B 448366

1
ARTES
LIBRARY
1837
SCIENTIA
VERITAS
OF THE
UNIVERSITY OF MICHIGAN
TCEBOR
CUARIS PENINSULAN-AMIENAM
SIRCUMSPICE
PROF.
THE GIFT OF
ALEXANDER ZIWET

i
•
QA
905
P87
·
7

5283
1.2.22
AN
Alexander Fired
ELEMENTARY TREATISE
ON

HYDROSTATICS.
FOR THE USE OF JUNIOR UNIVERSITY STUDENTS.
BY
RICHARD POTTER, A.M. F.C.P.S.
LATE FELLOW OF QUEENS' College, CambrIDGE; LICENTIATE OF THE Royal College
OF PHYSICIANS, LONDON; HONORARY MEMBER OF THE LITERARY AND
PHILOSOPHICAL SOCIETY OF ST. ANDREW'S;
PROFESSOR OF NATURAL PHILOSOPHY AND ASTRONOMY
IN UNIVERSITY COLLEGE, LONDON.
CAMBRIDGE:
DEIGHTON, BELL, AND CO.
LONDON BELL AND DALDY.
1859.
i
:
Cambridge:
PRINTED BY C. J. CLAY, M.A.
AT THE UNIVERSITY PRESS.
Prof. Alex. Zirvet
gt.
2-10-1923
Stacks
32 MEN
PREFACE.
THE present Treatise on Elementary Hydrostatics is
written to supply a text-book for the Author's
Junior Mathematical Class of Natural Philosophy,
and to include the various propositions which can
be solved without the Differential Calculus.
Within the prescribed limits it has been his
wish to make it as comprehensive as possible with-
out entering into the bounds of practical hydraulics,
more than is considered appropriate to a theoretical
treatise.
The Author has endeavoured to meet the wants
of Students who may look to hydraulic engineering
as their profession, as well as those who learn the
subject in a course of scientific education.
He hopes that in preparing a text-book for his
own teaching, he has also produced a treatise which
may be useful to others similarly engaged in tuition.
LONDON, 1859.
417635
:
:
:
CONTENTS.
1
INTRODUCTION.
Definitions-force of cohesion in solids-caloric the cause of fluidity-
atoms of fluids are in a state of neutral equilibrium under the
actions of opposing molecular forces-the liquids are not perfect
fluids-possess attraction of aggregation-viscid fluids-contra-
distinction of solids and liquids-characteristic distinction as to
the transmission of pressure-how it arises-non-elastic and elastic
fluids, 2; specific gravity-water as standard-unit of pressure-
pressure of a fluid on a given area, 3; force of gravity, how con-
sidered-relations of mass, density, volume, and weight of fluids,
4; liquids compressible-Canton's results, 5; law for change of
density-many gases have been reduced to liquids and some to
solids, 6.
CHAPTER I.
ON PRESSURE WITHIN FLUIDS
Transmission of pressure equally in all directions-experimental proof,
7; principle of virtual velocities holds good, 8; pressure on a
horizontal surface, varies as the depth-unit of pressure, 9; fluids
tend to find their level-a level surface, 10; pressure independent
of the quantity of fluid-horizontal pressures in equilibrium, 11;
upon a body in a fluid, 12; vertical pressures, resultant how ex-
pressed, 13; a body partly immersed, 14; law of the resultant
pressure on a body wholly or partly immersed-relative gravity—
the body ascends or descends in a fluid, 15; the conditions that a
body may float at the surface, 16; how deep a floating body sinks
-the weight of a body in a fluid of variable density, 17; a float-
ing body disturbed, 18; the equilibrium stable, unstable, or
neutral, 19; the metacenter-the whole pressure upon the sur-
face of a body, 20; the centre of pressure, 21; the pressure below
several fluid strata, 24; two fluids meeting in a bent tube, 25 ;
examples 26 to 32.
PAGE
1
vi
CONTENTS.
CHAPTER II.
HYDROSTATICAL INSTRUMENTS
The common hydrometer, 33; Sikes's-Nicholson's, 34; the specific
gravity bottle, 35; its uses, 36; the hydrostatic balance-specific
gravity, how found by it, 37; the knowledge of specific gravities
important-true weight of a body, 38; densities of alloys of
metals, how expressed, 39; condensation and expansion of alloys
and mixtures, ib.; proportions in alloys-allowance for condensa-
tion, 40; the spirit-level, its uses, 41; the depression in levelling
—the offing, 42; Bramah's press, 43; examples in hydrostatical
instruments, 45 and 46.
PAGE
33
CHAPTER III.
ON ELASTIC OR AERIFORM FLUIDS
Gases, vapours, temperature-no perfect thermometer, 47; the laws
of Boyle, Gay Lussac, and Amonton, 48; Boyle's law, how proved,
49; Gay Lussac's law, 50; how shown, 51; Regnault's results, 52;
Amonton's law discussed, ib.; on the atmospheric buoyancy, 53;
the barometer, how constructed, 54; its various uses, 55; it
measures pressure-the atmosphere not homogeneous, 56; the
density of the air decreases in a geometric progression as we
ascend, 57; the difference of heights found by the barometer, 58;
corrections, 59; height of earth's homogeneous atmosphere—
pressure on a square inch, 60; water and oil barometers, 61; air-
pump valves, 62; construction of the common air-pump, 63;
Newman's air-pump, 64; density of the air in the receiver—
siphon-gauge, 65; barometer-gauge, 66; the condenser-air-gun,
67; the siphon, 68; reciprocating springs-the suction-pump, 69;
the lifting-pump-the, forcing-pump, 70; the fire-engine, 71;
Archimedes' screw, 72; the pneumatic trough, 73; diving-bell,
74; mixtures of gases, 75; examples in pneumatics, 75 to 77.
47
CHAPTER IV.
ON HEAT
Caloric essential to bodies-change of bulk and form-latent heat,
how considered, 78; temperature, its measurement by thermo-
meters, 79; their defects-expansion of gases-absolute zero of
cold, 80; Dalton's views, their consequences, 81; the gases are
vapours of liquids and solids with great affinities for caloric—
78
CONTENTS.
vii
maximum density of water-specific heats or capacities for caloric
-volumes of bodies depend upon the caloric they contain and
the temperature-the converse of this-hammered iron becomes
red-hot, 82; fire-syringe-defective elasticity-heat by friction-
radiation of heat-conduction-radiant heat reflected, refracted
and polarized, 83; how shown in solar heat and dark radiant
heat-radiant heat various as light is coloured, 84; apparent
radiation of cold-conducting powers, how found-convection, 85;
construction of thermometers-standard mercurial, 86; scales of
thermometers, 87; comparison of scales-air thermometer, 88;
differential thermometer, 89; self-registering, 90; Ferguson's
pyrometer, 91; rates of expansion of some bodies, 92; compen-
sation bars-linear expansion, 93; Breguet's thermometer-Da-
niell's pyrometer, 94; fusing points of metals, 95; specific heats,
how determined, 96; table of specific heats of solids, 97; of gases,
98; Dalton's reflections, 99.
PAGE
CHAPTER V.
ON VAPOURS
They are subject to the same laws as gases-air saturated with vapour,
100; elastic force of dry and saturated air, 101; solids furnish
vapours-elastic force of vapours of various liquids-table, 102;
Alexander's formula, ib.; table of elastic force of steam, 103;
table of boiling points of water at different pressures, 104; method
of finding the elastic force of vapours, 105-106; Wollaston's
instrument an elementary steam-engine, 107; Carnot's view-
atmospheric steam-engine, 108; its disadvantages, 109; Watt's
condensing engine, 110; the box-valve, 111; Trevithick's high-
pressure engine, 112; rotatory steam engines, ib.; hygrometers,
ib.; imperfect hygrometers-the dew-point-wet and dry bulb
thermometer-Glaisher's tables, 113; varying dryness of the
air, 114.
100
CHAPTER VI.
ON THE RELATION OF LIQUIDS TO GASES AND SOLIDS
Liquids contain air, it escapes under the air-pump vacuum-springs
variously impregnated with gas-the laws determined by Henry
and Dalton, 115; how to be examined-capillary attraction and
repulsion, 116; how exhibited-circumstances that the one or the
other may arise, 117; investigation, 118; law of the ascent in a
capillary tube, 119; between parallel plates, 120; between plates
inclined at a small angle, 121; experimental results, 121.
115
viii
CONTENTS.
CHAPTER VII.
ON THE MOTION OF FLUIDS
Atoms of fluids capable of independent motion-motions of vibration
and translation, 123; velocity of issuing jets of fluids, 124; vena
contracta-steady motion, 125; time of a given quantity of liquid
flowing from a vessel kept always full, 126; of a cylinder allowed
to empty itself through a small orifice, ib.; forms of jets, 127; the
hydraulic ram, 128; Barker's mill-fireworks-the turbine-water
wheels, 129; form of the surface of a liquid in a rotating vessel,
130; the rarefaction in diverging streams of fluids, 131; adju-
tages—safety valves-on waves, 132; their interference-reflexion,
133; on sound, ib.; not transmitted through a vacuum-kaleido-
phone-shows superimposed vibrations, 134; velocity of sound in
air, at different temperatures, over water, 135; mathematical
investigation of the velocity in air-in water, 136; longitudinal
vibrations-loops and nodes, 137; pitch of musical notes from
strings-of notes from a tube closed at one end, 138; of notes
from a tube open at both ends, 139; vowel sounds-human voice
—harmonics—formation of the diatonic scale of notes, 140; tem-
perament, 141.
TABLE OF SPECIFIC GRAVITIES
PAGE
. 123
•
1 142
HYDROSTATICS.
INTRODUCTION.
In the Mechanical Sciences, which are called Hydrostatics and
Hydrodynamics, the matter which is subject to the action of forces
is said to be in a state of fluidity.
In solid bodies the particles are held together by the forces
of cohesion; but in fluids, the effect of an increase of caloric (the
cause of heat) has been to take away this force of cohesion, so
that the atoms of a perfect fluid are put in motion amongst each
other on the application of the slightest force. We must con-
sider that when at rest, the atoms of such fluids are in a state of
neutral equilibrium under the actions of the internal and external
forces to which they are subject; so that any additional force
applied to an atom puts it in motion. We must remember that
as bodies expand on being heated and contract in volume on
being cooled, the ponderable hard parts of the atoms must be
separated by considerable intervals, and possess in any given state
atmospheres of caloric and other imponderable fluids, which pro-
duce the repulsive forces counteracting the attractive forces of
the hard nuclei for each other. So that the actual state of a
quiescent body must be that of equilibrium amongst its atoms
under the action of opposing molecular or atomic forces.
In strictness the elastic or aeriform fluids alone fulfil the con-
dition of perfect fluidity, for the liquids or dense fluids possess
the attraction of aggregation by which small isolated portions of
them collect in spherical drops, and some degree of force is
P. H.
1
2
INTRODUCTION.
required to separate the parts of such drops which thus exhibit
a force of adhesion. In some problems this attraction of aggre-
gation or adhesion is an essential property to be considered; but
in the greater number of cases the properties of fluidity common
to liquids and gases alone affect the results. This is the case
even for the viscid fluids, such as tar, syrups, &c., which require
time to arrive at their state of equilibrium. The characteristic
state of fluid bodies, in contradistinction to that of solid bodies,
is that their particles are capable of motion amongst each other
on the application of the slightest force.
The characteristic distinction of solids and fluids as to the
transmission of pressure evidently arises from the mutual rela-
tions of their constituent atoms, so that solid bodies only trans-
mit pressure in the direction of its action; but fluids transmit
pressure equally in all directions from the absence of stable rela-
tions between contiguous atoms.
Fluids are subdivided into non-elastic fluids or liquids; and
elastic fluids or gases and vapours.
Definitions. The mass of a portion of fluid is the quantity of
matter which it contains as measured by its inertia, and is pro-
portional to its weight at the same place. The weight of a
body, as in dynamics, is the pressure produced by it under the
action of the force of gravity.
Def. The density of a body, solid or fluid, is the relation of
the quantity of matter it contains to its bulk, and is measured
by the mass or quantity of matter in a unit of volume, when uni-
form. It is however frequently expressed by reference to the
density of some fluid taken as a standard. Thus we say the
·density of mercury is about 13 times that of water; the density
of silver is about 10 times that of water. In these cases water
is evidently taken as the standard fluid. Sometimes the densi-
ties of the gases are referred to a standard gas, as when we say
that the density of hydrogen gas is only 4th that of atmospheric
air, and then we take atmospheric air for our standard fluid.
The two methods of expressing the density of a body are easily
convertible the one to the other in the mathematical results.
INTRODUCTION.
3
Def. The specific gravity of a body is the weight of a unit
of volume, and is generally expressed, like density, by reference
to a standard fluid. Thus, when we say that the weight of a
cubic foot of iron is nearly eight times that of a cubic foot of
water, we refer the specific gravity of iron to that of water as a
standard. Also we know the actual weight of a given volume
of a body when we know its specific gravity, since a cubic foot
of distilled water weighs 1000 ounces avoirdupois at the tempe-
rature 60° Fahrenheit very nearly.
The tables of densities and specific gravities are evidently
identical when the same standards are taken. A table will be
found at the end of this volume.
The term unit of pressure in hydrostatics has a different
meaning to that which it has in statics. In statics, pressure
being generally represented by weight, the unit of pressure is
the unit of weight which is taken, as one ounce, one pound, or
one ton; but in hydrostatics we have to consider the pressures of
fluids upon surfaces, and then we call the pressure upon a unit
of area of the surface, the unit of pressure, when it is a uniform
pressure. For example, if a cistern containing water has its
base horizontal, then the pressure on every square foot of the
base is the same, and the pressure on any area of it is propor-
tional to that area; so that if the pressure on a square foot is
known, the pressure on any given part of the base is known.
In this example a square foot is taken as the unit of area; in
other cases we might take a square inch, or square yard, as the
unit of area; and we can pass easily in calculation from one unit
to another, as the problem may require.
In this manner, if p be put for the unit of pressure, A the
area on which the pressure is P, and the pressure is uniform or
constant, we have
P=p.A.
When the pressure is not constant, then p is not constant,
and this formula does not apply; and the value of P requires to
be found from the rules of the science as investigated further on.
This will be the case whenever the pressure is required upon a
given area of the vertical side of a cistern containing water for
1-2
4
INTRODUCTION.
example; where the pressure increases with the depth below the
surface of the water.
The force of gravity is expressed in hydrostatics in the same
manner as in dynamics, being measured by the velocity produced
by its action in one second of time; and we put force of gravity
=g=32·19 feet velocity per one second. In hydrodynamics this
number has to be employed often, as it has in dynamics; but in
hydrostatics the expressions which contain it are easily changed
to others in terms of weights, since in dynamics weight equals
gravity multiplied by the mass or w=g.m.
The mass is proportional to the volume of a body when the
density is constant, or if we put m for the mass when the volume
is V and the density is p, we have
m∞ V when p is constant;
and again, the mass varies as the density when the volume is
constant, or
m xp when V is constant;
so that by the rules of Algebra, when V and p may both vary,
we have
mx p. V;
and if C be some constant, m = C.p.V.
1
To find the value of C, let m₁, P₁, V₁ be some given simul-
taneous values of m, p, and V, then
C =
m₁
When we put C = 1, as is usual, and then have
m=p.V,
we see that the units of mass, density, and volume, are not in-
dependent of each other, but the mathematical expressions must
be brought to a recognized form when we want to perform actual
computations; and this is easily accomplished, for we have
w=g.m
=g.p.V;
INTRODUCTION.
5
and therefore, in any formula where g.p.V occurs, we may
replace it by the weight w, of which the unit of measure will be
known from the data of the question: also, conversely, in order
to solve any question, we may replace w by its equivalent gp V,
when the solution turns upon the density and the volume. Ex-
amples of these will be found further on.
It was formerly thought that liquids were absolutely non-
elastic; but Canton proved, in the year 1762, that they diminished
slightly in bulk under pressure, and recovered their original
volume when the pressure ceased; and this has been confirmed
by Perkins, Colladon and Sturm, Ersted, Regnault, and others.
The law of Canton is this, that the diminution in volume of a
liquid is proportional to the pressure to which it is subject; also,
that the amount of diminution under a given pressure is different
for different liquids. The following table contains his results
when the barometer stood at 29 inches, and the thermometer
stood at 50° Fahrenheit.

Compression of a
volume unity un-
der the pressure
of the atmosphere.
Specific
gravity.
Spirit of wine
•000066
•846
Oil of olives...
⚫000048
•918
Rain-water....
•000046
1.000
Sea-water....
⚫000040
1.028
Mercury........
⚫000003
13.595
We see that the compressibility of liquids is so small that it
is only in particular cases it needs to be taken into account.
The law of Canton is expressed in a formula as follows. Let V
be the original volume of the liquid and the density p, V' the
volume under a pressure p measured in atmospheric pressures
upon a unit of area, and the density p'; let c be the compressi-
bility or the value of the numbers in the second column of the
table, that is, when p=1.
6
INTRODUCTION.
Then
V-V'=V.c.p,
or
V' = V (1 — cp),
and since
=
Ρ
V'.p' V.p, we have p'=
1 - cp
The volumes of aeriform fluids change greatly with changes
of pressure and temperature, and their peculiar properties in-
volve in a very great degree their relations to heat. The laws
to which they are subject will be found in the chapter treating
of gases and vapours. By a vapour is meant an elastic fluid
which is easily reduced to the liquid state by the application of
cold or pressure, or both. We say easily reduced to the liquid
state, for many of the gases have been reduced to the liquid
and some to the solid state.
Amongst those which have been brought to the solid state
are carbonic acid, ammonia, cyanogen, euchlorine, sulphureted
hydrogen, sulphurous acid, hydriodic acid, nitrous oxide, &c.
Others, as muriatic acid and olefiant gas, have been reduced to
the liquid state; whilst the following were found by Dr Fara-
day to remain in the gaseous state at the temperature 166° below
the zero of Fahrenheit's thermometer.
Hydrogen
Oxygen
Nitrogen
Nitric oxide
at 27 atmospheres pressure
27.
... 50
...
50.
Carbonic oxide ... 40
•
Coal-gas
32.
CHAPTER I.
ON THE PRESSURES WITHIN FLUIDS.
IN the introductory chapter it was stated that fluids at rest
transmit pressure equally in all directions, on account of the
state of neutral equilibrium which exists for each constituent
atom of the fluid, arising from their mutual pressures upon each
other, and being the results of the internal and external forces
to which the body is subject. The experimental proof of this
property is as follows.
PROP. 1. To describe the experimental proof that fluids trans-
mit pressures in all directions.
Let ABCDE represent
the horizontal section of a
close vessel filled with fluid,
having pipes in the sides with
tight-fitting pistons, and the
fluid filling the pipes up to
the pistons, of which the cen-
ters are in the horizontal sec-
tion of the figure 1.
Now if any pressure is
A
Fig. 1.

B
D
applied to any one of the pistons, then pressures must be
applied also to each of the others to keep them in their places;
and the pressures must be proportional to the areas of the
pistons respectively, so that the unit of pressure is the same for
all, and likewise for all parts of the surface of the vessel; or
fluids transmit pressure equally in all directions.
•
8
ON THE PRESSURES WITHIN FLUIDS.
X
PROP. 2. To shew that the equation of virtual velocities holds
good for fluids in equilibrium.
In the figure 1, let a₁, a, a,, &c. be the areas of the pistons,
and h₁, h₂, h,, &c. their distances respectively in the pipes from
the body of the vessel.
Let V be the volume of fluid in the vessel itself, so that the
whole volume in the vessel and pipes
=V+a₁.h₂+ a₂.h₂+az.h₂+&c.
2
After the pistons have received simultaneous displacements,
let h₁', h'', h'', &c. be their distances respectively in the pipes.
And since the volume of fluid is unchanged it is
= V + α₁•h₁' + α₂.h₂' + a¸.h3' + &c.,
or subtracting this value from the former one, we have
0 = a₁ (h₁— h‚') + a(h₂ − h₂') + α, (h,− h₂') + &c....... (1).
Now if P₁, P, P,, &c. are the pressures applied to the pis-
tons respectively in equilibrium, then
(h¸ — h¸'), (h₂— h₂'), (h¸-h¸'), &c.
are their virtual velocities respectively; and if p is the unit of
pressure, then
P₁ = p.α₁, P₂ = p.ɑ₂, P₁=p.α, &c.;
3
therefore multiplying the equation (1) by p, we have
0=P(-k)+P, (b − k)+Pg (k – ký) + &C.,
2
which is the equation of virtual velocities; and if some of the
pistons are moved inwards, others must be moved outwards, or
some of the terms will be positive and others negative.
PROP. 3. To find the pressure upon a horizontal plane surface
within a fluid of uniform density, and subject to the action of
gravity.
Imal
ON THE PRESSURES WITHIN FLUIDS.
9
Let AB be the horizontal plane
surface and its area a, upon which
the pressure P is to be found.
Suppose a vertical prism of the
fluid ABCD above AB up to the
surface CD of the fluid, and whose
base is AB, to be separated from
the rest of the fluid by an ima-
ginary rigid film, which would
C
Fig. 2.

D
B
a-b
not alter the state of the fluid; then if the outside fluid were
removed, we see that the whole weight of the internal fluid must
be supported by the base AB, since no part would be supported
by the vertical sides of the film; and the pressure upon AB
would be the weight of the fluid equal in volume to the prism.
Since before the external fluid was supposed to be removed
there was equilibrium; therefore the pressure on the under side
of the area AB equals the pressure on the upper side of it, and
each equals the weight of fluid of the volume of the prism.
Let the depth AC=z, p the density of the fluid, and Ρ the
unit of pressure on AB; then the volume of the prism V= a.z,
and P=gpV=gpaz.
Also since the pressure is uniform, .. P=p.a,
and p = gpz,
or the pressure varies directly as the depth z.
If the area had been indefinitely small, as ab in the figure,
and equal to a, we should have had the pressure upon it in like
manner equal to p.a. But when the area is indefinitely small
it will be all sensibly at the same depth when not horizontal, so
that on the surface of a body in a fluid, or on the surface of the
containing vessel, the pressure on any indefinitely small area is
p.a=gpz.a.
COR. 1. It is evident the pressures are equal at all equal
depths, and since fluids transmit pressure equally in all direc-
In the pas
شر
x
lever
visum
L
anim
be
the f
10
ON THE PRESSURES WITHIN FLUIDS.
靠
​tions, therefore in equilibrium the upper free surface where the
pressure vanishes is horizontal.
In viscid fluids the surface will become horizontal only in
some sensible time after it has been disturbed; also in fluids
generally waves will arise when the fluid is disturbed, and they
must have ceased before the fluid can be said to be in equi-
librium.
COR. 2. Within a fluid which is in equilibrium we may
suppose imaginary films to separate certain portions from the rest
without disturbing the respective parts, and all the parts of the
surface would remain in the same horizontal plane. Hence when
a fluid is contained in a system of vessels which communicate
with each other, the surfaces of the fluid in all the vessels are in
the same horizontal plane.
If fig. 3 represents a set of
vessels of different forms
communicating by a pipe
AB, then the surfaces C,
D, E, F, in the separate
vessels will be all in the
A
C
Fig. 3.

D
E
F
B
same horizontal plane. This in common language is stated by
saying that 'fluids always find their level,' or tend to do so.
This is shewn also by taking a horizontal plane in the pipe
AB, at which the pressures must be equal in equilibrium, and
the heights of the surfaces above it the same, whatever be the
form of the vessels.
COR. 3. From this we see that a level surface, when of
small extent, is a horizontal plane; but when large, as the sur-
face of a lake, it has the curvature of the earth's surface. These
must be examined when the uses of the spirit-level are discussed.
COR. 4. Since the pressure upon a horizontal area depends
upon the magnitude of the area, and upon the depth below the
surface of the fluid, the pressure on the bases of vessels resting
upon a horizontal table is the same when the bases are of the
same area, and the depth of the fluid is the same whatever be
the quantity in the vessel.
ON THE PRESSURES WITHIN FLUIDS.
11
Let AB be the base of a
cone ABC, or of a cylinder
ABDE, or of an inverted frus-
tum of a cone ABGF, or any
other form of vessel; then the
pressure on AB is the same
whichever of the vessels be
Fig. 4.

F
D
used, if the fluid be at the same height in all.
B
The pressure on the base AB of the cylinder ABDE is the
weight of the contained fluid, since the vertical sides can sup-
port no part of the weight; but the cone being one-third the
volume of the cylinder, the pressure on its base, when filled
with fluid, is three times the weight of the fluid it contains.
In the inverted frustum of a cone the pressure is evidently less
than the weight of the contained fluid.
The pressure upon the table in each case is the weight of the
vessel and the contained fluid. The pressure on the inner sur-
face of the base of the cone filled with fluid, neglecting the
weight of the vessel itself, is three times that of the outer sur-
face of it upon the table. The resultant vertical pressure will
be found discussed in Prop. 5, in explanation of this circum-
stance.
PROP. 4. To shew that the horizontal pressures upon the sur-
face of a vessel containing fluid are in equilibrium with each
other.
Let PQ be any indefinitely small horizontal prism in figures
5 and 6, taken in vessels containing fluid.
Fig. 5.
α
N
Fig. 6.


P
a
P
N
12
ON THE PRESSURES WITHIN FLUIDS.
Then the unit of pressure at P and Q will be the same, since
they are at the same depth; also the pressure is in every case
perpendicular to the surface at each point where there is equi-
librium; for if any component pressure parallel to the surface
existed, it would set the fluid in motion, contrary to the supposi-
tion of equilibrium existing.
Then if PN, QN are the normals to the surfaces at P and
Q respectively, the angles they make with the axis of the prism
are the same as the angles between the perpendicular section
ab of the prism and the oblique sections in which it meets the
surfaces of the vessels at P and Q, since these sections are per-
pendicular respectively to the former directions.
Let PN make an angle 0 with the axis of the prism PQ.
Let a = the perpendicular section ab of the prism PQ in the
figures.
a= the oblique section in which it meets the surface of
the vessel at P.
a" that at Q.
=
We have a = a' cos NPQ = a" cos NQP by the property of
projections.
Then p being the unit of pressure at P and Q, pa' is the
pressure on the oblique section at P, and its horizontal compo-
nent is pa' cos 0 = pa, since = cos 0.
α
α
Similarly the horizontal component of pa" at Q is pa, and
the same result will arise in whatever horizontal direction the
prism be taken, therefore the horizontal pressures are always in
equilibrium with each other.
COR. A similar method of proof shows that the horizontal
pressures upon a body immersed in a fluid are in equilibrium
with each other.
PROP. 5. To investigate the resultant vertical pressures on
the surfaces of a vessel containing fluid.
ON THE PRESSURES WITHIN FLUIDS.
13
Let PP' be an indefinitely small
vertical prism meeting the hori-
zontal base AB of the vessel ABC,
which is filled with fluid in the
indefinitely small area a at P'; then
if a' is the oblique area in which the
prism cuts the surface of the vessel
at P, and the angle between the A
normal PN, and the horizontal line
PQ is 0, as in the last Prop., we have now
Fig. 7.

