BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF 1891 .Ks.'dS.X'j.. ^^jU1± CORNELL UNIVERSITY LIBRARY 3 1924 101 157 836 The original of tliis bool< is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924101157836 A TREATISE ON HYDROSTATICS. A TREATISE HYDEOSTATICS BY ALFRED GEORGE GREENHILL, PROFESSOR OF MATHEMATICS IN THE ARTILLERY COLLEGE, WOOLWICHt MACMILLAN AND CO. AND NEW YORK. 1894. [All rights reserved.] PREFACE. The aim of the present Treatise on Hydrostatics is to develop the subject from the outset by means of illus- trations of existing problems, chosen in general on as large a scale as possible, and carried out to their numerical results ; in this way it is hoped that the student will acquire a real working knowledge of the subject, while at the same time the book will prove useful to the practical engineer. It is very important in Hydrostatics that the units employed should be kept constantly in view ; and for this reason the condensed notation proposed by M. Hospitaller at the International Congress of Electri- cians of 1891 has been adopted. In this notation the full length expression of so many " pounds per square inch " or " kilogrammes per square centimetre " is abbre- viated to Ib/in^ or kg/cm^ ; and so on for other physical quantities. The gravitation unit of force has been universally employed, except in a few problems of cosmopolitan vi PREFACE. interest, in which the variation of gravity becomes perceptible. In accordance with modern ideas of mathematical instruction, a free use is made of the symbols and operations of the Calculus, where the treatment requires it, although an alternative demonstration by elementary methods is occasionally submitted ; because, as it has well been said, "it is easier to learn the Differential Calculus than to follow a demonstration which attempts to avoid its use." Particular attention has been given to the applica- tions of the subject in Naval Architecture, and the Transactions of the Institution of Naval Architects have been ransacked for appropriate illustrations. The diagrams, which have been drawn by Mr. A. G. Hadcock, late Royal Artillery, are intended to represent accurately to scale the objects described. No attempt has been made to rival the beautiful shaded figures of the French treatises, for fear of obscuring essential principles. A type of uniform size has been employed through- out: although adding considerably to the bulk, it is hoped that this uniformity will prove acceptable to the eyes of the readers, and counterbalance the dis- advantage of the extra size of the book. CONTENTS. CHAPTER I. The Fundamental Principles, . . . 1 PAGE CHAPTER II. Hydrostatic Thrust, .... . .41 CHAPTER III. Archimedes' Principle and Buoyancy. Experimental Determination of Specific Gravity by THE Hydrostatic Balance and Hydrometer, . . 93 CHAPTER IV. The Equilibrium and Stability of a Ship or Floating Body, . .... 137 CHAPTER V. Equilibrium of Floating Bodies of Regular Form, and of Bodies Partly Supported. Oscillation of Floating Bodies, 189 CHAPTER VI. Equilibrium of Liquids in a Bent Tube. The Thermometer, Barometer, and Siphon, . . . 233 viii CONTENTS. CHAPTER VII. Pneumatics— The Gaseous Laws, . • • 279 PAGE CHAPTEE VIII. Pneumatic Machines, . • ■ 327 CHAPTER IX. The Tension of Vessels — Capillarity, . 385 CHAPTER X. Pressure op Liquid in Moving Vessels, . . 423 CHAPTEE XI. Hydraulics, . 461 CHAPTER XII. General Equations op Equilibrium, . ... 485 CHAPTER XIII. Mechanical Theory of Heat, . . 505 Appendix — Tables, ... . 528 Index, .... . . 528 ERRATA. P. 170, line 9 from the bottom, read " ... /O' = ajs, then /O' is the depth of a vessel of box form, supposed homogeneous and of a.o. s, which will float at the draft a." P. 436, last line, and p. 437, line 7, read P sin 8(0QI0L). HYDROSTATICS. CHAPTER I. THE FUNDAMENTAL PEINCIPLES 1. Introduction. Hydrostatics is the Science of the Equilibrium of Fluids, and of the associated Mechanical Problems. The name is derived from the compound Greek word vSpoa-TUTiKij, meaning the Science {e-wiaTriM) of the Statics of Water ; thus Hydrostatics is the Science which treats of the Equilibrium of Water, the typical liquid, and -thence generally of all Fluids. The Science of Hydrostatics is considered to originate with Archimedes (B.C. 250) in his work IlepJ oxovfievwv, now lost, but preserved in the Latin version of Guillaume de Moerbek (1269), " De lis quce vehuntur in humido" ; and recently translated into French by Adrien Legrand, " Le traits des corps fiottants d'Archimede," 1891. Archimedes discovered the method of determining the density and purity of metals by weighing them in water, and extended the same principles to the conditions of equilibrium of a ship or other floating body. 2 HISTORICAL IXTRODUCTION. Ctesibius, of Alexandria, and his pupil Hero (B.C. 120), the author of a treatise on Pneumatics, are considered the inventors of the siphon and forcing pump ; Vitruvius may- be consulted for these and other machines known to the Romans; while the leading principles of the flow of water as required in practical hydraulics are given by Frontinus in his work cle aquceductibus ivrbis Romce commentarius (a.d. 100). The writings of Pliny (lib. xxx. c. vi.) prove that the Romans were acquainted with the hydrostatical principle that water will rise in a pipe to the height of its source, and that lead pipes must then be employed, stone or brick conduits not being sufficiently watertight ; but being ignorant of the method of casting iron pipes strong enough to stand a considerable pressure or head of water, their large aqueducts were carried on the level, while leaden pipes were used only for the distribution of the water, specimens of which pipes have recently been discovered at Bath. A long detailed edict of Augustus concerning the waterworks of Venafrum is given in Mommsen's Corpus InsoriptionuTn Latinarum, vol. 10, part i. ; and allusions to the mode of water supply are found in Horace and Ovid — " Purior in viois aqua tendit rumpere plumbum, Quam quae per pronum trepidat cum murmure rivum ? " (Horace, Epist. I. x.) " Cruor emicat alte, Non aliter quam cum vitiato fistula plumbo Scinditur." (Ovid, Metamorphoses, iv. 122.) A great advance in the Theory of Hydraulics was made by Torricelli (1643), also the inventor of the barometer, who first enunciated the true theory of the HISTORICAL INTRODUCTION. 3 velocity and form of a jet of water, as deduced from the experiments of Galileo and himself with the orna- mental waterworks of the gardens of the Duke of Tuscany; repeated later in 1684 by Mariotte in the gardens of Versailles. In the writings of Stevinus of Bruges (c. 1600) we find many fundamental theorems of our science clearly enunciated and explained ; but the modern exact Theory of Hydrostatics is generally held to originate with Pascal (1653), in his two treatises, Traitd de V^quilihre des liqueurs and Traits de la pesanteur de la 'masse de I'air ; in which the fundamental principles are first clearly enunciated and illustrated, and the true theory and use of the barometer of Torricelli is explained. The elastic properties of a gas were investigated by Boyle and Mariotte, about 1660, and subsequently com- pleted by Charles and Gay Lussac ; and now the funda- mental principles of the equilibrium of fluids being clearly enunciated and established, the analysis was carried on and completed by Newton, Gotes, Bernoulli, d'Alembert, and other mathematicians of the 18th century ; while the applications of steam in the 19th century has been the cause of the creation of the subject of Thermodynamics, first placed on a sound basis by Joule's experiments, in which the relations are investi- gated between the heat expended and the work produced by means of the transformations of a fluid medium. Hydrostatics is a subject which, growing originally out of a number of isolated practical problems, satisfies the requirements of perfect accuracy in its applic'ation to the largest and smallest phenomena of the behaviour of fluids; and at the same time delights the pure theorist 4 THE DIFFERENT STATES OF MA TTER. by the simplicity of the logic with which the funda- mental theorems may be established, and by the elegance of its mathematical operations ; so that the subject may be considered as the Euclidean Pure Geometry of the Mechanical Sciences. Montucla's Histoire des MatMmatiques, t. iii., from which the preceding historical details are chiefly derived, may be consulted for a more elaborate account of the work of the pioneers in this subject of Hydrostatics and Hydraulics. 2. The Different States of Matter or Substance. A Fluid, as the name implies, is a substance which flows, or is capable of flowing ; water and air are the two fluids most universally distributed over the surface of the Earth. All substances in Nature fall into the two classes of Solids and Fluids ; a Solid substance (the land for instance), as contrasted with a Fluid, being a substance which does not flow, of itself. Fluids are again subdivided into two classes. Liquids and Gases, of which water and air are the chief examples. A Liquid is a fluid which is incompressible, or nearly so; that is, it does not sensibly change in volume with variations of pressure. A Gas is a fluid which is compressible, and changes in volume with change of pressure. Liquids again can be poured from one vessel into another, and can be kept in open vessels ; but gases tend to diffuse themselves, and must be preserved in closed vessels. THE DIFFERENT STATES OF MATTER. 5 The distinguishing characteristics of the three Kinds of Substances or States of Matter, the Solid, Liquid, and Gas, are summarized as follows in Lodge's Mechanics, p. 150 :— A Solid has both size and shape ; A Liquid has size, but not shape ; A Gas has neither size nor shape. 3. The Changes of State of Matter. By changes of temperature (and of pressure combined) a substance can be made to pass from one of these states to another; thus, by gradually increasing the temperature, a solid piece of Ice can be melted into the liquid state as Water, and the water again can be evaporated into the gaseous state as Steam. Again, by raising the temperature sufficiently, a metal in the solid state can be melted and liquefied, and poured into a mould to assume any required form, which will be retained when the metal is cooled and' solidified again; while the gaseous state of metals is discerned by the spectroscope in the atmosphere of the Sun. Thus mercury is a metal which is liquid at ordinary temperatures, and remains liquid between about — 40° C. and 357° C. ; the melting or freezing point being - 40° C, and the vapourizing or boiling point being 357° C. Conversely, a combination of increased pressure and of lowered temperature will if carried far enough reduce a gas to a liquid, and afterwards to the solid state. This fact, originally the conjecture of natural philoso- phers, has of late years, with the improved apparatus of Cailletet and Pictet, been verified experimentally with air, oxygen, nitrogen, and even hydrogen, the last of the gases to succumb to liquefaction and solidification. 6 THE DIFFERENT STATES OF MATTER. In Professor Dewar's lecture at the Royal Institution, June 1892, liquid air and oxygen were handed round in wine glasses, liquefaction in this case being produced by extreme cold, about — 192° C. All three states of matter of the same substance are simultaneously observable in a burning candle ; the solid state in the unmelted wax of the candle, the liquid state in the melted wax around the wick, and the gaseous state in the flame. Although the three states are quite distinct, the change from one to the other is not quite abrupt, but gradual, during which process the substance partakes of the qualities of both of the adjacent states, as for instance the asphalte pavement in hot weather ; metals and glass become plastic near the melting point, and steam is saturated with water at the boiling point. 4. Plasticity and Viscosity. All solid substances are found to be plastic more or less at all temperatures, as exemplified by the phenomena of punching, shearing, and the flow of metals, investigated experimentally by Tresca (vide fig. 8); but what dis- tinguishes the plastic solid from the viscous fluid is that the plastic solid requires a certain magnitude of stress (shear) to make it flow while the viscous fluid requires a certain length of time for any shearing stress, however small, to permanently displace the parts to an appreciable extent. (K. Pearson, The Elastical Researches of Barre de Saint Venant, p. 253.) According to Maxwell {Theory of Heat, p. 303), " When a continuous alteration of form is only produced by stresses exceeding a certain value, the substance is called a solid, however soft (plastic) it may be. DEFINITION OF A FLUID. 7 "When the very smallest stress, if continued long enough, will cause a constantly increasing change of form, the body must be regarded as a viscous fluid, how- ever hard it may be." Mallet, in his Construction of Artillery, 1856, p. 122, and Maxwell (Theory of Heat, chap. XXI.) illustrate this difference between a soft solid and a hard liquid by a jelly and a block of pitch ; also by the experiment of placing a candle and a stick of sealing wax on two sup- ports ; after a considerable time the sealing wax will be found bent, but the candle remains straight, at ordinary temperatures. A quicksand behaves like a fluid, and, in opposition to the process of melting and founding metals, it requires to be artificially solidified in tunnelling operations ; this is now affected either by a Freezing Process, in which pipes containing freezing mixtures are pushed into the quicksand, or else by the injection of powdered cement or lime-grouting, which solidifies in combination Math the sand. 5. We are now prepared to give in a mathematical form The 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 very easily divided along any plane (given plenty of time, if the fluid is viscous)." Corollary. It follows that, when the fluid is at rest, the tangential stress in any plane in its interior must vanish, and the stress must be entirely normal to the plane — this is the mechanical axiom which is the founda- tion of the Mathematical Theory of Hydrostatics. 8 THE STRESS IN A SUBSTANCE. The Theorems of Hydrostatics are thus true for all stagnant fluids, however viscous they may be ; it is only when we come to Hydrodynamics, the Science of the Motion of Fluids, that the effect of viscosity will make itself felt, and modify the phenomena ; unless we begin by postulating perfect fluids, that is, fluids devoid of viscosity. 6. Stress. We have used the word Stress in the Definition of a Fluid above ; a stress is defined as composed of two equal and opposite balancing forces, acting between two bodies or two parts of the same body. These two forces constitute the " Action and Eeaction " of Newton's Third Law of Motion, which acccording to this law "are equal and opposite." (Maxwell, Matter and Motion, p. 46.) The Stress between two parts of a body is either (i.) of the nature of a PuLL or Tension, tending to prevent separation of the parts, or (ii.) of the nature of a Theust or Pressure, tending to prevent approach, or (iii.) of the nature of a Shearing Stress, tending to prevent the parts from sliding on each other. In a Solid Substance all three kinds of Stress can exist, but in a Fluid at rest the stress can only be a normal Thrust or Pressure; a tensional stress would overcome the cohesion of the fluid particles. Nevertheless a column of mercury, many times the barometric height, may be supported in a vertical tube by its adhesion to the top of the tube, in which case the hydrostatic pressure is negative above the barometric height, or the mercury is in a state of tension; and Mr. Worthington has measured experimentally in ethyl THE PRESSURE IN A FLUID. 9 alcohol enclosed in a glass vessel a tension up to 17 atmospheres, or 255 pounds per square inch. {PMl. Trans., 1892.) The Stress across a dividing plane in a Solid can be resolved into two components, one perpendicular to the plane, of the nature of a tension or pressure, and the other component tangential to the plane ; and it is this tangential stress which is absent in a Fluid at rest. 7. The Measurement of Fluid Pressure. If we consider a fluid at rest on one side of any- imaginary dividing plane, the fluid is in equilibrium under the forces acting upon it and of the stress across the- plane, which is of the nature of a thrust (pouss^e), perpendicular to the plane. Definition. " The Pressure (pression) at any point of the plane is the intensity of the Thrust estimated per unit of area of the plane." Thus if a thrust of P pounds is uniformly distributed over a plane area of A square feet, as on the horizontal bottom of the sea or of any reservoir, the pressure at any point of the plane is P/A pounds per square foot, (but P/14i4:A pounds per square inch). If the thrust P is not uniformly distributed over the . area A, as for instance on the vertical or inclined face of a wall of a reservoir, then P/A represents the average pressure over the area, in pounds per square foot ; and the actual pressure at any point is the average pressure over a small area enclosing the point. Thus if AP pounds denotes the thrust on a small plane area AA square feet enclosing the point, the pressure there is the limit of AP/AA ( = dP/dA, in the notation of the Differential Calculus) pounds per square foot. 10 U^^TS OF MEASUREMENT. 8. Units of Length, Weight, and Force. As we are dealing with a Statical subject, we shall employ the statical gravitation unit of force, which is generally defined as the Attraction of the Earth on the Unit of Weight; but more strictly it is the tension of the plumb line when supporting the Unit of Weight, thus allowing for the discount in the Attraction of the Earth due to its rotation. The British Unit of Weight is the Pound, defined by Act of Parliament, so that our unit of force is the force which is equal to the tension of a thread or plumb line supporting a Pound Weight ; and we shall call this force the Force of a Pound. With a foot as Unit of Length, our pressures will be measured la. pounds per square foot ; this may be written as lb per D foot, or D', or ft^, or as Ib/ft^. The Metric Units of Length and Weight are the Metre and Kilogramme, or the Centimetre and the Gramme ; and with these units, pressure will be given in Idlo- gramm.es per square m,etre, or gra^mmes per square centimetre. According to the Act of Parliament, 8th August, 1878, Schedule III., 1 foot = 30-47945 centims = -304794 metres; 1 metre = 3-28090 feet =3^37079 inches; 1 pound =453-59265 grammes = •45359265 kg; 1 kilogramme = 2-20462 lb =15432-3487 grains. Therefore a pressure of one Ib/ft^ is equivalent to a pressure of -4536 x (3-2809)2 = 4-8826 kilogrammes per square metre (kg/m^); and a pressure of one kg/m^ is equivalent to a pressure of 2-2046 X (-3048)2 = 0-2048 Ib/ft^. UNITS OF MEASUREMENT. 11 A pressure of one kg/cm^ is thus 2048 Ib/ft^, or 14-2 Voj'vD?; so that the normal atmospheric pressure, called an atmosphere, being taken as 14| or 14-7 Ib/in^, is the same as 1"033 kg/cm^; and therefore, for practical pur- poses, the atmosphere may be taken as one kg/cm^. With the Gravitation Unit of Force, the weight of a body is at once the measure of the quantity of matter in the body, and also of the force with which it is apparently attracted by the Earth; and the word Weight may be used in either sense without ambiguity or con- fusion, when dealing with hydrostatical problems on the surface of the Earth. We must notice however that, in consequence of the variation of g, this unit of force will vary slightly in magnitude at different points of the Earth ; but the variation is so small that it makes no practical difference in engineering problems; the variation is only important when we consider tidal or astronomical phenomena, covering the Earth and extending to the Moon, Sun, and planets. 9. The Safety Valve. To measure the pressure of a fluid in a vessel, and to prevent the pressure from exceeding a certain amount, the Safety Valve was invented by Papin, 1681. ^ It consists essentially of a spherical or conical plug C, fitting accurately into a circular orifice in the vessel, and kept closed against the pressure of the fluid by a lever AB, with fulcrum at A ; carrying either a .sliding weight W lb, when used on a steady fixed vessel ; or else held down at the end 5 by a spiral spring S, which can be screwed to any desired pull of T lb, when the vessel is subject to shock and oscillation (fig. 1). 12 THE SAFETY VALVE. Then if the pressure of the fluid on the seat of the valve is p Ib/in^, and the orifice is d inches in diameter, the thrust on the valve is \Trd?p lb ; so that, taking moments about the fulcrum A of the lever AB, lird?p ^AG=WxAE OT Tx AB, when the valve is on the point of lifting. g^^w..//....v..//w/^/////////^/ p<^ Fig. I. Sometimes the valve is held down by a weight (fig. 2) or by a spiral spring, superposed directly without the intervention of a lever as in fig. 3, the form used in steamers and hydraulic machinery. The danger of the sticking of the valve in the seat is obviated in Ramsbottom's safety valve (fig. 4), consisting of two equal conical valves, held down by a bar and a spring midway between them; then one or the other valve, or both valves, will open when the thrust of the fluid on it is half the pull of the spring. THE SAFETY VALVE. 13 Where the pressure of a fluid is exerted over a circular area or piston, it is often convenient to estimate the pressure in pounds per circular inch, written as Ib/O in, or Ib/O"; and many pressure gauges attached to hydraulic machinery are graduated in this manner ; a pressure of p Ib/in^ being \irp or ■7854p Ib/O in- Then the thrust on a circular area d inches in diameter is obtained by multiplying this pressure in Ib/O" by d?. Fig. 2. Fig. 3. Fig. 4. It is important in steam boilers that the area of escape from the safety valve should be suflBciently large, so as to allow the steam to escape as fast as it is generated ; according to a rule given by Rankine the area of the valve in in^ should be 0006 times the number of lb of water evaporated per hour. If the orifice of the safety valve is d ins diameter at the top and conical, the semi-vertical angle of the conical plug being a, then a lift of x ins of the valve will 14 THE PRESSURE GAUGE. give an annular area of internal diameter (l—2x tan a ins, and therefore of area ttx tan a{d — x tan a) in^. But if we consider the valve as a flat disc, of d ins diameter, a lift of x ins will give -wdx in? area of escape sideways. 10. The Pressure Gauge. To measure pressures continually without blowing off at the Safety Valve, the simplest and most efficient instrument is Bourdon's Pressure Gauge (iig. 5). Fig. 5. This consists essentially of a tube AB, bent into the arc of a circle, closed at one end A, and communicating at the other end B with the vessel containing the fluid whose pressure is to be measured. The cross section of the tube AB is flattened or elliptical, the longer diameter standing at right angles to the plane of the tube AB, thus o. THE PRESSURE GAUGE. 15 The working of the instrument depends upon the principle, discovered accidentally by its inventor M. Bourdon (Proc. I. G. E., XI., 1851), that as the pressure in the interior increases and tends to make the elliptic cross section more circular, the tube AB tends to uncurl into an arc of smaller curvature and greater radius ; and the elasticity of the tube AB brings it back again to its original shape as the pressure is removed. The end B being fixed, the motion of the free end A is communicated by a lever and rack to a pointer on a dial, graduated empirically by the application of known test pressures. By making the tube AB of very thin metal, and the cross section a very flattened ellipse or double segment, the instrument can be employed to register slight varia- tions of pressure, such as those of the atmosphere ; it is then called Bourdon's Aneroid Barometer. But when required for registering steam pressures, reaching up to 150 or 200 Ib/in^, the tube is made thicker ; and when employed for measuring hydraulic pressures of 750 to 1000 Ib/in^, or even in some cases to 5 or 10 tons/in^, the tube AB must be made of steel, carefully bored out from a solid circular bar, and after- wards flattened into the elliptical cross section, and bent into a circular arc. Pressures in artillery due to gunpowder reach up to 35,000 or 40,000 lhjm\ and more, say up to 20 tons/in^; or from about 2,500 to 3,000 atmospheres, or kg/cm^; such high pressures require to be measured by special instruments called crusher gauges, depending on the amount of crushing of small copper cylinders by the pressure. 16 THE EQUALITY OF FLUID PRESSURE 11. The Equality of Fluid Pressure in, all directions. We may now repeat the Definition of a Fluid given in Maxwell's Theory of Heat, chap. V. ; Definition of a Fluid. " A fluid is a body the contiguous parts of which, when at rest, act on one another with a pressure which is perpendicular to the plane interface which separates those parts." From the definition of a Fluid we deduce the important Theorem. " The pressures in any two directions at a point of a fluid are equal." Let the plane of the paper be that of the two given directions, and draw an isosceles triangle whose sides are perpendicular to the two given directions respectively, and consider the equilibrium of a small triangular prism of fluid, of which the triangle is the cross section (fig 6). Let P, Q be the thrusts perpendicular to the sides and It that perpendicular to the base. Then since these three forces are in equilibrium, and since R makes equal angles with P and Q, therefore P and Q must be equal. But the faces on which P and Q act are also equal; therefore the pressures, or thrusts per unit area, on these faces are equal, which was to be proved. Generally for any scalene triangle ahc, the thrusts or forces P, Q, R acting through the middle points of the sides and perpendicular to the sides are in equilibrium if proportional to their respective sides, so that the pressure is the same on each face ; and a similar proof will hold if a tetrahedron or polyhedron of fluid is taken. If we consider the equilibrium of any portion of the fluid enclosed in a polyhedron when the pressure of the fluid is uniform, we are led to the theorem in Statics that IN ALL DIRECTIONS. 17 " Forces acting all inwards or all outwards through the centres of gravity of the faces of a polyhedron, each pro- portional to and perpendicular to the face on which it acts, are in equilibrium." iw The Fig. 6. Fig. 7. 12. The Transmissibility of Fluid Pressure. Hydraulic Press. Any additional pressure applied to the fluid will, if the fluid is an incompressible liquid, be transmitted equally to every point of the liquid : this principle of the " Transmissibility of Pressure " was enunciated by Pascal {Equilihre des liqueurs, 1653), and applied by him to the invention of The Hydraulic Press. This machine consists essentially of two communicating cylinders, filled with liquid, and closed by pistons (fig. 7); then if a thrust P lb is applied to one piston, of area B square feet, it will be balanced by the thrust W lb applied to the other piston of area A square feet such that P/5= W/A, the pressure of the liquid being supposed uniform and equal to P/B or W/A, Ib/ft^ ; and by making the ratio of A/B sufiiciently large, the mechanical advantage W/P can be increased to any desired amount. 18 THE HYDRAULIC PRESS. The difficulty of keeping the pistons tight against the leakage of the liquid prevented the practical application of Pascal's invention, until Bramah (in 1796) replaced the pistons by plungers (fig. 8) and made a water-tight joint by his invention of the cupped collar GO, pressed into U shape in cross section from an annular sheet of leather, which eifectually prevents the escape of the water. The applied thrust P can be applied, directly or by a lever, to the plunger of a force fwrnf, provided with a stuffing box, the invention of Sir Samuel Morland, 1675 ; and then repeated strokes of the pump will cause the thrust W exerted by the head of the ram to act through any required distance. In some portable forms, required for instance for punch- ing or rail bending, the pressure is produced and kept up by a plunger P which advances on a screw thread. For testing gauges Messrs. Schaffer and Budenberg employ an instrument consisting of a small ram working in a horizontal barrel full of water, the traverse of the ram being effected by its revolution in a screw. The gauge to be tested or graduated and the sta.ndard gauge are attached to the barrel and each registers the pressure of the water. The machine can even be used for testing vacuum gauges by turning the ram the reverse way, so as to diminish the pressure of the water below the atmospheric pressure. The Hydrostatic Bellows was devised by Pascal as a mere lecture experiment to illustrate his Principle of the Transmissibility of Pressure ; the large cylinder in fig. 7 is replaced by leather fastened to W, as in bellows, while the small cylinder is prolonged upwards by a pipe to a certain vertical height : and the thrust P is produced by THE HYDRAULIC PRESS. 19 the head of water poured in at the top of the pipe by a man standing on a ladder. In this way a small quantity of water poured in the pipe is shown lifting a consider- able weight W supported by the bellows, and leakage is \It=IZ] Fig. 8. avoided. For a diagram consult Ganot's Physique ; the instrument is of no practical use, except for Nasmyth's attempt to replace the Hydraulic Press by his patent Steel Mattress {Engineer, 23 xMay, 1890, p. 426). 20 ENERGT DUE TO PRESSURE. 13. The Principle of Virtual Velocities. Pascal's Principle of the Transmissibility of Pressure was applied by him to verify the Principle of Virtual Velocities in the case of an incompressible liquid, thus showing that a liquid can be made to take the place of a complicated system of levers, in transmitting and multi- plying a thrust. For taking a closed vessel, filled with incompressible liquid, and fitted with cylindrical openings closed by pistons, of areas A,B,G,... ft^ ; then if the pistons move inwards through distances a, &, c, ... feet respectively, the condition that the volume of liquid is unchanged requires that Aa+Bb+Gc+...=0, some of the quantities a,b,c,... being positive and some negative. But if P, Q, R,... denote the thrust in lb on the pistons, then P/A = Q/B = R/C=... = the uniform pressure in Ib/f t^ of the liquid, and therefore Pa + Qb+Rc+...=0, a verification of the Principle of Virtual Velocities. 14. The Energy of Liquid due to Pressure. We have supposed the fluid employed to be incom- pressible liquid : for if a compressible gas had been used to transmit power, part of the energy would be used up in compressing the gas, if used to transmit power; so that a gas would behave like a machine composed of elastic levers. But with an incompressible liquid the energy is entirely due to the pressure ; and if the pressure is p Ib/ft^, the energy of the liquid is p ft-lb per cubic foot (or p ft-lb/f t^). THE HYDROSTATIC PARADOX. 21 As a practical illustration of Pascal's Principle, applied to a closed vessel and a number of pistons, the Hydraulic Power Company of London supply water in mains for the purpose of lifts and domestic motors, at a pressure of 750 Ib/in^, or 108000 Ib/ft^, equivalent to an artificial head of 1728 ft, if a cubic foot of water is taken as weighing 1000 oz or 62-5 lb. This gives an energy of 1728 ft-lb per lb of water, or 17,280 ft-lb per gallon of 10 lb ; so that if water at this pressure is used at the rate of 2 gallons per minute, it furnishes energy at the rate of 34,560 ft-lb per minute, say one horse-power of 33,000 ft-lb per minute, allowing for friction in the pipe, estimated at a velocity of 5 f/s. With a consumption of 4 million gallons, or 640,000 cubic feet per 24 hours, this gives 640,000 X 108,000 = 6-912 x lO^" ft-lb per 24 hours, equivalent to nearly 1500 h.p. 15. The Hydrostatic Paradox. The fact that a thrust of P lb exerted on a piston of area A ft^, fitting into a vessel filled with incompressible liquid, produces a pressure p — P/A Ib/ft^ throughout the liquid and an energy of p f t-lb/ft^, or pv ft-lb in v f t^ was considered paradoxical by early writers on Hydro- statics; and numerous experiments, similar in principle to the Hydraulic Press, were "devised to exhibit this so-called Hydrostatic Paradox (Hon. R Boyle, Hydro- statical paradoxes made out of new experiments, for the most part physical and easy, 1666); and at the present time the Keeley Motor in America is a paradoxical instrument, devised with the intention of utilizing the hydrostatic energy of pressure. THE ACCUMULATOR. But this hydrostatic energy is unavailable for con- tinued use, unless replenished as fast as it is used, as by the force pump with the Hydraulic Press ; or unless the energy is stored up by the Accumulatoe (iigs. 9, 10), which consists of a vertical piston or ram B, loaded with weights W so as to produce the requisite pressure p. w F'S 9 Fig. 10. Then pv ft-lb of energy are stored up in the Accumu- lator when the ram is raised so as to displace v ft^ less of water ; that is, if the ram is raised vjA ft, where A is the cross section of the ram in ft^. In fig. 10, Mr. Tweddell's form of Accumulator, the area A must be taken as the horizontal cross section of the shoulder at DD. The Accumulator thus acts as the flywheel of Hydraulic Power ; so that an engine working continually and stor- ing up unused energy in the Accumulator can replace a larger engine working only occasionally. THE HTDRAULIG PRESS. 23 16. The Applications of the Hydraulic Press. Hydraulic Power is now used to a great extent on steamers, for hoisting, steering, and working the guns; an Accumulator however cannot well be carried afloat, on account of its great weight. On land Hydraulic Power is extensively used for cranes and lifts ; also on a large scale to replace steam hammers for forging steel by steady squeezing into shape, when a thrust up to 4000 tons is required, and on canals for locks. Fig. II. Fig. 12. On a small scale the Hydraulic Press is useful when applied to jacks, for lifting (fig. 11) or pulling (fig. 12), as manufactured by Tangyes of Birmingham; one great advantage of the machine being that the motion in either direction can be so easily controlled. The Bramah collar in these presses is seen to be replaced by a cupped piece of leather, pressed into shape from a circular sheet. 24 THE HYDRAULIC PRESS. An application to a certain form o£ weighing machine (Duckham's) may be mentioned here, consisting of a com- bination of a Bourdon Gauge and of a small Hydraulic Press, suspended from the chain of a crane (fig. 13). The pressure of the water in the Press is read off on the Bourdon Gauge, graduated so as to show the weight of the body suspended from the ram of the Press. Fig. 13. Fig. 14. In fig. 8 the Hydraulic Press is shown as employed for making elongated rifle bullets or lead pipe : a cylindrical wire or tube of lead, in a semi-molten state, is squeezed out through the hole in the fixed plunger D, which fills up the cavity in the ram of the Hydraulic Press, as the ram rises ; the length of wire or pipe formed will be to the length of the stroke of the ram as the cross section of the lead cavity to the cross section of the wire or pipe. The hydrostatic pressure in the molten lead is intensi- fied over the pressure in the water of the Press in the ratio of the cross section of the ram to the cross section of the lead cavity. THE AMAGAT GAUGE. 25 " Pressure Intensifying Apparatus " for other purposes, such as ri vetting and pressing cotton, applied in a similar manner to the Hydraulic Press, is described by R. H. Tweddell in the Proc. Inst. Mech. Engineers, 1872, 1878. For a general description of the applications of Hydraulic Power and of the Hydraulic Press, the reader is referred to the treatises on Hydraulic Machinery by Prof. Robinson and Mr. F. Colyer. 17. The Amagat Gauge. In this gauge (fig. 14) devised for measuring great pressures, the principle is the reverse of the Hydraulic Press, or Pressure Intensifying Apparatus ; in the Amagat Gauge {manometre a pistons libres) " unequal pressures act on unequal areas, producing equal thrusts " ; so that a pressure p(lb/m^) acting over an area a = ^7rd^in^ is measured by a balancing pressure q acting over an area A=lTrD^; and then pa = qA, or p = qA/a = qD^jd^. {Nature, 21 Feb., 1890 ; Challenger Scientific Reports on the Compressibility of Water, by Prof. P. G. Tait.) To allow for the friction in this gauge, or generally in a Hydraulic Press, suppose the collar is h inches high, and that /x is the coefficient of friction between the leather and the metal ; then for a pressure of q Vajin? the total normal thrust between the leather and the metal is -irDhq, and the frictional resistance to motion is fXTrDhq pounds. This gauge might be usefully employed either to test or even to replace the Crusher gauges used in artillery for measuring powder pressures ; a mechanical fit of the pistons, if made long and provided with cannelures, is found to oifer a sufficient frictional resistance to the leakage of the fluid, so that cupped leather or packing may be dispensed with. 26 EXAMPLES. Examples. (1) What must be the diameter of a safety valve, the weight at the end of the lever being 60 lb, and its distance from the fulcrum 30 in, the weight of the lever 7 lb and its C.G. at 16 in from the fulcrum, the weight of the valve 3 lb and its C.G. at 3 in from the fulcrum, for the valve to blow off at 701b/in2? Find also the leverage of the weight to allow the steam to blow off at 50 Ib/in^. (2) In a hydraulic press a thrust of 20 lb is applied at the end of a lever at 6 ft from the fulcrvim, actu- ating the plunger of the force-pump which moves in a line 1 ft from the fulcrum ; the plunger is in diameter 1 in and the ram is 10 in; find the thrust in tons exerted by the ram. (3) The plunger of a force pump is 10 in (8f) diameter, the length of the stroke is 42 in (30), and the pressure of the water acted upon is 50 Ib/in^. Find the number of ft-lb and ft-tons of work performed in each stroke. (4) The ram of a hydraulic accumulator is 10 ins in diameter, determine the load in tons requisite for a pressure of 700 Ib/in^. Find the fall in the ram in 1 minute, if water is not being supplied, and the water is working an engine of 9 h.p. (5) Give sketches and describe the construction of a hydraulic crane. Estimate the volume of ram necessary if a weight of 5 tons is to be lifted 20 ft, the water pressure being 700 Ib/in^ and efEciency of machine J. SURFACES OF EQUAL PRESSURE. 27 (6) The hydraulic lifts used in the construction of the Forth Bridge had a diameter of 14 inches and a range of 12 inches; the water was supplied at a pressure of 35 cwt/in^, and the lift took 5 hours. Find the lifting force of the ram and its rate of working in terms of a horse-power. (7) Prove that in consequence of the friction of the collar, the efficiency of the hydraulic press is reduced to 1 — 4/xn, where n denotes the ratio of the height of the collar to its diameter. 18. Theorem. " In a fluid at rest under gravity, the pressure at any two points in the same horizontal plane is the same ; in other words, the surfaces of equal pressure are horizontal planes." Suppose A and B are any two points in the same hori- zontal plane ; draw two horizontal planes a short distance apart, one above and the other below AB ; and consider the equilibrium of the stratum of fluid between these horizontal planes (fig. 15). Draw the two vertical planes through A and B per- pendicular to AB, and two vertical planes parallel to AB on each side of AB a short distance apart ; and consider the equilibrium of the prism of the fluid stratum cut out by these vertical planes. The fluid pressures being normal to the faces of the prism, and the weight acting vertically downwards, the conditions of equilibrium require the thrusts on the faces perpendicular to AB to be equal; and the faces also being equal, the pressures at A and B are equal. A similar proof holds when the prism is replaced by any thin cylinder on AB as axis, with ends at A and B perpendicular to the axis AB. 28 THE FREE SURFACE If ^C is drawn horizontally and perpendicular to AB, a similar proof shows that the pressures at A and G are equal ; as also if AC is drawn in any horizontal direction ; and therefore the pressure is the same at all points in the horizontal plane ABC; or in other words — "The surfaces of equal pressure in a fluid at rest under gravity are horizontal planes.'' Fig. 16. 19. If the fluid is a liquid, it can have a free surface, without diffusing itself, as a gas would; and this free surface, being a surface of zero pressure or more generally of uniform atmospheric pressure, will also be a surface of equal pressure, and therefore a horizontal plane. Hence the Theorem : " The free surface of a liquid at rest under gravity is a horizontal plane." The theorem can be proved experimentally with great accuracy by noticing that the image by reflexion in the surface of the liquid of a plumb line is straight with the line itself, and not broken; and this proves that the sur- face is perpendicular to the vertical or plumb line, which is the definition of the horizontal plane; also by the accuracy experienced in the use of the mercurial horizon in Astronomy and Surveying. Suppose it was possible for the free surface to be changed into a different form; for example -into a series IS A HORIZONTAL PLANE. 29 of waves at rest, like the hills and dales of dry land, or the surface of the Mer de Glace in Switzerland. An inclined plane PQ (fig, 16) could then be drawn, cutting off the top of a wave, and the stress across PQ being normal from the definition of a fluid, the plane PQ behaves like a smooth plane, and the top of the wave would begin to slide down PQ, and equilibrium would be destroyed. Thus the waves could not be at rest, but would move, as we see realized in nature. These are matters of common observation, as dis- tinguishing characteristics between a solid and a liquid ; as for instance between land and water. The surface of the land has hills and valleys, but the surface of water is a horizontal plane. A road or railway has inclines, but a canal is a level road, and locks are required for a change of level. 20. We have supposed in the preceding that no distri- buted forces, such as those due to gravity, are acting throughout the fluid; and thus the pressure in the fluid will be uniform, and the same in all directions ; and to prove this theorem we may consider the equilibrium of any finite portion of the fluid, in the form of a prism, tetra- hedron, or polyhedron. When distributed forces, gravity for instance, act throughout the fluid, the pressure will not be uniform but will vary from one point to another. Yet the same theorem that " the pressure of a fluid is the same in all directions about a point" can still be established in exactly the same manner, by taking a prism or tetrahedron and making it indefinitely small ; then the distributed forces, which are proportional to the 30 THE DISTRIBUTION OF PRESSURE weight or volume of the contained fluid, are indefinitely- small compared with the thrusts on the faces, which are proportional to the areas of the faces, and may therefore be neglected ; and the proof therefore proceeds as before. In determining the pressure at any point of a fluid at rest, the preceding theorem shows us that we need deter- mine it in one direction only, say in the horizontal or vertical direction when the fluid is at rest under gravity ; and this we shall now proceed to investigate. 21. Theorem. " The pressure in a homogeneous liquid at rest under gravity increases uniformly with the depth." Let Pq denote the pressure, in Ib/ft^, at any point of a horizontal plane, and p the pressure at the horizontal plane z feet lower down in the liquid ; and let w denote the weight in lb of a cubic foot of the liquid ; w then measures the density or heaviness of the liquid. Draw any vertical cylinder standing on a base of A square feet, and consider the equilibrium of the liquid filling the part of this cylinder cut off" by the two hori- zontal planes (fig. 17) ; the liquid is acted upon by a vertical downward thrust p^A lb on its upper end, by the vertical downward force wzA lb of its gravity, by a vertical upward thrust pA lb on the lower end ; and by the thrust of the liquid on the vertical surface of the cylinder. This last force contributes nothing to the support of the liquid ; so that resolving vertically, pA=p^A+'wzA, or p= p^ +WZ, which gives- the pressure at any point, and proves that it increases uniformly with the depth. IN A HOMOGENEOUS LIQUID. 31. If z ft is the depth below the free surface, then |3q denotes the atmospheric pressure on the surface ; and if this atmospheric pressure is absent, then p = wz, obtained as before from the consideration of the equi- librium of a cylinder of liquid with zero pressure at the upper end. Fig. 17. Fig. 18. To verify this experimentally, take a glass cylinder, with the lower edge ground smooth and greased, and a metal disc of given weight and thickness and of the same diameter as the cylinder, placed on the bottom of the cylinder so as to fit watertight, and held in position by means of a thread .(fig- 18). On submerging the cylinder vertically in liquid, it will be found that the thread may be left slack and the metal disc will be supported by the pressure of the liquid, when the depth of the bottom of the disc is to the thick- ness of the disc in a ratio equal to or greater than the density of the disc to the density of the liquid; or algebraically, if w denotes the density of the liquid, w' of the metal disc, and a the thickness of the disc, iv'aA=w2A, or z/a^w'kv. 32 PRESSURE IN A LIQUID. The effect of the atmospheric pressure will not sensibly modify the result, provided the thickness of the glass is inconsiderable; however, in the general case, with atmo- spheric pressure f^, and A denoting the area of the disc, B and C the external and internal horizontal sections of the glass cylinder, then w'aA = (pg + wz)A — (Pq + WZ — wa)(A — B) ^j\G, reducing, when A=B,io w'aA = wzA +p^{A — C). In the words of one of Boyle's Hydrostatical Paradoxes, " a solid body as ponderous as any yet known (that is 20 times denser than water, such as gold or platinum), though near the top of the water, can be supported by the upward thrust of the water." (Cotes, Hydrostatical and Pneumatical Lectures, p. 14.) If we suppose the atmospheric p^ is the pressure due to an increase of depth h in the liquid, then Po = wh, and p = w{h+z); so that now the pressure in the liquid is the same as if the free surface of zero pressure was at a height of h feet above the horizontal plane where the pressure Pd — wh. Again p^ might be the pressure due to liquid of depth h' and density w', so that 23o = w'h', and p = w'/i' + wz. So also for any number of superincumbent fluids which do not mix ; their surfaces of separation must be horizontal planes, for instance with air or steam on water, water on mercury, and oil on water, etc. THE HEAD OF A LIQUID. 33 22. The He-ad of Water or Liquid. The pressure wz at a depth z ft in liquid is called the pressure due to a head of z feet of the liquid. Thus a head of z feet of water, of density or heaviness w Ib/f t^ produces a pressure of wz Ib/f t^ or wz—lH Ib/in^ ; and a head of s inches of water produces a pressure of wz -7-1728 Ib/in^ ; and on the average, i« = 62-4. In round numbers a cubic foot of water weighs 1000 oz, and then it;=1000-;-16 = 62-5. In the Metric System, taking a cubic metre of water as weighing a tonne of 1000 kilogrammes, or a cubic deci- metre as weighing a kilogramme, or a cubic centimetre as weighing a gramme, a head of z metres of water gives a pressure of z tonnes per square metre (t/m^) or 1000 z kg/m^, or z/10 kg/cm^, or 100 grammes per square centi- metre (g/cm^), and a head of z centimetres of water gives a pressure of z g/cm^ ; thus a great simplification in prac- tical calculations is introduced by the Metric System of Units. The pressure of the atmosphere, as measured by the barometer, was taken in §8 as about 14f Ib/in^, or 2112 pounds (say 19 cwt, or nearly a ton) per square foot; with Metric Units the atmosphere was taken as one kg/cm^, or 10 t/m^) ; and an atmosphere is thus due, in round numbers, to a head of 30 inches or 76 centimetres of mercury, of specific gravity 136 ; a head of 10 metres or 33 to 34 feet of water ; or a head of 26,400 feet or 5 miles, or 8500 metres of homogeneous air of normal density, occupying about 12-5 ft^ to the lb, or 754 cm^ to the g, or 0-764 m^ to the kg. Any discrepancy in these results is due to taking the nearest round number in each system of units. 34 THE CORNISH PUMPING ENGINE. Regnault worked with a standard barometric height of 76 cm of mercury, and we may call this pressure due to a head of 76 cm of mercury a Regnault atmosphere ; but it is more convenient to take 75 cm ; thus a head of 300 m of mercury, in a tube up the Eiffel Tower, gives a pressure of 400 atmospheres. The density of sea water is generally taken as 64 Ib/ft^, so that an atmosphere of 14f Ib/in^ is equivalent to a head of 33 ft of sea water ; thus a diver at a depth in the sea of 27 J fathoms or 165 ft experiences a pressure of 5 atmospheres over the atmospheric pressure, in all a pressure of 6 atmospheres or 88 Ib/in^. In the previous discussions of the Hydraulic Press and Machines working by the Transmission of Pressure we supposed the pressure uniform and neglected the varia- tions due to gravity and difference of level ; but these variations are so slight compared with the great pressures employed as to be practically insensible. Thus a pressure of 7501b/in^ is due to a head of 1728 feet of water, compared with which an alteration of 10 feet, or even 100 feet, is insensible. 23. The Cornish Pumping Engine. Suppose M lb is added to W in fig. 7, the equilibrium is destroyed : the piston A will descend say x feet, and the piston B will be raised y feet, such that Ax = By; and now the pressure under the piston A will become (M+W)/A\h/it\ while under the piston B it will still be P/B Ib/f t^ ; and the difference between those pressures being due to a head 0^ x + y feet of the liquid, wix + y) = (M+ W)IA - P/B = Ml A, A LIQUID MAINTAINS ITS LEVEL. 35 since W/A=P/B, when P and W balance at the same level ; so that MB ^~wA{A+By M y~w{A+B)' This principle is employed in the Cornish Pumping Engine; the plunger or piston A of the pump at the bottom of the mine is weighted by M sufficiently to raise the column of water and B to the surface of the ground ; the action of the steam being employed merely to raise the piston A and the weight M at the end of a stroke so as to make the next stroke. 24. A Liquid maintains its Level. By alternate horizontal and vertical steps of appropriate magnitude, we can make the preceding theorems apply to homogeneous liquid contained in a vessel of any irregular shape, so as to be independent of the form of the containing vessel ; and thus we prove that the sur- faces of equal pressure are horizontal planes and that the pressure increases uniformly with the depth, even when the liquid is divided up into irregular channels, as in water mains; and that if left to itself the water will regain its original level, the principle applied in waterworks. It was not from ignorance of these hydrostatical principles, but of the art of making strong waterpipes that the Eomans constructed high stone aqueducts to carry water to cities on the level ; where nowadays iron pipes would be employed, laid in the ground, at great economy and with the additional advantage of escaping long continued frost. 36 TEE COMMON SURFACE OF TWO LIQUIDS. Coming to more recent times, the principle that "liquids maintain their level" was doubted by our engineers when they reported a diiference of level of 32J feet between the Mediterranean and the Red Sea, as making the Suez Canal impracticable. (Gomptes Mendus, 1858; British Association Repoii, 1875.) The statements that "a Liquid maintains its Level" but that " a Solid does not maintain its Level " may be taken as the fundamental distinguishing characteristics of a Liquid and a Solid ; it is proved experimentally by noticing that the isolated portions of the free surface of a homogeneous liquid, filling a number of communi- cating vessels of arbitrary shape, all form portions of the same horizontal plane. The principle is employed not only in the design of waterworks, but also in the theory of levelling instru- ments, and of the gauge glass of a boiler. 25. Theorem. " The common surface of two liquids of different densities, which do not mix, is a horizontal plane, when at rest under gravity." s.c. Fig. 19. Draw any horizontal planes AB, CD in the upper and lower liquids z feet apart and draw vertical lines AG, BD cutting the surface of separation of the liquids in P and Q ; and let AP = x,BQ = y; so that (fig. 19) PG=z-x,QD = z-y. STABILITY OF EQUILIBRIUM. 37 Let w denote the density or heaviness of the upper liquid, w' of the lower. Then p^ denoting the pressure at the level AB and p at the level CD, by considering the equilibrium of small vertical prisms or cylinders on A C and BB as axes, p —Pd = wx + w'{z — x), p —pa = ivy + wXz — y) ; so that, by subtraction, {w' — vj)(y — x) = 0. Now, since by supposition w — w' is not zero, therefore y — x = 0, or y = x; and this proves that P and Q are in the same level, at E and F; and generally that the common surface EF is a horizontal plane. 26. The Stability of Fquilibrium of Superincwmhent Liquids. K a number of liquids of different densities, such as mercury, water, and oil, are poured into a vessel, they will come to rest with their common surfaces horizontal planes ; and the stability of the equilibrium requires that the densities of the liquids should increase as we go down. For suppose a portion of two liquids to be isolated in a thin uniform endless tube APGDQB, EF representing the original level surface of separation ; and suppose P and Q the surfaces of separation when the liquid in the tube is slightly displaced, and kept in this position by a stopcock s.c. in CD or AB. Supposing the pressures at A and B equal, the pressure at G will exceed the pressure at B by wx + w\z — D; and the slope a of FB to the horizon cannot exceed e, the angle of repose. To determine the horizontal thrust Q on DE, suppose the wall BE to yield horizontally a slight distance, and in consequence the substance to crack along a plane of cleavage EM or EN, making an angle Q with the vertical wall BE. Fig. 21. We suppose that the prism of material BEM or BENF begins to slide down the plane EM or EN; and then as before, if W denotes the weight in lb of the material in the prism, Q=W cot(0 + e). If the plane EM meets the talus BF in M, and we put BE = a, then „,. , , „ sin 6 cos a "^ cos(0+a) , _ , , „ sin cos a cos(6 + e) and Q = hwLa^ yw-, — %•,/), x ' ^ ^ cos(0 + a)sin(0 + e) T • J. r, 1 7 o sin cose reducmgto Q=^wto^ ^■^^g_^^^ sin(0-e) | ■^sin(0 + e)J if the slope of the talus is the angle of repose, or = ^wla^ \ 48 THRUST OF A GRANULAR SUBSTANCE. As increases from zero, the thrust Q also increases from zero; and when Q + e = ^Tr, or the plane EM is parallel to the talus DF, Q = ^wla^cos^e, the same as for liquid of density w cos^e. But this implies that the wall DE is surcharged to an infinite height; but if surcharged to a finite height b, then when the plane of cleavage EN meets the horizontal level surface in N, W =wl{i(a + lf ta.n0-^¥cot a}, and the corresponding value of Q will become a maximum for a value of 6 depending not only on e, but also on the ratio of b to a. The determination of this maximum value must be deferred ; but now it is important to notice that P and Q are not equal, the difference between them being taken up by the frictional resistance of the ground BE. *32. The Thrust due to an Aggregation of Cylindrical Particles or of Spherules. An exact Theory of Earth Pressure can be constructed if we suppose the substance which is held up by a retain- ing wall to be composed of individual particles or atoms of cylindrical form, such as canisters, pipes, barrels, or cylindrical projectiles, regularly stacked; or else to be composed of spherules, such as lead shot, billiard balls, or spherical shot and shell, piled in regular order, as common formerly in forts and arsenals. It will be necessary to begin by supposing that the lowest layer of cylinders or spheres is imbedded in the ground ; as otherwise a wedging action takes place, due to the slightest variation of level, and the problem is tp a certain extent indeterminate, as in the preceding article on Earth Pressure. AGGREGATION OF CYLINDERS. 49 Now in the case of cylindrical bodies, regularly packed as close as possible (fig. 22), the slope of the talus BF is 60° ; and if EN is drawn through E the foot of the retaining wall DE parallel to the talus BF, the thrust between the cylinders across the plane EN will also make an angle of 60° with the horizon ; so that consider- ing the equilibrium of BENF, of weight W, the thrust Q on the retaining wall BE is given by Q=Trcot60°. Fig. 22. Also, if w denotes the apparent heaviness of the sub- stance, measured in Ib/ft^, W=iwl(h? - b^)coi 60° = ^vl{^a^ + ab)cot 60°, so that Q = wl(^a^ + ab)cot^QO'' = lwl(-ha^ + ab), the same as the hydrostatic thrust of liquid, of heaviness ^w, on the portion BE of a vertical wall, of which the top edge B is submerged to a depth b in the liquid. If the cylindrical particles are of diameter d, and composed of solid metal of density p, then since the triangular prism formed by the axes of three adjacent cylinders is of cross section ly/'3 d^S, vertical planes of contact come into existence ; and alternate vertical columns descend, so that tujp^^Trd/x. The thrust Q will become very large when the cylinders are nearly in square order. The thrust Q is theoretically infinite when the cylinders are in square order : but this arrangement being unstable, a seismic rearrangement takes place, and the original tri- angular order is regained, except that the talus now appears stepped. Suppose the wall AB or DE to yield horizontally a slight distance ; the cylinders in ABC or DENF will roll and wedge down along planes of cleavage BC or EN. The lowest layer of cylinders being imbedded, no further motion is possible ; but if they were free to roll sideways, a molecular rearrangement would take place, and the cylinders would appear wedged against the walls AB and DE in close order, except along two planes of cleavage. OF CYLINDERS AND SPHERES. 51 * S3. When the substance is composed of spherules or spherical atoms, we suppose the lowest horizontal stratum is embedded in the ground and arranged (i.) in square order : (ii.) in triangular order. In (i.) the spheres in the talus DF are seen in triangular order, in a plane having the slope a of the face of a regular octahedron ; and therefore '°'" = d^^" = 73' ^i°« = Vf. tana = V2; while the thrust across the parallel plane EN is also inclined at an angle a to the horizon ; so that Q=Trcota = wl{^a'^ + a6)cot^a = ^wl{^a? + ah), the same as for liquid of density Jw. In (ii.) the internal arrangement is essentially the same as in (i.), but now the talus BF may show the spheres arranged, either (ii., a) in triangular order, or (ii., h) in square order. In case (ii., a) the slope a of the talus DF is the slope of a face of a regular tetrahedron on a horizontal base, so that cosa = ^, sin a = 1^2, tan a = 2^2: while the reaction across the plane EN, parallel to DF, will be inclined to the horizon at the angle P, the slope of the edge of the regular tetahedron, so that cos^ = ;^, sin/3=Vl. tan/3 = V2- In case (ii., h) the values of a and ^ are inverted ; so that, in each case, (ii., a) and (ii., h), Q=Wcotacoi^ = lwl{la^-it-ab), the same as for liquid of heaviness \'w. 52 THRUST EXERTED BY AN If p denotes the density, real or apparent, of a single spherule, while w denotes the apparent density of an aggregation of a large number of spherules, we shall find p ^"J^ For if we suppose the horizontal layers in square order, and we take a volume consisting of a very large number n^ of spheres, standing on a square base whose side is of length nd, then the height will be ^J-lnd, and the volume \ijiin?d?; while the volume occupied by the f^ spheres will be ^ttw^cZ^ ; and therefore W _ TT So also if the horizontal layers are in triangular order, the length of the volume being nd, the breadth will be \J'^nd, and the height \^^nd ; so that the volume will be \iji'r^d^, as before. When the number of spheres is limited, the effect of the irregularity of the arrangement on the outside of the volume makes itself felt. Thus 1000 spheres, each one inch in diameter, can be packed in cubical order in a cubical box, the interior of which is 10 inches long each way ; but other arrangements are possible by which a larger number of spheres can be packed in the box ; the discovery of these arrangements is left as an exercise for the student. (Cosmos, Sep. 1887.) This problem of the packing of spheres is known of old as that of "the thirsty intelligent raven"; the story is given by Pliny, Plutarch, and ^lian ; it is quoted by Mr. W. Walton in the Q. J. M., vol. ix., p. 79, in the following form as due to Leslie Ellis : — AGGREGATION OF SPHERES. 