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Title: Encyclopaedia Britannica, 11th Edition, "Hydromechanics" to
"Ichnography"

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THE ENCYCLOPÆDIA 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 ‘ARABĪ
HYLOZOISM IBN ATHĪR
HYMEN IBN BATUTA
HYMENOPTERA IBN DURAID
HYMETTUS IBN FARADĪ
HYMNS IBN FĀRID
HYPAETHROS IBN GABIROL
HYPALLAGE IBN HAUKAL
HYPATIA IBN ḤAZM
HYPERBATON IBN HISHĀM
HYPERBOLA IBN ISHĀQ
HYPERBOLE IBN JUBAIR
HYPERBOREANS IBN KHALDŪN
HYPEREIDES IBN KHALLIKĀN
HYPERION IBN QUTAIBA
HYPERSTHENE IBN ṢA‘D
HYPERTROPHY IBN TIBBON
HYPNOTISM IBN ṬUFAIL
HYPOCAUST IBN USAIBI‘A
HYPOCHONDRIASIS IBO
HYPOCRISY IBRAHĪM AL-MAUṢILĪ
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 (ὑδρομηχανικά), 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 Περὶ τῶν ὀχουμένων, 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 ℔ 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 ℔ per sq. ft., or P/144A ℔ per sq. in.
(℔/ft.2 and ℔/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
ΔP ℔ acts on a small plane area ΔA ft.2 enclosing a point B, the pressure
p at B is the limit of ΔP/ΔA; and

p = lt (ΔP/ΔA) = 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 Fig. 1a.
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 ℔ is applied to
one piston of area A ft.2, it will be balanced by a thrust W ℔ 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
Fig. 1b.
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 α
ft.2, the pressure at A and By by p0 and p ℔/ft.2, and by w the density of
the liquid estimated in ℔/ft.3,

pα − p0α = wα·AB,
(1)

p = w·AB + p0.

(2)

Thus in water, where w = 62.4℔/ft.3, the pressure increases 62.4 ℔/ft.2,
or 62.4 ÷ 144 = 0.433 ℔/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 σ rises to the height h and of density ρ 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 σh + p0 and ρk + p0,
and these being equal we have

σh = ρk.
(1)

The principle is illustrated in the article Barometer, where a column of
mercury of density σ and height h, rising in the tube to the Torricellian
vacuum, is balanced by a column of air of density ρ, 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 = σ/ρ = 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 σh at a depth h ft. in
liquid of density σ 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 1⁄3πy3, 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 Fig. 2.

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 ρ, 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,

∫ ∫ lpdS = ∫ ∫ ∫ρX 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
dx dy dz,
(2)

thus leading to the differential relation at every point

dp dp dp
= ρX, = ρY, = ρZ.
dx dy dz (3)

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
ρ dx ρ dy ρ dz (4)

are the partial differential coefficients of some function P, = ∫ dp/ρ, 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,
dx dx (5)

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,
ρ dx ρ dy ρ dz (1)

P= ∫ dp/ρ = z + a constant.
(2)

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

p = ρz + p0.
(3)

Suppose the density ρ varies as some nth power of the depth below O,
then

dp/dz = ρ = μzn
(4)
zn+1 ρz ρ ρ 1/n
p=μ
n+1
=
n+1
=
n+1 ( ) μ
,
(5)

supposing p and ρ to vanish together.

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

p = ρk = R ρθ,
(6)

where θ denotes the absolute temperature; and then

dθ d p 1
R
dz
=
dz ( )ρ = n+1 ,
(7)

so that the temperature-gradient dθ/dz is constant, as in convective
equilibrium in (11).

From the gas-equation in general, in the atmosphere

1 dp 1 dp 1 dθ ρ 1 dθ 1 1 dθ
= − = − = − ,
ρ dz p dz θ dz p θ dz k θ dz(8)

which is positive, and the density ρ diminishes with the ascent, provided
the temperature-gradient dθ/dz does not exceed θ/k.

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

dp/dz = ρ = p/k, p = p0ez/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 = loge (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

ρ/ρ0 = (θ/θ0)n, p/p0 = (θ/θ0)n + 1,
(11)
dz 1 dp p 1
= = (n + 1) = (n + 1) R, γ = 1 + ,
dθ ρ dθ ρθ n

where γ 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 dθ a2
= (n + 1) =− dr,
ρ ρ0 θ0 r2 (12)

gravity varying inversely as the square of the distance r from the centre;
so that, k = p0/ρ0, denoting the height of the homogeneous atmosphere at
the surface, θ is given by

(n + 1) k (1 − θ/θ0) = a(1 − a/r),
(13)

or if c denotes the distance where θ = 0,

θ a c−r
= · .
θ0 r c−a (14)

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

p − p0 = k (ρ − ρ0), or ρ/ρ0 = 1 + (p − p0)/λ, λ = kρ0,
(15)

so that λ is the pressure due to a head k of the liquid at density ρ0 under
atmospheric pressure p0; and it is the gauge pressure required on this law
to double the density. Then

dp/dz = kdρ/dz = ρ, ρ = ρ0ez/k, p − p0 = kρ0 (ez/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

kez/k − k − z = ½z2/k, when k is large.
(17)

For sea water, λ 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 λ,

dp = λdρ/ρ, (p − p0)/λ = log (ρ/ρ0),
(18)
dp λ dρ 1 1 ρ0 z
zd
=
ρ dz = ρ, λ ( ρ0
−
ρ ) = z, 1− ρ =
k
, λ = kρ0,
(19)

and the lowering of the surface is

p − p0 ρ z z2
ρ0
− z = k log
ρ0 − z = −k log ( 1 − k ) − z ≈ 2k (20)
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= ∫ ∫ p dx dy,
(1)

and the coordinates x, y of the C.P. are given by

xR = ∫ ∫ xp dx dy, yR = ∫ ∫ 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 α + y sin α − h = 0,
(3)

R= ∫ ∫ ρ (h − x cos α − y sin α) dx dy,
(4)

xR = ∫ ∫ ρx (h − x cos α − y sin α) dx dy,
(5)

yR = ∫ ∫ ρy (h − x cos α − y sin α) dx dy.
Placing the new origin at the C.G. of the area A,

∫ ∫ xd x dy = 0, ∫ ∫ y dx dy = 0,
(6)

R = ρhA,
(7)

xhA = −cos α ∫ ∫ x2 dA − sin α ∫ ∫ xy dA,
(8)

yhA = −cos α ∫ ∫ xy dA − sin α ∫ ∫ y2 dA.
(9)

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

xh = −a2 cos α, yh = −b2 sin α,
(10)

where

∫ ∫ x2dA = Aa2, ∫ ∫ y2dA = 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
Fig. 3.
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 θ or a slope of 1 in m,

GM = GG1 cot θ = 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

GM = 240 × 42 × 20 2.24 ft.

10 9000 (2)

also

cot θ = 24, θ = 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 ΣydA tanθ, so that there is no change of displacement if
ΣydA = 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

Σwy dA tan θ·y = w tan θ Σ y2 dA = w tan θ·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 θ,
(5)

BM = BB1 cot θ = 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 θ 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.

The naval architect distinguishes between the stability of form,
represented by the righting couple W.BM, and the stability of ballasting,
represented by W.BG. Ballasted with G at B, the righting couple when the
ship is heeled through θ is given by W.BM. tanθ; but if weights inside the
ship are raised to bring G above B, the righting couple is diminished by
W·BG.tanθ, so that the resultant righting couple is W·GM·tanθ. Provided
the ship is designed to float upright at the smallest draft with no load on
board, the stability at any other draft of water can be arranged by the
stowage of the weight, high or low.

19. Proceeding as in § 16 for the determination of the C.P. of an area,
the same argument will show that an inclining couple due to the
movement of a weight P through a distance c will cause the ship to heel
through an angle θ about an axis FF′ through F, which is conjugate to the
direction of the movement of P with respect to an ellipse, not the
momental ellipse of the water-line area A, but a confocal to it, of squared
semi-axes

a2 − hV/A, b2 − hV/A,
(1)

h denoting the vertical height BG between C.G. and centre of buoyancy.
The varying direction of the inclining couple Pc may be realized by
swinging the weight P from a crane on the ship, in a circle of radius c. But

if the weight P was lowered on the ship from a crane on shore, the vessel
would sink bodily a distance P/wA if P was deposited over F; but
deposited anywhere else, say over Q on the water-line area, the ship
would turn about a line the antipolar of Q with respect to the confocal
ellipse, parallel to FF′, at a distance FK from F

FK = (k2 − hV/A)/FQ sin QFF′
(2)

through an angle θ or a slope of one in m, given by

1 P P V
sin θ = = = · FQ sin QFF′
m wA·FK W Ak2 − hV (3)

where k denotes the radius of gyration about FF′ of the water-line area.
Burning the coal on a voyage has the reverse effect on a steamer.

Hydrodynamics

20. In considering the motion of a fluid we shall suppose it non-viscous, so
that whatever the state of motion the stress across any section is normal, and
the principle of the normality and thence of the equality of fluid pressure can
be employed, as in hydrostatics. The practical problems of fluid motion, which
are amenable to mathematical analysis when viscosity is taken into account,
are excluded from treatment here, as constituting a separate branch called
“hydraulics” (q.v.). Two methods are employed in hydrodynamics, called the
Eulerian and Lagrangian, although both are due originally to Leonhard Euler.
In the Eulerian method the attention is fixed on a particular point of space, and
the change is observed there of pressure, density and velocity, which takes
place during the motion; but in the Lagrangian method we follow up a particle
of fluid and observe how it changes. The first may be called the statistical
method, and the second the historical, according to J. C. Maxwell. The

Lagrangian method being employed rarely, we shall confine ourselves to the
Eulerian treatment.

The Eulerian Form of the Equations of Motion.

21. The first equation to be established is the equation of continuity, which
expresses the fact that the increase of matter within a fixed surface is due to
the flow of fluid across the surface into its interior.

In a straight uniform current of fluid of density ρ, flowing with velocity
q, the flow in units of mass per second across a plane area A, placed in
the current with the normal of the plane making an angle θ with the
velocity, is ρAq cos θ, the product of the density ρ, the area A, and q cos θ
the component velocity normal to the plane.

Generally if S denotes any closed surface, fixed in the fluid, M the
mass of the fluid inside it at any time t, and θ the angle which the
outward-drawn normal makes with the velocity q at that point,

dM/dt = rate of increase of fluid inside the surface,
= flux across the surface into the interior
= − ∫∫ ρq cos θ dS,
(1)

the integral equation of continuity.

In the Eulerian notation u, v, w denote the components of the velocity q
parallel to the coordinate axes at any point (x, y, z) at the time t; u, v, w
are functions of x, y, z, t, the independent variables; and d is used here to
denote partial differentiation with respect to any one of these four
independent variables, all capable of varying one at a time.

To transfer the integral equation into the differential equation of
continuity, Green’s transformation is required again, namely,

dξ dη dζ
∫∫∫ ( dx
+
dy
+
dz ) dx dy dz = ∫∫ (lξ + mη + nζ) dS, (2)
or individually

dξ
∫∫∫ dx
dx dy dz = ∫∫ lξ dS, ...,
(3)

where the integrations extend throughout the volume and over the surface
of a closed space S; l, m, n denoting the direction cosines of the outward-
drawn normal at the surface element dS, and ξ, η, ζ any continuous
functions of x, y, z.

The integral equation of continuity (1) may now be written

dρ
∫∫∫ dt
dx dy dz = ∫∫ (lρu + mρv + nρw) dS = 0,
(4)

which becomes by Green’s transformation

dρ d(ρu) d(ρv) d(ρw)
∫∫∫ ( dt
+
dx
+
dy
+
dz ) dx dy dz = 0, (5)
leading to the differential equation of continuity when the integration is
removed.

22. The equations of motion can be established in a similar way by
considering the rate of increase of momentum in a fixed direction of the fluid
inside the surface, and equating it to the momentum generated by the force
acting throughout the space S, and by the pressure acting over the surface S.

Taking the fixed direction parallel to the axis of x, the time-rate of
increase of momentum, due to the fluid which crosses the surface, is

− ∫∫ ρuq cos θ dS = − ∫∫ (lρu2 + mρuv + nρuw) dS,
(1)

which by Green’s transformation is

d(ρu2) d(ρuv) d(ρuw)
− ∫∫∫ ( dx
+
dy
+
dz ) dx dy dz. (2)

The rate of generation of momentum in the interior of S by the
component of force, X per unit mass, is

∫∫∫ ρX dx dy dz,
(3)

and by the pressure at the surface S is

dp
− ∫∫ lp dS = − ∫∫∫ dx
dx dy dz,
(4)

by Green’s transformation.

The time rate of increase of momentum of the fluid inside S is

d(ρu)
∫∫∫ dt
dx dy dz;
(5)

and (5) is the sum of (1), (2), (3), (4), so that

dρu dρu2 dρuv dρuw dp
∫∫∫ ( dt
+
dx
+
dy
+
dz
− ρX +
dx ) dx dy dz =(6)0,
leading to the differential equation of motion

dρu dρu2 dρuv dρuw dp
+ + + = ρX − ,
dt dx dy dz dx (7)

with two similar equations.

The absolute unit of force is employed here, and not the gravitation
unit of hydrostatics; in a numerical application it is assumed that C.G.S.

units are intended.

These equations may be simplified slightly, using the equation of
continuity (5) § 21; for

dρu dρu2 dρuv dρuw
+ + +
dt dx dy dz
du du du du
=ρ ( dt
+u
dx
+v
dy
+w
dz )
dρ dρu dρv dρw
+u ( dt
+
dx
+
dy
+
dz , ) (8)

reducing to the first line, the second line vanishing in consequence of the
equation of continuity; and so the equation of motion may be written in
the more usual form

du du du du 1 dp
+u +v +w =X− ,
dt dx dy dz ρ dx (9)

with the two others

dv dv dv dv 1 dp
+u +v +w =Y− ,
dt dx dy dz ρ dy (10)
dw dw dw dw 1 dp
+u +v +w =Z− .
dt dx dy dz ρ dz (11)

23. As a rule these equations are established immediately by determining
the component acceleration of the fluid particle which is passing through (x, y,
z) at the instant t of time considered, and saying that the reversed acceleration
or kinetic reaction, combined with the impressed force per unit of mass and
pressure-gradient, will according to d’Alembert’s principle form a system in
equilibrium.

To determine the component acceleration of a particle, suppose F to
denote any function of x, y, z, t, and investigate the time rate of F for a
moving particle; denoting the change by DF/dt,

DF F(x + uδt, y + vδt, z + wδt, t + δt) − F(x, y, z, t)
= lt·
dt δt
dF dF dF dF
= +u +v +w ;
dt dx dy dz (1)

and D/dt is called particle differentiation, because it follows the rate of
change of a particle as it leaves the point x, y, z; but

dF/dt, dF/dx, dF/dy, dF/dz
(2)

represent the rate of change of F at the time t, at the point, x, y, z, fixed in
space.

The components of acceleration of a particle of fluid are consequently

Du du du du du
= +u +v +w ,
dt dt dx dy dz (3)
Dv dv dv dv dv
= +u +v +w ,
dt dt dx dy dz (4)
Dw dw dw dw dw
= +u +v +w ,
dt dt dx dy dz (5)

leading to the equations of motion above.

If F (x, y, z, t) = 0 represents the equation of a surface containing
always the same particles of fluid,

DF dF dF dF dF
= 0, or +u +v +w = 0,
dt dt dx dy dz (6)

which is called the differential equation of the bounding surface. A
bounding surface is such that there is no flow of fluid across it, as
expressed by equation (6). The surface always contains the same fluid
inside it, and condition (6) is satisfied over the complete surface, as well
as any part of it.

But turbulence in the motion will vitiate the principle that a bounding
surface will always consist of the same fluid particles, as we see on the
surface of turbulent water.

24. To integrate the equations of motion, suppose the impressed force
is due to a potential V, such that the force in any direction is the rate of
diminution of V, or its downward gradient; and then

X = −dV/dx, Y = −dV/dy, Z = −dV/dz;
(1)

and putting

dw dv du dw dv du
− = 2ξ, − = 2η, − = 2ζ,
dy dz dz dx dx dy (2)
dξ dη dζ
+ + = 0,
dx dy dz (3)

the equations of motion may be written

du dH
− 2vζ + 2wη + = 0,
dt dx (4)
dv dH
− 2wξ + 2uζ + = 0,
dt dy (5)
dw dH
− 2uη + 2wξ + = 0,
dt dz (6)

where

H = ∫ dp/ρ + V + ½q2,
(7)

q2 = u2 + v2 + w2,
(8)

and the three terms in H may be called the pressure head, potential head,
and head of velocity, when the gravitation unit is employed and ½q2 is
replaced by ½q2/g.

Eliminating H between (5) and (6)

Dξ du dw dv du dv dw
dt
−ξ
dx
−η
dx
−ζ
dx + ξ ( dx + dy + dz ) = 0, (9)
and combining this with the equation of continuity

1 Dρ du dv dw
+ + + = 0,
ρ dt dx dy dz (10)

we have

D ξ ξ du η dv ζ dw
dt ( ρ )− ρ dx
−
ρ dx
−
ρ dx
= 0,
(11)

with two similar equations.

Putting

ω 2 = ξ 2 + η 2 + ζ 2,
(12)

a vortex line is defined to be such that the tangent is in the direction of ω,
the resultant of ξ, η, ζ, called the components of molecular rotation. A
small sphere of the fluid, if frozen suddenly, would retain this angular
velocity.

If ω vanishes throughout the fluid at any instant, equation (11) shows
that it will always be zero, and the fluid motion is then called irrotational;
and a function φ exists, called the velocity function, such that

u dx + v dy + w dz = −dφ,
(13)

and then the velocity in any direction is the space-decrease or downward
gradient of φ.

25. But in the most general case it is possible to have three functions φ,
ψ, m of x, y, z, such that

u dx + v dy + w dz = −dφ − m dψ,
(1)

as A. Clebsch has shown, from purely analytical considerations (Crelle,
lvi.); and then

d(ψ, m) d(ψ, m) d(ψ, m)
ξ=½ , η=½ , ζ=½ ,
d(y, z) d(z, x) d(x, y) (2)

and

dψ dψ dψ dm dm dm
ξ +η +ζ = 0, ξ +η +ζ = 0,
dx dy dz dx dy dz (3)

so that, at any instant, the surfaces over which ψ and m are constant
intersect in the vortex lines.

Putting

dφ dψ
H− −m = K,
dt dt (4)

the equations of motion (4), (5), (6) § 24 can be written

dK d(ψ,m)
− 2uζ + 2wη − = 0, ..., ...;
dx d(x,t) (5)

and therefore

dK dK dK
ξ +η +ζ = 0.
dx dy dz (6)

Equation (5) becomes, by a rearrangement,

dK dψ dm dm dm dm
dx
−
dx (
dt
+u
dx
+v
dy
+w
dz )
dm dψ dψ dψ dψ
+
dx ( dt
+u
dx
+v
dy
+w
dz )
= 0, ..., ...,
(7)
dK dψ Dm dm Dψ
− + = 0, ..., ...,
dx dx dt dx dt (8)

and as we prove subsequently (§ 37) that the vortex lines are composed of
the same fluid particles throughout the motion, the surface m and ψ
satisfies the condition of (6) § 23; so that K is uniform throughout the
fluid at any instant, and changes with the time only, and so may be
replaced by F(t).

26. When the motion is steady, that is, when the velocity at any point
of space does not change with the time,

dK
− 2vζ + 2wη = 0, ..., ...
dx (1)
dK dK dK dK dK dK
ξ +η +ζ = 0, u +v +w = 0,
dx dy dz dx dy dz (2)

and

K = ∫ dp/ρ + V + ½q2 = H

(3)

is constant along a vortex line, and a stream line, the path of a fluid
particle, so that the fluid is traversed by a series of H surfaces, each
covered by a network of stream lines and vortex lines; and if the motion
is irrotational H is a constant throughout the fluid.

Taking the axis of x for an instant in the normal through a point on the
surface H = constant, this makes u = 0, ξ = 0; and in steady motion the
equations reduce to

dH/dν = 2vζ − 2wη = 2qω sin θ,
(4)

where θ is the angle between the stream line and vortex line; and this
holds for their projection on any plane to which dν is drawn
perpendicular.

In plane motion (4) reduces to

dH dQ q
dν = 2qζ = q ( dv
+
r ), (5)

if r denotes the radius of curvature of the stream line, so that

1 dp dV dH d ½q2 q2
+ = − = ,
ρ dν dν dν dν r (6)

the normal acceleration.

The osculating plane of a stream line in steady motion contains the
resultant acceleration, the direction ratios of which are

du du du d ½q2 d ½q2 dH
u +v +w = − 2vζ + 2wη = − , ...,
dx dy dz dx dx dx (7)

and when q is stationary, the acceleration is normal to the surface H =
constant, and the stream line is a geodesic.

Calling the sum of the pressure and potential head the statical head,
surfaces of constant statical and dynamical head intersect in lines on H,
and the three surfaces touch where the velocity is stationary.

Equation (3) is called Bernoulli’s equation, and may be interpreted as
the balance-sheet of the energy which enters and leaves a given tube of
flow.

If homogeneous liquid is drawn off from a vessel so large that the
motion at the free surface at a distance may be neglected, then Bernoulli’s
equation may be written

H = p/ρ + z + q2/2g = P/ρ + h,
(8)

where P denotes the atmospheric pressure and h the height of the free
surface, a fundamental equation in hydraulics; a return has been made
here to the gravitation unit of hydrostatics, and Oz is taken vertically
upward.

In particular, for a jet issuing into the atmosphere, where p = P,

q2/2g = h − z,
(9)

or the velocity of the jet is due to the head k − z of the still free surface
above the orifice; this is Torricelli’s theorem (1643), the foundation of the
science of hydrodynamics.

27. Uniplanar Motion.—In the uniplanar motion of a homogeneous
liquid the equation of continuity reduces to

du + dv = 0,

dx dy (1)

so that we can put

u = −dψ/dy, v = dψ/dx,
(2)

where ψ is a function of x, y, called the stream- or current-function;
interpreted physically, ψ − ψ0, the difference of the value of ψ at a fixed
point A and a variable point P is the flow, in ft.3/second, across any
curved line AP from A to P, this being the same for all lines in accordance
with the continuity.

Thus if dψ is the increase of ψ due to a displacement from P to P′, and
k is the component of velocity normal to PP′, the flow across PP′ is dψ =
k·PP′; and taking PP′ parallel to Ox, dψ = v dx; and similarly dψ= −u dy
with PP′ parallel to Oy; and generally dψ/ds is the velocity across ds, in a
direction turned through a right angle forward, against the clock.

In the equations of uniplanar motion

dv du d2ψ d2ψ
2ζ = − = + = −∇2ψ, suppose,
dx dy dx2 dy2 (3)

so that in steady motion

dH dψ dH dψ dH
+ ∇2ψ = 0, + ∇2ψ = 0, + ∇2ψ = 0,
dx dx dy dy dψ (4)

and ∇2ψ must be a function of ψ.

If the motion ia irrotational,

dφ dψ dφ dψ
u=− =− ,v=− = ,
dx dy dy dx (5)

so that ψ and φ are conjugate functions of x and y,

φ + ψi = ƒ(x + yi), ∇2ψ = 0, ∇2φ = 0;
(6)

or putting

φ + ψi = w, x + yi = z, w = ƒ(z).

The curves φ = constant and ψ = constant form an orthogonal system;
and the interchange of φ and ψ will give a new state of uniplanar motion,
in which the velocity at every point is turned through a right angle
without alteration of magnitude.

For instance, in a uniplanar flow, radially inward towards O, the flow
across any circle of radius r being the same and denoted by 2πm, the
velocity must be m/r, and

φ = m log r, ψ = mθ, φ + ψi = m log reiθ, w = m log z.
(7)

Interchanging these values

ψ = m log r, φ = mθ, ψ + φi = m log reiθ
(8)

gives a state of vortex motion, circulating round Oz, called a straight or
columnar vortex.

A single vortex will remain at rest, and cause a velocity at any point
inversely as the distance from the axis and perpendicular to its direction;
analogous to the magnetic field of a straight electric current.

If other vortices are present, any one may be supposed to move with
the velocity due to the others, the resultant stream-function being

ψ = Σm log r = log Πrm;
(9)

the path of a vortex is obtained by equating the value of ψ at the vortex to
a constant, omitting the rm of the vortex itself.

When the liquid is bounded by a cylindrical surface, the motion of a
vortex inside may be determined as due to a series of vortex-images, so
arranged as to make the flow zero across the boundary.

For a plane boundary the image is the optical reflection of the vortex.
For example, a pair of equal opposite vortices, moving on a line parallel
to a plane boundary, will have a corresponding pair of images, forming a
rectangle of vortices, and the path of a vortex will be the Cotes’ spiral

r sin 2θ = 2a, or x−2 + y−2 = a−2;
(10)

this is therefore the path of a single vortex in a right-angled corner; and
generally, if the angle of the corner is π/n, the path is the Cotes’ spiral

r sin nθ = na.
(11)

A single vortex in a circular cylinder of radius a at a distance c from the
centre will move with the velocity due to an equal opposite image at a
distance a2/c, and so describe a circle with velocity

mc/(a2 − c2) in the periodic time 2π (a2 − c2)/m.
(12)

Conjugate functions can be employed also for the motion of liquid in a
thin sheet between two concentric spherical surfaces; the components of
velocity along the meridian and parallel in colatitude θ and longitude λ
can be written

dφ 1 dψ 1 dψ dψ
= , =− ,
dθ sin θ dλ sin θ dλ dθ (13)

and then

φ + ψi = F (tan ½θ·eλi).
(14)

28. Uniplanar Motion of a Liquid due to the Passage of a Cylinder
through it.—A stream-function ψ must be determined to satisfy the
conditions

∇2ψ = 0, throughout the liquid;
(1)

ψ = constant, over any fixed boundary;
(2)

dψ/ds = normal velocity reversed over a solid boundary,
(3)

so that, if the solid is moving with velocity U in the direction Ox, dψ/ds =
−U dy/ds, or ψ + Uy = constant over the moving cylinder; and ψ + Uy =
ψ′ is the stream function of the relative motion of the liquid past the
cylinder, and similarly ψ − Vx for the component velocity V along Oy;
and generally

ψ′ = ψ + Uy − Vx
(4)

is the relative stream-function, constant over a solid boundary moving
with components U and V of velocity.

If the liquid is stirred up by the rotation R of a cylindrical body,

dψ/ds = normal velocity reversed

dx dy
= −Rx − Ry ,
ds ds (5)

ψ + ½R (x2 + y2) = ψ′,

(6)

a constant over the boundary; and ψ′ is the current-function of the relative
motion past the cylinder, but now

V2ψ′ + 2R = 0,
(7)

throughout the liquid.

Inside an equilateral triangle, for instance, of height h,

ψ′ = −2Rαβγ/h,
(8)

where α, β, γ are the perpendiculars on the sides of the triangle.

In the general case ψ′ = ψ + Uy − Vx + ½R (x2 + y2) is the relative
stream function for velocity components, U, V, R.

29. Example 1.—Liquid motion past a circular cylinder.

Consider the motion given by

ω = U (z + a2/z),
(1)

so that

a2 a2
ψ=U ( r + r ) cos θ = U ( 1 + r2 ) x, (2)
a2 a2
φ=U ( r + r ) sin θ = U ( 1 + r2 ) y. (2)

Then ψ = 0 over the cylinder r = a, which may be considered a fixed
post; and a stream line past it along which ψ = Uc, a constant, is the curve

a2
( r − r ) sin θ = c, (x2 + y2) (y − c) − a2y = 0 (3)

a cubic curve (C3).

Over a concentric cylinder, external or internal, of radius r = b,

a2
ψ′ = ψ + U1y = [ U ( 1 − b2 ) + U1] y, (4)

and ψ′ is zero if

U1/U = (a2 − b2)/b2;
(5)

so that the cylinder may swim for an instant in the liquid without
distortion, with this velocity U1, and ω in (1) will give the liquid motion
in the interspace between the fixed cylinder r = a and the concentric
cylinder r = b, moving with velocity U1.

When b = 0, U1 = ∞; and when b = ∞, U1 = −U, so that at infinity the
liquid is streaming in the direction xO with velocity U.

If the liquid is reduced to rest at infinity by the superposition of an
opposite stream given by ω = −Uz, we are left with

ω = Ua2/z,
(6)

φ = U (a2/r) cos θ = Ua2x/(x2 + y2),
(7)

ψ = −U (a2/r) sin θ = −Ua2y/(x2 + y2),
(8)

giving the motion due to the passage of the Cylinder r = a with velocity U
through the origin O in the direction Ox.

If the direction of motion makes an angle θ′ with Ox,

dφ dφ 2xy
tan θ′ =
dy / dx
=
x2 − y2
= tan 2θ, θ = ½θ′,
(9)

and the velocity is Ua2/r2.

Along the path of a particle, defined by the C3 of (3),

y2 y (y − c)
sin2 ½θ′ = = ,
x2 + y2 a2 (10)
dθ′ 2y − c dy
½ sin θ′ = ,
ds a2 ds (11)

on the radius of curvature is ¼a2/(y − ½c), which shows that the curve is
an Elastica or Lintearia. (J. C. Maxwell, Collected Works, ii. 208.)

If φ1 denotes the velocity function of the liquid filling the cylinder r =
b, and moving bodily with it with velocity U1,

φ1 = −U1x,
(12)

and over the separating surface r = b

φ U a2 a 2 + b2
φ1
=−
U1 ( 1 + b2 ) = a2 − b2 , (13)

and this, by § 36, is also the ratio of the kinetic energy in the annular
interspace between the two cylinders to the kinetic energy of the liquid
moving bodily inside r = b.

Consequently the inertia to overcome in moving the cylinder r = b,
solid or liquid, is its own inertia, increased by the inertia of liquid (a2 +
b2)/(a2 ~ b2) times the volume of the cylinder r = b; this total inertia is

called the effective inertia of the cylinder r = b, at the instant the two
cylinders are concentric.

With liquid of density ρ, this gives rise to a kinetic reaction to
acceleration dU/dt, given by

a 2 + b2 dU a 2 + b2 dU
πρb2 = M′ ,
a 2 − b2 dt a 2 − b2 dt (14)

if M′ denotes the mass of liquid displaced by unit length of the cylinder r
= b. In particular, when a = ∞, the extra inertia is M′.

When the cylinder r = a is moved with velocity U and r = b with
velocity U1 along Ox,

a2 b2 b2 a2
φ=U
b2 − a2 ( r + r ) cos θ − U1 b2 − a2 ( r + r ) cos(15)
θ,

a2 b2 b2 a2
ψ = −U
b2 − a2 ( r − r ) sin θ − U1 b2 − a2 ( r − r ) sin(16)
θ,

and similarly, with velocity components V and V1 along Oy

a2 b2 b2 a2
φ=V
b2 − a2 ( r + r ) cos θ − V1 b2 − a2 ( r + r ) cos(17)
θ,

a2 b2 b2 a2
ψ=V
b2 − a2 ( r − r ) sin θ + V1 b2 − a2 ( r − r ) sin(18)
θ,

and then for the resultant motion

a2 z a2b2 U + Vi
w = (U2 + V2) +
b2 − a2 U + Vi b2 − a2 z
b2 z a2b2 U1 + V1i
−(U12 + V12) − .
b2 − a2 U1 + V1i b2 − a2 z (19)

The resultant impulse of the liquid on the cylinder is given by the
component, over r = a (§ 36),

b2 + a2 2b2
X = ∫ ρφ cos θ·a dθ = πρa2 ( U b2 − a2 − U1
b2 − a2 ); (20)
and over r = b

2a2 b2 + a2
X1 = ∫ ρφ cos θ·b dθ = πρb2 ( U b2 − a2 − U1
b2 − a2 ), (21)
and the difference X − X1 is the component momentum of the liquid in
the interspace; with similar expressions for Y and Y1.

Then, if the outside cylinder is free to move

V1 2a2 b2 − a2
X1 = 0, = , X = πρa2U .
U b2 + a2 b2 + a2 (22)

But if the outside cylinder is moved with velocity U1, and the inside
cylinder is solid or filled with liquid of density σ,

U1 2ρb2
X = −πρa2U, = ,
U ρ (b2 + a2) + σ (b2 − a2)
U − U1 (ρ − σ) (b2 − a2)
= ,
U1 ρ (b2 + a2) + σ (b2 − a2) (23)

and the inside cylinder starts forward or backward with respect to the
outside cylinder, according as ρ > or < σ.

30. The expression for ω in (1) § 29 may be increased by the addition
of the term

im log z = −mθ + im log r,
(1)

representing vortex motion circulating round the annulus of liquid.

Considered by itself, with the cylinders held fixed, the vortex sets up a
circumferential velocity m/r on a radius r, so that the angular momentum
of a circular filament of annular cross section dA is ρm dA, and of the
whole vortex is ρmπ (b2 − a2).

Any circular filament can be started from rest by the application of a
circumferential impulse πρm dr at each end of a diameter; so that a
mechanism attached to the cylinders, which can set up a uniform
distributed impulse πρm across the two parts of a diameter in the liquid,
will generate the vortex motion, and react on the cylinder with an impulse
couple −ρmπa2 and ρmπb2, having resultant ρmπ (b2 − a2), and this
couple is infinite when b = ∞, as the angular momentum of the vortex is
infinite. Round the cylinder r = a held fixed in the U current the liquid
streams past with velocity

q′ = 2U sin θ + m/a;
(2)

and the loss of head due to this increase of velocity from U to q′ is

q′2 − U2 (2U sin θ + m/a)2 − U2
= ,
2g 2g (3)

so that cavitation will take place, unless the head at a great distance
exceeds this loss.

The resultant hydrostatic thrust across any diametral plane of the
cylinder will be modified, but the only term in the loss of head which
exerts a resultant thrust on the whole cylinder is 2mU sin θ/ga, and its
thrust is 2πρmU absolute units in the direction Cy, to be counteracted by a
support at the centre C; the liquid is streaming past r = a with velocity U
reversed, and the cylinder is surrounded by a vortex. Similarly, the

streaming velocity V reversed will give rise to a thrust 2πρmV in the
direction xC.

Now if the cylinder is released, and the components U and V are
reversed so as to become the velocity of the cylinder with respect to space
filled with liquid, and at rest at infinity, the cylinder will experience
components of force per unit length

(i.) − 2πρmV, 2πρmU, due to the vortex motion;

(ii.) − πρa2 dU/dt, − πρa2 dV/dt, due to the kinetic reaction of the
liquid;

(iii.) 0, −π(σ − ρ) a2g, due to gravity,

taking Oy vertically upward, and denoting the density of the cylinder by
σ; so that the equations of motion are

dU dU
πρa2 = − πρa2 − 2πρmV,
dt dt (4)
dV dV
πρa2 = − πρa2 + 2πρmV − π (σ − ρ) a2g,
dt dt (5)

or, putting m = a2ω, so that the vortex velocity is due to an angular
velocity ω at a radius a,

(σ + ρ) dU/dt + 2ρωV = 0,
(6)

(σ + ρ) dV/dt − 2ρωU + (σ-ρ) g = 0.
(7)

Thus with g = 0, the cylinder will describe a circle with angular velocity
2ρω/(σ + ρ), so that the radius is (σ + ρ) v/2ρω, if the velocity is v. With σ
= 0, the angular velocity of the cylinder is 2ω; in this way the velocity

may be calculated of the propagation of ripples and waves on the surface
of a vertical whirlpool in a sink.

Restoring σ will make the path of the cylinder a trochoid; and so the
swerve can be explained of the ball in tennis, cricket, baseball, or golf.

