Chapter I: Front Matter (1)

Transcriber's notes:

(1) Numbers following letters (without space) like C2 were originally
printed in subscript. Letter subscripts are preceded by an
underscore, like C_n.

(2) Characters following a carat (^) were printed in superscript.

(3) Side-notes were relocated to function as titles of their respective
paragraphs.

(4) Macrons and breves above letters and dots below letters were not
inserted.

(5) [root] stands for the root symbol; [alpha], [beta], etc. for greek
letters.

(6) The following typographical errors have been corrected:

ARTICLE HYDROMECHANICS: "... and [omega] the angular velocity
about it generated by an impulse couple M, and M' is the couple
required to set the surrounding medium in motion ..." 'impulse'
amended from 'impluse'.

ARTICLE HYMENOPTERA: "... see P. Cameron's British Phytophagous
Hymenoptera (4 vols., London, Roy. Soc., 1882-1893)." 'Roy'
amended from 'Ray'.

ARTICLE HYRCANUS: "During its later years his reign was much
disturbed, however, by the contentions for ascendancy which arose
between the Pharisees and Sadducees, the two rival sects or
parties which then for the first time (under those names at least)
came into prominence." 'disturbed' amended from 'distrubed'.

ARTICLE ICELAND: "Iceland is emphatically a land of proverbs,
while of folk-tales, those other keys to the people's heart, there
is plentiful store." 'people's' amended from 'poeple's'.

ENCYCLOPAEDIA BRITANNICA

A DICTIONARY OF ARTS, SCIENCES, LITERATURE
AND GENERAL INFORMATION

ELEVENTH EDITION

VOLUME XIV, SLICE II

Hydromechanics to Ichnography

ARTICLES IN THIS SLICE:

HYDROMECHANICS HYTHE
HYDROMEDUSAE I
HYDROMETER IAMBIC
HYDROPATHY IAMBLICHUS (Greek philosopher)
HYDROPHOBIA IAMBLICHUS (Greek romance writer)
HYDROSPHERE IANNINA
HYDROSTATICS IAPETUS
HYDROXYLAMINE IAPYDES
HYDROZOA IATROCHEMISTRY
HYENA IAZYGES
HYÈRES IBADAN
HYGIEIA IBAGUÉ
HYGIENE IBARRA
HYGINUS (eighth pope) IBERIANS
HYGINUS (Latin writer) IBEX
HYGINUS, GAIUS JULIUS IBIS
HYGROMETER IBLIS
HYKSOS IBN 'ABD RABBIHI
HYLAS IBN 'ARABI
HYLOZOISM IBN ATHIR
HYMEN IBN BATUTA
HYMENOPTERA IBN DURAID
HYMETTUS IBN FARADI
HYMNS IBN FARID
HYPAETHROS IBN GABIROL
HYPALLAGE IBN HAUKAL
HYPATIA IBN HAZM
HYPERBATON IBN HISHAM
HYPERBOLA IBN ISHAQ
HYPERBOLE IBN JUBAIR
HYPERBOREANS IBN KHALDUN
HYPEREIDES IBN KHALLIKAN
HYPERION IBN QUTAIBA
HYPERSTHENE IBN SA'D
HYPERTROPHY IBN TIBBON
HYPNOTISM IBN TUFAIL
HYPOCAUST IBN USAIBI'A
HYPOCHONDRIASIS IBO
HYPOCRISY IBRAHIM AL-MAUSILI
HYPOSTASIS IBRAHIM PASHA
HYPOSTYLE IBSEN, HENRIK
HYPOSULPHITE OF SODA IBYCUS
HYPOTHEC ICA
HYPOTHESIS ICE
HYPOTRACHELIUM ICEBERG
HYPSOMETER ICELAND
HYRACOIDEA ICELAND MOSS
HYRCANIA ICE-PLANT
HYRCANUS ICE-YACHTING
HYSSOP I-CH'ANG
HYSTASPES ICHNEUMON
HYSTERESIS ICHNEUMON-FLY
HYSTERIA ICHNOGRAPHY
HYSTERON-PROTERON

HYDROMECHANICS ([Greek: hydromêchanika]), the science of the mechanics of water and fluids in general, including _hydrostatics_ or the mathematical theory of fluids in equilibrium, and _hydromechanics_, the theory of fluids in motion. The practical application of hydromechanics forms the province of hydraulics (q.v.).

_Historical._--The fundamental principles of hydrostatics were first
given by Archimedes in his work [Greek: Peri tôn ochoumenôn], or _De
iis quae vehuntur in humido_, about 250 B.C., and were afterwards
applied to experiments by Marino Ghetaldi (1566-1627) in his _Promotus
Archimedes_ (1603). Archimedes maintained that each particle of a
fluid mass, when in equilibrium, is equally pressed in every
direction; and he inquired into the conditions according to which a
solid body floating in a fluid should assume and preserve a position
of equilibrium.

In the Greek school at Alexandria, which flourished under the auspices
of the Ptolemies, the first attempts were made at the construction of
hydraulic machinery, and about 120 B.C. the fountain of compression,
the siphon, and the forcing-pump were invented by Ctesibius and Hero.
The siphon is a simple instrument; but the forcing-pump is a
complicated invention, which could scarcely have been expected in the
infancy of hydraulics. It was probably suggested to Ctesibius by the
_Egyptian Wheel_ or _Noria_, which was common at that time, and which
was a kind of chain pump, consisting of a number of earthen pots
carried round by a wheel. In some of these machines the pots have a
valve in the bottom which enables them to descend without much
resistance, and diminishes greatly the load upon the wheel; and, if we
suppose that this valve was introduced so early as the time of
Ctesibius, it is not difficult to perceive how such a machine might
have led to the invention of the forcing-pump.