53 " A thirsty raven flew to a pitcher and found there was water in it but so near the bottom that he could not reach it. Seeing however plenty of equal spherical pebbles near the place, he cast them one by one into the pitcher, and thus by degrees raised the water up to the very brim and satisfied his thirst. Prove that the volume of the water must have been to that of the pitcher in a ratio of 3^2 — tt to 3^2, or more." If the lead shot were melted, the density would become changed from w to p, and the hydrostatic thrust would be increased in the ratio of 3^2 to ir for the same apparent head of the substance. Here again, in a substance composed of spherules, a complicated state of wedging action would take place if the lowest stratum of spheres were not imbedded, but were free to roll on a smooth horizontal floor, especially if the walls were to yield slightly. (Osborne Reynolds On the JDilatancy of Media cotu- posed of Rigid Particles in Contact. Phil. Mag., Dec. 1885. Rankine, Stability of Loose Earth ; Phil. Trans., 1867. Woven Wire, Segregation, and Spherical Packing. Engineering, Jime 1893.) Various simple illustrations of the thrust of spheres, leading to elegant statical theorems of the application of the Principle of Virtual Velocities, can be constructed with billiard balls, of given diameter d, placed in open canisters of various diameters D, so that thfe spheres arrange themselves in horizontal layers (i.) B < 2d, singly; (ii.) 2cZ-^ N V "^ "-"-\ \7* 1 — A^ M \P /- — ^^»- z ^ '- — Fig. 26. 39. The student of Dynamics will notice that the c.p. of the area A coincides with the centre of oscillation, or the centre of percussion of the area, with respect to the horizontal axis 00' in which the surface is cut by the plane of the area A ; also that it coincides with the C.G. of the lamina bounded by the perimeter of A, loaded so that the superficial density is proportional to the depth below the free surface of the liquid. A bullet striking at the C.P. (centre of percussion) K of a target formed of rigid sheet metal of uniform thick- ness, bounded by the perimeter of the area A, will cause 66 CENTRE OF PRESSURE the target to begin rotating about the axis 00', and if suspended from the axis 00', there will be no impulsive action on the axis; so that 00' is called the axis of spontaneous rotation with respect to the c.p. K. Afterwards the target will swing about 00' like a plummet at L, suspended from 0' by a thread so as to hang at the level of K. Returning to the plane area A in the liquid, as we turn the area about the axis 00' , the position of Kin the plane will not alter ; even when coincident with the sur- face, for the evanescent superincumbent film of liquid will vary in thickness as the distance from 00'. The effect of taking into account the atmospheric pressure on the surface of the liquid is equivalent to supposing that the plane area A is sunk without rotation in the liquid, so that G is submerged to an additional depth H, the head of liquid equivalent to the atmospheric pressure ; h now becomes changed to h + H, but F and D remain constant ; so that - F ])_ ^~h+W y~AQi+Hy , KH y D . , and ^PTr, = = = -rr5> a constant, so that K will describe a straight line through in the plane of A. 40. But if the area A is turned about G in its own plane, K will describe a curve in the plane of A. Suppose the plane turned through the angle Q, then the new depth of the element LA is h + ysinQ + zcoBQ; so that y, z denoting the coordinates of the new c.p. with respect to the old axes Gy, Gz, fixed in the area A, OF A PLANE AREA. 67 R = 'Ew(h + ysm9 + z cos 0) A J. = whA , and By = 1,w{h + ysin6 + z cos 6)AA .y = wAkghin 6 + wD cos Q, B,z = i:,wQi+ysxaQ+z cos Q)A.A .z = wD sin 6+wA ky^cos 6, Aky^ and Ak^^ now denoting the moments of inertia of the area A about Gy and Gz. These considerations show that the c.p. describes an ellipse in the plane A ; and that the axes of this ellipse are those for which B vanishes ; so that, changing to these axes, hy = /c/sin 9, hz = k/cos 6 ; which shows that the c.p. K is the antipole of the line 00' with respect to the ellipse the Tnomental ellipse or swing conic of the area A at G. Fig. 27. If the momental ellipse becomes a circle, as for instance in a square plate, or in a plate bounded by a regular poly- gon, the c.p. is always vertically below the CG. of the area, at a depth inversely proportional to the depth of the c.G. 68 EXAMPLES ON In fig. 27 the position of the CP. of some simple figures is indicated, whence F for the area about a horizontal axis through its C.G. can be inferred, and vice versa. As drawn in fig. 27, GK=^^OG in the rectangle or parallelogram, having a pair of sides horizontal ; and GK = ^00 in the triangles, having one side horizontal, and in the circle or ellipse. Thence we infer that, about a horizontal axis through G, ]F= 252-552, the true weight (in vacuo) of a cubic inch of water. 108 EXAMPLES. Examples. (1) Prove that an inch of rain over an acre weighs about 100 tons. (2) Find the weight of water in a lake whose area is 5 acres and average depth 10| feet, and also the number of gallons it contains, supposing a cubic foot of water to weigh 1000 ounces, and a gallon to contain 277"25 cubic inches'. (3) Taking the earth as a sphere, whose girth is 40,000 kilometres, or 360 x 60 nautical miles of 6080 feet, and of mean s.G. 5-576, prove that the weight is about 6-027 x lO'^i metric tonnes, or 5-932 x lO^i British tons. (4) Show how the mean transverse section of a piece of fine wire may be determined by weighing it first in vacuo and then in water. If the wire is ten yards long, find the greatest error in determining the mean transverse section if the weights are determined accurately to tenths of a grain and the weight of a cubic inch of water is 252-5 grains. (5) Investigate the conditions of equilibrium of a body floating partially immersed in a fluid. An iron shell one-eighth of an inch thick floats half immersed in water, the specific gravity of iron being 8 ; find the diameter of the shell. (6) Prove that the calibre d, in inches, of an n bore gun, is given by the relation (Z = log -1(0-2226 -J log 71); given that n spherical lead bullets, of s.G. 11-4, and diameter d inches, weigh one pound. EXAMPLES. 109 Hence prove that a 12 and 20 bore are 0729 and 0'615 inches in calibre. (7) The area of the base of a vessel with vertical sides containing water is 85 cm^. Find how much the pressure at each point of the base is increased if 1000 grammes of lead, specific gravity 11 •4, are suspended in the water by a thread. (8) Given the S.G. s of ice and s' of sea-water, prove that the volume and weight of an iceberg, of which V cubic feet is seen above the water, is Fs7(s'-s) fts aud DVss'l{s'-s) lb. (9) A body floats in a fluid of s.G. s with as much of its volume out of the fluid as would be immersed in a second fluid of S.G. s, if it floated in that fluid. Prove that the s.G. of the body is ss7(s + s'). (10) Prove -that, in selling iron of S.G. 7'8 by weight, in air of s.G. '00128, with a balance and standard brass weights of s.G. 8'4<, what is sold as 100,000 tons of iron is really about 1'2 tons more. (11) A piece of copper of s.G. 1"85 weighs 887 grains in water and 910 grains in alcohol ; required the S.G. of the alcohol. (12) Two cubic feet of cork, of S.G. 0'24, is kept below water by a rope fastened to the bottom. Prove that the tension of the rope is 95 pounds. (13) Prove that, if volumes A and B of two different substances equilibrate in vacuo, aud volumes A' and B' equilibrate when submerged in liquid, the densities of the substances and of the liquid are as A'_5' A'_£' A^_F A A' B B '■ A B' 110 THE HYDROMETER. (14) If the S.G. of the gold supplied by Hiero to the goldsmith was 19, and if the S.G. of the crown as debased was found to be 16, and the S.G. of the silver employed for this purpose was 11 ; then show that 33 parts in 128, or rather more than one-fourth part by weight, was silver. (15) The crown used by the Stuart sovereigns, which was destroyed in the seventeenth century, is said to have been of pure gold (s.G. 19'2) and to have weighed 7^ lbs. How much would it have weighed in water ? If it had been of alloy, partly silver (s.G. lO'S) and partly gold, and had weighed 7^\ lbs. in water, how much of each metal would it have contained ? 63. The Hydrometer. For determining the density and S.G. of a liquid, an instrument called a Hydrometer (French ardomUre) is employed, consisting of a bulb and a uniform stem. Hydrometers are of two kinds — (i.) the common or Sikes's hydrometer, of variable immersion but fixed weight, for determining the density of a liquid (fig. 36) ; (ii.) the Fahrenheit or Nicholson hydrometer, of fixed displacement or immersion but of variable weight, which can be used for determining the density of a liquid, or of a small solid, or to determine the weight of a small solid (fig. 37). As these instruments are usually small, the inch and ounce are used as British units of length and weight, but better still the centimetre and gramme in questions relating to their use; and then densities will be given THE COMMON HYDROMETER. m in oz/in^, or with the metric units the densities and specific gravities will be the same, a great numerical simplification. In the common hydrometer of variable immersion, let W denote its weight, in oz or g, V its volume, in in^ or cm^ and a the cross section of the stem, in in^ or cm^. Then the liquid of smallest density w^ in which the hydrometer will not sink, but just fioat with its highest point A in the surface, is given by w,= WIV; and if, when placed in a liquid of density w, the hydro- meter floats immersed to the point M, then if AM=x, w=WI{V-ax); J.1, i. w V so that — = tt- • w^ V — aX If we put V/a = a, then a is the length J.0 of the stem which would have the total volume V; and now w _ a , Wg a — x' so that, if we represent the density w^ by the ordinate AC, and draw the hyperbola CP having as asymptotes the vertical side of the stem AM, and the horizontal line through at a depth AO = a below A, then to the same scale the ordinate MP will represent the density w of the liquid in which the hydrometer floats immersed up to M; and the marks of graduation M may be numbered so as to give, for instance, the density in kg per litre, or lb per gallon. 64. The hydrometer then, in its simplest form, may be considered as consisting of the rod AO, of weight W and uniform cross section a, ballasted so as to float upright. 1 1 2 GRAD UA TION OF THE STEM The sensibility of the instrument, as measured by the distance between the graduations, is proportional to a or V/a, and therefore inversely proportional to the sectional area ; so that the longer and thinner the rod AO is made, the greater its sensibility in indicating differences of density ; but as such an instrument would be inconveniently long, the lower part of the rod is replaced by the bulb B, of equal volume and weight. With equal increments of density, starting from tw^ and A, the graduations on the stem proceed in harmonic progression, and become closer together. To graduate the instrument geometrically, we draw the horizontal and vertical lines AG and GE, AG re- presenting to scale the density w^; and now if AR represents any greater density w, and we draw the straight line OQR meeting GE in Q, then the horizontal line QM will cut the stem in the corresponding gradua- tion M, the point to which the hydrometer will sink in liquid of density w represented by AR. For if the vertical line RP and the horizontal line MQ meet in P, then w _AR_AR_OA w~ AG~'MQ''0M so that OM.MP^OA.AG, or rect. OP = rect. OG, and P therefore lies on the hyperbola GPD. Incidentally we notice that this construction gives the geometrical method of inserting a given number of har- monic means between two given quantities, OA and OL. 66. The greatest density which can be measured by the common hydrometer , is represented by LD, where L is the lowest division on the stem, just above the bulb B; and common hydrometers are of two kinds — OF THE COMMON HYDROMETER. 113 (i.) for heavy liquids (salts) denser than water, floating in water with A in the surface ; (ii.) for light liquids (spirits) less dense than water, floating in water with L in the surface. A C R Ci Ri Cz Fig. 37. Fig. 36. Suppose the readings of a conomon hydrometer are required for s.G.'s between 1 and s, and that I denotes the length of the stem AL ; then Y 1 -Yf f = s, or -' V—al s according as the hydrometer is required for salts or spirits, or as s is > or < 1 ; and then V-d 1 s giving the ratio of the volume of the bulb to the volume of the stem. &.H. H 114 SIKES'S HYDROMETER. Thus the marine hydrometer or salinometer is required to register s.G.'s ranging from 100 to 1'04 ; so that the volume of the bulb must be 25 times that of the stem. The S.G. of pure milk being 1-03125, the Lactometer requires a bulb of 32 times the volume of the stem. 66. Sikes's Hydrometer. To increase the range of the instrument, a series of weights W^, W^, ■■■, are provided of known volumes, which can be fixed on the stem, below the bulb B, or above at A ; this instrument is known as Sikes's hydrometer. It is convenient to make the volumes of these weights all equal, and to use the instrument with one of these weights always attached, so that the total volume of the hydrometer may be supposed to include the volume of the additional weight. Now when a weight W-^ is attached, the density curve becomes a similar hyperbola C^jD^, having the same asymptotes OA and OE; and it is convenient for con- tinuity of measurement to choose W-^ so that the initial ordinate AG^ of the new hyperbola is equal to the final ordinate LD of the former hyperbola; the substitution of the next weight TTg giving the new density hyperbola OgDg, in which AG^ = LD-^, and so on. Then, with OA = a, AL — l, a _ W+W ,_W+W^ _ Wn a-l W -W+W'~---~Wn.-,' so that the weights W, W+W^, W+W^, Tf + Fg, ..., are in G.P. ; and also the densities ^u^, w^, lu^, ... , at which the weights must be changed. In this way Sikes's hydrometer is an instrument capable of measuring densities over a considerable range, obviating the necessity of an inconveniently long stem. THE HYDROMETER OF FIXED IMMERSION. 115 67. But if the weights are of equal, density w, the addition of a weight Tfj will change the. centre of the hyperbola which represents the density from to 0-^, where and so on; and now when the weight is changed from W„ to Wn+-^, leaving ZZ)„ = J.C„+j, then lf+lf„+i = (F +^^)t«„. Eliminating w.^, W+Wn _ w(V-al)+Wr, _ w(V-al)-W W+W„+^ wV +lf„+i wF-F ' so that ^+^" =fl ^r, ■W V wV-WJ ' and therefore the weights W, W+ W^ W+ W^, ■■■ are in G.P., as before ; to which this case reduces by supposing the volume of the weights zero, or their density w infinite. 68. In a Fahrenheit or Nicholson Hydrometer of con- stant displacement V, in^ or cm^ (fig. 37), the instrument is loaded so as to bring a fixed mark if in the stem down to the surface of the liquid ; and then if W is the weight of the hydrometer, ly^ the density of the liquid in which it floats immersed to M, and if W is the weight which must be placed in the upper scale A to bring the hydrometer down to the mark M in the liquid whose density w is required, tu _ W+ W w, W ' so that, for densities in a.p., the weights W are in A.p. also. 116 NICHOLSON'S HTDROMETER. 69. To determine the weight, say a; g, of a small body, the hydrometer is placed in a liquid, and in the scale A the weight TT^ g is observed which is required to sink the hydrometer to M; W-^ is then removed, and the body is placed in the upper scale A, and the additional weight W^ g required to be added in A to bring the hydrometer down to M is observed ; then or x=W-^— Tf 2- 70. To determine the s.G. of the body, the liquid must be pure distilled water ; and the weight x g having been determined as above, the body is placed in the lower cage G (added by Nicholson to Fahrenheit's hydrometer), being tied down if it tends to float up; and now the weight W^g is observed, which placed in the upper scale A brings the hydrometer down to M; then TFj — If g is the weight of the body in water, W^-W^ „ „ „ air; and therefore the S.G. is (§ 53) 71. When the density of the air is rigorously taken into account, and we denote by fj and Fthe volumes of air and water displaced by the hydrometer, by D the density of water, and by s, B, o-, and p the S.G.'s of the body, the weights, the liquid, and the air, then D{pU+crV)=W+wJ^-^ = If +x(i-g+Tf 3(1-1). THE SPEGIFIO GRA VITT BOTTLE. 117 Therefore x{l-P^ = {-W^-W^)i\-^, An aqueous spirit is said to be x per cent, over proof when 100 volumes of this spirit diluted with water yields 100 + a; volumes of proof spirit; and it is said to be X per cent, under proof when it contains 100 — a; volumes of proof spirit in 100 volumes. Let s denote the s.G. of an aqueous spirit, weighing 100 g, composed of a g of pure alcohol, of s.G. S, and 100 — a g of water. Then 100/s cm^ is the volume of the aqueous spirit, a/S and 100 — a cm^ the volumes of the constituent alcohol and water; so that as/S and (100 — «)« are the percentages by volume of alcohol and water required for making this spirit, or the volumes in cm^ of alcohol and water required to make 100 cm^ of spirit. SPECIFIC VOLUME. 127 79. The Specific Volume of a Substance. The specific volume (s.v.) of a homogeneous substance is the number of units of volume occupied by the unit of weight ; for instance the volume is ft^/lb, or the volume is m^/t, or m^kg, or is cm^/g ; the specific volume is thus the reciprocal of the density, it is sometimes called the rarity (whence the French word areometre), or room- age of the substance. With very light compressible elastic substances, such as air, oxygen, hydrogen and gases generally the specific volume is better remembered and used than the density, which would be expressed by a small decimal ; and it is practically more' easily observed and measured when ex- pansion takes place. Thus the s.v. of ordinary air is about 13(ft^/Ib) meaning that 13 cubic feet of air weigh a lb, and then the s.G. so that, in Metric units, the s.G. of air is 1-23 kg/m^ or .g/litre, and the s.v. is 811 cm^g, or 0'81 m^/kg (litres/g) ; while the s.v. of hydrogen is about 200 ft^lb, or 12'5 m^/kg, at ordinary atmospheric pressure and temperature ; and one litre of hydrogen thus weighs 0-08 g, or one m^ weighs 0-08 kg. For the stowage of cargo the ton is taken as the unit of weight, and the specific volume or roomage is the volume in ft^ of a ton ; thus sea-water occupies about 35 ft^ to the ton, fresh water about 35'84, say 36. Cargo stowing closer than 40 ft^ to the ton is reckoned and charged by dead weight, while cargo of greater roomage is charged by volume of measurement, reckon- ing 40 fV as a ton. 128 SPECIFIC VOLUME OF MIXTURES. A Table in an Appendix gives the roomage of different kinds oi" cargo; convenient approximate rules are given by (i) roomage in ft^/ton =36-f-s, (ii) heaviness in tons/(yard)^ = f s, where « denotes the S.G. of the substance. 80. Given the weights Wj, W^, ..., Wn, in lb or tons, of volumes F^, V^, ... , F,j, ft^, of substances of S.V. Wj, ^2 '^n: so that V, = v,i W„ V, = rj W,, ..., u, = rj W, ; tlien the average S.v. of the mixture is given by v=VIW, where W= W^+ W^+ . . . + TF„, the sum of the weights; and generally also v=v,+ r,+... + v,„ the sum of the volumes, supposing there is no change of volume ; and then W,+ W,+ ...+ Wn W^+W,_+... + W,, ~ ■ -EW' Thus, for example, atmospheric air of s.v. 13 being a mechanical mixture of oxygen of s.v. 11| and nitrogen of S.V. 13 J, it follows from the above that in a given quantity of air the weights of oxygen and nitrogen are as one to four, and the volumes as 7 to 32. For putting (1 = 13, ^1 = 111, v^ = l-3l in the equations we find ^1 = 1, Z:_l W^ 4' F2~32' THE LACTOMETER. 129 81. Since the s.v. is the reciprocal of the density, it follows that the s.v. of a mixture of equal weights of different substances is the a.m. of the s.v.'s of the in- gredients; and the s.v. of a mixture of equal volumes is the H.M. of the s.v.'s of the ingredients, supposing no change of volume to take place. In the common hydrometer (iig. 36, p. 113), the hyper- bolic curves GP, OiP^ ... , representing graphically the densities, would become transformed into straight lines radiating from the centres 0, 0^, ... , of the hyperbolas, giving the corresponding specific volumes or rarities. 82. For a lactometer to give y the number of gallons of water to one gallon of milk, or reciprocally z the number of gallons of milk to one gallon of water, when a length X of the stem is shown above the mixture, then denoting by S and 1 the S.G. of pure milk and pure water and by s the S.G. of the mixture, yS+l = {y + l)s, or 8+z = {l+z)s, so that y = -Fi — ' 0— s s — 1 V a But by § 63, 8= V—aX a — x F ^ g V — al~a — l y J. xia — l) so chat y = -Tr — {■> a{i — x) a(l — x) ^ = -? TV x{a — i) and the curves for y and z are equal hyperbolas. G.H. I 1 30 ORA VIMETRIG DENSITY OF G UNPO WDEB. 83. The Gravimetric Density of Gunpowder. Artillerists employ both the specific gravity and the specific volume in measuring the density of the powder in the cartridge of a gun. The gravimetric density of the charge of powder is defi^ned to be the ratio of its weight to the weight of the water which would fill the chamber of the bore behind the projectile in the gun. The G.D. (gravimetric density) is therefore the s.G. of the powder, or powder gases when fired, which fill this powder chamber. The specific volume of the powder charge, or its gases, is also given by the number of in^ occupied by a lb; and a lb of pure distilled water having a specific volume of 2773 in^ the G.D. is obtained by dividing 27'73 by the number of cubic inches allotted to each lb of powder; this is equivalent to taking I) = 62'3, the density of water at about 68° F. Thus a gun charge expressed by 2 0-84 means 75 lb of P^ powder, with 33 in^ of space per lb of powder, and a consequent G.D. of 27'7S-:-33 = 0'84. (Mackinlay, Text Book of Gunnery, 1887, p. 22.) According to §§ 32, 33 the G.D. of a charge of lead shot will be i7rV'2 = 0-7403 of the s.G. of lead ; and the G.D. of a charge of the new cordite powder, composed of cylindrical filaments, will be 1^^3 = 0-9067 of the S.G. of the substance of the cordite. GENERAL EXERCISES. 131 General Exercises on Chapter III. (1) The diameters of two globes are as 2:3, and their weights as 1 : 5 ; compare their specific gravities. (2) The weight of a vessel when empty is 3 oz; when filled with water, it is 9 oz; and when filled with olive oil, 8-49 oz ; required the S.G. of the oil. (3) A vessel filled with water weighs 5| oz, and when a piece of platinum weighing 29| oz is placed in it, and it is filled up with water, it weighs 33 oz; prove that the s.G. of the platinum is 19-5. (4) The weight of a piece of cork in air is f oz, the weight of a piece of lead in water is 6f oz, and the weight of the cork and lead together in water is 4-07 oz. Prove that the S.G. of the cork is 0-24. (5) A piece of metal weighing 36 lb in air, and 32 lb in water, is attached to a piece of wood whose weight is 30 lb, and then the compound body is found to weigh 12 lb in water. Prove that the S.G. of the wood is 0-6. (6) The s.G.'s of platinum, standard gold, and silver being respectively 21, 17'5, and 10'5, and the values of an ounce of each 30s, 80s, and 5s respectively ; prove that the value of a coin composed of platinum and silver, which is equal in weight and magnitude to a sovereign, is 6s od. (7) A solid, whose weight is 250 grains, weighs 147 in water, and 130 in another fiuid. Prove that the s.G. of the latter is 1'262. (8) A solid, whose weight is 60 grains, weighs 40 grains in water, and 30 grains in sulphuric acid; required the s.G. of the acid. 132 GENERAL EXERCISES ON (9) The S.G. of gold being 19-25, and of copper 8-9, what are the weights pf copper and gold respectively in a compound of these metals which weighs 800 grains in air, and 750 in water ? (10) A piece of gun-metal was found to weigh 1057-9 grains in air, and 934-8 grains in water ; find the proportions of copper and of tin in 100 lb of the metal, the s.G. of the copper being 8-788 and of tin 7-291. (11) A body immersed in a liquid is balanced by a weight P, to which it is attached by a thread passing over a fixed pulley; and when half immersed, is balanced in the same manner by a weight 2P- Prove that the densities of the body and liquid are as 3 to 2. (12) It is found on mixing 63 pints of sulphuric acid, whose s.G. is 1-82, with 24 pints of water, that 1 pint is lost by their mutual penetration ; find the S.G. of the compound. (13) A piece of gold immersed in a cylinder of water causes it to rise a inches ; a piece of silver of the same weight causes it to rise 6 inches; and a mixture of gold and silver of the same weight c inches ; prove that the gold and silver in the com- pound are by weight as h — c:c — a. (14) The S.G. of lead is 11-324 ; of cork is 0-24; of fir is 0-45 ; determine how much cork must be added to 60 lb of lead that the united bodies may weigh as much as an equal volume of fir. (15) The s.G.'s of pure gold and copper are 19-3 and 8-62; required the s.G. of standard gold, which is an alloy of 11 parts pure gold and one part copper. DENSITY AND SPECIFIC GRAVITY. 133 (16) If the liquid employed with Nicholson's Hydrometer be water, the substance a mixture of two metals whose S.G.'s are 14 and 16, and the weights used are 16 oz, 1 oz, 2 oz; find the quantity of each metal in the mixture. (17) Show that the units may be chosen so that the specific gravity and the density of a substance are identical. A nugget of gold mixed with quartz weighs 12 (10) ounces, and has a specific gravity 6'4 (8-6); given that the specific gravity of gold is 19'35, and of quartz is 2-15, find the quantity of gold in the nugget. (18) Air is composed of oxygen and nitrogen mixed together in volumes which are as 21 to 79, or by weights which are as 23 to 77; compare the densities of the gases. (19) How many gallons of water must be mixed with 10 gallons of milk to reduce its S.G. from 1-03 to 1-02 ? (20) Bronze contains 91 per cent, by weight of copper, 6 of zinc, and 3 of tin. A mass of bell-metal (consisting of copper and tin only) and bronze fused together is found to contain 88 per cent, of copper, 4-875 of zinc, and 7-125 of tin. Find the proportion of copper and tin in bell-metal. (21) Two fluids are mixed together: first, by weights in the proportion of their volumes of equal weights ; secondly, by volumes in the proportion of their weights of equal volumes; compare the specific gravities of the two mixtures. 134 GENERAL EXERCISES ON (22) A mixture of gold with n different metals contains r per cent, of gold and r-^, r^, r^, ..., r„ per cent, of the other metals. After repeated processes, by which portions of the other metals are taken away, the amount of gold remaining unaltered, the mixture contains s per cent, of gold and Sj, Sg, S3, . . . , s„ per cent, of the other metals. Find what percentage of each metal remains. (28) A quart vessel is filled with a saturated solution of salt. A quart of water is poured drop by drop into the vessel, causing the solution to overflow, but is poured in so slowly that it may be sup- posed to diffuse quickly through the solution. Show that after the operation the amount of salt left in the solution in the vessel will be Ije of the original amount, where e is the base of the Naperian logarithms. (24) From a vessel full of liquid of density p is removed one-%th of the contents, and it is filled up with liquid of density cr. If this operation is repeated m times, find the resulting density in the vessel. Deduce the density in a vessel of volume V, originally filled with liquid of density p, after a volume U of liquid of density o- has dripped into it by infinitesimal drops. (25) The mixture of a gallon of A with TT^ lb of B has a S.G. s,^, with W^ lb of 5 a s.G. s^, with Tfg lb of 5 a S.G. S3 ; find the S.G.'s of A and B. (26) Find the chance that a solid composed of three substances whose densities are p^, p^, p^, will float in a liquid of density p^. DENSITY AND 8PEGIFIG GRAVITY. 135 (27) A vessel is filled with three liquids whose densities in descending order of magnitude are pj, p^, p^. All volumes of the liquids being equally liisely prove that the chance of the density of the mix- ture being greater than p is iPl- P2)(Pl- Ps)' or 1—- (P-Psf {P2- PsXPl- Ps)' according as p lies between pj and p^ or between < P2 and ps. (28) Describe some method of determining the absolute expansion of a liquid. A piece of copper is weighed in water at 16° and at 80°, the weights of water displaced being 50 g and 48'809 g ; find the mean coefficient of cubical expansion of copper between those temperatures ; given the s.G. of water at 16° and 80° as 0-999 and 0-972. (29) The hydrometer is used to determine the s.G. of a liquid which is at a temperature higher than that of water. When the hydrometer is transferred from water to the liquid the S.G. appears at first to be s, but afterwards to be s^. Show that, neglecting the density of the air, the true S.G. at the temperature of the water is «+-(Sx-s), where a and a' are the coefficients of expansion of the hydrometer and the liquid respectively. 136 GENERAL EXERCISES. (30) Show that the coefficient of expansion of a body may be found as follows : — Let s be tl^e s.G. of the body at zero temperature compared with water at its greatest density ; 1+e-^, 1 + 63 the volumes at temperatures t-^, t^ of a unit volume at zero temperature; l+E^, l+E^ the volumes at t^, t^ of a unit volume of water at its greatest density ; w the weight of the body in a vacuum ; Wj, w^ its apparent weights in water at temperatures t^, t^ ; then gj — 62 = -^1 — -£"2 — s(Wi — '^2)!'^ ^^^y nearly. (31) Prove that, if a hydrometer of weight W sinks to certain marks on the stem in a liquid at tempera- tures t^ and ^21 ^^^ to the same marks in the liquid at zero temperature, when weights w^ and W2 are fixed at the top of the hydrometer, the coefficients of cubical expression of the hydrometer and of the liquid are respectively h i and -h h (32) Determine the s.v. in cubic feet to the ton, and the density in lb per cubic foot of lead shot, cast iron spherical shot, and cast iron spherical shells with internal radius three-quarters the outside radius, given the s.G. of lead as 11-4, and of cast iron 7'2. Determine also the S.v. or roomage of earthen- ware pipes, and cylindrical barrels, of apparent density p. CHAPTEE IV. THE EQUILIBRIUM AND STABILITY OF A SHIP OR FLOATING BODY. 84. Simple Buoyancy. The Principle of Archimedes leads immediately, as in § 48, to the Conditions of Equilibrium of a hody supported freely in fluid, like a fish in water, or a balloon in air, or like a ship floating partly immersed in water (flg. 38, p. 148). The body is in equilibrium under two forces ; (i.) its weight W acting vertically downwards through 0, the C.G. of the body ; and (ii.) the buoyancy of the fluid, equal to the weight of the displaced fluid, and acting vertically upwards through B, the C.G. of the displaced fluid ; and for equilibrium these two forces must be equal and directly opposed. The Conditions of Equilibrium of a body, floating like a ship on the surface of a liquid, are therefore (i.) the weight of the body must be less than the weight of the total volume of liquid it can displace, or else the body will sink to the bottom of the liquid ; (ii.) the weight of liquid which the body displaces in the position of equilibrium is equal to the weight W of the body ; (iii.) the C.G. B of the displaced liquid and G of the body must lie in the same vertical line GB. 137 1 38 SIMPLE B UO TANCY. 85. In a ship the draft of water is a measure of the displacement and buoyancy of the water, while the free- hoard, or height of the deck above the water line, is a similar measure of the reserve of buoyancy, or of the extra cargo which the ship can carry without sinking. The Plimsoll mark is now, by Act of Parliament, painted on all British ships ; it is a mark which must not be submerged when the vessel is floating in a fresh water dock, before putting to sea ; and the mark is fixed at such a height as to give the vessel a reserve of buoyancy of 25 per cent, of its total buoyancy. The buoyancy of a pontoon or cask, employed as a support or buoy, is however generally used to mean its reserve of buoyancy, or the additional weight required to submerge it. Thus the (reserve of) buoyancy of a body, a life buoy for instance, of weight W lb and (apparent) s.G. s, and therefore displacing Wja lb of water, is (i-l) Fib. 86. When a ship loses its reserve of buoyancy, and is sunk in shallow water, it can be raised by building a caisson on the deck so as to bring the level of the bulwarks above the surface at low water. All leaks and orifices below water having been stopped by divers, the vessel is pumped out at low water by powerful steam pumps ; and thereby soon acquires suffi- cient buoyancy to rise from the bottom of the sea, so as to be moved into a dock for repair ; in this manner such large vessels as the Utopia, the Austral, and the Howe have been raised. THE CAMEL AND FLOATING DOCK. 139 When a vessel draws too much water for entering or leaving a port, as for instance Venice, the Zuyder Zee, or Chicago through the lakes of N. America, camels are employed to lessen the draft of water. These camels consist of large tanks, which are sub- merged by the admission of water, and then secured to the sides of the vessel by chains passing under the keel. On being pumped out the extra buoyancy of the camels raises the vessel and lessens the draft of water to the desired extent. The same principle is employed in floating docks : the dock is submerged by the admission of water, so that the vessel can be floated on to the blocks on the bottom of the dock and be there secured : the water is then pumped out of the dock and the vessel is thereby raised above the level of the water, and can then be deposited on staging ashore, or even repaired on the floating dock itself; in this case it is convenient to secure the dock to the quay wall by pivoted bars. The double power dock, designed by Messrs. Clark and ' Stansfield, consists of a central pontoon which supports the vessel, and two large side tanks or camels, which can float independently. The vessel is raised as far as possible by pumping out the central pontoon ; the camels are then submerged by the admission of water, and secured to the sides of the pontoon ; and now, the buoy- ancy of these camels, on being pumped out, is sufiicient to raise the vessel completely above the water. By this arrangement not only is economy of material secured, but the pontoon or the camel can be alternately raised completely out of the water for the purpose of examination and repair. {Trans. I. Naval Architects, xx.) 140 TONS PER INCH IMMERSION. 87. Deuoting by A the water line area (fiottaison) of a ship in square feet, that is, the area of the plane curve formed by the water line, then an additional load of P tons properly placed (that is, so that the C.G. of P is vertically over or under the C.G. of the water line area) will cause the ship to draw h feet more water, of density D Ib/ft^ suppose, given by the equation BAh = 224!0P. Strictly speaking this supposes either that the ship is wall-sided, meaning that the sides of the ship in the neighbourhood of the water line form part of a cylindrical surface ; or else that the mean water line area at the mean draft is A ft^; and thus, given P/h, we can deter- mine A, and vice versa. For sea water we take D = 64, so that the s.v. of sea water is 2240 -=-64 = 35 ft^ton ; and Ah=35P; or if h is given in inches, ^^ = 420P, A~420' and P/h is the number of tons required to immerse the ship one inch. Thus in a ship loading 10 tons to the inch, the water line area is 4200 ft^ ; and loading or consuming 300 tons of coal will change the draught 2 ft 6 in. For a ship L ft long and B ft broad at the water line, A = cLB, where c is called the coefficient of finetiess of the area. The following rules are given by Mr. W. H. White for a the coefficient of fineness, and n = P/h the number of tons per inch immersion (Naval Architecture) ; — TEE MERCURY WEIGHINO MACHINE. 141 1. For ships with tine ends, - 07, 2. For ships of ordinary form (including probably the great ma- jority of vessels), - - 0-75, 3. For ships of great beam in proportion to the length, and ships c= n = LB 600 LB 5C0 LB with bluff ends, - - 0-84 =^ 500 A Mercury Weighing Machine has been invented by Mr. Rutter {Industries, 16 Oct., 1891) in which the body to be weighed is placed in a scale pan which is suspended from a cylindrical plunger immersed in mercury, and the weight is read off on graduations corresponding to the weight of the extra quantity of mercury displaced. Thus the s.G. of mercury being 13'6, the vertical graduations will correspond to kilogrammes per centi- metre immersion if the cross section of the plunger is 1000-j- 13-6 = 73-53 cm^ in area, or 9'68cm in diameter. 88. Suppose the ship's weight and displacement is W tons, and that the draft of water increases by h inches as the density of the water diminishes from w to w' tons/ft^ ; then the original displacement being Fft^ the ship now acting like the common hydrometer; and these two equations are sufficient to determine V and W when A, w, and w' are known. Or denoting the s.v.'s in ft^ton by v and v', V=vW, V+i\Ah = i/W, ^\Ah = (v-v')W. 142 SINK AGE IN FRESH WATER. Thus if •y = 35 for sea water, and t;' = 35-84< for fresh water, ^Ah = Q-MW ; and if n denotes the number of tons per inch immersion, ,_ 10-08F _ W 420% "40%' approximately, giving the sinkage in inches of the ship in passing from salt to fresh water. For instance if a ship of 8500 tons displacement draws 25 ft of water at sea, and if the length on the water line is 330 ft and the breadth 65 ft, the sinkage in passing into fresh water is a little over 5 inches, and draft 25 ft 5 ins. 89. Large vessels are now built in compartments separ- ated by transverse watertight bulkheads, so as to localise and restrict the effect of a leak or perforation. Now if one of these compartments is bilged and becomes filled with water, the loss of buoyancy in ft^ is the volume of water which has entered, so that the sinkage in feet loss of buoyancy in ft^ ~ intact water line area in ft^' If the compartment is fitted with a watertight deck below the water line, the water line area of the vessel may be taken as unchanged, and the sinkage will be correspondingly diminished. If the compartment is occupied by cargo, such as coal, timber, casks, etc., the volume of water which enters is diminished by the volume of this cargo, so that the loss of buoyancy is that due to the unoccupied space in the compartment, up to the new water line. Thus, according to §§ 32, 33, the unoccupied space is 1— ^7r^3 or 1— ^■7r;^2 of the volume according as the cargo in the compartment is composed of equal cylindrical or spherical bodies, such as casks or grain closely packed. THE ENERGY OF IMMERSION. 143 The s.G. of a lump of coal being 1-4, while a coal cargo stows at a 40 to 45 ft7ton, the fraction of unoccupied space is about 04 or 40 per cent. 90. The Energy of Immersion. By the operation of plunging a body into a fluid a certain amount of energy is communicated to the fluid. If a liquid is contained in a vessel of finite size, and a body is made to displace a volume 7 ft', the gain of energy is the work required to raise Fft' of the liquid from the level of B, the O.G. of the volume V of the body, to the level of B\ the c.G. of the equal volume contained between the old and new surfaces of the liquid ; and the gain is therefore wV{z-z')it-\h, if w denotes the density of the liquid in lb/ft', and z, z' denote the vertical depths of B, B' below a given hori- zontal plane, say the new surface of the liquid. When the vessel containing the liquid is of practically unlimited size, the level of the liquid does not sensibly change by the operation of immersion of the body, and now 5' lies in the surface ; so that the gain of energy is the work required to lift Ffb^ of the liquid from B to the surface. If the body is completely submerged, the level of the surface of the liquid will not change for difierent positions of the body ; so that, if the C.G. of the body is depressed vertically through x ft, the gain of energy or work required to depress the body is equal to that required to raise of volume V of the liquid from the second to the first position occupied by the body, supposed to displace Fft' of liquid, and is therefore wYx ft-lb. 144 THE ENERGY OF IMMERSION The foi'ce required to depress the body, or the upward buoyancy of the liquid is thus a force of wV pounds, as before in § 45. 91. But if the body is only partially immersed, as at first, and is now depressed vertically through a small vertical distance x ft, the liquid will rise vertical distances on the side of the vessel, and on the side of the body, "'^ , , ax Bx and X + o or — /S — a' /3 — a /3 — a if a, j3 denote the (average) areas of the horizontal cross sections of the body and of the vessel made by the hori- zontal planes of the surface of the liquid; so that the extra volume displaced will be jj_a§x the C.G. of which will be raised through h-^ hw^ 01' i^ feet, p — a p — a and the consequent gain of energy is ^W a_ = ^w Ux, ft-lb. At the same time the depression through x ft of the volume V already immersed will give a gain of energy of wVx ft-lb, so that the total gain of energy is wVx + )iw^^ = w{V+^U)x ft-lb. The average resistance of the liquid to the depression of the body is thus a force of 'w{V+^'0) pounds, the buoyancy of a volume F-f-^CT of the liquid ; reducing as before, for an infinitesimal depression, to a force oi wV pounds, the buoyancy of a volume V of the liquid. OF A FLaATING BODY. 145 92. When a and /3 are constant, the body and the vessel are cylindrical, and the preceding expressions hold for finite values of the depression h. Thus if the body is a vertical cylinder floating freely in the liquid, the weight of the cylinder and the buoyancy •of the water is wV\h; and the work required to depress the cylinder vertically through x ft is the gain of energy of the liquid less the loss of energy of the body, and the work is therefore ^w Ux = hw-^ — ft-lb. If the cylinder is of height h and density w', the length hw'/w of the axis is submerged, and h (w — w')/w stands out of the liquid ; to immerse the body completely, it must be pushed down a vertical distance x, given by and the work required is therefore 93. Generally a floating body will come to rest in a position in which the energy of the system is a minimum ; and the preceding considerations show that the distance between 0, the O.G. of the body, and B, the C.6. of the liquid displaced, will then be a minimum ; the distance being a maximum for positions of unstable equilibrium. ExaTTi'ples. (1) At low water a gallon was found to weigh 10 lbs, and at high water to weigh 1025 lbs; and it required 25 tons to bring a vessel at high water down to the draught at low water ; prove that the ship weighed 1000 tons. 146 EXERCISES IN SIMPLE BUOYANCY. (2) A steamer loading 30 (25) tons to the inch in fresh water is found after a 10 days' voyage, burning 60 (52) tons of coal a day, to have risen 2 feet (25 inches) in sea water at the end of the voyage; prove that the original displacement of the steamer was 5720 (5000) tons, taking a cubic foot of fresh water as 62'5 pounds and of sea water as 64 pounds. (3) A steamer in going from salt into fresh water is observed to sink two inches, but after burning 50 tons of coal to rise one inch ; prove that the steamer's displacement was 6500 tons, supposing the densities of sea and fresh water are as 65 to 64. (4) A steamer of 5000 tons displacement drawing 25 feet of water has to discharge 300 tons of water ballast to lessen the draft one foot, to cross the bar into a river. Prove that after burning 50 tons of coal in going up the river the steamer will be drawing 24- 2 ft in fresh water ; and now the admission of 293 tons of water ballast will be sufficient to increase the draft one foot. (5) A sphere of radius r ft and weight Tf lb is let gently down into a vertical cylinder of radius R, con- taining water of twice the density of the sphere. Show that the work done on the water before the sphere begins to float is Fr(f-ir7E2), ft-lb. (6) Explain why it is that if a man puts his hand and arm into a bucket partly filled with water, potential energy is imparted to the water. A sphere of radius r is held just immersed in a cylindrical vessel of radius R containing water, EXERCISES IN SIMPLE BUOYANCY. 147 and is caused to rise gently just out of the water. Prove that the gain of potential energy of the sphere and the loss of potential energy of the water are respectively 2TFr(l-|r7i22), and W'r{l-lr^lB?),ii-Va, W and W being the weight in lb of the sphere and of the water displaced by the sphere. If the sphere be allowed to rise until it is half out of the water, prove that the loss of potential energy is to the loss in the previous case in the ratio of 39^2 -24r2: ^B,B?-^^t\ If the sphere be left to itself when under water, and if we could suppose the water to come to rest on the sphere leaving it, what would be the velocity with which the sphere would shoot out. (7) The arms of a balance are each of length a cm, and one of them at its end carries as a permanent counterpoise hanging from it a cylindrical vessel whose sectional area is a cm^, containing liquid of density w, in which dips a fixed vertical solid cylinder of sectional area ^ cm^. The beam is itself counterpoised for all inclinations, and the cylinder does not touch the vessel. Show that, when an addition of F g is made to the load on the other arm, the sine of the inclina- tion of the beam to the horizontal is altered by /i_i\f; \B aJwa 148 THE CENTRE OF BUOYANCY AND 94. The Conditions of Stability of a Ship. In additioa to satisfying the conditions of equilibrium of a floating body (§§ 45, 48, 84) it is necessary that a ship should fulfil the further condition of stability of equilibrium, so as not to capsize ; if slightly disturbed from the position of equilibrium, the forces called into action must be such as tend to restore the ship to its original position. Practically the stability of a ship is investigated by inclining it ; weights are moved across the deck and the angle of heel thereby produced is observed, and thence an estimate of the stability can be formed. Fig. 38. 95. Let the total weight of the ship be W tons, and let it displace V ft^ of water ; then W=wV, or 2240Tr=D7, where w denotes the density of the water in tons/ft^, and Dn lb/ft^ so that B = 22^0w. Now suppose a weight of P tons on board, originally amidships, is moved to one side of the deck, a distance of b ft ; and that the ship, originally upright, is observed to heel through an angle 6 (fig. 38). THE METACENTRE OF A SHIP. 149 The G.G. of the displaced water, called the centre of buoyancy (c.B.), will move on a curve (or surface) called the curve {or surface) of buoyancy, from B to B^, such that G2B2 is vertical iu the new position of equilibrium, G2 being the new c.G. of the ship when P tons is moved from g to g^, so that the ship will move as if the surface of buoyancy was supported by a horizontal plane. As P is moved across the deck from g to g^ a distance of b ft, so the C.G. of the body moves on a parallel line from Q to G^, such that OG2 = bP/ W ; this follows because the moments of P and W about Og must be the same ; and, if the new vertical £36^2 ^^^^ the old vertical BG in m, P P Gm = -^b cot 6, G^m = ^b cosec 6. The ultimate position ilf of m for a small angle of heel is the point of ultimate intersection of the normals at B and at the consecutive point B^ on the curve of buoyancy, and M is therefore the centre of curvature of the curve of buoyancy at B ; the point M is called the metacentre, and GM is called the metacentric height. In the diagram the ship is drawn for clearness in one position, and the water line is displaced; but the page can be turned so as to make the new water line horizontal. 96. For stability of equilibrium the metacentre M must be above G, for if M were below G then on bringing P back suddenly from g^ to g, the forces acting on the ship would form a couple tending to capsize the ship ; but if M is above G the forces would then form a couple, con- sisting of W acting vertically downwards through G, and W acting vertically upwards along B^tti, tending to re- store the ship to the upright position. 150 METACENTRIC HEIGHT. The angle of heel Q is measured either by a spirit level or by the deflection of a pendulum or plummet. If the ship is symmetrical and upright when P is amidships, and if moving the weight P tons across the deck through 26 ft causes the plummet to move through 2a ft when suspended by a thread I ft long, then sin Q = all ; so that ijr^rii = y^, and (rgiri may be taken as the metacentric height OM. Thus in H.M.S. Achilles, of 9000 tons displacement, it was found that moving 20 tons across the deck, a dis- tance of 42 ft, caused the bob of a pendulum 20 ft long to move through 10 inches. Here F=9000, P=20, 6 = 21, i = 20, a = ^; and therefore GM= 2-24 ft. Also sin 6 = 0-02083, 0=1" 12'. 97. The displacement W tons or V &? is determined by approximate calculations from the drawings of the ship, as also the O.B. B ; while G is determined from the weights of the diSerent parts of the structure, and from the distribution of the cargo and ballast. If the weight P was hoisted verticall}'' up the mast a distance h, B and M would not change, but Q would ascend to G-^, through a height GG-^ = hP/Wit. The metacentric height would be correspondingly di- minished ; so that if P is now moved along a yard on the mast a distance 6 feet, the ship will heel through an angle d^, such that Pb=W.GM sind= W. O^Msine^; and to produce the original angle of heel d, P requires to be moved through a less distance 6', such that h — h' = h sin Q. ALTERATION OF TRIM. 151 98. So far we have considered as the chief practical problem the transverse metacentre M in its relation to the heeling or rolling of a ship ; but a similar metacentre exists for any alteration of trim, caused either by change of stowage of cargo, or by press of sail and other pro- . pulsion ; this longitudinal metacentre is found by a similar experimental process, but from the shape of a ship it is necessarily much higher than the transverse metacentre. The trim of a ship is defined as the difference of draft of water at the bow and stern, and the change of trim is defined as the sum of the increase of draft at one end and the decrease of draft at the other. Suppose the trim is changed x inches in a vessel L ft long at the water line hj moving P tons longitudinally fore and aft through a ft, and that the ship turns through a small angle d, a gradient of one in 12Z/aJ. The moment to change the trim is Pa ft-tons, so that if M^ denotes the longitudinal metacentre, Pa=W.GM^.&me= W .GM^.xjl^L; thus the moment required to change the trim one inch is W. QMJ12L ft-tons. For instance, if a ship of 9200 tons, 375' ft long, has a longitudinal metacentric height of 400 ft, and a weight of 50 tons already on board is shifted longitudinally through 90 ft, the change of trim will be about 5^ inches. Practically it is found convenient to incline the ship by filling alternately the boats suspended on either side of the ship with a known vreight of water; and to change the trim by filling and emptying water tanks at the ends of the ship. 152 WEDGES OP EMERSION AND IMMERSION. 99. As the ship heels through an angle Q, and the water line changes from LU to L^L'^ (fig. 38), a certain volume U of water may be supposed removed from the wedge- shaped volume L'FL'^, called the wedge of emersion, to the volume LFL^, called the tuedge of iinmersion, so as to form the new volume of displacement L^KL'^. If 6p 62 denote the C.G.'s of the wedges of emersion and immersion, BB^ is parallel to h-p^ J and if BY, c-^c^ are the projections of BB^, h-p^, on the new water line ig-^-z' BB^ = hp^ .U/r, B Y= c^c^ . U/ V. We notice that when 6 is evanescent, the line b-fi^ coalesces with the water line LL' ; and therefore the tangent to the curve of buoyancy at B, being the ultimate direction of the line BB2, is parallel to the water line LL', a theorem due to Bouguer. If the C.G. of the ship is at G, and GZ is drawn per- pendicular to B^in, the moment in fb-tons of the couple tending to restore the ship to the upright position is W.GZ=W{BY-BG sine) = w(^YC^c^-BG sin e^ = w(U. CjC^ -V.BG sin 6), Atwood's formula {Phil. Trans., 1798). Curves are now drawn for ships by naval architects, called cross curves of stability, which exhibit graphically the value of the righting moment W . GZ for a given inclination 9, say an angle of 30°, 45°, or 60°, and for difierent drafts of water of the ship and displacements W or F ; the volumes U of the wedges of immersion and emersion are calculated and the corresponding values of CjO^; also the values of BG, and thence GZ is known for an assumed position of G. STABILITY OF A SUBMARINE VESSEL. 153 100. In the case of a body floating completely sub- merged, like a fish, submarine boat, or Whitehead torpedo, the buoyancy always acts vertically upwards through B, the c.G. of the displaced liquid; or the c.B. is a fixed point in the body. In the position of stable equilibrium, B must be above 0; and if displaced through an angle Q, the righting moment will be W.OBsiaO. When the water line area is very small, as is the case of a rod or hydrometer, then BM is small, so that the body behaves as if completely submerged, and requires ballasting to bring below B. Stability is secured with greater facility by increase of beam on the water line area ; a caisson as in fig. 2-3, p. 54, or a yacht of similar cross section, would require con- siderable ballast for stability. In a body of revolution (fig. 45, p. 1 90) such as a cork, a circular pontoon, a cigar ship, a spherical buoy, or a hydrometer floating with its spherical bulb partially immersed, the curves of buoyancy are circular so long as no break in the circular cross section meets the water line ; the metacentre M lies in the axis or centre, and the equilibrium is stable in the position in which G is verti- cally below M ; the righting moment at any inclination Q being W.GMsme. 101. To determine the metacentre theoretically, we sup- pose the angle Q through which the ship heels to be small, so that the C.B. moves from B to the adjacent point B^ on the curve of buoyancy BB^, without alteration of the displacement V. 154 THEORETICAL DETERMINATION The resultant buoyancy of W tons acting vertically upwards through B^ in the displaced position is equi- valent to an equal buoyancy W acting upwards through B, and a couple of moment W .BY (fig. 38) ; and this couple is due to the upward buoyancy through b^ of the wedge of immersion LFL^, and an equal downward force through h-^, due to the loss of buoyancy of the wedge of emersion L'FL'^. Suppose the ship turns about an axis through F per- pendicular to the plane of the paper; then, denoting by y the distance of an element AJ. of the water line area A from this axis of rotation, the element of volume of either wedge is ultimately y tan Q . LA. The equality of the wedges of immersion and emersion leads, on dropping the factor tan Q, to the condition so that the axis of rotation passes through the C.G. of the water line area, which we may denote by F. The righting couple of the wedges of immersion and emersion will be lAuy tan Q . A^l .y = w tan Q^y^LA = w tan Q .Ak^, ft- tons, where Alc^ denotes the moment of inertia in ft* (biquad- ratic feet) of the water line area A about the axis of rotation (§ 38) ; so that W .BY=wi&n6.Ak\ But W==wV, and BY= BM sin 6, ultimately ; so that, finally, with cos 0=1, or sin = tan B, BM = Ak^lV, the radius of curvature of the curve of buoj'ancy at B. OF THE METACENTRE. 155 The ship may thus be assimilated, in the neighbour- hood of the water line in the upright position, to a surface of revolution about a horizontal axis, as in fig. 45, p. 190, when BB^ will be the arc of a circle whose centre is M; and for an angle of heel Q the chord BB^=tm.fi\Q.BM\ and therefore the moment of the wedges of emersion and immersion will be accurately 102. We can prove these theorems by considering separately the wedges of immersion and emersion. Suppose the axis of rotation through F divides the water line area into two parts, of areas A-^ and ulg, and that A^ and h^ denote the distances of the C.G.'s of these areas from the axis of rotation. Then the volumes of the wedges of emersion and immersion may be taken as J^AjtanS and ^ 2'''2 ^^"^ ^ j and these volumes being equal and denoted by V , Ucot6 = A-Ji-^ = A^h^, which proves that the axis of rotation passes through the C.G. of the water line area. Also these wedges being equivalent to laminas A^ and A^, loaded so that the superficial density is proportional to the distance from the axis of rotation (§ 39), A-J'1'1 ' AJ-h^ and therefore 'Lz^^A. . Ak\ . c.^c^ = — jj — tan t^ = -j^ tan B; and BY=-^tB.n6, B3I=^, ultimately, as before. 156 STABILITY OF A WALL-SIDED VESSEL. 103. If the vessel is "wall sided" from the upright to an angle of heel Q, then up to this limit the curve of flotation reduces to a point F ; and the volume U of the wedges of emersion and immersion will he given by C/" cot = ^ j/ij = J. g/ig. If \e^, 6362 be the perpendiculars from fc^, h^ on the upright water plane A, then, as before (fig. 38, p. 148), ^1^2 = ~Tr ts.^ Q, NB^ = -rr tan 6 = BM tan Q, so that the subnormal Nm is constant and equal to BM\ and the curve of buoyancy BB^ is a parabola (fig. 44, p. 190) of which BM is the semi-latus-rectum, and therefore Mm = BN= INB^ tan = ^BM tan^ft Now if GZ is the arm of the righting couple W . OZ, GZ= Gm sin d = (GM+ ^BM i&n^d) sin d. The surface of buoyancy is now given accurately by the equation of § 107, and is therefore a paraboloid. For instance, if a cylinder, whose cross section is the water line area A, floats upright immersed to a depth h in liquid, V=Ah, and (§ 107), 2^ _ 2/^ 0^ is the equation of the surface of buoyancy, a paraboloid. 104. For diff'erent distribution of the same weight on board, the displacement V of the ship remains constant ; and drawing all the diSerent water planes of the ship for constant displacement V (isocarenes), these planes all touch a certain surface F fixed in the ship, called the sur- face of flotation, and the ship moves as if this surface rolls and slides on the plane surface of the water. We have just proved (§ 101) that the line of inter- section of any two such consecutive water planes passes CURVES OF FLOTATION AND BUOYANCY. 157 through the c.G. of the water line area, so that F, the point of contact of a water plane with the surface of flotation, is the C.G. of the corresponding water line area. We have also proved that the tangent line of the curve of buoyancy is parallel to the corresponding water plane ; and therefore the tangent plane of the surface of buoy- ancy B is parallel to the corresponding water plane, and the normal line is perpendicular. 105. Consequently the body can float in equilibrium wherever the normal to the surface of buoyancy passes through the C.G. of the body, with this normal vertical ; and therefore the determination of the positions of equi- librium depends on the geometrical problem of drawing normals from G to the surface of buoyancy B. A gradually contracting liquid sphere, with centre at G, is employed as an illustration by Reech (/. de I'ecole polytechnique, 1858); the free surface of the sphere cutting the surface 5 in a series of spherical contour lines, like those on the Earth. The positions of equilibrium correspond (i.) when un- stable to the top of a hill on B ; (ii.) stable to the bottom of a lake ; (iii.) stable-unstable to a pass or bar (§ 93) ; and the conditions of equilibrium are the same as if the surface B was placed on a smooth horizontal plane. 106. Fig. 26, p. 65, may be taken to represent the general horizontal water line area of a ship ; and now an inclining couple, due to moving a weight on board, will heel the ship about an axis perpendicular to the plane of the couple, only when this axis is a principal axis of the momental ellipse of the water line area A at its C.G. (§ 40.) 158 INCLINING COUPLE. For let the area A turn through a small angle about the line Gy ; then fe^Sg' ^^ c.G.'s of the wedges of emer- sion and immersion, being the c.p.'s with respect to Gy of the two parts into which the area A -is divided by Gy, will, according to the methods of §§ 38-40, lie in the line OGK; and therefore the vertical plane of the inclining couple is parallel to OG, the diameter conjugate to Gy with respect to the momental ellipse; and OG, Gy are at right angles only when they are the principal axes of the momental ellipse of the water line area. The varying directions of the inclining couple may be produced by swinging a weight of P tons suspended from a crane round in a circle of radius h, about G as centre suppose (fig. 26, p. 65) ; as, for instance, in a float- ing derrick crane, required for lifting and transporting great weights. When the weight P is over K, the inclining couple of Pb ft-tons will turn the water plane A about Gy through a small angle 0, given by sm 6 = PhlwAlc'^, a slope of one in wAk^/Pb. If the weight P was lowered on to the vessel from a crane on shore, and deposited over G, the C.G. of the water line area A, the vessel would sink bodily without heeling a distance P/n inches, n denoting the number of tons per inch of immersion. But if the weight P was deposited over K, the vessel would be depressed and inclined ; and the resultant effect will be equivalent to a heel through the same angle 6 about 00', the anti-polar of the point K with respect to the momental ellipse. DU PIN'S THEOREMS. 