Another explanation may be given of the sidelong force, arising from
the velocity of liquid past a cylinder, which is encircled by a vortex.
Taking two planes x = ± b, and considering the increase of momentum in
the liquid between them, due to the entry and exit of liquid momentum,
the increase across dy in the direction Oy, due to elements at P and P′ at
opposite ends of the diameter PP′, is

ρ dy (U − Ua2r−2 cos 2θ + mr−1 sin θ) (Ua2r−2 sin 2θ + mr−1 cos θ)
+ ρ dy ( −U + Ua2r−2 cos 2θ + mr−1 sin θ) (Ua2r−2 sin 2θ − mr−1 cos θ)
= 2ρdymUr−1 (cos θ − a2r−2 cos 3θ),
(8)

and with y = b tan θ, r = b sec θ, this is

2ρmU dθ (1 − a2b−2 cos 3θ cos θ),
(9)

and integrating between the limits θ = ±½π, the resultant, as before, is
2πρmU.

31. Example 2.—Confocal Elliptic Cylinders.—Employ the elliptic
coordinates η, ξ, and ζ = η + ξi, such that

z = c ch ζ, x = c ch η cos ξ, y = c sh η sin ζ;
(1)

then the curves for which η and ξ are constant are confocal ellipses and
hyperbolas, and

J = d(x, y) = c2 (ch2 η − cos2 ξ)

d(η, ξ)

= (1/2)c2 (ch 2η − cos 2ξ) = r1r2 = OD2,
(2)

if OD is the semi-diameter conjugate to OP, and r1, r2 the focal distances,

r1, r2 = c (ch η ± cos ξ);
(3)

r2 = x2 + y2 = c2 (ch2 η − sin2 ξ)

= ½c2 (ch 2η + cos 2ξ).
(4)

Consider the streaming motion given by

w = m ch (ζ − γ), γ = α + βi,
(5)

φ = m ch (η − α) cos (ξ − β), ψ = m sh (η − α) sin (ξ − β).
(6)

Then ψ = 0 over the ellipse η = α, and the hyperbola ξ = β, so that these
may be taken as fixed boundaries; and ψ is a constant on a C4.

Over any ellipse η, moving with components U and V of velocity,

ψ′ = ψ + Uy − Vx = [ m sh (η − α) cos β + Uc sh η ] sin ξ
- [ m sh (η − α) sin β + Vc ch η ] cos ξ;
(7)

so that ψ′ = 0, if

m sh (η − α) m sh (η − α)
U=− cos β, V = − sin β,
c sh η c ch η (8)

having a resultant in the direction PO, where P is the intersection of an
ellipse η with the hyperbola β; and with this velocity the ellipse η can be

swimming in the liquid, without distortion for an instant.

At infinity

m m
U=− e−a cos β = − cos β,
c a−b
m m
V=− e−a sin β = − sin β,
c a−b (9)

a and b denoting the semi-axes of the ellipse α; so that the liquid is
streaming at infinity with velocity Q = m/(a + b) in the direction of the
asymptote of the hyperbola β.

An ellipse interior to η = α will move in a direction opposite to the
exterior current; and when η = 0, U = ∞, but V = (m/c) sh α sin β.

Negative values of η must be interpreted by a streaming motion on a
parallel plane at a level slightly different, as on a double Riemann sheet,
the stream passing from one sheet to the other across a cut SS′ joining the
foci S, S′. A diagram has been drawn by Col. R. L. Hippisley.

The components of the liquid velocity q, in the direction of the normal
of the ellipse η and hyperbola ξ, are

−mJ−1 sh (η − α) cos (ξ − β), mJ−1 ch (η − α) sin (ξ − β).
(10)

The velocity q is zero in a corner where the hyperbola β cuts the ellipse
α; and round the ellipse α the velocity q reaches a maximum when the
tangent has turned through a right angle, and then

√(ch 2α − cos 2β)
q = Qea ;
sh 2α (11)

and the condition can be inferred when cavitation begins.

With β = 0, the stream is parallel to x0, and

φ = m ch (η − α) cos ξ
= −Uc ch (η − α) sh η cos ξ/sh (η − α)
(12)

over the cylinder η, and as in (12) § 29,

φ1 = −Ux = −Uc ch η cos ξ,
(13)

for liquid filling the cylinder; and

φ th η
= ,
φ1 th (η − α) (14)

over the surface of η; so that parallel to Ox, the effective inertia of the
cylinder η, displacing M′ liquid, is increased by M′th η/th(η- α), reducing
when α = ∞ to M′ th η = M′ (b/a).

Similarly, parallel to Oy, the increase of effective inertia is M′/th η th (η
− α), reducing to M′/th η = M′ (a/b), when α = ∞, and the liquid extends
to infinity.

32. Next consider the motion given by

φ = m ch 2 (η − α) sin 2ξ, ψ = −m sh 2 (η − α) cos 2ξ;
(1)

in which ψ = 0 over the ellipse α, and

ψ′ = ψ + ½R (x2 + y2) = [ −m sh 2 (η − α) + ¼Rc2 ] cos 2ξ + ¼Rc2 ch 2η,
(2)

which is constant over the ellipse η if

¼ Rc2 = m sh 2 (η − α);
(3)

so that this ellipse can be rotating with this angular velocity R for an
instant without distortion, the ellipse α being fixed.

For the liquid filling the interior of a rotating elliptic cylinder of cross
section

x2/a2 + y2/b2 = 1,
(4)

ψ1′ = m1 (x2/a2 + y2/b2)
(5)

with

∇2ψ1′ = −2R = −2m1 (1/a2 + 1/b2),

ψ1 = m1 (x2/a2 + y2/b2) − ½R (x2 + y2)
= −½R (x2 − y2) (a2 − b2) / (a2 + b2),
(6)

φ1 = Rxy (a2 − b2) / (a2 + b2),

w1 = φ1 + ψ1i = −½iR (x + yi)2 (a2 − b2) / (a2 + b2).

The velocity of a liquid particle is thus (a2 − b2)/(a2 + b2) of what it
would be if the liquid was frozen and rotating bodily with the ellipse; and
so the effective angular inertia of the liquid is (a2 − b2)2/(a2 + b2)2 of the
solid; and the effective radius of gyration, solid and liquid, is given by

k2 = ¼(a2 + b2), and ¼(a2 − b2)2 / (a2 + b2).
(7)

For the liquid in the interspace between α and η,

φ m ch 2 (η − α) sin 2ξ
=
φ1 ¼ Rc2 sh 2η sin 2ξ (a2 − b2) / (a2 + b2)

= 1/th 2 (η − α) th 2η;

(8)

and the effective k2 of the liquid is reduced to

¼ c2/th 2 (η − α) sh 2η,
(9)

which becomes ¼ c2/sh 2η = 1⁄8 (a2 − b2)/ab, when α = ∞, and the liquid
surrounds the ellipse η to infinity.

An angular velocity R, which gives components −Ry, Rx of velocity to
a body, can be resolved into two shearing velocities, −R parallel to Ox,
and R parallel to Oy; and then ψ is resolved into ψ1 + ψ2, such that ψ1 +
½Rx2 and ψ2 + ½Ry2 is constant over the boundary.

Inside a cylinder

φ1 + ψ1i = −½ iR (x + yi)2 a2 / (a2 + b2),
(10)

φ2 + ψ2i = ½ iR (x + yi)2 b2 / (a2 + b2),
(11)

and for the interspace, the ellipse α being fixed, and α1 revolving with
angular velocity R

φ1 + ψ1i = −1⁄8 iRc2 sh 2 (η − α + ξi) (ch 2α + 1) / sh 2 (α1 − α),
(12)

φ2 + ψ2i = 1⁄8 iRc2 sh 2 (η − α + ξi) (ch 2α − 1) / sh 2 (α1 − α),
(13)

satisfying the condition that ψ1 and ψ2 are zero over η = α, and over η =
α1

ψ1 + ½ Rx2 = 1⁄8 Rc2 (ch 2α1 + 1),
(14)

ψ2 + ½ Ry2 = 1⁄8 Rc2 (ch 2α1 − 1),
(15)

constant values.

In a similar way the more general state of motion may be analysed,
given by

w = m ch 2 (ζ − γ), γ = α + βi,
(16)

as giving a homogeneous strain velocity to the confocal system; to which
may be added a circulation, represented by an additional term mζ in w.

Similarly, with

x + yi = c√[ sin (ξ + ηi) ]
(17)

the function

ψ = Qc sh ½ (η − α) sin ½ (ξ − β)
(18)

will give motion streaming past the fixed cylinder η = α, and dividing
along ξ = β; and then

x2 − y2 = c2 sin ξ ch η, 2xy = c2 cos ξ sh η.
(19)

In particular, with sh α = 1, the cross-section of η = α is

x4 + 6x2y2 + y4 = 2c4, or x4 + y4 = c4
(20)

when the axes are turned through 45°.

33. Example 3.—Analysing in this way the rotation of a rectangle filled
with liquid into the two components of shear, the stream function ψ1 is to

be made to satisfy the conditions

(i.) ∇2ψ1 = 0,
(ii.) ψ1 + ½Rx2 = ½Ra2, or ψ1 = 0 when x = ±a,
(iii.) ψ1 + ½Rx2 = ½Ra2, ψ1 = ½R (a2 − x2), when y = ± b.

Expanded in a Fourier series,

32 cos (2n + 1) ½ πx/a
a 2 − x2 =
π3 a2 Σ (2n + 1)3
,
(1)

so that

16 cos (2n + 1) ½πx/a · ch (2n + 1) ½πy/a)
ψ1 = R
π3 a2 Σ (2n + 1)3 · ch (2n + 1) ½πb/a
,

16 cos (2n + 1) ½πz/a
w1 = φ1 + ψ1i = iR
π3 a 2 Σ ,
(2n + 1)3 ch (2n + 1) ½πb/a (2)

an elliptic-function Fourier series; with a similar expression for ψ2 with x
and y, a and b interchanged; and thence ψ = ψ1 + ψ2.

Example 4.—Parabolic cylinder, axial advance, and liquid streaming
past.

The polar equation of the cross-section being

r1/2 cos ½θ = a1/2, or r + x = 2a,
(3)

the conditions are satisfied by

ψ′ = Ur sin θ − 2Ua1/2r1/2 sin ½θ = 2Ur1/2 sin ½θ (r1/2 cos ½θ − a1/2),
(4)

ψ = 2Ua1/2r1/2 sin ½θ = −U √ [ 2a (r − x) ],

(5)

w = −2Ua1/2z1/2,
(6)

and the resistance of the liquid is 2πρaV2/2g.

A relative stream line, along which ψ′ = Uc, is the quartic curve

(4a2y2 − (y − c)4 4a2y2 + (y − c)4
y − c = √ [ 2a (r − x) ], x= , r= ,
4a (y − c)2 4a (y − (7)
c)2

and in the absolute space curve given by ψ,

dy (y − c)2 2ac
=− ,x= − 2a log (y − c).
dx 2ay y−c (8)

34. Motion symmetrical about an Axis.—When the motion of a liquid
is the same for any plane passing through Ox, and lies in the plane, a
function ψ can be found analogous to that employed in plane motion,
such that the flux across the surface generated by the revolution of any
curve AP from A to P is the same, and represented by 2π (ψ − ψ0); and, as
before, if dψ is the increase in ψ due to a displacement of P to P′, then k
the component of velocity normal to the surface swept out by PP′ is such
that 2πdψ = 2πyk·PP′; and taking PP′ parallel to Oy and Ox,

u = −dψ/ydy, v = dψ/ydx,
(1)

and ψ is called after the inventor, “Stokes’s stream or current function,” as
it is constant along a stream line (Trans. Camb. Phil. Soc., 1842;
“Stokes’s Current Function,” R. A. Sampson, Phil. Trans., 1892); and
dψ/yds is the component velocity across ds in a direction turned through a
right angle forward.

In this symmetrical motion

d dψ1 d 1 dψ
ξ = 0, η = 0, 2ζ =
dx ( dxy + dy )
y dy ( )
1 d2ψ d2ψ 1 dψ 1
=
y ( dx2
+
dy2
−
y dy = − y ∇2ψ, ) (2)

suppose; and in steady motion,

dH 1 dψ dH 1 dψ
+ ∇2ψ = 0, + ∇2ψ = 0,
dx y2 dx dy y2 dy (3)

so that

2ζ/y = −y−2∇2ψ = dH/dψ
(4)

is a function of ψ, say ƒ′(ψ), and constant along a stream line;

dH/dv = 2qζ, H − ƒ(ψ) = constant,
(5)

throughout the liquid.

When the motion is irrotational,

dφ 1 dψ dφ 1 dψ
ζ = 0, u=− =− , v=− = ,
dx y dy dy y dx (6)
d2ψ d2ψ 1 dψ
∇2ψ = 0, or + − = 0.
dx2 dy2 y dy (7)

Changing to polar coordinates, x = r cos θ, y = r sin θ, the equation (2)
becomes, with cos θ = μ,

d2ψ d2ψ
r2 + (1 − μ2) = 2 ζr3 sin θ,
dr2 dμ (8)

of which a solution, when ζ = 0, is

B dPn B dPn
ψ= ( Arn+1 + rn ) (1 − μ2) dμ = ( Arn − 1 + rn+2 ) y2 dμ (9),
φ = { (n + 1) Arn − nBr−n−1 } Pn,
(10)

where Pn denotes the zonal harmonic of the nth order; also, in the
exceptional case of

ψ = A0 cos θ, φ = A0/r;
ψ = B0r, φ = −B0 log tan ½θ = −½B0 sh−1 x/y.
(11)

Thus cos θ is the Stokes’ function of a point source at O, and PA − PB
of a line source AB.

The stream function ψ of the liquid motion set up by the passage of a
solid of revolution, moving with axial velocity U, is such that

1 dψ dy
= −U , ψ + ½Uy2 = constant,
y ds ds (12)

over the surface of the solid; and ψ must be replaced by ψ′ = ψ + ½Uy2 in
the general equations of steady motion above to obtain the steady relative
motion of the liquid past the solid.

For instance, with n = 1 in equation (9), the relative stream function is
obtained for a sphere of radius a, by making it

ψ′ = ψ + ½Uy2 = ½U (r2 − a3/r) sin2 θ, ψ = −½Ua3 sin2 θ/r;
(13)

and then

φ′ = Ux (1 + ½a3/r2), φ = ½Ua3 cos θ/r2,
(14)

dφ a3 dφ a3
− =U cos θ, − = ½U sin θ,
dr r3 r dθ r3 (15)

so that, if the direction of motion makes an angle ψ with Ox,

tan (ψ − θ) = ½ tan θ, tan ψ = 3 tan θ/(2 − tan2 θ),
(16)

Along the path of a liquid particle ψ′ is constant, and putting it equal to
½Uc2,

(r2 − a3/r) sin2 θ = c2, sin2 θ = c2r / (r3 − a3),
(17)

the polar equation; or

y2 = c2r3 / (r3 − a3), r3 = a3y2 / (y2 − c2),
(18)

a curve of the 10th degree (C10).

In the absolute path in space

cos ψ = (2 − 3 sin2 θ) / √ (4 − sin2 θ), and sin3 θ = (y3 − c2y) / a3,
(19)

which leads to no simple relation.

The velocity past the surface of the sphere is

1 dψ′ a3 sin2 θ
r sin θ dr = ½U ( 2r + r2 ) r sin θ = ⁄ U sin θ, when r =(20)
32 a;

so that the loss of head is

(9⁄4 sin2 θ − 1) U2/2g, having a maximum 5⁄4 U2/2g,
(21)

which must be less than the head at infinite distance to avoid cavitation at
the surface of the sphere.

With n = 2, a state of motion is given by

ψ = −½ Uy2a4 μ/r4, ψ′ = ½ Uy2 (1 − a4 μ/r4),
(22)

φ′ = Ux + φ, φ = −1⁄3 U (a4 / r3) P2, P2 = 3⁄2 μ2 − ½,
(23)

representing a stream past the surface r4 = a4μ.

35. A circular vortex, such as a smoke ring, will set up motion symmetrical
about an axis, and provide an illustration; a half vortex ring can be generated
in water by drawing a semicircular blade a short distance forward, the tip of a
spoon for instance. The vortex advances with a certain velocity; and if an
equal circular vortex is generated coaxially with the first, the mutual influence
can be observed. The first vortex dilates and moves slower, while the second
contracts and shoots through the first; after which the motion is reversed
periodically, as if in a game of leap-frog. Projected perpendicularly against a
plane boundary, the motion is determined by an equal opposite vortex ring, the
optical image; the vortex ring spreads out and moves more slowly as it
approaches the wall; at the same time the molecular rotation, inversely as the
cross-section of the vortex, is seen to increase. The analytical treatment of
such vortex rings is the same as for the electro-magnetic effect of a current
circulating in each ring.

36. Irrotational Motion in General.—Liquid originally at rest in a singly-connected space cannot
be set in motion by a field of force due to a single-valued potential function; any motion set up in the
liquid must be due to a movement of the boundary, and the motion will be irrotational; for any small
spherical element of the liquid may be considered a smooth solid sphere for a moment, and the
normal pressure of the surrounding liquid cannot impart to it any rotation.

The kinetic energy of the liquid inside a surface S due to the velocity function φ is given by

dφ 2 dφ 2 dφ 2
T = ½ρ ∫ ∫ ∫ [ ( dx ) + ( dy ) + ( dz ) ] dx dy dz,
dφ
= ½ρ ∫ ∫ φ dν dS (1)

by Green’s transformation, dν denoting an elementary step along the normal to the exterior of the
surface; so that dφ/dν = 0 over the surface makes T = 0, and then

dφ 2 dφ 2 dφ 2 dφ dφ dφ
( dx ) + ( dy ) + ( dz ) = 0,
dx
= 0,
dy
= 0,
dz
= 0.
(2)

If the actual motion at any instant is supposed to be generated instantaneously from rest by the
application of pressure impulse over the surface, or suddenly reduced to rest again, then, since no
natural forces can act impulsively throughout the liquid, the pressure impulse ῶ satisfies the
equations

1 dῶ 1 dῶ 1 dῶ
= −u, = −v, = ῶ,
ρ dx ρ dy ρ dz (3)

ῶ = ρφ + a constant,
(4)

and the constant may be ignored; and Green’s transformation of the energy T amounts to the theorem
that the work done by an impulse is the product of the impulse and average velocity, or half the
velocity from rest.

In a multiply connected space, like a ring, with a multiply valued velocity function φ, the liquid
can circulate in the circuits independently of any motion of the surface; thus, for example,

φ = mθ = m tan−1 y/x
(5)

will give motion to the liquid, circulating in any ring-shaped figure of revolution round Oz.

To find the kinetic energy of such motion in a multiply connected space, the channels must be
supposed barred, and the space made acyclic by a membrane, moving with the velocity of the liquid;
and then if k denotes the cyclic constant of φ in any circuit, or the value by which φ has increased in
completing the circuit, the values of φ on the two sides of the membrane are taken as differing by k,
so that the integral over the membrane

dφ dφ
∫ ∫ φ dν dS = k ∫ ∫ dν dS, (6)

and this term is to be added to the terms in (1) to obtain the additional part in the kinetic energy; the
continuity shows that the integral is independent of the shape of the barrier membrane, and its
position. Thus, in (5), the cyclic constant k = 2πm.

In plane motion the kinetic energy per unit length parallel to Oz

dφ 2 dφ 2 dψ 2 dψ 2
T = ½ρ ∫ ∫ [ ( dx ) + ( dy ) ] dx dy = ½ρ ∫ ∫ [ ( dx ) + ( dy ) ] dx dy
dφ dψ
= ½ρ ∫ φ dν ds = ½ρ ∫ ψ dν ds. (7)

For example, in the equilateral triangle of (8) § 28, referred to coordinate axes made by the base
and height,

ψ′ = −2Rαβγ/h = −½ Ry [ (h − y)2 − 3x2 ] /h
(8)

ψ = ψ′ − ½R [ ( 1⁄3 h − y)2 + x2 ]
= −½R [ ½h3 + 1⁄3 h2y + h) (x2 − y2) − 3x2y + y3 ] /h
(9)

and over the base y = 0,

dx/dν = −dx/dy = + ½R ( 1⁄3 h2 − 3x2) / h, ψ = −½R ( 1⁄9 h2 + x2).
(10)

Integrating over the base, to obtain one-third of the kinetic energy T,

h / √3
⁄ T = ½ρ
13 ∫ −h / √3 ¼R2 (3x4 − ⁄ h4) dx/h = ρR2 h4 / 135 √3
1 27
(11)

so that the effective k2 of the liquid filling the triangle is given by

k2 = T / ½ρR2A = 2h2 / 45
= 2⁄5 (radius of the inscribed circle)2,
(12)

or two-fifths of the k2 for the solid triangle.

Again, since

dφ/dν = dψ/ds, dφ/ds = −dψ/dν,
(13)

T = ½ρ ∫ φ dψ = −½ρ ∫ ψ dφ.
(14)

With the Stokes’ function ψ for motion symmetrical about an axis.

dψ
T = ½ρ ∫ φ y ds 2πy ds = πρ ∫ φ dψ. (15)

37. Flow, Circulation, and Vortex Motion.—The line integral of the tangential velocity along a
curve from one point to another, defined by

dx dy dz
∫ ( u ds + v ds + w ds ) ds = ∫ (u dx + v dy + z dz), (1)

is called the “flux” along the curve from the first to the second point; and if the curve closes in on
itself the line integral round the curve is called the “circulation” in the curve.

With a velocity function φ, the flow

−∫ dφ = φ1 − φ2,
(2)

so that the flow is independent of the curve for all curves mutually reconcilable; and the circulation
round a closed curve is zero, if the curve can be reduced to a point without leaving a region for which
φ is single valued.

If through every point of a small closed curve the vortex lines are drawn, a tube is obtained, and
the fluid contained is called a vortex filament.

By analogy with the spin of a rigid body, the component spin of the fluid in any plane at a point is
defined as the circulation round a small area in the plane enclosing the point, divided by twice the
area. For in a rigid body, rotating about Oz with angular velocity ζ, the circulation round a curve in
the plane xy is

dy dx
∫ ζ ( x ds − y ds ) ds = ζ times twice the area. (3)

In a fluid, the circulation round an elementary area dxdy is equal to

dv du dv du
u dx + ( v + dx dx ) dy − ( u + dy dy ) dx − vdy = ( dx − dy ) dx dy, (4)

so that the component spin is

dv du
½ ( dx − dy ) = ζ, (5)

in the previous notation of § 24; so also for the other two components ξ and η.

Since the circulation round any triangular area of given aspect is the sum of the circulation round
the projections of the area on the coordinate planes, the composition of the components of spin, ξ, η,
ζ, is according to the vector law. Hence in any infinitesimal part of the fluid the circulation is zero

round every small plane curve passing through the vortex line; and consequently the circulation
round any curve drawn on the surface of a vortex filament is zero.

If at any two points of a vortex line the cross-section ABC, A′B′C′ is drawn of the vortex filament,
joined by the vortex line AA′, then, since the flow in AA′ is taken in opposite directions in the
complete circuit ABC AA′B′C′ A′A, the resultant flow in AA′ cancels, and the circulation in ABC,
A′B′C′ is the same; this is expressed by saying that at all points of a vortex filament ωα is constant
where α is the cross-section of the filament and ω the resultant spin (W. K. Clifford, Kinematic, book
iii.).

So far these theorems on vortex motion are kinematical; but introducing the equations of motion of
§ 22,

Du dQ Dv dQ Dw dQ
+ = 0, + = 0, + = 0,
dt dx dt dy dt dz (6)

Q = ∫ dp/ρ + V,
(7)

and taking dx, dy, dz in the direction of u, v, w, and

dx : dy : dz = u : v : w,

D Du D dx
dt ( u dx + v dy + w dz ) = dt dx + u dt + ... = −dQ + ½ dq2, (8)

and integrating round a closed curve

D
dt ∫ (u dx + v dy + w dz) = 0, (9)

and the circulation in any circuit composed of the same fluid particles is constant; and if the motion
is differential irrotational and due to a velocity function, the circulation is zero round all reconcilable
paths. Interpreted dynamically the normal pressure of the surrounding fluid on a tube cannot create
any circulation in the tube.

The circulation being always zero round a small plane curve passing through the axis of spin in
vortical motion, it follows conversely that a vortex filament is composed always of the same fluid
particles; and since the circulation round a cross-section of a vortex filament is constant, not
changing with the time, it follows from the previous kinematical theorem that αω is constant for all
time, and the same for every cross-section of the vortex filament.

A vortex filament must close on itself, or end on a bounding surface, as seen when the tip of a
spoon is drawn through the surface of water.

Denoting the cross-section α of a filament by dS and its mass by dm, the quantity ωdS/dm is called
the vorticity; this is the same at all points of a filament, and it does not change during the motion; and

the vorticity is given by ω cosεdS/dm, if dS is the oblique section of which the normal makes an
angle ε with the filament, while the aggregate vorticity of a mass M inside a surface S is

M−1 ∫ ω cos ε dS.

Employing the equation of continuity when the liquid is homogeneous,

dζ dη d2 d2 d2
2 ( dy − dz ) = ∇2u, ... , ∇2 = − dx2 − dy2 − dz2 , (10)

which is expressed by

∇2 (u, v, w) = 2 curl (ξ, η, ζ), (ξ, η, ζ) = ½ curl (u, v, w).
(11)

38. Moving Axes in Hydrodynamics.—In many problems, such as the motion of a solid in liquid, it
is convenient to take coordinate axes fixed to the solid and moving with it as the movable trihedron
frame of reference. The components of velocity of the moving origin are denoted by U, V, W, and the
components of angular velocity of the frame of reference by P, Q, R; and then if u, v, w denote the
components of fluid velocity in space, and u′, v′, w′ the components relative to the axes at a point (x,
y, z) fixed to the frame of reference, we have

u = U + u′ − yR + zQ,
v = V + v′- zP + xR,
w = W + w′ − xQ + yP.
(1)

Now if k denotes the component of absolute velocity in a direction fixed in space whose direction
cosines are l, m, n,

k = lu + mv + nw;
(2)

and in the infinitesimal element of time dt, the coordinates of the fluid particle at (x, y, z) will have
changed by (u′, v′, w′)dt; so that

Dk dl dm dn
= u+ v+ w
dt dt dt dt
du du du du
+l (
dt
+ u′
dx
+ v′
dy
+ w′
dz )
dv dv dv dv
+m (
dt
+ u′
dx
+ v′
dy
+ w′
dz )
dw dw dw dw
+n (dt
+ u′
dx
+ v′
dy
+ w′
dz ). (3)

But as l, m, n are the direction cosines of a line fixed in space,

dl = mR − nQ, dm = nP − lR, dn = lQ − mP;

dt dt dt (4)

so that

Dk du du du du
dt = l ( dt − vR + wQ + u′ dx + v′ dy + w′ dz ) + m (...) + n (...)
1 dp 1 dp 1 dp
=l (X− p dx
)+m(Y− p dy
)+n(Z− p dz
), (5)

for all values of l, m, n, leading to the equations of motion with moving axes.

When the motion is such that

dφ dψ dφ dψ dφ dψ
u=− −m ,v=− −m ,w=− −m ,
dx dx dy dy dz dz (6)

as in § 25 (1), a first integral of the equations in (5) may be written

dp dφ dψ dφ dψ
∫ ρ + V + ½q2 − dt − m dt + (u − u′) ( dx + m dx )
dφ dψ dφ dψ
+ (v − v′) ( dy + m dy ) + (w − w′) ( dz + m dz ) = F(t), (7)

in which

dφ dφ dφ dφ
− (u − u′) − (v − v′) − (w − w′)
dt dx dy dz
dφ dφ dφ dφ
= − (U − yR + zQ) − (V − zP + xR) − (W − xQ + yP)
dt dx dy dz (8)

is the time-rate of change of φ at a point fixed in space, which is left behind with velocity
components u − u′, v − v′, w − w′.

In the case of a steady motion of homogeneous liquid symmetrical about Ox, where O is
advancing with velocity U, the equation (5) of § 34

p/ρ + V + ½q′2 − ƒ (ψ′) = constant
(9)

becomes transformed into

p U dψ
+ V + ½q2 − + ½U2 − ƒ (ψ + ½Uy2) = constant,
ρ y dy (10)

ψ′ = ψ + ¼U y2,
(11)

subject to the condition, from (4) § 34,

y−2 ∇2ψ′ = −ƒ′(ψ′), y−2 ∇2ψ = −ƒ′ (ψ + ½Uy2).

(12)

Thus, for example, with

ψ′ = ¾U y2 (r2a−2 − 1), r2 = x2 + y2,
(13)

for the space inside the sphere r = a, compared with the value of ψ′ in § 34 (13) for the space outside,
there is no discontinuity of the velocity in crossing the surface.

Inside the sphere

d 1 dψ′ d 1 dψ′ 15 y
dx ( y dx ) + dy ( y dy ) = 2
2ζ = U ,
a2 (14)

so that § 34 (4) is satisfied, with

15 15
ƒ′ (ψ′) = Ua−2, ƒ (ψ′) = Uψ′ a−2;
2 2 (15)

and (10) reduces to

p 9 x2 2 y2 2

ρ
+V−
8
U { ( a2 − 1 ) − ( a2 − ½ ) } = constant; (16)

this gives the state of motion in M. J. M. Hill’s spherical vortex, advancing through the surrounding
liquid with uniform velocity.

39. As an application of moving axes, consider the motion of liquid filling the ellipsoidal case

x2 y2 z2
+ + = 1;
a2 b2 c2 (1)

and first suppose the liquid to be frozen, and the ellipsoid to be rotating about the centre with
components of angular velocity ξ, η, ζ; then

u = − yζ + zη, v = − zξ + xζ, w = − xη + yξ.
(2)

Now suppose the liquid to be melted, and additional components of angular velocity Ω1, Ω2, Ω3
communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due
to a velocity-function

b2 − c2 c2 − a2 a 2 − b2
φ = − Ω1 yz − Ω2 zx − Ω3 xy,
b2 + c2 c2 + a2 a 2 + b2 (3)

as may be verified by considering one term at a time.

If u′, v′, w′ denote the components of the velocity of the liquid relative to the axes,

2a2 2a2
u′ = u + yR − zQ = Ω3y − Ω2z,
a 2 + b2 c2 + a2 (4)
2b2 2b2
v′ = v + zP − xR = Ω1z − Ω3x,
b2 + c2 a 2 + b2 (5)
2c2 2c2
w′ = w + xQ − yP = Ω2x − Ω1y,
c2 + a2 b2 + c2 (6)

P = Ω1 + ξ, Q = Ω2 + η, R = Ω3 + ζ.
(7)

Thus

x y z
u′ + v′ + w′ = 0,
a2 b2 c2 (8)

so that a liquid particle remains always on a similar ellipsoid.

The hydrodynamical equations with moving axes, taking into account the mutual gravitation of the
liquid, become

1 dp du du du du
+ 4πρAx + − vR + wQ + u′ + v′ + w′ = 0, ... , ... ,
ρ dx dt dx dy dz (9)

where

abcdλ
A, B, C = ∫∞0 (a2 + λ, b2 + λ, c2 + λ) P
P2 = 4 (a2 + λ) (b2 + λ) (c2 + λ).
(10)

With the values above of u, v, w, u′, v′, w′, the equations become of the form

1 dp
+ 4πρ Ax + αx + hy + gz = 0,
ρ dx (11)
1 dp
+ 4πρBy + hx + βy + fz = 0,
ρ dy (12)
1 dp
+ 4πρCz + gx + fy + γz = 0,
ρ dz (13)

and integrating

pρ−1 + 2πρ (Ax2 + By2 + Cz2)
+ ½ (αx2 + βy2 + γz2 + 2fyz + 2gzx + 2hxy) = const.,
(14)

so that the surfaces of equal pressure are similar quadric surfaces, which, symmetry and dynamical
considerations show, must be coaxial surfaces; and f, g, h vanish, as follows also by algebraical
reduction; and

4c2(c2 − a2) c2 − a2 2
α=
(c2 + a2)2 Ω22 − ( c2 + a2 Ω2 − η )
4b2(a2 − b2) a 2 − b2 2
−
(a2 + b2)2 Ω32 − ( a 2 + b2 Ω 3 − ζ ) ,
(15)

with similar equations for β and γ.

If we can make

(4πρA + α) x2 = (4πρB + β) b2 = (4πρC + γ) c2,
(16)

the surfaces of equal pressure are similar to the external case, which can then be removed without
affecting the motion, provided α, β, γ remain constant.

This is so when the axis of revolution is a principal axis, say Oz; when

Ω1 = 0, Ω2 = 0, ξ = 0, η = 0.
(17)

If Ω3 = 0 or θ3 = ζ in addition, we obtain the solution of Jacobi’s ellipsoid of liquid of three
unequal axes, rotating bodily about the least axis; and putting a = b, Maclaurin’s solution is obtained
of the rotating spheroid.

In the general motion again of the liquid filling a case, when a = b, Ω3 may be replaced by zero,
and the equations, hydrodynamical and dynamical, reduce to

dξ 2c2 dη 2a2 dζ 2c2
=− Ω2 ζ, = Ω1 ζ, = (Ω2 ξ − Ω2 η)
dt a 2 + c2 dt a2 + c2 dt a2 + c2 (18)
dΩ1 a2 + c2 dΩ2 a2 + c2
= Ω2 ζ + ηζ, = −Ω1 ζ − ξζ;
dt a2 − c2 dt a2 − c2 (19)

of which three integrals are

a2
ξ2 + η 2 = L − ζ2,
c2 (20)
(a2 + c2)2
Ω12 + Ω22 = M + ζ2,
2c2 (a2 − c2) (21)
a2 + c2
Ω1 ξ + Ω2 ηN = + ζ2;
4c2 (22)

and then

dζ 2 4c4
( dt ) =
(a2 + c2)
(Ω2ξ − Ω12η)2

4c4
= [ (ξ2 + η2) (Ω12 + Ω22) − (Ω1ξ + Ω2η)2 ]
(a2 + c2)2
4c4 (a2 + c2)2 a2 a2 + c2
(a2 + c2)2 [ LM − N2 + { 2c2 (a2 + c2) } ζ2
= −M −N
c2 2c2
(a2 + c2) (9a2 − c2)
−
16c4 (a2 − c2) ζ4 ] = Z, (23)

where Z is a quadratic in ζ2, so that ζ is an elliptic function of t, except when c = a, or 3a.

Put Ω1 = Ω cos φ, Ω2 = −Ω sin φ,

dφ dΩ1 dΩ2 (a2 + c2)
Ω2 = Ω2 − Ω1 = Ω2ζ − (Ω1ξ + Ω2η) ζ,
dt dt dt (a2 − c2) (24)
a 2 + c2
dφ (a2 + c2) N+
4c2
=ζ− · ,
(a2 + c2)2
dt (a2 − c2) M+ ζ2
2c2 (a2 − c2)
(25)
a 2 + c2
ζ dζ a2 + c2 N+ ζ dζ
4c2
φ= ∫ − ∫ (a2 + c2)2
· ,
√Z a2 − c2 M+ ζ2 √Z
2c2 (a2 − c2)
(26)

which, as Z is a quadratic function of ζ2, are non-elliptic integrals; so also for ψ, where ξ = ω cos ψ, η
= −ω sin ψ.

In a state of steady motion

dζ Ω1 Ω2
= 0, = ,
dt ξ η (27)

φ = ψ = nt, suppose,
(28)

Ω1ξ + Ω2η = Ωω,
(29)
dφ a2 + c2 ω
=ζ− ζ,
dt a2 − c2 Ω (30)
dψ 2a2 Ω
=− ζ,
dt a2 + c2 ω (31)
a2 + c2 ω 2a2 Ω
1− =− ,
a2 − c2 Ω a2 + c2 ω (32)

ω a2 + c2 2 (a2 − c2) (9a2 − c2)
( Ω − ½ a2 − c2 ) =
4 (a2 + c2)
,
(33)

and a state of steady motion is impossible when 3a > c > a.