N
α
cos NPP' = sin 0.
α
P
B
Now the pressure on the oblique area at P=pa', and its up-
ward vertical component
= pa' cos NPP'
=pa' sin 0
=pa
=gpz.a.
z being the depth of PQ below the surface, let z' be the depth
of P' below the surface of the fluid;
Then the pressure on the area a at P' is downwards, and
= gpz'.a.
The difference of the downward pressure at P', and the up-
ward pressure at P,
= gpa (z' — z)
=
weight of the fluid of the volume of the prism PP'.
The same result holds for all indefinitely small vertical
prisms which can be taken in the fluid; and the resultant of all
the vertical pressures, upwards and downwards, equals the
weight of the fluid contained in the vessel, of whatever form it
may be.
14
ON THE PRESSURES WITHIN FLUIDS.
COR. 1. If a body be immersed
Fig. 8.

in a fluid, such as PQP (fig. 8), and
a vertical small prism PP' were taken
in it, the difference of the vertical
pressures upon the oblique areas in
which it meets the surface of the
body will be as above
= gpa (2-2)
=
the weight of the fluid of the volume of the prism PP.
•
Now the same being true for all other small vertical prisms
which can be taken in the body, the resultant of the vertical
pressures acting upon the body equals the weight of an equal
volume of the fluid to it, or which it displaces, and acts vertically
upwards.
The resultant acts vertically downwards through the center
of gravity of the contained fluid in the first case (fig. 7), and
upwards through the center of gravity of the fluid displaced in
the latter case (fig. 8), since the vertical pressures are systems of
parallel forces represented by the weights of the prisms of fluid.
COR. 2. If the body were only partly immersed, we should
have the prisms terminating in the plane of the surface of the
fluid, and z = 0 for some parts of the body; but the sum of the
prisms of fluid would make up the whole fluid displaced, and the
resultant pressure would be the weight of the fluid displaced, as
before.
COR. 3. That the resultant pressure of a fluid upon a body,
wholly or partly immersed in it, equals the weight of the fluid
displaced by it, and acts vertically upwards through the center
of gravity of the fluid displaced, is easily shown at once by sup-
posing in fig. 8 and fig. 9, that before the body was immersed in
the fluid, the part displaced was separated from the rest by an
imaginary rigid film; then if the exterior fluid were removed,
that within the film would gravitate vertically downwards with
a pressure equal to its weight acting through its center of gravity;
and before the exterior fluid was removed, there being equili-
ON THE PRESSURES WITHIN FLUIDS.
15
brium, this downward pressure was balanced by an equal and
opposite upward pressure from the exterior fluid. The exterior
fluid would produce evidently the same pressure upon the sur-
face of the body, wholly or partly immersed in it, that it does
upon the imaginary film of the same form and position; and
hence we conclude, that the resultant pressure of a fluid upon a
body, wholly or partly immersed in it, equals the weight of the
fluid displaced, and acts vertically upwards through the center of
gravity of the fluid displaced.
PROP. 6. To find the accelerating force of relative gravity
with which a body immersed in a fluid ascends or descends.
Referring to fig. 8, the resultant moving force acting upon
the body is evidently the difference of its weight acting down-
wards through its center of gravity, and of the vertical pressure
of the fluid upwards through the center of gravity of the fluid
displaced. Now if the body is homogeneous, or its density uni-
form, these will be opposite vertical forces acting through the
same point, the center of gravity of the body or fluid displaced
in all positions of the body; but if its density is variable, there
will be only certain positions of the body when its center of
gravity and that of the fluid displaced are in the same vertical
line, and in the other positions there will be a resultant pres-
sure which generally will produce motion of rotation as well as
translation; and thus a body ascending or descending in a fluid
may at the same time have an oscillatory motion.
Taking now the case of the homogeneous body only,
let
P be the density of the fluid,
ρ
body,
V' be the volume of the body;
then the weight of the body = gp'V', and the weight of the fluid
displaced = gp V'.
Now supposing the body to descend in the fluid, we have the
resultant moving force = weight of the body - weight of
the fluid displaced = g V' (p' — p),
and the mass of the body = p'V';
16
ON THE PRESSURES WITHIN FLUIDS.
.. the accelerating force of relative gravity
g'
=
_
resultant pressure
mass moved
gV' (p'- p)
p'V'
=9(1-6),
which is a fractional part of gravity g, depending on the values
of p and p'.
We have supposed p' greater than p, or the body to descend;
and if p' were less than p, we should have
−1),
9=-9(-1),
g' g
and the body would ascend in the fluid.
If p'p, then g'= 0, and the body would remain at rest any
where in the fluid.
The case of the air-balloon ascending and descending in the
atmosphere fulfils these three conditions. When the balloon, its
contained gas, and appendages, weigh less than the air they dis-
place, the balloon rises; when they weigh more than the air dis-
placed, it descends.
PROP. 7. To find the conditions that a body may float in
equilibrium at the surface of a fluid.
In order that two forces may
balance, they must act through
the same point, be equal in mag-
nitude and opposite in direction;
therefore, in the case of the equi-
librium of floating bodies, the
weight of the body being sup-
ported by the upward fluid pres-
Fig. 9.

D
sure, we have, by Cor. 2, Prop. 5, the two conditions to be
fulfilled as follows.
ON THE PRESSURES WITHIN FLUIDS.
17
(1) The weight of the body equals the weight of the fluid
it displaces.
(2) The centers of gravity of the body and the fluid dis-
placed must be in the same vertical line.
DEF. The plane in which the surface of the fluid cuts the
floating body is called the plane of floatation.
PROP. 8. To find how deep a given body will sink in a given
fluid of greater density than itself, when floating in equilibrium.
In fig. 9, let V be the volume of the fluid displaced, p its
density, V' the volume of the body, and p' its density, then the
first of the conditions of the last proposition gives us
gp V = gp'V',
V
and therefore
é
V
The volume V must be also taken so as to fulfil the second
condition, or the plane of floatation AB must be perpendicular to
the line joining the centers of gravity of the body and the fluid
displaced.
In a few simple cases the computation for the value VV''
ρ
is easy; but in the more interesting cases of floating bodies it is
more complicated.
Examples will be found at the end of the chapter.
PROP. 9.
When the density of the fluid increases as the depth,
to find the weight of a body at a given depth.
σ
Let p be the density at the surface of the fluid, & the increase
of density at a depth unity, and therefore the density at a depth
z will be p+0%.
Let p' be the density of the body, and V' its volume, then its
weight at the depth = weight of the body - the weight of the
fluid it displaces
P. H.
=gp'V' − g(p+02) V!.
=gVi{p – (p+ a)},
Don't be wag
not th
is not
2
18
ON THE PRESSURES WITHIN FLUIDS.
and the body will float in the fluid at a depth z, such that
2=
p' - p
σ
PROP. 10. A floating body being slightly displaced vertically
from its position of equilibrium, to find the accelerating force
causing it to return to the position of equilibrium.
Let AB be the original plane of
floatation, of which the area is A;
let ab be the level of the surface of
the fluid, and x the depth of AB
below ab.
Fig. 10.

b
A
B
Then the volume of fluid displaced more than in equilibrium
will be Ax, nearly, and the weight of this volume is the moving
force bringing the body back to its position of equilibrium. Let
as before p = density of the fluid, V= volume of fluid displaced
in equilibrium, p′ = density of the body, and V' its volume.
Then the accelerating force bringing the body to its original
position
jo
weight of the volume Ax of the fluid
mass of the body
-
gp Ax
V'p
- JPAx
Vp
A
=977x,
and V'p'= Vp,
which varies with the depth x and area A, and changes sign
when AB is above ab.
Calculations for the motion of the body from this formula are
only approximate, because the disturbance of the floating body
will give rise to waves on the surface of the fluid, which are not
taken into account.
9. ough?
We
of way of ass finie
day bodyle
ON THE PRESSURES WITHIN FLUIDS.
19
COR. If the body were floating in a
vessel of finite extent, and the area of the
surface of the fluid in the vessel were A',
the rise of the surface of the fluid in the
vessel, by the depression of the body,
would be sensible. Let x' be the rise of
Fig. 11.

A
B
the surface of the fluid to ab, fig. 11, and x+x' the height of ab
above AB,
A
then (A'-A) = Ax, and x'=
x
•
A' — A
X,
and we have the accelerating force
weight of the volume 4(x+x') of the fluid
mass of the body
gp4x(1+^^~A)
A'A
V'p'
Α Α'
gpxAA,
Vp
Α Α'
x.
='9 V (A' — A) • ∞
PROP. 11. To investigate the conditions that the equilibrium
of a floating body may be stable, unstable, or neutral.
Let the figure 12 represent
a section of the floating body
through its center of gravity G.
Let AB be the surface of the
fluid, and the center of gravity
of the fluid displaced before the
angular disturbance, and when
the line HG was vertical. Let
H' be the center of gravity of
the fluid displaced after the dis-
Fig. 12.

M
A
HÁ H
turbance, as in the figure. Draw H'M a vertical line meeting
the line HG produced in M, then M is the metacenter of the
floating body.
Thi
2-2
20
ON THE PRESSURES WITHIN FLUIDS.
Now if, as in fig. 12, G is below M, there is a statical couple
bringing the body back to its first position with the line MGH
vertical, the equal and parallel forces being the weight of the
body acting vertically downwards through G, and the fluid pres-
sure acting vertically upwards through H'.
If, as in fig. 13, the center of gravity G of the floating body
were above M, the moment of the couple would evidently
tend to turn the body from the original position of equilibrium,
which would be therefore unstable.
If M and G coincided, the
equilibrium would exist in the
new position, and therefore would
be neutral.
In stable equilibrium then,
the center of gravity of the body
is below the metacenter.
In unstable equilibrium the
center of gravity is above the
metacenter.
B
Fig. 13.

M
A
HAH"
In neutral equilibrium the center of gravity is at the meta-
center.
PROP. 12. To find the whole pressure of a fluid upon the
surface of a body immersed in it.
Let BPCQ be the body, PQ an
indefinitely narrow horizontal ring
upon its surface, and let its area be
a, and depth AM=z.
Let p be the density of the fluid,
A the whole area of the surface of
the body equal to the sum of the areas
of all the elementary rings, as PQ,
which can be formed upon it, or, using
Σ for the sign of summation, A =Σ (a).
A
Fig. 14.

P
M
B
Let z be the depth of the center of gravity of the surface
A below the surface of the fluid, and
.. 7.A=Σ(a.z)
by the property of the center of gravity.
ON THE PRESSURES WITHIN FLUIDS.
21
Now the pressure of the fluid upon the ring PQ=gpaz, by
Prop. 3, and the whole pressure upon the surface of the body
= Σ(gpaz)
=
= gpΣ (a.z)
= gpАz,
=
which is the same as if the whole surface of the body were
horizontal in the fluid, and at the depth of its center of gravity.
It will be found in the examples that this proposition is of fre-
quent application to find the pressures on given surfaces in
fluids.
COR. If the surface on which the pressure is required be
the whole or part of the surface of a vessel containing fluid, the
same rule will evidently hold good.
PROP. 13. To find the center of pressure of a plane area
immersed in a fluid.
DEF. By the center of pressure we mean the point where
the whole pressure on the plane surface may be considered to act,
and would produce the same mechanical effect as the actual pres-
sures on the surface.
The center of pressure is found in a similar manner to the
center of gravity in statics; namely, by taking moments about
any given lines. The cases of different forms of floodgates are
easily solved by the Integral Calculus, but the case of a rect-
angular floodgate is of so much practical importance that it will
be here solved by Algebra alone.
Let ABCD be the rectangular
floodgate, with the side AB in A
the surface, and inclined at an
angle to the surface of the fluid,
and seen obliquely in fig. 15.
Draw OM bisecting the opposite
sides AB and CD, then the cen-
ter of gravity and the center of
Fig. 15.

B
m
D
M
pressure will both be in the line OM. Take PpQ an indefi-
nitely narrow elementary area parallel to AB, and draw from
22
ON THE PRESSURES WITHIN FLUIDS.
its middle point p the vertical line pm to the surface of the fluid.
Join O and m, then the line Om in the surface of the fluid is
perpendicular to AB, and the angle pOm=0 is the angle be-
tween the rectangle ABCD and the surface of the fluid; also
the depth pm Op sin 0.
=
b
Let AB= a, BC=b, and let the breadth of PQ= where
N
Then the area of the element
n is a very large number.
PpQ=a.
n
b
>
The pressure on PpQ
= gpx area x depth pm
b
=gpa..Op sin 0.
N
h
Now let Opm.
where the extreme values of the number
N
m are 0 and n;
.. the pressure on PpQ = gpab². m. sin 0,
n
and the moment of this pressure about the line AB is
pressure × arm Op=gpab³.
m²
n³
η
3
.sin 0.
Let X be the distance of the center of pressure from O, also
by Prop. 12 the whole pressure upon ABCD = gp. area × depth
of center of gravity
b
=gpab
•
sin 0,
2
and its moment about AB with the arm X= sum of the moments
of the pressures upon all the elements such as PpQ about the
same line; or, using again Σ for the sign of summation, we have
ab²
др
2
.sin 0xX= {gpab³.. sin 0}
m³
2
n
= gpab³ sin 0Σ
θ
m
8
n
ON THE PRESSURES WITHIN FLUIDS.
23
.. X=26Σ
'm'
3
n
where m is to have every integral value from 0 to n, or
Σ (m²) = 0 + 1ª + 2ª + 3² + &c. ... n²
3
= a + Bn + yn² + Sn³…….say,
where a, ẞ, y, & are constants independent of the number of the
B,
terms of the series, and a = 0, since Σ (m²) = 0 when n = 0.
Again, carrying the series one term further,
0+1* +2 +3*+& ....n+ (n+1)
= a + B. (n + 1) + y . (n + 1)² + S. (n + 1)³ ;
subtracting the former from this,
(n + 1)² = n³ + 2n + 1 = ß + y (2n + 1) + d. (3n² + 3n + 1),
or n² (38 − 1) + n (38 + 2y − 2) + (S + y + ß − 1) = 0,
which is to be true for all values of n, and the coefficient of each
power is therefore to be zero,
or
38-10, and 8= 13
38+2y2=0,
8+y+B-1 = 0,
and (m²
Σ
3
η
N
+
6 2
n
-12
y =
2'
1
...
В
6
2
+
3 1 1 1
+ +
6n² 2n 3
and when n is indefinitely great, this sum =
.. X=3b,
ક્રૂ,
and the center of pressure is below the center of gravity and
independent of 0.
If 0=0, and the surface ABCD is parallel to the surface of
the fluid at a given depth, then the center of pressure coincides
with the center of gravity, since the pressures are a system of
24
ON THE PRESSURES WITHIN FLUIDS.
parallel forces proportional to the areas upon the surface of the
floodgate.
The point found as the center of pressure is that where a
single force will balance the actual pressures, and it requires par-
ticular attention in the construction of vats, tanks, and reservoirs
of wood or metal to contain fluids.
t
If a single force is to be applied to the gates of a dock or
canal lock, which shall balance the fluid pressures and relieve the
hinges of all strain, it is clear that the center of pressure is the
point where it must be applied. If, again, the equivalent force
at any point of the gate is required, the moments of the force,
and of the whole pressure acting at the center of pressure about
the hinges, must be equated to each other.
PROP. 14. To find the pressure at a given depth when several
fluids which do not mix are in equilibrium.
When a number of fluids which do not
mix are placed in any vessel, then when in
equilibrium they will arrange themselves in
their order of densities; those which are
specifically lighter taking the higher posi-
tions, and their common surfaces where two
meet must be horizontal, in order that the
horizontal pressures may balance.
The unit of pressure at any point, such
as P, fig. 16, is that due to the different fluids
Fig. 16.

Air
Spirit
Oil
Water
P
Mercury
in the column above it.
Let the depth of P be
made up of a depth z, in a fluid whose density is P₁,
22
23
&c.
Par
Py
&c.
Then if p, is the pressure due to the upper stratum of fluid,
P₂
Ps
&c.
second
third
&c.
ON THE PRESSURES WITHIN FLUIDS.
25
8
and if p is the unit of pressure at P or at the depth 2₁ +%₂+,+
&c., we have
P = P₁+ P₂+ P3+&c.
2
= g(P₁ ²₁ + P₂ ≈ ₂ + P3≈3+&c.),
2
which result may be arrived at directly, by supposing a cylin-
drical vertical column to be separated by an imaginary rigid film
from the rest, as in Prop. 3, and the sum of the weights of the
various portions of the column makes up the weight at the base.
COR. The previous propositions suppose the existence of a
vacuum around the fluid which is under consideration, since the
fluid pressure was taken to arise from the fluid itself only.
When the circumstances occur in the atmosphere, there are cases
where the atmospheric pressure requires to be taken into ac-
count, whilst in others, being equal in all directions, it does not
affect the problem under discussion. The equivalent of the at-
mospheric pressure, in terms of that arising from any given fluid,
will be found further on.
PROP. 15. To find the conditions of equilibrium when two
fluids which do not mix meet in an inverted bent tube.
The heavier fluid will occupy the lowest portion of
the bent tube, as in fig. 17, say from A to D; and
the lighter one the other leg, say from A to B.
D
Fig. 17.

B
A
From A draw a horizontal line to C, and let
the vertical height of D above the level of AC be h,
and p the density of the fluid; let h' be the height of
B above AC, and p' the density of the fluid in the
leg AB. Then the portion of the fluid from A to C will balance
of itself, and the pressure at A must equal that at C, or we
must have
gph = gp'h',
h
and therefore
é
h
P
This result holds equally under the pressure of the atmo-
sphere and in a vacuum, since the pressure of the atmosphere
26
ON THE PRESSURES WITHIN FLUIDS.
will be the same at B as at D, if we neglect the small difference
of their heights.
COR. 1. The same result evidently holds good whatever be
the inclinations of the two legs to the horizon; and also if two
vessels of any form were connected by a pipe at their lower
ends.
COR. 2. If each of the tubes or vessels contained several
fluids which did not mix, the lowest portion would be occupied
by the heaviest; and drawing a level surface AC within it, the
condition of equilibrium will be that the whole pressures at that
level must be equal.
EXAMPLES.
Ex. 1. A cylinder with its axis vertical being filled with
water, find the pressure on the base and concave surface when
the height is 10 feet, and the diameter of the base is 2 feet.
ANS. The weight of a cubic foot of water being 1000
ounces, the pressure on the base is the weight of 31.416 cubic
feet of water, and equal to 1963-5 lbs. By Prop. 12, the pres-
sure on the concave surface equals the weight of 314.16 cubic
feet of water equal to 19635 lbs.; since the surface of the cylinder
= 10 × 2 × 3·1416 square feet, and its center of gravity is at the
middle point of the axis.
Ex. 2. Show that if the height of the cylinder in the last
example were 1 foot in place of 10 feet, the pressure on the base
would equal that upon the concave surface.
Ex. 3. Show that a cube with its base horizontal being
filled with fluid, the pressure on the base is twice that on any
one of the vertical faces.
Ex. 4. Show that if a rectangular parallelopiped, of which
the edges are a, b, and c inches, be set with that one of its faces
whose sides are b and c inches, horizontal, and be filled with
fluid, then the pressures on two contiguous vertical faces will be
ON THE PRESSURES WITHIN FLUIDS.
27
in the ratio of b to c; also the pressure on the base equals that
on one of the vertical faces multiplied by
upon the other multiplied by
2b
α
2c
2
and equals that
a
Ex. 5. The area of the surface of a sphere being that of
four great circles, show that if a sphere of one foot diameter be
immersed in water with its center 10 feet below the surface, then
the whole pressure upon its surface will be 1963-5 pounds.
Ex. 6. To find the weights upon the upper board of the
hydrostatic bellows when the surface of the water in the pipe is
at a given height above the upper board of the bellows.
This instrument may be made in a variety of
forms. Before the India-rubber cloth was in-
vented it was frequently made like fig. 18, where
AB is a vertical pipe from the center of the upper
circular board EF. The board EF was con-
nected by leather, strengthened by horizontal
metal rings, with the lower board CD. The two
boards CD and EF might be brought near to-
gether with the metal rings only between them,
or be separated to a considerable distance.
Fig. 18.

B
H
A
The usual method of now making the instru-
ment is to have the pipe AB at the side, and con-
nected with a pipe leading to the center of the lower board.
At the end of that pipe, in the center of the lower board, is
screwed the brass fitting in the neck of a bottle of India-rubber
cloth, of a cylindrical form, strengthened internally with hori-
zontal metal rings. The upper board EF then only rests upon
the circular end of the bottle.
'
Now if the bellows and pipe contained water to the height B,
it would raise, if free to do so, the upper board EF to the position
GBH, since fluids tend to find their level;' and if the upper
board is in equilibrium in the position EF, weights must be
placed upon it equal to the weight of the column of fluid EFGH,
which can therefore be easily calculated when the diameter of
EF and the height AB are known.
1
28
ON THE PRESSURES WITHIN FLUIDS.
Let the height AB be ten feet, and the diameter of the upper
board EF be two feet; then the weights put upon the upper
board, together with the board itself, the pipe AB and fluid in
it, must equal 1963-5 pounds, the calculation being that of Ex-
ample 1.
If the diameter of the upper board is one foot, and the height
AB is five feet, the weight supported is 245 pounds.
16
The effect being independent of the diameter of the pipe,
this experiment illustrates the great effect sometimes produced
in nature by thin high columns of water which open at their
lower ends into close cisterns of moderate dimensions.
Ex. 7. A surface of one square inch being immersed in
mercury at the depth of 30 inches, required the pressure upon it,
taking the specific gravity of mercury = 13.58.
The pressure will be evidently, by Prop. 3, the weight of
30 cubic inches of mercury = 13.58 x the weight of 30 cubic
1000
1728
inches of water = 13.58 × × 30 ounces = 14·735 pounds.
Ex. 8. What will be the height of a column of water which
will produce the same pressure on any given area with a column
of 30 inches of mercury?
=
If we put p = the density of water, p' the density of mer-
cury, and 'the respective depths, the pressure on any hori-
zontal area A being, by Prop. 3, = gpz. A = gp'z'. A;
..='.', and by the question
P
é
= 13.58, z' 30 inches;
P
and the height of the column of water required = 30 × 13.58
inches = 33.95 feet.
Ex. 9. To find the height of the column of oil, specific
gravity 92, which shall produce the same pressure as a column
of 30 inches of mercury.
The height of the column of oil required
13.58
30 X
=
inches 36.9 feet.
•92
ON THE PRESSURES WITHIN FLUIDS.
29
Ex. 10. A cone, with its base horizontal, being filled with
fluid, find the ratio of the height to the radius of the base, when
the pressure on the base equals that on the concave surface.
The center of gravity of the concave surface of the cone, as
found in statics, is in the line drawn from the vertex of the cone
to the center of the base, and at frds that distance from the
vertex.
The concave surface will develope or spread out into a cir-
cular sector, of which the arc is the circumference of the base,
and the radius is the generating line AB (fig. 19) of the surface
(called also the slant side).
Let_r=BC= radius of the base,
h=AC=height of the cone,
G the center of gravity of the
concave surface, and
AG=&h; also AB=√r²+h².
Then by Prop. 12, the pressure on
the concave surface
= gpπr √r² + h² × & h,
and the pressure on the base
= gρær². h,
=
and when these are equal we have
or
& √ r² + h² = r,
h=r.
5.
2
B
Fig. 19.
A

Ex. 11. Show that in the regular tetrahedron filled with
fluid, and one face for the base horizontal, the pressure on each
of the inclined faces equals two-thirds that on the base.
Ex. 12. If a cone resting on its base be filled with fluid,
show that when the height is three times the radius of the base,
30
ON THE PRESSURES WITHIN FLUIDS.
the pressure on the concave surface is (2.108 times) more than
twice that on the base.
Ex. 13. A cylinder, of which the density is p', floats in
equilibrium at the surface of a fluid whose density is p, with its
axis vertical; to find how deep it will sink. Also the density of
ice being that of water, to show that the thickness of a sheet
of ice floating at the surface of water equals nine times the
height of its surface above the water.
Let CMD be the level of the surface in
fig. 20, and let AM=x, to be found when
AB = h is given.
Fig. 20.

B
M
From Prop. 8,
V_e
x
V
P
h
;
.. x=h.
h. P
ρ'
A
;
P.
also BM=h
• - p'
which gives h when BM is known.
P
Ex. 14. A cone floats in a fluid with its axis vertical and
apex downwards; to find how deep it will sink.
Let AB=h be given, and
АМ
AM=x to be found, in fig. 21.
Since the plane of floatation cuts
from the cone a portion, as in the
figure, similar to the whole cone,
we have
V
хв
======
Fig. 21.

B
C
+M
A
3
n. Ve.
૨
.. x=h.
源
​Ex. 15. To find how deep a paraboloid of revolution will
sink in a given fluid, the volume of a segment of a parabo-
loid being one-half that of a cylinder of the same base and
altitude.
ON THE PRESSURES WITHIN FLUIDS.
31
Let AB=h, AM=x, DME being
in the surface of the fluid in fig. 22, and
DM² = 4mx, BC= 4mh.
Then the volume of the whole para-
bolic segment
½ π . BC² × AB={π.4mh²,
and the volume of the part immersed
=πDM² × AM = 2πmx²,
Fig. 22.

Br
E
M
D
A
and by the formula of Prop. 8,
V
p' __ x²
V
P
h²
h2,
and
x = h
P
Ex. 16. To find how deep a cylinder floating in a fluid with
its axis horizontal will sink,
The depth to which it will sink will be independent of the
length of the cylinder.
Let AOB be the vertical diame-
ter, and O the center of the perpen-
dicular section of the cylinder in
fig. 23, and DC the surface of the
fluid; also let the angle AOC= 0,
the radius 40= a, and
AM = x = a vers 0,
the area of the segment ACD area
=
of sector ACOD – area of triangle DOC
and
Fig. 23.
B
A
M
arc × rad. - DC × OM = a².0 — (2a sin 0. a cos 0)
= a² (0 – † sin 20),
area of segment __ 0 – 1 sin 20
V_L
T'
P
area of circle
π
1
"

32
ON THE PRESSURES WITHIN FLUIDS.
and if is found from the expression
0 - sin 20 = π
ρ'
P
then x is known from x = a vers 0.
The equation for finding 0 is
0 + sin 20
2
ρ
when the axis of the cylinder is below the surface of the fluid.
Ex. 17. If a vat, or a barrel of a cylindrical form, with its
axis vertical, the bottom closed and the top open, had the staves
parallelograms, where could a single hoop be placed to counter-
act the fluid pressure upon the staves when it was filled with
fluid?
Ex. 18. Suppose a single hoop were placed in the last
experiment higher than the center of pressure, what would be
the effect?
Ex. 19. Suppose the hoop were placed lower, what would
be the effect?
Ex. 20. Suppose the hoop placed in any given position,
and the vat were only partly filled with fluid, what would be
the result?
CHAPTER II.
ON HYDROSTATICAL INSTRUMENTS.
PROP. 16. To explain the construction and principle of the
common hydrometer.
The common hydrometer ABME consists of
two bulbs A and B blown upon a tube of glass
with the blowpipe. The bulb A is weighted to
cause the instrument to float with the stem ME
vertical, and the amount of weight in A is regu-
lated for the fluid to which it is to be applied;
its sensitiveness is increased as the diameter of
the stem is diminished with the same dimensions
of bulb.
D
Fig. 24.

M
C
A
B
A slip of paper being introduced at the end
E before it is closed, the instrument is placed
in fluids of known but different specific gravities,
and the depth to which it sinks in each is noted. These and
intermediate divisions being marked upon the slip of paper,
it is put into its place in the tube and the end E closed with
the blowpipe.
When the hydrometer is placed in any liquid and floats in
equilibrium, it displaces its own weight of the liquid; and the
volume displaced will be less as its density and specific gravity
are greater, and this is indicated by the depth to which it sinks,
as at the point M in the surface CMD of the liquid, and this is
easily read off.
Being made of glass, it has the advantage, besides its cheap-
ness, that it may be placed in acid and saline solutions without
injury.
P. H.
3
34
ON HYDROSTATICAL INSTRUMENTS.
PROP. 17. To explain the construction and uses of Sikes's
hydrometer.
Sikes's hydrometer is made of brass similar to
fig. 25, having a solid ball A at the lower end, con-
nected by a tapering stem with a hollow ball B;
from B is the flat brass rod BC on which the divi-
sions are marked. From the smallness of the rod
BC the instrument is very sensitive, and at the
same time a considerable range is obtained by its
being provided with a set of brass weights, such as
D, which will pass over the narrow upper part of
the stem between A and B, and rest upon the
ball A.
D
Fig. 25.
C

B
O
The way in which the specific gravity of liquids is found
by Sikes's hydrometer, is like that of the common hydrometer,
but the value of the divisions for the different weights, such as
D, must be known.
PROP. 18. To explain the construction and uses of Nichol-
son's hydrometer.
Fig. 26.
D
B
Fig. 27.
to
α
F
Nicholson's hydrometer is chiefly made
of brass, but of different forms. When
made like fig. 26, it has a cup at the lower
part A, in which bodies can be placed for
the purpose of weighing them in water.
This cup is connected with the hollow ball
B, from the upper part of which rises a
steel wire, and on the end of it is the sup-
port for the cup D. When in use the in-
strument is sunk in the liquid to the mark
Cupon the steel wire in fig. 26, or H in fig. 27. When made
of the form fig. 27, it has a solid ball at E, a hollow vessel
with a flat top at F, and the steel wire with mark H rises
from a bridge of brass wire abc. The cup & is supported on
the end of the steel wire as before.


