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diff --git a/old/66078-0.txt b/old/66078-0.txt deleted file mode 100644 index fc52592..0000000 --- a/old/66078-0.txt +++ /dev/null @@ -1,13322 +0,0 @@ -The Project Gutenberg eBook of A Treatise on Mechanics, by Henry Kater - -This eBook is for the use of anyone anywhere in the United States and -most other parts of the world at no cost and with almost no restrictions -whatsoever. You may copy it, give it away or re-use it under the terms -of the Project Gutenberg License included with this eBook or online at -www.gutenberg.org. If you are not located in the United States, you -will have to check the laws of the country where you are located before -using this eBook. - -Title: A Treatise on Mechanics - -Author: Henry Kater - Dionysius Lardner - -Release Date: August 17, 2021 [eBook #66078] - -Language: English - -Character set encoding: UTF-8 - -Produced by: Thiers Halliwell, deaurider and the Online Distributed - Proofreading Team at https://www.pgdp.net (This file was - produced from images generously made available by The Internet - Archive) - -*** START OF THE PROJECT GUTENBERG EBOOK A TREATISE ON MECHANICS *** -Transcriber’s notes: - -The text of this e-book has mostly been preserved in its original form -including some inconsistency of hyphenation and use of diacritics -(aeriform/aëriform). Three spelling typos have been corrected -(arrangment → arrangement, pully → pulley, dye → die) as have typos -in equations on pages 40 and 43. Some missing punctuation has been -corrected silently (periods, commas, incorrect quotes). Footnotes have -been positioned below the relevant paragraphs. - -In this plain text version, italic text is denoted by _underscores_ -and bold text by *asterisks*. A caret (^) indicates that the following -character is superscripted, e.g. D^2 (D squared). - - - - A - - TREATISE ON MECHANICS, - - BY - - CAPTAIN HENRY KATER, V. PRES: R.S. - - ----and---- - - DIONYSIUS LARDNER, D.C.L. F.R.S. &c. &c. - - A NEW EDITION REVISED & CORRECTED. - 1852. - - [Illustration: _H. Corbould del._ _E. Finder fc._] - - London: - PRINTED FOR LONGMAN, BROWN, GREEN & LONGMANS. PATERNOSTER ROW: - - - - -ADVERTISEMENT. - - -This Treatise on Mechanics, which was originally published in 1830, -is the work of Dr. Lardner, with the exception of the twenty-first -chapter, which was written by the late Captain Kater. The present -edition has been revised and corrected by Dr. Lardner. - - _London, January, 1852._ - - - - -CONTENTS. - - - CHAP. I. - - PROPERTIES OF MATTER. - - Organs of Sense.--Sensations.--Properties or Qualities.--Observation. - --Comparison and Generalisation.--Particular and general Qualities.-- - Magnitude.--Size.--Volume.--Lines.--Surfaces.--Edges.--Area.--Length. - --Impenetrability.--Apparent Penetration.--Figure.--Different from - Volume.--Atoms.--Molecules.--Matter separable.--Particles.--Force.-- - Cohesion of Atoms.--Hypothetical Phrases unnecessary.--Attraction. 1 - - - CHAP. II. - - PROPERTIES OF MATTER, CONTINUED. - - Divisibility.--Unlimited Divisibility.--Wollaston’s micrometric - Wire. --Method of making it.--Thickness of a Soap Bubble.--Wings of - Insects. --Gilding of Wire for Embroidery.--Globules of the Blood.-- - Animalcules.--Their minute Organisation.--Ultimate Atoms.--Crystals.-- - Porosity.--Volume.--Density.--Quicksilver passing through Pores of - Wood.--Filtration.--Porosity of Hydrophane.--Compressibility.-- - Elasticity.--Dilatability.--Heat.--Contraction of Metal used to - restore the Perpendicular to Walls of a Building.--Impenetrability of - Air.--Compressibility of it.--Elasticity of it.--Liquids not absolutely - incompressible.--Experiments.--Elasticity of Fluids.-- Aeriform - Fluids.--Domestic Fire Box.--Evolution of Heat by compressed Air. 9 - - - CHAP. III. - - INERTIA. - - Inertia.--Matter Incapable of spontaneous Change.--Impediments to - Motion.--Motion of the Solar System.--Law of Nature.--Language used - to express Inertia sometimes faulty.--Familiar Examples of Inertia. 27 - - - CHAP. IV. - - ACTION AND REACTION. - - Inertia in a single Body.--Consequences of Inertia in two or more - Bodies.--Examples.--Effects of Impact.--Motion not estimated by - Speed or Velocity alone.--Examples.--Rule for estimating the - Quantity of Motion.--Action and Reaction.--Examples of.