Notwithstanding these inventions of the Alexandrian school, its
attention does not seem to have been directed to the motion of fluids;
and the first attempt to investigate this subject was made by Sextus
Julius Frontinus, inspector of the public fountains at Rome in the
reigns of Nerva and Trajan. In his work _De aquaeductibus urbis Romae
commentarius_, he considers the methods which were at that time
employed for ascertaining the quantity of water discharged from
ajutages, and the mode of distributing the waters of an aqueduct or a
fountain. He remarked that the flow of water from an orifice depends
not only on the magnitude of the orifice itself, but also on the
height of the water in the reservoir; and that a pipe employed to
carry off a portion of water from an aqueduct should, as circumstances
required, have a position more or less inclined to the original
direction of the current. But as he was unacquainted with the law of
the velocities of running water as depending upon the depth of the
orifice, the want of precision which appears in his results is not
surprising.

Benedetto Castelli (1577-1644), and Evangelista Torricelli
(1608-1647), two of the disciples of Galileo, applied the discoveries
of their master to the science of hydrodynamics. In 1628 Castelli
published a small work, _Della misura dell' acque correnti_, in which
he satisfactorily explained several phenomena in the motion of fluids
in rivers and canals; but he committed a great paralogism in supposing
the velocity of the water proportional to the depth of the orifice
below the surface of the vessel. Torricelli, observing that in a jet
where the water rushed through a small ajutage it rose to nearly the
same height with the reservoir from which it was supplied, imagined
that it ought to move with the same velocity as if it had fallen
through that height by the force of gravity, and hence he deduced the
proposition that the velocities of liquids are as the square root of
the head, apart from the resistance of the air and the friction of the
orifice. This theorem was published in 1643, at the end of his
treatise _De motu gravium projectorum_, and it was confirmed by the
experiments of Raffaello Magiotti on the quantities of water
discharged from different ajutages under different pressures (1648).

In the hands of Blaise Pascal (1623-1662) hydrostatics assumed the
dignity of a science, and in a treatise on the equilibrium of liquids
(_Sur l'équilibre des liqueurs_), found among his manuscripts after
his death and published in 1663, the laws of the equilibrium of
liquids were demonstrated in the most simple manner, and amply
confirmed by experiments.

The theorem of Torricelli was employed by many succeeding writers, but
particularly by Edmé Mariotte (1620-1684), whose _Traité du mouvement
des eaux_, published after his death in the year 1686, is founded on a
great variety of well-conducted experiments on the motion of fluids,
performed at Versailles and Chantilly. In the discussion of some
points he committed considerable mistakes. Others he treated very
superficially, and in none of his experiments apparently did he attend
to the diminution of efflux arising from the contraction of the liquid
vein, when the orifice is merely a perforation in a thin plate; but he
appears to have been the first who attempted to ascribe the
discrepancy between theory and experiment to the retardation of the
water's velocity through friction. His contemporary Domenico
Guglielmini (1655-1710), who was inspector of the rivers and canals at
Bologna, had ascribed this diminution of velocity in rivers to
transverse motions arising from inequalities in their bottom. But as
Mariotte observed similar obstructions even in glass pipes where no
transverse currents could exist, the cause assigned by Guglielmini
seemed destitute of foundation. The French philosopher, therefore,
regarded these obstructions as the effects of friction. He supposed
that the filaments of water which graze along the sides of the pipe
lose a portion of their velocity; that the contiguous filaments,
having on this account a greater velocity, rub upon the former, and
suffer a diminution of their celerity; and that the other filaments
are affected with similar retardations proportional to their distance
from the axis of the pipe. In this way the medium velocity of the
current may be diminished, and consequently the quantity of water
discharged in a given time must, from the effects of friction, be
considerably less than that which is computed from theory.

The effects of friction and viscosity in diminishing the velocity of
running water were noticed in the _Principia_ of Sir Isaac Newton, who
threw much light upon several branches of hydromechanics. At a time
when the Cartesian system of vortices universally prevailed, he found
it necessary to investigate that hypothesis, and in the course of his
investigations he showed that the velocity of any stratum of the
vortex is an arithmetical mean between the velocities of the strata
which enclose it; and from this it evidently follows that the velocity
of a filament of water moving in a pipe is an arithmetical mean
between the velocities of the filaments which surround it. Taking
advantage of these results, Henri Pitot (1695-1771) afterwards showed
that the retardations arising from friction are inversely as the
diameters of the pipes in which the fluid moves. The attention of
Newton was also directed to the discharge of water from orifices in
the bottom of vessels. He supposed a cylindrical vessel full of water
to be perforated in its bottom with a small hole by which the water
escaped, and the vessel to be supplied with water in such a manner
that it always remained full at the same height. He then supposed this
cylindrical column of water to be divided into two parts,--the first,
which he called the "cataract," being an hyperboloid generated by the
revolution of an hyperbola of the fifth degree around the axis of the
cylinder which should pass through the orifice, and the second the
remainder of the water in the cylindrical vessel. He considered the
horizontal strata of this hyperboloid as always in motion, while the
remainder of the water was in a state of rest, and imagined that there
was a kind of cataract in the middle of the fluid. When the results of
this theory were compared with the quantity of water actually
discharged, Newton concluded that the velocity with which the water
issued from the orifice was equal to that which a falling body would
receive by descending through half the height of water in the
reservoir. This conclusion, however, is absolutely irreconcilable with
the known fact that jets of water rise nearly to the same height as
their reservoirs, and Newton seems to have been aware of this
objection. Accordingly, in the second edition of his _Principia_,
which appeared in 1713, he reconsidered his theory. He had discovered
a contraction in the vein of fluid (_vena contracta_) which issued
from the orifice, and found that, at the distance of about a diameter
of the aperture, the section of the vein was contracted in the
subduplicate ratio of two to one. He regarded, therefore, the section
of the contracted vein as the true orifice from which the discharge of
water ought to be deduced, and the velocity of the effluent water as
due to the whole height of water in the reservoir; and by this means
his theory became more conformable to the results of experience,
though still open to serious objections. Newton was also the first to
investigate the difficult subject of the motion of waves (q.v.).