E
The instrument is furnished with sets of weights by means
of which it is sunk to the marks C or H in the fluid. With
ON HYDROSTATICAL INSTRUMENTS.
35
1000 of the grain or half-grain weights in the upper cup, it will
float in distilled water with the mark C or H in the surface.
If
any body of less than that weight be placed in the cup,
the weights which are required in addition to sink it again to
the same mark, being taken from 1000, we have the difference,
the weight of the body in air. If the body is placed in the
cup A, or on the flat top ac, and the same process is followed,
the weight of the body in water is found, and then its specific
gravity can be calculated as with the hydrostatic balance.
From the smallness of the steel wire on which the marks C
and I are made, the instrument is exceedingly sensitive, and
will serve instead of a delicate balance; but it is very trouble-
some, being affected by the varying temperature of the water
with which it is used.
To find the specific gravity of liquids by Nicholson's hydro-
meter the instrument itself must be weighed. Then this weight,
together with the weights to be put in the cups D or G to sink
the instrument to the marks C or H, gives the weight of the
fluid displaced, and thus the weights of the same bulks of dis-
tilled water and any other liquid can be compared.
Various forms of the hydrometer have had the names areome-
ter, alcoholmeter, saccharometer, &c.
PROP. 19. To explain the uses of the specific gravity bottle.
The specific gravity bottle is made like
fig. 28.
It has a stopper AB made of a piece of
glass tube, which is ground into the neck
of the bottle until the distilled water, which
will fill the bottle and the pipe of the stop-
per, weighs exactly 1000 grains.
If the bottle be then filled with any
other liquid, its specific gravity is known
at once to great accuracy, by weighing again,
Fig. 28.
B

the volume being that of 1000 grains of distilled water.
The specific gravity bottle is also of use to determine the
specific gravities of powders, sand, and small crystals, &c., not
3--2
36
ON HYDROSTATICAL INSTRUMENTS.
soluble in water; for when a given weight of any of these is
put into the bottle, and then the remaining space filled with
distilled water, and the bottle then weighed, we have the weight
of distilled water of equal bulk to the bodies from the following
formula.
Let W be the weight of the bottle when holding 1000 grains
of water,
w the weight of powder, sand, &c. introduced into it,
W' the weight of the bottle when filled with sand, &c. and
water.
Then the weight of distilled water equal in bulk to the powder,
sand, &c.
=W+w-W',
and the specific gravity of the powder, sand, &c.
the weight of the sand, &c. in air
the weight of an equal volume of water
W
W+w-W' •
PROP. 20. To explain the uses of the hydrostatic balance.
The hydrostatic balance, as
in fig. 29, is merely the com-
mon balance, with one of the
scale-pans hung with shorter
cords for the convenience of
weighing bodies in water when
hung from it, as A in the figure,
by some fine thread, or best of
all, by a very fine hair.
ब
A
Let w be the weight of the body in air,
w
water.
Fig. 29.

▼
DI
ON HYDROSTATICAL INSTRUMENTS.
37
Then the weight of the water displaced by the body equals
the weight lost by weighing in water, by Prop. 6,
= w w',
and the specific gravity of the body
weight of the body
weight of an equal volume of water
W
w - w
where the specific gravity of water is taken unity.
In the above discussion it is supposed that the body sinks
in water. If the body is specifically lighter than water, it is
necessary to hang a heavy body to it to cause it to sink.
Let w = the weight of the light body in air,
2
w₁ =
w' =
w" =
heavy
heavy body in water,
both together..........
The weight of both in air minus the weight of both in
water equals the weight of water displaced by both
= w+w, — w",
—
the weight of water displaced by the heavier
=w, w' ;
—
therefore by subtracting, the weight of water displaced by the
lighter body
=w+w'-w",
and the specific gravity of the light body
w + w
20
27.
The specific gravity of a liquid is easily found with the
hydrostatic balance by weighing any heavy body in air, in
water, and in the liquid; since the weight lost by weighing in
water equals the weight of the water displaced, and the weight
38
ON HYDROSTATICAL INSTRUMENTS.
lost by weighing in the liquid equals the weight of the liquid
displaced; and the volumes displaced are equal.
Let w = weight of the body in air,
w
w"
water,
.... the liquid.
Then the specific gravity of the liquid
w-w'
w — w
The specific gravity of a body is one of the characters which
enable us in many cases to determine its nature. In mineralogy,
the form, crystalline or otherwise, the form of fracture, the
lustre, the colour, the hardness, the colour of the streak when
soft, and specific gravity are generally sufficient to determine
the mineral, if it is of a known species. The simple metals
are readily distinguished by their appearance and specific
gravity, and the composition of the alloys may be often known
approximately by the same means.
If the components in an alloy of two metals, or a mixture
of two liquids, are known, the proportions of each can be found
by taking the specific gravity of the alloy or mixture, and
allowing for the change of bulk which generally takes place on
the combination of liquids, or of metals in a state of fusion.
The alloys of some metals occupy less bulk than their compo-
nents did, and those of others occupy more space. The taking
the specific gravity of the alloy or mixture, when the pro-
portions are known, enables us to ascertain whether there
has been expansion of volume or condensation during the com-
bination.
The true weight of a body is its weight in a vacuum, its
apparent weight in air is its true weight minus the weight of an
equal bulk of air, since air acts like other fluids, producing a
resultant vertical pressure equal to the weight of the air dis-
placed. The air continually changing in density, when great
nicety of weighing of large bodies is attempted, the nature of
the weights, as well as of the body weighed, require to be con-
sidered.
ON HYDROSTATICAL INSTRUMENTS.
39
PROP. 21. To find the specific gravity of an alloy or mix-
ture of given composition, supposing no change of bulk to have
arisen.
Let P1 be the density of one of the components, V, its
volume, and w, its weight,
P₂
2
the density of the other component, V, its volume, and w
its weight,
ρ
p the density of the alloy or mixture, Vits volume, and w
its weight.
Then V=V₁+ V₂, and w = w₁+w0₂
1
1
=gpV;
•p=
w₁ + w₂
gV
1
w₁ + w z
g (V₁ + V₂)
1
w₁ + w₁
1+
W3
2
P₂
11
272
(w₁ + w₂) P1P2
W₁P₂+w2P1
which gives p when the weights and densities of the components
are known, and if by experiment it differs from this we know
whether they have contracted or expanded on being combined.
PROP. 22. An alloy or mixture having undergone condensa-
tion, to find the degree of condensation from the known specific
gravities.
Since the mass is unchanged, let V and p be as in the last
proposition, and p' the actual volume and density of the mix-
ture or alloy, then Vp = V'p',
and V-V=V(1-E);
V-V' _p' — p
therefore the condensation
V
40
ÓN HYDROSTATICAL INSTRUMENTS.
COR. Similarly for expansion,
the expansion
V'- V
V
PROP. 23. To find the proportions of the components in an
alloy of given metals, supposing no change of volume to have
taken place, when the specific gravity or density is known.
Let p₁V₁₁
2 27
P₂V₂w₂, pVw be as in Prop. 21, and now w、
and w, are to be found, supposing that V= V₁+ V₂;
but
w=w₁+w₂
1
=w₂+ GP₂V ₂
= w₁+gp₂ (V — V₁)
...
W
1
= w₁²+ JP₂
P2
w
W
1
др дру
=w/1
10 (11) 10 (11),
-
=
20₁ = 10 C1 (PA),
or w₁
Ꮅ
and w₂=
=w-w₁ = w
·P₂ (P₁ =P).
P₂
COR. These values of w, and w, will be only approxima-
tions, on account of the condensation or expansion, but would
serve to find the true values more nearly if the law of conden-
sation or expansion, depending upon the proportion
known. Thus let m be a given number, and
mV = V₁ + V₂,
and proceeding as before,
гог
1
20 R3 (p-mp3)
= 2
P\ P₁ - P₂
W
20 2
201
2
were
These formulæ apply in like manner to the mixtures of
liquids.
ON HYDROSTATICAL INSTRUMENTS.
41
PROP. 24. To explain the construction and uses of the spirit-
level.
The plumb-line hanging always perpendicular to the surface
of still water, it was formerly the general means of determining
the horizontal and vertical directions.
The first fluid-levels were
formed like fig. 30, by a tube
bent twice at right angles, and
filled partly with fluid, as in
E
ABC. Now A and C being the
Fig. 30.

A
B
free surfaces in the same level, an eye E looking from C past.
A will see the objects in the same level with them.
This method is not very accurate, and even when floats
with sights were applied, it was only applicable to very ordinary
purposes.
The spirit-level, as now applied
to astronomical and surveying in-
struments, is susceptible of very A
great accuracy. The essential part
a
Fig. 31.

B
consists of a glass tube, as AB, fig. 31, slightly curved, and
containing spirit of wine except a bubble at a in the figure.
The ends being closed with the blowpipe, the spirit and bubble
remain the same, and the bubble only rests at the highest part.
This instrument is more sensitive, as the tube is more nearly
straight, for then a small change of inclination to the horizon
causes the bubble to move through a larger space.
When the tube, fig. 31, mounted in an appropriate brass
tube with means of adjustment, is attached to a telescope, they
constitute the telescopic level, which is of continual use in sur-
veying for many purposes, as drainage, the laying out of canals,
railways, &c. A level surface in Prop. 3, Cor. 3, is defined to
be that in which water or any other liquid rests, and, when of
small extent, is a horizontal plane; but when larger, such as the
still surface of a lake or the sea, it has the curvature of the
earth's surface. This curvature has to be considered and al-
lowed for in such operations as levelling for carrying a canal over
a country, to determine the gradients of railways, &c.
42
ON HYDROSTATICAL INSTRUMENTS.
PROP. 25. To investigate an expression for the depression in
levelling.
We may consider the earth as a sphere
of 3956 miles radius. Let ADE represent a′
in fig. 32 the level surface through A and
D. Draw AB a horizontal line and tangent
to the level surface at A; also C being the
center, draw the secant CDB meeting the
tangent in B, then BD is called the depres-
sion of the point D below the horizontal
line AB.
Fig. 32.

B
D
E
G
By Euclid, Prop. 36, Book 3, BD (BC + CD) = AB²; and
since BD is small compared with CD, we have
BD
AB2
2. CD'
nearly.
Now CD being 3956 miles, the formula expresses BD and
AB in miles; therefore, bringing BD into feet, we have
BD in feet =
(AB miles)² × 3 × 1760
2 × 3956
= 2 (AB miles)², nearly;
or BD in inches = 8 inches x (AB miles);
or if AB is 1 mile, then BD is 8 inches,
AB is 2
AB is 3
&c.
BD is 32
BD is 72
&c.
We see that the depression soon amounts to a large quantity
on even the lake and canal surfaces which are met with in this
country. In actual surveys, the atmospheric refraction requires
to be taken into account in reducing the observations.
COR. If B were the place of the eye of an observer, then A
is called the offing or visible horizon; and if BD is given, by the
formula
3
AB miles =
BD feet.
ON HYDROSTATICAL INSTRUMENTS.
43
If BD is 6 feet, then AB is 3 miles; and if BD is 24 feet,
then AB is 6 miles, &c. If B' is a point seen from B beyond
the offing A, we have BB'= AB+ AB', which is found by the
above rule, if the depressions at B and B' are given.
PROP. 26. To find the mechanical advantage of Bramah's
press.
Bramah's press consists essentially of a powerful forcing
pump connected by a pipe with a strong hollow cast iron cylin-
der, in which works, water-tight, a solid cylinder or ram of the
same material. The hydrostatical principles in effect in the
press are, that fluids transmit pressure equally in all directions;
and that the pressures, on any plane areas within them, are pro-
portional to the areas.
Let ABC be the lever handle of the forcing-pump, and P the
power acting at A and in equilibrium on the lever with the
reaction Q from the rod of the plunger acting at B, then
P× AC = Q × BC,
Fig. 33.

W
K
Q
B
A
D
C
α
Let DE be the plunger of the forcing-pump working water-
tight in the cylindrical barrel of the pump, of which a and b are
44
ON HYDROSTATICAL INSTRUMENTS.
strong metal valves opening upwards; and let L be the cistern
supplying the water for the press.
Let FH represent the hollow iron cylinder in which the solid
ram GK works water-tight. On the stage at K, upon the head
of the ram, let the weight W be supported. It is required to
W
find the relation of W and P, or the mechanical advantage P'
neglecting the weights of the ram, the lever, and the plunger,
which, with a counterpoise on the lever, may balance each other.
Let r be the radius of the ram GK,
зав
.... plunger DE,
p the unit of pressure of the water, which, neglecting the
weight of the valve b, will be the same in the cylinders of the
pump and the ram.
Then in equilibrium
W=р × πr²,
12
Q=p × πr¹² = P×
AC
BC;
and dividing, 2 × BC'
W 22 AС
P
X the mechanical advantage required.
AC
Example, let r = 6 inches, r'= ½ inch, and
BC
10; then
W
P
36 × 4 × 10 = 1440;
a large advantage to be obtained with only small loss by
friction in the leather collars by which the plunger and ram are
kept water-tight, and thus, in respect of friction, it has a great
advantage over the screw press.
Examples in Hydrostatical Instruments.
Ex. 1. A common hydrometer has the cylindrical stem one
fifth the volume of the whole instrument, and floats with the
whole stem above the surface in water; what is the specific
gravity of the liquid in which it will sink to the top of the
stem?
ON HYDROSTATICAL INSTRUMENTS.
45
Since the weight of the fluid displaced equals the weight of
the instrument when it floats in equilibrium, we have gp V=gp'V';
and from the formula
required is ·8
V
=2, where p=1, the specific gravity
V ρ
Ex. 2. Show that if the hydrometer of the last question be
placed in a fluid of the specific gravity 9, it will float with ths
the length of the stem above the fluid.
Ex. 3. The specific gravity bottle which contains 1000
grains of water has 150 grains of sand put into it, and then being
filled with water it weighs 90 grains more than when filled with
water only; show that the specific gravity of the sand is 2·5.
Ex. 4. Equal parts by weight of water and strong sul-
phuric acid of specific gravity 1·846 being mixed, show that if
there had been no change of volume, the specific gravity of the
diluted acid would have been 1.2972.
Ex. 5. The specific gravity bottle being filled with the
diluted acid of the last question, it is found to weigh 1388-4
grains; show that the condensation in volume has been rather
less than one fifteenth part, being
1
15.2
th.
Ex. 6. A piece of a simple metal weighing 1131 grains in
air is found to lose 10 grains of weight on being weighed in
water; what is the metal?
Ex. 7. A crystal of barytic spar weighs in air 111 grains,
and in water 87; show that the specific gravity is 4.625.
Ex. 8. A piece of calcareous spar weighs in air 187 grains,
and in water 117 grains; show that the specific gravity is 2·67.
Ex. 9. A piece of mahogany weighs in air 372 grains, a
brass weight, hung to it to sink it, weighs 385 grains in water,
and the two together weigh in water 302 grains; show that the
specific gravity of the mahogany is 817.
Ex. 10. The specific gravity of pure gold being 19′3, and
that of copper 8.9, show that the specific gravity of standard
46
ON HYDROSTATICAL INSTRUMENTS.
gold, which consists in every 24 carats of 22 carats gold and 2
carats copper, would be 17-58, if there were no change of volume.
Ex. 11. The specific gravity of standard gold being found
by experiment to be 17.157, show that the expansion in volume
is rather less than one fortieth part, and is th.
1
40.56
Ex. 12. A pebble weighs in air 131 grains, in water 81
grains, and in pyroxilic spirit 88.9 grains; show that the spe-
cific gravity of the pyroxilic spirit is ⚫842, and that of the pebble
is 2.62.
Ex. 13. Some plumber's solder, composed of lead and tin,
has the specific gravity 8.878; determine approximately the
component parts of the solder, the specific gravity of lead being
11.35, and that of tin being 7·29.
Ex. 14. A Nicholson's hydrometer requires 1003 half grains
to sink it to the fixed mark in water; when a piece of granite is
put in the upper cup it requires 289 to sink it to the fixed mark,
and when it is put in the lower cup it requires 556-4; show that
the specific gravity of the granite is 2·66.
Ex. 15. A person standing 96 feet above the level of the
sea observes a part of the mast of a ship beyond the offing, which
is 24 feet above the water line; show that the distance of the ship
from the observer is 18 miles.
CHAPTER III.
ON ELASTIC OR AERIFORM FLUIDS.
It was explained in the Introduction how the fluidity of matter
arose from the caloric, which is an essential part of all bodies as
we know them; and in perfect fluids the molecular attractions
have ceased to produce any sensible effect of aggregation or ad-
hesion. The properties of gases and vapours involve essentially
the consideration of temperature, and can only be fully discussed
when the subject of heat has been studied. It is thus necessary
to take the definition and measure of temperature as known, in
order to discuss even the more prominent properties of elastic
fluids.
By temperature we mean the sensible heat which affects the
thermometer; and the thermometer being an instrument for
measuring temperatures, its scale of degrees is supposed to be
constructed so as to indicate equal increase or decrease of tem-
perature for each change of a degree by the index of the instru
ment. In the fluid thermometers, as the mercurial, spirit, and
air thermometers, the index of the scale is the free surface of
the fluid in the tube; but in those of solid materials, as Bre-
guet's thermometer, Daniell's pyrometer, &c., it is a pointer
moving along the scale of the instrument. In the investigation
of the ordinary laws of elastic fluids the accuracy of the thermo-
metric scales of degrees may be taken for granted, whilst the
higher parts of the subject will involve the theory of the ther-
mometer itself, to be afterwards discussed.
Again, the experiments being generally performed in the
atmosphere, its fluid pressure is involved in the results, and has
to be considered, together with other pressures of the experi-
ment. We may assume that it is known from the height of the
48
ON ELASTIC OR AERIFORM FLUIDS.
barometer, of which the theory will be discussed afterwards; and
thus can be expressed in terms of the pressure produced by a
column of any other given fluid.
The elastic fluids are those which require the constraint of
vessels, or external pressures, to keep them at given volumes,
and which expand when relieved from the reactions by which
they are retained at any given density. The elastic force with
which they tend to expand is measured by the pressure it pro-
duces upon a unit of area when constant or uniform, and is con-
nected in three primary laws with the volume and temperature
of the elastic fluid.
Firstly, the law of Boyle (or Mariotte) for the relation of the
elastic force and volume, when the temperature remains constant.
Secondly, the law of Gay Lussac for the relation of the
volume and temperature, when the elastic force is constant.
Thirdly, the law of Amonton, from which the relation of
the pressure to the temperature is found, when the volume
remains constant.
These are to be considered as only approximations, applying
with sensible accuracy in ordinary circumstances, but failing in
extreme cases.
Law 1. When the temperature remains constant the elastic
force of a gas is inversely proportional to the volume it occupies.
For the experimental proof of the law at
pressures greater than the atmospheric pressure,
let ABC represent, in fig. 34, a bent tube of
glass which is closed at A and open at C; then
a portion of gas occupying the closed leg from
A to a being separated by mercury or other
liquid in the lower bend of the tube aBb from
the atmosphere, if a and b are in the same
level the liquid in aBb is in equilibrium of it-
self, and the pressure upon the surface at a
equals that at b, or the elastic force of the gas
in the leg Aa, and acting upon the surface at
Fig. 34.

C
a, equals the pressure of the atmosphere acting upon the surface
ON ELASTIC OR AERIFORM FLUIDS.
49
of the liquid at b. Let the pressure of the atmosphere on
a unit of area at b be p. Let p be the density of the liquid
used in the experiment, and h the height of the column of it
which will produce by its weight the pressure p, then p=gph
by Prop. 3. Now if more liquid be poured into the tube by
the open end C, the gas in the closed leg will be found to be
compressed into a less space. Let c and d be the free surfaces
of the liquid in equilibrium, draw a horizontal line from c to c',
the portion cBc' will balance of itself, and the elastic force of
the gas in Ac now balances the pressure of the atmosphere on
the surface at d, together with that from the column of liquid
c'd. Let h' be the height of d above the level c'c, and the
pressure from it is gph'. The elastic force of the gas in Ac is
now balanced by the pressure p', such that
'p'=gph+gph'=gp (h+h').
Put e the elastic force of the gas when it occupied the volume
Vin Aa; e' the elastic force when it occupies the volume V' in
Ac; then it is found for experiments within a very large range,
and until a gas comes near its point of liquefaction, that
Ρ
e
h
é¯h + h
V
T'
or if p", e", V" were any other corresponding values of p, e,
and V, then
p"
"/
e
V'
p
e
or the elastic force of the gas is inversely proportional to the
space it occupies.
Since the quantity of gas is the same if p', p" are the densi-
ties when the volumes are V' and V", we have the mass
and
=V'p' = V"p",
V
V"
e
e" __p"
P
e Ρ
or the elastic force of a gas varies directly as its density.
Also p"
p
ρ
p”, and if the ratio be given for any one
ρ
case we may put for its value; and now omitting the distinc-
P. H.
4
50
ON ELASTIC OR AERIFORM FLUIDS.
tive marks we have a general expression as the result of Boyle's
law, p = кp.
B
Fig. 35.
B
The value of « will be found simply expressed further on.
For pressures less than that of the atmo-
sphere let BE, fig. 35, represent a vessel
containing liquid, and let AD represent a
tube within it, closed at A and open at D
and filled with the liquid. Let a portion of
gas be passed into the tube so as to occupy
the space Aa when the level of the surface
of the fluid outside is BB', and a the surface
inside is in the same level; then the elastic
force of the gas acting at a balances the
pressure of the atmosphere on the surface
BB', since B, a and B are in the same level
plane. Let part of the exterior fluid be removed until the sur-
face is at the level Cc C', then the gas inside will be found to
have expanded to a space as Ad, and if the height cd=h' and
p' is now the pressure at d, e' the elastic force of the gas, p' the
density, and V' the volume, we have

p' +gph' =gph.
E
D
From the result of experiments it is found that as before
P
h
e Ꮴ
p h = h = q = y;
Ꮴ ;
-
and the law applies to pressures both greater and less than that
of the atmosphere.
Law 2. When the pressure upon a gas remains the same,
the increase or decrease of volume is directly proportional to the
increase or decrease of the temperature respectively.
It was discovered by Gay Lussac`and Dalton at nearly the
same time, that the gases generally increase from a volume 8 at
the freezing point of water to a volume 11 at the boiling point,
and the expansion is uniform, under a constant pressure.
Let a be the expansion for each degree of the thermometer,
7 the number of degrees between the freezing and boiling
points. Then
ON ELASTIC OR AERIFORM FLUIDS.
51
11-8
· a T⁰
=*375,
8
•375
and
a
то
The value of a depends upon the number of degrees Tupon the
scale of the thermometer which is used. If T°= 100° as on the
centigrade scale, then a='00375, and if 7"-180° as on Fahren-
heit's scale, which is used in this country, then
3
1
α
8 × 180
480
Let V be the volume of a gas at the freezing temperature
when t° 0, the volume V at to above that point, then
=
V=V₂+V₁at°
= √(1+ at°);
or the volumes form an arithmetic progression when the tem-
peratures are in arithmetic progression; this is strictly Gay
Lussac's law. Dalton's views will be found discussed in the
chapter on Heat. The above law was found by Dulong and
Petit to hold with considerable accuracy from nearly the solidi-
fying temperature of mercury to the temperature of boiling
water, the temperatures being those shown by the common mer-
curial thermometer. From the boiling point of water to that of
mercury there was a considerable deviation from the law; which
was undoubtedly more due to the unequal expansion of the
mercury in the thermometer, than to that of the gas. We shall
be however far from justified in taking the law as strictly
a physical law, or, indeed, as anything more than a most useful
empirical law.
With a better method of experimenting it was found by
Rudberg that 100 measures of atmospheric air expand to 136-4
or 136.5 between the freezing and boiling points of water, instead
of to 137-5, as found by Gay Lussac; this gives
1
494'
4-2
52
ON ELASTIC OR AERIFORM FLUIDS.
which should be now used,
and V=V(1+
494/
where tº are the degrees above 32° the freezing point, on Fah-
renheit's scale; and to be taken negative below that point.
He
Again, with very superior methods of experimenting Regnault
has found that a is not exactly the same for all gases.
found that between the freezing and boiling points of water
100 measures of hydrogen gas
100
100
100
expand to 136.61
137.10
carbonic acid gas
sulphurous acid gas............ 139′03
cyanogen gas
138.77
from which the values of a for these gases must be calculated
where great accuracy is required.
Law 3. The general relations of the pressure, density, and
temperature of a gas are given by the formula p=kp (1+aťº)
obtained by compounding the two previous laws; and therefore
when the volume and density are constant the pressure varies as
(1 + atº).
0
Let Po, P., Vo be the commencing pressure, density, and
volume when = 0 at the freezing point respectively,
tº
then pop。, by Boyle's law.
0
Let the density change from p, to p', and the volume from
V₁ to V', when the temperature changes from the freezing point
to ťº above it, and the pressure remaining po,
0
V'
then =1+at° = 29, by Gay Lussac's law.
Po
P
Let again the pressure change from p. to p, whilst the
density changes from p' to p, and the temperature remains at
to above freezing,
then P
(1 + aťº),
Po P Po
ON ELASTIC OR AERIFORM FLUIDS.
53
or
Po
p=Pº. p (1+ atº) = xp (1 + at°),
Po
and when p or the volume is constant,
p ∞ (1 + aťº).
For temperatures below the freezing point, to must be taken
negative.
On the Atmosphere.
The atmosphere acts like
other fluids, producing a resul-
Fig. 36.

B
▼
A
tant pressure on a body im-
mersed in it, which equals the
weight of the air displaced and
acts vertically upwards through
the center of gravity of the air
displaced. It is found that 100
cubic inches of air weigh 31 grains very nearly at the average
heights of the barometer and thermometer, and these would be
contained in a cube of which the edges were 4.642 inches. We
see that this buoyancy from the air may soon become of a mag-
nitude distinctly sensible in ordinary balances and on bodies of
moderate dimensions.
An air-pump experiment, like fig. 36, shows this property
of the air very evidently. A closed hollow glass globe as A
in the figure, about 3 inches diameter, hanging from one end
of the beam of a small balance is counterpoised in air by a
metal weight B. When the instrument is placed under the
receiver of an air-pump and the air withdrawn, it is found that
A and B no longer balance, but that A is the heavier, which
becomes more evident as more air is withdrawn from the
receiver.
Such a question as-which is the heavier, a pound of iron
or a pound of lead?-requires it first to be stated what is
meant by the pound weight, and whether the absolute weights
54
ON ELASTIC OR AERIFORM FLUIDS.
in a vacuum, or the apparent weights in air, are intended, be-
fore an answer can be expected.
The barometer is an instrument which measures the pressure
of the atmosphere. In the simplest form it consists of a glass
tube AB, 33 or 34 inches long, closed at the end A and open at
B, with a cup CD, fig. 37, to hold mercury. The tube AB
with the end B upwards, being filled with pure mercury recently
boiled to free it from air and moisture, and all air-bubbles
being removed from the inside of the tube, the end B being
closed with the thumb it is inverted, as in fig. 37, with the
end B in the cup DC containing mercury, and
below the surface the
It is found that the
Fig. 37.
B
A
then the end B being
thumb is withdrawn.
mercury at the top falls to some point a and
there rests; the height of a above b, the level
of the surface of the mercury in the cup, is
called the height of the barometric column,
and measures the pressure of the atmosphere.
At the level of the surface b inside and outside
the tube the mercury would be in equilibrium
of itself, and therefore the pressures must be
equal from the column a b inside the tube and
from the atmosphere outside. The space from A to a is called
the Torricellian vacuum because Torricelli first maintained that
the height of the column a b was the measure of the atmospheric
pressure. Pascal demonstrated that to be the true explanation
by taking the instrument up the mountain Puy de Dome, and
observed that the surface a fell as the ascent was made up the
mountain, and the portion of the atmosphere above the instru-
ment became less. Thus one experiment settled the long con-
troverted point of the cause of suction and the phenomena
attributed to nature's abhorrence of a vacuum. The height ab
is found to be continually changing with the changing state of
the atmosphere; it increases, or the barometer rises, when a
cold dry north-easterly or easterly wind succeeds a warm moist
south-westerly or westerly wind, and the converse. This is the
direct consequence of the lower and more dense parts of the
atmosphere changing in density, which being connected with

ON ELASTIC OR AERIFORM FLUIDS.
55
changes of the weather, the barometer is often called the
weather-glass. As stated above, the barometer measures the
pressure of the atmosphere, which is the weight of the vertical
column of air at the place of the instrument, at that time. The
average height of the barometric column in this latitude at the
level of the sea is 29.98 inches, or 30 inches very nearly. There
are small diurnal oscillations of the height, connected with the
heat derived from the sun, which causes disturbance of the
density, and motions in the atmosphere.
Fig. 38.
A
The barometer being applied to three different uses; first, as
a weather-glass; secondly, as a meteorological instrument; and,
thirdly, as a means of determining the heights of mountains
with great accuracy; it is constructed accordingly. The wea-
ther-glass is generally an ornamental instrument with slight
pretensions to accuracy, made with a bent tube like Aab B,
fig. 38, closed at A and open at B. The sur-
faces of the mercury being a and b they move
in opposite directions, with a change of the ver-
tical height between them. A float of glass
resting upon the surface b in the tube, being
suspended by a thread attached to a wheel
with a counterpoise to it, the wheel is turned
round as the float rises or falls, and by means
of an index on the face of the instrument tells
the state of the barometric pressure, and the
changes which indicate generally changes of
the weather.

a
The barometers used for meteorological observations should
be read off with certainty to one-hundredth of an inch at least,
which is accomplished by a vernier applied to the scale at the
upper part of the tube.
The mountain-barometer requires the best workmanship,
for under a magnifying eye-glass, by the vernier and estimation,
it should read to one-thousandth of an inch, and be so con-
structed as to be portable without derangement. To obtain the
requisite accuracy, in all the better barometers now made there
is a method of allowance for the change of the height of the
56
ON ELASTIC OR AERIFORM FLUIDS.
mercury in the cistern as well as in the tube. The best
methods have the bottom of the cistern movable with a crew,
and the level of the mercury within it is thus brought to the
fixed standard height before an observation is made.
There have been other instruments invented for measuring
the pressure of the atmosphere which have had their chief
advantages in portability; such as the sympiesometer of Adie,
the aneroid barometer of Vidi, and the excellent manometer of
Bourdon. For the requisite tables to be used with a barometer,
see a little treatise upon it by Mr Belville, of the Royal Obser-
vatory, Greenwich.
The atmosphere consists of about four-fifths nitrogen gas, one-
fifth oxygen gas in volumes, with about one-thousandth part its
volume of carbonic acid gas, and aqueous vapour in very various
but small proportions. Sulphurous acid gas in places where
pit-coal is largely used, and considerable quantities of carbonic
acid gas in crowded rooms, are amongst the causes of local
variations. From these considerations the air at any place not
being of uniform composition its density is not given accurately
from the pressure where great nicety is involved. The density
is shown strictly only by the buoyancy produced by it, as in
the experiment of fig. 36. Dr Prout found sensible changes of
density, from unknown causes, affecting the atmosphere in 1832,
which, as he suggested, might be connected with the cause of
Asiatic cholera, then very virulent. The Aurora borealis is
most probably caused by vaporous matter like the vaporous
comets, and of like composition to the meteoric stones, which,
coming into contact with the higher regions of the atmosphere
in its motion through the planetary spaces, is made luminous by
the earth's electro-magnetism, and, taking magnetic forms, be-
comes mixed with the atmosphere, and may slightly affect its
density.
In simple gases the nucleus of each atom being surrounded
by its atmosphere of caloric, electricity, &c. in equilibrium they
must take a symmetrical or cubical arrangement. Since the
cube which must be attributed to each atom is of different
magnitude in different gases, therefore on being mixed the
ON ELASTIC OR AERIFORM FLUIDS.
57
equilibrium cannot be complete until they are uniformly diffused
through each other. This takes place even when a light gas
is placed over a heavier one, differing from the property of
liquids discussed in Prop. 14, and also with respect to mixtures
of gases and vapours.
Dalton expressed this property by saying that each gas
acted as a vacuum to the others, so that there is a tendency to
make a uniform mixture; but the complete mingling of the
constituents is only accomplished after some time. From the
continual disturbances from acting causes the atmosphere cannot
be considered an absolutely homogeneous mixture.
PROP. 27. To show that the density of the air decreases in
a geometric progression for a series of heights in arithmetic
progression.
If we take a vertical column of the
atmosphere above any place as A, fig. 39,
and suppose it separated from the rest of the
atmosphere by an imaginary rigid film, and
take the area of the perpendicular section
of the column equal to unity, we see that
the pressure at any points, as A, P, or B, is
the weight of the column above those points,
and becomes less as we ascend. At first
we suppose the temperature of the column
the same everywhere.
Fig. 39.