--Velocity - of two Bodies after Impact.--Rule for finding the common Velocity - after Impact.--Magnet and Iron.--Feather and Cannon Ball impinging. - --Newton’s Laws of Motion.--Inutility of.--Familiar Effects - resulting from Consequences of Inertia. 34 - - - CHAP. V. - - COMPOSITION AND RESOLUTION OF FORCE. - - Motion and Pressure.--Force.--Attraction.--Parallelogram of Forces. - --Resultant.--Components.--Composition of Force.--Resolution of - Force.--Illustrative Experiments.--Composition of Pressures.-- - Theorems regulating Pressures also regulate Motion.--Examples.-- - Resolution of Motion.--Forces in Equilibrium.--Composition of Motion - and Pressure.--Illustrations.--Boat in a Current.--Motions of Fishes. - --Flight of Birds.--Sails of a Vessel.--Tacking.--Equestrian Feats. - --Absolute and relative Motion. 48 - - - CHAP. VI. - - ATTRACTION. - - Impulse.--Mechanical State of Bodies.--Absolute Rest.--Uniform and - rectilinear Motion.--Attractions.--Molecular or atomic.--Interstitial - Spaces in Bodies.--Repulsion and Attraction.--Cohesion.--In Solids - and Fluids.--Manufacture of Shot.--Capillary Attractions.--Shortening - of Rope by Moisture.--Suspension of Liquids in capillary Tubes.-- - Capillary Siphon.--Affinity between Quicksilver and Gold.--Examples - of Affinity.--Sulphuric Acid and Water.--Oxygen and Hydrogen.--Oxygen - and Quicksilver.--Magnetism.--Electricity and Electro-Magnetism.-- - Gravitation.--Its Law.--Examples of.--Depends on the Mass.-- - Attraction between the Earth and detached Bodies on its Surface.-- - Weight.--Gravitation of the Earth.--Illustrated by Projectiles.-- - Plumb-Line.--Cavendish’s Experiments. 63 - - - CHAP. VII. - - TERRESTRIAL GRAVITY. - - Phenomena of falling Bodies.--Gravity greater at the Poles than - Equator.--Heavy and light Bodies fall with equal Speed to the Earth. - --Experiment.--Increased Velocity of falling Bodies.--Principles of - uniformly accelerated Motion.--Relations between the Height, Time, - and Velocity.--Attwood’s Machine.--Retarded Motion. 84 - - - CHAP. VIII. - - OF THE MOTION OF BODIES ON INCLINED PLANES AND CURVES. - - Force perpendicular to a Plane.--Oblique Force.--Inclined Plane.-- - Weight produces Pressure and Motion.--Motion uniformly accelerated.-- - Space moved through in a given Time.--Increased Elevation produces - increased Force.--Perpendicular and horizontal Plane.--Final - Velocity.--Motion down a Curve.--Depends upon Velocity and Curvature. - --Centrifugal Force.--Circle of Curvature.--Radius of Curvature.-- - Whirling Table.--Experiments.--Solar System.--Examples of centrifugal - Force. 85 - - - CHAP. IX. - - THE CENTRE OF GRAVITY. - - Terrestrial Attraction the combined Action of parallel Forces.-- - Single equivalent Force.--Examples.--Method of finding the Centre of - Gravity.--Line of Direction.--Globe.--Oblate Spheroid.--Prolate - Spheroid.--Cube.--Straight Wand.--Flat Plate.--Triangular Plate.-- - Centre of Gravity not always within the Body.--A Ring.--Experiments. - --Stable, instable, and neutral Equilibrium.--Motion and Position of - the Arms and Feet.--Effect of the Knee-Joint.--Positions of a Dancer. - --Porter under a Load.--Motion of a Quadruped.--Rope Dancing.-- - Centre of Gravity of two Bodies separated from each other.-- - Mathematical and experimental Examples.--The Conservation of the - Motion of the Centre of Gravity.--Solar System.--Centre of Gravity - sometimes called Centre of Inertia. 107 - - - CHAP. X. - - THE MECHANICAL PROPERTIES OF AN AXIS. - - An Axis.--Planets and common spinning Top.--Oscillation or Vibration. - --Instantaneous and continued Forces.--Percussion.--Continued Force. - --Rotation.--Impressed Forces.--Properties of a fixed Axis.--Movement - of the Force round the Axis.--Leverage of the Force.--Impulse - perpendicular to, but not crossing, the Axis.--Radius of Gyration.-- - Centre of Gyration.--Moment of Inertia.--Principal Axes.--Centre of - Percussion. 128 - - - CHAP. XI. - - OF THE PENDULUM. - - Isochronism.--Experiments.--Simple Pendulum.--Examples illustrative - of.--Length of.--Experiments of Kater, Biot, Sabine, and others.-- - Huygens’ Cycloidal Pendulum. 145 - - - CHAP. XII. - - OF SIMPLE MACHINES. - - Statics.--Dynamics.--Force.--Power.--Weight.--Lever.--Cord.-- - Inclined Plane. 160 - - - CHAP. XIII. - - OF THE LEVER. - - Arms.