An experiment was devised by Lord Kelvin for demonstrating this, in which the difference of steadiness
was shown of a copper shell filled with liquid and spun gyroscopically, according as the shell was slightly
oblate or prolate. According to the theory above the stability is regained when the length is more than three
diameters, so that a modern projectile with a cavity more than three diameters long should fly steadily when
filled with water; while the old-fashioned type, not so elongated, would be highly unsteady; and for the
same reason the gas bags of a dirigible balloon should be over rather than under three diameters long.

40. A Liquid Jet.—By the use of the complex variable and its conjugate functions, an attempt can be
made to give a mathematical interpretation of problems such as the efflux of water in a jet or of smoke from
a chimney, the discharge through a weir, the flow of water through the piers of a bridge, or past the side of a
ship, the wind blowing on a sail or aeroplane, or against a wall, or impinging jets of gas or water; cases
where a surface of discontinuity is observable, more or less distinct, which separates the running stream
from the dead water or air.

Uniplanar motion alone is so far amenable to analysis; the velocity function φ and stream function
ψ are given as conjugate functions of the coordinates x, y by

w = ƒ(z) where z = x + yi, w = φ + ψi,
(1)

and then

dw dφ dψ
= +i = −u + vi;
dz dx dx (2)

so that, with u = q cos θ, v = q sin θ, the function

dz Q Q Q
ζ = −Q = = (u + vi) = (cos θ + i sin θ),
dw (u − vi) q2 q (3)

gives ζ as a vector representing the reciprocal of the velocity q in direction and magnitude, in terms
of some standard velocity Q.

To determine the motion of a jet which issues from a vessel with plane walls, the vector ζ must be
Constructed so as to have a constant direction θ along a plane boundary, and to give a constant skin
velocity over the surface of a jet, where the pressure is constant.

It is convenient to introduce the function

Ω = log ζ = log (Q/q) + θi
(4)

so that the polygon representing Ω conformally has a boundary given by straight lines parallel to the
coordinate axes; and then to determine Ω and w as functions of a variable u (not to be confused with

the velocity component of q), such that in the conformal
representation the boundary of the Ω and w polygon is made
to coincide with the real axis of u.

It will be sufficient to give a few illustrations.

Consider the motion where the liquid is coming from an
infinite distance between two parallel walls at a distance xx′ Fig. 4.
(fig. 4), and issues in a jet between two edges A and A′; the
wall xA being bent at a corner B, with the external angle β = ½π/n.

The theory of conformal representation shows that the motion is given by

√ (b − a′·u − a) + √(b − a·u − a′) 1/n
ζ= [ √ (a − a′·u − b)
] , u = ae−πw/m;
(5)

where u = a, a′ at the edge A, A′; u = b at a corner B; u = 0 across xx′ where φ = ∞; and u = ∞, φ = ∞
across the end JJ′ of the jet, bounded by the curved lines APJ, A′P′J′, over which the skin velocity is
Q. The stream lines xBAJ, xA′J′ are given by ψ = 0, m; so that if c denotes the ultimate breadth JJ′ of
the jet, where the velocity may be supposed uniform and equal to the skin velocity Q,

m = Qc, c = m/Q.

If there are more B corners than one, either on xA or x′A′, the expression for ζ is the product of
corresponding factors, such as in (5).

Restricting the attention to a single corner B,

Q n √ (b − a′·u − a) + √ (b − a·u − a′)
ζn = (q) (cos nθ + i sin nθ) =
√ (a − a′·u − b)
,
(6)
Q n Q n
ch nω = ch log (q) cos nθ + i sh log (q) sin nθ

b − a′ u−a
= ½(ζn + ζ−n) = √ a − a′ √ u − b , (7)
Q Q n
sh nΩ = sh log ( )
cos nθ + i ch log
q q
(sin nθ )
b−a u − a′
= ½(ζn + ζ−n) = √
a − a′ u−b √,
(8)

∞ > a > b > 0 > a′ > −∞
(9)

and then

dΩ 1 √ (b − a′·b − a′) dw m
=− , =− ,
du 2n (u − b) √ (a − a·u − a′) du πu (10)

the formulas by which the conformal representation is obtained.

For the Ω polygon has a right angle at u = a, a′, and a zero angle at u = b, where θ changes from 0
to ½π/n and Ω increases by ½iπ/n; so that

dΩ A √ (b − a·b − a′)
= , where A = .
du (u − b) √ (u − a·u − a′) 2n (11)

And the w polygon has a zero angle at u = 0, ∞, where ψ changes from 0 to m and back again, so
that w changes by im, and

dw B m
= , where B = − .
du u π (12)

Along the stream line xBAPJ,

ψ = 0, u = ae−πφ/m;
(13)

and over the jet surface JPA, where the skin velocity is Q,

dφ
= −q = −Q, u = aeπsQ/m = aeπs/c,
ds (14)

denoting the arc AP by s, starting at u = a;

b − a′ u−a
ch nΩ = cos nθ = √ a − a′ √ u − b , (15)
a−b u − a′
sh nΩ = i sin nθ = i √ a − a′ √ u − b , (16)

∞ > u = aeπs/c > a,
(17)

and this gives the intrinsic equation of the jet, and then the radius of curvature

ds 1 dφ i dw i dw dΩ
ρ=−
dθ
=
Q dθ
=
Q dΩ
=
Q du / du
c u − b √ (u − a·u − a′)
= · ,
π u √ (a − b·b − a′) (18)

not requiring the integration of (11) and (12)

If θ = α across the end JJ′ of the jet, where u = ∞, q = Q,

b − a′ a−b
ch nΩ = cos nα = √ a − a′ , sh nΩ = i sin nα= i √ a − a′ , (19)

Then

cos 2nα − cos 2nθ = 2 a − b·b − a′ = ½sin2 2nα a − a′

a − a′·u − b u−b
√ (a − b.b − a′) √ (u − a·u − b′)
sin 2nθ = 2
a − a′·u − b (20)
√ (a − a·b − a′)
= sin 2nα ;
u−b
2n c b √ (a − b·b − a′)
φ ρ ( 1+ u−b )
√ (u − a·u − a′) (21)
a − a′ + (a + a′) cos 2nα − [ a + a′ + (a − a′) cos 2nα ] cos 2nθ cos 2nα − cos 2nθ
= × .
(a − a′) sin2 2nα sin 2nθ

Along the wall AB, cos nθ = 0, sin nθ = 1,

a > u > b,
(22)
Q n b − a′ a−u
ch nΩ = i sh log (q) =i √ a − a′ √ u − b , (23)
Q n a−b u − a′
sh nΩ = i ch log ( q
) =i √ a − a′
√ u−b
,
(24)
ds ds dφ m c Q
= = =
du dφ dt πqu π qu (25)
AB Q du √ (a − b) √ (u − a′) + √ (b − a′) √ (a − u) 1/n du
π
c = ba∫q u ∫[ √ (a − a′) √ (u − b′) ] u
.
(26)

Along the wall Bx, cos nθ = 1, sin nθ = 0,

b>u>0
(27)
Q n b − a′ a−u
ch nΩ = ch log (q) = √ a − a′ √ b − u , (28)
Q n a−b u − a′
sh nΩ = sh log (q) = √ a − a′ √ b − u . (29)

At x where φ = ∞, u = 0, and q = q0,

Q n b − a′ a a−b −a′
( q0 ) = √ a − a′ √ b + √ a − a′ √ q .
(30)

In crossing to the line of flow x′A′P′J′, ψ changes from 0 to m, so that with q = Q across JJ′, while
across xx′ the velocity is q0, so that

m = q0·xx′ = Q·JJ′
(31)
JJ′ q0 b − a′ a a−b −a′ 1/n

xx′
=
Q
[ √ a − a′ √ b − √ a − a′ √ b ] ,
(32)

giving the contraction of the jet compared with the initial breadth of the stream.

Along the line of flow x′A′P′J′, ψ = m, u = a′e−πφ/m, and from x′ to A′, cos nθ = 1, sin nθ = 0,

Q n b − a′ a−u
ch nΩ = ch log (q) = √ a − a′ √ b − u , (33)
Q n a−b u − a′
sh nΩ = sh log (q) = √ a − a′ √ b − u . (34)

0 > u > a′.
(35)

Along the jet surface A′J′, q = Q,

b − a′ a−u
ch nΩ = cos nθ = √ a − a′ √ b − u , (36)
a−b u − a′
sh nΩ = i sin nθ = i √ a − a′ √ b − u . (37)

a′ > u = a′eπ/sc > −∞,
(38)

giving the intrinsic equation.

41. The first problem of this kind, worked out by H. v. Helmholtz, of the efflux of a jet between
two edges A and A1 in an infinite wall, is obtained by the symmetrical duplication of the above, with
n = 1, b = 0, a′ = −∞, as in fig. 5,

u−a −a
ch Ω = √ u , sh Ω = √ u ; (1)

and along the jet APJ, ∞ > u = aeπs/c > a,

a
sh Ω = i sin θ − i √ u = ie−1/2 πs/c, (2)
c c
PM = ∫∞s sinθ ds = ∫ e−½πs/c ds = ½π e−1/2 πs/c = ½π sin θ, (3)

so that PT = c/½π, and the curve AP is the tractrix; and the coefficient of contraction, or

breadth of the jet π
= .
breadth of the orifice π+2 (4)

A change of Ω and θ into nΩ and nθ will give the solution for two walls converging symmetrically
to the orifice AA1 at an angle π/n. With n = ½, the reentrant walls are given of Borda’s mouthpiece,
and the coefficient of contraction becomes ½. Generally, by making a′ = −∞, the line x′A′ may be
taken as a straight stream line of infinite length, forming an axis of symmetry; and then by

duplication the result can be obtained, with assigned n, a, and b, of the efflux from a symmetrical
converging mouthpiece, or of the flow of water through the arches of a bridge, with wedge-shaped
piers to divide the stream.

Fig. 5. Fig. 6.

42. Other arrangements of the constants n, a, b, a′ will give the results of special problems
considered by J. M. Michell, Phil. Trans. 1890.

Thus with a′ = 0, a stream is split symmetrically by a wedge of angle π/n as in Bobyleff’s problem;
and, by making a = ∞, the wedge extends to infinity; then

b n
ch nΩ = √ b − u , sh nΩ = √ b − u . (1)

Over the jet surface ψ = m, q = Q,

u = − eπφ/m = − beπ2/c,

1 eπ2/c
ch Ω = cos nθ = √ eπ /c + 1 , sh Ω = i sin nθ = i √ eπ /c + 1 ,
2 2
(2)
½π ds 2n
e½π2/c = tan nθ, = .
c dθ sin 2nθ (3)

For a jet impinging normally on an infinite plane, as in fig. 6, n = 1,

e½π2/c = tan θ, ch (½πs/c) sin 2θ = 1,
(4)

sh ½πx/c = cot θ, sh ½πy/c = tan θ,

sh ½πx/c sh ½πy/c = 1, e½π(x + y)/c = e1/2 πx/c + e1/2 πy/c + 1.
(5)

With n = ½, the jet is reversed in direction, and the profile is the catenary of equal strength.

In Bobyleff’s problem of the wedge of finite breadth,

ch nΩ = √ b √ u − a , sh nΩ = √ b − a √ u ,

a u−b a u−b (6)
b a−b
cos nα = √ a , sin nα = √ a
,
(7)

and along the free surface APJ, q = Q, ψ = 0, u = e−πφ/m = aeπs/c,

eπ2/c − 1
cos nθ = cos nα √ eπ /c − cos2 nα ,
2

cos2 nα sin2 nθ
eπ2/c = ,
sin2 nθ − sin2 nα (8)

the intrinsic equation, the other free surface A′P′J′ being given by

cos2 nα sin2 nθ
eπ2/c = ,
sin2 nα − sin2 nθ (9)

Putting n = 1 gives the case of a stream of finite breadth disturbed by a transverse plane, a
particular case of Fig. 7.

When a = b, α = 0, and the stream is very broad compared with the wedge or lamina; so, putting w
= w′(a − b)/a in the penultimate case, and

u = ae−w ≈ a − (a − b)w′,
(10)
w′ + 1 1
ch nΩ = √ w′ , sh nΩ = √ √ w′
,
(11)

in which we may write

w′ = φ + ψi.
(12)

Along the stream line xABPJ, ψ = 0; and along the jet surface APJ, −1 > φ > −∞; and putting φ =
−πs/c − 1, the intrinsic equation is

πs/c = cot2 nθ,
(13)

which for n = 1 is the evolute of a catenary.

43. When the barrier AA′ is held oblique to the current, the stream line xB is curved to the branch
point B on AA′ (fig. 7), and so must be excluded from the boundary of u; the conformal
representation is made now with

dΩ √ (b − a·b − a′)
=−
du (u − b) √ (u − a·u − a′) (1)
dw m 1 m′ 1
=− − ,
du π u−j π u−j

m + m′ u−b
=− · ,
π u − j·u − j′
mj′ + m′j
b= ,
m + m′ (2)

taking u = ∞ at the source where φ = ∞, u = b at the branch
point B, u = j, j′ at the end of the two diverging streams where
φ = −∞; while ψ = 0 along the stream line which divides at B
and passes through A, A′; and ψ = m, −m′ along the outside
boundaries, so that m/Q, m′/Q is the final breadth of the jets, Fig. 7.
and (m + m′)/Q is the initial breadth, c1 of the impinging
stream. Then

b − a′ u−b b−a u − a′
ch ½Ω = √ a − a′ √ u − b , sh ½Ω = √ a − a′ √ u − b , (3)
2b − a − a′ N
ch Ω = − ,
a − a′ u−b
√ (2·a − u·u − a′)
sh Ω = √ N ,
u−b
a − b·b − a′
N=2 .
a − a′ (4)

Along a jet surface, q = Q, and

chΩ = cos θ = cos α − ½sin2 α(a − a′) / (u − b),
(5)

if θ = α at the source x of the jet xB, where u = ∞; and supposing θ = β, β′ at the end of the streams
where u = j, j′,

u−b ½sin2 α u−j cosθ − cosβ
= , ½sin2 α ,
a − a′ cos α − cos θ a − a′ (cos α − cos β) (cos α − cos θ)
u − j′ cos θ − cos β′
= ½sin2 α ;
a − a′ (cosα − cos β′) (cos α − cosθ) (6)

and ψ being constant along a stream line

dφ dw ds dφ dw du
= ,Q = = ,
du du dθ dθ du dθ
πQ ds π ds (cos α − cos β) (cos α − cos β′) sin θ
= = ,
m + m′ dθ c dθ (cos α − cos θ) (cos θ − cos β) (cos θ − cos α′)
sin θ cos α − cos β′ sin θ
= + ·
cos α − cos θ cos β − cos β′ cos θ − cos β
cos α − cos β sin θ
· ,
cos β − cos β′ cos θ − cos β′ (7)

giving the intrinsic, equation of the surface of a jet, with proper attention to the sign.

From A to B, a > u > b, θ = 0,

Q a − a′
ch Ω = ch log = cos α − ½ sin2 α
q a−b
Q √ (a − u·u − a′)
sh Ω = sh log = sinα
q u−b
Q (u − b) cos α − ½ (a − a′) sin2 α + √ (a − u·u − a′) sin α
=
q u−b (8)
ds ds dφ Q dw
Q =Q =−
du dφ du q du
m + m′ (u − b) cos α − ½ (a − a′) sin2 α + √ (a − u·u − a′) sin α
= ·
π j − u·u − j′
AB (2b − a − a′) (u − b) − 2(a − b) (b − a′) + 2√ (a − b·b − a′·a − u·u − a′)
π
c = ba ∫ a − a′·j − u·u − j′
du,
(10)

with a similar expression for BA′.

The motion of a jet impinging on an infinite barrier is obtained by putting j = a, j′ = a′; duplicated
on the other side of the barrier, the motion reversed will represent the direct collision of two jets of
unequal breadth and equal velocity. When the barrier is small compared with the jet, α = β = β′, and
G. Kirchhoff’s solution is obtained of a barrier placed obliquely in an infinite stream.

Two corners B1 and B2 in the wall xA, with a′ = −∞, and n = 1, will give the solution, by
duplication, of a jet issuing by a reentrant mouthpiece placed symmetrically in the end wall of the
channel; or else of the channel blocked partially by a diaphragm across the middle, with edges turned
back symmetrically, problems discussed by J. H. Michell, A. E. H. Love and M. Réthy.

When the polygon is closed by the walls joining, instead of reaching back to infinity at xx′, the
liquid motion must be due to a source, and this modification has been worked out by B. Hopkinson
in the Proc. Lond. Math. Soc., 1898.

Michell has discussed also the hollow vortex stationary inside a polygon (Phil. Trans., 1890); the
solution is given by

ch nΩ = sn w, sh nΩ = i cn w
(11)

so that, round the boundary of the polygon, ψ = K′, sin nθ = 0; and on the surface of the vortex ψ = 0,
q = Q, and

cos nθ = sn φ, nθ = ½π − am s/c,
(12)

the intrinsic equation of the curve.

This is a closed Sumner line for n = 1, when the boundary consists of two parallel walls; and n = ½
gives an Elastica.

44. The Motion of a Solid through a Liquid.—An important problem in the motion of a liquid is
the determination of the state of velocity set up by the passage of a solid through it; and thence of the
pressure and reaction of the liquid on the surface of the solid, by which its motion is influenced when
it is free.

Beginning with a single body in liquid extending to infinity, and denoting by U, V, W, P, Q, R the
components of linear and angular velocity with respect to axes fixed in the body, the velocity
function takes the form

φ = Uφ1 + Vφ2 + Wφ3 + Pχ1 + Qχ2 + Rχ3,
(1)

where the φ’s and χ’s are functions of x, y, z, depending on the shape of the body; interpreted
dynamically, C − ρφ represents the impulsive pressure required to stop the motion, or C + ρφ to start
it again from rest.

The terms of φ may be determined one at a time, and this problem is purely kinematical; thus to
determine φ1, the component U alone is taken to exist, and then l, m, n, denoting the direction
cosines of the normal of the surface drawn into the exterior liquid, the function φ1 must be
determined to satisfy the conditions

(i.) ∇2φ1 = 0. throughout the liquid;

(ii.) dφ1/dυ = −l, the gradient of φ down the normal at the surface of the moving solid;

(iii.) dφ1/dυ = 0, over a fixed boundary, or at infinity;

similarly for φ2 and φ3.

To determine χ1 the angular velocity P alone is introduced, and the conditions to be satisfied are

(i.) ∇2χ1 = 0, throughout the liquid;

(ii.) dχ1/dυ = mz − ny, at the surface of the moving body, but zero over a fixed surface, and at
infinity; the same for χ2 and χ3.

For a cavity filled with liquid in the interior of the body, since the liquid inside moves bodily for a
motion of translation only,

φ1 = −x, φ2 = −y, φ3 = −z;
(2)

but a rotation will stir up the liquid in the cavity, so that the χ’s depend on the shape of the surface.

The ellipsoid was the shape first worked out, by George Green, in his Research on the Vibration of
a Pendulum in a Fluid Medium (1833); the extension to any other surface will form an important step
in this subject.

A system of confocal ellipsoids is taken

x2 y2 z2
+ + =1
a2 + λ b2 + λ c2 + λ (3)

and a velocity function of the form

φ = xψ,
(4)

where ψ is a function of λ only, so that ψ is constant over an ellipsoid; and we seek to determine the
motion set up, and the form of ψ which will satisfy the equation of continuity.

Over the ellipsoid, p denoting the length of the perpendicular from the centre on a tangent plane,

px py pz
l= , m= , n=
a2 + λ b2 + λ c2 + λ (5)
p2x2 p2y2 p2z2
1= + + ,
(a2 + λ)2 (b2 + λ)2 (c2 + λ)2

p2 = (a2 + λ) l2 + (b2 + λ) m2 + (c2 + λ) n2,
= a2l2 + b2m2 + c2n2 + λ,
(7)
dp dλ
2p = ;
ds ds (8)

Thence

dφ dx dψ
= ψ+x
ds ds ds
dx dψ dp
= ψ + 2 (a2 + λ) l ,
ds dλ ds (9)

so that the velocity of the liquid may be resolved into a component -ψ parallel to Ox, and −2(a2 + λ)l
dψ/dλ along the normal of the ellipsoid; and the liquid flows over an ellipsoid along a line of slope
with respect to Ox, treated as the vertical.

Along the normal itself

dφ dψ
ds { ψ + 2(a2 + λ) dλ } l, (10)

so that over the surface of an ellipsoid where λ and ψ are constant, the normal velocity is the same as
that of the ellipsoid itself, moving as a solid with velocity parallel to Ox

dψ
U = −ψ − 2 (a2 + λ) ,
dλ (11)

and so the boundary condition is satisfied; moreover, any ellipsoidal surface λ may be supposed
moving as if rigid with the velocity in (11), without disturbing the liquid motion for the moment.

The continuity is secured if the liquid between two ellipsoids λ and λ1, moving with the velocity U
and U1 of equation (11), is squeezed out or sucked in across the plane x = 0 at a rate equal to the
integral flow of the velocity ψ across the annular area α1 − α of the two ellipsoids made by x = 0; or
if

dα
αU − α1U1 = ∫λλ ψ dλ dλ,
1
(12)

α = π√ (b2 + λ.c2 + λ).
(13)

Expressed as a differential relation, with the value of U from (11),

d dψ dα
dλ [ αψ + 2 (a2 + λ) α dλ ] − ψ dλ = 0, (14)
dψ d dψ
3α
dλ
+ 2 (a2 + λ)
dλ ( α dλ ) = 0, (15)

and integrating

dψ
(a2 + λ)3/2 α = a constant,
dλ (16)

so that we may put

M dλ
ψ= ∫ (a2 + λ) P , (17)

P2 = 4 (a2 + λ) (b2 + λ) (c2 + λ),
(18)

where M denotes a constant; so that ψ is an elliptic integral of the second kind.

The quiescent ellipsoidal surface, over which the motion is entirely tangential, is the one for which

dψ
2 (a2 + λ) + ψ = 0,
dλ (19)

and this is the infinite boundary ellipsoid if we make the upper limit λ1 = ∞.

The velocity of the ellipsoid defined by λ = 0 is then

dψ0
U = −2a2 − ψ0
dλ
M M dλ
=
abc − ∫∞0 (a2 + λ)P

M
= (1 − A0),
abc (20)

with the notation

abc dλ
A or Aλ = ∫∞λ (a2 + λ) P
d dλ
= −2abc
da2 ∫∞λ P , (21)

so that in (4)

M UxA xAλ
φ= xA = , φ1 = ,
abc 1 − A0 1 − A0 (22)

in (1) for an ellipsoid.

The impulse required to set up the motion in liquid of density ρ is the resultant of an impulsive
pressure ρφ over the surface S of the ellipsoid, and is therefore

∫ ∫ ρφl dS = ρψ0 ∫ ∫ xl dS = ρψ0 (volume of the ellipsoid) = ψ0W′,
(23)

where W′ denotes the weight of liquid displaced.

Denoting the effective inertia of the liquid parallel to Ox by αW′. the momentum

αW′U = ψ0W′
(24)
ψ0 A0
α= = ;
U 1 − A0 (25)

in this way the air drag was calculated by Green for an ellipsoidal pendulum.

Similarly, the inertia parallel to Oy and Oz is

B0 C0
βW′ = W′, γW′ = W′,
1 − B0 1 − C0 (26)
abc dλ
∫
Bλ, Cλ = ∞λ (b2 + λ, c2 + λ) P ;
(27)

and

A + B + C = abc / ½P, A0 + B0 + C0 = 1.
(28)

For a sphere

a = b = c, A0 = B0 = C0 = 1⁄3, α = β = γ = ½,

(29)

so that the effective inertia of a sphere is increased by half the weight of liquid displaced; and in
frictionless air or liquid the sphere, of weight W, will describe a parabola with vertical acceleration

W − W′
g.
W + ½W′ (30)

Thus a spherical air bubble, in which W/W′ is insensible, will begin to rise in water with acceleration
2g.

45. When the liquid is bounded externally by the fixed ellipsoid λ = λ1, a slight extension will give
the velocity function φ of the liquid in the interspace as the ellipsoid λ = 0 is passing with velocity U
through the confocal position; φ must now take the form x(ψ + N), and will satisfy the conditions in
the shape

abc abcdλ
A + B 1 + C1
a1b1c1 + ∫λλ (a2 + λ) P
1

φ = Ux = Ux ,
abc abcdλ
B0 + C 0 − B 1 − C 1 1−
a1b1c1 − ∫ λ01
(a2 + λ) P (1)

and any confocal ellipsoid defined by λ, internal or external to λ = λ1, may be supposed to swim with
the liquid for an instant, without distortion or rotation, with velocity along Ox

Bλ + C λ − B 1 − C 1
U .
B0 + C 0 − B 1 − C 1

Since − Ux is the velocity function for the liquid W′ filling the ellipsoid λ = 0, and moving bodily
with it, the effective inertia of the liquid in the interspace is

A 0 + B1 + C1
W′.
B0 + C 0 − B 1 − C 1 (2)

If the ellipsoid is of revolution, with b = c,

A + 2B1
φ = ½Ux ,
B0 − B 1 (3)

and the Stokes’ current function ψ can be written down

B − B1
ψ = − ½ Uy2 ;
B0 − B 1 (4)

reducing, when the liquid extends to infinity and B1 = 0, to

A B
φ = ½ Ux , ψ = − ½ Uy2 ;
B0 B0 (5)

so that in the relative motion past the body, as when fixed in the current U parallel to xO,

A B
φ′ = ½Ux ( 1 + B0 ) , ψ′ = ½Uy2 ( 1 − B0 ) . (6)

Changing the origin from the centre to the focus of a prolate spheroid, then putting b2 = pa, λ =
λ′a, and proceeding to the limit where a = ∞, we find for a paraboloid of revolution

p B p
B=½ , = ,
p + λ′ B0 p + λ′ (7)
y2
= p + λ′ − 2x,
p + λ′ (8)

with λ′ = 0 over the surface of the paraboloid; and then

ψ′ = ½ U [ y2 − p √ (x2 + y2) + px ];
(9)

ψ = −½ Up [ √ (x2 + y2) − x ];
(10)

φ = −½ Up log [ √ (x2 + y2) + x ].
(11)

The relative path of a liquid particle is along a stream line

ψ′ = ½ Uc2, a constant,
(12)
p2y2 − (y2 − c2)2 p2y2 − (y2 − c2)2
x= , √ (x2 + y2) =
2p (y2 − c2) 2p (y2 − c2) (13)

a C4; while the absolute path of a particle in space will be given by

dy r−x y2 − c2
=− = ,
dx y 2py (14)

y2 − c2 = a2 e−x/p.
(15)

46. Between two concentric spheres, with

a2 + λ = r2, a2 + λ1 = a12,
(1)

A = B = C = a3 / 3r3,

a3/r3 + 2 a3/a13 a3/r3 − a3/a13
φ = ½ Ux , ψ = ½ Uy2 ;
1 − a4/a12 1 − a3/a13 (2)

and the effective inertia of the liquid in the interspace is

A0 + 2A1 a13 + 2a3
W′ = ½ W′.
2A0 − 2A1 a13 − a3 (3)

When the spheres are not concentric, an expression for the effective inertia can be found by the
method of images (W. M. Hicks, Phil. Trans., 1880).

The image of a source of strength μ at S outside a sphere of radius a is a source of strength μa/ƒ at
H, where OS = ƒ, OH = a2/ƒ, and a line sink reaching from the image H to the centre O of line
strength −μ/a; this combination will be found to produce no flow across the surface of the sphere.

Taking Ox along OS, the Stokes’ function at P for the source S is μ cos PSx, and of the source H
and line sink OH is μ(a/ƒ) cos PHx and −(μ/a)(PO − PH); so that

a PO − PH
ψ=μ ( cos PSx + ƒ cos PHx − a ), (4)

and ψ = −μ, a constant, over the surface of the sphere, so that there is no flow across.

When the source S is inside the sphere and H outside, the line sink must extend from H to infinity
in the image system; to realize physically the condition of zero flow across the sphere, an equal sink
must be introduced at some other internal point S′.

When S and S′ lie on the same radius, taken along Ox, the Stokes’ function can be written down;
and when S and S′ coalesce a doublet is produced, with a doublet image at H.

For a doublet at S, of moment m, the Stokes’ function is

d y2
m cos PSx = −m ;
dƒ PS3 (5)

and for its image at H the Stokes’ function is

d a3 y2
m cos PHx = −m ;
dƒ ƒ3 PH3 (6)

so that for the combination

a3 1 1 y2 a3 ƒ3
ψ = my2 ( ƒ3 PH3
−
PS3 ) = m ƒ3 ( PH3 − PS3 ), (7)

and this vanishes over the surface of the sphere.

There is ao Stokes’ function when the axis of the doublet at S does not pass through O; the image
system will consist of an inclined doublet at H, making an equal angle with OS as the doublet S, and
of a parallel negative line doublet, extending from H to O, of moment varying as the distance from
O.

A distribution of sources and doublets over a moving surface will enable an expression to be
obtained for the velocity function of a body moving in the presence of a fixed sphere, or inside it.

The method of electrical images will enable the stream function ψ′ to be inferred from a
distribution of doublets, finite in number when the surface is composed of two spheres intersecting at
an angle π/m, where m is an integer (R. A. Herman, Quart. Jour. of Math. xxii.).

Thus for m = 2, the spheres are orthogonal, and it can be verified that

a13 a23 a3
ψ′ = ½ Uy2 ( 1 − r13 − r23 + r3 ), (8)

where a1, a2, a = a1a2/√ (a12 + a22) is the radius of the spheres and their circle of intersection, and r1,
r2, r the distances of a point from their centres.

The corresponding expression for two orthogonal cylinders will be

a12 a22 a2
ψ′ = Uy ( 1 − r12 − r22 + r2 ). (8)

With a2 = ∞, these reduce to

a5 x a4 x
ψ′ = ½Uy2 ( 1 − r5 ) a , or Uy ( 1 − r4 ) a , (10)

for a sphere or cylinder, and a diametral plane.

Two equal spheres, intersecting at 120°, will require

x a3 a4 (a − 2x) a3 a4 (a + 2x)
ψ′ = ½Uy2 [ a − 2r13 + 2r15
+
2r23
−
2r25 ], (11)

with a similar expression for cylinders; so that the plane x = 0 may be introduced as a boundary,
cutting the surface at 60°. The motion of these cylinders across the line of centres is the equivalent of
a line doublet along each axis.

47. The extension of Green’s solution to a rotation of the ellipsoid was made by A. Clebsch, by
taking a velocity function

φ = xyχ
(1)

for a rotation R about Oz; and a similar procedure shows that an ellipsoidal surface λ may be in
rotation about Oz without disturbing the motion if

[ 1/ (a2 + λ) + 1/ (b2 + λ) ] χ + 2 dx/dλ
R=− ,
1 / (b2 + λ) − 1 / (a2 + λ) (2)

and that the continuity of the liquid is secured if

dχ
(a2 + λ)3/2 (b2 + λ)3/2 (c2 + λ) ½ = constant,
dλ (3)
N dλ N Bλ − A λ
χ= ∫∞λ (a2 + λ) (b2 + λ) P = abc · a2 − b2 ; (4)

and at the surface λ = 0,

[ (1/a2 + 1/b2) · N/abc · (B0 − A0)/(a2 − b2) ] − N/abc · 1/a2b2
R=− ,
1/b2 − 1/a2 (5)
N 1/b2 − 1/a2
=R ,
abc 1/a2b2 − [ (1/a2 + 1/b2) · (B0 − A0) / (a2 − b2) ] (6)
(a2 − b2)2 / (a2 + b2)
=R
(a2 − b2) / (a2 + b2) − (B0 − A0)

The velocity function of the liquid inside the ellipsoid λ = 0 due to the same angular velocity will
be

φ1 = Rxy (a2 − b2) / (a2 + b2),
(7)

and on the surface outside

N B0 − A 0
φ0 = xyχ0 = xy ,
abc a 2 − b2 (8)

so that the ratio of the exterior and interior value of φ at the surface is

φ0 B0 − A 0
= ,
φ1 (a2 − b2) / (a2 + b2) − (B0 − A0) (9)

and this is the ratio of the effective angular inertia of the liquid, outside and inside the ellipsoid λ = 0.

The extension to the case where the liquid is bounded externally by a fixed ellipsoid λ = λ1 is
made in a similar manner, by putting

φ = xy (χ + M),
(10)

and the ratio of the effective angular inertia in (9) is changed to

a12 − b12 abc
(B0 − A0) − (B1 − A1) +
a12 + b12 a1b1c1
.
a 2 − b2 a12 − b12 abc
− − (B0 − A0) + (B1 − A1)
a 2 + b2 a12 + b12 a1b1c1 (11)

Make c = ∞ for confocal elliptic cylinders; and then

ab ab b2 + λ
Aλ = ∫∞λ (a2 + λ) √ (4a2 + λb2 + λ) = a2 − b2 ( 1 − √ a2 + λ ), (12)
ab a2 + λ
Bλ =
a 2 − b2 ( √ b2 + λ − 1 ), Cλ = 0;

and then as above in § 31, with

a = c ch α, b = c sh α, a1 = √ (a2 + λ) = c ch α1, b1 = c sh α1
(13)

the ratio in (11) agrees with § 31 (6).

As before in § 31, the rotation may be resolved into a shear-pair, in planes perpendicular to Ox and
Oy.

A torsion of the ellipsoidal surface will give rise to a velocity function of the form φ = xyzΩ,
where Ω can be expressed by the elliptic integrals Aλ, Bλ, Cλ, in a similar manner, since

Ω=L ∫∞λ dλ / P3.
48. The determination of the φ’s and χ’s is a kinematical problem, solved as yet only for a few
cases, such as those discussed above.

But supposing them determined for the motion of a body through a liquid, the kinetic energy T of
the system, liquid and body, is expressible as a quadratic function of the components U, V, W, P, Q,
R. The partial differential coefficient of T with respect to a component of velocity, linear or angular,
will be the component of momentum, linear or angular, which corresponds.

Conversely, if the kinetic energy T is expressed as a quadratic function of x1, x2, x3, y1, y2, y3, the
components of momentum, the partial differential coefficient with respect to a momentum
component will give the component of velocity to correspond.

These theorems, which hold for the motion of a single rigid body, are true generally for a flexible
system, such as considered here for a liquid, with one or more rigid bodies swimming in it; and they
express the statement that the work done by an impulse is the product of the impulse and the
arithmetic mean of the initial and final velocity; so that the kinetic energy is the work done by the
impulse in starting the motion from rest.

Thus if T is expressed as a quadratic function of U, V, W, P, Q, R, the components of momentum
corresponding are

dT dT dT
x1 = , x2 = , x3 = ,
dU dV dW (1)
dT dT dT
y1 = , y2 = , y3 = ;
dP dQ dR

but when it is expressed as a quadratic function of x1, x2, x3, y1, y2, y3,

dT dT dT
U= ,V= ,W= ,
dx1 dx2 dx3 (2)
dT dT dT
P= ,Q= ,R= .
dy1 dy2 dy3

The second system of expression was chosen by Clebsch and adopted by Halphen in his Fonctions
elliptiques; and thence the dynamical equations follow

dx1 dT dT
X= − x2 + x3 , Y = ..., Z = ...,
dt dy3 dy2 (3)
dy1 dT dT dT dT
L= − y2 + y3 − x2 + x3 , M = ..., N = ...,
dt dy3 dy2 dx3 dx2 (4)

where X, Y, Z, L, M, N denote components of external applied force on the body.

