1911 Encyclopædia Britannica/Hydromechanics
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 forcingpump were invented by Ctesibius and Hero. The siphon is a simple instrument; but the forcingpump 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 forcingpump.
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 wellconducted 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 eightytwo (fiftyone on the velocity of water in conduit pipes, and thirtyone 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 waterwheel 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 reexamined 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. 
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 sealingwax; after a considerable time the sealingwax 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
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.
 
Fig. 1a. 
It follows that the pressure of a fluid requires to be calculated in one direction only, chosen as the simplest direction for convenience.
7. The Transmissibility of Fluid Pressure.—Any additional pressure applied to the fluid will be transmitted equally to every point in the case of a liquid; this principle of the transmissibility of pressure was enunciated by Pascal, 1653, and applied by him to the invention of the hydraulic press.
This machine consists essentially of two communicating cylinders (fig. 1a), filled with liquid and closed by pistons. If a thrust P ℔ 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
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.
 
Fig. 1b. 
Hence the theorem.—The free surface of a liquid at rest under gravity is a horizontal plane. This is the characteristic distinguishing between a solid and a liquid; as, for instance, between land and water. The land has hills and valleys, but the surface of water at rest is a horizontal plane; and if disturbed the surface moves in waves.
9. Theorem.—In a homogeneous liquid at rest under gravity the pressure increases uniformly with the depth.
This is proved by taking the two points A and B in the same vertical line, and considering the equilibrium of the prism by resolving vertically. In this case the thrust at the lower end B must exceed the thrust at A, the upper end, by the weight of the prism of liquid; so that, denoting the cross section of the prism by α ft.^{2}, the pressure at A and By by p_{0} and p ℔/ft.^{2}, and by w the density of the liquid estimated in ℔/ft.^{3},
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 p_{0} denotes the atmospheric pressure, the pressure in the liquid at the level of the surface of separation will be σh + p_{0} and ρk + p_{0}, and these being equal we have
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,
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/cm^{2}; and one ton/inch^{2}, 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 114 : 477 : 221 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).
 
Fig. 2. 
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 13πy^{3}, where y is the depth of metal, the horizontal sections being equal so long as y is less than the radius of the outside hemisphere. Afterwards, when the metal has risen above B, to the level KK′, the additional thrust is the weight of the cylinder of diameter KK′ and height BH. The upward thrust is the same, however thin the metal may be in the interspace between the outer mould and the core inside; and this was formerly considered paradoxical.
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,
where l, m, n denote the direction cosines of the normal drawn outward of the surface S.
But by Green’s transformation
∫∫ lp dS = ∫∫∫  dp  dx dy dz, 
dx 
thus leading to the differential relation at every point
dp  = ρX,  dp  = ρY,  dp  = ρZ. 
dx  dy  dz 
The three equations of equilibrium obtained by taking moments round the axes are then found to be satisfied identically.
Hence the space variation of the pressure in any direction, or the pressuregradient, is the resolved force per unit volume in that direction. The resultant force is therefore in the direction of the steepest pressuregradient, 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 
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 
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 gasequation, 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  = 0,  1  dp  = 0,  1  dp  = 1,  
ρ  dx  ρ  dy  ρ  dz 
When the density ρ is uniform, this becomes, as before in (2) § 9
Suppose the density ρ varies as some nth power of the depth below O, then
p = μ  z^{n+1}  =  ρz  =  ρ  (  ρ  )  ^{1/n}  , 
n + 1  n + 1  n + 1  μ 
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 gasequation
where θ denotes the absolute temperature; and then
R  dθ  =  d  (  p  ) =  1  , 
dz  dz  ρ  n + 1 
so that the temperaturegradient dθ/dz is constant, as in convective equilibrium in (11).
From the gasequation 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 
which is positive, and the density ρ diminishes with the ascent, provided the temperaturegradient dθ/dz does not exceed θ/k.
With uniform temperature, taking k constant in the gasequation,
so that in ascending in the atmosphere of thermal equilibrium the pressure and density diminish at compound discount, and for pressures p_{1} and p_{2} at heights z_{1} and z_{2}
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
dz  =  1  dp  = (n + 1)  p  = (n + 1) R, γ = 1 +  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  = (n + 1)  p_{0}  dθ  = −  a^{2}  dr,  
ρ  ρ_{0}  θ_{0}  r^{2} 
gravity varying inversely as the square of the distance r from the centre; so that, k = p_{0}/ρ_{0}, denoting the height of the homogeneous atmosphere at the surface, θ is given by
or if c denotes the distance where θ = 0,
θ  =  a  ·  c − r  . 
θ_{0}  r  c − a 
When the compressibility of water is taken into account in a deep ocean, an experimental law must be employed, such as
so that λ is the pressure due to a head k of the liquid at density ρ_{0} under atmospheric pressure p_{0}; and it is the gauge pressure required on this law to double the density. Then
and if the liquid was incompressible, the depth at pressure p would be (p − p_{0})/p_{0}, so that the lowering of the surface due to compression is
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  =  λ  dρ  = ρ, λ (  1  −  1  ) = z, 1 −  ρ_{0}  =  z  , λ = kρ_{0},  
zd  ρ  dz  ρ_{0}  ρ  ρ  k 
and the lowering of thesurface ia, ' ' ~, f .. U th ... e;  (J) .g gf
z klogpu z klogr E, z 2E (zo)
as beforem 17). . a  A16.
Centre of Pressure.—A plane area exposed to fluid pressure on one side experiences asingle 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 resultantthrust R=ff1>dxdy, (F).
and the coordinates x, y of the are iven by * &R =fé'xpdxdy, 5»R =f§ 'ypdxdy. ' (2) The C.P. is thus the .G. of aplairelamitia 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 ine of intersection of the plane of the area with the free surface of the liquid. fIf
the equation of this line, referred to. new coordinate axes in the plane area, is written
xcos a{ysin iz.—h=0, (3)
R=f p(hxcos ay sin'a)dxdy,  " (4)
5cR= px(h<;xcos a—ysina/)dxdy, (5)
5/R = fpy(h ~x cos a.y sin u)dxdy.
Placing the new ori in at the C.G. of the area A, ' " '§ :¢dxdy=0, fJydxdy=0, ' i 'f(6)
R=Ph» »; (7)
xhA= cos affx'dA sin affxydA, (8)
yhA= 4cos af{xydA >~sin ¢ffy°dA.1 < ' (§)f Turning the axes to make t em coincide with the principal axesof the area A, thus making ffxydA=0, 1 = ' ' 2 aEh=via"cos»a., yh=b'sina; . i » (io) < where »
f {x'dA=A4', ffy'dA=Ab', (II)
a and b denoting the semiaxes 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 5 of the depth of the lower side; of triangle with a vertex in the surface and base horizontal is % of the depth o the basei 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 'tolthe 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 oran ellilpse is a concentric circle or ellipse of one quarter the Size: he C.P. of water lines passing through a fixed point lies on a straight line, the anti polar 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. sln the design of a structure such as a tall reservoir dam 1t 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 M etacenlrer'The. principle of Archirnedes, in § 12 leads Mintmediately.. to the
conditions of equilibrium
of a. body supported
freely in fluid,
like a iishiu water or
a balloon in the air
or like a ship (fig, gl
floating partly imJl
 G 61 .
f
5»
I1"
t . mersed inwater and
the rest in a, ir. The(
body is in equilibrium
under two
forces:(i.) its
weight W acting
verticallyvdownward
through G, the C.G. of the body, and (il.) 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;
FIG. 3.
for equilibrium Vthese two; forces must be equaland opposite in the same' line; ' ~ A
The conditions. of equilibrium of a body, floating like aship on the surface of a liquid, are therefore:(i.) the weight of thebody 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 liq1iid'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 liein the same vertical line GB; 18. In addition to satisfying these conditions of equilibrium, a ship must fulfil the further condition of stability, soas to keep upright; if displaced slightly from this position, the forces called into play must besuch as to restore the ship to the upright again. The stability of a shipis investigated practically by inclining it; a weight is moved across the deck and the angle is observed of the heel produced. ~ A
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 Gfthe' reduced distance G»Cu=c(P/W);and' if B, called the centre of buoyancy, moves to Bi .along the curve of buoyancy BB1, the normal of this curve at B1 will be the new vertical B1Gly meeting the old vertical in a point M, the centre of curvature of BB1, called the meta centre. < 'If the'ship'heelsfthrqugh an angle 0 or a slope of I in m,  ' ' 'GM=GGicot6=mc(P/W), ' ' (1)
and GM is called the' meta centric hei ht; and the shi' 'must be ballasted, so that G lies below M. If 5 was above M, the tangent drawn (rpm G to the évolute of B, and normal to the curve of buoyanc, would: give the vertical in a new position of equilibrium. ¢"I'hus<lh M.S. ' ' Achilles T' of,9000 tons displacement i twas found that movirig~2o'tons' across the deck, a distance of 42 ft., causedtlie bob ofla pendulum 20lf1i:'l0l'l'g' to move 'through' to in., so that * » ' *V iGMf%>§ 42>§ % =2'~24ft.i ~ iii) 1°
alS9 ' .  A I f
y cot a ==24, 0 =2°24/. (3), ,
Ina diagram it isconduciveto clearness to draw the.ship in one position, and to incline the wa.terline; and the pagecan be turned if it is desired to bring the new waterline horizontal. Suppose the ship turns about an axis.through F inthewaterline area, perpendicular to the plane of the paper; denoting by y the distance of an elementdA 1f'the waterline area from the axis of rotation; the change of displacement is 2ydA tan 6, so that there is no change of displacement if 2ydA=jo, , that 1s, 'if the axis passes through the 'C.G. of the watéréline area, which we denote by F and call the centre of flotation.
The righting couple of the wedges of immersion and 'emersion willbe  i » “
~ EwydA tan'9Ly sw tan 0Zy”dA =iw tan 6.Ak2 ft. tons, ' (4) w denoting the density of water in tons/ft.3, and W=wV, x for a displacement of V ft.3 <
his couple, combined with the original buoyancy*W through B, is equivalent to the new buoyancy through B, so that. W.BB;l='wAk2tan6, A » ' (5)
»BM=BB, cot6=Ak?/V, (6)
giving the radius of curvature BM of the curve ofibuoyancy B, in terms of the displacement V, and Akz the moment of inertia of the waterline area about 'an axis through F, perpendicular to the plane of displacement.,  »= . ,
An inclining couple due to moving a weight about in a ship wgillheel the, ship aboutan axis perpendicular to the plane of the coupletouly when this axis is a 'principal axis at F of themomental ellipse. of the waterli ne area A. For if the ship turns through a smallvangleid about the l1ne Fl, then bl, bzythe C.G. of the .wedge of .immersion and emersion, will be the C.P. with respect 'to F F ' of the two partsof the waterline area, so that bib; will be conjugate to FF' with respect to the momental ellipse at F. Q =;
The naval architect distinguishes between the stability of, form, represented by the righting couple W.BM, and the stability»of'balla§ ling, represented by W, BG. Ballasted with G at B, the righting couple when the ship is heeled through 0 is given by W.BM. tan 6; but if weights inside the ship are raised to bring Q above B, the righting couple is diminished by W§ BGJ tan 0, so t 'at the resultant righting couple is W.GM. tan P, 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 waterline area A, but a confocal to it, of squared semiaxes
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 waterline 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
through an angle θ or a slope of one in m, given by
sin θ =  1  =  P  =  P  ·  V  FQ sin QFF′ 
m  wA·FK  W  Ak^{2} − hV 
where k denotes the radius of gyration about FF′ of the waterline area. Burning the coal on a voyage has the reverse effect on a steamer.