--Fulcrum.--Three Kinds of Levers.--Crow Bar.--Handspike.--Oar. - --Nutcrackers.--Turning Lathe.--Steelyard.--Rectangular Lever.-- - Hammer.--Load between two Bearers.--Combination of Levers.-- - Equivalent Lever. 167 - - - CHAP. XIV. - - OF WHEEL-WORK. - - Wheel and Axle.--Thickness of the Rope.--Ways of applying the Power. - --Projecting Pins.--Windlass.--Winch.--Axle.--Horizontal Wheel.-- - Tread-Mill.--Cranes.--Water-Wheels.--Paddle-Wheel.--Rachet-Wheel.-- - Rack.--Spring of a Watch.--Fusee.--Straps or Cords.--Examples of.-- - Turning Lathe.--Revolving Shafts.--Spinning Machinery.--Saw-Mill.-- - Pinion.--Leaves.--Crane.--Spur-Wheels.--Crown-Wheels.--Bevelled - Wheels.--Hunting-Cog.--Chronometers.--Hair-Spring.--Balance-Wheel. 178 - - - CHAP. XV. - - OF THE PULLEY. - - Cord.--Sheave.--Fixed Pulley.--Fire Escapes.--Single moveable - Pulley.--Systems of Pulleys.--Smeaton’s Tackle.--White’s Pulley.-- - Advantage of.--Runner.--Spanish Bartons. 199 - - - CHAP. XVI. - - ON THE INCLINED PLANE, WEDGE, AND SCREW. - - Inclined Plane.--Effect of a Weight on.--Power of.--Roads.--Power - Oblique to the Plane.--Plane sometimes moves under the Weight.-- - Wedge.--Sometimes formed of two inclined Planes.--More powerful - as its Angle is acute.--Where used.--Limits to the Angle.--Screw. - --Hunter’s Screw.--Examples.--Micrometer Screw. 209 - - - CHAP. XVII. - - ON THE REGULATION AND ACCUMULATION OF FORCE. - - Uniformity of Operation.--Irregularity of prime Mover.--Water-Mill. - --Wind-Mill.--Steam Pressure.--Animal Power.--Spring.--Regulators. - --Steam-Engine.--Governor.--Self-acting Damper.--Tachometer.-- - Accumulation of Power.--Examples.--Hammer.--Flail.--Bow-string.-- - Fire Arms.--Air-Gun.--Steam-Gun.--Inert Matter a Magazine for - Force.--Fly-Wheel.--Condensed Air.--Rolling Metal.--Coining-Press. 224 - - - CHAP. XVIII. - - MECHANICAL CONTRIVANCES FOR MODIFYING MOTION. - - Division of Motion into rectilinear and rotatory.--Continued and - reciprocating.--Examples.--Flowing Water.--Wind.--Animal Motion.-- - Falling of a Body.--Syringe-Pump.--Hammer.--Steam-Engine.--Fulling - Mill.--Rose-Engine.--Apparatus of Zureda.--Leupold’s Application - of it.--Hooke’s universal Joint.--Circular and alternate Motion.-- - Examples.--Watt’s Methods of connecting the Motion of the Piston - with that of the Beam.--Parallel Motion. 245 - - - CHAP. XIX. - - OF FRICTION AND THE RIGIDITY OF CORDAGE. - - Friction and Rigidity.--Laws of Friction.--Rigidity of Cordage.-- - Strength of Materials.--Resistance from Friction.--Independent of - the Magnitude of Surfaces.--Examples.--Vince’s Experiments.--Effect - of Velocity.--Means for diminishing Friction.--Friction Wheels.-- - Angle of Repose.--Best Angle of Draught.--Rail-Roads.--Stiffness - of Ropes. 260 - - - CHAP. XX. - - ON THE STRENGTH OF MATERIALS. - - Difficulty of determining the Laws which govern the Strength of - Materials.--Forces tending to separate the Parts of a Solid.--Laws - by which Solids resist Compression.--Euler’s theory.--Transverse - Strength of Solids.--Strength diminished by the Increase of Height. - --Lateral or Transverse Strain.--Limits of Magnitude.--Relative - Strength of small Animals greater than large ones. 272 - - - CHAP. XXI. - - ON BALANCES AND PENDULUMS. - - Weight.--Time.--The Balance.--Fulcrum.--Centre of Gravity of.-- - Sensibility of.--Positions of the Fulcrum.--Beam variously - constructed.--Troughton’s Balance.--Robinson’s Balance.--Kater’s - Balance.--Method of adjusting a Balance.--Use of it.--Precautions - necessary.--Of Weights.--Adjustment of.--Dr. Black’s Balance.-- - Steelyard.--Roman Statera or Steelyard.--Convenience of.--C. Paul’s - Steelyard.--Chinese Steelyard.--Danish Balance.--Bent Lever Balance. - --Brady’s Balance.--Weighing Machine for Turnpike Roads.--Instruments - for Weighing by means of a Spring.--Spring Steelyard.--Salter’s - Spring Balance.--Marriott’s Dial Weighing Machine.--Dynamometer.-- - Compensation Pendulums.--Barton’s Gridiron Pendulum.--Table of linear - Expansion.--Second Table.--Harrison’s Pendulum.--Troughton’s - Pendulum.--Benzenberg’s Pendulum.--Ward’s Compensation Pendulum.-- - Compensation Tube of Julien le Roy.--Deparcieux’s Compensation.-- - Kater’s Pendulum.--Reed’s Pendulum.--Ellicott’s Pendulum.--Mercurial - Pendulum.--Graham’s Pendulum.