A
B
P
C
Let any height AB=z, and let this be divided into a very
great number m of parts, and each equal to 8, or 8=
2
m
If we take two neighbouring perpendicular sections P and
Q distant 8, and let the height
AP=nd, AQ= (n + 1) . 8,
Pn
and the densities at these sections and P₁₁ respectively, and
Pn, Pr the corresponding pressures: then the difference of the
pressures PnP is the weight of the elementary volume be-
tween the sections, and on account of the smallness of 8 we
n+1
58
ON ELASTIC OR AERIFORM FLUIDS.
might take its density Pn or P. Since the temperature is con-
sidered constant, Boyle's law applies, and we have
PnPn+1=9Pnd = K (Pn - Pr+1)
друб=к (Pn-Pat1),
до
or dividing by Pn›
PR+1
=1- a constant ratio.
K
Pn
Putting B=1-90, if p is the density at the lowest station,
K
we have the densities at the succession of heights 8, 28, 38, &c.
equal to p, ẞp, B²p, ß³p, &c. forming a geometric series with the
common ratio ß.
PROP. 28. To find an expression for the difference of the
pressures at A and B; and the difference of heights of two places,
or the height AB, when the difference of pressures is known.
Let Ρ be the pressure at A, fig. 39, p' that at B, then the
difference p-p' is the weight of the column between A and B,
or equal to the sum of the weights of all the elements;
or
p-p' = gpd (1+ B+ B² + ß³ + &c. ... + Bm-1)
=
= gpd (1) by summation,
and substituting for B its value 1-
gd
we have
K
p—p' = gpd
(1 — ßm)
98
K
= Kρ (1-ẞm)
= p(1 − ẞ"),
p'
or
Br
8″ – (1 – 99)™.
K
Ρ
Taking the logarithms on each side and substituting the expan-
sion of log. (1 – 2º) and md = z, we have
-
K
ON ELASTIC OR AERIFORM FLUIDS.
59
loge
-(2) = m log. (1-96)
(98,
-m {26 +975 +95 + &c...)
K 2K2 3K
g²
- - 2 2 1 + 28 +93 + &c...});
ย
K
2K 3K2
and when 8 is indefinitely small,
K
z = log. (2).
2
9
;
Let h and h' be the heights of the barometer at A and B
h
respectively, then P
and
h' p
K
* == log. (*)
2=
-
g
This formula requires several corrections. First the tempera-
ture of the atmosphere diminishes 1º for every 100 yards of alti-
tude in the lower atmosphere; secondly, the heights of the mer-
cury in the barometer at the upper and lower stations, or h' and h,
require correction for the difference of temperatures; thirdly, a
correction is required for the moisture in the air; and, fourthly,
the force of gravity g is slightly different in different latitudes.
As to the height to which the atmosphere reaches there
have been widely different conclusions. From the duration of
twilight it has been concluded that the height of the atmosphere
in these latitudes does not exceed 45 miles; and this is the gene-
rally received height.
In the lower atmosphere the barometer is found to fall about
one-tenth of an inch for every 30 yards increase of elevation,
which may give an idea of the accuracy with which the heights
of mountains may be determined by properly constructed baro-
meters.
PROP. 29. To find the height the earth's atmosphere would
reach if everywhere of the same density as at the earth's sur-
face.
60
ON ELASTIC OR AERIFORM FLUIDS.
The pressure of the atmosphere being the same on a square
inch as that produced by a column of mercury of 30 inches
height, if p be this pressure, p, the density of mercury, pa the
density of the air at the earth's surface, H the height of the
atmosphere supposed homogeneous,
we have
P
p=9pm × 30 inches = gpå¤;
... H=Pm x 30 inches.
Pa
Pa
Taking the specific gravity of mercury = 13'6, and that of air
= .0013, we have
13.6
H=2.5
= 10461·5 × 2.5 feet
⚫0013
PROP. 30.
= 8718 yards,
= 5 miles nearly.
To find the pressure of the atmosphere on a
square inch at the earth's surface.
The pressure of the atmosphere being measured by the baro-
metric column, the pressure on each square inch equals the
weight of a cylinder of mercury whose base is one square inch,
and whose height is the height of the barometer, and it therefore
is different at different times.
Taking the mean height of the barometer to be 30 inches,
and the specific gravity of mercury = 13.6, the pressure required
equals the weight of 30 cubic inches of mercury, and equals
13.6 multiplied by 30 cubic inches of water,
1000
= 13.6 × 30 ×
ounces,
1728
= 236·1
= 14.7 pounds,
or nearly 15 pounds.
This result is frequently required to be employed where
pressures are referred to that of the atmosphere.
The Magdeburgh hemispheres are two hemispheres to which
handles can be screwed, one being furnished with a pipe and
ON ELASTIC OR AERIFORM FLUIDS.
61
stopcock, and the two fitting air-tight, they are exhausted by
the air-pump, and the stopcock being then turned and the
handles applied, it is found that great force is required to sepa-
rate them.
Ex. Let the area of the circular section of the hemispheres
be 10 square inches, or the diameter 3.56 inches, nearly, the
force at each handle must be more than 147 pounds in order to
separate them.
PROP. 31. To find the height of the barometric column when
the liquid employed is water; or to find the height of a water-
barometer.
=
Since the pressure of the atmosphere equals that produced
by the column of water on a unit of area, let h' be the height
of the water-barometer, and the density of water p', when
the height of the mercurial barometer is h, and the density of
mercury = p, also the pressure of the atmosphere on a unit of
area = p,
then p=gph=gp'h',
and h'=h.º.
P
If h = 30 inches, and ²=13·6,
P
then h'= 2·5 × 13.6 feet = 27·2+6·8
= 34 feet.
This result will be required in discussing the theory of the
siphon, the pumps, and other instruments.
PROP. 32. To find the height of an oil-barometer.
This question is to be worked as the last, and if the specific
gravity of the oil were ⚫94, then
13.6
the height of the oil barometer = 2·5 × feet
•94
36.2 feet nearly.
The results of questions analogous to the last three propo-
sitions will be found in examples 7, 8, 9, to Chapter I. Baro-
62
ON ELASTIC OR AERIFORM FLUIDS.
meters have been made of water as well as oil, for the purpose
of having enlarged scales of the changes of height of the
columns, but they have the disadvantage of requiring an allow-
ance to be made for the elastic force of their vapours in the
Torricellian vacuum. The elastic force of the vapour of mercury
in that of the ordinary barometer is inappreciable, although
there is no doubt of its existence.
On Pneumatic Instruments.
There have been numerous different constructions of air-
pumps invented, but two will perhaps only remain in general
use in this country; namely, the common table air-pump for
ordinary experiments, and Newman's simple and effective air-
pump where a high degree of exhaustion is required.
Fig. 40.
The common table air-pumps have their valves made in
a very simple manner, by tying a strip of oiled silk or thin
bladder, half an inch broad, over an aperture such as A in figure
40, of the small pipe AB passing
through a plug of brass which screws
into the bottom of the barrel of the
pump for the lower valve and into the
piston for the higher valve of the pump.
The pressure under the oiled silk will
cause it to rise and allow air to escape through AB from below,
but falling upon the aperture it prevents the air returning.
Its lightness is of advantage in procuring a considerable
degree of exhaustion, and it is strong enough to bear the
pressure of the atmosphere on the small area of the pipe; but it
requires to be occasionally renewed to keep the instrument in
effective condition.

PROP. 33. To explain the construction of the common double-
acting air-pump.
Let AB and CD in fig. 41 represent the cylindrical barrels
of the pumps open to the air at B and D, and connected at A
ON ELASTIC OR AERIFORM FLUIDS.
63
and C with the pipe EFG leading to G the middle of the plate
M
Fig. 41.

B
A
E
ર
L
K
D
H
H
of the air-pump HI, on which the glass receiver as KL, or other
apparatus, is placed, which it is required to exhaust of air.
Let a, b, c, d represent the valves opening upwards, of which
b and d are in the air-tight pistons which are raised and lowered
by the toothed rods on which the toothed wheel of the figure
acts, as it is turned round backwards and forwards by means of
the handle M.
Let the piston be be at the bottom of the barrel, and then
being raised the valve b closes the aperture and prevents the
external air from entering, whilst the air in the barrel being
rarefied, the elastic force of the air under the valve a raises it up
and the air rushes from the receiver and pipes to fill the barrel
AB. When the piston descends again the valve a closes and
prevents the air in the barrel from returning to the receiver, and
the air in the barrel being compressed by the descent of the
piston it raises the valve b and escapes outwards. The same
takes place in the barrel CD as the piston is raised and lowered,
and in this manner a portion of air is withdrawn from the
receiver at each stroke of the pistons, as long as there is elastic
force sufficient in the remaining air to raise the valves. When
the elastic force becomes insufficient to raise the valves the
pumps cease to act, and no further exhaustion can be obtained.
64
ON ELASTIC OR AERIFORM FLuids.
The lightness of the valves and accurate fitting of the pistons
are evidently points of consequence in the construction; and
the labour of working the pumps is reduced by having two
pumps acting, so that the pressures of the atmosphere on the
pistons counterbalance each other upon the wheel and the winch
by which it is turned. The common air-pumps are in good
order when they will exhaust the air tooth part.
PROP. 34. To explain the construction and mode of action of
Newman's air-pump.
Newman's air-pump has a
single large cylinder as AB, fig.
42; it is open to the air at the
top B, but has a separation CD
though which the piston-rod
works, and with a metal valve b
in its opening upwards; and above
the separation is a quantity of oil.
The piston is solid as EF, and
has a metal valve a in it also
opening upwards. The pipe G
leading to the plate of the air-
H
Fig. 42.

B
D
G
A
pump is at such a distance from the bottom of the cylinder
that the piston EF passes below it at each stroke. The piston
being raised and lowered by means of the lever with handle
H, and toothed arc acting upon the toothed rack of the piston-
rod, when brought up to the separation CD the air above EF
is forced through the valve b and the oil above it, and when
moved down again the air below may raise the valve á and pass
into the barrel, or when it becomes very rare may only fill the
barrel when EF passes below the pipe G; but the pump will
continue to act as long as any air can be lifted through the valve
b and the oil above it. When in good order this pump will
exhaust the air to less thanth part of the original air in the
receiver.
PROP. 35. To find the quantity of air in the receiver after
a given number of strokes of the piston of an air-pump.
ON ELASTIC OR AERIFORM FLUIDS.
65
Supposing the valves to act perfectly, let a be the volume of
the air in the receiver and pipes at commencing, b the volume of
the barrel. Let p be the original density of the air,
P₁ the density after the first stroke of the piston,
Pa
&c.
... second
&c.
Then since the volume a is expanded to the volume a+b by
raising the piston,
P₁(a+b)=p.a, or p₁ =→↓·
1+
a
ρ
Similarly
P₂ (a + b) = p₁ α, or P₂ =
1+
(179);
and so onwards, or after the nth stroke we have
Pn
(1
1+
a
a
and we see that the density of the air in the receiver decreases
in a geometric progression, but never becomes zero even if the
valves were perfect.
PROP. 36. To explain the construction of the siphon and
barometer-gauges of the air-pump.
The air-pump gauges are instruments
for showing the degree of exhaustion which
has been attained, by exhibiting the elastic
force of the air remaining in the receiver
through the height of the mercurial column
which it will support.
For the common air-pump the siphon-
gauge is generally used, and it is made of a
bent tube of glass like fig. 43, closed at one
end as A and open at the other B. It is
filled with mercury from A to some point a,
and being screwed air-tight in a vertical
position to the pipes of the air-pump, with
P. H.
Fig. 43.

5
A
66
ON ELASTIC OR AERIFORM FLUIDS.
which its open end B is in communication, then, when the elastic
force of the remaining air in the receiver will not support the
height of the column of mercury between the levels of A and a,
say about 2 inches, the mercury falls from the end A, and as
the exhaustion proceeds the levels of A and a come nearer and
nearer together. If the level of the surface in the closed leg is
only 4th inch above that in the other, the elastic force of the
remaining air is only oth that of the original quantity if the
barometer stands at 30 inches.
The siphon-gauge should not be depended upon in accurate
experiments, because by usage small quantities of air will pass
into the closed leg and may be often seen forming a small bub-
ble at A by means of a magnifying eye-glass.
Fig. 44.
CO
For accurate experiments the barometer-gauge should be
always employed. It consists of a straight tube of glass 31
inches or more in length, the upper end being cemented into
a brass cap which communicates with the pipes of the air-pump
by its open end C, fig. 44. The lower end dips
into a cup of mercury as B in the figure. When
the air-pump is used the internal air of the receiver
and pipes being rarefied, the mercury rises from
the
cup into the tube, say to some height A. If a
perfect vacuum could be obtained in the receiver
the height AB would be the same as in the baro-
meter, and when any degree of exhaustion is pro-
duced we know the elastic force of the remaining
air by the difference between the height AB and
the height the mercury stands in a good baro-

meter.
B
A
th of an inch, then
When the difference between the height AB and the height
of the barometer, standing at 30 inches, is
the air remaining in the receiver is
quantity.
th of the original
PROP. 37. To explain the construction of the condensing
syringe.
ON ELASTIC OR AERIFORM FLUIDS.
67
The condensing syringe consists of a cylin-
drical barrel as AB, fig. 45, in which an air-tight
piston works by a rod from the handle C. The
valves a and b, the former at the further end of
the barrel and the latter in the piston, both open
outwards, so that as the piston is forced down, the
valve b closing, the contained air is forced through
the valve a into a receiver screwed to the end D.
and is prevented returning by the valve a closing
as the piston is drawn up again, and a fresh supply
of air enters the barrel through the valve b, which
opens to admit it.
Fig. 45.
C

-1ª
B
If the syringe acts perfectly and the same quantity is forced
into the receiver at each stroke of the piston, then the quantity
of air in the receiver will evidently be in an arithmetic pro-
gression as the syringe is worked.
If the receiver be furnished with a stop-cock, by weighing it
when filled with condensed air, and measuring the air which
escapes on the stop-cock being opened, and then weighing the re-
ceiver again, we obtain the weight of the air which had escaped.
The weight of a given volume of gas is also obtained by
having a spherical receiver filled with the gas of which the weight
is known, then exhausting with the air-pump and weighing
again. In experiments to determine the weight of hydrogen gas
many precautions are necessary on account of its great lightness.
The condenser or pump, and the receiver, of the air-gun are
now made differently to the above. The pump has a solid
metal plug-piston, which requires only oil to keep it air-tight in
the barrel, and neither the barrel nor piston have any valve in
them, the fresh supply of air entering the barrel through a hole
in its side near the end when the piston is drawn back. The
receiver has a strong metal valve to prevent the air which is
forced into it from escaping. When the air-gun is used the
charged receiver is screwed to it, and a strong bent steel spring
being let off by the trigger it forces a steel pin against the valve
of the receiver, so as to open it and allow a part of the con-
densed air to escape, and so to drive the charge from the barrel.
5—2
68
ON ELASTIC OR AERIFORM FLUIDS.
PROP. 38. To explain the construction and mode of action of
the siphon.
The siphon however varied in form consists essentially of a
bent tube of glass or metal, and its use is to remove a liquid
from one vessel to another.
Fig. 46.
C
Let a Cb, fig. 46, be the bent tube or siphon, AB the surface
of the liquid in the vessel. Then the
siphon being in the first place filled
with the liquid, and the ends a and b
kept closed, until it is put into the
position of the figure, if we draw a
horizontal line ABd and the height of
the highest point C above it is less
than the height of a barometer formed
of that liquid, then the fluid will not
break at C, but the portions on each

A
d
B
side of the highest point to the horizontal line ABd will balance
each other. The liquid in the tube from d to b will be however
unbalanced, and will by its weight flow out of the tube, and
since there will be no break at the highest point, the pressure of
the atmosphere, which is nearly equal at the surface AB and the
orifice b, will cause an equal quantity to enter the tube to that
which flows out, and thus the tube being always kept full the
flow will continue until the surface AB descends below the
level of either a or b, in either of which cases the flow will
cease.
It is clear that the limits of the height on the average of
a siphon for mercury cannot exceed 30 inches, nor of one for
water, 34 feet; but for alcoholic spirit might be higher accord-
ing to its specific gravity. No siphon of course can act in
a vacuum, the pressure of the air or other elastic fluid being
necessary in order to keep the tube filled with liquid.
The experiment called Tantalus' cup is a vessel containing
a siphon, of which one end opens into the vessel near the bottom,
and the other end passing through the bottom of the vessel
opens below. When filled above the bend of the siphon the
liquid flows out again and the vessel empties by the action of
ON ELASTIC OR AERIFORM FLUIDS.
69
the siphon. Tides-wells, or reciprocating springs, are caused by
natural siphons in the rocks connecting the well and its source
of supply; the flow ceasing when the siphon has drained the
source of supply and recommencing when the water rises again
above the bend of the siphon.
In the different kinds of pumps for raising water or other
liquids, the pressure of the atmosphere acts to keep the barrel
filled with liquid from the well or reservoir as they are ordinarily
constructed. The suction-pump is one which depends chiefly
for its mode of action on the pressure of the atmosphere. The
lifting pump is one which acts chiefly by lifting a column of
water. And the forcing pump acts chiefly by forcing upwards
a quantity of water.
PROP. 39. To explain the construction and mode of action of
the suction-pump.
Fig. 47.
In figure 47 let AB represent the barrel of the suction-pump,
C the opening of its pipe below the sur-
face of the water in the well or cistern,
D the spout or exit-pipe, and EF the
lever by which the piston or bucket of
the pump is raised and lowered. Let a
be the valve at the bottom of the barrel
opening upwards, and b a like valve in
the piston. On the piston being worked
up and down, it will if air-tight first ex-
haust the air from the barrel and pipe
leading to the cistern, in like manner to
the air-pump, and the pressure of the
atmosphere upon the surface of the water
in the cistern will cause it to rise up the
pipe into the barrel, provided it be not more than about 34 feet,
on the average, above the water in the cistern. When the water
enters the barrel its return is prevented by the valve a falling
and closing the opening, and as the piston descends into the

70
ON ELASTIC OR AERIFORM FLUIDS.
water the valve b opens to allow it to pass above the piston;
then on the ascent of the piston the water is lifted in the barrel
and finally flows out by the pipe D; a fresh supply filling the
barrel from the pressure of the atmosphere upon the surface in
the cistern.
PROP. 40. To explain the construction and mode of action
of the lifting pump.
Let AB be the level of the surface of the water
in the well or cistern, CD the cylindrical barrel
of the pump, having a pipe descending below the
surface AB, as in fig. 48. Let a be the valve at
the bottom of the barrel, b that in the piston, and
c another in the exit-pipe, all opening upwards.
The top of the barrel being closed, the piston-rod
EF works through a water-tight stuffing box d.
The pump with its valves and piston acts as in
the cases before described; the water passing up
the exit-pipe GĦ is prevented from returning by
the valve c falling and closing the aperture, and
the water may thus be lifted to any height when
a sufficient force is applied to the piston-rod
EF.
D
Fig. 48.

E
To
G
PROP. 41. To explain the construction and mode of action
of the forcing pump.
Fig. 49.
nf
D
The barrel of the forcing pump is open at the top, with a
pipe descending below the surface AB of the water
in the cistern, as in fig. 49. It has two valves
a and b opening upwards, and an exit-pipe CD.
The piston E is solid, without a valve, being
raised and lowered by the piston-rod EF. The
air is forced from the barrel on the descent of the
piston through the valve b, and then the barrel
being exhausted on the ascent of the piston, the
pressure of the atmosphere on the surface AB will
cause the water to rise in the pipe and barrel to
some height, not exceeding about 34 feet; and
being prevented returning from the barrel by the

2
ON ELASTIC OR AERIFORM FLUIDS.
71
valve a falling and closing the aperture, then on the descent
of the piston again it is forced through the exit-pipe CD, and
prevented returning by the valve b closing the aperture.
By this pump water may be forced to any elevation when
sufficient force can be applied.
PROP. 42. To explain the construction and mode of action
of the common fire-engine.
The fire-engine consists of two forcing pumps and an air-
vessel to maintain a continuous jet of water from the nozle of
the exit-pipe.
Let AB be the surface in fig. 50 of the water in the cistern,
from which the pumps are supplied; CD and EF the barrels of
Fig. 50.

K
DI
a
¡E
the forcing pumps, the pistons of which are worked by the lever
GH. Let a, b, c, d be the valves of the forcing pumps, of which
b and d open into the air-vessel K. Then the exit-pipe coming
to near the bottom of the air-vessel, when the pumps are work-
ing the contained air becomes compressed, and the water as in
the figure occupies a considerable part of the vessel, and thus
the elastic force of the compressed air acting upon the water
produces the air-spring, which maintains a continuous jet of
water from the nozle L of the pipe.
72
ON ELASTIC OR AERIFORM FLUIDS.
If the pumps communicate by a rigid pipe with the source of
water supply they will draw up by the pressure of the atmo-
sphere, as other pumps do, the water they require; but if con-
nected only by a flexible pipe there must be a pressure of the
water supplied to keep the pipe distended, and to cause the
water to flow along it, from either another engine or from a
higher level or head of water.
There have been many forms of machines invented for
raising water and called pumps, which are more properly dis-
cussed in treatises on practical mechanics than in hydrostatics,
such as the centrifugal pump and the chain-pumps, but Ar-
chimedes' screw may properly find a place amongst the ordinary
pumps.
PROP. 43. To explain the construction and mode of action
of Archimedes' screw.
Archimedes' screw consists of
a pipe wrapped in a spiral
round a cylinder, as in fig. 51,
and the axis of the cylinder is
inclined to the horizon. Let
AB be the surface of the water
in the cistern, a the lower open
end of the pipe, which on the
cylinder being turned round
Fig. 51.

A
passes into the water, which then fills the lower part of the pipe.
On the end a being turned to the place b, the water will have
moved to the lowest part of the cylinder, but will have reached
a higher part of the pipe, and on the cylinder being continuously
turned in the right direction, the water will ascend and finally
flow from the upper open end c. In this manner Archimedes'
screw raises water from a lower to a higher level without the
friction of the pistons of other pumps and without valves, but
the quantity raised is not large compared with the size of the
machine. There is evidently a limit to the inclination of the
cylinder to the horizon compared with the inclination of the
pipe to the axis of the cylinder, in order that the parts of
the pipe at the lower side of the cylinder in any part of a revo-
ON ELASTIC OR AERIFORM FLUIDS.
73
lution may be lower than those on each side of them on the
upper side of it.
PROP. 44. To explain the principle of the pneumatic trough
and diving-bell.
Let ABCD be a vessel in fig. 52,
containing a liquid, of which the sur-
E
F.
Fig. 52.
H

a
A
K
D
face is AB. If a vessel as EF or GH
open at one end and closed at the
other be filled with the liquid, and
the closed end be turned upwards, it
may be brought above the surface,
and the vessel will remain filled with
the liquid, provided its length is not
more than the barometric column for that liquid, arising from
the pressure of the atmosphere. A vessel of glass, for instance,
as GH, may be placed with its open end G upon a shelf of the
trough GK below the surface AB, and remain full of the liquid.
If air or any gas be now forced under the open end at G it will
rise through the liquid, and occupy the upper part of the vessel,
as for instance from H to ab. If the vessel is graduated, the
volume of the gas at the atmospheric pressure is known by
bringing the surfaces-of the liquid inside and outside the vessel
GĦ to the same level, and reading off the graduation at the
level. If the surface inside as ab is higher than AB, the con-
tained gas is subjected to less than atmospheric pressure, and if
at cd in the vessel EF it is below AB, it is subjected to greater
than atmospheric pressure. The volumes Hab and Fcd can be
calculated by Boyle's law, when the volume at the atmospheric
pressure and the heights of ab and cd above and below AB are
given, together with the density of the liquid.
If the vessel EF or GH filled with air or gas were immersed
directly in the liquid with the closed end upwards, we should
find the volume, the air or gas, occupied by Boyle's law when
the depth of immersion was known, and EF would then repre-
sent the case of the diving-bell.
It is easy to see that gases may be kept in vessels within
troughs of liquids which do not absorb them or on which they
74
ON ELASTIC OR AERIFORM FLUIDS.
do not act, and may be measured with accuracy, mixed in given
proportions, and the results of such mixtures examined.
PROP. 45. When a diving-bell is of a prismatic or cylin-
drical form, to find the part of the bell which will be free from
water when sunk to a given depth, and no fresh supply of air
has been admitted.
Let AB be the surface of the water in
fig. 53, CD the diving-bell, and AC the
depth of C, the top of the bell below AB
equal to h feet. Let CD the height of the
axis of the bell in feet = a, and CM=x the
part of the axis which is vertical, above the
water in the bell at M. If V is the whole
volume of the bell, V' the volume the air
occupies at the depth AM=h+x, then if we
Fig. 53.
A
C
M
B