20. In considering the motion of a fluid we shall suppose it nonviscous, 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 outwarddrawn 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, 
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 = ∫∫ (lξ + mη + nζ) dS, 
dx  dy  dz 
or individually
∫∫∫  dξ  dx dy dz = ∫∫ lξ dS, ..., 
dx 
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 outwarddrawn 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  dx dy dz = ∫∫ (lρu + mρv + nρw) dS = 0, 
dt 
which becomes by Green’s transformation
∫∫∫ (  dρ  +  d(ρu)  +  d(ρv)  +  d(ρw)  ) dx dy dz = 0, 
dt  dx  dy  dz 
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 timerate of increase of momentum, due to the fluid which crosses the surface, is
which by Green’s transformation is
− ∫∫∫ (  d(ρu^{2})  +  d(ρuv)  +  d(ρuw)  ) dx dy dz. 
dx  dy  dz 
The rate of generation of momentum in the interior of S by the component of force, X per unit mass, is
and by the pressure at the surface S is
− ∫∫ lp dS = − ∫∫∫  dp  dx dy dz, 
dx 
by Green’s transformation.
The time rate of increase of momentum of the fluid inside S is
∫∫∫  d(ρu)  dx dy dz; 
dt 
and (5) is the sum of (1), (2), (3), (4), so that
∫∫∫ (  dρu  +  dρu^{2}  +  dρuv  +  dρuw  − ρX +  dp  ) dx dy dz = 0, 
dt  dx  dy  dz  dx 
leading to the differential equation of motion
dρu  +  dρu^{2}  +  dρuv  +  dρuw  = ρX −  dp  , 
dt  dx  dy  dz  dx 
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  +  dρu^{2}  +  dρuv  +  dρuw 
dt  dx  dy  dz 
= ρ (  du  + u  du  + v  du  + w  du  ) 
dt  dx  dy  dz 
+ u (  dρ  +  dρu  +  dρv  +  dρw  ), 
dt  dx  dy  dz 
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  + u  du  + v  du  + w  du  = X −  1  dp  ,  
dt  dx  dy  dz  ρ  dx 
with the two others
dv  + u  dv  + v  dv  + w  dv  = Y −  1  dp  ,  
dt  dx  dy  dz  ρ  dy 
dw  + u  dw  + v  dw  + w  dw  = Z −  1  dp  .  
dt  dx  dy  dz  ρ  dz 
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 pressuregradient, 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  = lt·  F(x + uδt, y + vδt, z + wδt, t + δt) − F(x, y, z, t) 
dt  δt 
=  dF  + u  dF  + v  dF  + w  dF  ; 
dt  dx  dy  dz 
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
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 “,2;'='37+ua;+1J@+wa;» (3) Dr dv dv dv dv 7=3;+ 1454v3;+1v, ;;. (4) Dw dw dw dw dw F=ji7+uE+r1;5+wE;» (5) leading to the equations of motion above. If F (x, y, z, t) =o represents the equation of a surface containing always the same particles of fluid, D F F Q? =o, or%+u%E+1/gg;+w¢%=o, (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 an art of it. Y P But turbulence in the motion will vitiate the principle that a bounding surface will always consist of the same fluid particles, as we sec on the suiface of turbulent water. 24. To integrate the equations of motion, suppose the impressed force is due to a potential V, such that theiforce 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 E, &=2E» 5;, =211, E52r. (2) dE d dy Tx " or the equations of motion may be written du dH E;  211§ '+2wn+Q; =0» (4) %2105f2u§ '+%I;=0, (5) %2uq+2v£+l(gl=o, (6) where H =fdP/p+V+i9', (7) 9” = u'+v' +w', (8) and the three terms in may be called the pressure head, potential head, and head of velocity, when the gravitation unit is employed and sq' is replaced by %q'/eEliminating H between (5) and (6) . % ¢§ ¢ @@: du dr/dw dt idx "7dx'f°dx+f — (9) and combining this with the equation of continuity LPQ a+a, a» p dt +dx dy dz "0' (Io) 1 2 5 'EQ nd” Cdw we have dt (P)pdxZi;5 Ec~ 0, (U) with two similar equations. Putting w' 9+ + = rr' § '. (12) a vortex line is defined to be such that the tangent is in the direction Of w. th€ resultant of E, vp, § ', called the components of molecular rotaton. A small sphere of the fluid, if frozen suddenly, would retain this angular velocity. If w vanishes throughout the fluid at any instant, equation (11) shows that it will always be zero, and the fluid motion is then called 1rro ia1z onal; and a function ¢ exists, called the velocity function, suc t at udx+vdy+wdz==do, (13) and then the velocity in any direction is the spacedecrease or downward gradient of gb. 25. But in the most general case it is possible to have three functions ¢, ip, m of x, y, z, such that udx+vdy+wdz = d4>md¢, (1) as A. Clebsch has shown, from purely analytical considerations (Crelle, lvi.); and then d(~P~m) d(¢.m) d(¢, m) f'*?d<y, z>' "'i3l"z, x)' "'i'”'F'd<x.y (2) and d d d e£+»3§ +:, }§ =<>, sff§ +»'§§ +r%';'f=<>. (3) so that, at any instant, the surfaces over which up and m are constant intersect in the vortex lines. Putting the equations of motion (4), (5), (6) § 24 can be written and therefore H§ 4§ m'§ '§ =K, (4) %21¢§ '+2wq =O, . . ., .; (5) e'§ , }§ +»'§ }§ +r§ =0. <6> Equation (5) becomes, by a rearrangement, . dK dilf dm dm dm @ EW 71T+“1i;+"@+“¢Z) dm .za 11./, iz./, 11 +35 '5+14Z§ +1/;§ +w£ =0, —. —, (7) dK diff Dm dm Di# Ia;EC(E'+E;'W=0, ..., ..., (S) and as we prove subsequently (§ 37) that the vortex lines are composed of the same fluid particles throughout the motion, the surface m and (P 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, gi2'U§ "i'2'£U11=O, ..., . . (1) £%Lf+n?1§ +I'?§ =0, 1¢'?§ +v?;§ +w'?§ =0. (2) and I 2 K='f<lP/DlV+%q'=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 ix for an instant in the normal through a point on the surface H=constant, this makes u=o, £=o; and in steady motion the equations reduce to ~ dlI/dv =2vg'2w1, =2qw sin6, (4) where 0 is the angle between the stream line and vortex line; and this holds for'the1r projection on any plane to which dv is drawn perpendicular. In plane motion (4) reduces to dH d
 =29§ '=9(;%+§)» (5)
if r denotes the radius of curvature of the stream line, so that 1 <1 if! 45 £93; sl.” p£+dv'dv dv ri (6) the normal acceleration. The osculatmg plane of a streain line in steady motion contains the resultant acceleration, the direction ratios of which are Q du du @' dlq2 @ udx+va;+wE ix 21/5+2w1;—5; 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 balancesheet 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 mption at the free surface at a distance maybe neglected, then Bernoulli's equation may be written ~ H=P/n+2+9'/2£=P/P+h, (5) 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. ' V, In particular, for a jet issuing into the atmosphere, where p=P, Q'/2g=h2, ' (9) or the velocity of the jet is due to the head kz of the still free surface above 'the orifice; this is Torricellifs theorem (1643), the foundation of the science of hydrodynamics. 27. Uniplanar M ation.In the uni planar motion of a homogeneous liquid the equation of continuity reduces to %+%=o, (1) so that we can put u= dw/dy, v=d/f/dx, (2) where il is a function of x, y, called the stream or currentfunction; interpreted physically, ¢~¢0, the difference of the value of it at a fixed point A and a variable point P is the'flow, in ft.'*/ second, across any curved line AP from A to P, this being the same for all lines in accordance with the continuity.
Thus if dip is the increase of up due to a displacement from P to P', and' le is the component of velocity normal to PP", the flow across PP' is dd/=k.PP; and taking P ' parallel to Ox, di;/ = vdxpand similarly dt= udy with PP' parallel to Oy; and generally dgb/dS is the velocity across ds, in a direction turned throug a right angle forward, against the clock.,
In the equations of uni planar motion ' " 'dv du:Pup d'l/
2i'=E§ .E; = W +W = V*¢.SUP1f>0Se, (3) so that in steady motion
i£+v“lf§ 5;=0. '%I+v'~t%§ = 0. 'g;+v?»l»=0, (4) and VRD must be a function of it.
If the motion is irrotational, » ~£'= @ = d¢ § L/'
u' dx dy' U 35 '° dx' (5)
so that il/ and d> are conjugate functions of x andy, , ¢+I»i=f(x+yi). VW/=0. v'¢=0; (6)
or putting¢+¢1=w.
x+y1=2. w=f(2)The
curves ¢=constant and i/»=constant form system; and the interchange of 4> and up willxgive a.new state of uni planar motion, ” in which the velocity at every point is turned through a right angle without alteration of magnitude. For instance, in a uni planar flow, radially inward towards 0, the flow across any circle of radius 1' being the' same and denoted by 21rm, the velocity must be m/1, and » ¢='MlOg r, w//=m0, ¢+/»i=m log reiff, w=im log z.' (7) Interchanging these values ' ' ..
¢=m log r, ¢=m0, ¢+¢i==m log rdf (8) gives a state of vortex mqtion, '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 direcf tion; analogous to the magnetic field of a straight electric current. If other vortices are present, any one may be SUPP($¢d, .to, move Bith the velocity due to the others, the resultant streamsfurlction Qing
¢»¢2m log r=log Ilrfm; (9)
the path of a vortex is obtained by equating the value of it 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, molving on a line parallel to a plane boundary, wil have a corresponding pair of images, forming a rectangle of vortices, and the pat of a vortex will be the Cotes spiral
r sin 29==2a, or x'2+y"=af2; ., . (lo) an orthogonal
this is therefore the path of a single vortex in a 'rightangled' corner; and generally. if the angle of the corner is ir/n, the path is the Cotes spiral '
r sin nl) =na. (II)
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 aidistance zz*/c, and so describe a circle with velocityhzc/(0)c2)in the periodic time'21r(a3cf)/m. ' (12) Conjugate functions can be employed also' for the motion 'dl' liquid in a thin sheet between two concentric spherical surfaces; the components of velocity along the meridian and parallel in co latitude, 0 and longitude) can be written '
d § .= .£Q . ..E. .i?== éL[/
tw Sinodx' Sinadi da', 3
and then
¢l¢i=F(tan éti. e/i). (14)
28. Uniplanar Motion of a Liquid due to the Passage of a Cylinder through it.A streamfunction it must be determined to satisfy the conditions ,
v'~»//=o, throughout the liquid; ' i (1) I vb ='constant, over any fixed boundary; M (2) dit/ds=»normal velocity reversed over a solid boundary, (3) so that, if the solid is moving with velocity U in the direction Ox, dit/d.s=Udy/ds, or ¢'l'Uy =constant over the moving cylinder; 0DYNAMlC
andi¢+Uy=sU/"is:the stream function of the relative motion of the liquid past the cylinder, and similarly i//Vx for the component velocity V along Oy; and generally
V/=!/+UyVx (4)
is the relative streamfunction, 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, dgl/his == normal velocity reversed dx ~ dy
!/+iR(x”+y') =¢/Y U (6)
a constant over the boundary; and it/' is the currentfunction of the relative motion past the cylinder, but now ~ l, V'=;/+2R=o, (7)
throughout theuliquid. O A
inside an equilateral triangle, for instance, of height h,  it' = r 2R~1I3“//h. (3)
where a, B. 'V are thepenpendiculars on the sides of the triangle. In the general case 4/ =¢+Uy Vx+a}R(x¢4yf) is the relative stream function for velocity compo neg its, U, V, R. 1 29.:Example I.jluquid motion past a circular cylinder. f Consider the motion given by »
w=U(2+¢1'/2). (I)
so that i ¢=U (fPg) COS 0 =U (1 rg) N. (2) ¢=U<f '¥)>sin 6=U<I lg) y.
Then it =o over the cylinder r =a, which may be considered a fixed post; and a stream line past it along which t=Uc, a constant, is the curve ' '(r
 sin 0=r, (x'+y')(y c)  a”y=o, (3) a cubic curve (Ca).