These equations are proved by taking a line fixed in space, whose direction cosines are l, m, n,
then

dl dm dn
= mR − nQ, = nP − lR, = lQ − mP.
dt dt dt (5)

If P denotes the resultant linear impulse or momentum in this direction

P = lx1 + mx2 + nx3,
(6)
dP dl dm dn
= x1 + x2 + x3
dt dt dt dt
dx1 dx2 dx3
+l +m +n ,
dt dt dt
dx1
=l (dt − x2R + x3Q )
dx2
+m ( dt − x3P + x1R )
dx3
+n ( dt − x1Q + x2P )
= lX + mY + nZ,
(7)

for all values of l, m, n.

Next, taking a fixed origin Ω and axes parallel to Ox, Oy, Oz through O, and denoting by x, y, z
the coordinates of O, and by G the component angular momentum about Ω in the direction (l, m, n)

G = l (y1 − x2z + x3y)
+ m (y2 − x3x + x1z)

+ n (y3 − x1y + x2x).
(8)

Differentiating with respect to t, and afterwards moving the fixed origin up to the moving origin O,
so that

dx dy dz
x = y = z = 0, but = U, = V, = W,
dt dt dt
dG dy1
dt
=l ( dt − y2R + y3Q − x2W + x3V )
dy2
+m ( dt − y3P + y1R − x3U + x1W )
dy3
+n ( dt − y1Q + y2P − x1V + x2U )
= lL + mM + nN,
(9)

for all values of l, m, n.

When no external force acts, the case which we shall consider, there are three integrals of the
equations of motion

(i.) T = constant,

(ii.) x12 + x22+ x32 = F2, a constant,

(iii.) x1y1 + x2y2 + x3y3 = n = GF, a constant;

and the dynamical equations in (3) express the fact that x1, x2, x3 are the components of a constant
vector having a fixed direction; while (4) shows that the vector resultant of y1, y2, y3 moves as if
subject to a couple of components

x2W − x3V, x3U − x1W, x1V − x2U,
(10)

and the resultant couple is therefore perpendicular to F, the resultant of x1, x2, x3, so that the
component along OF is constant, as expressed by (iii).

If a fourth integral is obtainable, the solution is reducible to a quadrature, but this is not possible
except in a limited series of cases, investigated by H. Weber, F. Kötter, R. Liouville, Caspary,
Jukovsky, Liapounoff, Kolosoff and others, chiefly Russian mathematicians; and the general solution
requires the double-theta hyperelliptic function.

49. In the motion which can be solved by the elliptic function, the most general expression of the
kinetic energy was shown by A. Clebsch to take the form

T = ½p (x12 + x22) + ½p′x32

+ q (x1y1 + x2y2) + q′x3y3
+ ½r (y12 + y22) + ½r′y32
(1)

so that a fourth integral is given by

dy3 / dt = 0, y3 = constant;
(2)
dx3
= x1 (qx2 + ry2) − x2 (qx1 + ry1) = r (x1y2 − x2y1),
dt (3)
1 dx3 2

r2 ( dt ) = (x12 + x22) (y12 + y22) − (x1y1 + x2y2)2

= (x12 + x22) (y12 + y22) − (FG − x3y3)2
= (x12 + x22) (y12 + y22 + y32 − G2) − (Gx3 − Fy3)2,
(4)

in which

x12 + x22 = F2 − x32, x1y1 + x2y2 = FG − x3y3,
(5)

r (y12 + y22) = 2T − p(x12 + x22) − p′x32
− 2q (x1y1 + x2y2) − 2q′x3y3 − r′y32
= (p − p′) x32 + 2 (q − q′) x3y3 + m1,
(6)

m1 − 2T − pF2 − 2qFG − r1y32
(7)

so that

1 dx3 2

r2 ( dt ) = X3
(8)

where X3 is a quartic function of x3, and thus t is given by an elliptic integral of the first kind; and by
inversion x3 is in elliptic function of the time t. Now

(x1 − x2i) (y1 + y2i) = x1y1 + x2y2 + i (x1y2 − x2y1) = FG − xy3y3 + i √ X3,
(9)
y1 + y2i FG − x3y3 + i √ X3
= ,
x1 + x2i x12 + x22 (10)
d
(x1 + x2i) = −i [ (q′ − q) x3 + r′y3 ] + irx3 (y1 + y2i),
dt (11)
d FG − x3y3 + i √ X3
log (x1 + x2i) = −(q′ − q) x3 − r′y3 + rx3 ,
dti F2 − x32 (12)

d x1 + x2i Fy3 − Gx3
dti
log √ x1 − x2i = −(q′ − q) x3 − (r′ − r) y3 − Fr F2 − x32 , (13)

requiring the elliptic integral of the third kind; thence the expression of x1 + x2i and y1 + y2i.

Introducing Euler’s angles θ, φ, ψ,

x1 = F sin θ sin φ, x2 = F sin θ cos φ,
x1 + x2i = iF sin θε−ψi, x3 = F cos θ;
(14)
dψ
sin θ = P sin φ + Q cos φ,
dt (15)
dψ dT dT
F sin2 θ = x1 + x2
dt dy1 dy2

= (qx1 + ry1) x1 + (qx2 + ry2) x2
= q (x12 + x22) + r (x1y1 + x2y2)
= gF2 sin2 θ + r (FG − x3y3),
(16)
FG − x3y3 Fr dx3
ψ − qFt = ∫ F2 − x32 √ X3
,
(17)

elliptic integrals of the third kind.

Employing G. Kirchhoff’s expressions for X, Y, Z, the coordinates of the centre of the body,

FX = y1 cos xY + y2 cos yY + y3 cos zY,
(18)

FY = −y1 cos xX + y2 cos yX + y3 cos zX,
(19)

G = y1 cos xZ + y2 cos yZ + y3 cos zZ,
(20)

F2(X2 + Y2) = y12 + y22 + y32 − G2,
(21)
Fy3 − Gx3 + i √ X3
F(X + Yi) = εψi.
√ (F2 − x32) (22)

Suppose x3 − F is a repeated factor of X3, then y3 = G, and

p′ − p q′ − q
X3 = (x3 − F)2 [ r
(x3 + F)2 + 2
r G (x3 + F) − G2 , ] (23)

and putting x3 − F = y,

dy 2 p′ − p q′ − q p′ − p q′ − q p′ − p
( dt ) = r2y2 [4 r
F2 + 4
r
FG − G2 + 2 (2 r
F+
r
G ) y + (24)
r
y2 ],

so that the stability of this axial movement is secured if

p′ − p q′ − q
A=4 F2 + 4 FG − G2
r r (25)

is negative, and then the axis makes r√(-A)/π nutations per second. Otherwise, if A is positive

dy
rt = ∫ y √ (A + 2By + Cy2)
1 sh−1 √ A √ (A + 2By + Cy2) 1 ch−1 A + By
= = ,
√A ch−1 y√ (B2 ~ AC) √A sh−1 y √ (B2 ~ AC) (26)

and the axis falls away ultimately from its original direction.

A number of cases are worked out in the American Journal of Mathematics (1907), in which the
motion is made algebraical by the use of the pseudo-elliptic integral. To give a simple instance,
changing to the stereographic projection by putting tan ½θ = x,

(Nx eψi)3/2 = (x + 1) √ X1 + i (x − 1) √ X2,
(27)
X1
= ± ax4 + 2ax3 ± 3 (a + b) x2 + 2bx ± b,
X2 (28)

N3 = −8 (a + b),
(29)

will give a possible state of motion of the axis of the body; and the motion of the centre may then be
inferred from (22).

50. The theory preceding is of practical application in the investigation of the stability of the axial motion
of a submarine boat, of the elongated gas bag of an airship, or of a spinning rifled projectile. In the steady
motion under no force of such a body in a medium, the centre of gravity describes a helix, while the axis
describes a cone round the direction of motion of the centre of gravity, and the couple causing precession is
due to the displacement of the medium.

In the absence of a medium the inertia of the body to translation is the same in all directions, and is
measured by the weight W, and under no force the C.G. proceeds in a straight line, and the axis of rotation
through the C.G. preserves its original direction, if a principal axis of the body; otherwise the axis describes
a cone, right circular if the body has uniaxial symmetry, and a Poinsot cone in the general case.

But the presence of the medium makes the effective inertia depend on the direction of motion with
respect to the external shape of the body, and on W′ the weight of fluid medium displaced.

Consider, for example, a submarine boat under water; the inertia is different for axial and broadside
motion, and may be represented by

c1 = W + W′α, c2 = W + W′β,
(1)

where α, β are numerical factors depending on the external shape; and if the C.G. is moving with velocity V
at an angle φ with the axis, so that the axial and broadside component of velocity is u = V cos φ, v = V sin
φ, the total momentum F of the medium, represented by the vector OF at an angle θ with the axis, will have
components, expressed in sec. ℔,

u V v V
F cos θ = c1 = (W + W′α) cos φ, F sin θ = c2 = (W + W′β) .
g g g g (2)

Suppose the body is kept from turning as it advances; after t seconds the C.G. will have moved from O to
O′, where OO′ = Vt; and at O′ the momentum is the same in magnitude as before, but its vector is displaced
from OF to O′F′.

For the body alone the resultant of the components of momentum

V V V
W cos φ and W sin φ is W sec. ℔,
g g g (3)

acting along OO′, and so is unaltered.

But the change of the resultant momentum F of the medium as well as of the body from the vector OF to
O′F′ requires an impulse couple, tending to increase the angle FOO′, of magnitude, in sec. foot-pounds

F·OO′·sin FOO′ = FVt sin (θ − φ),
(4)

equivalent to an incessant couple

N = FV sin (θ − φ)
= (F sin θ cos φ − F cos θ sin φ) V
= (c2 − c1) (V2 / g) sin φ cos φ
= W′ (β − α) uv / g.
(5)

This N is the couple in foot-pounds changing the momentum of the medium, the momentum of the body
alone remaining the same; the medium reacts on the body with the same couple N in the opposite direction,
tending when c2 − c1 is positive to set the body broadside to the advance.

An oblate flattened body, like a disk or plate, has c2 − c1 negative, so that the medium steers the body
axially; this may be verified by a plate dropped in water, and a leaf or disk or rocket-stick or piece of paper
falling in air. A card will show the influence of the couple N if projected with a spin in its plane, when it
will be found to change its aspect in the air.

An elongated body like a ship has c2 − c1 positive, and the couple N tends to disturb the axial movement
and makes it unstable, so that a steamer requires to be steered by constant attention at the helm.

Consider a submarine boat or airship moving freely with the direction of the resultant momentum
horizontal, and the axis at a slight inclination θ. With no reserve of buoyancy W = W′, and the couple N,
tending to increase θ, has the effect of diminishing the metacentric height by h ft. vertical, where

c1 u2
Wh tan θ = N = (c2 − c1) tan θ,
c2 g (6)
c 2 − c1 c1 u2 1+α u2
h= = (β − α) .
W c2 g 1+β g (7)

51. An elongated shot is made to preserve its axial flight through the air by giving it the spin sufficient for
stability, without which it would turn broadside to its advance; a top in the same way is made to stand upright on
the point in the position of equilibrium, unstable statically but dynamically stable if the spin is sufficient; and the
investigation proceeds in the same way for the two problems (see Gyroscope).

The effective angular inertia of the body in the medium is now required; denote it by C1 about the axis of
the figure, and by C2 about a diameter of the mean section. A rotation about the axis of a figure of
revolution does not set the medium in motion, so that C1 is the moment of inertia of the body about the axis,
denoted by Wk12. But if Wk22 is the moment of inertia of the body about a mean diameter, and ω the
angular velocity about it generated by an impulse couple M, and M′ is the couple required to set the
surrounding medium in motion, supposed of effective radius of gyration k′,

Wk22ω = M − M′, W′k′2ω = M′,
(1)

(Wk22 + W′k′2) ω = M,
(2)

C2 = Wk22 + W′k′2 = (W + W′ε) k22,
(3)

in which we have put k′2 = εk2, where ε is a numerical factor depending on the shape.

If the shot is spinning about its axis with angular velocity p, and is preceding steadily at a rate μ about a
line parallel to the resultant momentum F at an angle θ, the velocity of the vector of angular momentum, as
in the case of a top, is

C1pμ sinθ − C2μ2 sin θ cos θ;
(4)

and equating this to the impressed couple (multiplied by g), that is, to

c1
gN = (c1 − c2) u2 tan θ,
c2 (5)

and dividing out sin θ, which equated to zero would imply perfect centring, we obtain

c1
C2μ2 cos θ − C1pμ + (c2 − c1) u2 sec θ = 0.
c2 (6)

The least admissible value of p is that which makes the roots equal of this quadratic in μ, and then

C1
μ=½ p sec θ,
C2 (7)

the roots would be imaginary for a value of p smaller than given by

c1
C12p2 − 4 (c2 − c1) C2u2 = 0,
c2 (8)
p2 c1 C2
= 4 (c2 − c1) .
u2 c2 C12 (9)

Table of Rifling for Stability of an Elongated Projectile, x Calibres long, giving δ the Angle of Rifling, and n
the Pitch of Rifling in Calibres.

Cast-iron Common Shell Palliser Shell Solid Steel Bullet Solid Lead Bullet
ƒ = 2⁄3, S.G. 7.2. ƒ = ½, S.G. 8. ƒ = 0, S.G. 8. ƒ = 0, S.G. 10.9.
x β−α δ n δ n δ n δ n
1.0 0.0000 0° 0′ Infinity 0° 0′ Infinity 0° 0′ Infinity 0° 0′ Infinity
2.0 0.4942 2 49 63.87 2 32 71.08 2 29 72.21 2 08 84.29
2.5 0.6056 3 46 47.91 3 23 53.32 3 19 54.17 2 51 63.24
3.0 0.6819 4 41 38.45 4 13 42.79 4 09 43.47 3 38 50.74
3.5 0.7370 5 35 32.13 5 02 35.75 4 58 36.33 4 15 42.40
4.0 0.7782 6 30 27.60 5 51 30.72 5 45 31.21 4 56 36.43
4.5 0.8100 7 24 24.20 6 40 26.93 6 32 27.36 5 37 31.94
5.0 0.8351 8 16 21.56 7 28 23.98 7 21 24.36 6 18 28.44
6.0 0.8721 10 05 17.67 9 04 19.67 8 56 19.98 7 40 23.33
10.0 0.9395 16 57 10.31 15 19 11.47 15 05 11.65 13 00 13.60
Infinity 1.0000 90 00 0.00 90 00 0.00 90 00 0.00 90 00 0.00

If the shot is moving as if fired from a gun of calibre d inches, in which the rifling makes one turn in a
pitch of n calibres or nd inches, so that the angle δ of the rifling is given by

tan δ = πd / nd = ½ dp / u,
(10)

which is the ratio of the linear velocity of rotation ½dp to u, the velocity of advance,

π2 d2p2 c1 C2d2
tan2 δ = = = (c2 − c1)
n2 4u2 c2 C12
W′ W′ k1 2
W′ 1+
W
α (1+ W ε)( d )
= (β − α) · .
W′ k1 4
W 1+
W
β (W) (11)

For a shot in air the ratio W′/W is so small that the square may be neglected, and formula (11) can be
replaced for practical purpose in artillery by

π2 W′ k2 2 k1 4
tan2 δ =
n2
=
W
(β − α) ( d ) /( d ) ,
(12)

if then we can calculate β, α, or β − α for the external shape of the shot, this equation will give the value of δ
and n required for stability of flight in the air.

The ellipsoid is the only shape for which α and β have so far been determined analytically, as shown
already in § 44, so we must restrict our calculation to an egg-shaped bullet, bounded by a prolate ellipsoid
of revolution, in which, with b = c,

ab2 dλ ab2 dλ
A0 = ∫∞0 (a2 + λ) √ [ 4 (a2 + λ) (b2 + λ)2 ] = ∫∞0 2 (a2 + λ)3/2 (b2 + λ) , (13)

A0 + 2B0 = 1,
(14)
A0 B0 1 − A0 1
a= ,β= = = .
1 − A0 1 − B0 1 + A0 1 + 2α (15)

The length of the shot being denoted by l and the calibre by d, and the length in calibres by x

l / d = 2a / 2b = x,
(16)
x 1
A0 = ch−1x − ,
(x2− 1)3/2 x2 − 1 (17)
−x x2
2B0 = ch−1x + ,
(x2 − 1)3/2 x2 + 1 (18)
x sh−1 √ (x2 − 1) x
x2A0 + 2B0 = = log [ x + √ (x2 − 1) ].
√ (x2 − 1) √ (x2 − 1) (19)

If σ denotes the density of the metal, and if the shell has a cavity homothetic with the external ellipsoidal
shape, a fraction f of the linear scale; then the volume of a round shot being 1⁄6 π d3, and 1⁄6 π d3 x of a shot x
calibres long

W = 1⁄6 πd3 x (i − ƒ3) σ,
(20)
d2
Wk12 = 1⁄6 πd3 x (1 − ƒ5) σ,
10 (21)
l 2 + d2
Wk22 = 1⁄6 πd3 x (1 − ƒ5) σ.
20 (22)

If ρ denotes the density of the air or medium

W′ = 1⁄6 πd3 xρ,
(23)
W′ 1 ρ
= ,
W 1 − ƒ3 σ (24)
k12 1 1 − ƒ5 k22 x2 + 1
= , = ,
d2 10 1 − ƒ3 k12 2

(25)
ρ x2 + 1
tan2 δ = (β − α) ,
σ ⁄ (1 − ƒ5)
15
(26)

in which σ/ρ may be replaced by 800 times the S.G. of the metal, taking water as 800 times denser than air
on the average, in round numbers, and formula (10) may be written n tan δ = π, or nδ = 180, when δ is a
small angle, and given in degrees.

From this formula (26) the table following has been calculated by A. G. Hadcock, and the results are in
agreement with practical experience.

52. In the steady motion the centre of the shot describes a helix, with axial velocity

c1
u cos θ = v sin θ = ( l + c2 tan2 θ ) u cos θ ≈ u sec θ, (1)

and transverse velocity

c1
u sin θ − v cos θ = ( l − c2 ) u sin θ ≈ (β − α) u sin θ; (2)

and the time of completing a turn of the spiral is 2π/μ.

When μ has the critical value in (7),

2π 4π C2 2π
= cos θ = (x2 + 1) cos θ,
μ p C1 p (3)

which makes the circumference of the cylinder on which the helix is wrapped

2π 2πu
(u sin θ − v cos θ = (β − α) (x2 + 1) sin2 θ cos θ
μ p

= nd (β − α) (x2 + 1) sin θ cos θ,
(4)

and the length of one turn of the helix

2π
(u cos θ + v sin θ) = nd (x2 + 1);
μ (5)

thus for x = 3, the length is 10 times the pitch of the rifling.

53. The Motion of a Perforated Solid in Liquid.—In the preceding investigation, the liquid stops dead
when the body is brought to rest; and when the body is in motion the surrounding liquid moves in a uniform
manner with respect to axes fixed in the body, and the force experienced by the body from the pressure of
the liquid on its surface is the opposite of that required to change the motion of the liquid; this has been
expressed by the dynamical equations given above. But if the body is perforated, the liquid can circulate
through a hole, in reentrant stream lines linked with the body, even while the body is at rest; and no reaction
from the surface can influence this circulation, which may be supposed started in the ideal manner
described in § 29, by the application of impulsive pressure across an ideal membrane closing the hole, by
means of ideal mechanism connected with the body. The body is held fixed, and the reaction of the

mechanism and the resultant of the impulsive pressure on the surface are a measure of the impulse, linear ξ,
η, ζ, and angular λ, μ, ν, required to start the circulation.

This impulse will remain of constant magnitude, and fixed relatively to the body, which thus experiences
an additional reaction from the circulation which is the opposite of the force required to change the position
in space of the circulation impulse; and these extra forces must be taken into account in the dynamical
equations.

An article may be consulted in the Phil. Mag., April 1893, by G. H. Bryan, in which the analytical
equations of motion are deduced of a perforated solid in liquid, from considerations purely hydrodynamical.

The effect of an external circulation of vortex motion on the motion of a cylinder has been investigated in
§ 29; a similar procedure will show the influence of circulation through a hole in a solid, taking as the
simplest illustration a ring-shaped figure, with uniplanar motion, and denoting by ξ the resultant axial linear
momentum of the circulation.

As the ring is moved from O to O′ in time t, with velocity Q, and angular velocity R, the components of
liquid momentum change from

αM′U + ξ and βM′V along Ox and Oy

to

αM′U′+ ξ and βM′V′ along O′x′ and O′y′,
(1)

the axis of the ring changing from Ox to O′x′; and

U = Q cos θ, V = Q sin θ,
U′ = Q cos (θ − Rt), V′ = Q sin (θ − Rt),
(2)

so that the increase of the components of momentum, X1, Y1, and N1, linear and angular, are

X1 = (αM′U′ + ξ) cos Rt − αM′U − ξ − βM′V′ sin Rt
=(α − β)M′Q sin (θ − Rt) sin Rt − ξ ver Rt
(3)

Y1 = (αM′U′ + ξ) sin Rt + βM′V′ cos Rt − βM′V
= (α − β) M′Q cos (θ − Rt) sin Rt + ξ sin RT,
(4)

N1 = [ −(αM′U′ + ξ) sin (θ − Rt) + βM′V′ cos (θ − Rt) ] OO′
= [ −(α − β) M′Q cos (θ − Rt) sin (θ − Rt) − ξ sin (θ − Rt) ] Qt.
(5)

The components of force, X, Y, and N, acting on the liquid at O, and reacting on the body, are then

X = lt. X1/t = (α − β) M′QR sin θ = (α − β) M′VR,
(6)

Y = lt. Y1/t = (α − β) M′QR cos θ + ξR = (α − β) M′UR + ξR,
(7)

Z = lt. Z1/t = −(α − β) M′Q2 sin θ cos θ − ξQ sin θ = [ −(α − β) M′U + ξ ] V.
(8)

Now suppose the cylinder is free; the additional forces acting on the body are the components of kinetic
reaction of the liquid

dU dV dR
−αM′ ( dt − VR ), −βM′ ( dt + UR ), εC′ dt ,
(9)

so that its equations of motion are

dU dU
M ( dt − VR ) = −αM′ ( dt − VR ) − (α − β) M′VR, (10)
dV dV
M ( dt + UR ) = −βM′ ( dt + UR ) − (α − β) M′UR − ξR, (11)
dR dR
C = −εC′ + (α − β) M′UV + ξV;
dt dt (12)

and putting as before

M + αM′ = c1, M + βM′ = c2, C + εC′ = C3,
(13)
dU
c1 c2VR = 0,
dt (14)
dV
c2 + (c1U + ξ) R = 0,
dt (15)
dR
c3 − (c1U + ξ − c2U) V = 0;
dt (16)

showing the modification of the equations of plane motion, due to the component ξ of the circulation.

The integral of (14) and (15) may be written

c1U + ξ = F cos θ, c2V = − F sin θ,
(17)
dx F cos2 θ F sin2 θ ξ
= U cos θ − V sin θ = + − cos θ,
dt c1 c2 c1 (18)
dμ F F ξ
dt = U sin θ + V cos θ = ( c1 − c2 ) sin θ cos θ − c1 sin θ, (19)
d2θ F2 F2 Fξ dμ
C3
dt2 = ( c1 − c2 ) sin θ cos θ − c1 sin θ = F dt , (20)
dθ F2 cos2 θ F2 sin2 θ Fξ
C3
dt
= Fy = √ [− c1
−
c2
+2
c1 cos θ + H ; ] (21)

so that cos θ and y is an elliptic function of the time.

When ξ is absent, dx/dt is always positive, and the centre of the body cannot describe loops; but with ξ,
the influence may be great enough to make dx/dt change sign, and so loops occur, as shown in A. B.

Basset’s Hydrodynamics, i. 192, resembling the trochoidal curves, which can be looped, investigated in § 29
for the motion of a cylinder under gravity, when surrounded by a vortex.

The branch of hydrodynamics which discusses wave motion in a liquid or gas is given now in the articles
Sound and Wave; while the influence of viscosity is considered under Hydraulics.

References.—For the history and references to the original memoirs see Report to the British
Association, by G. G. Stokes (1846), and W. M. Hicks (1882). See also the Fortschritte der Mathematik,
and A. E. H. Love, “Hydrodynamik” in the Encyklöpadie der mathematischen Wissenschaften (1901).
(A. G. G.)

HYDROMEDUSAE, a group of marine animals, recognized as belonging to the Hydrozoa (q.v.) by the
following characters. (1) The polyp (hydropolyp) is of simple structure, typically much longer than broad, without
ectodermal oesophagus or mesenteries, such as are seen in the anthopolyp (see article Anthozoa); the mouth is
usually raised above the peristome on a short conical elevation or hypostome; the ectoderm is without cilia. (2)
With very few exceptions, the polyp is not the only type of individual that occurs, but alternates in the life-cycle of
a given species, with a distinct type, the medusa (q.v.), while in other cases the polyp-stage may be absent
altogether, so that only medusa-individuals occur in the life-cycle.

The Hydromedusae represent, therefore, a sub-class of the Hydrozoa. The only other sub-class is the
Scyphomedusae (q.v.). The Hydromedusae contrast with the Scyphomedusae in the following points. (1) The
polyp, when present, is without the strongly developed longitudinal retractor muscles, forming ridges (taeniolae)
projecting into the digestive cavity, seen in the scyphistoma or scyphopolyp. (2) The medusa, when present, has a
velum and is hence said to be craspedote; the nervous system forms two continuous rings running above and
below the velum; the margin of the umbrella is not lobed (except in Narcomedusae) but entire; there are
characteristic differences in the sense-organs (see below, and Scyphomedusae); and gastral filaments (phacellae),
subgenital pits, &c., are absent. (3) The gonads, whether formed in the polyp or the medusa, are developed in the
ectoderm.

The Hydromedusae form a widespread, dominant and highly differentiated group of animals, typically marine,
and found in all seas and in all zones of marine life. Fresh-water forms, however, are also known, very few as
regards species or genera, but often extremely abundant as individuals. In the British fresh-water fauna only two
genera, Hydra and Cordylophora, are found; in America occurs an additional genus, Microhydra. The paucity of
fresh-water forms contrasts sharply, with the great abundance of marine genera common in all seas and on every
shore. The species of Hydra, however, are extremely common and familiar inhabitants of ponds and ditches.

In fresh-water Hydromedusae the life-cycle is usually secondarily simplified, but in marine forms the life-cycle
may be extremely complicated, and a given species often passes in the course of its history through widely
different forms adapted to different habitats and modes of life. Apart from larval or embryonic forms there are
found typically two types of person, as already stated, the polyp and the medusa, each of which may vary
independently of the other, since their environment and life-conditions are usually quite different. Hence both
polyp and medusa present characters for classification, and a given species, genus or other taxonomic category
may be defined by polyp-characters or medusa-characters or by both combined. If our knowledge of the life-
histories of these organisms were perfect, their polymorphism would present no difficulties to classification; but

unfortunately this is far from being the case. In the majority of cases we do not know the polyp corresponding to a
given medusa, or the medusa that arises from a given polyp.1 Even when a medusa is seen to be budded, from a
polyp under observation in an aquarium, the difficulty is not always solved, since the freshly-liberated, immature
medusa may differ greatly from the full-grown, sexually-mature medusa after several months of life on the high
seas (see figs. 11, B, C, and 59, a, b, c). To establish the exact relationship it is necessary not only to breed but to
rear the medusa, which cannot always be done in confinement. The alternative is to fish all stages of the medusa in
its growth in the open sea, a slow and laborious method in which the chance of error is very great, unless the series
of stages is very complete.

At present, therefore, classifications of the Hydromedusae have a more or less tentative character, and are liable
to revision with increased knowledge of the life-histories of these organisms. Many groups bear at present two
names, the one representing the group as defined by polyp-characters, the other as defined by medusa-characters. It
is not even possible in all cases to be certain that the polyp-group corresponds exactly to the medusa-group,
especially in minor systematic categories, such as families.

The following is the main outline of the classification that is Adopted in the present article. Groups founded on
polyp-characters are printed in ordinary type, those founded on medusa-characters in italics. For definitions of the
groups see below.

Sub-class Hydromedusae (Hydrozoa Craspedota).
Order I. Eleutheroblastea.
” II. Hydroidea (Leptolinae).
Sub-order 1. Gymnoblastea (Anthomedusae).
” 2. Calyptoblastea (Leptomedusae).
Order III. Hydrocorallinae.
” IV. Graptolitoidea.
” V. Trachylinae.
Sub-order 1. Trachomedusae.
” 2. Narcomedusae.
Order VI. Siphonophora.
Sub-order 1. Chondrophorida.
” 2. Calycophorida.
” 3. Physophorida.
” 4. Cystophorida.

Organization and Morphology of the Hydromedusae.

As already stated, there occur in the Hydromedusae two distinct types of person, the polyp and the medusa; and
either of them is capable of non-sexual reproduction by budding, a process which may lead to the formation of
colonies, composed of more or fewer individuals combined and connected together. The morphology of the group
thus falls naturally into four sections—(1) the hydropolyp, (2) the polyp-colony, (3) the hydromedusa, (4) the
medusa-colonies. Since, however, medusa-colonies occur only in one group, the Siphonophora, and divergent
views are held with regard to the morphological interpretation of the members of a siphonophore, only the first
three of the above subdivisions of hydromedusa morphology will be dealt with here in a general way, and the
morphology of the Siphonophora will be considered under the heading of the group itself.

1. The Hydropolyp (fig. 1)—The general characters of this organism are described above and in the
articles Hydrozoa and Polyp. It is rarely free, but usually fixed and incapable of locomotion. The foot by
which it is attached often sends out root-like processes—the hydrorhiza (c). The column (b) is generally
long, slender and stalk-like (hydrocaulus). Just below the crown of tentacles, however, the body widens out

to form a “head,” termed, the hydranth (a), containing a stomach-like
dilatation of the digestive cavity. On the upper face of the hydranth the
crown of tentacles (t) surrounds the peristome, from which rises the
conical hypostome, bearing the mouth at its extremity. The general
ectoderm covering the surface of the body has entirely lost the cilia
present in the earlier larval stages (planula), and may be naked, or clothed
in a cuticle or exoskeleton, the perisarc (ps), which in its simplest
condition is a chitinous membrane secreted by the ectoderm. The perisarc
when present invests the hydrorhiza and hydrocaulus; it may stop short
below the hydranth, or it may extend farther. In general there are two types
of exoskeleton, characteristic of the two principal divisions of the
Hydroidea. In the Gymnoblastea the perisarc either stops below the
hydranth, or, if continued on to it, forms a closely-fitting investment
extending as a thin cuticle as far as the bases of the tentacles (e.g. Bimeria,
Fig. 1.—Diagram of a typical
see G. J. Allman [1],2 pl. xii. figs, 1 and 3). In the Calyptoblastea the Hydropolyp.
perisarc is always continued above the hydrocaulus, and forms a cup, the a, Hydranth;
hydrangium or hydrotheca (h, t), standing off from the body, into which b, Hydrocaulus;
the hydranth can be retracted for shelter and protection. c, Hydrorhiza;
t, Tentacle;
ps, Perisarc, forming in the region
of the hydranth a cup or
hydrotheca(h, t),—which,
however, is only found in polyps
of the order Calyptoblastea.

From Allman’s Gymnoblastic Hydroids, by permission of the Council of the Ray Society.
Fig. 2.—Stauridium productum, portion of the colony magnified; p, polyp; rh, hydrorhiza.

The architecture of the hydropolyp, simple though it be, furnishes a long series of variations affecting
each part of the body. The greatest variation, however, is seen in the tentacles. As regards number, we find
in the aberrant forms Protohydra and Microhydra tentacles entirely absent. In the curious hydroid
Monobrachium a single tentacle is present, and the same is the case in Clathrozoon; in Amphibrachium and
in Lar (fig. 11, A) the polyp bears two tentacles only. The reduction of the tentacles in all these forms may
be correlated with their mode of life, and especially with living in a constant current of water, which brings

food-particles always from one direction and renders a complete whorl or
circle of tentacles unnecessary. Thus Microhydra lives amongst Bryozoa, and
appears to utilize the currents produced by these animals. Protohydra occurs
in oyster-banks and Monobrachium also grows on the shells of bivalves, and
both these hydroids probably fish in the currents produced by the
lamellibranchs. Amphibrachium grows in the tissues of a sponge, Euplectella,
and protrudes its hydranth into the canal-system of the sponge; and Lar
grows on the tubes of the worm Sabella. With the exception of these forms,
reduced for the most part in correlation with a semi-parasitic mode of life, the
tentacles are usually numerous. It is rare to find in the polyp a regular,
symmetrical disposition of the tentacles as in the medusa. The primitive
number of four in a whorl is seen, however, in Stauridium (fig. 2) and
Cladonema (Allman [1], pl. xvii.), and in Clavatella each whorl consists
regularly of eight (Allman, loc. cit. pl. xviii.). As a rule, however, the number
in a whorl is irregular. The tentacles may form a single whorl, or more than
one; thus in Corymorpha (fig. 3) and Tubularia (fig. 4) there are two circlets;
Fig. 3.—Diagram of
in Stauridium (fig. 2) several; in Coryne and Cordylophora the tentacles are Corymorpha. A, A
scattered irregularly over the elongated hydranth. hydriform person giving
rise to medusiform persons
As regards form, the tentacles
by budding from the margin
show a number of types, of which
of the disk; B, free
the most important are (1) filiform,
swimming medusa
i.e. cylindrical or tapering from
(Steenstrupia of Forbes)
base to extremity, as in Clava (fig. detached from the same,
5); (2) capitate, i.e. knobbed at the with manubrial genitalia,
extremity, as in Coryne (see (Anthomedusae) and only
Allman, loc. cit. pl. iv.); (3) one tentacle. (After
branched, a rare form in the polyp, Allman).
but seen in Cladocoryne (see
Allman, loc. cit. p. 380, fig. 82). Sometimes more than one type
Fig. 4.—Diagram of Tubularia indivisa. A of form is found in the same polyp; in Pennaria and Stauridium
single hydriform person a bearing a stalk (fig. 2) the upper whorls are capitate, the lower filiform. Finally,
carrying numerous degenerate medusiform as regards structure, the tentacles may retain their primitive
persons or sporosacs b. (After Allman.) hollow nature, or become solid by obliteration of the axial cavity.

The hypostome of the hydropolyp may be small, or, on the
other hand, as in Eudendrium (Allman, loc. cit. pls. xiii., xiv.), large and trumpet-shaped. In the curious
polyp Myriothela the body of the polyp is differentiated into nutritive and reproductive portions.

Histology.—The ectoderm of the hydropolyp is chiefly sensory, contractile and protective in function. It
may also be glandular in places. It consists of two regions, an external epithelial layer and a more internal
sub-epithelial layer.

The epithelial layer consists of (1) so-called “indifferent” cells secreting the perisarc or cuticle and
modified to form glandular cells in places; for example, the adhesive cells in the foot. (2) Sensory cells,
which may be fairly numerous in places, especially on the tentacles, but which occur always scattered and
isolated, never aggregated to form sense-organs as in the medusa. (3) Contractile or myo-epithelial cells,
with the cell prolonged at the base into a contractile muscle-fibre (fig. 6, B). In the hydropolyp the

ectodermal muscle-fibres are always directed longitudinally. Belonging primarily to the epithelial layer, the
muscular cells may become secondarily sub-epithelial.

From Allman’s Gymnoblastic Hydroids, by permission of the Council of the Ray Society.
Fig. 5.—Colonies of Clava. A, Clava squamata, magnified. B, C. multicornis, natural size; p, polyp; gon, gonophores; rh, hydrorhiza.

