , . .
dt M + m^ ^ M + m^ '
M^ = — : [mu + Mv] + — (v - w)coso)<
dt M + m^ ^ M + m^ '
Thus the springs have their maximum compression when --^ = i^, t .«
dt dt
when (1)1 = ir/2, or < = -^ ; at this instant the energy stored in the Erst
spring
Mm
/* jj\- "'^ + ^' M + m
while the energy in the second spring is equal to
ip,
1 P' 1 / \» M Mm
/z' /I + /*' M + m
At the instant of greatebt compression the amounts of energy stored
in the two springs are inversely as the strengths of the springs.
The springs regain their original length and the collision cea^ics
when P = — i.e., when utt = w, or
. _ TT _ / Mm /1 4- ft'
"~ w ~ ^ V M + m uu'
this is the time the collision lasts. Wo see that it increases as the masses
of the carriages increase and diminishes as the strengths of the springs
increase. It is independent of the relative velocity of the cairiagee
before impact.
In the case of the collision between elastic bodies the elasticity of tb#
material aerveji fofite^^ pf the springs in the preceding example. Th^
PROPERTIES OF MATTEI^
bodies when they come into coRI&ion fliitten at the point of contact so
that the bodies have & finite area in common. In the neighbourhood of
this area each body is compressed ; the compression attains a maximum,
then diminishes and vanishes when the bodies separate. The tlieor; of
the collision between elastic bodies has be«n worked out from this point
of view by Hertz (eee Collected Papers, English Translation, p. 146), who
finds expressions for the area uf the surface in contact between the
colliding bodies, the duration of the contact and the maximum pressure.
The duration of contact of two equal spheres was proved by Herts to
be equal to
2D43:
V tt{r-«>2'
where R is the radius of either of the spheres, « the density of the
sphere, q and ^ respectively Young's modulus and Poiseon's ratio for
the substance of which the spheres are made. Hamburgel has measured
the time two spheres are in cont&ct by making the spheres close an
electric circuit whilst they are in contact and measuring the time the
current is flowing. The results of his experiments are given in the
following table. They relate to the colliiian of brass spheres 1'3 cm.
in radius:
K«taU™ValocttTln™.i»rMfl. 1 TS7
™
lati 1 ES'S 1
DoratiOD oreolliaion (colcalated)
„ „ „ (observed) .
■000 IBB
■oooiya
■oooier
■000173
■000153
■000157
-OOOMO
-0001 IS
The duration of the impact is several time* the gravest time of vibra-
tion of the body. In order to start such vibrations with any vigour
the time of colli'-ion would have to be small eompared with the time
of vibration. We conclude that only a smaU put of the energy is spent
in setting the sphere.^ in vibration.
I example of the order of magnitude of the quantities involved
of spho
quote the results given by Herts for two
radms meeting with a. relative velocity of 1 cm.
of the surface of contact is 'OlS cm. The time
■conds. The maximum total pressure is ^■47
pressure per unit area is 7300 kilogrammea
in the collision
steel spheres 2
perset'ond. The radii
of contact is '00038
kilo^rummes and the
per aqtiare centimetre.
In this theoi-y and in the example of the carriages with springs wa
have supposed that the work dene on the springs is all stored up as
available potential enci'gy and is ultimately reconverted into kinetic
energy, ao that the total kinetic energy at the end of the impact is the
same as at the beginning. This is the case of the impact of what are
called perfectly elastic bodies, for which the co-eflicient of restitution is
equal to unity. In other cases we see by equation (3) that, instead of the
whole work done on the springs being reconveitod into kinetic energy,
only the constant fraction e' of it is so reconverted, the rest being ulti-
mately converted ijito heat. Now our study of the elastic properties of
bodies has shown many examples id which it is impossible to convert the
energy due to strain into kinetic energy and the kinetic energy baok
again into energy due to strain without dissipation. We may mention
the phenomena of elastic fatigue or viscosity of metala (see pa^ §7),
IMPACT.
113
as exemplified by the tor^ionnl vibraticns of a metal wh-e, wliero tlie
Buccei=sive transformations of the energy were accompanied by o. con<
tinued loss of available energj-. Again, the elnaliu after-eflect would
prevent a total conversion of strain energy into mei^banical enei^.
For example, if wa laid a. wire up to a cerlnin point, and moai'uro
the extension cori'osponding to any load, then griiitiially unload the
wire, if the straining has gone beyond the elastic limit the extensioua
during unloading will not be the aatue ns during loading; and in this case
there will in any complete cycle be a los.s of mechaDicttl energy proportional
to the area included between the curves for laiding and unloading. The per-
centage lots in this case would depend upon tlie intensity of the maximum
stress; if this did not strain the body beyond its elastic limit there would
be no loss from this cause, while if the maximum strain exceeded this limit
the lofs might be considerable. This may be the reason why tbe value
of e diminiiihes as the relative veloi^ity at the moment of collision increase?,
for Hertx has shown that the m.aximum pressure increases with the
relative velocity being proportional to the 2/M\s power of the velocity,
while it is independent of the size of the balls. Thus the greater the
relative velocity the more will the maximum pressure exceed the elastic
limit and the larger the amount of heat produced. In addition to
the loss of energy by the viscasity of metals and hysteresis tbcre in
in many caaes of collision permajient deformation of the stirface in
the neighbourhood of the surface of contact. This is very evident
in the case of lead and brass. The harder tbe body the greater tbe value
of e. We can 6ee the reason for this if we remember that the hardness
of a body is measured by tbe maximum stress it can sufTer without
being strainnl beyond the elastic limit, while any stiiiin beyond tbe
elastic limit would increase the amount of heat pro<lucod and so dimini-sh
the value of e.
When we consider the vario'ja ways in which imperfections in the
elastic pi'0[iei ty can prevent the complete transformation of the energy due
to strain into hinelic energy and vice v^rsii, it is somenhat surprising that
the laws of the collision of imperfectly elastic bodies are as simple -aa
Newton's and Hodgkinson's experiments show them to be, for these laws
express the fact that in the collision a constant fraction, t', of the initial
kinetic energy is converted into heat, and that this fraction is independent
of the size of the rphercs and only varies very slowly with the relative
velocity at impact, t'oi- example, Hodgkinson's experiments show that
when the relative velocity at impact was increased threefold the value of a
in the case of the collision between cast-iron spheres only diminished fittm
'G9 to *&0, A series of experiments on tbe impact of bodies meeting with
very small relative velocities would be very interesting, for with small
velocities the 6tres.s©s would diminish, and if these did not exceed those
corresponding to elastic limits some of tbe causes of the dissipation of
energy would be eliminated, and it is possible that the value of e might
be considerably increased.
We find, too, from experiment that bodies require time to recover even
fiwm small strain, so that, if the rise and fall of the stress is very rapid,
there may bo dissipation of energy in cases where the elastic limit for
slowly varying forces is not overstepped.
Hodgkinson gives the following formula for the value of e^^, when two
different bodies A and B collide, in terms of the values of «aa ^^'^ Lhe
114
PROPERTIES OF MATTER.
collision between two bodies each of matenal A and ^bb* ^^^ value for the
collision between two bodies each of material B.
•ab —
£aA I ^BB
71 9i
and he finds this formula agrees well with his experiments.
The following considerations would lead to a formula giving e^^ in
terms of e^A ^^^ ^bb* Hertz has shown that the displacements of the
bodies A and B in the direction of the common normal to the two surfaces
over which the bodies touch are proportional to
Iz^'andlz^'
1i 7,
where (t,, <t, are the values of Poisson's ratio for the bodies A and B and
5^,, 7, the values of Young s modulus. Now the stresses are equal, so that,
assuming that the quantities of work done on the two bodies are in the
ratio of the displacements, then, if E is the whole work done,
l^E
1-
^E
1 -
^ +
1-<T.
land
1-
r ' 1 _ ^ *
9i 7,
9i <7«
will be the amounts done on the two bodies. Now the first body converts
1 - 6*^ and the second 1 - e^oB of this work into heat ; hence the energy
converted into heat will be
(1 - «•**)
LzZl' + (i_«.3,)1z3'
and this must equal
l-<^l-<r.-
(1-«'ab)E
£
A A +
» BB
l-cr»
hence
^ AB —
!7i 7f
The following table of the values of e is taken from Hodgkinson's
Report to the British Association^ 1834 :
Cast-iron balls .
Cast-iron — lead .
Cast-iron — boulder stone .
Boulder stone— brass .
Boulder stone — lead .
Boulder stone — eltn .
Blm balls ....
3oft brass (16 pt. Gn. and 1 pt. tin)
Bell metal (16 pt. Gu. and 4 pt. tin)
Lead
Lead—elm
IBlm— sof t brau • • • •
•66
•13
•71
•62
•17
•56
•60
•36
•59
•20
•41
•52
Clay .
Clay — soft brass
Glass .
Cork .
Ivory .
Lead — glass
iSoft brass— glass
Bell metal — glass
Cast-iron— glass
Lead — ivory
Soft brass— ivory
Bell metal— i?oi7
•17
•16
•94
•65
•81
•25
•78
•87
•91
•44
•78
•nr.
IMPACT. 115
The case where a permanent deformation is produced has recently been
investigated by Vincent (Proceedings Cambridge Philoaophical Society^
vol. X. p. 332). The case taken is that of the indentation produced in lead
or pamffin by the impact of a steel sphere. He finds that the volume of
the dent is pi opoi-tional to the energy (-f the sphere just before impact;
that during the impact (t.6., while the lead is flowing) the pressure bt tweon
the sphere and the lead is constant and varies from 6xl0^tol3x K/dync s
per square centimetre for diffVrent specimens of lead ; for paraiiin the
corresponding frersure is about 10* dynes per square centimetre.
CHAIT'ER XL
COMPRESSIBILITY OF LIQUIDS.
Contents.— Changes in Volume of a Tube nndcr Internal and External Prcsiure—
Measurements of Compressibility of Liquids by methods of Jamin, Regnault,
Buchanan and Tait, Amagat— Comprcsfibility of Water— Effects of Temporatore
and Pressure— Compressibility of Mercury and other Liquids— Tensile Strength
of Liquids.
The fact thai water is compressible under pressure was established in 1762
by OantoOy aacl ainoe then measurements of the changes of volume of
liquids under pressure have been made by many physicists.
The problem is one beset with experimental difficulties, some of which
may be illustrated by considering the case of a liquid inclosed in a vessel
such as a thermometer; when pressure is applied to the liquid, the
depresi^ion of the liquid in the stem will be due partly to the contraction
of the liquid uadar praiMune and partly to the expansion of the bulb of the
thermoineter. Iq cmr, then, to be able to determine from tho depression
of the liquid the compressibility of water we must be able to estimate the
alteration in volume of the tube under pressure. We shall therefore
consider in some detail the alteration in volume of a vessel subject to
internal and external pressure. We shall take the case of a long cylindrical
tube with flat ends exposed to an external pressure p^ and an internal
pressure p^. The strain in aueh a cylinder has been shown by Lam6
to be (1) a radial dii^laoement p given by the equation
p = Ar + -
r
where r is the distance of the point under consideration from the axis of
the cylinder and A and B constants, and (2) an extension parallel to the
axis of the cylinder.
The radial displacement p involves an elongation along the radius equal
to dp/dr and an elongation at right angles to p in the plane at right angles
to the axis of the cylinder equal to p/r. Let the elongations along the
radius, at right angles to it and to the axis of the cylinder, and along the
axis be denoted by 6, /, g respectively, and let P, Q, R be the normal
stresses in these directions ; then by equation (1), p. 72, we can easily prove
(0
where k is the bulk modulus and n the coefficient of rigidity.
COMPRESSIBILITY OF LIQUIDS. 117
Since e = ^ancl/=e
dr r
we have e = A-^. /«A + ^
r r
Thus the radial stress is equal to
If a and h are respectively the internal and external radii of the tube,
then when r = a the radial stress is equal to -j?^ and when r = 6 the radial
stress is equal to -/>|, hence we have
-,..»a4"(a- »).(*- 1), ,.)
The whole force parallel to the axis tending to stretch the cylinder is
hence the stress in this direction is equal to
vo^Po — if^^Pi
The stress parallel to the axis is, however, equal to
(».V"W(*4>
hence we have
From (2), (3) and (4) we get
Since the radial displacement is Ar + — , the internal volume of the
tube when strained is tr( a + Aa + — J*/(l +^)
where I is the length of the tube ; hence, retaining only the first powers
of the small quantities A, B and g, we have, if Sv. is the change in the
internal volume,
* \(6«-a«) ife^6*-a* n J
118 PROPERTIES Of MATTER.
and if 2r, is the change in the external volume,
Methods of Measuring: Compressibility of Liquids— There are
two cases of special importance in the determination of the compressibility
of fluids : the flr.^t is when the internal and external pressures are equal ;
in this case p^ =/7p and we have
Thus the diminution of the volume is independent of the thickness of the
walls of the tube. Some experimenters have been led into error by supposing
that, if the walls of the tube were very thin, there would be no appreciable
diminution in the volume of the tube. If the vessel had been filled with
liquid which was subject to the pressure p,,, the diminution in the volume
(>f the liquid would be iraHpJKy where K is the bulk modulus of the liquid.
The diminution of volume of the liquid minus that of the vessel is
therefore
•ra'^^-l)
thus by experiments with equal pressures inside and out, which was
Regnault'd method, we deter uiine
1 1
60 that to deduce K we must know k.
Another method^ used by Jamin, was to use internal pressure only, when
the apparent change in the volume of the liquid is the sum of the changes
of volumes of the liquid and of the inside of the vessel. Jamin thought
that he determined the change of volume of the vessel by placing it in an
outer vessel full of water and measuring the lise of the water in a gradu-
ated capillary tube attached to this outer vessel ; by subtracting this change
in volume from the apparent change he thought he got the change in
volume of the liquid without requiring the values of the elastic constants
of the material of which the vessel is made. A little consideration will
show, however, that this is not the c€ise. Let ^v be the change in the
volume of the liquid, di\ the change in the internal volume, 2r, that in the
external volume; it is 5 1?, that is measured by the rise of liquid in the
capillary tube attached to the vessel containing the tube in whioh the
liquid is compressed.
Observations on the liquid inside the tube give
St? + dv^
if we subtract Jamin*s correction we get
hv + h\ - Sr,
substituting the values of ^i\ and d-, when p^^oy^e find
COMPRESSIBILITY OF LIQUIDS. 119
Hence, nfter applying Jamin's correction, we get jr(i'/ji^,l ^ - j- j thesame
quantity aa was determined by Regnauit's method, eo tliat to get K by
Jamin's method we require to know k.
The oppnrntus used by Regnault in his experiments on the compressi-
bility of liquids (J/emoi"jM de FlngtiCut de France,
vol. zxi. p. 420) was sioiilar to that represented in
Fig. 82, The piriometer was filled with the liquid
whose compre-ssibility wna to Ikj measured, the
prcate-st care being taken to get rid of air-bubbles,
Ttie h'qnid reached up into the graduated stem of tbe
piezometer, the volume betiveen successive marks on
the stem being accurately known. Tlie piezometer
was placed in an outer vessel which was tilled with
watei' nnd the whole system placed in a. large tank
filled with water, the object being to koep the
temperatura of tbe system constant. The tubes
shown in the system were connected with a vessel
full of compresGed nir, tbe pressure of which wiis
measured by a carefully tested manometer ; the
tubes were so arranged that by turning on the
proper taps pi'essure could bo applied (I) to the
outside of the pie£ometer and not to the inside ; {i)
simultaneously to the outside and the inside ; (St) to
the inside and not to tbe outside. Tbe picEonieter
used by Regnaull was in the form of a cylindrical
tube with heniiapberical ends. Fur simplicity let
us take the case (represented in the figure) of a pieEometer in the form of
a cylinder with flat ends, to which the foregoing investigation applies.
If w,, w,, u, ore the apparent diminution in the volume of the liquid in
tbe three cases re^pL-ctively, the pressure being the same, we have by the
prcceditig theory
_ ^o?vi n
I'lo. S2.
Hence w, + w, = wj
a relation by which we can check to some extent the validity of the
theoretical investigation. Such a check ia very desirable, aa in this investi-
gation we have assumed that the material of which the piezometer is made
is iBotropic and that the walls of the piezometer are of uniform thickness,
conditions which are very difficult to fulfil, while it is important to
ensure that a failure in any one of them has not been sufficient to
appreciably impair the uccui'acy of the tbeoi-etical investigations, Ilegnault
in his investigations adopted Lamp's assumption that Poisson'e ratio ia
equal to 1/1; ou this assumption n'=jk,Bo that the measurement of w^
130
PROPERTIES OF MATTER.
gives the value of k, and tlicn the measurement of w, the value of E, the
bulk modtiluE for the liquid. This -was the method adopted by Renault.
It lA, however, open to objection, la the first place, the delei-mlnn lions
which have been made of the value of Puisson'a ratio for glass range from
•33 to ■22, instead of the assumed value '25, while, seoondly, the equntion by
which h is determined from measurements of ui, is obtained
on the assumption of perfect uniformity in the material
which it is difficult to verify. It is thus desirable to
determine k for the material of which the piezometer is
made by a separate investigation, and then to determine thia
compressibility of the liquids by using the simplest relation
obtained between the apparent change in volume of the liquid
and the pressure ; this is when the inside and outside of the
piezometer are exposed to equal pressures. The most direct,
and probably the most accurate, way of finding k for a solid is
) the longitudinal contraction under pressure. An
which enables this to be done with great
accuracy is described by Amagat in the Journal de Phyeique,
Series 2, vol, viii. p. 869. The method was lirst used by
Buchanan and Tail. Another method of determining k for
a solid is to make a tube of the solid closed by a graduated
capillary tube as in Fig. 83. The tube and part of the
capillary being filled with water, a tension P is applied to the
tube, the tube stretches and the internal volume increases, the
inciease in volume being measured by the descent of the liquid
ia the capillary tube ; if v is the original internal volume, iv
the incrense in this volume, then wc see by the investigation,
p. 72, that
^ P
If we have found h, then K can be found by means of the
piezometer.
If we can regard the compressibility of any liquid, eay
mercury, as known, the most accurate w,iy of finding tha
compressibility of finy other liquid would be to fill the
pieeonietcr first with mercury, and determine the apparent
change of volume when the inside and outside of the
piezometer are exposed to the same pressure; then Gil the
piezometer with the liquid and again find the appBirent change
in volume. We shall thus get two equations from which we
can find the value of K for the liquid and it for the piezometer.
Results of Experiments- — The results of experiments made by
dillfirent observers on the compressibility of water are given below.
Eegnault." — Temperature not specified ; pi-essures from 1 to 10 atmo-
spheres —
compressibility per atmosphere = 0.000048,
• Hfii'oirt* dt VlntliM (It Franer, Tol. isL p. 420.
COMPRESSIBILITY OF LIQUIDS.
1^1
ORAaSI.^
PAGLIANIand VICENTINI.f
RONTGEN and SCHNEIDER. I
Temp.
Compreuibility
per atmosphere.
Temp.
Compressibility
per atmosphei ..
Temp.
Compressibility
per atmospheie.
00
1-5
40
max. density
pt.
10-8
13-4
18-0
25.0
34-5
43
530
503 X 10-7
515
499
480
477
462
456
453
442
441
0.0
2-4
15-9
49-3
61-1
66-2
77-4
99.2
503 X 10-7
496
450
403
389
389
398
409
9
18-0
512x10-7
4.S1
462
Tait§ has found that the elfceb of temperature and pressure, for
temperatures between 6° 0. and 15° C. to pressures from 150 to 500
atmospheres, may be represented by the empirical formula
r«-«
pv^
= 0-0000489 - 0-00000025^ - 0.0000000067p
where v is the volume at IP C. under the pressure of p atmospheres and v^
the volume at tP under one atmosphere. Thus the compressibility diminisho:5
as the pressure increases.
The numbers given above, from Grassi's experiments, indicate that
water has a maximum compressibility at a temperature between 0° and 4^0.:
this result has not, however, been confirmed by subsequent observers. The
results of Pagliani and Yicentini indicate a minimum compressibility at
a temperature between 60° and 70° 0.
The results of various observers on the compressibility of mercury are
given in the following table :
Observer.
Golladon and Sturm |t ,
Aim^f . . . •
Kegnault**
Amaury and Descampstt
TaitU . • . .
Amagat§§
Be Metzl
Mean
Compressibility
per atmosphere.
85-2 X 10-'
390 X 10-'
35-2 X 10-'
38-6 X 10-»
860x10-'
390x10-'
37-4x10-'
37-9 X 10
-7
The compressibility of mercury, like that of most fluids, increases as the
• Grass!, Annalet de Chimie et de Physique [3], 31, p. 437, 1851.
t Pagliani and Vicentini. Nuovo Cimento [3], 16, p. 27, 1884.
t Rontgen and Schneider, Wied, Ann., 33, p. 644. 1888.
§ Tait, Properties of Matter, Ist ed. (1885), p. 190.
II CoUadon and Sturm, Ann. de Chimie et dc Physique, 36. p. 137, 1827.
5 Aim6, Annates de Chimie et de Physique [3], 8, p. 268, 1843.
♦• Kegnault, Mimtnres de VJnstitut de France, 21, p. 429, 1847.
ft Amaury and Descamps, Compt. Rend., 68, p. 1564, 1869.
iX Tait, Chailengtr Report, vol. ii. part iv.
§§ Amagat, Jowmal de Physique \2\ 8, p. 203, 1889.
De Mets, Wied, Awn., 47, p. 731, 1892.
122
PROPERTIES OF MATTER.
temporature increases. According to De Metz, the compressibility at (" 0.
b given by
87-4 X 10-' + 87-7 xlO-'«<
Tlie compressibilities of a number of liquids of frequent occurrence are
given below.
Fluid.
Compressibility per
atmosphere.
Temp.
Observer.
Sea- water ....
436x10-''
17 •5"*
Gras^i
Ether • •
. I 1156x10-'
0*
Quincke
%$ • •
1110x10-'
0"*
Grass!
Alcohol
828x10-'
0"*
Quincke
♦» • <
959x10-'
17 -i"
II
»» • '
828x10-'
7-3'
Grassi
Methyl alcohol ,
913x10-'
13 5'
II
Turpentine .
682x10-'
0*
Quincke
It • '
779x10-'
18-6
fi
Chloroform .
625x10-'
8-6"
Grassi
GUcerine .
252x10-'
0"
Quincke
Olive oil .
486x10-'
0*
fi
Carbon bisalphide
639x10-'
0"
II
II II *
038x10-'
17*
fi
Petroleum .
050x10-'
0*
ft
«i ...
746x10-'
19-2'
*i
water »
§
Quincke's paper is in Wiedemanns Annalen, 19, p. 401 , 1883. References to
the papeis by the other observers have already been given. An ei^ten-
sive series of investigations on
the compressibility (^ solutions
has been made by lU>ntgen
and SchneidiBr {Wied, Ann,^ 29,
p. 165,andSlyp. 1000), who have
shown that the compressibility
of aqueous solutions is less
than that of water. For the
details of their results we must
refer the reader to th^r paper.
Tensile Strettsrth of
Liquids* — Liquids from which
the air has been carefully ex-
pelled can sustain a Considerable
pull without rupture. The best
known illustration of this is
wa tQt vapour, *^® sticking of the mercury at
' the top of a barometer-tube.
If a barometer-tube filled with
\j mercury be carefully tilted op
Fio. 84. to a vertical position, the mer«
. cury sometimes adheres to the
top of the tube, and the tube remains filled with mercury, although the
length of the column is greater than that which the norinal bunometrio
pressure would support, and the extra length of mercttry is in a state of
tension. Another method of showing that liquids oan woMask tensioii
COMPRESSIBlLl'n' OF LIQUIDS. les
wittiout rupture is to use a tube tike that la Fig. 84, filled vitii water uud
the vapour of w&ter, and [ram which the air has lieen carefully expelled
by boiling the water and driving the air out by the steam.* If the water
occupies the position iiidiciited io the figure, the tube mounted on a bo.ird
may be moved rapidly forward in the direction of the arrow, and then
brought suddenly to rest by striking the board against a table without the
water coli'mu breaking, although the column must have experienced a
conaidetiible impulsive tension. If the column does break, a small bubble
of air can generaily he observed at the place of rupture, and until this
bubble has been removed Ihe column will break with great ease. On the
removal of the bubble by tapping, the column can again sustain a con-
iddenible shock without lupliire.
Piofes-ior Osborne Reynuld.s used the followiDg method for measuring
the tension liijuida would stand without brenking. ABCD, Kg. 85, is a
glass U-tube, closed at both ends, containing air-fi-ee liquid ABC and
vapour of the liquid CD. The tube is fixed to a boai-d and whirled by a
lathe about an axis a little beyond the end A aod perpendicular to the
plane of the ba-u-d. If CE is an arc of an circle with centre 0, then when
ihe board is rotating the liquid KA is io a state of tension,
the tension increasing from E to A, and being easily -^^^
calculable if we know the velocity of rotation. By this
inetbod Professor Osborne Eeynolda found that watei' could ^|i;
sustain a tension of 72-5 pounds to the square inch witliout
ruptui-e, and Professor Worth ington, using the same method,
found that alcohol could sustain lUi and strong sulphuiic
acid 173 pounds per square inch. This method meaeuies the
stress lirjuids can sunttia without rupture. Betthelot has
used a method by which the strain ia measured. The liquid
freed from air by leng boiling nearly filled a straight thick-
walled glass tube, tile iTst oF the space being occupied by the
vapour of the liquid. The liquid was slightly heated until it ~ o
occupied the whole tube; on cooling, the liquid continued for i'iq. gj.
some time to fill the tube, finally bi'Oaking with a loud
metallic click, and the bubble of vapour reappeared : the length of this
bubble measured the exienston of the liquid. M. Berthelot in this way
gftt extensions of volume of J/120 for water, 1/03 for alcohol, and 1/59 for
ether. Professor Woithinglon has improved this method by inserting iu
the liquid an ellipsoidal bulb filled witli mercuiy and pi-ovided with a
nai-row giaduated capillniy stem ; wlieii the liquid is in n stale of tension
the volume of the bulb expands and the meicuiy sinks iu the stem ; fmm
the amount it sinks the leusiim can be measured. The extension was
measured in the same way as in Beitbelot's experiments. In this way
Professor Worthington showed {P/.il. Trava. A. 18:12, p. 3!>5) that the
absolute coefficient of volume elasticity for alcohol is the same for
extension as for compression, and is constant between pi-essurea of +12
and —17 atmospheres.
* Df ion anil .To!» [Phil. Tmn». Jl. 1805, p. BC8) have shown Ihat air or other franes
belli in foliition do not iilTvcl these expuilments. The boillDg is prubalily eHivaci'Joi
oul^ iu ri'moving babbtck ur ttee giu^a.
CITAPTEE XIT.
THE RELATION BETWEEN THE PRESSURE AND VOLUME
OF A GAS.
CoKTENxa— Bojie's taw— Deviations from Dovle'a Law— Be? n null's Esperlinrnts—
Amag&fa EiperimenlB -Expo rim BQts at Lliw Prasjuria— Van der Waals Eqoation.
In this cliapter ve rIihII confioe oiir^IveB to the discusGion of the relation
between the pressure and the voLiime of a gas when the temperature ia
conetant and no change of state taken place ; the liquefaction of gasea
will be dealt with in the volume on Heat.
The relation between the pressure and the volume of a given mass of
f;a£ was first stated by Jioyle in a paper communicated to the Royal Society
in 1661. The esperiment which led to this law is thus described by him.
" We took then a long glass tube, which by a dexterous hand and the help
of a lamp was io such a manner crooked at the bottom, that the part
turned up was almost parallel to the I'est of the tube, and the onfice of
this shorter leg of the siphon (if I may so call the whole instrument) being
hermetically seale<l, the length of it was divided into inches (each of which
was subdivided into eight part^) by a straight list of paper, which, con-
taining those divisions, was carefully pa.sted 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 j ust 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
cyhndec should be of the name 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
strengthen the included air, and continuing this pouring in of quicksilver
till the air in the shorter leg was by condensation reduced to take up but
half the space it possessed (I say, possessed not filled) before, we cast our
eyes upon the longer leg of the glass, on which was likewise pasted a list
of paper carefully 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 . , . the same air
being brought to a degree of density about twice as great as that it hod i
before, obtains a spring twice as strong as formerly." Boyle made a series
of raeasurenients with greater compressions until he had reduced the
volume to one quarter of its original value, and obtained a close agreement
between the pi-easiu-e observed and " what that pressure should be according
to the hypothesis that supposes the pressures and expansions* to be in J
reciprocal proportions." Although Mariotte did not state the law until J
fourteen years after Boyle had published his discovery, " the hypothesis ■
u Ent^'liati.
THE PRESSURE AND VOLUME OF A OAS. 125
tbat supposes the pressures and espnnsions to be ia reciprocal proporliotis "
is often on the Continent called Maiiotte's Law.
If V is the volume of a given mass of gas unci j> the pressure to which
it is subjected, then Boyle's Law states that wtien the temperature is
coDstiint
/)ti = eciiistaut.
Another way oE stating this li;
pressure />,
I is that, if p is the density ot a gaa undei
P-Kf.
where R is a constant whtsD the tstnperature is coDstant. Later reseorclies
made hy Chai'W and Oay-Liissac bave shown bow K varies with the
teuipeiuture and with the nntiire of the gas. These will be described in
the volume OD Heat J it will suffice to say here that the pressure of a perfect
gas is given by the equatiun
p = KNT,
where T is the absolute temperatura, N the number of molecules of the gas
in unit volume, and K a constant which is the same for all gases.
From the ei|uiition pi- = c we see that if Ap, Ar are con'esponding incre-
ments in the preci^ure and volume of a gas whose teuiperature is. coniitaut.
=Pi
but the left-band Kide is by definition the bulk modulus of elasticity,
hence the bulk modulus of elasticity of a gas at a constant temperature ia
equal to the pi-essure.
The woik I'equired to diminish tlie volume ot a gas by in is plv ; the
work which has to bo done to diminish the volume from t>, to 0, is there-
fore
', since by Boyle's Law p = c/i', when the tcmperatui
« that in this case the work is
?1 r
where p. is the pressure when the volume is ti,.
Deviations from Boyle's Law.— The lii-st to estabtieh in asitis-
factoi'y manner the exislenpe in some gases, at any rate, of a departure from
Boyle's Law was DespreU, who, in 1827, enclosed a number of difl'eront
gnaes in barometer -tubeij of the same length standing in the same cistern.
The quantity ot the different gases was adjusted so that initially the mercury
stiiod at the same height in the diSetent tubes; pressure was then sp[)!ied
to the merciii'y in the cistern, so that mercuiy whs forced up the tubes.
It wBs then found that the volumes occupied by the gases were no longct
isfj Properties d^ lifAffEtt.
equal, the volumes of carVonic acid and ammonia were less than Ihnt of
air, while that of hj-drogen was greater. This showed that some nf tlifi
gases did not obey Boyle's Law ; it left open the question, however, as to
whether any gases did obey it. The next great advarce was made by
Regnault*«ho in 1847 settled the question as to the behaviour of certain
gase* for pressures between 1 and about 30 atmofpheie?. Upgcauk's
method was to start with a certain quantity of gas occupying a volume v
in a tube (^ealed at the upper end, and with the lower end opening into a
closed vessel fidl of mercoiy, and then by pumping meTCury up a long
mercury column rising from the cloned vessel to increase the pressure until
the volume was halved. By ueosuiing the dilTerence of height of
mercury in the column and in the tube the prepsuro lequired to do this
c^ulil be determined. Air under this pressure was now pumjed into the
closeil tube until the volume orcupied by the gas was again v ; mercury
was again pumped up the column until (he volume had again been halveil
end a new reading of the pressure taken ; air wne pi'mped in again until
the volume wa<i again v, and then the pressui'e increased again until the
volume was lialved. Tn this way the values oF pe ht a series of different
pressures could be compared. Tbe results are shown in the following
table; /■, la given in millimetres of mercury, j>^t^ \s the value of pv at Ihe
pressure givfn in the table, ^i^ri the value at doulja this pressure. Tbe
esperimenta were made at lerapei-alures lietween 2° C. and 10' 0.
AIR-
MITROaEN.
CAEBOS.C ACID.
HTDROCES. 1
Po
P.'Wp,'',
To
F,v,Jp,v^
P.
I>="Jpi".
p.
P.*-.!Pi^
73872
1-OOHU
753-B6
1-001 01 3
764-03
1-007597
206S-20
1 -002708
1158 -43
1-001074
1012313
4319-05
1 -003836
2159-22
i-ooion7
2164-81
8770-16
1-004286
8030-22
1-0O19G0
3186-13
1-028494
3989-47
0-996961
D33fl-lt
i-ooesee
49G3'S2
l-002a63
4879-77
1-045825
5845-18
0-996121
11*72 1)0
1006619
5957 sa
1-003271
6820-22
1-068137
7074 -iie
0-994697
1-003770
8393-68
1-08427S
0H7-6!