Over a concentric cylindexqxexternal or internal, of radius r=b, ~t'=¢+U1y=iU(1  '§) + Uay, <4>
F
and vp is zero if
Ui/ U = (42 °' bf)/112; (5)
so that the cylinder may swim for an instant in the liquid without distortion, with this velocity Ui; and w in (1) will givethe liquid motion in thejnterspace between the fixed cylinder r==a and the concentric cylinder r=b, moving with velocity Ui. ' ~When b=o, Ui=oo; and when l>=oo, U;= U., so that at infinitely the liquid is streaming in the direction x0 with velocity U. If the liquid is reduced to rest at infinity by the superposition of an opposite stream given by w = Uz, we are left with w==Ua“/Z, (6)
4> = U(¢1”/f) COS 0 = Uazx/(xf +y'). (7) = U(“2/t) Sin 0 = ~U¢l'y/(x'+y'). (8) givin 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 0' with Ox, fl¢ /df# 2x;V
tail 0 =€5l/'~i§ =;:, ;;'2=lZ3.Il 20, 0=fl'0, , and the velocity is Ua'/r'.
Q 'Along the path of a particle, defined by the C3 of (3), 2
SW i0'=xT?iTT==2Q5TQ' . <'°)
dw ~ d ”
= ~ ' %.sinBa;=3%1753% (ix)
on the radius of curvature is ia'/(yic), which shows that the curve is an Elastica or Lintearia. (J. C, Maxwell, Collected Works, ii. 208,) If ¢, d'enotes the velocity function of the liquid filling the cylinder r =b, and moving bodily with it with velocity Ui, ¢l = 'Uxa (lg)
and over the separating surface r=b ¢ U Q azlb
" 3;"Ur'(l Tb2) "a2~' bil A (13)
and this, by § 36, is also the ratio of the kinetic energy in the annular inter space between the two cylinders to the kinetic energy of the liquid moving bodily inside 1=b.
Consequently the inertia to overcome in moving the cylinder 1=b, solid or liquid, is its own inertia, increased by the inertia of liquid (a”+b')/(41/*~b') 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
acceleration dLT/dt, given by
22+b2 dU a2+b2, gg
"f””zF:5far"a1"sM an U4)
liflvid Of density P, this gives rise to a kinetic reaction to if M' denotes the mass of liquid displaced by 'unit length of the; cylinder r=b. In particular, when a =oo, the extra inertia is M'. When the cylinder r=a is moved with velocity U and r=b with velocity U, along Ox, ,
¢= Ugg?<§ ;2+r)cos0U1£ 37l;(r+'¥)cos0, (15) ~//=U5;'¥;l;<?~r)sin0»UiI¢ I:p(r£;)sin0; (II6) A and similarly, with velocity components V and V1 along Oy i A 4>= VE; (lglV) sin0Vb;€% <11gi) sin 0, (17) 2 2 'l
i//= V5%<% r) cos0+V1=5, Ii?(r§ ;>cos0; (18) and then for the resultant motion
az z a2b2 lVi
iw (U2 + V2)b'a”UIVi + b” ~ (12 z l>f 5 a'b'U+V£
(U"+V )b2  a2U, +v, ¢ ' U9)
The resultant impulse of theiliquid on the cylinder is given by the component, over r=a (§ 36), V A
X =fp¢ C05 was =2p22 U QQ1§ ¢U, ,§ %); (20) A and over r=b A
x, =fp¢¢0So.z2dv=2, , b2 U Z§ %u.'§ ff; C, (21) and the difference XX1 is the component' momentum of the liquid in the inter space; with similar expressions for Y and Yi. Then, if the outside cylinderiis free to move x, =0, %g% X=»p22Uf§ Ig. (22)
But if the outside cylinder is moved with' velocity U), and the inside cylinder is solid or filled with liquid of density a, U1 2pb2
X—¥Ua?U1 U =p(b2+a2)+a2(b2 ag 1
UUi: (p'¢r)(b3az) (2)
U1 P(b'+q')+v(5¢1”)' 3
and the inside cylinder starts forward or backward with respect to the outside cylinder, according as p> or <¢r. 30. The expression for w in (1) § 29 may be increased by the addition of the term,
im log z=m0 + im log r, V (1)
representing vortex motion circulating round the annulus 'of li uid.
qConsidered by itself, with the cylinders 'held fixed, the vortex sets up a circumferential velocity m/f on a radius r, so that the angular momentum of a circular fi ament of annular cross section dA is pmdA, and of the whole vortex is pmr(b'¢az). Any circular filament can be' started from rest by the application of a circumferential impulse vrpmdr at each end. of a diameter; so that a mechanism attached td the cylinders, which can set up a uniform distributed impulse vrpm across the two parts of a diameter in the liquid, will generate the vortex motion, and react on the cylinder with an impulse couple—pmra* and pmfbz, having resultant pm1r(b' »~a'), and this couple is infinite when b=oo, as the angular momentum of the vortex is infinite. Round the cylinder 1 =a held fixed in the U current the liquid streams past with velocity q' =2U sin 0+m/a; (2)
and the loss of head due to this increase of velocity from U to Q' is q”'~ U* =(2U sin 0+m/a.)'  U2 ii
23 23 V y
to
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 cylrnclexyis 2mU sin 0/ga, and its thrust is 2rrpmU 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 21pmV 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 perunit length ' 2 (i.) *2fpmV, , 2rrpmU, due to the yortex motion: (ii.) iirpazgéi, vrpa'%¥, due to the kinetic reaction of the liquid; (iii.) o, —1(irp)a'g, due to gravity, ' ' V taking Oy vertically upward; and denoting the density of the cylinder by tr; so that the equations of motion are ~ 1r¢ra2%&g =  1rpl1.%~ 21rpmV, (4) dV dV, ,
1:10225 = 1rpcL' 5—l21rpmV  1r(n  p)ag, (5) or, putting m=a'w, so that the vortex velocity is due to an angular velocity w at a radius a,
(0Ip)tiU/ritjl2pwV =O, (6)
(<f+P)¢iV/df'2pwU +(<1r>)g =0' (7) Thus with g=o, the cylinder will describe a circle with angular velocity 2P¢»1/(vrlp), so that the radius is (a+p)v/zpw, if the velocity is v.. With ¢==0, the angular 'velocity of the cylinder is 2¢o; in this way the velocity may be calculated of the propagationof ripples and waves on the surface of a vertical whirlpool in a sink. Restoring u 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 pdy (U  Uazf” cos 2 0+mr'1 sin 0) (Ua'r“2 sin 2 01mr°1cos0) + pdy (UlUa2r'“ cos 20lm7'x sin 0)(Ua2r°2 sin 20 mr* cos0) =2pdymUf'1(C0S0 a'r“2 cos 30), (8) “ and with y=b'ta.n 0, rsb sec 0, this is 2pmUd0(»1 a'b"'cos 30 cos 0), V (9) and integrating between the limits 6= =1==}1r, the resultant, as before, is 21rpmU.
31. Example 2.Confocal Elliptic Cylinders.—Employ the elliptic coordinates 11, £, and § '='qlEi, such that z=cch§ ', x=cch1;cosE, y=cshqsinf; (I) then the curves for which 17 and £ are constant are confocal ellipses and hyperbolas, and
J =V%% = ¢”(Ch'n  C0S'£) C
= %C2(Ch2'l7 cos2.E) = rlrz = OD2,
if OD is the semidiameter conjugate to OP, and rl, rg the focal (2)
distances,
V r1, r2=c(ch1;=l=cos Q); ' (3)
rf = x' lyi = c” (ch'1;  siniij)
= iv”(Ch 211 lCOS 2E) (4)
Consider the streaming motion given by w =mCh(s'1), vialBi, (5)
¢>=m ch(n¢)c0S(£/3), Vlpbtm Sh(na)Si11(£B) (6) j Then W 30 over the ellipse 'r=a, and the hyperbola £=H, so that these may be taken as fixed boundaries; and ti/'15 a constant on a C4. Over any ellipse 11, moving with components U and V of velocity, V xl/=1PIUyVx'=[msh(11a)cos;3lUcsh1;]sin§ [msh(qa) sinB+Vcchn]cos£; (7)
so that gli' =o, ifU=
% cos B, V= 7% sin 5, (8)
having a resultant in the direction PO, where P is the intersection of an ellipse 11 with the hyperbola B; and with this velocity the ellipse 1 can be swimming in the liquid, without distortion for an instant. At infinity
U = '¥e'°cos B =  3%c0S B, ~ V
V= ¥e““sin B = »5%~sin B, (9)
a and b denoting the semiaxes of the ellipse a; so that the liq uid is streaming at infinity with velocity Q=m/(a.~lb) in the direction of the asymptote of the hyperbola 1 1 An ellipse interior to 17=¢ will move in a direction opgosite to the exterior current; and when 71 ==o, U =eo, but V= (m/c) s e. sin 13. Negative values of 1; must be interpreted by a streaming rnotion on a parallel plane at a level slightly different, as on adouble Riemann sheet, the 'stream passing from one sheet to the other across a cut SS' joining the foci S, 5'. 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 11 and hyperbola £, are mJ"Sh(11f=)C0S(E~l9), m]"Ch(1r¢)Si11(£B)~ (10) The velocity q is zero in a corner where the hyperbola B cuts the ellipse a; and round the ellipse a the velocity q reaches a mafriimum when the tangent has turned through a right angle, and then g = QW# (ch;>a.;c:s 28); (I I)
and the condition can be inferred when cavitation begins. With 13=o, the stream is parallel to xo, and 41 = m ch (1ra)c0sE
=  Uc ch(11a)sh11cos 2/sh(11~a) (12) over the cylinder 11, and as in (12) § 29, 4>i=Ux=Ucch11cos E, (13)
for liquid filling the cylinder; and Q Z th 11 (I)
¢1 th (111) ' 4
over the 'surface of 11; so that parallel to Ox, the effective inertia of the cylinder 11, displacing M' liquid, is increased by M'th11/th(11a), reducing when a.=oo to I/I'th11=M'(b/a). Similarly, parallel to Oy, the increase of effective inertia is M'/th 11 t (1111), reducing to M'/th 11=M'(a/b), when a=co, and the liquid extends to infinity.
32. Next consider the motion given by 41=mch2(11a)sin2£, 1l=msh2(11a)cos2£; (1) in which 1]/='o over the ellipse o., and //' =l/+%R(x'+;v')
=[m sh 2(1)a.)I}Rc2]cos 2£+}Rc' ch 211, (2) which is constant over the ellipse 11 if }Rc'=m sh 2(11a); (3)
so that this ellipse can be rotating with this angular velocity R. for an instant without distortion, the ellipse a. being fixed. For the liquid filling the interior of a rotating elliptic cylinder of cross section
x”/H'+y”/b* = I, (4)
11/1' =m1(x'/U'+y'/b') (5)
with V'¢1'=2R=2m1 1 a' 1 bi
(/ + / Q,
 Pi =m1(x”/¢1'+y'/b')  %R(x'+> )
=~%R(x'y“) (02  b')/ (112l112). (6) ¢1=Rxy(G''b“)/(¢1'+b'),
wi =4>1 +1011 =  %iR(x+yi)“(¢l' ' 52)/(f1'+b')~ 1 The velocity of a liquid particle is thus (a” bf)/(a*+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 (a'b')'/(aflb')f of the solid; and the effective radius of gyration, solid and liquid, is given by
k' =i(a”+b'), and Ha'b')'/(a'+b'). (7) For the liquid in the inter space between a and 11, $ m ch 2(11a) sin 25
41, %R¢:' sh 211 sin 2.E(a2b*)/la'1b') = I/th 2(1)a)th 211; (8)
and the effective k' of the liquid is reduced to tb'/th 2(11¢)Sh 211. (9)
which becomes § c'/sh 211=%(a2bf)/ab, when a=¢o, and the liquid surrounds the ellipse 11 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 , l is resolved into 1]/, +»l/2, such that 1[q+%Rx2 and 1//1l%Ry' is constant over the boundary.
Inside a cylinder
4>1+l/1i=  %iR(x+yi)'11'/(¢1'+b2), (10) ¢1+l/1i= =liR(r+yi)”b'/ (112lb°'), (1 I) and for the inter space, the ellipse a being fixed, and ai revolving with angular velocit R
Y
¢1+l/1i=  § iRc'sh 2(11'G+£i) (ch 2aI)/sh 2(a.1a), (12) 4>2+1l/2i= § iRc'sh 2(11a+£i) (ch 2a. 1)/sh 2(u.,  u.), (13) satisfying the condition that gl, and 1//2 are zero over 11 =a., and over fl = 111
»/f1+%Rx'=tRv”(ch 2¢1+1). (14)
»l/1l%Ry'= %Rv“(Ch 2111  1), (15) constant values.