The sub-epithelial layer consists primarily of the so-called interstitial cells, lodged between the narrowed
basal portions of the epithelial cells. From them are developed two distinct types of histological elements;
the genital cells and the cnidoblasts or mother-cells of the nematocysts. The sub-epithelial layer thus
primarily constituted may be recruited by immigration from without of other elements, more especially by
nervous (ganglion) cells and muscle-cells derived from the epithelial layer. In its fullest development,
therefore, the sub-epithelial layer consists of four classes of cell-elements.

Fig. 6 A.—Portion of the body-wall of Hydra, showing ectoderm cells above, separated by “structureless
lamella” from three flagellate endoderm cells below. The latter are vacuolated, and contain each a
nucleus and several dark granules. In the middle ectoderm cell are seen a nucleus and three nematocysts,
with trigger hairs projecting beyond the cuticle. A large nematocyst, with everted thread, is seen in the
right-hand ectodermal cell. (After F. E. Schulze.)

The genital cells are simple wandering cells (archaeocytes), at first minute and without any specially
distinctive features, until they begin to develop into germ-cells. According to Wulfert [60] the primitive
germ-cells of Gonothyraea can be distinguished soon after the fixation of the planula, appearing amongst
the interstitial cells of the ectoderm. The germ-cells are capable of extensive migrations, not only in the
body of the same polyp, but also from parent to bud through many non-sexual generations of polyps in a
colony (A. Weismann [58]).

The cnidoblasts are the mother-cells of the nematocysts, each cell
producing one nematocyst in its interior. The complete nematocyst
(fig. 7) is a spherical or oval capsule containing a hollow thread,
usually barbed, coiled in its interior. The capsule has a double wall,
an outer one (o.c.), tough and rigid in nature, and an inner one (i.c.)
of more flexible consistence. The outer wall of the capsule is Fig. 6 B.—Epidermo-muscular cells
of Hydra. m, muscular-fibre
incomplete at one pole, leaving an aperture through which the thread
processes. (After Kleinenberg, from
is discharged. The inner membrane is continuous with the wall of the
Gegenbaur.)
hollow thread at a spot immediately below the aperture in the outer
wall, so that the thread itself (f) is simply a hollow prolongation of
the wall of the inner capsule inverted and pushed into its cavity. The entire nematocyst is enclosed in the
cnidoblast which formed it. When the nematocyst is completely developed, the cnidoblast passes outwards
so as to occupy a superficial position in the ectoderm, and a delicate protoplasmic process of sensory nature,
termed the cnidocil (cn) projects from the cnidoblast like a fine hair or cilium. Many points in the
development and mechanism of the nematocyst are disputed, but it is tolerably certain (1) that the cnidocil
is of sensory nature, and that stimulation, by contact with prey or in other ways, causes a reflex discharge of
the nematocyst; (2) that the discharge is an explosive change whereby the in-turned thread is suddenly
everted and turned inside out, being thus shot through the opening in the outer wall of the capsule, and
forced violently into the tissues of the prey, or, it may be, of an enemy; (3) that the thread inflicts not merely
a mechanical wound, but instils an irritant poison, numbing and paralysing in its action. The points most in

dispute are, first, how the explosive discharge is brought
about, whether by pressure exerted external to the capsule
(i.e. by contraction of the cnidoblast) or by internal pressure.
N. Iwanzov [27] has brought forward strong grounds for the
latter view, pointing out that the cnidoblast has no contractile
mechanism and that measurements show discharged capsules
to be on the average slightly larger than undischarged ones.
He believes that the capsule contains a substance which
swells very rapidly when brought into contact with water, and
that in the undischarged condition the capsule has its opening
closed by a plug of protoplasm (x, fig. 7) which prevents
access of water to the contents; when the cnidocil is
stimulated it sets in action a mechanism or perhaps a series of
chemical changes by which the plug is dissolved or removed;
as a result water penetrates into the capsule and causes its
contents to swell, with the result that the thread is everted
violently. A second point of dispute concerns the spot at
which the poison is lodged. Iwanzov believes it to be
contained within the thread itself before discharge, and to be
introduced into the tissues of the prey by the eversion of the
thread. A third point of dispute is whether the nematocysts are
formed in situ, or whether the cnidoblasts migrate with them
Fig. 7.—Diagrams to show the structure of to the region where they are most needed; the fact that in
Nematocysts and their mode of working. (After Hydra, for example, there are no interstitial cells in the
Iwanzov.) tentacles, where nematocysts are very abundant, is certainly
a, Undischarged nematocyst. in favour of the view that the cnidoblasts migrate on to the
b, Commencing discharge. tentacles from the body, and that like the genital cells the
c, Discharge complete. cnidoblasts are wandering cells.
cn, Cnidocil.
N, Nucleus of cnidoblast.
The muscular tissue consists primarily of processes from
o.c, Outer capsule.
the bases of the epithelial cells, processes which are
x, Plug closing the opening of the outer capsule.
contractile in nature and may be distinctly striated. A further
i.c., Inner capsule, continuous with the wall of
the filament, f.
stage in evolution is that the muscle-cells lose their connexion
b, Barbs.
with the epithelium and come to lie entirely beneath it,
forming a sub-epithelial contractile layer, developed chiefly in the tentacles of the polyp. The evolution of
the ganglion-cells, is probably similar; an epithelial cell develops processes of nervous nature from the base,
which come into connexion with the bases of the sensory cells, with the muscular cells, and with the similar
processes of other nerve-cells; next the nerve-cell loses its connexion with the outer epithelium and
becomes a sub-epithelial ganglion-cell which is closely connected with the muscular layer, conveying
stimuli from the sensory cells to the contractile elements. The ganglion-cells of Hydromedusae are
generally very small. In the polyp the nervous tissue is always in the form of a scattered plexus, never
concentrated to form a definite nervous system as in the medusa.

The endoderm of the polyp is typically a flagellated epithelium of large cells (fig. 6), from the bases of
which arise contractile muscular processes lying in the plane of the transverse section of the body. In
different parts of the coelenteron the endoderm may be of three principal types—(1) digestive endoderm,
the primitive type, with cells of large size and considerably vacuolated, found in the hydranth; some of

these cells may become special glandular cells, without flagella or
contractile processes; (2) circulatory endoderm, without vacuoles and
without basal contractile processes, found in the hydrorhiza and
hydrocaulus; (3) supporting endoderm (fig. 8), seen in solid tentacles
as a row of cubical vacuolated cells, occupying the axis of the
tentacle, greatly resembling notochordal tissue, particularly that of
Amphioxus at a certain stage of development; as a fourth variety of From Gegenbaur’s Elements of

endodermal cells excretory cells should perhaps be reckoned, as seen Comparative Anatomy.

in the pores in the foot of Hydra and elsewhere (cf. C. Chun, Fig. 8.—Vacuolated Endoderm Cells
Hydrozoa [1], pp. 314, 315). of cartilaginous consistence from the
axis of the tentacle of a Medusa
The mesogloea in the hydropolyp is a thin elastic layer, in which
(Cunina).
may be lodged the muscular fibres and ganglion cells mentioned
above, but which never contains any connective tissue or
skeletogenous cells or any other kind of special mesogloeal corpuscles.

2. The
Polyp-
colony.—All
known
hydropolyps
possess the
power of
reproduction
by budding,
From Allman’s Gymnoblastic Hydroids, by permission of
and the buds
the Council of the Ray Society.
produced
may become Fig. 9.—Colony of Hydractinia echinata, growing on the
either polyps Shell of a Whelk. Natural size.
or medusae.
The buds may all become detached after a time and give rise to
separate and independent individuals, as in the common Hydra, in
which only polyp-individuals are produced and sexual elements are
developed upon the polyps themselves; or, on the other hand, the
polyp-individuals produced by budding may remain permanently in
connexion with the parent polyp, in which case sexual elements are
never developed on polyp-individuals but only on medusa-
individuals, and a true colony is formed. Thus the typical hydroid
From Allman’s Gymnoblastic Hydroids,
colony starts from a “founder” polyp, which in the vast majority of
by permission of the Council of the Ray
cases is fixed, but which may be floating, as in Nemopsis,
Society. Pelagohydra, &c. The founder-polyp usually produces by budding
polyp-individuals, and these in their turn produce other buds. The
Fig. 10.—Polyps from a Colony of
polyps are all non-sexual individuals whose function is purely
Hydractinia, magnified. dz, dactylozoid;
nutritive. After a time the polyps, or certain of them, produce by
gz, gastrozoid: b, blastostyle; gon,
budding medusa-individuals, which sooner or later develop sexual
gonophores; rh, hydrorhiza.
elements; in some cases, however, the founder-polyp remains
solitary, that is to say, does not produce polyp-buds, but only

medusa-buds, from the first (Corymorpha, fig. 3, Myriothela, &c.). In primitive forms the medusa-
individuals are set free before reaching sexual maturity and do not contribute anything to the colony. In
other cases, however, the medusa-individuals become sexually mature while still attached to the parent
polyp, and are then not set free at all, but become appanages of the hydroid colony and undergo
degenerative changes leading to reduction and even to complete obliteration of their original medusan
structure. In this way the hydroid colony becomes composed of two portions of different function, the
nutritive “trophosome,” composed of non-sexual polyps, and the reproductive “gonosome,” composed of
sexual medusa-individuals, which never exercise a nutritive function while attached to the colony. As a
general rule polyp-buds are produced from the hydrorhiza and hydrocaulus, while medusa-buds are formed
on the hydranth. In some cases, however, medusa-buds are formed on the hydrorhiza, as in Hydrocorallines.

In such a colony of connected individuals, the exact limits of the separate “persons” are not always
clearly marked out. Hence it is necessary to distinguish between, first, the “zooids,” indicated in the case of
the polyps by the hydranths, each with mouth and tentacles; and, secondly, the “coenosarc,” or common
flesh, which cannot be assigned more to one individual than another, but consists of a more or less
complicated network of tubes, corresponding to the hydrocaulus and hydrorhiza of the primitive
independent polyp-individual. The coenosarc constitutes a system by which the digestive cavity of any one
polyp is put into communication with that of any other individual either of the trophosome or gonosome. In
this manner the food absorbed by one individual contributes to the welfare of the whole colony, and the
coenosarc has the function of circulating and distributing nutriment through the colony.

The hydroid colony shows many variations in form and architecture which depend simply upon
differences in the methods in which polyps are budded.

After Hincks, Forbes, and Browne. A and B modified from Hincks; C modified from Forbes’s Brit. Naked-eyed Medusae.
Fig. 11.—Lar sabellarum and two stages of its Medusa, Willia stellata. A, colony of Lar; B and C, young and adult medusae.

In the first place, buds may be produced only from the
hydrorhiza, which grows out and branches to form a basal
stolon, typically net-like, spreading over the substratum to
which the founder-polyp attached itself. From the stolon the
daughter-polyps grow up vertically. The result is a spreading or
creeping colony, with the coenosarc in the form of a root-like
horizontal network (fig. 5, B; 11, A). Such a colony may
undergo two principal modifications. The meshes of the basal
network may become very small or virtually obliterated, so
that the coenosarc becomes a crust of tubes tending to fuse
together, and covered over by a common perisarc. Encrusting
colonies of this kind are seen in Clava squamata (fig. 5, A) and
Hydractinia (figs. 9, 10), the latter having the perisarc
calcified. A further very important modification is seen when
the tubes of the basal perisarc do not remain spread out in one
plane, but grow in all planes forming a felt-work; the result is a Fig. 12.—Colony of Bougainvillea fruticosa,
natural size, attached to the underside of a
massive colony, such as is seen in the so-called
piece of floating timber. (After Allman.)
Hydrocorallines (fig. 60), where the interspaces between the
coenosarcal tubes are filled up with calcareous matter, or
coenosteum, replacing the chitinous perisarc. The result is a stony, solid mass, which contributes to the
building up of coral reefs. In massive colonies of this kind no sharp distinction can be drawn between
hydrorhiza and hydrocaulus in the coenosarc; it is practically all hydrorhiza. Massive colonies may assume
various forms and are often branching or tree-like. A further peculiarity of this type of colony is that the
entire coenosarcal complex is covered externally by a common layer of ectoderm; it is not clear how this
covering layer is developed.

In the second place, the buds may be produced from the hydrocaulus, growing out laterally from it; the
result is an arborescent, tree-like colony (figs. 12, 13). Budding from the hydrocaulus may be combined
with budding from the hydrorhiza, so that numerous branching colonies arise from a common basal stolon.
In the formation of arborescent colonies, two sharply distinct types of budding are found, which are best
described in botanical terminology as the monopodial or racemose, and the sympodial or cymose types
respectively; each is characteristic of one of the two sub-orders of the Hydroidea, the Gymnoblastea and
Calyptoblastea.

In the monopodial method (figs. 12, 14) the founder-polyp is, theoretically, of unlimited growth in a
vertical direction, and as it grows up it throws out buds right and left alternately, so that the first bud
produced by it is the lowest down, the second bud is above the first, the third above this again, and so on.
Each bud produced by the founder proceeds to grow and to bud in the same way as the founder did,
producing a side branch of the main stem. Hence, in a colony of gymnoblastic hydroids, the oldest polyp of
each system, that is to say, of the main stem or of a branch, is the topmost polyp; the youngest polyp of the
system is the one nearest to the topmost polyp; and the axis of the system is a true axis.

Fig. 13.—Portion of colony of Bougainvillea fruticosa (Anthomedusae-Gymnoblastea) more magnified. (From Lubbock, after Allman.)

Fig. 14.—Diagrams of the monopodial method of budding, shown in five stages (1-5). F, the founder-polyp; 1, 2, 3, 4, the succession of
polyps budded from the founder-polyp; a′, b′, c′, the succession of polyps budded from 1; a2, b2, polyps budded from 2; a3, polyp budded
from 3.

In the sympodial method of budding, on the other
hand, the founder-polyp is of limited growth, and
forms a bud from its side, which is also of limited
growth, and forms a bud in its turn, and so on (figs.
15, 16). Hence, in a colony of calyptoblastic hydroids,
the oldest polyp of a system is the lowest; the
youngest polyp is the topmost one; and the axis of the
system is a false axis composed of portions of each of
the consecutive polyps. In this method of budding
there are two types. In one, the biserial type (fig. 15),
the polyps produce buds right and left alternately, so
that the hydranths are arranged in a zigzag fashion,
forming a “scorpioid cyme,” as in Obelia and Fig. 15.—Diagram of sympodial budding, biserial type,
Sertularia. In the other, the uniserial type (fig. 16), the shown in five stages (1-5). F, founder-polyp; 1, 2, 3, 4, 5,
buds are formed always on the same side, forming a 6, succession of polyps budded from the founder; a, b, c,

“helicoid cyme,” as in Hydrallmania, according to H. second series of polyps budded from the founder; a3, b3,

Driesch, in which, however, the primitively uniserial series budded from 3.

arrangement becomes masked later by secondary
torsions of the hydranths.

In a colony formed by sympodial budding, a polyp
always produces first a bud, which contributes to the
system to which it belongs, i.e. continues the stem or
branch of which its parent forms a part. The polyp
may then form a second bud, which becomes the
starting point of a new system, the beginning, that is,
of a new branch; and even a third bud, starting yet
another system, may be produced from the same Fig. 16.—Diagram of sympodial budding, uniserial type,
polyp. Hence the colonies of Calyptoblastea may be shown in four stages (1-4). F, founder-polyp; 1, 2, 3,
complexly branched, and the budding may be biserial succession of polyps budded from the founder.
throughout, uniserial throughout, or partly one, partly
the other. Thus in Plumularidae (figs. 17, 18) there is
formed a main stem by biserial budding; each polyp
on the main stem forms a second bud, which usually
forms a side branch or pinnule by uniserial budding.
In this way are formed the familiar feathery colonies
of Plumularia, in which the pinnules are all in one
plane, while in the allied Antennularia the pinnules
are arranged in whorls round the main biserial stem.
The pinnules never branch again, since in the
uniserial mode of budding a polyp never forms a
second polyp-bud. On the other hand, a polyp on the
main stem may form a second bud which, instead of
forming a pinnule by uniserial budding, produces by
Fig. 17.—Diagram of sympodial budding, simple
biserial budding a branch, from which pinnules arise unbranched Plumularia-type. F, founder; 1-8, main axis
as from the main stem (fig. 18—3, 6). Or a polyp on formed by biserial budding from founder; a-e, pinnule

the main stem, after having budded a second time to formed by uniserial budding from founder; a¹-d¹, branch

form a pinnule, may give rise to a third bud, which formed by similar budding from 1; a2-d2 from 2, and so

starts a new biserial system, from which uniserial forth.

pinnules arise as from the main stem—type of
Aglaophenia (fig. 19). The laws of budding in hydroids have been worked out in an interesting manner by
H. Driesch [13], to whose memoirs the reader must be referred for further details.

Individualization of Polyp-Colonies.—As in other cases where animal colonies are formed by organic
union of separate individuals, there is ever a tendency for the polyp-colony as a whole to act as a single
individual, and for the members to become subordinated to the needs of the colony and to undergo
specialization for particular functions, with the result that they simulate organs and their individuality
becomes masked to a greater or less degree. Perhaps the earliest of such specializations is connected with
the reproductive function. Whereas primitively any polyp in a colony may produce medusa-buds, in many
hydroid colonies medusae are budded only by certain polyps termed blastostyles (fig. 10, b). At first not
differing in any way from other polyps (fig. 5), the blastostyles gradually lose their nutritive function and
the organs connected with it; the mouth and tentacles disappear, and the blastostyle obtains the nutriment
necessary for its activity by way of the coenosarc. In the Calyptoblastea, where the polyps are protected by
special capsules of the perisarc, the gonothecae enclosing the blastostyles differ from the hydrothecae
protecting the hydranths (fig. 54).

Fig. 18.—Diagram showing method of branching in the Fig. 19.—Diagram showing method of branching in the Aglaophenia-type.
Plumularia-type; compare with fig. 17. Polyps 3 and 6, Polyp 7 has produced as its first bud, 8; as its second bud, a7, which starts a
instead of producing uniserial pinnules, have produced uniserial pinnule; and as a third bud I7, which starts a biserial branch (II7-
biserial branches (31, 32, 33, 34; 61-63), which give off VI7) that repeats the structure of the main stem and gives off pinnules. The
uniserial branches in their turn. main stem is indicated by-·-·-·, the new stem by ······.

In other colonies the two functions of the nutritive polyp, namely, capture and digestion of food, may be
shared between different polyps (fig. 10). One class of polyps, the dactylozoids (dz), lose their mouth and
stomach, and become elongated and tentacle-like, showing great activity of movement. Another class, the
gastrozoids (gz), have the tentacles reduced or absent, but have the mouth and stomach enlarged. The
dactylozoids capture food, and pass it on to the gastrozoids, which swallow and digest it.

Besides the three types of individual above mentioned, there are other appendages of hydroid colonies, of
which the individuality is doubtful. Such are the “guard-polyps” (machopolyps) of Plumularidae, which are
often regarded as individuals of the nature of dactylozoids, but from a study of the mode of budding in this
hydroid family Driesch concluded that the guard-polyps were not true polyp-individuals, although each is
enclosed in a small protecting cup of the perisarc, known as a nematophore. Again, the spines arising from
the basal crust of Podocoryne have been interpreted by some authors as reduced polyps.

3. The Medusa.—In the Hydromedusae the medusa-individual occurs, as already stated, in one of two
conditions, either as an independent organism leading a true life in the open seas, or as a subordinate
individuality in the hydroid colony, from which it is never set free; it then becomes a mere reproductive
appendage or gonophore, losing successively its organs of sense, locomotion and nutrition, until its
medusoid nature and organization become scarcely recognizable. Hence it is convenient to consider the
morphology of the medusa from these two aspects.

(a) The Medusa as an Independent Organism.—The general structure and characteristics of the medusa
are described elsewhere (see articles Hydrozoa and Medusa), and it is only necessary here to deal with the
peculiarities of the Hydromedusa.

From Allman’s Gymnoblastic Hydroids, by permission of the Council of From Allman’s Gymnoblastic Hydroids, by permission of the
the Ray Society. Council of the Ray Society.
Fig. 20.—Cladonema radiatum, the medusa walking on the basal Fig. 21.—Clavatella prolifera, ambulatory medusa. t,
branches of its tentacles (t), which are turned up over the body. tentacles; oc, ocelli.

As regards habit of life the vast majority of Hydromedusae are pelagic organisms, floating on the surface
of the open sea, propelling themselves feebly by the pumping movements of the umbrella produced by
contraction of the sub-umbral musculature, and capturing their prey with their tentacles. The genera
Cladonema (fig. 20) and Clavatella (fig. 21), however, are ambulatory, creeping forms, living in rock-pools
and walking, as it were, on the tips of the proximal branches of each of the tentacles, while the remaining
branches serve for capture of food. Cladonema still has the typical medusan structure, and is able to swim
about, but in Clavatella the umbrella is so much reduced, that swimming is no longer possible. The
remarkable medusa Mnestra parasites is ecto-parasitic throughout life on the pelagic mollusc Phyllirrhoe,

attached to it by the sub-umbral surface, and its tentacles have become rudimentary or absent. It is
interesting to note that Mnestra has been shown by J. W. Fewkes [15] and R. T. Günther [19] to belong to
the same family (Cladonemidae) as Cladonema and Clavatella, and it is reasonable to suppose that the non-
parasitic ancestor of Mnestra was, like the other two genera, an ambulatory medusa which acquired louse-
like habits. In some species of the genus Cunina (Narcomedusae) the youngest individuals (actinulae) are
parasitic on other medusae (see below), but in later life the parasitic habit is abandoned. No other instances
are known of sessile habit in Hydromedusae.

The external form of the Hydromedusae varies from that of a deep bell or
thimble, characteristic of the Anthomedusae, to the shallow saucer-like form
characteristic of the Leptomedusae. It is usual for the umbrella to have an even,
circular, uninterrupted margin; but in the order Narcomedusae secondary down-
growths between the tentacles produce a lobed, indented margin to the umbrella.
The marginal tentacles are rarely absent in non-parasitic forms, and are typically
four in number, corresponding to the four perradii marked by the radial canals.
Interradial tentacles may be also developed, so that the total number present may
be increased to eight or to an indefinitely large number. In Willia, Geryonia, &c.,
however, the tentacles and radial canals are on the plan of six instead of four
(figs. 11 and 26). On the other hand, in some cases the tentacles are less in
number than the perradii; in Corymorpha (figs. 3 and 22) there is but a single After E. T. Browne,
tentacle, while two are found in Amphinema and Gemmaria (Anthomedusae), from Proc. Zool. Soc.
and in Solmundella bitentaculata (fig. 67) and Aeginopsis hensenii (fig. 23) of London.
(Narcomedusae). The tentacles also vary considerably in other ways than in
Fig. 22.—Corymorpha
number: first, in form, being usually simple, with a basal bulb, but in
nutans, adult female
Cladonemidae they are branched, often in complicated fashion; secondly, in Medusa. Magnified 10
grouping, being usually given off singly, and at regular intervals from the margin diameters.
of the umbrella, but in Margelidae and in some Trachomedusae they are given off
in tufts or bunches (fig. 24); thirdly, in position and origin, being usually implanted on the extreme edge of
the umbrella, but in Narcomedusae they become secondarily shifted and are given off high up on the ex-
umbrella (figs. 23 and 25); and, fourthly, in structure, being hollow or solid, as in the polyp. In some
medusae, for instance, the remarkable deep-sea family Pectyllidae, the tentacles may bear suckers, by
which the animal may attach itself temporarily. It should be mentioned finally that the tentacles are very
contractile and extensible, and may therefore present themselves, in one and the same individual, as long,
drawn-out threads, or in the form of short corkscrew-like ringlets; they may stream downwards from the
sub-umbrella, or be held out horizontally, or be directed upwards over the ex-umbrella (fig. 23). Each
species of medusa usually has a characteristic method of carrying its tentacles.

After O. Maas, Die craspedoten Medusen der Plankton After O. Maas, Craspedoten Medusen der Siboga-Expedition, by
Expedition, by permission of Lipsius and Tischer. permission of E. S. Brill & Co.
Fig. 23.—Aeginopsis hensenii, slightly magnified, showing the
Fig. 24.—Rathkea octonemalis.
manner in which the tentacles are carried in life.

The sub-umbrella invariably shows a velum as an inwardly
projecting ridge or rim at its margin, within the circle of
tentacles; hence the medusae of this sub-class are termed
craspedote. The manubrium is absent altogether in the fresh-
water medusa Limnocnida, in which the diameter of the mouth
exceeds half that of the umbrella; on the other hand, the
manubrium may attain a great length, owing to the centre of the
sub-umbrella with the stomach being drawn into it, as it were, to
form a long proboscis, as in Geryonia. The mouth may be a
simple, circular pore at the extremity of the manubrium, or by
folding of the edges it may become square or shaped like a After O. Maas, Medusae, in Prince of
Monaco’s series.
Maltese cross, with four corners and four lips. The corners of the
mouth may then be drawn out into lobes or lappets, which may Fig. 25.—Aeginura grimaldii.
have a branched or fringed outline (fig. 27), and in Margelidae
the subdivisions of the fringe simulate tentacles (fig. 24).

The internal anatomy of the Hydromedusae shows numerous variations. The stomach may be altogether
lodged in the manubrium, from which the radial canals then take origin directly as in Geryonia
(Trachomedusae); it may be with or without gastric pouches. The radial canals may be simple or branched,
primarily four, rarely six in number. The ring-canal is drawn out in Narcomedusae into festoons
corresponding with the lobes of the margin, and may be obliterated altogether (Solmaris). In this order the
radial canals are represented only by wide gastric pouches, and in the family Solmaridae are suppressed
altogether, so that the tentacles and the festoons of the ring-canal arise directly from the stomach. In

Geryonia, centripetal canals, ending blindly, arise from the ring-canal and run in a radial direction towards
the centre of the umbrella (fig. 26).

Histology of the Hydromedusa.—The histology described above for the polyp may be taken as the
primitive type, from which that of the medusa differs only in greater elaboration and differentiation of the
cell-elements, which are also more concentrated to form distinct tissues.

Fig. 26.—Carmarina (Geryonia) hastata, one of the Trachomedusae. (After Haeckel.)

a, Nerve ring. k, Dilatation (stomach) of the manubrium.
a′, Radial nerve. l, Jelly of the disk.
b, Tentaculocyst. p, Manubrium.
c, Circular canal. t, Tentacle (hollow and tertiary, i.e. preceded by six perradial
e, Radiating canal. and six interradial solid larval tentacles).
g″. Ovary. u, Cartilaginous margin of the disk covered by thread-cells.
h, Peronia or cartilaginous process ascending from the v. Velum.
cartilaginous margin of the disk centripetally in the outer
surface of the jelly-like disk; six of these are perradial, six
interradial, corresponding to the twelve solid larval
tentacles, resembling those of Cunina.

The ectoderm furnishes the general epithelial covering of the body, and the
muscular tissue, nervous system and sense-organs. The external epithelium is
flat on the ex-umbral surface, more columnar on the sub-umbral surface,
where it forms the muscular tissue of the sub-umbrella and the velum. The
nematocysts of the ectoderm may be grouped to form batteries on the
tentacles, umbrellar margin and oral lappets. In places the nematocysts may
be crowded so thickly as to form a tough, supporting, “chondral” tissue,
resembling cartilage, chiefly developed at the margin of the umbrella and

forming streaks or bars supporting the tentacles (“Tentakelspangen,” peronia) After O. Maas in Results of
or the tentaculocysts (“Gehörspangen,” otoporpae). the “Albatross”
Expedition, Museum of
The muscular tissue of the Hydromedusae is entirely ectodermal. The Comparative Zoology,
muscle-fibres arise as processes from the bases of the epithelial cells; such Cambridge, Mass., U.S.A.
cells may individually become sub-epithelial in position, as in the polyp; or,
Fig. 27.—Stomotoca
in places where muscular tissue is greatly developed, as in the velum or sub-
divisa, one of the Tiaridae
umbrella, the entire muscular epithelium may be thrown into folds in order to
(Anthomedusae).
increase its surface, so that a deeper sub-epithelial muscular layer becomes
separated completely from a more superficial body-epithelium.

In its arrangement the muscular tissue forms two systems: the one composed of striated fibres arranged
circularly, that is to say, concentrically round the central axis of the umbrella; the other of non-striated fibres
running longitudinally, that is to say, in a radial direction from, or (in the manubrium) parallel to, the same
ideal axis. The circular system is developed continuously over the entire sub-umbral surface, and the velum
represents a special local development of this system, at a region where it is able to act at the greatest
mechanical advantage in producing the contractions of the umbrella by which the animal progresses. The
longitudinal system is discontinuous, and is subdivided into proximal, medial and distal portions. The
proximal portion forms the retractor muscles of the manubrium, or proboscis, well developed, for example,
in Geryonia. The medial portion forms radiating tracts of fibres, the so-called “bell-muscles” running
underneath, and parallel to, the radial canals; when greatly developed, as in Tiaridae, they form ridges, so-
called mesenteries, projecting into the sub-umbral cavity. The distal portions form the muscles of the
tentacles. In contrast with the polyp, the longitudinal muscle-system is entirely ectodermal, there being no
endodermal muscles in craspedote medusae.

Fig. 28.—Muscular Cells of Medusae (Lizzia). The uppermost is a purely After O. Maas, Craspedoten Medusen der Siboga
muscular cell from the sub-umbrella; the two lower are epidermo-muscular Expedition, by permission of E. S. Brill & Co.
cells from the base of a tentacle; the upstanding nucleated portion forms part Fig. 29.—Tiaropsis rosea (Ag. and Mayer) showing the
of the epidermal mosaic on the free surface of the body. (After Hertwig.) eight adradial Statocysts, each close to an Ocellus. Cf. fig.
30.

The nervous system of the medusa consists of sub-epithelial ganglion-cells, which form, in the first place,
a diffuse plexus of nervous tissue, as in the polyp, but developed chiefly on the sub-umbral surface; and
which are concentrated, in the second place, to form a definite central nervous system, never found in the
polyp. In Hydromedusae the central nervous system forms two concentric nerve-rings at the margin of the
umbrella, near the base of the velum. One, the “upper” or ex-umbral nerve-ring, is derived from the
ectoderm on the ex-umbral side of the velum; it is the larger of the two rings, containing more numerous but
smaller ganglion-cells, and innervates the tentacles. The other, the “lower” or sub-umbral nerve-ring, is
derived from the ectoderm on the sub-umbral side of the velum; it contains fewer but larger ganglion-cells
and innervates the muscles of the velum (see diagram in article Medusae). The two nerve-rings are
connected by fibres passing from one to the other.

The sensory cells are slender epithelial cells, often with a cilium or stiff protoplasmic process, and should
perhaps be regarded as the only ectoderm-cells which retain the primitive ciliation of the larval ectoderm,
otherwise lost in all Hydrozoa. The sense-cells form, in the first place, a diffuse system of scattered sensory
cells, as in the polyp, developed chiefly on the manubrium, the tentacles and the margin of the umbrella,
where they form a sensory ciliated epithelium covering the nerve-centres; in the second place, the sense-
cells are concentrated to form definite sense-organs, situated always at the margin of the umbrella, hence
often termed “marginal bodies.” The possession of definite sense-organs at once distinguishes the medusa
from the polyp, in which they are never found.

The sense-organs of medusae are of two kinds—first, organs sensitive to light, usually termed ocelli (fig.
29); secondly, organs commonly termed otocysts, on account of their resemblance to the auditory vesicles
of higher animals, but serving for the sense of balance and orientation, and therefore given the special name
of statocysts (fig. 30). The sense-organs may be tentaculocysts, i.e. modifications of a tentacle, as in
Trachylinae, or developed from the margin of the umbrella, in no connexion with a tentacle (or, if so
connected, not producing any modification in the tentacle), as in Leptolinae. In Hydromedusae the sense-
organs are always exposed at the umbrellar margin (hence Gymnophthalmata), while in Scyphomedusae
they are covered over by flaps of the umbrellar margin (hence Steganophthalmata).

Modified after Linko, Traveaux Soc. Imp. Nat., St. Modified after O. and R, Hertwig, Nervensystem und
Petersbourg, xxix. Sinnesorgane der Medusen, by permission of F. C. W. Vogel.

Fig. 30.—Section of a Statocyst and Ocellus of Tiaropsis
Fig. 31.—Section of a Statocyst of Mitrocoma annae.
diademata; cf. fig. 29.

ex, Ex-umbral ectoderm. sub, Sub-umbral ectoderm.

sub, Sub-umbral ectoderm. c.c, Circular canal.
c.c, Circular canal. v, Velum.
v, Velum. st.c, Cavity of statocyst.
st.e, Cavity of statocyst. con, Concrement-cell with otolith.
con, Concrement-cell with otolith.

The statocysts present in general the structure of either a knob or a closed vesicle, composed of (1)
indifferent supporting epithelium: (2) sensory, so-called auditory epithelium of slender cells, each bearing at
its free upper end a stiff bristle and running out at its base into a nerve-fibre; (3) concrement-cells, which
produce intercellular concretions, so-called otoliths. By means of vibrations or shocks transmitted through
the water, or by displacements in the balance or position of the animal, the otoliths are caused to impinge
against the bristles of the sensory cells, now on one side, now on the other, causing shocks or stimuli which
are transmitted by the basal nerve-fibre to the central nervous system. Two stages in the development of the
otocyst can be recognized, the first that of an open pit on a freely-projecting knob, in which the otoliths are
exposed, the second that of a closed vesicle, in which the otoliths are covered over. Further, two distinct
types of otocyst can be recognized in the Hydromedusae: that of the Leptolinae, in which the entire organ is
ectodermal, concrement-cells and all, and the organ is not a tentaculocyst; and that of the Trachylinae, in
which the organ is a tentaculocyst, and the concrement-cells are endodermal, derived from the endoderm of
the modified tentacle, while the rest of the organ is ectodermal.

Modified after O. and R, Hertwig, Nervensystem und Modified after O. and R, Hertwig, Nervensystem und
Sinnesorgane der Medusen, by permission of F. C. W. Vogel. Sinnesorgane der Medusen, by permission of F. C. W. Vogel.

Fig. 32.—Section of a Statocyst of Phialidium. Fig. 33.—Optical Section of a Statocyst of Octorchis.

ex, Ex-umbral ectoderm. con, Concrement-cell with otolith.
sub, Sub-umbral ectoderm. st.c, Cavity of statocyst.
v, Velum.
st.c, Cavity of statocyst.
con, Concrement-cell with otolith.

In the Leptolinae the otocysts are seen in their first stage in Mitrocoma annae (fig. 31) and Tiaropsis
(figs. 29, 30) as an open pit at the base of the velum, on its sub-umbral side. The pit has its opening turned
towards the sub-umbral cavity, while its base or fundus forms a bulge, more or less pronounced, on the ex-
umbral side of the velum. At the fundus are placed the concrement-cells with their conspicuous otoliths
(con) and the inconspicuous auditory cells, which are connected with. the sub-umbral nerve-ring. From the
open condition arises the closed condition very simply by closing up of the aperture of the pit. We then find
the typical otocyst of the Leptomedusae, a vesicle bulging on the ex-umbral side of the velum (figs. 32, 33).
The otocysts are placed on the outer wall of the vesicle (the fundus of the original pit) or on its sides; their
arrangement and number vary greatly and furnish useful characters for distinguishing genera. The sense-
cells are innervated, as before, from the sub-umbral nerve-ring. The inner wall of the vesicle (region of

closure) is frequently thickened to form a so-called “sense-cushion,” apparently a ganglionic offshoot from
the sub-umbral nerve-ring. In many Leptomedusae the otocysts are very small, inconspicuous and
embedded completely in the tissues; hence they may be easily overlooked in badly-preserved material, and
perhaps are present in many cases where they have been said to have been wanting.