0-993258
8628 64
1-004768
9620-06
1'0»9830
10361-88
9767-42
1-005147
1098M2
1-00C45S
I
It will be Been from these figurps that between preraurea of from about
1 to 30 atmoBpheros the product pv constantly diminishes For air, nitrogen,
and carbonic acid, as the pressure increases, the diminution being most
marked for carbonic acid ; on the other hand in hydrogen pv iccreases with
the pressure. Natforer, who in 1850 published the results of expeiimeotB
on the relation between tbe pressure and volume oE a gas at very high
pressure, showed that after passing certain pressuiespv for air and nitrogen
begins to increase, so that po has a minimum value at a certain pressure ;
after passing this pressure air and nitrogen resemble hydrogen, and pv
continually increa.'es as the pressure increases. This retult was confirmed
by the researches of Amagat imd Cailletet. Eacii of these physicists worked
at tbe bottom of a mine, and produced their pressures by long columns of
mercury in a tube going up the shaft of the mine. Amagat's tube wu
SOO metres long, Cailletet's 250. Amagat found that the minimum value
of pv between 18° and 22' 0. occurred at the following pressures:
* Mimoirci dt I'/mtilul dt Franct, vol. III. p. 329.
THE PRESSURE AND VOLUME OF A GAS.
The results of liU espei i-
meats are exhibited in the Fal-
lowing figutes ; the ordinates ure
the values oipv, and the abscissie
the pressure, the unit of pressure
being the atmosphere, whicb is
the pressure due to a column i)f
mercuiy 760 mm. high at 0" C,
and at the latitude of Ftaia.
The numbers on the curves indi-
oste tbe tempernture at which
the experiments were nmdc. It
will be noticed that for nitrogen
the pressure at which pv is a
minimum diminishes us the tem-
perature increases, so much so
that at a temperalure of about
100° 0. the minimum value of
pv is hardly noticeable in the
curve. Thia ia shown clearly by
the following results given by
Amagat:
FlO. 86,-Elli)]en«,
au-i-c
M-i- C.
7.va- (.■
iiBvi:
pe
j»'
1"
pi
V
aOmetrcB .
2745
2875
3080
3S30
3575
60
2740
2875
SIOO
S360
3910
100 „ ...
2790
2B30
3170
3445
3B95
200
3075
3220
316S
3760
4020
320
3875
3915
i'JlO
447G
Ama^t extended his experiments to very mud) higher pressures, and
obtained the results shown in tbe following table; the temperature was
15° 0., and/'C was equni to 1 under the pressure of 1 atmosphere:
Air. Mitrogen.
p> pv
Oiygon.
750
1000
1500
2000
2500
3000
1'6.10 l'8ft65
l'i'71 2-oa2
•i-im 2-814
3-132 3 228
3-672 3-787
4-203 4-333
1-735
2-238
2-746
3-235
3-705
1-683
2-0)6
2-323
2-617
2-892
A question of consider.tble impoi-tiinru in these experiments, and one
which we have hardly sufficient information to answer (ati^fiictorily, arisps
from tbe condensation of gas on the walls of the manometer, and possibly
K penetration of tbe gas into the tubstance of these walls. It is well known
I SI
PROPERTfES OF MATTER.
t
tlint when we attempt to exhaust a g\ass vessel a considerable amount of
gnd caaten off the giiiss, and if th« vessel contains pieces of metal the
(iifficulty of getting ii viicuiim is still further iccreased, as gna for some time
continues to como from tho met,-)!. !Uucli of this is, no doubt, condensed on
the surface, but when we
remember that water can
be forced through gold it
tieems not improbahle that
at high pressure the gaa
viLiy be forced some dis-
tance int) the metnl iis
well as condensed on its
surface.
Boyle's Law at Low
Pressures. — Tlie diiii-
ciilty nrisiii!,' fiijm giiscom-
itig off il.e walls oF the
mil no meter hecoaies spe-
cially acute when tho pres-
sure is low, as here the
deviations from Boyle's L'lw are so small that any trilling error may
completely vitiate tha exi>eriments. This ia probably one of the reasons
why our knowledge of the relation tetween the pressure and volume of
Riisea at low pressures is bo unsatisfactory, and the results of different
experiments so contr:iiliftory. According to MeiideleeH", nnd his result has
been coutirmed by 'Fufli.-.', j>b for iiir at piossures below an atmosphere
, diminishes as the pressure
diminishes, the value of po
changing by about 3-5 per
cent, between the pressure
of 76U and 14 mm. of
mercury. If thi? is the
i^ise, then pv for air has a
iiiaxiiiium as well as a mini-
mum value. On the other
liiiiid, Amagat, who made
a ?cries of very careful
experiments at low pres-
sures, was not able to detect
itny departure from Boyle's
Law. According to Bohr,
and h\s result has been
confirmed by Baly nnd Ramsay, the law connecting p and v for oxygen
changes at a pressure of about '75 mm. of mercury. It has been
suggested that this is due to the formation of ozone. The recent
investigations by Lord Rjiyleigh on the relation between the pressure and
volume of gases at low pi-cssui-as do not show any departure from
Boyle's Law even in tho case of oxygen.
The results of Amiigat's experiments are in fair accordance with
the relation between p and v, arrived at by Van der Waals from
the Kinetic Theory of Oases, This relation is expressed by the
equation
THE PRESSURE AND VOLUME OF A GAS.
1S9
(p+^y-b)=-RT
^ here a, 5, R are constants and T is the ahsolute temperature. Thus p in
Boyle's equation is replaced by p + a/v^ and vhy v-b. The term a/v* or
Of)', where p is the density, arises from the attractions between the molecules
of the gas; this attraction assists the outside pressure to diminish the
volume of the gas. If we imagine the gas divided by a plane into two
portions A and B, then ap' is the attraction of A on B per unit area of the
plane of separation ; it is the quantity we call the intrinsic pressure in the
pv
Fie. S9.
theory of Capillarity {see chap. xv). The v of Boyle's Law is replaced by
v-b. Since the molecules are supposed to be of a finite although very
small size, only a part of the volume '' occupied " by the gas is token up
by the molecules, and the actual volume to be diminished is the difference
between the space ''occupied '' by the gas and that filled by itB molecules;
b is proportional to the volume of a molecule of the gas.
Van der Waals' equation may be written :
so that if
we have
(^..«) (l-5).EI
pv = y And -.=p = «,
(t/ + ax){l-bx) = 'RT
ThuF, if the temperature is constant, the curve which represents the rela*
tion between pv and p is the hyperbola
(y + €us) (1 - bx) »» constant.
ISO PROPERTIES OF MATTER.
The asymptotes of this hyperbola are y + cix^o, l^bx'^O, There is a
minimum value of pv at the point P (Fig. 89) where the tangent is horizontal.
The value of x at this point Is easily shown to be given by the equation
a(l-6a;V = 6RT.
If 6RT/a is less than unity there is a positive value of x given by this
equation. This corresponds to the minimum value for pv in the cases of air,
nitrogen, and carbonic acid. We see, too, from the equation that as T
increases x diminishes, that is, the pressure at which the minimum value
of pv occurs is lower at high temperatures than at low. This agrees with
the results of Amagat's experiments on nitrogen. When T gets so large
that 6RT/a is unity x = 0; at all higher temperatures it is negative — i.e.,
P is to the left of the vertical axis, there is thus no minimum value of pv^
and the gas behaves like hydrogen in that^n; continually inci eases* •& the
pressure ina-eadtui.
CIIAFfEH Xiri.
REVERSIBLE THERMAL EFFECTS ACCOMPANYING
ALTERATIONS IN STRAINS-
COHTESTH.— ApplicaL
ies— Ratio ot Adiabalic to Isotliei
Ip the coefficients of elasticity of a siibstince depend upon the tempera-
ture an altemtion in tbe state ol strain oE a body will be accompanied
by & change in ita temperature, IE the body is stiffer at a high tem-
pei-ature tbnn at a low one, then, if the strain ia increased, there will
be an increase in the temperature of the strained body, while if the body
ia elifftrata low temperature than at a high one, there will be a fall
in the temperature when the strain iBioereased, Thua, if the changes in
strain in any experiment take place so rapidly that the heat due to these
changes has uot time to escape, the coefficients of elasticity determiaed
by these experiments will be larger than the values determined by a
method in which tbe strains are maintained constant for a sufficiently long
time for the temperature to become uniform; this follows from the fact
that the thermal changes which take place when the strains are variable
are always such as to make the body stifier to resist the change in strain.
In those experiments by which the coefficients of elasticity are determined
by acoustical methods — i.e., by methods which involve the audible vibration
of the substance (see Sound, p. 125) — the heat will not have time to diffuse,
and we should expect such methods to give higher values than the statical
ones we have been describing. "When we calculate the ratio of the two
coefficients we find that the theoretical diflerence is far too small to
explain the considerable excess of tbe values of theconstanls of elasticity
found by Wertheim by acoustical methods over those found by statical
methods.
We can easily calculate by the aid of Thermodynnmica the thermal
I to a change of strain. To fix our ideas, suppose we have two
o maintained at a temperature Tj, the other at the tempera-
i temperatures ai-e supposed to ho absolute temperatures, and
To to be less than T,. Let us supposo that we have in the cool chamber a
stretched wire, and that we inci-onse the elongation e by £e ; then if P is the
tension rei^uiratl to keep the wire stretched, the woi'k done on the wire is
FalSe
vhere a is the area of the cross-section and I the length of the wire. Tfow
transfer the wire with its length -unaltered to the liot chamber, and for
simplicity suppose the thermal capacity of the wire exceedingly small, so
that we can neglect the amount of heat required to heat up the wire;
if the stiffness of the wire changes with temperature the tension P*
ret^uiitid to keep it stratcbed will not be tbe same as P. Let the wire
effects due
chambers,
ture T,; thes
132 PROPERTIES OF MATTER.
contract in the hot chamber until its elongation dimioishes by ie^ then the
work done by the wire is
P'aWe.
Now transfer the wire with its length unaltered back to the cold chamber ,
it will now be in the same state as when it started. The work done by
the wire exceeds that done on it by
(F-P)aB«;
hence the arrangement constitutes a heat engine, and since it is evidently
reversible it must obey the laws of such engines. These engines work
by taking heat ^H from the hot chamber and giving ^h out in the colder
chamber, and from the Second Law of Thermodynamics we have
1\ T, T,-T,
Now by the Conservation of Energy
2H - 2A = mechanical work done by the engine
-(F-P)aZa«;
hence ih^T^^^^a
T.-T
lie
o
TMXylie
\^J-/« constant
Now ih is the amount of heat given out by the wire when the elongation
is increased by ie^ and al is the volume of the wire ; hence the mechanical
equivalent of the heat given out per unit volume, when the elongation is
measured by ie, is equal to
'(S);
'e constant
If this heat is prevented from escaping from the wire it will raise the
temperature, and if 20 is the rise in temperature due to the elongation
20, we see that
2e=?4f^'"'"""'x8. (1)
where p is the density of the wire, K its specific heat, and J the mechanical
equivalent of heat. We soo that this expression proves the statement
made above, that the tempei-ature change which tikes place on a change in
the strain is always such as to make the body stiflen to resist the change.
We can i-eadily obtain unothoi eipicssioii i or cDj which is often more
convenient than that just given. In that I'oiuiuia wc havetiie expression
{dF/dT)e constant. Now, suppose that, instead of keeping constant all
through, we first allow the body to expand under constant tension ; if a; is
the coefficient of linear expansion for heat, and 5T the change in
temperature, the increase in the elongation is w^T ; now keep the
temperature constant, and diminish the tension until the shortening due
THERMAL EFFECTS ACCOMPANYING STRAINS.
to the diminution in tension just compensates for the lengthening due to (
the rise in temperatures. In order to diminish the elongation by uJT W8 f
must diminish the tension by qui£T where g U Young's modulus for the
wire, hence
JP = - q«,ST
hence by e^juation (1)
_ _ T„7wSa
But qde is the additional tension SF required to produt
hence the increase in temperature 00 produced by an
iF is given by the equation
I the elongation its,
ncrease of teDsion
(2)
Equations (1) and (2) are due to Lord Kelvin.
Dr. Joule {Phil. TraiM. cxlix. 1859, p. 91) has verified equation f2) liy
experiments on cylindrical bars of various substances, and the reeults of his
esperimentfi are given in the following table. The changes in temperature
were meaauied by thermo-electric couples inserted in the bars.
T
P
.
K
«P
ts
WlCUlBlod.
Iron . .
280 '3
1-h
l-21xl0-»
■no
1-OBxlO'
- -1007
-■107
Hara «tecl .
2747
7'0
1-23 xia-'
■102
1-09x10'
-■1620
-125
Cast ii on .
2S2-3
6 01
lllxlO-»
■120
M0x10»
-■14S1
-■lis
Copper
an -2
8-95
i-Tisaxio-'
■095
108x10'
-■174
-■151
A qualitative experiment can easily be tried with a piece o! india-
rubber. If an indiorubber band be loaded sufficiently to produce a
considerable extension and if it be then warmed by bringing a hot body
near to it, it will contract and lift the weight ; hence the iudiarubber gets
BtifTer by a rise in temperature ; by the rule we have given, it ought to
increase in temperature when stretched, since by so doing it becomes
stiller to resist stretching. That this is the case can easily be verified by
suddenly stretching a rubber-band and then testing its temperature by
placing it against a thermopile, or even between the lips, when It will be
found perceptibly warmer than it was before stretching.
We can easily calculate what effect the heat produced will have on the
apparent elasticity if it is not allowed to escape. The modulus of elasticity,
when the change in strain takes place so rapidly that the heat has not
time to escape, is often called the adiabatic modulus.
Ratio of Adiabatic to Isothermal Elasticity.— Suppose we take
the case of a wire, and suppose the teusiou increased by £P, if the heat
does not escape the increase le in the elongation will be due to two causes
— one from the increase in the pull, the other from the increase in the
temperature. The first part is equal (o SP/7, where q is Young's modulus
134
PROPERTIES OF MATTER.
for steady strain ; the second part is equal to hdut where ^6 is the change in
temperature, oi the coefficient of linear expansion ; hence
but by equation (2)
hence
C6 = — + W
JKp '
q JKp
or
h
hF
1
9
(i)
rr.
JKp
But if j' is the adiabatic " Young's Modulus,"
q' a"P
1
w
rr.
JKp
(3)
It follows from this equation that l/q' is always less than l/q or q
is always greater than q, as we saw from general reasoning must be
the case. By equation (3) we can calculate the value of q'/q. The
results for temperature 15°C are given in the following table, taken from
Lord Kelvin's article on ** Elasticity " in the Enct/clopcedia Britannica :
Substance.
Zino
Tin
Silver
Copper
Lead
Glass
Iron
Platinum
p
K
7-008
•0927
7-404
-0514
10-369
-0557
8-933
-0949
11-215
•0293
2-942
•177
7-553
•1098
21-275
•0314
Cd
•0000249
•000022
•000019
•000018
•000029
•0000086
•000013
•0000086
^/lOii
8-56
4-09
7-22
12-20
1-74
6 02
18-24
16-7
fflq deduced
(rum etiuat. 8.
1-008
1-00362
r00316
1 ^00325
1-00310
1 -000600
1-00259
1-00129
Thus we see that in the case of metals q' is not so much as 1 per
cent, greater than q. In Wertheim*s experiments, however, the excess
of q determined by acoustical methods over q determined by statical
methods exceeded in some cases 20 per cent. This disci-epancy has never
been satisfactorily accounted for.
CHAPTER SIV.
CAPILLARITY.
CosTENTS.— Surface Tension and Surface Energy— Rise of Liqnj,! in a CapiUarj
Tnbe— Relation between Pressure and Curvjlure of a Surfaoe— Stubilitj o(
Cyliodiical Film— Attractions and Repnlsionaduo to Snr face TeoBion— Methods
of Meas urine Sarfaca Tension— Temperature Coefficient of Surface Tension-
Cooling of Film on Stretching— Tension o( rerj Thin Films— Vapoar PresBnra
over Carved a orf ace —Effects oE Contamination of a Surface.
TiiERB are many phenomena which show that liqui.Ia faehava as if they
were enclosad in a strelched membrane. Thus, if we take a piece of bent
wire with a flexible silk thread stretching from oce side to the other and
dip it into a solution of soap and wtiter ao aa to get the part between the
silk and the wire covered with a film of the liquid, the silk thread will be
drawn tight as in Fig. 90, just aa it would be if the film were tightly
4
Btretched and endeavouring to contratt so that its area should be aa smaA
Hs pos.<tib]e. Or if we take a framework with two threads and dip it into
the fioap and water, both the threads will ba pulled tight as in Fig. 91, the
hquid again behaving as if it were in a state of ten^ion. If we take a
ring of wire with a liquid film upon it and then place on the film a closed
loop of silk and pierce the flloi inside the loop, the film outside will pull
' the silk into a tircle as in Fig. 93. The efiect is again just the same as it
would be if the films were in a state of tension trying to assume aa small
an area as possible, for with a given circumference the circle is the curve
which hna the largest area ; thus, when the silk is dragged into the circular
form, the area of the film outside is as small as po£sihl&.
Another method of illustrating the tendon in the skin of a liquid is
to watch the changes in shape of a drop of water forming quietly at tba
end of a tube before it finally breaks away. The observation is rendered
136
PltOPERTIES OF MATTER.
much easier if the water drops ar« ulloweil to form in a mixture of
paniffiD oil and bisulphide of carbon, as tlie dioiis bF' laiger and form
more gradually. The shape of the drop at c>ri( -I i.,f i-^ --II' » n in 1' ig "IS
It we moust a tlun indiai'ubber uitniliuii < mi l 1 i i jj in 1 --ii-pend
it as in Fig. 94, and giadiinlly fill the vessel with water and watch the
changes in the shape of tlie iiiembi ane, these will be found to correspond
closely to thoee in the drop of water falling from the tube ; the stage
corresponding to that immediately preceding the falling away of the drop
is especially interesting; a very marked waist forms in the membrane at
this stage, and the water in the hag falls rapidly and looks as if it were
going to burst away ; the
membrane, however,
^__^ reaches another figure of
etjuilibriiim, and if no
more water is poured in
remains as in Fig. 94.
Again, liquids behave
US if the tension in Ibeir
outer layers was different
Fio, iio. for different liquids. Tliis
may eiasily be shown by
covering a white flat-bottomed dieh with a thin layer of coloured water
and then touching n part of its surface with a glass rod which has been
dipped in alcohol ; the liquid will move from tlie part touched, leaving the
white bottom of the dibli dry. This f,how6 that the tension of the water is
greater than that of the mixture of alcohol and water, the liquid bein*
dragged awaj' froir. places where the teneion is weak to places where it is
CAPILLARITY.
137
ThereiBone very impoi'taDt difference between tLe behaviour of ordinary
stretched eketic membranes and that of liquid films, for while the tension
in a membrane increases with the amount of stretching, the tension in
a liquid film is independent of the stretching, provided that this is not no
great as to reduce the thickness of the film below about five millionths
of a centimetre. This can be shown by the following experiment : bend
a piece of wire into a closed plane curve and dip this into a solution of
tOiip and water so as to get it covei'«d with a film, then hold the wire in
a nearly vertical position eo its to allow the liquid in the him to drain
down ; this will cause the film, to be thinner at the top than at the
bottom ; the difTerence in thickness is very apparent when the film gets
thin enough to show the coIoui-b of thin plates, yet though the film
I shows that the tension
very uneven thicknetis the equilibrium of the fill
is the same throughout,* for if the tension in
the thill part were greater than that in the -i
thick, the top of the film would drag the
bottom part up, while if the tension of the
thick part were greater than that of the thin
the lower part of the tilm would drag the top
part down.
Definition of Surface Tension.— Sup-
pose that wo have a film fatietched on the /)
framework AB<_;D, Fig. 96, of which the sides
AB, BC and AB ore fixed while CD is
movable ; then, in order to keep CD in
equihbrium, a force F must be applied to it
at right angles to its length. This force is
required to balance the tensions exerted by
each face of the film ; if T is this tension,
then
2T.Cr-F;
the quantity T defined by this equation is called the surface tension of the
liquid ; for water at 18°0. it is about 73 dynes per centimetre.
Potential Energy of a Liquid arising from Surface Tension.—
If we pull the bar CD out through a distance x, the work done is Fx, and
this is equal to the increase in the potential energy of the film, but
Fa = 2T.CDa;=Tx,(increa6e of area of film). Thus the increase in the
potentialenergy of thetilmisequaltoT multiplied by the increase in area,
BO that in consequence of surface tension a liquid will possess an amount of
potential energy equal to tho product of the surface tension of the liquid and
the area of the surface. Staiting from this result we can, as Gauss showed,
deduce the consequences of the existence of surface tension from the
principle that when a mechanical system is in equilibrium the potential
energy is a minimum. Suppose that we take, aa Plateau did, two liquids of
the same density, say oil and a mixture of alcohol and water, and consider
the equilibrium of a mass of oil in the mixture. Since the density of the
oil is the same as that of the surrounding fluid, changes in the shape of the
mass will not affect the potential energy due to gravity; the only change
* If tbe film it vertfnal tbe tensiaii at the top ia ver; slightly greater than that at
the bnttom, »o as to allow tbe diffciccce of tension to balanue tbe exceedingly small
weight q[ the filu).
I3« PROPERTIES OF MATTER.
in the potential energy will be tlie change in tlie energy due to surfaca
tension, and, by the principle just stated, the oil vill assume the ebape ill
which this pateotial energy is a minimum— i.e., the shape in which the
area oi the Burface is a. minimum. The sphere is tha surface which for a
given volume has the smallest surface, so that the drops of oil in the liquid
will be spherical. This experiment can easily be tried, and the spherical
form of the drops is very evident, especially if the oil is made more
di^itinct by the addition of a little iodine.
If a drop of liquid is not surrounded by Quid of the same density,
but is like a drop of mercury on n plate wliieh it does not wet, then any
change in the shape of the drop will affect the potential energy due to
gravitation as well aa that due to surface tension, and the shape of the
drop will be determined by the condition that the total potential energy is
to bo as small aa possible ; if the drop is very large, the potential enej-gy
due to the surface tension is insignificant compared with that due to gravity,
and the drop spreads out
B centre of
. though
gravity low, ei
this involves an
the potential energy due to
the surface -tension. If,how-
ever the drop is very small,
the potential energy duo to gravity ^s irsignifiutnt in comparison with
that due to surface-tension and the drop take? the shape in which the
potential energy due to surfato tension is as small as possible ; this shape,
as we have seen, is the spber cal and thus surface tension will cause all
very small drops to be sj hencal Dew drops and raindrops are very
conspicuous examples of this othei ex'imples are afforded by the
manufacture of spherical pellets by the fall of molten lead from a shot
tower and by the apher cal form of soap bubbles W e shall show later on
that if the volume of hquid m a drop is the same as that of a sphere of
radius a the liquid will remain very nearly spherical if a' is small compared
with T/gp where T is the snrfaoe-tension and p the density of the liquid.
Thus, in the case of water, where T is about 73, drops oE less than 2 or 3
millimetres in radius, will be appioximately spherical.
Another impoilant problem which we can easily trtat by the method of
energy is that of the spreading of one liquid over the Kurf.ire o( another.
Suppose, for example, we place a drop of liquid A on another liquid B
(Fig. 117), we want to know whether A will spread over B like oil over
water, or whetlier A will contract and gather itself up into a drop. The
condition that the potential energy is to bo as small as possible shows that
A will spread over B if doing so involves a diminution in tho potential
energy; while, if the spreading involves an incro.ite in the potential
energy, A will do the reverse of spreading and will gather itself up in a
drop. Let us consider the change in the potential energy due to an
increase S in the area of contact of A and B where A is a flat drop. We
have three surface -tensions to consider: tliat of the surface of contact
between A and the aii', which we shall call T ; that of the surface of
contact between B and the air, which we shall call T,; and that of the
surface of contact of A and B, which we shall call T,,. Now when we
increase the gurfaoe of osntact between A and B by S we Increase the
energy due to the surface-tension between these two fluids by T„x8, we
CAPILLAKITY. tSf)
increase thnt due to the eurface- tens ion between A nnil the air by T, x S
snd diminiKh that due to the surface-tension between B and the air by
~ n the potentidl energy is
(T, + T„-T,)R,
and it this is negative S will in<
dition for this to be negative in
II spread o
T.>T,+T,„
Eo that if this condition is fulfilled the liquid A will spread out into a thin
film and cover B, and there will be no place where three liquid Slirfiica^
meet. If, on the otiier band, nny one of the tensions is lees than the sum
of the other two — i.e., it we can construct a trisnele whose xides are
proportional to T,, T, and Tj,. then a drop of one liquid can exist on the
Gurface of the other, and we should have the three liquid surfnces meeting
at the edge of a drop. Tha triun^-Ie whose sides are proportionnt to
T„ T„ T[, ia often called Neumann's triangle; the experiments of
Quincke, Marangoni and Van Mensbrugghe, show that for all the liquids
hitherto investigated this triangle cannot be drawn, as one of the tensions
is always greater than the sum of the other two, and hence that there can
be no position of equilibrium in which three liquid surfaces meet.
Apparent exceptions to this are due t« the fouling of the eutface of one of
the liquids. Thus, when a drop of oil stands on water, the water surface
is really covered with a thin coating of oil which has spread over the
surface ; or again, when a drop of water stands on mercury, the mercury
surface is greasy, aiid the greaiie has spi'sad over the water. Quincke has
shown that a drop of pure water will spread over the surface of purn
mercury.
Tlioogh three liquid sui-fnces cannot be in equilibrium when there is a
line along which all three meet, yet a solid and two liquid mrfoceH can be
in equilibrium ; this is shown by the equilibrium of water or of mprcury
in glass tubes when we have two fluids, water {or mercury), and air,
both in contact with the gUuss. Th-e consideration of the condition of
equilibrium in this ease natutally suggests the question as to whether
there is anything corresponding to surface-tension at the surface of
separation of two subatancei<, one of which is a solid. Though in this case
the idea of a skin in a state of tension is not so easily conceivable as for a
liquid, yet there is another way of regarding surface-tension which is as
readily applicable to a solid as to a liquid. We have Keen that the
existence of surface-tension implies the possession by each unit area of the
liqu'd of an amount of potential energy numerically equal to the surface^
tension : we may from this point of view regard surface-tension as sm'face
energy. There is no difficulty in conceiving that part of the energy of &
solid body may be proportional to its sulfate, and that in this sen&e the
body has a siii'face- tension, this tension being measured by the energy pei
unit area of the surface.
Let us now consider the equilibrium of a liquid in contact with aii' and
both resting on ii solid, and not acted upon by any forces escept those due
to surface-tension. Suppose A, Fig, 3S, represents the solid, B the liquid,
C the air, FO the surface of separation of liquid and air, ED the sur-
face of the solid. Let the angle FGD he denoted by 0; this angle is
140
PROPERTIES OF MATTER.
called the angle of contact of the liquid with the solid. Let the surface
of separation FG come into the position FG' parallel to FG. Then if FG
represented a position of equilibrium, the potential energy due to surface-
tension must be a minimum in this position, so that it will be unuficcted
'" ; • Fio. 93.
by any small displacement of the substances; thus the potential energy
must not be altered by the displacement of FG to F'G'. This displace-
ment of the surface causes B to cover up a long strip of the solid, the
breadbh of the strip being GG'. Let S be the area of this strip. Then
if Tp T, and T„ are respectively the stirfaccWbensions between A and C, B
and 0, and A and B, the changes in the energy due to the displacement are :
(1) An increase T„S due to the increase S
in the surface between A and B.
(2) An increase T,S cos 6 due to the
increase S cos in the surface between B
and C.
(3) A diminution T^S due to the diminu-
tion S in the surface between A and C.
\^ Ilence the total increase in the energy is
S(T„ + T,cose-Tj)
and as this must vanish when we have
equilibrium we have
Ti, + T,cos0 = Tj;
or
cos
T -T
e=ii ill
T.
Thus, if T, is greater than T„, cos is
positive and is less than a right angle ; if
T, is less than Tj,, cos Q is negative, and is
greater than a right angle; mercury is a
case of this kind, as for this substance is
liQ 9 J. about 140°. The angle is termed the
angle of contact. Since cos cannot exceed
unity, the greater of the two quantities T, or T„ must be less than the
sum of the other two. Ir this condition is not fulfilled the liquid B will
spread over the surface A.
Rise of a Liquid in a Capillary Tube.— We can apply the result we
have just obtained to find the elevation or depression of a liquid in a tube
which it does not wet and with which it has a finite angle of contact.
Suppose A is the height of the fluid in the tube above the horizontal
surface of the fluid outside, when there is equilibrium ; and suppose that
r is the radius of the tube at the top of the fluid column. Let T^ be the
CAPILLARITY. Ul
Bui-Face-tension between the tube and air, T, that between the liquid ami air
and T„ that between tbe tube and the liquid. Then, if there is equilibrium,
a slight displacement of the fluiS up tbe tube will not alter the potential
energy. Suppose then that the fluid ri^a a short distance x in the tube,
thus covering an additional area 2irncof the tube, and diminishing the area
of the tube in contact with the air by this amount. This increases the
potential energy due to surface-tension by 27ri'r(T„ — T,).
The increase in the potential energy due to gravity is the work done
(1) by lifting the mass Trr'xpxa;, where p is the density of the liquid,
against gravity through a height h — this is equal to ypAirr'*; and (2) by
lifting the volume v of the meniscus through a height x — this work is equal
O ffpVX.
3 tbe total ii
1 potential energy is
2frra;(T„ - TJ + gphirr'x + gpvis,
and as this muat vanish we have
,,^ <.^ 2(T,-T,.)
but it e ia tho angle of c.
e have just proved that
ise=T,-Tu
T'-' ypr
When tbe fluid wets the tube 6 ia zero and cos 9 = 1, If the meniscus
is so small that it may be regarded as bouuded by a hemisphere, v ia the
difference between tho volume of a hemisphere and that of tbe circum-
scribing cylinder — i.e.,
If is greater than a right angle A is negative, that is, the level of tha
liquid in the tube is lower than the horizontal surface; this is strikingly
shown by mercury, but by no other fluid. The angle of contact between
mercury and glass was measured by Gay Lussac hy causing mercury to
flow up into a spherical glass huib ; when the mei'cury is in the lower part
of the biilb the surface near the glfiss will he very much curved ; as the
mercury rises higher in the bulb the curvature will get less; the surface
of the mercury at different levels ia represented by tbe dotted lines in
Fig. 100. There is a certain level at which tbe surface will be horizontal;
at this place the tangent plane to the sphere makes with a hoiizontal plane
an angle equal to the supplement of the angle of contact between mercury
and glass. A modification of this method is to make u piece of clean
142
PROPERTIES OF MATTER.
plate glass dipping into mercury rotate about a horizontal axis until the
surface of the mercury on one side of the plate is flat ; the angle made by
the glass plate with the horizontal is then the supplement of the angle of
contact between mercury and glass.
The angle of contact between mercury and glass varies very widely
under different circumstances; thus the meniscus of the mercury in a
thermometer may not be the same when the mercury is rising as when it
is falling. "We should expect this to be the ca^^e if the meixjury fouls the
glass, for in this case the mercury when it falls is no longer in contact
with clean glass but with glass fouled by mercury, and we should expect
the angle of contact to be very different from that with pure glass. Quincke
found that the angle of contact of a drop of mei-cury on a «];las8 plate
steadily diminished with the time ; thus the angle of contact of a freshly
formed drop was 148° 55', and this steadily diminished, and after two days
Fio. loo.
Fio. 101.
was only 137° 14'; on tapping the plate the angle rose to 141° 19', and
after another two days fell to 140^.
If we force mercury up a narrow capillary tube and then gradually
diminish the pressure, the mercury at first, instead of falling in the tube,
adjusts itself to the diminished pressure by altering the curvature of its
meniscus, and it is only when the fall of pressure becomes too large for such
an adjustment to be possible that the mercury falls in the tube ; the con-
sequence ia that the fall of the mercury, instead of being continuous, takes
place by a series of jumps. This effect is illustrated by the old experiment
of bending a piece of capillary tubing into a U-tube (Fig. 101), pouring
mercury into the tube until it covers the bend and stands at some height
in either leg of the tube ; if the tube is vertical, the mercury can be made
by tapping to stand at the same height in both legs of the tube. Now slowly
tilt the tube so as to cause the mercury to run up the left leg of the tube ;
if the tube is slowly brought back to the vertical, the mercury will be
found to stand at a higher level in the left leg of the tube than in the
right, while the meniscus will be flatter on the left than on the right.
This principle explains the action of what are called Jamin's tubes, which
are simply capillary tubes containing a large number of detached drops of
liquid ; these can stand an enormous difference of pressure between the
ends of the tube without any appreciable movement of the drops along tho
CAPILLARITY. 143
tube. Thus, suppose that AB, CD, EF (Fig. 102) repreeeot three consecu-
Uve drops along the tube, then in consequence of the different curvatures
normal to its Rurfuce thi'ougU
oFABatAandB the pressure ill Iho air at A will be greater than that at B,
while the pressure at will he greater than that at D, and so od ; thus
each drop tran.sniits a emaller pressui'O than it receives ; if we have a largo
number of drops in the tube tiie difference of pressure at the ends arising
in this way may amount to several atmospheres.
Relation between Pressure and Curvature of a Surface. — K
we have a curvtd liquid surfnte in a state of tension the jjressure on the
concave side of the siirfaoe must be greater than that on the convex ; we
shall proceed to find the relation between the difference of pressure on the
two sides and the curvature of the surface.