In a similar way the more general state of motion may be analysed, given by
w=m¢h2(I1). ~/=a+Bi. (16)
as giving a homogeneous strain velocity to the confocal s stem to which may be added a circulation, represented by an additional term mf in w.
Similarly, with
x+;vi =¢~/ lSln(£+111)l (17)
the function
¢=Qv Sh%(11 ¢)S1n%(£B) (IS)
will give motion streaming past the fixed cylinder 11 =a, and dividing along £=/S; and then
 x2y'=c' sinfch 11, 2xy=c2 cos£sh 11. (19) In particular, with sh a.= I, the crosssection of 11 = a is x*I6x'y'Iy'=2c“, or x4Iy4=c' (20) when the axes are turned through 45°. 33. Examlple 3.Analysing in this way the rotation of a rectangle filled with iquid into the two components of shear, the stream function 1]/1 is to be made to satisfy the conditions (L) V2¢1 = or
(ii.) 1//, +%Rx'= § Ra', or gli =o when x= =*= a, (iii.) ¢, 1§ Rx' = %Ra', nh = %R(a2xf), when y = # b» Expanded in a Fourier series,
/
az.xz=%¢1,2 C—°5(f;'nJQ—3;§ '°°1 a. (1) so that
RE: 2 cos(2n+1)§ 1rx/a ch (2n+1)%1ry/a “ N' ¢2n+1)=* . ¢h(2n+11s1fz»/ti 16 cos(2n+1)l1rz/a
w' =¢l+b" =1R;5a2 (2n+1)3ch(2nii I)1}1rbla' (2) an ellipticfunction Fourier series; with a similar expression for 1,01 with x and y, a and b interchanged; and thence IP =¢i +1//2. Example 4.—Parabolic cylinder, axial advance, and liquid streaming past.
The polar equation of the crosssection being 1% cos § 0=al, orr + x=2a, (3)
the conditions are satisfied by
gb' = Ur sin 9 2Ualfi sin £6 =2Url sin § 0(1} cos £0  ai), (4) ul=2Ualr} sin § 0= 'U'f[2G(f0C)], (5) w ézuaia, (6)
and the resistance of the liquid is 21rpaV2/2g. A relative stream line, along which 1l'=Uc, is the quartic curve 1¢=v [2110111. x=4“f, %§ 2f, 1= 4'f, f, § ”“§ ', ,§ ')'. (1) and in the absolute space curve given by ip, QL QQ Q
dx lay 1, xy c2alog (y c). (8) 34. Illotion symmetrical abozlt an Axis.When the motion of a liquid is the same for any plane passing through Ox, and lies in the plane, a function 11/ can be found analogous to that emplo ed 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 21r(lgba); and, as before, if dip is the increase in 1/1 due to a displacement of P to P', then k the component of velocit normal to the surface swept out by PP' is such that 21rd¢/=21ryk.¥'P'; and taking PP' parallel to Oy and Ox, , 14 = di///ydy. v #dv//ydx. (I)
and il/ is called after the inventor, “ Stol»:es'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 dy!//yds is the component velocity across ds in a direction turned through a right angle forward. In this symmetrical motion
LQ i(;d~#)
£'°' "'°' 2'”'dx(ydx) +dy ya?
I dal/ d*1// I dip I
1 =§ 3;l@f§ g§ =§ V'¢. (2)
suppose; and in steady motion,
!%'l'§ ;%cV2¢=09 '((%I'l"§ , E%¥V2'//=0r » so that
1 2?/y =  y 2V2b =dH/dxf/ ' (4) is a function of 10, say f' (1//), and constant along a stream line; dH/dv=2q§ ', H f(//) =constant, (5) throughout the liquid.
When the motion is irrotational, f1¢» 1 Q Que#
r'°°' “ dx ydy' v' dy ydx' (6)
1 Q 'dll L Q./
Vw' 91 01' dxz'l'dyz y dy 9° (7) Changing to polar coordinates, x =r cos 0, y=r sin 0, the equation (2) becomes, with cos 0 =;4,
r'%+(1p')%=2 fr' sin 0, (8)
of which a solution, when g'=o, is ” ¢ = Af~+1+§ <1 wg? = A7”*l+%z y='§ ff. <9> 4>={(n+1)Ar"nBr'"“'}P, ., (IO)
where P.. denotes the zonal harmonic of the nth order; also, in the exceptional case of
//=A» cos 0, d>=A@/1;
up = Bur, ¢ =  Bolog tan ge
=  § B0 sh'1x/y. (1 1)
Thus cos 0 is the Stokes' function of a point source at O, and PA PB of a line source AB.
The stream function up of the liquid motion set up by the passage of a solid of revolution, moving with axial velocity U, is such that 5/%;=U%, ¢/+%Uy° =constant, (12)
over the surface of the solid; and p must be replaced by¢' =¢+}Uy2 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 1/»' =¢+§ Uy' = %U(r2af*/r) sin' 0, l/ =  %Ua° sin' 0/r; (13) and then
4>' = Ux(1 }£03/r2), qS= § Ua“ cos 0/12, (14) d 3 5 .
§ =U'% Cosa, %*;=;U%sm a, (15)
so that, if the direction of motion makes an angle nk with Ox, tan (//6) =§ tan 0, tan ¢=3 tan 0/(2 tan' 0). (16) Along the path of a liquid particle tk' is constant, and putting it equal to %Uc',
(rz a'/r) sm'0 =c2, sm” 0 =c'r/(13 113), (17) the polar equation;' or
y” =v'f“/(f“f1°), 1° =¢1"y'/(y'62), (IS) a curve of the 10th degree (Cm).
In the absolute path in space
cos (L = (2 3 sin“0)// (4sin”0), and sin°0 = (y'c2y)/al, (19) which leads to no simple relation.
The velocity past the surface of the sphere is 75% = %U(2r+§ >%}¥, =§ U sin 0, when r=a; (zo) so that the loss of head is
(X sin20 I)U2/2g, having a maximum QU'/zg, (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 10 = %Uy”¢1'u/f', W' = %Uy'(I  f1'n/f'). (22) ¢'=Ux+¢. ¢= § U(a'/f“)P2, P2=%/Pi, (23) representing a stream past the surface r* =a'p. 3 5. 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 co axially 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 leapfrog. 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 crosssection of the vortex, is seen to increase. The analytical treatment of such vortex rings is the same as for the electromagnetic effect of a current circulating in each ring.
36. Irrotatirmal lllotion in General.—Liquid originally at rest'in a singlyconnected space cannot be set in' motion y a eld of force due to'a singlevalued potential function; any motion set up in the liquid must be due to a movement of the boundary rand, 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 anv rotation.
The kinetic energy of the liquid inside a surface S due to the velocity function ¢ is given by
T=%»fffl(i%) 2+ (5%) ”+ (ti) ']dxdydz» =%, , ff¢%:ds (1)
by Green's transformation, dv denoting an elementary .step along the norlinal to the exterior of the surface; so that d¢/dv=o over the surface makes T=o, and then
<%;>2+ (§ l;>2+ <%>2=o, ?;=o, 5i;=o, ?%=o. (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 m satisfies the equations
1das 1 dw 1da
 'E£' ur vl p7i'E'°"wx
m=p¢+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 ¢, the liquid can 'circulate in the circuits independently of any motion of the surface; thus, for example, ¢ = m0 = m tan'1y/x (5)
will, give motion to the liquid, circulating in any ringshaped figure of revolution round Oz.
To lifnd 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 rp in any circuit, or the value by which 4: has increased in completing the circuit, the values of ¢> on the two sides of the membrane are taken as differing by k, so that the integral over the membrane ff gas =kff%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 =21rm. In plane motion the kinetic energy per unit length parallel to Oz “ef I l (%) 2+ (515) ildeeeffl (%£)”+ (%) ”ldxdy =%r> %'l~'=%P //%fdS (7)
Pior example, in the equilateral triangle of (8) § 28, referred to coordmate axes made by the base and height,
~/1' = 2Ra6~//h =  lR;v[(hy)”3x21/h (8) iv =¢' aR1eh.~y>2'+x21
gi =  § R[!;h3+§ h'y+h)x2y') 3x'yIy“]/h (9) and .over the base y=o,
dx/dv = dx/dy = +%R(%h'*3x”)/(W =  %R(%h”+x') (10) Integrating over the base, to obtain onethird of the kinetic energy T,
h N/ ¢
§ T=a» Q/j3§ R'<3x4afh4>dx/h
=1>R'h'/13543 (II)
so that the effective la' of the liquid filling the triangle is given by k' = T/%pR'A = 2h”/45
=§ (radius of the inscribed circle)', (12) or twofifths of the k' for the solid triangle. Again, since
d4>/du =d¢//ds, 'd¢/ds =  dt///dv, (13) T = ipfduixlf =  %/>f~//1i¢ (14)
with the Stokes' function , L for motion symmetrical about an axis.
T = § p 4> %21rydS = 1rpf¢d¢. (15)
37. Flow, Circulation, and Voriex Motion.¥The line integral of the tangential velocity along a curve from one point to another, defined by,
rf(1¢%+ if $32I ds= /(udxIvdy+zdz), (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 ¢, the How =¢l " 4721 (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 ¢» is single valued.
If through every point of a small closed curve the vortex lines are drawn, a tube is obtained, and the f'luid'contained is called a vortex jilament.
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 § ', the circulation round a curve in the plane xy is fi' < x Q 41% ds=g' times twice the area. (3) In a fluid, the circulation round an elementary area 'dxdy is equal to
1¢(11A~ !, <, +£d ») dy~ (u+'§ ;dy dxpay: (gc 11%) dxdy, (4) so that the component spin is
dv du
2 (dx dy) *iv (5)
in thelprevious notation of § 24; so also for the other two .components E and 17.
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, 5, " gr, is according to thevvector 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 crosssection ABC, 'B'C' is drawn of the vortex filament, joined by the vortex line A.', then, since the flow in AA' istalren in opposite directions in the complete circuit ABC AA'B'C' A'A, the resultant How 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 wa is constant where a is the crosssection of the filament and w 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, . D ¢Q Dv dQ D do; ldi¢+dx
0' di+<zy”°' diU+¢z'°', . (6)
Q=fflP/p+V, i (7)
and taking dx, dy, dz in the direction of u, 1/, w, and dx:dy:dz=u:p:w,
2 udxfvdy +wdz =%¢dx . . o
= dQ+sdq2, ' ' <8>
and integrating round a closed curve $4 (udxiV1/dylwdz)= 0, 'i(f))
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 crosssection of a vortex Filament is constant, not changing with the time, it follows from the previous kinematical theorem that aw is constant for all time, and the same for every crosssection 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.
Dcnoting the crosssection a of a filament by dS and its mass by dm, the quantity wdS/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 wcos edS/dm, if d§ is the oblique scrtion of which the normal makes an angle e with the filament, while the aggregate vorticity of a mass M inside a surface S is M'fo cos eds.
Employing the equation of continuity when the liquid is homogeneous.
 .» » fi',
dr di 2, <Z; ¢; fb .
2 Vnll, . .., ..., V ~ dxi dag. which is expressed by g ' A ' 1
V"(1¢, '1h¢f»') '2 CUYI (S, 11, I), (E, 11, s') ='i' curl (14/Ui W)' (U) . 38. Moving Axes in Hydradynamics.—In many problefnsfsuch as the motion of a solid in liquid, , it is convenient to take coordinate axes fixed to the solid and moving with it as themovabie 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 lu'~yR {zg, (1)
v=V HJ' zl' +x,
w=W iw'xQ IyP.