take the atmospheric pressure equal to that from a column of
34 feet of water, we have by Boyle's law, whatever be the form
of the bell,
။
V
V 34+h+x
34
V'
When the bell is cylindrical or prismatic
V
the quadratic equation
X
and we have
"
a
to determine x in feet.
x (34+h+x) = 34a
PROP. 46. If two gases which do not act chemically upon
each other are mixed in a vessel in the pneumatic trough, to
show that the product of the elastic force multiplied into the whole
volume of the mixture equals the sum of the products of the elastic
forces into the volumes of the components.
The gases having formed a complete mixture, let p, e, V be
the density, elastic force, and volume of the mixture; p', e', V'
those for one of the components, p", e", V" those for the other.
Then since the mass is constant, we have
pV=p'V' + p"V";
ON ELASTIC OR AERIFORM FLUIDS.
75
and by Boyle's law the elastic force is proportional to the
density;
.. eV e'V'+e"V".
COR. The same may be continued to any number of gases, or
eV=Σ (e'V').
Examples in Pneumatics.
Ex. 1. A cylindrical tube closed at one end, and 20 inches
long, with its perpendicular section one square inch, has an
air-tight piston at the open end. When the barometer stands at
30 inches and the specific gravity of mercury is 13.6, shew that
a force of 26.2 ounces must be applied at the piston to force it
down 2 inches into the tube, and that a force of 221 pounds
must be applied to force it to 8 inches from the closed end.
=
é
V
By Boyle's law we have therefore e'-e=e(−1)
e
₁,
= the pressure to be applied to change the volume from V to V'.
Applying the result of Prop. 30 to the first case, we have
20
e' — e = 236·1
18
-1)
- 1) ounces = 26-23 ounces.
In the second case
/20
e'-e236'1
8
1) ounces = 22′13 pounds.
Ex. 2. If the tube in the last question had been 30 inches
long, shew that the results would have been respectively 16·86
ounces and 40.57 pounds.
1
100
Ex. 3. If an air-bubble, which is a sphere of th inch
diameter at a depth of 102 feet in water, ascends to a depth of
1
68 feet, show that its diameter is then th inch, and when it
90.8
ascends to a depth of 34 feet its diameter is
1
th inch.
79.4
76
ON ELASTIC OR AERIFORM FLUIDS.
Ex. 4. If the volume of a gas is 100 cubic inches at the
temperature 60° Fahrenheit, what will be its volume under the
same pressure at 16° below the zero of Fahrenheit's scale?
If V be the volume at the freezing point,
V
0
60° temperature,
- 16⁰
V'
then
V= (1+
Vo (1 + at³) = Vo (1 +
V₁ V. (1 + 281),
V' = V。 (1 − at®) = V。 ( 1 −
V'
—
1 — at'⁰
and since by dividing 1+ at
494/'
16)
3216
494
.. V'V
223
261
85.44 cubic inches,
since V100 cubic inches.
Ex. 5. If the density of the atmospheric air is called unity
at the freezing point, what is its density at the boiling point of
water?
Since generally
p' _ _V
also 100 measures of air at the
Ρ
V'
>
freezing point become 136 4 at the boiling point; therefore if
p = 1, then p' at the boiling point is
ρ'
100
136'4
='733.
Ex. 6. If the volume of a gas is 100 cubic inches at the
freezing point of water when the barometer stands at 30 inches,
what will be the volume when the barometer stands at 29 inches
and the thermometer at 50°?
By Amonton's law_p=кp (1+aťº), p′ = кp′ (1+ať"º) ;
V'
p 1 +at'⁰
V
P
p' 1+at⁰
30
To find V', let V=100 cubic inches,
t° = 0, t'° 18°,
=
·
p
29
1
α=
then V' = 107·21 cubic inches.
494
ON ELASTIC OR AERIFORM FLUIDS.
77
Ex. 7. A tube which is cylindrical and 40 inches long, with
one end closed, has mercury poured into it to fill 30 inches of
its length, then the open end being covered with the thumb, it is
inverted with the covered end below the surface of mercury in
a cup, and the thumb is withdrawn; required the height the
mercury stands in the tube, the height of the barometer being
30 inches.
Let x be the height the mercury stands in the tube in inches,
then 40-x is the height occupied by the air. The barometric
column was 30 inches when the volume of the air occupied
10 inches of the tube, and when inverted the pressure is that
from 30-x inches of mercury upon the contained air. There-
V'
fore since
P by Boyle's law, we have
V P
40-x
30
10
30-x'
which gives x = 16.973 inches.
Ex. 8. A cylindrical tube closed at one end has an air-tight
piston at 14 inches from the end, and it requires a force of
131 pounds to draw the piston to 18 inches from the end when
the pressure of the atmosphere is 15 pounds upon the square
inch; show that the area of the surface of the piston is 4 square
inches.
If the receiver of an air-gun has a volume five times
that of the barrel of the pump, show that if the pump acted
perfectly it would require forty-five strokes of the piston to
charge the receiver to ten atmospheres.
Ex. 10. Shew that in a mine upon a mountain where the
barometer stands at 26 inches of mercury, the height of the
lower valve of a pump cannot be more than 29 feet above the
water in the mine.
CHAPTER IV.
ON HEAT.
In the introductory chapter it was stated, that caloric, as the
cause of heat, was an essential part of all bodies as we meet with
them; and that changes of the state of dense matter accompany
changes of the amount of caloric which it contains. Sometimes
the change is from a solid to a liquid form, or from either to a
vaporous form, and the converse; but generally, though not
universally, increase of the caloric of a body is attended by an
increase of its bulk, whether it is solid, liquid, or gaseous. The
exceptions occur in liquids coming near their temperatures of
solidification, as water and some metals, which expand again as
they cool before they become solid. Crystals are found to ex-
pand unequally in different directions on being heated.
It was discovered by Dr. Black that heat disappeared on the
change of matter from the solid to the liquid, or from the liquid
to the vaporous state, and he called the heat which had disap-
peared, latent heat. Thus it requires a large quantity of heat to
convert ice at the freezing temperature into water at the freezing
temperature, and also to convert water at the boiling temperature
into steam at the boiling temperature. It is in this respect that
the protoxide of hydrogen becomes ice, water, or steam, accord-
ing to the quantity of caloric it possesses, and which is essential
to each state. What would be the state of dense matter without
caloric we cannot tell; but, reasoning from analogy, we imagine
that bodies would be exceedingly small compared with their
bulks as we meet with them, and of a hardness compared with
which that of diamond is softness. The theory of latent heat
must be considered in conjunction with the known fact that it
will leave the body on the temperature diminishing to a certain
amount, as when water turns into ice at temperatures below the
ON HEAT.
79
freezing point, and has then given out its latent heat of liquidity,
and in other like cases. In this way if we call the latent heat
combined caloric, we must understand the combination of caloric
and dense matter as essentially of a different nature to that of an
acid and a base, which produce a salt with properties of its own
differing from those of its components. We have no evidence to
conclude that caloric forms definite compounds with dense mat-
ter, although it may do so, and give to it peculiar properties in
cases which are at present beyond our means of research. That
it is held by affinity for dense matter in bodies we cannot doubt,
but from its atoms being highly repulsive of each other, it leaves
such dense matter, unless restrained by the opposite repulsion of
other caloric. We shall see further on that this repulsion must
be considered in connexion with the radiation and the tempera-
ture of the body.
By the temperature or heat of a body we mean that degree
of heat which affects the senses, and which is shown by the
thermometer or pyrometer; the degree of heat being measured
by the bulk, or relative bulks, of some bodies of which the in-
strument is made. In the common mercurial thermometer, the
mercury in the bulb and stem expanding and contracting much
more than the glass in which it is contained, from an increase or
diminution of heat respectively, the change of temperature is
shown by the portion of the tube in the stem which is filled
with the mercury. By experiments we find easily that the equal
changes of temperature produce equal or nearly equal changes in
the height of the mercury in the thermometer, and hence we re-
ceive it as an instrument for measuring the temperatures within
its range. For all ordinary purposes a well constructed mer-
curial thermometer gives correct indications of the temperatures
of its scale, and its degrees show equal changes of temperature
at different parts of the scale for such uses. The thermometers
made with spirit of wine in glass, and those made with air in
glass, however, show some small but sensible differences from
the mercurial thermometer, and it becomes a question of some
importance to find an accurate thermometric scale of degrees
which shall indicate equal increase of temperature by the equal
increase of degrees on the scale. The gases are found to ex-
80
ON HEAT.
pand more uniformly than liquids and solids; and when these
latter are tested by the air-thermometer, they are found to ex-
pand more at the higher temperatures than at the lower ones for
equal increase of temperature; and hence the mercurial and spirit-
thermometers are not rigidly accurate instruments, if graduated
upon the supposition of uniform expansion of the mercury or
spirit.
If the expansion of the gases is to be tested, we need an ab-
solutely correct scale for comparison with them, and this is not
to be found. We have, however, the alternative of comparing
them with each other, as, for instance, by forming two air-ther-
mometers, one with a gas which has never been liquefied by cold
and pressure, and the other with one which is easily reducible to
the liquid state. If the latter showed a divergence from uniform
expansion, as compared with the former, we might take the
result as applying to others in like circumstances.
Regnault, by weighing the volumes of the gases contained in
known glass spheres, found the result given at the beginning of
Chapter II. The formula of Gay Lussac's law for the volumes
of the gases at different temperatures being as in Chapter II.
V=V。(1 ± atº), where V, is the volume at the freezing point of
water, and V the volume at t° above or below that point, the
values of V will evidently form an arithmetic series for equal
increments and decrements of temperature. We have V=0 when
1
α
1
494
now taking a = for atmospheric air, then when
t°=-462°, we have V=0, or the gas has become annihilated in
volume, and therefore in existence, whatever the pressure might
be. We should call this the absolute zero of the thermometric
scale; its distance, however, below the freezing point of water
would be different for the different gases, since a is different for
each of them; and we must conclude that Gay Lussac's law is
only an approximation sufficient for use within moderate limits
of temperature.
Dalton considered the volumes of the gases to form a geo-
metric series for temperatures in an arithmetic series, and such a
series will differ very slightly from an arithmetic one, when the
common ratio differs only slightly from unity. Let V be the
•
ON HEAT.
81
SV
V
volume at any temperature, &V the increase of volume for 1°
increase of temperature; then, according to Dalton's view,
constant. Putting this expression into a differential form and
integrating, we find V= V₁eato, where a is the constant, and e is
the base of the Napierian logarithms. Expanding the expo-
nential eat, we have
V= V₂ (1 + at® +
V=V₂(1+
Let t° 1°, and then
=
V=V1+a+
V₂(1+a+
V-V
1
and for air
powers of a.
V 494
0
a².t⁰2
a³.tos
+
1.2
1.2.3
+ &c....).
a²
a³
+
+&c....);
1.2 1.2.3
a, since we may neglect the higher
For moderate ranges of the scale near the freezing
point of water, the formula becomes
V=V。(1+aťº) nearly,
or agrees nearly with Gay Lussac's; but it leads to wide differ-
ences for extreme cases, and does not involve the absurdity of
the air becoming of volume zero at 494 degrees below the
freezing point of water; of which the connexion with nitrogen
and other gases is inconceivable.
If we take the expression V V₁eat when t=0, we have
V=V₁; and when t° is negative V= Ve-at, which is less than
V₁, but only becomes zero when tº is minus infinity. It appears
that we may take any point for starting point; for if t° were the
temperature when the volume is V', then V'Veato; let t=t'+t",
then V=Veato V₁cat.cat
Ve
eat'⁰ = V'eat”, of the same form as the ori-
ginal one; and thus we avoid the absurdity of the gas vanishing
at a particular temperature depending upon the value of a for
that gas and the freezing point of water.
=
0
The formula V=V₁eat would give a greater degree of ex-
pansion at the higher temperatures, as seen in the series of the
expansion; for the terms involving the higher powers of to will
then have sensible values.
P. H.
6
82
ON HEAT.
The gases are evidently the vapours of solids or liquids
which have great affinities for caloric, and far removed from
their dew-points, or the temperatures where they would become
liquids exhibited in dew.
It is probable that liquids may have their volumes expressed
with a like law, showing greater expansion at higher tempera-
tures; but if they crystallize, or their atoms take peculiar ar-
rangements with regard to each other when they become solid,
then there will be an additional term in the expression, as on
approaching the solid state and losing their liquidity, their atoms
commencing peculiar arrangements, the volume may increase for
diminishing temperatures from the temperature of maximum
density. Water is found to be at the greatest density at 7°1
above freezing, or at 39°1 Fahrenheit. A strict formula for
liquids must evidently involve this consideration, and also a
term for the value of their attraction of aggregation.
Though we may not know the absolute quantities of caloric
in bodies, yet we have the means of finding the relative quanti-
ties which they give out and absorb in passing through given
temperatures; these will be found discussed under 'specific heats;'
and it is presumed that the quantities of caloric which they con-
tain are represented relatively by the same numbers which repre-
sent their specific heats. This supposes that the specific heats,
or capacities for caloric, are constant at all temperatures, which
there is great reason to doubt.
>
As the volume of a body depends, when free, upon the tem-
perature and quantity of caloric which it contains, we have the
reciprocal result of a change of these when we forcibly change
the volume of it. It is a well-known fact that iron when briskly
hammered becomes red hot; but when once hammered and
condensed in bulk, it does not exhibit the same result on being
hammered again, unless it has been brought to its original state
by annealing in the fire. Other metals also show like results;
the drawing of wire and the rolling of plates of metal produce
heat. Other bodies generally become heated on being com-
pressed, but iron affords the advantage of large specific heat,
and strong tenacity to bear the blows of the hammer. The
gases give out heat on being greatly compressed, and in the fire-
ON HEAT.
83
syringe it is sufficient to fire tinder. We here meet with con-
siderations of the defective elasticity of bodies, which was dis-
covered by Professor Eaton Hodgkinson to be a general property
of solids; that is, they never recover immediately and perfectly
their original form after a strain, however small; and he considers
liquids and gases to be also subject to the same law, but in a
much less degree. If the hammered iron, in the above-men-
tioned experiment, had recovered its original form by virtue of
its elasticity, after each blow of the hammer, we can have no
doubt it would never have become heated. The minute sparks
of red-hot steel, struck by a flint from steel, occur in the same
way; but experiments have been tried where the condensation
was only temporary, yet heat was given out to the condensing
body, without sensible change of structure in either body. Thus
a smooth metal disc rotating, with a piece of smooth metal
pressing upon it, the latter becomes heated without any sensible
abrasion from either surface. In this case the pressure produces
a slight compression of the disc, and the rubbing body becomes
heated by each successive part of the disc, which recovers its
lost heat during the remainder of the revolution. There has
been no case brought forward where condensation may not have
produced the heat which is witnessed, like that of the hammered
iron, which is the normal experiment.
We conclude that the temperature of a body, its capacity for
caloric, and the amount of caloric in it, are connected together.
If the amount of caloric remains the same, the temperature of a
body is increased when its capacity for caloric is diminished, and
the converse.
The caloric may pass from one body to another in two dif-
ferent ways; first, by radiation, that is, by rays of heat (such as
the rays of light from a shining body) coming from one body to
another; secondly, by conduction, when the heat passes from one
body to another in contact with it, or from one particle to a
neighbouring one of the same body.
The laws of radiant heat are found to be the same as those of
light; that is, it is reflected, refracted and polarized, according
to the same laws. These are easily shown with regard to the
heat accompanying the light of the sun, but require more care in
6—2
84
ON HEAT.
their demonstration for the heat of a common fire, and of non-
luminous bodies.
With regard to the conduction of heat, it is very different
for different bodies; and the metals have been shown to conduct
heat in the same order in which they conduct electricity.
That the radiant light and heat from the sun are reflected
and refracted according to the same laws, is shown by the focus
of a concave mirror held in the sun's light being the same for
the heat as for the light, or they are reflected alike; and when
a convex lens is held in the sun's light, the focus of the heat is
at the same point or very near it, as that of the light, or they are
refracted by the glass in a like manner. That the radiant heat
of the sun may be polarized and doubly refracted can be shown
by similar experiments to those used for light, but with delicate
thermometers in place of the eye, to ascertain the state of polariza-
tion or double refraction of the beam of heat. These facts show
that radiant light and radiant heat must be of the same nature,
and that radiant heat comes from the sun in about 8 minutes and
13 seconds, and moves with the velocity of 192,500 miles per
second, in like manner with light.
In order to show that dark heat, or that which radiates from
hot bodies unaccompanied by light, follows the same laws as
that which accompanies light, we require sensitive methods of
measuring temperatures, such as the differential and air-thermo-
meters, and the thermo-electric multiplier of Nobili. The former
instruments are sufficient to prove the laws of the reflexion and
refraction of dark heat to be identical with those of light; and
by means of the thermo-multiplier Professor Forbes and M. Mel-
loni have proved its polarization and double refraction.
As the light from the sun consists of all the colours of the
solar spectrum with their different degrees of refrangibility, so
the heat which accompanies it has different degrees of refran-
gibility, but it passes through a plate of transparent glass equally
with the light, which is shown by the burning glass. The heat,
however, from a furnace or vessel of hot water, or from a ball of
metal heated below redness, passes only in a very small degree
through a plate of glass, and the glass becomes heated. This is
ON HEAT.
85
an effect similar to what we see with coloured glasses, which
transmit some colours of light, and absorb the others. Rock-
salt or chloride of sodium is found to transmit all kinds of radi-
ant heat and light equally, or it has nearly perfect diathermancy
as well as transparency. From this property prisms and lenses
of rock-salt are important pieces of apparatus in the experi-
ments upon dark heat.
When a thermometer is placed near a cold body it is found
to fall, or there appears a radiation of cold rays. This arises
from the thermometer giving off more radiant heat than it re-
ceives in return, and so becomes cooled. We must consider
bodies to be always giving off radiant heat, and receiving it also
from other bodies. If they receive more than they give off their
temperature rises, and if less it falls, and if they receive heat
equal to what they lose their temperature is stationary.
Fig. 54.
The conducting powers of bodies for heat can be compared
by taking equal rods of them, inserting one end in a vessel in
which water can be kept boiling, as at A, fig. 54, and at the
other end attaching balls with soft wax, or
putting pieces of phosphorus at equal dis-
tances on the rods. Then the time occu-
pied by the heat in passing from the hot
water to the wax or phosphorus, so as to
melt the one or inflame the other, shows
the relative conducting powers of the rods.
The results for the conducting powers of

A
the metals show that for silver it is far the best, then for copper,
then for many intermediate metals; and then in iron and platina
it is small comparatively to silver.
The liquids have been found to have very little, if any, con-
ducting power for heat; for if heat be applied above still water
or other liquid containing a thermometer, that instrument is so
slightly affected that the result may be due to radiation. If,
however, hot water is poured down a pipe so as to pass into cold
water, the two mix, because the hot water is specifically lighter,
and thus communicates its heat to the cold water by what is
called the convection of heat. A hot body in the air cools by
both radiation and convection.
86
ON HEAT.
On Thermometers.
In the uses of thermometers, which are very various, the
instrument should be constructed according to the purpose to
which it is to be applied. Thus, for meteorology, the scale
should extend to the utmost limits of heat and cold which may
be experienced in the climate of the place of observation; in
other instruments, the accurate graduation about the temperature
of boiling water may be most desirable; whilst in others, again,
the most complete range of the scale may be needed. The best
thermometers are supposed to accord with a carefully prepared
standard thermometer, and to be graduated by comparison with
such an instrument.
PROP. 47. To explain the construction of a standard mer-
curial thermometer.
A tube of glass of proper length and bore being found, the
first thing is to determine its cali-
ber in different parts of the bore,
which is done as follows: pass some
mercury into the tube as AB, fig. 55,
a
Fig. 55.

d e
0
B
to occupy some convenient space as ab, which should be noted;
then force the mercury forward until it occupies another portion.
as bc, then cd, de, &c.; which being all registered, when the in-
strument is graduated these spaces must each contain the same
number of degrees, since they are of equal volumes.
The end of the tube being melted with the
blowpipe, a bulb as A, fig. 56, is blown upon it,
and often drawn into a phial shape to procure
sensitiveness. The skill of the artist enables him
to regulate the capacity of the bulb to that of the
tube, so that the required range of scale may be
obtained. The bulb and tube are then filled with
mercury, which requires some dexterity of mani-
pulation, in order that no particle of air may re-
main in the bulb. Then the bulb being heated up
to the highest temperature for which it is required
Fig. 56,

B
I W
H
A
ON HEAT.
87
beyond the boiling point of water, the mercury filling the tube,
the end c is closed by melting with the blowpipe, and often
finished by bending, as in the figure, or formed with an indented
ring round it.
The graduation is obtained by immersing the ball and stem
in melting snow, noting the height of the mercury in the stem,
say at F in the figure, called the freezing point; then placing
them in the steam of boiling water when the barometer stands
at 30 inches, as the average height at the level of the sea; let
B be the place the mercury reaches to, which is called the boil-
ing point. It is finally required to divide the interval FB into
the number of degrees between the freezing and boiling points,
allowing for the varying caliber of the tube; and the expansion
of the mercury compared with that of the glass being con-
sidered uniform.
Fahrenheit considered that the mixture of snow and common
salt produced the greatest degree of cold to be obtained, and
marked that temperature the zero of his scale.
In Fahrenheit's scale the freezing point is marked 32º, and
the boiling point 212º.
In Celsius's, or the centigrade scale, the freezing point is
marked 0º, and the boiling point 100º.
In Reaumur's, the freezing point is marked 0°, and the boiling
point 80º.
From the ascertained graduation between F and B the scale
is continued above and below those points. In the best standard
thermometers the degrees are etched upon the glass stem by
fluoric acid. The thermometers filled with coloured spirit of
wine cannot be employed at temperatures above that at which
the spirit boils, but they have the advantage of showing the
temperatures below the point when mercury congeals, as pure
alcohol has never yet been solidified. They are graduated by
comparing them with a standard mercurial thermometer, like
the meteorological thermometers.
PROP. 48. To investigate formula for comparing the corre-
sponding degrees in Fahrenheit's, Celsius's, and Reaumur's ther-
mometers.
1
88
ON HEAT.
In figure 56, let Hbe the height the mercury stands in the
stem at any time, F the freezing point, and B the boiling point.
Let F be the degrees on Fahrenheit's scale for the point H, C°
those on the centigrade scale, and R° those on Reaumur's scale.
Then we have the following ratios:
space FH F-32°
C⁰ Ro
space FB
180°
100 ¯¯ 80⁰'
or
F¹º - 32º
9
C⁰
Rº
5
4
From which, when any one of the three quantities, Fº, Cº, R' is
given, the others may be found. Fahrenheit's scale has the
advantage of not requiring the mention of negative degrees in
ordinary atmospheric temperatures.
Ex. 1. When the temperature is 60° in London, what
would be the degrees named in Paris and Vienna?
Then F 60°. The Co named in Paris would be 15°5, and
the Rº in Vienna would be 12°4.
Ex. 2. When the temperature is 20º in London, what would
it be called in Paris and Vienna?
PROP. 49. To explain the construction and properties of an
air-thermometer.
4
Fig. 57.
A tube of glass with a bulb A, fig. 57, blown on the end of
it, having its open end placed in some vessel of
liquid, such as coloured water, and being supported
by a cork through which it passes in the neck of the
vessel, will form an air-thermometer for comparative
experiments. Let BC be the level of the surface of
the liquid in the vessel, then applying heat to the
bulb A, some of the contained air will pass out of the
open end through the liquid, and when the instru-
ment is again cooled to the temperature of the atmo-
sphere, the liquid in the tube will stand at some
point as a in the figure. Graduation may be ob-
tained by taking the instrument into a cold place of known tem-

C
ON HEAT.
89
perature, and then into a hot one of known temperature; and
having noted the heights of the liquid in the two cases, dividing
the interval into the requisite degrees, and noting at the same
time the height of the barometer.
The air-thermometer, when constructed for accurate measures,
is on a like principle to the above, but requires many precau-
tions; and, being affected by the atmospheric pressure, can only
be used with certainty when the barometer is stationary, unless
laborious calculations are submitted to. The liquid used must
be mercury, so that vapour in the bulb may not disturb the
results, and the air or gas filling it must be dried at commencing
the construction. In reading the scale the liquid inside and out-
side the tube must be brought to the same level, after allowing
for the capillary depression of the mercury in the tube; and this
requires a high vessel to hold the mercury.
PROP. 50. To explain the construction of the differential air-
thermometer.
Fig. 58.
A
B
The differential thermometer has two bulbs as A and B,
fig. 58, connected by a tube, and before the
opening to the atmosphere is closed, some coloured
sulphuric acid is passed into the tube to occupy
some part as acb, and then the connexion with
the external air is closed with the blowpipe.
When the two bulbs A and B are equally heated,
the terminations a and b of the sulphuric acid
remain at rest; but if one bulb is heated more
than the other, the increased elastice force of the
air in it drives the liquid acb towards the other
bulb. The instrument in this manner shows dif-
ferences of temperatures, and the bulbs may be at the same
height, or at different heights.

PROP. 51. To explain the construction of self-registering
thermometers.
The self-registering thermometers of Dr Rutherford and
Mr Six have been the most used. The former will be here
described. The most complete self-registering meteorological
90
ON HEAT,
and magnetic instruments are those of Mr Brooke, acting by
photography with gas-light and taking continuous results, but
requiring to be attended to frequently.
Fig. 59.
In the figure 59 a mercurial thermometer has its stem hori-
zontal, which being of a wide
bore, a short length of a steel
needle as a is placed within
it. As steel does not amalga-
mate with mercury, the latter,
as it advances, pushes the
needle a before it and leaves it at the highest point it reaches,
and marks the maximum temperature which has occurred.
In the figure 60 a spirit-of-wine thermometer, with its stem
horizontal, has a short length

b of black glass or enamel in-
side the spirit. The attrac-
tion of the spirit for the enamel
causes it to be carried back as
the temperature falls, and it is
Fig. 60.

left at the lowest temperature which occurs, and thus marks the
minimum. After noting the results, the observer brings the
enamel b to the end of the spirit in the stem by raising the end
of the instrument which has the bulb of the spirit for the mini-
mum thermometer, and brings the steel needle a to touch the
mercury in the maximum thermometer by means of a small
magnet. The instruments are then prepared to register the
highest and lowest temperatures in another interval of time.
The maximum instrument requires care, for if the steel a
gets into the mercury it becomes soiled, and must be taken out
and cleaned before the instrument will act well again.
The minimum instrument is not liable to get out of order,
for the enamel b does not easily get out of the spirit; and if it
should get out, it is easily brought to its place again inside the
spirit.
Thermometers are sometimes made advantageously of solid
materials entirely. Wedgewood's pyrometer consisted of pieces
of porcelain clay of known size, which being submitted to the
ON HEAT.
91
heat of a furnace were found to contract in bulk according to the
heat to which they had been exposed; and thus furnished a
means of finding the temperature. The indications of this in-
strument were, however, uncertain, from the same effect being
produced by a high temperature acting for a short time, as by a
lower temperature acting for a longer time.
Ferguson's pyrometer is more adapted to compare the expan-
sions of different substances than to ascertain temperatures; but
Breguet's thermometer and Daniell's pyrometer are valuable
instruments in particular cases for determining temperatures,
better than any others for those cases.
PROP. 52. To explain the construction of Ferguson's pyro-
meter.
A bar of metal, or other substance, as AB, fig. 61, is placed
between the ends of two rods of glass; the end A of one rod
being screwed up to its place and stationary, the other Bb is free
to move in its support when the bar AB expands on being
heated in a bath of hot water or otherwise. The end b of the
rod Bb presses at b near to C on the lever Ca, which turns about
Fig. 61.

B
A
a pivot at C, and this lever Ca presses at a against another lever
DE near its fulcrum or center D. The effect of the expansion is
thus magnified, and the point E of the second lever acts as an
index on a graduated arc, and by its motion along the arc shows
the expansion of the rod AB, when the mechanical advantages
92
ON HEAT.
of the levers have been found, and the effect of the heat upon the
glass rods. By placing different rods of metals at AB their
expansions can be found.
There have been many other methods employed to determine
the expansions of bodies. The following tables contain the
expansions of a few principal substances.

Substance.
Expands between
32⁰ and 2120.
Expands between Expands between
212º and 392º. 392º and 572º.
1
1
1
Mercury ......
55.50
54.25
53.00
1
1
1
Glass
387.00
363.00
329.00
Mercury in
glass......
1
1
1
64.80
63.78
63.18

Substance.
Silver
Copper
Brass
Gold ....
Expands between
32º and 212º.
1
...
174
1
194
1
185
1
224
1
Iron ...
282
1
Platina...
377
From the latter table it will be seen that when a thin strip
of brass is rivetted or brazed to a like strip of iron, then if the
compound bar is straight at one temperature, it will not be so at
#
ON HEAT.
93
other temperatures. In fig. 62, if the bar
were straight as in the upper figure, it
would become convex on the side of the
brass at higher temperatures as in the lower
figure, but concave at lower ones. Such
compound bars have been used to produce
Fig. 62.
IRON
BRASS
thermometers, self-regulating shutters to buildings, &c.; but
their most important use is in chronometers, where such small
compound strips of brass and steel used in the construction of
the compensation balance-wheel counteract the effect of change
of temperature on the balance-spring, and so the chronometer
maintains the same rate of going at different temperatures.
When the expansion of a body in volume is known, as in
the above tables, the linear expansion is found by a simple rule,
as in the next proposition; and the converse.
PROP. 53. To show that the expansion of a body in volume
is three times its linear expansion, nearly.
Let V be the original volume, and the distance of any two
points in it.
Let V' be the volume when the distance of the same points
is 7+x, and x is very small;
V'
(1+x)³
then
V
73
therefore the expansion in volume
V' – V
Ꮴ
= (1 + x)³ — l³°
3x
73
3x2 x3
T +7
τ
Xx
十​色
​X
= 3, nearly, since x is very small,
= 3 × linear expansion.
94
ON HEAT.
PROP. 54. To explain the construction of Breguet's ther-
mometer.
Fig. 63.
A
The essential part of Breguet's thermometer is a spiral AB,
fig. 63, formed of a compound flattened
wire consisting of silver, gold, and platina,
with the gold between the silver and pla-
tina, in accordance with its expansibility, as
seen in the table. The compound wire is
rolled to be very light and thin, and there
are twenty-three revolutions in the spiral
of the instrument. The end A is. fixed by
a piece of brass with a screw to a part AC,
which is supported by the brass arm CD,
but can be turned round to procure adjust-

at
α
ment of the index ab at the other end of the spiral. The index
ab is a light metallic index fastened to a needle soldered to the
lower free end of the spiral, and moves horizontally, with varia-
tion of the temperature, over a divided circle on the base or
pedestal of the instrument.
The advantage of this instrument is the rapidity, almost in-
stantaneous, with which it shows the temperature of the air in
which it is placed, and can be thus used where the sluggishness
of the other thermometers is a serious objection.
Fig. 64.
B
PROP. 55. To explain the construction of Daniell's pyrometer.
Daniell's pyrometer consists of a rectangular rod of plumbago
AB, fig. 64, with a hole down its
axis to receive a rod ab of thick
iron or platina wire. The end a
of the rod ab resting against the
bottom of the hole, a piece of por-
celain or tobacco-pipe be is pushed
against the other end b of the rod,
and is prevented slipping easily
by a band of platina, with a tight-
ening wedge of porcelain, passing
round it and a projecting part of
the plumbago.