159 107. The section BB^ of the surface of buoyancy made by a vertical plane parallel to GK will have a radius of curvature Ak^/V; so that, referred to three coordinate axes with B as origin, Bx in the normal to the surface of buoyancy, and By, Bz in the tangent plane parallel to the principal axes of the momental ellipse of the area A, the equation of the surface of buoyancy will be represented, approximately, by ^'^~Ah^^Alcy^^--- The vndicatrix of the surface of buoyancy, or a section of the surface made by a plane parallel and close to the tangent plane, is thus an ellipse, similar and similarly situated to the momental ellipse of the water line area A ; and the lines of curvature of the surface of buoy- ancy are therefore parallel to the principal axes of the corresponding water line area. 108. Dwpin's Theorems. I. If planes A cut off a constant volume V from a surface 8, these planes touch a surface {the surface of flotation) such that the point of contact F is at the c.g. of .the area A of section of the surface 8. II. .The surface described by the C.G.'s of the volume V {the surface of buoyancy) has the tangent plane at any point B parallel to the corresponding plane of section A. III. The indicatrix of the surface at B is similar and similarly situated to the momental ellipse at F of the plane area A of section of the surface 8 ; and the lines of curvature at B are therefore parallel to the principal axes at F of the area A (§ 40). 160 CURVES OF STATICAL It was in tlie ])receding maaner, from mechanical and hydrostatical considerations in connexion with Naval Architecture, that Dupin {Applications de Geometrie, 1814), was led to the discovery of these geometrical theorems, which now go by his name. Cor. The surface of buoyancy is thus necessarily a synclastic or rounded surface ; but the surface of flota- tion F may change from being synclastic to anticlastic or saddle-shaped in parts, especially where the water plane cuts the edge of the deck of a ship, or other edge of a floating body, as illustrated in fig. 39. Fig. 39. This fig. 39 is copied by permission from Mr. W. H. White's Course of Study at the R.N. College, Greenwich (Trans. I.N. A., 1877), and represents the cross section of an actual vessel, with the corresponding sections of the surfaces of flotation and buoyancy, represented by the curves FF and BB, and also the curve MM of meta- centric evolutes of the curve BB. A. cusp occurs on the curve of flotation FF in con- sequence of the immersion of the edge of the deck. AND DYNAMICAL STABILITY. 161 109. Curves of Statical and Dynamical Stability. In these curves, drawn in fig. 40 for the vessel in fig. 39, the abscissa represents the angle of inclination in degrees, while the ordinate in the curve of statical stability represents the arm GZ of the corresponding right- ing couple at this particular displacement (fig. 38), and the ordinate in the curve of dynamical stability re- presents the work in ft-tons required to heel the vessel slowly over from the vertical to the inclined position. T 4.0- 88'\<)6 Fig. 40. Since the work done by a constant couple is the pro- duct of the couple and of the circular measure of the angle through which it works, the ordinate of the curve of dynamical stability will be proportioned to the area of the curve of statical stability bounded by the final or- dinate at the corresponding inclination. Conversely the tangent of the inclination of the curve of dynamical stability is proportional to the ordinate of statical stability. In a vessel of circular cross section, as in fig. 45, p. 190, the metacentre is a fixed point, and the ordinates of the curves of statical and dynamical stability are therefore proportional to the sine and versed sine of the abscissa, representing the inclination (§ 100). O.H. L 162 POLAR CURVES OF STABILITY. 110. Mr. Macfarlane Gray suggests the use of polar curves of stability {Trans. I.N. A., 1875); the polar curve of statical stability will now be the curve described by Z in the ship (fig. 38), while the polar curve of dynamical stability will be the pedal of the curve of buoyancy BB^ with respect to G, or the locus of the feet of perpendiculars drawn from Q to the tangents of the curve BB^- By a well-known theorem of the Difiierential Calculus, if Q denotes the inclination of the ship in radians (of ISO/tt or 57'3 degrees), and p denotes the length of the perpen- dicular, ZB^, from G on the tangent at B^ of the curve of buoyancy. Thus W{dp/d6) is the righting moment in ft-tons, and /' w^de=wip,-p,) is the dynamical stability in ft-tons, or work done in heeling the ship from the first to the second position ; so that, as in § 93, the difference of energy in the two posi- tions is equal to the difference of vertical distances between G and B in the two positions multiplied by the weight W. Reckoned from the upright position, the dynamical stability in ft-tons = W(ZB^ - OB) = W{B^Y- GB vers Q), Moseley's formula {Phil. Trans., 1850). If the position of G is changed, say to B, a distance BG = a suppose, then the righting moment and dynamical stability are changed to W{GZ+GB sin 6) and W{ZB^ - GB cos 6) ; so that the polar curves of stability with respect to the TANGENTS TO CURVES OP STABILITY. 163 new pole B can be deduced from the curves with respect to the former pole by describing a circle on GB as diameter, and increasing or diminishing the former lengths of OZ and ZB^=p by the lengths of the inter- cepts made by this circle on these lines. Thus in the circular pontoon of fig. 45, p. 190, the polar curves of statical stability are circles, and of dynamical stability are limagons or cardioids, the pedals of a circle. 110. Since (d . GZ)/de = ZM^, the tangent ZT at Z of the curve of statical stability in fig. 40 is constructed by measuring a length ZN of 573 graduations of degrees, to represent the radian, and erect- ing the perpendicular NT = ZM^ ; and initially ZM^ = GM ; we are thus enabled to draw the tangent to the curve of statical stability. Similarly, if QF in fig. 40 is drawn to represent W. GZ to scale, then P V will be the tangent at P to the curve of dynamical stability (Jenkins, Trans. I.N.A., 1889). The curves of statical and dynamical stability are use- ful in showing how far a vessel may heel with safety ; a steamer will recover the upright position if heeled to any extent short of the angle of vanishing stability; but a sailing ship, heeled over by the wind, must not be allowed to incline so far as the angle corresponding to the point of maximum value of righting arm GZ. If the initial part OZ of the curve of statical stability is taken as a straight line, the curve OP of dynamical stability will be a parabola, the polar diagram of Z will be a Spiral of Archimedes, and the curve MM^ of pro- metacentres the involute of the circle with centre G and radius GM ; the curve of buoyancy BB^ will then be an involute of the involute of a circle. 164 INITIAL DYNAMICAL STABILITY. 111. Any small angular displacement of the ship about an axis through F the C.G. of the water line area may be resolved into component angular displacements, Q^ and Q^ suppose, about the principal axes of the momental ellipse of the water line area A. Replacing sin Q^ and sin Q^ by their circular measure Q^ and 02, the corresponding righting couples can be written F(pi-A)0iand W{p^-h)e^ ft-tons, where p^, p^ denote the principal radii of curvature of the surface of buoyancy, and h is the distance between the C.G. and the C.B. of the ship. e \ ^ M ■)> ^ } C / s, / N \.^- « f' ^ ^ H"^ i/y \ f B — —- Fig. 41. The work done by a gradually increasing couple is the product of the average value of the couple and of the circular measure of the angle through which it works ; and therefore the work done in heeling the ship is, in ft-tons, \W{{p^-}i)Q^^{p^-K)Qi\ LECLERT'S THEOREM. 165 112. Leclert's Theorems. A simple relation connecting r, the radius of curvature of the transverse curve of buoyancy, with r^ the radius of curvature of the corresponding parallel curve of flotation, has been discovered by M. Emile Leclert (1870). Let B-y, B^ denote the c.b.'s of a ship in two consecutive inclined positions, when the displacement is V, so that B-^B^ is a small arc of the curve of buoyancy; and let F^F^ be the corresponding parallel arc of the curve of flotation (fig. 41). Produce F■^B■^^, F^B^ to meet in ; and let the normals at B^, B^ to the curve' of buoyancy intersect in Jfj, and the normals at F^ F^ to the curve of flotation intersect in G^ ; so that B-^M-^, F^G^ become ultimately r and rj. Then since, by Dupin's Theorems, the normals at B^ and F-^ are parallel, and also at B^ and F^, therefore B^_B^_qB, F,G,-F,F-OF^' and therefore Jfj lies in the straight line OG^. Now suppose the displacement of the ship is changed from 7 to F— ^AFand F+|AF; and that in consequence Bj^ changes to b^ and /Sp B^ to b^ and jSg, F^ to /^ and c w laW' (4) A right circular pontoon, 50 feet long and 16 feet in diameter, is just half immersed on an even keel. The centre of gravity is 4 feet above the bottom. Calculate, and state in degrees, the transverse heel that would be produced by shifting 10 tons 3 feet across the vessel. State, in inches, the change of trim produced by shifting 10 tons longitudinally through 20 feet. 184 EXERCISES ON STABILITY Trace the curves of buoyancy and of prometa- centres of a raft or life-boat, supported by two parallel circular pontoons, half immersed in water. (5) It was found that filling the boats, suspended on each side of a vessel of 6000 tons displacement, alter- nately with 6 tons of water caused the vessel to heel so that the bob of a pendulum 6 ft long moved through 3 inches. Given that the distance between the centre lines of the boats was 40 ft, prove that the metacentric height was 1"152 ft. (6) A vessel of 6000 tons displacement heels over under sail through an angle of 5° ; show that its meta- centric height is about 2 feet : assuming that the component of the wind pressure perpendicular to the keel is a force equal to 26 tons acting at a point 25 feet above the deck, the c.G. of the ship and cargo being 15 feet below the deck. (7) In a cargo-carrying vessel, the position of whose c.G. is known, show how the new position of the c.G. due to a portion of the cargo shifting, may be found. A vessel of 4000 tons displacement, when fully laden with coals, has a metacentric height of 2| ft. Suppose 100 tons of the coal to be shifted so that its c.G. moves 18 feet transversely, and 4J feet vertically ; what would be the angle of the vessel if upright before the coal shifted ? (8) A ship is 220 feet long, has a longitudinal meta- , centric height of 252 feet, and a displacement of 1950 tons. If a weight of 20 tons (already on board) were shifted longitudinally through 60 feet, what would be the change of trim ? OF FLOATING BODIES. 185 (9) A ship is floating at a draft of 18 ft forward, and 20 ft aft, when the following weights are placed on board in the positions named : — Wt. in tons. Distance from c.o. of water plane in feet. 10 - - 901, . h before ; 30 . - 30j [-abaft. 30 - - 45/ What will be the new draft forward and aft, the " moment to change trim one inch " being 700 foot-tons, the " tons per inch " being 30 ? (10) A bridge of boats supports a plane rigid roadway AB in a horizontal position. When a small moveable load is placed at G the bridge is depressed uni- formly ; when the load is placed at a point C the end A is unaltered in level ; when at D the end B is unaltered in level ; and when at P the point Q of the roadway is unaltered in level. Prove that AG.GC=BG .GD = PG .OQ; and that the deflection produced at a point R by a load at P is equal to the deflection produced at P by the same load at M. (11) If a plane rigid raft is supported in a horizontal position by a number of floating bodies, a weight placed on the raft vertically over the centre of inertia of the planes of flotation will sink the raft vertically, while a weight placed anywhere else will cause the raft to turn about an axis, the antipolar of the weight with respect to the mo- mental ellipse of the planes of flotation. Compare this with the theory of a table resting on a number of elastic supports. 186 EXERCISES ON STABILITY (12) Find the dynamical stability in foot-tons at 30° of a rectangular pontoon 100 ft X 20 ft X 10 ft draft, having a gm. of 2 ft. (13) The curve of statical stability of a vessel is a seg- ment of a circle of radius twice the ordinate of maximum statical stability, which is 2500 tons- feet ; estimate the total dynamical stability of the vessel, the angle of vanishing stability being 85°. (14) The curve of stability of a vessel is a common para- bola, the angle of vanishing stability is 70°, and the maximum moment of stability 4000 ft-tons. Prove that the statical and dynamical stabilities at 30° are 3918 and 1283 ft-tons. (15) A cylindrical vessel with a flat bottom is free to turn about a liorizontal axis through its C.G. Prove that if a little molten metal be poured in, the vertical position is unstable ; that it does not become stable until the depth of the metal exceeds c — rjo^ — o?, where a is the radius of the cylinder, and c the height of the centre of gravity above the base; and that it is again unstable when the depth exceeds c-^ si (^ — c^- Determine the weight which must be fixed to the bottom of the vessel so as to make the equi- librium stable at first. (16) A canister containing water floats in a liquid, with its axis vertical. Prove that its stability for angular displacements will be unaffected if a certain weight of water is removed and a spherical ball of equal weight is placed in the cylinder so as to float in the water partially immersed, even though the ball touch the cylinder. OP FLO A TING BODIES. 1 87 (17) A vessel is constructed to carry petroleum in tanks formed by the sides of the vessel and a middle line and transverse bulkheads. If the water plane of the vessel be rectangular, and the tanks extend over frds the length, and are not full, investigate the stability at a small angle of heel, having given - — Length of vessel 240 ft, breadth 86 ft, c.G. of laden vessel 14 ft from top of keel, c.B. of laden vessel 8 ft from top of keel, displacement 2500 tons, and s.G. of petroleum OS. (18) A vessel is of box form, 300 ft x 50 ft, and draws 20 ft of water when intact. A bunker, 10 ft wide, 10 ft deep (6 ft below, 4 feet above the water line), containing coal, extends a length of 100 ft amid- ships at each side of the vessel. The c.G. of the vessel is 18 ft above the keel, find the GM. — (1) In the intact condition. (2) With both bunkers riddled, the inner and end bulkheads remaining intact. (19) A vessel in the form of a cube of side 12a containing liquid is placed so as to rest on the top of a fixed sphere of radius 5a. Neglecting the weight of the vessel prove that there will be stability provided the depth of the liquid is between 4ct and 6a. (20) Prove that a cylindrical kettle of radius a and height h will be in stable equilibrium on the top of a spherical surface of radius c, when the water inside occupies a height intermediate to the roots of the equation the weight of the kettle being n times the weight of water which fills it. 188 EXERCISES ON STABILITY. (21) A cup whose outside surface is a paraboloid of revolution of latus-rectum I, and whose thickness measured horizontally is the same at every point and very small compared with I, has a circular rim at a height h above the vertex, and rests on the highest point of a sphere of radius r. If water be now poured in until its surface cuts the axis of the cup at a distance -^h from the vertex, and if the weight of water be four times that of the cup, the equilibrium will be stable, if r + t (22) Prove that if a thin conical vessel of vertical angle 2a and weight W, whose C.G. is at a distance h from the vertex, is resting upright in a horizontal circular hole of radius c, it will become unstable when a weight P of liquid is poured into it to a depth x, so as to make f Ptc — 2(P+ W)c cot a + Wh cos^a positive. (23) A cylindrical vessel, floating upright in neutral equilibrium, will really be stable if the radius of curvature at the water line of the vertical cross section is greater than the normal cut off" by the medial plane. (24) Prove that the metacentric height given by (P/TF)&cosec0 (§ 94) can be made correct to the second order for the ship, when P is removed, by adding to it (§ 112) (P/F)(c-r,), where n\ denotes the radius of the curve of flota- tion, and c the height of P above the water line. CHAPTEE V. EQUILIBRIUM OF FLOATING BODIES OF REGULAR FORM AND OF BODIES PARTLY SUPPORTED. OSCILLATIONS OF FLOATING BODIES. 126. The Equilihrvwm of a floating Cylinder, Gone, Paraboloid, Ellipsoid, Hyperboloid, etc. When the body has the shape of one of these regular mathematical forms, the curves of flotation F and of its evolute G, of buoyancy B, and of the prometacentres M, or the metacentric evolute, can be determined by various theorems introducing interesting geometrical applica- tions of the properties of these curves and surfaces. For a prismatic or cylindrical body like a log, floating horizontally in water, the various surfaces are cylindrical and we need only consider their curves of cross section. If the section is an ellipse, these curves of flotation and of buoyancy are also ellipses; and the determination of the position of equilibrium will depend on the problem of drawing normals from the C.G. of the body to the ellipse of buoyancy, or tangents to its metacentric evolute ; and two or four normals or tangents can be drawn according as the C.G lies outside or inside this evolute. If the sides of the log in the neighbourhood of the water line are parallel planes, the curve of flotation reduces to a point, and the curve of buoyancy becomes a parabola (§ 103). 189 190 CURVES OF BUOYANCY If the submerged portion of the log is triangular, or more generally if the log is polygonal or if the sides in the neighbourhood of the water line are intersecting planes, the curves of flotation and of buoyancy are similar hyper- bolas of which the cross section of these planes are the asymptotes. Fig. 44. Fig. 45. When the cross section of the log is rectangular or triangular, the curves of flotation and of buoyancy are composed of parabolic and hyperbolic arcs, interesting figures of which, by Messrs. White and John, will be found in the Trans. Inst. Naval Architects, March, 1871 ; also by M. Daymard, I.N.A., 1884-. If the outside shape of the body is an ellipsoid or other quadric surface, then according to well-known theorems the surfaces of flotation and buoyancy are similar coaxial surfaces ; just as in the sphere, from which the ellipsoid jnay be produced by homogeneous strain. If the surface of the body is a quadric cone, the sur- faces of flotation and of buoyancy will be portions of hyperboloids of two sheets, asymptotic to the cone. OF THE UPRIGHT CYLINDER. 191 127. The Cylinder, floating upright. When a cylinder of s.G. s floats in water, the surface of flotation F reduces to a fixed point on the axis, so long as an end plane of the cylinder does not cut the surface of the water, and the surface of buoyancy is a paraboloid (§ 103). If h denotes the height of the cylinder and x the length of axis immersed, then x — sh; and for displacements in a vertical plane from the upright position of equilibrium the curve of buoyancy is a parabola (fig. 44;), and BM= Alc^l F= k^/x = k^/sh. The equilibrium is stable in the upright position if the C.G. G lies below M, or if BM>BO, or ^>l{h-x), p>Kl-s)- But if s2_, + 2g = (s-i)^-i(l-§') is negative, or ' h^J' the upright position of the cylinder is unstable in the corresponding vertical plane of displacement, and the cylinder " lolls " to one side. In this case the greatest value of k'^^jh^ is \, and then s = ^; so that the cylinder will float upright in any liquid if h? < Sk\ When the horizontal cross section of the cylinder is a rectangle of breadth h, k^ = ^-^¥ (§ 40) ; and this prismatic log cannot float upright, if i+i,j{i-^)>^>i-i^{^-~^- 192 CURVES OF BUOYANCY But if 67^,2 > f , hjh > ^^6, the log will float upright in any liquid. In a log of square vertical section b = h; so that it cannot float with faces horizontal and vertical, if i + W3( = 0-79) > s > I - i V3( = 0-21). When the cross section of the cylinder is a circle of radius a, A^ = J^a^ (§ 40) ; and this cylinder cannot float upright, if But if h/a < y/2, this cylinder will always float up- right, like a bung. As an exercise, the student may prove that the body in flg. 44, if floating in two liquids of S.G.'s s^ and s^, will be in stable equilibrium in the upright position if h? 2 {s^-s^f ■ If the body comes to rest when floating in water with its axis at an inclination B, then m must coincide with 0. But the curve BB^ being a parabola (§ 103), M hi 0'm=OB + BM+Mm = ^x+~+^ia.x)?e\ and therefore, if Om=OG, x^-hx + 2F + ^^tan^a = 0. Thus, for instance, if fig. 44 represents the cross section of a rectangular log of breadth h, and if JE just reaches the surface of the water in the position of equilibrium, x = h-ibta,ne, ¥ = ^-^h\ h l+2tan20 3-cos20 and 6 3 tan 3 sin 29 OF THE HORIZONTAL CYLINDER. 193 128. The Cylinder or Prism, floating horizontally. Now suppose that fig. 44 represents the side elevation of the same circular cylinder, when floating with its axis horizontal, and let fig. 45 represent the end elevation; so that FP' = 2y is the breadth and LL' = h is the length of the rectangular water line area, when the cylinder is floating at the draft OF=x. The centre G is the metacentre for displacements in the plane of fig. 45, and the equilibrium is therefore stable for these displacements. But for displacements in the plane of fig. 44, the equi- librium is stable if M lies above 0, or if Ak^=V.BM>V.BG. But Ak^==2hy.^h^ and by a well-known theorem, F.5(? = |%^ and therefore the equilibrium is stable if 2hy . r2-/i^>f %^ or h>2y. The cylinder will therefore float, like a cork, with its axis horizontal, if its length h is greater than the breadth 2y of the water line area ; but it will float, like a bung, with its axis vertical, if alh>J{2s{l-s)}, the least value of ajh being J^2 ; with intermediate dimensions the cylinder will float in an inclined position. When a cylindrical canister of thin material, whose C.G. H is at a height OH=h from the bottom, floating in water when empty with a length c of the axis immersed, is ballasted with liquid of S.G. s to a depth Of=x, a length OF=c + soc of the axis will become immersed; and as in iig. 43, p. 175, the system is equivalent to weights propor- tional to c, sx, and — (c-f-scc), concentrated at H, m, and M. 194 EQUILIBRIUM OF THE CONE The upright position of equilibrium will therefore be unstable or stable, according as c.OH+sx.Om-{c-\- sx)OM = ch+(s-l)k^ + \sx^ - Kc + sxf is positive or negative ; and h^ = la^ for a circular cylinder, F = ^-^h^ for a prismatic canister. If, as in § 115 and fig. 42, for the wall-sided ship, the line of draft OF is drawn in fig. 44, the curve of B will be the straight line OB, and the curve of metacentre M a hyperbola, with OD and OB as asymptotes. The curve of G for homogeneous cargo will be a hyper- bola, reducing to a straight line OG, if the weight of the vessel itself is insensible; and hence a graphical con- struction can be made for the conditions of stability in the upright position at different draft. The stability of a cylinder floating upright in two or more liquids (§ 125) would be illustrated by a cylindrical caisson of the Forth Bridge, floating partly in air, partly in water, and partly in the mud or quicksand at the bottom of the water. 129. The Gone, and the Triangular Prism. Unless otherwise stated, a homogeneous right circular cone on a circular base is intended when we speak of a cone ; and we denote the S.G. of the cone by s, the altitude OD by h, the radius of the base DE by a, and the semi- vertical angle by a. When s = J, the cone can evidently float with its axis horizontal and on the surface of the water ; the C.B.'s B aud B' of the equal immersed and unimmersed volumes AJSTD. TRIANGULAR PRISM. 195 V and V lying in a straight line BOB' perpendicular to the axis of the cone. Now, by well-known theorems of Mensuration, 0(? = P, BG = ^,~ = -- 4 tJTT TT and V. BM= A¥- = ah . ^\h^ (§ 40) ; so that this position of equilibrium is stable, if V.BM>V.BG, ah.^^h^>^Tra^h.—, or h^>3a^; TT that is, if the vertical angle 2a of the cone is less than 60°. 130. If the cone of S.G. s floats with its base horizontal and axis vertical and vertex downwards with a length 0F= X of the axis immersed (fig. 46), then x^ = sh^, since the volumes of the similar cones OLL' and OEE' are as the cubes of their altitudes x and h. Then, denoting FL by y, BG=l(h-x) = iO—si)h; BM=-yj^= 7 Y =T - = |a!tan2a; V t'^y X 4 X so that M is found geometrically by drawing Bb hori- zontally to meet the cone in b, and bM perpendicular to the surface of the cone to meet the axis in M. Also 0M= \x sec^a ; and the upright position of equi- librium is stable if OM > OG, or x sec^a > h ; or x> h cos^a, s > cos^a. The equilibrium is neutral when tana=(s-*-lA x = h cos^a, h — x = h sin^a ; and now oc(h — x) = x^ tan^a = y^, 196 EQUILIBRIUM OF THE CONE or the radius of the water line circle is the geometric mean of the segments of the axis of the cone made by the water line ; so that the sphere described on the axis OD as diameter will cut the cone in the circle of flotation LL' when the equilibrium in this position is neutral. Fig. 46. When a = ^x, then s = cos^a = \ ; so that a right-angled cone cannot float in water with its vertex downwards and axis permanently vertical, unless its s.G. s is greater than \. We infer, as in § 124, that the same conditions hold for the equilibrium and stability of the cone, floating inverted with its base submerged, if the S.G. of the cone is changed to 1 — s. When s = I, the cone can float in both positions ; and the equilibrium is neutral if cos^a = J. AND TRIANGULAR PRISM. 197 131. We may also take fig. 46 to represent the vertical cross section of an isosceles triangular log, floating horizontally ; and now „„ AW- 2y,iy^ 2 y^ 2 ^ „ V yx 3 X S so that M is found geometrically as before ; and OM = %xsec^a. The equilibrium is stable in this position if a; > ^ cos^a, as before ; but now s = x^/h^ > cos*a. If the cross section of the log is an equilateral triangle a = ^7r; and s>y*V> or 1 — 8 o^ generally h, where a = lir, while x^ = 2sh^ ; so that s>- ^S' 1 S < -g-^, giving the limits of the S.G. for which a diagonal of the log is vertical in a stable position of equilibrium. Fig. 47. 135. The Paraboloid and Parabolic Cylinder. In a parabolic cylinder floating with its axis horizontal the curves of flotation F and of buoyancy B are equal parabolas (fig. 47) ; while in the paraboloid generated by the revolution of the parabola about its axis OD, the surfaces of flotation and buoyancy are equal paraboloids ; also BM is equal to I, the semi-latus-rectum, and the equation connecting x = OF and y=FL is y^ = 2lx. AND PARABOLIC CYLINDER. 201 By well-known theorems of Mensuration, the area of the parabola LOL' is ^xy = %{nx^)^, two-thirds of the area of the circumscribing rectangle, and its O.G. B divides OF in the ratio of 3 to 2, so that OB = %OF; while the volume. of the paraboloid LOL' is ^TTxy^ — TrlxS one-half the volume of the circumscribing cylinder, and its C.G. B is situated so that OB = ^OF. The equation of the surface of buoyancy of the para- boloid, in the general case when the cross sections are elliptic, is now, according to § 103, 4 ir^/c. (8) A conical shell, vertex downwards, is filled to one- ninth of its depth with a fluid of density p', and then floats with one-third of its axis submerged in a fluid of density p ; flnd a relation between p and p', in order that the equilibrium may be neutral. (9) A frustum of a right circular cone, cut off" by a plane bisecting the axis perpendicularly, floats with its smaller end in water and its axis just half im- mersed. Prove that the s.G. is ^, and that the equilibrium will be neutral if the semi-vertical angle of the cone is tan"^— — - — 206 EXERCISES ON (10) A frustum of a cone floats with its axis vertical in a liquid of twice its own densitj'. Prove that the equilibrium will be stable if ri''<^ ^, where m = — ^ /> m-1' (a^ + b^)i h being the height of the frustum, and a and b the radii of its ends. Also if it floats with its axis horizontal, the equilibrium will be stable if a^+4!ab + b'^ (11) From a solid hemisphere, of radius a, a portion in the shape of a right cylinder, of radius b and height h, coaxial with the hemisphere and having the centre of its base at the centre of the hemi- sphere, is removed. Into this portion is fitted a thin tube which exactly fits it. The solid is placed with its vertex downwards in a fluid, and a fluid of S.G. s is poured into the tube. Find how much must be poured in, in order that the equi- librium may be neutral ; and if the tube be filled to a height 2h, show that s _ a'-2b%^ s'~ b' ' s' being the s.G. of the solid. (12) If a vessel be of thin material in the shape of a paraboloid of revolution, show that the equilibrium will be always stable, provided the density of the fluid inside be greater than that without ; the weight of the vessel being neglected. FLOATING BODIES. 207 (13) A thin hollow shell in the form of a paraboloid of revolution floats in water with its axis vertical and its vertex at a distance \;JU below the surface. Prove that if the equilibrium become neutral when water whose weight is equal to three times that of the shell is poured into it, the height of the centre of gravity of the shell above the vertex is is/'^^> where 21 is the latus rectum of the gen- erating parabola. (14) Prove that the work required to heel a ship of para- bolic cross section (§ 135) through an angle from the upright position is TTsin 0(iaan - 6 tan |0), where 6 denotes the distance GB. (15) A regular solid tetrahedron floats in water with a face horizontal and not immersed ; prove that the equilibrium is stable if the s.G. of the tetrahedron is greater than 0'512. Determine the limits of the S.G. between which the tetrahedron can float permanently with two edges horizontal. (16) A regular octahedron, S.G. \, floats with one diagonal or with two edges vertical ; find the metacentre. (17) To one angular point of a homogeneous regular tetrahedron of S.G. s is attached a particle of one- quarter (generally one-w*'') its weight. Prove that the tetrahedron can float in water with one edge vertical and another edge horizontal, if s = 0-2048, or, generally, s = l{\+n)-\ (18) A tetrahedron of weight W, whose base is a triangle ABC, and the angles at whose vertex are right 208 EXERCISES ON angles, has weights P, Q, R, whose sum is W, attached at A, B, G, respectively, and floats in a fluid with its base upwards and horizontal, the vertex being at depth h. Prove that ^ = 2m- 1 + (7 - 6m)cot 5 cot G, with two similar expressions for Q and R, where A,2= 4mV^cos A cosB cos G, and r is the radius of the circle circumscribing ABG. (19) Prove that a prism whose section is the acute angle triangle ABG cannot have three positions of equi- librium with the edge G alone immersed unless the S.G. is intermediate to {{a+h){a+b+c){a+b-c)}^-{(a-b){a-b + c){a + b-c)}^ 2bc J a^ + b^ — c^ and --—5 ^-r-„- 2a^ or 2¥ (20) Prove that the isosceles triangular log of § 134, if of density np, can float at an inclination 6 in two liquids of densities p and 2p, the upper liquid being of depth fji, with the base not immersed, provided (1+cos 20){w(cos 20+cos2a)-/x2}{TO(cos 20+cos 2a) + 2^2}2 = 2%2cos2a(cos 20 + cos 2a)* (21) Investigate the equilibrium and stability for finite displacements of the cone of Ex. (22), p. 188, con- taining liquid and resting in a circular hole ; or of a wedge-shaped prismatic vessel resting on two smooth parallel horizontal bars. FLOATING BODIES. 209 (22) A thin conical vessel, inverted over water, is depressed by a weight attached to the rim so as to be completely submerged, with a generating line vertical and the enclosed air on the point of beginning to escape. Prove that the ratio of the weight of the cone to the weight of water it can hold is f (cos 2a)i (23) Prove that the equilibrium of a homogeneous ellipsoid of s.Q. 0'5 floating in water is stable if the least axis is vertical, stable-unstable if the mean axis is vertical, and unstable altogether if the greatest axis is vertical. (24) A homogeneous ellipsoid floats in a liquid with its least axis GOG' vertical, and a weight W, f of that of the ellipsoid, fixed at the upper end G, such that the plane of flotation passes through the centre. Prove that, if it be turned about the mean axis (6) through a finite angle Q, the moment of the couple which will keep it in that position will be W{c- ae^cos 0(1 - e^cos^^) - i}sin 0, where e is the eccentricity of the section (a, c). (25) A light ellipsoidal shell partly filled with water is free to rotate round its centre which is fixed, and the shell, when in its position of stable equilibrium, is turned through any angle. Prove that the work necessary to eflect the dis- placement varies as a—p, where p is the perpen- dicular from the centre of the ellipsoid on the tangent plane parallel to the new surface of the water, and 2a the longest axis. 210 NUMERICAL CALCULATIONS 140. Ship Design and Calculation. The drawings of the outside surface of a ship give the equidistant contour lines in (fig. 48) (i.) the side elevation (sheer plan) ; (ii.) the end elevation (body plan) ; (iii.) the plan (half-breadth plan) ; these curves and planes may be supposed referred, as in Solid Geometry, to the coordinate axes. Ox vertical in the stem, Oz horizontal along the keel, and Oy perpendicular to the medial plane. 10 9}$ 9 Fig. 48. Considering that a ship is symmetrical with respect to this medial plane, a representation of one half of the ship is sufficient. The curves and surfaces employed here are not in general such as those just investigated, which are given by analytical conditions (curves et surfaces analytiques) ; but they are fair curves and surfaces drawn through guiding points and lines (curves et surfaces topogra- phiques) ; so that in the determination of the correspond- ing areas, volumes, C.G.'s, and moments of inertia, mechanical Planimeters must be employed, or else the methods of Approximate Quadrature. (Pollard et Dudebout, TMorie du navire.) IN NAVAL ARCHITECTURE. 211 Denoting the half breadths and half areas in the plane oiyOz by y and ^A, then in the notation of the Integral Calculus, \A=fydz, \Ay=f\yHz, \Az=fyzdz, \Ahl=f\yHz, ^Ahl=fyzHz ■ and denoting by V the displacement up ' to any given water line, y=fAdx, 'Vx=JxAdx, — But now, according to the methods of approximate quadrature, the half area ^A is found by dividing it up by equidistant ordinates 2/1, 2/2> Vsi-'-Vn, at intervals Az ; and then (i.) by the trapezoidal rule, in which the boundary curve is replaced by the broken line composed of the chords joining the tops of the ordinates 1^ = (^2/1 + 2/2+3/3+ • •• +2/«-i + 42/»)Az ; (ii.) by Si/mpson's rule, in which an odd number of ordinates and an even number of intervals must be taken, the curve joining the tops of three adjacent ordinates is replaced by a parabola whose axis is parallel to the ordinates. The area between the ordinates 2/2^-1 and j/sm+i is therefore composed of the trapezoidal part K2/2m-l + 2/2»+l)2A2:, and of the parabolic segment, equal to f of the circum- scribing parallelogram, and therefore § {2/2m- KVim - 1 + 2/2m+i) } 2 A^; = -- Kj/a™ - 1 - 22/2,„ + yi^+i)Az ; so that the whole strip is (2/2™-i+42/2m+2/2™+i)JAs! ; and therefore, by Simpson's rule, I A = (2/1 + 42/2 + 22/3 + • • ■ + 22/2n - 1 + 42/2» + 2/2«+i)i As. 212 NUMERICAL CALCULATIONS The other integrals required for finding the moments, first and second (moments of inertia), and for finding the volumes and their moments, being represented graphically by the areas of curves whose ordinates are ^y^, yz, ^y^, yz^, A and Ax, can be evaluated in the same manner bj'- Simpson's rule. Similar calculations for determining V are performed for plane sections perpendicular to the keel Oz {square sections), and used to check the former results ; some- times also, but rarely, for plane sections parallel to the medial plane xOz. Thus, for instance, if the half breadths of the water plane of a vessel are 3, 4-5, 9-2, 12-4, 139, 14-5, 14-3, 13-4, 11'6, 8'0, 2-4 ft, at an interval of 16 ft, the area of the half plane is 1678 ft^, and the c.G. of the half area is 100- ft from one end, and 4 ft from the middle line. So also, if the areas of the water line sections, reckoned downwards, are 4000, 4000, 3200, 2500, 1500, 600, 100 ft^, at an interval of 2 ft, the displacement is 27933 ft^ or 798'1 tons, and the c.b. is at a depth 3"98 ft below the load water section. As additional exercises the student may work out the half area and its c.G. for half breadths (1) 14, 33, 49, 57, 50, 40, 25 ft, at intervals of 30 ft ; (2) -5, 6, 10, 12-4, 12-5, 12-5, 12-5, 12-4, 123, 11, 8, -5 ft, at intervals of 12 ft. (3) Determine the displacement in tons and the posi- tion of the C.B. for the different water planes 3 ft apart, at which the tons per inch immersion are 27, 26, 24-8, 22-8, 20-5, 17-5, 13 ; the displacement below the lowest plane being 50 tons, with a C.B. 15 inches below this plane. IN NAVAL ARGHITEGTURE. 213 The Examination Papers for Science Schools and Classes, Part IV., Naval Architecture, may be consulted for additional exercises. Various empirical rules have long been in existence for giving the displacement or tonnage of a vessel, given the length L, breadth B at the water line, and D the depth or draft of water (Moorsom, Trans. I.N.A., vol. I.). The simplest rule gives iX-BxI>-^ 100 = displacement in tons of 100 ft^, which is afterwards multiplied by an arbitrary coefficient, ranging from 0'5 to 0"8, according to the fineness of the shape of the vessel. The ratio of the true displacement in ft^ to the volume LxBxD of the box-shaped vessel of the same dimensions is called the "block coefficient" or "coefficient of fineness"; thus for an ellipsoid half immersed this coefficient is i7r = 0-5236. 141. The Conditions of Equilibrium of a Floating Body partly supported. When a floating body is partly supported, for instance, a body suspended by a thread or fine wire and weighed in the hydrostatic balance, a ballcock in a cistern, a bucket lowered by a rope into water, a boat partly hoisted up by a rope, a ship aground or on the launching ways, or a diving bell, the body is in equilibrium under three parallel forces: (i.) the weight, (ii.) the upward buoyancy, and (iii.) the thrust of the support or pull of the rope, which must also be vertical ; and then the conditions of equilibrium of three parallel forces must be applied, generally by taking moments about any point in the line of the reaction of the support. 214 FLOATING BODY If the body can move freely on an axis fixed at a given inclination, the resolved part of the weight and buoyancy parallel to this axis must be equal, and there- fore the weight and buoyancy must be equal, as in a body floating freely; and, in addition, the moments of the weight and buoyancy about the axis must be equal. Sometimes a body is pushed or held down, or sinks to the bottom ; and when it is held completely submerged, as, for instance, a submarine mine, or a body floating up against the under side of a sheet of ice, the C.B. coincides with the centre of figure of the body. If the body weighs W lb and its s.g. is s, the buoyancy is a force of Wjs pounds ; the reaction TF/s— W of the ice therefore passes through a fixed point in QB produced, such that OB W s BG Wjs-W 1-s' and the positions of equilibrium are those in which the normal from to the surface of the body is vertical. Thus an elliptic cylinder of s.G. s, floating horizontally against the under side of a sheet of ice, or resting on the bottom of the water, has four or two positions of equi- librium according as lies inside or outside the evolute of the ellipse of cross section (§ 126). 142. Suppose a vessel, drawing a ft of water, to take the ground along the keel K, and that the tide falls from LL' to L-Jj-^, a fall of x ft, the buoyancy thus lost being P tons (fig. 42, p. 168). If the vessel heels through a small angle Q, an up- setting couple P . Kh . sin Q ft-tons is introduced, bj' which the metacentric height GM, as in § 118, will be diminished by {PIW)Kh ft. PARTLY SUPPORTED. 215 If the vessel is wall-sided between these two water lines, P/W=xA/V, and Kh=a-^x, so that the loss of metacentric height is (ax-^x^)Airit; and when this length exceeds GM at this draft, the vessel cannot stand upright. Precautions are necessary in launching and in docking and undocking a vessel, to prevent it from falling on one side, for the reasons given above. Suppose, for instance, a vessel which is "trimming considerably by the stern" is being docked; at the moment before the keel takes the blocks prepared to receive it, the thrust on the keel at the stern post can be calculated from the " tons per inch of immersion " and "the moment in ft-tons to change the trim one inch " ; thence the value of P can be inferred, and an estimate made of the stability of the vessel. (F. K. Barnes, Annual of the Royal School of Naval Architecture, 1874; Statics of Launching, Proc. U.S. Naval Institute, Julj^ 1892.) If the ship heels over to a new position of equilibrium by turning about the keel -ST as a fixed horizontal axis, the displacement does not keep constant, as the water planes will touch a circular cylinder fixed in the ship ; and if KF' is perpendicular to the water plane, the line of contact will pass through F'. Supposing that B^N denotes the depth of the new C.B. B^ below the water plane, and OH the height of the C.G. above, and that the displacement has diminished from V to Fg ft*, then in the position of equilibrium, K . F'N= V. F'H. 216 FLOATING BODY 143. When a body is lowered into water by a rope, the point K must be taken as the point of attachment to the body ; for instance, in lowering a bucket into water, K will lie in the axis of the pivots of the handle, say at D in fig. 44, p. 190. Suppose the bucket is cylindrical, and of height h from the bottom to the pivots of the handle, and that it would float upright in unstable equilibrium with a length 6 of the axis immersed, and that Z) is at a height c above G. The bucket will begin to leave the upright position when a length x of the axis is immersed, such that W downwards through and the buoyancy Wxlh upwards through M have equal moments round />, the tension of the rope being W{\ — xlh) ; and therefore 6\ ^ xJ or x^-2hx + 2¥ + '2J3C = Q, which gives the length of axis out of the water in the position of neutral equilibrium ; a greater length out of water will make the equilibrium stable, and vice versa ; also 2k^ = |a^ for a circular bucket of radius a (§ 40). In a long thin cylinder, like a spar, lowered by one end, we may put r = ; and now 6 = sh, if s denotes the S.G. of the spar ; also c = ^li, and then x-h = hj{l—s); so that the vertical position is unstable, and the spar will assume an inclined position if the vertical length out of water is less than ^(1 — s) times the length of the spar. 144. If the body in fig. 44 is supported or pushed down at any other point, say at 0, E, or E', and comes to rest at an inclination Q to the vertical, moments must be PARTLY SUPPORTED. 217 taken about this point, remembering that the weight and buoyancy are proportional to sh and x, the weight acting through G and the buoyancy through m, where Om is given in § 127. Similar investigations will hold for the other bodies shown in figs. 46 and 47 ; the reasoning is similar to that employed in § 118, and the point of support may now replace m, the metacentre of the liquid inside. 145. Suppose the cone in fig. 46 is fioating in water of depth OK = Xh, with its vertex touching the bottom ; then if the axis is inclined at to the vertical in the position of equilibrium, equating moments round of the weight acting downwards through G, and of the buoyancy JDV of the volume V of the cone OL^L^', acting upwards through B^ (§ 133), gives the equation W4h sin d = I)r.^KF^, or s tan^a sin 6 = X*sin cos sin^a cos a { sec(0 + a)sec(0 — a) } ,"* or X* = s sec 6 sec^a{cos{9+a)cos(9 — a)}*. If = 0, X* = scos2a, and the equilibrium is apparently neutral in the upright position, but really stable. If a generating line is vertical, 6 = a, and X* = s sec*a(cos 2a)^ ; thus if a = 30°, and X = |, then s = iV2. 146. If the body in fig. 46 is pushed down at A", the condition of equilibrium is obtained by taking moments about U' ; and, as in § 124, the same condition of equi- librium holds for the cone, of S.G. 1— s, floating in water with its vertex above the surface, and its base touching the bottom at U'. 218 FLOATING BODY As an exercise, the student may prove that if the cone is pushed down at E' till E' is brought to the surface of the water, 2s cos^(0 — a)(sin cos a + 4 cos 6 sin a) = cos^(0 + a)(sin 20 + 4 sin 2a). Similarly, if fig. 46, p. 196, represents a triangular log, touching the bottom of the water at 0, with L^L'^K as the water line, we shall find \^ = s sec 6 sec^a{cos(0 + a)cos(0 — a)}^ ; and so on. If the log is pivoted about a horizontal axis through 0, then the inclined position of equilibrium will be stable if B^O, and therefore also FJD is vertical, or KF^ = hsmQ. If the axis lies in the surface of the water, and therefore (§ 133) f cos^0 = cos(0 + a)cos(0 — a) = co&^Q — sin^a, or cos^0 = 3 sin^a. Similarly for a cone cos Q = 2 sin a. But if the angle Q given by this equation is not real, the stable position of equilibrium is vertical. So also for the other bodies, if supported or submerged by the tension of a rope, or if resting on the bottom, as in the case of a ship with a rounded bottom, ashore on a sandbank or convex rock. In this case the point of application of the lost buoy- ancy P in the neighbourhood of the upright position of the ship must be changed from the keel K to the point Q where the vertical through the point of contact with the rock cuts the circle of inflexions in the ship of the two surfaces in contact (§ 121). PARTLY SUPPORTED. 219 147. The Isobath Inkstand. In this ingenious instrument (fig. 49) the ink is main- tained at a constant level in the dipping well A by means of a semicircular or hemispherical float F, pivoted about a diameter as axis, in the reservoir of ink G. If the S.G. of the float is exactly ^, the s.G. of the ink being unity, it is easily seen that the float will be in equilibrium when it has turned so as to raise the surface of the ink to the level of the axis F. For drawing FB, equally inclined as OE at an angle 6 to the horizon, the part DFE' of the float, supposed homo- geneous, will balance over the axis F; so that the moment of the buoyancy of EFL must balance the moment of the weight of the part DFE of the float ; and these moments are equal if the s.G. of the float is J. Fig. 49. If however the S.G. s of the float is a little greater than I, the level of the ink will be raised slightly above F. If the rise of level above F is x for a semicircular float of length I and radius a, the moment alx of the extra buoyancy of the ink equated to the moment of the extra weight in DFE' gives 220 EXERCISES ON FLOATING Balx . |a = (s — \)BaHO . fa — h-. or a; = (s— J)f a sin 0; and if s < J, the level will fall slightly below the axis F ; the level will also fluctuate to a slight extent. A similar investigation will hold for a hemispherical float. The same principle has been employed in some oil lamps, for maintaining a constant level of the oil in the wick; as in Hooke's or Milner's oil lamp, described in Young's Lectwres on Natural Philosophy, and De Mor- gan's Budget of Paradoxes, p. 149. Exam/pies. (1) A uniform rod rests in a position inclined to the vertical, with half its length immersed in water, and can turn about a point in it at a distance equal to one-sixth of the length of the rod from the extremity below the water. Prove that the s.G. of the rod is 0'125. (2) Three uniform rods, joined so as to form three sides of a square, have one of their free extremities attached to a hinge in the surface of water, and rests in a vertical plane with half the opposite side out of the fluid. Prove that the s.G. of the rods is 0775. (3) AB and BG are two rods hinged at B, the former being heavier and the latter lighter than water. The two rods float in water with the end A freely hinged at a flxed point in the surface of the water. Prove that the S.G. of the rod AB is equal to ah + ^hy-'iy'^ ab BODIES PARTLY SUPPORTED. 221 where 2a, 26 are the lengths of the rods, and 2y the length of the immersed portion of the lighter rod. (4) A hollow metal sphere of radius 6 inches would float freely in water with half its surface immersed. It is attached rigidly by means of a weightless rod to a tap, so that the distance of the centre of the sphere from the tap is 20 inches. The water rises to a height of one inch above the centre of the sphere before the tap turns, and the tap is then 11 inches above the surface of the water. Find the couple necessary to turn the tap. (5) A rectangular parallelepiped, moveable about one edge fixed horizontally, is partly immersed in water. If it can rest with another edge in the surface, and having one of its faces containing the fixed edge bisected by the liquid, find the s.G. of the solid ; and if the section perpendicular to the fixed edge be a square, prove that the s.G. of the solid is 0'75. Prove that, if the body rests with half its volume immersed and its faces equally inclined to the ver- tical, the S.G. is l + i (breadth by height). (6) A square log rests with one edge immersed in water, and partly supported along two parallel edges in the surface of the water, a distance c apart. Prove that the inclination of a diagonal to the vertical in a position of equilibrium is given by the equation cos 20 = 6s^'(l -IJ"^ sec q), where s denotes the S.G. and h the side of the square cross section of the log. 222 EXERCISES ON FLOATING (7) A square log of density np floats in two liquids of densities p and 2p respectively ; one edge is fixed in the surface of separation of the liquids, about which the log is capable of rotating, the whole of it being immersed. Show that, if ^n > 3, the angle 6 which the side through the fixed axis, which lies in the upper liquid, makes with the horizon is given by tan^e + (3% - 4)tan + 3(w - 2) = 0. (8) A homogeneous log, the cross section of which is a regular hexagon ABGDEF, can turn freely about a horizontal edge at A which is in the surface of water. If in the position of equilibrium AB lies above the water and half of BG is immersed, prove that the S.G. of the log is \\ 1 2- (9) A regular tetrahedron has one edge fixed in the surface of water. Show that it will be in equilibrium with the other edge inclined to the vertical at an angle cosec"^3, if the s.G. of the tetrahedron is 0-294375. (10) Prove that, if the bodies represented in figs. 46 and 47, floating in the upright position, are divided symmetrically by a vertical plane through DO into two parts, which are hinged together at 0, the parts will not remain in contact unless (i.) in the triangular prism or cone (fig. 46), x>h sin^a, sin a < s* or s* , (ii.) in the parabolic cylinder or paraboloid (fig. 47) p/?>s-i-s-S, or 3^/Z>s-«_s-*. BODIES PARTLY SUPPORTED. 223 (11) A spherical shell is floating in water, and is divided by a vertical plane into two halves, which are hinged at the lowest point. Show that, if the parts remain in contact, sin a(2 + cos^a) — 3a cos a > 7r(2 + cos a)sin*Ja, where a is the angle subtended at the centre of the sphere by any radius of the water line. Find the condition when the sphere sinks to the bottom. (12) Prove that half a paraboloid, cut off by a plane through its axis, just immersed in water against a rough vertical wall with its base in the surface, will be in equilibrium if the height is greater than three and a half times the semi-latus-rectum. (13) Prove that the equilibrium of a non-homogeneous sphere of radius h, whose C.G. is at a distance h below the centre, resting partly submerged in water on the top of a fixed rough spherical surface of radius a and depth of highest point c, will be stable if its weight is less than 1 c\2,h-c) 4x, where to denotes the uniform cross section of the tube, the column will oscillate like a pendulum of length ,_ W _ pk + pa + ah 2p'cio 2p' where W denotes the weight of the liquids, and a the length of the filament in the bend AB. Thus if p = p' = a; l = ^{a + h). {Princifia, lib. II., Pi-op. XLIV.) Suppose, however, that the branches of the tube are not vertical, but curved, so that the inclinations to the vertical at the points A, B, H, K are a, /8, Q, <{>. Then to push the column through a small distance x from its position of equilibrium by a piston at H will require a thrust reaching from zero to po}(x cos (p + k — x cos /3) + p'uix cos /3 + x cos a) — o-u){x cos a + h — x cos 9) = {p cos (p + {p' — p)cos^ + (p' — ar)cos a + o-cos djcox, so that, if the piston is removed, the column will oscillate like a pendulum of length i^ m^ . pcos + cos 6 Similarly for the oscillations of any number of liquids in a uniform bent tube ; but if the bore of the tube changes, as in a marine barometer, the problem is com- plicated by the variations of velocity in the tube. HARE'S HYDROMETER. 237 162. Hare's Hydrometer. This is an application of the principle of the Theorem of § 158 ; it consists of two vertical glass tubes AH and BK, dipping into vessels at A and B, containing two liquids whose densities, a and p, are to be compared (fig. 51). The upper ends of the tubes are cemented into a receptacle G, from which the air can be partially exhausted by an air pump or other means ; and the liquids now rise in AH and BK to heights h and h above their level at A and B, which heights are inversely as the densities, or such that (rh = pk, or p/(7 = h/k. An apparent tension draws up the liquid columns in the tubes ; for this reason any small pressure below the atmospheric pressure is sometimes called a tension, because the difference between this pressure and that of the atmosphere is a negative pressure, or a tension thus it is usual to speak of the tension of aqueous and other vapours; but the word tension is sometimes im- properly applied to very high pressures, such as those due to the gases of fired gunpowder. Examples. (1) Two equal vertical cylinders of height I stand side by side and there is free communication between their bases. Quantities of two liquids of densities Pj^, pj which would fill lengths a and c respectively of the cylinder, are poured in, and rest in stable equilibrium, each liquid being continuous. A given quantity of a liquid of density yog, intermediate between yo^ and p^ is poured slowly into one of the cylinders. 238 EXAMPLES ON EQUILIBRIUM Find the position of equilibrium, noticing the different cases which may occur; and show that if the liquid reaches the top of both cylinders at the same time, either {p-i_-Pi){^l-a-c) = (p.^-p^c, or (pi-/32)a = (/32-/53)c. (2) If two equal vertical cylinders in communication at the base are partly filled with mercury, and closed by pistons which are allowed to descend slowly, prove that air will not pass from one cylinder to the other if the difference of weights of the pis- tons is less than the weight of mercury ; and find the position of equilibrium. (3) A vessel, in the form of a cylinder with its axis verti- cal, is partially filled with water. The lower part of the vessel communicates with a reservoir of infinite extent; and a body, in the form of a cylinder, floats with its axis vertical, in the vessel. Fluid of less density than water is now slowly poured into the vessel. Find the quantity which must be poured in before the floating cylinder begins to move — (i.) when the density of the second fluid is greater, and (ii.) less than that of the cylinder. (4) A small uniform tube is bent into the form of a circle whose plane is vertical, and equal volumes of two fluids whose densities are p, a, fill half the tube ; show that the radius passing through the common surface makes with the vertical an angle tan-i^:::^- p + a- and find the period of a small oscillation. IF A BENT TUBE. 239 (5) A circular tube contains columns of two liquids whose densities are p, p, the columns subtending angles 2Q, 20' at the centre of the circle. If a be the angle which the portion of the tube intercepted between its lowest point and the common surface of the liquids subtends at the centre of the tube, prove that p sin Q sin(0 ± a) = p'sin 0'sin(0' + a) ; and find the period of a small oscillation. (6) A tube in the form of an equilateral triangle is filled with equal volumes of three fluids which do not mix, and whose densities are in Arithmetical Pro- gression ; prove that in equilibrium in a vertical plane the straight line joining the ends of the fluid of mean density will be vertical. Generally, if a fine tube of uniform bore in the form of a closed regular polygon of n sides is filled with equal volumes of n liquids, and held in any vertical position, the lines joining the surfaces of separation of the liquids will form another poly- gon, the sides of which have fixed directions ; and if the c.G. of the liquids is at the centre of the polygon, the liquids will rest in any position. (Wolstenholme, Proc. London Math. Society, VI.) Find the period of a small oscillation. (7) A fine tube ABG, of uniform bore, having the parts AB, EG straight and inclined at an angle 2y to one another, is held in a vertical plane, and con- tains several liquids of different densities. If the tube is turned about the point B in its own plane, and if a, j3 are the inclinations to the vertical of the straight line bisecting the angle 240 EXAMPLES. ABC in the two positions, prove that the weight of liquid which has passed from one branch to the other bears to the weight of the whole the ratio |(tan a — tan ^)tan y to 1. (8) A fine tube bent into the form of an ellipse is held with its plane vertical, and is filled with n liquids whose densities are p^ p^, ■■■, pn taken in order round the elliptic tube. If r.^, rg, ..., r„ be the distances of the points of separation from either focus, prove that ''l (Pl - Pi) + '^2^/'2 - Ps) + ■ • ■ + '^nipn - Pi) = 0. State the corresponding theorem, if the fiuids do not fill the tube ; and find the period of a small oscillation. (9) A cycloidal tube contains equal weights of two liquids, occupying lengths a and h; if it be placed with its axis vertical, prove that the heights of the free surfaces of the fluids above the vertex of the tube are as (3a +6)2 to (.36 + a)2. (10_) An uniform tube is bent into the form of a cycloid, and held with its vertex downwards and axis vertical. It is then partly filled with mercury, S.Q. 13'5, and chloroform, s.G. 1'5. Show that if the volume of the chloroform be three times that of the mercury, their common surface will be at the lowest point of the tube. (11) Prove that the finite oscillations of a filament of liquid in a circular or cycloidal tube, or a tube of any shape, can be compared with those of a particle at the middle point of the filament. DILATATION BY HEAT. 241 163. Dilatation and Coefficients of Expansion. The usual formulas and approximations employed in the measurement of dilatation may be explained at this stage. By the application of heat, liquids and solid bodies expand in general, by a fraction which is sensibly pro- portional to the rise of temperature. If a homogeneous solid body expands equally in all directions, so as to remain always similar to itself, and if corresponding lengths, areas, and volumes become changed from I, A, and V to l + Al, ^-t-A^, and F+AF, then Aljl, AA/A, and A F/F are called respectively the linear extension, the areal expansion, and the cubical expansion or dilatation. Since the body is supposed to remain similar to itself, therefore but considering tliat Al/l is in general so small that its square is insensible to the number of decimals to which Aljl is given in the formulas, we may put AA_^M AV_ Al A "^r F"'^; ' so that the areal and cubical expansions are respectively twice and thrice the linear extension. As it is found experimentally that the linear, areal, and cubical expansions are sensibly proportional to the increase of temperature t, therefore Aljlr, AA/At, and AF/Ft are sensibly constant; denoting them by X, fi, and c, and calling them the coefficients of linear exten- sion, of areal, and of cubical expansion, then /j. = 2\, c = 3X. G.H. , y = ic. Then, approximately, h = l{l+^(t-T)-y{t-32)} The correction therefore vanishes when the temperature 32y-^y . it is subtractive for higher and additive for lower tem- pei'atures. Taking 2'= 62, and the coefficients of cubical expansion of mercury c or y, and of linear expansion of brass b or /3, as given by c = 0-00018018, y = 000010()l, 6 = 000001878, /3 = 0-0000104, c/6 = y/^=9-6, then the correction vanishes when t = 28'3 F, and is subtractive for higher temperatures. STANDARDS OF LENGTH. 