V Now if k denotes the component of absolute velocity in a direction fixed in space whose direction cosines are 1, m, n, k=lu+mvInw; (2)
iand 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 Dkgdf l dm (54
E? di” Eivi' df”
du, Q, dii, Q
H E+" dx +0 dy+w dz
dv, dv, dv, dv
tm ifi1¢ a§ +v H3, +w H;
+n ii;;J+u'(%;+ v'%;+w%i5 . (3) But as l, m, n are the direction cosines of a line fixed in space, %= }nRnQ, (§ l£"[= nPIR, %=lQmP; (4) so that
%§ ==l gi'1/R +wQ+1/gf;if v'£%, +'1.¢/'egg i1n(. . .)+n(. . .) LQ liz:df
“'fX'pdx> +'”iY po) +"<Z poi' (5)
for all values of l, m, n, leading to the equations of motion with moving axes.
When the motion is such that
£l ¢3 'if 'l2 Q 'i'?€ W
“"dx mdx' v' dy mdyyw dz md? (6)
as in § 25 (I), a first integral of the equations in (5) may be written JP 1 2 ¢i¢> IN/ d¢ W
7iVliq rEj'ma? 'l'(U"14');3'E+mE; d d d .
+<v1/> 3#§ +mg +<ww'> %§ +m;§ =P<»>. <7> . in which
dab, d¢ 1 ¢&, Q
n”<“““>a5a°<”'”>;zy ff” “Wag
¢i¢ . 4 <B it d¢
E (U yR+~Q)dx (V zP+xR)dy (W xQ+yP)E2 (8) is the timefrate of change of 4: at a point fixed in space, which is left behind with velocity components uu', vv', ww'. 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 .
=consranr (9)
becomes transformed into
f'5+v i iqz%% éU2f(l/ +%Uy') =constant, (Io) xp/= ¢, +;Uy2' (II)
subyect to the condition, fro m (4) 4§ 34, V"V'P' = ~'f'(P')» iV'2V”P= '"f'('#+%Uy') (12) Thus, for example, With
, ~// =%U;v2<f'f1” 1), f' =x2+y2, (13) for the space inside the sphere r=a, compared with the value of ip? in § 34 (13) for the space outside, there is no discontinuity of the velocit in crossing the surface.
lnsidle the sphere
fi lit' iid!:li 2
2'“'3&(y dx) +dy(y dy) ZUGZ' (14)
so that § 34 (4) is satisfied, with J'(P')=5§ 'Ua'”, f<¢'>=2 5U¢'a~2; (rs) and (Io) reduces to ' A 1£4V§
U§ <§ x)'~(§ az; =c0nstant; (16)
this gives the state of motion in M. j. M. Hill's spherical vortex, advancing through the surrounding liquid with uniform velocity). ggi As an application of moving axes, consider the motion of liquid. filling. the ellipsoidal case 2
D § +§ , i§ +§ , ;=1; W in
and first suppose the liquid to be frozen, and the ellipsoid to be rotating about the centre with components of angular velocity E, and then Uthen
U 1¢=ytl~2n. v=e=E+xlT»'" F "“"+9f (2) (Fi) "la'+¢"5*m“£ “my N ow suppose the liquid to be melted, and additional components of angulanvelocity Sh. Sh. Sh communicated to the ellipsoidal 'case =(, i f}ni(e+ ~»=>(Qf+n:)(nf+n, »)=1 the additional velocity communicated to the liquid, will be due to ~ i a i = 6) z,2 2 z J 1 2
a velocityfunction I ' s [LM N'+{L M%Ng; £4 V bz cz cz azzx*na2 bgx () . I 4 2 62, 2 2 es '“1e'»;?=y= '% '=+.»2 1* 3 " l t t l '“ 2 l '?i?°';»] =z, (23) as may be verified by considering one term at a time., ,V ' X C 6 C) If u', v', w' denote the components of the velocity of the liquid where Z is a quadratic in ff, so that Q' is an elliptic function of 1, relative to the axes, except when c=a, or 30. u, =u+yR ZQ=aégfp9, y Z§ 4%9a, , (4) V Put sz, =n cos 4», o, = sz sin ¢~, ~ ( 1WP xR=V2”' t, , w2'>' at <5> § »~"a1i'=:z'%f>¢fi>»:¢'%'=f1“f—Z2”£Z=<f>1ff+<»»»~>f» (24) " " +202 ' a +2 ' A N+, ,2, , 62{2 g ' A f ~ d .  2 2 ' "T
w'=w+”QyP=2f%='¢”'r5i°3°=“"» i (6) g zi=fi% ~ I (25) Th P=ni+s, Q=n, ~+», R=n, +r. (1) ' 2°“<“"'” us, V, a*l52
urf +v/%+w/§ =0' (8) ¢= § 'Q§ ' a2+¢;2 'l' 45; fl .gdf' I ag 2 63 D ' '/Z az c2 (a2+c2)2 '/Z so that a liquid particle remains always on a similar ClllpS0ld. t M+ F (25) The hydrodynamical 'equations with moving axes, taking into ' ' ' ' 'account the mutual gravitation of the liquid, become d d d d
Eggf41rpAx+l%1/R+wQ+u'.i;+vfa%+wf7£=o, . . ., . . ., where A
~ abcdk V
A' B'C'=f> Za'+x. § +>~. ¢=+))P
P°=4(G”+7)(b'+>)(¢'+7) V V UO); With the values above of u, v, w, u', v', w', the equations become of the form A
 7'£+41rnKx+ax+hy+gZ=6, i (H)
f;§§ +4f»By+h»+ay+f2=<>, » ~ (12) lgg+41rnCZ+gx+fy+12=0» (13)
9
and integrating " V '
pp" + 3''P(Ax2 + By' 'lr Cf) V ° (14)f
+§ (ax2+;Sy*+~/z' +a fyalfzgzx Izhxy) »=»const., so that the surfaces of equal pressure are similar quadric surfaces, which, symmetry and dynamical considerations show, must be coaxial surfaces; and V f, g, h vanish, as follows also by algebraical reduction; and f
z 2 c2 a2 2, , 1 .
“="%5Jf#l“i 'lame'I a
4bwb=>, ,, <»=bf = l (15)
—~  — Shr V
(az+b2)2 8 a2+bZ I
with similar equations for B and 7. j lf we can make »,
(NFPA la)x' = (411/JB +l9)b' = (41/'C +1)¢". (16) the surfaces of equal pressure are similar to the external case, which can then be removed ivithout"affecting the motion, provided a, B, ~y onstant. V
regliliiiil is so when the axis of revolution is, a principal axis, say Oz; wh
W en 91:01 Ezov "l=0If
Q, =o or 03=§ ' 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 th t ting spheroid.lriil;
general motion again of the liquid filling a case, when a =b, KZ; may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to, ' g,
d5 252 @ aa' df; 262 ';,
V rrrzfgeb at a'f¢a"1f» m G"=+¢=@2f 919) “Sl , ~ 2 I I
%=sm+'§§§ »f, %=nit%fr; (19)
af which three integrals are j
2
ew =L»f;%r2, (zo)
Qi +125 =1i+2§ “2, , flip", (21) ols'+mnN é +'ff;§ r*: ~ ' (22)
which, 'as Z is a quadratic function of ff, are nonelliptic integrals; so also for tl/, where E =w cosilf, 71 = w sin ip; ' f in a state of steady motion i i » V dl' '$i 94,
I ggfo. Q7 (27)
¢=v//=nt, suppose, f V (28)
g ld 9i£+S7i'0==9w, A Q(29)
2 2
V ¢§ =§ °%;i '£2 gf! V (30)
V ed
=, i%, § r, (31)
2 2 2
Q an ca 2 ' "(a2 61) (902 cz) QA l5a""z * c2 ""°" 4(a2+c'z')" v (33) and a stateof steady motion is impossible when 3a> c >a. An experiment wasidevised by Lord Kelvin for demonstrating this, in which the difference of steadiness was shown of a copper shell filled with liquid and spun gyroscopic ally, 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' oldfashioned 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. 4o. A Liquid I et.4By the use of the complex variable and its conjugate functions, an(attempt can be madeto give a mathematical interpretation of problems such as the efilux of waterin a jet or of smoke from ag chimney, the discharge through a weir, the ilow 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, moreor 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 ¢» and stream function il/ are given as conjugate functions of the coordinates x, y by w=f(2)'§ where 2=x+;vi, w=¢+bi, '(1) and then F » .
Q§ =§ ~§ '+f§§ = uw; i (2)
so that, with u =q cos 0, v=g sin 0, the function dz Q Q Q
§ '= Q§ v1'=, ,Q, , i=ge(14fl'5'fl f;(cos 9+i Sill 0) (3) gives g' as a vector representing the reciprocal of the velocity 1 in direction and magnitude, in terms of some standard velocity Q. To determine the motion of a jet which issues from avessel with plane walls, the vector (must be constructed so as to have a constant direction 0 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
S2 =1<>§ I = l°g=<=o;and u=co, ¢=oo across the end jg' of the jet, bounded by the curved lines AP], A'P"I', . over which the skin velocity is Q. The stream lines xBA], xAjI' are given by ul/=0, m; so that if c denotes the ultimate breadth ]' 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 Q' is the product of corresponding factors, such as in (5).
Restricting the attention to a single corner B, "  '.  b .fl (Cog 713Q1 Sin ch nf! =ch lcg “cos n6+i sh log “sin 116 = %<f~+@~»> = <1> sh nS2=Sh log cos n0+i ch log "sin 110, V =%<r~¢> = ' <8> °o>a>b>o>a'> OO: (9). and then § 2 l / ba.ba' ¢§ u n1 du " 21l(u"b){ (ua.ua ' du " ru' U?) the formulas by which the conformal representation is obtained.
For the (Z polygon has a right angle at u =a, a', and a zero angle at uh=b, where H changes from o to sir/n and (2 increases by surgz; so t at ' d9 A /(ba.ba') d?¢'(ubN(ua.ual)'whereA' zn »' (H) And the w polygon has a zero angle at u=o, ob, where gb changes from 0 to m and back again, so that w changes by im, and gl; =§ , where B = 2. ' (12) Along the stream line xBAP], , 1 ¢=o, u=a¢*f¢/"'; (13) and over the jet surface JPA, where the skin velocity is Q, 5% = q =  Qy u 5ae1Y.l'Q/7" =a¢vrs/c, (14) denoting the arc AP by s, starting at u=a; ' 4 j g ch nQ=cosn8¥J§ E€;/ki, A (15) shjnQ=i sinn0=i/E1 g, (16) <=° >u=¢w*"'> fr, (17) and this gives the intrinsic equation of the jet, and then the radius of curvature ds xd¢ idw idw dn "= '35 =QTo='Q'J§ 2=Q'8T¢/322 c ub~/ ua.ua') 2 ab.b—a) not requiring the integration of (II) and (12) n =§ ' ""T/( If 0 =a across the end jj' of the jet, where u = so, q=Q, I chnSl=cosna= ggv, sh n§ z=isin na=i f%, (19)
Then A ab.ba', . .2 aa COS 21l."C0s 2M2 isln 21la5:5 V x/(a bb a')/ Si" 2“=2 (20) =sin Znav*"°'**"(a';lizZ a')i 2n§ b /(ab.ba') Tp' (I+ub)w/(ua.ua') (21) =aa'l(a+a') cos zna[a+a'+(aa') cos 2na]cos 2710 (aa') sin' zna Xcos znacos 2710 sin 2110 Along the wall AB, cos n6=o, sin n0=I, a>u> b, (22) <24> £ f § fL>i i fQ. A du~°°d4> dt * 1rqu 1r qu ' (25) QE f “Q Q 7' c "' b q u Pl(f1b)~/(1¢0')~l/(b¢')~/(affu) '/“rg 6) " ' v(aa')»/(ulf) u' (2 Along the wall Bx, cos n8= 1, sin n0=o, b> u>o (27) n f ' ¢hv=¢h1°g(%) =/f';v'11/#Ei <28> sh nQ=sh log n = {¥~§ 1/%. (29) Atxwhere ¢=oo, u=o, and q=q0, A Q " @;'1'/2 /'Ib/:E (qt) r /a E' z>+ 517' Q  430)
In crossing to the line of flow x'A'P']', nl/ changes from 0 to m, so thatwith g ==Q across j]', while across xx' the velocity is go, so that 1 m=sz°xx'=QJJ' 1 (31) 1 '.Q. Nba'/<2 / G1»/;@]=/» § “g$“ Z b' Qa' b » (32) giving the contraction of the jet compared with the initial breadth of the stream.