ON HEAT.
95
The difference of the expansions of the plumbago and me-
tallic rod ab is ascertained by an instrument like C in the figure,
which has a lever turning about a pivot with the end e as index
moving along a graduated arc, and an edge at d to be applied to
a notch at c on the end of the porcelain rod, when the instru-
ment C is put in its place at the end of the rod of plumbago.
The values of the divisions on the graduated arc can be
found by comparison with a mercurial thermometer below the
boiling point of mercury, and then extended to higher tem-
peratures.
As the materials of which the pyrometer is constructed bear
very high temperatures unchanged, the temperatures of furnaces
and the melting points of various metals are determined by it
with considerable accuracy.
Below is Professor Daniell's table of these temperatures.

Metals.
Fusing points.
Tin
442º
Lead....
612
Zinc
773
Silver
1873
Copper
1996
Gold….
2016
Cast iron ......
2786
On the Specific Heat of Bodies.
It has been explained that by the specific heats of bodies we
mean the relative amounts of caloric which they require to be
communicated to or taken away from them to raise or depress
them respectively a given number of degrees of temperature, and
that their capacities for caloric are supposed to be represented by
the same numbers as their specific heats, when the same standard
is adopted.
Let c=the capacity for caloric of a body, or the amount of
caloric required to raise a unit of mass one degree of temperature.
96
ON HEAT.
Then if c be constant, the amount required to raise a mass m one
degree will be mc; and to raise the mass m, t degrees of tempera-
ture will be m.c.t.
Let c', m', t' be like quantities for another body, then the
amount of caloric required to raise or depress it t' degrees will be
in like manner m'.c'.t'.
When these amounts are equal, we have
m.c.t=m'.c.t',
C m'.ť
or
;
m.t
C
t'
and if m = m', then
Suppose any weight, as one pound of iron, to be quickly
transferred from boiling water into an equal weight of water at 56º,
then they will soon have acquired a common temperature of about
72º, or the caloric which iron has given out in falling 212º — 72°
140°, has raised the equal weight of water 72° 56° = 16º.
C
Therefore if d' the capacity of water be taken unity, we have
the capacity of iron from the formula
ť
16
c = c
1 x
∙114,
t
140
or the capacity of iron is about th that of water for equal
weights.
9
10
To obtain the capacity for equal bulks we must multiply
this by the specific gravity of iron, say 7·8, and have the capacity
of iron = *8892; or in equal bulks iron contains about ths the
caloric which water contains. If the iron were at any lower
temperature than 212º the value of the capacity would be found
to be the same, unless exceedingly great accuracy of manipula-
tion and observation is used.
The capacities of solid bodies generally are easily found by
this method of immersion, but in practice several precautions are
required, in order to obtain exact results; for the effect of the
vessel in which the water is contained must be ascertained and
allowed for, and also care must be taken to avoid loss of heat
in the manipulation.
ON HEAT.
97
If equal weights of water or of mercury at different tempera-
tures are mixed, the resulting temperature is very nearly the
mean; but the careful experiments of MM.. Dulong and Petit
showed that the capacities of the metals increase sensibly with
the temperatures; and the same property has been found to hold
generally with liquids and gases, when they have been very care-
fully examined.
There have been other methods employed to determine the
specific heats of bodies, which give nearly the same results as
that of immersion: one of these is by measuring the quantity of
ice at the freezing point, which is melted by a body of given
weight and temperature in an instrument called a calorimeter;
and another is by noting the time occupied in the cooling of the
bodies from given temperatures. No method of mixing together
liquids which act chemically upon each other can of course be
applicable to determine their specific heats.

Specific heats for
equal weights.
1.000
Water
Iron
•114
Copper
⚫096
•
Zinc
·094
Arsenic
•081
Silver
⚫059
Tin
•056
Cadmium
·057
Antimony
⚫052
Gold coin
·034
•
Bismuth
Platina....
Mercury
⚫033
⚫033
⚫033
Lead...
Sulphur
Flint glass
⚫032
•190
•190
The determination of the specific heats of the gases and
vapours is a subject of great importance, and was not accurately
P. H.
7
98
ON HEAT.
accomplished until MM. Delaroche and Berard undertook the
investigation. By passing the heated gases through a long
spiral tube in a vessel of water, and observing the temperature
at entering the tube in the water and at leaving it, together
with the heat communicated to the water, they obtained the
results in the table below.

Specific Heats.
For equal bulks.
For equal weights.
Air
1.0000
1.0000
Hydrogen gas
•9033
12.3401
...
Carbonic acid
1.2583
•8280
Oxygen...
Nitrogen
•9765
.8848
1.0000
1.0318
Nitrous oxide
1.3503
•8878
Olefiant gas....
1.5530
1.5763
Carbonic oxide...
1.0340
1.0805
Aqueous vapour.
1.9600
3.1360
When water is taken as the standard, the results become as
in the next table.

Specific heats
for equal weights.
Water
1.0000
Air
•2669
Hydrogen gas
3.2936
Carbonic acid
•2210
Oxygen....
2361
Nitrogen
•2754
Nitrous oxide
•2369
...
Olefiant gas
•4207
Carbonic oxide...
•2884
Aqueous vapour.
.8474
ON HEAT.
99
Upon these results, Dalton, in the Appendix to the first part
of Volume II. of his New System of Chemical Philosophy, has
the following important remarks: "From the foregoing detail of
experiments on elastic fluids, it appears evident that such fluids
exhibit matter under a form in which it has the greatest possible
capacity for heat, when capacity is understood to denote the
total quantity of heat connected with the fluid; but if the capa-
city or specific heat is meant to denote the quantity of heat
necessary to raise the body a given number of degrees of tem-
perature, then the elastic-fluid form of matter is that which has
the least capacity for heat of any known form of the same matter.
When therefore we use the term specific heat as applied to
elastic fluids, we should henceforward carefully distinguish in
what sense they are used; but the terms may still be indif-
ferently used in the one or the other sense as applied to liquids
and solids, till some more decisive experiments show that a
distinction is required. Probably the anomalies that have oc-
curred in investigating of the zero of cold, or point of total
privation of heat, are in part due to the want of accordance
between the ratio of the total quantities of heat in bodies, and
the ratio of the quantities producing equal increments of tem-
perature," &c.
When the gases are suddenly condensed in the fire-syringe
it was stated that the heat produced is often sufficient to fire
tinder. This arises from the condensed gas having less capacity
for caloric than it had before the condensation, so that the tem-
perature is raised by the condensation. The law of this change
of capacity for caloric in gases is of importance in the theory of
sound, and some experiments seem to show that the instanta-
neous change, as found in experiments by Mr Joule and the
author, may be different from that which takes place after some
interval of time, when the caloric combined with the dense
matter has attained a new statical condition.
7-2
CHAPTER V.
ON VAPOURS.
IT has been stated before that the vapours differ from the
gases only in being easily reduced to liquids by cold or pressure,
or both. Whilst the vapour retains its elastic state it is subject
to the same laws as the gases, that is, Boyle's, Gay Lussac's,
and Amonton's laws apply to them, at temperatures which are
distant from their dew-points or points of liquefaction, but are
found to fail in accuracy near those points: there can be little
doubt but that the gases are subject to like failures near their
points of liquefaction.
When a vessel of any liquid is placed under the exhausted
receiver of an air-pump an amount of the liquid rises in vapour
which depends upon the temperature. When the evaporation
ceases the space in the receiver is said to be saturated with the
vapour, which thus attains a certain degree of elastic force and
density varying with the liquid and the temperature. When
such a space thus saturated with vapour is reduced in tempera-
ture, a portion of the vapour becomes liquid again, and is ex-
hibited in mist through it, or as dew on the vessel; and so also
if it is subjected to additional pressure and allowed to return to
its original temperature.
It is found that, if the vessel of liquid were placed under the
same receiver filled with dry air or gas, the same amount of
liquid would rise in vapour as in the exhausted receiver, only
more slowly, and the elastic force of the saturated air or gas
would be the elastic force of the dry air or gas plus that of the
vapour.
ON VAPOURS.
101
Let p be the pressure on a unit of area due to the elastic
force of the dry air or gas;
f be the pressure on a unit of area due to the elastic
force of the vapour;
p' be the pressure on a unit of area due to the elastic
force of the saturated air or gas;
then p'=p+ƒ when the volume is that of the original dry air
or gas, and if any two of the quantities p, p', ƒ are given, the
remaining one is known; but f will be known from tables of the
elastic force of vapours at different temperatures, and therefore
p=p'-ƒ for air or gas if dry will be known.
Secondly, let the pressure remain the same, and let V' be
the volume of the saturated air or gas when the pressure is p,
V the volume the dry air or gas would occupy at the same
pressure.
Then, by Boyle's law,
elastic force of the air or gas in volume V'
elastic force of the air or gas in volume V

V
e - f
T'
p
and V = V' 'p-f
or V': V
Ρ
p-f
which give either V or V' when the other with p and ƒ are
known.
That solids may furnish vapours as well as liquids is shown
in the case of ice and snow, which may be easily noticed to
diminish during a long frost; and such substances as camphor
disappear in vapour quickly when exposed to the air: so that
steam or aqueous vapour exists of an elastic force, which can be
measured far below the zero of Fahrenheit's scale, as has been
shown by M. Regnault. The subject of the elastic force of
steam at different temperatures is so important that many phi-
losophers have directed their attention to it; and a collection of
their results will be found in the Philosophical Magazine for
January 1849, in a reprint of a paper by J. H. Alexander, Esq.
102
ON VAPOURS.
As containing results for the vapours of other liquids as well
as water, a table of some of Dalton's results is inserted here.

Elastic force of the Vapours in inches of Mercury.
Temperature
by common
Thermometer.
Ether.
Sulphuret of
Carbon.
Alcohol.
Acetic
Acid.
Water.
70
3.75
3.134
•193
•11
35
7.5
6.20
•560
.27
•29
65
15
12.26
1.51
.69
.75
97
30
24.26
4.07
1.77
1.95
133
60
48.
11.00
4.54
5.07
173
120
29.70
11.7
13.18
220
240
80.2
30.
34.2
272
88.9
340
231.
It is evident that the elastic force of vapours increases in a
very high ratio to the temperatures. Many attempts were made
to find the relation between them, but we must not expect
greater conformity between calculation and experiment than is
shown by the formula of Mr Alexander, which does not differ,
in the whole range of temperatures which have been investigated,
more from the experiments of the different investigators than
they do from each other.
Mr Alexander's formula is
p
11148500
t 990)6
+
1695
where
p is the pressure due to the elastic force of steam in inches
of mercury, and t is the temperature on Fahrenheit's scale.
The following table contains a few selected results from those
occupying four pages of Mr Alexander's paper.
ON VAPOURS.
103
Temperature
Fahrenheit.
Pressure in inches of Mercury of Steam.
By the formula.
By observation.
Observers.
- 27°.112
•0066
•0106
Regnault.
- 13.
•0180
•0205
4.504
•0305
⚫0284
+ 1·706
•0437
•0457
9.41
•0664
•0638
32
•1956
•1811
40
•275
•250
Ure.
70
•849
•726
80
1.184
1.010
100
2.192
1.860
130
4.969
4.366
160
10.214
9.600
173
13.612
13.18
Dalton.
34.20
222
34.73
34.95
Taylor.
59.12
250
58.99
61.19
Ure.
139.70
300
130.02
133.75
Taylor.
340
228.74
231.00
Dalton.
372
346.95
325.
Arzberger.
403-88
511.43
514.22
French Acade-
micians.
435-227
731.92
716.13
The specific heat of steam at different temperatures is a
subject of great importance in the theory of the steam-engine.
Mr Watt found that the heat which became latent at the boiling
temperature when water was converted into steam at the same
temperature was sufficient to have raised the water 950° Fah-
renheit; and by Mr Southern's experiments it is nearly the
same in steam at different temperatures and degrees of elastic
force. This shows that when steam is condensed into water
104
ON VAPOURS.
again by cold applied to it, it gives out heat which would raise
an equal weight of water 950° of temperature.
The boiling points of liquids are the temperatures at which
the elastic force of their vapours equals the pressure to which
they are subject, and thus becomes lower as the pressure is less.
The following table from Dalton's Meteorological Essays gives
the boiling temperatures of water under different pressures of
air in the receiver of an air-pump.

Heat of
the water when
boiling.
Pressure upon its
surface in inches of
Mercury.
Rarefaction of
the air.
212º
30.0
1.
200
22.8
1.3
190
18.6
1.6
180
15.2
2.0
170
12.2
2.45
160
9.45
3.2
150
7.48
4.0
140
5.85
5.1
130
4.42
6.8
120
3.27
9.2
110
2.52
11.9
100
1.97
15.2
90
1.47
20.4
80
1.03
29.0
M. De Saussure found the heat of boiling water upon the
summit of Mont Blanc, 186°; the height of the mountain is near
three miles above the level of the sea; the barometer was 16
inches 144 of a line (a little above 17 English inches).
160
Dr Wollaston proposed to use the thermometer, graduated
accurately with large divisions about the boiling point, to deter-
mine the heights of mountains. Such an instrument is much
more portable than the barometer, and a small quantity of water
can be boiled by spirit of wine very quickly and easily in the
vessels to be used with the thermometer.
ON VAPOURS.
105
In determining the elastic force of the vapour in contact with
the liquid from which it arises, there are two cases, as the
elastic force of the vapour is less or greater than that of the
atmosphere.
PROP. 56. To explain the method of determining the elastic
force of a vapour when it is less than that of the atmosphere.
a
Fig. 65.
버
​A
Barometer tubes and well boiled mercury being
prepared, let a tube filled with mercury, and freed
from all air-bubbles, have its open end covered
until it is placed below the surface AB of the
mercury in the cup, fig. 65, and it will then be a
barometer, with the upper surface of the mercury
in the tube resting at some point a, as in the
figure. Let another tube be filled in the same
way, except a small portion at the open end,
which being filled with the liquid of which the vapour is to be
examined, let the end be then closed with the thumb until that
end is below the surface AB of the mercury in the cup. The
liquid being specifically lighter than mercury rises through it to
the upper closed end of the tube, and the tube being vertical,
when the thumb is withdrawn the mercury falls to some point
as c with a portion be of the liquid resting upon it. The portion
of the tube above b is filled with the vapour of the liquid, and
its elastic force is measured by the column of mercury whose
height is the difference of the heights of a and c above the level
of AB. The liquid be and the space above it being brought to
a variety of temperatures, the elastic force of the vapour becomes
known for those temperatures, as long as it does not exceed the
pressure of the atmosphere when the point c has come down to
the level of AB.

PROP. 57. To explain the methods of determining the elastic
force of a vapour when greater than that of the atmosphere.
Figures 66 and 67 represent the two instruments of the
methods of determining the elastic force of vapours when greater
than that of the atmosphere.
106
ON VAPOURS.
Fig. 66.
C
A
Fig. 67.
B
C
ABC, fig. 66, represents a bent tube of glass, closed at the
end A, and open at the end C.
Mercury being passed into the tube
to fill the leg AB and part of BC,
some of the liquid whose vapour is
to be examined is passed through
the mercury to occupy a small por-
tion of the tube near A when placed
vertical. Heat being applied to the
liquid near A by a vessel of heat-
ed oil surrounding it, or otherwise,
when vapour is formed above the
liquid let the surfaces of the mercury
in the two legs be a and b. Draw
a horizontal line from a to a', then
the elastic force of the vapour at A supports the column of mer-
cury ba' together with the pressure of the atmosphere; and
the elastic force is expressed in inches of mercury by the height
of the barometer plus the height a'b.


α
B
α
Figure 67 represents Marcet's boiler, which consists of a
strong spherical metal vessel supported on a tripod. It has
three apertures to which are adapted, by screws and steam-tight,
first, a thermometer a b, with its bulb inside the boiler to show
the temperature of the vapour; secondly, an aperture with a
stuffing-box at d to admit a long straight tube of glass, open at
both ends as BA; and thirdly, an aperture with a pipe and
stop-cock c as in the figure. In using the instrument, a known
quantity of mercury is poured into the boiler, and the lower end
of the tube of glass is passed below its surface A in the figure.
A quantity of the liquid whose vapour is to be examined is then
poured upon the mercury, when the thermometer and pipe with
the stop-cock are screwed in their places. Now, the heat of a
lamp being applied under the boiler, when the liquid boils, if
the stop-cock c is left open, the atmospheric air in the boiler
will be forced out by the ascent of the vapour, and when the
vapour only issues through it the stop-cock can be closed, and,
the heat being still applied, the temperature of the vapour is
shown by the thermometer ab, and its elastic force by the
ON VAPOURS.
107
height, as AC in the figure, to which the mercury is forced in
the glass tube AB. The elastic force of the vapour at the tem-
perature shown is measured, in inches of mercury, by the height
of the barometer plus the height of the column AC.
If the stop-cock c is opened when the elastic force of the
vapour is considerable, the hand may be held near the aperture
and will only feel a cold jet of condensed vapour to strike it.
This arises from the sensible heat of the vapour having become
latent during the expansion of the jet after issuing into the air.
It is thus sometimes said that high-pressure steam blows cold,
whilst a jet of low-pressure steam will scald severely.
PROP. 58. To explain the mode of action of Dr Wollaston's
instrument as an elementary steam-engine.
This instrument consists of a cylindrical
tube of glass AB, with a bulb BC, blown
on one end of it, as in fig. 68. In the cy-
linder there moves a steam-tight piston
acb, with its piston-rod ced, a tube which
can be closed at the end d by screwing on
the cap d, as in the figure; it passes loosely
through an aperture e in the brass cover to
the end of the cylinder A, which is con-
nected with a handle f. If the bulb BC
be filled with water, and the piston be then
pushed into the tube with the end d open,
the air will pass out through the tubular
Fig. 68.

d
10
A
Do
piston-rod ced, and when the piston is pushed to the surface of
the water the cap d may be screwed on, and so as to close the
end at d. If the bulb BC be now held over the flame of a lamp,
when the water in BC comes to the boiling temperature, and the
elastic force of the steam equals the atmospheric pressure, the
steam will rise and fill the cylinder AB, forcing the piston to
the top. If the instrument be now removed from the flame the
heat will pass away by radiation and the convection of the air,
and the steam in AB will condense into water, when the pres-
sure of the atmosphere on the upper side of the piston will force
it down again; and the same process may be repeated.
108
ON VAPOURS.
If we consider this experiment in its simplest form with
M. Carnot, we consider a cylindrical tube with tight piston
closed at one end and open at the other, the piston being at the
closed end, but with a quantity of water at the end, which, when
converted into steam at the atmospheric pressure, will fill a
given volume of the cylinder. Let sufficient heat be now com-
municated gradually to the water to convert it into steam, the
piston will move up the cylinder until it comes to its position of
equilibrium, with the pressures on each side of it equal. If
the heat be now abstracted again the piston will return to its
first position, and then the same operations may be repeated
continuously. In these operations the work done, neglecting
friction, momentum of the piston, &c. is measured by the quan-
tity of air forced from the cylinder, and this equals the quantity
of steam at the atmospheric pressure which is formed by the
heat communicated and abstracted; which ought to be propor-
tional to the amount of fuel consumed in order to produce that
heat, and to the quantity of oxygen gas of the atmosphere which
is employed to support the combustion.
PROP. 59. To explain the construction and mode of action
of the atmospheric pumping engine.
Newcomen's atmospheric steam-engine had an iron cylinder
AB, open to the atmosphere at the top, but closed at the bottom
Fig. 69.

A
OG
FO
except where three pipes entered, called the steam-pipe, the cold-
ON VAPOURS.
109
water pipe, and the condensed water-pipe respectively, as in
fig. 69. The pipes had each a stop-cock, which could be opened
and shut by an attendant. The cylinder had an air-tight piston
CD within it, and the piston-rod EF was connected with the
beam turning about an axis G. The pump-rods, as HK, were
connected with the other end of the beam. Let a represent the
stop-cock in the steam-pipe coming from the boiler, b that in the
cold-water pipe coming from an elevated cistern, and c that in
the condensed water-pipe.
Then if the piston CD were at the bottom of the cylinder
and the stop-cock a were opened, the steam from the boiler would
enter the cylinder and the pump-rods preponderating the piston
would rise to the top of the cylinder; the stop-cock a being then
shut and b opened a jet of cold water entering the cylinder, as in
the figure, the steam would be condensed and a vacuum formed
under the piston, when the pressure of the atmosphere being
nearly 15 pounds on each square inch of the area of the piston,
it would be forced down and raise the pump-rods at the other
end of the beam. The stop-cock b being closed when the con-
densation was complete, that at c would be opened to allow
the water from the injection and condensed steam to escape
from the cylinder, and then c being closed, a would be opened
again and another stroke of the piston take place, and so
onwards.
The disadvantages of this steam-engine were, that when the
cylinder and piston were cold there was a loss of steam, which
entered the cylinder when a was opened, by condensation, and,
on the other hand, when they were heated the cold water injected
into the cylinder did not completely condense the steam to pro-
duce a vacuum. Mr Watt's improvements were first directed
to remedy these defects by performing the condensation in
a separate vessel, and keeping the cylinder always heated, and
afterwards he contrived both single- and double-acting engines
with closed cylinders.
110
ON VAPOURS.
PROP. 60. To explain the principle of the double-acting con-
densing steam-engine.
Let ABCD represent the cast-
iron cylinder of the engine, EF
the induction-pipe or steam-pipe
from the boiler, GH the eduction-
pipe leading to the condenser K,
into which enters the cold-water
pipe k, and from which a pipe 7
leads to the air-pump of the en-
gine. Let abcd represent valves
opening into the cylinder, which
can be opened and shut; egf the
steam-tight piston, and gh the
piston-rod passing through the
steam-tight stuffing-box i.
Suppose the cylinder to be
heated and filled with steam, by
D
ja
Fig. 70.
h

C
G
c
Ja
H
A
B
k
K
the operation called 'blowing through' by the engineers, then
if the valves a and c be opened the steam above the piston rush-
ing through c to the condenser and meeting the jet of cold water
from the pipe k, will be condensed, and there will be a vacuum
above the piston and the pressure of the steam below, which
entering by a forces the piston to the top of the cylinder, when the
valves a and c are shut and those at d and b are opened. The
steam below the piston now rushes through b and becomes con-
densed, leaving a vacuum below the piston; and the steam from
the induction-pipe entering by d forces the piston to the bottom
of the cylinder, and then the process goes on as before. The
reciprocating motion of the piston is communicated to the main
beam of the engine by the piston-rod, and from it the connecting
rod goes to the fly-wheel, which produces an equalized rotatory
motion by its moment of inertia.
The figure 70 being only for illustration and explanation
the practical arrangement of the valves would always be very
different, and there have been numerous different constructions
used, as the D valves of Bolton and Watt, the box-valves, the
plug-valves, the three-way cock, &c.
ON VAPOURS.
111
PROP. 61. To explain the mode of action of the box-valve of
a steam-engine.
B
Fig. 71.
Let A be the induction-pipe, BC a chamber into which it
leads, having three pipes from it, as in fig. 71, of
which a leads to the bottom of the cylinder, d
to the top, and c leads to the eduction-pipe and
condenser. A box be slides upon the smooth
face of the chamber BC into which the three
pipes open, and is sufficiently long to cover two
of them only at the same time. It is moved
up and down by the rod ef passing through
a stuffing-box, and connected with the engine.
In the position of the box be, as in the figure,
we see that the induction-pipe is in communica-
tion through a with the lower part of the cylin-
der, and that the upper part is in communication
through d and c with the condenser.
When the box eb is pushed down to cover the openings of
c and a, we see that the induction-pipe will be in communication
with the upper part of the cylinder, whilst the lower part will
be in communication with the condenser. In this way the action
of the four valves of fig. 70 can be performed by the sliding up
and down of a box like be, fig. 71, which is kept close to the
face of the chamber by having a vacuum on its inside and a
pressure of steam on its outside.

PROP. 62. To explain the construction of the high-pressure
steam-engine.
The condensing engine will evidently work with steam of
the elastic force of the atmosphere. Mr. Trevithick saw that by
using high-pressure steam a much simpler engine could be em-
ployed, dispensing with the condenser, air-pump, &c., and the
loss of power which was thus avoided might nearly compensate
for the want of condensation. A sufficient supply of cold con-
densing water is also frequently difficult to obtain. In such
cases Mr Watt's condensing engines would be of little value,
and could not, for instance, be used on railways. The high-
pressure engines are now exceedingly numerous; and when
112
ON VAPOURS.
high-pressure steam super-heated, with condensation, and when
the entrance of the steam into the cylinder is cut off at part,
say one half or one third the stroke, and then acts expansively
by its elastic force, the engine becomes the most economical
of all.
The high-pressure engine is constructed like the condensing
engine, fig. 70, but without a condenser, and the eduction-pipe
opens into the air, or into the chimney of the furnace.
Many plans of rotatory steam-engines have been invented to
produce rotatory motion at once, but from the difficulty of keep-
ing the working parts steam-tight in such arrangements, they
have not established themselves, although they promised at the
first sight a great saving of power.
The steam-hammer consists of a heavy hammer or ram,
which is raised by high-pressure steam acting upon the under-
side of a piston in a cylinder, and on the steam being allowed
to escape the hammer falls directly upon the object to be ope-
rated upon.
On Hygrometers.
The hygrometer is an instrument for showing the degree of
moisture or dryness of the atmosphere. There are very many
substances which possess the property of being as it is termed
weatherwise, or of being affected by the amount of vapour in
the air where they are placed. Sponge, seaweed, hair, strips
of whalebone, the awn of the wild oat and feathergrass, cords
formed of animal and vegetable fibres, and many other substances
have been used for hygrometers. Though very sensitive to
changes of moisture the hygrometers formed of these substances
have the disadvantage of not furnishing a scale which can be
compared at distant times. Dalton observed for a long time
the indications of a hygrometer formed with about 6 yards
of whipcord fastened to a nail at one end, and thrown over
ON VAPOURS.
113
a small pulley, being stretched by a weight of 2 or 3 ounces
at the free end. It had a scale divided into tenths of inches.
In different states of the air in a room without fire but with a
moderate circulation of air, it varied in length above 13 inches,
being longer when the air was drier. It was found that the ob-
servations of different years could not be compared, as the cord
continually increased in length with the time it was used. The
hygrometers of like principle are liable to the same objection.
Leslie's hygrometer consisted of a differential thermometer,
having one of its bulbs covered with thin cloth, which was
always kept moist by a few threads leading to a vessel of water,
and it showed the degree of evaporation from the cold produced
by the indication of the thermometer. Daniell's hygrometer
was used with æther, and required some attention.
The ascertaining the dew-point directly is a certain but labo-
rious method of determining the moisture in the atmosphere.
It consists of using cold water with freezing mixture in solution
when necessary, which is poured from one glass vessel to another
until dew is only perceptibly formed upon the glass, when the
temperature of the water is noted. This temperature is more
below that of the air as the air is the drier. This laborious
method, which had been practised by some meteorologists for
years, is now unnecessary, since the wet and dry bulb thermo-
meter gives equally correct indications by inspection; and by
means of Mr Glaisher's tables, given in his pamphlet on the
instrument and its uses, the quantity of water in a given bulk
of air is easily found, and thus a great service has been rendered
to the science of meteorology by his investigations.
PROP. 63. To explain the uses of the hygrometer called the
wet and dry bulb thermometer.
The wet and dry bulb thermometer, as in fig. 72, has two
mercurial thermometers placed near together, with their bulbs as
at A and B, of which A is naked but B is covered with soft
cotton cloth, from which there passes a band of a few soft
cotton threads to a small vessel of water below the instrument.
The bulb B has thus always a wet covering around it, from
P. H.
8
114
ON VAPOURS.
which evaporation is continually going on, unless
the air is saturated with moisture. As the vapour
rises from the covering of B it absorbs the heat
necessary to convert it from a liquid into vapour,
and the temperature of B falls according to the
evaporation going on; and thus is generally
below the thermometer A. The difference of
the temperatures of the two thermometers, or of
the points a and b in the figure, thus shows
the effect of the wetness or dryness of the air.
Fig. 72.