Let the small poi-tion of a liquid film, represented in Fig, 103by ABCD
where ABand CD are equal and parallel and at right angles to ADand BU,
be in equilibrium under the surface tension and a difference of pressure^
between the two sides of the film. 'When a system of forces acting on a
body are Jn equilibrium we know by Mechanics that the algebi'aical
sum of the work done by these forces when the body suffers a small dis-
placement is zero. Lot the film ABCD (Fig. 103) be displaced so that
each point of the film moves outward along the norm
a small distance ir, and let A'B'C'D'be the
displaced position of ABCD. Then the
work done by the pressure is equal to
p X area ABCD x x ;
the work done against the surface tension
is T X increase in ai-ea of the Furface ; luid
since a film bas two sides the increase in
the area of the film is twice the difference
between the ai'eas A'B'C'D' and the area
ABCD. Hence the work done against sur- '
face tension is equal to
2Tx (area A'B'C'D'-area ABCD)
Hence by the mechanical principle referred to
y X area ABCD x a: - 2T{area. A'B'C'D' -
if we are conwdering a drop of water instead of a film we must write T
instead of 2T in this equation.
Spherical Soap-bubble.— in this case ABOD will be a portion of a
spherical surface and the normals AA', BB', CC, DD' wiSl all pas.'* through
0, the centre of llie sphere. Let 11 be the radius of the sphere, then by
similar triangles
A'B' = AB^' = AUM + ^'
a ABOD)
0)
OA
B'C'-B0'^' = i(Cf
\' + nJ
144
PROPERTIES OF MATTER.
The area A'B'C'D' = A'B'. BC = AB. BC
.B0(..|)
as we suppose xfR is so small that its square can be neglected.
Hence
Fio. 104.
area A'B'CT)'= area ABOD
Fia. t05.
('4')
(2)
substituting this value for the area A'B'C'D' in equation (1)^ the equation
becomes
4T
BO that the pressure inside a spherical soap-bubble exceeds the pressure
outside by an amount which is inversely proportional to the radius of the
bubble.
General Case of a Curved Soap-bubbIe«— If the element of the
film ABGD forms a portion of a curved surface, we know from the theory
of such surfaces that we can find two lines AB, BC at right angles
to each other on the surface such that the normals to the surfacx)
CAPILLARITY. 145
at A and B intersect in O, while those at B and intersect
iu a point O'. The lines AB, BC are said to be elements of the
mrves of Principal Curvature of the surface, and AO and BC are called
the Kadii of principal curvature of the surface. We must now distinguish
between two classes of surfaces. In the first class, which includes spheres
and ellipsoidsi the two points and 0' are on the same side of the surface,
and the surfaces are called synclastic surfaces ; in the second class, which
includes surfaces shaped like a saddle or a dice-box, O and O' are on
opposite sides of the surface ; and the surfaces are called anti-clastic
surfaces. We shall consider these cases separately, and take first the
case of synclastic surfaces. In this case (Fig. 104) we have by similar
triangles
OA' / x\
A'B' = AB-— = AB/ 1 + g j if R is the radius of principal curvature OA.
Similarly B'C = BC^l + ^\ if R' is the radius of principal curvature O'B.
Hence area A'B'C'b' » area
a4))
-area ABCD[l+a:
as we suppose a;/R, x/Bf both so small that we can neglect the product of
these quantities in comparison with their first powers. Substituting this
value for the area A'B'C'iy in equation (1) we get
^(k4') <«)
Let us now take the case of an anti-clastic surface, represented in
Fig. 105. In this case we have
a'B'=ab(i + -)
FC'=BC^ = BC(1-J)
hence area A'B'0'D' = area ABCD/l+a;^i -I>\^
Substituting this value of the area A'B'CD' in equation (1) we get
We can include (3) and (4) in the general formula
'-"{k^i)
if wu make the convention that the radius of curvature is to be taken as
positive or negative according aa the corresponding centre of curvatuiit
146
PROPERTIES OF MATTER.
is on the side of the sarfaoe where the pressure is greatest or on the
o| p mie side.
When a soap film is exposed to equal pressures on the two sides />» 0,
and we must therefore have
i+1
0.
In this case the curvature in any normal section must be equal rn 1 opf o ite
to the curvature in the normal section at right angles to the tii-st. Bj
^MJ A ^^m
^
Fio. 106.
stretching a film on a closed piece of wire and then bending the wire we
can get an infinite number of surfaces, all of which pa^sess this property;
we can also get surfaces with this propeity by forming a film between the
rims of two funnels open at the end, as in Fig. 106. By moving the
funnels relatively to each other we get a most interesting seiies of
snifacci, all of which have their princi|>al curvatures equal and opposite.
'■* .
Fio. 107.
If the film is in the shape of a sui*face of revolution — t .«., one which can be
traced out by making a plane curve rotate about a line in its plane — we
know from the geometry of such surfaces that (Fig. 107)
11- PC
R'-PG
where is the centre of curvature of the plane curve at P, and G the
point where tlie normal at P cuts the axis AG about which the curre
rotates.
CAPILLARITY. 147
11 the pressures on the two sides of the film are equal we must have
PO=-PG.
The only curve with this property is the catenary, the curve in which
a uniform heavy string hangs under gravity, and this, therefore, is the
shape of the cross-section of a soap film forming a surface symmetrical
about an axis, when the pressures on the two sides are equal.
Stability of Cylindrical Films.— Let us consider the case of a
symmetrical film whose sui-face approaches closely that of a right circular
cylinder. Let EPF be the curve which by its rotation about the straight
line AB generates the surface occupied by the film. EPF will not differ
much from a straight line, and PG, the normal at P, will be very nearly
equal to PN where PN is at right angles to AB. Hence, if R is the
radius of curvature at P and p the constant difference of pressure between
the inside and outside of the film, we have
"^'(h-^Fn) C)
Let y be the height of P above the straight line EF and a the distance
between the lines EF and AB, then
PN=a+y
and as ^ is very small compared with a wc have approximatelly
PN a a'
Substituting this value of 1/PN in equation (1) we get
K 2T a a' a*\^ ^^T ajj a* ^"^
if y' is the distance of P from a horizontal line at a distance
»•(&-!)
below EF. Since the film is very nearly cylindrical, p ia very nearly
equal to 2T/a, so that the distance between this line and EF will be
veiy small.
Hence we see from equation (2) that the reciprocal of the radius of
curvature at a point on the curve is proportional to the distance of the
point from a straight line. Kow we saw (p. 9G ) that the path
148
PROPERTIES OF MATTER.
described by a point fixed near to the centre of a circle when the circle
rolls on a straight line possesses this property, hence we conclude that the
cross-section of a nearly cylindrical film is a curve of this kind. The curve
possesses the following properties : it cuts the straight line, which is the path
of the centre of the circle, in a series of points separated by half the
circumference of the rolling circle, its greatest distance from this line
P>'
t
^ •
"" -- .ff
■ --..
1
M L K
Fio. 109.
is equal to the distance of the point from the centre of the rolling
circle, while the reciprocal of the radius of ciurature at a point is pro-
portional to its distance from this line.
Let us now consider what is the pressure in a nearly cylindrical
bubble with a slight bulge. Let us suppose that the len<(th of the bubble
is less than the distance between two points where the curve which
generates the surface crosses the path of the centre of the rolling
circle. The section of the bubble must form a part of this curve.
Let A and C, Fig. 109, be the ends of the bubble APC, the
section of the film. Let the dotted h'ne denote the completion of the
curve of which APO forms a part. Then if ;> is the excess of pressure
.«••
•»
*
'v
%
■A
A
^'**— ^^^^^'^^'^
c
I
M
IC
FxQ. 110.
inside the bubble over the outside pressure and P any point on the
curve,
^-K^k)
where p is the radius of curvature of the curve at P. Now if we take
P at Q^ a point whero the curve crosses its axis 1/p » 0, hence
^ QK
CAPILLARITY.
UJ)
Now if the film were straight between A and the excess of pressure
f' would be given by the equation
, 2T
^=AM'
As QK is less than AM, p is greater than p', benoe the pressure in the
film which bulges out is greater than the pressure in the straight film.
We can prove in the same way that in a film that bends in, as in Fig. 110,
if the distance between the ends. is less than the dibtance between the
••.'
nfc
^
c
^^**"**<.*^,^^P^ ^^-.^-"'''''^^^^
2
/
1
M
L
rio. 111.
K
points Q and Q' on the curve , that is, if the length of the film is less than
half the circumference of its ends, the pressure is less than the pressure
in the straight film.
If the distance between the ends of the film is greater than half
the circumfei-ence of the ends of the film these conditions are reversed.
For let Fig. Ill repre-
sent such a film bending
in ; as before, the excess of
pressure p wiii be given by
the equation
_2T
^ QK
where Q is the point where
the curve of the film crosses
its axis. If the film were
straight between A and C,
p', the excess of pressure,
would be given by the
equation
^""AM Fio. 112.
Since in this case AM is greater than QK, p is less than p. Hence
the pressure in the film which bends in is greater than that in the
straight film. In a similar way we can prove that in this case the
pressure in a film which bulges out is less than the pressure in a straight
film. Hence we arrive at the result tbit, if the length of the film is
less than half the circumference of its end, the pressure in a film
that bulges out is greater than that in a film which bends in, while
150
PROPERTIES OF MATTER.
if the length of the film is greater than its semi-circoDferenre thci
prenmre in the film that bulges out is less than the jn^ssure in one
that bends in. Mr. Boys has devised a very beautiful experiment which
illustrates this point. The arrangement is represented in Fig. 112.
A and B are pieces of glass
tubing of equal diameter com-
mumcating with each other
through the tube C ; this com-
munication can be opened or
closed by turning the tap. £
and F are pieces of glass tubing
of the same diameter as A; they
are placed vertically below A
and B respectively. The distance
between A and £ and B and F
-^ can be altered by raising or
lowering the system ABC. First
begin with this distance less
than half the circumference of
the glass tube, Fig. 113, close
the tap and blow between A and
£ a bubble which bulges out,
■nd between B and F, one that bonds in. Now open the tap ; they will
both tend to straighten, air going from the one at A to he)p to fill up
that at B, showing that the pressure in the one at A is greater than in
that at B. Now repeat the experiment after increasing the distance
between A and £ and B and F to
~| f 1 more than half the circumference of
I the tube. We now find on opening
<^f> <^ the tap that the film which bulges
out is blown out still more, while the
one that bends in tends to shut itself
up, showing that air has gone from
B to A or that now the pressure at
B is greater than that at A.
It follows from this result that
the equilibrium of a cylindrical film
is unstable when its length is greater
than its circumference, while shorter
films are stable.
For let us consider the equilibrium
of a cylindrical film between two
equal fixed discs, A and B, Fig. 115,
and consider the behaviour of a
movable disc C of the same size placed
between them. Suppose the length
of the film is less than its circum-
ference and that C is midway between A and B ; move C slightly towards
B, then the film between B and will bulge out while that between A
and will bend in. As the distance between each of the films is less than
half the circumference the pressure in the film which bulges out will be
greater than in that which bends in, thus will be poshed back to its
Fio. 114.
CAPlLr.ARITY. 151
original poBition and tbe equilibrium will bo staUe. If C is not miiln-ay
between AB but nearer to B than to A, then even if AC ie greattr tlmn
the Bemi-circumference bo that wh«ii ia pushed towards D the pressuie
in AO ia greater than when
1
the Kim la straight, yet it is
easy to prove that the excess
of pressure in BO is, in
consequence of its greater
curvature, greater than that
in AC, so that C is again
pushed back to its old posi-
1
tion and the film ia again *■ c s
stable. Fw. lis.
Suppose now that the
distance between A ond B is greater than the circumference of the film,
and that C, originally midway between A and B, is slightly displaced
towards B. OB will bulge out aad CA will bend in; as the length of
1
than the semi-circumferenre
of the film the prepsure in
BO will be leas than that in
AC, and will be pushed
still further from its original
will be unstable. The film
will contract at one paiK
and expand in another until
its two sides come into con-
tact and the film breaks up
into two separate spherical
portion H.
These results apply to lluid
cylinders as welt as to cylin-
drical films. Such cylinders
are unstable when their length
IS greater than their circum-
ference. Examples of this
unstabilitjareafibrded by the
breaking up of a liquid jet
into drops. The development
of inequalities in the thickness
of the jet is shown in Pigs.
116andU7takenfrominstati-
taneous photographs. Thelittle
drops between the big ones
are made from the narrow i
breaks up. Another in.stance
a ghiss fibre in water, the wab
beautiful illustration of the %a
shown in Fig. 118, when again
beads.
If the liiid is very viscous
Fio, 118.
eck3 which form before the jot finally
of this instability ia aflbrded by dipping
Br gathers itself up into beads. A very
me effect ia that of a wet spider's web,
the water gathers itself up into spherical
the efi'ect of viscosity may counterbalancs
PROPERTIES OF MATTER.
B possible to get long tbin
15S
the instiiViility due to surface tenaioD;
threads ot^treacle or of niolten glass and qtwrtz.
Foroo' between two Plates due to Surface-tension.— Let A
ftndB(Fig. I Il))be two parallel plates
separated by a. film of water or some
!i()iiid which wets them; then, if d
is the distance between the plates
and D the diameter of the area of
the plate wet by the liquid, the
liiilii of curvature at the free sur-
r.Lce of the liquid ore approximately
dl2 and D/2, hence the presaui-o
I side the
The:
D the difference of pressure JH approximately -
film
. ten
than the
pressure by
eiy small compared with
Now the plate A is pressed towards B by the atmosphi
ftnd away from B by a pi
tlmn thin by iiT/(/ ; lience, if
The force variea inversely as the distance between the plates ; thus,
if a drop of water in placed between two plates of glass the plates are
forced together, and this still further increases the pull betwwn the plataa
as the area of the wettod surface increases while the distance between the
plates diminishes.
CAPILLARITY.
15S
Attractions and Repulsions of small Floating^ Bodies.— Smnii
bodies, such as eti-aw or pieces of cork, floating on the surfuce of
a liquid often attract each other and collect together in clusteiv ; tbi:4
occurs when the bodies are all wet hy the liquid, and also when none of
them are wet; if one body ia wet and one is not wet they repel each other
when they come cloae together. To investigate the theory of Ibia effect,
let us suppose that A and B are two parallel vertical plates immersed in a
liquid which wets both of them, the liquid will stand at a higher level
between the plates than it does outside. We shall begin by showing that
the horizontal force exerted on a
plate by a meoisciia such as FKQ,
UVW is the eame as tlie force
which would be exerted if the
meniscus were done away with and
the liquid continued horizontally
up to the surface of the plate. For
consider the water in the meniscus
PQR ; it is in equilibrium uinler
the horizontal tension at P, the
vertical tension at Q, the force
eserted by the plate on the liquid,
the vertical liquid pressure over
PR, and the pre-'sure of the atmo-
sphere over PQ. The resultant
pressure of the atmosphere over
PQ, which we shall call r, in the
hoiizontal direction is equal to the
pressure which would be exerted on
(JR, the part of the plate wet by
the meniscus, if this were exposed
directly to the atmospheric pressure without the intervention of the
liquid. The horizontal forces acting from left to right on the meniscus
» — T — force exerted by plate on menisciis.
Since the meniscus is in equilibrium the horizontal forces mu^t be in
equilibrium ; hence
force exerted by
iip]ate = T-
but this is precisely the force which would he exerted if the meniscus were
done away with and the horizontal surface of the liquid prolonged to meet
,he plate. Hence, as far as the horizontal forces are concerned, we may
ippose the surfaces of the liquid flat, and reprB^euted by the dotted lines
Fig. 130. Oonsidering now the forces acting on the plate A, the pulls
:erted by the surface- tension at K and U are equal and oppasite ; ou the left
,he plate ie acted on by the atmospheric pressure, on the riglit hy the pressure
m the liquid. Now the pressure in the liquid at any point is less than the
atmospheric pressure by an amount proportional to the heigbt of the point
above the level of the undisturbed liquid; thus the pressure on A tending
push it towards B is greater than the pressure tending to push it away
)m B, and thus the plates are pulled together.
Now suppose neither of the plates is wet by the liquid^a case rcpro-
154
PROPERTIES OF MATTER.
sented in Fig. 121. We can prove, as before, that we may arappopo the
fluid to be prolonged horizontally to meet the plates. The force tending
to push the plate A towards £ is the pressure in the liquid, the force
1 I
-^ts^
r>
-V'
Kio. 121.
Fio. 1D2.
\
tending to push it awn}' is the atmo^^plicric pressure. Now the piessure
at any point in the liquid is greater than the atmospheric pressure by an
amount proportional to the depth of the point below the undisturbed
surface of the liquid ; hence, the pressure tending to push A to B will be
greater than that tending to push it away from
B, so that the plates will again appear to
attract each other.
Now take the case where one plate is wet
by the liquid while the other is not. The
section of the liquid surface will be as in Fig.
122, the curvature of the surface being of one
sign against one plate, and of the oppo^te sign
against the other. When the plates are a
y considerable distance apart, the surfaces of the
liquid will be like that shown in Fig. 122;
between the plates there is a flat horizontal
surface at the same level as the undisturbed
liquid outside the plates ; in this case there is
evidently neitherattraction nor repulsion between
Fio. 12a the plates. Now suppose the plates pushed
nearer together, this flat surface will diminish,
and the last trace of it will be a horizontal tangent crossing the liquid.
Since the curvature changes sign in passing from A to B, there must be a
place between A and B where it vanishes, and when the curvature
vanishes, the pressure in the liquid is equal to the atmospheric pressure ;
this point, at which the tangent crosses the surface, must be on the
prolongation of the free surface of the liquid. Now suppose that the
plates are so near together that this tangent ceases to be horizontal, and
the liquid takes the shape shown in fig. 128. We can show, by the
CAPILLARITY.
155
^
method given on p. 153, thntthe action on lliu pUite A of llio meniscua
inside A is the wime as if llie meiiiKeus were removed and the liquid
Burface stretched horizontally belween the plates, tlie purface-tensiou in
this Burface being equal to the horitontal cwi/Jwewt of the Kiirfuce tension
at the point of inflection. Now oonaitter the plnte A; it ia pulled fiom
B by tbe siirfRce-tengiou and towiirda it by only the horizontal component
of thli. The force pulling it away is thni grouter than the other, nnd the
plates will therefore repel each other. If the plates are piishetl very near
together so that the point of inflection on the surface gets suppressed the
liquid may rise betwtcn the plates and the ropulsioa be replaced by an
attraction.
Methods of Measuring' Surface-tension.
By the Ascent of the Liquid in a Capillary Tube.— A finely
divided glass lii^ile ir placpil in a vt'rtical position by incnns of a plumb
line, tho lower end of the scale
dipping intoa vessel V, which contains
some of the liquid whose surfaca
tension is to be determined. The
capillary tube is prepared by di'awing
out a piece of carefully cleaned glass
tube until the internal diameter in
considerably lexs than a millimetre ;
the bore of the tube Hhould be as
uniform as popsible, for althoogh the
hi^ight to which the fluid rises in tha
capillary tube depends only on ths
ratlius of the tube nt the top of the
meniscus, yet when we cut the tuho
at this point to determine its radius,
if the tube is of uniform bore, no
error will ensue if we fail to cut it at
exnctly the right place. Atlacb tha
capillary tube to the scale by two
elastic bandf, and have a good light
behind the scale. Dip the capillary
tube in the liquid, and it will rush
up the tube ; then raise the capillary
tube, keeping its end beiow the tluid
in V. This will make the meniscua
sink in the tube and ensure that the
tube above the meniscus ia wetted by
the liquid. Now read ofV on the
scale the levels of the liquid in V
and the capillary tube, and the dif- Kiu. Ui.
erence of levels will give the height
to which the liquid rises in the tube. To measure r, the nidius of the
tube at the level of the meniscus, cut tbe capilt]Liy tube carefully across at
this point and then measure the internal radius by a good microscope with
a micrometer scale in the eyepiece. If the section, when observed in the
microscope, is found to be far from circular, the experiment should bs
156
PROPERTIES OF MATTER.
repeated with another tube. The surface tension T is determined by the
equation (p. 141).
T = ip(/(hr+ -Q ) where p is the density of the fluid.
If the angle of contact is not zero a knowledge of its value is required
before T can be determined by I! is method.
By the Measurements of Bubbles and Drops.—This method is
due to Quincke. The theory is as follows: suppose that AB, Figs. 125
r
Fio. 125.
and 126, represents the section of a large drop of mercury on a horizontal
glass plate or, when turned upside down a large bubble of air under a glass
plate in water. Let a central slab be cut out of the drop or bubble by two
parallel vertical planas unit distance apart, and suppose that this slab is
cut in half by a vertical plane at right angles to its length ; consider the
equilibrium of the portion of this slab above the horizontal section BO of
greatest area in the case of the drop, and below it in the case of the bubble.
Fio. 126.
The horizontal forces acting on the upper portion are the surface tension
T, and the horizontal pressures acting over the flat section ADEC and the
curved surface. If the drop is so large that the top may be considered as
plane there will be no change of pressure as we pass from the air just above
the surface of the drop to the mercury just below it;* in this case the
diflference in the horizontal components of the pressure oyer ADEO and
the pressure of the atmosphere over the curved surface is, since AD is
unity, equal to
As this must be balanced by the surface tension over AD we must have
T = i</pDE' (1)
By considering the equilibrium of the portion ABFGHD of the drop we
T(l 4- cos w) =" igph*
(2)
have
where h is the thickness of the bubble or drop, and to the angle of contact
at F between the liquid and the plate. From equation (2) we have
* If the drops are not largo enoagh for this assumption to be tme, a correction
has to be applied to allow for the difference in pressure on the two sides of the
surface through A*
ten dent of
r h, and uBiog
CAPILLARITY.
Thua the tLiokness of all large drops or bubbles in n liqni
the size of the drops or bubbles. By measuring either I
eqmition (1) or (5) we can determine T. In the cflfie of bubbles it i
convenient to use, instead of a flat piece of glass, the concave surface of a
large lenn, as this facilitates greatly the manipulation of the bubble. In
this case, if we use equation (2), we must remember thnt h ia the depth of the
bottom of the bubble below the horizontiil plane through the circle of
contact of the liquid with the glass. Thus, in Fig. 127, A is equal toNE and
nottoAE. It b more convenient to measure AE and then tocalculateNE
from the radius of curvature of the lens and the radius of the circle of
contact of the glass and the liquid. Determinations of the surface tension
of liquids by this method have been made by Quincke, Magie, and Wilbei-
Magie used this method to determine the
evident from equations (1) and (2) tbat
e of contact, as it if
2 ^'DE
By this method M
gie
{Phil.
Mag., vol, xi\
i. 1888) found the
following values for the angl
of con
tact with gloss :
ADRle lero.
Angle flnite.
u
Ethyl alcohol .
Water (?) .
. smnll
Methyl alcohol
Acetic acid .
. 20°
Chloroform .
Turpentine .
. 17"
Formic acid .
Petroleum .
. 26"
Benzine .
Ether .
. 1(1°
Determination of the Surface-tension by Means of Ripples. —
The velocity with which waves travel over the surface of a liquid depends
on the surface-tension of the liquid. The relation between the velocity and
surface -tenaion may be found as follows : Let Fig. 126 represent the section
of a harmonic wave on the surface of the liquid, the undisturbed level of
the liquid being xij. If gravity were the only force acting, the increase in
vertical prawure at N due to the dieturbance produced by the wave would
be equal to 3/iPN, when p is the density of the liquid.
The surface tension will give rise to an additional normal, and therefore
approximately vertical, pressure equal per unit s
"fi'"
e B b the
i5S
PROPERTIES OF MATTER,
radius of curvature of the section of the wave by the plane of the paper ;
the radius of curvature in the normal plane at right angles to the plane of
the paper is infinite. Now if the amplitude of the wave is very small
compared with the wave leneiih, the wave curve may be generated by a
point fixed to a circle rolling m a straight line ; the amplitude is equal
to the distance of the (A>int from the centre of the circle, and the
Fio. 128.
wave length is equal to the circumference of the rolling circle. The linn
xy is the path of the centre of the rolling circle. Now we saw (p 9C) that
for such a curve
1 PN
K
a'
where a is the radius of the rolling circle; but if X is the wave length
2ira = X, so that
I ^4-'^
II ' X»
ITT^PN
Thus the pressure at N, due both to gravity and surface-tension, is
(
hence we see that the effects of surface-tension are the same as if gravity
were increased by 47HT/X'p. Now the velocity of a gravity wave on deep
water is the velocity a body would acquire under gravity by falling
vertically through a distance X/4t, where X is the wave length — i.e., the
velocity is Jgkj^ir. Hence t?, the velocity of a wave propagated under
the influence of siu-face-tension as well as gravity, is given by the equation
The velocity of propagation of the wave is thus infinite both when
the wave length is zero and* when it is infinite ; it is proportional to the
square root of an expression consisting of the sum of two terms whose
product is constant. It follows from a well-known theorem in algebra
that the expi'ession will be a minimum when the two terms are equal.
Thus the velocity of propagation of the waves will be least when
4.r'T
X^'
or when
in this case tlie velocity is equal to
2^1
9P
^^{?)'
CAPILLARITY.
In tlie cnse of water, for wliich T = 7o,
X = 1'7 cm., and e = 23 en
i./soo.
i pi'opngated chiefly
of the foriniilion
?eii on tlio fiiii-fuce
l//L^
Hence no waves can travel ovei* the Burfuce of water with a Bmaller
velocity than S3 cm. pci* Boeond. Foi' any velocity greater timn this it
13 possible to llud a wave length X such t!mt wAves of this length will
travel with the given velocity. Waves wliase lengths are smaller than
that cotTesponding to the minimum velocity are collcil "ripples," ihtxa
whose lengths exceed this value "waves." A
by gravity, a ripple chiefly by surface tension.
The velocity of a '"wave" incrertses as the wave ]en|
that of a "ripple" diminishes. Interesting example
of ripples are fiiruishetj by the stnniling patterns often
of running water near an obatui^le, such ns ft
btone or a fishing-lino. Thus, let A_B represent
a stooe in a stream running from right to left,
the disturbance caused by the flow of the wnter
against the stone wit! give rise to ripples which
travel up stream with a velocity depentliiig iijio.i
their wave length. 01oe« to the slone the
velocity of the water is xero, so that the n)>]>les
travel rapidly away from the stone. When,
however, we get bo fur away from the stone, soy
at P, thut the velocity of the water is greater
than 23 cm./sec., it is possible to find a ripple of
such a wavelength that its velocity of propiigation ^'"^ '"■'■
over the water ia equal to the veWity of the
stream, the ripple will be atationai-y at P, and will form thei-e a pattern of
create and hoilowa. As the velocity of the water incrensoB as we recede
from the stone the ripplei which appear f.t)itionaiy must get shorter and
shorter in \\a\e length, and thus tha create in the pattern \vill get nearer
and nearer togetlier as we proceed up stream. We see that the condition
that the pattern should be formeil at all is that the velocity of the stream
must exceed 23 cm./sec. Fig. 129 is taken from a photograph of the
ripples behind a stone in running water. A similar explanation aiiplieu to
the pattern in front of a body moving through the liquid.
Lord Itajleigh was the fii-st {I'/iil. Ma;;., xxx. p. SSC) successfully to
apply the menaurement of ripples to the determination of the surface-
tension, and his method was used by Dr. Doraey (I'htL Mag., xliv. p. SCO)
to determine the Kurface-tension of a large number of solutions. Lord
Rayleigh's method is to generate the rijiples by the motion of a glass plate
attached to the lower prong of an electricnlly driven tuning-fork, and
dipping into the liquid to be examined. To remler the ripples (which for
the theory to apply have tobeof very smalt amplitude) visible, light reflected
from the surface is brought to a focus near the eye of the observer. On
account of the rapidity with which all plwaes of the waves are presented
in succession it is necessary, in order to see the waves distinctly, to use
intermittent illumination, the period of the illumination being the same
as that of the waves. The illumination can bo made intermittent by placing
in front of the source of tight a piece of tiiiplate rigidly attached to the
prong of a tuning-fork, and so aiTaoged that once on each vibration tha
I
I
leo PROPEHTIRS OF MATTRH.
light is intercepted by the interposition of the plate. This fork is
in unison with the one dipping into the liquid, lb is driven electi-o-
magneticaliy, and the intermittent current furnished by thia fork is
uaed to excite the vibrations of the dipping fork. By thia means the
ripples can be diatinctly seen, the number between two points at a known
distance apart counted, and the wave-lengtli X determined. If r is the
time of \'ibratioD of the fork vr = X,
and since t? = f^.?^
Sir *p
p 2nT' JjT*
an equation from which T can be determined, Tlio second term in this
esptession is in these experiments small compared with the fir^t.
Determination of Surface Tension by Oscillations of a
Spherical Drop of Liquid. — When the drop is in equillhrium under
surface-ten&ion it is spherical ; if it is slightly deformed, so as to assiune
any other form, and then left to itself, the surface-tension will pull it
back until it again becomes spheri<?al. When it has reached this state
the liquid in the drop is moving, and its inertia will carry the drop through
the spherical form. It will continue to depart from this form until the
surface-tension is able to overcome the inertia, when it is again pulled back
to the spherical form, passes through it and sgftin returns ; the drop will
thus vibrate about the spherical shape. We can find how the time of
vibration depends upon the size of the drop by the method of dimensions,
nnd the problem forms an escelleDt example of the use of this method,
t^uppose the drop free from the action of gravity, then [, the time of
vi'ji'ation of the drop, may depend iipcn a the radius, p the density, and
S the surface -tension of tiie liquid ; let
where C is a numerical constant not depending upon the units of mass,
length, or time. The dimensions of the left-hand side are one in time,
none in length, and none in mas?, which, adopting the usual notation, we
denote by [T]' [I-J'fM]'; the right-hand side must therefore be of the
s.-.mB dimensions, h'ow a is of dimensions [T]' [L]' [Ml"; p, ^^[LJ-'fMJ';
niid S, since it is energy per unit area, [TJ-' [L]" [MJ ; hence the dimen-
sions of ayS'are, [T]-"[L]-'»" [M]""'. As this is to be of the dimensions
of a time, we have
-2a = l, -Sy-)-a!=0, y+z = Q
therefore « = !, ff = i, a--J
So that (, the time of vibration, varies as •/poFJS; i.e., it varies as th«
square root of the mass of the drop divided by the surface-tension; a more
complete investigation, involving considerable mathematical analysis, shows
that r - — Y ^ ^ ^here , jg the time of the gravest vibration of the drop.
The reader can easily calculate the time of vibration of a drop of any size
if he remembers that the time of vibration of a drop of water 25 cm. in
radius is very nearly 1 second. The vibrations of a sphere under surface-
I.APILLARITY.
I6l
U
tension can easily btt followed by the eye if a Iarf;e spherical drop of water
is formed in a mixture of petroleum and biEulpliide of carbon of the same
deii>ity, Lenard {Wiedfmaiin'a Autiakn, xxx. p. 209) applied the
osL'illniion of a drop to detci-mine the surface-tension of a liquid. He
determined the time of vibration by taking instantiiueooa pliotogmpha of
the drops, and from this time deduced the surface-tension by the aid of
the preceding formula;.
Determination of Surface-tension by the Size of Drops.— The
surface-tension is Kometimes measured by deU'rmining the weight of a drop
of the liquid falhng from a tube. It we treat the problem as a statical
one and suppose that the liquid wets the tube from which it falls, then
just on the point of frilling the drop below the section J fl (Fig. 130) is to be
regarded as in equilibrium under the surface-tension noting
upwards, the weight of the drop acting downwards, the
pressure of the air on the surface of the drop acting upwards,
and the pressure in the liquid acting downwnrds across the
section AB. If a is the radius of the tube, T the surface-
tension, then the upward pull is 2?raT. If we suppose at
the instant of falling that the drop is cylindrical at the end of
the tube, the pressure in the hquid inside the drop will be
greater than the atmospheric pressure by TJa {see p. 145).
Hence the effect of the atmospheric pressure over the suifoce Tta. ISO.
of the drop and the fluid pressure across the section AB is a
downwards force equal to iritT/a or iraT. Hence, if w is the weight of
the drop we have, equating the upwa.rdB and downwards forces,
2waT = w + K^aT; ornaT=M.
The detachment of the di'op is, however, essentially a dynamical pheno-
menon, and no statical treatment of it can be complete. We should not
therefore expect the preceding expression to accord exactly with the results
of experiment. Lord Rayleigh* finds the relation 3'8aT=w to he sufficiently
exiict for many purposes. Most observers who have used this method
seem t-o have adopted the relation SiraTou), a. formula which gives little
more than half the true surface-tension ; the error comes in by neglecting
the change of pressure inside the drop produced by the curvature of its
Willielmy's Method. t— This consists in measuring the downward pull
exerted by a liquid on a thin plate of glass or met-al p;irtly immersed in the
liquid ; the liquid is supposed to wet the plate. The pull con be readily
measured by suspending the plate fi-om one of the arms of the balance and
observing the additional weight which must be placed in the other scale-fwin
to balance the pull on the plate when it isfiiirlially immersed in the hquid,
allowance being made if necessary for the eQect of the water displaced. If
i is the length of the wuter-Iine on the plate, T the surface-tension, then if
the liquid wets the pliife the doivnwn.rd pull due to surface-tension is Tl.