257 176. Temperature Correction in Standards of Length. No temperature correction is required in Standards of Weight, but an accurate scale of Length has engraved upon it the temperature at which its indications are correct. Thus the British Standard Yard is correct at 62° F, and the French Mfetre des Archives at 0° C ; and if the brass scale of the barometer is engraved with true inches at 62° F or 16|°C, and with true millimetres at 0° C,then at the higher temperature 62 F a scale millimetre division has stretched by (62 -32)/3 = 0-000312 mm, = 0'312/x (microns) or 312 /xpi (micromilli metres). In scientific work the relation 1 m = 39-3704.32 (39-37) ins, 1 in = 2-539977 (2-54) cm, is generally employed; but if, in accordance with the Act of Parliament, 1878 (§ 8), we take 1 metre = 39-37079 ins, 1 inch = 2-53995 cm, then, at 62° F, the scale metre = 39-37079 -=-1-000312 = 39-3.5851 ins; and, as the scale elongates or contracts uniformly with the temperature, this will be the invariable relation con- necting the scale metres and inches ; thus 30 scale ins = 762-22 scale mm. For instance, in fig. 58, a simultaneous barometric reading in the two graduations gave 29-482 ins, and 748-70 mm. Now 29-482 scale ins = 29-482-r-0-0393.585 = 749-06 scale mm. 748-70 scale mm = 748-70 X 0-0393585 = 29468 scale ins ; so that the mm scale reads 0014 ins or 036 mm below the inch scale, and is therefore this distance too high, if the inch scale is correct; but of this discrepancy, 0-22 mm 258 THE VERNIER. may be accounted for by the appearance of the graduation of 30 ins at the level of 762 mm, instead of 762-22 mm. (Standards of Length, Pratt and Whitney Co., 1887 ; Unites et ^talons, C. E. G-uillaume, 1893.) 177. The Vernier. To make the scale reading I to fractions of a scale division, the Vernier is employed (fig. 68, p. 2.52). This consists of a sliding piece of metal, the zero of which is brought to the level of the top of the mercury column ; and now if the vernier is to read upwards to one nth of a scale division, the length of vernier is made equal to to — 1 scale divisions, and it is then divided into n equal parts. Each scale division is therefore greater than a vernier division by one n-th. of a scale division ; so that if the i'-th vernier division coincides with a scale division, the extra fractional part of I is r/n of a scale division ; the number r is called the least count of the vernier. In fig. 58 the scale is shown to full size, divided into inches and twentieths of an inch on the right and into centimetres and millimetres on the left ; and 24 parts on the right are taken and divided into 25 parts to form the right hand vernier, while 19 parts on the left are divided into 20 for the left hand vernier ; the verniers therefore read to two-thousandths of an inch, or five-thousandths of a centimetre : and the reading of the verniers is 29-482 inches, or 748-70 mm. The vernier might also be made to read downwards to one n-th of a scale division, by taking n+l scale divisions and dividing them into n equal parts to form the vernier ; and now each scale division is less than a vernier division by one nth of a scale division. MAGNIFICATION OF THE MERCURY. 259 178. It will be noticed that while in a thermometer the mercury in the stem appears like a filament, the spherical bulb of the thermometer and the tube of the barometer appear like solid mercury ; so also the tube of the gauge glass of a boiler appears like solid water for the part occupied by water; and this water will appear coloured if a thin line of red glass is fused into the back of the tube. This appearance is due to the Law of Eefraction in Optics ; from which it is easily seen that the magnifica- tion of the mercury column is equal to the index of refraction ; so that the tube will have the appearance of a thermometer or barometer according as the ratio of the external diameter to the diameter of the bore is greater or less than the index of refraction of the glass ; the value of this index is about 1'5. So also for a spherical shell of glass, filled with mercury or liquid, like the bulb of a thermometer. 179. The siphon barometer (Gay-Lussac's) can also be employed for scientific purposes if the level of each sur- face at B and H is measured on the scale 8, which is made vertical ; and now there is no need to take account of any variation in the cross section of the tube, although the double correction for capillarity at the two surfaces B and H is troublesome. The ordinary Weather Glass is a siphon barometer in which a float in the mercury at B turns a dial hand by means of a vertical rack or by a thread and counterpoise, actuating the axle of the dial (fig. 67, p. 252) ; but this instrument is not capable of scientific accuracy, because of the influence of the varying cross section of the tube and cistern. 260 THE WEATHER GLASS ANT) Thus if a denotes the cross section of the tube and /3 of the cistern and if the float B descends a distance x while the surface H rises a distance y, then, the volume of the mercury remaining unchanged, ^x = ay, and the change in barometric height and X and y thus depend on ,8/a, which may change with the shape and temperature of the tube. The Marine Barometer is constructed on this system, and suspended from gimbals so as to hang vertically ; the graduations for inches on the scale being shortened by the factor y8/(a + /3) ; and the intermediate portion of the tube AH 'w, contracted to a small bore, so as to make the oscillations of the mercury sluggish, and to prevent the so-called " pumping," due to the motion of the ship. With a uniform bore the oscillations of the mercury would synchronize with a pendulum of half the length of the mercury column (§ 161). 180. In Sir Samuel Morland's steelyard or balance barometer (1670) the tube is suspended freely in the cistern from the arm of a balance (fig. 59, p. 252) ; and it is found necessary to counterpoise not only the weight of the tube, less its buoyancy in the cistern, but also the barometric column of mercury in the tube, or its equivalent thrust of air on the top of the tube. Measuring upwards from the bottom of the cistern, let X cm denote the height of the lower end of the tube, y cm the height of the surface of the mercury in the cistern, and y+hcra the height of the mercury in the tube, corresponding to a barometric height h cm. THE STEELYARD BAROMETER. 261 Reckoning pressure in cm of barometric height, and weight in cm^ of mercury, let the glass tube weigh ilf cm^ of mercury and contain V cm^ of mercury when full. Then, if the internal length of the tube is I cm, and its internal cross section at the top is a cm^, the tube now contains V— {x -\- 1 — y — h)a cm^ of mercury; and there- fore the equilibrium of the tube, if counterpoised by a weight of W cni^ of mercury, requires W=M+h^+V-{x+l-y-h)a-{h + y-x)^,{l) if yS denotes in cm^ the external cross section of the lower submerged part of the tube. But if the total quantity of mercury is U cm?, and the cross section of the cistern is y cm^, U^yy-{y-x)^+V-{x + l-y-h)a, (2) and therefore W=M+U-yy (3) We may suppose the counterpoise W to be constant, and then y is also constant, so that the level of the mercury in the cistern does not change. But if Kx denotes the change in x due to a change of Mh in h, then, from (1) or (2), a^h-{a-fi)^x = ^), Aa;_ a and in this way a continuous magnified mechanical register of the fluctuations of barometric height can be obtained by a pen attached to the counterpoise, tracing a line on a uniformly revolving drum. By sufficiently increasing the length y — xoi the sub- merged part of the tube, the buoyancy of the mercury could be made sufficiently large to dispense with the 262 THE WATEH AND GLYCERINE BAROMETER. counterpoise, and the tube would now float freely; but the stability would now require separate attention. 181. A barometer consisting of a column of mercury in a straight vertical tube, slightly conical in the bore, was suggested by Amontons (1695). As the atmospheric pressure increases the column of mercury rises slightly in the tube, and at the same time elongates, so as to come to a new position of equilibrium ; but the instrument is not of practical utility, as a shake is liable to spill out the mercury. We may notice here that, as the pressure in the mer- cury is greater than the atmospheric pressure in the cistern, a crack or leak in the cistern if below the level of the mercury will allow the mercury to escape till the level is lowered below the crack ; but as the pressure in the tube is less than the atmospheric pressure, a craCk in the tube will admit the air, and destroy the column. It may happen that the air which entered will drive the mercury above the leak to the top of the tube, thus filling up the Torricellian vacuum space. If membranes are used to cover these leaks, they will be found correspondingly bulged outwards or inwards. 182. The Water and Glycerine Barometer. If the mercury is replaced by some lighter liquid, say water or glycerine, the height of the barometer and its fluctuations are correspondingly increased, in the ratio of the S.V. of the liquid employed. A water barometer, constructed by Prof. Daniell {Phil. Trans., 1832) stood formerly in the hall of the Royal Society, and was afterwards placed in the Crystal Palace ; the mean height of the water column was 33J ft or 400 ins. HUrGENS'S BAROMETER. 263 Glycerine is now employed instead of water, as less liable to vitiate the Torricellian vacuum. In the Jordan glycerine barometer, at the Times office, employed for the meteorological records, a height of about 329"2 ins of glycerine corresponds to 30'61 of mercury; the S.G. of glycerine is thus about 1'262, the S.G. of mer- cury being 1S669. 183.- Huygens's Barometer. Huygens made a suggestion (1672) for magnifying the fluctuations of the barometer without greatly increas- ing its height by the combination of a column of mercury and of» water superposed, the common surface of the mercury and the water being placed at an enlarged part of the tube, so that a small increase of the height of the mercury column should cause a magnified motion of the upper surface of the water. Suppose then that A.^ denotes the cross section of the upper tube of water, A2 of the middle enlargement where -the mercury and water are in contact, J., of the lower tube of mercury, and A of the free surface of the cistern (fig. 60, p. 252). Then if x-^ and x.^ denote the heights above the top of the cistern of the upper surfaces of the water (or glycerine) and the mercury, and x denotes the depth below the top of the cistern of the free surface of the mercury ; if h denotes the height of the standard barometer, a- the density of mercury, and p the density of the water (or glycerine), a-h = p(x.^ — x^) + a-{x.^ + x). Denoting by Ah any change in k, and by Ax the coi- responding change in x, A-^Ax^ = A^Ax^ = AAx ; 264 HUYGENS'S BAROMETER. and therefore giving the fJuctuation of level in the cistern, which would be recorded by a float ; while -.=A"x/Wi-i;)-(i.-l)}. the fluctuation of the top of the barometric column. Thus for example, if the cistern is sufficiently large for IjA to be neglected, and if we take cr//o = 13, then we shall find that the ratio A^^'iOA.^ will make A£C^ = (5AA. about, so that the fluctuations of the top of the column are magnified six times. 184. Generally in a barometric column composed of n superincumbent liquids of densities Pv Pi- ■■■ Pn, reckoned from the top ; if denote the lieights above the top of the cistern of their upper surfaces, and A A 4 V 2' • • * -^^«» the cross section of the column at these levels ; if a; denotes the depth below the top of the cistern of the free surface of the liquid, of density p„, and A the area of this free surface, then compared with a standard mercury baro- meter, of height h and density cr, the corresponding fluc- tuations of level are given by the relations A.^Ax^ = A^A.i:2= ... =^„Aa;„ = J.Aa; crAh this is left as an exercise. THE WEIGHT OF THE ATMOSPHERE. 265 185. The Aneroid Barometer. This is an application of the principle of Bourdon's Pressure Gauge, described in § 10, to the measurement of small variations of external atmospheric pressure ; to make the instrument sufficiently sensitive the flattened curved tube must now be made very thin ; but in other respects the arrangement remains the same. The Aneroid Barometer is now more often made with a corrugated box, exhausted of air. A Brief Historical Account of the Barometer, by William Ellis, Q. J. Meteorological Society, July 1886, maj' be consulted for further details concerning the Baro- meter ; also the Smithsonian Meteorological Tables. 186. The Weight of the Atmosphere. If the two branches of the bent tube in fig. 50, p. 234<, are vertical and of the same uniform bore, we notice that the weight of each liquid above the horizontal plane of separation AB is the same, as in Hare's Hydrometer (fig. 51, p. 234). Thus if the bore of the siphon barometer (fig. 57, p. 252) is uniform, the weight of superincumbent air in the upward prolongation of the cistern as an imaginary vertical tube reaching to the limit of the atmosphere will be practi- cally equal to the weight of the mercury in the column AH ; consequently the " weight of the atmosphere " is practically the same as that of an ocean of mercury cover- ing the Earth, of uniform depth h, about 30 ins. or 76 cm, the average barometric height, and hence the name baro- meter, as measuring the weight of the air. This weight is the same as that of an ocean of fresh water about 34 ft or 10'33 m deep, or of sea water about 33 ft or 10 m deep. 206 THE WEIGHT OF THE ATMOSPHERE. Professor Dewar remarks in his Eoyal Institution Lecture, June 1892, that if the Earth were cooled down to about —200 C, the atmosphere would form a liquid ocean about 35 ft deep, of which about 7 ft at the top would be oxygen. Taking the atmospheric pressure as 1 kg/cm^, there will be about one kg of air per cm^ of the Earth's suiface ; with a quadrant of 10^ cm, or a radius of IO^-j-^tt cm, the surface of the Earth will be 47r X 1018 H-|7r2= lOis X 16-=-7r = 10i8 X 5-093 cm2; and this number v\ ill be practically the number of kg of air in the atmosphere. More accurately, with an average barometric height of 76 cm, and a density of mercury 13'6 g/cm^, the atmos- pheric pressure is 1'0336 kg/cm^ ; and the weight of the atmosphere is about 10^® x 5'264 1. According to Cotes (p. 94) this is the weight of a sphere of lead about 60 miles in diameter. If R cm denotes the radius and p the mean density of the Earth, the weight of the Earth _ I '^''jO-R^ _ p-B _ , /^5 i.toq the weight of the atmosphere 47ro--R% 'iah, ' on putting ie = 108 -^^7r,/i = 76, o-= IS'fi, p = 5-5. 187. A calculation is given in the Traite de I'dquilibre des liqueurs et de la fesanteiir de la masse de I'air, Blaise Pascal, 1653 ; he takes the 90° of the quadrant of the meridian as 1800 leagues, a degree = 50,000 toises, a toise = 6 ft, a ft^ of water =62 livres, and the mean height of the water barometer as 31 Paris feet ; and thence finds that the atmosphere weighs 10^^ x 8'284 livres. Mascart asserts {Gomptes Rendus, 18 Jan. 1892) that allowing for the curvature of the Earth, and for adiabatic METEOROLOGY OF THE BAROMETER. 267 expansion, the atmosphere really weighs about one-sixth more than a sea of mercury 76 cm deep, or of water about 10 m deep. 188. Meteorology of the Barometer. By simultaneous observations at a number of widely scattered meteorological observatories the height of the barometer, corrected by the thermometer, is registered ; and thence the isobars, or isobaric lines, lines on the Earth along which the barometer stands at the same height at a given time, can be plotted, as in the weather charts issued from the Meteorological Office. The gradient of the barometer, that is its most rapid rate of rise or fall in the direction perpendicular to the isobar, usually measured in hundredths of an inch per 15 miles, can thence be calculated ; and from these observa- tions much weather lore and prophecy valuable in Navi- gation can be deduced. (A Barometric Manual for the use of Seamen, issued by Authority of the Meteorological Council, 1890.) Given the height of the barometer a, b, c at three diffe- rent stations A,B,C, not too far apart, the gradient of the barometer is the same as that of the inclined plane which passes through the top of three posts of height a, b, c erected at sea level at A, B, G ; and the isobars will be the lines of intersection of parallel planes with the hori- zontal plane. The geometrical construction of these isobars depends on the problem of dividing an angle into two parts whose sines are in a given ratio; for instance to draw the isobar ^Ay through A, where the barometer is highest suppose, we must determine ^Ay so that the perpendiculars on it B^,Cy from B, C are in the ratio a — b : a — c. 268 ISOBARIC LINES. Otherwise, P and Q being the points on the lines AB and AG where the height of the barometer is x, then PQ is an isobar, and the parallel isobar /3y through A is such that amPA^ AP_ABa-c sin QAy~ AQ~ AG a-b' , AP a — x. AQ a-x because -irf, = r, -aTi= ^■ AB a—h AG a—c 189. If the gradient of the barometer exists over a sheet of water like an inland lake, then since the surfaces of equal pressure in the water are horizontal planes (§ 19) it follows that the free surface will no longer be horizon- tal, but will have an opposite gradient 13'6 times the gradient of the barometer, 136 being the S.G. of mercury. To find the rise and fall of the water in the lake at any point we must draw the isobar through the C.G. of the surface of the lake ; and now the free surface will be a plane passing through this nodal isobar, at the incline of the enlarged gradient opposite to that of the barometer; for in this way, according to the theorem of § 101, the total quantity of water in the lake will be preserved unchanged. Thus if the gradient of the barometer is two-hundredths of an inch per 15 miles, the surface of the water will have an opposing gradient of 272 hundredths ; so that in a lake 180 miles in diameter this will amount to a rise at one end and a fall at the other of 1044 in. The sheet of water over which there is a variation of barometric pressure must be of considerable size, like the American Lakes, for an appreciable rise and fall of the water to be produced. EXAMPLES. 269 Over the Ocean it is the variations of the gravity gradient due to the perturbation of the Moon and Sun, which produce such marked phenomena as the Tides, although insensible to the most delicate plumb-line ob- servations ; and when the tidal current is constrained in narrow waters, the average gradient of the sea at any instant between two places may become easily perceptible, being the difference of height of water above mean sea level divided by the distance between the places. Examples. (1) A straight pipe 40 feet long 6 inches in diameter, closed at the top and full of ice, is inverted in a barrel a yard in diameter. Given the specific gravity of ice 09, and the height of the water barometer, 30 feet, show that when the ice melts the water will rise 2 inches in the barrel. (2) A barometer has a dial attached to it, and if the tube were cylindrical the markings on the dial would be at equal distances, but the small arm is really a cone of small angle. If a^, a^, a^ be three con- secutive angular intervals on the dial, show that a\{a, — a,) + a\{ar — U',) + 2aj,a,a^ = a\{ap + a,.) . Verify this result when the tube is cylindrical. (3) From the following data obtain the true reading, the barometer being placed at a height of 20 feet above sea level. Barometer reading, 29'5 in ; attached thermometer, 20°C ; ratio of area of sec- tion of tube to section of cistern = 1 : 41. Capillary action -|- '04 in ; fall in barometric height for each foot above sea-level, -001 in; co- efficient of expansion of mercury for 1°C = ■00018. 270 EXAMPLES. When the mercury in the cistern is at the zero of the scale, supposed marked on the tube, the mer- cury in the tube stands at 30 in. (4) A thin weightless cylindrical shell of radius a, closed at the top, stands in a large basin which contains a depth h of mercury ; the mercury stands in the cylinder at the barometric height h. Prove that, if the cylinder be turned about a point in the rim of its base, it will tend to return to its original position so long as the inclination of the axis to the vertical is less than . , 2a '''' h+^' provided that no part of the rim has reached the level of the mercury in the basin, and that the mercury in the cylinder has not reached the top. (5) A very wide cylindrical tube, closed afc the upper end, rests on the bottom of a flat dish of mercury, with the air inside the tube partially exhausted ; find the condition that it be not lifted up by buoyancy. Show also that when the tube is bent over from the vertical it will tend to come back again so long as the centre of gravity of the tube, together with the portion of the mercury above the open level in the dish, less the mercury below that level which is displaced by the tubes, does not fall outside the point of support. (6) A barometer consists of a vertical tube closed at the top, the diameter of which changes at the middle, so that the area of the transverse section of the upper portion of the tube is A and that of the lower portion B. THE SIPHON. 271 The tube contains a volume V of mercury, which is supported by the pressure of the air on its lower surface, the space above the mercury being free from air, and V being greater than hA and less than hB, where h is the height of the mercurial barometer. Show that the height of the upper surface of the mercury above the point where the diameter of the tube changes is {Bh-V)I{B-A)', and that if the barometer falls an inch, this surface will fall through B/(B — A) inches. If the mercury be slightly displaced, the oscilla- tions will synchronize with a pendulum of length hA/(B-A). 190. The Siphon. The siphon (Greek, ai(f>u>v) is a bent tube ABC, origin- ally designed for drawing off liquid from a cask or jar, and now employed on a large scale for going over or under an obstacle in carrying liquid to a lower level. As the action of the siphon depends essentially on the same principle as that of the barometer, the discussion of it is introduced here, and first in connection with draw- ing off mercury. The tube is first filled with mercury, and then the ends A and C being closed by the fingers or by corks, or the stopcock s.c. being closed, the tube is inverted with the ends in the vessels of mercury at A and C, at different levels, and the fingers or corks removed. The pressure at A and G being that of the atmosphere, the pressure in the mercury above the stopcock, when closed, exceeds the pressure below by az, where z denotes the difference of level of A and G; so that, on opening 272 THE SIPHON. the stopcock, the mercury flows through the tube ABC from A to G in & continuous stream (fig. 61). If the density p of the surrounding medium is taken into account, the pressure of the liquid above the stop valve exceeds the pressure below the valve by {(T-p)z. Thus if jo > (T, the action of the siphon is reversed, as, for instance, in transferring hydrogen by a siphon tube ; and now the siphon and the vessels at A and G must be inverted, and the fluid will be transferred from the lower to the upper level. Fig 61. Fig 62. The vertical height of the highest point B of the tube above A must not, however, exceed h, the height of the barometer; otherwise, as in AB'G, the mercury in the branch AB' will subside to the barometric height AH, when the finger is removed from A ; and now, when the stopcock in B'G is opened, the mercury in the branch B'G will subside to the barometric height GH'. DYNAMICS OF THE SIPHON. 273 This supposes that the mercury column divides where the pressure vanishes or becomes negative ; but if, as in Mr. Worthington's experiments (§ 6), we suppose that the mercury column can support a tension of a certain amount, ah suppose, without breaking, the siphon can still work, so long as the height of B' above H does not exceed h. 191. In its dynamical action the siphon may be assimi- lated to a chain coiled up at A, and led over a pulley at B so that the end hangs at G; the preponderating length z will set the chain in motion, so that the coil at A will become gradually transferred into a coil at G. If X cm of chain have passed over in t seconds, and the moving part ABG, of length I cm suppose, has then acquired a velocity v cm/sec, if tu denotes the weight in g/cm of the chain, and T g denotes the tension of the chain at A, the equation of motion is wl dv rn and r=?^ 9 the momentum in second-grammes generated per second ; so that 1^=9^ — t^'^; (1) and the chain therefore starts with an initial acceleration gz/l, and tends to a terminal velocity ^{gz), just like a body falling under gravity in a medium in which the resistance varies as the square of the velocity. In the siphon there will be no loss of energy at A due to the continuous series of impacts, so that we may halve the above value of T; and now the equation of motion in the siphon becomes 27-4 STARTING THE ACTION ^1=^^-^^' ; ^^> so that the terminal velocity in the siphon is.^(2c/2:). By integration of equation (2), or v = J{2gz)i2.u\i^^^^t; r Ivdv , 1 gz Jgz-\v^ ^gz-^v^ or ^v'^ = gz{l — e~''l''). 192. A leak in the tube ABC, if above the level of A, will admit air, and vitiate the action of the siphon, even to the extent of stopping the flow if the leak is sufficiently large ; but liquid will escape from a leak in the tube below the level of A. The large siphons or standpipes of waterworks are designed to reach the altitude of the service reservoirs, so that the water in passing through may be cleared of air, which tends to accumulate in the mains. In the distiller's siphon (fig. 62, p. 272) the action is started by opening the stopcock s.c, closing the end G with the hand, and sucking the air out by the curved mouthpiece at G'; as soon as the spirit passes the highest point of the bend at B, the action of the siphon com- mences, and it can be stopped and restarted by closing and opening the stopcock. The preceding methods are not desirable with noxious liquids, such as acids, which cannot be handled or tasted with impunity ; the siphon is then started by first closing the stopcock s.c, and filling the branch BG OF THE SIPHON. 275 through one of two small funnels at B, the other funnel permitting the escape of air ; on shutting off these funnels by plugs or stopcocks and opening S.C., the liquid is made to flow through the siphon. ] 93. If however the length of the longer branch BG is insufficient, the liquid will not be drawn up to the level of the bend B in the branch AB, but will rest at a lower level B' a certain vertical height x above A, and at a certain distance y from A if the branch AB is curved ; and the siphon will not start. The vertical height of the liquid GG' left in the longer branch BG will also be x, in consequence of the equality of the pressures at A and G, and at B' and C; and the pressures at A and G being due to a head h of the liquid, the pressure in B'G' will be due to a head h — x. Denoting by a, b the lengths of the branches AB, BG, and supposing' for simplicity that BG is inclined at an angle a to the vertical, the air which originally occupied the length a of the shorter arm AB now occupies the length a + b — y — x sec a. Therefore by Boyle's Law (Chap. VII.), which asserts that the product of the volume and pressure of a given quantity of air remains the same at the same temperature, ah = (a + b — y—xseca){h — x), or "^ + ""^"3 — \-y + xseca. In the critical case when the liquid just reaches the bend B, y = a, and x denotes the vertical height of B above A ; so that , ah , = 5 \-xseca ; h — x and a greater value of b will start the siphon. 276 SIPHONS ON A LARGE SCALE. In the siphons of fig. 61 the branches are vertical ; and now if a, b denote the lengths of the vertical branches, and c the length of the horizontal part, then in the critical case {a+c)h = (b + c — a)(h — a), , (a + c)h , or b = - , ^ +a — c. h — a 194. As employed for drawing off water over an em- bankment, the siphon is shown in fig. 63 ; for example, over the reservoir dam of water works (fig. 20), or in draining a fen or inundation. {Proc. Inst. Civil Engineers, XXII.) An automatic valve, opening inwards, is placed at A and a stop valve at C. The siphon is filled either through a funnel by means of a hand pump, or else by exhausting the air by an air pump at JB. On opening the stop valve G, the water flows through the siphon ; and on closing the stop valve, the siphon remains filled for an indefinite time, the valve at A preventing the return of the water in AB. In this, as in all other cases, the height of B above the upper level of the liquid must be kept below the head of liquid corresponding to the atmospheric pressure. Sometimes the siphon is inverted, as required for carrying a water main across the bed of a river; and now there is no limitation of depth to its working. A water main, or a pipe line for conveying oil, carried in an undulating line in the ground, may be considered as a series of erect and inverted siphons ; and on an emer- gency, the pipe may be carried over an obstacle, which is higher than the supply source or hydraulic gradient by something under the atmospheric head of the liquid. THE INTERMITTENT SIPHON. 277 195. An intermittent siphon is shown in fig. 64; the vessel is gradually filled up to the level' of B, when the action of the siphon suddenly commences, and the vessel is rapidly emptied; and so the operation goes on periodically. Fig. 63. Fig. 64. The Gwp of Tantalus, invented by Hero, depends on this principle ; and it is also used for securing an inter- mittent scouring flow of water. The action of natural intermittent springs and geysers is explained in this manner ; and the underground flow of certain rivers, such as the Mole, by subterranean inverted siphons. Examples. (1) If a vessel contains liquids of various densities, will the action of the siphon be impeded ? Two equal cylindrical pails of horizontal section A are placed, one on the ground, and the other on a stand of height A ; the former is empty, and the latter contains masses mj, m,^ of two diflferent homogeneous liquids ; a fine siphon tube of negligible volume has its two ends at the bottoms of the two pails and through it flows liquid until equilibrium is attained, a mass m,^ of density p remaining in the upper pail ; prove that mj + THj — 2m3 = J. A.p. 278 EXAMPLES. (2) A siphon tube with vertical arms filled with mercury, of s.G. a, and closed at both ends is inserted into a basin of water. When the stoppers are removed, examine what will ensue, and prove the following results if the barometer is sufficiently high : — (1) If h, the whole length of the outside arm, exceeds a, the whole length of the immersed arm, the mercury will flow outwards and the water will follow it. (2) If a > b, the end of the immersed tube must be at a depth below the free surface of the water exceeding (a — b)(T in order that the mercury may not flow back into the basin. (3) Two equal cylinders side by side contain mercury, one quite full and open at the top, the other full to 20 inches from the top and closed, the 20 inches being occupied by air at the atmospheric pressure, which is 30 inches of the barometric column. If the two vessels are connected by a siphon dipping into the two liquids, prove that, when the siphon is put in action, 5 inches of mercury will flow from one of the cj'linders into the other. What takes place when the leg of the siphon which is in the closed cylinder is not long enough to reach the mercury in that cylinder ? CHAPTEE VII. PNEUMATICS. THE GASEOUS LAWS. 196. Hitherto we have dealt with the properties of Liquids or Incompressible Fluids like Water; and now we proceed to consider Air and Gases, or Compressible Fluids, and their properties, a branch of Hydrostatics sometimes called Pneumatics, from the Greek word Trvev- fxariKri, meaning the science which concerns Trveu/xa, air or gas. A given quantity of a Gas ("a parcel of gas" in Boyle's words) requires to be kept in a closed vessel, to prevent diffusion ; and by changing the volume of the vessel and the temperature, the pressure of the gas is altered. Given the volume and the temperature, the pressure of a given quantity of a gas is determinate ; so that the pressure p is a function of the volume v or density p, and of the temperature t. Expressed analytically P =/('": t), or F{p,v,t) = 0; and to determine this function, two new Laws, based upon experiment, are required, which are called 279 280 THE GASEOUS LAWS 197. The Gaseous Lmvs. Law I. — Boyle's Law. "At constant temperature the pressure of a given quantity of a Gas is inversely proportional to the volume, or directly to the density." This law was enunciated by Boyle in his Defence of the Doctrine touching the Spring and Weight of the Air in answer to Linus, 1662 ; abroad it is attributed to Mariotte, who did not however publish it till 1676. Thus if p denotes the pressure and v the volume of unit quantity of the gas, one gramme suppose, and p de- notes the density, so that p = l/v, then p = kp, or pv = k, where k depends only on the temperature : so that, on the (p, v) diagram, an isothermal is a hyperbola (fig. 65), along which the hydrostatic energj'' pv (§ 14) is constant. For instance, a gunner, who can push with a force of P pounds, can, under an atmospheric pressure of p Ib/in^, introduce an airtight sponge into a closed cannon, d ins in calibre and I ins long in the bore^ a distance x ins, given by P + {-rrd?p = \-,rd^p-L, orf = . ^ l-x I P+lTrd^p Thus, if P=100, p = 15, d = o, Z=120, we find that cc = 30-42. Law II. — Chaeles's or Gay-Lussac's Law. " At constant pressure the volume of the Gas increases uniformly with the temperature, and at the same rate for all gases." Combining this with Boyle's Law we find that the product of the pressure and volume of a given quantity of OF BOYLE AND CHARLES. 281 any Gas increases uniformly with the temperature at the same rate ; so that we may write pv = lc = h^{l + aT), where a is a constant coefficient of expansion, the same for all gases. On the Centigrade scale of temperature a = 0003665 = ^4^-^; and now putting k^^Rja (the height of the homogenous atmosphere at Cj. MV = ijf- + Tl=i?0, where = - + t; \a / a and Q is called the absolute temperature, and — 1/a the absolute zero; this is therefore —273 C, or about —460 F, since Ija — ^y. 273 = 492 on the Fahrenheit scale. But —274 0, or —461 F is sometimes taken as nearer to the correct value of the absolute zero of temperature. At this absolute zero the pressure of a given quantity of gas would be zero, whatever the volume. In an experiment by Robins {Nexv Principles of Gun- nery, Prop, v., p. 70) a gun barrel, which would contain about 800 grains of water, was raised to a white heat, and plunged into water, when it was found that about 600 grains of water had entered the barrel. This proves that the air left in the barrel had been ex- panded to four times its volume ; so that, if the water was at 15° C, or 288 absolute, the temperature of the white heat was about 1152 absolute, or 880° C, or 1552° F. 198. The equation 'pv = Re (A) connecting p the pressure, v the volume, and d the absolute temperature of a given quantity, say one g, is called the Characteristic Equation of a Perfect Gas. •282. THE CHARACTERISTIC SURFACE. It may be illustrated geometrically by the surface shown in fig. 65, in which the isothermals, along which 6 is constant, are hyperbolas, while the isometrics, v constant, and the isobars, p constant, are straight lines. This model surface can be constructed of pieces of card- board, as made by Brill of Darmstadt. Denoting by P, V, the pressure volume, and temper- ature in any given initial state, then (A) may be written embodying the Laws of Boyle and Charles in a form suit- able for calculation from experiments. Regnault found that at Paris a litre of dr'y air at C and a barometric height 76 cm is l'293187g; so that measuring pressure in millimetres of mercury head, we find, for a gramme of air at Paris, P = 760, p = l/F= -001293187, = 273, and ie = PF/e = 215-3. Taking the density of mercury as 13'59, this makes the height at Paris of the homogeneous atmosphere at C k^ = ha- /p = 7Q867Q-o cm, say 8000 m. The weight in g of F litres of dry air at a temperature T C and a pressure of h mm of mercury is therefore a formula required in exact weighings, in allowing for the buoyancy of the air. 199. It must be noticed that the gravitation measure of force is employed in these formulas, so that in accurate comparisons the local value of g must be allowed for. Thus if g changes to g' in going from Paris to any other locality, Greenwich for instance, the absolute pres- GRAVITATION UNITS. 283 sure due to a given head of mercury changes in the ratio of g to g', and therefore also the density of a litre of dry air defined by these conditions. Fig. 65. Thus, with the centimetre and second as units of length and time, ^' = 980-94 at Paris, / = 981-l7 at Greenwich, 6^ = 980'61 at sea level in latitude 45° ; so that the weight of a litre of dr)'- air at 0°C and under 760 mm of mercury head changes from 1'293187 g at Paris to l-293187(cy7^ or (?/£?) = 1-293559 g at Greenwich, or to 1-292752 g in latitude 45". But the head h of mercury or h of homogeneous air which will produce the same absolute pressure as 760 mm of mercury or 8000 m of air at Paris is at Greenwich (760, or 8000)g/g' = 759-82 mm of mercury, or 79982 m of air; and at sea level in latitude 45° is (760, or 800{})g/G = 760-26 mm of mercury, or 8002-7 m of air. 284 VA RIA TION OF GRA VITY. Suppose, for iustance, that g is doubled or halved, as might appear to be the case in a lift, or the cage of a mine; the pressure due to a given head of liquid is doubled or halved, but the head corresponding to a given pressure is halved or doubled ; and generally the pressure due to a head h of mercury or h of air is proportional to g ; but, for a given pressure, gh or gh remains constant. These variations are due to the employment of the gravitation unit of force; but as the variations on the surface of the Earth do not amount to 03 per cent, they are insensible in most practical problems. All physical measurements of force are primarily made in gravitation units, from their convenience, intelligi- bility, and precision ; and these measurements can after- wards be converted into absolute units by multiplying by the local value of g, when it is required to compare delicate measurements made in different localities. 200. With British units, the foot and the pound, and with the Fahrenheit scale, a ft^ of air at 55 F and a barometric height of 30 ins, equivalent to a pressure of 14|lb/in2 or 2112 Ib/ft^, is found to weigh about 1-25 oz, so that about 13 ft^ weigh one lb; and therefore, putting P = 2112, F=13, = 460-1-55 = 515, ie = PF/e = 53'3, for one lb of air; then \ =R/a= 5S-3X 492 = 26,224 ft. The work required to compress the air at constant temperature from volume F to v is represented by the area of the hyperbolic isothermal pv = PV on the {p, v) diagram, cut off by the abscissas Fand v; it is therefore, by a well known formula (§ 233), PFlogF/v, DALTON'S LA W. 285 expressed in ft-lb, if V is given in ft** and P ia Ib/ft^. Thus if a cubic yard of atmospheric air is compressed to a cubic foot, P = 2112, F=27, v = l; and the work required is 2112 X 27 X log,27 = 188,000 ft-lb. 201. A third law, sometimes called Dal ton's law, is added for a mixture of gases which do not act chemically on each other ; thus atmospheric air is a mechanical mix- ture of oxygen and nitrogen. Law III. — Dalton's Law. The pressure of a mechanical mixture (not a chemical mixture) of gases all at the same temperature in a closed vessel is the sum of the separate pressures each gas would have if it alone was present in the vessel. This law again must be accepted as based upon experi- mental proof Granted Dalton's law, Boyle's law is seen to follow immediately ; for if a pound of a gas is introduced into an exhausted chamber and produces a certain pressure, the introduction of a second pound of the gas will by Dalton's law double the pressure and will also double the density ; a third pound of the gas will treble the pressure and density, and so on ; so that the pressure is propor- tional to the density. Take W^, W^ ... TF„, lb or g, of n perfect gases, having pressures Pv Pi' ■ ■ ■ f ™' when the volumes are v-^, v^, ... v^, and the absolute temperatures are Q^, 62, ••■ On- 286 EXPERIMEXTAL VERIFICATION Now if all these gases are brought to the same volume Fanrl temperature 0, the new pressures Wj, ©2, . . . CT„, will be given by -g-= =-^™' suppose. If these gases are now mixed together mechanically in a closed vessel of volume V, the pressure P of the mix- ture will, by Dalton's law, be given by P = STl-l-CT2+--- + CT^; SO that ____.+ __ + ...+_, or, if PV/Q is denoted by S, ^=72^+^2+ •••+P)i- If Pj, p2' •■• Ptc denote the original density of each gas, and ky kp ... /<;«, the head of each gas which produces its original pressure ; then 7 Pti ^ Pri^n ^vPn and if p denotes the density, and K the pressure-head of the mixture, or S==-^1W, and E,= wJ^, SO that K^'ZknW^^ ■nWnfJ^Wn 202. The Experimental Verification of Boyle's Law. Boyle took a bent tube OABK with vertical branches (fig. 66, p. 296) and filled the bend with mercury so that it stood at the same level AB in the two branches ; the OF BOYLE'S LA W. 287 end being closed, the air in OA is at the atmospheric pressure, measured by the height h of the mercurial barometer BC. Now if mercury is poured into the branch BK up to the level K, then mercury will rise in OA to a point H, such that the head of mercury due to the difference of level of H and K measures the excess of pressure of the air in OH over the atmospheric pressure ; or the head h+KH of mercury measures the pressure in OH. Boyle found that, if the tube OA is cylindrical, AH_HK OA_ h+KH _HL 0H~ h ' °^ 0H~ h ~ h' if KL = h ; so that the pressure in OH is inversely as the volume, and therefore proportional to the density. Thus if HK=li, it is found that OH = ^OA. This apparatus of Boyle is not susceptible of very great accuracy, as the graduations for H become very close together when the pressure is great; and this militates against the use of this arrangement as a pressure gauge, as has been suggested. 203. An apparatus was devised by Regnault (Jamin, Gours de Physique, t. I.) in which the tube BK was carried to a height of 30 metres up a tower and mast, and the mercury was forced in by a pump from below ; at the same time a known quantity of air, carefully dried, was forced into OA, so as to keep the level of the mer- cury nearly constant at A, the tube OA being surrounded by water to keep it at a constant temperature. More recently Amagat has obtained great pressures by the head of a column of mercury in narrow flexible steel tubes, carried down the shaft of a mine some 400 m in depth, or up the Eifiel tower, 300 m high. 288 EXPERIMENTAL VERIFICATIONS These experiments were recorded by plotting the value of pv, the hydrostatic energy of a given quantity of the gas, corresponding to values of p; and if Boyle's Law was accurately true, the points plotted at constant temperature should range themselves in a straight line parallel to the axis oi p; this was found to be approxi- mately the case. Examined more closely the points were found, for great values of p, to lie very nearly in a sti-aight line slightly inclined to the axis of ^, indicating 'the law pv = hp + k, or p{v — b) = k; so that the isothermals on the (p, v) diagram are still hyperbolas and now h is called the co-volume. A full account of the experiments upon which the Gaseous Laws are based will be found in two Reports to the British Association, 1886 and 1888, Experimental Knovjledge of the Properties of Matter with respect to Volume, Pressure, Temperature, and Specific Heat. By P. T. Main. Boyle's Law will be found to hold when the air in OA is expanded, by drawing oif mercury in the bend AB through a stopcock at the lowest point ; and generally a convenient mode of varying the level of K is by means of a large vessel of mercury which can be raised or lowered by a winch, and which communicates by a flexible tube with the stopcock (fig. 66, p. 295). 204. The Law for rarefaction of the air in OA can also be shown by the apparatus of fig. 67, p. 295, also devised by Boyle, A cylindrical glass tube OA, closed at 0, is partly filled with mercury, and sunk vertically in a deep vessel OF BO YLE'S LA W. 289 of mercury ; the point A is marked where the mercury- stands at the same level inside and outside the tube, and now the air in OA is at atmospheric pressure. On raising the tube vertically, the pressure of the air in OA will be diminished and the air will expand, so that the mercury will be drawn up to a level H, such that the pressure of the air in OH is due to a head h — AH or DH of mercury, if GJD is the true barometric height ; and it is found experimentally that OH.HI) = OA.GD, so that Boyle's law is verified. The same apparatus could be employed to compress the air by depressing the tube ; but now the level H of the mercury inside the tube OA would be below the level of the mercury in the vessel, and its position would be diffi- cult to observe, except by the mark left by the liquid in rising in the tube, unless transparent water and glass were employed. It was in this way that Charles II. won his wager that he would demonstrate the compression of air in a hollow cane, the Koyal Society being appointed referees (Phil. Trans., Jan. 1671) ; the other story of him concerning a fish in a bucket of water appears spurious, as no record of it is to be found. If the hollow cane was 21 in long, closed at one end, and depressed vertically in water till it was just sub- merged, the water would rise one inch in the cane, if the water barometer stood at 400 ins ; or, if the cane was 44 ins long, and just submerged vertically, the water would rise 4 ins in the cane. The principle is employed in the Deep Sea Sounding Machine ; a tube of length a, closed at the top, is lowered G.H. T 290 EXPERIMENTAL ILLUSTRATIONS vertically into the sea, and the length y of the interior marked by the entrance of the water being measured, the depth X reached by the lower end of the tube is given by the equation of the two expressions of the pressure in atmosphei'es of the imprisoned air, H+x—y _ a H ~a-y' where H denotes the height of the water barometer ; and x = y{H-\-a-y)l(a-y), so that the depths corresponding to the graduations of the tube are the ordinates of a hyperbola, which can be con- structed geometrically as in §115; and the graduations can be made uniform by giving a hyperbolic shape to the tube. 205. A simple experimental illustration is described in Weinhold's Experimental Physics ; in this the tube OA, closed at 0, is of fine uniform bore, and contains a fila- ment AB of mercury of length k suppose, less than h, the barometric height. The tube may be suspended from a fixed point at 0, and now, when held in the horizontal position, the pressure in OA is equal to the atmospheric pressure. When the tube is held vertically upwards, the air in OA is compressed by a head h+k of mercury, and will therefore occupy a length ah/(h+k), if Boyle's Law is true, the original length OA being denoted by a. When the tube hangs vertically downwards, the air in OA will expand to a length ahj(h—k); and generally, if OP is the length of the air column when the tube makes an angle Q with the vertical, the pressure in OP will be due to a head h+k cos Q of mercury ; so that OP = ahlih+kcose), OF BOYLE'S LA W. 291 and P therefore describes a conic section with focus at 0, excentricity hjh, and semi-latus-rectum a. This arrangement may be used as a barometer, for the determination of h, and it is then called a Sympiezometer. If read in the two vertical positions, the effect of tem- perature is eliminated ; for if x, y denote the lengths of the air column in the two positions ah = x(h — k) = y(h + k), h x + y or =_:l^; Ic x — y and X and y expand at .the same rate, and their ratio is therefore independent of the temperature, while the mercury columns h and k have the same coefficient of expansion. A similar theory holds when both ends of the tube are closed ; now if a, h denote the lengths of the air columns, in the horizontal position, and k the length of the separat- ing mercury filament, then when inclined at an angle to the vertical the filament will travel through a distance r given by the equation , „ ah bh kcosd = r~, — a—r o+r Thus if k = h = 20 ins, and the lengths of the air fila- ments when the tube is held vertical are 10 and 5 ins, then in the horizontal position their lengths will be 6f and 8J ins. 206. If the open tube is of insufficient length, some ot the mercury will run out as the tube is inclined. Thus if the tube, of length I, is brim full of mercury when in the upright position, the mercury occupying a length k, and the air underneath a length l — k under a mercury head h + k; then at an inclination 6 the air 292 THE PIPETTE. column will expand to a length o\ and the fraction of mercury spilt is \ — (l — rjjh, given by 'r{h+{l-r)oose} = {l-h)Qi + h), r = lQisece + l) + s/{lQisece-lf+lcB&ce{h+h-l)}, Z_r = J(Z-Asec0)-V{K^-'^sec0)2 + /<;sec0(^+/(;-i)}. If c denotes the length of the mercury filament in the horizontal position, when the contained air is at atmos- pheric pressure, h{l-c) = {l-h)Qi+lc), or ch = ]c(h+k-l). When the tube hangs vertically downwards, cos 0= —1, and l-r = i(h + l)-J{l(h + iy-ch}. This can be illustrated experimentally with a tumbler of height I, filled with water to a depth c, and closed by a card; on carefully inverting the tumbler, a certain fraction of the water will leak out, so that the depth of the water is reduced to i{h+l)-^{i{h+lf-ch}, where h now denotes the height of the water barometer. The same principle applies to the pipette; if it is dipped into liquid to a depth c cm, and a volume Fcm^ of air is imprisoned by the finger, while the area of the surface of the liquid inside is a cm^ ; then, on raising the pipette out of the liquid, the surface will sink x cm, given by the positive root of the equation h-c+x _ V h ~ V+ ax 207. Say's Stereometer. This instrument, the invention of a French officer Say, is intended for the measurement of the density of gun- powder and other substances, which must not be con- taminated with moisture or contact with liquids. SAY'S STEREOMETEB. 293 As the action of the instrument is an application of Boyle's Law, the description is introduced here, and illus- trated by reference to the arrangement in fig. '66, p. 295. The stop-cock is replaced by a three-way cock, which can establish communication between the branches AO and BK, between either branch and the reservoir of mercury, or between all three. Two fixed marks are made on the tube AO,&tA and H suppose, and the mercury filling AH is drawn off into the reservoir and weighed, and thence the volume, Uevc?, of AH is inferred. The mercury is now brought to the same level at A and B in the two branches, and in the reservoir, and a globe is screwed on at 0, so that the air in J. and the globe is at atmospheric pressure. The reservoir is now raised till the level of the mercury in the branches rises to H and K ; and now denoting the vertical height HK by k, and the volume of the globe, reaching to H, by Fcm^, then by Boyle's Law V+JU_'h±k r_h V ~ h ' ^"^ V~k' which determines V. The globe is now unscrewed, and a known weight W g of the substance, whose volume x cm^ and density W/x is required, is placed in it ; the globe is again screwed on, and the operation is repeated. Now if b' is the difference of level of the mercury in the two branches when the level in the branch J. rises to H, V—x_h X _h h ~ir^k" .^"^ U~h~¥ As an increase of pressure tends to make the gunpowder absorb air, the process may be reversed by starting with 294 GRAPHICAL REPRESENTATIONS the level of the mercuiy in the two branches at H, and drawing off mercury by lowering the reservoir till the mercury stands at the level of A in J.O. 208. Graphical Representations of Boyle's Law. Putting OA = a, OH==x, OK=y, KL = 00^ = 00^=h, in fig. 66, then according to Boyle's Law 0H.HL=0A.002, or x(x + y+h) = ah; a relation of the second degree in x and of the first in y. Describe a circle on AO2 as diameter, cutting the hori- zontal line through in Q. Now if the level of K is given, take C the middle point of KO2, and with centre G and radius CQ describe a circle cutting the outside line OA in H and R'; then the point H in OA will be the corresponding level of the mercury. For OO^^KL, and therefore 06'= CZ, or OH'^HL; and OA.OO^^^ OQ^ = OH . OH' = OH . HL. The other point H' is such that the air which, occupies a length OA of the tube at atmospheric pressure, or under a head 00 ^ of mercury, will occupy a length OH' under a head LH or OH of mercury. To determine K geometrically when H is given, take OH' the third proportional to OH and OQ, and mark oft H'K equal to HO^. 209. The geometrical method of § 114 may be employed to give a graphical representation of the relative levels of H and K in the branches of the tube in fig. 66, and of the relative heights of and H in fig. 67. Draw through 0, 0^, Og straight lines sloping at 45° to the horizon. If H is projected horizontally to H, on the sloping line through 0, and the vertical line K^H-^M^ is drawn to meet the horizontal line through K in K^ and the OF BOYLE'S LA W. 29.= sloping line through 0^ in M-^, then x + y + h=:-Mj^Kj^, so that Ojilfi . M^K.^ is constant; and the curve described by Z"j, while fij describes the sloping line OH.^, is a hyperbola with the vertical and the sloping line through 0^ for asymptotes. Fig. 66. Fig. 67. But projecting K horizontally to K^ on the sloping line through 0, drawing the vertical L^K^B.^ to meet the sloping line through Og in L^, and drawing the horizontal line through H to meet this vertical in JS^ and the sloping line through 0^ in N^, then, as before, so that OH . j&2-^2 ^s constant, and as K^ describes the sloping line OK^, H^ describes a hyperbola with the horizontal line through and the sloping line through Og for asymptotes. These hyperbolas can be constructed geometrically in the manner previously explained in §§ 64, 115^ 296 VITIATED TORRICELLIAN 210. Similarly in fig. 67, assuming Boyle's Law to hold, OH.HD = OA.GD, or {x — y){h — y) = ah, if X and y denote the heights 6f and H above the level of the mercury in the cistern ; so that if the sloping line OE is drawn through C, meeting the horizontal line through D in E, and if the points and H are projected as before, then as ifj describes GE, 0-^ will describe a hyperbola with the vertical and sloping line through E for asymptotes ; and as 0^ describes GE, H^ will describe a hyj)erbola with the horizontal and the sloping line through E for asymptotes. The value of h can be inferred from the observation of two positions of the tube OA; for if I, I' denote the lengths of the air column OH corresponding to heights y, y' of the mercury column AH, then l(y-h) = l'{y'.-h)-ah, or a ^_^, . 211. Vitiated Vacuum of a Barometer. If the tube OA is fixed in position with at the same level as the top of the barometric tube GB, the relative fluctuations of H and D will represent the barometric fluctuations of a barometer in which a small quantity of air has vitiated the Torricellian vacuum, compared with the indications of a standard barometer. Now if b denotes the constant height of above the mercury in the cistern, (6 - y)(h — t/) = a constant = ab suppose, where h and y fluctuate, assuming that the temperature is constant. VACUUM OF A BAROMETER. 297 Thus if the vitiated barometer reads 29'5 ins when the standard barometer reads 30 ins, the correction to be added to the reading y of the vitiated barometer is 125 32-3/- Also, if a vitiated siphon barometer reads 31 ins when the true reading is 32, and the length of the vacuum is then one inch, the correction to be added to the observed height of 29 ins is 0'5 inch; and generally to an observed height 3/ is 2/(23 -i/). Referred as before to the sloping line through (fig. 68) B^ will describe a hyperbola with the vertical and sloping line through for asymptotes, as H^ describes the line OH^, whence the barometric correction DJS^ for the height y in the vitiated barometer can be measured off from the diagram; and conversely, as D^ describes the sloping line, H.^ will describe a hyperbola with the hori- zontal and sloping line through for asymptotes. 212. Denoting by j/i, J/a the readings of this vitiated barometer when the true barometric heights are A^, h^, {b -y){h-y) = {h- 2/i)(^i - 2/i) ={h- yi){K - yd ; i- yi{K-yi)-y'lK-y^ (h-yd-iK-y^) and the correction to be added to the height y is ;^ (^1-2/1X^2-2/2X2/1-2/2) (2/1 - 2/X^i - 2/1) - (2/2 - 2/X^2 - 2/2)' Thus if the readings of a vitiated barometer are 29'9 and 29'4 ins when the true barometric heights are 30'4 and 29"8, then & = 31'9, so that the length of the vitiated vacuum when the reading is 29 is 2'9 ins; and the correction to be added to the reading y is 1/(31"9 — 1/). 298 VITIATED TORRICELLIAN But to deduce the barometric correction from the com- parison with another vitiated barometer, of height c, in which the corresponding readings are z, z^, z^, the additional equations (c -z){h-z) = {c- %)(Ai - Zj) = (c - Z2)(h - z^ will determine h^ and h^ ; and thence we find that the corrections to the observed heights y, z are I ,, _ (^ - 2/1) (^ - Vi){ (g - ^i)(yi - ^1) - ('^ - ^2)Cy2 - ^2)} ^ ~ - (^ ~ ^i)(<' - ^2) { (^ - yi)(yi - ^1) - (^ - ^2)(y2 - %)} (c - 2;){ (6 - y^{c - z^) -{b- y^){c -z^)] If the temperature varies, and is denoted by t, tj, Tg C at the three observations, then (b-y){h-y) ^ {h-y^){\-y^ _ {h-y^){h^-y^) _ 273 + T 273-l-Ti 273 + T2 ' so that, eliminating 6, (yi-y,)(273+T) ^ (y,-y)(273+T,) (2/-2/i)(273+t,) ^q Merely inclining the tubes would cause the true and vitiated barometers to fluctuate diflFerently, and the arrangement has been suggested as a gravimeter for measuring a change in g (Gomptes Rendus, June-July, 1893) ; for if h' and y' denote the barometric heights when g has changed to g', and the temperature does not change, g'{h'-y')(b-y')=g{h-y){b-y}. Work out, for example, the value of g' when g = 981, h = h' = 7e, 6 = 100, 2/ = 38cm, y -y' = 10 fji.fji. 213. Suppose the vacuum of the barometer OA was originally perfect, and that the air which vitiates the vacuum was introduced as a small bubble, which at atmospheric pressure occupied a length AB of the tube. As the bubble PQ rises slowly and steadily in the tube. VACUUM OF A BAROMETER. 299 by the gradual transfer of mercury from above to below the bubble, the height of the superincumbent mercury QR is the mercury head of the pressure in the bubble, and is therefore equal to the difference of level of P and -D, and inversely proportional to PQ. Fig. 68. Fig. 69. If therefore P is projected horizontally to Pj, on to the sloping line GE (fig. 69), the corresponding point Qj will describe a hyperbola with the vertical and sloping line through E for asymptotes ; and the corresponding point iJj will describe a rectangular hyperbola, with horizontal and vertical asymptotes through E. _ When R reaches 0, the column QR may remain fixed ; but practically the mercury will gradually trickle down the sides of the tube, or the bubble will otherwise make its way to the top, until the air bubble is entirely above the mercury, and we have the vitiated barometer. Suppose for instance that a barometer stands at 30 ins, and that the length of the Torricellian vacuum is then 2 ins, so that h = 32. 300 EQUILIBRIUM OF BUBBLES. If a bubble of air, which at atmospheric pressure would occupy half an inch of the tube, is now introduced, the barometric column will fall 3 inches ; and the correction to be added to a reading y is 15/(32 — 3/). 214. The behaviour of rising bubbles can be studied in soda-water, champagne, or water boiled in a glass vessel ; the pressure diminishes and the bubbles expand as they rise, until they reach the surface and burst. Denoting by V the volume of a bubble at the surface and by h the height of the water barometer (about 400 ins or 10 m), then at a depth h the pressure is doubled ; and therefore, by Boyle's Law, the volume of the bubble is halved and the density of the air in it is doubled ; at a depth (rb—V)h the pressure will be increased n fold and the volume will become Vjn; and generally at a depth z the pressure will be l+zjh atmospheres, and the volume of the bubble will be rh/(h + z). Denoting by k the height of the homogeneous atmo- sphere (or of the air barometer) so that h/k is the S.G. of air, then when h+z = k or z = k — h, the density of the bubble will be the same as that of water, and the bubble will be in equilibrium. Taking ^ = 10 m, fc = 8000 m, then /c- A = 7990 m, the depth in the sea at which a bubble of air will be in equilibrium; also k/h = 800, the pressure in the bubble in atmospheres. The density of hydrogen being about one-fourteenth of that of air at the same pressure, a bubble of hydrogen will be in equilibrium at about 14 times this depth. THE CARTESIAN DIVER. 