B
"The difference between the readings of the dry bulb and
wet bulb in this country, between the months of April and
September, will frequently be 9° to 12°, less frequently 12º to 15º,
and occasionally will amount to 18°; and during the other months
of the year it will frequently be between 4° and 9°."
CHAPTER VI.
ON THE RELATION OF LIQUIDS TO GASES AND SOLIDS.
WHEN a vessel of water is placed under the receiver of an air-
pump and the pump is worked, we see, as the exhaustion pro-
ceeds, that numerous air bubbles form in the water, and burst on
rising to the surface. It is found that about th of its bulk of
air escapes from ordinary spring water under the air-pump va-
cuum. This contained air also escapes with the steam in the
boiling of water; and hence the need of the air-pump of the
condensing steam-engine to remove the air as well as the water
from the condenser.
Springs are found in various places in which the water is
strongly impregnated with sulphureted hydrogen gas, others
with nitrogen gas, others again with a large proportion of car-
bonic acid gas. From these waters generally the gas escapes
when the pressure to which it has been subject is removed. The
necessity of boiling mercury before using it in the construction
of a barometer has been before mentioned.
The following laws have been established by Henry and
Dalton.
1. The gas in a liquid is retained by the external pressure,
and when this pressure is removed the gas escapes. It is also in
a great measure expelled by boiling.
2. The pressure arising from one species of gas or vapour
will not retain another gas in the liquid, for a portion of the ab-
sorbed gas escapes until there is equilibrium: the proportions
in and out of the liquid having the particular ratio for each gas.
3. The quantity of gas absorbed by a liquid is proportional
to the pressure; and the temperature of the liquid rises during
the absorption.
4. The absolute quantity of gas which a liquid will absorb
under any given pressure is very different for different gases.
8-2
116
ON THE RELATION OF LIQUIDS
2
Thus water absorbs its own bulk of carbonic acid, and nitrous
oxide, 24 times its bulk of sulphureted hydrogen gas, about th
of its bulk of oxygen gas, about th of its bulk of nitrogen, and
th of its bulk of hydrogen gas.
40
5. Different liquids absorb the various gases in very dif
ferent proportions. Thus alcohol is found to absorb twice its
bulk of carbonic acid, whilst water absorbs only its own bulk.
In verifying these laws, pure liquids free from air or gas at
the commencement are needed, and brisk agitation with the
liquid, with fresh supplies of the gas added as the absorption
goes on, in a proper apparatus, is required in order to find the
effect.
Capillary Attraction and Repulsion.
When a solid body is partly immersed in a liquid, they are
found at the places where they come together to have an action
upon each other, which is said to be due to capillary attraction
or repulsion, arising from forces which are sensible only at in-
sensible distances.
If a flat plate of glass whose perpen-
dicular section is CD, fig. 73, be placed
vertically in water of which the surface is
AB, it is found that the water will rise.
around it to some height as ab, about th
of an inch above AB, or the water is said
to have a capillary elevation as it approaches
the glass.
A
Fig. 73.

C.
D
If a flat plate of glass whose perpendicular section is CD,
fig. 74, be similarly immersed in mercury of
which the level of the surface is AB, then
the surface of the mercury becomes convex
near the glass,.and is only in contact with it
at some depth ab, about 4th of an inch be-
low AB, so that the mercury near the glass
is said to have a capillary depression.
17
A
Fig. 74.

C
D
B
b
TO GASES AND SOLIDS.
117
There is said to be a capillary attraction between water and
glass, and a capillary repulsion between mercury and glass; but
in both sensible only at insensible distances, since the capillary
actions cease at the slightest separations of the liquids from the
solids. Wax and greasy bodies show capillary depression when
partly immersed in water, whilst the metals which are wetted
by mercury show capillary elevation when immersed partly in
it; so that the terms capillary attraction and repulsion are only
relative, there being only attraction between the atoms of dense
matter directly.
When a solid body is wetted by a liquid, we conclude that
the attraction of the solid for the particles of the liquid is greater
than their attraction for each other, so that some portion of the
liquid adheres to the solid. When a solid body is not wetted by
a liquid, the attraction of the particles of the liquid for each
other is greater than their attraction for the solid. The former
case of the wetted surface will always give capillary elevation;
but the latter case does not necessarily give capillary depression,
as will be seen below.
PROP. 64. To show that if the attraction of a solid partly
immersed in a liquid for a particle of the liquid at the surface in
contact with it, is more than half that of the liquid for the same
particle, there will be capillary elevation; and if less, there will be
capillary depression.
Fig. 75.
100
R 2
B
Let CAD be the surface of the solid in fig. 75, which is
partly immersed in a liquid of which
the surface AB is in the first in-
stance supposed horizontal, and A
the particle in the surface in contact
with the solid. Since the forces we
have to consider are sensible only at
insensible distances, if we take an
indefinitely small wedge in the solid
whose section is aAb, and its edge
at A perpendicular to the plane of the
figure, its attraction on the particle

a
R₁
D
118
ON THE RELATION OF LIQUIDS
A will be the same in whatever direction it be taken from A in
the solid, and the sum of the equal attractions of all such wedges
makes up the attraction of the solid. Taking BAB' a straight
horizontal line, and CAD vertical, the resultant attraction of the
portion of the solid whose section is B'AD will make an angle
of 45° with the horizontal and vertical directions, let it equal R
acting in AR₁. In the same way the resultant attraction of the
portion whose section is CAB' will be R, acting in a line AR,
whose direction makes an angle of 45° with the lines AB' and
AC, as in the figure.
In like manner, again, the resultant attraction of the fluid,
whose section BAD will be in the direction AR', making the
angles R'AB and RAD each 45°. Let its magnitude be R'.
The particle A being supposed to be in equilibrium, by resolving
horizontally we have
(2R- R') cos 45° = 0
:0
therefore 2R=R'.
The resultant of the forces acting in AR₁, AR, and AR′ is
therefore vertical, as well as the fluid pressure and the force of
gravity; and hence the surface of the fluid is horizontal, since,
as in Prop. 4, the resultant force is always perpendicular to the
surface of a fluid when there is equilibrium.
If R is greater than
R'
>
2
then the resultant lying nearer
always to the greater force, and being perpendicular to the capil-
lary surface, this surface will be concave, as in fig. 73.
If R is less than
R
2
then the resultant will be nearer to the
line AR, fig. 75, and the surface will be convex, as in fig. 74.
The angle BAD, fig. 75, is called the angle of contact, and
equals 90° when R Between mercury and glass the angle
Ꭱ
2
of contact is found to be about 140°. Between water and glass
it is very small, since glass is wetted by water, and R greater
than R.
TO GASES AND SOLIDS.
119
When two plates are brought near together in a liquid, the
form of the surface of the liquid between them is nearly circular;
and when a tube of small bore is placed in the liquid, the surface
of the liquid within is nearly spherical. The capillary eleva-
tions or depressions for different distances of the plates or dif-
ferent radii of the tubes can be found from the consideration
that the force exerted is proportional to the line of contact of the
liquid and solid, and it is measured by the weight of the
column of liquid which is supported above the level of its
general surface.
PROP. 65. To investigate the law of the ascent of a liquid in
small tubes of different radii.
Let CD represent a tube, as of glass,
in fig. 76; let AcB be the level of the
surface of the liquid which wets it. Let
ab be the capillary surface inside the tube,
of which the diameter ab=2r, and ac=h
the height of ab above the level of AB.
Let P be the power due to a unit of length
of the surface of contact, then the power
exerted within the tube
A
D
Fig. 76.

B
= P.2πr
and this is balanced by the weight of the column supported
above the level of AcB. Let p be the density of the liquid, and
the force of gravity, then the weight of the column
g
=gp.r.h, very nearly;
and equating these expressions, we have
2P 1
h
C
gpr
or the height varies inversely as the radius of the tube, which is
in accordance with experiment; and it is found for water and
glass that the height h is one inch when r is th inch, from
which the heights for other radii can be calculated. It is from
120
ON THE RELATION OF LIQUIDS
the result so conspicuous in small or capillary tubes that the
name of capillary attraction has been given.
PROP. 66. To investigate the law of the ascent of a liquid
between two parallel plates near together.
Let CD and EF represent the two plates in fig. 77, whose
distance is d. Let ab be the surface be-
tween them, and ach the height above.
the level of the liquid AcB. Let P, p, g
be as before. Let the breadths of the
=
plates and length of the line of contact on
each. Then equating the power along the
line of contact to the weight of the ele-
vated column, we have
2Pl=gph.d.l, very nearly;
or h
2P 1
gpd
ac
d
A
Fig. 77.

C
E
F
B
or the height varies inversely as the distance of the plates. This
is in accordance with experiment, and with the result of the pre-
vious Proposition; the height being the same when the radius of
the tube equals the distance of the plates.
133
COR. When there is capillary depression the converse rules
hold good. The depression of mercury is an inch between par-
rallel plates of glass when their distance is rd of an inch, and
in a tube when its diameter is th of an inch, from which the
results for other cases can be calculated; and it must be taken
into account when barometers are made with tubes of glass
whose internal diameters are not very large.
67
PROP. 67. Two plates meet in a vertical line, and are in-
clined at a very small angle; required the form of the capillary
surface when their lower edges are immersed in a liquid which
wets them.
Let AOB, fig. 78, be the vertical line in which the plates
CABD, EABF' meet at the small angle CAE.
TO GASES AND SOLIDS.
121
Let OMH be the level of the
axis of x, with OA the axis of y.
Fig. 78.
surface of the liquid, and
Let aPb be the surface of

E
y
P
a
H
O
M
D
B
F
the liquid between the plates, and P any point, drawing PM
parallel to OA; let OM=x, PM=y. Now if D is the dis-
tance of the plates at a unit of distance from O, and d is the
distance at P,
we have
D d
1 XC
or d= D.x
1
1
and the height PM=y, by the last Prop., cc ã Dx'
and xy= constant=m, say,
which is the equation to the rectangular hyperbola referred to
the asymptotes, as seen in experiments.
From the properties above discussed
it arises that when we dip a needle in a
liquid that wets it, the drop which hangs
from the needle when withdrawn from
the liquid takes a position of equilibrium,
as at A, fig. 79, and does not fall to
the point.
Fig. 79.
When a liquid which wets a substance is placed between two
plates of it, as at A in fig. 80, it has con-
cave surfaces and moves up to the angle
Fig. 80.
where they meet; and if small in quantity
they may be turned with the open part
downwards without its falling out.
A
122 ON THE RELATION OF LIQUIDS TO GASES AND SOLIDS.
When there is capillary repulsion be-
tween the substance and liquid, a small
quantity of the latter between plates, as
at A, fig. 81, takes a flattened convex
form, and falls out when they are turned
with the open end below the horizontal
direction.
Fig. 81.
A
CHAPTER VII.
ON THE MOTION OF FLUIDS.
THE problems of hydrodynamics which can be solved with-
out the aid of the differential calculus are not very numerous.
In fluids the constituent atoms being free to move indepen-
dently of each other on the application of the slightest force,
except that in liquids the attraction of aggregation requires to be
considered for the strict solution of most cases, the motion of the
fluid must be considered as originating in the motion of its atoms
individually. In the motion of a fluid, when no crystallic ar-
rangement like that in water near the freezing temperature
exists, the laws of fluidity in regard to the symmetrical arrange-
ment of the atoms have to be recognized. If circumstances
different to those which would exist in the case of equilibrium
are impressed upon one or more atoms of a fluid, then motion
must ensue; and it may be a motion of translation of the atoms
amongst each other or with respect to other bodies, involving a
motion of masses of the fluid; or it may be that a vibratory
motion of the nuclei or centers of the atoms about their places of
rest only exists, whilst the most general and most frequent case
will involve both motions of vibration and translation. Admit-
ting that both kinds of motion do really exist, it requires, in
most problems, that we take notice of one only as affecting the
result which is under investigation.
A force may act upon a whole mass of fluid, and yet a part
of it only may receive motion at any instant, because the action
of the force upon the other parts is counteracted by the re-
sistance of the vessel in which the fluid is contained. The case
of a condensed gas in a receiver with an aperture in it which is
124
ON THE MOTION OF FLUIDS.
opened, and the case of water flowing from an opening in the
containing vessel, are such cases: the portion at and near the
opening at any time being that which receives motion at any
instant.
Fig. 82.
B
α
Torricelli first ascertained the law of the velocity with which
a liquid flows from an opening in a
vessel which contains it. Let AB
be the surface of the liquid in the
vessel, fig. 82, C a small opening
in an arm at the lower part; then
when the jet from C was vertical,
as in the figure, it attained a height
at the highest point a nearly the
level of AB, and he concluded that
if the resistance of the air and fric-
tion had not existed it would have
attained accurately to the level of
AB, and therefore when it issued

C
EO
FO
from the aperture at C it must have had a velocity equal to that
which a heavy body would acquire in falling from the level of
AB to the point C; for it is known in the science of dynamics,
that when a heavy body is projected directly upwards it loses
the velocity in ascending which it acquires in descending again.
Let h be the height of AB above C, v the velocity of issuing
at C, and g the force of gravity; then by dynamics we have
v² = 2gh.
Since fluids transmit pressure equally in all directions, the velo-
city of the jet at any orifice will be the same whether it issues
vertically upwards, vertically downwards, horizontal or inclined,
being always that due to the height of the surface above the
orifice. This law has been found to hold more accurately for the
gases than for liquids, as might be expected from their more
perfect fluidity. The gas contained in a vessel or receiver having
a small aperture which is suddenly opened into a vacuum, the
velocity of the issuing jet is that due to the height of the column
of the gas, which, if of uniform density, would produce the
pressure or elastic force which it exerts in the receiver. Thus
ON THE MOTION OF FLUIDS.
125
hydrogen gas issues into a vacuum with much greater velocity
than atmospheric air, and carbonic acid gas with less. Professor
Graham found that if the gas issued into a space containing a
moderate portion of gas instead of into a vacuum, a similar law
held good, but with a height of the gas due to the difference of
the pressures inside and outside the receiver.
It was a long time after Torricelli's discovery before the
science of hydrodynamics was sufficiently advanced to afford a
strict mathematical proof of the above law; and as it involves
the differential calculus, it cannot be admitted here.
When a liquid issues into the atmosphere from an aperture
in a thin plate, the jet is found to contract after leaving the
aperture, and the narrowest part is called the vena contracta.
It is found that if the area of the vena contracta be taken as the
effective orifice, the quantity of liquid which issues in a given
time comes near to that given by calculation. Newton con-
sidered the area of the vena contracta to bear the ratio of 1 to
√2 to that of the real aperture, but it is generally considered to
be that of 5 to 8. When to the aperture a cylindrical, conical,
or other form of tube is fixed, it is called an adjutage, and its
form affects considerably the quantity of liquid discharged in a
given time.
When the velocity of a fluid is the same at the same point
at all times, it is called a case of steady motion. When the
velocity is not always the same, it is called a case of variable
motion. In Prop. 67 the motion is steady, in Prop. 68 it is
variable.
PROP. 68. A cylindrical vessel containing liquid, having a
small aperture in the base, to find the time of a quantity flowing
out which is equal to the content of the vessel, when it is kept
continually full.
Let CB be the surface of the fluid in the vessel in fig. 83,
A the aperture in the base, let the height of BC above A be
equal to h; then if v is the velocity of the fluid issuing at A,
we have
v² = 2gh.
126
ON THE MOTION OF FLUIDS.
Let a be the area of the vena contracta of the
issuing stream, s the space any part would move
through in a time t if v were constant; then the
quantity flowing out in the time t is
sxa = vt.a
= tx√2gh.
Fig. 83.

C
B
Now if r is the radius of the cylindrical vessel with the height
h, we have its volume = r².h, and, equating this to the last
expression, we have
α
h
2g
which gives the time t as required.
PROP. 69. To find the time the vessel of the last question will
take to empty itself when no fresh liquid is added.
As the surface CB descends, the velocity of issuing at A will
be continually diminishing until the vessel is empty, and will
be always that due to the height of the surface above A. Now
when a heavy body is thrown vertically upwards, its velocity at
any point in the ascent equals that at the same point in the
descent, namely, that due to the height fallen through in the
descent; and the time occupied in ascending to the highest point
equals that of descending again to the same point. In dyna-
mics we have the space s fallen through by a heavy body = gť²
vt, if v is the velocity acquired in falling for a time t from
rest;
:. t = 28.
But with a constant velocity the space described = vel. × time,
and
the time
space
velocity
Therefore the time of the ascent or descent of the body acted
on by the force of gravity equals twice the time required to pass
ON THE MOTION OF FLUIDS.
127
through the same space if the body moved with the first or last
velocity continued constant.
Hence, comparing with the cases of the emptying vessels of
liquid, the time of the cylindrical vessel emptying when no fresh
liquid is added equals twice that of the same quantity flowing
out when it is kept always full, for the velocity of the surface in
α
×
the vessel = velocity at the orifice.
πr2
PROP. 70. When a jet of liquid issues from an aperture in a
vessel containing it, to find the form of the jet.
If the jet be vertical it will remain vertical, but if it issues
in any other direction the curve which it takes will be the
common parabola, since each particle of the liquid may be con-
sidered as projected from the orifice, and its path will be that of
a projectile, as investigated in dynamics. For the directions of
projection horizontal the forms of the jets at D, E, and F are
represented in fig. 82. The maximum range on a horizontal
plane through the orifice occurs when the jet issues at an
angle of 45° with the horizon, as found by theory and ex-
periment.
When any mass of fluid is in motion it possesses momentum
as a solid would, but its effect upon any obstacle it should strike
would be very different, from its wanting the attraction of cohe-
sion possessed by the solid. If the fluid is moving in a pipe, the
momentum may be easily seen to produce a very great effect
when suddenly checked, since it is restrained from diverging
laterally by the resistance of the pipe. When a stop-cock upon
a pipe from a cistern is left open some time, until the water in
the pipe has acquired its full velocity, and is then suddenly
shut, we hear a succession of blows within the pipe, which may
burst it, if it is weak. This property was made by Montgolfier,
in his hydraulic ram, the means to raise water from a lower to a
higher level by very simple machinery. This machine is still
in use in many places, performing the work for which it was
invented.
128
ON THE MOTION OF FLUIDS.
PROP. 71. To explain the construction and mode of action of
Montgolfier's hydraulic ram.
Let AB, fig. 84, be the level of the surface of the water in
the cistern, from which the
b
pipe CD passes and termi-
nates in a chamber which
has two openings fitted with
strong metal valves, as at a
and b. Of these opens
downwards, and a opens up-
wards into an air-vessel, into
which the exit-pipe passes
air-tight, to near the bottom.
If the valve b is shut the
Fig. 84.

F
1-
pressure of the water in the E
pipe will raise the valve a,
D
condense the air in the air-vessel, and find its level with AB
in the exit-pipe. If the valve b be now pushed down a jet of
water will issue through the opening and the water in the pipe
will acquire a certain velocity and momentum, and when this
becomes sufficient, it will lift the valve b and close the opening
suddenly. The momentum of the water will now raise the
valve a, and a portion of water will enter the air-vessel, con-
densing the air still more, and then the valve a will fall and
prevent the water escaping again, when the elastic force of the
condensed air will force it up the exit-pipe until it escapes at F.
When the water in the pipe has come to rest the valve b will
fall again and the same result take place as before.
PROP. 72. To explain the construction and mode of action
of Barker's mill.
Barker's mill has a vertical pipe AB connected with a hori-
zontal one CD, fig. 85. They turn together about a pivot E
and in a bearing at F. The horizontal pipe CD is closed at
the ends but has two openings at a and b, at the opposite sides
of it. When water is poured into the upper part and escapes
through the openings a and b, with a velocity depending upon
the height of the surface in the pipe AB, there is an unbalanced
ON THE MOTION OF FLUIDS.
129
reaction opposite each of the holes
a and b, and these cause rotation
in the opposite direction to that of
the issuing streams.
a
Fig. 85.

B
F
E
60 ||
Moving fireworks generally act
upon the same principle with Bar-
ker's mill; that is, an unbalanced
reaction from the heated gases which:
issue from the openings in them,
gives them motion in the opposite direction. The rocket as-
cending with great velocity requires a very strong case to con-
tain the composition with which it is charged, especially as the
case is choked near its lower end and a large surface of the com-
position is on fire at the same time, so that the produced gases
issue with great velocity and produce a large unbalanced reaction.
The Turbine is a modification of Barker's mill, with a hollow
horizontal chamber between two circular discs in place of the
horizontal pipe. This chamber has openings along its circum-
ference and partitions of a particular curved form in its interior.
It is now constructed so as to exhaust the power of a stream of
water as completely as the best water-wheels, and can be used
with a supply of water to which they could not be applied.
The undershot water-wheel consists of a vertical wheel turn-
ing about an axle, with boards attached to its circumference
perpendicular to its plane, and called float-boards. These float-
boards dipping at their lowest positions into a stream of water
are carried along by its momentum, which thus gives rotation
to the wheel.
The breast-wheel has a trough of masonry of the form of the
wheel, in which the water strikes the float-boards above the
lowest point, and acts partly by its weight and partly by its
momentum.
The overshot wheel has the water laid on the upper part of
the wheel; which thus acts chiefly by its weight. It is some-
times considered identical with the bucket-wheel.
The bucket-wheel has a series of troughs formed along its
circumference of peculiar shape to retain the water as long as
P. H.
9
130
ON THE MOTION OF FLUIDS.
possible during the revolution of the wheel. The water is laid
on, or enters the troughs, near the upper part of the wheel, and
acts almost entirely by its weight. This is the most effective of
all the forms of water-wheels, and exhausts most completely the
power of the fall of water, of which it renders available for
work about 75 per cent.
PROP. 73. A vessel turning about a vertical axis contains
a quantity of liquid; required the form which the surface of the
liquid assumes.
Fig. 86.
Let ABCD in fig. 86 represent the vessel rotating round the
vertical axis EF, and containing liquid. It is
found that the surface of the liquid, which was
horizontal before the rotation commenced,
becomes curved and highest at the side of the
vessel after it has commenced. The rotation
having become uniform, the surface of the
liquid takes a form such as bac, which re-
mains the same, and therefore there is equili-
brium amongst the forces acting upon the
liquid. Now the forces acting upon any par-

A
M
IN ENT
ticle are the fluid pressure, the force of gravity, and the centri-
fugal force arising from the rotation.
If P be a particle in the surface bac, and we draw the
normal PN, the tangent to the curve PT, and the radius of the
circle which P describes PM, then the pressure of the atmo-
sphere and the fluid pressure both act in the normal PN; in
which direction the resultant of the vertical force of gravity 9,
(velocity)2
and the horizontal centrifugal force =
radius
= radius × (an-
gular velocity)²=a².PM must also act, if a= angular velocity
of rotation.
Resolving in the direction of the tangent PT, we have for
equilibrium
or
2
α
a². PM sin PTM – g cos PTM = 0;
tan PTM
g
a. PM
1
PM'
ON THE MOTION OF FLUIDS.
131
which is the property of the parabola. Let y²=4mx be its
equation with origin a and axis of x the axis of rotation; then
by conic sections tan PTM= therefore, comparing with the
2m
y
above equation, we have the semiparameter of the generating
parabola of the surface equal to 2.
PROP. 74. To explain the cause of the rarefaction in diverg-
ing streams of fluids.
DO
Fig. 87.
A
The discovery of the remarkable effects of diverging streams
of air and their explanation were made by
Mr Roberts of Manchester. An instrument
for showing these effects is often made, as in
figures 87 and 88, of tin-plate; A being a cir-
cular disc, say 1 inch in diameter, and BC
another equal disc perforated in its center to
admit the end of the pipe D, which is soldered
to it. Some small prominences are generally
made round the edge of BC to prevent the
disc A sliding laterally away.
When a stream of air is forced along the
pipe D, the disc A is not then blown away,
but takes at some small distance a position of
equilibrium, from which it requires some force
to remove it.

B
3
Fig. 88.
A

Fig. 89.

C
This effect arises as follows: let a be the
opening of the pipe in the disc CB, fig. 89,
and let the stream of air diverge on all sides
from it as it issues from the pipe D on striking the disc A; then
in the divergence it becomes rarefied, and at a short distance
from a becomes of less density than the atmosphere. Let bac
represent the sector which a portion of air would describe on
passing from the center to the circumference of the disc BC; we
see that if by the first law of motion, the air continued to move
with the velocity with which it issued from a, an elementary
portion, such as pq, would soon occupy many times the space
9-2
132
ON THE MOTION OF FLUIDS.
which it did on leaving the opening a, and become more rarefied
as it was further from the center. If there were a vacuum around
the edge, this would be really the case; but with the atmospheric
pressure at the edge the velocity is checked gradually, and
a certain amount of rarefaction only takes place. When the
disc A is in a position of equilibrium, the atmospheric pressure
on its outer face equals the sum of the variable pressures on the
inner face.
COR. If a stream of air is forced along a conical pipe, a like
result arises if it moves from the small to the wide end, or it
becomes rarefied as it passes along. If, however, it passes from
the wide end towards the small end, the converse of condensation
takes place.
The effects of adjutages to apertures in vessels of liquids
which empty through them, depend on the same principles, and
the important subjects of the best forms of the chimneys of
furnaces, the construction of safety valves, and the forms of
pipes for conveying steam or liquids, involve the same con-
siderations.
On Waves.
If a disturbance is impressed on the surface of a liquid at
rest, there are waves formed around the place where the disturb-
ance is made. The simplest case is when the waves are formed
around a center, as when a stone is thrown into a pond of still
water.
A
Fig. 90.
a b
Suppose a heavy round body, as a, fig. 90, to be thrown into
still water, it will carry before it the
water at the surface where it falls,
and on account of the little com-
pressibility and high degree of elas-
ticity of water, elevations as b, c
will be formed around a. Then the
heavy body falling below the sur-
A
B
face, the water which had been depressed below the level rises
above it at d, whilst depressions take the places of the elevations
at b and c, and fresh elevations arise at e and ƒ, and so onwards;
ON THE MOTION OF FLUIDS.
133
so that a series of circular waves diverge around the center,
whilst the space within continues for some time in a state of
oscillation and undulation. If we observe the surface at any
point by watching some small floating body, we see that each
point in the surface has an oscillatory motion, rising above and
falling below the original level of the surface alternately, and
thus gives rise to an undulatory or wave motion diverging from
the center.
If two sets of waves be formed by throwing two bodies into
water, it is easily seen that they exist together, but interfere
whilst passing through each other, so that where each set would
cause an elevation we have produced a higher elevation, and
where each would produce a depression we have a deeper de-
pression produced; but where one set would produce elevation
and the other depression, we have a less elevation or depression,
and if the tendencies were equal the surface remains level.
When two stones are thrown simultaneously into a pond of still
water, the waves being alike round each center, at each instant
the interference is regular, so that the places where they strengthen
each other, and the places where they neutralize each other, are
seen to occur in regular hyperbolic curves.
If a series of waves strike directly against a plane surface
they are reflected, and the series of reflected waves pass through
the original waves as if they came from an origin at an equal
distance behind the plane to what the real origin is in front
of it.
On Sound.
Sounds are affections of the organ of hearing, which may
arise in various ways, but the most frequent are those which
arrive through the air.
That the air is the medium through which the sound from
a distant sounding body comes to the ear, is shown by an expe-
riment with the air-pump; for if a bell be rung in the receiver,
the sound diminishes as the exhaustion is produced, and at
length in the best air-pump vacuum is scarcely audible; so
134
ON THE MOTION OF FLUIDS.
that we conclude no sound could be transmitted through a per-
fect vacuum.
A sound may be produced by a single impulse transmitted
through the air, as when a body is struck by a hammer; but
a continuous sound requires a succession of impulses for its
production, and when these occur with regularity they constitute
a musical sound.
Fig. 91.
That sounding bodies, such as bells, springs, stretched cords,
&c. are in a state of vibration whilst sounding can be shown by
various experiments. A simple and beautiful experiment with
Wheatstone's kaleidophone shows this for springs producing a
musical note, in an instructive manner. The kaleidophone con-
sists of a disc of wood AB, fig. 91, into
which are stuck wires of about th inch
in diameter and 9 to 12 inches long or
more, of which one should be bent at right
angles, as in the figure. A small convex
mirror is stuck at the free end of each by
black sealing-wax, as at a, b and c. What
are called steel beads, but are really silvered