Method of Detachment of a. Plate. — Some observers have deter-
mined the surface-tension of liquids by measuring the pull required to drag
a plate of known area away from the surface. The theory of this method
resembles in many respects that by which we determined the thickness of
a drop or air bubble {tee p. 15G). Xet us take tlie case of a rectangular
i.Sl-
162 PROPERTIES OF MATTER.
plate being pulled away from the surface, and let the figure represent a
section by a plane at right angles to the length of the rectangle. Considering
the equilibrium of the portion whose section is EBCF, and whose length
perpendicular to the paper is unity, the horizontal forces acting upon it
are : (1) the forces due to surface-tension — i.e., 2T acting from left to right ;
(2) the atmospheric pressure on the curved surface BC acting from right to
left, which is equal to l\d
where 11 is the atmos-
pheric pressure and d is
the height of the lower
"p ^ surface of the plate above
Fio. 181. the undisturbed level of
the liquid ; and (3) the
fluid pressure acting across the surface EF from left to right. The
pressure in the liquid at i'' is equal to IT, and therefore the resultant fluid
pressure across EF is equal to lid - \gpd*, where p is the density of the
liquid. Hence, equating the components in the two directions, we have
4.T
2T + nci - Ji7f>^ = nc^, or ci» »= _
99
Now the fluid pressure just below the surface is less than the atmospheric
pressure by gpd, hence the upward pull P required to detach an area of the
plate equal to A is equal to Agpd, and substituting for d its value, wo find
P = 2An/T^
Jaegrer'S Method. — In this method the least pressure which will force
bubbles of air from the narrow orifice of a capillary tube dipping into the
liquid is measured. The pressure in a spherical cavity exceeds the pressure
outside by 2T/a where a is the radius of the sphere, hence the pressure
required to detach the bubble of air exceeds the hydrostatic pressure at the
oriflce of the tube by a quantity proportional to the surface-tension. Tiiis
method, which was used by Jaeger, is a very good one when relative and
not absolute values of the surface-tension are required ; when, for example,
we want to find the variation of surface-tension with tempei*ature.
The following are the values of the 8urface-tension at 0® 0., and the
temperature coefficients of the surface-tension for some liquids of frequent
ooeurrence. The surface-tension at t° C. is supposed to be equal to T^ - /5(.
Liquid T^ Q
Ether (0,H,,0) . . 19-3 . . .115
Alcohol (0,H,0) . . 25-3 . . . .087
Benzene (OgH,) . . 30-6 . . . -132
Mercury .... 527*2 . . . -379
Water .... 75*8 . . . -152
The 6urface-ten.sion of salt solutions is generally greater than that of
pure water. If T^ is the surface-tension of a solution containing n gramme
equivalents per litre, T^ the surface-tension of pure water at the same
temperature, Dorsey* has shown that T„ = T^ + Rn, where R has the
f oUowing values^ NaCl (1-58); KCl (1-71); *(Na,OOJ (200): i(K.OO.)
(177);i(ZnSOJ(l-86). / -x -» ./ v /> ?v t •/
* Dorsej, PhiL Mag,, 44, 1897, p. S69^
CAPILLARITY. l6s
On the Effect of Temperature on the Surface-tension of
Liquids. — i'lio sui'tHce-lensiou of aUIiijuids tJiiului?lies as the temperatura
iijcitiisea. This caa be eliown in the case of water by the following
eliperiment : A pool of water in formed on a hori^iontal platd of clean
metal ; powdered sulphur is dusted over the surface of the water and heat
applied locally to the uoder surface of the metal by a fine jet. On the
application of the heat the portion of the water immediately over the flame is
rapidly swept cleav of the sulphur; this b due to the greater tension in
the cold liquid outside pulling the sulphur away against the feebler tension
in the warmer water.
EiJtTOB (IFied. Ann. 27, p. 448) has pointed out that for mauy liquid;)
d{Tvi)ldl is tnjual tu— 3'I, being independent of the nature of tlie liquid and
the temperature; here Tis the surface-ten.sion of the liquid, i; the "uiolecuUr
volume" — t.a., the molecular weight iji\ided by the density- — and t the
temperature. It is clear that, if we assume that d{Tv})/dt has this value
for a liquid whose density and EUi'fac«-tencion at different temperatures
are known, we can determine the molecular weight of the liquid. Tha
method has been applied for this purpose, and eome intoresting results
have been obtained ; for exninple, water is a liquid for which Eiitvcs' rulo
does not hold, if we suppose the molecular weight of water to be 18.
If, however, we ansume tiie molecular weight of water to be .16 — i.e., that
each molecule of water haa the coinjiosition ^H,0, then Eotvos' rule is
found to hold attemi^ratures between lUO* and200°0.; below the lower
of these tempeiiituies the molecubr weight would have to be taken as
greater thnn 3G in order to make Edtvois' rule apply- Hence, E^tvos con-
cluded that the molecules of water, or at any tate the moieculta of the
surface layers, have the composition 2H,0 above 100^ 0., while below that
temperature they have a still more complicated compoeition.
It follows that if Eiitvos' rule is true,
T«i = 2-1 ((,-()
where t, is moiue ron^tatit temperature, which can be determined if we
know the value of T and ti at any one temperature ; l^ in the temperature at
which the sui'fuce- tension vauishei:, it is therefore a temperatui'e which
piobahly does not differ much from the critical temperature ; the values of
(| for ether, alcohol, water, are roiij,'hly about 180°, SO-t", 5G0° 0. Their
critical temperatures aie ealimiited by Van der Waais to be 100°, 256',
fiao= 0.
Coolinjf due to the Stretching of a Fllm.-Since the surface-
tens iou changes with the temperature, any elmngea in the area of a film will,
as they involve work done by or against surface-tension, be Accompanied hy
thermal changes. Wa ran calculate the amount of these thermid changes
if we can imn^Mue a little heat engine which works by the change of
EUI face-tensiou ivith temperature. A very simple engine of this kind is as
follnwR : Suppose that we have a rectangular linmework on which a lilni
is etivtcheil, and that one of the sides of the framework can move nt right
angles tu itd length. Let the mass of the framewoik nnd lilm be so small
that it hofl no aiipreciable heat capacity. Suppose we have a hut chamber
Bud n cold chamber, maintained respectively at the absolute temperaturen
e^ and 0„ where t), and B, are so near together that (bo amount of he.'it
required to raise the body from 8, to fl, is small compaied with tha
thermal effect due to cbaDj|;e of area Let us place the Cilm in the hot
164 PROPERTIES OF MATl'ER.
chamber, and streteh it so that its area increases by A, then take it out
of the hot chamber and place it in the cold one, and allow^ the film to
contract by the amount A ; the film has thus recovered its original area.
Let it be now placed again in the hot chamber. If the surface-tension of
the film when in the cold chamber is greater than when in the hot, then
the film when contracting may be made to do more work than was
required to stretch it, so that there will be a gain of work on the cycle ;
the process is plainly reversible, so that the film and its framework and
the two chambers constitute a reverf>ible engine. Hence, if H, is the
heat absorbed in the hot chamber, H, that given out in the cold, both
being measured in mechanical units, we have by the Second Law of
Thermodynamics,
H,_H,_H,-H,
fl, 6), 0, - e.
(1)
If T^, T^ are respectively the surface-tensions at the temperatures 6^
and Oft then the work done in stretching the film = 2T^A, while the work
done by the film when contracting is 2T^A, hence the mechanical work
gained = 2(T^-T^) A. By the principle of the Conservation of Energy
the mechanical work gained must equal the difierence between the
mechanical equivalents of the heat taken from the hot chamber and given
up to the cold ; hence
H,-H, = 2(T^-T^)A
and from ( 1 ) H^ = 20^ A^^^ " f ^^
If /3 is the temperature coefficient of T, then
/3 =
_ T^i - T^
hence H|=-20jA/3
Thus H, is positive when (i is negative, so that when the surface-
tension gets less as the temperature increases, heat must be applied to the
film to keep the temperature constant when it is extended — i.e,, the film
if left to itself will cool when pulled out. This is an example of the rule
given on page 132 that the temperature change which takes place is such
as to make the system stifier to resist extension. For water /3 is about
T/550, so that the mechanical equivalent of the heat required to keep
the temperature constant is about half the work done in stretching
the film.
Surface-tension of very thin Films.— The fact that a vertical
soap film when allowed to drain shows different colours at different
pliwes and is yet in equilibrium shows that the thickness of the film
may vaiy within wide limits without any substantial change in the
surface-tension. The connection between the thickness of the film
and the surface-tension was investigated by Riicker and Reinold.* The
method used is repi-esented diagrammatically in Fig. 132. Two cylindrical
films were balanced against each other, and one of them was kept thick by
()a6sing an electric current up it ; this keeps the film from draining, the
• Rticker and Reinold, PhiL Tram. 177, part IL p. 627, 188«,
CAPHXARTTY.
!(J5
Tut. 181
Other film was alluwed to diain, and a difference of surface-tension vraa
indicated by a bulging of one of the cylinders and a shrivelling of the
other. When films are fiiftt formed the value of their surface-tension is
very irregular ; but Riiclier and Reinold found that, if they were allowed
to get into a Rteajjy state, then a direct comparison of the surface-tension
over a range of thickness extending from 1S50^/i (/i./i is 10"' cm.) down t<3
the stage of extreme tenuity, when the film shows the black of the firat order
of Newton's scale of colour,
showed no appreciable
change in surface- tension,
although, had the difference
amounted to as much as
one- half per cent., EeinoM
and Riicker believed they
could have detecfed it A
large number of determina-
tions of the thickness of
the black films were made,
some by determining the
electrical resistance and
then deducing the thick-
ness, on the assamption
that the specific resistance
is the same as for the —
liquid in bulk, others by
determining the retarda-
tion which a beam of light suffers on paRsiug through the film
assuming the refraction index to be that of the liquid in mass: all the^e
determinations gave for the lliickness of the black films a constant value
about V2 fi.fi. At first sight it appears as if the surface -tension suffered
no change until the thickness is less than 12 ^.^. The authors have
shown, however, that this U not ihe right interpretation of their results,
for they find that the black and coloured parts of the film are separated
by a sharp line showing that there is ft discontinuity in the thickness.
In extreme cases the rest of the film may be as muuli as 2.10 times thicker
than the black part with which it is in
contact. The Eection of a film showing , |~ |
a black part is of the kind shown in
Fig. 133. The st«biiity of the film Fio i3a.
shows that the tension in the thin
part is e<[ua1 to that in the thick. It; is remarkable that in these films
there are never any parts of the film with a thickness anywhere between
12 fi.ft. and something between 45 and 95 /!./<.; films whose thicknesses
are within this range are unstable. This is what would occur if the
surface-tension first begins to diminish at the upper limit of the unstable
thickness, and after diminishing for some time, then begins to increase as
the thickness of the film gets les.'', until at 13 ^i.^i. it has regained its
original value ; after this it increases for some time, and then diminishes
indefinitely as the thickness of the film gets smaller and smaller. The
changes in surface-tension are represented graphically by the curve in
Fig. 1S4, where the ordinatea represent the surface- tension and the
' ' e the thickness of the film. For suppose we have a film thinning, it
166 PROPERTIES OF MATTER.
will be in equilibrium until the upper part gets the thickness corresponding
to the point P on the curve ; as the tension now gets less than in the thicker
part of the (ilm, the thicker parts pull the thin part away, and would certainly
break it, were it not that after the film gets thinner than at R the tension in-
creases until, when the film reaches the thickness corresponding to (?, the
tension is the same as in the thick film, and there is equilibrium between the
thick and the thin pieces of the film. This equilibrium would be stable,
for if the film were to get thinner the tension would get greater, and the
film would contr.icb and thicken again, while if it got thicker the tension
would fall and the film would be pulled out until it regained its original
thickness. Thus all the films which are in contact with thick films must
have the constant thickness corresponding to (?. The equilibrium at *S',
when the tension has the same value as at (?, is unstable, for any
extension of the film lowers the tension, and thus makes the film yield
more readily to the extension. The region between R and P ifl unstable.
Thickness
Fio. 184.
SO is that between T and 0. The region TR would be stable, but would
be very difficult to realise. If we start with a thick film and allow it to
thin, the only films of thickness less than that at P which will endure will
be those whose thickness is constant and equal to the thickness at Q,
Johannot (Phil, Mag., 47, p. 501, 1899) has recently shown that a black
film of oleate of soda may consist of two portions, one having a thickness
of 12 fi.fi, the other of 6 fi.fi. In this case there must bd another dip
between S and R in the curve representing the relation between surface-
tension and thickness.
Vapour Pressure over a Curved Surface.— Lord Kelvin was the
first to show that in consequence of sui-face-tension the vapour pressure in
equilibrium with a curved surface is not the same as the pressure of the
vapour in equilibrium with a flat one. We can see from very general
considerations that this must be the case, for when water evaporates from
a flat surface there is no change in the area of the surface and therefore
no change in the potential energy due to surface-tension ; in the case of a
carved surface, however, such as a spherical drop, when water evaporates
there will be a diminution in the area of the surface and therefore A
CAPILLARITY.
167
diminution In the potential energy due to surFaoe-tendon. Thos the
surface-tension will promote evaporntion in this case, as evaporatian is
accompanied by a dtminiitian in the potential energy. Thus eTaporation
will go on Further from a spherical drop than from a plane autface ; that is;
the pressure of the wiiter vnpoiir in equilibrium with the spherioal drop is
greater than for the plane area.
Lord Kelvin's determination of the effect of curvature on the vapour
pres.<ure id aa fullows : liet a fine capillary
tube be placed in a liquid, let the liquid rise y-^ ^~~^^
to A in the tube, and let B be the level of the / \^
hquid in the outer vessel. Then there mu&t / \
be a state of equilibrium between the liquid j \
and its vapour both at A nod B, otherwise
evaporation or condensation would go on and
the system would not attain a steady state.
Let;; p' be the pressures of the vapour of the
liquid at B and A respectively, h the height
of A above B,
where a is the density of the vapour. If r
the radius of the surface of the liquid at A«
then T being the surface-tension,
2T M difference of pressure on the two sides of \
r the meniscus.
Now the pressure on the liquid side of the |
meniscus is equal to ^. — gph where ^ is the
density of che liquid and II the pressure at ^"i. IBS.
the level of the liquid suifaca in the outer
vessel ; the pressure on the vapour side of the meniscus is M—goh\ thus
the difference of pi-essures is equal to ^(p - s)h, so that
,.K.^^
Hence by equation (1)
* In the inveatlgatioD of the oapillaiTuceDt Id tubes given on p. l4l,r is neglected
Id comparlEOD with p.
■^ Ttie fornmUin tha text gives the Talneforp'-p when this is small comfared
withp; tha Kcaeial equation for p' may be proved to Le (Degleeting a in (Mmparisoo
whera 8 is the abiolate temperatore and R the constant in tha equation for a porleot
168
PROPERTIES OF MATTER.
hence the equilibrium vapour pressure over the concave hemispherical
surface is less than that over a plane surface at the same temperature by
2T<r/{p - a)r. We may write this as u«l(j> - a) where u is the amount by
which the pressure below the curved surface ia leas than that below the plane.
Iftheshape of the liquid surface had been convex, like that of a dewdrop,
instead of conovve, the pressure below the curved surface of liquid would be
greater than that in the phine surface instead of beiog less, and the pressure
of the water vapour over the surface would be great er than that over a plane
surface. It can be shown that if an external pressure u were applied to a plane
surface the vapour pressure would be increased by uajp (see J. J. Thomson,
ApplicalioM of Dynamics p. 171). Unless the drops are exceedingly small,
the effect of curvature on the vapour pressure is inappreciable ; thus if the
radius of the drop of water is one -thousandth part of a millimetre the
chan);e in the vapour pressure only amounts to about one part in nine
hundred. As the effect is inversely proportional to the I'adius, it in-
creases rapidly as the size of the drop diminishes, and for a drop 1 fi.fi
in radius the vapour pretuure over the drop when in equilibrium would be
more tlian double that over a plane surface. Thus a drop of this size
would evnporate rapidly in an atmosphere from which water would condeDfie
on a plane surface. This has a very important connection with the
phi-nomena attending the formation of rain and fog by the precipitation of
water vapour. Suppose that a drop of water had to grow from an
indefinitely email drop by precipitation of water vapour on its surface ;
since the vapour pressure in eijuilibrium with a very small drop is much
CAPILLARITY. 169'
grenter than tliB normnl, tbe dt-op, unless plikced in a spnce in wliich tbe
water vapour is in a very supersiktunited condition, will evaporate and
dimiDish in size instead of being tbe seat of condensation and increaeing in
radius. Tbus these email drops would be unstable and would quickly
disappear. Hence it would seem as if this would be an insuperable difficulty
to the formation of drops of rain or cloud if these drops have to pass
through an initial stage in which their size is very small. Aitken has
shown that as a matter of fact these drops are not formed under ordinaiy
conditions when water and water vapour alone are present, even though
the vapour is considerably oversaturnted, and that for the formation of
rain and fog the preseuce of dust ia necessary. As the water is deposited
around tbe particles of duet, the drops thus commence with a finite radius,
and so avoid the dilfioultiea connected with their early stages. The effect
of dust on the formation of cloud can be shown very easily by the following
experiment. A and ]i are two vessels connected with each other by a
flexible pipe ; when B is at the upper level indicated in the diagram the globe
A is partly filled with water ; if the vesjwl B is lowered the water runs out of
A, the volume of the gas in A increases, and the cooling cauned by the
expansion causes the region to be overaaturated with water vapour. If A is
filled with the ordinary dusty air from a room, a cloud is formed in A
whenever B ia lowered ; this cloud falls into the water, carrying some dust
with it ; on repeating the process a eecond time more dust it earned down,
and so by continued expansions the air can be made dust free. We find
that, after we have made a considerable number of expansions, the cloud
ceases to be formed when the expansion takes place; that the absence of
the cloud is due to the absence of dust can be proved by admitting a little
dii^t through tbe tube; on making the gas expand again a cloud is at once
formed.
It was supposed for some lime that without dust no clouds could bo
formed, but it has been shown by C, T. E. Wilson that gaseous ions can
act as nuclei for cloudy condensation if tbe supersatu ration exceeds a.
certain value, and he has also shown that if perfectly dust-free air has its
volume suddeidy vncreoKd 1*4 time a dense cloud is produced. Though
dust is not absolutely essential for tlie formation of clouds, yet the
conditions under which clouds can be formed without duat are very
exceptional, inasmuch as they require a very considenible degi'ee of super-
saturation.
Movement of Camphor on Water.— if a piece of camphor is
scraped and the shavings allowed to fall on a clear water surface they
dance about with great vigour. This, as Marnngoni has bhowa, is due b>
tLe camphor dissolving in the water, the solution having a smaller surface-
tension than pure water ; thus each little ptit^h of surface round a particle
of camphor issurrounded by a film havingasti'ongcr^iurfnce-tension than its
own, it will therefore be pulled out nnd the surface of the w.iter near the
bit of uampbor set in motion. For the moveiiRtits to take place the
surface-tension of the water suifiico must be greater than that of tho
camphor solution ; if tbe surfjwe is greasy the surf ace-ten siou is less than
that of pure water, and may be so much reiluced that it is no longer suflicient
to produce the camphor movements. Lord Rayleigh has measured the
thickness of the thinnest film of oil which will prevent the motion of the
camphor ; the thickness was determined by weighing a drop of oil which
was allowed to spread over a known area. Ha found that to stop
170
PROPERTIES OF MATTER.
the camphor movements (which involved a reduction of the surface-tension
by about 28 per cent.) a layer of oil 2 m/x thick was required (1 ^/x = 10"' cm.),
and that with thinner films the movements were htill perceptible. This
thickness is small compared with 12 ^fi tJie thickness found by Riicker
and Reirold for black films, but it must be remembered that the surface
which stops the camphor movements is still far from acting as a surface
of oil ; the surface-tension, though less than that of water, is greater than
that of oil. The manner in which the tension of a contaminated water
surface vaiies with the amount of contamination has been investigated
by Miss Poctkels and also by Lord Rayleigh {Phil, Mag.^ 48, p. 321). Miss
Pockels determined the surface-tension by measuring the force required to
detach a disc of known area from the surface; Lord Kayleigh used
Wilhelmy's method. The amount of contamination was varied by confining
the greased surface between strips of glass or metal dipping into the water;
by pulling these apart the area of the greased surface was increased and
therefore the thicknesri of it diminished, while by pushing them together
the thickness could be increased.
The way in which the surface-tension is affected by the thickness of the
layer of grease is shown by the carve (Fig. 137) given by Lord Bayleigh.
d
o
a
H
CQ
\
\c
Tliickuesd o£ Oil Film
Fio. 137.
In this curve the ordinates are the values of the surface-tension, the absdsssa
the thicknesses of the oil film ; both of these are on an arbitrary scale. It
will be seen that no change in the surface-tension occurs until the thickness
of the oil film exceeds a certain value (about Ifc./x); at this stage the surface-
tension begins to fall rapidly and continues to do so until it reaches the
thickness corresponding to the point G (about 2.;x./x.) ; this is called the cam-
phor point, being the thickness required to stop the movemt nts of the cam*
phor particles. After passing this point the variation of the surfac-tension
with the thickness of the film becomes much less rapid. Lord Rayleigh
gives reasons for thinking that the thickness l/i./i is equal to the diameter
of a molecule of oil.
CAPILLARITY.
171
Thus, wlien tlie fimount oFTOntfimmation is between the limits corre-
Bpoiiding tu a ihickiiets of tlie surface Uyei- between 1 ,i./. and the emallest
thickness required to give the surface-tension of oil, any diminution in the
contamination such as would be produced by an extenKion of the surface
would result in an increase in the surface-tension. Tliis is a principle of
preat importance ; it seems first to have been clearly Ktated by Klai'angoni.
Suppose we push a strip of metel along a surfacein this condition, the metal
will heap up the j^rease in front and scrape the surface behind, thus tlie
surface-tension behind the htrip will be greater than that in front, so that
the strip will be pulled back ; there will thus be a force resisting the motion
of theatiipdue to the variation of the suif ace-tension. This is Marangoni's
expltination of the phenomenon of superlicial viscosity discovered by Plateau.
Plateau found that ii i vibrating body sucli as a compass-needle was
disturbed from its position of equilibrium and then allowed to return to it
(1) with ita EUiface buried beneath the surface of the liquid, (2) with
its face on the surface of the liquid, then with certain liquids, of whicli
water was one, the time taken in the second case is considerably greater
than that in the fii-st. We see that it must be so if the surface of the
liquid ia contaminated by a foreign substance which lowers its surface-
tension.
Calming* of Waves by Oil. — Similm- considerations will explain the
action of oil in stilling trouble<l wnters. Let us suppose that the wind
acts on a portion of a contaminated surface, blowing it forward; the
motion of the surface film will make the liquid behind the patch cleaner
and therefore increase ita surface-tension, white it will heap up the oil in
front and so diminish the surface-tension; thus the puU back will be
greater tlian the pull forward, and the motion of the surfiice will be
returiled in a way that could not occur if it wei* perfectly clean. The
oiled surface acts so as to check any relative motion of the various parts
of the surface layer and so prevents any heaping up of the water. It is
these heaps of water which, under the action of the wind, develop into a
high sea ; the oil act-* not so much by smoothing them down after they
have grown as by stifling them at their birth.
A contaminated surface has a power of self-adjustment by which the
surface-tension can adjuRt itself within fairly wide limits; a film of such a
liquid can thus, as Lord Rayleigh points out, adjust itself so as to be in
equilibrium under circumstances when a film of a pure liquid would have
to break. Thus, to take the case of a vertical film, if the surface-tension
were absolutely constant, as it is in the case of a pure liquid when the Him
172 PROPERTIES OF MATl'ER.
is not too thin, thU film would break, since there would be nothing to
balance the weight of the film. If, however, the film were dirty, a very
slight adjustment of the amount of dirt at different parts of the surface
would be sufficient to produce a distribution of surface-tension which would
ensure equilibrium. It is probably on this account that films to be durable
have to be made of a mixture of substances, such as soap and water.
Collision of Drops. — If a jet of water be turned nearly vertically
upwards the drops into which it breaks will collide with each other; if the
water is clean the drops will rebound from each other affer a collision, but
if a little soap or oil is added to the water, or if an electrified rod is held
near the jet, the drops when they strike will coalesce instead of
rebounding, and in consequence will grow to a much larger size. This can
be made very evident by allowing the drops to fall on a metal plate ; the
change in the tone of the sound caused by the drops striking against the
plate when an electrified rod is held near the jet is very remarkable.
The same thing can be shown with two colliding streams. If two
streams of pure water strike against each other in dust-free air, as in
Fig. 188, they will rebound ; if an electrified rod is held near, however,
they coalesca.
CHAPTER XV.
LAPLACE'S THEORY OF CAPILLARITY.
Contents. — Intrinsic Pressure in a Pluid — Work required to move a Particle from
the Inside to the Outside of a Liquid— Work required to produce a new Liquid
Surface — Effect of Curvature of surface— ThicTcness at which Surface-tension
changes effect of abruptness of transition between two Liquids in contact.
Laplace's investigations on surface-tension throw so much light on this
subject, as well as on the constitution of liquids and gases, that no account
of the phenomena associated with surface-tension would be complete without
an attempt to give a sketch of his theory. Laplace started with the assump-
tion that the foi*ces between two molecules of a liquid, although very intense
when the distance between the molecules is very small, diminish so rapidly
when this distance increases that they may be taken as vanishing when the
distance between the molecules exceeds a certain value c : c is called the
range of molecular action. We shall find that we can obtain an ezplana-
3 5-
, i
A
Fio. 189.
tion of many surface-tension phenomena even although we do not know
the law of force between the molecules. Let the attraction of an infinite
flat plate of the fluid bounded by a plane surface on a mass m at a point
at a distance z above the surface be vi(ry\/{z\ where a is the density of the
fluid ; in accordance with our hypothesis y\/{z) vanishes when z is greater
than c. It is evident, too, that ma\p(z) will be the attraction at a point ou
the axis of any disc with a flat face whose thickness is greater than c and
whose diameter is greater than 2c.
Suppose we imagine a fluid divided into two portions A and J5 by a
plane ; let us find the pull exerted on Bhy A, Divide B up into thin layers
whose thickness is dz ; then if 2? is the height of one of these layers above the
surface of separation the force on unit area of the layer is equal to (r\(/{z)(rdz ;
hence the pull of -i on B per unit area is equal to <r* / \ly{z)dzy
o
which, since \l/(z) vanishes when 2;> c,is the same aa a^ I ;//(5j)cfe.
This pull between the portions ^ and ^ is supposed to be balanced by a
pressure called the "intrinsic pressure," which we shall denote by K. K then
m
ifi equal to «r* / \l4js)dz
174 PROPERTIES OF MATTER
We Bhall find that the phenomena of capill irity require ns to snppoee
that, in the case of water, the intrinsic pressure is very large, amounting
on the lowest estimate to sevei-al thousand atmospheres. We may remark
in passing that the intrinsic pressure plays a very important part in
Van der Waals' Theory of the Continuity of the Liquid and Gaseous States ;
it is the term ajt^ which occurs in his well-known equation
^/> + 5^(t;-6) = RT (^p.l29)
We see, too, at once from the preceding investigation that K is equal
to the tensile strength of the liquid, so that if the common supposition
that liquids are as " weak as water," and can only bear very small tensile
stresses without rupture, were true, Laplace's theory, which, as we have
seen, requires liquids to possess great tensile strength, would break down
at the outset. We have seen, however, p. 122, that the rupture of
liquids under ordinary conditions gives no evidence as to the real ten<^ile
strength of the liquids, for it was shown that when water and other
liquids are carefully deprived of gas bubbles — in fact, when they are
not broken before the tension is applied — they can stand a tension of a
great many atmospheres without rupture ; thus on this point the properties
of liquids are in accordance with Laplace's theory.
There is another interpretiition of K given by Dupr^ which enables
us to form an estimate of its value. Consider a film of thickness A
(where A is small compared with o) at the top of the liquid ; the work
required to pull unit of area of this film off the liquid and remove it
out of the sphere of its attraction \& evidently
■/
a»A / yi^{z)dz or KA
Thus the work required to remove unit volume of the liquid and
scatter it through space in the form of thin plates whose thickness is
small compared with the range of molecular attraction is K. Now the
work required to take one of these films and still further disintegrate
it until each molecule is out of the sphere of action of the others will
be small compared with the work required to tear the film off the surface
of the liquid ; hence K is the work required to disintegrate unit volumes
of the liquid until its molecules are so far apart that they no longer
exert any attraction on each other ; in other words, it is the work required
to vaporise unit volume of the gas. In the case of water at atmospheric
temperature this is about 600 thermal units or GOO x 4-2 x 10" = 2.')*2 x 10^
mechanical units; or since an atmosphere expressed in these units is 10*
this would make K equal to about 25 000 atmospheres *
Work required to move a Particle ttoxa the Inside to the
Outside of a Fluid. — Consi<ier th^ force on a particle P at a depth z
below the surface ; the force due to the stratum of fluid above P will be
balanced by the attraction of the stratum of thickness z below P ; thus
tiie force acting on P will be that due to a slab of liquid on a particle at
* Van der Waals gives the following value of K deduced from his equation ;
water 10,500-10,700, ether 1300-1430, alcphol ^100-2400, carbon bisulphide 2900-2899
atmospheres.
LAPLACE'S THEORY OF CAPILLARITY. 175
a dL^fcance z above its surface — ».«., mir\{z). Hence the work done in
bringing the particle to the surface is
w / (T^\{z)dz = m(K/<r) ;
o
as an equal amount of work will be required to take the particle from the
surface out of the range of molecular attraction, the total amount of work
required is thus 2m(K/op).
Hence, if a particle moving with a velocity v towards the surface starts
from a depth greater than c it cannot cross the surface unless
, , 2mK , 4K
In the case of water, for which er = 1 and K on the preceding estimate
is 25,000 atmospheres or 2 5 x 10^", we see that a particle would not cross
the surface unless its velocity were greater than 3*2 x 10*. The aveiuge
velocity of a molecule of water vjipour at 0° C. is about 6 x 10*, so that if
the water contained molecules of water vapour it would only be those
possessing a velocity considerably greater than the mean velocity, which
would be able to escape across the surface.
Work required to produce a new Liquid Surface. —Let u« con-
aider the amount of work required to separate the two portions A and B
into which a plane G divides the liquid. Dividing B up, as before, into
slices parallel to the interface, then the work done in removing the slice,
whose thickness is dz and whose height above the plane is &, is i)er unit
of area equal to
a'dz I \p(x)dx = a'dzVjif v= I \p{x)(lx
z z
hence the work required to remove the whole of the liquid B standing on
unit area away from A\a j <Pvdz ;
integrating this by parts we see that it is equal to
00 Of)
O O
Now the term within brackets vanishes at both limits, and -, " - >/'(«)>
dz
hence the work required is a^ I z4^{z)dz
For this amount of work we have got 2 units of area of new surface,
hence the energy corresponding to each unit of area (t.«., ttfe surface-
tension), which we shall denote by T is given by the equation
T»i^/
zyl^{z)dz (I)
176
PROPERTIES OF MA'ITER.
Young, at the beginning of the century, showed how from T and K it
was possible to calculate the range of molecular forces. He did this by
assuming a particular value for the force, but his argument is applicable
even when we leave the force undetermined.
If \Uz\ is always positive, then, since c is the greatest value of z for
which \plz) has a finite value, we see from equation (1) that
T<i(7-c A(5
)dz
hence
c>
2T
For water T = 75, and if we take K = 25,000 atmospheres = 2 5 x 10^^ then
the above relation shows that c>(lx lO'*. In this way we can get an
D
\
\
\
\
/
/
Fio. 140.
inferior limit to the range of molecular action. This method, which was
given by Young, was the first attempt to estimate this quantity, and it
seems to have been quite overlooked until attention was recently called to
it by Lord Rayleigh.
It is instructive to consider another way of finding the expression for
the surface-tension. Consider a point P inside a liquid sphere (Fig. 140).
Then, if P is at depth d^ below the surface, greater than c, the forces acting
on it, due to the attraction of the surrounding molectdes, are in
equilibrium ; if the distance of P below the surface is less than c, then
to find the force on F describe a sphere with radius c and centre P, and
LAPLACE'S THEORY OF CAPILLARITY. 177
the force on P, acting towivrds the centre of the larger sphere, will be
equal to the attrection which would be exerted on P by a quantity of the
fluid placed eo as to till BA CD, the portion of the sphere wliose centre ia
F, which is outside the larger sphere. This portion may be regarded as
coDBisting of two parts — (1) the portion above the timgent plane at A, the
poiut on the large sphere nearest to P, and (2) the lenticular portion
between this plane and the
sphere. Now the attraclion
of the portion above the
tangent plane is the same oh
that, of a slab of the liquid
extending to infinity and
having the tangent plane for
lis, lower face, for the por-
tions of liquid which have
to be added to the volume
ADEF to make up this dab '
are at a greater distance ^'°- '*■■
from P than c, and so do
not esert any attraction on matter at P. Thus, if AP = z, the attraction
of A FDE on unit mass at P, using the previous notation, is o^^s) ; the
attraction of the lenticular portion at P can bo shown to be^4'{s)where
K is the radius of the liquid sphere. Hence the total force at P acting
on unit mass in the direction AP is equal to
"+(-)+ 5«») (s)
Consider now the equilibrium of a thin cylinder of the fluid, the axis of
the cylinder being PA ; divide this cylinder up into thin discs, then if dz
is the thickness of n disc.s its distance from A and a the area of the crosa-
Boctiou of the cylinder, the force acting on this disc is equal to
{^«.)»J+W}.4i
Tl>i3 force has to be balanced by the escess of pressure on the lower face
of the disc over that on the upper face; this excess of pressure is, tf p
represents the pressure, equal to aj-dz',
heuce, equating this to the force acting on the disc, we get
Thus the excess of pressure at a point at a distance c, below A over th«
pressure aH A a etjual to
2T
or with our previous notation K + — ~ .