--Compensation Pendulum of Wood and - Lead.--Smeaton’s Pendulum.--Brown’s Mode of Adjustment. 278 - - - - -THE - -ELEMENTS OF MECHANICS. - - - - -CHAP. I. - -PROPERTIES OF MATTER--MAGNITUDE--IMPENETRABILITY--FIGURE--FORCE. - - -(1.) Placed in the material world, Man is continually exposed to the -action of an infinite variety of objects by which he is surrounded. The -body, to which the thinking and living principles have been united, -is an apparatus exquisitely contrived to receive and to transmit -impressions. Its various parts are organised with obvious reference to -the several external agents by which it is to be effected. Each organ -is designed to convey to the mind immediate notice of some peculiar -action, and is accordingly endued with a corresponding susceptibility. -This adaptation of such organs to the particular influences of -material agents, is rendered still more conspicuous when we consider -that, however delicate its structure, each organ is wholly insensible -to every influence except that to which it appears to be specially -appropriated. The eye, so intensely susceptible of impressions from -light, is not at all affected by those of sound; while the fine -mechanism of the ear, so sensitively alive to every effect of the -latter class, is altogether insensible to the former. The splendour of -excessive light may occasion blindness, and deafness may result from -the roar of a cannonade; but neither the sight nor the hearing can be -injured by the most extreme action of that principle which is designed -to affect the other. - -Thus the organs of sense are instruments by which the mind is enabled -to determine the existence and the qualities of external things. The -effects which these objects produce upon the mind through the organs, -are called _sensations_, and these sensations are the immediate -elements of all human knowledge. MATTER is the general name which -has been given to that substance, which, under forms infinitely -various, affects the senses. Metaphysicians have differed in defining -this principle. Some have even doubted of its existence. But these -discussions are beyond the sphere of mechanical philosophy, the -conclusions of which are in nowise affected by them. Our investigations -here relate, not to matter as an abstract existence, but to those -qualities which we discover in it by the senses, and of the existence -of which we are sure, however the question as to matter itself may be -decided. When we speak of “bodies,” we mean those things, whatever they -be, which excite in our minds certain sensations; and the powers to -excite those sensations are called “properties,” or “qualities.” - -(2.) To ascertain by observation the properties of bodies, is the -first step towards obtaining a knowledge of nature. Hence man becomes -a natural philosopher the moment he begins to feel and to perceive. -The first stage of life is a state of constant and curious excitement. -Observation and attention, ever awake, are engaged upon a succession -of objects new and wonderful. The large repository of the memory is -opened, and every hour pours into it unbounded stores of natural facts -and appearances, the rich materials of future knowledge. The keen -appetite for discovery implanted in the mind for the highest ends, -continually stimulated by the presence of what is novel, renders torpid -every other faculty, and the powers of reflection and comparison are -lost in the incessant activity and unexhausted vigour of observation. -After a season, however, the more ordinary classes of phenomena cease -to excite by their novelty. Attention is drawn from the discovery of -what is new, to the examination of what is familiar. From the external -world the mind turns in upon itself, and the feverish astonishment -of childhood gives place to the more calm contemplation of incipient -maturity. The vast and heterogeneous mass of phenomena collected by -past experience is brought under review. The great work of comparison -begins. Memory produces her stores, and reason arranges them. Then -succeed those first attempts at generalisation which mark the dawn of -science in the mind. - -To compare, to classify, to generalise, seem to be instinctive -propensities peculiar to man. They separate him from inferior animals -by a wide chasm. It is to these powers that all the