Along the line of, flow x'A'P' I', ¢=m, u=a'e~'f¢/"', and from x' to A', cos n0=1, sin n&=o, G. U Q»¢sq chnilchlog  a af b u, (33) it Q "=JllbJ1l"'0.I A sh 710 sh log E17 B:I, <>>~>f1' on Along the jet surface A']', q =Q, gn ' 1 g A ch nn=cos no=/f% %7/5;ji, (35) sh n9=isinn0=i'!l%:§ /9l: 3, (37) al>u =ale, r/c> my 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=I, b¢0, a'=°o, asinfig.5, V chS2=?, shS2= ff; ' (1) and along the jet APJ, oo > u =ae"'/°> a, Shn=isana=¢ § =¢¢4"'°, (2) me mf 8) 0 PM= T°sin6ds= ei /ds=§ ¢i /=§ sin0, (3) so that P'I'=c/ir, and the curve AP is the tractrix; and the cotfiicient of contraction, or 4 breadth of the 'et 'ir . ', breadth of the orifice '1+2' (4) A change of fl and 0 into nfl and nb will give the solution for two walls converging symmetrically to the orifice'AA1 at an an le i/n. With 1l==§ , the reentrant walls are given of Borda's moutiipiece, and the coefficient of contraction becomes Q. Generally, by making af = oo, 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 ob tained, with assigned n, a, g and b. of the efiiux from § a symmetrical converging f M i ' J' A
 3
z 1 ~ Al 8 s B Fic.. 5. Fig. 6. mouthpiece, or of the flow of water through the arches of a bridge, with wedgeshaped piers to divide the stream., 42 Other arrangements of the constants n, a, b, a' will ive the results of special problems considered by j. M. Micheg, Phil. Trans. 1890. Thus with a'=o, a stream is split symmetrically by a wedge of angle vr/n as in Bobyleff's problem; and, by making a=oo, the wedge extends to infinity; then b 1 ch 1lQ= Q, Sh nQ= (I) Over the jet surface //=m, q=Q, ', u= r1f¢/'"= bw'/', /a ch fZ=cos n0= l;7§ ;i, sh§ 'l=isinn0=i 55%, (2) =*""'=f““ '"'» ii=;%7r ' <9 For a jet impinging normally on an infinite plane, as in fig. 6, n = I, elf"/==tan 6, ch (hs/c) sin 20 = I, (4) sh évrx/c =cot 0, sh § 1ry/c =tan 6, Sh ifx/C Sh ivry/¢=1, el"<**">/°=e%"'/'+e¥"”/°+I (5) With n = ~§ , 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 nQ= dwg, sh n$2= /i?/é, (6) n  E sin mx= '£ £ () cos a a, a, 7 and along the free surface AP ], q =Q, ¢=o, u=e"¢/"'éae*'/°, j e'f'/°1 C05 M=cos11a, le, " /C coszna sin'nB ' (8) = — T»1s1n“n0  sin'na. the intrinsic equation, the other free surface A'P']' being given by cos'na sin'n0 1rd/=, ... . .., . . 6 = sinfna  sin“n8 (9) Putting n = 1 gives the case of a stream of finite breadth disturbed by a transverse plane, a particular case of Fig. 7. A When a=b, a.=o, and the stream is very broad compared with the wedge or lamina; so, putting 'w=w' (ab)/a in the penultimate case, and, u=ac“"2:a(ab)'w', (Io) chnSZ==; %fl, shnt':= 7%1, (Il) in which we may write w' =4>+¢i. (12) Along the stream line xABP], ¢=o; and along the jet surface APJ, 1 >¢»> so; and putting 4>= *FS/C"I, the intrinsic equation isrs/c = wana, i ' (13) which for n= I is' the evolute of a catenary. xxv. 5 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 re  ¢ presentation is made now with <§ Q ~, / (ba.ba') () 'du
 Z'Tb), / (ua.u'  )a I . A,
dwl 2; E'; .: aT. ' '1l'uj 7ruj mlm' ub, "  “T , f 21 1;=' EJ lJ (2) » m+m ' si taking u=w at the source where FiG~7¢==oo, u=b at the branch point B, u=j, j' at the end of the two diverging streams where ¢=foo; while nl/=0 along the stream line which divides at B and passes through A, 'A'; and 'L =m, m along the outside boundaries, so that m/Q, m'/ is the final breadth of the jets, and (m+m')/Q is the initial breaclgi, c, of the impinging stream. Then °*' *Q* sh iS2= (Sf 2b0.a' N °h“~Tr'm Q/(2.au.ual) shS21/N=1;3—, b.b ' ~ N =2'iT;f*l"' c (47 Along a jet surface, q=Q, and chSZ=cos0=cosu.iesin'a.(aa')/(u.b), (5) if 0=a at the source x of the jet xB, where 1¢=°°; and supposing 0 =S, B' at the end of the streams where u =j, j', ub § sin'a. uez' . cos'0cos 8aa cos a.cos 0' aa é Smal (cos acos B) fcos acos 05 uj' ' cos0cos B aa"'% Sm 2°'(cos acos ¢s'i (cos a.cos 05' '(6) and il/ being constant along a stream line do dw ds do dw du Ein Qffw ='2irr>1rQ ds 1rds ' (cos afcos B) (cos acos/S') sin 0, m+m @'c 35° (cos a.cos 9f(cos 9cos B)"(cos Qcos i') sin 0 cos szcos B' sin 0
 cos a.cos 9+cos 5cos /3"cos 0cos B
cos acos'B sin 0 n cos Bcos B'°cos 0cos B ' V (7) giving the, intrinsic equation of the surface of a jet, with proper attention to the sign. From A to B, a>u>b, 0=0, ch Sl=ch log %=cos a5 sin 'ag sh f2==sh log%= sin a f I % (ub) cos a%(aa'):in;q+{ (a.u.ug')sin;x (8), ds ds dd> A H Q¢F¢ Q¢f1»@' q du m+ng' (ub) cos a.%(a  Q/) sin 2aI V (au u»a') sin q ( 1r ' " julujr ' 9 AB “(2boa')(u11)2(ab)(ba')l2l(ab.ba'.au.ua')du 0 7' 5 °' b' aa'.ju .uj' ° (I) 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, a.=B=B', 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=x, will give the solution, by duplication, of a jet issuing by a re entrant 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 byj. H. Michel, ,A. Love and M. Réthy. I I! When the polygon is closed by the walls joining, insteadofreaching back to in nity at'aqx', the liquid motioninust=be'due'to'a source, and this modification has beenfworked out by B'. 'Hoplcinson in the Proc. Land. Math. Soc., 1898.' ' ' »* ' ' ' ' ff Michell has discussed also the hollow 'vortex stationary inside a polygon'(Phil. Trans., 1890); the solution is givenby > if ch nS)'= sn w, sh n(2=i cn rv
so that, round' the boundary of the polygon, and on the surface of the vortex ]/ =o, q =Q, andcosn0=sn ¢>, m9=§ 1ram 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 =% gives an Elastica. 44. The Motion of gz Solid through a Liquid:An. important problem in the motion of a liquid is the determination of the state (11)
Jl¢=K', , sin nH==o;
of velocity set up by the passage of a solid through it;~and thence j of the pressure and reaction of the liquid ontliesurface of the solid, by which its motion is influenced when it is free. ' ' ' V Beginning with a single body in liquid extending jto'infinity*, 'an'd denoting by U, V, W, P, Q, R] the components of linear and angular velocity with respect to axes fixed in the body, the velocityfunctlon O takes the form ' ~ »~
¢=U¢1+V4>2'l'fV4?a'l'PX1+QX2'l"RXa, (X) where the ¢'s and. fs ate functions of,3c, ' y, "z depending on the shape of the body; interpreted dynamically, Cpo represents the impulsive pressure required to stop the motion, or C+p4> to start it again from rest.
The terms of ¢ ma be determined one at a time, and this problem is purely kinematicai; thus to determine ¢1, 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 41, must be determined to satisfy the conditions (i.) V'¢, =o, throughout the liquid; ' ~ .., ~, (ii.) $35 = l, 'the gradient of ¢ down the normalat the surface of the moving solid; ' ' ' "
(iii.) %' ==o, over a fixed boundary, or at infinity; similarly for np, and os. ' ' ” To determine X, the angular velocity¢P alone is introduced, and the conditions to be wtisfied are, (i.) V'X'=0, throughout the liquid; (ii.) %Zff = mz  ny, at the surface oithe moving bodyfbut zero: over a fixed surface, and at inlinity; the same for xg and Xa. For a cavity filled with liquid in theinterior of the body, since the liquid' inside moves bodily for a motion of 'translation'only, 4n=x, d>z='7. ¢a= '*2?' . (2) but a rotation will stir up the liquidin the cavity, so that the'X's de nd on the shape of the surface. Tehe ellipsoid was the shape first worked out, by George Green, in his'Research on the Vibration of a Pendulum in a Fluid Me inrn (1833); the extension to any other surface will form an 'important step in this subject. 1 ~ »
A system of confocal ellipsoids is taken x»+y2+z= I
E2+" x" BM'> ¢2+x= ' (3)
and a velocity function of the form ' V » ¢ gxlpr
where up is function of Xonlyfiso that rl/ is constantover an ellipsoid; and we seek to determine the motion set up, and the form of 11/ 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 = v = z
I a'+7' '” " FQFX (5)
2: Ex! Zz?,
<a2+>~>2+<b='+»>=+<c2+»>" (6)
P' = (<1'+7ll'+(5'+7)"if+(C'+))1'» I (7) = azlz lwbim' lczn' +A,
wa
Thence M d d 1
= x '/
as a;"*+'?ar
»=§ ¢+2ca=+;>§§ z§§ , '“ ~ (cg) I0 that't'he'velocity of the liquid ma be resolved into at component ¢' parallel to Ox, and' 2(a2+7)l1iiZ 7d> alongthe norinalf of lthe ellipsoid; and the liquid flows over an ellipsoid along a line of slope 7 with respect to Ox, treated as the vertical. Along the normal itself ~ ¢3 1
i"f= ¢+¢<a+n@ 1, no
~ d§ , dk Y
Sothat. overthe surfaconf an ellipsoid where 7 and W are constant. the nozrnal velocxtygs the same as that of the ellipsoid itself, moving as azsolid withwelocxty parallel to Ox ~ V ~, P ' fuew¢'z<<»”+»>§ . T ml
and sothe boundary condition is satisfied; moreover, any ellipsoidal surface X may be supposed' moving as if rigid with the velocity in (1 I), without disturbing the liquid motion for the moment.
 The Continuity is secured if the liquid between two ellipsoids 7
and)~, , ~moving with the velocity U and U1 of equation (11), is squeezed* out or sucked in across the plane x =o'at a'rate equal to the integral #ow of the velocity:P across the annular 'area a, a of the two ell1pqoids xnafl e by x=0; or if, f T tu¢, U, = iv gat, i (12)
<»=~ <f>“+~f+»> 1 <1s>
Expressed as a differential relation, with the value off; U from (I 1), gi a, lf+2 (a'+A)a% ¢%§ =o, V (14) “ of i T H 4 4 'r
3aE¥+ a(a's{7)aX<aaL£) =o, (15) and ~mte'gratxng .
1 (a'+7$)°”a§§ =a constant, (16) so that we may put ' ' ' ' O
Md) (175
(13)
y 1 1 'f' ' 'P<a”+»> ' t
P”=4(G'+>)(1>“+>)(¢>'+7),
where M denotes a constant; so that np is an elliptic integral of the second kind.