In 1738 Daniel Bernoulli (1700-1782) published his _Hydrodynamica seu
de viribus et motibus fluidorum commentarii_. His theory of the motion
of fluids, the germ of which was first published in his memoir
entitled _Theoria nova de motu aquarum per canales quocunque
fluentes_, communicated to the Academy of St Petersburg as early as
1726, was founded on two suppositions, which appeared to him
conformable to experience. He supposed that the surface of the fluid,
contained in a vessel which is emptying itself by an orifice, remains
always horizontal; and, if the fluid mass is conceived to be divided
into an infinite number of horizontal strata of the same bulk, that
these strata remain contiguous to each other, and that all their
points descend vertically, with velocities inversely proportional to
their breadth, or to the horizontal sections of the reservoir. In
order to determine the motion of each stratum, he employed the
principle of the _conservatio virium vivarum_, and obtained very
elegant solutions. But in the absence of a general demonstration of
that principle, his results did not command the confidence which they
would otherwise have deserved, and it became desirable to have a
theory more certain, and depending solely on the fundamental laws of
mechanics. Colin Maclaurin (1698-1746) and John Bernoulli (1667-1748),
who were of this opinion, resolved the problem by more direct methods,
the one in his _Fluxions_, published in 1742, and the other in his
_Hydraulica nunc primum detecta, et demonstrata directe ex fundamentis
pure mechanicis_, which forms the fourth volume of his works. The
method employed by Maclaurin has been thought not sufficiently
rigorous; and that of John Bernoulli is, in the opinion of Lagrange,
defective in clearness and precision. The theory of Daniel Bernoulli
was opposed also by Jean le Rond d'Alembert. When generalizing the
theory of pendulums of Jacob Bernoulli (1654-1705) he discovered a
principle of dynamics so simple and general that it reduced the laws
of the motions of bodies to that of their equilibrium. He applied this
principle to the motion of fluids, and gave a specimen of its
application at the end of his _Dynamics_ in 1743. It was more fully
developed in his _Traité des fluides_, published in 1744, in which he
gave simple and elegant solutions of problems relating to the
equilibrium and motion of fluids. He made use of the same suppositions
as Daniel Bernoulli, though his calculus was established in a very
different manner. He considered, at every instant, the actual motion
of a stratum as composed of a motion which it had in the preceding
instant and of a motion which it had lost; and the laws of equilibrium
between the motions lost furnished him with equations representing the
motion of the fluid. It remained a desideratum to express by equations
the motion of a particle of the fluid in any assigned direction. These
equations were found by d'Alembert from two principles--that a
rectangular canal, taken in a mass of fluid in equilibrium, is itself
in equilibrium, and that a portion of the fluid, in passing from one
place to another, preserves the same volume when the fluid is
incompressible, or dilates itself according to a given law when the
fluid is elastic. His ingenious method, published in 1752, in his
_Essai sur la résistance des fluides_, was brought to perfection in
his _Opuscules mathématiques_, and was adopted by Leonhard Euler.

The resolution of the questions concerning the motion of fluids was
effected by means of Euler's partial differential coefficients. This
calculus was first applied to the motion of water by d'Alembert, and
enabled both him and Euler to represent the theory of fluids in
formulae restricted by no particular hypothesis.

One of the most successful labourers in the science of hydrodynamics
at this period was Pierre Louis Georges Dubuat (1734-1809). Following
in the steps of the Abbé Charles Bossut (_Nouvelles Experiences sur la
résistance des fluides_, 1777), he published, in 1786, a revised
edition of his _Principes d'hydraulique_, which contains a
satisfactory theory of the motion of fluids, founded solely upon
experiments. Dubuat considered that if water were a perfect fluid, and
the channels in which it flowed infinitely smooth, its motion would be
continually accelerated, like that of bodies descending in an inclined
plane. But as the motion of rivers is not continually accelerated, and
soon arrives at a state of uniformity, it is evident that the
viscosity of the water, and the friction of the channel in which it
descends, must equal the accelerating force. Dubuat, therefore,
assumed it as a proposition of fundamental importance that, when water
flows in any channel or bed, the accelerating force which obliges it
to move is equal to the sum of all the resistances which it meets
with, whether they arise from its own viscosity or from the friction
of its bed. This principle was employed by him in the first edition of
his work, which appeared in 1779. The theory contained in that edition
was founded on the experiments of others, but he soon saw that a
theory so new, and leading to results so different from the ordinary
theory, should be founded on new experiments more direct than the
former, and he was employed in the performance of these from 1780 to
1783. The experiments of Bossut were made only on pipes of a moderate
declivity, but Dubuat used declivities of every kind, and made his
experiments upon channels of various sizes.

The theory of running water was greatly advanced by the researches of
Gaspard Riche de Prony (1755-1839). From a collection of the best
experiments by previous workers he selected eighty-two (fifty-one on
the velocity of water in conduit pipes, and thirty-one on its velocity
in open canals); and, discussing these on physical and mechanical
principles, he succeeded in drawing up general formulae, which
afforded a simple expression for the velocity of running water.

J. A. Eytelwein (1764-1848) of Berlin, who published in 1801 a
valuable compendium of hydraulics entitled _Handbuch der Mechanik und
der Hydraulik_, investigated the subject of the discharge of water by
compound pipes, the motions of jets and their impulses against plane
and oblique surfaces; and he showed theoretically that a water-wheel
will have its maximum effect when its circumference moves with half
the velocity of the stream.