After O. and R, Hertwig, Nervensystem und Sinnesorgane After O. and R, Hertwig, Nervensystem und Sinnesorgane
der Medusen, by permission of F. C. W. Vogel. der Medusen, by permission of F. C. W. Vogel.

Fig. 34.—Tentaculocyst (statorhabd) of Cunina solmaris. n.c,
Fig. 35.—Tentaculocyst of Cunina lativentris.
Nerve-cushion; end, endodermal concrement-cells; con, otolith.

ect, Ectoderm.
n.c, Nerve-cushion.
end, Endodermal concrement-cells.
con, Otolith.

In the Trachylinae the simplest condition of the otocyst is a freely projecting club, a so-called statorhabd
(figs. 34, 35), representing a tentacle greatly reduced in size, covered with sensory ectodermal epithelium
(ect.), and containing an endodermal core (end.), which is at first continuous with the endoderm of the ring-
canal, but later becomes separated from it. In the endoderm large concretions are formed (con.). Other
sensory cells with long cilia cover a sort of cushion (n.c.) at the base of the club; the club may be long and
the cushion small, or the cushion large and the club small. The whole structure is innervated, like the
tentacles, from the ex-umbral nerve-ring. An advance towards the second stage is seen in such a form as
Rhopalonema (fig. 36), where the ectoderm of the cushion rises up in a double fold to enclose the club in a
protective covering forming a cup or vesicle, at first open distally; finally the opening closes and the closed
vesicle may sink inwards and be found far removed from the surface, as in Geryonia (fig. 37).

Fig. 36.—Simple tentaculocyst of Rhopalonema velatum. The After O. and R, Hertwig, Nervensystem und Sinnesorgane der
process carrying the otolith or concretion hk, formed by endoderm Medusen, by permission of F. C. W. Vogel.
cells, is enclosed by an upgrowth forming the “vesicle,” which is not Fig. 37.—Section of statocyst of Geryonia (Carmarina
yet quite closed in at the top. (After Hertwig.) hastata).

st.c, Statocyst containing the minute tentaculocyst.
nr1, Ex-umbral nerve-ring.
nr2, Sub-umbral nerve-ring.
ex, Ex-umbral ectoderm.
sub, Sub-umbral ectoderm.
c.c, Circular canal.
v, Velum.

The ocelli are seen in their simplest form as a pigmented patch of ectoderm, which consists of two kinds
of cells—(1) pigment-cells, which are ordinary indifferent cells of the epithelium containing pigment-
granules, and (2) visual cells, slender sensory epithelial cells of the usual type, which may develop visual
cones or rods at their free extremity. The ocelli occur usually either on the inner or outer sides of the
tentacles; if on the inner side, the tentacle is turned upwards and carried over the ex-umbrella, so as to
expose the ocellus to the light; if the ocellus be on the outer side of a tentacle, two nerves run round the base
of the tentacle to it. In other cases ocelli may occur between tentacles, as in Tiaropsis (fig. 29).

The simple form of ocellus described in the foregoing paragraph may become folded into a pit or cup, the
interior of which becomes filled with a clear gelatinous secretion forming a sort of vitreous body. The distal
portion of the vitreous body may project from the cavity of the cup, forming a non-cellular lens as in Lizzia
(fig. 28). Beyond this simple condition the visual organs of the Hydromedusae do not advance, and are far
from reaching the wonderful development of the eyes of Scyphomedusae (Charybdaea).

Besides the ordinary type of ocellus just described, there is found in one genus (Tiaropsis) a type of
ocellus in which the visual elements are inverted, and have their cones turned away from the light, as in the
human retina (fig. 30). In this case the pigment-cells are endodermal, forming a cup of pigment in which the
visual cones are embedded. A similar ocellus is formed in Aurelia among the Scyphomedusae (q.v.).

Other sense organs of Hydromedusae are the so-called sense-clubs or cordyli found in a few
Leptomedusae, especially in those genera in which otocysts are inconspicuous or absent (fig. 39). Each
cordylus is a tentacle-like structure with an endodermal axis containing an axial cavity which may be
continuous with the ring-canal, or may be partially occluded. Externally the cordylus is covered, by very
flattened ectoderm, and bears no otoliths or sense-cells, but the base of the club rests upon the ex-umbral
nerve-ring. Brooks regards these organs as sensory, serving for the sense of balance, and representing a

primitive stage of the tentaculocysts of Trachylinae; Linko, on the other hand, finding no nerve-elements
connected with them, regards them as digestive (?) in function.

The sense-organs of the two fresh-water medusae Limnocodium and Limnocnida are peculiar and of
rather doubtful nature (see E. T. Browne [10]).

The endoderm of the medusa shows the same general types of structure as in
the polyp, described above. We can distinguish (1) digestive endoderm, in the
stomach, often with special glandular elements; (2) circulatory endoderm, in
the radial and ring-canals; (3) supporting endoderm in the axes of the tentacles
and in the endoderm-lamella; the latter is primitively a double layer of cells,
produced by concrescence of the ex-umbral and sub-umbral layers of the
coelenteron, but it is usually found as a single layer of flattened cells (fig. 40);
in Geryonia, however, it remains double, and the centripetal canals arise by
parting of the two layers; (4) excretory endoderm, lining pores at the margin of
the umbrella, occurring in certain Leptomedusae as so-called “marginal
tubercles,” opening, on the one hand, into the ring-canal and, on the other
hand, to the exterior by “marginal funnels,” which debouch into the sub-umbral
cavity above the velum. As has been described above, the endoderm may also
Fig. 38.—Ocellus of
contribute to the sense-organs, but such contributions are always of an
Lizzia koellikeri. oc,
accessory nature, for instance, concrement-cells in the otocysts, pigment in the
Pigmented ectodermal
ocelli, and never of sensory nature, sense-cells being in all cases ectodermal. cells; l, lens. (After
Hertwig.)
The reproductive cells may be regarded as belonging primarily to neither
ectoderm nor endoderm, though lodged in the ectoderm in all Hydromedusae.
As described for the polyp, they are wandering cells capable of extensive migrations before reaching the
particular spot at which they ripen. In the Hydromedusae they usually, if not invariably, ripen in the
ectoderm, but in the neighbourhood of the main sources of nutriment, that is to say, not far from the
stomach. Hence the gonads are found on the manubrium in Anthomedusae generally; on the base of the
manubrium, or under the gastral pouches, or in both these situations (Octorchidae), or under the radial
canals, in Trachomedusae; under the gastral pouches or radial canals, in Narcomedusae. When ripe, the
germ-cells are dehisced directly to the exterior.

After W. K. Brooks, Journal of Morphology, x., by Fig. 40.—Portions of Sections through the Disk of Medusae
permission of Ginn & Co. —the upper one of Lizzia, the lower of Aurelia. (After
Fig. 39.—Section of a Cordylus of Laodice. Hertwig.)

c.c, Circular canal. el, Endoderm lamella.
v, Velum. m, Muscular processes of the ectoderm-cells in cross section.
t, Tentacle. d, Ectoderm.
c, Cordylus, composed of flattened ectoderm ec covering a en, Endoderm lining the enteric cavity.
large-celled endodermal axis en. e, Wandering endoderm cells of the gelatinous substance.

Hydromedusae are of separate sexes, the only known exception being Amphogona apsteini, one of the
Trachomedusae (Browne [9]). Moreover, all the medusae budded from a given hydroid colony are either
male or female, so that even the non-sexual polyp must be considered to have a latent sex. (In Hydra, on the
other hand, the individual is usually hermaphrodite.) The medusa always reproduces itself sexually, and in
some cases non-sexually also. The non-sexual reproduction takes the form of fission, budding or sporogony,
the details of which are described below. Buds may be produced from the manubrium, radial canals, ring-
canal, or tentacle-bases, or from an aboral stolon (Narcomedusae). In all cases only medusa-buds are
produced, never polyp-buds.

The mesogloea of the medusa is largely developed and of great thickness in the umbrella. The sub-
epithelial tissues, i.e. the nervous and muscular cells, are lodged in the mesogloea, but in Hydromedusae it
never contains tissue-cells or mesogloeal corpuscles.

(b) The Medusae as a Subordinate Individuality.—It has been shown above that polyps are budded only
from polyps and that the medusae may be budded either from polyps or from medusae. In any case the
daughter-individuals produced from the buds may be imagined as remaining attached to the parent and
forming a colony of individuals in organic connexion with one another, and thus three possible cases arise.
The first case gives a colony entirely composed of polyps, as in many Hydroidea. The second case gives a
colony partly composed of polyp-individuals, partly of medusa-individuals, a possibility also realized in
many colonies of Hydroidea. The third case gives a colony entirely composed of medusa-individuals, a
possibility perhaps realized in the Siphonophora, which will be discussed in dealing with this group.

The first step towards the formation of a mixed hydroid colony is undoubtedly a hastening of the sexual
maturity of the medusa-individual. Normally the medusae are liberated in quite an immature state; they
swim away, feed, grow and become adult mature individuals. From the bionomical point of view, the
medusa is to be considered as a means of spreading the species, supplementing the deficiencies of the
sessile polyp. It may be, however, that increased reproductiveness becomes of greater importance to the
species than wide diffusion; such a condition will be brought about if the medusae mature quickly and are
either set free in a mature condition or remain in the shelter of the polyp-colony, protected from risks of a
free life in the open sea. In this way the medusa sinks from an independent personality to an organ of the
polyp-colony, becoming a so-called medusoid gonophore, or bearer of the reproductive organs, and losing
gradually all organs necessary for an independent existence, namely those of sense, locomotion and
nutrition.

In some cases both free medusae and gonophores may be produced from the same hydroid colony. This is
the case in Syncoryne mirabilis (Allman [1], p. 278) and in Campanularia volubilis; in the latter, free
medusae are produced in summer, gonophores in winter (Duplessis [14]). Again in Pennaria, the male

medusae are set free in a state of maturity, and have ocelli; the female medusae remain attached and have no
sense organs.

Modified from Weismann, Entstehung der Sexualzellen bei den Hydromedusen.

Fig. 41.—Diagrams of the Structure of the Gonophores of various Hydromedusae, based on the figures of G. J. Allman and A.
Weismann.

A, “Meconidium” of Gonothyraea. s.c, Sub-umbral cavity.
B, Type of Tubularia. t, Tentacles.
C, Type of Garveia, &c. c.c, Circular canal,
D, Type of Plumularia, Agalma, &c. g, Gonads.
E, Type of Coryne, Forskalia, &c. sp, Spadix.
F, G, H, Sporosacs. e.l, Endoderm-lamella.
F, With simple spadix. ex, Ex-umbral ectoderm.
G, With spadix prolonged (Eudendrium). ect, Ectotheca.
H, With spadix branched (Cordylophora).

The gonophores of different hydroids differ greatly in structure from one another, and form a series
showing degeneration of the medusa-individual, which is gradually stripped, as it were, of its characteristic
features of medusan organization and finally reduced to the simplest structure. A very early stage in the
degeneration is well exemplified by the so-called “meconidium” of Gonothyraea (fig. 41, A). Here the
medusoid, attached by the centre of its ex-umbral surface, has lost its velum and sub-umbral muscles, its
sense organs and mouth, though still retaining rudimentary tentacles. The gonads (g) are produced on the
manubrium, which has a hollow endodermal axis, termed the spadix (sp.), in open communication with the

coenosarc of the polyp-colony and serving for the
nutrition of the generative cells. A very similar
condition is seen in Tubularia (fig. 41, B), where,
however, the tentacles have quite disappeared, and the
circular rim formed by the margin of the umbrella has
nearly closed over the manubrium leaving only a small
aperture through which the embryos emerge. The next
step is illustrated by the female gonophores of
Cladocoryne, where the radial and ring-canals have
become obliterated by coalescence of their walls, so
that the entire endoderm of the umbrella is in the
condition of the endoderm-lamella. Next the opening
of the umbrella closes up completely and disappears,
so that the sub-umbral cavity forms a closed space
surrounding the manubrium, on which the gonads are
developed; such a condition is seen in the male
gonophore of Cladocoryne and in Garveia (fig. 41, C),
where, however, there is a further complication in the
form of an adventitious envelope or ectotheca (ect.)
split off from the gonophore as a protective covering,
and not present in Cladocoryne. The sub-umbral cavity
(s.c.) functions as a brood-space for the developing
embryos, which are set free by rupture of the wall. It is
After Allman, Gymnoblastic Hydroids, by permission of
evident that the outer envelope of the gonophore
the Council of the Ray Society.
represents the ex-umbral ectoderm (ex.), and that the
inner ectoderm lining the cavity represents the sub- Fig. 42.—Gonophores of Dicoryne conferta.

umbral ectoderm of the free medusa. The next step is A, A male gonophore still enclosed in its ectotheca.
the gradual obliteration of the sub-umbral cavity (s.c.) B and C, Two views of a female gonophore after
by disappearance of which the sub-umbral ectoderm liberation.
t, Tentacles.
comes into contact with the ectoderm of the ov, Ova, two carried on each female gonophore.
manubrium. Such a type is found in Plumularia and sp, Testis.
also in Agalma (fig. 41, D); centrally is seen the spadix
(sp.), bearing the generative cells (g), and external to these (1) a layer of ectoderm representing the
epithelium of the manubrium; (2) the layer of sub-umbral ectoderm; (3) the endoderm-lamella (e.l.); (4) the
ex-umbral ectoderm (ex.); and (5) there may or may not be present also an ectotheca. Thus the gonads are
covered over by at least four layers of epithelium, and since these are unnecessary, presenting merely
obstacles to the dehiscence of the gonads, they gradually undergo reduction. The sub-umbral ectoderm and
that covering the manubrium undergo concrescence to form a single layer (fig. 41, E), which finally
disappears altogether, and the endoderm-lamella disappears. The gonophore is now reduced to its simplest
condition, known as the sporosac (fig. 41, F, G, H), and consists of the spadix bearing the gonads covered
by a single layer of ectoderm (ex.), with or without the addition of an ectotheca. It cannot be too strongly
emphasized, however, that the sporosac should not be compared simply with the manubrium of the medusa,
as is sometimes done. The endodermal spadix (sp.) of the sporosac represents the endoderm of the
manubrium; the ectodermal lining of the sporosac (ex.) represents the ex-umbral ectoderm of the medusa;
and the intervening layers, together with the sub-umbral cavity, have disappeared. The spadix, as the organ

of nutrition for the gonads, may be developed in various ways, being simple (fig. 41, F) or branched (fig.
41, H); in Eudendrium (fig. 41, G) it curls round the single large ovum.

The hydroid Dicoryne is remarkable for the possession of gonophores, which are ciliate and become
detached and swim away by means of their cilia. Each such sporosac has two long tentacle-like processes
thickly ciliated.

It has been maintained that the gonads of Hydra represent sporosacs or gonophores greatly reduced, with
the last traces of medusoid structure completely obliterated. There is, however, no evidence whatever for
this, the gonads of Hydra being purely ectodermal structures, while all medusoid gonophores have an
endodermal portion. Hydra is, moreover, bisexual, in contrast with what is known of hydroid colonies.

In some Leptomedusae the gonads are formed on the radial canals and form protruding masses
resembling sporosacs superficially, but not in structure. Allman, however, regarded this type of gonad as
equivalent to a sporosac, and considered the medusa bearing them as a non-sexual organism, a
“blastocheme” as he termed it, producing by budding medusoid gonophores. As medusae are known to bud
medusae from the radial canals there is nothing impossible in Allman’s theory, but it cannot be said to have
received satisfactory proof.

Reproduction and Ontogeny of the Hydromedusae.

Nearly every possible method of reproduction occurs amongst the Hydromedusae. In classifying methods of
generation it is usual to make use of the sexual or non-sexual nature of the reproduction as a primary difference,
but a more scientific classification is afforded by the distinction between tissue-cells (histocytes) and germinal
cells, actual or potential (archaeocytes), amongst the constituent cells of the animal body. In this way we may
distinguish, first, vegetative reproduction, the result of discontinuous growth of the tissues and cell-layers of the
body as a whole, leading to (1) fission, (2) autotomy, or (3) vegetative budding; secondly, germinal reproduction,
the result of the reproductive activity of the archaeocytes or germinal tissue. In germinal reproduction the
proliferating cells may be undifferentiated, so-called primitive germ-cells, or they may be differentiated as sexual
cells, male or female, i.e. spermatozoa and ova. If the germ-cells are undifferentiated, the offspring may arise from
many cells or from a single cell; the first type is (4) germinal budding, the second is (5) sporogony. If the germ-
cells are differentiated, the offspring arises by syngamy or sexual union of the ordinary type between an ovum and
spermatozoon, so-called fertilization, of the ovum, or by parthenogenesis, i.e. development of an ovum without
fertilization. The only one of these possible modes of reproduction not known to occur in Hydromedusae is
parthenogenesis.

(1) True fission or longitudinal division of an individual into two equal and similar daughter-individuals is not
common but occurs in Gastroblasta, where it has been described in detail by Arnold Lang [30].

(2) Autotomy, sometimes termed transverse fission, is the name given to a process of unequal fission in which a
portion of the body separates off with subsequent regeneration. In Tubularia by a process of decapitation the
hydranths may separate off and give rise to a separate individual, while the remainder of the body grows a new
hydranth. Similarly in Schizocladium portions of the hydrocaulus are cut off to form so-called “spores,” which
grow into new individuals (see Allman [1]).

Much modified from C. Chun, “Coelenterata,” in Bronn’s
Tierreich. Fig. 44.—Diagrams of Medusa budding with the formation of an
entocodon. The endoderm is shaded, the ectoderm left clear.
Fig. 43.—Direct Budding of Cunina.

A, B, C, E, F, In vertical section. A, B, C, D, F, Successive stages in vertical section.
D, Sketch of external view. E, Transverse section of a stage similar to D.
st, Stomach. Gc, Entocodon.
m, Manubrium. s.c, Cavity of entocodon, forming the future sub-umbral cavity.
t. Tentacle. st, Stomach.
s.o, Sense organ. r.c, Radial canal.
v, Velum. c.c, Circular canal.
s.c, Sub-umbral cavity. e.l, Endoderm lamella.
n.s, Nervous system. m, Manubrium.
v, Velum.
t, Tentacle.

(3) Vegetative budding is almost universal in the Hydromedusae. By budding is understood the formation of a
new individual from a fresh growth of undifferentiated material. It is convenient to distinguish buds that give rise
to polyps from those that form medusae.

(a) The Polyp.—The buds that form polyps are very simple in mode of formation. Four stages may be
distinguished; the first is a simple outgrowth of both layers, ectoderm and endoderm, containing a
prolongation of the coelenteric cavity; in the second stage the tentacles grow out as secondary diverticula
from the side of the first outgrowth; in the third stage the mouth is formed as a perforation of the two layers;
and, lastly, if the bud is to be separated, it becomes nipped off from the parent polyp and begins a free
existence.

(b) The Medusae.—Two types of budding must be distinguished—the direct, so-called, palingenetic type,
and the indirect, so-called coenogenetic type.

The direct type of budding is rare, but is seen in Cunina and Millepora. In Cunina there arises, first, a
simple outgrowth of both layers, as in a polyp-bud (fig. 43, A); in this the mouth is formed distally as a
perforation (B); next the sides of the tube so formed bulge out laterally near the attachment to form the
umbrella, while the distal undilated portion of the tube represents the manubrium (C); the umbrella now
grows out into a number of lobes or lappets, and the tentacles and tentaculocysts grow out, the former in a
notch between two lappets, the latter on the apex of each lappet (D, E); finally, the velum arises as a growth
of the ectoderm alone, the whole bud shapes itself, so to speak, and the little medusa is separated off by
rupture of the thin stalk connecting it with the parent (F). The direct method of medusa-budding only differs
from the polyp-bud by its greater complexity of parts and organs.

The indirect mode of budding (figs. 44, 45) is the commonest method by which medusa-buds are formed.
It is marked by the formation in the bud of a characteristic structure termed the entocodon (Knospenkern,
Glockenkern).

The first stage is a simple hollow outgrowth of both
body-layers (fig. 44, A); at the tip of this is formed a
thickening of the ectoderm, arising primitively as a
hollow ingrowth (fig. 44, B), but more usually as a solid
mass of ectoderm-cells (fig. 45, A). The ectodermal
ingrowth is the entocodon (Gc.); it bulges into, and
pushes down, the endoderm at the apex of the bud, and
if solid it soon acquires a cavity (fig. 44, C, s.c.). The
cavity of the entocodon increases continually in size, Fig. 45.—Modifications of the method of budding
while the endoderm pushes up at the sides of it to form shown in fig. 44, with solid Entocodon (Gc.) and
a cup with hollow walls, enclosing but not quite formation of an ectotheca (ect.).
surrounding the entocodon, which remains in contact at
its outer side with the ectoderm covering the bud (fig. 44, D, v). The next changes that take place are chiefly
in the endoderm-cup (fig. 44, D, E); the cavity between the two walls of the cup becomes reduced by
concrescence to form the radial canals (r.c.), ring-canal (c.c.), and endoderm-lamella (e.l., fig. 44, E), and at
the same time the base of the cup is thrust upwards to form the manubrium (m), converting the cavity of the
entocodon into a space which is crescentic or horse-shoe-like in section. Next tentacles (t, fig. 44, F) grow
out from the ring-canal, and the double plate of ectoderm on the distal side of the entocodon becomes
perforated, leaving a circular rim composed of two layers of ectoderm, the velum (v) of the medusa. Finally,
a mouth is formed by breaking through at the apex of the manubrium, and the now fully-formed medusa
becomes separated by rupture of the stalk of the bud and swims away.

If the bud, however, is destined to give rise not to a free medusa, but to a gonophore, the development is
similar but becomes arrested at various points, according to the degree to which the gonophore is

degenerate. The entocodon is usually formed, proving
the medusoid nature of the bud, but in sporosacs the
entocodon may be rudimentary or absent altogether.
The process of budding as above described may be
varied or complicated in various ways; thus a
secondary, amnion-like, ectodermal covering or
ectotheca (fig. 45, C, ect.) may be formed over all, as
in Garveia, &c.; or the entocodon may remain solid
and without cavity until after the formation of the
manubrium, or may never acquire a cavity at all, as
described above for the gonophores.

Phylogenetic Significance of the Entocodon.—It is
seen from the foregoing account of medusa-budding
that the entocodon is a very important constituent of
the bud, furnishing some of the most essential portions
of the medusa; its cavity becomes the sub-umbral
cavity, and its lining furnishes the ectodermal
epithelium of the manubrium and of the sub-umbral
cavity as far as the edge of the velum. Hence the
entocodon represents a precocious formation of the
sub-umbral surface, equivalent to the peristome of the
polyp, differentiated in the bud prior to other portions
of the organism which must be regarded as antecedent
to it in phylogeny.

If the three principal organ-systems of the medusa,
namely mouth, tentacles and umbrella, be considered
in the light of phylogeny, it is evident that the
manubrium bearing the mouth must be the oldest, as
representing a common property of all the Coelentera,
even of the gastrula embryo of all Enterozoa. Next in Fig. 46.—Diagrams to show the significance of the
order come the tentacles, common to all Cnidaria. The Entocodon in Medusa-buds. (Modified from a diagram

special property of the medusa is the umbrella, given by A. Weismann.)

distinguishing the medusa at once from other I, Ideally primitive method of budding, in which the
morphological types among the Coelentera. If, mouth is formed first (Ia), next the tentacles (Ib), and
lastly the umbrella.
therefore, the formation of these three systems of
II, Method. of Cunina; (a) the mouth arises, next the
organs took place according to a strictly phylogenetic umbrella (b), and lastly the tentacles (c).
sequence, we should expect them to appear in the order III, Hypothetical transition from II to the indirect method
with an entocodon; the formation of the manubrium is
set forth above (fig. 46, Ia, b, c). The nearest approach
retarded, that of the umbrella hastened (IIIa, b).
to the phylogenetic sequence is seen in the budding of IV, a, b, c, budding with an entocodon (cf. fig. 44).
Cunina, where the manubrium and mouth appear first, V, Budding with a solid entocodon (cf. fig. 45).
but the umbrella is formed before the tentacles (fig. 46,
IIa, b, c). In the indirect or coenogenetic method of budding, the first two members of the sequence
exhibited by Cunina change places, and the umbrella is formed first, the manubrium next, and then the
tentacles; the actual mouth-perforation being delayed to the very last (fig. 46, IVa, b, c). Hence the budding

of medusae exemplifies very clearly a common phenomenon in development, a phylogenetic series of
events completely dislocated in the ontogenetic time-sequence.

The entocodon is to be regarded, therefore, not as primarily an ingrowth of ectoderm, but rather as an
upgrowth of both body-layers, in the form of a circular rim (IVa), representing the umbrellar margin; it is
comparable to the bulging that forms the umbrella in the direct method of budding, but takes place before a
manubrium is formed, and is greatly reduced in size, so as to become a little pit. By a simple modification,
the open pit becomes a solid ectodermal ingrowth, just as in Teleostean fishes the hollow medullary tube, or
the auditory pit of other vertebrate embryos, is formed at first as a solid cord of cells, which acquires a
cavity secondarily. Moreover, the entocodon, however developed, gives rise at first to a closed cavity,
representing a closing over of the umbrella, temporary in the bud destined to be a free medusa, but usually
permanent in the sessile gonophore. As has been shown above, the closing up of the sub-umbral cavity is
one of the earliest degenerative changes in the evolution of the gonophore, and we may regard it as the
umbrellar fold taking on a protective function, either temporarily for the bud or permanently for the
gonophore.

To sum up, the entocodon is a precocious formation of the umbrella, closing over to protect the organs in
the umbrellar cavity. The possession of an entocodon proves the medusa-nature of the bud, and can only be
explained on the theory that gonophores are degenerate medusae, and is inexplicable on the opposed view
that medusae are derived from gonophores secondarily set free. In the sporosac, however, the medusa-
individual has become so degenerate that even the documentary proof, so to speak, of its medusoid nature
may have been destroyed, and only circumstantial evidence of its nature can be produced.

4. Germinal Budding.—This method of budding is commonly described as budding from a single body-layer,
instead of from both layers. The layer that produces the bud is invariably the ectoderm, i.e. the layer in which, in
Hydromedusae, the generative cells are lodged; and in some cases the buds are produced in the exact spot in which
later the gonads appear. From these facts, and from those of the sporogony, to be described below, we may regard
budding to this type as taking place from the germinal epithelium rather than from ordinary ectoderm.

(a) The Polyp.—Budding from the ectoderm alone has been described by A. Lang [29] in Hydra and
other polyps. The tissues of the bud become differentiated into ectoderm and endoderm, and the endoderm
of the bud becomes secondarily continuous with that of the parent, but no part of the parental endoderm
contributes to the building up of the daughter-polyp. Lang regarded this method of budding as universal in
polyps, a notion disproved by O. Seeliger [52] who went to the opposite extreme and regarded the type of
budding described by Lang as non-existent. In view, however, both of the statements and figures of Lang
and of the facts to be described presently for medusae (Margellium), it is at least theoretically possible that
both germinal and vegetative budding may occur in polyps as well as in medusae.

(b) The Medusa.—The clearest instance of germinal budding is furnished by Margellium (Rathkea)
octopunctatum, one of the Margelidae. The budding of this medusa has been worked out in detail by Chun
(Hydrozoa, [1]), to whom the reader must be referred for the interesting laws of budding regulating the
sequence and order of formation of the buds.

The buds of Margellium are produced on the manubrium in each of the four interradii, and they arise
from the ectoderm, that is to say, the germinal epithelium, which later gives rise to the gonads. The buds do
not appear simultaneously but successively on each of the four sides of the manubrium, thus: and

secondary buds may be produced on the medusa-buds before the latter are set free as medusae. Each bud
arises as a thickening of the epithelium, which first forms two or three layers (fig. 47, A), and becomes

separated into a superficial layer, future ectoderm, surrounding a central mass, future endoderm (fig. 47, B).
The ectodermal epithelium on the distal side of the bud becomes thickened, grows inwards, and forms a
typical entocodon (fig. 37, D, E, F). The remaining development of the bud is just as described above for
the indirect method of medusa-budding (fig. 47, G, H). When the bud is nearly complete, the body-wall of
the parent immediately below it becomes perforated, placing the coelenteric cavity of the parent in
secondary communication with that of the bud (H), doubtless for the better nutrition of the latter.

Especially noteworthy in the germinal budding of Margellium is the formation of the entocodon, as in the
vegetative budding of the indirect type.

5. Sporogony.—This method of reproduction has been described by E. Metchnikoff in Cunina and allied genera.
In individuals either of the male or female sex, germ-cells which are quite undifferentiated and neutral in character,
become amoeboid, and wander into the endoderm. They divide each into two sister-cells, one of which—the spore
—becomes enveloped by the other. The spore-cell multiplies by division, while the enveloping cell is nutrient and
protective. The spore cell gives rise to a “spore-larva,” which is set free in the coelenteron and grows into a
medusa. Whether sporogony occurs also in the polyp or not remains to be proved.

6. Sexual Reproduction and Embryology.—The ovum of Hydromedusae is usually one of a large number of
oögonia, and grows at the expense of its sister-cells. No regular follicle is formed, but the oöcyte absorbs nutriment
from the remaining oögonia. In Hydra the oöcyte is a large amoeboid cell, which sends out pseudopodia amongst
the oögonia and absorbs nutriment from them. When the oöcyte is full grown, the residual oögonia die off and
disintegrate.

Fig. 47.—Budding from the Ectoderm (germinal epithelium) in Margellium. (After C. Chun.)

A, The epithelium becomes two-layered. G,H, Formation of the medusae.
B, The lower layer forms a solid mass of cells, which (C) becomes a s.c, Sub-umbral cavity.
vesicle, the future endoderm, containing the coelenteric cavity r.c, Radial canal.
(coel), while the outer layer furnishes the future ectoderm. st, Stomach, which in H acquires a secondary communication with
D, E, F, a thickening of the ectoderm on the distal side of the bud the digestive cavity of the mother.
forms an entocodon (Gc). cc, Circular canal.
v, Velum.
t, Tentacle.

The spermatogenesis and maturation and fertilization of the germ-cells present nothing out of the common and
need not be described here. These processes have been studied in detail by A. Brauer [2] for Hydra.

The general course of the development is described in the article Hydrozoa.
We may distinguish the following series of stages: (1) ovum; (2) cleavage,
leading to formation of a blastula; (3) formation of an inner mass or
parenchyma, the future endoderm, by immigration or delamination, leading to
the so-called parenchymula-stage; (4) formation of an archenteric cavity, the
future coelenteron, by a splitting of the internal parenchyma, and of a
blastopore, the future mouth, by perforation at one pole, leading to the gastrula-
stage; (5) the outgrowth of tentacles round the mouth (blastopore), leading to
the actinula-stage; and (6) the actinula becomes the polyp or medusa in the
manner described elsewhere (see articles Hydrozoa, Polyp and Medusa). This
is the full, ideal development, which is always contracted or shortened to a
greater or less extent. If the embryo is set free as a free-swimming, so-called
planula-larva, in the blastula, parenchymula, or gastrula stage, then a free
actinula stage is not found; if, on the other hand, a free actinula occurs, then
there is no free planula stage.

The cleavage of the ovum follows two types, both seen in Tubularia (Brauer
[3]). In the first, a cleavage follows each nuclear division; in the second, the
nuclei multiply by division a number of times, and then the ovum divides into
as many blastomeres as there are nuclei present. The result of cleavage in all
cases is a typical blastula, which when set free becomes oval and develops a
flagellum to each cell, but when not set free, it remains spherical in form and
has no flagella.

The germ-layer formation is always by immigration or delamination, never
by invagination. When the blastula is oval and free-swimming the inner mass is
formed by unipolar immigration from the hinder pole. When the blastula is
spherical and not set free, the germ-layer formation is always multipolar, either
by immigration or by delamination, i.e. by tangential division of the cells of the
blastoderm, as in Geryonia, or by a mixture of immigration and delamination,
as in Hydra, Tubularia, &c. The blastopore is formed as a secondary
perforation at one spot, in free-swimming forms at the hinder pole. Formation
of archenteron and blastopore may, however, be deferred till a later stage
(actinula or after).

The actinula stage is usually suppressed or not set free, but it is seen in
Tubularia (fig. 48), where it is ambulatory, in Gonionemus (Trachomedusae),
and in Cunina (Narcomedusae), where it is parasitic.

In Leptolinae the embryonic development
culminates in a polyp, which is usually formed
by fixation of a planula (parenchymula), rarely
by fixation of an actinula. The planula may fix
itself (1) by one end, and then becomes the
hydrocaulus and hydranth, while the hydrorhiza
grows out from the base; or (2) partly by one Modified from a plate by L.
side and then gives rise to the hydrorhiza as Agassiz, Contributions to
well as to the other parts of the polyp; or (3) Nat. Hist. U.S., iv.
entirely by its side, and then forms a recumbent
Fig. 48.—Free Actinula of
hydrorhiza from which a polyp appears to be
Tubularia.
budded as an upgrowth.

In Trachylinae the development produces always a medusa, and there is no
polyp-stage. The medusa arises direct from the actinula-stage and there is no
entocodon formed, as in the budding described above.

Life-cycles of the Hydromedusae.—The life-cycle of the Leptolinae consists
of an alternation of generations in which non-sexual individuals, polyps,
produce by budding sexual individuals, medusae, which give rise by the sexual
process to the non-sexual polyps again, so completing the cycle. Hence the
alternation is of the type termed metagenesis. The Leptolinae are chiefly forms
belonging to the inshore fauna. The Trachylinae, on the other hand, are above
all oceanic forms, and have no polyp-stage, and hence there is typically no
alternation in their life-cycle. It is commonly assumed that the Trachylinae are
forms which have lost the alternation of generations possessed by them
ancestrally, through secondary simplification of the life-cycle. Hence the
Trachylinae are termed “hypogenetic” medusae to contrast them with the
metagenetic Leptolinae. The whole question has, however, been argued at
length by W. K. Brooks [4], who adduces strong evidence for a contrary view,
that is to say, for regarding the direct type of development seen in Trachylinae

as more primitive, and the metagenesis seen in Leptolinae as a secondary
complication introduced into the life-cycle by the acquisition of larval budding.
The polyp is regarded, on this view, as a form phylogenetically older than the
medusa, in short, as nothing more than a sessile actinula. In Trachylinae the
polyp-stage is passed over, and is represented only by the actinula as a
transitory embryonic stage. In Leptolinae the actinula becomes the sessile
polyp which has acquired the power of budding and producing individuals
either of its own or of a higher rank; it represents a persistent larval stage and
remains in a sexually immature condition as a neutral individual, sex being an
attribute only of the final stage in the development, namely the medusa. The
polyp of the Leptolinae has reached the limit of its individual development and
is incapable of becoming itself a medusa, but only produces medusa-buds;
hence a true alternation of generations is produced. In Trachylinae also the
beginnings of a similar metagenesis can be found. Thus in Cunina octonaria,
the ovum develops into an actinula which buds daughter-actinulae; all of them,
both parent and offspring, develop into medusae, so that there is no alternation
of generations, but only larval multiplication. In Cunina parasitica, however,
the ovum develops into an actinula, which buds actinulae as before, but only
the daughter-actinulae develop into medusae, while the original, parent-actinula
dies off; here, therefore, larval budding has led to a true alternation of
generations. In Gonionemus the actinula becomes fixed and polyp-like, and
reproduces by budding, so that here also an alternation of generations may
occur. In the Leptolinae we must first substitute polyp for actinula, and then a
condition is found which can be compared to the case of Cunina parasitica or
Gonionemus, if we suppose that neither the parent-actinula (i.e. founder-polyp)
nor its offspring by budding (polyps of the colony) have the power of
becoming medusae, but only of producing medusae by budding. For further
arguments and illustrations the reader must be referred to Brooks’s most
interesting memoir. The whole theory is one most intimately connected with
the question of the relation between polyp and medusa, to be discussed
presently. It will be seen elsewhere, however, that whatever view may be held
as to the origin of metagenesis in Hydromedusae, in the case of
Scyphomedusae (q.v.) no other view is possible than that the alternation of
generations is the direct result of larval proliferation.