30] In Metric units, if the volume of the bubble at the surface is Fcm*, its weight will be practically Vhjh g; and at a height x cm above the position of equilibrium, the volume or buoyancy will be Vh/(Jc — x) cm^ org; so that the upward moving force on the bubble will be y(_h h\_ Vhx \k — x kJ ~ k{lc — x) g- 215. Allowing, as in § 149, for the effective inertia of the displaced water, spherical in form, as half the weight of water displaced, the upward acceleration of the bubble is Vhx k(k — x) _ X Vh , , Vh'^~ik^^' k +*jfc-cc The equilibrium of the bubble at a depth k — h is con- sequently unstable ; if slightly depressed, the bubble is compressed and sinks to the bottom ; but if it rises the bubble expands and reaches the surface. 216. By imprisoning the bubble in a glass bottle which is ballasted so as to be on the point of sinking, we can study the equilibrium and stability near the surface, and this arrangement is now called a Cartesian Diver; it illustrates the action of the bladder in a fish, and in- cidentally the dangerous instability of a submarine boat. Denoting hy W g the weight of the bottle, its S.G. by s, and by V cm^ the volume at atmospheric pressure of the imprisoned air, or the weight of water in g the bottle will hold, then at a depth z cm the volume and buoyancy of the bubble is Vh/(h + z) ; and this is equal to W— W/s, and the bottle is in equilibrium, when 302 THE LA WS OF EBULLITION When z is greater than the depth of water, the Car- tesian Diver will float on the surface, and to depress it, we must have means for increasing h; this is effected by covering the vessel of water with a bladder, which can be pressed in by the hand ; and now if h is increased to h', the volume of the bubble at a depth 2^ becomes Vh/(h'+z'); so that in the position of equilibrium h'+z=h + z, or z' diminishes as h' increases, by an equal amount ; and when z' is negative, the diver will sink. A Cartesian Diver is readily constructed of a glass bottle, ballasted by lead wire wound round the neck; thus a pint bottle can be so weighted that it just floats in water when one quarter of an ounce is placed in it; and now if inverted neck downwards over water, it is in unstable equilibrium when the level of the water inside it is at a depth of a little over 5 inches, the height of the water barometer being 400 inches; and increasing' this to over 405 inches will cause the diver to sink. 217. Ebullition and the Laws of Vapou7- Pressure. When water or a liquid boils, bubbles of vapour are formed in the interior, which rise to the surface and burst; and the pressure of the vapour in a bubble may be taken as that of the surrounding liquid. Any cause which tends to the formation and dis- entanglement of bubbles, as for instance the shaking of a locomotive boiler, is of assistance to ebullition, and improves the performance of the boiler. At the surface the pressure of the vapour given off during ebullition is equal to that of the external air or vapour. A saturated vapour is the name given to a gas formed AND OF SATURATED VAPOURS. 303 in contact with the liquid from which it is derived by ebullition ; and the " Law of Saturated Vapours " asserts that " the pressure of a saturated vapour is a function of the temperature alone, and not of the volume " ; for if the volume of the vapour in contact with its liquid is increased or diminished, more vapour is evaporated or condensed, so that the pressure is unaltered, if the tem- perature does not change. The pressure of the vapour is determined experi- mentally by observing the depression of the column GH in fig. 68 at different temperatures when a small bubble of the liquid just small enough to evaporate completely is introduced into the tube. A Table has been constructed and a curve drawn out from experiments by Eegnault, giving the pressure of aqueous vapour or steam, expressed in millimetres of mercury head for temperatures below the standard boiling point, reaching from about 4'6 mm at 0° C to 760 mm at 100" C ; and expressed in atmospheres for temperatures above the standard boiling point, reaching to about 5 atmospheres at 200 G. In determining the boiling point of a thermometer it is thus important to observe the barometric height, as it is found that the boiling point varies about 1° for 27 mm of mercury head; and as the pressure and therefore the temperature of the bubbles increase with the depth in the water, the bulb of the thermometer should be in the steam, and not immersed in the water. From Kegnault's Table we can calculate the depth at which water of given temperature can be made to boil, when the pressure on the surface is suddenly diminished, as for instance in the receiver of an air pump. 304 PRESSURE AND DENSITY OF VAPOUR. Thus at 100 C the vapour pressure is one atmosphere of 760 mm ; so that if the pressure is suddenly reduced to a fraction m of an atmosphere, the water will boil to a depth (1 — m) of the height of the water barometer. 218. According to Dal ton's Law (§ 201), if h denotes the total pressure of the air, in mm of mercury head, and / mm the pressure of the aqueous vapour present, then h—f will be the pressure of the dry air alone. The density of the aqueous vapour is found experi- mentally to be about | of the density of dry air at the same pressure; so that, if a litre of dry air weighs 1"2932 g at 0° C and 760 mm of mercury head, then at a temperature of t C, a barometric height h mm, and an aqueous pressure of/ mm, a litre of air weighs H-aTV760^8 760y ^^' ' and the density p of this damp air, in g/cm*, is given by _ 00012932 A- 1/ , P~ l + ar 760 ' so that damp air is perceptibly lighter than dry air. It is important therefore that the air should be care- fully dried in all experiments on Boyle's Law; so that the mode of experimenting with a gas in a vessel inverted over water is unsuitable for accurate measurements. 219. Corrections for Weighing in Air. When a weight is marked, say P g, this means the absolute weight, as determined in vacuo. But if B denotes the density of the metal of the weight, and p the density of the air, the apparent weight is diminished, by the buoyancy of the air PpjB, to CORRECTION FOR WEIGEINQ IN AIR. 305 At the temperature t where c denotes the coefficient of cubical expansion of the metal, and Bq the density at 0° C ; also p is given by the formula in the last article. When therefore, as in § 57, a body of some other density cr is being weighed in air, and its true weight is TTg when equilibrated by the weight marked P g, ^-p=pg-|)/(i-^). and (r = o-o/(l + CT), if G denotes the coefficient of expansion of the body. We might, for instance, be weighing W g of hydrogen, in which case the negative value obtained, for P would mean that the' weight P would have to be placed in the same scale as the " parcel " of hydrogen. If the body is now weighed by the Hydrostatic Balance in water, of density D^ at t° C, and equilibrated by a weight marked P'g, then and therefore =^ = ^^^^ — -, Jr cr—p a^=cr{l + C(T-T)}, the density at standard temperature T. Here D^ may be calculated from the formula of § 170, or preferably taken from Table I., in consequence of the abnormal dilatation of water. 306 CRITICAL TEMPERATURE. 220. Vapours and their Critical Temperature. When a saturated vapour is isolated from contact with the liquid from which it is formed, and the temperature is raised, the vapour is said to be superheated ; and the vapour now begins to obey approximately the Gaseous Laws of §197. If not too much superheated, the vapour can by com- pression be brought again to the saturated state ; and further compression liquefies the vapour in part. But' Andrews found (1871) that beyond a certain critical temperature of the vapour liquefaction by compression became impossible; and the Gaseous Laws were now obeyed very closely. For this reason it is advisable to divide the gaseous state of matter (§ 2) into two classes : (i.) true gases, which cannot be liquefied, because heated above the critical temperature, and which obey the Gaseous Laws of § 197 very closely ; (ii.) vapours, which can be ultimately liquefied by com- pression, the temperature being below the critical point ; and which show increasing divergence from these Gaseous Laws. Thus, for instance, the Critical Temperatures of water, chlorine, oxygen, and nitrogen are, respectively, 370, 141, -113, and -146C. The Characteristic Equation where a and h are small constants, has been proposed as a generalised form of the Characteristic Equation of a Perfect Gas, (A) § 198, by Van der Waals in his Essay on the Continuity of the Liquid and Gaseous State of Matter, THE AIR THERMOMETER. 307 1873, a translation of which is published by the Physical Society of London. This equation is intended to serve for large variations above and below the critical temperature of a substance. The pressure p is infinite when v = h, so that according to the formula h, the covolume (§ 203), would appear to be the ultimate volume of the substance, below which it cannot be compressed; but Van der Waals assigns reasons for supposing that b is about four times the ultimate volume. Thus there is evidence that liquids cannot be compressed below 02 to O'S of their volume at ordinary atmospheric pressure. 221. The Air Thermometer. Charles's Law is really equivalent to the definition of equal degrees of temperature as being those for which a perfect gas (say hydrogen) receives equal increments of volume under constant pressure. The Air Thermometer difiiers only from the ordinary mercury thermometer in having the bulb filled with air instead of mercury, the air being confined by a small filament of mercury in the stem. As with the hydrometer (§ 64) the bulb may be supposed replaced by a length of the stem of equivalent volume, sealed at one end ; and now if the positions of the mercury at the freezing and boiling points are marked, the air being at the same pressure, and this interval is divided in 100 degrees Centigrade, then it will be found that the distances of these points from the sealed end contain 273 and 373 degrees, called absolute degrees; and, according to Charles's Law, the Air Thermo- meter will give the same records when filled with any other so-called perfect gas (hydrogen for instance). 308 DIFFERENTIAL AIR THERMOMETER. Heating the air from to 200 C would cause the volume to expand from 273 to 473, so that the density of the air would be about that of coal gas, as employed in balloons. The mercury thermometer is preferred for ordinary purposes, as its indications are independent of the height of the barometer, which requires to be observed and allowed for in a reading of the Air Thermometer. For if the barometer changes from the standard height h to K, the reading of absolute temperature Q on the air thermometer corresponds to a true temperature Qh'jh. Thus if the barometer falls from 30 to 29 ins, the read- ing 300 corresponds to a true absolute temperature 290. If, however, the open end is made to dip into a vessel of mercury, like the tube OA in fig. 67, p. 295, in which the level of the mercury can be raised or lowered, the air in the thermometer can be easily brought to a standard atmospheric pressure, measured by a fixed head HD of mercury. 222. The Differential Air Thermometer consists of two equal glass bulbs containing air, connected by a horizontal uniform tube in which a small filament of mercury or liquid separates the air in the two bulbs. A small difference of temperature of the two bulbs is recorded by the displacement of the filament. Denoting by V^ and V^ cm^ the volumes of the two portions of air when the bulbs are at the same tempera- ture T, and by w cm^ the cross section of the tube, then when the bulbs are raised to temperature tj and T2C, the filament will be displaced from zero through a dis- tance X cm, such that, the pressure in each bulb being changed from P to p, ISOTHERMAL ATMOSPHERE. 309 273+Ti 273+t' 273+T2 273+t' Fi-toa! _ 273 + Ti Fi_ei P^ ""^ F2+coa!~273+T2 F^ 0^ F/ if 01, 02 denote the corresponding absolute temperatures. If, as usual, Fi= F2= F suppose, wx Kt2~ti) — ^2~^- F~273 + KT2+ri) 02 + 0i' and now if the temperature of each bulb is increased by t', the filament will reach a distance y, given by o(«:-y) Kt2-Ti) ■ _ ._g2llgi , F ~273 + Kt2+Ti+2t') 02+0i+2t' or 33-3/ 02 + 01 a; ~02 + 0i+2t" so that the graduations are in H.P. for equal increments in t'. 223. Isothermal Equilibrium of the Atmosphere. In ascending or descending in a quiescent atmosphere of uniform temperature it is found that the logarithm of the barometric height or of the pressure changes uni- formly with the height; or expressed analytically, the barometric height h, the pressure p, and the density p at any height z are connected with the corresponding quantities, kg, p^, p^ at the ground by the formula K Po Po where /c = £ = £9 P Po is the height of the homogeneous atmosphere ; so that at a height k the barometer stands at l/e of its height on the ground. 3 1 ISOTHERMAL A TMOSPHERE. Taking logarithms, z = k logeQi Jh) = k logipjp) Zi-Z2 = fikQ-og^Ji^ - logjo^i), where fi = logJO = 2-30258509 ^ 2B, the modulus which converts common into Naperian logarithms. Expanded in powers of Q>'Q — h)/(h^+h), z = k log,^ = 2^ tanh - ^^^ ° h hg+h of which the first term only is retained in Babinet's barometric formula, employed when the barometric change is small. Thus if the barometer falls from 30 to 27 inches in ascending 2800 ft, we find that A; = 26,600 ft; and the barometer will fall to 21'87 inches at a height of 8400 ft. With British units, at a temperature 32 F, /<;= 26,214. ft, and iuik= 60,300 ft ^ 10,000 fathom ; so that the difference of the common logarithms of the barometric heights multiplied by 10,000 gives the differ- ence of level in fathoms. For a height of 7 miles or 7 X 880 = 6160 fathom, log{hJh) = -6160, hjh ^ 4. On Snowdon, 3720 ft high, log(yA) = 0-062, log(h/ho) = T-938, h/h^ = 0-867 ; so that if ^0 = 30, ^=26, h^-h^A, a. fall of four inches. Down a mine the pressure will be n atmospheres at a depth klogeU; thus for n — 2, the depth would have to be about 18,720 ft. LAPLACE'S BAROMETRIC FORMULA. 311 With Metric units, at a temperature rG, A; = 8000(1 + aT), where a = 0-003665, 2^1 — 02 = 18,400(1 + aT)log^^QiJh^, metres. 224. Laplace's Barometric Formula. The atmosphere is supposed of uniform temperature t, the arithmetic mean ^{t^ + t^ of the temperatures tj and Tg at the two stations ; and since a is slightly increased by the presence of aqueous vapour, Laplace puts a = 0-004, so that 1 + «'^=1 + ^T0W^' To allow for the variation in g, given in latitude X and at height h, a fraction hjR of the Earth's radius R, by Bouguer's and Clairaut's formula £f = (?(l - ^ - 000266 cos 2x), Laplace introduces the factor - = 1 + ^ + 0-00266 cos 2A, because k is inversely proportional to g (§ 199) ; and now he writes the formula z- , - 02= 18400(l +|:+000266 cos2x){l +%^)logio|. in which h may be replaced by the mean height ^(z-^+z^), supposed known approximately. This gives the height in metres ; and to give it in feet, 18400 must be replaced by 60369, the equivalent number of feet. Also on the Fahrenheit scale, the temperature factor becomes ^ 900 ■ 312 ISOTHERMAL AND CONVEGTIVE 225. To prove the formula of § 223, employing the Integral Calculus, suppose that —dp denotes the diminu- tion of pressure in ascending a height dz in a stratum of density p ; then dp— — pdz. But if p = kp, dp _ dz p k ' and, integrating, logp — logp(,= —r, or ^ = e"*. K Pf^ The proof of this theorem is not easy by elementary methods, not involving the use of the Calculus; the following elementary proof is submitted : — Suppose the air between two horizontal planes at a distance z, at which the pressures are p and p^, is divided into a large number n of strata, of equal depth z/n; and that the density in each stratum is taken as uniform, the density in the rth stratum being denoted by pn and the pressure being denoted by pr at the bottom and by pr+i at the top of the stratum. Then considering the equilibrium of the vertical column of these strata, standing on a base of unit area, z z Pr -Pr+1 = Pr- =l^r;^/ or ^r+i = i_A; Pr nk' so that B=&=a=...=^i=...=^^ = i-4, Po Pi Pi Pr Pn-T. nk the pressure decreasing in g.p. as we ascend in A.P. by equal vertical steps zjnk ; and therefore Pg \ nkJ EQUILIBRIUM OF THE ATMOSPHERE. 313 Now making n indefinitely great, and putting Pn=P, £=it(i-4r k^2hJ SAk) +■•• = e~^ or exp(-|). We notice that p and p are zero when z is infinite ; so that on our theoretical assumptions the height of the atmosphere of uniform temperature is infinite. 226. Gonvective Equilihriwrn of the Atmosphere. According to the experience of mountaineers and aeronauts, the temperature of a quiescent atmosphere is not uniform, but diminishes slowly with the height at a rate which may be supposed uniform ; so that we put connecting Q the absolute temperature at the height z with 0(, the absolute temperature at the ground; and c will now represent the theoretical height of the atmo- sphere, the absolute temperature and the pressure being zero at this height. Then the characteristic equation of the air becomes £=£-«|- = ^(l-?); p Po % V c/ so that in gravitation units, as in § 225, dp _ pdz _ 1 dz _ p~ p , _z k ' c and integrating, logp-logpo = llog(l-^, 3 1 4 GONVECTIVE EQ UILIBRIVM Po v^ c; \ej' or P_ Po Then for two stations, at altitude ^^ and Zg. c 00 ^W Vo^ Vo^ Vo^ If we put _£^ = y or ^ = -^, c—k ' k y— 1 then p oc p''', or pi)'^ is constant, if V denotes the s.v. of the air ; and, as will be shown hereafter, this is the relation connecting p and v when the air expands adiabatically, that is without parting with any of its heat, if y denotes the ratio of the specific heats of air at constant pressure and constant volume. 227. By Conduction of Heat a quiescent atmosphere tends to Isothermal Equilibrium; but considering the extreme slowness of conduction in air compared with the rapidity with which the air is carried to different heights by the wind, Sir W. Thomson has proposed the Gonvective Equilibrium as more representative of normal conditions {Manchester Memoirs, 1865). According to his hypothesis the interchange of the same weight W lb of gas at any two places A and B without loss or gain of heat by conduction (adiabatically) would simply interchange the pressure, density, and temperature, so that no real cfiange would ensue ; W lb of gas moving from A to B through an imaginary non- conducting pipe AGB, and an equal weight TFlb from B replacing it at A through another imaginary pipe BBA. OF THE ATMOSPHERE. 315 In this manner the condition p = Xp'', or pv'^ = constant, is realised; and, as above, the temperature diminishes uniformly with the height, and the theoretical height of the atmosphere is y/(y— 1) times the height of the homogeneous atmosphere at the ground. As the result of experiment we may put y=l"4, so that y/(y-l) = 3-5. This makes the height of the atmosphere about 28,000 m, 28 km, or 17 miles. 228. The best experimental determination of y is from the observed velocity of sound ; denoting it by a f/s, then theory shows that the velocity of sound is that acquired in falling vertically through ^yk ft ; so that a^=gyk=gyplp, y = a?lgh. At the freezing temperature, a = 1093, /<;= 26214; so that, with ^=32-19, log a =3-0386, log a2 = 6-0772, •loggr =1-5077, log/c =4-4185, log^^ = 5-9262,. logy =0-1510, y =1416, say 1-4. Now, if zero suffixes refer to the ground level, and 6 denotes the absolute temperature at a height z, then the differential equation of equilibrium is dp_ pdz_ PoOg, __ dadz_ gyOp. This leads, on integration, to losP^-SfyOo r^ 316 CONVECTIVE EQUILIBRIUM. 229. Starting with the experimental relation we can employ the elementary method of § 225 to deter- mine the pressure, density, and temperature, at any height z ; and now the equation 'PT-Pr+l = Pr- becomes p'^r-p'^r+l = Pr~^> {Pl+lY = A L_ A 7-1 \pr J \ py-^ nxJ y ^1 y-1 1 y pr''^'^ n\' neglecting squares and higher powers of z/n ; 7-1 7-1 y~^ ^ • so that the (y — l)th powers of the density, and there- fore the f(y— l)/yth powers of the pressure diminish uniformly with the height; and finally -'--'-=^^-v ^•^-^"^-^fl Also i='PPs=^py-i, 00 p Po Jo so that eo-6 = ^leo=^6„ and p, p, 6 vanish when y F" c yk Z = G — —^ the theoretical height of the atmosphere. COMPRESSIBILITY OP WATER. 317 230. Denoting the depth c — z below the free surface of the atmosphere by x, "V- 1 -v-1 y — 1 oa — — y— 1 X • r V ^ - y ,f or putting y — 1 = 1 /%, p^Jn+iy^^'^' suppose' _ gi"+^ _ jua;"+^ _ pee ^~"(to + 1)»+iX»~%+1 ~w + 1" We thus obtain the expression for the pressure p when the density p varies as some power n of the depth below the free surface; and if the fluid is now supposed in- compressible, a pressure P applied at the free surface is transmitted without change throughout the fluid ; and ^ n+1 Suppose for instance that y = 2, so that the density is proportional to the square root of the pressure, or the excess of the pressure over a standard pressure P ; then p = P + V> the law employed by Laplace in calculating the density of the strata of the Earth; then 71 = 1, and the density varies as the depth. 231. When the compressibility of water is taken into account in determining the pressure and compression in a deep ocean, we employ the experimental law % P connecting v^, the volume of a lb of water at atmo- spheric pressure and p^ the density, with v the volume and p the density at a pressure of p atmospheres more ; also 318 COMPRESSIBILITY OF MERCURY. \ is the coejfficient of cubical corrvpression per atmosphere, and we may put \ = 000005. Then if h denotes the height of the water barometer, and X denotes the depth to which water of depth cCq is reduced by the compressibility, ^ h pji where p denotes the average density, which may be taken as the density at the mean depth ^Xf^. Then ^ = P^=\-})p; OCf, p fi so that the surface is lowered by the compression Xf) — x=^XXo^/h. Thus if a;u= 100 h, and A = 33 ft, the surface is lowered about 8 ft. Similarly it is calculated that the depth of an ocean 6 miles deep is lowered about 620 ft by compression, corresponding to X = 0"00004 ; and that, if incompressible, the Ocean would have its surface 116 ft higher, and cover two million square miles of land. So also, in allowing for the compression of the mercury in his experiments on Boyle's Law (§ 203) Regnault took X = 0"000004!628 per metre head of mercury; and now p = Xf) and Xg — x = ^Xx^\ Thus a column of mercury 25 m high is shortened 1"45 mm, which is negligible. 232. Reckoning x downwards from the free surface, and using the gravitation unit of force, dp = pdx or p = P+ypdx, CURVES OP PRESSURE AND DENSITY. 319 so that if the density p is represented on a diagram by the horizontal ordiuates of a vertical axis Ox, the pressure p will be represented by the area of the curve of density ; and a separate curve of pressure can be plotted from this condition, just as curves of tons per inch immersion and curves of displacement in fig. 42, p. 168. Thus if the density is uniform, the curve of pressure is a straight line, as in fig. 20, p. 43; if the density in- creases uniformly with the depth, the curve of pressure is a parabola, and the density varies as the square root of the pressure ; and so on. Take the case of the curve of pressure in going from water into air ; at a depth x in water represented by a straight line ; at a height z in an atmo- sphere in Thermal Equilibrium, p = Pexp( — 2:/A;), represented by the Exponential Curve, in which the subtangent is constant and equal to k ; and in an atmo- sphere in Convective Equilibrium, y — 1 z\y^ p = p(l- y 233. The work required to compress a substance from volume v + Av to vit^ by the application of an average external pressure of p Ib/ffc^ is pAv ft-lb. Thus if Av ft* of atmospheric air is forced into a receiver of volume v, filled with air at atmospheric pressure p and density p, the increase of density is p(Av/v), and of pressure is p(Av/v), when thermal equi- librium is established; and the average increase of pressure being ^p(Avjv) Ib/ft^, the energy is increased by ^piAvf/v ft-lb. 320 ENERGY OF COMPRESSION. For a Unite range of compression, from v-^ to v^ ft^, the work is therefore, in the notation of the Integral Calculus, / 'pdv. where p is given as some function of v from the physical properties of the substance. Thus in the adiabatic compression (§ 226) of a given quantity, say one lb, of air, so that the work required to compress it from v-^ to ■Vg ft^ is, in ft-lb. /" .-1' the diflference of the hydrostatic energies divided by y — 1. This result follows geometrically from the graphical representation on the (p, v) diagram of the adiabatic curve QiQ'QQ^- The tangent at Q, the limit of the chord QQ' through the consecutive point Q', cuts the axis Op in T, where pT=y.Op; and therefore the elementary rectangles Qp' and Qv' are in the ratio of y to 1; and therefore also the whole areas P1Q1Q2P2 ^^^ ''^iQiQ2'^2i while their difference is p^Vi—PiVi. For isothermal compression we must put y = l, and the work required is PF logKM); this may be obtained either by integration, or by the Exponential Theorem, as the limit, when y=l, of CHIMNEY DRAUGHT. 321 In an atmosphere in Convective Equilibrium the work required to compress 1 lb of air at altitude z-^^ to its density at a lower level z^ is, in ft-lb, f^^-jp^^k e^-d -, Jc(z^-z^) ^ z^-z^ y-1 00 y-1 c(y-l) y or 1/y times the work required to raise 1 lb of air from the level z^ to the level z^; and when y=l, as in an Isothermal Atmosphere, the work is the same. 234. The Draught of a Chimney. The currents of air in the atmosphere are primarily due to inequalities of temperature and thence of density ; a familiar instance of the artificial production of a current of air is seen in the draught of a chimney. Considering the draught through the closed furnace of a steam engine boiler, the air makes its way through the grate bars and the fire, as through a porous plug, and acquires with the gases of combustion a certain average temperature, which we shall denote by t C or Q' absolute, T or Q denoting the temperature of the outside cold air. It is calculated that about 20 lb of air is required to burn 1 lb of coal ; and denoting by p the density of the cold air, then the density of the hot air issuing from the top of the chimney at the same pressure may be taken to be pdlQ' ; so that h ft denoting the vertical height of the top of the chimney above the fire, the pressure of the cold air outside will exceed the pressure of the hot air inside the furnace, taking their densities as uniform, by )ph; ('4> and this will be felt as a pressure on the furnace door. This will also be the upward pressure on a lid at the top of the chimney, if the furnace door is opened. 322 MA XIMUM DBA UOHT To measure the draught a glass inverted siphon gauge filled with water (fig. 71, p. 345) is placed in the side of the chimney, and now if z inches is the diflference of level of the surface of the water in the two branches, and D denotes the density of water, In round numbers, D/p = 800 ; so that h 200V 07 If the horizontal cross section of the chimney is A ft^, then the weight of cold air which fills the chimney is Ahp lb ; and the lieight of the column of hot air of equal weight is h6'/0, and their difi'erence of height, 0' .\, T is taken as the head producing the velocity v of the hot air up the chimnej'. Or, otherwise, if x ft denotes the head of hot air equivalent to z inches of water, T. ,v, e^ 1 2a;/j|, = Z)0 = 1 2(1 - 1) V, The rate of flow of the air through the chimney de- pends very much on the state of the fire; it is assumed that the average velocity v of the hot air up the chimney is either due to this head x, or to some fraction of it, depending on the resistance of the fire and the friction of the flues ; but putting OF A CHIMNEY. 323 the weight of air in lb which flows up the chimney per second is =^/>VC2^/^)v{J-(|-i/}. a maximum ^Apy/{2gh), wlien 6'= 2d. Thus if the outside air is at 17 C, = 290, and 6' = 580, t' = 307 C, nearly the temperature of melting lead ; and z^ 3 h 400' so that a chimney 100 ft high produces a draught of f inch of water, when the flow through it is a maximum. In Metric units, a litre of air at C weighs 1'29 g ; so that a chimney h m high will produce a draught of z cm of water, given by 1 = 0-129 x273g-l) = 35Q-^,); for instance, if 9' = 26, and t = 0, = 0-0645 h. Variations of. barometric height or of temperature will cause air to enter or leave a given space, such as a room or a mine. This is illustrated by a "whistling well," in which a whistle placed in the lid is blown by the current of air which enters or leaves the well ; also by the liberation of gas in a coal mine when the barometer falls. Thus if the barometer falls from h to h', or the tem- perature rises from 9 to 9', the density of the air falls from p to p', where P~h9" 324 EXAMPLES and the percentage of air which leaves a room is ioo(i-5)-ioo(i-||). A room 10 m by 6 m by 5 m will, with a barometric height 76 cm and a temperature C, contain 10 X 6 X 5 X 1-2932 = 388 kg of air ; and now, if the barometer falls to 75 cm and the tem- perature rises to 15° C, the room will lose about 6^ per cent or 25 kg of air. Exaimples. (1) Prove that if volumes Fj and V^ of atmospheric air are forced into vessels of volume f/j and fJj, and if communication is established between them, a quantity of air of volume at atmospheric pressure will pass from one to the other. (2) Prove that, if a partially exhausted siphon with equal arms is dipped into mercury; and if the sum of the heights to which the mercury rises in an arm, when first this and then the other arm is unstopped, is equal to the height of the baro- meter; then the original pressure of the air in the siphon is due to a head of mercury whose height is equal to the difference of the lengths of the mercury in the siphon in the intermediate and final stages. (3) Prove that, if a piston of weight TTlb is in equi- librium in a vertical cylinder with a ft of air beneath it, and if it is depressed a small distance ON PNEUMATICS. 325 X ft, the energy of the system when the tempera- ture is unaltered is increased by about ^ Wx^/a ft-lb. (4) The mouth of an inverted cup is submerged to a depth of 6 ins in warm mercury, and it is found that no air escapes. Prove that, if the barometer stands at 30 ins, the mercury cannot be more than 100° F. warmer than the atmosphere. (5) A gas saturated with vapour, originally at a pressure p, is compressed without change of temperature to pne-%th of its volume, and the pressure is then found to be pn- Prove that the pressure of the vapour and of the gas in its original state is respectively n — 1 n — 1' (6) Prove that, if the height of the column of mercury in § 211 is reduced from y to y' by the introduction of a bubble of water into the vitiated Torricellian vacuum, just small enough to evaporate com- pletely, the pressure of the vapour, in mercury head /, is given by f=.(P+h-y-y')^,; reducing to y — y' when the vacuum is perfect, or y = h; and that now (b — y)(h — y—f) is constant, when h, h, and y change, the temperature remain- ing constant. E.g., If A = 29-81 ins; and y=29 when 6 = 32,-^, y'=28-5 when 6'= 30^; then /=0-47 in, and the dry air would occupy a length 004 in of the tube at atmospheric pressure. 326 EXAMPLES. (7) A piston, of weight t^A, in a closed vertical cylinder o£ height a and section A is in equilibrium at a height ajn from the base, the pressure of the air underneath it being p. Prove that a small rise t C of the temperature of the air underneath will raise the piston through a height approximately equal to n n'p — r;j 273 (8) An air thermometer is made of a bulb and tube inverted vertically in a reservoir of mercury of depth h so that the tube rests on the bottom. Prove that if the volume of the bulb and tube is equal to a length c of the tube, and h is the height of the barometer, the graduation for the temperature is at a height above the bottom of the tube where 9o is the absolute temperature at which the enclosed air begins to escape. (9) Assuming that the relative distribution of oxygen and nitrogen at different heights in an atmosphere in equilibrium follows the law that one is not affected by the other, find at what height in an isothermal atmosphere the proportion of oxygen would be reduced to half what it is at sea level, where the proportions by weight may be taken to be 80 parts of nitrogen to 20 of oxygen, and where the densities are in the ratio of 14 to 16. CHAPTEE VIII. PNEUMATIC MACHINES 235. The Montgolfier Hot-Air Balloon. This balloon, invented by the Montgolfiers in 1783, is historically interesting as the first employed by the aeronauts Pil^tre de Eozier and d'Arlandes to make an ascent in the atmosphere. The principle is the same as that of the ordinary hot- air toy balloon ; the air in the balloon is rarefied by heat to such an extent that the total weight of the balloon, of the hot air it contains, of the car and of the aero- nauts is equal to or less than the weight of the external cold air displaced, when the balloon begins to rise. Denote by W lb the weight of the balloon, car, and aeronauts, as weighed in vacuo, or corrected for the buoyancy of the air ; and denote by W lb the weight of cold air they displace, so that TT— TF' lb is the apparent weight when weighed in air; denote also by Fft* the capacity of the balloon, so that Jf = Vp lb denotes the weight of cold air which fills the balloon, p denoting the density, in Ib/ft^, of the surrounding cold air. Then when the air inside is raised in temperature from Q to Q' degrees absolute, part of the air will flow out, 327 328 THE MONTGOLFIER leaving the remainder at the same pressure but at density pQjQ', and therefore of weight Fp|=ilf I lb. By Archimedes' principle the balloon will float in equilibrium when the weight of the balloon and the hot air it contains is equal to the weight of cold air displaced ; that is, when F+Jf|>= W' + M, e _ M-W+W' 6' -6 W-W "'' e'~ M ' 9 -M-W+W" ^^ giving 0' — 6, the requisite increase of temperature. The balloon will now be in unstable equilibrium, like a bubble of air in water, and will begin to rise, as it cannot descend. 236. The balloon will continue to rise and the hot air to escape, till another stratum of air is reached, of height z suppose, and of density p^ and absolute tem- perature 0j, and therefore of pressure ^j, given by Pz^PzOz p P 9' p denoting the pressure at the ground. The pressure of the hot air in the balloon being also Pz, the quantity of hot air left in the balloon, supposed always at the absolute temperature 9', is F4 = i.f^^ fib. while the weight of cold air displaced is (M+ W'f~ lb ; P EOT AIR BALLOON. 329 so that, for equilibrium, pi) p or P^- ^ p w'+M(\-es')' But the barometer and thermometer carried by the aeronauts give p^i the pressui'e and Q^ the absolute tem- perature, compared with p and Q, the pressure and temperature at the ground ; and by the Gaseous Laws P__Pz6__ W__ ,. P. p erw'+M(i-e,ie') ;^ ^ If Q' denotes the temperature which is just suiRcient for levitation, as given by equation (1), then in (2) „, .-. ^"-^-H^-'D ,3, Pz W so that with this temperature 6' the balloon will not rise unless 6z/6 is less than unity, or unless the temperature of the air diminishes as we ascend in the atmosphere. Thus in an atmosphere in thermal equilibrium of uni- form temperature, 6' must be increased beyond the value given by (1) for the balloon to ascend. In such an atmosphere it has been shown that, with p p where h denotes the height of the homogeneous atmo- sphere at the temperature 6 ; so that taking this height at the freezing temperature as 26,214 ft, 330 THE HYDROGEN The temperature Q' of the hot air required to ascend to a height z ft is now given by equation (2), ^-M+ w'~ Wp/p; ^ ^ where 6z = 0, and p;i = pe^T^{ — z/Jc), in an atmosphere of uniform temperature ; but 1 d~ y ^ k' p~\ y h) ' in an atmosphere in Convective Equilibrium (§ 226). 237. The Hydrogen or Gas Balloon. In this balloon the requisite levitation is secured by filling the balloon with hydrogen, as first carried out by Charles and Robert in 1783, a few months after the first ascent in the Montgolfier balloon ; or nowadays with coal gas, which is specifically lighter than air. With hydrogen the balloon can be made of much smaller dimensions ; but this advantage is counter- balanced hj the difficulty of the manufacture of the gas and its great speed of diffusion ; so that a balloon is now generally made of larger dimensions and filled with coal gas from the nearest gasworks. For military purposes, however, where the balloon is required to be held captive at a height of about 1000 ft, it is important to keep down the size, so as to reduce the effect of the wind ; so that military balloons are now filled with hydrogen, carried highly compressed in steel flasks, at a pressure of about 100 atmospheres. With the same notation as for the Montgolfier balloon, suppose the gas employed has a specific volume nv and a density pjn, n times and one-Tith that of air at the same pressure and temperature. OR COAL GAS BALLOON. 331 Then the balloon will be in unstable equilibrium on the ground when U ft^ of the gas, weighing P = Upln lb, has been placed in the balloon, given by the equation, derived from Archimedes' principle, W+Upln=W'+Up, or 'W+P=W'-\-nP, W-W jj_ W- W n n-l ' p n-\ ^^ Thus to lift a ton with hydrogen, 14 times lighter than air, one-13th of a ton of hydrogen is required, occupjang 31,360 ft^, taking the specific volume of air as 13 ft^lb. The great Captive Balloon of Chelsea had a capacity of 424,000 ft^ ; so that the gross weight lifted by hydrogen could be about 13^ tons. The aeronaut's practical rule is that "1000 ft^ of coal gas will lift 40 lb " ; so that putting F-F' = 40, ?7=1000, v = \Ip==\-2-5, XX A n 1000 we find T = Tr, — rs-f = 2, n=2. 71—1 40x125 Denoting generally by A the lift or ascensional force, in kg/m^, of a gas n times lighter than the air, or o£ density p' kg/m^, p denoting the density of the air, then A = p-p' = p(l-l), (2) where |0 = 1'293; and thus for hydrogen, ^ = 1'2. Now if the balloon is on the point of rising when inflated with U m^ of gas, weighing P kg, W-W'=UA = {n-l)P (3) A m^ of air at C and 76 cm of barometer weighs r293 kg, of hydrogen 0'088 kg, of coal gas at C or of 332 THE THEORY air at 200 C about 078 kg ; so that a m^ of hydrogen will lift 1-293 -0088 = 1-205 kg, and a cm^of air at 200 C, or of coal gas will lift 0-513 kg; this makes to = 14-7 for hydrogen, and % = l-8 for the hot air or coal gas. But coal gas, especially from the latest products of distillation, can be made of density from 0-33 to 0-37 of the density of the air, so that now n is about 3. Gas of this lightness was employed in the celebrated ascent by Coxwell and Glaisher on 5th September, 1862, from Wolverhampton, when an altitude was attained at which the barometer stood at 7 inches and the thermo- meter at about —12 F. 238. The balloon in unstable equilibrium on the ground is like a bubble of air compressed to the density of water at a great depth in the ocean (§ 214) ; and being unable to descend, it will rise when let go. To carry the balloon rapidly clear of the neighbouring obstacles it is advisable that the volume TJ or quantity P of gas should be increased so as to give an ascensional force, which at starting will be a force of {n-\)F-{W-W') pounds. As the balloon rises, the gas contained in it will expand until the balloon is completely inflated, and will now occupy Fft^; and this will take place where the density of the air is pUjV, and the density of the gas is pUjnV, the temperature being supposed unaltered. The ascensional force will now be (n-l)P-(w-W'^ pounds, (4) or W— W'UjV pounds less than at starting; and there- fore practically unaltered, since W is small. OF THE BALLOON. 333 239. The balloon will still continue rising; but now it is very important that the neck of the balloon should be left open, to allow gas to escape as the balloon rises into the more rarefied air, and thus to equalize the pressure of the gas and the air ; otherwise the pressure of the imprisoned air might burst the balloon, especially if the rays of the sun should suddenly strike upon it. At a height z the ascensional force will now be ""^k-^-^-K. = {('n,-l)Q+F'}^-TF pounds, (5) on putting Q=Vpjn=Mln, (6) where Q denotes the weight in lb of the gas and M of air which would fill the balloon at the ground : also VA = {n-l)Q, .' (7) A denoting the lift of the gas, in Ib/ft^. This ascensional force is zero, and the balloon comes to rest, when p~pe, (n-l)Q+W ^"^^ The quantity of gas now left in the balloon is, in lb, n~'^p~{n-l)Q+W' (n-l)Q-(W-W') ~^ {n-l)Q+W' ^' ^^^ so that the number of lb of gas lost is (n-l)Q+W' ^^^ ^' ^ "^ or, if the balloon started full, with P = Q, the gas lost is (n-l)Q-iW-W') . in-l)Q+W' ^^^ ^ ^ 334 THE THEORY To pull the balloon down to the ground without any further loss of gas will require a force, gradually in- creasing to = F'(l-^) pounds (12) at the ground ; this is such a very small quantity that a very slight loss of gas is sufficient to cause the balloon to descend, and ballast must be thrown out to restore equilibrium. In consequence of the balloon losing gas in the ascent, and collapsing in the descent, a captive balloon, raised and lowered slowly, requires less work to pull it down than the work required to resist its ascent. The gradual loss of gas by diffusion brings the balloon down again, generally in about two hours' time; and if it is required to descend rapidly, a valve is opened at the top of the balloon, to let the gas escape quickly. On the other hand, by throwing out ballast the height lost can be recovered and even exceeded, or the balloon can be guided into a favourable current of air. A free balloon is always rising or falling, and it must be steered in a vertical plane either by throwing out ballast or letting off' gas ; but it can be kept at a moderate average elevation by the areonaut Green's invention of a rope trailing on the ground, acting as a spring. 240. In the absence of knowledge of the distribution of pressure, density, and temperature in the upper strata of the atmosphere, we must suppose that the true state is something intermediate to the states of Thermal and of OF THE BALLOON. 335 Convective Equilibrium, and calculate on these two hypotheses; these calculations are required, not only in ballooning, but also in gunnery M'ith high angle fire at long ranges, to allow for the tenuity of the atmosphere and consequent reduction in the resistance of the air. In an atmosphere of uniform temperature z=k log, ^ = /iA; logio^ g: , (13) with modulus p = 1d, giving the height z attainable. But with Convective Equilibrium also ^ = ^(^1_|) = 336(0-0,), (15) with y=l"4, and /c=960; so that the thermometer falls 1° C for every 336 ft ascended, or 1°F for every 186 ft. 241. Suppose the balloon is completely inflated on the ground, and prevented from rising by B kg of ballast ; then, neglecting W, W+B=VA (16) The removal of the ballast B (delestage) will allow the balloon to rise to a stratum of density p^, where P_^ = J^ = 1_A. (17) p VA VA' ^'-'^ and to Teach an additional height Az, where the density has diminished Api, additional ballast AB must be thrown out, given by AB=rA{Ap,lp) (18) 336 THE THEORY A,^k^ = k^^ (19) In an isothermal atmosphere, Pz pz Thus if V= 500 m^, and if hydrogen is used, A = 1-2, VA = 600 kg ; and to rise to a level where p^ = %p, or z= 18400 logio(p//o^) = 2300 m, the ballast removed must be 5=JFA = 150kg; and throwing out 10 kg more ballast will make the balloon rise an additional height of 178 m. 242. Neglecting the weight of everything except the envelope of the balloon, supposed a sphere of volume V w?, and diameter d = 4/(6 yjir) m, and of supeivficial density m kg/m^, then if filled with gas of ascensional force A kg/m^, the lift at the ground is VA — irdHi kg. A small hole being left open for the escape of the gas, the lift is reduced to {VAIq)-Trd''m'kg in a stratum where the density is reduced to one-gth of the density on the ground; and the balloon will just rise to this stratum if VA „ ^6F\f q \ TT / V^'-^, (20) called " the equation of the three cubes." In an isothermal atmosphere z = 8000 logeq = 18400 log^^g = 6133 log F- M, where if = 12600 + 18400(logm-log J.); (21) so that z increases by 6133 m when F is multiplied ten- fold. OF TEE BALLOON. 337 ? 5 10 40 200 F 1-25 10 640 80,000 z 12,900 18,400 29,500 42,300 For hydrogen A = \-1\ and the superficial density of the envelope is found to range from 300 down to 50 g/m^ ; so that putting, on the average, m = 01, we find F=0-01g5; and hence the table (Cosmos, April, 1893, La pratique des ascension^ a^rostatiques). 500 1,250,000 m8 49,700 m For copper one 100th of an inch or ^ mm thick, m=2'2, so that if a copper sphere of this thickness, 100 ft or 30*5 m in diameter, is filled with hydrogen, the ascensional force at the ground is about 11,400 kg or 25,000 1b; also logg = 0-4429, so that ;s = 8150m, oi over 5. miles. By similar calculations the Jesuit Francis Lana (1670) first demonstrated the possibility of aeronautics ; but the copper sphere exhausted of air which he considered would not bear the external pressure without collapse. Valuable information concerning the state of the atmo- sphere has been obtained by free balloons, carrying self- recording meteorological instruments. (Gomptes Mendus, April, 1893, Q. Hermite, L'explora- tion de la partie atmospherique au moyen des ballons.) 243. To attain a height of 30,000 to 40,000 ft, as pro- posed by Bixio and Barral in 1850, let us put 2 = 35,000; also /<; = 27,800, corresponding to 15 C or 60 F, and a barometric height of 30 ins at the" ground. On the isothermal theory pIP^ = pIpz = exp zjh = 3-5, giving at the altitude z a barometric height of 8'6 inches, G.H, T 338 THE THEORY AND PRACTICE On the convective equilibrium theory, 0^/0 = -640, so that t2= -88 C ; /02/p = '331, or p/p^ = 3; PzIp = '^^^, a barometric height of 64 inches. Now, from the condition of equilibrium (8), p. 333, according as we assume the isothermal or convective equilibrium state. The quantity W'/W is so small that it may be neglected; and taking n = S for very light coal gas, we have QjW=V^5 or 1-5, a mean value being 1'625, for a height of 35,000 feet to be attained; and F/Q = -6154. An ordinary balloon of 680 cubic yards capacity will lift about 750 lb gross ; and Coxwell employs a gross weight Tr=1254 1b of a balloon with a capacity of 32,000 cubic feet, to be filled with gas of specific gravity 0-440 of that of the air, so that n = 2-27 about. Then at the maximum height attainable with this balloon p/p^ = {n-\)QIW =1-1 about; and in an isothermal atmosphere ^ = fclog;i-l = 2650; and when there is convective equilibrium, 244. In Coxwell's balloon F= 90,000 ft^ ; and supposing the coal gas to have a specific volume of 40 cubic feet to the lb, Q = 90,000-h40 = 2250 lb, about a ton; and then TF=1385 lb, the gross weight of balloon, car, ropes, ballast, instruments and aeronauts the balloon can take to a height of 35,000 feet. OF AERONA UTICS. 339 The lowest barometric height recorded by Glaisher in the ascent of 5th September, 1862, was 7 inches; this would give, in an atmosphere of uniform tempera- ture 15 C or 60 F, a height z = 'k log,30/7 = 41,000 feet ; and, when in convective equilibrium, from (8), with a temperature 03 = 191, t3=— 82C, or -116 F; the mean of 60 F and - 116 F being - 28 F. 245. If Coxwell's balloon had been used as a Mont- golfier fire balloon, with a capacity F= 90,000 ft^, we must employ equation (4) to determine the requisite temperature to ascend to a height 35,000 ft ; and then from (2), p. 329, neglecting W'jM, e'~ M p; Here M is the weight of 90,000 ft^ of air at 60 F ; and, supposing 13 ft* to weigh a lb, if= 90,000-^1 3. Supposing W=1400 1b, and putting t = 15, = 288, pi p^ = So in the isothermal atmosphere, we find 0'/9='3, and therefore 0' = 96O, t' = 687 C. In convective equilibrium 0^= 181, p/pi,= S ; and (§ 236) 0/0' = 0-39, 0' = 46O, t' = 187°C, not as excessive temperature. Similarly it can be shown that for a balloon of 15,000 ft* capacity, and gross weight 500 lb, to rise to a height of 1000 ft, the temperature of the air inside must be raised to about 250° C, on either hypothesis of Thermal or Con- vective Equilibrium. 340 THE HISTORY OF AEROXAVTICS. The fire balloon increases in efficiency as the tempera- ture diminishes in the upper strata of the atmosphere, and it can keep the air for as long as the fuel lasts, so that it is suitable for long voyages, especially now that the former risk of catching fire, by which Pilatre de Kozier lost his life, is obviated by Mr. Percival Spenser's adoption of asbestos for the material (Times, 31 .Jan. 1890). On the other hand, its great size oijmjjar&d with the hydrogen balloon is a disadvantage fin* mSSxaxj purposes, where the wind is the great difficaltj :»: be encountered in filling and in holding captive the l3a]!t>j«. For the History and Practice of Aeratsntie* the follow- ing works may be consulted : — Faujas de Saint Fond. Description des experiences de la machine a^rostatique de Mm. de Jlontgolfier. Paris, 1783-4. Vincent Lunardi. Aerial Voyages. 17S4. Tiberius Cavallo, F.R.S. The Hiitory and Practice of Aerostation. London, 1785. Thomas Baldwin, M.A. Airopaidia. 1786. Monck Mason. Aeronautica. 1838. Hatton Turnor. Astra Contra. 1865. Glaisher. Travels in the Air. 1871. Tissandier. Histoire des ballons. 1887. Stevens. The History of Aeronautics. Scientific American Supplement, March, 1890. The Scientific American. Hot air ballooning, p. 143, 5 Sep., 1891. Cleveland Abbe. The Mechanics of the Earth's Atmo- sphere. Smithsonian Institution, 1891. Eevue maritime et coloniale. Ballons. May-Nov., 1892. THE GASHOLDER. 341 Examples. (1) What must be the dimensions of a balloon the whole weight of which, with its appendages, is 700 pounds, that it may just rise half a mile high ; supposing air to have 14 times the specific gravity of the gas under the same pressure, that 5 cubic feet of air at the earth's surface weighs 6 ounces, and that the density at the earth's surface is 4 times as great as at the height of 7 miles ? (2) A balloon ascends 1000 feet above the height at which it is fully inflated ; determine the fraction of gas which escapes, with the temperature uniform. (3) A w'eightless balloon, filled with hydrogen, is held by a string, and has a small hole on the lower side. The tension of the string is 3'79 g ; while after the barometer has fallen 8 mm the tension becomes 3'75 g. Find the height of the barometer and the volume of the balloon, given that the temperature of the air is 15° C, and that at 0° C and 760 mm pressure a litre of air weighs 1'293 g, and of hydrogen '089 g. 246. The Gasholder. Gasholders, formerly called Gasometers, are vertical cylindrical vessels of sheet iron, used for storing gas as it is manufactured. The cylinder is closed at the top, but open at the bottom and floats inverted over water contained in a circular tank, in which the cylinder can rise or fall (fig. 70.) The buoyancy of the water displaced by the sheet iron may be considered inappreciable ; and now if W lb is the weight of the gasholder and r ft the radius, the gas inside 34.2 THE THEORY OP must be at a pressure WIttt^ Ib/ft^ (over atmospheric pressure); and this can be measured in a siphon gauge (fig. 71) by a column of water TF/D7rr2ft or 12 WjBTrr^uis high, the diiFerence of level of the water inside and outside the gasholder. Denoting the weight of the iron plate in Ib/ft^ by m, and the height of the cylinder by a ft, then W= m(27rra + irr^). This supposes that the top of the cylinder is flat ; but it is generally slightly dome-shaped, for strength. Fig 70 To keep down the pressure of the gas, W must be made as small as possible for a given volume F= irr^a ; and this is secured by making a = r, or the height equal to half the diameter ; the proof of this is a simple exercise in the Differential Calculus. Tlien, with these proportions, 'W=^Trr^m, or Tf/7rr^ = 3m, so that the pressure of the gas is three times the weight of the sheet iron, in Ib/ft^. THE GASHOLDER. 343 For sheet iron plates one 16th inch thick, m = 2-5 ; and then the minimum pressure of the gas is 7-5 Ib/ft^, indicated on the siphon gauge by 12x7-5-T-624 = 7-5^5-2=l-5 ins of water; one inch of water giving a pressure of 52 Ib/ft^. If the pressure is too great the weight W is partly relieved by counterbalance weights or by buoyancy chambers (fig. 73). 247. The vertical stability of the Gasholder against wind is secured by rollers running against outside guid- ing columns, braced together. Large gasholders are made telescopic, with two or more lifts, to avoid a great depth in the tank ; and as tiie gas- holder is filled, the innermost section rises first, and when fully inflated it raises the next outside section by a joint, . made gastight by a channel of water (fig. 72), and so on. The largest gasholders attain a capacity of over 12 million ft*; the one at Greenwich, 300 ft in diameter and 180 ft high in six lifts of 30 ft, will contain 12,730,000 ft* or 220 tons of gas, of specific volume 25 ft*/lb, the produce of about 1500 tons of coal. With sheet iron one 16th inch thick, the pressure will be 9*5 Ib/ft^ or about 1"8 ins of water; and the weight of the sheet iron will be about 300 tons. If the gasholders are placed in communication, the gas will flow into the one of lowest pressure, generally the largest; just as soapbubbles in communication gradually exhaust into the largest bubble. It is important that a gasholder should not be filled too full by day, as the Sun's rays will cause the gas to expand and escape, and the gasholder is said to blow; 344- THE GOVERNOR OF the increase of volume or pressure for a given rise of tem- perature can be calculated by the Gaseous Laws of § 197. 248. The, Governor of a Gasholder. Denoting by p and p' the density of the air and of the gas, then in ascending a moderate height h ft vertical along a gas pipe, the pressure of the air has diminished by ph and of the gas by p'h, treating their densities as uniform (§ 173). The pressure of the gas over atmospheric pressure has therefore increased by (p-p')h = Dz^l2, if measured by z ins of water, of density D, in a siphon gauge ; and therefore It is estimated that a difference of level of 20 ft corre- sponds to a change of pressure of one 10th of an inch of water, so that h/z=200 ; and putting D/p = 800, this makes P =ip- It is important then that the gas from the gasholder should be delivered into the gas mains leading to different levels at an appropriate pressure, and this is effected by the Governor (fig. 73). It consists of a miniature gasholder, provided with counterbalance weights W and buoyancy chambers B ; and the flow of the gas is regulated by a conical plug G in a circular hole, which rises and falls with a governor according to the increase or diminution of the pressure, and thereby checks or stimulates the flow. The pressure in the main is adjusted by alteration of the counterbalance weight. 249. The theory of the governor is therefore the same A GASHOLDER. 345 as that of the old fashioned wooden reservoirs of gas, in \vhich the buoyancy of the water was appreciable. Starting with a volume of U&? of gas at atmospheric pressure, when the upward buoyancy of the water is equal to the weight of the gasholder less the counter- balance weights; suppose that Fft^ of gas at atmospheric pressure is introduced into the gasholder, and that it rises x ft in consequence, with respect to the outside level of the water. o V^ k "vwww Fig. 71. Fig. 72. Fig. 73. . Denoting' by |8 ft^ the cross section of the buoyancy chamber, the loss of buoyancy is D^x lb ; so that if r is the internal radius of the vessel, the gas is at a pressure B^xjirr^ lb/ft2, over atmospheric pressure. If the siphon gauge records y ft of water, then 2/ = /3cc/7rr2; and y will be the difference of the levels of the water inside and outside the gasholder. 346 OSCILLATIONS OF A GASHOLDER. The volume of the gas will now be U + 'n-T\x + 'y)ii^, under a head of H+y ft of water, H denoting the height of the water barometer, about 34 ft. Therefore by Boyle's Law, U+V ^ H+y U+-wr\x+y)~ H ' V=U^+^r%x+y)(l+^); and the gasholder in its descent can give out Vii^ of gas at atmospheric pressure. 250. If this gasholder is depressed from its position of equilibrium through a small vertical distance z ft, and if the level of the water inside and outside rises through p and q ft, then a and y denoting in ft^ the interior and exterior water line areas /3z = ap + yq, (1) expressing the condition that the quantity of water in the tank is unchanged. The pressure of the gas inside rises from DiH+y) to BiH+y-p + q). while the volume of the gas diminishes from aa to a{a — z—p), if a denotes the length of the cylinder occupied by the gas. The increase of upward buoyancy and air thrust is therefore P = D^(z+q) + Da{q-p), (2) and this is the force in lbs required to depress the vessel through a distance z. If no gas can escape, then, by Boyle's law, a{H+y) = (a-z-p)(H+y-p + q), q=pHH+y)^^. THE DIVING BELL. 347 But as z, p, q are small, we may put Therefore, from equation (1), and, according to the usual theory of § 148, if the gas- holder and its counterpoise weigh W lb, the oscillations will synchronize with a pendulum of length Wz_ W { a + yyi+y{H+y ) P ~ I)(a + l3 + y) Ba + (a + ^){H+yY '251. But if the counterbalance weights are replaced by an equal gasholder, and there is free communication between their interiors, so that there is no change of volume or pressure in the gas, then p = q; and the length of the equivalent pendulum is Wz _ W(\ 1 \ D/3(z + q)~ B\I3 a + 13+yr as in the vertical oscillations of a ship in a dock (§ 152). When /3 is small, the oscillations are slow, and the force P is small. 252. The Diving Bell and Diving Dress. With the aid of the Balloon we are able to mount up in the air, and the Diving Bell is an instrument by which we can descend in water and make a prolonged stay. The Diving Bell is a bell -shaped body made of cast iron (or lead; Evelyn's Diary, 19th July, 1661) sufficiently thick for it to sink in water, even when full of air, on being lowered mouth downwards by a chain (fig. 74, p. 355). Denoting by s the S.G. of the material (cast iron) by Uit^ the volume of air which fills the bell, and by WUii^ the total volume of water displaced by the bell 348 THE THEORY OF when submerged and full of air, so that (F— 1)P" is the volume of the metal ; then for the bell to sink, neglecting the weight of the air inside the bell, (P -\)sU> It? U, or Jc > \Ij^> determining the linear dimensions of the interior of the bell, supposing it similar to the exterior. Taking s=7'2 for cast iron, then k> 105, so that the thickness of the bell should exceed one 40th the external diameter, for the bell to sink in water when full of air. As the bell descends the pressure in the interior in- creases, and the air would be compressed and the water would rise in the bell ; but by pumping in air from above the water is kept out, and fresh air is introduced, the foul air escaping under the lower edge of the bell. Taking i) = 64 for sea water, and an atmospheric pressure of 14f Ib/in^, the head IT of water which pro- duces the pressure of an atmosphere is given by ir= 14| X ] 44H-64 = 33 feet= 5J fathoms ; and 9 ft or IJ fathoms of depth gives an increased pressure of 4 Ib/in^. When the bell is lowered to a depth of S3 ft or 10 m in sea water, air must be pumped in against a pressure of 2 atmospheres, and a barometer in the bell would stand at double its ordinary height; at a depth of 66 ft the pressure rises to 3 atmospheres, and so on in proportion, up to a depth of 165 feet or 27|^ fathoms, when the pressure would be 6 atmospheres, which is about the extreme limit endurable, and then only for a few minutes by the most practised divers. 253. If a diving bell is lowered without pumping in air, as in the Sounding Machine of § 204, then when the THE DIVING BELL. 349 depth of the base is x ft, and the water has risen y ft in the bell, so that x — y is. the depth of its surface, the pressure of the air is (H+x — y)/ff atmospheres ; and therefore the volume of the air is UII/(H+x-y){t^ But if air is pumped in so that the volume in the bell is Fft^ and if 7/ still denotes the height of the water in the bell, then at atmospheric pressure this air would occupy a volume V(H+x-y)/H{t^; so that V^'^^~y -U{t^ of atmospheric air (that is, of air at atmospheric density) has been pumped in. Thus, putting y = 0, and F= U, we find that Ux/H ft^ of atmospheric air must be pumped in to clear the bell of water at a depth of £c ft ; and if the bell is lowered at a uniform rate v ft/sec, the pressure and density of the air increase uniformly, so that atmospheric air must be pumped in at a uniform rate, Vv/Hit^/sec, to keep the water from rising in the bell. If the bell is made of metal of s.G. s and weighs Wlh, the tension of the chain by which it is lowered will be yib, where W W W'V s s U W denoting the weight of water which will fill the bell. 254. Suppose now that the water barometer rises from H to H+ AH, while the temperature of the air in the bell changes from t to t C, and that the water rises an additional distance Ay in the bell; then A ft^ denoting 360 THE THEORY OF the water line area inside, the new pressure, expressed in the former atmospheres, is given by H-\-^H+x-y-b.y _H.