B
glass, may be used as convex mirrors, which answer the pur-
pose. A string of these beads can be often purchased at the
cheap toy shops for a few pence.
When this instrument is placed in the sun's light we see
a minute image of the sun in the convex mirror; and if the
wire be struck sharply we see when it is straight that the image
forms vibrations of changing ellipses, and when bent at right
angles changing lemniscates. These may exist without sound
being heard, but when a violin bow is applied near the fixed
ends of the wires to produce musical notes, there arise small
vibrations upon the larger ones. These are called superimposed
vibrations, and are produced by the vibrations of the wire giving
the musical note. They appear variously upon the larger ones,
and often look like the teeth upon a fine saw, being finer and
closer as the note produced is higher.
We learn from such experiments that the form of the vibra-
tions may be very various, but the frequency of the vibrations
changes as the musical note changes in pitch.
ON THE MOTION OF FLUIDS.
135
The vibrations of the wires produce vibrations in the atoms
of air near them, and these are transmitted by the elasticity of
the air to other atoms at a distance, and thus form sound waves
diverging around the sonorous origin.
These sound waves travel through the air with a velocity of
about 1100 feet per second, which can be shown by noting the
interval of time between the flash and report of a gun at a
known distance. By a comparison of various experiments, it is
concluded that the velocity of sound over the ordinary surface of
a country is 1090 feet per second at the freezing temperature of
water; and that it is 1.14 foot per second more for every degree
of temperature Fahrenheit, above freezing, and the same quan-
tity less for every degree below it. The velocity of sound does
not change with changes of the barometer, because the density
of the air changes with its elastic force. The velocity of sound
is the same for notes of different pitch; for we hear the notes
from a peal of bells at a distance in their regular order, and in
like manner the tune played by a band of musicians sounds the
same at different distances.
Ordinary sounds cease to be heard over very moderate dis-
tances of the rough surface of a country, but are audible very
much further over still water or smooth surfaces. It is probable
that sound is not only lost in passing over rough surfaces, but
that its velocity is slightly retarded, as waves at the surface of
water are in passing up a rough channel. In artillery practice,
where the distance of an object to be fired at is often found by
the interval between the flash and report of a gun, it is found
that a little more elevation must be given to the gun when fired
over water than when fired over land, since "the water attracts
the shot” more than the land! A more philosophical explanation
is, that the velocity of the sound was rather more over the water
than over the land, and hence the distance it would travel in the
same time rather more.
The history of the mathematical investigation of the velocity of
sound is singular. Sir Isaac Newton, from insufficient methods,
found that the velocity of sound in air was equal to √g. H, where
g is put for the force of gravity, and H the height the earth's
atmosphere would reach if homogeneous. This is the same
136
ON THE MOTION OF FLUIDS.
expression as is found by the more correct solutions of Euler and
Lagrange; and it gives the velocity of sound to be about 916.
feet per second, or nearly th less than that found by experi-
ment. Laplace suggested that the discrepancy arose from heat
being suddenly developed in the rapid condensation of the air;
and Poisson, assuming a particular form for such effect, found
that the velocity is represented by the expression √g.y.H, if
Y , where c is the specific heat of air under a constant pres-
C
C₁
sure, and c, the specific heat under a constant volume. By
4
methods liable to serious objections the values 1.3748, 1.4061,
3
1·421 have been found for y. The author, from other methods,
found that y= 1. for small condensations and rarefactions; and he
has shown that if we consider the condensations and rarefactions
to take place in all directions when the sound wave passes
through air, that the expression for the velocity of sound is
3
√g.H. This gives the velocity = 1122′2 feet per second at
the freezing temperature; and from the experience of the ar-
tillery, it is probably the true velocity when sound passes over
still water.
The expression, velocity = √g. H supposes the condensations
and rarefactions to take place only in the direction of the wave
motion. This supposition gives very nearly the velocity of
sound passing through water, as determined by MM. Colladon
and Sturm, in their experiments through the water of the Lake
of Geneva, who found it to be 4708 feet per second. Using
Canton's and Ersted's value of the compressibility of water, the
velocity is found to be by calculation 4876 feet per second, which
is not greatly different from experiment.
..
Fig. 92.
If we note what takes place when a flat spring or the side of
a bell is in a state of vibration
producing sound, we see that
a series of atoms of air which
were originally at equal dis-
tances, as in the line ab, fig. 92,
b
f
ON THE MOTION OF FLUIDS.
137
will be put into states of alternate condensation and rarefac-
tion, as at cd and ef respectively, and the vibrations will be in
the direction in which the sound travels. The vibrations are
from this called longitudinal vibrations. The distance from c to
d or from e to fis called the breadth of the sound wave, and the
letter is frequently put for it. It is convenient to compare
these waves with those formed at the surface of water, though
so differently produced; for in-
stance, in fig. 93, ad is the
Fig. 93.

breadth of a waveλ, and the
a
d
particles at a and d are in the
same state or phase of vibra-
tion; therefore if t be the time of an atom describing a complete
vibration and of the wave travelling from a to d, v the velocity
of sound = 1100 feet per second, nearly, we have λ = v. t, from
which either λ or t can be found when the other is given.
Fig. 94.
Vibrating columns of air in wind instruments, and vibrating
bodies generally, have what are called loops and nodes. The
meanings of these terms are perhaps best illustrated by their
places in vibrating strings fixed at the
ends, as AB, fig. 94. It may vibrate as
a whole or between a and b, and then
produces its lowest or fundamental note,
or it may vibrate as between c and d,
in two halves with a point e stationary,
and then called a node, whilst the
points p and q are in the state of great-
est vibration, and are called loops. The
a
9
Z
h
Z
note then sounded is the octave of the fundamental note. The
same string may also vibrate as between ƒ and g, with loops at r,
s, and t, and nodes at h and 7. It then sounds the fifth note
above the octave, and so onwards.
The places of the nodal lines upon plates of glass are seen
in a great variety of forms by putting dry sand upon the plate
and causing it to sound, when held at some point, by applying
violin bow to the edge.
The pitch of a note depends on the frequency of the vibrations,
138
ON THE MOTION OF FLUIDS.
1
t
or it is inversely as the time of a vibration, or as and
1
therefore as
but v is constant, therefore it varies as
i
λ
λ
V
PROP. 75. To find the pitches of the notes which can be ob-
tained from the same stretched string of given length.
When the tension of a string is the same, the time of vibra-
tion of the lowest note is proportional to its length; and we
saw that the string AB, fig. 94, might vibrate as a whole, in
halves, thirds, &c.; and therefore from the same string the
notes whose pitches are as 1, 2, 3, 4, &c. can be produced; and
by the property of superimposed vibrations two or more of these
may be sounded together.
PROP. 76. To find the pitches of the notes which can be ob-
tained from a cylindrical tube closed at one end and open at the
other.
Let AB represent, in fig. 95, the tube,
closed at A and open at B. When we blow
across the open end the air will be set in
vibration, and there will be a loop in the
column of air in the tube at B; but the air at
the closed end must be at rest, and there
must be a node at A. Therefore the sound
wave of the lowest note will have for its
breadth four times the length of the tube or
distance AB.
Any notes may be obtained from the same
tube which fulfil the required conditions of a
loop at B and a node at A; as, for instance, in
fig. 96, there may be nodes at A and a with
loops at B and b. The sound wave will have
its length one third that of fig. 95. In the
same way the conditions will be fulfilled when
the sound wave has its length one fifth, one
seventh, &c. of that of the fundamental one,
and the pitches of the notes which can be ob-
tained will be as 1, 3, 5, 7, &c.
一
​Fig. 95.

B
A
Fig. 96.

ON THE MOTION OF FLUIDS.
139
We see that by blowing stronger over the tube the next note
to be obtained above the fundamental one is the fifth above the
octave, and the octave cannot be obtained; this is in accordance
with experiment.
The closed organ pipes or stopped diapason belong to this
class.
PROP. 77. To find the pitches of the notes which can be ob-
tained from a tube open at both ends.
When the tube is open to the air at both
ends, as in fig. 97, and we blow across one end,
setting the air in vibration, there will be a loop
there and at the other end also, since it is open
to the air; but there must be a node between
them, as at a in the figure. The sound wave of
the lowest note is therefore twice the length of
the tube, and the note produced is the octave
above that which would be produced if the
end A were closed. This agrees with the
experiment.
Fig. 97.

B
A
α
Fig. 98.
The same conditions will be fulfilled if we have loops at A, c,
and B, with nodes at a and b, and the sound
wave, as in fig. 98, of the note produced is the
length of the tube, being the half of that of
fig. 97, and the note produced is the octave of
the fundamental note.
Similarly the conditions will be fulfilled if
there are three, four, five, &c. nodes in the
tube; and the pitches of the notes to be ob-
tained are as 1, 2, 3, 4, &c., which is in ac-
cordance with experiment.

B
C
A
a
The flageolet, flute, bugle, open diapason pipes of the organ,
&c. belong to this class of musical instruments.
The reed stops of organs are formed with vibrating brass
springs, and the note produced depends upon the length and
elasticity of the spring; but it is modified by the length and
form of the tube to which it is attached, so as to imitate the
140
ON THE MOTION OF FLUIDS.
hautboy, trumpet, violin, &c. With a seraphine reed and a tube
which can be lengthened by sliding a tube over the one holding
the reed, we obtain the series of vowel sounds with the same
note, and the order is i, e, a, o, u, pronounced in the continental
manner as ee, ai, aa, o, u. These are formed in the human voice
by altering the cavity of the mouth and throat.
The human voice is formed in the larynx behind the promi-
nence in front of the neck, by the vibrations of the tendinous
chordæ vocales, which are under the power of muscles acting
under the influence of the will.
When two notes are sounded together, they harmonize or
please the ear in proportion as the lengths of their waves are
commensurable; thus the fundamental note and its octave may
be said to form a unison. The next most perfect accordance is
3
when the ratio is 1 to; the next again when it is 1 to
4
; the
next again when it is 1 to 5; and the next again when it is
5
1 to
These in the series of natural notes are as follows:

Fundamen-
Second
tal note
C
Major
third
Fourth Fifth
Major
sixth
Seventh Octave
D
F
G
B
C
E
A
3
5
pitch 1
2
4
3
2
3
The interval being large between C and E, the note D is
made the fifth of the octave below the fifth, and its pitch is
3 3 9
X
4 2-8'
which though it will not accord with the fundamental
note, will do so with the fifth, its best chord.
The interval between A and C octave is also large, and the
note B is made the major third to the fifth, with which it will
ON THE MOTION OF FLUIDS.
141
therefore accord, and its pitch is
scale of notes is as follows:
3
5 15
X
Thus the diatonic
2
8

C
D
E
F
G
A
B
C
pitch 1
0100
5
4
3
5
15
2
8
4
2
8
The note C is called the key-note of the scale. It is clear
that if any other note were taken for key-note, the relations of
the sound waves would not hold the same.
In instruments with fixed notes the tuning requires tempera-
ment, so that in playing in the most usual different keys the
least possible discord may arise; and the ear does not require
perfect concords.
A TABLE OF SPECIFIC GRAVITIES.
A cubic foot of water at 60° Fahrenheit weighs 997·137 ounces avoir-
dupois; 100 cubic inches of atmospheric air, barom. 30 in., therm.
60°, weighs 31.0117 grains.
Specific gravities of gases, air being
taken as standard fluid.
Specific gravities of some bodies, water
being taken as standard fluid.
Atmospheric air…...
Hydrogen gas...
Nitrogen
""
Oxygen
1.0000
Flint
2.59
0688 Cornelian
2.61
9757 Rock crystal
2.65
1-1026 Emerald ……..
2.77
وو
Carbonic acid gas
1-5245 Calc spar
2.72
Diamond
3.52
Topaz, Oriental ...
4.01
Specific gravities of some bodies, water
being taken as standard fluid.
Beryl
3.55
""
Ruby
4.28
""
Water....
1.0000
Garnet, Bohemian.
4.19
Sea-water
...... 1·0263 to 1·0295|| Barytic spar
4.43
Cork
•2400 Granite ....
2.65 to 2.75
Pine wood
•5500 Potassium
•865
Maple,
Mahogany
·820 to 1.063
Ebony
1.209
7550 Sodium
Aluminium
Antimony
•972
2.560
6.702
Box wood ......... 912 to 1.328
Zinc ..
.... 6·861 to 7.100
Oil of turpentine
•869
Tin ....
7.291
Camphor....
•989
Iron
7.788
Olive oil ...
915 Nickel
8.279
Linseed oil....
•940
Cobalt.....
8.538
Alcohol...
....
793 to 829
Copper
8.895
Nitric æther
•909
Bismuth
9.822
Sulphuric æther... 632 to 739
Silver
10.474 to 10.530
Tallow
•942
Lead
11.352
Bees'-wax
964
Mercury
13.568
Ivory
1.825
Gold....... ... 19-275 to 19-340
Gum Arabic
1.452
Platinum
20.980
Sulphur
1.991
Plate glass ......... 2:48 to 2.52
Phosphorus....
1.714
Crown glass
2.54
Pit coal
1.329
Flint glass
.2.80 to 3.72
CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS.
}
Cambridge, NOVEMBER, 1859.
WORKS
EDITED FOR THE SYNDICS
OF THE
Cambridge University Press.
SOLD BY
DEIGHTON, BELL & CO. CAMBRIDGE.
AND AT THE CAMBRIDGE WAREHOUSE,
32 PATERNOSTER ROW, LONDON.
Pearson's Exposition of the Creed, edited by TEMPLE
CHEVALLIER, B.D. Professor of Mathematics in the University
of Durham, and late Fellow and Tutor of St Catharine's College,
Cambridge. Second Edition. Demy 8vo. 108. 6d.
Select Discourses, by John
Smith, late Fellow of
Queens' College, Cambridge. Edited by H. G. WILLIAMS,
B.D. Professor of Arabic in the University. Royal 8vo. 108. 6d.
The Works of Isaac Barrow, compared with the
Original MSS. enlarged with Materials hitherto unpublished. A
new Edition, by A. NAPIER, M.A. of Trinity College, Vicar
of Holkham, Norfolk. 9 Vols. Demy Octavo. £4 148. 6d.
A Treatise of the Pope's Supremacy, and a Discourse
concerning the Unity of the Church, by ISAAC BARROW. One
Volume, 8vo.
I28.
Examination of Students who are not members of the
University. Examination Papers, Class Lists, &c. for December
1858, with the Regulations for the Examination in 1859. Demy
Octavo. 28. 6d. By Post 28. 8d.
The Class Lists for 1858 may be had separately.
Price 6d., or by Post for Seven Stamps.
First Annual Report of the Syndicate presented to the
Senate. Price 18. or by Post for Thirteen Stamps.
Wheatly on the Common Prayer, edited by G. E.
CORRIE, D.D. Master of Jesus College, Examining Chaplain to
the Lord Bishop of Ely. 128. 6d.
2
Works edited for the Syndics
Sancti Irenai Episcopi Lugdunensis Libros 'quinque
adversus Hæreses textu Græco in locis nonnullis locupletato,
versione Latina cum Codicibus Claromontano ac Arundeliano
denuo collata, præmissa de placitis Gnosticorum prolusione,
fragmenta necnon Græce, Syriace, Armeniace, commentatione
perpetua et indicibus variis edidit W. WIGAN HARVEY, S.T.B.
Collegii Regalis olim socius. 2 Vols. Demy Octavo. 368.
The Gospel according to Saint Matthew in Anglo-
Saxon and Northumbrian Versions, synoptically arranged: with
Collations of the best Manuscripts. By J. M. KEMBLE, M.A.
and Archdeacon HARDWICK, Christian Advocate.
108.
Cambridge Greek and English Testament, in Parallel
Columus on the same page. Edited by J. SCHOLEFIELD, M.A.
late Regius Professor of Greek in the University. A new
Edition printed on Demy 4to. Writing Paper, with large margin
for MS. notes.
128.
Cambridge Greek and English Testament, in Parallel
Columns on the same page. Edited by J. SCHOLEFIELD, M.A.
late Regius Professor of Greek in the University. Fourth
Edition. Small Octavo.
78. 6d.
Cambridge Greek Testament. Ex editione Stephani
tertia, 1550. Small Octavo. 38. 6d.
A Catalogue of the Manuscripts preserved in the
Library of the University of Cambridge. Vol. I.
Octavo. 308. Vol. II. 208. Vol. III. 258.
Vol. IV. In the Press.
Demy
Catalogus Bibliotheca Burckhardtianæ, cum Appen-
dice librorum aliorum Orientalium in Bibliotheca Academiæ
Cantabrigiensis asservatorum. Jussu Syndicorum Preli Aca-
demici confecit T. PRESTON, A.M. Collegii SS. Trinitatis
Socius. Demy 4to. 58.
M. T. Ciceronis de Officiis Libri tres, with Marginal
Analysis, an English Commentary, and copious Indices, by
H. A. HOLDEN, M.A. Head Master of Ipswich School,
late Fellow and Assistant Tutor of Trinity College, Cambridge.
Post vo.
98. 6d.
of the Cambridge University Press.
3
M. T. Ciceronis Oratio pro Tito Annio Milone,
with a Translation of Asconius' Introduction, Marginal Analysis,
and English Notes, by J. S. PURTON, M.A. President and Tutor
of St Catharine's College. Post Octavo. 38. 6d.
M. Minucii Felicis Octavius. The text newly revised
from the Original MS. with an English Commentary, Analysis,
Introduction, and Copious Indices. Edited by H. A. HOLDEN,
M.A. Head Master of Ipswich School, late Fellow and
Assistant Tutor of Trinity College, Cambridge. Crown Octavo.
98. 6d.
Cæsar Morgan's Investigation of the Trinity of Plato,
and of Philo Judæus, and of the effects which an attachment to
their writings had upon the principles and reasonings of the
Fathers of the Christian Church. A new Edition, revised by
H. A. HOLDEN, M.A. Head Master of Ipswich School, late
Fellow and Assistant Tutor of Trinity College, Cambridge.
Crown Octavo. 48.
Theophili Episcopi Antiochensis Libri Tres ad Auto-
De
lycum. Edidit, Prolegomenis Versione Notulis Indicibus
instruxit GULIELMUS GILSON HUMPHRY, S.T.B. Collegii
Sanctiss. Trin. apud Cantabrigienses quondam Socius.
Post Octavo. 68.
Obligatione Conscientiæ Prælectiones Decen
Oxonii in Schola Theologica habitæ a ROBERTO SANDERSON, SS.
Theologiæ ibidem Professore Regio. With English Notes,
including an abridged Translation, by W. WHEWELL, D.D.
Master of Trinity College. Octavo. 98.
Grotius de Jure Belli et Pacis, with the notes of
Barbeyrac and others; accompanied by an abridged Translation
of the Text, by W. WHEWELL, D.D. Master of Trinity College.
Three Volumes, 8vo, 428. The translation separate, 148.
The Homilies, with Various Readings, and the
Quotations from the Fathers given at length in the Original
Languages. Edited by G. E. CORRIE, D.D. Master of Jesus
College, Examining Chaplain to the Lord Bishop of Ely.
Octavo. 108. 6d.
Archbishop Usher's Answer to a Jesuit, with other
Tracts on Popery. Edited by J. SCHOLEFIELD, M.A. late
Regius Professor of Greek in the University. Octavo. 138. 6d.
Cambridge University Press Books.
Wilson's Illustration of the Method of explaining the
New Testament, by the early opinions of Jews and Christians
concerning Christ. Edited by T. TURTON, D.D. Lord Bishop
of Ely. Octavo. 88.
Lectures on Divinity delivered in the University of
Cambridge. By John Hey, D.D. Third Edition, by T.
TURTON, D.D. Lord Bishop of Ely. 2 vols. 8vo. 308.
Theophylacti in Evangelium S. Matthæi Commen-
tarius. Edited by W. G. HUMPHRY, B.D. Prebendary of St
Paul's, and Vicar of St Martin's-in-the-Fields, London, late
Fellow of Trinity College. Octavo.
148.
Tertullianus de Corona Militis, de Spectaculis, de
Idololatria, with Analysis and English Notes, by GEORGE
CURREY, B.D. Preacher at the Charter House, late Fellow and
Tutor of St John's College. Crown Octavo. 78. 6d.
Astronomical Observations, for the Years 1849, 1850,
and 1851, made at the Observatory of Cambridge, by the Rev.
JAMES CHALLIS, M.A. Plumian Professor of Astronomy and
Experimental Philosophy in the University of Cambridge, and
late Fellow of Trinity College. Royal 4to. Ready. 258.
Astronomical Observations, for the Years 1846, 1847,
and 1848, made at the Observatory of Cambridge, by the Rev.
JAMES CHALLIS, M.A. Plumian Professor of Astronomy and
Experimental Philosophy in the University of Cambridge, and
late Fellow of Trinity College. Royal 4to, 420 pages, 258.
The Publishers are directed to offer a limited number of Copies
of the Cambridge Observations of former years at the following
reduced prices :—
For the years
For the years
1828 and 29
............. at 48. each
1839...
at 11s. 6d.
*
1830, 81, 82, 83, 34, and 85 ... 5s.) year
1840 and 1841
178. 6d.
1836 ..........
8s. 6d.
1887
10s. 6d.
1842...
1843
178.
15s.
1888
*****. 148.
1844 and 1845 .............
88.
Cambridge: DEIGHTON, BELL AND CO.
London :
CAMBRIDGE WAREHOUSE, 32 PATERNOSTER ROW.
Mathematical Works
PUBLISHED BY
DEIGHTON, BELL AND CO.
(AGENTS TO THE UNIVERSITY.)
ARITHMETIC AND ALGEBRA.
Principles and Practice of Arithmetic. By the Rev. J. HIND. Eighth
Edition, with Questions, 48. 6d.
KEY, with Questions for Examination. Second Edition. 58.
Principles and Practice of Arithmetical Algebra, with Examples. By
the Rev. J. HIND. Third Edition. 12mo. 5s.
Elements of Algebra. By the Rev. J. HIND. Sixth Edition, revised.
540 pp. 8vo. 10s. 6d.
Treatise on Algebra. By the Rev. GEORGE PEACOCK, late Dean of Ely.
2 vols. 8vo. 17. 11s. 6d.
Treatise on the Theory of Algebraical Equations. By the Rev.
J. HYMERS, D.D. Third Edition. 8vo. 10s. 6d.
Collection of Examples in Arithmetic, Algebra, Geometry, &c. &c.
with Answers and Occasional Hints. By the Rev. A. WRIGLEY. Fifth Edition. 8vo. 88. 6d.
Elementary Course of Mathematics. By the Rev. HARVEY GOODWIN,
D.D. Dean of Ely. Fifth Edition. 8vo. 15s.
Collection of Problems and Examples, adapted to the Elementary
Course of Mathematics. By the Rev. HARVEY GOODWIN, D.D. Dean of Ely. Second Edi-
tion. 8vo. 68.
TRIGONOMETRY.
Elements of Plane and Spherical Trigonometry. By the Rev. J. HIND.
Fifth Edition. 12mo. 6s.
Solutions of the Trigonometrical Problems proposed at St John's Col-
lege, Cambridge, from 1829 to 1846. By the Rev. T. GASKIN. 8vo. 98.
Syllabus of a Course of Lectures upon Trigonometry and the Applica-
tion of Algebra to Geometry. 8vo. 78. 6d.
MECHANICS AND HYDROSTATICS.
Physical Optics, Part. II. The Corpuscular Theory of Light: Dis-
cussed Mathematically. By RICHARD POTTER, A.M. F.C.P.S. 8vo. 78. 6d.
Of Motion. An Elementary Treatise. By the Rev. J. R. LUNN, M. A.
[Nearly ready.
8vo.
The Propositions in Mechanics and Hydrostatics which are required
for those who are not Candidates for Honours. By A. C. BARRETT, M.A. Second Edition.
Crown 8vo. 6s.
Mechanical Euclid. Containing the Elements of Mechanics and Hy-
drostatics. By W. WHEWELL, D.D. Fifth Edition, 5s.
Elementary Statics. By C. J. ELLICOTT, B.A. 8vo. 4s. 6d.
Elementary Statics. By the Rev. HARVEY GOODWIN, D.D. Dean of
Ely. Crown 8vo. 68.
A Treatise on Statics. By the Rev. S. EARNSHAW, M.A. Fourth
Edition. 8vo. 10s.
Elementary Dynamics. By the Rev. H. GOODWIN, D.D. Dean of
Ely. Crown 8vo. 5s.
Dynamics, or, a Treatise on Motion. By the Rev. S. EARNSHAW.
Third Edition. 8vo. 148.
A Treatise on the Dynamics of a Rigid Body. By the Rev. W. N.
GRIFFIN. 8vo. 6s. 6d.
*** SOLUTIONS OF THE EXAMPLES. 8vo. 63.
Problems in illustration of the Principles of Theoretical Mechanics.
By W. WALTON, M.A. Second Edition. 8vo. 188.
Treatise on the Motion of a Single Particle and of two Particles
acting on one another. By A. SANDEMAN. 8vo. 8s. 6d.
Treatise on Hydrostatics and Hydrodynamics. By W. H. BESANT,
M.A., 8vo. 98.
[Just Published.
WORKS PUBLISHED BY DEIGHTON, BELL AND CO. CAMBRIDGE.
The Principles of Hydrostatics. By T. WEBSTER, M.A. 8vo. 78. 6d.
Problems in illustration of the Principles of Theoretical Hydrostatics
and Hydrodynamics. By W. WALTON, M.A. 8vo. 10s. 6d.
Collection of Elementary Problems in Statics and Dynamics. De-
signed for Candidates for Honours, first three days. By W. WALTON, M.A. 8vo. 108. 6d.
A Collection of Elementary Problems in Hydrostatics and Optics.
Designed for Candidates for Honours, first three days. By W. WALTON, M.A. [Preparing.
CONIC SECTIONS AND ANALYTICAL GEOMETRY.
Conic Sections. Their principal Properties proved Geometrically.
By W. WHEWELL, D.D. Master of Trinity. Third Edition. 8vo. 2s. 6d.
The Geometrical Construction of a Conic Section.
T. GASKIN. 8vo. 38.
By the Rev.
Treatise on Conic Sections. By the Rev. J. HYMERS, D.D. Third
Edition. 8vo. 98.
A Treatise on the Application of Analysis to Solid Geometry. By
D. F. GREGORY, M.A. and W. WALTON, M.A. Second Edition. 8vo. 12s.
The Elements of Conic Sections. By J. D. HUSTLER, B.D. Fourth
Edition. 8vo. 4s. 6d.
A Treatise on Plane Co-ordinate Geometry. By the Rev. M. O'BRIEN.
8vo. 98.
A Treatise on Analytical Geometry of Three Dimensions. By
J. HYMERS, D.D. Third Edition. 8vo. 10s. 6d.
Problems in illustration of the Principles of Plane Co-ordinate Geo-
metry. By W. WALTON, M.A. 8vo. 168.
Solutions of the Geometrical Problems proposed at St John's College,
from 1830 to 1846. By the Rev. T. GASKIN, M.A. 8vo. 128.
DIFFERENTIAL AND INTEGRAL CALCULUS.
An Elementary Treatise on the Differential Calculus. By W. H.
MILLER, M.A. Third Edition. 8vo. 6s.
Elementary Treatise on the Differential Calculus, in which the method
of Limits is exclusively made use of. By the Rev. M. O'BRIEN, M.A. 8vo. 108. 6d.
A Manual of the Differential Calculus. With Simple Examples.
By HOMERSHAM COX, B.A. 4s.
Treatise on the Differential Calculus. By W. WALTON, M.A. 8vo.
10s. 6d.
A Treatise on the Integral Calculus. By the Rev. J. HYMERS, D.D.
8vo. 108. 6d.
Geometrical Illustrations of the Differential Calculus. By M. B. PELL.
8vo. 2s. 6d.
Examples of the Principles of the Differential and Integral Calculus.
Collected by D. F. GREGORY. Second Edition. Edited by W. WALTON, M.A. 8vo. 18s.
ASTRONOMY.
Elementary Chapters on Astronomy, from the "Astronomie Phy-
sique " of BIOT. By the Rev. HARVEY GOODWIN, D.D. Dean of Ely. 8vo. 3s. 6d.
Elements of the Theory of Astronomy. By the Rev. J. HYMERS, D.D.
8vo. 148.
Newton's Principia. First Three Sections, with Appendix, and the
Ninth and Eleventh Sections. By the Rev. J. H. EVANS, M.A. Fourth Edition. 8vo. 6s.
A Treatise on Heat. By PHILIP KELLAND, M.A. 8vo. 98.
Series of Figures Illustrative of Geometrical Optics. From SCHELL-
BACH. By the Rev. W. B. HOPKINS. Plates. Folio. 108. 6d.
A Treatise on Crystallography. By W. H. MILLER, M.A. 8vo. 7s. 6d.
Mathematical Tracts. By the Rev. M. O'BRIEN, M.A. On Laplace's
Coefficients; the Figure of the Earth; the Motion of a Rigid Body about its Centre of Gravity,
Precession and Nutation. 8vo. 48. 6d.

UNIVERSITY OF MICHIGAN
3 9015 06536 9228
QA
905
.P87
Potter
An elementary...
treatise on hydro-
417635
statics