To complete our survey of life-cycles in the Hydromedusae it is necessary to
add a few words about the position of Hydra and its allies. If we accept the
view that Hydra is a true sexual polyp, and that its gonads are not gonophores
(i.e. medusa-buds) in the extreme of degeneration, then it follows from
Brooks’s theory that Hydra must be descended from an archaic form in which
the medusan type of organization had not yet been evolved. Hydra must, in
short, be a living representative of the ancestor of which the actinula-stage is a
transient reminiscence in the development of higher forms. It may be pointed
out in this connexion that the fixation of Hydra is only temporary, and that the
animal is able at all times to detach itself, to move to a new situation, and to fix
itself again. There is no difficulty whatever in regarding Hydra as bearing the
same relation to the actinula-stage of other Hydromedusae that a Rotifer bears
to a trochophore-larva or a fish to a tadpole.

The Relation of Polyp and Medusa.—Many views have been put forward as to the
morphological relationship between the two types of person in the Hydromedusae.
For the most part, polyp and medusa have been regarded as modifications of a
common type, a view supported by the existence, among Scyphomedusae (q.v.), of
sessile polyp-like medusae (Lucernaria, &c.). R. Leuckart in 1848 compared
medusae in general terms to flattened polyps. G. J. Allman [1] put forward a more
detailed view, which was as follows. In some polyps the tentacles are webbed at the
base, and it was supposed that a medusa was a polyp of this kind set free, the
umbrella being a greatly developed web or membrane extending between the
tentacles. A very different theory was enunciated by E. Metchnikoff. In some
hydroids the founder-polyp, developed from a planula after fixation, throws out
numerous outgrowths from the base to form the hydrorhiza; these outgrowths may
be radially arranged so as to form by contact or coalescence a flat plate. Mechnikov
considered the plate thus formed at the base of the polyp as equivalent to the
umbrella, and the body of the polyp as equivalent to the manubrium, of the medusa;
on this view the marginal tentacles almost invariably present in medusae are new
formations, and the tentacles of the polyp are represented in the medusa by the oral
arms which may occur round the mouth, and which sometimes, e.g. in Margelidae,
have the appearance and structure of tentacles. Apart from the weighty arguments
which the development furnishes against the theories of Allman and Mechnikov, it

may be pointed out that neither hypothesis gives a satisfactory explanation of a
structure universally present in medusae of whatever class, namely the endoderm-
lamella, discovered by the brothers O. and R. Hertwig. It would be necessary to
regard this structure as a secondary extension of the endoderm in the tentacle-web,
on Allman’s theory, or between the outgrowths of the hydrorhiza, on Mechnikov’s
hypothesis. The development, on the contrary, shows unequivocally that the
endoderm-lamella arises as a local coalescence of the endodermal linings of a
primitively extensive gastral space.

The question is one intimately connected with the view taken as to the nature and
individuality of polyp, medusa and gonophore respectively. On this point the
following theories have been put forward.

1. The theory that the medusa is simply an organ, which has become
detached and has acquired a certain degree of independence, like the well-
known instance of the hectocotyle of the cuttle-fish. On this view, put forward
by E. van Beneden and T. H. Huxley, the sporosac is the starting-point of an
evolution leading up through the various types of gonophores to the free
medusa as the culminating point of a phyletic series. The evidence against this
view may be classed under two heads: first, comparative evidence; hydroids
very different in their structural characters and widely separate in the
systematic classification of these organisms may produce medusae very similar,
at least so far as the essential features of medusan organization are concerned;
on the other hydroids closely allied, perhaps almost indistinguishable, may
produce gonophores in the one case, medusae in the other; for example,
Hydractinia (gonophores) and Podocoryne (medusae), Tubularia (gonophores)
and Ectopleura (medusae), Coryne (gonophores) and Syncoryne (medusae),
and so on. If it is assumed that all these genera bore gonophores ancestrally,
then medusa of similar type must have been evolved quite independently in a
great number of cases. Secondly, there is the evidence from the development,
namely, the presence of the entocodon in the medusa-bud, a structure which, as
explained above, can only be accounted for satisfactorily by derivation from a
medusan type of organization. Hence it may be concluded that the gonophores
are degenerate medusae, and not that the medusae are highly elaborated
gonophores, as the organ-theory requires.

2. The theory that the medusa is an independent individual, fully equivalent
to the polyp in this respect, is now universally accepted as being supported by
all the facts of comparative morphology and development. The question still
remains open, however, which of the two types of person may be regarded as
the most primitive, the most ancient in the race-history of the Hydromedusae.
F. M. Balfour put forward the view that the polyp was the more primitive type,
and that the medusa is a special modification of the polyp for reproductive
purposes, the result of division of labour in a polyp-colony, whereby special
reproductive persons become detached and acquire organs of locomotion for
spreading the species. W. K. Brooks, on the other hand, as stated above, regards
the medusa as the older type and looks upon both polyp and medusa, in the
Hydromedusae, as derived from a free-swimming or floating actinula, the
polyp being thus merely a fixed nutritive stage, possessing secondarily
acquired powers of multiplication by budding.

The Hertwigs when they discovered the endoderm-lamella showed on
morphological grounds that polyp and medusa are independent types, each
produced by modification in different directions of a more primitive type
represented in development by the actinula-stage. If a polyp, such as Hydra, be
regarded simply as a sessile actinula, we must certainly consider the polyp to
be the older type, and it may be pointed out that in the Anthozoa only polyp-
individuals occur. This must not be taken to mean, however, that the medusa is
derived from a sessile polyp; it must be regarded as a direct modification of the
more ancient free actinula form, without primitively any intervening polyp-
stage, such as has been introduced secondarily into the development of the
Leptolinae and represents a revival, so to speak, of an ancestral form or larval
stage, which has taken on a special role in the economy of the species.

Systematic Review of the Hydromedusae

Order I. Eleutheroblastea.—Simple polyps which become sexually mature and
which also reproduce non-sexually, but without any medusoid stage in the life-cycle.

The sub-order includes the family Hydridae, containing the common fresh-water
polyps of the genus Hydra. Certain other forms of doubtful affinities have also been
referred provisionally to this section.

Hydra.—This genus comprises fresh-water polyps of simple structure. The
body bears tentacles, but shows no division into hydrorhiza, hydrocaulus or
hydranth; it is temporarily fixed and has no perisarc. The polyp is usually
hermaphrodite, developing both ovaries and testes in the same individual.
There is no free-swimming planula larva, but the stage corresponding to it is
passed over in an enveloping cyst, which is secreted round the embryo by its
own ectodermal layer, shortly after the germ-layer formation is complete, i.e. in
the parenchymula-stage. The envelope is double, consisting of an external
chitinous stratified shell, and an internal thin elastic membrane. Protected by
the double envelope, the embryo is set free as a so-called “egg,” and in Europe
it passes the winter in this condition. In the spring the embryo bursts its shell
and is set free as a minute actinula which becomes a Hydra.

Many species are known, of which three are common in European waters. It
has been shown by C. F. Jickeli (28) that the species are distinguishable by the
characters of their nematocysts. They also show characteristic differences in the
egg (Brauer [2]). In Hydra viridis the polyp is of a green colour and produces a
spherical egg with a smooth shell which is dropped into the mud. H. grisea is
greyish in tint and produces a spherical egg with a spiky shell, which also is
dropped into the mud. H. fusca (= H. vulgaris) is brown in colour, and
produces a bun-shaped egg, spiky on the convex surface, and attached to a
water-weed or some object by its flattened side. Brauer found a fourth species,
similar in appearance to H. fusca, but differing from the three other species in
being of separate sexes, and in producing a spherical egg with a knobby shell,
which is attached like that of H. fusca.

The fact already noted that the species of Hydra can be distinguished by the
characters of their nematocysts is a point of great interest. In each species, two
or three kinds of nematocysts occur, some large, some small, and for specific
identification the nematocysts must be studied collectively in each species. It is
very remarkable that this method of characterizing and diagnozing species has

never been extended to the marine hydroids. It is quite possible that the
characters of the nematocysts might afford data as useful to the systematist in
this group as do the spicules of sponges, for instance. It would be particularly
interesting to ascertain how the nematocysts of a polyp are related to those
possessed by the medusa budded from it, and it is possible that in this manner
obscure questions of relationship might be cleared up.

Protohydra is a marine
genus characterized by the
absence of tentacles, by a great
similarity to Hydra in
histological structure, and by
reproduction by transverse
fission. It was found originally
in an oyster-farm at Ostend.
The sexual reproduction is
unknown. For further
information see C. Chun
(Hydrozoa [1]. Pl. I.).

Polypodium hydriforme
Ussow is a fresh-water form
parasitic on the eggs of the
sterlet. A “stolon” of unknown
origin produces thirty-two
buds, which become as many
Polypodia; each has twenty-
four tentacles and divides by Fig. 49.—Diagram showing possible modifications
fission repeated twice into four of persons of a gymnoblastic Hydromedusa. (After
individuals, each with six Allman.)
tentacles. The daughter- a, Hydrocaulus (stem).
individuals grow, form the full b, Hydrorhiza (root).
number of twenty-four c, Enteric cavity.
d, Endoderm.
tentacles and divide again. The e, Ectoderm.
f, Perisarc, (horny case).

polyps are free and walk on g, Hydranth (hydriform person) expanded.
g′, Hydranth (hydriform person) contracted.
their tentacles. See Ussow [54]. h, Hypostome, bearing mouth at its extremity.
k, Sporosac springing from the hydrocaulus.
Tetraplatia volitans Viguier k′, Sporosac springing from m, a modified
is a remarkable floating marine hydriform person (blastostyle): the genitalia are
form. See C. Viguier [56] and seen surrounding the spadix or manubrium.
l, Medusiform person or medusa.
Delage and Hérouard m, Blastostyle.
(Hydrozoa [2]).

Haleremita Schaudinn. See F. Schaudinn [50] and Delage and Hérouard
(Hydrozoa [2]).

In all the above-mentioned genera, with the exception of Hydra, the life-
cycle is so imperfectly known that their true position cannot be determined in
the present state of our knowledge. They may prove eventually to belong to
other orders. Hence only the genus Hydra can be considered as truly
representing the order Eleutheroblastea. The phylogenetic position of this
genus has been discussed above.

Order II. Hydroidea seu Leptolinae.—Hydromedusae with alternation of
generations (metagenesis) in which a non-sexual polyp-generation (trophosome)
produces by budding a sexual medusa-generation (gonosome). The polyp may be
solitary, but more usually produces polyps by budding and forms a polyp-colony.
The polyp usually has the body distinctly divisible into hydranth, hydrocaulus and
hydrorhiza, and is usually clothed in a perisarc. The medusae may be set free or may
remain attached to the polyp-colony and degenerate into a gonophore. When fully
developed the medusa is characterized by the sense organs being composed entirely
of ectoderm, developed independently of the tentacles, and innervated from the sub-
umbral nerve-ring.

The two kinds of persons present in the typical Hydroidea make the
classification of the group extremely difficult, for reasons explained above.
Hence the systematic arrangement that follows must be considered purely
provisional. A natural classification of the Hydroidea has yet to be put forward.
Many genera and families are separated by purely artificial characters, mere
shelf-and-bottle groupings devised, for the convenience of the museum curator

and the collector. Thus many subdivisions are diagnosed by setting free
medusae in one case, or producing gonophores in another, although it is very
obvious, as pointed out above, that a genus producing medusae may be far
more closely allied to one producing gonophores than to another producing
medusae, or vice versa, and that in some cases the production of medusae or
gonophores varies with the season or the sex. Moreover, P. Hallez [22] has
recently shown that hydroids hitherto regarded as distinct species are only
forms of the same species grown under different conditions.

Sub-Order 1. Hydroidea Gymnoblastea (Anthomedusae).—Trophosome without
hydrothecae or gonothecae, with monopodial type of budding. Gonosome with free
medusae or gonophores; medusae usually with ocelli, never with otocysts. The
gymnoblastic polyp usually has a distinct perisarc investing the hydrorhiza and the
hydrocaulus, sometimes also the hydranth as far as the bases of the tentacles
(Bimeria); but in such cases the perisarc forms a closely-fitting investment or
cuticule on the hydranth, never a hydrotheca standing off from it, as in the next sub-
order. The polyps may be solitary, or form colonies, which may be of the spreading
or encrusting type, or arborescent, and then always of monopodial growth and
budding. In some cases, any polyp of the colony may bud medusae; in other cases,
only certain polyps, the blastostyles, have this power. When blastostyles are present,
however, they are never enclosed in special gonothecae as in the next sub-order. In
this sub-order the characters of the hydranth are very variable, probably owing to the
fact that it is exposed and not protected by a hydrotheca, as in Calyptoblastea.

Fig. 50.—Sarsia (Dipurena) gemnifera. b, Fig. 51.—Sarsia prolifera. Ocelli are seen at the base of the
The long manubrium, bearing medusiform tentacles, and also (as an exception) groups of medusiform
buds; a, mouth. buds.

Speaking generally, three principal types of hydranth can be distinguished,
each with subordinate varieties of form.

1. Club-shaped hydranths with numerous tentacles, generally scattered
irregularly, sometimes with a spiral arrangement, or in whorls (“verticillate”).

(a) Tentacles filiform; type of Clava (fig. 5), Cordylophora, &c.

(b) Tentacles capitate, simple; type of Coryne and Syncoryne; Myriothela
is an aberrant form with some of the tentacles modified as “claspers”
to hold the ova.

(c) Tentacles capitate, branched, wholly or in part; type of Cladocoryne.

(d) Tentacles filiform or capitate, tending to be arranged in definite
whorls; type of Stauridium (fig. 2), Cladonema and Pennaria.

2. Hydranth more shortened, daisy-like in form, with two whorls of
tentacles, oral and aboral.

(a) Tentacles filiform, simple, radially arranged or scattered irregularly;
type of Tubularia (fig. 4), Corymorpha (fig. 3), Nemopsis,
Pelagohydra, &c.

(b) Tentacles with a bilateral arrangement, branched tentacles in addition
to simple filiform ones; type of Branchiocerianthus.

3. Hydranth with a single circlet of tentacles.

(a) With filiform tentacles; the commonest type, seen in Bougainvillea
(fig. 13), Eudendrium, &c.

(b) With capitate tentacles; type of Clavatella.

4. Hydranth with tentacles reduced below four; type of Lar (fig. 11),
Monobrachium, &c.

The Anthomedusa in form is generally deep, bell-shaped. The sense organs are
typically ocelli, never otocysts. The gonads are borne on the manubrium, either
forming a continuous ring (Codonid type), or four masses or pairs of masses
(Oceanid type). The tentacles may be scattered singly round the margin of the
umbrella (“monerenematous”) or arranged in tufts (“lophonematous”); in form they
may be simple or branched (Cladonemid type); in structure they may be hollow
(“coelomerinthous”); or solid (“pycnomerinthous”). When sessile gonophores are
produced, they may show all stages of degeneration.

Classification.—Until quite recently the hydroids (Gymnoblastea) and the
medusae (Anthomedusae) have been classified separately, since the connexion
between them was insufficiently known. Delage and Hérouard (Hydrozoa [2])
were the first to make an heroic attempt to unite the two classifications into
one, to which Hickson (Hydrozoa [4]) has made some additions and slight
modifications. The classification given here is for the most part that of Delage
and Hérouard. It is certain, however, that no such classification can be
considered final at present, but must undergo continual revision in the future.

With this reservation we may recognize fifteen well-characterized families and
others of more doubtful nature. Certain discrepancies must also be noted.

1. Margelidae (= medusa-family Margelidae + hydroid families
Bougainvillidae, Dicorynidae, Bimeridae and Eudendridae). Trophosome
arborescent, with hydranths of Bougainvillea-type; gonosome free medusae or
gonophores, the medusae with solid tentacles in tufts (lophonematous).
Common genera are the hydroid Bougainvillea (figs. 12, 13), and the medusae
Hippocrene (budded from Bougainvillea), Margelis, Rathkea (fig. 24), and
Margellium. Other hydroids are Garveia, Bimeria, Eudendrium and
Heterocordyle, with gonophores, and Dicoryne with peculiar sporosacs.

2. Podocorynidae (= medusa-families Thamnostomidae and Cytaeidae +
hydroid families Podocorynidae and Hydractiniidae). Trophosome encrusting
with hydranths of Bougainvillea-type, polyps differentiated into blastostyles,
gastrozoids and dactylozoids; gonosome free medusae or gonophores. The
typical genus is the well-known hydroid Podocoryne, budding the medusa
known as Dysmorphosa; Thamnostylus, Cytaeis, &c., are other medusae with
unknown hydroids. Hydractinia (figs. 9, 10) is a familiar hydroid genus,
bearing gonophores.

3. Cladonemidae.—Trophosome, polyps with two whorls of tentacles, the
lower filiform, the upper capitate; gonosome, free medusae, with tentacles solid
and branched. The type-genus Cladonema (fig. 20) is a common British form.

4. Clavatellidae.—Trophosome, polyps with a single whorl of capitate
tentacles; gonosome, free medusae, with tentacles branched, solid. Clavatella
(fig. 21), with a peculiar ambulatory medusa is a British form.

5. Pennariidae.—Trophosome, polyps with an upper circlet of numerous
capitate tentacles, and a lower circlet of filiform tentacles. Pennaria, with a
free medusa known as Globiceps, is a common Mediterranean form.
Stauridium (fig. 2) is a British hydroid.

6. Tubulariidae.—Trophosome, polyps with two whorls of tentacles, both
filiform. Tubularia (fig. 4), a well-known British hydroid, bears gonophores.

7. Corymorphidae
(including the medusa-
family Hybocodonidae).—
Trophosome solitary
polyps, with two whorls of
tentacles; gonosome, free
medusae or gonophores.
Corymorpha (fig. 3), a
well-known British genus,
sets free a medusa known
as Steenstrupia (fig. 22).
Here belong the deep-sea
genera Monocaulus and
Branchiocerianthus,
including the largest
hydroid polyps known,
both genera producing
sessile gonophores.

8. Dendroclavidae.—
Trophosome, polyp with
filiform tentacles in three
or four whorls.
Dendroclava, a hydroid,
produces the medusa After Haeckel, System der Medusen, by permission of
known as Turritopsis. Gustav Fischer.

9. Clavidae (including Fig. 52.—Tiara pileata, L. Agassiz.
the medusa-family
Tiaridae (figs. 27 and 51). Trophosome, polyps with scattered filiform
tentacles; gonosome, medusae or gonophores, the medusae with hollow
tentacles. Clava (fig. 5), a common British hydroid, produces gonophores; so
also does Cordylophora, a form inhabiting fresh or brackish water. Turris
produces free medusae. Amphinema is a medusan genus of unknown hydroid.

10. Bythotiaridae.—Trophosome
unknown; gonosome, free medusae, with
deep, bell-shaped umbrella, with interradial
gonads on the base of the stomach, with
branched radial canals, and correspondingly
numerous hollow tentacles. Bythotiara,
Sibogita.

11. Corynidae (= hydroid families
Corynidae, Syncorynidae and
Cladocorynidae + medusan family
Sarsiidae).—Trophosome polyps with
capitate tentacles, simple or branched,
scattered or verticillate; gonosome, free
medusae or gonophores. Coryne, a common
British hydroid, produces gonophores;
Syncoryne, indistinguishable from it,
produces medusae known as Sarsia (fig.
51). Cladocoryne is another hydroid genus;
Codonium and Dipurena (fig. 50) are
medusan genera.
After Haeckel, System der
Medusen, by permission of 12. Myriothelidae.—The genus
Gustav Fischer. Myriothela is a solitary polyp with scattered
capitate tentacles, producing sporosacs.
Fig. 53.—Pteronema darwinii.
The apex of the stomach is 13. Hydrolaridae.—Trophosome (only
prolonged into a brood pouch known in one genus), polyps with two
containing embryos.
tentacles forming a creeping colony;
gonosome, free medusae with four, six or
more radial canals, giving off one or more lateral branches which run to the
margin of the umbrella, with the stomach produced into four, six or more lobes,
upon which the gonads are developed; the mouth with four lips or with a folded
margin; the tentacles simple, arranged evenly round the margin of the umbrella.
The remarkable hydroid Lar (fig. 11) grows upon the tubes of the worm

Sabella and produces a medusa known as Willia. Another medusan genus is
Proboscidactyla.

14. Monobrachiidae.—The genus Monobrachium is a colony-forming
hydroid which grows upon the shells of bivalve molluscs, each polyp having
but a single tentacle. It buds medusae, which, however, are as yet only known
in an immature condition (C. Mereschkowsky [41]).

15. Ceratellidae.—Trophosome polyps forming branching colonies of which
the stem and main branches are thick and composed of a network of
anastomosing coenosarcal tubes covered by a common ectoderm and supported
by a thick chitinous perisarc; hydranths similar to those of Coryne; gonosome,
sessile gonophores. Ceratella, an exotic genus from the coast of East Africa,
New South Wales and Japan. The genera Dehitella Gray and Dendrocoryne
Inaba should perhaps be referred to this family; the last-named is regarded by
S. Goto [16] as the type of a distinct family, Dendrocorynidae.

Doubtful families, or forms difficult to classify, are: Pteronemidae, Medusae
of Cladonemid type, with hydroids for the most part unknown. The British
genus Gemmaria, however, is budded from a hydroid referable to the family
Corynidae. Pteronema (fig. 53).

Nemopsidae, for the floating polyp Nemopsis, very similar to Tubularia in
character; the medusa, on the other hand, is very similar to Hippocrene
(Margelidae). See C. Chun (Hydrozoa [1]).

Pelagohydridae, for the floating polyp Pelagohydra, Dendy, from New
Zealand. The animal is a solitary polyp bearing a great number of medusa-
buds. The body, representing the hydranth of an ordinary hydroid, has the
aboral portion modified into a float, from which hangs down a proboscis
bearing the mouth. The float is covered with long tentacles and bears the
medusa-buds. The proboscis bears at its extremity a circlet of smaller oral
tentacles. Thus the affinities of the hydranth are clearly, as Dendy points out,
with a form such as Corymorpha, which also is not fixed but only rooted in the
mud. The medusae, on the other hand, have the tentacles in four tufts of (in the

buds) five each, and thus resemble the medusae of the family Margelidae. See
A. Dendy [12].

Perigonimus.—This
common British hydroid
belongs by its characters to the
family Bougainvillidae; it
produces, however, a medusa
of the genus Tiara (fig. 52),
referable to the family
Clavidae; a fact sufficient to
indicate the tentative character
of even the most modern
classifications of this order.

Sub-order II. Hydroidea
Calyptoblastea (Leptomedusae).
—Trophosome with polyps
always differentiated into
nutritive and reproductive
individuals (blastostyles)
enclosed in hydrothecae and
gonothecae respectively; with
sympodial type of budding.
Gonosome with free medusae or Fig. 54.—Diagram showing possible modifications
gonophores; the medusae of the persons of a Calyptoblastic Hydromedusa.
typically with otocysts, Letters a to h same as in fig. 49. i, The horny cup or

sometimes with cordyli or ocelli hydrotheca of the hydriform persons; l, medusiform
person springing from m, a modified, hydriform
(figs. 54, 55).
person (blastostyle); n, the horny case or gonangium
The calyptoblastic polyp of the enclosing the blastostyle and its buds. This and the
nutritive type is very uniform in hydrotheca i give origin to the name
character, its tendency to Calyptoblastea. (After Allman.)

variation being limited, as it
were, by the enclosing hydrotheca. The hydranth almost always has a single circlet

of tentacles, like the Bougainvillea-
type, in the preceding sub-order; an
exception is the curious genus
Clathrozoon, in which the hydranth has
a single tentacle. The characteristic
hydrotheca is formed by the bud at an
early stage (fig. 56); when complete it
is an open cup, in which the hydranth
develops and can be protruded from the
opening for the capture of food, or is
withdrawn into it for protection.
Solitary polyps are unknown in this
Fig. 55.—View of the Oral Surface of one of
the Leptomedusae (Irene pellucida, Haeckel), sub-order; the colony may be creeping
to show the numerous tentacles and the
otocysts.
or arborescent in form; if the latter, the
budding of the polyps, as already
ge, Genital glands.
M, Manubrium.
stated, is of the sympodial type, and
ot, Otocysts. either biserial, forming stems capable
rc, The four radiating canals. of further branching, or uniserial,
Ve, The velum.
forming pinnules not capable of further
branching. In the biserial type the
polyps on the two sides of the stem have primitively an alternating, zigzag
arrangement; but, by a process of differential growth, quickened in the 1st, 3rd, 5th,
&c., members of the stem, and retarded in the 2nd, 4th, 6th, &c., members, the
polyps may assume secondarily positions opposite to one another on the two sides of
the stem. Other variations in the mode of growth or budding bring about further
differences in the building up of the colony, which are not in all cases properly
understood and cannot be described in detail here. The stem may contain a single
coenosarcal tube (“monosiphonic”) or several united in a common perisarc
(“polysiphonic”). An important variation is seen, in the form of the hydrotheca itself,
which may come off from the main stem by a stalk, as in Obelia, or may be sessile,
without a stalk, as in Sertularia.

In many Calyptoblastea there occur also reduced defensive polyps or
dactylozoids, which in this sub-order have received the special name of sarcostyles.

Such are the “snake-like zoids” of
Ophiodes and other genera, and as
such are generally interpreted the
“machopolyps” of the
Plumularidea. These organs are
supported by cuplike structures of
the perisarc, termed
nematophores, regarded as
modified hydrothecae supporting
the specialized polyp-individuals.
They are specially characteristic
of the family Plumularidae.

The medusa-buds, as already
stated, are always produced from
blastostyles, reduced non-nutritive
polyps without mouth or
tentacles. An apparent, but not
real, exception is Halecium
halecinum, in which the
blastostyle is produced from the
After Allman, Gymnoblastic Hydroids, by permission
side of a nutritive polyp, and both of the council of the Ray Society.
are enclosed in a common theca Fig. 56.—Diagrams to show the mode of formation of
without a partition between them the Hydrotheca and Gonotheca in Calyptoblastic
Hydroids. A-D are stages common to both; from D
(Allman [1] p. 50, fig. 24). The arises the hydrotheca (E) or the gonotheca (F); th,
theca; st, stomach; t, tentacles; m, mouth; mb,
gonotheca is formed in its early medusa-buds.
stage in the same way as the
hydrotheca, but the remains of the hydranth persists as an operculum closing the
capsule, to be withdrawn when the medusae or genital products are set free (fig. 56).

The blastostyles, gonophores and gonothecae furnish a series of variations
which can best be considered as so many stages of evolution.

Stage 1, seen in Obelia. Numerous medusae are budded successively within
the gonotheca and set free; they swim off and mature in the open sea (Allman

[1], p. 48, figs. 18, 19).

Stage 2, seen in Gonothyraea. Medusae, so-called “meconidia,” are budded
but not liberated; each in turn, when it reaches sexual maturity, is protruded
from the gonotheca by elongation of the stalk, and sets free the embryos, after
which it withers and is replaced by another (Allman [1], p. 57, fig. 28).

Stage 3, seen in Sertularia.—The gonophores are reduced in varying degree,
it may be to sporosacs; they are budded successively from the blastostyle, and
each in turn, when ripe, protrudes the spadix through the gonotheca (fig. 57, A,
B). The spadix forms a gelatinous cyst, the so-called acrocyst (ac), external to
the gonotheca (gth), enclosing and protecting the embryos. Then the spadix
withers, leaving the embryos in the acrocyst, which may be further protected by
a so-called marsupium, a structure formed by tentacle-like processes growing
out from the blastostyle to enclose the acrocyst, each such process being
covered by perisarc like a glove-finger secreted by it (fig. 57, C). (Allman [1],
pp. 50, 51, figs. 21-24; Weismann [58], p. 170, pl. ix., figs. 7, 8.)

Stage 4, seen in Plumularidae.—The generative elements are produced in
structures termed corbulae, formed by reduction and modification of branches
of the colony. Each corbula contains a central row of blastostyles enclosed and
protected by lateral rows of branches representing stunted buds (Allman [1], p.
66, fig. 30).

The Leptomedusa in form is generally shallow, more or less saucer-like, with
velum less developed than in Anthomedusae (fig. 55). The characteristic sense-
organs are ectodermal otocysts, absent, however, in some genera, in which case
cordyli may replace them. When otocysts are present, they are at least eight in
number, situated adradially, but are often very numerous. The cordyli are scattered
on the ring-canal. Ocelli, if present, are borne on the tentacle-bulbs. The tentacles
are usually hollow, rarely solid (Obelia). In number they are rarely less than four,
but in Dissonema there are only two. Primitively there are four perradial tentacles, to
which may be added four interradial, or they may become very numerous and are
then scattered evenly round the margin, never arranged in tufts or clusters. In
addition to tentacles, there may be marginal cirri (Laodice) with a solid endodermal
axis, spirally coiled, very contractile, and bearing a terminal battery of nematocysts.

The gonads are developed
typically beneath the radial
canals or below the stomach or
its pouches, often stretching as
long bands on to the base of
the manubrium. In
Octorchidae (fig. 58) each
such band is interrupted,
forming one mass at the base
of the manubrium and another
below the radial canal in each
radius, in all eight separate
gonad-masses, as the name
implies. In some
Leptomedusae excretory After Allman, Gymnoblastic Hydroids, by permission of
“marginal tubercles” are the council of the Ray Society.

developed on the ring-canal. Fig. 57.—Diagrams to show the mode of formation of an
Acrocyst and a Marsupium. In A two medusa-buds are
seen within the gonotheca (gth), the upper more advanced
Classification.—As in the than the lower one. In B the spadix of the upper bud has
Gymnoblastea, the difficulty protruded itself through the top of the gonotheca and the
acrocyst (ac) is secreted round it. In C the marsupium (m)
of uniting the hydroid and is formed as finger-like process from the summit of the
medusan systems into one blastostyle, enclosing the acrocyst; b, medusa-buds on the
blastostyle.
scheme of classification is
very great in the present state of our knowledge. In a great many Leptomedusae
the hydroid stage is as yet unknown, and it is by no means certain even that
they possess one. It is quite possible that some of these medusae will be found
to be truly hypogenetic, that is to say, with a life-cycle secondarily simplified
by suppression of metagenesis. At present, ten recent and one extinct family of
Calyptoblastea (Leptomedusae) may be recognized provisionally:

1. Eucopidae (figs. 55, 59).—Trophosome with stalked hydrothecae;
gonosome, free medusae with otocysts and four, rarely six or eight, unbranched
radial canals. Two of the commonest British hydroids belong to this family,
Obelia and Clytia. Obelia forms numerous polyserial stems of the
characteristic zigzag pattern growing up from a creeping basal stolon, and buds

the medusa of the same name. In Clytia the polyps arise singly from the stolon,
and the medusa is known as Phialidium (fig. 59).

2. Aequoridae.—Trophosome only known in one genus (Polycanna), and
similar to the preceding; gonosome, free medusae with otocysts and with at
least eight radial canals, often a hundred or more, simple or branched.
Aequorea is a common medusa.

3. Thaumantidae.—Trophosome only known in one genus (Thaumantias),
similar to that of the Eucopidae; gonosome, free medusae with otocysts
inconspicuous or absent, with usually four, sometimes eight, rarely more than
eight, radial canals, simple and unbranched, along which the gonads are
developed, with numerous tentacles bearing ocelli and with marginal sense-
clubs. Laodice and Thaumantias are representative genera.

4. Berenicidae.—Trophosome unknown; gonosome, free medusae, with four
or six radial canals, bearing the gonads, with numerous tentacles, between
which occur sense-clubs, without otocysts. Berenice, Staurodiscus, &c.

After Haeckel, System der Medusen, by permission of Gustav Fischer.
Fig. 58.—Octorchandra canariensis, from life.

5. Polyorchidae.—Trophosome unknown; gonosome, free medusae of deep
form, with radial canals branched in a feathery manner, and bearing gonads on
the main canal, but not on the branches, with numerous hollow tentacles
bearing ocelli, and without otocysts. Polyorchis, Spirocodon.

6. Campanularidae.--Trophosome as in Eucopidae; gonosome, sessile
gonophores. Many common or well-known genera belong here, such as
Halecium, Campanularia, Gonothyraea, &c.

7. Lafoëidae.—Trophosome as in the preceding; gonosome, free medusae or
gonophores, the medusae with large open otocysts. The hydroid genus Lafoëa
is remarkable for producing gonothecae on the hydrorhiza, each containing a
blastostyle which bears a single gonophore; this portion of the colony was

formerly regarded as an independent parasitic hydroid, and was named
Coppinia. Medusan genera are Mitrocoma, Halopsis, Tiaropsis (fig. 29, &c.).

(So far as the characters of the trophosome are concerned, the seven
preceding families are scarcely distinguishable, and they form a section apart,
contrasting sharply with the families next to be mentioned, in none of which
are free medusae liberated from the colony, so that only the characters of the
trophosome need be considered.)

After E. T. Browne, Proc. Zool. Soc. of London, 1896.
Fig. 59.—Three stages in the development of Phialidium temporarium. a, The youngest stage, is
magnified about 22 diam.; b, older, is magnified about 8 diam.; c, the adult medusa, is magnified.

8. Sertularidae.—Hydrothecae sessile, biserial, alternating or opposite on the
stem. Sertularia and Sertularella are two very common genera of this family.

9. Plumularidae.—Hydrothecae sessile, biserial on the main stem, uniserial
on the lateral branches or pinnules, which give the colony its characteristic
feathery form; with nematophores. A very abundant and prolific family; well-
known British genera are Plumularia, Antennularia and Aglaophenia.

10. Hydroceratinidae.—This family contains the single Australian species
Clathrozoon wilsoni Spencer, in which a massive hydrorhiza bears sessile
hydrothecae, containing hydranths each with a single tentacle, and numerous
nematophores. See W. B. Spencer [53].

11. Dendrograptidae, containing fossil (Silurian) genera, such as
Dendrograptus and Thamnograptus, of doubtful affinities.

Order III. Hydrocorallinae.—Metagenetic
colony-forming Hydromedusae, in which the
polyp-colony forms a massive, calcareous
corallum into which the polyps can be retracted;
polyp-individuals always of two kinds,
gastrozoids and dactylozoids; gonosome either
free medusae or sessile gonophores. The
trophosome consists of a mass of coenosarcal
tubes anastomosing in all planes. The
interspaces between the tubes are filled up by a
solid mass of lime, consisting chiefly of calcium Fig. 60.—Portion of the calcareous
corallum of Millepora nodosa,
carbonate, which replaces the chitinous perisarc showing the cyclical arrangement
of ordinary hydroids and forms a stony corallum of the pores occupied by the
“persons” or hydranths. About
or coenosteum (fig. 60). The surface of the twice the natural size. (From
Moseley.)
coenosteum is covered by a layer of common
ectoderm, containing large nematocysts, and is
perforated by pores of two kinds, gastropores and dactylopores, giving exit to
gastrozoids and dactylozoids respectively, which are lodged in vertical pore-canals
of wider calibre than the coenosarcal canals of the general network. The coenosteum
increases in size by new growth at the surface; and in the deeper, older portions of
massive forms the tissues die off after a certain time, only the superficial region
retaining its vitality down to a certain depth. The living tissues at the surface are cut
off from the underlying dead portions by horizontal partitions termed tabulae, which
are formed successively as the coenosteum increases in age and size. If the
coenosteum of Millepora be broken across, each pore-canal (perhaps better termed a
polyp-canal) is seen to be interrupted by a series of transverse partitions,
representing successive periods of growth with separation from the underlying dead
portions.