-\-x-y V 273+t H ~ H F-^Ai/ 273 + t" or, denoting the volume of atmospheric air in the bell by Fq, so that 273+T The diminution AT in the tension of the chain is given by AT=W'AAyjU; and the tension is therefore unaltered i{ Ay = 0, or Ag_/, .x-y\ t'-t H ~V^ H /'273+T Neglecting squares and products of Ay and AH, AAy_V_ AH-Ay _ t-t V V, If ~273+t' so that, i{ AH=0, '^y V^+V^AH 273 + t' but if there is no change of temperature, Ay _ V^ AH~r'+ V,AH' Since AVo=^Ar-^Ay, a.nd AV= -AAy, a volume AF„ of atmospheric air pumped into the bell will lower the water a distance —Ay, given by Ay _ VH AV~V^+V^H' TEE DIVING BELL. 351 We may suppose that LV^ is the volume of atmo- spheric air liberated by opening a bottle of aerated water; conversely, a flask which has been opened and screwed up again inside the bell will contain the con- densed air, and be liable to burst on reaching the surface. Suppose the bell is lowered an additional depth Ace, without pumping in air; then the new pressure in the former atmospheres H+x + i^x — y — i^y _ H+x — y V H ~ H V-ALy' or, approximately, -^ = y^^y^ff - 25 o. If the bell is cylindrical and of height a, then U=Aa, V=A(a — y); and if the mercurial barometer in the bell has risen from h to h' in descending to a depth x, and no air has been pumped in, Fq= U, and h' _Vo_ a _ x-y , h~V~a-y~^ H ' and then y = a\l — r-,), =(*'-»)(f+p) if aR} 273 + t' If the bell is stationary and a volume AF,, of atmo- spheric air is pumped in, AAy _ { a—y)H ~~Ey~a~ {a-yf + aH' and therefore, if air is pumped in at a uniform rate, the water descends in the bell with velocity inversely pro- portional to a—y ' ^ ^ If the bell descends a small distance Aa;, ^y^ (fl-yf . Ate {a-yf + aH' and if the velocity v of descent is constant, / H \ x = vt = v( h 1 )• 256. The Stability of the Diving Bell. Although it is convenient to make the apparent weight of the bell in water, as measured b}' the tension of the chain, as small as possible for ease of manipulation, still a certain preponderance is requisite to ensure the, stability of the bell in its vertical position hanging downwards, so as to prevent it from turning mouth upwards when the air escapes under the lower edge. AND ITS STABILITY. 353 The investigation of the stability is similar to that required for a ship aground (§ 142) ; the bell is supposed slightly displaced through an angle 9 from the vertical position, represented in fig. 74, p. 355, by drawing the surface of the water and the vertical forces in their displaced position relative to the bell. Now, if K denotes the point of attachment of the chain, G the c.G. of the metal of the bell, supposed homo- geneous, B and B^ the C.G.'s of the volume occupied by the air in the upright and inclined positions, and M the metacentric centre of curvature of the curve of buoyancy BB^ ; then, as in § 101, BM^ABjV, where AW denotes the moment of inertia, in ft*, of the water area A about its c.G. F, and F denotes the volume, in ft^, of the air in the bell ; but now the metacentre M lies below the centre of buoyancy B. The forces acting upon the bell in the displaced position W (i.) W lb, acting downwards through G; s (ii.) DF lb, acting upwards through M; fiii) W DF lb, the tension of the chain, acting ^ '' s upwards through K ; form a couple, whose moment round K, dropping the factor sin 9, is in ft-lb, w(i-'^)kg-dv.km ^ w(i--)kG-DV.KB-I)AB ; and the equilibrium of the bell is stable if this moment is positive. Q.H. 2 354 THE DIVING BELL Suppose the bell is initially full of water, and that a small volume of air is pumped in ; the bell will not continue to hang vertically unless w(\-^^KG-BAk'^ is positive. But in working the bell the water is expelled, and the air escapes under the lower edge ; although the air would force its way out through a hole at the top of the bell, but then the level of water in the bell could not be kept steady. The stability of the bell, as measured by the above righting moment, diminishes as air is pumped in and V increases, Al? remaining constant if the interior of the bell is cylindrical ; so that stability must be secured when the bell is full of air. At a great depth, where the density p of the air in the bell becomes appreciable, D in the formula must be repilaced hy B — p. 257. The original idea of the Diving Bell is of great antiquity (Berthelot, Annales de Ghimie et de Physique, XXIV., 1891) ; but Smeaton was the first to use it for Civil Engineering operations in 1779 ; and it was extensively employed on the wreck of the Eoyal George, in 1782 and 1817 ; and in Kamsgate Harbour by Rennie in 1788. But the Diving Bell is now generally superseded by the Diving Dress (fig. 75), which is an india-rubber suit for the diver, provided with a copper helmet fitted with small circular windows and an air valve. The diver has thick lead soles on his boots, and carries leaden weights round his neck, so adjusted that his apparent weight in water and stability are nearly the same as ordinarily on land. AND DIVING DRESS. 355 Fresh air is pumped down to him through a tube as in a diving bel], the air escaping through the valve at the back of his head; if the diver wishes to rise he partly closes the valve, which causes the dress to be inflated and increases his buoyancy; so also sunken vessels are raised nowadays by large india-rubber bags placed in the hold and pumped full of air. In the Fleuss system the diver carries with him a vessel of compressed air, and he is thus independent of the pipe and can travel long distances ; as was required, for instance, during the construction of the Severn tunnel, when the water burst in and flooded the workings. Figs. 74, 75, 76, 77, A form of diving bell is shown in flg, 76 which is useful for the construction of harbours ; the neck of the bell reaches the surface of the water, and entrance is made to the bell through an air lock; a large diving bell of this nature was employed at the construction of the New Port of La Rochelle (Cosmos, 26th April, 1890). 356 SUBAQUEOUS OPERATIONS. The upward thrust of the air being equal to the weight of a cylindrical column of the water on an equal base, the weight of this bell must exceed the weight of water it displaces. The same principles are employed in sinking caissons for underwater foundations, as in the Forth Bridge (fig. 77); or in driving a tunnel under a river through muddy soft soil, as in the Hudson River and the Blackwall tunnels, now in progress. Air is forced in to equalize the pressure of the head of water, and to prevent its entrance, being retained by air locks through which the workmen and materials can pass ; a slight diminution of air pressure allows the water to percolate sufiiciently to loosen the ground, but an increase of pressure is apt to allow the air to blow out in a large bubble. This system, due to Mr. Greathead, has overcome the difficulties of subaqueous tunnelling; but if employed in the projected Channel Tunnel, a pressure of about 10 atmospheres would be required, to which the work- men are not yet accustomed. {The Diving Bell and Dress, J. W. Heinke, Proc. Inst. Civil Eng. XV. ; Diving Apparatus, W. A. Gorman, Proc. Inst. Mechani- cal Engineers, 1882 ; The Forth Bridge, Engineering, Feb., 1890.) Examples. (1) Two thin cylindrical gasholders which will hold four times their weight of water, and one of which just fits over the other, will float mouth downwards half immersed in water. If the larger one is now placed over the smaller, determine the position of equilibrium. EXAMPLES. 357 (2) Two equal cylindrical gasholders, of weight W and height a, float with a length ma occupied by gas, which at atmospheric pressure would occupy a length a. If a weight P is placed upon one ot them, and gas is transferred to the other till the top of the first just reaches the water, prove that the other rises a height (3) Prove that gas of constant pressure, measured by a height h of water, can be delivered by a gasholder in the form of a truncated cone, whose sides are inclined at an angle a to the vertical, if the thick- ness of the sides is h sin a. (4) Coal gas, of density 0^6 of that of the air, is delivered to the pipes at a pressure of 2 ins of water ; prove that 30U ft higher the pressure will be given by 3 '8 ins; the temperature being 10° 0. (5) Prove that the small vertical oscillations of a cylindrical solid, closed at the top and inverted over mercury in a wide basin, will synchronize with a pendulum of length M 1 o-yS , a h + z "•"Z? h + Z+Vja where M g denotes the weight of the body, a and /3cm^ the horizontal cross sections of the interior and of the material of the body, cr and h the density of mercury and the height of the baro- meter, Fcm^ the volume of air in the cavity, and z cm the difference of level of the mercury inside and outside. 358 EXAMPLES. (6) A caisson, closed at the top and divided in the middle by a horizontal diaphragm, whose weight is half that of the water it will contain, is floating over water. Prove that the draft of the caisson will be doubled when a hole is opened in the diaphragm. (7) A diving bell with a capacity of 125 ft^ is sunk in salt water to a depth of 100 ft. If the s.G. of salt water is 1'025, and the height of the fresh water barometer 34 ft, find the volume of atmospheric air required to clear the bell of water. (8) Two cylindrical caissons closed at the top, of equal cross section and heights \H and ^H, are placed in water so that the first is just submerged, and the second at a depth such that the air occupies the same volume in each ; prove that A.^2 is the depth of the water surface in the second. What will happen if communication is made by a pipe between the air spaces in the two caissons ? (9) Find how deep a cylindrical diving bell of height a and radius c, with a hemispherical top, must be sunk so that the water rises inside to the base of the hemisphere; and prove that the volume of atmospheric air now required to clear the bell of water is a 3 a times the volume of the bell. (10) Prove that if two equal cylindrical diving bells of height a, whose air spaces communicate by a pipe, are sunk so that their tops are at depths z-i and z^, and if a volume of atmospheric air is forced in, which would occupy a length h of either bell, the THE PUMP. 359 surface of the water in each bell is lowered (11) Determine the effect on the level of the water in a diving bell, on the pressure of the air, and on the tension of the chain, due to a floating body inside, according as it has come from the exterior, or has been detached from the interior ; or due to a workman leaving his seat in the bell to work on the bottom of the water. Prove that if a bucket of water weighing P lb is drawn up into the bell, then (§ 254), (i) the fall of water level in the bell, (ii) the diminution of volume of the air, (iii) the increase of tension of the chain are respectively PV H .. PV^ ... PV^ ^^^ D(V^+V,AHy ^"^ D(W+V^AH)' ^"^^ V'+V^AH' Write down the values of these expressions for a cylindrical bell. 258. Pumps. The simplest form of water pump is the common syringe, consisting of a piston rod and piston, working in a cylinder, which is dipped into water. If in contact with the lower surface of the piston, the water will, in consequence of the atmospheric pressure, follow the piston to a height which is only limited by the barometric head of water; the cylinder thus becomes filled with water, which is ejected on reversing the motion of the piston; this is the earliest form of fire- engine. By the addition of valves, as in fig. 8, p. 19, the cylinder may be fixed in position, and the piston with its packing may be replaced by a plunger working through a stufiing 360 THE FORCING PUMP box, as easier of manufacture and of adjustment in work- ing, and now this machine is called a force pump (§ 12) ; it is used on a large scale in Cornish pumping engines (§ 23) for driving water to a high reservoir in water works, and in draining mines ; the water lifted being often 30 times the weight of the coal raised. Fig. 78. Two such force pumps, placed side by side, and worked in alternate opposite directions by a lever, constitute the modern manual fire engine, which does not, however, differ essentially from the machine invented by Ctesibius, described in Hero's Tlvev/ijiaTiKa, B.C. 120 ; the pumps discharge into an air vessel, in which the cushion of air preserves a steady continuous stream of water in the hose. In a steam fire engine the piston rod of the steam cylinder actuates the piston of a double acting force pump (fig. 78), by which the continuous stream is produced. AND FIRE ENGINE. 361 The Worfchingtou pumping engine is of similar design ; tile ratio of the piston area of the steam cylinder to that of the pump being made somewhat greater than the inverse ratio of the steam and water pressures, according to the speed at which the pump is to be worked. Water may be used instead of steam to actuate the pump ; and now the product {Ah) of its head Qi ft) and of the area of piston on which it acts {A ft^) must exceed the product {Bk) of the height to which the water is forced {k ft) and of the area of the pump plunger {B ft^) ; the delivery (Q ft^/sec) depending on this excess. For, denoting the length of stroke by I ft, the moving force D{Ah — Bk) lb on the piston, acting through x ft suppose, will for l — x ft, the remainder of the stroke, be changed to a resisting force of DBk lb; and therefore if we take W lb as the inertia of the piston and the moving water in the pump, and v f/s as the maximum velocity' acquired, I Wv^/g ^D{Ah- Bk)x = DBk{l - x). The average velocity of the piston being ^v, the de- livery Q = \vB ; and hence we find which increases from zero to ^(^glB^J)k/W), as Ah increases from Bk to infinity. If the water pressure is used to check the motion of the piston, then ^Wv^lg = D{Ah-Bk)x = D{Ah+Bk)(i-x); and therefore, as before, ^ //. ,r,DAh(. BVc' 362 WORCESTER'S AND SA VERY'S ENGINE. The adjustment of the valves must be very accurate to secure this motion without shock ; or else a connect- ing rod, crank, and fly-wheel must be added, as in the ordinary steam engine. In the inventions of the Marquis of Worcester (1663) and of Savery (1696) steam acted directly upon the surface of the water, without the intervention of pistons ; and considerable waste by condensation took place. Fig. 79. Fig. 80. This method is nevertheless employed nowadays in the Pulsator or Pulsometer pump (fig. 79) where economy is not of so much importance as rapidity and certainty of action; an automatic spherical valve G admits steam to act alternately on the surface of the water in the vessels A and B, by which the water is forced to the required level; entrance and exit valves being provided to each chamber, as in a pump. THE SUCTION AND LIFTING PUMP. 363 269. In the common auction pump of domestic use the upper valve is placed in the bucket or piston, so that water passes through the bucket and is lifted by it, when the upper valve closes, to the desired level (fig. 80). If the piston rod is thickened so that its cross section is about half that of the pump barrel, half the water will be ejected during the down stroke of the bucket, and a more equable flow is thereby secured. The suction of a pump is the height reckoned from the surface of the water supply to the lower valve, but the height of the discharge above the lower valve is the height to which the water is forced or lifted. When the lift of the pump is considerable, a relief valve, opening upwards, is placed in the discharge pipe, and the barrel is closed with a cover and stuffing box, through which the piston rod works; the lower fixed valve may now be dispensed with, and this arrangement is called a Lifting Pump; but if the bucket valve is suppressed, the water is raised in the down stroke, and this is called a Forcing Pump (fig. 8). The suction is limited theoretically by the barometric head of water, about 33 ft or 10 m, but water can be lifted or forced to an indefinite height ; the suction and forcing pumps of a mine or of the pumping engines of water works must therefore be placed at a low level, very nearly that of the water supply. Instances are recorded in which the suction of a pump has reached even 40 ft; but in such cases the water must be highly aerated ; so that we may consider the column in the suction pipe as composed of alternate strata of liquid and air, as in the Sprengel pump (§ 275), instead of continuous solid water. 364 LIMIT ATIOXS OF WORKING Denote by a and ^ the cross section, in ft^, of the suction pipe AO and barrel OB of a vertical suction pump, by a the height of the suction pipe, and by 6 and c the greatest and least height, OB and OG, of the lower side of the bucket P above the lower fixed valve (fig. 80). Then in the Tith stroke, while the pump is sucking, the water rises in the suction pipe AO from a height Xn-\ to Xn ft above the level A of the supply, and the air above the column changes in densitj' from p„_i to pn, where pn-\ H—Xn-\ and the tension of the pump rod increases from D^Xn-l to DjSXn^h. Also by Boyle's law, a{a — Xn)pn + ^bpn = a(a — Xn-l)pn-l + fiCp J so that a{{a-x„.i){H-x.n^i)-{a~Xn){H-x^} = ^{b{H-Xn)-cH}, a quadratic equation for Xn in terms of Xn-\, of which the positive root must be taken, the negative root corre- sponding to a different physical problem. This equation may be written and the second member being positive, it follows that a{Xn — Xn^-^, the volume which enters the suction pipe in the nth stroke, is less than /3(h — c), the volume swept out by the bucket. The water will reach the barrel in the first stroke if OJj = a, Xq being zero ; and therefore if ±_PSb-c) H aH+'Bh OF TEE PUMP. 865 It will just reach the barrel in the second stroke, if Xq^ = a; and the condition is obtained by eliminating Xj between the equations a{aH-(a-x^){B-Xj)=^{b{H-x{)-cH}; and so on. If the pump will not suck, then x^ is always less than a ; and the greatest height x to which the water rises in the suction pipe is obtained by putting Pn ^^ Pn - 1) "^n ^^ •^n -l^^X l a(a-x){H-x) + ^cH=a(a-x){H-x) + ^b{H-x), sc ^ c The first requisite for the pump to work is therefore a<(l-j)^; so that water can be drawn through the lower valve. 260. Now if Xm-i and Xm denote the height above the level of the supply of the water in the barrel at the beginning and end of the mih stroke, the air which occupied a length a + c — Xm-i of the barrel under a head If at the beginning of the stroke will at the end occupy a length a + b — Xm under a head H—Xm.; so that, (a + b-Xm)(H — Xm) = (a + c-x^.i)H, a quadratic for determining x^ in terms of a;,„_i. Also the tension of the pump rod, due to the pressure of the air, increases from zero to D^x^ during the stroke ; and if the lower valve opens when the pressures above and below are equal, the bucket has then risen a distance z, given by - + Xm — a = H — a, or - = c + z^ "" ' c U- 366 THE AIR PUMPS Similarly the position of the bucket in the down stroke when its valve opens can be determined. The greatest height to which the water can be drawn in the barrel, provided it does not reach the bucket, is obtained by putting Xm-i = oi^m.==^\ and therefore x = Ua+h) + J{\{a+hy-{h-c)H] ■ and therefore a requisite condition is {h-c)H<\{a + hf, or BG and the upper fixed valve opens when the air of volume B—G and density a-n-i has been compressed to atmo- spheric density p, and therefore to volume (B — G)(rn -il p- 268. The weight of the valves is another cause tending to limit the rarefaction; suppose then that cr and vf denote the pressures required to lift the fixed and piston valve in Hauksbee's pump. So long as the valves operate, the pressures in the barrels at the end of the nth in and out stroke are respectively fcp + CT' and kpn—'^. During the nth stroke the air which occupied the receiver A at pressure lcpn-\ and the clearance G at pressure hp + '^' has expanded to air of volume A and pressure Icpn and of volume B and pressure kpn — '^; and therefore Akpn+B{kpn--!;r):=Akp^.i+G{kp + rn'); ^ G{ ^V5'\ f A W rs Gf TS'W °' P^^-k-B\P^Tc) = \A + B)V''-'~Tc~B\P^Tc)] _( A W/ G\ CT OcT'l -\a+b) \V-b)p~Tc~bTcj- The piston valve is lifted in the down stroke when the volume B of air of pressure kp„-i — rs is compressed to pressure kp + zs' and volume to xB, given by kpn-l — '!S = x{kp + T:!'); OF THE WEIGHT OF THE VALVES. 373 and the lixed valve is opened in the up stroke when the volume C of air of pressure kp + rn' is expanded to volume 2/Cand pressure kpn-i — '^s, given by kp+rs' = y(Jcpn_-^-nj); so that xy = l. Generally y kp + vf Jcp+^f A y-i Cj ( A y-i-) kp+vf\A+B) '^bX \a^b) y The piston valve opens after the fixed valve, if B + C-yOxB, or if B{l-x)-G{y-\) l—xfJcp->r''^ ^\/ A Yi^-^l^B) X \ kp + i is positive, as is generally the case ; and the valves cease to act when _Q _B ^~B' y~G' 269. To measure the rarefaction, a glass tube maj'^ be led down from the bottom of the receiver to a cistern of mercury ; and the height of the column, drawn up as in Hare's hydrometer (§ 162) will measure the rarefaction, the difference of the height of this column and of the barometric height being the meicurj'^ head of the pressure in the receiver. The pressure may also be measured by a barometer inside the receiver; and, to keep down the height, a shortened siphon barometer is employed, in which the Torricellian vacuum does not begin to appear till the pressure is considerably reduced, say to a head of 2 ins of mercury. 374 THE COMPRESSING OR 270. The Condensing Pump. If the direction of motion of the valves of an air pump is reversed, air will be forced into the receiver by the motion of the piston, and the pump is called a com/pressing or condensing pump or a condenser. Thus Smeaton's pump can be used as a condenser if the receiver is fixed over the top valve, air being drawn from the atmosphere through the lower valve, which may be dispensed with. Fig. 82. A simple form of condensing pump, employed for in- flating pneumatic tires of bicycles, is shown in fig. 82 ; and fig. 78 may be taken to represent the condensing or exhausting pump required for the Westinghouse or vacuum brake on railway carriages. Condensing pumps are required to supply fresh air, condensed to the requisite pressure, to submarine divers and diving bells, to caissons and other subaqueous operations; also in the transmission of power to drive machinery (Popp's system), to the Whitehead torpedo, and to boring machines in mines, and in the construction of long tunnels such as the St. Gothard, the air in its return assisting the ventilation. CONDENSING PUMP. 375 With no clearance, a volume B of atmospheric air is forced every stroke into the volume A of the receiver, so that after n complete strokes the density pn is given by Apr, = {A+nB)p, ,„ = (l+™^^. so that the density (and pressure) mounts up in a.p. The thrust P,j lb required to make the wth stroke complete is given by Pn={pn-l^)l3 = np/3BjA. A gauge for the condenser may be made of a horizontal glass tube OA, closed at 0, and containing a filament of mercury AB, exposed to the pressure in the receiver. Now as the pressure mounts up in n strokes from p to Pn> the filament will move from A towards 0, to An sup pose, such that, by Boyle's law, OAn_P_ A _ TTh AAn_n OA ~ pn~ A+nB~ m + n' OAn~m' if m = A/B. Sometimes this gauge is held vertical, dipping into a cistern of mercury; and now, if OA = a, at atmospheric pressure, and OA^ = yn, the pressure in atmospheres after n strokes is yn h A so that the graduations are given graphically by equi- distant ordinates of a hyperbola. 271. With clearance 0, the volume B of atmospheric air of density p and the volume A of air of density p^-i becomes condensed in the nth stroke to the volume A + G of air of density />„, so that (A + C)p„ = Ap„^i + Bp. 376 EFFECT OF CLEARANCE Writing this equation as a Geometi'ical Progression, B A ( B \ Pn or Pn_ P '\A + G. B_(B ''G \G {p-cp)' A A + Gj The ultimate compression, when %=oo, is BjG atmo- spheres ; and this is otherwise evident, if we notice that air will cease to enter the receiver A when its pressure is that of the air compressed to the volume G from the volume B of atmospheric air. Writing this relation Pn_ B fB P G 'G \G VV A + Gj expanding by the Binomial Theorem, reducing, and lastly putting C=0, we obtain, as before, But practically, as in diving operations, the pumps are set to compress the air to the requisite number of ON THE COMPRESSION. 377 atmospheres B/C, by an adjustment which gives the clearance G the appropriate magnitude. If ts and ct' denote the pressures required to lift the fixed and moving valves, and p^ the limiting pressure ; then air of pressure ^^ +ct in the clearance Owill have been compressed from volume B and pressure f) — CT' ; so that G{f^+vs) = B{p-r,'), B. If it is assumed that in a diving bell or dress at a depth z, and under a pressure of -z^ or 1 + rf atmospheres, the air escapes at a I'ate of Flog^ or l^logfl+-5-j ft7minute, the pump must make n strokes a minute, given by F, B F, / , z\ 272. A graphical construction of the working of the condenser is given in fig. 83 ; here OA represents the volume of the receiver, OB of the barrel, OG of the clearance, while ordinates represent the pressure of the air. In the first stroke the atmospheric air filling AB is compi-essed f)-om atmospheric pressure Bb or Gp along the isothermal hyperbola bp-^, centre A, until it cuts the ordinate Gp in pj, and then Gp.^ represents the pressure in the receiver at the end of the first stroke. In the return stroke the pressure in the barrel OB falls along the hyperbola Pid-^, centre 0, until it reaches atmo- spheric pressure ; after which the piston valve opens. The work done in the fii'st complete stroke is there- fore represented by the area bp-^dy 378 TEE AIR PUMP AND In the second stroke the atmospheric air in OB is compressed along the hyperbola hf^, centre 0, until the pressure becomes pj; and then the valve at opens, and the pressure in OA mounts up along the hyperbola filPi' centre A. In the return stroke the pressure in OG falls along the hyperbola p^d^, centre 0, and the work done in the second stroke is represented by the area hf^p^d^^ ; and so on. To construct these points geometrically, draw Af to meet Bh in \, then \pi is parallel io AB; draw Op to meet h^ y>i '^^ "i- then Cjdj is parallel to Gp ; draw 06^ cutting bp in e^, then e^f^ is parallel to Cp; draw Ap.^^ to meet e^'f^ in g^, then g^p^ is parallel to AB ; and if g^Pi meets Op in c^, then c^d^ is parallel to Gp ; and so on. The ultimate compression in the receiver, represented by the pressure Cp^, is obtained by producing Op to meet Bh in c, and drawing cp^ parallel to AB. A similar construction can be employed for the air- pump, or for the combined condenser and air-pump. 273. The Air Pump and Gondensmg Pump coTuhined. Tate's air pump can be made to act as a condensing pump by screwing the receiver which is to be filled on the end of the barrel. Suppose then, as the most general case, that air is pumped from a vessel of volume A and forced into a vessel of volume A' by means of a Smeaton pump of volume B, leaving clearances G and G' at the ends. Starting with all the air at atmospheric density p and the piston close to A, and denoting by p„_i and p'n-i the densities in the vessels A and J.' at the end of the ti — 1th stroke towards A' ; then {A+B-G')pn-i + iA'+G')p\^r = (A+A' + B)p, (1) CONDENSER COMBINED. 379 an equation expressing the constancy of the total quantity of air enclosed. During the nth stroke towards A the fixed valves are closed and the piston valve opens, when the air in the barrel B assumes the uniform density a-n-i, given by Ba-n-l = {B—C')pn-l-{-G'pn-l, (2) the valve opening when the piston divides the volume B into two parts P and P', such that P a-n-\ = {B—C')pn_i\ ,„. P' ri' ' f ^ ' During the nth return stroke towards A' the piston valve is closed, and the fixed valves open; and at the end of this stroke (^ + £-C>„ = ^/,„_i + C'cr„-i, (4) (A' + C")p'„ = ^y„-i + (5-0K_i; (5) and the addition of equations (2), (4), and (5) leads as a verification to equation (1). Eliminating an-i between (2) and (4), B{A+B-G')pr. = ABp,,.^+G{B-G')p^.r+GG'p\.^ = {AB+BG-GG')p^_^ rirt> +^f^,{{A+A'+B)p-{A + B-G')p^_^}, or B{A'+G'){A+B-G')pn = {B{A + G){A'+G')-GG\A+A'+B)}p^_^ + GG'{A+A'+B)p, which may be written in the form Pn — I^P = MPn-l — f'-p) = '^"'(^— f^)p, (6) ^ B(A + C)(A'+ G') - GG'jA + A'+B) where A- B{A'+G'){A+B-G') GG\A+A'+B) ^'~{A'+G'){B- G){B - G') + GG'{A +B-G'}' and the ultimate exhaustion of J. is to a density /xp. 380 ADIABATIG EXPANSION AND COMPRESSION. Similarly p'n - /m'p = A(/o'„_ i - fx.'p) = X"(l - m')p> ,_ iB-C){B-G'){A+A'+B) ^^^'^^ ^ ~{A'+ G'){B -C){B- C") + CG'{A +B- O')' and the ultimate compression in A' is to a density ix'p. In Smeaton's air pump, A'=cc; and in Hauksbee's air pump, B and C" are infinite, but i? - C" is finite ; so that, putting £-C" = B', A G A= J— -g7, ^ = j7, as before (§ 266). In the condensing pump, ^ = 00; and now, putting B — C=B", so that B" is the volume swept out by the piston, and making B and G infinite, gives the result for the ordinary condensing pump, with clearance G' (§ 271). 274. In the preceding investigations the temperature of the air has been assumed constant ; but if the pumps are worked rapidly, the adiabatic laws employed in § 226 show that the temperature rises and falls with the density and pressure. Eefrigerating machinery depends to a great extent in its action on this lowering of temperature with rare- faction; while on the other hand the compressed air supplied to the diver is warmed to an appreciable extent. The expressions obtained for the density of the air will not be altered; and the change in pressure will only affect the points at which the valves operate and the work required for exhaustion or compression ; and when after a lapse of time thermal equilibrium' has been restored by conduction of heat, the pressure will assume the value that has been employed. 275. Mercurial Air Pumps. The rarefaction is limited by the leakage of the piston and valves, and by the air absorbed and given off by THE MERCURIAL AIR PUMP. 381 the oil ; so that the vacuum which can be produced by an ordinary air pump is not sufficiently good for in- candescent electric lamps. In the Fleuss air pump the passages of the pump are filled with oil which circulates and fills up the clearance, and the oil is freed from air in a duplicate pump alongside. Sometimes mercury is employed to fill up the clearance, as in Kravogl's air pump ; and one of the earliest and best methods of exhausting a vessel is to make it into a Torricellian vacuum, as in Torricelli's original method, during which he discovered the barometer (§ 171); but this method requires a large quantity of mercury. In Sprengel's mercurial pump (fig. 84) the exhaustion of the air from a globe G is performed automatically by an intermittent flow of mercury by drops from the reservoir A, which gradually sweep out the air, and dis- charge it in bubbles in the cistern B\ a pinch cock E on a short length of india-rubber tube controlling the flow of the mercury. This is the essential part of the instrument; but a duplicate arrangement FG is now generally placed alongside, the vessel G to serve as an air trap for the bubbles in the mercury (fig. 85). To make a joint perfectly airtight, it is sealed by mercury surrounding it, as shown in the joint above D in fig. 85. If the bubbles at B are discharged into a receiver, par- tially exhausted of air by a pump, the apparatus can be considerably shortened below its normal height of about 40 ins; three or four fall tubes may be employed, to make the exhaustion more rapidly. 382 THE MERCURIAL AIR PUMP. The barometric column HK measures the rarefaction ; this is also measured by the M^Leod gauge, consisting of a graduated tube closed at the top which can be filled up with mercury to a given head, as in fig. 66 ; the compres- sion of the rarefied air imprisoned in the gauge measures the rarefaction, to a millionth of an atmosphere, which would be quite insensible on the barometric column HK. If the drops of mercury occupy equal lengths a of the fall tube, and if c denotes the length of the air -bubble when at the level of the cistern B, and therefore at atmospheric pressure, then the lengths of the successive bubbles above the cistern are ch ch ch h — a h — 2a' h—Sa where h denotes the height of the mercury barometer ; these lengths of air increasing in H.P. till the junction D is reached, so that they can be represented by the ordinates of a rectangular hyperbola. An inverted Sprengel tube LMA, driven by compressed air at L, can be employed to raise the mercury again from the cistern B to the reservoir A. (Rev. F. J. Smith, Phil. Mag., 1892; Nature, Aug. 1893; Mercurial Air Pumps, S. P. Thompson: Journal Soc. of Arts, 1887.) Examples. (1) Prove that if the height of the spout of a suction pump above the water supply is the height of the water barometer, and if at the commencement of any stroke the water in the suction pipe is m and n ft below the spout and the fixed valve, the water will rise ^m{^m— ^n) ft in the next stroke, if there is no clearance and if the pump is of uniform section throughout. EXAMPLES. 383 (2) Examine the effect of taking alternate strokes of an air pump and of a condenser attached to a receiver. Prove that if the barrel of each pump is one twentieth of the receiver, and the condenser be worked for 20 strokes and then the air pump for 14 strokes, the density of the air will be practi- cally unaltered. Fig. 84. Fig. 85. (3) Prove that if a bladder occupies one nih. of the volume of the receiver of an air pump, and if it bursts when the pressure is reduced to one mth of an atmosphere, the mercurial gauge will fall /i(m— l)/m?i, when h is the height of the barometer. 384 EXAMPLES. (4) Prove that if the temperature is constant the work required to increase q fold, or to diminish to one- gth, the density of atmospheric air of pressure P in a receiver of volume V is, respectively, PV{\Qgqi-q + V) and PF(l---log ^(7). Calculate the work required and the change of temperature in these cases when the compression or rarefaction takes place adiabatically (§ 233). Workout F=lm3, P=10%g/m2, ^=100. (5) • Examine the change in the indications of the siphon barometer of § 179, placed in the receiver of an air pump or condenser, when the dimensions of the barometer are taken into account ; and prove that, in one stroke of the air pump, the barometric column falls a distance, approximately, J3h / A a§h\ A+BV (A + Bfa + ^y while, in n strokes of the condenser, it rises, approximately, Bhf, a/3h 1/ , B\\ '"aV-^a['+''a)\' where A denotes the original volume of atmo- spheric air in the receiver. Prove also that if v denotes the volume and a the height of the air pump gauge in § 269, the mercury will rise in one stroke (neglecting the square of v), Bh Bhv / A h\ A + B {A+B)A^'^A+Ba)' CHAPTEE IX. THE TENSION OF VESSELS. CAPILLARITY. 276. The vessels employed for containing a fluid under great pressure are generally made cylindrical or spherical for strength ; and it is important to determine the stress in the material for given fluid pressure, or the maximum pressure allowable for given strength of material, for the purpose of calculating the requisite thickness. The simplest case of a vessel in tension is a circular pipe or cylindrical boiler, exposed to uniform internal pressure, so that thei'e is no tendency to distortion from the circular cross section. With an internal pressure p, a circumferential pull will be set up in the material of magnitude T per unit of length suppose, acting across a longitudinal section or seam of the cylinder ; and to determine T we suppose a length I of the cylinder to be divided into two halves by a dia- metral plane, and consider the equilibrium of either half. Denoting by d or 2r the internal diameter of the tube, the resultant fluid thrust on the curved semicircular sur- face is equal to the thrust across the plane base, and is therefore pld; and this thrust being balanced by the pull Tl on each side of the diametral plane, therefore 2Tl=pld, or T=\pd=pr (1) Q.H. 2b 385 886 TENSION OF CYLINDRICAL, SPHERICAL, 277. When the cylindrical vessel is closed there is in addition a longitudinal tension in the material ; denoting by T' the longitudinal pull per unit length across a cir- cumferential seam or section, the fluid thrust ^ird^p on the end of the cylinder must be balanced by the longi- tudinal pull TrdT across a circumferential seam, and TrdT' = \Trd?p, or T = l'pd = lpr = \T. (2) Thus the longitudinal tension is half the circum- ferential tension; or in the cylindrical shell of a boiler the circumferential joints and rows of rivets which resist the longitudinal pull need be only half the strength of the longitudinal joint, which resists the circumferential pull. 278. A spherical surface in tension is beautifully illustrated by a soap bubble as a complete sphere in air, or as a hemisphere on the surface ; denoting the internal diameter of a spherical vessel by d or 2r, the tension per unit length across a diametral section, due to internal pressure p, is also T' or ^pr ; for considering the equilibrium of either hemisphere into which the sphere is divided, the fluid thrust on the hemisphere is equal to the thrust on the base, or ^ird^p ; and this is balanced by irdT, the resultant pull round the circumference ; so that, as before, 7rdr = l7rd;% or r' = ip(^ = Jpr. Thus if a cylindrical boiler is made with hemispherical ends, these ends need have only half the thickness of the cylindrical shell ; but they will weigh the same as flat ends of the same thickness as the shell. The same results are obtained by considering the equi- librium of the part cut off by any plane parallel to the AND CONICAL VESSELS. 387 axis of the cylinder; if this part subtends an angle W at the axis, the fluid thrust on it, 2Zr sin .f, is balanced by the components of the pull on each side, perpendicular to this plane, 2iTsinO; and sin divides out; so also for the spherical surface. 279. If the cylinder has a conical end or shoulder, of vertical angle 2a, the stresses in the surface will be no longer uniform ; taking a circular cross section PMQ, of centre M and diameter 2y, cutting off the conical end POQ, and denoting by 2\ and T^ tlie tensions per unit length across this section and across the straight section of the surface, then from the equilibrium of POQ, 2x2/ TjCOs « = T^y^P) 01^ ^1 = hpy sec a. To determine T^, consider the equilibrium of either half of the surface cut off by two adjacent circular sec- tions pmq, p'm'q', equidistant from PMQ ; therefore 22\.'pp'=p.PQ.7n7n'+^ptsi,na-p7n.pq-^pta.na.p'7n'.p'q', or T„ —py cos a +py tan a sin a =py sec a = 2Tj. Thus 2\ and T^ become large when the conical end is nearly flat ; so that the ends require strengthening with longitudinal stays. Suppose however that there is a conical shoulder, as in the Coney Island Stand Pipe (§ 42) ; then if 2a denotes the diameter of the upper small end, and p the average pressure at the shoulder, 27r2/TiCOS a = upward thrust = -wiy^ — a^)p. 280. If the thickness of the material is e, then Tje and T/e are the average circumferential and longitudinal tensions, per unit of area, in the cylinder ; denoting them by t and t', t = W=prle. 388 AVERAGE TENSION AND When the thickness e is small compared with r, these average values will differ only slightly from their maxi- mum or minimum values ; t may be supposed limited by the working tension or tenacity of the material, as deter- mined in the testing machine ; and the requisite thickness e is given by e = rfjt. Thus a locomotive boiler 4 ft iu diameter, of steel of working tenacity 6 tons/in^, should be 027 inch thick to carry a pressure of 150 Ib/in^ ; . and water mains 6 ft in diameter, of cast iron of tenacity 1 ton/in^, to carry water under a head of 200 ft, should be 1"4 inches thick. 281. If I denotes the length of a cylindrical vessel of radius r and thickness e, required to contain a volume v of gas at a pressure p, the volume of metal of tenacity t required in the cylindrical part is ^irrle = 2TrrHp/t = 2vp/t, which is independent of the proportions of the cylinder. Thus in the Herresschoff boiler, composed of a long spiral copper tube, or in vessels required to carry gas like oxygen or hydrogen at a great pressure, the pro- portions may be varied without altering the weight. Vessels for holding compressed air for pneumatic guns are now made spherical ; and if a volume v at pressure p is to be carried in n spherical vessels of radius r and thickness e, e = ipr/t, for tenacity t ; and the volume of metal i-n-nr^e = 2Trnr^p/t = §vpjt, which again is independent of the radius of the vessels, and shows a saving of 25 per cent, of material over the cylindrical shape. PRESSURE IN VESSELS. 389 282. For an external collapsing pressure p, such as is experienced by the tubes and flues of a boiler, we take d or 2r to denote the external diameter ; and the tensions t and t' become changed into equal pressures. Thus in the Severn tunnel, a brick cylinder 30 ft external diameter and 2 ft thick, the average crushing pressure in the brickwork due to a head of 100 ft of water outside would be 325 Ib/in^ ; this excessive pressure would crush the mortar and cause the bricks to fly, and requires to be kept down by incessant pumping of the water in the neighbouring ground. The longitudinal thrust and average pressure in a curved dam can be calculated in the same way ; for example, in a concrete dam in Australia, with a radius of 1400 ft, 110 ft high and wide at the bottom, and 14 ft wide at the top. It is asserted that the Exeter canal, one of the earliest in this country, was purposely made winding, from the supposed extra stability of the curved banks. So also with spherical surfaces ; thus the Magdeburg hemispheres, a Magdeburg ell or 2 ft in diameter, would require a force of about 3400 lb to pull them apart, when half exhausted of air ; and the thrust at the joint would be about 45 Ib/in^. 283. The Stress Mlipse. Sometimes a tube is made with a winding spiral seam ; and to determine the stress across this seam, or generally across any oblique section, we must investigate the distribution of stress in the material, due to given circumferential and longitudinal tensions (or pressures), t and t' : and this introduces the theory of the stress ellipse. 390 THE STRESS ELLIPSE. Taking OA in the direction of the circumference and OB of the axis at any point of the surface (fig. 86), the stress, represented by the vector OP, across a short length aO/3 of an inclined section will be balanced by the stress t across /3y, represented by OA, and by the stress t' across ay, represented by OB suppose, where ay and /3y are drawn parallel to OA and OB. V \ H (3 o^ 1 y^/ \^ ^""^^^^ ^ Y 1 "^:^%^ a\ ' 5g^ m\a T K, B ""'xr / " K u, Fig. 86. Therefore, if OM and ON are the projections of OP on OA and OB, OM.al3=OA.I3y, 0N.al3=0B.ay; or, denoting the angle between a/3 and OB by 6, and OA, OB by a, b, OM = a cos e, ON =b sin 6; and therefore P describes an ellipse, with a and b as semi-axes, and 6 is the excentric angle of P ; this is called the stress ellipse. In fig. 86, OPi = a, OP^ = b, OR = i(a + b), RP = Ua-b), PRP^ = 2d; CONJUGATE STRESSES. 391 TPV is the tangent and HP I the normal at P ; also KJP = a, PK^ = b, PL^ = a, PL^ = h, OH=a + b, OI=a-b, etc.; thus various mechanical desci'iptions of the ellipse are in- ferred (Prof. T. Alexander, Trans. R. Irish Academy, 29). 284. Conjugate Stresses. It is easy to see that if OP is the stress across the plane OQ, then OQ is the stress across OP, by consider- ing the equilibrium of the parallelogram pqq'p', of which OP, OQ are the median lines ; for the resultant forces across pq and p'q' balance; and therefore also the re- sultant forces across qq' and pp', which cannot be the case unless they pass through the centre of the parallelo- gram, and are therefore parallel to pq. The stresses OP and OQ are called conjugate stresses; but OP and OQ are not conjugate diameters ; and denot- ing the angle BOQ by ~~z I — z. — • '1 '3 '2 G.H. 2C 402 SURFACE TENSION Also, as an exercise, the student may prove that sin^g 1 2 cos g 1 sin^/3 _ 1 2 cos /3 1 sia^y_ 1 2 cos y . 1 For instance with liquid films, a = /3 = y = f x, and 4^,2 ^^2 ^^^^ ^^2 ,,^2 ^^^^ ^^2 ^,^2 ^^^.^ j.^2 296. If a third sphere of another liquid was brought into contact, a compound body would be formed, bounded by portions of spherical surfaces; and the same condi- tions would have to be satisfied at the interfaces and edges of intersection, with the additional conditions of equilibrium of the points of intersection ; and so on for an aggregation of any number of liquids. The equilibrium of the edges and of their points of intersection can be studied in the arrangement of the liquid films in froth, especially when imprisoned in a glass vessel; the tension being the same everywhere, three films meet in an edge at angles of 120°; and four edges and six faces meet in a point, at equal angles. According to Maxwell {Mathematical Tripos, 1869), the number of regions and edges is equal to the number of faces and points. 297. Suppose the regular arrangement of the spheres in § 33 was subjected to a uniform squeezing pressure; the spheres if plastic would be flattened into rhombic dodecahedrons, in which any two adjacent faces are in- OF LIQUID FILMS. 403 clined at 120°, as with the liquid films, and also in the honeycomb. The corners of the rhombic dodecahedron are of two kinds, where (i) three, (ii) four edges and faces meet ; and these corners can be constructed by planes through the edges of a tetrahedron or cube, meeting in the centre. This can be realised practically with liquid films pro- ceeding from the edges of a wire tetrahedron or cube, which has been dipped in soapy water ; but whereas the arrangement is stable in the tetrahedron, a small cubelet, with curved edges and faces, is generally formed at the centre of the cube ; and on examination the interior arrangement of the liquid film in froth will be found to be composed of these elementary arrangements. If a right prism whose ends are equilateral triangles is dipped into the liquid, two corners of the first kind can be formed if the height is greater than ^^6 times a side of the triangle. 298. If the substance B is made solid, with a plane face, the liquid G will form a drop on it in the shape of a spherical segment or meniscus, meeting the plane at an angle a, called the angle of contact, given by TM \a V — — w?-' /:/-'5\ Fig. 90. 303. For the rise h between two parallel, vertical, plane plates, a distance d apart, we have 2TcQ^a = iudh, or h — 2Tcosa/wd, half the rise in a circular tube, as is easily observed. If the plates are vertical, but not quite parallel, then d varies as x the distance from the line of intersection of THE CAPILLAR Y CUR VE. 407 the plates; so that the elevation y is inverselj' as x, showing that the upper surface of the liquid between the plates will be a curve in the form of a hyperbola, with vertical and horizontal asymptotes, one the line of inter- section of the planes, and the other the line of intersection with the free liquid ; this is easily verified experimentally. 304. The Capillary Curve. The vertical cross section of the cylindrical surface formed by the free surface of a liquid in contact with a plane boundary, or of a broad drop on a horizontal plane, is called the capillary curve. Supposing the angle of contact is obtuse, as with mercury, the free surface is depressed below the asymp- totic horizontal plane with which it coalesces at a distance from the edge; and the depression x and the slope ^ of the capillary curve are connected by the simple relation, x — 2c sin ^) = rfsm% the tension T would become changed into a thrust or pressure ; this arrangement would be unstable unless the flexibility of AP was destroyed; and now Eankine's Hydrostatic Arch is realised, in which the thrust is uniform, when the load is due to material of uniform density reaching to the level Ox. 308. The differentiation of these equations gives dx „ . ^dd) „ 'WX-j-=is\n(p-T^, or xp = c'. This can be proved directly by considering the equi- librium of the elementary arc pp', whose middle point is P and centre of curvature Q; the hydrostatic thrust wx.pp' on the chord pp' is balanced by 2T sin PQp, the 410 THE EQUILIBRIUM OF component of the tensions along the normal, so that W P The curvature 1/p is thus proportional to the distance X from Oy, so that the curve AP is also the Elastica, the curve assumed by a bow, of uniform ^eccwra? rigidity B, bent hy a tension F ui Oy ; the bending moment Fx at P is then equal to the moment of resilience B/p, and FIB = T/w = c\ lb' h\ bl Fig. 91. Fig. 92. 309. If fig. 90 represents the vertical section of a circular tube, and of the corresponding capillary sur- faces of revolution, the curves are of a much more complicated analytical nature. The general property of all such capillary surfaces, separating two fluids at different pressures, is expressed by the fact that " the difference of pressure on the two sides is equal to the product of the surface tension and of the total curvature of the surface," the total curvature being defined as the sum of the reciprocals of the two principal radii of curvature of the surface. To prove this, take a small element of the surface at P cut out by lines equidistant from P and parallel CAPILLARY SURFACES. 411 to the lines of curvature qPq', rPr', of which Q, R are the centres of curvature (fig. 91). Then if p denotes the difference of pressure on the two sides and T, T' the surface tensions in the directions Pq, Pr, resolving along the normal PQR, p . 2Pq . 2Pr =2T sin PQq . 2Pr + 2 T'sin PRr . 2Pq, a m PQq , in PRt_ T T This reduces to the above when T=T' ; but a tangential stress U will exist in the lines of curvature, if they do not coincide with the axes of the stress ellipse at P. 310. Thus if T=T', a constant, then ^3 is proportional to the total curvature ; so that a pressure p varying in this manner over an open sheet of a surface can be balanced by a uniform tension T round the edge at right angles to it ; and a closed surface is in equilibrium. Again, if p is proportional to {PQ . PR)'\ the Gaussian measure of curvature, then T.PR + T'.PQ = constant ; and this can be satisfied by making T and T' inversely proportional to PR and PQ ; in this case the variable pressure p acting over an open sheet will be balanced by the tensions T and T' acting on a serrated edge consist- ing of elements of the lines of curvature ; and the closed surface is in equilibrium. If p = 0, the total curvature is zero, the characteristic property of minimuTn surfaces ; these are realised experimentally by liquid films, sticking to various boundaries, straight, circular, helical, or twisted. 311. In cylindrical surfaces one of the radii of curva- ture, PB, is infinite and its reciprocal zero, and we obtain the preceding relations for the Elastica (§ 308). 412 THE CONSTRUCTION OF In a conical surface, PQ is infinite, and the value of T^ in § 279 is obtained immediately. In a surface of revolution, AP (fig. 91) about Ox as axis, the principal radii of curvature are PQ and PG, the radius of curvature and normal to Ox of the meridian curve AP ; so that, if filled to the level LL', ..lp=t{1-^+-^), J__LP^ L °^ PQ~ c^ PG' The complete integration of this intrinsic relation is intractable ; but the curve AP can be drawn, as Young pointed out in 1804, by means of successive small arcs, struck with Q as centre ; this method was first put into operation by Prof. John Perry, acting under the instructions of Sir W. Thomson, in 1874; and now Mr. C V. Boys has constructed a celluloid scale with recip- rocal graduations (fig. 90), by means of which the curves can be drawn with ease and rapidity. The scale carries a glass pen at P, and is pivoted instantaneously at Q by means of a brass tripod, pro- vided with three needle points, two of which stick in the paper, and the third acts as the centre at Q. 312. Sir W. Thomson illustrates the form of a liquid drop, and generally of a fiexible elastic surface, by means of a sheet of indiarubber fastened to a horizontal circular ring ; water is poured into the sheet, by which it is dis- tended and assumes a variety of forms of revolution about a vertical axis (fig. 91). Denoting by T and 2" the tension per unit length of the surface in the direction of the meridian AP and per- pendicular to it, then at the section PP', of diameter 2y, CAPILLARY SURFAOES. 413 where the surface makes an angle ■^ with the vertical, 2'rryT cos \l/- = w . volume LPAP'L' = w I iry^dx + wiry^Qi — x), if the vessel is filled to a height h above the lowest point A with liquid of density w ; and the value of T is then given by the characteristic equation (§ 309), As an exercise the student may prove that dT T'-T . , -^r-= sini/f ; as y. so that T is a max. or min., either when ■>/r = 0, that is at the widest or narrowest part of the surface ; and then d^T T-T T'-T —r^ = or ; ds^ yp yp or else when T= T. If, however, T' should turn out negative, the surface would tend to pucker. Thus, as exercises, the student may prove that if the surface is a paraboloid, generated by i/^ = 4ax, then T'=wJ(-^){\^{lQh?+Sali+M'')-{x-lh+iaf]> so that T' is negative and the surface will pucker where while in a sphere, of radius a, ^ 2a — X which is always positive if 3a>h>^a; but if h<^a, T is negative where x and J)' = (^p + --j -,, by Boyle's law, and therefore OTT If 'p=p', then J.^=I67ra2', and the electric energy is ^EA = ^A^a = 8wa^T=2ST, or double the surface energy of the bubble. 316. If a circle of wire is dipped into a basin of soapy water, and raised gently in a horizontal position, a sur- face of revolution is formed by the film sticking to the wire, orthogonal to the surface of the water, and the meridian curve is a catenary. For neglecting the weight of the film, the condition of equilibrium of the zone cut ofi" by an upper horizontal circle FP', of radius y, and by the circle CO' on the water, of radius c suppose, is 2TryT cos-\}/- = 2TrGT, or y cos \p- = c, where \/r denotes the angle MPG between the ordinate PM and the normal PG. Dropping the perpendicular MB on the normal, then PH=G, a constant, a property of the catenary. 317. The film always breaks when the height of the wire above the water exceeds a certain amount, about one-third the diameter of the wire cii'jle; this maybe accounted for as follows — ■ While the wire is at a moderate distance from the water, two catenoids can theoretically be drawn, satisfy- ing the conditions, and corresponding to the two loops or festoons in which an endless chain will hang over two smooth pullies at the same level, not too far apart. G.H. 2d 418 THE GATENOID, These catenoids are similar surfaces, and their common tangent cone will have its vertex on the surface of the water ; and thus the tangent cones along the junction BB' with the wire will have their vertices, one below and the other above the water, the second surface being unstable and therefore non-existent. At a certain distance of the wire from the water, these two catenoids coalesce, just as the two festoons of the endless chain coalesce when the pullies exceed a certain distance apart ; and the liquid film always breaks at this distance, that is, when the vertex of the tangent cone round the edge of the wire reaches the surface. In fact, the plane film formed on the wire circle is now found to have a smaller area than the corresponding catenoid. 318. If air is blown into the space bounded by the catenoid and the plane film, sticking to the wire circle, or if air is removed ; the pressure in this space exceeds or falls below the atmospheric pressure bj' a certain amount, which we can denote by Tja; so that 2a is a length, the radius of the sphere into which the plane film is bulged ; and the meridian curve GPB of the sur- face will change, still however cutting the surface of the water at right angles at G (tigs. 93, 94). Considering the equilibrium of the zone of the film, bounded by the horizontal circles PP' and GG', 1-KyTQ.o%^\f'AircT=Tr{y'-e)- or -rric^-y^)-, tre^ " 2a(y cos\p- — c) = y^ — c^, or c^ — y^, according as the pressure inside is increased or diminished; therefore, writing n for the length of the normal PG, 2ac-c^ 2a , 2ac+c^ 2a , 2 — = 1, or = — = hi. UNDULOID, AND NOBOID. 419 319. The meridian curve GPB defined by this relation is of transcendental nature, and the surface is that which contains a given volume with minimum area; but the above relation proves that the curve GPB can be generated by the focus P of an ellipse or hj'perbola AA', rolling on the axis OQ. Fig. 93. Fig. 94. For writing p and r for the lengths PM and PG, then a well-known relation gives, with the usual notation, ^' = ^'* + l in the ellipse or hyperbola ; and here When a is infinite, then, as at first, 7/ = c sec i/r, or 7/^ -en in the curve GP \ and therefore p^ = cr in the rolling curve AG, which is therefore a parabola; and therefore 420 THE HELICOID. the roulette of the focus of a parabola is a catenary, as is well known. The surface formed by the revolution of the roulette of the focus of an ellipse is called the unduloid, and by the roulette of the focus of a hyperbola the nodoid, the surface formed by the catenary has already been called the catenoid. The unduloid becomes a cylinder when the rolling ellipse is a circle ; and it becomes a sphere, or a series of contiguous spheres, when the ellipse degenerates into a finite straight line. The portion of the nodoid formed by the revolution of a loop must be taken as having the internal pressure Tja over atmospheric pressure ; this can be produced by blowing a bubble between two plates of glass, propped open at a small angle by a piece of wood at the node. If the portion of the catenoid bounded by the planes BB' and CO' could be solidified without losing its flexibility, then on cutting it open along a meridian GPB, and pulling CC out straight, the surface will take the form of the helicoid, the surface assumed by a film sticking to a uniform helix and its axis. If any section PP' of the Catenoid is replaced by a thread or wire, the tension due to the film above, or the thrust due to the film below will be yT sin \^ or sT, if s denotes the arc GP ; and the stress in the corresponding spiral on the Helicoid will be found to be <»+?)• where s is the radius of the cylinder on which the spiral is wound (C. V. Boys, Soap Bubbles.) THICKNESS OF THE FILM. 421 320. In these investigations the weight of the liquid film has been taken as insensible ; otherwise the weight would influence the result, as in tents and marquees. But the thickness of the liquid films, as determined by Reinold and Rucker from optical measurements, may be as small as 12^/x; and half a litre of oil poured on the surface of the sea has been found to cover 10^ cm^, or 100 m square, without losing its continuity, implying a thickness of oQ/xix ; but this thin film is still effective for checking the ripples and small waves and calming the surface of the sea in a storm. Many interesting maximum and minimum problems can be solved in a simple manner from the mechanical considerations involved in the theory of flexible surfaces under tension ; as for instance : — The circle has the greatest area for given perimeter; illustrated by an endless thread in a plane film, when the film in its interior is broken. The sphere has the greatest volume for given surface ; illustrated by the soap bubble. Two segments on given bases and of given perimeter will enclose a maximum area when they are arcs of equal circles, realised by passing the endless thread through rings at the ends of the bases ; and so also for spherical segments on given circles, realised by blowing bubbles on the ends of a frustum of a cone ; etc. Examples. (1) Prove that the height of a flat drop of mercury is a mean proportional between the diameter of a capillary tube, and the depth to which mercury is depressed in it, supposing the angle of contact 180°. 422 EXAMPLES. (2) Investigate the coefficient of expansion of the radius of a soap bubble, supposing that the surface ten- sion diminishes uniformly with the temperature. (3) A soap bubble of radius a is blown inside another of radius h ; and the radii change to a' and b' when the atmospheric pressure changes from p to p'. Prove that p__V b{a^- a'^)(b'^ -a'^) + a%'^ - a'^b^ p'~b b'(a^ - a'^){¥ -a^) + a?b'^ - a'%^' (4) Prove that a flexible surface, of superficial density w Ib/ft^, hanging as a horizontal cylinder the , vertical cross section of which is a catenary, is changed by a pressure diflference p Ib/ft^ on its sides into a cylinder in which the tension across a generating line is still wy, where y is the height above a fixed horizontal plane, and in which the radius of curvature is changed to w y^ _ p + w a ii y = a where the slope \/f = 0; also that (p + w cos ■\p-)y = {p + iv)a. Determine the equation of this curve. (5) Prove that if r, r' denote the radii of curvature of a pair of perpendicular normal sections of the sur- face in § 309, making an angle (p with the lines of curvature ; and if t, t', and u denote the corre- sponding normal tensions and tangential stress, due to a pressure difference p ; then where E, R denote the principal radii of curvature. CHAPTER X. PRESSURE OF LIQUID IN MOVING VESSELS. 321 . When a vessel containing liquid is moving steadily, as for instance a locomotive engine, with given accelera- tion a (ft/sec^), an attached plumb line is deviated from the vertical ; and the surfaces of equal pressure and the free surface will be perpendicular to this plumb line, when the liquid is moving bodily with the vessel. If the liquid fills the vessel completely so that there is no free surface, the liquid will move bodily with the vessel, provided the vessel has no rotation. If however a vacant space is left, which may be supposed filled with some other liquid of a difierent density, oscillations will be set up in the free surface or surface of separation ; but these oscillations die out rapidly in consequence of viscosity (§ 4), until the liquid and vessel move together bodily. 