J. N. P. Hachette (1769-1834) in 1816-1817 published memoirs
containing the results of experiments on the spouting of fluids and
the discharge of vessels. His object was to measure the contracted
part of a fluid vein, to examine the phenomena attendant on additional
tubes, and to investigate the form of the fluid vein and the results
obtained when different forms of orifices are employed. Extensive
experiments on the discharge of water from orifices (_Expériences
hydrauliques_, Paris, 1832) were conducted under the direction of the
French government by J. V. Poncelet (1788-1867) and J. A. Lesbros
(1790-1860). P. P. Boileau (1811-1891) discussed their results and
added experiments of his own (_Traité de la mésure des eaux
courantes_, Paris, 1854). K. R. Bornemann re-examined all these
results with great care, and gave formulae expressing the variation of
the coefficients of discharge in different conditions (_Civil
Ingénieur_, 1880). Julius Weisbach (1806-1871) also made many
experimental investigations on the discharge of fluids. The
experiments of J. B. Francis (_Lowell Hydraulic Experiments_, Boston,
Mass., 1855) led him to propose variations in the accepted formulae
for the discharge over weirs, and a generation later a very complete
investigation of this subject was carried out by H. Bazin. An
elaborate inquiry on the flow of water in pipes and channels was
conducted by H. G. P. Darcy (1803-1858) and continued by H. Bazin, at
the expense of the French government (_Recherches hydrauliques_,
Paris, 1866). German engineers have also devoted special attention to
the measurement of the flow in rivers; the _Beiträge zur Hydrographie
des Königreiches Böhmen_ (Prague, 1872-1875) of A. R. Harlacher
(1842-1890) contained valuable measurements of this kind, together
with a comparison of the experimental results with the formulae of
flow that had been proposed up to the date of its publication, and
important data were yielded by the gaugings of the Mississippi made
for the United States government by A. A. Humphreys and H. L. Abbot,
by Robert Gordon's gaugings of the Irrawaddy, and by Allen J. C.
Cunningham's experiments on the Ganges canal. The friction of water,
investigated for slow speeds by Coulomb, was measured for higher
speeds by William Froude (1810-1879), whose work is of great value in
the theory of ship resistance (_Brit. Assoc. Report._, 1869), and
stream line motion was studied by Professor Osborne Reynolds and by
Professor H. S. Hele Shaw. (X.)

HYDROSTATICS

Hydrostatics is a science which grew originally out of a number of isolated practical problems; but it satisfies the requirement of perfect accuracy in its application to phenomena, the largest and smallest, of the behaviour of a fluid. At the same time, it delights the pure theorist by the simplicity of the logic with which the fundamental theorems may be established, and by the elegance of its mathematical operations, insomuch that hydrostatics may be considered as the Euclidean pure geometry of mechanical science.

1. _The Different States of a Substance or Matter._--All substance in nature falls into one of the two classes, solid and fluid; a solid substance, the land, for instance, as contrasted with a fluid, like water, being a substance which does not flow of itself.

A _fluid_, as the name implies, is a substance which flows, or is capable of flowing; water and air are the two fluids distributed most universally over the surface of the earth.

Fluids again are divided into two classes, termed a liquid and a gas, of which water and air are the chief examples.

A _liquid_ is a fluid which is incompressible or practically so, i.e. it does not change in volume sensibly with change of pressure.

A _gas_ is a compressible fluid, and the change in volume is considerable with moderate variation of pressure.

Liquids, again, can be poured from one open vessel into another, and can be kept in an uncovered vessel, but a gas tends to diffuse itself indefinitely and must be preserved in a closed reservoir.

The distinguishing characteristics of the three kinds of substance or states of matter, the solid, liquid and gas, are summarized thus in O. Lodge's _Mechanics_:--

A solid has both size and shape.
A liquid has size but not shape.
A gas has neither size nor shape.

2. _The Change of State of Matter._--By a change of temperature and pressure combined, a substance can in general be made to pass from one state into another; thus by gradually increasing the temperature a solid piece of ice can be melted into the liquid state of water, and the water again can be boiled off into the gaseous state as steam. Again, by raising the temperature, a metal in the solid state can be melted and liquefied, and poured into a mould to assume any form desired, which is retained when the metal cools and solidifies again; the gaseous state of a metal is revealed by the spectroscope. Conversely, a combination of increased pressure and lowering of temperature will, if carried far enough, reduce a gas to a liquid, and afterwards to the solid state; and nearly every gaseous substance has now undergone this operation.

A certain critical temperature is observed in a gas, above which the liquefaction is impossible; so that the gaseous state has two subdivisions into (i.) a true gas, which cannot be liquefied, because its temperature is above the critical temperature, (ii.) a vapour, where the temperature is below the critical, and which can ultimately be liquefied by further lowering of temperature or increase of pressure.

3. _Plasticity and Viscosity._--Every solid substance is found to be plastic more or less, as exemplified by punching, shearing and cutting; but the plastic solid is distinguished from the viscous fluid in that a plastic solid requires a certain magnitude of stress to be exceeded to make it flow, whereas the viscous liquid will yield to the slightest stress, but requires a certain length of time for the effect to be appreciable.

According to Maxwell (_Theory of Heat_) "When a continuous alteration of form is produced only by a stress exceeding a certain value, the substance is called a solid, however soft and plastic it may be. But when the smallest stress, if only continued long enough, will cause a perceptible and increasing change of form, the substance must be regarded as a viscous fluid, however hard it may be." Maxwell illustrates the difference between a soft solid and a hard liquid by a jelly and a block of pitch; also by the experiment of supporting a candle and a stick of sealing-wax; after a considerable time the sealing-wax will be found bent and so is a fluid, but the candle remains straight as a solid.

4. _Definition of a Fluid._--A fluid is a substance which yields continually to the slightest tangential stress in its interior; that is, it can be divided very easily along any plane (given plenty of time if the fluid is viscous). It follows that when the fluid has come to rest, the tangential stress in any plane in its interior must vanish, and the stress must be entirely normal to the plane. This mechanical axiom of the _normality of fluid pressure_ is the foundation of the mathematical theory of hydrostatics.

The theorems of hydrostatics are thus true for all stationary fluids, however viscous they may be; it is only when we come to hydrodynamics, the science of the motion of a fluid, that viscosity will make itself felt and modify the theory; unless we begin by postulating the perfect fluid, devoid of viscosity, so that the principle of the _normality of fluid pressure_ is taken to hold when the fluid is in movement.

5. _The Measurement of Fluid Pressure._--The pressure at any point of
a plane in the interior of a fluid is the intensity of the normal
thrust estimated per unit area of the plane.