178
PROPERTIES OF MAITER
The pressure has the same value at all points whose depth below the
Aurface is greater than c. The term 2T/R represents the excess of
pressure due to the curvature of the surface ; we obtained the same value
by a different process on
p. 145. If the sui'face of
the liquid sphere had been
concave instead of convex,
an inspection of the figure
shows that to obtain the
force on P we should
have to subtract the attrac-
tion due to the lenticular
portion fiom the attrac-
tion due to the portion
ADE instead of adding
it; this would make the
pressure at a point in the
mass of the fluid less than
that at a point in the
fluid but close to the
1 surface by 2T/11.
Thickness at which
the Surface-tension
Changfes. — We can determine the point at which the surface-tension
begins to change by finding the change of pi*essure which takes place as
we cross a thin film. Let Fi^. 143 represent the section of such a film,
bounded by spheres ; if the thickness of the film is small, the radii of these
spheres may be taken as approximately equal. Let P be a point in the film,
ABP a line at right angles to both surfaces, then the investigation just
given shows that ii AP^z, BP = z\ the force on unit mass at i'is equal to
when R is the radius of one of the films. We see, too, from the last paragraph
that the pressure at B must be greater than that at A by
t t ^ t
J{^i{z) + q^\{z)]dz - J WW) - ^y^{z')]dz ^ ^^Jz^[z)dM
o
where t is equal to AB, the thickness of the film. Hence, from the for-
mula (p. 145) for the diflference of pressure inside and outside a soap
bubble, we may regard
FiQ. 142.
'^Jzxl^z)di
as the surface-tension of a film of thickness t. Since \f/{z) vanishes when
z is greater than c, the surface-tension will leach a constant value when t
is as great as c ; hence c, the range of molecular action, is the thickness of a
LAPLACE'S THEORY OF CAPILLARITY.
179
film when the surface-tension begins to fall off. When t is less than c we
see from the preceding expression that, T being the surface-tensioni
Now if T is represented by a curve like Fig. 134, dT/Ji is zero down to P,
positive from P to R, negative from R to T, and positive again for all
thinner films; hence, siuce the force of a slab is attractive when \// is
positive, repulsive when i// is negative, this would imply, on Laplace's
theory, that the molecular forces due to a slab of liquid at a point outside
are at first attractions ; then, as the point gets nearer the slab, they change
to repulsions, and change again
to attractions as the point ap<
proaches still nearer to the slab.
If t is so small that i//(<) can be
regarded as constant, we see
that T will vary as t', so that
ultimately the surface - tension
will diminish very rapidly as the
film gets thinner.
On the Effect of the Ab-
ruptness of Transition be-
tween two Liquids on the
Surface-tension of their Interface.— Laplace ascumed that the range
of molecular forces was the same for all bodies, and that at equal distances
the force was proportional to the density of the substance. This implies
that the function \l4js) is the same for all bodies. This hypothesis is
certainly not general enough to cover all the facts; it is probably,
however, sufficiently general to give the broad outlines of capillary
phenomena. Let us calculate on this hypothesis the surface-tension
between two fiuids A and B. Let o*, (r, be the densities of these fluids ;
then to separate a sphere whose area is S from the liquid A requires the
expendituie of work equal to
Fio. 143.
JS(r,' jzi{z)dz
{see p. 175)
Let us make a spherical hole of equal size in B. To do this will require
the expenditure of an amount of work equal to
t^'^h
z\l^{z)dz
Let us place the sphere A in the hole in B, and let the fluids come into
contact under their molecular forces ; during this process the amount of
work done by these forces is
S<
h^tj^iil^)^
180 PROPERTIES OF MATTER.
Hence the total expenditure of work required to produce an area S of
interface of A and B is
iS(r,» j%yi{z)dz + JS<r,» fz\P{z)dz - Str^tr, jz\l,{z)d»
• o •
But this work is by definition equal to T^bS where T^d is the surface-
tension between A and B ; hence we see that T^^ = {^i - 0'^» where
C^iJzxP{z)dz
is a constant for all substances. This result is not a complete representa-
tion of the surface-tension, for if it were there would always be surface-
tension between liquids of different densities, so that two such liquids
could not mix; it would also require that the surface-tension between
fluids of equal density should be zero, and that
vTab = vTac + vTci
where Txb» T^o ^^^ '^cb are respectively the surface -tensions between fluids
A and B, A and 0, and B and respectively. None of these results are in
accordance with experiment. Let us, however, on the Assumption that the
surface-tension is represented by an expression of this kind, calculate
(following Lord Rayleigh) the effect of making the transition between
A and B more gradual ; we can do' this by supposing that we have between
A and B a layer of a third fluid whose density is the arithmetical mean
between the densities of A and B; then T^c = i T^u = Tcb> Hence, though
now we have two surfaces of separation instead of one, the energy per unit
area of each is only one quarter of that of unit area of the original surface ;
hence the total energy due to surface-tension is only one half of the
energy when the transition was more abrupt. By making the transition
between A and B still more gradual by interposing n liquids whose
densities are in arithmetical progrcFs, we reduce the energy due to surface-
tension to l(n-t- 1) of its original value. Thus we concludf^ that any dimi-
nution in the abruptness will diminish the energy due to surface tentdoti.
This result may have important bearings on the nature of chemical action
between the surface layers of liquids in contact, for if a layer of a chemical
compound of A and B were interposed between A and B the transition
between A and B would be less abrupt than if they were directly in contact,
and therefore the potential energy, as far as it results from surface-tension,
would be less. Chemical combination between A and B would result in a
diminution of this potential energy. Now anything that tends to increase
the diminution in potential energy resulting from the chemical combina-
tion promotes the combination; the forces that give rise to surface-
tension would, therefore, tend to pi*omote the chemical combination. Thus,
in the chemical combination between thin layers of liquid there is a factor
present which is absent or insignificant in tV*^ case of liquids in bulk, and
LAPLACE'S THEORY OF CAPlLLAftlTY. 181
we may expect that chemical combination between thin layers of liquids
might take place even though it were absent in ordinary cases.
Similar considerations would lead us to expect changes in the strength of a
solution near the surface whenever the surface-tension of the solution depends
upon its strength : if the suiface-tension inci-eased with the strength there
would be a tendency for the salt to leave the surface layers, while if the
surface-tension diminished as the strength of the solution increased the
salt would tend to get to the surface, so that the surface layers would Le
stronger solutions than the bulk of the liquid. The concentration or
dilution of the surface layers would go on until the gradient of the
osmotic pressures resulting from the variation in the strengths of different
layers is so great that the tendency to make the pressure equal just
balances the effects due to surface-tension.
CHAPTER XVI.
DIFFUSION OF LIQUIDS.
Contents. — General Law of Diflfusion— Methods of determining the Co efficient of
Diffusion — Diffusion through Membranes. Osmosis — Osmotic Pie-sure — Vapour
Pressure of a Solution— Elevation of the Boilirg-point of Solutions — Depression
of the ricezing-point— Dissociation of Electrolytes.
^^
If two liquids are left in contact with each other and are free from the
action of external forces, then if they can mix in any proportion they will
of themselves go on mixing until the whole mass is uniform in composi-
tion. This process may be illustrated by taking a vertical glass tube and
filling the lower part with a strong solution of a coloured salt, such as
copper sulphate. On the top of this clear water is poured very slowly
and carefully, so as not to give rise to any currents in the liquid. The
coloured part will at first be separated from the clear by a sharply marked
surface, but if the vessel is left to itself it will be found that the upper
part will become coloured, the colour getting fainter towards the top,
while the colour in the lower part of the
tube will become fainter than it was origin-
ally. This change in colour will go on until
ultimately the whole of the tube is of a
uniform colour. There is thus a gradual
transference of the salt from the places
where the solution is strong to those where
it is weak and of water in the opposite
direction, and equilibrium is not attained
until the strength of the solution is uniform.
This process is called diffusion. In liquids
it is an exceedingly slow process. Thus, if
the tube containing the copper sulphate
solution were a metre long and the lower
half were filled with the solution, the upper
half with pure water, it would take con-
siderably more than ten years before the
mixture became approximately uniform ; if the height of the tube were a
centimetre, it would take about ten hours, the time required being
proportional to the square of the length of the tube.
The first systematic experiments on diffusion were made by Graham in
]851. Th^ method he used was to take a wide-necked bottle, such as is
shown in Fig. 144, and fill it to within a shoi*t distance of the top with
the salt solution to be examined ; the bottle was then carefully filled up
with pure water pressed from a sponge on to a disc of cork floating on the
top of the solution ; the bottle was placed in a larger vessel filled with
pure water to about an inch above the top of the bottle. This was left
undisturbed for several days, and then the amount of salt which had
escaped from the battle into the outer vessel was determined. Graham
was in tbis way able to show that solutions of the same strength of
Fio. 14i.
V DIFFUSION OF LIQUID^ 183 ^H
^1 difTerent substances diJTused witli different velocitiee ; that solutions of the ^^|
^M saDie salt of difierent streugths diQused with velocities proporttooal to the ^^|
^M strength ; that the rate of diil'itsiau increased with the temperature, und ^^H
^M that the proportion of two salts in a mixture was altered by diffusion, and ^^H
^M thiit in some citses a decomposition or separation of the constituents of ^^H
^M coinpltcated salts, such as bisulphate of potash and potash oluio, could be '
^M brought about by diffusion. Though Graham's experiments proved many
^M important and interesting properties of diffusion, they did not lead to
^M sufficiently definite laws to enable us to calculate the state of a mixtore at
■ any futui-e time from its state at the present time. This step was made
^M by Pick, who, guided by Fourier's law of the conduction of heat— the
^B diffusion of tempera lure— enunciated in 1855 the law of diffusion, which i
I
■1
1
\
-
1
1
\
u
\V
^
\
y
\
.^
^
\
\
\
v^
\
<-
>
^^H
\
\
s-
.^
N
s
V-
\
^H
\
s
.-^
^
^
^
^
=
■
V.
V
!!;;;
:~
has been abundantly veria^d by subsequent experiments. Ficlt's law may ^^M
be stated iia follows : Imauiue a mixture of silt and water airanged so that ^^H
layei-a of equal density ai-e hoi-izontal. Let the stale of the mixture ^^1
be such timt in the layer at a heiRlit x above a fixed pkne there are ^^1
n gi-ammes of salt per cubic centimetre ; then across unit area of this ^^1
plane It'-^ grammes of salt will pass in unit time from the side on which ^^M
the solution ia stronger to that on which it is weaker. R is oalle-J the ^^M
diH'u.sivity of the substance ; it depends on the nature of the salt and the ^^1
solvent, on tlie temperature, and to a slight extent on the strength of ^^H
the solution. This law Lt analogous to Fourier's law of the conduction of ^^H
heat, and the some mathematical methods which gtvo the solution of the ^^H
thermal problems can be applied to determine the distribution of salt ^^H
through the liquid. The curves in Figs. 145 and 14(i represent the solution ^^H
of two important problemx. The fii'st i-epresents the diffusion of salt from ^^H
a saturated solution into a vertical column of water, the surface of sepora- ^^H
tion being initially the plane ;e=o. The oi'dinates represent the amount ^^H
of folt in the solution at a distance from the original surface of separation ^^H
re[ii*e8ented by the abscissa). The times which have elapsed since the ^^H
commencement of diffusion are proportioual to the sijuares of the number* ^^1
rsi
PROPERTIES OF MATTER.
tm the curve ; thus, if the first curve reprosents the stat« of things after
time T, the second represeiita it after n time 2T, the third afler a time
ST, aod Eo on ; for the same ordiuate the abaci?-sa on curve 2 is twice
that on curre 1, on curve S three times that on curve 1, luid ?o on; thus
the time required for diBusion through a given length is proportional to
the square of the length. The curves are copied from Lord Kelvin's
Collected Papers, vol. iii. p. 432 : for copper sulphate through water
T-= 25,700 seconds, for pugar through water T=17,100,and for sodium
chloride through water T = ."j3'J0. 'I'liq second figure. Fig, Hd, represents
the diffusion when we have initially n,tliin layer of fait solution at the bottom
nf a vertical vessel, the rest of the vessel being filled with pure water; the
ordinates represent the aranunt of salt at a distance from the bottom of
the vessel represented by the abscissa;. The times which have elapsed
since the commencement are
proportional to the squares of
the numbers on the curvM.
By stirring up a solution
of a salt with pure water we
bring thin layers of the solvenb
and of the salt near together;
OB the time required for difius-
ing through a given distance
varies as the square of the
distance, the time required
tor the salt and water to
become a uniform mixture is
greatly diminished by drawing
out the liquid into these thin
layers by stirring, and as
much diffusion will take much
in a few seconds as would
take place in as many hours
ig. We can see in a general way why the time required
lal to the square of the thickness of the layers; for if we
la of tiie layers wG not only halve the distance the salt
double the gradient of the strength of the solution,
aw double the speed of dilFusion ; thus, as we halve
the distance and double the speed, the time required is reduced to one
quarter of its original value.
Methods of Determining' the Coefficient of Diffusion.— if we
know the original distribution of the salt through the water and the value of
R, we con, by Fourier's mathematical methods, calculate the distribution of
salt after any intei'val T ; conversely, if we know the distribution after this
interval, we can use the Fourier result to determine the value of R.
Thus, if we have any means of measuring the amount of salt in the
different parts of the solution at successive inten*als, wo can deduce the
value of R. It is not advisable to withdraw a sample from the solution
and then determine its composition, as the withdmwul of the .'ample
might produce currents in the liquids whose effects might far outweigh any
due lo pure difi'usion ; it b, therefore, necessary to sample the compoaitioQ
of the solution when in ailu, and this has been done by measuring some
physical property of the solution v-*hich varies in a known way with tb«
without the mi]
will be proporti
halve the thicknes
has to travel but '
and thus by Fick;
DIFFUSION OF LIQUIDS. 185
strength of the solution. In Lord Kelvin's melliod the upecifie gravity is
the property investigated : the lower half of n vertical vessel is filled with
a solution, the upper halt with pure water, (-lass beads of different densities
are placed in the solution ; at first they float at the junction of the solution
and the water, but as diffusion goes on they separate out, the heavier ones
sink and the lighter ones rise. By noting the position of the beads of
known density we lan get the distribution of salt in the solution, and
thence deduce the value of R. The objection to the method is that air
bubbles are apt to form on the beads when salt will crystallise out on them,
and thus alter their buoyancy. In the oise of sugar solutions the strength
of the different layers can be determined by the rotation of the plane of
polarisation. H. F. Weber verified Fick'a law in the case of zinc sulphate
solution by measuring the eleGtrom.otive force between two amalgamated
sine plates; he had previously determined how the electromotive force
depends on the strength of the solutions in contact with the plates. The
diffu8ionofdifferent8altawascomparedbyLong{H'i«i. iinn. 9, p. Gl3)bythe
method shown in Fig. 147. A stream of pure water flows through the bent
tube, a wide tube fastened on to the bent tube establishes communication with
the solution in the beaker ; after the wat«r has flowed through the bent tube
for some time the amount of salt it carries over in a given time becomea
constant. As the wat«r in the tube is continually being renewed, while the
strength of the solution in the beaker may be regarded as constant, since in
the experiments only a very small fraction of the salt is carried over, the
gradient of concentration in the neck will be proportional to the etrengtb of
the solution ; so that the amount of salt carried off by the stream of water
in unit time is proportional to the product of the diliusivity and the
strength of the solution. By measuring the amount of salt earned over by
the stream in unit time the diffusivities of different salts can be compared.
As a result of these expevimente it has been found that as a general rule
the higher the electrical conductivity of a solution of a salt the more
rapidly does the salt diffuse. The relative values of the diffusivity for some
of the commoner salts and acids are given in the table on p. 18C. The
Fohitions contain the same number of gramme equivalents per litre, and
the numbers in the table are proportional to the number of molecules of
the salt which ci-ofs unit sui^ace ia unit time under the same gradient of
strength of solutioii.
tiuli'-tanee.
KCl
NH.Ol .
NaOl .
LiCl
KCy ,
BaC), ,
SnCl, ,
OoCI,
M|.'C1, .
COCI, ,
NiCl, ,
KBr
PROPERTIES OF MATTER.
KI
0U9
Nal
NH.NO,
KNUj .
N.1NO, .
LiNO, .
BnN.U, .
SrN,0„ ,
(NH.),SM
Nu,f!U, .
MgSO, .
ZdSO, .
CiiSO, .
MnSU. .
298
Thexe nuinbei'ii show tbat as a general rule the Raits which diUuso the most
rnpidly are those whose solutions hnre the highest electrical conductivity.
The absolute values of the diffusivity foralni-ge number of subatftucea have
been determined by SchuhmeLster (irieit. Akad. 70, p. fi03) and Scbeffer
(CAem. Bei: xv. p. 788, svi, p. 1908). The largest value of the diffusivity
found by Scheft'er was for nitric acid ; tbe difitisivity varied with tlie
concentration and with the temperature; for very dilute solutions at
90° C. it was 2 X 10'* (cm.y/sec. — i.-e.,if the strength of solution varied by
one per cent, in 1 cm. the amount of acid crossiog unit area in one second
would be about one Gve-millionths of the acid in 1 c.c. of the solution.
For Bolutions of NaCl the diffusivity was only about one half of this value.
Graham found that the velocity of diffusion of N^aOl through gelatine was
about the s.ime as through water.
Diffusion throug'h Membranes. Osmosis.— Graham was led by
his esperiments on ditlusion to divide substances into two classes — (.'rystaf-
loid and colloid. The crystalloids, which include mineral acids and salts,
and which as a iTile can bo obtained in definite ctystalline forms, diH'use
much more rapidly than the siibstaDces called by Graham colloids, such as
the guma, albumen, starch, glast, which are amorphous and bhow no signs
of crystallisation. The crystalloids when dissolved in water change in a
marked degree its properties : for eKample, they diminish the vapour
pressure, lower the freezing- and raise the boiling-point, Colloidal sub-
6t-anees, when dissolved in water, hardly produce any efl'ects of this kind,
in fact, many colloidal solutions seem to be little more than mechanical
mixtures, the colloid in a very finely divided state being susFiended in the
Uuid, Tbe properties of solutions of this class are very interesting ; thd
pai'liclea move in tho electric field, in some cases as if they were positively,
in others as if they were negatively, charged. The addition of a trace of
acid or alkali is often sufficient to produce precipitjition. Tlie reader will
find an account of the properties of these solutions In papers by Ficton
and Linder {Journal vf Ckemieal SoeUti/, vol. 70, p. 5G8, 18'J7 ; vol. Gl,
p. U8, isa?); Stoeckj and Vanino {Ztitgekrifi f. Phya. Cktm., vol. SO,
p. 98, 1899); ^axAy (Proeetdings of Royal Soaely, Cli, p. 110; Journal of
Physiology, 24, p. 288). Colloidal sut^tances when mixed with not t«0
much water form jellies ; the structure of these jelHes is sometimes on a
Guflicientty coarse ecaXe to be visible under thv microscope (m4 Uurdy,
DIFFUSION OP LIQUIDS.
187
Pronedit^ffs lioyal SocUUj, PC, p. 05. 1000), Rnd apparently eonsistB of a
more or JufeB solid fraiuewoik tlii-ouiili which the liquid is diapereed.
Through many of these jellies crystalloida ore aLle to diffuse with a
velocity approaching that through pure wat^r ; the colloids, on the other
hand, are stopped by such jellies. Gi"aham founded on thia a method for the
Beparation of crj'sta Holds and colloids, called dialy^ia, In tliia melhoU a film of
a colloidal substance, tuch aa parchment paper r-
(paper ti-enled with Kulphurio arid) or a piece of
bladder ie fastened round the end of a gWs tube,
the lower end of the tube dippingin water which is
fi'equeotly changed, and the solution of cryttalloida
iind eolloids is put in the tube above the parchment
paper. I'he oi-j's-tulloitls diffuse tlii'oiif;h inlo the
water, and the colloids remain behind ; if time be
given and the water into which the crystiilloids
diiluse be kept fi-esh, tlieciysialbids can be entirely
eepHi-ated from the colloids.
The passage of liquids tlirougli films of this
kind is called osmosia. The fii'sC example of it
Bcoms to have been observed by the Abb6 NoUet.
in 1 74S, who found that when a blnddef full of
alcohol was immersed in water, the wat«r entered
the bladder more rapidly than the alcuhol escaped,
ED that the bladder swelled out and almost bur.^t.
If, on the other hand, a bladder containing water
was placed in aloohoJ the bladder shrank,
The motion of fluids through these membranes
can be observed with very simple apparatus: all
that is necessai'y ia to attach a piece of parchment-
paper firmly on the end of a glass tube, the upper
poiliou of which is drawn out into a line capillary
tube. If this tube ia fflled with a solution of sugar
and immersed in pure water, the top of the liquid
in the capillary part of the tube moves upwards
with sensible velocity, showing the entrance of
water through the p;xichment-papcr. Grrtham
regarded this trnn^port of water through the
membrane as due to this colloidal substance being
able to hold more water in combination when in
contact with pure water than when in contact with
a salt solution; thus, when the hydration of the
membmne corresponding to tlie side next the water
extendsto the side next the solution, the membrane Fra. 1-iS,
cannot hold all the water in combination, and some
of it is given up ; in this way water is transported from one side oi the
membrane to the other.
Membranes of parchment-paper or bladder are permeable by crystailloida
as well as by water. There are other membranes, however, which, while
permeable t« water are impermeable to a large numher of salts; these
membranes are called semi -permeable membranes. One of these, which
lias been extensively need, ia the gelatinous precipitate of ferrocyanide of
copper, which is produced when copper sulphate and potassium ferro-
[)
I8S PROPEttTIES OF MATTEII.
cyaniile come into contAct. Thia [recipilate is m ecli an itally exceedingly
weak, but PEeffer made serviceable membrnTies by precipitating it in the
pores of a porous pot. If eudi a pot ifi tilled with a very dilute solution
of copper sulphate and immersed in one of ferrocyanide of putnesiuin the
two solutions will difliise into the walls of the pot, nnd where they meet
the gelatinous precipitate of ferrocyanide of copper will be formed; in this
way a continuous membrane may l>c obtained. For details as to the pre-
cautions which must be taken in tbe preparation of these membranes tlie
reader is referred to a paper by Adie {Proeeedinga of Chemical Society,
lix. p. S44). If a membrane of this kind be deposited in a porous pot
fitted with a pressure gauge, as in Fig. 118, and the pot be filled with a
dilute solution of a salt and immersed in pui'e water, water will flow into
the pot and compress the air in the gauge, the
pressure in the pot increasing until a definite
pressure is reached depending on the strength
of the solution. When this pressure ia
reached there is equilibrium, and there is no
further increase in the volume of water in
the |Jot.
Osmotic Pressure. — Thus the flow of
water through the membrane into tha
stronger solution can be prevented by apply-
ing to the solution a definite pressure ; this
pressure is ctilled the osmotic pressure of the
solution. It is a quantity of fundamental
importance in considering tbe properties of
the solution, as many of these properties,
such as the diminution in the vapour pres-
sure, and the lowering of the freesiog-point,
are determinate as sood as tbe osmotic
pressure is known.
The work done when a volume v of
water passes across a semi permeable mem-
brane from pure water into a solution whera
the osmotic pressure is P is equal to Pb.
For, lot tbe solution be enclosed in a vertical
Water tube closed at the bottom by a semi-peiineabla
membrane, then when there is equilibrium
the solution ia at such a height in the tube
that the pressure at the membrane due to
tbe head of the solution is equal to the
osmotic pressure. When the system ia is
equilibrium we know by Mechanics that the total work done during any
small alteration of the system must be zero. Let this alteration consist in
a volume V of water going through the semi- permeable membrane. This
will raise the level of the solution, and the work done against gravity is
the same as if a volume v of the solution were liaised from the level of the
membrane to that of the top of the liquid in the tube. Thus the work done
against gravity is vgph, where A ia the height of the solution in the tube
and p the density of tlie solution ; since the pressure due to the head of
solution is equal to the osmotic pressure, ffph=P. Hence the work done
against gravity by this alteration ia Pv, and dnce the total work dona
'■/'"ibrane
Flo 149.
DIFFUSION OF LIQUIDS.
189
iDUBt be zero, the work done oti the liquid when it crosses the membrane
must be Pr.
The values of the osmotic pressures for dilTerent Bolutiona was first
determined by Pfeffer," who found the very remarkable result that for
weak solutions which do not conduct electricity the osmotic pressure 19
equal to the gaseous pressure which would be exerted by the molecules of
the salt if these were in the gaseous state and occupying a volume equal
to that of the solvent in which the salt is dissolved. Thus, if 1 gramme
equivalent of the salt were disaolved in a litre of water the osmotic pressure
would be about 22 atmospheres, -which is the pressure exertMi by 2
gramiiiea of hydrogen occupying a litre. Pfeffer's esperimentfl showed
that aiiproximntely, at any rate, the osmotic pi-essure was, like the procure
of a gas, proportiiiHiil to the absolute temperature. If the cell is placed in
another solution instead of pure water, water will tend to run into the cell
if the osmotic pressure of the solution in the cell is greater than that of
the solution in which it i:i immersed, while if the osmotic pi-essure in the
cell is less than that out.-ide the volanie of wfiter in the cell will decrease;
if the osmotic pressure is the sama inside and outside there will be no
change in the volume of the water inside the cell. Solutions which have
the siimo osmotic pressure are called isotonic solutions. A convenient
method of finding the strenf<ths of solutions of different salts which are
isotonic was iuveuted by De Vries.t He showed that the membrane lining
the cell-wall of the leaves of some plants, such as TradebCanlia disculor,
Cui-cuma nibricaTilh, and Bei/miia inanicata, is a semi- permeable membrane,
being permeable to water but not to salts, or at any rate not to many
salts. Thecoutentsof tlie cells contnin salts, and so have a definite osmotic
pressure. If tMese cells are placed in a solution having a greater osmotic
pressure than their own, water will run from the cells into the solution,
the cells will shrink and will present the appearance shown in Fig, 150 b.
Kig. IfiO a sliows the appearance of the cells when surrounded by water;
the HAdkest solution which produces a detachment of the cell will be
ui'l'rosimately isotonic with tlio contents of (he cell. In this way a seriefl
Brr.
190 PROPERTIES OF MATTER.
of solutions can be prepared which are isotonic with each other. De Vries
found that for non-electrolytes isotonic solutions contained in each unit of
volume a weight of the salt proportional to the molecular weight ; in other
words, that isotonic solutions of non-electrolytes contain the same number
of molecules of the salt. This is another instance of the analogy between
osmotic pressure and gaseous pressure, for it is exactly analogous to
Avogadro's law, that wlien the gaseous pressures are the fame all gases
at the same temperature contain the same number of molecules per unit
volume. Although the direct measurements on osmotic pressure hitherto
made may seem a somewhat slight base for the establishment of such an
important conception, an immense amount of experimental woik has been
done in the investigation of such phenomena as the lowering of the vapour
pressure, the raising of the boiling- and the lowering of the freezing-point
produced by the solution of salts in water. The conception of osmotic
pressure enables us to calculate the magnitude of these effects from the
strength of the solution;
Slfa^"""^ the agreement between the
values thus calculated and
the values observed is fo
Woler vapour/ /^ ^\ ^^aUr ^xxpoixr close as to furnish strong
evidence of the truth of
this conception.
Vapour Pressure of
a Solution.— The change
« . . ^ x>^^s.^^^ ^xxxvxvxxxxv ill the vapour pressure due
to the presence of salt in
the solution can be calcu-
lated by the following
M^mhranB mothod duo to Van t*
Fio. 151. Hoff : Suppose the salt
solution J[, Fig. 151, is
divided from the pure water J5 by a semi-peimeable membrane — i.e., one
which is permeable by water and not by the salt; transfer a small
quantity of water whose volume is v from A to B by moving the
membrane from right to left. If II is the osmotic pressure of the solu-
tion the work required to effect this transference is IIv ; now let a volume
V of water evaporate from B and pass as vapour through the membrane into
the chamber A and there condense. If V is the volume of the water vapour,
5/> the excess of the vapour pressure of the water over B above that over Ay
the work done in this process is IjN, The process is clearly a reversible one,
and hence by the Second Law of Thermodynamics, since the temperatures
of the two chambers are the same, there can be no loss or gain of mechanical
work. Thus, since the work spent in one part of the cycle must be equal
to that gained in the other, we have
nv = SpV
Suppose p is the vapour pressure over the water, let V be the volume
occupied at atmospheric pressure 11, by the quantity of water vapour which
at the pressure ^ occupies the volume V ; then by Boyle's Law,
IIoV'-/)V
DIFFUSION OF LIQUIDS.
191
80 that
but for water vapour vfY'
1/1200, hence
lp__ Tl 1
p llo 1200
The osmotic pressure in a solution of 1 gramme equivalent per litre
of a salt which does not dissociate when dissolved is about 22 atmospheres ;
thus for such a solution
«/)__ 22
p 1200
or the vapour pressure over the solution is nearly 2 per cent, less than
over pure water.
If the surface of the solution is subjected to a pressure equal to
the osmotic pressure the vapour pressure over the solution will increase
and will be equal to the
pressure over pure water.
For let Fig. 152 represent a
vessel divided by a dia-
phragm permeable only by ^^ ^^^
tf/vt au*
fioUfiicri,,
MfaJier vapour
' carut our
\^<Usr
M^mbrojnM
Fio. 152.
water and by water vapour,
and let the salt solution in
A be subject to a pressure
equal to the osmotic pres-
sure. Under this pressure
the liquids will be in equi-
librium, and there will be
no flow of water across the diaphragm. If the vapour pressure of the
water is greater than that of the salt solution, then water vapour from B
will go across the diaphragm and will condense on ii ; this will make
the solution in A weaker and reduce the osmotic pressure. Since the
external pressure on ^ is now greater than its osmotic pressure, water
will flow from ^ to ^ across the diaphragm; thus there would be a
continual circulation of water round the system, which would never be
in equilibrium. As this is inadmissible, we conclude that the vapour
pressure of the water is not greater than that of the solution ; similarly if
it were less we could show that there would be a continual circulation in
the opposite direction ; in this way we can show that the vapour pressure
of the solution when exposed to the osmotic pressure is equal to that of
pure water. This is an example of the theorem proved in J. J. Thomson's
Applicationa of Dynamics to Physics and Chemistry , p. 171 (see also Poyu-
ting, Phil. Jfa^r., xii. p. 39), that if a pressure of n atmospheres be applied
to the surface of a liquid the vapour pressure of the liquid, /?, is increased
by Ipi where
bp density of the vapour at atmospheric pressure
p density of the liquid
Raising of the Boiling-point of Solutions.— The determina-
tion of the vapour pressure is attended with considerable difficulty, and it
is much easier to measure the effect oE salt on the boiling- or freeziog-poiut
of the solution.
Let A tuid S be vessels containing respectively Bait solution and pure
water, eeparat«d by a semi-permeable membrane, and let the temperatures
of the vessels be such that the vapour preiisure over the solution is the
same as that over pure water. Let be the absolute temperature of the
water, d + cO that of the solu-
tion. Now suppose a volume
V of water flows from £ to A
across the diaphragm ; if II is
the oKDiotic pressure of the
solution, mechanical woik lit)
will be done in this operation
Let this quantity of water be
evaporated from 1 and pass
through the w alls of the
diaphragm and coudeu^ in
- B As the vapour pressures
are the same in the two
cases, no mechanical work is
Flu I^J. gained or spent in this opera-
tion. Ihe system is now in
its original state, and the operation is evidently a reversible one, so that
we can apply the Second Law of Thermodynamics. Now by that law we
Heat taken from the boiler
Absolute temperature of boUi
Mechanical wjrkti(
Heat given up in the refrigerator
Absolute temperature of refrigerator
by the engine
Difference ol" the temperatures of boiler and refrigerator.
In our case the mechanical work done is IIo. The heat given up in tha
refrigerator is the heat given out when a volume v of water condenses
from steam at a temperature 0; if X is the heat given out when unit mase
of steam coadenses and a the den-sity of the liquid, the heat given out in
the refrigerator is \aD ; hence by the Second Law we have
Uv ,
n'
Let us apply this to find the change in the boiling-point produced by
dissolving 1 gramme equivalent of a salt in a litre of water; here II
is 22 atmospheres, or in C.G.S. units 22 x I0». A is the latent heat of
steam in mechanical unite — i.e , 536 x 4'2 >c 10', a is unity, and d = S73 ;
Sfl:
873x29x10"
= 'S? of a degree.
The experiments of Eaoult and others on the raising of the boiling-
point of solutions of organic salts which do not dissociate have uhown
DIFFUSION OF LIQUIDS. \Qs
that the amount of the rise in the boiling-point is almost exactly -37 of
ft degree for each gramme equivalent per litre, a result which ia strong
confirmation of the truth of the theory of oemotic pr«esnra.