322. No oscillations however need be set up in the free surface by a vertical motion of the vessel (although Lord Rayleigh asserts that the horizontal free sur- face may become unstable), nor will the plumb line be deviated; this we may suppose realised in Atwood's machine, or else, initially, in the scales of a common balance, when equilibrium is destroyed. 423 424 PRESSURE IN ASCENDING Suppose then that a bucket A and a counterpoise B, or else two buckets A and B, are suspended by a rope over a pulley, and that equilibrium is destroyed and motion takes place, in consequence of the inequality of the weights of A and B. Denoting these weights in lb by W and W, by T pounds the tension of the rope, by a the vertical ac- celeration of A and B, and by g the acceleration of gravity, in ft/sec^ ; then by the principles of Elementary Dynamics and by Newton's Second Law of Motion, g~ W W ~W+W'' and thereioi-e J = ■Txrj~-uf" the n.M. of W and W. We suppose the preponderating bucket A to be reduced to rest by applying to it an upward acceleration a; so that now the pressure at any depth 2; in the water in the bucket becomes changed from Bz to Bz(l-^. If B was also a bucket of water, the pressure at a depth z in it would be changed from Dz to Dz{l+"^. If the buckets are cylindrical and of weight negligible compared with the water they contain, then the hydro- static thrust on the bottom of the buckets is F(l-p or Tf'(l+p, each equal to T, as is otherwise evident. AND DESCENDING BUCKETS. 425 Barometers attached to A and B, standing at a height li when at rest, would now have heights h , h and \-ajg l + a/g' So also with a bucket attached to a spring, performing vertical, simple harmonic oscillations, or placed on a vessel performing dipping oscillations ; or with the water on the top of the piston of a vertical engine ; a horizontal plane of cleavage may make its appearance when the amplitude and speed of the oscillations is sufficiently increased. 323. Suppose now that in each bucket a part, of the weight, W or W, consists of a piece of cork of S.G. s. If the corks ai'e floating freely no change will take place in consequence of the motion. But if cornpletely submerged by a thread attached to the bottom of the bucket, then denoting the tensions of the thread in J. by P lb, and the weight of the cork by ilf lb, the buoyancy of the cork at rest will be M/slh; and therefore in motion will be M/ may s\ a J lb. Therefore p=^(i-«)(j-i)=-^,a-o=fG-i>. and if s > 1, P becomes negative and the body must be supposed suspended by a thread from the top of the bucket. For the tension of the thread in B the sign of a must be reversed. Suppose W= W, so that the buckets balance ; then if the thread holding down the cork M \n A is cut, the 426 IMPULSIVE PRESSURE equilibrium is destroyed ; and the student may prove as an exercise that the bucket A will descend, and the cork will rise through the water, with accelerations respectively 9' M , fl \ 2W (i-')-4r-. »^ Ki-0 2TF+ilfQ-l) 2W-\-M(^--i) This treatment, as in § 148, ignores the motion of the water due to the passage of the cork; but, as in §149, the result can be corrected for a small spherical or cylin- drical cork in a large vessel. 324. If the bucket A strikes the ground with velocity V and is suddenly reduced to rest, an impulsive pressure is set up for an instant in the water. Suppose, however, that the impact takes an appreciable time, t seconds. To stop the body A weighing TTlb, moving with velocity v ft/sec, in a short time t seconds, requires an average resistance R of the ground, given in pounds by R = ^. gt The product Rt of the force of R pounds and of the t seconds for which it acts is called the impulse, in second-pounds; and its mechanical equivalent Wv/g is called the momentum, also in sec-lb, of W lb moving with velocity v ft/sec. To stop the water in the bucket reaching to a depth z ft requires therefore a force Dzav/gt pounds, a ft^ de- noting the cross section of the bucket, or a pressure Dzv/gt Ib/ft^, compared with which the pressure due to gravity is insensible when t becomes small, so that the bucket runs the risk of bursting when it strikes the IN MOVING LIQUID. 427 ground, a tension Drzv/gt pounds per unit depth being set up in the circumferential hoops, if of radius r. As for the bucket B, the rope becoming slack, it moves freely under gravity, and the pressure of the water in it is reduced throughout to zero, or atmospheric pressure. 325. When a closed vessel, completely filled with water, and moving bodil}'^ in a given direction with velocity v, is suddenly stopped, the water comes to rest simul- taneously ; so that the impulse on any portion of the water is equal and opposite to the momentum of this liquid. Suppose this portion of water is removed and replaced bj' an equal solid, of S.G. s and weight M lb; the momentum of this solid, moving bodily with the liquid with momentum Mvjg sec-lb, is changed by the impulse Mv/gs of the surrounding liquid into — ( — 1 sec-lb g \s ) in the direction opposite to the original motion ; so that if the body is lighter than the surrounding water, the body will recoil with this momentum. If the body is like the cork attached by a thread to the bottom of the bucket A, the impulsive tension of the thread will be — — 1 ,sec-lb. g\s I The iTupulsive pressure at depth z in the water on impact of the bucket is therefore (in sec-lb/ft^) Dzv/g; or, if the velocity is suddenlj'^ reduced from v to v', the impulsive pressure is I>z{v—v')/g. 428 THE HYDRAULIC RAM. 326. So also in shutting off quickly, in t seconds, a stop valve in a water pipe or main, I ft long, filled with water flowing with velocity v ft/sec, the pressure in the neighbourhood of the valve is increased by Dlvlgt Ib/ft^ ; this may become excessive if t is made too small. The increase of pressure due to a sudden check of motion in a pipe is called water ram; so also in the impact of sea waves, the spray is sent to a great height. To diminish the shock of water ram, an air vessel must be provided, as in fig. 78, p. 360 ; and if ^ denotes the aver- age pressure of the cushion of air, the water is stopped in t = Dlv/gp seconds, during which the average velocity is ^v, so that q = A.^vt = ^ADlv^jgp ft^ of water enters the air vessel, if the cross section of the main is A ft^; and this water can be delivered at a pressure p, or to a head h=p/D ft, if its return into the main is prevented by a valve. This is the principle of Montgolfier's hydraulic ram, in which the main is laid at a slope, with a fall of H feet suppose, and a valve at the lower end opens and shuts automatically, to start and check the flow. If the valve is open T seconds for the column of water in the main to acquire the velocity v, then v^gTH/l; and the water which flows out of the valve is, in ft^, Q = A.ivT=^Alv'/gM. Since q ft^ is thereby lifted to an effective height h — H ft, the efficiency is q(h-H)_ H QH ~ V THE GA VQE GLASS. 429 327. Consider now the pressure in liquid which is moving bodily in a vessel with given acceleration a in a fixed direction, like the water in the boiler or tender of a locomotive engine. It is convenient to reduce Dynamical problems to a question of Statics by the application of D'Alembert's principle, which asserts that " the reversed eflfective forces and impressed forces of a system are in equilibrium," the effective force of any particle or body being defined as the force required to give it the acceleration which it actually takes. Thus if the weight of a particle is m lb, and if it has an acceleration a ft/sec^ in a given direction, its effective force is ma/g pounds (ma poundals) in that direction ; in c.G.s. units the effective force of m g moving with acceleration a spouds (cm/sec^) is ma dynes. If the particle is carried along steadily by the vessel as a plumb bob at the end of a short thread, this thread will therefore assume the direction of the resultant of g and of a reversed. The surfaces ot equal pressure in the liquid will be parallel planes perpendicular to this direction of the plumb line ; and therefore the free surface, if it exists, will also have this direction, when the liquid moves bodily with the vessel. Any floating body will occupy the same position as before, relative to this new free surface, all the forces being changed in the same ratio. 328. The level of the surface of the water in the boiler is marked in the gauge glass (§ 24); and now if the engine, originally on the level, is standing or running steadily with uniform velocity on an incline a, the mean 430 CHANGE OF LEVEL water level will still be a horizontal plane FL^, making an angle a with its original plane FL, the nodal line of the two surfaces passing through F, the C.G. of the water line area (§ 108) ; so that if c is the distance of the glass tube from this nodal line, the change of level in the gauge glass is c tan a (fig. 95). But if the engine and train is moving freely with acceleration g sin a down the incline, the direction of the attached plumb line and of the normal to the surfaces of equal pressure will be that of the resultant of g and g sin a reversed, and will therefore be perpendicular to the rails ; so that the water in the gauge glass will return to its normal position. 329. When steam is turned on, the engine will receive an additional acceleration a, which we may suppose constant and represented by fa, so that the reversed acceleration up the plane is a+g sin a; and the plumb line will now make an angle with the perpendicular to the rails, given by tan Q = ajg cos a- The water level FL^ will be perpendicular to the plumb line, so that the height of the water in the glass will change by c tan Q = cajg cos a, and will be the same whether the engine is going up or down the incline. Conversely when steam is shut off and the brakes fully applied, the retardation produced with a coefficient of friction n will be ixg cos a ; and now the plumb line and surfaces of equal pressure will be deviated with respect to the gauge glass through the angle of friction ^ = tan-^y(;i ; IN A LOCOMOTIVE BOILER. 431 and the water in the gauge will fall c tan ; and the triangles OJO', OQR are similar, as also the triangles OQJ, .ORO', homologous sides being inclined at an angle ^. The tension of the plumb line is thus T=P{JQIOJ) pounds, in a direction making an angle with JQ ; and thus the rolling of the ship converts the steady vertical lines of force due to gravity alone into variable equi- angular spirals round J as pole, of radial angle , , _, sin0 ci = tan ^^ 3 r. ^ 2(cos t^ — cos a) The surfaces of equal pressure in the bucket ot water will be cylinders, the sections of which are orthogonal equiangular spirals; and the free surface, if so small- that the waves on it die out rapidly, will be perpen- dicular to the plumb line PQ. 340. When the angle between the plumb line PQ and the normal to the surface of the sand in the box exceeds e the angle of repose (§ 30), the sand' will slip ; and thus the preceding investigation will enable us to determine 438 STRESSES DUE TO ROLLING the tendency of a grain cargo to shift, of the water in the boilers, or of petroleum cargo to wash about, and to a certain extent the tendencj' to produce sea-sickness at any point of the ship. Hence the necessity of loading grain cai'go up to the beams, and of filling petroleum tanks up to a height in the expansion chamber; while the disturbing effect of the rolling is seen to diminish in descending towards the keel. A marine barometer at Q, if free to swing in gimbals, will hang in the direction QP, and the mercury column, if mobile, will stand at a height y instead of /i, and register the pressure so that y = h{OJIJQ); to prevent this pumping action, the tube of the marine barometer is contracted to a fine bore for part of its length (§ 179). 341. At the end of a roll

==0, J coincides with 0', and the plumb line PQ sets itself in the direction QO', and the water in the bucket will be perpendicular to QO'. 342. When the weight P of the plummet becomes sensible compared with W the weight of the rest of the ship, and when the length I of the plumb line is considerable, the oscillations of the ship will be modified thereby; but now the treatment becomes intractable, except for small oscillations. Suppose then that the plumb line is suspended from the mast at a point at a height OG=h, and that it swings through a small angle 6' from the vertical as the ship rolls through a small angle 6; the ship will also move sideways a small distance, y suppose. The approximate equations of motion of the ship and of P are now, putting OM=a, g dt^" ^' ^ %= -(ir+P}a0+PA(0+0'), Ptd^y d^e jd'6'\_p 440 ANALYSIS OF THE COMBINED The combined motion will be compounded of small oscillations of simple pendulum type, of equivalent length X, given by r cos ^/{ W+ ^TTWa^^l - cos /3)'^(2 + cos /3)}, where .2/3 is the angle which the surface of the liquid left in the bowl subtends at its centre. (3) Prove that in liquid filling a closed box, which is suspended by equal parallel chains of length I and oscillating through an angle 2a, the surfaces of equal pressure are parallel planes perpendicular to the chains ; and that when the chains make an angle d with the vertical, the rate of increase of pressure is ^y (3 cos 0—2 cos a). ROTATING LIQUID. 443 344. Liquid in a Vessel rotating about a Vertical Axis. When the axis of rotation of the vessel is vertical, the surfaces of equal pressure are surfaces of revolution about this axis and therefore do not move in the vessel, so that a free surface is possible ; and when the liquid moves bodily with the vessel this surface is a paraboloid. For the plumb line supporting a particle at P, at a distance PM=y from the axis, will assume the direction of the resultant of gravity g and the reversed acceleration 2/to^ acting along MP ; so that if the plumb line cuts the axis in G (fig. 99), GM_j_ MP~ya)^' and therefore the surface of equal pressure AP through P, which cuts the plumb line at right angles, will have the subnormal GM=gju?, a constant, a characteristic property of the paraboloid, generated by the revolution of a parabola of semi-latus-rectum l=glu?. The velocity at the end of the latus-rectum is lui = ^{gl), the velocity due to the head ^l ; and I is the height of the conical pendulum which has the same period of revolution as the liquid. If the angular velocity is expressed by R, the revolu- tions per second, then w = 'i'KR, and (§ 154) Z = ^ = /^i?-2 = iXi?-2 = 0-8165xi?-2ft. 345. Taking any point Pj vertically below P in the surface, and considering the equilibrium of a thin prism about PP^ as axis, then in liquid of density w, the pressure at Pj must exceed the pressure at P by w . PPi, exactly as when at rest ; so that the surface of equal pressure through P^ will be another paraboloid. 444 LIQUIDS ROTATING BODILY If the vessel is filled with liquids of different densities, for instance, air, oil, water, and mercury, then, as in a state of rest, the liquids will distribute themselves in strata of densities increasing as they go down ; but the surfaces of separation of any two liquids and the free surface will be equal paraboloids, of latus-rectum '2,gjw^, instead of horizontal planes, the vertical depth of each stratum remaining the same as when at rest. -E' T 1 * X G M Q yC 1 * 1 ■p C' p; k <^' /./ c y N^Sw ip, B' - - x^ r^-^^- £ Fig. 98. Fig. 99. Plaster of Paris on the top of mercury can thus be made to assume the form of a paraboloid ; in this way Mendeleef proposes to form the speculum of a telescope. 346. If the free surface AP is in contact with air, the air must also be supposed to rotate bodily with the liquid when its density p is taken into account ; for if the air is at rest, the free surface AP is no longer a surface of equal pressure, as the pressure at A exceeds the pressure at P by px, if AM=x. Generally, if it was possible for two liquids of different densities p and o-, water over mercury for instance, to rotate bodily with different angular velocities w and fi, slipping smoothly over each other at their surface of separation, then along this surface ABOUT A VERTICAL AXIS. 44,5 so that the surface is a paraboloid of latus-rectum 347. The surfaces of separation of liquids in rotation can be shown experimentally in a glass vessel swung round at the end of a thread, like a conical pendulum ; also in a glass spinning top, filled with water and mer- cury, a bright silver belt being formed by the mercury. Cream is now separated from milk in an iron vessel, poised on the rounded top of a rapidly rotating vertical axle (fig. 98), the cream rising through the side pipe reaching to the broadest part of the interior, while the refuse milk escapes past the upper edge ; cloth, etc., is dried in the same way in Hydroextractors. The C.G. of the vessel is placed as close as possible under the centre of the support, by means of adjusting screws; this tends to diminish the vibration of the spindle, which is held in position at the upper end by an india- rubber ring ; for if the depth of the C.G. below the centre of the support is h, the angular velocity must exceed ^{g/h) before the vessel will tend to wobble (§ 349); thus if i^ = 0'02 inch, the critical number of revolutions is about 22 per second. The rise of the liquid round the inside of the vessel will also diminish h, and tend to increase the dynamical stability. 348. When the revolutions are very great, the stresses due to the rotation will exceed the elastic limit, and the metal will be deformed ; in this manner teapots are spun out of ductile pewter. 446 SURFACES OF EQUAL PRESSURE A ring of metal of radius r ft, cross section a ft^, and density w Ib/ft^, moving with peripheral speed v f/s, will experience a centrifugal force wav^jgr lb/ft ; and this will (§ 276) set up a circumferential tension T lb/ft, given by 2Ta = X 2r, or T= ; gr g so that T is independent of the radius ; and if T denotes the tenacity, the ring will burst if v==J(gT/w). Thus, in a steel flask a foot in diametei', making 16,000 revs/min, and taking w = 500, we find T is about 11 million Ib/ft^, or 34 tons/in^ 349. If the plummet is sunk to P^, its plumb line will take the direction G-^^P^, normal to A-^P.^ the surface of equal pressure through P^, but not normal to the free surface. The plumb line GP can also hang vertically from G, but the equilibrium will be unstable until its length is shortened to less than g/w^, the radius of curvature of the paraboloid at the vertex ; and if the plummet is lighter than the liquid it displaces, like a cork, the plumb line must be in the prolongation of GP, and fastened at a point beyond P ; and the equilibrium in the axis will be stable. The resultant thrust of the liquid on a finite portion whose C.G. is Pj, at a distance y from the axis, has a vertical component W, the weight of this portion of liquid, and a horizontal component towards the axis W'yw^/g or W'y/l. Thus, if the submerged plummet is of finite size, of weight Tf lb and s.G. s, the vertical and horizontal effective forces on it are IN ROTATING LIQUID. 447 Wfl — -J downwards, and Tffl — -W away from the axis, where Pj now denotes the c.G. of the plummet ; and these forces must be balanced by the tension of the thread. If s < 1, the body will be urged towards the axis of this whirlpool ; and if it floats freely it cannot be in equi- librium unless the C.G. is in the axis, or coincides with the c.G. of the displaced liquid. A floating body thus tends to the central depression of a whirlpool as a position of stable equilibrium. 350. If the floating body has a vertical axis of symmetry, like a cylinder, cone, etc., the water line is given by the condition that the weight is equal to the weight of liquid displaced, bounded by the parabolic free surface, continued through the body. Consider, for instance, a vertical cylinder of radius a, height h, and S.G. s, and let o denote the circumferential velocity of its curved surface. For moderate values of v the top of the cylinder will be out of water, and the bottom will be covered by water ; the water line will rise on the side, and the vertex A of the free surface will descend through equal distances ; also the vertical oscillations will be the same as in the liquid at rest. As V is increased the water will first flow over the top, or the vertex A will first come below the bottom of the cylinder, according as s is > or < J ; and finally, for greater values of v, when both the top is partly covered and the bottom uncovered, we shall find that the depth of A below 0, the centre of the cylinder, is {\-s)lv^lg. 448 THE PARABOLIC SPEED MEASURER For if this depth OA is denoted by x, the volume of water displaced is (§ 135) 7ra% -■kI{{x + Ihf -{x- Ihf } = 7r(a2 - 2Zcc)A ; and this is equal to Trsa^'h in the position of equilibrium. Also the increase of buoyancy due to a small addi- tional vertical displacement x' is 2irlkx', so that (§ 148) the vertical oscillations synchronize with a pendulum of length ^sa'^jl. If the water is contained in a coaxial cylinder of radius h, the cylinder sinks J(6^ — a^}wi^/g, until the water reaches the top ; the modification when the angular velocity w is still further increased is left as an exercise. So also for a cone, of height h and radius of base a, floating vertex downwards; it can be shown that the water line reaches the base when o>' = i(1-s)gh/aK 351. The Parabolic Speed Measurer. This consists essentially of a glass cylinder or cell, partly filled with mercury, the remaining volume being occupied by air or water ; it is placed on a vertical shaft Ox, the angular velocity w of which is to be measured by noting the rise of the mercury on a graduated scale. Suppose Oy is the level of the mercury when at rest ; the mercury will rise on the walls to the level PMP', while the vertex of its paraboloidal surface will sink to A suppose, where (fig. 99) PM^_2g AM ~ u?' Putting AM=x, then in a cylindrical vessel AO = OM=lx, from the fact that the volume of the paraboloid swept AND ITS GRADUATION. 449 out by AMP is half the volume of the circumscribing cylinder (§ 135) ; so that, denoting the radius by a, a^ zg a^ and therefore the angular velocity is proportional to the square root of OM, the rise of the mercury. The free surface of the mercury will cut the hori- zontal plane 00 in a circle at D, such that OI) = \iJia\ and this is therefore a fixed circle. In another form of speed indicator the rotating vessel is closed at the top by a fixed cover GC , provided with a cylindrical part GP.^.[G' dipping into the mercury ; and the mercury can rise through holes in the circumference P-^( into a fine glass tube Ox in the vertical axis. The pressure in the rotating mercury gliding round P-^( is measured by the column at rest in Ox, and the column will therefore rise to M at the same level as would the free surface PP', while the vertex A will now remain near 0, if the closed vessel is filled with mercury ; and the graduations for M are the same as before. The graduations for equal increments of angular velocity can be made geometrically by dividing HE and GE into the same number of equal parts n; and now, if N and R are corresponding divisions, such that EE _ GR_r HE~GE~n the vertical NQ and OB will meet in a point Q at the level of P, such that GP_GP^ CR_r^ GE'GB GE^n^' so that the angular velocity registered at P is r/n of the angular velocity registered at the level of E. G.H. 2p 450 THE PARABOLIC SPEED MEASURER The depth of the directrix plane below a minimutn, ^^2a, when 00 = sjiijigja,), and then is the focus of the paraboloid. 352. As the speed w is increased, P will meet the top EE' (and the liquid will spill out unless the top is closed) or A will meet the bottom BB' first, according as the cylinder is more or less than half full ; and finally, for still greater values of 00, the top EE' will be partly covered with mercury, and the bottom BB' partly un- covered (along pp'), as in a cream separator. Taking for simplicity the cylinder as half full of mercury, and of height h, the depth x of the vertex A below will now be given by the equation (§ 135) ■Kl[{x + \lif-{x-\Kf} = \7ra%, or x = la^/l, where l^gjoj^-^ and the areas of the circular edges of the mercury are 2'?rlix±^h). If the bottom BB' is still covered while the mercury meets the top EE' in a circle PP' of radius y, and if OH=b, AH=x, then the volume of the paraboloid PAP ^TTXy^ = -n-a^b, ,, , 2g t/2 2a^h i/* w^ X x'- 2a^o and y could be observed on a transparent cover to the cylinder; while x, if it could be observed, would be proportional to w. The depth of the directrix below is now a minimum whou w = g''(r'''b~^. AND ITS GRADUATIONS. 451 So also when the bottom BB' is uncovered before the mercury reaches the top. 353. A parabolic speed measurer, showing the form of the free surface, can be made by replacing the glass cylinder by a rectangular box or cell, of which two vertical sides are sheets of plate glass a small distance apart. The mercury will- now spread out into a film, bounded above by a parabola, or parabolic cylinder as we may consider it, when the thickness of the mercury film is small. In this case we shall find, since the area of the parabola AMP is two-thirds of the rectangle AP, that ii AM=x, then AO = \x, OM=%x; so that the graduations for P and A are proportionately the same as before ; and now OD = \iJ'ia, and this is the minimum depth of the directrix below 0. When the speed is sufiiciently increased for the para- bola AP to meet the top EE', then as before, %xy = ab, and J = ^ = ^t^' or M varies as x'^. The cases may also be worked out when, as before, the bottom is partly uncovered, and when the top is also partly covered. 354. Even when the thickness of the mercury film and the distance between the glass plates is appreciable, we shall find no material difierence in the results ; for the vertex A of the paraboloid will sink below half the distance which MP rises above (fig. 99), so that the graduations for P will be proportionately the same as before, increasing as the square of the angular velocity. 452 LIQUID IN ROTATING This follows from the geometrical theorem that the volume of mercury above the horizontal tangent plane at A is one-third of the volume of the vessel included between the horizontal planes through A and MP, when the cross section of the vessel is rectangular. 355. In finding the thrust on a plane wall of a rotating vessel of liquid and its C.P., we notice that the pressure at a point at a distance y from the axis exceeds the pressure wz at the same level on the axis, at a depth z below the vertex of the free surface, by ^luoo^y^/g, or ^wy^/l ; and it is convenient to calculate the thrust and C.P. due to wz and ^wy^/l separately, the thrust due to the latter being ^wAk^/l, where Ak^ denotes the moment of inertia of the wetted area A about the axis of rotation. Consider as an exercise the side of a rectangular vessel. So also the average pressure over the surface S of a vessel filled with liquid, due to the rotation, is ^wWjl, ■where k denotes the radius of gyration of the surface 8 about the axis. But if the axis of rotation does not pass through the highest point of S, the small bubble left at the top moves to the point where the highest surface of equal pressure touches S, and where a small drop of liquid can be re- moved without altering the pressure, and the average pressure due to gravity is diminished by the head equivalent to the vertical descent of the bubble. 356. If the vessel is composed of a tube of small bore, the paraboloids of equal pressure must be drawn through the ends and through the points of separation of the liquids; and now the pressures are equal at the points where one of these paraboloids cuts the tube ; so that is now no longer generally true that CURVED TUBES. 453 "the heights of the free surfaces above their common surface of two liquids in a bent tube are inversely as their densities" (§ 158). When u> is sufficiently increased a free surface will reach an end of the tube, and liquid will be spilt, unless the end is closed ; and now the pressure of the imprisoned air modifies the result. If, as in § 161, a filament of length c makes angles Q, (j>, with the normals to the free surface through the ends, and if these normals make angles &, <^ with the vertical, the same reasoning will show that the small oscillations of the filament will synchronize with a pen- dulum of length c/(sec 0' cos + sec ^' cos ^). The same principles will apply to the liquid in a rotating vessel of any shape, by means of which the student can prove the results of the following theorems or exercises. Exardfles. (1) A fine tube bent into three sides of a square, each of length a, is filled with liquid and rotated about a vertical axis bisecting the middle side at right angles. Prove that no mercury will escape until the angular velocity exceeds where h denotes the height of the barometer. If the horizontal part of the tube alone contains mercury and the vertical parts atmospheric air, the tube being closed at the upper ends, determine the angular velocity required to make the mercury rise a height c in the vertical branches. 454 EXAMPLES ON (2) A straight tube closed at the lower end is rotated about a vertical axis through this end at a con- stant angle a to the vertical ; prove that \il=gju)^, the greatest length of mercury it can contain is {icota + «y(2Z^)}coseG a. (3) If a circular tube of radius a rotates about a vertical diameter, and contains a filament of mercury sub- tending an angle 2a at the centre, the filament divides at the lowest point for angular velocity ^ = \/i9l«-)sec |a. For a greater angular velocity w the filament divides into two equal halves, the centres of which subtend at the centre of the circle an angle 2 cos-i{(fi/w)^ cos Ja}. (4) If the tube rotates about a vertical tangent, a filament of length (TT + Oja just reaches the highest point if u,^ = ^ |(isini9-siu^6). (5) If an open vertical cylinder of height h and radius a, full of liquid, is set spinning about its axis with peripheral velocity v, the vertex of the free sur- face will sink to a depth ^v^/g below the rim, and a volume ^ira^v^jg of liquid is spilt ; so that if the cylinder is stopped, the surface will be at a depth \v^jg below the rim. When ^v^lg is greater than Ti, the bottom of the cylinder will be uncovered in a circle of area '«■('-¥) ^ and when the cylinder is stopped, the liquid stands at a depth ^gh^jv^. ROTATING LIQUID. 455 (6) If the cylinder is closed at the top by a piston, the upward thrust on the piston is if Ji' = ^v^/g, the head of the velocity v ; and the downward thrust on the bottom is WTra%h+lh'). The average pressure over the curved surface will be due to a head h' + ^h, and will therefore be l + 2h'/h times its value when the liquid is at rest. (7) The piston will be lifted when the upward thrust on it is greater than its weight P ; or, denoting the weight of liquid by W, when 2h W and the piston will rise a distance s, given by ^h~i2h iw (8) Prove that if the circular cylinder is rotated about a parallel vertical axis, the edge of the free sur- face will be a plane ellipse. (9) If the vessel is a rectangular box rotated about a vertical axis through its centre, so that v is the velocity of its vertical edges, the volume spilt will be ■^abv^jg = \ahh', provided the bottom is not uncovered, which will happen when v^ = 8gh, or h' = '^h; and now f of the liquid is spilt. (10) If the vessel is a closed sphere rotated about a vertical diameter, the greatest pressure is at the lowest point so long as to < y/(g/a)- 456 EXAMPLES. For greater values of w prove that the greatest pressure is along, the circle where the paraboloidal surface of equal pressure touches the sphere, at a depth glu? below the centre; and determine the paraboloid orthogonal to the sphere. If W denotes the weight of the liquid filling the sphere, the liquid thrusts on the upper and lower hemispheres are (11) When the vessel is an open hemisphere, the bottom is uncovered when w > ^(gfa) ; and the volume of liquid left in the vessel is f rf = |-7r(c//ft)^/, even with a hole at the lowest point. The average pressure over the surface, before the bottom is uncovered, is diminished from \wa to ^wa\l — -^\ (12) In an open paraboloid, one-%th of the liquid is spilt if g/aj^ = nL, where L denotes the semi-latus- rectum; and all is spilt if g/ui^ = L, the surfaces of equal pressure being now similar to the paraboloid ; and thus if a hole is made anywhere in this vessel, all the liquid must flow out. If the horizontal cross sections of the paraboloid at a depth z below the rim are the ellipses the water line of the liquid will sink to a depth ; \(o- c/ THE SEA LEVEL. 457 (13) If the paraboloid is closed and filled with liquids of various densities, originally in horizontal strata, the arrangement of the strata is inverted when the angular velocity exceeds ^/{g/L). (14) In an open cone, the upper part of the surface becomes uncovered when the depth of the vertex of the free surface exceeds half the depth of the cone ; and the volume the cone can now hold is ^-w it / Consider also an inverted cone on a whirling table, and find when the liquid will escape. (15) If a closed surface filled with liquid is rotated with angular velocity w about a fixed axis at an angle a to the vertical, the surfaces of equal pressure are paraboloids of revolution round a parallel axis at a distance t/sin a\u^, of latus-rectum 2g cos alw^, the same as for liquid under the diminished gravity in the direction of the axis. 357. The Free Surface of the Ocean. Careful measurement shows that the surface of the Ocean, which we may take as the mean surface of the Earth, is not exactly spherical, as drawn by Archimedes, but of the spheroidal shape the surface would assume in consequence of the diurnal rotation. The normal to the free surface of the Ocean is at any point in the direction of the plumb line ; and now if we assume that the solid part of the Earth is spherical or rather centrobaric, the plumb line will take the direction of the resultant g of the attraction of pure gravitation G to the centre of the Earth, and of the centrifugal force due to the whirling motion of rotation. 458 TEE FREE SURFACE It is more convenient now to employ the absolute unit of force ; so that the attraction of pure gravitation on a plummet weighing TFg is WG dynes, where G denotes the acceleration of gravity on a projectile or freely falling body ; and the centrifugal force at a distance y cm from the polar axis is Wyo)^ dynes, where w denotes the angular velocity of the Earth, in radians per second. —2 ^Wy w^ S Fig. 100. Fig. 101. Producing the plumb line to meet the equator EQ in Q, then by the triangle of force EQP (fig. 100) EQ_ Wya? EP' EQ = %,.EP. WG' MP~G Now in the case of the surface of the Earth, G is so nearly constant and equal to g (980) and EP is so nearly equal to R the mean radius of the Earth, 10^-4-|7rcm, that we may, as a close approximation, put EQ^eKMP, where e^^Rw^jg, so that PQ is the normal to an ellipse of excentricity e passing through P; and the surface of the Ocean may be taken as the spheroid generated by the revolution of this ellipse about the polar axis. This would be accurately true if the acceleration of pure gravity at a distance r from the centre was G(r/R) ; and g at any point P would now vary as the normal PQ. OF THE OCEAN. 459 The time of a sidereal revolution of the Earth being T=23h 56 m 4 s = 86164 s, CO = 27r/y= log- 15-8629. logi? = 8'8039, log 0,2 = 97258, log i?ft,2 = 0-5297, log ^ = 2-9912, log £f/i?ft,2 = 2-4615, Sf/i?a)2 = 289-4«:.17^ e = l/17. 358. Denoting the equatorial and polar semi-axes of the spheroidal surface by a and 6, then (a — &)/a is called the ellipticity and denoted by e; and The above value of e^ gives e = 1/578; but geodetic measurements make e= 1/300 about; the discrepancy is due to the fact that the solid Earth is not spherical or centrobaric, but that its shape partakes of the ellipticity of the Ocean, and to precisely the same amount ; showing that the solid Earth was once in a molten viscous con- dition, during which time it took the present shape. Suppose we increase w to Q,, so that Ril^jg = 1 ; then with the above value of e^, Q, = 17to ; and now the plumb line would be parallel to the Earth's axis, and would point to the Pole Star; at the equator water would fly off into space and bodies too, unless fastened down to the ground ; and the water would collect in lakes with the free surface always parallel to the equator. This implies however that the solid part of the Earth is rigid and does not change its spherical shape; but practically the solid form would be deformed into a spheroid, to a much greater extent than at present. 460 TEE SEA LEVEL. 359. Plateau has devised an apparatus by which these phenomena may be imitated ; a vertical axis is fixed in a vessel of water, and oil, of equal density, is placed on a solid nucleus fixed to the axis; the spheroidal form is closely imitated when the axis is revolved (fig. 101). Here the constraining cause is the capillarity tension of the surface of separation, T dynes/cm suppose ; and it can be proved as an exercise that if the mean radius is R cm and the density a- g/cm^, the ellipticity due to a small angular velocity w is ^auP'R^IT. 360. According to astronomical definitions the angle PGQ is called the latitude of the place; not the angle PEQ, which is distinguished as the geocentric latitude, the angle EPO being called the angle of the centre ; the tangent plane at P perpendicular to the plumb line OP is called the sensible horizon, and the parallel plane through the centre E the rational horizon of P. The angle of the centre EPG is the gradient of the free level surface with respect to the mean spherical surface through P ; denoting it by 0, and the geocentric latitude by 6, sin tj) = {EG/EP)sm 6 = e^cos sin = e sin 26. Thus in latitude 45°, where ^ is greatest, a river flow- ing south is running away from the centre of the Earth on an apparent gradient of about one in 300. The Mississippi rises in latitude 75°, and flows nearly due south into the sea in latitude 30°, a distance of 900 geographical miles, at an average gradient with respect to the Earth's centre of one in 320 ; so that if the source is one-quarter of a mile above sea level, the mouth will be about 2 J miles farther from the centre of the Earth. CHAPTER XI. HYDRAULICS. 361. The word Hydraulics means primarily the science of the Motion of Water in Pipes ; but it is now extended to cover the elementary parts of the practical science of the Motion of Fluids. This includes the Discharge from Orifices, the Theory of Hydraulic Machinery, such as Water Wheels, Turbines, Paddle Wheels and Screw Propellers, Injectors, etc., which can be treated by the aid of Torricelli's and Bernoulli's Theorems ; and the Motion in Pipes, Canals, and Rivers, taking into account the effect of Fluid Friction so far as it can be treated in an elementary manner. 362. Torricelli's Theorem. The velocity v of discharge of water from a small orifice a depth h below the free surface was given by Torricelli (1643) as the velocity v acquired in falling from the level of the free surface, so that iv^=gh, or v = J{2gh); and V is then called the velocity due to the head h. This is argued by asserting that the hydrostatic energy of the water, Dh ft-lb per ft^, or h ft-lb per lb, becomes converted on opening the orifice into the kinetic energy ^Dv^lg ft-lb/fls, or ^v^jg ft-lb/lb. 461 462 TORRICELLI'S THEOREM ON Thus the jet of water, if directed nearly vertically upwards, would nearly reach the level of the free sur- face ; and if directed in any other direction will form a parabolic jet, of which the directrix lies in the free surface of the still liquid. The cross section of the jet OVR, while continuous and not shattered into drops, will be inversely as the velocity; and the horizontal component of the velocity being constant, equidistant vertical planes will intercept equal quantities of water, so that G the C.G. of the water will coincide with the C.G. of the parabolic area cut off by the chord ; and the height of the C.G. of the jet cut off by a horizontal chord OR will be two-thirds of the height of the vertex (fig. 104, p. 469J. If the jet could be instantaneously reduced to rest and frozen, it could stand as an arch, without shearing stress across normal sections. If the vessel is in motion, the velocity of efilux v is still taken as due to the head of the pressure p ; in this way the eillux from an orifice in a rotating vessel (Barker's Mill or a Turbine) is given (§ 345) by v = Ji^gz + y^u?), or from an orifice in an ascending or descending bucket ; balanced by a counterpoise at the end of a rope over a pulley by 1) = ^{2{g ± a)z] (§ 322) ; the student may work out the motion of the buckets completely as an exercise. 363. The velocity of efflux v must be reckoned not exactly at the orifice, but a little in front at the point where the jet is seen to contract to its smallest cross section; this part is called the vena contracta, and the ratio of the cross section of the vena contracta to that of the orifice is called the coefficient of contraction, and denoted by c-^. FLOW AND JETS OF WATER. 463 Practically, in consequence of friction, the velocity v at the vena contracta is a little less than ^{2gli), and the ratio of v to ^(2gh) is called the coefficient of velocity, and denoted by Cg- Now if the area of the vena contracta is A ft^, and of the orifice is B f t^, A = c^B ; and the flow of water is Av = c-fiv = c-^c^Bi^{1gh) ft^/sec ; and the product c-^c^ is denoted by c, and called the coefficient of discharge. The flow of water through a standard vertical orifice one in^ in section, under a head of 6J ins, is called the miner's inch; since £ = 1 -f- 144 ft^, A = 0'54ft, and we may put on the average c = 0'62, this flow is about 1'5 ft^/rninute. 364. Torricelli's Theorem is still employed when the head varies, as in filling or emptying, a reservoir or lock, in sinking a ship by a hole under water, or in pouring out water from a vessel through a spout; and now, if X denotes the area of the surface of the water at a height X above the orifice B, X^= -cBv= -cBJ{2gx), Xdx --A BJ{2gxy X giving the time t of filling or emptying the vessel between the levels x and h ; this may be worked out for vessels of various form, as the cylinder, cone, sphere, etc. Thus, for example, if an orifice of one ft^ be opened in the bottom of a sheet iron tank, 30 ft long, 20 ft broad, and 9 ft deep, drawing 4 ft of water, the tank will sink in about 26 minutes, taking c = 0'6. 464 FLOW WITH VARIABLE HEAR If the orifice in a vertical wall is large, and the varia- tions of head over its area is taken into account, and if y denotes the breadth of the orifice at a depth x below the surface, the efflux in ft^/sec is, with c= 1, Q =yy^(2gx)dx and t =fXdx\Q,. Thus if h, h' denote the depth of the top and bottom of a rectangular orifice of breadth h, Q = hJ{tg)J'\Mx = ihJ{2g)Qfi - 0) ; h' so that the average velocity of efflux is due to the head )V h-} 9V h-h' and this is ^h, if h' = 0. For example, the time of draining to a depth of 3 inches the ditch of a fortress, one mile long, 30 ft broad, and 9 ft deep, by a vertical cut 2 ft broad, is 13f hours ; and to lower the depth to one inch will take 12 hours more. 365. The flow of water is DAv = DA. J{2gh) lb/sec, possessing momentum DAv^jg — 2BAh second-lb /sec; this will therefore be the thrust in lbs of the jet against a fixed plane perpendicular to its direction. This thrust is double the hydrostatic thrust due to the head h ; thus, for instance, the water of Niagara, falling 162 ft, can balance a column of water 324 ft high in a J -shaped tube, with its lower mouth under the fall. The energy of the jet is BAv . \v^jg = DAvh ft-lb/sec ; and therefore the H.p. (horse-power) is \DAv^l5^0g = BAvhlbm. With a metre and kilogramme as units, Z) = 1000, (/ = 981 ; and 75 kg-m/sec is the oheval-vapeur. MOMENTUM AND ENERGY OF A JET. 465 Thus a jet of water 10 ins in diameter, issuing under a head of 600 ft has 7300 h.p. ; these large jets are used for hydraulic mining in California, the nozzle being con- trolled by an apparatus called a hydraulic giant. 366. Denoting hj p the hydrostatic pressure Bh Ib/ft^ due to the head h, then v = ^{2gp/D); and the jet discharges AJ(2gp/D) ft3/sec, or A^(2gpD) lb/sec, possessing momentum 2Ap sec-lb, and energy and h.p. ^»!ft-lb, and ^^. Thus the velocity of efflux from the Hydraulic Power main (§ 14) would be 333 f/s, and the flow through a hole I inch diameter, with c = 0-65, would be 40,000 gallons in 24 hours. Again the pressure in the air vessel in fig. 78, for a steady flow of Fft^sec of water through a delivery pipe A ft2 in section, is ^DV^/gA^ Ib/ft^. 367. Suppose that two fluids, water and steam for instance, are issuing by two nearly equal nozzles, of cross sections A and a ft^, from a vessel in which the (gauge) pressure is p ; denoting the density of the steam by S, then, according to Torricelli's theorem, the velocity of the steam jet_ ID the velocity of the water jet y §' the delivery in lb of the steam jet _ a IS the delivery in lb of the water jet Ay D' the momentum of the steam jet_^ the momentum of the water jet A' the energy or h.p. of the steam jet _ a ID the energy or h.p. of the water jet Ay 8 a.H. 2(J 466 GIFFARD'S INJECTOR. If the area of the water surface in the boiler is Oft^, the time required to lower the surface one inch is V7^ —taK;^ — ) seconds. For instance, if a piston of weight W lb is placed in a vertical cylinder of cross section C ft^ resting on the surface of water, and a small vertical nozzle of area A ft^ is opened in the piston, the water will flow through the orifice with velocity ^{^WgjDG) f/s; and the piston will descend with velocity ^ (2A^Wg/DG^). •V C Btk Fig. 102. Fig. 103. 368. The superior energy of the steam jet enables it, even when mixed and condensed with water, to over- come the water jet and to enter the boiler; the maximum water fed in being the difference between the quantities of water and steam blown out, and therefore A^(2gpD)-aJ(2gpS) lb/see. BERNOULLI'S THEOREM. 467 In this way the action of Giffard's Injector may be popularly explained; also of the jet pump, working with two liquids, say water and mercury ; the injector is shown in fig. 102, in the form patented by Hall. Thus for a pressure of 100 Ib/in^, and a jet \ inch in diameter, taking (5 = 0-23, D = C2-4 Ib/ft^, the water fed in is 2 '4 lb/sec or 14'4 gallons/minute. A good average value of ^{DjS) is 16, so that one lb of steam forces 15 lb of water; but in a steam pump (fig. 78, p. 360) one ft^ of steam will force nearly one f¥ of water, or 380 times its weight; the injector has there- fore a mechanical eflBciency much inferior to that of the pump, but it has the advantage, of working when the engine is still, and of heating the feed water. If the requisite diameter of the vena contracta A at the throat of the injector is d inches, so that A = ;|7rd^-H 144 ft^, when Q denotes the given number of ft^ of water and condensed steam to be injected in one hour against a gauge pressure of p atmospheres, of 147 Ib/in^, then n Q«nni'^'^' //2 X32X147X144 \ or i7rcZ^ = iJ^, where ilf=1165. Rankiue replaces M by 800, to allow for the steam used in the injector, for the air sucked in, and for the friction in the pipes ; the discrepancy is also partly due to the erroneous hypothesis concerning the flow of the steam jet, and to thermodynamic influences left out of account. 369. Bernoulli's Theorem. In Bernoulli's Theorem the gradual interchange of the energies due to pressure, head, and velocity in a 468 BERNOULLI'S THEOREM. stream line filament in the interior of the liquid, or in a smooth pipe of gradually varying section is, expressed by the equation p + Dx+^Bv^lg = Dh, a constant, or 7s + a; + g- = /i, a constant, 1) Zg where p denotes the pressure, D the density, v the velocity and X the height above a fixed horizontal plane. Thus with British units, the total constant energy Bh along a stream line is in ft-lb/ft^, and composed of p due to the pressure, Dz to the head, and ^Dv^/g due to the velocity ; or in ft-lb/lb, the energy or head h is composed of p/D due to the pressure, z to the head, and ^v^/g to the velocity. 370. Bernoulli's Theorem is illustrated experimentally in fig. 104 by an apparatus devised by Froude (British Association Report, 1875) ; a tube of varying section carries a current of water between two cisterns filled with water to nearly the same level, and the pressure is measured by the height of water in small vertical glass tubes called piezometer tubes ; and it is found, in accord- ance with Bernoulli's Theorem, that the water stands higher where the cross section of the current is greater, and the velocity consequently less. If the velocity at the throat E is that given by Torri- celli's Theorem, the pressure there is reduced to atmo- spheric pressure, and the tube can be removed in the neighbourhood of E, as at the throat of the injector jet. At M the cross section is less than at E and the pressure is below atmospheric pressure, so that water will be drawn up in a curved piezometer tube like a siphon. THE VENA GONTRACTA. 469 By the observation of the heights in piezometer, at L and N as well, the velocity of flow can be inferred, knowing the cross section of the current ; this is the principle of the Venturi Water Meter, invented by Mr. Clemens Herschel ; also of the aspirator. Fig. 104. 371. As an application of Bernoulli's Theorem, Lord Rayleigh {Phil. Mag., 1876) determines the area A of the vena contracta of a jet, issuing from a re-entrant pipe at (figs. 103, 104), of cross section B, inserted in a pipe of cross section C; this ajutage is called a ring nozzle, and it is employed sometimes with a fire engine jet. Then if Ff/s denotes the velocity, and p Ib/ft^ denotes the gauge pressure in the pipe G, the equation of con- tinuity gives Av=CV, and Bernoulli's equation gives p v^— V^ 470 THE COEFFICIENT OF CONTRACTION. A third equation is given by the principle of momen- tum; taking the momentum which enters and leaves the space cut off by two planes A and C (fig. 103, p. 466) the momentum which leaves this part, in sec-lb/sec, is D{Av^-GV^)lg. But if p denotes the (average) pressure in Ib/ft^ over the annular end of the tube G, pC-p'{G-B) sec-lb is the momentum per second due to the pressure ; and therefore, in the state of steady motion, pG-pXG-B) = D{Av^-CV^)lg. If we assume p=p', then „ Av^-GV^ pB= , and therefore 2_ v^-yi _i 1 ^ B~ Av''-GV'-~ A^ G' so that B is the h.m. of A and G. In particular, if G is infinite, A = ^B; so that the inferior limit of the coefficient of contraction Cj is 0'5. By the additioti of mouthpieces or ajutages of various shapes, Cj can be increased, and even made greater than unity, a fact known to the Romans and prohibited by the law (Galix devexus amplius rapit, Erontinus). In this case a partial vacuum, of head h', is formed in the throat of the ajutage, so that the effective head becomes h + h'; and the jet now acts as an aspirator, for creating a partial vacuum, or as a lifting injector. 372. The principle of momentum shows that if water is flowing with velocity v through a tube of cross section, the axis of the tube being in the form of a circle, the tension of the tube is thereby increased by BAv^jg lb. RADIATING CURRENT. 471 As this is independent of the curvature of the tube, it follows that the flow of the water has no tendency to move the tube ; so that if the tube assumes a definite shape under external forces when the water is at rest, it will preserve the same form when the water is flowing with ponstant velocity v; but the tension is increased by BAv^lg lb, or by the weight of water which will fill twice the length of tube which is equal to the head of the velocity. For this reason the siphon tubes of figs. 61, 62 are liable to be lifted by the flow of the liquid, if the weight of the tube is small. Similar results hold if the tube and water are replaced by a uniform chain (§ 191). 373. As another application of Bernoulli's theorem, consider plane motion symmetrical about a central axis Ox, between two parallel horizontal planes, a distance a ft apart (Rankine, Applied Mechanics, §§ 629-633). I. Suppose the water to be flowing radially from the axis, so that the velocity is v fjs, at a distance r ft ; then the volume Q which crosses any circle of radius r is Q = 2-jrarv ft^sec ; and since Q is constant, therefore Q 1 zirar T Then Bernoulli's equation gives £=/,_^ e^, so that the surfaces of equal pressure are given by r^(h — x) = & constant ; and these surfaces are thus generated by the revolution of Barlow's Curve (§ 289). 472 FREE CIRCULAR AND SPIRAL VORTICES. II. Turn the direction of motion at each point through a right angle ; the surfaces of equal pressure are now unchanged, and we obtain what is called a free circular vortex, in which \rv, the area swept out by the radius of a particle in one second, is the same for all particles. Since the circular lines of flow now lie on the surfaces of equal pressure, a free surface can exist ; this state of motion is easily produced in a hemispherical basin when the plug at the bottom is removed. Generally in any state of vortical motion about an axis, if two circular filaments of radii r and r', of velocities V and v', and of equal weight W, are made to change place, their new velocities will become u and u', given by ri; = r'u and r'v' = ru' ; so that the work required is equal to the increase of kinetic energy, \ W 1 Tf/1 1 \ and this is positive, so that the motion is stable, if r'v' > rv when r' > r. Thus if v^jr is constant, we obtain a stable vortex, in which the surfaces of equal pressure are cones. III. By superposing the states of motion in I. and II. in given proportions we obtain Rankine's free spiral vortex {Applied Mechanics, § 631) in which the lines of flow are equiangular spirals ; this is useful in the dis- cussion of certain turbines and centrifugal pumps. The surfaces of equal pressure remain the same as before; but in this case the lines of flow cross the surfaces of equal pressure, so that a steady free surface is not possible; this is observable when the basin is nearly emptied. WHIRLPOOLS AND CYCLONES. 473 374. Whirlpools and cyclones are vortices of the nature of II. and III. ; hut the central part, where the velocity would be very great, soon assumes in consequence of viscosity a bodily rotation, as in § 344, called a forced vortex; thus a comparative calm is found in the centre of a cyclone, but the barometer is very low. The combination of a free circular vortex with a central forced vortex is called a compound vortex ; the pressure is due to the head from a point up to the free surface, which is formed by the revolution of the combination of a Barlow curve and of a parabola. Thus the centrifugal pump consists of a wheel with curved blades, which draws water in at its axis, forms the water into a forced vortex, and delivers it into an external whirlpool chamber as a free spiral vortex, where its dynamic head enables the water to overcome a certain head and rise to the additional height. 375. The Water Wheel and Paddle Wheel. If the jet of water impinges normally on a series of plane plates, following in regular procession with velocity u on the circumference of a wheel (the Under- shot Water Wheel), then in t seconds BAvt lb of water is reduced in velocity from v to u, and the change of momentum is BAvt{v — u)lg sec-lb; so that the thrust of the jet is DAv(v — u)jg lb, and the wheel is working with horse-power H.p. = DAvu{v — u)/550g. The ratio of this H.P. to the h.p. of the jet is called the efficiency of the wheel, and denoted by e ; so that vu(v—u) 1 „/« IV ' = —^v^ = ^-\v-2)' a maximum J or 50 per cent., when u = ^v. 474 THE PELTON WHEEL. 376. By replacing the plane plates by cupped vanes, as in the Pelton wheel, the efficiency is greatly increased ; and it is claimed, may reach 90 per cent. The velocity of the water relatively to the vane is supposed constant and equal io v — u; so that if the cupped vane is hemispherical, the water leaves the lip of the cup with relative velocity v — u backwards, and therefore with velocity u — {y — u) or 2u — v in the forward direction relatively to the ground. The H.p. of the wheel is now doubled; so that the efficiency a maximum 1, when u — ^v. The water now drops out of the wheel with no velocity, and there is theoretically no loss of energy; but practically a small amount of divergence must be given by the vanes to make the water run away clear of the wheel, so as not to be carried over. For instance, under a head of 1100 ft of water a Pelton wheel should run with a peripheral speed of 183 f/s ; and would give 3 h.p. for every ininer's inch, of 1'6 ft^ per minute, with an efficiency of 90 per cent. 377. The Paddle Wheel may be assimilated to the undershot wheel with flat floats or plates, working faster than the jet ; and now if the circumference of the wheel measured through the centre of the floats is p ft, and the wheel makes n revolutions per second, the water which was flowing past the sides of the ship with velocity v will be driven by the paddles with velocity up, v denoting the velocity of the ship through the water. THE PADDLE WHEEL. 475 Then if the area of a pair of floats is A ft^ (or of a single float in a stern wheel), every second a volume Anf i\? of water has its velocity changed by n'p — v; so that the thrust in pounds of the paddles is given by T= DAnp{np — v)/g. The effective horse power (e.h.p.) is Tv /550 = DAnpv(np — v)/550g, while the indicated horse power (i.h.p.) of the engines is Tnpl 550 = DAnYinp - v)/550g ; and therefore the efiiciency e, the ratio of the e.h.p. to the I.H.P., is given by e = v/np. The steamer advances as if a toothed wheel of cir- cumference ep engaged in a horizontal rack, so that the velocity of advance is v; the paddle wheel is thus always slipping a certain percentage, and not merely at starting, as in a locomotive engine. The velocity np — v is called the slip velocity, and the ratio (np — v)/np is called the slip ratio, and denoted by s, while 100s is called the slip percentage ; so that s = l—e, or e+s = l; 378. Oblique impact on a Sail. The Windmill and Screw Propeller. The previous method will serve to determine the thrust and power of wind on a sail moving obliquely, if we assume that the air behaves like an incompressible fluid, or like a dust cloud of inelastic particles. Let a and ^ denote the angles between the normal of the sail and the directions of the wind and of the ship ; let V denote the velocity of the ship and u of the wind, in f/s; and let the density of the air be S Ib/ft^, and the area of the sail A ft^. 476 SAILS AND WINDMILLS. Then, resolving perpendicular to the sail, a weight 8A{u cosa — v cos ^)t lb of air strikes the sail in t seconds, and is reduced in velocity in that direction by ucosa — v cos /3 ; the nor- mal thrust R on the sail is given in pounds by SA {u cosa — v cos ^'f/g, and the propulsive thrust on the ship is T=SA cos /3(w cos a — ■?; cos ^f/g ; so that the E.H.p. of the sail is Tv/550 = SAv cos ^(u cos a — i; cos ^fjooQg. Thus the propulsive h.p. of the sails of a ship, spread- ing 30,000 ft^ of canvas, set at an angle (S = 60°, and sailing S.W. at 10 knots in a trade wind blowing S.E. at 12 knots is about 322'4, taking ,5 = 0-078 Ib/ft^. 379. In an ordinary windmill the sail moves at right angles to the wind, so that a+/3 = Jtt; and now the E.H.P. = 8Av cos ^{u sin /3 — -y cos ^YJbbQg ; which is a maximum when v = \u tan /3, or one-third the speed at which the wheel would revolve without load. Here A denotes the area of an element of the sail at a given distance r from the axis ; so that v = ^irrn, if the wheel revolves n time a second ; and it is assumed that the sails do not interfere with each other. But when the sails nearly overlap, as in the Canadian windmill, A must be taken to represent an annular strip of mean radius r, intercepting 8Au lb/sec of air ; so that now the h.p. of the element A is SAuv cos ,S(u sin ^ — v cos (8)/5o0^, a maximum when v = ^uia,n^, or one-half the speed of the wheel when unloaded. (Smeaton, Experimental Enquiry on the Power of Water and Wind to turn Mills, Phil. Trans., 1759.) THE PRESSURE TURBINE. 