Thus, if a thrust of P lb. is distributed uniformly over a plane area
of A sq. ft., as on the horizontal bottom of the sea or any reservoir,
the pressure at any point of the plane is P/A lb. per sq. ft., or
P/144A lb. per sq. in. (lb./ft.² and lb./in.², in the Hospitalier
notation, to be employed in the sequel). If the distribution of the
thrust is not uniform, as, for instance, on a vertical or inclined
face or wall of a reservoir, then P/A represents the average pressure
over the area; and the actual pressure at any point is the average
pressure over a small area enclosing the point. Thus, if a thrust
[Delta]P lb. acts on a small plane area [Delta]A ft.² enclosing a
point B, the pressure p at B is the limit of [Delta]P/[Delta]A; and

p = lt([Delta]P/[Delta]A) = dP/dA, (1)

in the notation of the differential calculus.

6. _The Equality of Fluid Pressure in all Directions._--This
fundamental principle of hydrostatics follows at once from the
principle of the normality of fluid pressure implied in the definition
of a fluid in § 4. Take any two arbitrary directions in the plane of
the paper, and draw a small isosceles triangle abc, whose sides are
perpendicular to the two directions, and consider the equilibrium of a
small triangular prism of fluid, of which the triangle is the cross
section. Let P, Q denote the normal thrust across the sides bc, ca,
and R the normal thrust across the base ab. Then, since these three
forces maintain equilibrium, and R makes equal angles with P and Q,
therefore P and Q must be equal. But the faces bc, ca, over which P
and Q act, are also equal, so that the pressure on each face is equal.
A scalene triangle abc might also be employed, or a tetrahedron.

It follows that the pressure of a fluid requires to be calculated in
one direction only, chosen as the simplest direction for convenience.

7. _The Transmissibility of Fluid Pressure._--Any additional pressure
applied to the fluid will be transmitted equally to every point in the
case of a liquid; this principle of the _transmissibility of pressure_
was enunciated by Pascal, 1653, and applied by him to the invention of
the _hydraulic press_.

This machine consists essentially of two communicating cylinders (fig.
1a), filled with liquid and closed by pistons. If a thrust P lb. is
applied to one piston of area A ft.², it will be balanced by a thrust
W lb. applied to the other piston of area B ft.², where

p = P/A = W/B, (1)

the pressure p of the liquid being supposed uniform; and, by making
the ratio B/A sufficiently large, the mechanical advantage can be
increased to any desired amount, and in the simplest manner possible,
without the intervention of levers and machinery.

Fig. 1b shows also a modern form of the hydraulic press, applied to
the operation of covering an electric cable with a lead coating.

8. _Theorem._--In a fluid at rest under gravity the pressure is the
same at any two points in the same horizontal plane; in other words, a
surface of equal pressure is a horizontal plane.

This is proved by taking any two points A and B at the same level, and
considering the equilibrium of a thin prism of liquid AB, bounded by
planes at A and B perpendicular to AB. As gravity and the fluid
pressure on the sides of the prism act at right angles to AB, the
equilibrium requires the equality of thrust on the ends A and B; and
as the areas are equal, the pressure must be equal at A and B; and so
the pressure is the same at all points in the same horizontal plane.
If the fluid is a liquid, it can have a free surface without diffusing
itself, as a gas would; and this free surface, being a surface of zero
pressure, or more generally of uniform atmospheric pressure, will also
be a surface of equal pressure, and therefore a horizontal plane.

Hence the _theorem_.--The free surface of a liquid at rest under
gravity is a horizontal plane. This is the characteristic
distinguishing between a solid and a liquid; as, for instance, between
land and water. The land has hills and valleys, but the surface of
water at rest is a horizontal plane; and if disturbed the surface
moves in waves.

9. _Theorem._--In a homogeneous liquid at rest under gravity the
pressure increases uniformly with the depth.

This is proved by taking the two points A and B in the same vertical
line, and considering the equilibrium of the prism by resolving
vertically. In this case the thrust at the lower end B must exceed the
thrust at A, the upper end, by the weight of the prism of liquid; so
that, denoting the cross section of the prism by [alpha] ft.², the
pressure at A and By by p0 and p lb./ft.², and by w the density of the
liquid estimated in lb./ft.³,

p[alpha] - p0[alpha] = w[alpha]·AB, (1)

p = w·AB + p0. (2)

Thus in water, where w = 62.4lb./ft.³, the pressure increases 62.4
lb./ft.², or 62.4 ÷ 144 = 0.433 lb./in.² for every additional foot of
depth.

10. _Theorem._--If two liquids of different density are resting in
vessels in communication, the height of the free surface of such
liquid above the surface of separation is inversely as the density.

For if the liquid of density [sigma] rises to the height h and of
density [rho] to the height k, and p0 denotes the atmospheric
pressure, the pressure in the liquid at the level of the surface of
separation will be [sigma]h + p0 and [rho]k + p0, and these being
equal we have

[sigma]h = [rho]k. (1)

The principle is illustrated in the article BAROMETER, where a column
of mercury of density [sigma] and height h, rising in the tube to the
Torricellian vacuum, is balanced by a column of air of density [rho],
which may be supposed to rise as a homogeneous fluid to a height k,
called the height of the homogeneous atmosphere. Thus water being
about 800 times denser than air and mercury 13.6 times denser than
water,

k/h = [sigma]/[rho] = 800 × 13.6 = 10,880; (2)

and with an average barometer height of 30 in. this makes k 27,200
ft., about 8300 metres.

11. _The Head of Water or a Liquid._--The pressure [sigma]h at a depth
h ft. in liquid of density [sigma] is called the pressure due to a
_head_ of h ft. of the liquid. The atmospheric pressure is thus due to
an average head of 30 in. of mercury, or 30 × 13.6 ÷ 12 = 34 ft. of
water, or 27,200 ft. of air. The pressure of the air is a convenient
unit to employ in practical work, where it is called an "atmosphere";
it is made the equivalent of a pressure of one kg/cm²; and one
ton/inch², employed as the unit with high pressure as in artillery,
may be taken as 150 atmospheres.

12. _Theorem._--A body immersed in a fluid is buoyed up by a force
equal to the weight of the liquid displaced, acting vertically upward
through the centre of gravity of the displaced liquid.