The quiescent ellipsoidal surface, over which the motion is entirely tangential; is 'the one for which ~ f d ~
2(a'+})a¥ +[/=o, (19)
and this is the infinite boundary ellipsoid if we make the upper limit = oo. V ,
I The velocity of the ellipsoid defined by A =o is then V U = *2G2 *¢q
= M '° Mdk
abc. f, , 'ia2+A)P .
with the notation
Am;A~=f“' abcdh
so that in (4)
M
=f%<1A.>. ' ' <=<>>
" A (a2lASP
1 =2abc& ~ AQ, (21)
¢=rb52xA=iU§ . '§ , i¢=?§  %, O (22) in (I) for an ellipsoid. ' »
The impulse required to setup the motion in liquid of density p is the resultant of an impulsive pressure p¢ over the surface S of the ellipsoid, and is therefore .
ff, ,¢zds =p~/f., f {x1ds T
=p¢0 volume of the ellipsoid) =//uW', (23) where W: denotes the weight of liquid displaced. Denotmg the eliectiveinertia of the liquid parallel to Ox by ¢W', the momentum »
°W'U=¢.»w" s <2 4>
==%3=;;i%; (25),
in this way the air drag was calculated by Green for an ellipsoidal pendulum. .
Similarly, the inertia 'parallel to Oy and Oz is B, , C
BW' TTEW, 7W — I % W', ~ (26) " ' BA»~C>, '=, ? (27)
and V
A+B+C==abc/=}P, A¢+Bo+Cu=I. (28) For a sphere
0=b=€.Ao"Bo*==Co==i» ¢=~'B=~/=§ » (20) 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 VY, 'l,5%g no
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 )=>, , a slight extension will give the velocity function ¢ of the liquid in the inters ace as the ellipsoid)=o is passing with velocity U through the conffxcal position; ¢» must now take the form x(//~{N), 5 and will satisfy the conditions in the shape Q B abc ' M abcdk
A+ 1'f'C1, = abc)(a2'l'))P
¢“ U"B., +C..B.C. U'°x db; ' m, abcdx ' (0 615161 0 (012 'i')>P
and any confocal ellipsoid defined bi '), internal or external to. ).=), , may be supposed to swim wit the liquid for an instant, without distortion or rotation, with velocity along Ox 2 Bx+C.B1Ci s?
4 U BTW# C., ~ BT Ci
Since  Ux is the velocity function for the liquid W' filling thel ellipsoid A=o, and moving bodily with it, the effective mertiaof the liquid in the inter space isA..+B.+c. Bo'l'Eo " Ei " C1W, ' (2) I
If the ellipsoid is of revolution, with b=C, j A +281
45" Bl:
and the Stokes' current function up can be written down ¢=  iUy2§ § €3. <4>
reducing, when the liquid extends to 'infinity and B1=o, to ¢=iUx§ . ¢= rUy=gi. <5> =
so that in the relative motion past the body, as when fixed in the g current U parallel to xO, f
¢'==sUx(1+§ ';,). ¢'=sUf (1 § ;) 4 <61 E Changing] the origin from the centre to thefocus of a prolatel spheroi, t en putting b“=pa, J=>'a, and proceeding to thelimité where a =oo, we find for a paraboloid of revolution 1, =; = 1L EL: ' l ~ T 1 ~ ' 2
A;, %f5/=1>+>~'2x. ' .<8> §
with X =o over the surface of the paraboloid; and then ¢'=%Uly'in/(r*+y?)'f12x1; oi, ¢ =%U1>l/ (x'+;v')xl; (10) 5
¢=%Ui> log li(x'+y')+#l» (H)
The relative path of a liquid particle is along a stream line A 4/= § Uc', a constant, (12)
S 2y2 (2 c2)2 g = , 62)2 ' 1 K
" L=T><y=' L”ci) ' *'<"'+y') ==z>o= is "9 a Ci; while the absolute path of a particle in space'will be given by d z 2
ifif ='%1>i» <'4>
y'c'=a'e"/P. (15)
46. Between two concentric spheres, with G2+X=f2, U2+7=a12;
A==B =C =a°/3r°,
Ia' ai” 2 1ar' af"
and the effective inertia of the liquidin the inter space Ani2A, °+ 3,
2A0 2A1W = ixatils 2:25 W '
When the spheres are not concentric, an expression for the effective inertia can be found by the method of'*irnages (W. M. Hicks, Phil, Trans., 1880). »The
image of a 'source of strcn h /4 at S outside a s here of radius a is a source of strength pa/gat H, where OS =f, OpH=a'/f, and a line sink reaching from the image H to the centre O of line strength  ii/a; this combination 'wil be found to produce no How across the surface of the sphere. v Taking Ox along OS, the Stokes' function at P for the source S is it cos PSx, and of the source H and line sink OH is »;¢(a, [f).cos Plfx and  (/1/a)(PO  PH); so that, , ip = p (cos P'5x+?cosPHx'QQEE, (4) and il/ = p, 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 ipaage system; to realize physically the condition of zero How 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 caube written clown; 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
im 3] cos PSx= f m%; i (5)
and for its imageat H the Stokes' function is m Iéjcos PHx =m%!Fifi; (6)
so that for the combination .. ., s g 2 3, :s
~°='"y" i'1rm"1%=1)='"f'§ ('i§ n@°ifs=i~ <7> and this vanishes oven the surface of 'the sphere. There is no Stokes' Lunction when the axis of the doublet at S does not pass through O; the imager system will, consist of an inclined doublet at H, making an equa 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 movinfg in the presence of a. fixed sphere, or, inside it. The method o electrical images will enable the stream function ¢ to be inferred from a distribution of doublets, finite in number when 'the surface is composed of two, spheres intersecting at an angle wr/m, where m is an integer (R. A. Herman, Quart. Jour. of Math. xxii.).
hThus for m=2, the spheres are orthogonal, and it can be' verified t at ' i ' »
a 3 a as
¢, =§ .Uy2 1 is':¢;+;, , (8)
the radius pf the spheres, and
rl, rs, r the' distances of, a point where, qi, wi. afaiaz/V (ai'+a¢.*') is their clrcle of intersection, and from their centres.
The correspondiugexpression for two orthogonal cylinders will be ~ ' ' '2 ' '2' y
¢'=Uy "ft7é+%=, <9>.;
With a, =°o, these 'reduce to'   i, 'I 5, i », 5 ' .
u 'P'=%Uy”<1gs)§ ».01' U95 (f gl A (lo) for a sphere or cylinder, and a diarnetral #plane 'O ' Two equal spheres, intersecting at Iso, will require O ..¢g 02 a“(a2.9 313, a'(¢%lfx), ip " a. 2n3'T 2715 +2723 2736 r '(I with fa similar expression for cylinders; so that the plane x=o rnayibe introduced as a boundary, cutting the surface at 6092 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; . (~) f 4>=J¢x., A
for a rotation R about Oz; and ai similar procedure shows, that an ellipsoidal surface A may be in rotation about Oz without disturbing the motion if
I I — dx
R 3 a2 +A' +b2+>.> *HEX
W ' I/(b"'l°?)” I/(1124X)
and that the continuity of the liquid is secured 'if (2)
(a'+7)3/2(b2+})3/'(¢2+})%2§ =constant, A (3) °°, Nd> N B.A,
X i 'f'X ""P<f»+ ><1f+»> Tb? WT' 4 and at the surface }=0, : j V 'N, B>A Q, .
R Gi'2+7;§ >El'i ao?b“0 a.bcifiF?',  " rib* 1/Ei, , ' (5)
N I/l>*  I/Q'
T=R ' .
° 's for (a+i=)*”..2 1.=° <9
a2 b2' z/(a2+ bs)
“Rm ~ 2>/<a2+1»2> c 0Ao The velocity function of the liquid inside the ellipsoid λ = 0 due to the same angular velocity will be
and on the surface outside
φ_{0} = xyχ_{0} = xy  N  B_{0} − A_{0}  ,  
abc  a^{2} − b^{2} 
so that the ratio of the exterior and interior value of φ at the surface is
φ_{0}  =  B_{0} − A_{0}  , 
φ_{1}  (a^{2} − b^{2}) / (a^{2} + b^{2}) − (B_{0} − A_{0}) 
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 = λ_{1} is made in a similar manner, by putting
and the ratio of the effective angular inertia in (9) is changed to

.  

Make c = ∞ for confocal elliptic cylinders; and then
Aλ = ∫∞λ  ab  =  ab  ( 1 − √  b^{2} + λ  ), 
(a^{2} + λ) √ (4a^{2} + λb^{2} + λ)  a^{2} − b^{2}  a^{2} + λ 
Bλ =  ab  ( √  a^{2} + λ  − 1 ), Cλ = 0; 
a^{2} − b^{2}  b^{2} + λ 
and then as above in § 31, with
the ratio in (11) agrees with § 31 (6).
As before in § 31, the rotation may be resolved into a shearpair, 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 A_{λ}, B_{λ}, C_{λ}, in a similar manner, since
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 x_{1}, x_{2}, x_{3}, y_{1}, y_{2}, y_{3}, 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
x_{1} =  dT  , x_{2} =  dT  , x_{3} =  dT  , 
dU  dV  dW 
y_{1} =  dT  , y_{2} =  dT  , y_{3} =  dT  ; 
dP  dQ  dR 
but when it is expressed as a quadratic function of x_{1}, x_{2}, x_{3}, y_{1}, y_{2}, y_{3},
U =  dT  , V =  dT  , W =  dT  , 
dx_{1}  dx_{2}  dx_{3} 
P =  dT  , Q =  dT  , R =  dT  . 
dy_{1}  dy_{2}  dy_{3} 
The second system of expression was chosen by Clebsch and adopted by Halphen in his Fonctions elliptiques; and thence the dynamical equations follow
X =  dx_{1}  − x_{2}  dT  + x_{3}  dT  , Y = ..., Z = ..., 
dt  dy_{3}  dy_{2} 
L =  dy_{1}  − y_{2}  dT  + y_{3}  dT  − x_{2}  dT  + x_{3}  dT  , M = ..., N = ..., 
dt  dy_{3}  dy_{2}  dx_{3}  dx_{2} 
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  = mR − nQ,  dm  = nP − lR,  dn  = lQ − mP. 
dt  dt  dt 
If P denotes the resultant linear impulse or momentum in this direction
dP  =  dl  x_{1} +  dm  x_{2} +  dn  x_{3} 
dt  dt  dt  dt 
+ l  dx_{1}  + m  dx_{2}  + n  dx_{3}  , 
dt  dt  dt 
= l (  dx_{1}  − x_{2}R + x_{3}Q ) 
dt 
+ m (  dx_{2}  − x_{3}P + x_{1}R ) 
dt 
+ n (  dx_{3}  − x_{1}Q + x_{2}P ) 
dt 
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 (y_{1} − x_{2}z + x_{3}y) + m (y_{2} − x_{3}x + x_{1}z) + n (y_{3} − x_{1}y + x_{2}x). 