Lowering- of the Freezingr-point of Solutions.— a similar in-
vestigation enables us to calculate the depression of the freexing-point
due to the addition of salt. Let A, B (Fig. 16J} represent two vessels
separated by a semi-permeable membrane, A containing the salt solution
at ita freezing-point and B pure watei- at its freezing-point. Let avolume
t> of water pass across the semi- perm cable membrane from Bio A; if n is
the oemotic pressure of the solution, mechitnical work (In will be gained by
this process. Let this quantity of waiei- be frozen in A, the ice produced
tak< n fiom A placed in S, and there melted. The system has now returned
to its original condition, and the process is plainly reversible ; hence we can
apply the Second Law of Tbermo dynamics. If S is the absolute tempera-
ture of the freezing-point of pure water, — 20 that of the freezing-point
of the solution, if X is the latent beat of water, and a its density ; the
heat taken from the hot chamber B at the temperature is X^v; hence
by the Second Law we have
fl "30 \e
Thus in the case of waUr for which 9 = 273, X = 80 x 4-2 x 10^ a- 1 and
when the strength of the solution is 1 gramme equivalent per litre,
n = 22 X 10» ; hence SB = 1-79''.
This has been verified by Baoult in the case of solutions of oi^anio
salts and acids. The result of the comparison of theory with experi-
ment for a variety of solvents is shown in the following table:
Lowering of freeiin
K point for ornoic Falts,
eolTEDt
1 gramme n
olecule iliBEohed in a litre
Observed
Acetic acid .
. 3'9
. J(«8
Formic acid .
. 28
. 2-8
Benzene
. 49
. 5-1
Nitro-benzene
. 705
. 6-9
Et hy lene- d ibromide
. 11-7
. 11-9
194
PROPERTIES OF MATfEn.
Dissociation of Electrolytes.— The precedmg tlieoiy gives a
satisfactory account of the effect upon the boiling- nud freezing-points
produced by organic salts and aciJs when the osmntic pressure is
calculated on the assumption that it is eqnal to the gaseous pressure
which would he produced by the Bnme weight of the salt if it were
gasified iind confined in a volume equal to that of the solvent. When,
however, mineral salts or acida are dissolved in water, the effect on the
boiling- and freeiirg-points produced by n gramme equivalents per liti* is
greater than that produced by the same nuuit)er of gramme equivalents of
«n organic mUt, although if the osmotic pressure were given by the same
rule, the effects on the freezing- and boiling- points ought to be the same
in the two cases. The osmotic pressure then in a solution of a mineral
salt or acid is gi'eater than in one of equivalent strength (i.e., one
for which n is the same) of an organic Kilt or acid ; this lias been
verified by direct measurement of the osmotic pi-essure by the methods
of Pfeffer and De Vriea. This increase in the osmotic presaura ia
sxplained by Arrhenius as being due to a partial dissociation of the
molecules of the salts into their constitutents ; tlnis some of the
molecules of NaOl are supposed to split up into separate atoms of
Ka and 01. Since by this dissociation the number of individual
particles in nuit volume is increased, the osmotic pressure, if it foliows
the law of gaseous pressui'e, will also be increased. According to
Arrhenius, the atoms of Na and Ul into which the molecule of the salt
ia split are charged respectively with positive and negative electricity,
which, as they move under electric forces, will make the solution a
conductor of electricity. In this way he accounts for the fact that
tho^e solutions in which the osmotic pressure is abnormally large are
conductors of electricity, and tliat, as a rule, the greater the conduc-
tivity the greater the excess of the osmotic pressure. This view, of
which an account will be given in the volume on Electricity, has been
very successful in connecting the various pi'operties of solutions.
Though the of-motic pressuie plnys such an important part in the
theory of solution, there ia no generally accepted view of the way in which
the salt produces this pressui-e. One view is that tlie salt oxiste in the
interstices between the molecules of the solvent in the state corresponding
to a perfect gas. If the volume of these interstices bore a constant
proportion to the volume of the eolvent, then, whatever Ibis ratio may h^
we should get the ordinary relation between the quantity of salt and
the osmotic pressure to which it gives rise. For, suppose p is the
pressure of the gaseous salt, f the volume of the interstices, V the volume
of the solvent; then if a semi-permeable membrane be pushed so that &
volume iiV of wat«r passes thi'ough it, and n is the osmotic pressure)
then the work done is IIoV; but if So is tlio diminution in the volume
vi (he interstices, the work done is pSv ; hence
Dutif tlie volume occupied by the intei-stices bears a const4int ratio to 1
that of the solvent
I
I
where V is the volume of the solvent ; hence
DIFFUSION OF LIQUIDS. 195
nV=pi? or 11 = ^;
that 19, the osmotic pressure is the same as if the gaseous salts oocu[»)cil
the whole volume of the solvent.
Another view {see Poynting, Phil. Mag. 42, p. 289) is that the
phenomenon known as osmotic pressure arises from the molectiles of salt
clinging to the molecules of the water, and so diminishing the mobility and
therefore the rate of diffusion of the latter. Thus, suppose we have puie
water and a salt solution separated by a semi-permeable membrane, since the
water molecules in the solution are clogged by the salt they will not be able
to pass across the membrane as quickly as those from the pure water, ani
there will be a flow of water across the membrane from the pure water
to the solution. Poynting shows that the mobility of the molecules of
a liquid is increased by pressuie, so that by applying a proper pressure
to the solution we may make the mobility of the molecules of water in
it the same as those of the pure water, and in this case there will be no
flow across the membrane; the pressure required is the osmotic prca-ure.
Poynting shows that this view will explain the properties of inorganic
Baits if we suppose that each molecule of salt can completely destroy the
mobility of one molecule of water.
cuArrEii xvir.
DIFFUSION OF GASES.
OONTENTa. — Co-enident of Diffusion— Diftusion of Vaponra— Esplnnntlon nt Diffa-
dion on KinoElo Theory of Gases— ESects of a Perforated Diaplirapni -Pa-sage of
Gases through Poroas Bod ins— Thermal EfTuBion- AtmoljsiB— I'asB»ge of (iase*
IbrougU luiliitriibber, Liquids, Hot Metals— Di final o& of Uetsla tliiougb Uetal.
If a mixture of two gases A and B is confined in a vessel the gasea
will mix aDd each will ultimately be uniformly diffused through the vessel
as if the other were not present. If they are not iiniformly mixed to
begin with, there will be a flow of the gas A from the places where the
density of A is great to those where it is email. The law of this diffusion
is aunlogous to that of the conduction of heat or to the diffusion of litiuids
and may be expressed mathematically as follows : Suppose the two gases
are arranged so that the layers of equal density are horizontal planes, and
let p be the density of A at a height x above a fixed horizontal plane ; then
in unit time the mass of A which passes downward through unit area of a
horizontal plane at a height ir is proportional to the gradient of p and is
equal to E-^ where K is the interditTusity of the gases A and S. The
vnlue of K has been measured by Loschmidt* and Obermayert for a
considerable number of pairs of gases. The method employed by these
observers was to take a long vertical cylinder separated into two parts by a
disc in the middle. The lower half of the cylinder was filled with the
heavier gas, the upper half with the lighter. The disc was then removed
with great care so as not to set up air currents, and the gases were then
allowed to diffuse into each other; after the lapie of a certain time the
disc was replsced and the amount of the heavier gas io the upper half of
the cylinder detei'mined. From this the value of K was determined on
the assumption (which is probably only approximately true) that the
value of K does not change when the proportions of the two gases are
altered. Woitz^ used a different method to determine the coefficient of
interdiffuaion of air and carbonic acid ; beginning with the carbonic acid
below the air he measured by means of Jamin's interference refractometer
the refractive index of various layers after the lapse of definite intervals of
time ; from the refractive index lie could calculate the proportion of air and
carbonic acid gas, and was thus able to follow the coui-se of the diQ'usion,
He found that the coefficient of diffusion depended to some extent on tha
proportion between the two gases, the values of K at atmospheric pressure
at 0° C. varying between -1288 and -ISCG cra.Vsec. The values found by
LoBchmidt and v. Obermayer are given in the following table. They are
for 76 cm. pressure and U^G:
• LoBchmidl, IFim. Jlerichu. 81. p. 307, 1870, 82, p. «3, 1870.
♦ Obonoayer, Wim. Btrichtt, 81, p. 1B2. 1880.
I WuiU, Wicdanann't AmxaLci, 17, p. 201, 18S2.
DIFFUSION OF GASES.
CO,
-N,0
CO,
-00.
CO,
-0, .
00,
-Air
CO,
-OH,
CO
-H,
CO,
-C,H. .
00
-0, .
CO
-H, .
00
-C,H,
SO,
-K-
0,-
H, .
0.-
N, .
Air .
M,-
Air.
H,-
CH,
H,-
N,0
H -
C,H, .
H,-
CH .
We may, perhaps, gain some
that the rate of equalisation ii
air is about half that of the
:om.>o.
■09831
■1405ft
■UO'db
■14231 . . . -13433
■15*!ii0 . . . 14650
-55585 . . . ■53400
. ■1U061
. -18717
. ■G4«8i
. -11630
'. '. -66550
. -17875
. -17778
. -63405
. -62544
. ■58473
. -45933
. '18627
) idea of the rapidity of difTusion by saying
mposition of a mixture of hydrogen and
' Q of temperature in copper.
mple of the rate at which difTuBion goee on we may quote the
result of an experiment by Graham on the diffusion of 00, into air.
Carbonic acid was poured into a vertical cylinder 57 cm. high until it Blled
ooe-tenth of the cylinder. The upper nine-tenths of the vessel was
filled with air and the gases weru left to diffuse. They were found to be
very approximately uniformly distributed throughout the cylinder after
the lapse of about two hours. As the time taken to reach a state of
approximately uniform distribution is proporlioual to the square of the
length of the cylinder, if the cylinder were only one centimeti'e long
approximately uniform distribution would be attained after the lapse of
about two seconds.
The interdiffuaity is inversely proportional to the pressure of the
mixed gas; it increases with the temperature. According to the experi-
menta of Loschmidt and v. Obermayer it is propoitional to 0" where B is
the absolute temperature and n a quantity which for different pairs of
gasea varies between 1^75 and 2.
Diffusion of Vapours. — The cnse when one of the diffusing gaaee
is the vapour of a liquid is of special importance, as it is on the rate
of diffusion that the rate of evaporation depends. The methods which
have been employed to measure the rate of diffusion of the vapour of a
liquid consist esseutially in having some of the liquid at the bottom of &
cylindrical tube and directing a blast of vapour-free gas across the mouth
of the tube. When the blast has been blowing for some time a uniform
gmdient of the density of the vapour is established in the tube, the value
of this is &/t where S is the maximum vapour pressure of the liquid at the
temperature of the experiment and C the distance of the eurfaoe of the
liquid from the mouth of the tube. The mass of rapour which in ooit
196
PROPERTIES OF MATTBK.
time flowsoutof the tube— (i.e., the amDiint of the liquid wliich, evaporate* 1
in unit time and which can therefore be easily ine.isiired), is KS/i where K 1
i.s the (liffitKivity of the vapour into the gng; as o is known we can readily I
determine K by this method. A few of the losulta of experiments mada I
by Stefan' and Winkelmannt ai>e given in the following table:
ViLUB OF K IS cm. '/sec. AT
HydioKcn.
Air.
OiH".nic a
Wntev vapour .
. BBT
. -198
. -isi
Ether
. ^ae
. -O—'a ..
. -0552
Carbon-bisiilpbiilB
. -369
. -rnxs
. -0G29
»enzoI
. '234
. -07.^1 .
, 0527
MethyUlohol .
. vODl
. -IS^;) .
. mm
Ethyl-alcoiu.1 .
. ■•6806
. -oyn .
. -oGiia
Explanation of Diffusion on the Kinetic Theory of Gases.—
The kinetic theory according to "which a gns consists of a gi'est number of
individual particles called molecules in rapid motion, afibrds a ready ex-
planation of diffusion. Suppose we have two layei-s A and B in a mixture
oE gases and that these layers are separated hy a plane C. Let there he
more molecules of some gas y in A than in B, then since the molecules are
in motion they will be continually ci-os.<ing the plane of separation, some
^'oing from A to B-nnd some from B to A, but inasmuch as the molecules
of 7 in A are more numerous thitn those in B, more will pass from A to B
than from B to A. Thus, A wi 11 lose and B gain some of the gas y ; this
will go on until the quantities of y in unit volumes of the layers A and B
are equal, when as many molecuIeH will pnss from A to B as from B to A«
and thus the equality, when once establit^hed, will not be diaturhed by the
motion of the molecules. It follows from the kinetic theoiy of gasoa
(gee Boltzmann, VorUaiingen iiher GanLlteorie, p. 91) that, if there are M
moleuulesof y in unit volume '.f B, n + hi in a unit volume of A at a
distance ex from that in B, and if x be measured at right angles to the
plane separating the layers, thpu the escess oF the number of molecules
of y which go across unit area of from A to B over those which go from
I
A to B is equal to -SSOSXc-
whe
is then
lan free path of the molecules
i the quantity Xo is evidently
of y and c, their average velocity of translatio:
proportional to the difi'usity.
Now c only depends upon the temperature, being proportional to the
aquare root of the abwlute temperature, while X is inversely proportional
to the density, and if the density is given ib does not, at least if the
molecules are regarded as hard elastic spheres, depend upon the tempera-
ture. If the pressure is given, then the density will be inversely, and
X therefore directly proportional to the absolute tomperKture. Thus, on
this theory the coefficient of difi'usion should vary as fl' where fl is the
absolute tempemtuie. The expei'iments of Loschmidt and von Obermayer
seem to show tha.t it vaiies somawhat more rapidly with the temperature.
Another method of regarding the process of difiusion, which for soma
purpasefl is of great utility, is as follows ; The diffusion of one gas A 1
through another B when the layers of equal density are at right angles to J
• Stefan, Witn. Akail.. Ber., 86, p. 323. 1872.
I Wukelmann, Witd. Ann., 22, pp. 1 and li>2, 1881.
DIFFUSION OF GASES.
199
the axis x maj be regaiiUd as due to a current of the gaa A. moviiif;
parallel to the axis of x mth a cei'tain velocity m througli & current of B
Btreaming with the velocit.y vin tlie opposite direction. To move a current
of one gaa through another rei|tiire9 tbo applicattoa of a force to one gas in
one direction and an eijual foivo to the other gas in the opposite direction.
This force will be proportional (1) to the relative velocity w + « of the two
curi'eiitK, (2) to the number of molecules of A per unit volume, and (3) to
that of the molecules of B. Let it then per unit volume of gaa be equal
to A„/)|P, {« + f), where A„ ia a quantity depending on the nature of the
gases A and B, but not upon their densitiea nor upon the velocity with which
they are sti'oaming through each other; p, and p, ure respectively the
densities of the giises A and B — i.0.,tbeir masses per unit volume. Hence,
to eufctain the motion of the gases a force A„ p^ p, (u + 1>) parallel to x must
act on each unit of volume of A aud an equiil force in the opposite
direction on each unit volume of B. These forces may arise in two ways;
there may be external forces acting on the gases, and there may also be
forces arising from variations in the partial pressures due to the two
gases. Iiet X,, X, be the extertial forces per unit mass acting on the gases
A and B respectively, and p,,p, the partial pressures of the gases A and B
respectively. Considering the forces acting parallel to x on unit volume
of A, the external force is X,/),, and the force due to tho variation of the
partial pi-essure is — dpjdx ; hence the total force is equal to — dpjdx + X|p,,
and as this is the force driving A through B we have
--ii-^^iPi=^i.J'^'i^'^^)
similarly,
+ X^,= -A,jP,p,(i* + k)
(1)
m
Let us consider the case when there are no external forces and when
the total pressure p, + p, is constant throughout the vessel in which
difl'usion is taking place. In this cose the number of molecules of A
which cross unit area in unit time must equal the number of mojeculea of
B wliich oi-oss the same area in the tame time in the opposite direction,
Let this number be q; then if ii, si, are respectively the numbera »t
iBolecutes of A and B per unit volume,
e the mnsses of the molecules of A and B respectively
hence A,,pj.,(" + f) = A,,'",**)/", + b,)?
Now n, + n, is proportional to the total pressure, and ae Ihw ta
ronstant throui;hout the volume, »<, + «, will be constant. Putting X - in
^equation (1) and writing N for ?', + n„ we get
PROPERTIES OF MATTER.
, is tlie number of molec
pressure j; :
hence
Kp^A„'H,m, dx
Now 1 is the number of molecules of A pissing unit surf.
time and diijdx ia the gradieut of tlie nmuber per unit voliii
fi-om the definition of K, the inCeidifTusity, given on p. I'^JC, wt
■ if P is the toUl pre
^:PA,;
Thus, if A,, is confitant, K vai-icB inversely as P, find directly as (pj»^.
Since the pressure of a. given number of molecules per unit volume is
proportional to the absolute temperature, K, if Aj, is constant, varies
directly afi the square of the absolute tempemture.
We can determine A,, if we know the velocity acquired by one of the
gases A when acted upon by a known foi-ce. Suppose that the gas A ia
uniformly distributed, so that rf/j,/rfj; = 0, and that when acted upon by a
known force it moves through B with a velocity u; suppose, too, that B is
very largely in excess and is not acted upon by the force, we have then v
very small compared with u, and from ei^uation (1) we have
Thufi, if we know «, the velocity acquired under a, known force X, we can
find A„, Aud hence E, the diffusivity. This result is of great importance
in the theory of the diffusion of ions in electrolytes, and Nemst has
developeii an electrolytic theoiy of diffusion in fluids on this basis.
Another important application of this result is to determine X from
measurements of K and u. Thus, to take an example, if the particles of
the gas A are charged with electricity and placed in an electric field of
known strength, the force X will depend upon the charge ; hence, if in this
case we measure (as has been done by Townscnd) the values of K and u,
we can deduce the value of X, and hence the charge canied by the
particles of A.
On the Obstruction offered to the Diffusion of Gases by a
perforated Diaphrag'm.— if a jwrforated diaphragm is placed across a
cylinder it doL's not diminish the diiTusion of gases in the cylinder in the
ratio of the area of tiie openings in the diaphragm to the whole area of
the diaphragm, hut in n much staaller degree, for the eOect of the per-
foration is to make the gradient in the density of the gases in the neigh-
bourhood of the hole grenter than it would have been if the diaphragm
had been removed, and therefoi'e the flow thi'ough the hole greater than
thi'ough an equal area when there is no diaphragm. Thus, to take a casa
investigated by Dr. Horace Brown and 5Ir. Esoombe (Proceedings Royal
Society, vol. 67, p. 124), suppose we have CO, in a cylinder, and place
ftcross the cylinder a disc wet with a solutiou of caustic alkali which
DIFFUSION OF GASES,
SOI
absorbs the CO,, so that the density of the CO, nest the disc ia zero.
Then if p id tiie density of the CO, at, the top of the cylinder, the density
gradient ia pjl where I is, the distance between the dijic and the top of the
cylinder, so that the amount of CO, absorbed by unit area ol the disc
will be iWi where k is the difl'usivity of 00, through itself. Now siippoae,
insteJid of a disc extending completely across the cylinder, we have a much
smaller disc of radius a, then at the disc the density of the CO, will be
zero, but it will recover its normal value p at a distance from the disc
proportional to o; thus the gradient of density in the neighbourhood of
ihe disc will be of the order p/a and not pjt, and the nmount of CU,
absorbed by the disc will be proportional to A (p/a) va- — i.e.. will be
proportional to a; so that the absorption of the CO, will only diminish as
the radius of the disc and not iis the area. This was vended by Brown
and E'^combe, and it baa very important applifiitiows to the piissage of
gases through tho openiiigs in the leaves of plants.
Passage of Gases through Porous Bodies.— Theie are three
processes by which gas may pass through a solid perforated by a series of
holes or canals ; the size of the holes or pores determining 'the method by
which the gas escapea, Jf tho plate is thin and the [lores are not
exceedingly fine, the gas escapes by what is called effusiou ; this is the
process by which water or air escapes from a vessel in which a hole ia
bored. The rate of est.'ape is given by Torricelli's theorem, so that the
velocity with which a gas streams through an aperture into a vacuum is
proportional to the square root of the quotient of the pressure of the gas
by its density, and thus for different gases under the same pressure the
velocity will vary inversely as the square root of the density of the gas.
Bunsen founded on this result a metbod of (inding the density of gases.
This case, strictly speaking, is not one of ditl'iision at all, but merely the
flow of the gas as a whole thiTiiigh the aperture. If the gas is a mixture
of dilTerent gases its composition will not be altered when the gas passes
through an aperture of this kind.
The second method is the one which occurs when the holes are not too
fine, and when the thickness of the plate ia large compared with the
diameter of the holes. In this case the laws are the same as when a gas
flows through long tubes; they dejieud on the viscosity of the gas, and are
discussed in the chapter relating to that property of bodies. No change
in the composition of a mixture of gases is produced when the gases are
forced through apertures of this kind ; this is again a motion of the gas
as a whole, and not a true case of dilTusion. The third method occurs
when the pores are exceedingly fine, such as those found in plates of
meerschaum, stucco, or a plnte of graphite prepared by squeezing together
powdered graphite until it forms a coherent mass. In this ciise, when we
have a mixture of two gases, eath finds its way through the plate
independently of the other, and the composition of the mixture is in
general altered by the passage of the gas through the plate. The laws
governing the passage of gases through pores of thb kind were investi-
gated by Graham, who found that the volume of the gas (estimated at a
standard pressure) passing through a porous plate was directly propor-
tional to the difierance of the pressures of the gas on tho two sides, and
inversely proportional to the square root of the molecular weight of the
gas. Thus for the same difference of procure hydrogen was found to
escape through a plate of compressed graphite at four times the rate of
^
a02 PROPERTI^ OF MATTER.
oxygen. Thus, if we havemistiires of equal volumes of hydrogen and oxygen
and allow tlieni to pas8 thi-ough a poi-oua diaplirngm, einte the hydrogen
gets through at four time* the v^te of the oxygen, the mixture, after pass-
ing through the plate, will be much richer in liydrogen thnn in osygen.
The rate of dia'uaion can be meaawred by an instrument of the following
kind (Fig. 155); A porous plate ia fastentd on the top of a. tube which can
be ustd as a barometer tube. A vessel for holding the gas being attached
to the upper part of the tube, this and the space above the mercury are
exhausted ; gas at a definite pressure is then let into the vessel, and the
tuto at which it passes through the diaphragm into tbe vacuum over the
mercury is measured by the rate of
depression of the mercury column.
The laws of diffusion of gases
thiough fine pores are readily explained
by the Kinetic Theoiy of Gases ; for if
the pores are so fine that the molecules
pass through them without coming
into collision with other molecules, the
rote at which the molecules pass through
will be proportional to the average
velucity of translation of the molecules.
According to the Kinetic Theory of
Gn^^es this average velocity is inversely
proportional to tlie square root of ths
molecular weight of the f;as and directly
proportional to tbe sijuaie root of the ab-
Bolute temperature. Henceata given
temperature the velocity with which
the gas streams through the apertureft
will be inversely proportional to the
square root of the molecular weight;
this is the result discovered by Graham.
Thermal Effusion.— The sama
renaoning will explain another pheno-
menon sometimes called thermal eflii-
Kion. Suppose we have a vessel divided
nto two portions by a porous diuphriiigm ; let the pressures in the two
;>ortions be equal but their temperatures difi'erent, then gas will stream
from the cold to the hot part of tlie vessel through the diaphragm. For
since the pressures are equid the densities in the two parts of the vessel
are inversely proportional to the absolute temper.vtures wliile the velocities
are directly proportional to the square roots of the absolute temperatures.
Hence the number of molecules passing from the gas through the
diaphragm, which is proportional to the product of the density and the
velocity, will be inversely proportional to the square root of the absolute
temperature; thus more gas will p»ss fi-om the cold side than from the
hot, and thei'o will bo a stieain of gas from the cold to the hot portion
through the diiiphiagui.
Atniolysis. — The diS'u&ion of gases through porous bodies was applied
by Graham to produce the separation of a mixture of gases ; this
separation was called by him atmolysis, and to effect it he used
instrument of the kind shown in Fig. I5G. A long tube made from tha
Tio. 158.
DIFFUSTOS OF GASES.
203
sterna of clay tobacco-pipes is fixed by meaiiR of corks in a glass or
metal tube. A glass tube is inserted in one of the end corks, and ia
connected with an air-pump so that tlie annular s]>iice between the
tobacco-pipeH and tlie outer tube can be exhausted. The mixed gases
whose constituents have to be separated is made to flow through the clay
pipes. Some of the gasea escape through the walls and can be pumped
jiway and collecteii while the I'est flows on through the tube. In the gas
which passes through the walls of the tube there is a greater proportion
of thp lighter gas than there was io the mixture originally, white in the
gris which flows along the tube thei-o is a greater projwrtion of the
Fio. 158.
heavier constituent. If the constituents of ihe mixture difier much in
density a considerable Bopar;itioii o£ the gases may be produced by this
arrangement.
Passage of a Gas through India-rubher.— The fact that gases
can pass through thin india-rubber wit's discovered in 1831 by Mitchell, who
found that india-rubber toy-balloons collapsed sooner when inflated with
carbonio acid than with hydrogen or air, and sooner with hydrogen than
air. The subject was investigated by Graham, who gave the following
table for the volumes of diffoi'ent gaies which pass through india-rubber
in the same time :
N,
1
O,
2-556
CH,
2-US
The speed with which the gosoa pass through the rubber i
very rapidly with its temperature. There is no simple relation between
these volumes and the densities of the gas ae there is in the case of
diffusion through a porous plate, and the raeclianism by which the gases
efl'ect their passage is pro)>ably quite dilTerent in the two cases. The
passage of gdsos through rubber seems lo have many points of resem-
blance to the passage of liquids tiirough colloidal membranes such aa
piu-ch men t- paper or bladder. The lubbei- is able to absorb and retain a
ceitain amount of carbonic acid gas, this amount increasing with the
pressure of the gas in contact with the burfoce of the rubber. Thus the
layers of rubber next the CO, first get saturated with ^he gas, and this
state of saturation gets transmitted from layer to layer ; but as on the
other side of the sheet of nibbei' the pressure of the CO, is less, the outer
layers cannot retain the whole of their CO, so that some of the gaa
gets free.
Passage of a Gas through Liquids.— TJiis is probably analogous
to the last case ; the gases which ai'e most readily absorbed by the liquid
are those which pass through it most rapidly.
204
PROPERTIES OF MAITER.
Passage of Gases througrh red-hot Metal.— Devilie and Troost
found that hydrogen passed readily through red-hot platinum and iron.
No gas besides hydrogen is known to pass through platinum. Troost
found that oxygen diffused through a red-hot silver tube ; quartz is said
to be penetrable at high temperatures by the gases from the oxy hydrogen
flame.
Diffusion of Metals througrh Metals. — Baniell showed that
mercury diffused through lead, tin, zinc, gold, and silver. Henry proved
the diffusion of mercury through lead by a very striking experiment : he
took a bent piece of lead and placed the lower part of the shorter arm in
contact with mercury ; after the lapse of some time he found that the
mercury trickled out of the longer arm. He also showed the diffubion of
two solid metals through each other by depositing a thin layer of silver
on copper ; when this was heated the silver disappeared, but on etching
away the copper surface silver was found. A remarkable series of ex>
periments on the diffusion of metals through lead, tin and bismuth has been
made by Sir W. Roberts- Austen*; his results are given in the following
table. K is the diffusivity :
DifTasing Metal.
Solvent.
1
Femperature.
K cm.Ysec.
Gold
Lead
... 492° .
.. 3-47xl0-»
„
9>
. 492° .
.. 3-55xl0-»
Platinum
Jf
. 492°
... l-96xl0-»
„
))
492° .
l-96xlO-»
Gold
»>
. 555
... 3-69x10-*
„ •••
Bismuth
555 ,
... 523x10-*
„ ...
Tin
555 .
... 5-38 X 10-*
Silver
>»
555
... 4-77x10-*
Lead
»
. 555 .
... 3-68x10-*
Gold
Lead
. 550
... 3-69x10-*
Rhodium
19
550 .
... 3-51x10-*
It will be seen from these results that the rate of diffusion of gold
through lead at about 500° is considerably greater than that of sodium
chloride through water at 1 8° C. Sir W. Roberts- Austen has lately shown
that there is an appreciable diffusion of gold through soUd lead kept at
ordinary atmospheric temperatures.
• Roberts* A\Lbtcn,PAt/. Trans, A., 1896, p. 893.
CHAPTER XVIIL
VISCOSITY OF LIQUIDS.
CosTESTS. —Definition ot Viacogity— Flow of Liqnii] tlirongli Cn(iillary Tube— Flow
of Gas throuK)! Cnpillary Tube —Methods of UeasaremcDt of Co-efflcientn of
Viscosity— Effect of TompeiBiureainJ Pressoro on Viscosity of Liioids— Via-
coflity of Solutions aod Mixtures— Lubrication— Ei plan at ion o£ Vincosity of
Gases on Kinetic Tlieorv— Mean-free Palh- ElfectsoCi'emperuture and I'ressDie
on Viscosity of Grvsca— Viscoslry of Oaeeous Uiitures- Keiistance to Motion
of a SoliU through a Viscous fluid.
A n.niD, wliether liquid or gaseous, when not acted on by extern:i1
forces, moves like a rigid body when in a steady state of rootion. "When
in this state there can be no rootion of one part oE the liquid rehitiva
to another ; if such relative motion is produced, nay by atirring the
liquid, it will die away soon after tb« stirring ce<isea. Thus, for example,
when a stream of water Hows over a fixed horizontal plane, since the
top layers of the stream are moving while the bottom layer in conlsct
with the plane is at rest, one part of the stream is moving relatively
to the other, but this relative motion can only bo m^ntained by the
action of an external force which makes the pressure increase as we go
up stream. If this fori'e were withdi-awn the whole of the stream
Tia. 157.
would come to rest. Tlie slowly moving liquid near the bottom of
the stream acts as a drag on the more rapidly moving liquid near the top,
and there are a series of tangential forces acting between the horizontal
layors into which we may 6uppo,ie the stream divided ; thus the force
noting along a surface such as AB tends to retard the more rapidly
moving liquid above it and accelerate the motion of the liquid below
it; it thuB tende to equalise the motion, and it there were no external
forces these tangential stresses would soon reduce the fitiid to rest.
The property of a liquid whei'eby it resist* the relative motion of ita
parts is called viscosity. The law of this viscous resistance was formu-
lated by Newton {Prineipia, Lib. II., Sec. 9). It may be stated as
follows: Suppose that a stratum of liquid of thickuess e is moving
horizontally fi-om left to right and that the horizontal velocity, which
is nothing at OD, increases uniformly with the height of the liquid,
and let the top layer be moving with the velocity V ; then the
tangential stresi which may be supposed to net across each unit of a
Bui-faue such as AB i^ proportional to the gradient of the velocity — i.e.,
to V/c— and tends to stop the relative motion, the tangential stress on the
liquid bebw AB being fiom left to right, that on the liquid above AB
from right to left. The ratio of the stress to the velocity gradient is called
the CO efficient of viscosity of the fluid ; we shall denote it by the sym-
bol If, The viseasity may be defined in terms of quantities, which may»be
directly measured m follows; "The viscosity of a substance is measured
206 PROPERTIES OF MATTER.
by the tangential force on unit area of either of two horizontal planes
at unit distance apart, one of which is iixed, while the other moves
with the unit of velocity, the sp& . e between being filled with the viscous
substance " (MaxweU's Theory of Heat),
It will be seen that there is a close analogy between the viscous
stress and the shearing stress in a strained ela^stic solid. If a stratum
of an elastic solid, such as that in Fig. 157, is strained so that the hori-
zontal displacement at a point P is proportional to the height of P
above the plane CD, the tangential stress is equal to n x (gradient of
the displacement) where n is the rigidity of the substance. The viscous
stress is thus related to the velocity in exactly the same way as the
shearing stress is related to the displacement. This analogy is brought
out in the method of regarding viscosity introduced by Poisson nnd
Maxwell. According to this view, a viscous liquid is regarded as able
to exert a certain amount of shearing stress, but is continually breaking
down under the influence of the stress. We may crudely represent
the state of things by a model formed of a mixture of matter in
states A and B, of which A can exert shearing stress while B cannot,
while under the influence of the stress matter is continually passing
from the state A to the state B. If the rate at which the shear
disappears from the model is proportional to the shear, say X9, where
d is the shear, then, when things are in a steady state, the rate at
which unit of volume of the substance is losing shear must be equal
to the rate at which shear is supplied to it. If ^ is the horizontal
displacement of a point at a distance x from the plane of reference, then
sa _z. The rate at which shear is supplied to unit volume is ddjdi or ;
doc CbX CLk
but d^jdt is equal to v, the horizontal velocity of the particle, hence the
rate at which the shear is supplied is dvjdx. Thus, in the steady state,
ax
If n is the coefficient of rigidity, the shear Q will give a tangential
stress equal to nQ or
n dv
X dx.
If If is the coefficient of viscosity, the viscous tangential stress is equal to
do
dx.
Hence, if the viscous stress arises from the rigidity of the substance,
ri = n/X.
The quantity 1/X is called the time of relaxation of the medium ; it
measures the time taken by the shear to disappear from the substance
when no fresh shear is supplied to it.
This view of the viscosity of liquids is the one that naturally suggests
itself when we approach the liquid condition by starting from the solid
state ; if we approach the liquid condition by starting from the gaseous
state we aid led {see p. 218) to regard viscosity as analogous to dilTu&ioii
VISCOSITY OF LIQUIDS.