higher mental -attributes may be traced, and it is from their right application that -all progress in science must arise. Without these powers, the phenomena -of nature would continue a confused heap of crude facts, with which the -memory might be loaded, but from which the intellect would derive no -advantage. Comparison and generalisation are the great digestive organs -of the mind, by which only nutrition can be extracted from this mass of -intellectual food, and without which, observation the most extensive, -and attention the most unremitting, can be productive of no real or -useful advancement in knowledge. - -(3.) Upon reviewing those properties of bodies which the senses -most frequently present to us, we observe that very few of them are -essential to, and inseparable from, matter. The greater number may be -called _particular_ or _peculiar qualities_, being found in some bodies -but not in others. Thus the property of attracting iron is peculiar to -the loadstone, and not observable in other substances. One body excites -the sensation of green, another of red, and a third is deprived of all -colour. A few characteristic and essential qualities are, however, -inseparable from matter in whatever state, or under whatever form it -exist. Such properties alone can be considered as tests of materiality. -Where their presence is neither manifest to sense, nor demonstrable by -reason, _there_ matter is not. The principal of these qualities are -_magnitude_ and _impenetrability_. - -(4.) _Magnitude._--Every body occupies space, that is, it has -magnitude. This is a property observable by the senses in all bodies -which are not so minute as to elude them, and which the understanding -can trace to the smallest particle of matter. It is impossible, by any -stretch of imagination, even to conceive a portion of matter so minute -as to have no magnitude. - -The _quantity_ of space which a body occupies is sometimes called its -_magnitude_. In colloquial phraseology, the word _size_ is used to -express this notion; but the most correct term, and that which we shall -generally adopt is _volume_. Thus we say, the volume of the earth is so -many cubic miles, the volume of this room is so many cubic feet. - -The external limits of the magnitude of a body are _lines_ and -_surfaces_, lines being the limits which separate the several surfaces -of the same body. The linear limits of a body are also called _edges_. -Thus the line which separates the top of a chest from one of its sides -is called an edge. - -The _quantity_ of a surface is called its _area_, and the _quantity_ -of a line is called its _length_. Thus we say, the _area_ of a field -is so many acres, the _length_ of a rope is so many yards. The word -“magnitude” is, however, often used indifferently for volume, area, -and length. If the objects of investigation were of a more complex -and subtle character, as in metaphysics, this unsteady application of -terms might be productive of confusion, and even of error; but in this -science the meaning of the term is evident, from the way in which it is -applied, and no inconvenience is found to arise. - -(5.) _Impenetrability._--This property will be most clearly explained -by defining the positive quality from which it takes its name, and -of which it merely signifies the absence. A substance would be -_penetrable_ if it were such as to allow another to pass through the -space which it occupies, without disturbing its component parts. Thus, -if a comet striking the earth could enter it at one side, and, passing -through it, emerge from the other without separating or deranging any -bodies on or within the earth, then the earth would be penetrable by -the comet. When bodies are said to be impenetrable, it is therefore -meant that one cannot pass through another without displacing some or -all of the component parts of that other. There are many instances of -apparent penetration; but in all these, the parts of the body which -seem to be penetrated are displaced. Thus, if the point of a needle be -plunged in a vessel of water, all the water which previously filled the -space into which the needle enters will be displaced, and the level of -the water will rise in the vessel to the same height as it would by -pouring in so much more water as would fill the space occupied by