For if the body is removed, and replaced by the fluid as at first,
this fluid is in equilibrium under its own weight and the thrust of
the surrounding fluid, which must be equal and opposite, and the
surrounding fluid acts in the same manner when the body replaces the
displaced fluid again; so that the resultant thrust of the fluid acts
vertically upward through the centre of gravity of the fluid
displaced, and is equal to the weight.

When the body is floating freely like a ship, the equilibrium of this
liquid thrust with the weight of the ship requires that the weight of
water displaced is equal to the weight of the ship and the two centres
of gravity are in the same vertical line. So also a balloon begins to
rise when the weight of air displaced is greater than the weight of
the balloon, and it is in equilibrium when the weights are equal. This
theorem is called generally the _principle of Archimedes_.

It is used to determine the density of a body experimentally; for if W
is the weight of a body weighed in a balance in air (strictly _in
vacuo_), and if W´ is the weight required to balance when the body is
suspended in water, then the upward thrust of the liquid or weight of
liquid displaced is W - W´, so that the _specific gravity_ (S.G.),
defined as the ratio of the weight of a body to the weight of an equal
volume of water, is W/(W - W´).

As stated first by Archimedes, the principle asserts the obvious fact
that a body displaces its own volume of water; and he utilized it in
the problem of the determination of the adulteration of the crown of
Hiero. He weighed out a lump of gold and of silver of the same weight
as the crown; and, immersing the three in succession in water, he
found they spilt over measures of water in the ratio 1/14 : 4/77 :
2/21 or 33 : 24 : 44; thence it follows that the gold : silver alloy
of the crown was as 11 : 9 by weight.

13. _Theorem._--The resultant vertical thrust on any portion of a
curved surface exposed to the pressure of a fluid at rest under
gravity is the weight of fluid cut out by vertical lines drawn round
the boundary of the curved surface.

_Theorem._--The resultant horizontal thrust in any direction is
obtained by drawing parallel horizontal lines round the boundary, and
intersecting a plane perpendicular to their direction in a plane
curve; and then investigating the thrust on this plane area, which
will be the same as on the curved surface.

The proof of these theorems proceeds as before, employing the
normality principle; they are required, for instance, in the
determination of the liquid thrust on any portion of the bottom of a
ship.

In casting a thin hollow object like a bell, it will be seen that the
resultant upward thrust on the mould may be many times greater than
the weight of metal; many a curious experiment has been devised to
illustrate this property and classed as a hydrostatic paradox (Boyle,
_Hydrostatical Paradoxes_, 1666).

Consider, for instance, the operation of casting a hemispherical bell,
in fig. 2. As the molten metal is run in, the upward thrust on the
outside mould, when the level has reached PP´, is the weight of metal
in the volume generated by the revolution of APQ; and this, by a
theorem of Archimedes, has the same volume as the cone ORR´, or 1/3
[pi]y³, where y is the depth of metal, the horizontal sections being
equal so long as y is less than the radius of the outside hemisphere.
Afterwards, when the metal has risen above B, to the level KK´, the
additional thrust is the weight of the cylinder of diameter KK´ and
height BH. The upward thrust is the same, however thin the metal may
be in the interspace between the outer mould and the core inside; and
this was formerly considered paradoxical.

_Analytical Equations of Equilibrium of a Fluid at rest under any
System of Force._

14. Referred to three fixed coordinate axes, a fluid, in which the
pressure is p, the density [rho], and X, Y, Z the components of
impressed force per unit mass, requires for the equilibrium of the
part filling a fixed surface S, on resolving parallel to Ox,
_ _ _ _ _
/ / / / /
| | lp dS = | | | [rho]X dx dy dz, (1)
_/ _/ _/ _/ _/

where l, m, n denote the direction cosines of the normal drawn outward
of the surface S.

But by Green's transformation
_ _ _ _ _
/ / / / / dp
| | lp dS = | | | -- dx dy dz, (2)
_/ _/ _/ _/ _/ dx

thus leading to the differential relation at every point

dp dp dp
-- = [rho]X, -- = [rho]Y, -- = [rho]Z. (3)
dx dy dz

The three equations of equilibrium obtained by taking moments round
the axes are then found to be satisfied identically.

Hence the space variation of the pressure in any direction, or the
_pressure-gradient_, is the resolved force per unit volume in that
direction. The resultant force is therefore in the direction of the
steepest pressure-gradient, and this is normal to the surface of equal
pressure; for equilibrium to exist in a fluid the lines of force must
therefore be capable of being cut orthogonally by a system of
surfaces, which will be surfaces of equal pressure.

Ignoring temperature effect, and taking the density as a function of
the pressure, surfaces of equal pressure are also of equal density,
and the fluid is stratified by surfaces orthogonal to the lines of
force;

1 dp 1 dp 1 dp
----- --, ----- --, ----- --, or X, Y, Z (4)
[rho] dx [rho] dy [rho] dz

are the partial differential coefficients of some function P, =
[int]dp/[rho], of x, y, z; so that X, Y, Z must be the partial
differential coefficients of a potential -V, such that the force in
any direction is the downward gradient of V; and then

dP dV
-- + -- = 0, or P + V = constant, (5)
dx dx

in which P may be called the hydrostatic head and V the head of
potential.

With variation of temperature, the surfaces of equal pressure and
density need not coincide; but, taking the pressure, density and
temperature as connected by some relation, such as the gas-equation,
the surfaces of equal density and temperature must intersect in lines
lying on a surface of equal pressure.

15. As an example of the general equations, take the simplest case of
a uniform field of gravity, with Oz directed vertically downward;
employing the gravitation unit of force,

1 dp 1 dp 1 dp
----- -- = 0, ----- -- = 0, ----- -- = 1, (1)
[rho] dx [rho] dy [rho] dz
_
/
P = | dp/[rho] = z + a constant. (2)
_/

When the density [rho] is uniform, this becomes, as before in (2) § 9

p = [rho]z + p0. (3)

Suppose the density [rho] varies as some nth power of the depth below
O, then

dp/dz = [rho] = [mu]z^n (4)

z^(n+1) [rho]z [rho] /[rho]\^1/n
p = [mu]------- = ------ = ----- ( ----- ) , (5)
n + 1 n + 1 n + 1 \[mu] /

supposing p and [rho] to vanish together.