807
and as arising from the movement of the molecules from one pai't of the
substance to another. This point of view will be considered later.
Flow of a Viscous Fluid througrh a Cylindrical CapUlary
Tube. — When the fluid is driven through the tube by a constant
difference of pressure it settles down into a steady state of motion such
that each particle of the fluid moves pai-allel to the axis of the tube,
provided that the velocity of
the fluid through the tube does
not exceed a certain value de-
pending on the viscosity of the
liquid and the radius of the
tube. The relation between
the difference of pressure at
the beginning and end of the
tube and the quantity of liquid
flowing through the tube in
unit time can be determined as
follows :
Let the cross-section of the
tube be a circle of radius OA = a,
let V be the velocity of the fluid
parallel to the axis of the tube
at a point P distant r from this
axis. Then dv/dr is the gradient
of the velocity, and the tangen-
tial stress due tx) the viscosity
is rfdv/dr: this stress acts parallel
to the axis of the tube. Consider the portion of fluid bounded by two
coaxial cylinders through F and Q and by two planes at right angles to
the axis of the tube at a distance A^ apart. Let r, r + Ar be the radii of
the cylinder through F and Q respectively. The tangential stress due to
viscosity acting in the direction to diminish t? is at F equal to 17-7- ; the
dr
area of the surface of the cylinder through F included between the two
planes is 2irrAZf hence the total stress on this surface is
2ir»;r— A«
dr
Similarly the stress acting on the surface of the cylinder throngh Q
include(i between the two planes is
Fio. 15S.
H'i'
dr\ dr ) J
Az
and this acts in the dii*ection to increase v ; hence the resultant stress
tending to increase v is equal to
Besides these tangential forces there are the pressures acting over the
plane ends of the ring; if n denote the pressure gradient — t.e., the
increase of pressure per unit length in the direction of v^ then the
208 PROPERTIES OF MATrER.
effect of the pressures over the ends of the ring is equivalent to
a force 27rrAr.nA2; tending to diminish v. Since the motion is steady there
is no change in the momentum of the fluid, hence the force tending to
diminish v must be equal to that tending to increase it ; we thus get
ar\ dr
•«• V.,-ry-)=rU (1)
Now since the liquid is moving parallel to the axis of the tube the
pressure must be the same all over a cross section of the tube; hence
IT does not depend upon r. Again, v must be the same for all points
at the same distance from the axis, if the fluid is incompressible, for if
V changed as we moved parallel to the axis down the tube, the volume of
liquid flowing into the ring through P and Q would not be the same as
that flowing out. Since 11 does not defend upon r, and the left-hand side
of equation (1) does not depend upon anything but r, we see that n must
be constant ; hence, integrating (1), we get
where C is a constant; we have therefore ,
dr r
Integrating again we have
i7V = Jr'n + Clogr + C' (2)
where 0' is another constant of integration. Since the velocity is not
infinite along the axis of the tube — i.e., when r = 0, C must vanish. To
determine C we have the condition that at the surface of the tube
the liquid is at rest, or that there is no slipping of the liquid past
the walls of the tube. This has been doubted ; indeed, Helmholtz and
Piotrowski thought that they detected finite effects due to the slipping
of the liquid over the solid. Some very careful experiments made by
Whethani seem to show that under any ordinary conditions of flow no
appreciable slipping exists, at least in the case of liquids. We shall
assume then that v = at the surface of the tube — i,e,f when r = a; this
condition reduces equation (2) to
j;v = J(7-'-a»)n (3)
Now if p, is the pressure where the liquid enters the tube, p, the
pressure where it leaves it, / the length of the tube,
II = _ 0^ " P*)
I
the negative sign is taken because the pressure gradient was taken
positive when the pressure increases in the direction of v. Substituting
this value for n, equation (3) becomes
,^=aj_£.(«'-0 (4)
VISCOSITY OF LIQUIDS.
^09
The volume of liqtdd Q which passes in unit time across a section of
the tube
a
I
2irrvdr.
olri
(5)
This is the law discovered by Poiseuille for the flow of liquids through
capillary tubes. We see that the quantity flowing through such a tube
is proportional to the square of the area of cross-section of the tube.
When the liquid flows through the capillary tube from a large vessel,
as in Fig. 159, the pressure p^ at the orifice A of the capillary tube
diflfers slightly from that due to the head of the liquid above A, for this
B
Fio. 169.
head of liquid has not merely to drive the liquid through the capillary
tube against the resistance due to viscosity, it has also to communicate
velocity and therefore kinetic energy to the liquid, so that part of the
head is used to set the liquid in motion. We can calculate the cor-
rection due to this cause as follows : let A be the height of the surface
of the liquid in the large vessel above the outlet of the capillary tube, p
the density of the liquid ; then if Q is the volume of the liquid
flowing through the tube in unit time, the work done in unit time is
equal to gphQ. This work is spent (1) in driving the liquid through the
capiUary tube against viscosity, and this part is equal to (pj -/?,) Q if
j9, and Pf are the pressures at the beginning and end of the capillary tube
(2) in giving kinetic energy to the liquid. The kinetic energy given to
the liquid in unit time is equal to
iPp
xvx 2irrdr
where v is the velocity of exit at a distance r from the axis of the capillary
tube. If we assume that the distribution of velocity given by equation (4)
holds right up to the end B of the tube, then by the help of the equation (5)
we have
210 PROPERTIES OF MATTER.
Substituting this value in the integral we find that the kinetic energy
possessed by the fluid issuing from the tube in unit time is f)Q7^*»*t
hence, equating the work spent in unit time to the kinetic energy gained
plus iihe work done in overcoming the viscous resistance, we have
IT a
or
K*"^)=^'"*^'
Thus the head which is spent in overcoming the viscous resistance is not A,
but A-
TT^aV
This correction has been investigated by Hagenbach,* Oouette,t and
Wilberforce,t and hafj been shown to make the results of experiments
agree more closely with theory. It is probably, however, not quite accu-
rate on account of the assumption made as to the distribution of velocity
at the orifice.
Viscosity of Gases* — The viscosity of gases may be measured in
the same way as that of liquids, but the case of a gas flowing through a
capillary tube differs somewhat from that investigated on p. 208, where
the liquid was supposed incompressible and the density constant ; in the
case of the gas the density will, in consequence of the variation in
pressure, vary from point to point along the tube. Using the notation of
the previous investigation, instead of v being constant as we move parallel
to the axis of the tube, the fact that equal massas pass each cross-section
requires pv to be constant as long as we keep at a fixed distance from the
axis of the tube. Since p is proportional to p, where p is the pressure of
the gas, we may express this condition by saying that pv must be
independent of z where 2; is a length measured along the axis of the tube.
Thus, since p varies along the tube, v will not be constant as z changes ;
this variation of v will introduce relative motion between parts of the gas
at the same distance from the axis of the tube, and will give rise to
viscous forces which did not exist in the case of the incompressible liquid.
We shall, however, neglect these for the following reasons : if V^, is the
greatest velocity of the fluid, the gradient of velocity along the tube is of
the order YJly where I is the length of the tube ; the gradient of velocities
acioss the tube is of the order VJa, where a is the radius of the tube ; as
a is very small compared with I, the second p^radient, and therefore the
viscous forces due to it are very large compared with those due to the firsi^
We shall therefore neglect the effect of the first gradient. On this supposition
equation (1) still holds, and, since n = ^, we have
dz
d I dv\ „ d
dp
dz
• Hagenbach, Poggendorjfa Jnnalen, 109, p. 885.
{Couette, Armales de Chimie et de Physique, [6], 21, p. 4U|
Wilberforce, Philosophical Magazine^ (5) 31," p. 407,
VISCOSITY OF LIQUIDS. 211
or, Tegarding p aa constant over a cross-section of the tube, we Lave
d ( A}»,)\ dp , V
Since po is indepeadeiit of z, we see that -'' is constant and tqual to
.1
Tia. 160.
Solving the diOerential equation in the snine way as that on p. S
Sid
PROPEnrtES OP MATTER.
and if V, is the volume entering, V, that leaving the tube per second, w«
hkra
Measurement of the Coefflcient of Viscosity.— The Tiscosity ^
has most frequently been detei'iuined by metis urements of the rate of flow
of the fluid through capillary tubes. An app,)ratii3 by which thin can be
done is shown in Fig. 160. ff is a closed vessel containing air under
pressure ; the pressure in this vessel is kept constant by means of the tube
D, which connects G with a Mariotte's bottle ; the pressure in G is aiways
that due to a column of water whose height is the height of the bottom of
the air tubes in the Moriotte's bottle above the end of the tube D. Tha
glass vessel abcdef, in which tk is a capillary tube, contains the fluid whose i
coefficient of viscosity is to be determined ; this vessel communicatee with i
FiQ. 161.
Flo. 162.
G by means of the tube LKJ ; the pressure nets on the liquid in ahcdej",
and causes it to flow through the capillary tube fi-om left to right ; two
marks are made at b and c, and the volume between these marks ia
carefully determined. Let us call it V ; then, if T is the time the level of
the liquid takes to fall from 6 to c, Q = V/T. The area of cross-section of
the tube has to be determined with great core, and precautions must be
taken to prevent any dugt getting into the capillary tube. As the
viaeoaity varies very rapidly with the temperature, it is necessary to .
m<iintiun the tempei'ature constant; for this purpose the vessel a&%fe^ia
placed in a bath filled with water,
"With an apparatus of this kind Poiseuille's law can be verified, and '
the viscosity determined. It is found that, although Poiseuille's law holds
with great exactness when the rata of flow is slow, yet it breaks down
when the mean velocity Q/ira* esceeds a certain value de£>eading on the
ahte of the tube and the viscosity of the liquid. This point has been
investigated by Osborne Reynolds, who finds that the state of Sow we
have postulated in deducing Poiseuille's law — {,«., that the liquid m
straight lines parallel to the axis of the tube — cannot exist when the mean
velocity exceeds n critical value; the steady flow is then I'SpIaced by a
irregular turbulent motion, the poi'ticles of liquid moving from side to side
of the tube. This is beautifully shown by one of Reynolds' experimenta.
"Water is made to flow through a tube such as that sliownin Fig. 161, and a
little colouring matter is introduced at a point at the mouth of the tube : if
tho velocity is small the coloui-ed water forms a stniight band parallel to
the axis of the tube, as in Fig. IGl ; when the velocity is increased this band
becomes sinuous and finally loses all deflniteness of outline, the colour
filling the whole of the tube, as in Fig. 162. Reynolds conduded from his
dxperiments that the steady motion cannot exist if the mean velocity ia
greater than 1000 n!f.a whore i\ is the viscosity, p the density of the liquid,
uidathe radius of the tube. The unitsare centimetre, gramme and second.
Meuuremonts of the viscosity of fluids both liquid and gaseous, have b
VISCOSITY OF LIQUIDS. 213
made by determiatng the c&uple which must be applied to a cylioiler to
keep it fixed when a coaxial cylinder is rotated with uniform velocity, the
space between the cylinders bein^ filled with the liquid whose viscosity
has to be determined. This method has been used by Couette and Mallock.
The theory of the method is as follows ; the paiiicles of the fluid will
describe circles round the common axis of the cylinders. Let PQ be poicta
on a mdius of the cylinders;
after a time T, let P come to F,
Q to Q', let OP' produced cut QQ'
in Q". Then the velocity gradient
at P will be equnl to (Q'Q"fl')-i-
P'Q"; if M is the angular velocity
with which the particle at P de-
Gcribes its orbit, u-\-cia that of the
particle at Q, then Q'Q" = OQ'c-Ji.
Let OP=r,OQ = r+ir,t)iiin since
P'Q'' = h' the velocity gradient at
P is (r + Ir)— , or when Sr is very
small, r-j- ; hence the tangential
stress acting on unit area of the
"""'""''" "' Fid. 1C3.
the portion of liquid bounded by
coaxial cylinders through P and R and by two parallel planes at right
angles to the axes of the cylinders and at unit distance apart. This
annuluB is rotating with constant angular velocity round the axis of the
cylinders, hence the moment about this aiis of the forces acting upon the
annuhis must vanish. Now the moment oF the forces acting on the inner
face of this anuuliu is
= -'7r.(r
<ir
and this must he equal and oppoiiite to the moment of those acting on the
outer sill face of the cylinder; now A may be taken anywhere; hence we
see that this expiei'sion must be constant und equal to the moment of the
couple acting on unit length of the outer cylinder, wliieb is, of con i-se, equal
and opposite to the moment of that on the inner. Let us coll this moment
r, then Sfl-iji-'-— = p
dr
Integrating this equation we Cod
where C is a constant. If the radii of the in
a and b respectively, and if the inner cylinder i
rotates with an angular velocity Q, then since
ler and outer cylinders a
, at rest and the outer oi
1^0, when r*-o, and j ^
214 PROPERTIES OF MATTER.
Henoe, if we measure r for a given velocity O, we can deduce the value of if.
This case presents the sime peculiarities as the flow of a viscous liquid
through a capillaiy tube ; the law expressed by the preceding equation is
only obeyed when O is less than a certain critical value When Q exceeds
this value the motion of the fluid becomes turbulent, and for values of Q
just above this value the relation between r and CI becomes irregular; it
becomes regular again when Q becomes con-^iderably greater , but r is no
longer proportional to il, but is of the form qO + /Sft' where a and /3 are
constants. These facts aie well shown by the cuive given in Fig. 164,
Fio. 164.
which represents the results of Couette's* experiments on the viscosity of
water. The abscissae are the values of O and the ordinates the values
of r/O. Tiie instability set in at B when the outer cylinder made about
one revolution jier second ; the radii of the cylinders were 14'G4 and 14*39
rm. respectively.
Tliis method can be applied to determine the viscosity of gasos as well
as of liquids.
Method of the Oscillating' Disc— Another method of determining
i;, which has been used by Coulomb, Maxwell, and O. E. Meyer, is that of
measuring the logarithmic decrement of a horizontal disc vibrating over a
flxed parallel disc placed at a short distance away, the space between the
diflos being filled with the liquid whose viscosity is required. The viscosity
^ Coaette, AnnaUt de Chimii et de Phif$iQM« [6], 21, p, 438.
VISCOSITY OF LIQUIDS. 815
of tlie liquid gives rise to a couple tending to retard the motion of tlio
disc proportioDul to the proiiuct of the angular velocity of the disc and
the viscosity of the liquid; the calculation of this couple is somenbat
dilfictilt. We shall refer the reader to the solution given by Majcwell
i:
{Collected Paie^-a, vol. ii. p. 1). "IIhs method, as well as the preceding one,
cm he used for gases as well »» for litpiids.
Among other methods for meneuring jf we may mention the determina-
tion of the logarithmic deciemenl for a pendulum vibrating in the fluid
(Stokea) ; the logarithmic decrement of a sphere vibrating about a diameter
216
PROPERTIES OF MATTER.
in an ocean of the fluid ; the logarithmic decrement of a hollow sphere
filled with the liquid and vibrating about a diameter (Helmholtz and
Piotrowski, HelmhoUz Collected Papers, vol. i. p. 172).
Temperature Coefficient of Viscosity.— in all experiments on
viscosity it is necessary to pay great attention to the measurement of the
temperature, as the coefficient of viscosity of liquids diminishes very
rapidly as the temperature increases. This is shown by the curve (Fig.
165) taken from the paper by Thorpe and Bodger (Phil, Trans,, 1894, A.
Part ii. p. 897), which shows the relation between the viscosity of water
and its temperature. It will be seen that the viscosity of water at 80° C
is only about one-third of its value at 10° 0, Thorpe and Rodger, who
determined the co-efficients of viscosity of a large number of liquids, found
the formula given by Slotte, 17=0/(1 + 6«)", where 17 is the co-efficient of
viscosity at the temperature t and C, b and n are constants depending on
the nature of the liquid, was the one that agreed best with their experi-
ments. For water they found that
•017941
»7
5423
(1 + ordi^oty-
where i is the temperature in degrees Centigrade.
The following table, taken from Thorpe and Rodger's paper (Phil.
Trans., A. 1894, p. 1), gives the value of 1; in C.G.S. units for some liquids
of frequent occurrence. The table gives the value of the constants C, b, n
in Slotte's formula
, = C/(l + 6«)»
SUBSTANCB
b
A
Bromine
•012535
•008935
1^4077
Chloroform
•007006
•006316
1-8196
Carbon tetrachloride ....
•013466
•010521
1-7121
Carbon bisulphide ....
•004294
•005021
1-6328
Formic acid
•029280
•016723
1-7164
Acetic acid
•016867
•008912
2 0491
Ethjl ether
•002864
•007332
1-4644
Benzene
•009055
•011963
15554
Tolaene
•007684
•008850
1 6522
Methyl alcohol
•008083
•006100
2-6793
Bthyl alcohol
•017753
•004770
4-3731
Propyl alcohol
•038610
•007366
8*9188
Butyl alcohol :
O'to62*
•051986
•007194
4-2452
62'toll4''
•056959
•010869
8-2160
Inactive amyl alcohol :
*^ «■ jk ^# *^
0*to40'
•085358
•008488
4-3249
40" to 80"
•093782
•012520
8-3395
80" to 128"
•152470
•026540
2-4618
Active amyl alcohol :
0"to85"
•111716
•009851
4-3786
85' to 78*
•124788
•015463
8-2542
78" to 124*
•147676
•127583
2-0050
AUyl alcohol
•021736
•009139
2-7925
Nitrogen peroxide ....
•005267
•007098
17349
VISCOSITY OF LIQUIDS. SI7
Warburgfoundthat^formertTiiryat IT^" is equal to -016323. A later
determinntioTi by Umani (jVikh'. Cim. [4] 3, p. 151) gives k = 'Ol.'iTTat 10".
The value of q for liquid carbonic a-cid is very fcinall, being nt 15° only
1/U-Gof that of water.
Effect of Pressure on the Viscosity.— The viscosity of wnter
diminishes slightly under iucreiised pressure, while that of benzol Bn<)
ether increases.
Viscosity of Salt Solutions.— a large number ot experiments
have been made on the viscosity of eoliitions, but no simple laws eoB-
Dfiictiug the viscosity with the strength of the solution have been arrived
at. la some cases the viscosity of the solution is less than that of water,
Euid in many cases the viscosity of thesulution isamnximtmi foraparticulur
strength,
Viscosity of Mixtures.— Here again no general results have been
arrived at, although considerable attention has been paid to this subject.
In many cnses the viscosity of a miitture of two liqiii<Is A, B is less thaa
that calculated by the formula
where rr,^, ij^ ■i''^ respectively the viscosities of A and B, and a, h are the
volumes of A and B In a volume a + i of the misture.
Lubrication. — When the surfaces of two solids are covered with oil
or some other lubricant they are not in contact, and the friction between
tfaem, which is much less than when they are in contact, is due to fluid
friction. The laws of fluid friction discussed in this chapter show that,
if we have two parallel plaiies at a distance d apart, the interval between
them being filled with a liquid, then if the lower plane i.t at rest and
the upper one moving parallel to the lower one with the velocity V,
if V is not too great there is a retarding tangential force acting on the
moving plane, and equal per unit area to qV/<j, where i] is a quantity
called the coetEcient of viscosity of the liquid. If we regard this as a
frictioaal force acting on the moving plate we see that the friction would
depend upon the velocity, and would only depend upon the pressure between
the bodies in so far as the pressure affected the thicknajs of the liquid
layer and the viscosity of the lubricant.
The laws of friction, when lubricants are used, are complicated, depending
largely upon the amount of lubncatian. When the lubricant is present
in Bulliciently large amounts to fill tbe Kpaces between the moving parts
the friction seems to he proportional to the relative velocity of these parts.
When the supply of lubricant is insufiicieDt, part of it collects as a pad
between the moving parts, as in Fig. 1€6; here the lower surface is at
rest and the upper one rotating from left to right. Professor Osborne
Beynolds* has shown that, as the breadth and thickness of this pad
depend upon the pressure and relative velocity, it would be possible to get
friction proportional to the pressure and independent of the relative
velocity, even when the friction was entirely caused by the viscosity of a
thin layer of liquid between the moving parts.
Viscosity of Gases. — Uases possess viscosity, and the forces called
into play by this property are, as in the case of liquids, proportional to
the velocity gradient; in fact, the deflnition of viscosity given on p. 2l>5,
■ Itejnolds, PhU. Trant., 1886, pt. i. p. 1S7.
218 PROPERTIES OF MATTER.
applies to gases as well as to liquids. The most remarkable property of
the viscosity of gases is that within wide limits of pressure the viscosity
is independent of the pressure, being under ordinary circumstances the
same at a pressure of a few millimetres of mercury as at atmospheric
pressure. This is known as Maxwell's Law, as it was deduced by Maxwell
from the Kinetic Theory of Gases; it has been verified by numerous
experiments. Boyle has some claim to be regarded as the discoverer
•V^•.•7??^^t///////////, . . . , iillllllW^'.'
Fio. 166.
of this law, for about 1660 he experimented on the effect of diminishing
the pressure on the vibrations of a pendulum, and found that the vibrations
died away just as quickly when the pressure was low as when it wai
high. This law follow^s very readily from the view of viscosity supplied
by the Theory of Gases. Thus, suppose we have two layers of gas A
and B at the same pressure, and that A has a motion as a whole from
left to right, while B is either at rest or moving more slowly than A in
this direction. According to the Kinetic Theory of Gases, molecules of
the gas will be continually crossing the plane separating the layer A from
B
Fio. 167.
the layer B, Some of these molecules will cross the plane from A to B
and an equal number, since the pressure of the gas remains uniform, from
^ to -4. The momentum parallel to the plane of those which leave A
and cross over to B is greater than that of those which replace them
coming over from Bio A ; thus the layer A is continually losing momentum
while the layer B is gaining it. The effect is the same as if a force panUlel
to the plane of separation acted on the layer J, so as to tend to
stop the motion from left to right, while an equal and opposite force acted
on B, tending to increase its motion in this direction ; these forces are
the viscous forces we have been discussing in this chapter. If the distri-
bution of velocity remains the sjime, the magnitude of these forces will
be proportional to the number of molecules which cross the plane of sepa-
ration in unit time.
The molecules are continually striking against each other, the average
free run between two collisions, called the mean free path of the molecules,
being extremely small, only about 10"* cm. for air, at ixtmospherio
pressure. This free path varies, however, inversely as the pressure, and at
the extremely low pressures which can be obtained with modern air* pumpa
VISCOSITY OF LIQUIDS.
S19
Cfin atiafn a 1en;^b of several centimetres. When one molecdie sti ikes
against another its course is deflected, so that, although it is travelling at
a great speed, it makes but little progiess in any assigned direction. The
consequence of this is that the molecules which cross in unit time the
plane of separation between A ard B can all be re«^avded as coming from
a thin layer of gas next this plane, a definite fi-action of the molecules
in this layer crossing the plane. The longer tlie free path of the molecules
the thicker the layer, the
tiiickness being directly
proportional to the mean
free path. If n is the
number of molecules per
unit volume and t the
thick nc&s of the layer,
the number of molecules
which in unit time cross
unit area of the plane
separating A and B will
be proportional to nt.
Let us consider the effect
on this number of halving
the pressure of the gas.
This halves n but doubles
t ; 1 18 proportional to the
free path, which varies
inversely as the pressure,
hence the product nt, and
therefore the viscosity,
remains unaltered. This
rasoning holds until the
thickness of the layer from
which the molecules cross
the plane of separation
gets so large that the layer
reaches to the sides of
the vessel containing the
gas. When this is the
case no further diminu-
tion in the pressure can
increase t, and as n dimin-
ishes as the prcvssure
diminishes, the product
nt and, therefore, the viscosity, will fall as the pressure falls. Thus in a
vessel of given size the viscosity remains unaffected by the pressure until
the pressure refiches a certain value, which depends upon the size of the
vessel and the nature of the gas; when this pressure is passed the
viscosity diminishes rapidly with the pressure. This is shown very clearly
by the curves in Fig. 1G8, based on experiments made by Sir William
Crookes {Phil, Trans,, 172, pt. ii. 387). In these curves the ordinates
represent the viscosity and the abscisssd the pressure of the gas.
The diminution in viscosity at low pressures is well shown by an incan-
descent electric lamp with a broken filament. If this be shaken while the
Miliionths of on AtmQtph9r9»
FlO. 168.
220
PROPERTIES OF MATTER,
lamp is exhausted it will be a long time before the oscillations die away ;
if, however, air is admitted into the lamp through a crack made with a
file the oscillations when started die away almost immediately.
Another reason why the effects of viscosity are less at very low pressures
than at higher ones is the slipping of the gas over the surface of the solids
with which it is in contact. In th^ case of liquids, no effects due to slip
have been detected. Kundt and Warburg* have, however, detected such
effects in gases even up to a pressure of several millimetres of mercury.
The law of slip {see Maxwell, " Stresses in a Rarefied Gas,'' PhiL Trans,^
187) may be expressed by saying that the motion in the gas is the same
as if a certain thickness L were cut off the solids, and that the gas in
contact with this new surface were at rest. This thickness L is propor-
tional to the mean free path of the molecules of the gas. According to
the experiments of Kundt and Warburg it is equal to twice the free
path ; hence, as soon as the free path gets comparable with the distance
between the solids in the gas, the slip of the gas over these solids will
produce appreciable effects in the same direction as a reduction in
viscosity.
Mean Free Path. — If we know the value of the viscosity we can
calculate the mean free path of the molecules of a gas : for if we calcu-
late, from the principles of the Kinetic Theory of Gases, the rate at which
momentum is flowing aa*oss unit area of the plane Af B^ Fig. 167, we find
that it is equal to
cue
where v is the velocity of the stratum at a height x above a fixed plane,
X is the mean free path, p the density of the gas, c the '' velocity of mean
square '' (this can be calculated from the relation p = \pc? where p is the
pressure in the gas). The rate of flow of momentum across unit area
is equal to the tangential stress at the plane AB ; hence, if 17 is the viscosity
of the gas, ri = * 350cpX. Let us calculate from this equation the value
of X for air; taking for the viscosity at atmospheric pressure and at
15° C. J7 = l-9xl0"\ p at pressure 10® and temperature 15° C,
1-26 X 10-», we get c = 4«6 x 10*, and X = -00001 cm. At the pressure of a
millionth of an atmosphere the mean free path in air is 10 cm.
The values of ri for a few of the most important gases are given in
the following table ; the temperature is about 15° 0. These numbers
are given by 0. E. Meyer ; they are deduced from his own experiments
on the viscosity of air by the method of the oscillating disc and the expe-
riments made by Graham on the relation between the rates of flow of
different gases through capillary tubes :
Gm
1|X104
Gas
,X104
Air .
. 19
Sulphuretted hydrogen .
► 1-3
Hydrogen
. -93
Hydrochloric acid .
1-56
Marsh-gas
. 1-2
Carbonic acid .
. 1-6
Water- vapour .
. -975
Nitrous oxide (N^O)
. 1-6
Ammonia
. 108
Methyl ether . . ,
, 1-02
Carbonic oxide
. 1 -84
Methyl chloride • .
. 116
Ethylene .
. 1-09
Cyanogen
. 1-07
Nitrogen .
. 1 -84
Solphuroas acid (SO.J
, 1-38
Oxygen .
. 212
Ethyl chloride
. 1-05
Niuic oxide (NO) .
. 1 -86
Chlorine • e • •
. 141
* Pogg. Ann,, 155, p. 857*
VISCOSITY OF LIQUIDS. 531
Effect of Temperature upon the Viscosity of Gases.— Tncieiu^e
of tempeiuture has opposite efl'ects on the viscosities of liquids aod of gases,
for while, as we have seen, it diminishes the viecosity of liquids it increases
that of gases. If ij is the coefficient of viscosity, and if this is assumed
to boprDportioTialtoT''whei'eTistheal>8olute temperature, then, according
to Lord Rnyleigli'fi* experiments, we have the follo^viIlg values for n :
Oxygen
Hydrogen
Helium
The values of e i-elute to
-754
-681
1 11 3
128-2
72-2
72-2
to which 1} =
■c/T-
150-2
formula suggested by SulLerliind, according
thus, at very high tern pertita res, if this relation
is true, i) would vai-y as the square root of the RbKoIute temperature.
According to Koch,t the viscosity of mercury vapour varies much more
rapidly with the temperature than that of uny other known gas. Ha
concluded from his experiments that for this gas »; = aT''°. The results
given above for helium end argon, both, like mercury vapour, monatomic
elements, show that a rapid variation with temperature is not a necessarv
charactei-istic of monatoniic gases. Lord Kayleigh found that the viscosity
of argon was 1-21, and of helium 0>90 that of air.
Coefficient of Viscosity of Mixtures.— Graham made an extensive
series of experiments on the coelficients of viscosity of mixtui-es of gases
by meisunng the time taken by a known volume of gas to flow through
a capillary tube. He found that for mixtures of oxygen and nitrogen, and
of osygen and carbonic acid, the rata of flow through the tubes of the
mixture was the arithmetical mean rata of the gases mixed ; with mixtures
containinghydrogen the results were very different ; how diflerent is shown
the ratio of the transpiration time of
by the following table,
which gives
the mixtui-es to tliat of
pure oxygen
Hydrogen and Carbonic Ac
d.
100
. -mi
87-6 ...
2-5
. -m*
GG
G
. -61 S7
SO
10
. -5722
76 ...
2&
. 'ersfl
CO
GO
. -7839
25
76
. -7636
eo
. -7521
100
. -7470
It will be
seen fio
n this teble
of air to pure
hydroge
n alters the
. while the addition of 5 per cent.
alters the time of effusion by about 20 per cent,
the mixture of half hydrogen, half air, has a time of eflTusion wbiL-h only
diffei-s from that oF pure air by about 8 per cent. Thus the addition of
hydrogen to air has tittle influence on the viscosity, while the addition
of air to hydrogen has an enormous influence.
Resistance to a Solid moving- through a Viscous Fluid.— When
ft solid moves through a tluid the portions of the fluid next the solid are
• Biifleiph, Pr<K. Suy. Soc . 66, p. 68,
t Koch, Wial. Ann., 19. p. fi87.
S22 PROPERTIES OP MATTER.
moving with the name velocity as the solid, while the portions of tbe fluid at
some distance off are at rest. The movement of the solid thus involves
relative motion of the fluid ; the vLsco^ity of the fluid resists this motion,
so that there is a force acting on the solid tending to resist its motion.
Sir George Stokes has shown that in the case of a sphere moving with
a very small uniform velocity V through the fluid the force resisting the
motion is equal to (dynjaV wh(ire a is the mdius of the sphere, if the
viscosity of the fluid through which it is falling. Consider now the case
of a sphere falling through a viscous fluid ; just after starting from rest the
velocity will be small and the weight of the sphere will be greater than
tne viscous resistance; the velocity of the sphere, and therefore the
resistance, will increase until the resistance is equal to the weight cf the
sphere. When this velocity, which is called the critical velocity, is reacned,
the forces acting on the spliere will be in equilibrium, and the sphere will
fall with a uniform velocity which may also be called the terminal velocity.
Since the effective weight of the sphere is equal to 47ra'(p - o')^/3, where p is
the density of the sphere and a that of the liquid through which it is moving,
if V is the terminal velocity,
or V = ^^ -' -^ — ^ (1)
so that the terminal velocity is proportional to the square of the radius
of the sphere. In the ca.se of a drop of water falling through air for which
rj = l'Sx 10'*, we find, if the radius of the drop is 1/100 of a millimetre,
V = 1*2 cm ./sec. This result explains the slow rate at which clouds con-
sisting of fine drops of water fall. Since rj is independent of the pressure,
the terminal velocity in a gas will, since a in this case is small compared
with p, be independent of the pressure.
As an application of this formula we may mention that the size of small
drops of water has been determined by measuring the rate at which they
fell through air ; from this the value of the radius can be determined by
equation (1). The expression for the resistance experienced by the sphere
falling through the viscous lic^uid is obtained on the supposition that the
motion of the liquid is so slow that terms depending upon the squares of
the velocity of the licjuid can be neglected in comparison with those re-
tained. Now, if V is the velocity, p the density of the liquid, the forces on
the liquid depending iii)on the squares of the velocity, are proportional to
the gradient of the kinetic energy per unit volume — i.e., to the gradient of
^pV ; the forces due to viscosity are proportional to the gradient of the
viscous stress. If a is the radius of the sphere, the distance from the
sphere at which the velocity may be neglected is proportional to a, hence
the velocity gi-aiient is of the order (V/a), and the viscous stress »;V/a.
Hence, if we can reject the efiects depending on the squares of the
velocity in compaiison with the effects of viscosity, pV* must be small
compared with rjY/a, or pYa must be small compared with ly. Hence, if
the preceding solution holds, we see, by substituting for V the value of
the limiting velocity, that -g ^ ^^ " ff^ must be small. Loixl Bayloigh *
^ Lord Kayicigb, PhiL Mag., [b] i)6, p. 861.