477 380. Suppose this wheel was enclosed in an annular pipe, of external and internal radii r^, r^, and areas J.J, J.2 ; and suppose the wheel was driven by water, as a Pressure Turbine ; then if the axial flow u is supposed to remain unchanged, the water after passing through the wheel will be rotating bodily with n — u/p revs/sec, or with angular velocity (a = 2Tr{n — u/p) radians/sec. The turning moment L of the wheel is equal to the angular momentum generated per second ; and therefore, L^Wk^ai/y (ft-lb), and the h.p. of the wheel is a maximum when n = ^u/p. 381. By supposing the water at rest and the wheel to advance with velocity u, we can thus obtain a pre- liminary idea of the action of the Screw Propeller ; but now np, the speed of advance of the screw in a solid nut with n revs/sec must be greater than u, for the screw propeller to exert a forward propulsive thrust T; so that while, the reaction being normal on a smooth screw, Tp = 2-!rL. Now the efficiency E.H.P. Tu u e = i.H.P. 2TrnL np as in the paddle wheel. = l-f 478 THE SCREW PROPELLER. Of the work wasted by the propeller, one half is carried away by the energy of the wake, rotating as a forced vortex, and the other half is lost by the shock at the leading edge of the blades. The first loss can be minimised by making the pitch p small and correspondingly increasing n the revolutions, provided fluid friction is left out of account ; but the second loss can be suppressed by employing a screw of gaining pitch, increasing from ujn to any final value p ; and the eifective pitch will now be found to be the h.m. of ujn and p, while 382. A second screw on the same axis, of opposite pitch and revolutions, can be employed to utilise the energy of rotation of the wake of the first screw, as in the Whitehead torpedo ; and now the efiiciency is perfect. Such an arrangement of propellers enclosed in a tube, and running in reverse order, could be employed as an axial flow Pressure Turbine, one screw being fixed to act as guide blades. The Theory of the Turbine is perfect because the water is compelled by the guide blades to take the best course; the general principle to be followed is to make the water enter without shock and leave with little or no velocity. But the screw propeller works in water already set in motion by the ship, and the water is free to follow its easiest course, so that the vortex formed may be anything intermediate to the free vortex and the forced vortex assumed above, while the axial flow may be accelerated ; and the theory is correspondingly com- plicated (Eankine, Trans. I. N. A., 186.5). THE IMPULSE TURBINE. 479 383. In the Imfipulse Turbine jets of water are re- ceived without shock on appropriately curved blades, and guided by them so as to leave the wheel radially ; and if Q ft^sec of water moving with velocity v enter the wheel at an angle ,8 on a circumference of radius r, moving with velocity V, the H.p. of the wheel is BQVv cos ^jb^dg; this follows from the mechanical principle that the rate of change of angular momentum about an axis is equal to the impressed couple. 384. Fluid Friction in Pipes. The Hydraulic Gradient. So long as the water in a main is at rest, the hydro- statical theorems laid down in §§ 21-24 hold good; and if stand pipes are erected at different points, the water will rise to the level of the supply reservoir in all of them, while the pressure in the main will be that due to the head in the adjacent stand pipe. Small stand pipes of this nature, inserted in a tube for measuring the pressure, are called piezometers. But when the water is in motion, fluid friction absorbs the energy of the water at a uniform rate; so that the water levels in the piezometers will no longer lie in a horizontal line, but will slope downwards in the direc- tion of motion at an incline called the hydraulic gradient; this gradient is observable in the strata of the earth and in the sands of the desert. In contradistinction to Morin's Laws of Friction for Solids, Fluid Friction is found to be » (i.) independent of the pressure ; (ii.) proportional to the surface ; (iii.) proportional to the square of the sliding velocity. 480 FLUID FRICTION AND If water is flowing bodily with velocity v f/s through a main of diameter d ft, the frictional drag on a length, iftis ■wdlkv^ pounds, where k is a coefficient found by experiment ; a good average value in iron pipes is 0-008. Thus the frictional drag in a pipe line for the con- veyance of petroleum, 30 miles long and 6 inches in diameter, with a delivery one ft^ per second would be about 26,000 pounds, requiring a pumping pressure of 920 Ib/inl 385. The loss of pressure on the hydrostatic pressure is Trdlkv^ _4' a maximum when x = \h, and the efficiency is -|. The loss of head is proportional to the length, so that for a straight pipe the hydraulic gradient through the upper levels in the piezometers is a straight line, sloping at an angle Q, given for a level main by tane = ^|^. d 2g THE HYDRA ULIC GRADIENT. 481 If the pipe is required to deliver Q ft^/sec, then giving the requisite diameter d for a given delivery Q and hydraulic gradient 6 in water works; for instance, between a reservoir and a cistern at a lower level. The slight deviations of a main from a straight level line do not sensibly aifect the results of this formula, unless the pipe rises above the hydraulic gradient, when it acts in the manner of a siphon (§ 194). 386. The Resistance of Ships. The resistance to the motion of a vessel through the water is initially zero, but the resistance mounts up as the velocity increases. Taking the knot as a speed of 100 ft a minute, one h.p. is equivalent to 330 knot-pounds, or 83-^224 knot-tons; so that if a steamer of W tons displacement is propelled at a speed of F knots, it experiences a resistance the same as that of a smooth gradient of one in 224WF-7-33 h.p. ; in the Paris and iVew YorJc, for instance, if 17=10000, F=20, and h.p. = 20000, the gradient is about one in 68. Although no theory is in existence which will enable us to predict with certainty the resistance at a given speed of a vessel of given design, still the experiments of Froude enable us to assign this resistance from the measured resistance of a model of the vessel run at a correspondingly reduced speed. According to Fronde's Law, " The resistance of similar vessels at speeds as the square root of the length, or as the sixth root of the displacement, is as the displacement or as the cube of the length." G.H. 2h 482 THE RESISTANCE OF SHIPS. Then if L, n^L are the lengths, and D, n^D the dis- placements of a vessel and its model, the resistances at speeds V, wF will be R, n^R; and therefore the h.p.'s will be as 1 to n''. If the resistance is supposed to be due to skin friction, and this again is supposed to be proportional to the wetted surface, or as 1 to -n.*, then the remaining factors of the resistance are as 1 to n^, or proportional to the square of the velocity, as in fluid friction. Thus, for instance, we may take n = 0'l; and if a vessel is designed to have a length L and a speed of V knots, a reduced model of 100th the length is run at one- tenth the speed, and the resistance r pounds is measured; then Froude's Law asserts that the full-sized vessel will experience a resistance lOV pounds at V knots, and the effective h.p. required will be lOVF/330. The coal required per hour is proportional to the h.p. or to n', but the coal per mile is as n^ or as the displace- ment ; so that over the same length of voyage the coal endurance is the same. In popular language, to increase the speed one per cent, over a given voyage, we must increase the length two per cent, or the tonnage and coal capacity 6 per cent., and the horse power, boiler capacity, and daily consumption of fuel by 7 per cent. Thus taking the Paris and New York as the model for a new design of a steamer to cross the Atlantic, 2800 miles, at a speed of 21 knots, then ■)i = l-05, ■}i«=l-34, n''=l-W; so that the new steamer would have about 13,400 tons displacement and require 28,000 H.P. ; this is approxi- mately the case in the Gampania and Lucania. EXAMPLES. 483 The voyage would take 2800-=-21 = 133J hours; but if 5 hours is deducted for longitude difference on the west- ward voyage, when running before the sun, the apparent time is 128J hours, so that the apparent speed is raised to 21-82 knots. ExaTU'ples. (1) A bucket of water in a balance discharges 4 lb of water per minute through an orifice in its base at 45° to the vertical, and is kept constantly full by a vertical stream which issues from an orifice 8 ft above the surface with velocity 30 f/s. Prove that the bucket must be counterpoised by about 0066 lb more than its weight. (2) The bucket valve in fig. 80 (p. 362) has a small leak, one-800th of the cross section of the barrel, and the height of the water barometer is taken as 32 ft, the height ^40 as 16 ft, and the specific volume of the air 800 times that of water. Prove that the pump will not suck unless the bucket is moved with a velocity greater than W2 = l-13 f/s; but that afterwards water will be lifted if the velocity is greater than 0'04 f/s. (3) Prove that a hydraulic engine (fig. 78), in which water under pressure is admitted through small orifices to actuate the piston, will do most work when the speed is ^^3 of the unloaded speed, and the load is f of the maximum load, and that the efficiency is then f . (4) Discuss the influence of inertia and of fluid friction in the pipe, when the Hydi-aulic Press (§ 12) is actuated by the Accumulator (§ 15). (Cotterill, Applied Mechanics, § 256.) 484 EXAMPLES. (5) Prove that the H.P. of the feed pump of a boiler, which evaporates TTlb/min of water at a gauge pressure p lb/in", must exceed 144 Tfp-7-33000. (6) Prove that if the jet of §371 delivers Q ft^/sec, and the hose is I ft long, the pumping H.P. of the fire engine is \l()Og\A^ (7-2 "^ Gi (7) If water is scooped up from a trough between the rails into a locomotive tender to a height of h ft, determine the minimum velocity required, and the delivery at a given extra speed, taking the frictional losses as represented by a given fraction of the head. (8) Show how liquid may be raised through a siphon tube, made to revolve about its longer branch, which is held vertical ; and determine the delivery and the mechanical efficiency for a given angular velocity. (9) Show how to determine the elements of a cyclone from observations at three points. What is the direction of rotation in the N. and S. hemispheres ? CHAPTER XII. GENERAL EQUATIONS OF EQUILIBRIUM. 387. It was proved in Chapter I., §§ 19, 20, that the surfaces of equal pressure and the free surface of a liquid at rest under gravity are horizontal planes; but this assumes that gravity acts in parallel vertical lines. When we examine more closely the surface of a large sheet of water like the open sea, we find it uniformly curved, so that the surface is spherical ; showing that the lines of force of gravity converge to the centre of the Earth ; and Archimedes in his diagrams of floating bodies represents them immersed in a spherical ocean. If three posts are set up, a mile apart in a straight canal, to the same vertical height out of the water, the visual line joining the two extreme posts will, in the absence of curvature by refraction, cut the middle post 8 ins lower; hence it is inferred that the diameter of the Earth in miles is the number of 8 ins in 1 mile, or 7920. If I miles apart, the visual line cuts at a depth 8P ins ; for instance, the Channel tunnel 20 miles long, if made level, would rise in the middle 800 ins from the straight chord ; but if made straigid, it would have a gradient of about one in 400 at the ends, and water reaching to the ends would have a head of 800 inches in the middle. 485 486 LEVEL SURFACES OF EQUILIBRIUM. 388. Careful measurements of a degree of the meridian in different latitudes reveal that the mean level surface of the Ocean is not exactly spherical, but slightly ellip- soidal and bulging at the Equator ; an effect attributable to the Earth's rotation, and investigated in the theory of the Figure of the Earth. Lastly, the imperceptible deflections of the plumb line, due to the perturbative attraction of the Moon and Sun, are rendered very manifest by the phenomena of the Tides, due to the same cause of perturbation. 389. All these manifestations are examples of the general principle, enunciated in § 24 as " Liquids tend to maintain their Level," but now the level surface must be taken to mean the surface which is everywhere perpendicular to the result- ant force of gravity at the point, as indicated by the plumb line. To prove the theorem that " The surfaces of equal pressure in a fluid at rest under given forces are at every point perpendicular to the line of resultant force " ; draw two consecutive surfaces of equal pressure PP", QQ', on which the pressures are p and p + Ap suppose; and consider the equilibrium of a cylindrical element of cross section a, the axis PR of which is normal to the surfaces of equal pressure. The resultant thrust on the curved side of the cylinder of the surrounding liquid, of uniform pressure in planes perpendicular to the axis, being zero, the resultant thrust on the ends must be balanced by the resultant impressed force, which must therefore act along the normal PR to the surface of equal pressure. SURFACES OF EQUAL PRESSURE. 487 Denoting this force per unit volume by F, and the element PR of the normal by Ai/, Fa. Ai/=aA23, F^\i^ = ^, (1) Ai/ dv ^ ' or the resultant force per unit volume is the space varia- tion, or gradient, of the pressure f in its direction. This is true also for any other direction PQ, making an angle Q with the normal PR ; for if it meets the consecutive surface QQ' in Q, and PQ = As, COS0 = lt-p^ = ^, and Fcose = ^^=f, (2) dv ds ds so that "the component force in any direction is the space variation of the pressure in that direction; and the resultant force is the greatest space variation, and therefore normal to the surface of equal pressure." Thus if the force F has components X, Y, Z parallel to three fixed rectangular axes Ox, Oy, Oz, dx ' dy ' dz ' or in the notation of differentials, dp = Xdx+Ydy + Zdz; (3) and the lines of force must be capable of being cut orthogonally by a system of surfaces, of equal pressure. 390. The impressed forces of gravity and inertia (but not of electricity or magnetism) are proportional at any point to the density p ; so that it is usual to multiply F by p, and thus measure F per unit of mass, lb or g, instead of per unit volume, ft^ or cm^ ; and now we write dp = p(Xdx+Ydy+Zdz) (4) 488 LINES OF FORCE PERPENDICULAR This equation may be deduced from the equilibrium of the fluid filling a fixed closed surface S; denoting by I, m, n the direction cosines of the outward drawn normal at a point of the surface, and resolving parallel to Ox, i/l'pdS = / / / pXdxdydz, the integrations extending over the surface and through- out the volume of 8. But, by Green's transformation, and therefore % = pX, ^^ = pY, ^£ = pZ. (5) Also taking moments round the axes, //p(ny — m,z)dS — 1 1 1 p{yZ—z Y)dxdydz ; and these equations are now satisfied identically. 391. From equations (4) and (5), d?j) _dpy dY_dp „ dZ dydz dz ^ dz dy "dy fdZ dY\_ ydp „dp ^\dy dz) dz dy with two similar equations ; so that, eliminating p, „ 2gk 'pg on putting gjauy^ = 289, ajk = 800. IN THE ATMOSPHERE. 493 This proves that the Earth cannot be spherical; for here the surface of equal pressure p^ is given by 1 _ ? — ^^y^ . r 2ga ' so that, if the equatorial radius of this surface is b, a^. (o^6^_, w^a_577 b~ 2ga 2g " 578" In convective equilibrium yJc dd (a\^j , 0,2 , y-lV ej~h\ r) 2gk' 398. Equation (4) is employed to determine the pressure in the interior of a spherical liquid mass due to its own gravitation, for instance in the Earth when in a molten condition. We assume the well-known theorems that the attrac- tions of a homogeneous spherical shell (i.) is zero in the interior cavity, and (ii.) is the same as if the matter is condensed at the centre for an exterior point. With C.G.s. units, denoting by y the constant of gravitation, that is the attraction in dynes between two homogeneous spheres each weighing one g when their centres are one cm apart, and first supposing the liquid of uniform density p g/cm^ then (1) becomes dp , since the attraction at a distance r from the centre in the interior of the liquid sphere is the same as that of the mass ^-rrpr^ condensed at the centre, the attraction of the shell exterior to the radius r being zero. 494 GRA YITATINO SPHERE OF LIQUID. Denoting by V the gravitation potential of the liquid, the rate of increase of which in any direction is the force per unit mass in that direction, then in the interior of a sphere of radius a, ~=-i-^ypr, V=2^yp(a^-ir^), if a constant is added, so that there is no abrupt change in V in passing into exterior space, where V=iTrypa^/r, the same as for the whole mass collected at the centre of the sphere. Supposing the pressure zero at the exterior radius a, where V= V^ = ^-ny pa^, then V = p{V- ^o) = I'^yp'Ca' - '^')- Hence the thrust across a diametral plane, or the attraction between the two hemispheres, is /" p . Iirrdr = ^ir^yp^a'^ = lyM^ja^, if ilf g denotes the mass of the liquid sphere. 399. The gravitation constant y is determined by the Cavendish experiment; it is found that, in g.g.s. units, 7=10-8x6-69. Now if a denotes the radius (19^-i-|7r cm), p g/cm^ the mean density of the Earth, and g spouds (cm/sec^) the mean acceleration of gravity on the surface, g = yM/a^ = i-rry pa = f yp x 10" ; so that, with g = 981 spends, yp = 10-7x3-68, p = 5-5. Thus at the centre of the Earth, if of uniform density p, the pressure p = f TrypW = ypa. LAPLACE'S LA W OF DENSITY. 495 This is the pressure due to a head \a, with uniform surface gravity g ; also p = 109 X ^p/7r = 10i2 X 1-718 harads (dynes/cm^) = 1,718,000 atmospheres, or megabarads, of a million barads. 400. Considering that the surface density of the Earth is about 2 only, the density cannot be uniform, but must increase towards the centre. Equation (1) of § 389 now becomes so that p must be some known function of r, or else the relation connecting p and p must be known, for this equation to be integrated. Thus, if p oc r~^ p—p'x p — p'. Laplace assumed that the density p was proportional to the square root of the pressure j» (or oip-\-p^, equivalent to assuming that the cubical elasticity (§ 422) is double the pressure, or pdpjdp = 2p ; and now, putting p+p^ = 2-Kyp\^, the student may prove, as an exercise, that, at any radius r, K . r p = a- sm -, where o- denotes the density at the centre. The mean density of the Earth being denoted by p, and its mass in g by E, /a ^irprHr = iiTTtTK I r sin -ctr = 4 tto-k"! sin cos - I- J K \ K K K / 496 DENSITY OF STRATA OF THE EARTH. Denoting the surface density by p^, then K . a so that ^^sf^Vfl-'^cot^Y whence k is determined : for instance, by putting p = o-5, po=2- With Po = 0, and surface density /Oq = 0, we must put a/K = TT ; and now we find 0- = i7rV 401. If homogeneous liquid fills a spherical case of radius a, and if a spherical cavity of radius b cm is made anywhere in the interior of the liquid, with its centre B in Ox at a distance c from the centre of the liquid sphere, the cavity will collapse unless kept distended by a rigid spherical surface ; and now the potential at a point P in the interior of the cavity is 27ryp{a^ - iOJP^) - 2Try p(b^ - iBP^) = constant - ^Trypcx ; so that the field of force in the cavity is of uniform intensity ^irypc, parallel to xO. If the cavity is filled by a solid spherical nucleus of density o-, the attraction on this sphere is iwypo . ^TTo-b^ = i^ypa-b^c dynes, in the direction BO. The potential in the interior of the liquid is now due to a complete sphere of centre 0, radius a, and density p, and to a sphere of centre B, radius b, and density (r — p; and therefore in the liquid ^-i^ypia'-r')+i^y{cr-pW{-^-^), the pressure vanishing at the end A of the diameter Ox remote from B. GRAVITATING LIQUID SPHERES. 497 402. The hydrostatic thrust on the nucleus, over which BP is constant, is in the direction OB ; and integrating over the surface and throughout the volume of the nucleus, this thrust in dynes is given by = III ^iryp^xclxdydz = ^iryp^ . -iirb^c. The resultant of gravitation and hydrostatic thrust is therefore \'^-Tr\ p{a- — p)b^c dynes in the direction BO ; so that, if o- > p, the nucleus will be in stable equilibrium at the centre of the sphere, with B and in coincidence ; but, if cr < p, the nucleus will tend to float up to the svirface ; so also the Earth would float up to one side out of the Ocean, if its mean density were less than the density of the water of the Ocean. (Frincipia, lib. iii., prop, x.) 403. The same result can be proved by calculating the variable part of the potential energy of the liquid and the solid nucleus ; this is the same as that of a complete liquid sphere of density p and of a solid nucleus of density cr — p, and is therefore a constant — %Tr^y p(cr — p)Wc^ ergs ; and the variable part is the work required to displace the nucleus to a distance c from the centre 0. The same method holds when the external vessel and the nucleus are of any shape; and now the pressure in the liquid is yp{V— Fq), where F^ is the minimum value of the potential F on the equipotential surface which touches the vessel; and the nucleus is in equilibrium when the potential energy of the system is a maximum. O.H. 2 1 498 GRAVITATING LIQUID SPHERES. 404. The potential at a point P external to the case is and drawing the equipotential surfaces in the neighbour- hood of the surface, these will represent the surfaces of equal pressure and the free surface of an Ocean composed of a small quantity of liquid poured over the surface. Suppose, for instance, that the Ocean just covers the surface ; it will be shallowest near A, and its greatest depth h at the point A' nearest to B will be given by or pa'h ^ (cr-p)b\2b-h) a + h {a + b)(a-b + h)' 405. If the nucleus B is liquefied, but kept spherical in shape by a fixed rigid case, and if a solid sphere of radius b' and density o-' is placed in this interior liquid with its centre B' in OB at a distance x from B, the potential energy of the system is a constant — -Y'-TT^ypo-'ft'^ca^ — fTr^cro-'t'^a;^ ergs; and this is a maximum when pC + (TX = 0, X= — Cp/a; provided however that x\y' + z^) = iiry p^a' - cc2) - (f ,ry/>%2 - yc,^){y\+ z% the pressure being zero at the poles of the axis of revolu- tion ; and now the surfaces of equal pressure are similar quadric surfaces of revolution, ellipsoids or hyperboloids as a? 5 ^iryp, and parallel planes if w^ = ^iryp. Over the spherical surface, p = \pw\y'^+z% so that the force tending to split the shell across a meridian circle is II yo)^y^+z^)mdS= HI puj^ydccdydz = lTrpa}^a\ 500 EQUILIBRIUM OF ROTATING LIQUID. 407. If the case is removed, the rotating liquid will lose the spherical form ; and as the angular velocity w is gradually increased from zero, the free surface of the liquid when rotating bodily will first assume the form of an oblate spheroid (Maclaurin's), and afterwards of an ellipsoid (Jaeobi's). We assume that the potential Fof the homogeneous ellipsoid of density p, is given by (Minchin, Statics, II., 308), J V a' + \ b^ + X c^ + \)^(aHX.b^+X.c^+\) or 2iry pabc{P' -^A'a?- ^E'l/ - \G'z') suppose, where and then A' + B' + C'=^, A+B+G^^iryp. Then in the rotating liquid £= V+W{%/ + z^) + a. constant P = a constant- J^a;^-|(5-a)%^-i(a-a)>2 . so that the surfaces of equal pressure are similar to the exterior surface, if Aa2 = (i? _ 0,2)62 = (C- «'V o)^ = ^'b^'-A'a ^ ^ CV - A'a^ . Trypabo b^ ~ c^ ' /'"/ &2g2 (i2 \ ^^ MACLAURIN'S AND JACOBI'S FIGURES. 501 408. One solution is obviously h'^ = c^ (Maclaurin's spheroid) ; and then -/, d\ 2 (a2 + x)V + X) V(c^-«T , ,, dP /a a 5-^ = - ^777. = 7772 79^ - 77; ^COS -1 a '' cdG~ (c2-aV (c2-a2/°^ ? _ 2aH^ ^a _3a2 _3+/2 3 on putting c^ja^ = 1 +/^. 409. The ellipticity (§ 358) of the spheroid is given hj and then expanding in powers of/, 2 "* _ J_ V'2 _ 8_ /4 ^. 8 _ Thus if we assume that the Earth is homogeneous, and of mean radius R, then ^irypR^g, and l = l|p^=J_ = 4 239^231-2, e low'' Sitft)'' 5 agreeing very nearly with Newton's estimate of 230. {Principia, lib. iii., prop, xix.) But as the true ellipticity of the Earth is about 1/300, the density must be greater in the interior; for if we assume that there is a concentric centrobaric nucleus of 502 EQUILIBRIUM OF ROTATING LIQUID. density ^yo and radius k, the additional potential at the pole and the equator is A7ry/)(;ix-l)- and i'7^yp{^i.-l)--■, SO that equation (A) of surface equilibrium becomes 7.3 hs Uvpil^ -W-- hAa? = i■,^yp{^, - l)'f - \ Cc? + JcoV, , ,2 7^3 / 1 1 or J^— = iG'ac^-\A!a^ + ^{^,-\)~r--' l-wyp ^ ^ ^^ ^c^\a c and now g = f,'7rypR + ^i!-yp{ix-l)-^^, so that 1- ^^ . I+Km-IX/'^/J !)^ 410. The integral which is the remaining factor of (A), § 407, gives the relation connecting hja and cja for a Jacobian ellipsoid. Putting & = c in this integral gives G*d\ r a'^dX p c'd\ r = 0, a transcendental relation which becomes (3 + 14/2 + 3/*)tan-y= 3/+ 13/^ and gives, approximately, /= 1-395, - = 0-584, -^=0-187. C Lnryp At this critical angular velocity the stable figures of equilibrium of the rotating liquid will pass from Maclaurin's spheroids into Jacobi's ellipsoids (Thomson and Tait, Natural Philosophy, §§ 771-778). JACOBI'S ROTATING ELLIPSOID. 503 411. A plummet, weighing W g, at the end of a plumb line on the surface of Jacobi's ellipsoid, will experience an apparent attraction of gravitation, having- components WAx, W(B-wyj, W(Q-w^)z dynes; and these may be written WAa^/px py pz^\ where 2^ denotes the length of the perpendicular from the centre on the tangent plane ; so that the plumb line will take the direction of the normal to the ellipsoid ; and denoting the polar gravity by G, and the length of the n.ormal to the equatorial plane by v, the tension in dynes of the plumb line, Wg= WAay'p= WAv= WGv/a. An ocean of small depth would spread itself over this ellipsoid, so that the depth at any point is inversely as g, and therefore directly as p. 412. If this Jacobian ellipsoid is enclosed in a rigid case, and rotated with new angular velocity Q, then p = constan t - ^pAx^ - y{B - Q^)/ - i/)(0 - Q^y ; so that at the surface the change of pressure is If there is a liquid nucleus of density p + pi, it can assume the form of the coaxial ellipsoid of semi-axes ftj, 6i, Cj, determined by the condition that Tryp abc {P' -iA'x^ - ^By - |GV) + 7ryp,a,b,c,iP\ - hA'^ - ^B'^ - lG\z^) + W(y'+^^) is constant over its surface, the suflBxes referring to this interior ellipsoid ; and therefore aiWypahcA' + -n-ypia^-iC^A'^ = hj^^iiry pabcB' + Trypia^b^c^B\ — u?) = c^i-TrypabcC + iryp^a^-fi^C\ — w^), 504 ROTATING CYLINDERS OF LIQUID. equations for determining A{, B(, G-[, etc., when A', B , C and o)^ are given. Thus if the outer case is spherical, A' = B' = C', and abcA' = %. It might even be possible for the interior nucleus to rotate bodily as a concentric but not coaxial ellipsoid, when the outer case is made to rotate about an axis not a principal axis. 413. When a.= oo, the ellipsoidal case becomes an elliptic cylinder ; and now A = 0, „_ r °° 2-7ry pbcdX _ 4nrypc „_ r'^ 27ry pbcd\ _ ^iry ph . ~J (6HX)*(c2 + X)*~ c + 6 ' so that if filled with one liquid rotating bodily, the surfaces of equal pressure are the quadric cylinders given by c-f&-4:^y+(^&-4^y=^°"«*^^^*; and if there is a central nucleus of density yo+pi, bounded by the coaxial elliptic cylinder of serai-axes a-^, \, the condition of equilibrium of the surface is • CHAPTER XIII. THE MECHANICAL THEOEY OF HEAT. 414. When work is done by the expansion of a gas, as, for instance, by the powder gases in the bore of a gun, or by the steam in the cylinder of a steam engine, a certain amount of heat is found to disappear; and according to the First Law of Thermodynamics, the heat which disappears bears a constant ratio to the work done by the expansion. Thermodynamics is the science which investigates the relations between the quantities of heat expended and work given out in the Conversion of Heat into Work, and vice versa; and for a complete exposition of the subject, the reader is referred to the treatises of Clausius, Tait, Verdet, Maxwell, Shanii, Baynes, Parker, Alexander, Anderson, etc. ; also to the Smithsonian Index to the Literature of Thermodynamics. In measuring quantities of heat, the unit employed is either the Bi'itish Thermal Unit (b.t.u.) or the calorie. The B.T.U. is the quantity of heat required to raise the temperature of one lb of water through 1° F. The calorie is the quantity of heat required to raise the temperature of one g of water through 1° C. 505 506 SPECIFIC HEAT. This is the small calorie, also called the therm; as the calorie is sometimes defined as the quantity required to raise one kg of water through 1° C ; this is 1000 therms. To be precise the water should be at or near its maximum density, or at a temperature of 4° C. 415. Different substances require different quantities of heat to raise their temperatures through the same number of degrees ; and are thus distinguished by their specific heat. The specific heat (s.H.) of a substance is the number of B.T.u. required to raise the temperature of one lb of the substance through 1° F, or of calories required to heat one g through 1° C. In other words, the specific heat is the ratio of the quantity of heat required to heat the substance to the quantity required to heat an equal weight of water through the same number of degi-ees ; the specific heat is thus the same in any system of units. With solid or liquid substances the specific heat is practically independent of the pressure or temperature, so that the above definition is sufficient for them; and now if weights Fi, F^, ..., Wn (lb org), of substances (solid or liquid) of S.H.'s Cj, Cj, . . . , fi,„ at temperatures t-^, Tg, . . . , t,i (F or C) are placed in a vessel impervious to heat, the final uni- form temperature T assumed by conduction is given by ( l^lCl+ PF"2C2 + ■ • • + WnCn)T= FiCjTi + W^C^T^ + . . . + F„C„T„, or T=:XWct/^Wc. But substances in the gaseous state absorb or give out heat in a manner depending on the relation between the LATENT HEAT. 507 volume, pressure, and temperature, and the specific heat may be made to assume any value by a properly assigned relation, which must therefore be specified in defining the specific heat ; for instance, the assigned relation may be of constant volume, or of constant pressure. 416. In melting a lb or g of a solid substance, although the temperature does not vary, a certain number of units of heat disappear, called the latent heat of fusion; and again, in converting the substance into vapour, the number of units of heat required is called the latent heat of vaporisation. The latent heat of fusion of ice into water is found to be 144 B.T.u. or 80 calories ; and of vaporisation into steam at 212 F or 100 C is found to be about 966 b.t.u. or 537 calories. Suppose for instance that a meteor weighing 3 tons, of S.H. 0-2, heated to 3,000 F, fell into a pond containing 10 tons of water at 60 F ; then x tons of water would be boiled away, given by 966a;+ 10(212 - 60) = 3 x 02 x (3000 - 212), x = OloS. If the water was at the freezing point, and one ton was frozen into ice, the temperature would be raised by the meteor to 210 F ; and if the meteor weighed 4 tons, about 0"3 tons of water would be boiled away. According to Eegnault's experiments, the latent heat of steam at any other temperature F or C is 1091-7 -0-695(F- 32), B.T.U., or 606-5 -0-695 C, calories; so that to lieat one lb or g, respectively, of water from the freezing point, and to evaporate it into steam at tem- perature F or C requires 1091-7 -|-0-305(F- 32), b.t.u., or 606-5 + 0-305 C, calories; this is called the total heat of steam at that temperature. 508 MECHANICAL EQUIVALENT OF HEAT. 417. The constant factor which, according to the First Law of Thermodynamics, converts units of heat which disappear into the equivalent units of work performed is called the Mechanical Eq%b%valent of Heat, and is denoted by J. According to Joule's experiments, as revised recentlj"^ by Rowland and Griffiths, 1 B.T.u. =779 ffc-lb (at Manchester) ; 1 large calorie = 427 kg-m (at Paris) ; 1 small calorie = 4-19 x lO'' ergs = 4-19 joules. The reciprocal of J is the Heat Equivalent of Work ; it is generally denoted by A. In the Thermodynamical equations, unless expressly stated otherwise, we adopt one of two systems of units : — (i.) The British (f.p.s) system of the foot, pound, second, and Fahrenheit scale, and the gravitation measure of force ; measuring volume v in ft^, pressure p in Ib/ft^ work in ft-lb, heat H in B.T.U. ; and thus take J =779. (ii.) The C.G.s. system of the centimetre, gramme, second, and Centigrade scale, and the absolute measure of force ; measuring volume v in cm^, pressure p in barads (dynes/cm^), work in ergs, heat in small calories or therms; and take J= 419 X 10^ A = 2-386 X 10-^. As an application, consider the theory of the Injector on Thermodynamical Principles ; then if W lb of water is injected by S lb of steam against a pressure head of h ft of water, and if the water injected is raised in temperature from F^ to F^, and if H denotes the total heat of one lb of steam at the boiler temperature F; then the heat which disappears in the Injector is, in B.T.U., SH- S(F^ - 32) - W{F^ - F^), where jy= 1091-7 + 0-305(1'- 32); THERMODYNAMICS OF THE INJECTOR. 509 and if the water is lifted \ ft, and the frictional losses are denoted by L ft-lb, the work done is {W+S)h+ Wh^+L. Therefore, by the First Law of Thermodynamics, (Tf+ySf)^+ Wh,+L = J{SH-8{F^-Z^)- W{F^-F^)}. Suppose for example that the boiler pressure is 100 Ib/in"^, and F = 328 ; suppose also F^ = 50, F^ = 120 ; then, neglecting ^^ and L, we find TF'/>Si = 16, about. 418. The simplest thermodynamic machine is a gun or cannon ; it is a single-acting engine which completes its work in one stroke, and does not work in a continuous series of cycles like most steam engines. When the gun is fired, the shot is expelled by the pressui-e of the powder gases ; the pressure is represented on a {p, v) diagram (§ 197) by the ordinate MP of the curve GPD, OM representing to scale the volume of the powder gases when the base of the shot has advanced from A to M; the curve GPD starts from a point G, such that the ordinate AG represents the pressure when the shot begins to move (fig. 105). The area AMPG then represents the work done bj'' the powder (per unit area of cross section of the bore) when the base of the shot has advanced from A to M, the area ABDG representing the total work done by the powder as the base of the shot is leaving the muzzle B. If OM represents cubic inches and MP represents tons per square inch, then the areas represent inch-tons of work, reducible to foot-tons by dividing by 12. Suppose the calibre of the gun is d inches and the shot weighs TFlb; and that it acquires velocity v f/s at M; then equating the kinetic energy and the work done, ^ff^2/^ = 2240 X iTTC?-^ X area J.ilfPO-r-12. 510 GRAPHICAL REPRESENTATION^ OF WORK This supposes the bore is smooth ; but if it is rifled with a pitch of 6 feet, the angular velocity at M is ^Trvjh ; so that if the radius of gyration of the shot about its axis is k feet, the kinetic energy is replaced by 2^ V + h^ /■ To allow for the friction of the bore an empirical deduction, say of | ton/in^, is made from the pressure represented by MP. 419. Such a diagram is called the Indicator Diagram of the shot ; and if the gun is free to recoil, there is a similar indicator diagram for the gun, representing the pressure on the base of the bore at corresponding points of the length of recoil. The recoil can be measured at any instant by Sebert's velocimeter ; the travel of the shot is measured by electric contacts at equal intervals along the bore, and the corre- sponding pressures are recorded by crusher gauges (§ 10) fixed in the side of the gun; the muzzle velocity is found from electric records outside the gun, and tlience is inferred the average pressure in the bore, represented by the ordinate AH, such that the rectangle AB, AH is equal to the area ABDG. A comparison of these diflerent records affords an independent check on the work done by the powder gases, inferred from the experiments of Noble and Abel, and enables us to assign the pressure deduction due to the friction of the bore. As in fig. 42, the curve AQE is drawn, such that its ordinate MQ represents to scale the work done by the powder or the kinetic energy acquired by the shot, each proportional to the area AMPG ; and therefore the IN THE BORE OF A GUN. 511 velocity at M may be represented by the ordinate Mv of the curve AvV, where Mv is proportional to ^MQ. Thus if, as in the pneumatic gun, we may take the pressure as uniform and represented by the line HK of average pressure, then AQE will be a straight line, and AvV a parabola; in this case the gun may be made of uniform thickness, calculated by § 290, and great economy in weight is secured. p V ,,? ■ - — " E c: A- I ,Ps, (2) where is a certain function, called the entropy ; then dE=dH—A'pdv = 9d(p-A2Jdv, (3) embodying the First and Second Laws. The internal energy E depends only on the state of the gas as given by p, v, 6, its pressure, volume, and temperature, connected by the Characteristic Equation, F{p,v,e)=--0; so that a change in E is independent of the intermediate states; or, in other words, dE is a perfect differential, and so also is d(f), according to the Second Law. The First and Second Laws of Thermodynamics are thus expressed by the relations /dH=AW, /dH/e = 0, W denoting the work done, and the integrals being taken round a closed cycle in which there is no escape of heat by conduction ; the quantity dH/6 is sometimes called the heat-weight of the heat dH. If H units of heat pass from a body at a temperature 02 to another body at a lower temperature 6^, the entropy of the first body falls HjO^ and of the second rises H/G^ ; so that the entropy of the system rises 1 1^ \0^ 02- The entropy is thus unchanged if no heat passes except between bodies at the same temperature; but conduction of heat between bodies of different temper- ature raises the entropy, and the entropy thus tends to a maximum. 518 THERMODYNAMICAL RELATIONS. 424. Of the four quantities f, v, 6, <{>, two only are independent ; and any pair may be taken as independent variables. Prof Willard Gibbs selects v and as variables, so that from (3), with Clausius's notation for partial differ- ential coefficients, But denoting by x, y any pair of independent variables ^, , 'dE „dy

dx ?« ^dx dy dy ^ ay 3^ dxdy ~'dy'dx~^ ^dxdy ^ ""dy dx ^^^'dx'dy and ;^^^ =.5— ^ + 0:^"^""^^ 5 ^P- ^30 3^ 3V_^9E3-_^ dxdy dxdy dx dy -^dx^y' 30 3^_39 30_ ./3p 3u_^ 'dv\ dx dy dy dx \dx dy dy dxJ' |M = ^?iE^) (4) d{x, y) 3(33, y) This proves that if the plane of the {p, v) diagram is covered by isothermal lines, for which Q is constant, and isentropic or adiabatio lines, for which ^ is constant, then integrating round any closed cycle, ffdQd(^ = AffdfdbV = A times the area of the cycle ; or the area of the cycle is JffdQdf. 425. A cycle a^y^ which is bounded by two iso- thermals 0j and Q^, and two adiabatics 0j and 02' ^^ called a Oarnbt cycle, fig. 107 ; and it thus encloses an area J(02-0i)(0,-0i) (5) THE GARNOT CYCLE. 519 With Q and ^ as variables, the Carnot cycle on the (0, i) along ^y, the tempera- ture 02 is constant, and the heat absorbed is thus ^2 = ^2( Carnot assumed that the efficiency of an engine working in this cycle between the temperatures 0^ and 02, was 0(02-01); and he supposed that G was constant ; but we see now that 0, called Carnot's function, is the reciprocal of the absolute temperature of the source of heat. 520 REVERSIBILITY OF THE CARA^OT CYCLE. The Carnot cycle a/3yJ is reversible ; that is, if described in the reverse direction aSy^, as in a refrigerating machine, the heat H^ absorbed at temperature 0j is given out as heat H^ at a higher temperature Q^, at the. expense of the work represented by the area of the cycle. Carnot's principle asserts that the efficiency of a reversible cycle is a maximum ; for if it were possible to obtain a greater efficiency by another arrangement, this could be made to drive the Carnot cycle backwards and thus create energy, and realise " Perpetual Motion." Thus a thermodynamic engine, for instance a low pressure engine, working between the extreme tempera- tures of the freezing and boiling points, 0°C and 100° C, gives away at least 273 out of 373 units of heat to the condenser ; so that its efficiency falls short of 0'27. 426. By taking x, y to represent any pair of the variables p, v, 6, (jt, we obtain various thermodynamical relations ; thus with independent variables (i.) p, v; 3(0, ^)_ 3(^3, v) ' = A- (ii.) e, ; d{p, v) 3(0, 4>y = J; (iii.) d,f; dp dpv '^30' or dpV dd" jde

; 300 dp ^t- or dpV_ d •^ dp ' (V.) e, V ■ dm ^dd' or d,p dd~ jdei>, dv ' (vi.) V, (j> ; 300 dv d(j> or d^p _ d(p jd^e dv ' THERMODYNAMICAL RELATIONS. 521 These relations are proved geometrically in Maxwell's Theory of Heat, by taking the Carnot cycle a^yS so small that it may be considered a parallelogram ABGD ■ and now an inspection of fig. 108 shows that the area of the parallelogram, or JA6A,p = AK.Ak = AL.Al^AM.A')n = AN.A'n. Fig. 107. Fig. 108. Then the relation (iii.) is equivalent to (iii.) AK A0' for AK is the dilation of v at constant pressure, while Ak is — A„p, is the diminution of pressure corresponding to the increment A

dy ^^W~ de~ "d^ de^^dy Te' and if the change of state is given by the relation f{x,y) = 0, then ?/^+§/^ = 0, dx de dy dd ' and W^ Wd^ dx dd^dy dd ' so that = 0^-^ I ^-^^d) .. (7) Taking Q and v as variables, and denoting the S.H. at constant volume by c^, then . ^ _(fiB dv_ . 3„p dv Thus if Cp denotes the S.H. at constant pressure, when 'd^ dep dv_ dO "^ dv de ' ..-.=- ; so that CpA6_AK dA(p~AL' Therefore, for all substances, Cp_AK AN _E^ ,-.„. ^,~zz ■ AJi~Eg ^ ' Maxwell proves equations (8) and (9) geometrically from fig. 107, as follows: — E -E -v(^^ Ak\ _ &r&ei ABGD _^AN ^ ^~ \An Am/ Am,. An Am,' _ ^AM A(p ''"'^ANAe' o.{E^-Ee)=vB^^ aI = ^^JMA^ A? = Ave{^y=Ave(^)" = Ee{Cp-c^), since c^ £'0 = CpE^. 524 THERMODYNAMIC AL RELATIONS 428. Applying these formulas to air, for which pv = R6, then M = ^=P, M=_^=_£; and Cp — Cy=:AR= R/J. Also Cp = yc„, where y is 1'4, about (§ 228); so that _ yAR _ AR ''p-y-V ''"-y-r With British units, J =119, R=5SS (§ 200); so that ^i? = 0-068, c^ = 0-238, c„= 0-170; and the numbers are the same with metric units and the Centigrade scale; these numbers were obtained in this manner by Rankine in 1850, before they had been de- termined experimentally. If we divide the S.H. Cv by the s.v. v of the gas, we obtain the thermal capacity per unit volume; this is found to be very nearly the same number for all gases at the same temperature. The numerical value of y is determined most accurately from the observed velocity of sound (§ 228) ; another mode of determination, due to Clement and Desormes, is to compress air into a closed vessel, and to observe the pressure p, when the temperature 6 is the same as that of the atmosphere. A stopcock is then opened, and suddenly closed when the air ceases to rush out; and it is assumed that the enclosed air has expanded adiabatically to atmospheric pressure p. After a time the air inside will regain the surrounding temperature 6, and its pressure p^ is again observed; so that 02> the temperature at the instant of closing the stopcock, is given by 0^ = Oplp^- OF A PERFECT GAS. 525 If V denotes the volume of the vessel, then the air left inside, at pressure p^ and temperature 6, originally occupied a volume Vpjp^ at pressure p^^; and in ex- panding adiabatieally to volume V it assumed the atmo- spheric pressure p ; so that py^=Pi{ypiiPiV> or pjp-=(pjp^)-y, iogPi-logp log Pi- log K 429. Taking 6 and v as variables with a perfect gas, dH= edd> = c„de + e^'^dv av = c^dd + A6 -Za^v = c^dd + Apdv, so that we may put E = c„e = ABe/{y-l). Thus the internal energy of the gas, in heat units S, in the state represented by the point a in the diagram of fig. 107, is A times the area of the indefinitely extended adiabatic curve aaSv, cut ofi" by the ordinate aa. The increase in internal energy E in passing from the state a to the state j8 by any path a^ is thus A times the area vaah^v ; and this area is made up of aa6/3, repre- senting the work done in compressing the gas from a to /3, and of va^v, representing the mechanical equivalent of the heat supplied in going from a to /3. , dd , , r,dv Also dtp = c„-^- +AIi — ; and, integrating, ^ = c„ log -t- {Cp — c„)log v + a constant = c„ log0i;'>''^-fa constant, ^-9^o=«viog|Qy"=c.iog^^(jj. 526 EXAMPLES. "With Q and

, v) diagram are the similar hyperbolas p^ — yv^ = constaii t. EXAMPLES. 527 Prove that the isothermals and adiabatics cut at a maximum angle cot"^ 2^y on the line p^y — v = Q. Discuss the same problem for the isometrics and isobars on the (6, ) diagram. (2) Prove that, if a perfect gas expands along the curve yu*^ = constant, the work done by expansion is (y — l)/(y — /c) of the mechanical equivalent of the heat absorbed. (3) Prove that the specific heat of a perfect gas, expanding along the curve f{p, v) = 0, is (4) Prove that, iipv = Rd"; Cp-Cy=nm6''-\ (5) Determine the heat equivalent of the kinetic energy of rotation of the Earth, supposed homogeneous and of S.H. c ; and determine the number of degrees which this heat would raise the temperature of the Earth, taking c = 0-2. (6) Find what fraction of the coal raised from a mine 500 fathoms deep is used in the engine raising the coal, and 30 times its weight of water, supposing the heat of combustion of 1 lb of coal is 14,000 RT.U., and the efiiciency of the engine is j-. (7) Compare the work done and the work given out when V ft^ of atmospheric air is compressed adia- batically to n atmospheres, cooled down to the original temperature, and expanded adiabatically to atmospheric pressure ; for instance, in a White- head torpedo. 528 TABLES. TABLE I.— DENSITY OF WATER (MENDELEEF). c. s(g/cm3). i)(lb/i"ts). j;(oni'/g). C. s(g/cm'*). i)(lb/ft3). v{cm^lg). 0° 0-999873 62-4162 1-000127 40° 0-992334 61 -9456 1 -007725 5° 0-999992 62-4237 1-000008 50° 0-988174 61 -6860 1-011967 10° 0-999738 62-4078 1 000262 60° 0-983356 61-3852 1-016926 15° 0-999152 62-3712 1-000849 70° 0-977948 61 -0476 1-022549 20° 998272 62-3163 1-001731 80° 0-971996 60-6760 1-028811 ■25° 0-997128 62-2449 1 -002881 90° 0-965537 60-2729 1-035693 30° 0-995743 62-1584 1-004275 100° 0-958595 59-8395 1-043193 TABLE II —SPECIFIC GRAVITY. Platinum, 22 Aluminium, 2-6 Pure Gold, 19-4 Stone, Brickwork, or Earth 2 Standard Gold, 17-5 Glycerine, 1-26 Mercury, 13-6 Sea Water, 1-026 Lead, 11-4 Pure Distilled Water, 1 Silver, 10-5 Ice, 0-92 Copper, 8-8 Oak, 0-93 Brass, 8 Petroleum, 0-88 Wrought Iron or Steel, - 7-8 Pure Alcohol, 0-79 Cast Iron, 7-2 Cork, 0-24 TABLE lU.- -ROOMAGE. Salt Water, 35 Wjton. Cast Iron, 4-6 ft-Vton. Fresh Water, 36 „ Wheat or Grain, 45 Coal, 40 to 46 Timber, 66 Pig Iron, 9 Tea, 90 1 INDEX. Absolute dilation of mercury 243 Absolute temperature . . 242 Accumulator . . . .22 Adiabatic expansion . . 266 ^lian 52 Aggregation of cylindrical particles . . . .48 Airlock . . . .355 Air pumps .... 366 Air pump and condensing pump combined . . . 378 Air thermometer . . . 307 Alexander .... 391 Alloys and mixtures, . . 120 Amagat 287 Amagat gauge . . .25 Andrews . . . 242, 306 Aneroid barometer . 15, 265 Angle of contact . . . 403 Angle of repose . . .45 Angle of the centre . . 460 Angular oscillations of a float- ing body . . . 228 Anticlastic . . .160 Aral Sea . . 79 Archimedes . . 1, 93, 484 Archimedes' principle . 77, 93 Ar^omfetre . . . .127 Aristotle . . . . 105 Ascending and descending buckets .... 424 Ascensional force . . . 331 Atmosphere . . . .11 Atmospheric air . . .128 PAGE Atwood 152 Average density . . .98 Average pressure over a surface 81 Axis of spontaneous rotation 66 Babinet's. barometric formula 310 Barads .... 489 Barker's mill ... 462 Barlow curve . . 395 Barometer . . . 251 Baroscope .... 105 Bear Valley dam . . .55 Bending moment . . . 414 Bernoulli's lintearia . 409 Bernoulli's theorem . . 467 Berthelot » . .95, 354 Biquadratic feet . . .64 Bixio and Barral . . . 337 Blackwall Tunnel . . .77 Block coefficient . . . 213 Body plan . . . .210 Bourdon's pressure gauge 14, 265 Boyle ... 3, 21, 366 Boyle's law . . 280, 286, 294 Boyle's statical baroscope . 105 Boys 412 Bramah ... .18 British thermal unit . . 505 Bubble electrified . . . 416 Budenberg . . . .18 Buoyancy . . . 94 Buoyancy, centre of . .149 Buoyancy, curve (or surface) of 149 Buoyancy, simple . . . 149 2l 529 530 INDEX. PAGE Cailletet .... 5 Caissons .... 356 Caissons of the Forth Bridge . 182 Calorie ... . 505 Camels ... . 139 Capillary attraction . . 398 Capillary curve . . . 407 Captive balloon . . .331 Cardioids . . .163 Carnot cycle . . 518 Camot's function . . . 519 Cartesian diver . . 301 Caspian Sea . . .79 Catenoid . . .418 Cathetoraeter . . . 243 Celsius . ... 246 Centigrade . . . 246 Centimetre . . .10 Centre of buoyancy ■ . . 149 Centre of oscillation . . 65 Centre of percussion . . 65 Centre of pressure . 43, 60 Centrifugal pump . . 473 Centrobario .... 457 Change of level in a loco- motive boiler . . . 430 Change of trim . . .151 Channel Tunne . . 356, 484 Characteristic equation of a perfect gas .... 281 Charles 3 Charles II. . . 100, 289, 398 Charles or Gay Lussac's law . 280 Chicago. . . . 139 Chimney draught Chisholm Circle of inflexions Circular inch . Clark and StansSeld Clearance Clemens Herschel . CoeflScient of contraction 462, 470 CoeflScient of cubical compres- sion 318 Coefficient of discharge . . 463 Coefficient of expansion . 241 Coefficient of fineness . 140, 213 Coefficient of velocity . . 463 Common surface of two liquids . . . .36 321 103 177 13 139 370 469 Component horizontal thrust of a liquid . . . .80 Component vertical thrust . 73 Compound vortex . . . 473 Compressibility of mercury . 318 Compressibility of water . 317 Condensing pump . . . 374 Conditions of stability of a ship 148 Cone .... 189, 194 Conemaugh Dam . . 45 Coney Island stand pipe 76, 387 Conical pendulum . . . 224 Conjugate stresses . . 391 Conservatoire des Arts et Mi^tiers . . . .104 Conveotive currents . . 38 Conveolive equilibrium of the atmosphere . . .313 Cornish pumping engine . 34 Corrections for weighing in air 304 Cosmos . . . 337 Cotes . . .100 Coxwell. . . . 332 Coxwell's balloon . . . 338 Cream separator . . 445 Critical temperature . 242, 306 Cross curves of stability 152, 167 Crown of Hiero . . .99 Ctesibius . . 2, 360 Cubical compression . . 515 Cup of Tantalus . . .277 Curve (or surface) of buoy- ancy 149 Curve of dynamical stability . 161 Curves of pressure and density 319 Curves et surfaces analytiques 210 Curves et surfaces topographi- ques 210 Carve of statical stability . 161 Cylinder .... 189 Cylinder floating upright . 191 Cylinder or prism floating horizontally . . ,193 D'Alembert's principle . . 429 Dalton's law .... 285 Daniell .... 262 D'Arlandes . . . 327 INDEX. 531 Daymard Dead Sea PAGE . 190 79, 98, 123, 125, 234 Ueep-sea sounding machine . 289 Delestage . . . 335 De Morgan .... 220 Density . . 30, 96 Dewar . . . . 6, 266 Differential air thermometer . 308 Dilation by heat . . . 241 Diving bell and diving dress . 347 Draft 138 Draught of a chimney . . 321 Druitt Halpin . . .435 Duokham . . 24 Dulong and Petit . . 243 Dupin . . . . 160 Dupin's theorems . . .159 Dynamical stability . . 230 Dynamics of the Siphon . 273 515, Earth' pressure Earthwork dam Ebullition Effective force Eiffel Tower . Elastica Elasticity Elastic limit . Electrified bubble . Elgar . Ellipsoid Elliptic functions . EUipticity Ellis Energy of compression Energy of immersion Energy of liquid . Entropy Equality of fluid pressure Equation of the three cubes Equilibrium of bubbles . Equilibrium of liquids in ' bent tube . Equilibrium of rotating liquid 499 Equilibrium of the atmosphere 489 Espaoe nuisible . . . 370 Evelyn 347 Exeter Canal .... 389 Experimental verification of Boyle's law 392 43 302 429 287 409 522 . 445 . 416 . 180 189, 202 . 230 . 459 . 265 . 320 . 143 . 20 . 517 16 336 300 233 286 PAGE Fahrenheit ... 247 Fahrenheit or Nicholson hydrometer . . 110, 115 Fire engine .... 360 First law of dynamics . . 505 Fleuss 355 Fleuss and Jackson air pump 381 Flexaral rigidity . . . 410 Floating body partly sup- ported .... 214 Flottaison . . . .140 Flow with variable head . 464 Fluid 4, 7 Fluid friction . . 480 Force pump . . 360 Forced vortex . . . 473 Forth Bridge . . . 77 Freeboard . . .138 Free circular vortex . 472 Free spiral vortex . . . 472 Free surface of a liquid . . 28 Free surface of the ocean . 457 Friction brake . . . 435 Frontinns . . 2, 470 Froude . . 468, 481 Galileo . Garnett . Gas Gaseous laws . Gasholder Gauge glass . Gaussian measure of curvature 411 Gay Lussac . . 3, 259, 280 General equations of equilibrium Geocentric latitude Giffard's injector . Gorman .... Governor of a gasholder Gradient of the barometer Gramme Graphical construction of the working of a condenser Graphical representations of Boyle's law Gravimetric density of gun powder .... Gravitating spheres Gravitation measure of force . 3 . 126 4, 279 . 280 . 341 . 429 484 460 466 356 344 267 10 377 294 130 497 282 532 INDEX. Greathead Great Salt Lake . Green Green's transformation Griffiths Guillaume Guillaume de Moerbek Gulf Stream . Guthrie . PAGE . 356 79, 98 . 334 . 487 . 508 . 258 . 1 38, 489 . 118 220, Half-breadth plan . . .210 Hare's hydrometer . . 237 Hauksbee . . . .366 Hauksbee air pump . . 367 Head of water . .33 Heat equivalent of work . 508 Heat weight .... 517 Heaviness . . .30 Heeling effect of the screw propeller .... 178 Height of the homogeneous atmosphere Heinke ..... Helicoid .... Hermite Hero Herresschoflf . Hiero of Syracuse . H.M.S. "Achilles" Hooke . Horace . Hudson River Tunnel Huygen's barometer Hydraulics . Hydraulic gradient Hydraulic Power Company Hydraulic press . . .17 Hydraulic ram . . . 428 Hydrodynamics . . 8 Hydrogen or gas balloon . 330 Hydrometer 110,114,115,125,237 Hydrostatic balance . 98, 99 Hydrostatic bellows . 18 Hydrostatic paradox . . 21 Hydrostatic thrust . .41 Hydrostatic thrust in a mould 74 Hyperboloid . . . 189, 202 Imperial measures of capacity 106 Impulse turbine . . . 479 252 . 556 . 420 337 2, 277, 360 . 388 . 95 150 366 2 77 263 461 276, 481 21 Impulsive pressure Inclining couple Indicatrix . . . . Indicator diagram . Injector on thermodynamical principles . . . . Interchange of buoyancy and reserve of buoyancy . Intermittent siphon Internal energy Inverted siphons Involute Isobars . Isobath inkstand Isooar&nes Isometrics Isothermals . Isothermal equilibrium of the atmosphere PAOE 426 158 159 510 508 180 . 277 . 516 . 276 163, 432 . 282 219 . 156 . 282 . 282 309 Jacobi's rotating ellipsoid ' . 502 Jamin . . . 103, 245, 287 Jenkins .... 163, 438 Jordan's glycerine barometer 263 Joule ..... 3 Keeley . Kilogramme . 21 10, 104 Lactometer . . .114, 129 Lana . . . . 337 Laplace ..... 399 Laplace's barometric formula 311 Laplace's law of the density of the strata of the earth . 494 La Rochelle .... 355 Latent heat of fusion . . 507 Latent heat of vaporisation . 507 Latitude . . . .460 Laws of vapour pressure . 302 Leclert's theorems 165, 173, 202 Legrand ... 1 Leslie Ellis .... 52 Level surfaces of equilibrium 485 Lift 331 Lifting pump . . . 363 Limapons .... 163 Lines of curvature . 159, 41 1 Liquid 4 Liquid ballasting . . . 203 INDEX. 533 PAGE Liquid films . . . 415, 417 Liquid in communicating ves- sels 79 Liquid in rotating curved tubes 452 Liquid maintains its level . 35 Liquid rotating about a verti- cal axis . . . 443 Little . . .176 Lodge . . . . 5 Longitudinal Metaoentre . 151 M'Leod gauge . . .382 Macfarlane Gray . .162 Mackinlay . . . .130 Maolaurin's and Jacobi's fig- ures of equilibrium of rotat ing liquids .... Magdeburg hemispheres Main . . Mallet . Marine barometer . Mariotte .... Marquis of Worcester . Mascart .... Maximum draught of a chim- ney Maxwell 6, 401, 402, 416 Mean depth .... Mechanical equivalent of heat 508 Mechanical theory of heat . 505 Mediterranean . . . 234 Mendeleeff, . 97, 102, 242, 444 Mercurial air pump . . 380 Mercury weighing machine . 141 Metacentre .... 149 Metacentric height . . 149 Meteorology of the barometer 267 Metre 10 Metric system . . 33 Milner 220 Miner's inch . . . 463 Minimum surfaces . .411 Mississippi . • ■ 460 Mixtures and alloys . . 120 Mole 277 Momental ellipse . . 67, 157 Moment of inertia . . .64 Momentum and energy of a jet 465 500 366 242, 288 . 7 . 438 . 3 . 362 . 266 322 520 64 PAGE 2 213 327 428 4 18 Mommsen Moorsom Montgolfier hot air balloon Montgolfier's hydraulic ram Montucla Morland Morland's diagonal barometer 254 Morland's steelyard or balance barometer . . . 260 Moseley . . .162 Mount Everest . 253 Nasmyth . .19 Niagara. . . . 464 Nicholson . . . .172 Nicholson hydrometer . 110, 115 Nodoid 419 Noble and Abel . . .510 Numerical calculations in naval architecture . . 210 Obliquity of the stress . . 392 Osborne Reynolds . . .53 Oscillations of a gasholder . 346 Oscillations of a ship . . 441 Otto von Guericke . . 366 Ovid 2 Packing of spheres . . 52 Paddle wheel. . . .475 Palaemon . . . .95 Papin 11 Parabolic cylinder . . . 200 Parabolic speed measurer, . 448 Paraboloid . . . 189, 200 Pascal . . . 3, 17, 253, 266 Pascal's vases . . .78 Pelton wheel . . . .474 Pearson 6 Period 224 Perry . ... 412 Pictet ... .5 Piezometer .... 468 Pilatre de Eozier . . . 327 Pipette ... .292 Planimeters . . . 210, 514 Plasticity .... 6 Plateau .... 400, 460 PlimsoU. . . . 138 Plimsoll mark . . . 168 534 INDEX. PAGE Pliny .... 2 Plutarch .... 52 Pneumatics . . . 279 Pneumatic machines . . 327 Polar curves of stability . 162 Pollard et Dudebout . 210 Pound .... 10 Pratt and Whitney Co. . 258 Pressure .... 9 Pressure and density of va- pour 304 Pressure in a homogeneous liquid 30 Pressure intensifying appa- ratus 25 Pressure in the interior of a gravitating sphere of li- quid 492 Pressure of liquid in moving vessels .... 423 Pressure turbine . . . 477 Principal axis . . 1 57 Principal radii of curvature 410 Principia .... 236 Principle of virtual velocities 20 Product of inertia . . .65 Pulsator or pulsometer pump 362 Pumps . . . . 359 Quaker Bridge Dam 44 Radiating current . . 471 Radius of gyration . 64 Rainbow . . . .400 Ramsbottom's safety valve . 12 Rankine 13, 53, 396, 471, 478 Rankine's hydrostatic arch . 409 Rarity Rational horizon Rayleigh Reaumur Reed Refraction in optics Regelation of ice . Regnault Eegnault atmosphere Regnault's table Reinold and Rucker Rennie . Reservoir wall 127 460 398, 401, 469 247 180 259 404 34, 243, 282 34 303 421 354 41 Resistance of ships . 481 Resultant thrust . . 82 Rise in capillary tube . . 405 Robins .... 281 Roomage . 127 Rotating cylinders of liquid . 503 Rotating liquid . . . 433 Rotating sphere of liquid . 498 Rowland . . . .508 Royal society . . 262, 289 Rutter . . .141 Safety valve . 11 Sails and windmills 476 Salinometer . 121 Salt Lake 125 Saturated vapour 302 Savery . 362 Say's Stereometer . 292 Schaffer . 18 Schadlicher Raum . 370 Screw propeller 178 Sennett . 124 Sensible horizon 460 Severn Tunnel 355 389 Shearing stress 8 Sheer plan 210 Ship aground . 232 Ship design and calculation . 210 Sikes's hydrometer 110 114 Simple buoyancy . 137 Simple harmonic motion 435 Simple harmonic vertica oscillations . 224 Simpson's rule 21 1' 514 Siphon . 271 Siphon barometer . 252 Siphon gauge 322 Smeaton. 354 Smeaton's air pump 367 Smith . 382 Smithsonian Meteorologica tables .... 265 Soap bubbles . 416 Solids . 4 Specific gravity 96 Specific gravity bottle 117 Specific heat . 506 Specific volume 127 Spenser .... 340 INDEX. 535 PAGE 163 125 50 381 382 231 Spiral of Archimedes Spirit hydrometer . Spherical atoms Sprengel's Mercurial Pump . S. P. Thompson . Stability Stability of a vessel with liquid cargo . . 174 Stability of equilibrium . 37 Stability of the diving bell . 352 Standards' department . . 107 Standards of length . . 257 Statical and dynamical sta- bility 161 Steadiness .... 231 Stevinus . . . 3 Stewart and Gee . . 103, 118 Stiffness. . . . 172, 231 Stress 8 Stresses due to rolling . . . 438 Stress ellipse . . . 389 Submarine boat . . . 153 Suction Pump . . 363 Suez Canal . . .36 Superficial tension . . . 399 Surcharged retaining walla . 46 Surface of buoyancy . . 159 Surface of flotation . 156, 159 Surfaces of equal density . 38 Surfaces of equal pressure 27, 486 Surfaces of equal pressure in a swinging body . . . 436 Swing conic . . . 67 Synolastic . . .160 Syringe . ... 359 Tangye 23 Tenacity . . . 446 Tension 8 Tension of vessels . . 385 Theory of earth pressure . 45 Thermal capacity per unit volume Thermodynamics . Thermometer . The screw propeller Thick tube . Thomson Thomson and Tait . Thrust . . 524 . 505 . 246 . 478 . 393 314, 400, 412 . 433 8 PAGE Thrust of • a liquid under gravity . . . .58 Thrust on a lock gate . . 54 Thurot . . . 95 Torricelli ... 2, 251 Torricelli's theorem . .461 Torricellian vacuum . .251 Total curvature . . 410 Total heat of steam . . 507 Tower of Pisa . . 393 Trade winds . . .38, 489 Transmissibility of fluid pres- sure ..... 17 Transmission of pressure Trapezoidal rule Trim Triangular prism . Tweddell Unduloid Vacuum brake Van der Waals Vapours . Variation of gravity Vena contracta Venafrum Venice . . Venturi water meter Vernier . Vertical oscillations Vertical oscillations of ing body Viscosity Vitruvius Vitiated vacuum of s meter . Vyrnwy Dam . Walker, J. . Walker's steel-yard. Wall-sided . Walton, W. . Water and glycerine Water line area Water ram Water wheels . Watt's Indicator Weather glass Wedge of emersion float- . 34 . 211 151 . 194 22 . 419 . 374 . 306 . 306 . 284 462 2 . 139 . 469 . 258 225 baro- 224 6 2, 95 296 44 . 103 . 100 140, 169 . 52 barometer 262 . 140 . 428 434, 473 . 513 259 . 152 536 INDEX. ,PAOE Wedge of immersion . . 152 Weights and Measures Act . 103 Weight thermometer . . 245 Weight and weighing . . 103 Weight of the atmosphere . 265 Weinhold . . . .290 Whirlpools and cyclones . 473 AVhite, W. H. . 140, 160, 228 White and John . . .190 Whistling well . . .323 Whole normal pressure . . 82 WillardGibbs . . .518 PAGE Wolstenholme, . . .239 Work done by a couple . .164 Work done by powder . . 509 Work required to exhaust the receiver .... 369 Worthington ... 8, 273 Worthington pumping engine 361 Young . Zuyder Zee . 220, 399 . 139 GLASGOW ; PRINTED AT THE UNIVERSITY PRESS, BY ROBERT MAOLEHOSE.