These equations can be made to represent the state of convective
equilibrium of the atmosphere, depending on the gas-equation

p = [rho]k = R[rho][theta], (6)

where [theta] denotes the absolute temperature; and then

d[theta] d / p \ 1
R-------- = -- ( ----- ) = -------, (7)
dz dz \[rho]/ (n + 1)

so that the temperature-gradient d[theta]/dz is constant, as in
convective equilibrium in (11).

From the gas-equation in general, in the atmosphere

1 dp 1 dp 1 d[theta] [rho] 1 d[theta] 1 1 d[theta]
----- -- = --- -- - ------- -------- = ----- - ------- -------- = --- - ------- --------, (8)
[rho] dz p dz [theta] dz p [theta] dz k [theta] dz

which is positive, and the density [rho] diminishes with the ascent,
provided the temperature-gradient d[theta]/dz does not exceed
[theta]/k.

With uniform temperature, taking k constant in the gas-equation,

dp/dz = [rho] = p/k, p = p0e^(z/k), (9)

so that in ascending in the atmosphere of thermal equilibrium the
pressure and density diminish at compound discount, and for pressures
p1 and p2 at heights z1 and z2

(z1 - z2)/k = log e (p2/p1) = 2.3 log10 (p2/p1). (10)

In the convective equilibrium of the atmosphere, the air is supposed
to change in density and pressure without exchange of heat by
conduction; and then

[rho]/[rho]0 = ([theta]/[theta]0)^n, p/p0 =
([theta]/[theta]0)^(n+1), (11)

dz 1 dp p 1
-------- = ----- -------- = (n + 1)------------R, [gamma] = 1 + ---,
d[theta] [rho] d[theta] [rho][theta] n

where [gamma] is the ratio of the specific heat at constant pressure
and constant volume.

In the more general case of the convective equilibrium of a spherical
atmosphere surrounding the earth, of radius a,

dp p0 d[theta] a²
----- = (n + 1)------ -------- = - --- dr, (12)
[rho] [rho]0 [theta]0 r²

gravity varying inversely as the square of the distance r from the
centre; so that, k = p0/[rho]0, denoting the height of the homogeneous
atmosphere at the surface, [theta] is given by

(n + 1) k (1 - [theta]/[theta]0) = a(1 - a/r), (13)

or if c denotes the distance where [theta] = 0,

[theta] a c - r
-------- = --- · -----. (14)
[theta]0 r c - a

When the compressibility of water is taken into account in a deep
ocean, an experimental law must be employed, such as

p - p0 = k([rho] - [rho]0), or [rho]/[rho]0 = 1
+ (p - p0)/[lambda], [lambda] = k[rho]0, (15)

so that [lambda] is the pressure due to a head k of the liquid at
density [rho]0 under atmospheric pressure p0; and it is the gauge
pressure required on this law to double the density. Then

dp/dz = kd[rho]/dz = [rho], [rho] = [rho]0e^(z/k),
p - p0 = k[rho]0(e^(z/k) - 1); (16)

and if the liquid was incompressible, the depth at pressure p would be
(p - p0)/p0, so that the lowering of the surface due to compression is

ke^(z/k) - k - z = ½z²/k, when k is large. (17)

For sea water, [lambda] is about 25,000 atmospheres, and k is then
25,000 times the height of the water barometer, about 250,000 metres,
so that in an ocean 10 kilometres deep the level is lowered about 200
metres by the compressibility of the water; and the density at the
bottom is increased 4%.

On another physical assumption of constant cubical elasticity
[lambda],

dp = [lambda]d[rho]/[rho], (p - p0)/[lambda] = log([rho]/[rho]0), (18)

dp [lambda] d[rho] / 1 1 \ [rho]0 z
-- = -------- ------ = [rho], [lambda]( ------ - ----- ) = z, 1 - ------ = ---, [lambda] = k[rho]0, (19)
zd [rho] dz \[rho]0 [rho]/ [rho] k

and the lowering of the surface is

p - p0 [rho] / z \ z²
------ - z = k log ------ - z = k log ( 1 - --- ) - z [approx] --- (20)
[rho]0 [rho]0 \ k / 2k

as before in (17).

16. _Centre of Pressure._--A plane area exposed to fluid pressure on
one side experiences a single resultant thrust, the integrated
pressure over the area, acting through a definite point called the
centre of pressure (C.P.) of the area.

Thus if the plane is normal to Oz, the resultant thrust
_ _
/ /
R = | |pdxdy, (1)
_/_/

and the coordinates [=x], [=y] of the C.P. are given by
_ _ _ _
/ / / /
[=x]R = | | xp dx dy, [=y]R = | | yp dx dy. (2)
_/_/ _/_/

The C·P. is thus the C·G. of a plane lamina bounded by the area, in
which the surface density is p.

If p is uniform, the C·P. and C·G. of the area coincide.

For a homogeneous liquid at rest under gravity, p is proportional to
the depth below the surface, i.e. to the perpendicular distance from
the line of intersection of the plane of the area with the free
surface of the liquid.

If the equation of this line, referred to new coordinate axes in the
plane area, is written

x cos [alpha] + y sin [alpha] - h = 0, (3)
_ _
/ /
R = | | [rho](h - x cos [alpha] - y sin [alpha]) dx dy, (4)
_/_/
_ _
/ /
[=x]R = | | [rho]x(h - x cos [alpha] - y sin [alpha]) dx dy, (5)
_/_/
_ _
/ /
[=y]R = | | [rho]y(h - x cos [alpha] - y sin [alpha]) dx dy.
_/_/

Placing the new origin at the C.G. of the area A,
_ _ _ _
/ / / /
| | xd x dy = 0, | | y dx dy = 0, (6)
_/_/ _/_/

R = [rho]hA, (7)
_ _ _ _
/ / / /
[=x]hA = -cos [alpha] | | x² dA - sin [alpha] | | xy dA, (8)
_/_/ _/_/
_ _ _ _
/ / / /
[=y]hA = -cos [alpha] | | xy dA - sin [alpha] | | y² dA. (9)
_/_/ _/_/

Turning the axes to make them coincide with the principal axes of the
area A, thus making [int][int] xy dA = 0,

[=x]h = -a² cos [alpha], [=y]h = -b² sin [alpha], (10)

where
_ _ _ _
/ / / /
| | x² dA = Aa², | | y² dA = Ab², (11)
_/_/ _/_/

a and b denoting the semi-axes of the momental ellipse of the area.