Differentiating with respect to t, and afterwards moving the fixed origin up to the moving origin O, so that
x = y = z = 0, but  dx  = U,  dy  = V,  dz  = W, 
dt  dt  dt 
dG  = l (  dy_{1}  − y_{2}R + y_{3}Q − x_{2}W + x_{3}V ) 
dt  dt 
+ m (  dy_{2}  − y_{3}P + y_{1}R − x_{3}U + x_{1}W ) 
dt 
+ n (  dy_{3}  − y_{1}Q + y_{2}P − x_{1}V + x_{2}U ) 
dt 
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.) x_{1}^{2} + x_{2}^{2}+ x_{3}^{2} = F^{2}, a constant, (iii.) x_{1}y_{1} + x_{2}y_{2} + x_{3}y_{3} = n = GF, a constant;
and the dynamical equations in (3) express the fact that x_{1}, x_{2}, x_{3} are the components of a constant vector having a fixed direction; while (4) shows that the vector resultant of y_{1}, y_{2}, y_{3} moves as if subject to a couple of components
and the resultant couple is therefore perpendicular to F, the resultant of x_{1}, x_{2}, x_{3}, 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 doubletheta 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 = 12p (x_{1}^{2} + x_{2}^{2}) + 12p′x_{3}^{2}+ q (x_{1}y_{1} + x_{2}y_{2}) + q′x_{3}y_{3}+ 12r (y_{1}^{2} + y_{2}^{2}) + 12r′y_{3}^{2} 
so that a fourth integral is given by
dx_{3}  = x_{1} (qx_{2} + ry_{2}) − x_{2} (qx_{1} + ry_{1}) = r (x_{1}y_{2} − x_{2}y_{1}), 
dt 
1  (  dx_{3}  )  ^{2}  = (x_{1}^{2} + x_{2}^{2}) (y_{1}^{2} + y_{2}^{2}) − (x_{1}y_{1} + x_{2}y_{2})^{2} 
r^{2}  dt 
= (x_{1}^{2} + x_{2}^{2}) (y_{1}^{2} + y_{2}^{2}) − (FG − x_{3}y_{3})^{2} = (x_{1}^{2} + x_{2}^{2}) (y_{1}^{2} + y_{2}^{2} + y_{3}^{2} − G^{2}) − (Gx_{3} − Fy_{3})^{2}, 
in which
r (y_{1}^{2} + y_{2}^{2}) = 2T − p(x_{1}^{2} + x_{2}^{2}) − p′x_{3}^{2}− 2q (x_{1}y_{1} + x_{2}y_{2}) − 2q′x_{3}y_{3} − r′y_{3}^{2}= (p − p′) x_{3}^{2} + 2 (q − q′) x_{3}y_{3} + m_{1}, 
so that
1  (  dx_{3}  )  ^{2}  = X_{3} 
r^{2}  dt 
where X_{3} is a quartic function of x_{3}, and thus t is given by an elliptic integral of the first kind; and by inversion x_{3} is in elliptic function of the time t. Now
y_{1} + y_{2}i  =  FG − x_{3}y_{3} + i √ X_{3}  , 
x_{1} + x_{2}i  x_{1}^{2} + x_{2}^{2} 
d  (x_{1} + x_{2}i) = −i [ (q′ − q) x_{3} + r′y_{3} ] + irx_{3} (y_{1} + y_{2}i), 
dt 
d  log (x_{1} + x_{2}i) = −(q′ − q) x_{3} − r′y_{3} + rx_{3}  FG − x_{3}y_{3} + i √ X_{3}  , 
dti  F^{2} − x_{3}^{2} 
d  log √  x_{1} + x_{2}i  = −(q′ − q) x_{3} − (r′ − r) y_{3} − Fr  Fy_{3} − Gx_{3}  , 
dti  x_{1} − x_{2}i  F^{2} − x_{3}^{2} 
requiring the elliptic integral of the third kind; thence the expression of x_{1} + x_{2}i and y_{1} + y_{2}i.
Introducing Euler’s angles θ, φ, ψ,
x_{1} = F sin θ sin φ, x_{2} = F sin θ cos φ, x_{1} + x_{2}i = iF sin θε−ψi, x_{3} = F cos θ; 
sin θ  dψ  = P sin φ + Q cos φ, 
dt 
F sin^{2} θ  dψ  =  dT  x_{1} +  dT  x_{2} 
dt  dy_{1}  dy_{2} 
= (qx_{1} + ry_{1}) x_{1} + (qx_{2} + ry_{2}) x_{2} = q (x_{1}^{2} + x_{2}^{2}) + r (x_{1}y_{1} + x_{2}y_{2}) = qF^{2} sin^{2} θ + r (FG − x_{3}y_{3}), 
ψ − qFt = ∫  FG − x_{3}y_{3}  Frdx_{3}  ,  
F^{2} − x_{3}^{2}  √ X_{3} 
elliptic integrals of the third kind.
Employing G. Kirchhoff’s expressions for X, Y, Z, the coordinates of the centre of the body,
F(X + Yi) =  Fy_{3} − Gx_{3} + i √ X_{3}  εψi. 
√ (F^{2} − x_{3}^{2}) 
Suppose x_{3} −F is a repeated factor of X_{3}, then y_{3} = G, and
X_{3} = (x_{3} − F)^{2} [  p′ − p  (x_{3} + F)^{2} + 2  q′ − q  G (x_{3} + F) − G^{2} ], 
r  r 
and putting x_{3} − F = y,
(  dy  )  ^{2}  = r^{2}y^{2} [ 4  p′ − p  F^{2} + 4  q′ − q  FG − G^{2} + 2 ( 2  p′ − p  F +  q′ − q  G ) y +  p′ − p  y^{2} ], 
dt  r  r  r  r  r 
so that the stability of this axial movement is secured if
A = 4  p′ − p  F^{2} + 4  q′ − q  FG − G^{2} 
r  r 
is negative, and then the axis makes r√(−A)/π nutations per second. Otherwise, if A is positive
rt = ∫  dy 
y √ (A + 2By + Cy^{2}) 
=  1  sh^{−1}  √ A √ (A + 2By + Cy^{2})  =  1  ch^{−1}  A + By  ,  
√ A  ch^{−1}  y√ (B^{2} ~ AC)  √A  sh^{−1}  y √ (B^{2} ~ AC) 
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 pseudoelliptic integral. To give a simple instance, changing to the stereographic projection by putting tan 12θ = x,
X_{1}  = ± ax^{4} + 2ax^{3} ± 3 (a + b) x^{2} + 2bx ± b, 
X_{2} 
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
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. ℔,
F cos θ = c_{1}  u  = (W + W′α)  V  cos φ, F sin θ = c_{2}  v  = (W + W′β)  V  . 
g  g  g  g 
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
W  V  cos φ and W  V  sin φ is W  V  sec. ℔, 
g  g  g 
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. footpounds
equivalent to an incessant couple
N = FV sin (θ − φ) = (F sin θ cos φ − F cos θ sin φ) V = (c_{2} − c_{1}) (V^{2} / g) sin φ cos φ = W′ (β − α) uv / g. 
This N is the couple in footpounds 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 c_{2} − c_{1} is positive to set the body broadside to the advance.
An oblate flattened body, like a disk or plate, has c_{2} − c_{1} 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 rocketstick 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 c_{2} − c_{1} 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
Wh tan θ = N = (c_{2} − c_{1})  c_{1}  u^{2}  tan θ,  
c_{2}  g 
h =  c_{2} − c_{1}  c_{1}  u^{2}  = (β − α)  1 + α  u^{2}  .  
W  c_{2}  g  1 + β  g 
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 C_{1} about the axis of the figure, and by C_{2} 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 C_{1} is the moment of inertia of the body about the axis, denoted by Wk_{1}^{2}. But if Wk_{2}^{2} 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 ′,
in which we have put k ′^{2} = εk^{2}, where ε is a numerical factor depending on the shape.
 Table of Rifling for Stability of an Elongated Projectile, x Calibre; long, giving 5 the Angle of
If the shot is spinning about its axis with angular velocity p, and is precessing steadily at a rate it about a line parallel to the resultant momentum F at an angle 0, the velocity of the vector of angular momentum, as in the case of atop, is Clpn sin 0 Cui sin 0 cos 0; (4)
and equating this to the impressed couple (multiplied by g), that is, to KN = (61 603142 tan 0. (5)
and dividing out sin 0, which equated to zero would imply perfect centring, we obtain 1
Cui' cos 0 Clpp l (czc1)§ u' sec 0 = 0. (6) The least admissible value of p is that which makes the roots equal of this quadratic in yi, and then
C
 =%@§ 1>Se¢0, (7)
the roots would be imaginary for a value of p smaller' than given by Cgp—4<¢, ¢, >§ c, u2 =0, (8)
%=4(C2C1)2 (9)
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 5 of the rifling IS given by tan 6 = 1rd/nd = édp/u, (10)
If udenotes the density of the metal, and if the shell has a cavity homo the tic with the external ellipsoidal shape, a fraction ' f of the linear scale; then the volume of a round shot being %'ll'd3, and %1l'd3x of a shot x calibres long
W = 2, 1rd3x(1 f“)¢, (20)
W/6,2 =, § 4d3x§ l;<1f>¢, (21)
Wk; = éirdixlig 5dE(1 j5)a. (22) If p denotes the density of the air or medium W' =%1f1i°x/>. (23)
W' 1 p
Wim? . <20
k1° 1 1~j5 kg” x2l1
22rat?' EFT (25)
2
tan” a = fins 1.) (26)
in which ¢r/p may be replaced by 800 times the S.G. of the metal, taking water as Soo times denser than air on the average, in round numbers, and formula (IO) may be written n tan 5=1r, or n5= 180, when 6 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.
Rzflzng, andnthe Pitch of Rzfling in Calibres. Castiron Common Shell Palliser Shell Solid Steel Bullet 1 Solid Lead Bullet f= § , S.G. 72. f=§ , S.G. 8. » f=o, S.G. 8. f=o, S.G. I0'9. x B  a 5 n 5 n 5 n 5 n
1 0 00000 0° 0' Infinity Infinity 7 0° 0' Infinity Intinity 20 04942 2 49 6387 71 08 2 29 7221 8429 2'5 06056 3 46 47'9I 53232 3 19 54'17 53'24 3'0 06819 4 41 33'45 42'79 4 09 43'47 50'74 3'5 07370 5 35 3213 3575 4 58 3653 4240 40 o7782 6 30 2760 3072 5 45 3121 3643 45 08100 7 24 2420 2693 6 32 2736 3194 50 o8351 8 16 2156 2398 7 21 2436 28'44 6o 0°872I IO 05 1767 1967 8 56 1998 2333 100 049395 16 57 IO'3I 1 I47' 15 05 1 165 1360 Infinity 1 oooo 90 oo 0oo 000 90 oo 0oo 0oo which is the ratio of the linear velocity of rotation édp to u, the velocity of advance,
52. In the steady motion the centre of the shot describes a helix with axial velocity
tan'6=%;=°;j3;=(c2ci)g(%i;,
W' W' k' 2
=N4(B)llWa. V(I'l'W'e> . (H)
W iz bpm (k1)i
W il
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
tan2 6=;é=l%(Ba) 7 ' a (12)
if then we can calculate B, a., or 3a for the external shape of the shot, this equation will give the value of 6 and n required for stability of Flight in the air..
The ellipsoid is the only shape for which a and B have so far been determined analytically, as shown already in»§ 44, so we 'must restrict our calculation to an eggshaped bullet, bounded by a prolate ellipsoid of revolution, in which, with b=c, A = °° ab2d>§ * vpv * °° V abldk " (13) ° 0 (¢12'l'})/ l4(¢l2+>)(b'“lf>)2l o 2(<1'+?)”/°(b“+>) A0'l'2Bo= 1, (14)
JL B Liu;
°”1Ao' 8 1hn'1+A0"Y'+2a' (15)
The length of the shot being denoted by l and the calibre by d, and the length in calibres by x
V l/d=2a/2b=x, (16)
x, 1
A°“ef1'fn°“ '°'“a=f', “"
x, ae', . 1
“B” 1 11>f“°h '”+r.+1 “Sl
h'W(x¢1
x"A.>+2B@ 3°Hgg) 7(fi:7310sl4=+~/<42 1>l. (19) ucos0+vsin0 = 1lEitanzli ucos (izusecd, (1) and transverse velocity
usin0vc0s6.=<I%)usin0>'§ =(Ba.)usin0; (2) and the time of completing a turn of the spiral is 2ir/ii. When;.¢ has the critical value in (7), 2, Er=5g%cos0 =%'(x2+1)cos0, (3) which makes the circumference of the cylinder on which the helix is wrapped
2/§ (usin0vcos0) =€%'(;3 a)(x2I1) sin20 cos 0 =nd(Ba)(x2l1)sin0¢os6» (4)
and' the length of one turn of the helix %'(u cos 9l11 sin 6) =nd(x2+1); (5) thus for x=3, the length is IO times the pitch of the rifiing. 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 re entrant 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 theaplplication of impulsive pressure across an ideal membrane closing the hole, by 'means of ideal mechanism connecteclwith 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 E, 17, § ', and angular 71, p, v, 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 ringshaped 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
the axis of the ring changing from Ox to O′x′; and
so that the increase of the components of momentum, X₁, Y₁, and N₁, linear and angular, are
The components of force, X, Y, and N, acting on the liquid at O, and reacting on the body, are then
Now suppose the cylinder is free; the additional forces acting on the body are the components of kinetic reaction of the liquid
so that its equations of motion are
and putting as before
M + αM′ = c₁, M + βM′ = c₂, C + εC′ = C₃,
c₁dUdt − c₂VR = 0,
c₂dVdt + (c₁U + ξ)R = 0,
c₃dRdt − (c₁U + ξ − c₂U)V = 0;
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
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.)