VISCOSITY OF LIQUIDS. 223
has pointed out how much this restricts the application of Stokes' result ;
thus, for example, in the case of drops of water falling through air, the
theory does not apply if the drops are more than about one-tenth of a
millimetre in radius. When the velocity of the falling body exceeds a
certain critical value the motion of the surrounding fluid becomes
turbulent, just as when the velocity of a fluid through a capillary tube
exceeds a certain value the flow ceases to be regular (see p. 212). When
this turbulent stage is reached the resistance becomes proportional to the
square of the velocity. Mr. Allen,* who has recently investigated the
resistance experienced by bodies falling through fluids, flnds that this can
be divided roughly into three cases — (a) where the velocity is very small,
when the preceding theory holds, and the resistance is proportional to the
velocity ; (6) a stage where the velocity is great enough to make the forces
depending on the square of the velocity comparable with those depending
on viscosity ; in this stage the resistance is proportional to the velocity
raised to the power of 3/2 ; (c) a stage where the velocity is so great that
the motion of the fluid becomes turbulent; in this stage he flnds the
resistance to be proportional to the square of the velocity. When the
resistance is proportional to the square of the velocity the method of
dimensions shows that it does not for a given velocity depend upon the
viscosity of the liquid. For, suppose the resistance is proportional to
a*p*'?/'V*, this expression must bo of the dimensions of a force — t.tf., 1 in
mass, 1 in length, and - 2 in time ; hence we have
I'^y + z
-2-= -2-?l
so that a; =w 9i, y = w - 1, « = 2 - n,
and the resistance is proportional to {^ap/nYi^^/p)* thus, if ns2 the
resistance is proportional to Y*a*p, and is independent of viscosity. The
energy of the body is spent in producing turbulent motion in the liquid
and not in overcoming the viscous resistance.
A great deal of attention has been given to the resistance of bodies
moving with high speeds, such as bullets. It is doubtful, however, if the
viscosity of the fluid through which the bullet moves has any efl!ect upon
the resistance ; we shall not, therefore, enter into this subject, except to
say that the most recent researches, those by Zahm, seem to indicate that
lor velocities loss than about 30000 cm./sec. the resistance may be repre-
sented by av' 4- ^y', where a and h are constants.
• AUeu, PhU, Mag,, Sept. and Nov. 1900.
INDEX
Acceleration due to gravity, 7-24
Air, deviaiions from Boyle's law as to, 126
Airy, hydrostatic theory of earth's cru&t,
23
Dolcoath experiment, 35
Harton pit experiment, 35
Amagat, minimum valae of jot;., 120, 127
Angle of shear, 66
Arc, correction for pendalam swing, 10
Atmolysis, 202
Baily'8 Cavendish experiment, 89
Bailie and Coma*s experiment, 39
Bars, bending of, 85-102
vibration of, 94
Barymeter, von Stemeck's, 26
Bending of rods or bars, 85-102
Bemouilli's correction for arc of swing
of pendulam, 10
Boiling-point, depression of, in solutions,
191
Borda's pendulam experiments, 10
Boug^er's pendulum experiments, 10
experiments on determination of
density of earth, 32
rule and exceptions, 22-3
Boyle's law, 125
at low pressures, 128
deviations of various gases from, 126
Boys's Cavendish experiment, 40
Braun's Cavendish experiment, 41
Breaking-point of stretched wires, 55
Bubbles and drops, measurement of
surface tension by, 156, 161
Camphor, movements of on surface of
water, 169
Capillarity, 135-181
Laplace's theory of, 173-181
Capillary tubes, rise of fluids in, 140
Carbonic acid, deviation of, from Boyle*8
law, 126
Carlini's pendulum experiment-, 35
Cassini's and Borda's pendulum experi-
ment, 10
Cavendish experiment, 36
by other observers, 39
see Earth, determination of density qf
Clairaut'B theorem, 22
Cullision, 109
duration oF, on impact, 112
of drops, 172
s^e also Impact
Colloids, 186
Compressibility of liquidi^, see Liquids
Computed times of pendulums, 15
Contamination of films, 170
Critical velocity in viscous fluids, 222
Crystalloids, 186
DfiFFORQES' pendulum, 19
Degree of latitude, measurement of a,
21
Diaphragm, diffusion through, 186, 200
Differential gravity balance, 26
Diffusion of gases see Gases
of liquids, see Liquids
of metald, 204
Dilatation under strain, 64
Dissociation of electrolytes, 194
Earth, determination of density of, 31
by fUry, 35
Baily, 39
Bouguer, 32
Boys<, 41
Braun, 41
Carlini, 35
Cavendisli, 36
Cornu and Bailie, 39
von Jolly, 42
Maakelyne, 33
Mendenhall, 35
Poynting, 43
Richarz and Kri^ar-Htcnzcl, 42
von Sterneck, 36
Wilsing, 41
Effusion, thermal, 202
Elastic after-effect, 55
curve, 95
fatigue, 57
limit, 53, 69
Elasticity, 53
modulus of, 69, 102
see also Young's Modulus
Electrolytes, dissociation of, 194
Ellipticity of earth, 23, 24
Elongation under strain, 64
226 INDEX
Eqailibriam of liqnids in contact, 139
Eqaivalent simple pendulum, 18
Fatigue, elastic, 57
Faye's rule, 23
Films, contamination of, 170
cooling effects, on stretching, 168
stability of cylindrical, 147
Flexure, 99
Floating bodies, forces acting on, 153
Fluid motion, effect of, on pendulums, 14
surfaces, disruption of, 174
Formulas for pendulum motion, 13-24
Freezing-point, depression of in solu-
tions, 193
Galileo*8 observations respecting pen-
dulums, 8
Gaseous pressures and volumes, 124
Gases, diffusion of, 196
kinetic theory as applied to the, 198
obstruction to, offered by perforated
diaphragms, 200
through porous bodies, 201
Gases, passage of, through india-rubber,
203
throagh liquids, 203
through red-hot metals, 204
Gafes, viscosity of, 210, 218
influence of temperature upon, 221
Giavitation, constant, 29
Newton's law, 28
qualities of, 45-52
tee also Earthy density of
Gravity, acceleration of, 7
history of research, as to, 7
Clairaut's theorem, 22
Newton's theory of, 20
Hicher's observations on, 20
bwedish and Peruvian expeditions
of investigation, 21
Gravity balance, Tlirelfall and Pollock'^,
27
Gravity meters, differential, 26
Half-seconds pendulum, von Stemeck,
24
Hodgkinson*s table of values of e on
impact. 114
Homogeneous strain, 62
Hooke's law, 69
Hydrogen, deviations of, from Boyle's
law, 126
Hydrostatic theory, 23
Huygens' pendulum clock, 9
theory of pendulums, 9
Indian survey, experiments on pendu-
lums, 23
Impact, 109
duration of colliMon on, 112
kinetic energy of, 110
tnvariable pendSam, 28
Jabobb'b method of determining mean
surface-tension, 162
Jolly, von, experiments on gravitation
42
Eater's convertible pendolam, 12
and Sabine's experiments, 23
Kelvin's table of thermal effects ao-
companying strain, 184
Kinetic theory of gases, 218
explanation of diilasion by the,
198
Laplace's theory of capillarity, 178
Latitude, determination of length of 1*
of, 21
Liquids, capillarity of, 135
compressibility of, 116, 122
diffusion of, 183
determination of co-efficient of
184
through membranes, 186
in contact, 139
films, stability of, 147
flow of viscous, through cyllndrica]
capillary tubes, 207
potential energy of, due to surface
tension, 137
rise of, in capillary tubes, 140
surface-tension of, 137
relation between curvature and
pressure of surface, 142
methods of measuring, 155
by bubbles and drops, 156, 161
by ripples, 157
temperature, effects on, 163
table of compressibility of various,
122
tensile strength of, 122
vapour-pressure over curved surface
of, 166
viscosity of, 205
Loaded pillar, stability of, 97
wires, anomalous effects in, 68
Lubrication, 217
Mass, 3
constancy of, 5
definition of, 4
unit of, 5
Max weirs law of gaseous viscosity, 218
Mean free path, 218, 220
Menden hall's gravitation experiment, 85
Mercury, compressibility of, 121
Metals, diffusion of, through metals^ 204
elastic properties of, 53, 57
viscosity of, 57
Michell, ^ev. J., 36
Microstructure of metals under stresa
58
Modulus of elasticity, 69, 102
Young's, 70, 73, 74, 76
of rigidity, 7
I
Newtos'b iheory of gtaviWlioo, 2S
ihcory of gravity, 20
NitroBcn, deviation of, from Bojle's law.
Oil, effect of, on waves, 171
Osmosla, 186
Oamotio piessutc, IBS
Pkndvlvus, Besset's experiments, 18
Bonla and Cas&ini'a, 10
clock. 9
DetTorgcB, 19
formulsifor, 13-24
Hnlf-tecoDds pendulum. 24
Huygens' theory of. 9
Indian survey experiments. 23
invariable. 23
Kater'a convertible, 12
Newton's use of, 9
Papers on the theory of, 7
Kepaold's, If)
von Sterneck's, 24
U.S. aorvcy, 20
variation in length of seconds, 2
vieldiog of support of, IS
reraanent set, G3
I'icBrd's peadulnni expeiiments, 9
I'iezometer (ihe), 119
I'oieeuille'B lair, 209
Poiason's ratio, 13, 87, 120
Poynting'a Kravitation eiperimentu, 43
" , effect ot, on viacosiiy, 217,
219
__. 124
from Boyle's law at low.
Reich'b Cavendish experiment, 39
Rapsold's pendnlnm, 18
geaolalioo of stmin, SB
Reversible pendnlum. theory of, 13
Reiergible thermal effects aocompanjing
strain, 131
Richer, observations on gravity, 20
Rigidity, co-eHlcient of, S3
modal us of, 70
Ripples, tueaaarement of surface-tension
by, 167
Rods, stresses snd strains of, 71, 73, 79,
83, f
-103
Sabine's pendDlam. 23
Salt solntioDS, viscosity of, 217
Scbiehallion eiperimeul, 32
SolutioD-i, depression of boiling-point ot,
13t
of freezing-point of, IS3
vapour pressure of, 100
Spiral springs, 101-108
energy of, 104-108
Stability of cylindrical films, 147
of loaded pillar, 97
Stemeck, von, Barymeter. 2S
half- seconds pendnlum. 24
pendulum experinienta, 30
Strain. 62
anomalous effects of alternating, on
wire. 68
alteration of m
homogeneous, 62
resolution ot a. 66
in relation to work, 70
thermal effects acoompanjing, 131
Stresses, 68
on bars, 71
Stretched film. 144
cooling due to stretching. 163
Stretched nire, anomalous effects on
loading. GS
Surface-tension, 137
effccta between two liquids, 179
in thick films. 178
forces between 3 plates, due to, 162
Surface-tcns^ion. Jaeifer's method ot
measuring. 1S2
OBClllaiions of a spherical drop
under. 160
of thin Rims, 164
meascremeDt of by detachment of
a plate, 161
Ripple method, 157
Wilhelmy's method, 161
Swedish and Peruvian expeditions to
det«rmine length ot 1* of lati-
tude, 21
Table of moduli of olastioity. 102
thermal effects of strain, 131
Tangential streaa, 63
Temperature, co-efficient of viscosity, 219
effects of, on soif ace- tension, 163
on breaking stress of wires, 61
Tensile strength of liquids, 123
Terminal velocity in viscous floida, 222
Thermal effecta ot strain, 131
Kelvin's Uble of, 134
Thermal effusion, 202
Thickness of films, influence of, on
snrfaoe-tension, 178
Thin films, surface-tension of. 114
Threlfall and Pollock's gravity balance,
37
Torsion, 78
in cylindrical tubes, 78
In solid rods, 79
228
(T.£f« SUBYBT pendnlams, 20
INDEX
VArouB, diffuBion of, 197
Vapour pressure, of solutions, 190
on ourved surfaces, 166
Vibration of bars, 95
Viscosity, 60
temperature co-efficient of, 216
determination of co - efficient of,
212
by oscillating disc, 214
effects of pressure upon, 217
gaseous, effect of temperature on,
221
of gases, 218
of liquids, 205
of metals, 57
of mixtures, 221
of salt solutions, 217
Viscous fluids, resistance o^ to motion of
soUds, 221
velocity in, 222
Volume and pressure of gases. 124
Water, compressibility of, 121
Waves, calming of, by oil, 171
Weight, 1
standards of, 5
Wilhelmy's method of measuring sur-
face-tension, 162
Wilsing's gravitation experiments, 41
Work in relation to strain, 70
Yield pointy 53
Young's modulus, 70, 73
determination of, 74
by flexure, 99
by optical measurement, 76
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Bxwninationg — Bin and Frogteu of Stenm FropnlBion — Developmaot of
Free Trade- Shipping Irfigialatiiin, 1S62 to 1676— " Locksler Hall" Cm*—
Bhipmutun' SiwUtii»— LoadiuK of Ships— Sbipmng Legiilati<ni, 18B4 to ISH—
aullatiQa of Shipping. Thb Febbonnei. : Shipomten— Offioen— MaiinaiB—
Datiea and Piwent PodtioiL Edvcatiok : A Seunan'a Education i what It
■iumld be— Prewot Meau of Education— Hinta. DiBclFLuri Ami Ddtt —
Pcetacript— Th« Serioai Becr«ue in tba Niunber of Britieb & " " " "
ig tha Attention of tlis Nation
ELEMENTARY SEAMANSHIP.
D, WILSON-BAEKER. MiHTEs Marinkr; F.K.S.E., P.K.G.S.,&0.,&o.
With Frantiapiece, Nui
General Costknts.— The Building oi a Ship; Parts of Hull. Maat%
ftc^— EopoB, Knota, Splioingn &c. — Gear, Lead and Log, Ao. — Biggiog,
Anchors- Sflilmaking — The Sails, &e, — Handling of Boata under Sail —
Signals and Signollinf— Rule of the Road— Keeping and Relieving Watch-
Points of Etiquette- GloHarr of Sea Terms and Phrues— Index.
I
IDHDOK : CHAHLES CRIFFIN < CO., LIMITED. EXETEH STFIEET, STUND.
NAUTICAL WOHKa. 41
GRIFFIN'S UAUTICAL SERIES.
Second Edition, RevUed and lUustraled. Price 3a. 6d.
IsTAVIGA-TIOISr:
PRACrrXCAXd and T^aEOREIXICAXd.
Bt DAVID WILSON- BARKER, R.N.R., F.R.S.B., 4c., Ac,
WILLIAM ALLINGHAM,
T.lomlM
HOlltb numecoue Sllustiatlons aii& Ctamlnatton stuestlone.
Genbkal Contentb. — DeGcitioiia — Lii,tJtQ<le and Longitude—Iaatmineiila
ol NavigatioD—CorrectiDD of Connw*— Plane Suline — Tntverae Swluig^DftT'i
Work— ParttUal Sailing — Midiila Latitudo tWing— Merartur's Chart-
Mercfttor Sailing — Corrent Soling — Piaition by Beninjn! — Great Circle Sailing
—The Tidea— Qneatiaiu— Appendii : 0>mp>BB enoc— Nmneroiu U*efnl Hinta
'- PHDlmi IbB kind ot work tequlrol (or Ib-e Naw Certlflvu
(Tom Second Miila lo on™ Mi
iliIlT ulapled to Ibe
wi In the bigbeit prot
8nl8iideorN«vlg»tlooi
I College, H.U
MARINE METEOROLOGY,
FOE OFFICERS OF THE MERCHANT NAVT.
By WILLIAM ALLINGHAM,
JoiuL AuUior ol " Ksvjgslloii. Theoretical and Practical."
SUMMAKY OF CONTSINTS.
iNTBOPrfTORT.— InBtrnmeati dial it EcK for Meteorological Purpoaei.— Uetwni-
nfniii._tfliid Foroe l^alet.-HlBlorr ol uieldv ot atonni,— llnrrl.'au»7a«uDBi,°Bnd
acorai Traclw.— Solution of the Oynlone Problem.— Ocean CnrreDU.— leobiim.-Sjn.
ebronutu Cliaru.— Dew, Mlitu, Kogi, and Haie.— Clouda.— RalD, Bdov. antT Hall.—
Mlriee, Ralnbowi, Coronaa, Halo*, ami MoUote.— Lightning, CorpoMBU, and Anmrai.—
(JnBBTioiis.— .' '
iUdio Niuilcil ia»o''—Skipri'V Bamat.
*,■ For Complete List of Ghifti
A NiDTiOAL Sbriis, >ee p, 39,
lOODOK: CHARLES GRIFFIH ft CO.. LIMirEO, EXETER STREET, STRAND.
«i OmAMLm GMiFTiM s oolv MnucATnmB.
GRirmrs haptical sketkr
teoon> Bdisos, Rittskd. With Km in o — UkmAnaitm^ Pkioe JiL M.
Practical Mechanics:
Applied to the Bequirements of the Sailor,
Bt THOS. MACKENZIE,
MamB
CowilSf^ — £«aciihrxa.;«i a&d CcHDniHtkA «{ F<
and LfiTinp AercBX»— 1^ Mfmaiinl F
B«st I>eTeE»— I^ WlM«a and Axle
tbe "Oid
TW CeuXrt <£ Gcmrxtr cf a Sfti^ azid Cui^ — BcQsIzvp ^toeBctk «{ Bflpe
flteriWire, lUmDik. ficnm. Codr— I^csxiokB and Skam-CbUitfan erf ~
C k M aUiia tag Stnm cf 1^ S?iu^-Oeni3T of fiffot cf
ixcsmarT bocc . . . cvsitunt a i.aagb Aaorsr «{ ~
** Well woxss t^ nkouer . . . wiE be fcmad xsaDmcsvLT
Ko Ships' Omcxas' k<vecjlsi miD beskcwlnt^ be
GaRa23( M^ccscxii;''^ * Pbactical Msceaxx&. ' y«wiiiiimii?ii% sy
pan' cxpenenoe al aea. it ha^ tcild rut hat mmck amwy ffcgre ■• to
iLrtter to tbe Psli2iii>en ircm. a lianter ILaziMr.
** I voit ex^sn* ilt thank* to vac fir tbe laibcmr and
hi ' Pm^mc-Ai M.rrHAyio5v.' . * . . It b a izrtCf-
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and aoodestr U> iqnars. 4^^ 4^ ! * PiLAmcxi MscaASics ' wonxp &aTX
1 TBI&.'' — XfVbat %C' tbe Axitbor frcstL tOkctSbet VTawwr Itlasaneri.
WORKS BT RICHABD C BUCK,
o! (te ThMiMw- Nftosio^ T^ruim^ CViUefn. & X^ * WoniM
A Manual of Trigt>noinetry
With Diagrams, Examples, and Exerci^s. Price &. Atf.
T&iiLD EiiiTiiiN, Kc'r]a^d and C<arrect^
\* Mr. Bsck't TcjLi-Book has Wen st'Ktiaixt TKCTAKErwix^
to iii« Xr«r Exai&inaiinw aS the Bnard of Traae^ in
it aA oKHgainry acbie^nL.
A Manual of Algpebra.
Dmign^il tn mitrt the PeoufrtimerTs a'' Saiton atul
jsBTvtNr ET>n7rt>. KcriwHA. IVkie 5fe. ftd.
%* Tbtwr n tw um wary worin' at. txaaoLA axtt ti
wbr wil. bft^Y Httlr nravfromir* at co-iiBiohiiif: « TrM:th<r Ther w book* tor **i
* Al. hoi ibe flAnnleii: lornlutfttwui*^ hi^rK UYMVim hWR. sToidiK* maA
kTeprwo; ax;t |K*''«oii: Tnv 'i^Muli't b^ Aftn%tii utaMtv huncoae mMMr of '
lie T^ fenuni^lfw »i«c R»f«<Tt:'«^ m*^ imlnv. trou sbr
Ihe OmIiii» >tf tbr * 'V a^.-Wmc '
^ClMa-'C anrnnfTML an4i ««r. foi iq. K ttrm-
Fur ooBRtiHir aJuh ic (^^vimnv"* >:*mouki finaw. we ti M
LOMDOK : CMItL£S BfilFFlK A CO.. inmn. DXIB STTSH, STIiAllfi.
IfAUTWAL WORKS. 43
GRIFFIN'S NAUTICAL SERIES.
Skcond Edction, TliOToiighly ReWaeil and Eitcnded. In Crown 8to.
Handsome Cloth. Prico 4e. 6d.
THE LEGAL DUTIES OF SHIPMASTERS.
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[ WORKS BT PROF. ROBERT H. SMITH, Assoc. M.I.C.E..
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THE CALCULUS FOR ENGINEERS
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CHEMISTRY FOR ENGINEERS.
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Pabt I. — Alkalikh and Alkauni Earth Metals : Magpeitium,
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the Carhidea of the Alkaliue Earth MetAle.
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In Larijc Vo, Lilirarii Siyln. Beauli/uUy lUtitlrated with SO PlaJcf, many
in Coloar», aiul 94 Fiijurra in Hit Trxl. £2, 29. n'f.
PRECIOUS STONES s.
Their Ppopertlea. OocuprenoeB, and Uaes. J
A Treatise for Dealers, Manufacturers, Jewellera, and for all fl
Collectora and others interested in Qems. S
By Dr. MAX BAUER,
ProfeuoT In tho Unlvenlty of Uartaurg,
TB4NSLATKD BY L. J. SPENCER, M.A. (C4NTAB.), F.D.8.
" Tbo plitei irs nmBrlublo lor Ihelr bcKutT, iluDdn^y, uid tratMnloeu. A bIwim at
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wlDduvr. " — AtheruKU m.
In Large Oroion Svo. With A'ui
u lUwitratiam, S«. 6d,
The Art of the Goldsmltli and Jeweller
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Abbibtbd by J, H. STANSBIE, B.Sc. (Lond.), F.I.O.,
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luirllon.— Hie Anslenl Qaldimlth'i An UetBlIUIgy <
[eltlDE, Kolling, anil Stittlng Qold,— The worinbop '-
_ _. „>. — Chslni ind Initenli AnllqiiH J(— "■ — -'
RgvLvil Elnucao Woik.— Frsoiods Srons.— Cutting. — Follihlng
- - - - - . _ . „ . ^^ J „_,.w,^.
— alldlng
Gold.— PrIOM,
L CONMlltS.— "
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ig Bud nnb
Mu-llUig. — MlHXllHaeoiu. — Appei
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LECTURES ON IRON-FOUNDING.
By TIIOMAS TUKNER, M.Sc, A.R.S.M., F.I.C.,
rroftfBsor Qt Motallurnj In the ljiiivi!r»lly uf BlnnlnfftiBiu!
CoSTBHTd.— Vuietlei ot Imn and StueL—Aiiiillcitloii of Cut. Iron.^HlgtoiT.— F>o-
dDcllcHi. — Iron OrsL— CompositlOD.— The BluC Fuman,— UaterinlB. — Rsutloni.—
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Anenlc, Copper, and Titaolnm The PDandrr.—Oeneral Arrangement.— Ra-malUiia
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Inflnence uf Shape aud Size ou SlreDglh of CaiUngL— Te§tx.
■' IroDtonnden will Ond mudi latumuUoii Id the book."— Jrnn Trada Cim l tr
(Rtriano-t).
/n JfediujH Sro. Haiidsome Cloth. Fvlty lUustraletl.
GENERAL FOUNDRY PRACTICE:
A Practical Handbook for Iron, Steel and Brass Founders,
Metallurgists, and Students of Metallurgy.
By a. C. M'WILLIAM. A.R.S.M., and PEKCV LONGMUIE,
LONDON: CHARLES GRIFFIN A CO.. LIMITED. EXETER STREET, STRKND.
4
■ CHEMISTRY AND TBCHNOLOOT.
69
1
^^^— Criffln'8 Chemical and Teohnological Piiblfcat
on8.
KK, To
^^^^B Inorganic Chemistry,
Profs. Duprb and H
^^^H Quantitative Analysis. .
Pbof, Humboldt Seiton, 70
^^1
^^^H Qualitative
70
^^1
^^^H Chemistry for Engrineers,
Blount aud Bloxah,
46
^H
^^^^P „ Manufacturers,
71
^^^^r Foods and Poisons, ■
A. WysTSR Bltth,
73
^^1
Tables for Chemists,
Prof. Oastbll-Evans,
79
^^1
Dairy Chemistry,
H. D. KiCHMOND,
73
^H
Dairy Analysis, .
„
73
^^1
Milk,
E. F. WiLLODGHBY,
73
^^H
Flesh Foods,
C. A- Mitchell,
74
Practical Sanitation,
Dr. 0. Reid,
78
^^H
Sanitary Engineering:, -
F. Wood, .
78
^^1
Technical Mycology,
Lafar and Saltkr,
74
^^H
Ferments
C. Oppbshbimer,
75
^^H
Toxines and Antitoxines,
11 11
74
^^1
Brewing, ....
Dr. W. J. SiTKES,
7ft
^^H
Bacteriology of Brewing,
W. A. ElLEY, .
75
^H
Sewage Disposal,
Santo Chimp, .
76
Trades' Waste. .
W. Naylob,
76
^^H
Smoke Abatement, .
Wm. Nicuolson,
76
^^H
Paper Technology, .
B. W. Sindall, .
81
^H
Cements
G. R. Rkdoravk,
76
Water Supply. .
R. E. MiDDLETON,
77
^H
Road Making, -
Thob. Aitken, .
79
^^H
Gas Manufacture,
W. Atkinson Butterf
ELD, 77
^^1
Acetylene, ....
Leeds and Botterfie
D, 77
^^1
Fire Risks,
Dr. Schwartz, .
77
^^H
Petroleum,
Sib Bovertos Rbdwo
OD, 61
^^H
(Handbook),
Thohbos and Redwoo
0, 61
^^H
Ink Manufacture, .
Mitchell AND Hepwo
KTH, 81
^^H
Glue, Gelatine, &c., .
Thos. Lambert, .
81
^^1
Oils, Soaps, Candles,
Wright k Mitchell
71
^^1
Lubrication & Lubricants,
Archbutt and Dkelk
, 32
^^H
India Rubber, .
Dr. Carl 0. Websr,
81
^^H
Painters' Colours, Oils, &c..
G. H. Hdkbt. .
80
^^H
Painters' Laboratory Guide,
80
^^1
Painting and Decorating,
W."j. Pkar'cb, '.
80
^^1
Dyeing, ....
Enecht and Rawbon,
82
Dictionary of Dyes.
Haweon and Gardner
82
^^1
The Synthetic Dyestuffs, ■
Cain and Thorpe,
82
^^H
Spinning. ....
Textile Printing,
H. R. Cahtbr, .
83
^^H
Srtuodr Rothwell,
83
^^H
Textile Fibres of Commerce
W. I. Hannah, .
83
^^H
Dyeing and Cleaning. .
G. H. HUR8T, .
84
^H
Bleaching, Calico- Printing,
LONDON t CHARLES GRIFFIN AGO..
Geo, Duerr,
84
J
LIMITED. EXETER STREET. S
TRAND.
I
A SHO&T MANUAL OF
INORGANIC CHEMISTRY.
A. DUPRE, Ph.D., F.R.S.,
WILSON HAKE, PhD.. F.I.O., F.C.S,,
Of IhB WeAmiiutsc HoipiuJ Medical ScfanoL
" A mU'VnttcD, ctav uid acci
Wfe Jlfrc« hirarulv wCth the aritem
Drv Dupi^ Had UakE Will hiiici &
I
mali HuubIi fcv Sniduu. — .4ru^»'.
LABORATORY HANDBOOKS B7 A. HUMBOLDT SEXTOM,
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ELEMENTARY METALLURGY:
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CHEMISTRY FOR ENGINEERS
AND MANUFACTURERS.
A PRACTICAL TEXT-BOOK.
BERTRAM BLOUNT, F.I.C., t A. G. BLOXAM, F.I.C.
voi:.itwie: k.
ploe IOb. Sd.
0HEMI8TRY OF ENGINEERING, BUILDING, AND
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THE CHEMISTRY OF MANUFACTURING
PROCESSES.
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OILS, FATS, BUTTERS, AND WAXES :
THEIfi PREPARATION ANO PROPERTIES, ANO MANUFACTURE THERE-
FROM OF CANDLES. SOAPS, AMD OTHER PRODUCTS.
By C. R. alder WRIGHT, D.Sc, F.R.S.,
iMt Lcctuni im ChciriSrY. St. Muv'i Hottnlal McdiciU SchocI : Emninci
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Thoroughly Revised, Eniarged, and b Part Rcwritlen
By C. AINSWORTH MITCHELL, B.A., F.I.C, F.C.S.
"wm n
e found A>»l
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UmDOK: CHARLES GRIFFIII > CO. UUITED, EXETER STREET, STRIND.
OBASLSS GRIFFIN * OO.'S FUBUOATIOSB.
liorouKhly Reviaed. Greatly Eulu
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FOODS;
THEIR COMPOSITION AND ANALYSIS.
By A. WYNTER BLYTH, M.R.C.S., F.IC, F.O.S.,
Hedlul'oacer or HeiUlb for 8l Mu^UboDS.
And M. WYNTER BLYTH, B.A., B.Sc., F.C.S.
Gehkbal CoNTKNia. — Biitory of AdnlteratioD. ^ LegiaUtion. — Ap-
paratim. — " A«h."— Sngar. —Confectionery. — Hooey. — Ireacle, -~JunB
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BitterAlinonds.—Aniiatto.— Olive Oil, —Water Analysis. —Appendix:
Adulteration Acta, &c.
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POISONS:
THEIR EFFECTS AND DETECTION.
By a. WYNTER BLYTH, M.R.Ca, F.I.C, F.O.S.,
OBNERAI. OONTBNTS.
I, — Eiitorical Introduction. II. — Claaaitication — Statiatict— Connection
between Tooiic Action and Chemical Composition — life Teste — GmenI
Method of Prooedure — The Spectroacope — Eiaminstion of Blood and Blood
Stains. m.-Poiaonons Gases. IV.— Aoida and Alkalies. V.— Mow
or less Volatile Poisonous Substances. VI. — Alkaloids and Poiwmona
Vegetable Princinles. VIL — Poisons derived from Living or Dead Animal
Snbatancea. VIlI.^ — The Oialic Acid Group. IX. —loorganio Poiiona.
Apptndix : Treatment, by Antidotes or otherwise, of Cases of PiHKaiing.
" [TndaabUdlr TBI aoin coMri.m itdu on Taiicologr in unr LAi]EU>in."-TM ^asl^ri r«i
tONDON : CHARLES GRIFFIN 1 DO.. LIMITED. EXETER STREET, STRAKa
CHEMISTRY AND TECHNOLOGY.
73
With Numerous Tables, and ji llluslralions. l6s,
DAIRY CHEMISTRY
FOR DAIRY MANAGERS. CHEMISTS, AND ANALYSTS
A Practical Handbook for Dairy Chemists and others
liaving Control of Dairies.
Bv H. DROOP RICHMOND, F.I.C.,
Conlenli.—l. InlToducloiy.— The Conslituenls of Milk, IT. The Analysis ol
Milk in. Normal Milk: its Adulterations and Alleralians, and their Detection.
IV The Chemical Control of the Dairy. V, Biological and Sanitary Mallen.
VI. Butler. VII. Olher Milk Products. VIII. The Milk of Mammals Mher
than the Cow. — Appendices. — Tables. — Index,
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Fully Illufltrated. With Photogrophs
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MILK: ITS PRODUCTION & USES.
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Hygiene and Control of Supplies.
Bv EDWARD F. WILLOUGHBY,
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Late InipecCor ol VannB and Oflnanl ScienlUlc AdvfiHfr to Wdlford and Sous, Ltd.
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the prlcB of» novel a work which wLU itind goiid u a wurti iif reference tor loniB yean
Ut E^M^^''—Agr\ndt. OaatU-
i - We cordlillj recommend it lo everyone who hiu anything at nil lo do with milk."—
Aairv Vorti. ^
DAIRY ANALYSIS.
Bv H. DROOP RICHMOND, F.I.C.,
Auily!^! la [he Aylubury Dairy Co., Lid.
Contests.— Composition of Milk and its Products.— Analysis a
Analysis of Liquid Products.— Application of Analyses lo the Solut.
Problems.— The Ana,lysij of Butler.— Analysis of Cheese.— Tables for C»lcu-
l Milk.-
laiioi
" Wiib
— Siandord Solutions. — Index.
1 to Ibe I
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AGRICULTURAL CHEMISTRY AND ANALYSIS :
A PRACTICAL HANDBOOK FOft THE US£ OF AGRICULTURAL STUDENTS.
By J. M. H. MUNRO. D.Sa, F.I.O., F.O.S.,
ProftsMor of OhemiBIry. DownMo College of Agrfcolinre.
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lion of Flesh.— Melhodl of Examining Animal Fai.— The Preservalion of Flesb.
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Fl«b Peptones,— The Cooking of Flesh.- Poisonous Flesh.- The Animal P«ni-
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Bt WALTER J. SYKES.
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GARMENT
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Bv GEORGE H. HURST, F.C.S.,
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By R. LLOYD PRAEGER. B.A., M.RI.A,
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GaHKBiL CotiTii!iT9.— A Dainy-Starred Pasturo— Under the Hawthonu
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OPEIl-fllR STUDIES Ijl GEOLOGY:
An Introduction to Geology Out-of-doors.
Br GEENVILLE A. J. COLE, F.G.S., M.R.LA.,
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OiNKKAL Contents.— The MatariaU of the Earth— A Mountain Hollow
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Folds of the Mountains.
r PaoF,
yss/'
JeoioffteaJ Unyiat
OPEH-AIR STUDIES III BIRD-LIFE:
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By CHARLES DIXON.
The Spadoiu Air.— Tha Open Fialda and Downs,— In the Hedgerows.- On
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Copseand Woodland.— By Stream and Fool.— The Sandy Wastes a^ Und-
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utimlir Rn-t
»iih c
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ml siriDf that a Haadbook at lUi tabieel wUJ bg in liiai
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