Fig. 61.—Enlarged view of the surface of a living Millepora, showing five dactylozooids surrounding a
central gastrozooid. (From Moseley.)

Besides the wider vertical pore-
canals and the narrower, irregular
coenosarcal canals, the
coenosteum may contain, in its
superficial portion, chambers or
ampullae, in which the
reproductive zoids (medusae or
gonophores) are budded from the
coenosarc.

The gastropores and
dactylopores are arranged in
various ways at the surface, a
common pattern being the
formation of a cyclosystem (fig.
60), in which a central gastrozoid Fig. 62.—Diagrams illustrating the successive stages
in the development of the cyclosystems of the

Stylasteridae. (After Moseley.)
is surrounded by a ring of
dactylozoids (fig. 61). In such a 1, Sporadopora dichotoma.
2, 3, Allopora nobilis.
system the dactylopores may be
4, Allopora profunda.
confluent with the gastropore, so 5, Allopora miniacea.
that the entire cyclosystem 6, Astylus subviridis.
7, Distichopora coccinea.
presents itself as a single aperture
s, Style.
subdivided by radiating partitions, dp, Dactylopore.
thus having a superficial gp, Gastropore.
b, In fig. 6, inner horseshoe-shaped mouth of
resemblance to a madreporarian
gastropore.
coral with its radiating septa (figs.
62 and 63).

The gastrozoids usually bear short capitate tentacles, four, six or twelve in
number; but in Astylus (fig. 63) they have no tentacles. The dactylozoids have no
mouth; in Milleporidae they have short capitate tentacles, but lack tentacles in
Stylasteridae.

The gonosome consists of free medusae in Milleporidae, which are budded from
the apex of a dactylozoid in Millepora murrayi, but in other species from the
coenosarcal canals. The medusae are produced by direct budding, without an
entocodon in the bud. They are liberated in a mature condition, and probably live but
a short time, merely sufficient to spread the species. The manubrium bearing the
gonads is mouthless, and the umbrella is without tentacles, sense-organs, velum or
radial canals. In the Stylasteridae sessile gonophores are formed, always by budding
from the coenosarc. In Distichopora the gonophores have radial canals, but in other
genera they are sporosacs with no trace of medusoid structure.

Classification.—Two families are known:—

1. Milleporidae.—Coenosteum massive, irregular in form; pores scattered
irregularly or in cyclosystems, without styles, with transverse tabulae; free
medusae. A single genus, Millepora (figs. 60, 61).

2. Stylasteridae.—Coenosteum arborescent, sometimes fanlike, with pores
only on one face, or on the lateral margins of the branches; gastropores with
tabulae only in two genera, but with (except in Astylus) a style, i.e. a conical,

thorn-like projection from the base of the pore,
sometimes found also in dactylopores; sessile
gonophores. Sporadopora has the pores scattered
irregularly. Distichopora has the pores arranged in
rows. Stylaster has cyclosystems. In Allopora the
cyclostems resemble the calyces of Anthozoan
corals. In Cryptohelia the cyclosystem is covered by
a cap or operculum. In Astylus (fig. 63) styles are
absent.

Affinities of the Hydrocorallinae.—There can be
no doubt that the forms comprised in this order bear
a close relationship to the Hydroidea, especially the
sub-order Gymnoblastea, with which they should
perhaps be classed in a natural classification. A
hydrocoralline may be regarded as a form of
hydroid colony in which the coenosarc forms a felt-
work ramifying in all planes, and in which the
Fig. 63.—Portion of the chitinous perisarc is replaced by a massive
corallum of Astylus
subviridis (one of the calcareous skeleton. So far as the trophosome is
Stylasteridae), showing concerned, the step from an encrusting hydroid such
cyclosystems placed at
intervals on the branches, as Hydractinia to the hydrocoralline Millepora is
each with a central
gastropore and zone of not great.
slit-like dactylopores.
(After Moseley.) Hickson considers that the families Milleporidae
and Stylasteridae should stand quite apart from one
another and should not be united in one order. The nearest approach to the
Stylasteridae is perhaps to be found in Ceratella, with its arborescent
trophosome formed of anastomosing coenosarcal tubes supported by a thick
perisarc and covered by a common ectoderm. Ceratella stands in much the
same relation to the Stylasteridae that Hydractinia does to the Milleporidae, in
both cases the chitinous perisarc being replaced by the solid coenosteum to
which the hydrocorallines owe the second half of their name.

Order IV. Graptolitoidea (Rhabdophora, Allman).—This order has been
constituted for a peculiar group of palaeozoic fossils, which have been interpreted as
the remains of the skeletons of Hydrozoa of an extinct type.

A typical graptolite consists of an axis bearing a series of tooth-like projections,
like a saw. Each such projection is regarded as representing a cup or hydrotheca,
similar to those borne by a calyptoblastic hydroid, such as Sertularia. The supposed
hydrothecae may be present on one side of the axis only (monoprionid) or on both
sides (diprionid); the first case may be conjectured to be the result of uniserial
(helicoid) budding, the second to be produced by biserial (scorpioid) budding. In
one division (Retiolitidae) the axis is reticulate. In addition to the stems bearing
cups, there are found vesicles associated with them, which have been interpreted as
gonothecae or as floats, that is to say, air-bladders, acting as hydrostatic organs for a
floating polyp-colony.

Since no graptolites are known living, or, indeed, since palaeozoic times, the
interpretation of their structure and affinities must of necessity be extremely
conjectural, and it is by no means certain that they are Hydrozoa at all. It can only be
said that their organization, so far as the state of their preservation permits it to be
ascertained, offers closer analogies with the Hydrozoa, especially the
Calyptoblastea, than with any other existing group of the animal kingdom.

See the treatise of Delage and Hérouard (Hydrozoa, [4]), and the article
Graptolites.

Order V. Trachylinea.—Hydromedusae without alternation of generations, i.e.
without a hydroid phase; the medusa develops directly from the actinula larva,
which may, however, multiply by budding. Medusae with sense-organs represented
by otocysts derived from modified tentacles (tentaculocysts), containing otoliths of
endodermal origin, and innervated from the ex-umbral nerve-ring.

This order, containing the typical oceanic medusae, is divided into two sub-
orders.

Sub-order 1. Trachomedusae.—Tentacles given off from the margin of the
umbrella, which is entire, i.e. not lobed or indented; tentaculocysts usually enclosed
in vesicles; gonads on the radial canals. The medusae of this order are characterized

by the tough, rigid consistence of the umbrella, due partly to the dense nature of the
mesogloea, partly to the presence of a marginal rim of chondral tissue, consisting of
thickened ectoderm containing great numbers of nematocysts, and forming, as it
were, a cushion-tyre supporting the edge of the umbrella. Prolongations from the
rim of chondral tissue may form clasps or peronia supporting the tentacles. The
tentacles are primarily four in number, perradial, alternating with four interradial
tentaculocysts, but both tentacles and sense-organs may be multiplied and the
primary perradii may be six instead of four (fig. 26). The tentacles are always solid,
containing an axis of endoderm-cells resembling notochordal tissue or plant-
parenchyma, and are but moderately flexible. The sense-organs are tentaculocysts
which are usually enclosed in vesicles and may be sunk far below the surface. The
gonads are on the radial canals or on the stomach (Ptychogastridae), and each gonad
may be divided into two by a longitudinal sub-umbral muscle-tract. The radial
canals are four, six, eight or more, and in some genera blindly-ending centripetal
canals are present (fig. 26). The stomach may be drawn out into the manubrium,
forming a proboscis (“Magenstiel”) of considerable length.

The development of the Trachomedusae, so far as it is known, shows an actinula-
stage which is either free (larval) or passed over in the egg (foetal) as in Geryonia;
in no case does there appear to be a free planula-stage. The actinula, when free, may
multiply by larval budding, but in all cases both the original actinula and all its
descendants become converted into medusae, so that there is no alternation of
generations. In Gonionemus the actinula becomes attached and polyp-like and
reproduces by budding.

The Trachomedusae are divided into the following families:

1. Petasidae (Petachnidae).—Four radial canals, four gonads; stomach not
prolonged into the manubrium, which is relatively short; tentaculocysts free.
Petasus and other genera make up this family, founded by Haeckel, but no
other naturalist has ever seen them, and it is probable that they are simply
immature forms of other genera.

2. Olindiadae, with four radial canals and four gonads; manubrium short;
ring-canals giving off blind centripetal canals; tentaculocysts enclosed.
Olindias mülleri (fig. 64) is a common Mediterranean species. Other genera are

Aglauropsis, Gossea and
Gonionemus; the last named bears
adhesive suckers on the tentacles.
Some doubt attaches to the position
of this family. It has been asserted
that the tentaculocysts are entirely
ectodermal and that either the
family should be placed amongst
the Leptomedusae, or should form,
together with certain
Leptomedusae, an entirely distinct
order. In Gonionemus, however,
the concrement-cells are
endodermal.

3. Trachynemidae.—Eight radial
canals, eight gonads, stomach not
prolonged into manubrium;
tentaculocysts enclosed.
Rhopalonema, Trachynema, &c.

4.
Ptyc
hoga
strid
ae After Haeckel, System der Medusen, by
(Pect permission of Gustav Fischer.

yllid Fig. 64. Olindias mülleri.
ae).
—As in the preceding, but with suckers on the
tentacles. Ptychogastria Allman (= Pectyllis), a
deep-sea form.

5. Aglauridae.—Eight radial canals, two, four
or eight gonads; tentacles numerous;

After E. T. Browne, Proc. tentaculocysts free; stomach prolonged into
Zool. Soc. of London. manubrium. Aglaura, Aglantha (fig. 65), &c.,
with eight gonads; Stauraglaura with four; Persa
Fig. 65.—Aglantha rosea
(Forbes), a British medusa.
with two. Amphogona, hermaphrodite, with male
and female gonads on alternating radial canals.

6. Geryonidae.—Four or six radial canals; gonads band-like; stomach
prolonged into a manubrium of great length; tentaculocysts enclosed. Liriope,
&c., with four radial canals; Geryonia, Carmarina (fig. 26), &c., with six.

7. Halicreidae.—Eight very broad radial canals; ex-umbrella often provided
with lateral outgrowths; tentacles differing in size, but in a single row.
Halicreas.

Sub-order 2. Narcomedusae.—Margin of the umbrella-lobed, tentacles arising
from the ex-umbrella at some distance from the margin; tentaculocysts exposed, not
enclosed in vesicles; gonads on the sub-umbral floor of the stomach or of the gastric
pouches.

The Narcomedusae exhibit
peculiarities of form and structure
which distinguish them at once from all
other Hydromedusae. The umbrella is
shallow and has the margin supported
by a rim of thickened ectoderm, as in
the Trachomedusae, but not so strongly
developed. The tentacles are not
inserted on the margin of the umbrella,
Fig. 66.—Cunina rhododactyla, one of the
but arise high up on the ex-umbral Narcomedusae. (After Haeckel.)
surface, and the umbrella is prolonged
c, Circular canal.
into lobes corresponding to the h, “Otoporpae” or centripetal process of the
interspaces between the tentacles. The marginal cartilaginous ring connected with
tentaculocyst.
condition of things can be imagined by
k, Stomach.
supposing that in a medusa primitively l, Jelly of the disk.
of normal build, with tentacles at the r, Radiating canal (pouch of stomach).
tt, Tentacles.
margin, the umbrella has grown down

past the insertion of the tentacles. As a tw, Tentacle root.

result of this extension of the umbrellar
margin, all structures belonging to this region, namely, the ring-canal, the nerve-
rings, and the rim of thickened ectoderm, do not run an even course, but are thrown
into festoons, caught up under the insertion of each tentacle in such a way that the
ring-canal and its accompaniments form in each notch of the umbrellar margin an
inverted V, the apex of which corresponds to the insertion of the tentacle; in some
cases the limbs of the V may run for some distance parallel to one another, and may
be fused into one, giving a figure better compared to an inverted Y. Thus the
ectodermal rim runs round the edge of each lobe of the umbrella and then passes
upwards towards the base of the tentacle from the re-entering angle between two
adjacent lobes, to form with its fellow of the next lobe a tentacle-clasp or peronium,
i.e. a streak of thickened ectoderm supporting the tentacle. Similarly the ring-canal
runs round the edge of the lobe as the so-called festoon-canal, and then runs
upwards under the peronium to the base of the tentacle as one of a pair of peronial
canals, the limbs of the V-like figure already mentioned. The nerve-rings have a
similar course. The tentaculocysts are implanted round the margins of the lobes of
the umbrella and may be supported by prolongations of the ectodermal rim termed
otoporpae (Gehörspangen). The radial canals are represented by wide gastric
pouches, and may be absent, so that the tentacles arise directly from the stomach
(Solmaridae). The tentacles are always solid, as in Trachomedusae.

The development of the Narcomedusae is in the main similar to that of the
Trachomedusae, but shows some remarkable features. In Aeginopsis a planula is
formed by multipolar immigration. The two ends of the planula become greatly
lengthened and give rise to the two primary tentacles of the actinula, of which the
mouth arises from one side of the planula. Hence the principal axis of the future
medusa corresponds, not to the longitudinal axis of the planula, but to a transverse
axis. This is in some degree parallel to the cases described above, in which a planula
gives rise to the hydrorhiza, and buds a polyp laterally.

In Cunina and allied genera the actinula, formed in the manner described, has a
hypostome of great length, quite disproportionate to the size of the body, and is
further endowed with the power of producing buds from a stolon arising from the
aboral side of the body. In these species the actinula is parasitic upon another

medusa; for instance, Cunoctantha octonaria upon Turritopsis, C. proboscidea upon
Liriope or Geryonia. The parasite effects a lodgment in the host either by invading it
as a free-swimming planula, or, apparently, in other cases, as a spore-embryo which
is captured and swallowed as food by the host. The parasitic actinula is found
attached to the proboscis of the medusa; it thrusts its greatly elongated hypostome
into the mouth of the medusa and nourishes itself upon the food in the digestive
cavity of its host. At the same time it produces buds from an aboral stolon. The buds
become medusae by the direct method of budding described above. In some cases
the buds do not become detached at once, but the stolon continues to grow and to
produce more buds, forming a “bud-spike” (Knospenähre), which consists of the
axial stolon bearing medusa-buds in all stages of development. In such cases the
original parent-actinula does not itself become a medusa, but remains arrested in
development and ultimately dies off, so that a true alternation of generations is
brought about. It is in these parasitic forms that we meet with the method of
reproduction by sporogony described above.

In other Narcomedusae, e.g. Cunoctantha fowleri Browne, buds are formed from
the sub-umbrella on the under side of the stomach pouches, where later the gonads
are developed.

Classification.—Three families of Narcomedusae are recognized (see O.
Maas [40]):

1. Cunanthidae.—With broad gastric pouches which are simple, i.e.
undivided, and “pernemal,” i.e. correspond in position with the tentacles.
Cunina (fig. 66) with more than eight tentacles; Cunoctantha with eight
tentacles, four perradial, four interradial.

2. Aeginidae.—Radii a multiple of four, with radial gastric pouches
bifurcated or subdivided; the tentacles are implanted in the notch between the
two subdivisions of each (primary) gastric pouch, hence the (secondary) gastric
pouches appear to be “internemal” in position, i.e. to alternate in position with
the tentacles. Aegina, with four tentacles and eight pouches; Aeginura (fig. 25),
with eight tentacles and sixteen pouches; Solmundella (fig. 67), with two
tentacles and eight pouches; Aeginopsis (fig. 23), with two or four tentacles and
sixteen pouches.

3. Solmaridae.—No gastric
pouches; the numerous tentacles arise
direct from the stomach, into which
also the peronial canals open, so that
the ring-canal is cut up into separate
festoons. Solmaris, Pegantha,
Polyxenia, &c. To this family should
be referred, probably, the genus
Hydroctena, described by C.
Dawydov [11a] and regarded by him
as intermediate between
Hydromedusae and Ctenophora. See
O. Maas [35].

After O. Maas, Craspedoten Medusen der
Appendix to the Trachylinae.
Siboga Expedition, by permission of E. S.
Of doubtful position, but Brill & Co.

commonly referred to the Trachylinae,
Fig. 67.—Solmundella bitentaculata
are the two genera of fresh-water (Quoy and Gaimard).
medusae, Limnocodium and
Limnocnida.

Limnocodium sowerbyi was first discovered in the Victoria regia tank in the
Botanic Gardens, Regent’s Park, London. Since then it has been discovered in
other botanic gardens in various parts of Europe, its two most recent
appearances being at Lyons (1901) and Munich (1905), occurring always in
tanks in which the Victoria regia is cultivated, a fact which indicates that
tropical South America is its original habitat. In the same tanks a small hydroid,
very similar to Microhydra, has been found, which bears medusa-buds and is
probably the stock from which the medusa is budded. It is a remarkable fact
that all specimens of Limnocodium hitherto seen have been males; it may be
inferred from this either that only one polyp-stock has been introduced into
Europe, from which all the medusae seen hitherto have been budded, or

perhaps that the female medusa is a sessile gonophore, as in Pennaria. The
male gonads are carried on the radial canals.

Limnocnida tanganyicae was discovered first in Lake Tanganyika, but has
since been discovered also in Lake Victoria and in the river Niger. It differs
from Limnocodium in having practically no manubrium but a wide mouth two-
thirds the diameter of the umbrella across. It buds medusae from the margin of
the mouth in May and June, and in August and September the gonads are
formed in the place where the buds arose. The hydroid phase, if any, is not
known.

Both these medusae have sense-organs of a peculiar type, which are said to
contain an endodermal axis like the sense-organs of Trachylinae, but the fact
has recently been called in question for Limnocodium by S. Goto, who
considers the genus to be allied to Olindias. Allman, on the other hand, referred
Limnocodium to the Leptomedusae.

In this connexion must be mentioned, finally, the medusae budded from the
fresh-water polyp Microhydra. The polyp-stages of Limnocodium and
Microhydra are extremely similar in character. In both cases the hydranth is
extremely reduced and has no tentacles, and the polyp forms a colony by
budding from the base. In Limnocodium the body secretes a gelatinous mucus
to which adhere particles of mud, &c., forming a protective covering. In
Microhydra no such protecting case is formed. In view of the great
resemblance between Microhydra and the polyp of Limnocodium, it might be
expected that the medusae to which they give origin would also be similar. As
yet, however, the medusa of Microhydra has only been seen in an immature
condition, but it shows some well-marked differences from Limnocodium,
especially in the structure of the tentacles, which furnish useful characters for
distinguishing species amongst medusae. The possession of a polyp-stage by
Limnocodium and Microhydra furnishes an argument against placing them in
the Trachylinae. Their sense-organs require renewed investigations. (Browne
[10] and [10a].)

Order VI. Siphonophora.—Pelagic floating Hydrozoa with great differentiation
of parts, each performing a special function; generally regarded as colonies showing

differentiation of individuals in correspondence with a physiological division of
labour.

A typical Siphonophore is a stock or
cormus consisting of a number of
appendages placed in organic connexion
with one another by means of a
coenosarc. The coenosarc does not differ
in structure from that already described
in colonial Hydrozoa. It consists of a
hollow tube, or tubes, of which the wall
is made up of the two body-layers,
ectoderm and endoderm, and the cavity is
a continuation of the digestive cavities of
the nutritive and other appendages, i.e. of
the coelenteron. The coenosarc may
consist of a single elongated tube or
stolon, forming the stem or axis of the
cormus on which, usually, the
appendages are arranged in groups
termed cormidia; or it may take the form
of a compact mass of ramifying,
anastomosing tubes, in which case the
Fig. 68.—Diagram showing possible
cormus as a whole has a compact form modifications of medusiform and hydriform
and cormidia are not distinguishable. In persons of a colony of Siphonophora. The
thick black line represents endoderm, the
the Disconectae the coenosarc forms a thinner line ectoderm. (After Allman.)
spongy mass, the “centradenia,” which is n, Pneumatocyst.
partly hepatic in function, forming the k, Nectocalyces (swimming bells).
so-called liver, and partly excretory. l, Hydrophyllium (covering-piece).
i, Generative medusiform person.
The appendages show various types of g, Palpon with attached palpacle, h.
e, Siphon with branched grappling tentacle,
form and structure corresponding to f.
different functions. The cormus is always m, Stem.
differentiated into two parts; an upper
portion termed the nectosome, in which the appendages are locomotor or hydrostatic

in function, that is to say, serve for swimming or floating; and a lower portion
termed the siphosome, bearing appendages which are nutritive, reproductive or
simply protective in function.

Divergent views have been held by different authors both as regards the nature of
the cormus as a whole, and as regards the homologies of the different types of
appendages borne by it.

The general theories of Siphonophoran morphology are discussed below, but
in enumerating the various types of appendages it is convenient to discuss their
morphological interpretation at the same time.

In the nectosome one or more
of the following types of
appendage occur:—

1. Swimming-bells, termed
nectocalyces or nectophores (fig.
68, k), absent in Chondrophorida
and Cystophorida; they are
contractile and resemble, both in
appearance, structure and
function, the umbrella of a
medusa, with radial canals, ring-
canal and velum; but they are
After A. Agassiz, from Lankester’s Treatise on without manubrium, tentacles or
Zoology. sense-organs, and are always
bilaterally symmetrical, a
Fig. 69.—Porpita, seen from above, showing
peculiarity of form related with
the pneumatophore and expanded palpons.
the fact that they are attached on
one side to the stem. A given cormus may bear one or several nectocalyces, and
by their contractions they propel the colony slowly along, like so many
medusae harnessed together. In cases where the cormus has no pneumatophore
the topmost swimming bell may contain an oil-reservoir or oleocyst.

2. The pneumatophore or air-bladder (fig. 68, n), for passive locomotion,
forming a float which keeps the cormus at or near the surface of the water. The
pneumatophore arises from the ectoderm as a pit or invagination, part of which
forms a gas-secreting gland, while the rest gives rise to an air-sack lined by a
chitinous cuticle. The orifice of invagination forms a pore which may be closed
up or may form a protruding duct or funnel. As in the analogous swim-bladder
of fishes, the gas in the pneumatophore can be secreted or absorbed, whereby
the specific gravity of the body can be diminished or increased, so as to cause it
to float nearer the surface or at a deeper level. Never more than one
pneumatophore is found in a cormus, and when present it is always situated at
the highest point above the swimming bells, if these are present also. In Velella
the pneumatophore becomes of complex structure and sends air-tubes, lined by
a chitin and resembling tracheae, down into the compact coenosarc, thus
evidently serving a respiratory as well as a hydrostatic function.

Divergent views have been held as to the morphological significance of the
pneumatophore. E. Haeckel regarded the whole structure as a glandular
ectodermal pit formed on the ex-umbral surface of a medusa-person. C. Chun
and, more recently, R. Woltereck [59], on the other hand, have shown that the
ectodermal pit which gives rise to the pneumatophore represents an entocodon.
Hence the cavity of the air-sack is equivalent to a sub-umbral cavity in which
no manubrium is formed, and the pore or orifice of invagination would
represent the margin of the umbrella. In the wall of the sack is a double layer of
endoderm, the space between which is a continuation of the coelenteron. By
coalescence of the endoderm-layers, the coelenteron may be reduced to vessels,
usually eight in number, opening into a ring-sinus surrounding the pore. Thus
the disposition of the endoderm-cavities is roughly comparable to the
gastrovascular system of a medusa.

The difference between the theories of Haeckel and Chun is connected with
a further divergence in the interpretation of the stem or axis of the cormus.
Haeckel regards it as the equivalent of the manubrium, and as it is implanted on
the blind end of the pneumatophore, such a view leads necessarily to the air-
sack and gland being a development on the ex-umbral surface of the medusa-
person. Chun and Woltereck, on the other hand, regard the stem as a stolo

prolifer arising from the aboral pole, that is to say, from the ex-umbrella,
similar to that which grows out from the ex-umbral surface of the embryo of
the Narcomedusae and produces buds, a view which is certainly supported by
the embryological evidence to be adduced shortly.

In the siphosome the following types of appendages occur:—

1. Siphons or nutritive appendages, from which the order takes its name;
never absent and usually present in great numbers (fig. 68, e). Each is a tube
dilated at or towards the base and containing a mouth at its extremity, leading
into a stomach placed in the dilatation already mentioned. The siphons have
been compared to the manubrium of a medusa-individual, or to polyps, and
hence are sometimes termed gastrozoids.

2. Palpons (fig. 68, g), present in some genera, especially in Physonectae;
similar to the siphons but without a mouth, and purely tactile in function, hence
sometimes termed dactylozoids. If a distal pore or aperture is present, it is
excretory in function; such varieties have been termed “cystons” by Haeckel.

3. Tentacles (“Fangfäden”), always present, and implanted one at the base of
each siphon (fig. 68, f). The tentacles of siphonophores may reach a great
length and have a complex structure. They may bear accessory filaments or
tentilla (f′), covered thickly with batteries of nematocysts, to which these
organisms owe their great powers of offence and defence.

4. Palpacles (“Tastfäden”), occurring together with palpons, one implanted
at the base of each palpon (fig. 68, h). Each palpacle is a tactile filament, very
extensile, without accessory filaments or nematocysts.

5. Bracts (“hydrophyllia”), occur in Calycophorida and some Physophorida
as scale-like appendages protecting other parts (fig. 68, l). The mesogloea is
greatly developed in them and they are often of very tough consistency. By
Haeckel they are considered homologous with the umbrella of a medusa.

From G. H. Fowler, after A. Agassiz, Lankester’s Treatise on Zoology.

Fig. 70.—Diagram of the structure of Velella, showing the central and peripheral thirds of a half-
section of the colony, the middle third being omitted. The ectoderm is indicated by close hatching, the
endoderm by light hatching, the mesogloea by thick black lines, the horny skeleton of the
pneumatophore and sail by dotting.
BL, Blastostyle. M, Medusoid gonophores.
C, Centradenia. PN, Primary central chamber, and PN′,
D, Palpon. concentric chamber of the pneumatophore,
EC, Edge of colony prolonged beyond the showing an opening to the exterior and a
pneumatophore. “trachea.”
G, Cavity of the large central siphon. S, Sail.

6. Gonostyles, appendages which produce by budding medusae or
gonophores, like the blastostyles of a hydroid colony. In their most primitive
form they are seen in Velella as “gonosiphons,” which possess mouths like the
ordinary sterile siphons and bud free medusae. In other forms they have no
mouths. They may be branched, so-called “gonodendra,” and amongst them
may occur special forms of palpons, “gonopalpons.” The gonostyles have been
compared to the blastostyles of a hydroid colony, or to the manubrium of a
medusa which produces free or sessile medusa-buds.

7. Gonophores, produced either on the gonostyles already mentioned or
budded, as in hydrocorallines, from the coenosarc, i.e. the stem (fig. 68, i.).
They show every transition between free medusae and sporosacs, as already
described, for hydroid colonies. Thus in Velella free medusae are produced,
which have been described as an independent genus of medusae, Chrysomitra.
In other types the medusae may be set free in a mature condition as the so-
called “genital swimming bells,” comparable to the Globiceps of Pennaria. The
most usual condition, however, is that in which sessile medusoid gonophores or
sporosacs are produced.

The various types of appendages described
in the foregoing may be arranged in groups
termed cormidia. In forms with a compact
coenosarc such as Velella, Physalia, &c., the
separate cormidia cannot be sharply

From G. H. Fowler, after G. distinguished, and such a condition is described
Cuvier, Lankester’s Treatise technically as one with “scattered” cormidia. In
on Zoology. forms in which, on the other hand, the
Fig. 71.—Upper surface of
coenosarc forms an elongated, tubular axis or
Velella, showing stem, the appendages are arranged as regularly
pneumatophore and sail. recurrent cormidia along it, and the cormidia
are then said to be “ordinate.” In such cases the
oldest cormidia, that is to say, those furthest from the nectosome, may become
detached (like the segments or proglottides of a tape-worm) and swim off, each
such detached cormidium then becoming a small free cormus which, in many
cases, has been given an independent generic name. A cormidium may contain
a single nutritive siphon (“monogastric”) or several siphons (“polygastric”):

The following are some of the forms of cormidia that occur:—

1. The eudoxome (Calycophorida), consisting of a bract, siphon, tentacle and
gonophore; when free it is known as Eudoxia.

2. The ersaeome (Calycophorida), made up of the same appendages as the
preceding type but with the addition of a nectocalyx; when free termed Ersaea.

3. The rhodalome of some Rhodalidae, consisting of siphon, tentacle and
one or more gonophores.

4. The athorome of Physophora, &c., consisting of siphon, tentacle, one or
more palpons with palpacles, and one or more gonophores.

5. The crystallome of Anthemodes, &c., similar to the athorome but with the
addition of a group of bracts.

Embryology of the Siphonophora.—The fertilized ovum gives rise to a
parenchymula, with solid endoderm, which is set free as a free-swimming
planula larva, in the manner already described (see Hydrozoa). The planula has
its two extremities dissimilar (Bipolaria-larva). The subsequent development is
slightly different according as the future cormus is headed by a pneumatophore
(Physophorida, Cystophorida) or by a nectocalyx (Calycophorida).

(i.) Physophorida, for example
Halistemma (C. Chun, Hydrozoa
[1]). The planula becomes
elongated and broader towards
one pole, at which a pit or
invagination of the ectoderm
arises. Next the pit closes up to
form a vesicle with a pore, and so
gives rise to the pneumatophore.
From the broader portion of the
planula an outgrowth arises
which becomes the first tentacle
of the cormus. The endoderm of
the planula now acquires a cavity,
and at the narrower pole a mouth
is formed, giving rise to the
primary siphon. Thus from the
original planula three appendages
are, as it were, budded off, while
the planula itself mostly gives rise
to coenosarc, just as in some
hydroids the planula is converted
chiefly into hydrorhiza.

(ii.) Calycophorida, for
example, Muggiaea. The planula
Fig. 72.—A, Diphyes campanulata; B, a group
develops, on the whole, in a
of appendages (cormidium) of the same
similar manner, but the
Diphyes. (After C. Gegenbaur.)
ectodermal invagination arises,
not at the pole of the planula, but a, Axis of the colony.
m, Nectocalyx.
on the side of its broader portion,
c, Sub-umbral cavity of nectocalyx.
and gives rise, not to a v, Radial canals of nectocalyx.
pneumatophore, but to a o, Orifice of nectocalyx.
t, Bract.
nectocalyx, the primary
n, Siphon.

swimming bell or protocodon g, Gonophore.
i, Tentacle.
(“Fallschirm”) which is later
thrown off and replaced by
secondary swimming bells, metacodons, budded from the coenosarc.

From a comparison of the two embryological types there can be no doubt on
two points; first, that the pneumatophore and the protocodon are strictly
homologous, and, therefore if the nectocalyx is comparable to the umbrella of a
medusa, as seems obvious, the pneumatophore must be so too; secondly, that
the coenosarcal axis arises from the ex-umbrella of the medusa and cannot be
compared to a manubrium, but is strictly comparable to the “bud-spike” of a
Narcomedusan.

Theories of Siphonophore Morphology.—The many theories that have been
put forward as to the interpretation of the cormus and the various parts are set
forth and discussed in the treatise of Y. Delage and E. Hérouard (Hydrozoa [4])
and more recently by R. Woltereck [59], and only a brief analysis can be given
here.

In the first place the cormus has been regarded as a single individual and its
appendages as organs. This is the so-called “polyorgan” theory, especially
connected with the name of Huxley; but it must be borne in mind that Huxley
regarded all the forms produced, in any animal, between one egg-generation
and the next, as constituting in the lump one single individual. Huxley,
therefore, considered a hydroid colony, for example, as a single individual, and
each separate polyp or medusa budded from it as having the value of an organ
and not of an individual. Hence Huxley’s view is not so different from those
held by other authors as it seems to be at first sight.

In more recent years Woltereck [59] has supported Huxley’s view of
individuality, at the same time drawing a fine distinction between “individual”
and “person.” The individual is the product of sexual reproduction; a person is
an individual of lower rank, which may be produced asexually. A Siphonophore
is regarded as a single individual composed of numerous zoids, budded from
the primary zoid (siphon) produced from the planula. Any given zoid is a
person-zoid if equivalent to the primary zoid, an organ-zoid if equivalent only

to a part of it. Woltereck considers the
siphonophores most nearly allied to the
Narcomedusae, producing like the buds
from an aboral stolon, the first bud
being represented by the
pneumatophore or protocodon, in
different cases.

Contrasting, in the second place,
with the polyorgan theory are the
various “polyperson” theories which
interpret the Siphonophore cormus as a
colony composed of more or fewer
individuals in organic union with one
another. On this interpretation there is
still room for considerable divergence
of opinion as regards detail. To begin
with, it is not necessary on the
polyperson theory to regard each
appendage as a distinct individual; it is
still possible to compare appendages
with parts of an individual which have
become separated from one another by
a process of “dislocation of organs.”
Thus a bract may be regarded, with
Haeckel, as a modified umbrella of a
medusa, a siphon as its manubrium,
After C. Gegenbaur. and a tentacle as representing a
Fig. 73.—Physophora hydrostatica. medusan tentacle shifted in attachment
from the margin to the sub-umbrella; or
a′, Pneumatocyst.
a siphon may be compared with a
t, Palpons.
a, Axis of the colony. polyp, of which the single tentacle has
m, Nectocalyx. become shifted so as to be attached to
o, Orifice of nectocalyx.
the coenosarc and so on. Some authors
n, Siphon.

g, Gonophore. prefer, on the other hand, to regard
i, Tentacle.
every appendage as a separate
individual, or at least as a portion of an
individual, of which other portions have been lost or obliterated.

A further divergence of opinion arises from differences in the interpretation
of the persons composing the colony. It is possible to regard the cormus (1) as a
colony of medusa-persons, (2) as a colony of polyp-persons, (3) as composed
partly of one, partly of the other. It is sufficient here to mention briefly the
views put forward on this point by C. Chun and R. Haeckel.

Chun (Hydrozoa [1]) maintains the older views of Leuckart and Claus,
according to which the cormus is to be compared to a floating hydroid colony.
It may be regarded as derived from floating polyps similar to Nemopsis or
Pelagohydra, which by budding produce a colony of polyps and also form
medusa-buds. The polyp-individuals form the nutritive siphosome or
trophosome. The medusa-buds are either fertile or sterile. If fertile they become
free medusae or sessile gonophores. If sterile they remain attached and
locomotor in function, forming the nectosome, the pneumatophore and
swimming-bells.

Haeckel, on the other hand, is in accordance with Balfour in regarding a
Siphonophore as a medusome, that is to say, as a colony composed of
medusoid persons or organs entirely. Haeckel considers that the Siphonophores
have two distinct ancestral lines of evolution:

1. In the Disconanthae, i.e. in such forms as Velella, Porpita, &c., the
ancestor was an eight-rayed medusa (Disconula) which acquired a
pneumatophore as an ectodermal pit on the ex-umbrella, and in which the
organs (manubrium, tentacles, &c.) became secondarily multiplied, just as they
do in Gastroblasta as the result of incomplete fission. The nearest living allies
of the ancestral Disconula are to be sought in the Pectyllidae.

After Haeckel, from Lankester’s Treatise on Zoology.
Fig. 74.—Stephalia corona, a young colony.

p, Pneumatophore lo, Orifice of the aurophore.
. s, Siphon.
n, Nectocalyx. t, Tentacle.
l, Aurophore.

2. In the Siphonanthae, i.e. in all other Siphonophores, the ancestral form
was a Siphonula, a bilaterally symmetrical Anthomedusa with a single long
tentacle (cf. Corymorpha), which became displaced from the margin to the sub-
umbrella. The Siphonula produced buds on the manubrium, as many
Anthomedusae are known to do, and these by reduction or dislocation of parts