This shows that the C.P. is the antipole of the line of intersection
of its plane with the free surface with respect to the momental
ellipse at the C.G. of the area.

Thus the C.P. of a rectangle or parallelogram with a side in the
surface is at 2/3 of the depth of the lower side; of a triangle with a
vertex in the surface and base horizontal is ¾ of the depth of the
base; but if the base is in the surface, the C·P. is at half the depth
of the vertex; as on the faces of a tetrahedron, with one edge in the
surface.

The _core_ of an area is the name given to the limited area round its
C.G. within which the C·P. must lie when the area is immersed
completely; the boundary of the core is therefore the locus of the
antipodes with respect to the momental ellipse of water lines which
touch the boundary of the area. Thus the core of a circle or an
ellipse is a concentric circle or ellipse of one quarter the size.

The C.P. of water lines passing through a fixed point lies on a
straight line, the antipolar of the point; and thus the core of a
triangle is a similar triangle of one quarter the size, and the core
of a parallelogram is another parallelogram, the diagonals of which
are the middle third of the median lines.

In the design of a structure such as a tall reservoir dam it is
important that the line of thrust in the material should pass inside
the core of a section, so that the material should not be in a state
of tension anywhere and so liable to open and admit the water.

17. _Equilibrium and Stability of a Ship or Floating Body. The Metacentre._--The principle of Archimedes in § 12 leads immediately to the conditions of equilibrium of a body supported freely in fluid, like a fish in water or a balloon in the air, or like a ship (fig. 3) floating partly immersed in water and the rest in air. The body is in equilibrium under two forces:--(i.) its weight W acting vertically downward through G, the C.G. of the body, and (ii.) the buoyancy of the fluid, equal to the weight of the displaced fluid, and acting vertically upward through B, the C.G. of the displaced fluid; for equilibrium these two forces must be equal and opposite in the same line.

The conditions of equilibrium of a body, floating like a ship on the surface of a liquid, are therefore:--

(i.) the weight of the body must be less than the weight of the total volume of liquid it can displace; or else the body will sink to the bottom of the liquid; the difference of the weights is called the "reserve of buoyancy."

(ii.) the weight of liquid which the body displaces in the position of equilibrium is equal to the weight W of the body; and

(iii.) the C.G., B, of the liquid displaced and G of the body, must lie in the same vertical line GB.

18. In addition to satisfying these conditions of equilibrium, a ship must fulfil the further condition of stability, so as to keep upright; if displaced slightly from this position, the forces called into play must be such as to restore the ship to the upright again. The stability of a ship is investigated practically by inclining it; a weight is moved across the deck and the angle is observed of the heel produced.

Suppose P tons is moved c ft. across the deck of a ship of W tons
displacement; the C.G. will move from G to G1 the reduced distance
G1G2 = c(P/W); and if B, called the centre of buoyancy, moves to B1,
along the curve of buoyancy BB1, the normal of this curve at B1 will
be the new vertical B1G1, meeting the old vertical in a point M, the
centre of curvature of BB1, called the _metacentre_.

If the ship heels through an angle [theta] or a slope of 1 in m,

GM = GG1cot[theta] = mc(P/W), (1)

and GM is called the metacentric height; and the ship must be
ballasted, so that G lies below M. If G was above M, the tangent drawn
from G to the evolute of B, and normal to the curve of buoyancy, would
give the vertical in a new position of equilibrium. Thus in H.M.S.
"Achilles" of 9000 tons displacement it was found that moving 20 tons
across the deck, a distance of 42 ft., caused the bob of a pendulum 20
ft. long to move through 10 in., so that

240 20
GM = --- × 42 × ---- = 2.24 ft.; (2)
10 9000

also

cot [theta] = 24, [theta] = 2°24´. (3)

In a diagram it is conducive to clearness to draw the ship in one
position, and to incline the water-line; and the page can be turned if
it is desired to bring the new water-line horizontal.

Suppose the ship turns about an axis through F in the water-line area,
perpendicular to the plane of the paper; denoting by y the distance of
an element dA if the water-line area from the axis of rotation, the
change of displacement is [sum]ydA tan[theta], so that there is no
change of displacement if [sum]ydA = 0, that is, if the axis passes
through the C.G. of the water-line area, which we denote by F and call
the centre of flotation.

The righting couple of the wedges of immersion and emersion will be

[Sigma]wy dA tan [theta]·y = w tan [theta] [Sigma] y² dA
= w tan [theta]·Ak² ft. tons, (4)

w denoting the density of water in tons/ft.³, and W = wV, for a
displacement of V ft.³

This couple, combined with the original buoyancy W through B, is
equivalent to the new buoyancy through B, so that

W.BB1 = wAk² tan [theta], (5)

BM = BB1 cot [theta] = Ak²/V, (6)

giving the radius of curvature BM of the curve of buoyancy B, in terms
of the displacement V, and Ak² the moment of inertia of the water-line
area about an axis through F, perpendicular to the plane of
displacement.

An inclining couple due to moving a weight about in a ship will heel
the ship about an axis perpendicular to the plane of the couple, only
when this axis is a principal axis at F of the momental ellipse of the
water-line area A. For if the ship turns through a small angle [theta]
about the line FF´, then b1, b2, the C·G. of the wedge of immersion
and emersion, will be the C·P. with respect to FF´ of the two parts of
the water-line area, so that b1b2 will be conjugate to FF´ with
respect to the momental ellipse at F.