the -needle. - -(6.) _Figure._--If the hand be placed upon a solid body, we become -sensible of its impenetrability, by the obstruction which it opposes to -the entrance of the hand within its dimensions. We are also sensible -that this obstruction commences at certain places; that it has certain -determinate limits; that these limitations are placed in certain -directions relatively to each other. The mutual relation which is found -to subsist between these boundaries of a body, gives us the notion of -its _figure_. The _figure_ and _volume_ of a body should be carefully -distinguished. Each is entirely independent of the other. Bodies having -very different _volumes_ may have the same _figure_; and in like manner -bodies differing in _figure_ may have the same _volume_. The figure of -a body is what in popular language is called its _shape_ or _form_. The -volume of a body is that which is commonly called its _size_. It will -hence be easily understood, that one body (a globe, for example) may -have ten times the volume of another (globe), and yet have the same -figure; and that two bodies (as a die and a globe) may have _figures_ -altogether different, and yet have equal _volumes_. What we have here -observed of volumes will also be applicable to lengths and areas. The -arc of a circle and a straight line may have the same length, although -they have different figures; and, on the other hand, two arcs of -different circles may have the same figure, but very unequal lengths. -The surface of a ball is curved, that of the table plane; and yet the -_area_ of the surface of the ball may be equal to that of the table. - -(7.) _Atoms--Molecules._--Impenetrability must not be confounded -with inseparability. Every body which has been brought under human -observation is separable into parts; and these parts, however small, -are separable into others, still more minute. To this process of -division no practical limit has ever been found. Nevertheless, many -of the phenomena which the researches of those who have successfully -examined the laws of nature have developed, render it highly probable -that all bodies are composed of elementary parts which are indivisible -and unalterable. The component parts, which may be called _atoms_, are -so minute, as altogether to elude the senses, even when aided by the -most powerful scientific instruments. The word _molecule_ is often used -to signify component parts of a body so small as to escape sensible -observation, but not ultimate atoms, each molecule being supposed to -be formed of several atoms, arranged according to some determinate -figure. _Particle_ is used also to express small component parts, but -more generally is applied to those which are not too minute to be -discoverable by observation. - -(8.) _Force._--If the particles of matter were endued with no property -in relation to one another, except their mutual impenetrability, the -universe would be like a mass of sand, without variety of state or -form. Atoms, when placed in juxtaposition, would neither cohere, as in -solid bodies, nor repel each other, as in aeriform substances. On the -contrary, we find that in some cases the atoms which compose bodies -are not simply placed together, but a certain effect is manifested in -their strong coherence. If they were merely placed in juxtaposition, -their separation would be effected as easily as any one of them could -be removed from one place to another. Take a piece of iron, and -attempt to separate its parts: the effort will be strongly resisted, -and it will be a matter of much greater facility to move the whole -mass. It appears, therefore, that in such cases the parts which are in -juxtaposition _cohere_ and resist their mutual separation. This effect -is denominated _force_; and the constituent atoms are said to cohere -with a greater or less degree of force, according as they oppose a -greater or less resistance to their mutual separation. - -The coherence of particles in juxtaposition is an effect of the same -class as the mutual approach of particles placed at a distance from -each other. It is not difficult to perceive that the same influence -which causes the bodies A and B to approach each other, when placed at -some distance asunder, will, when they unite, retain them together, -and oppose a resistance to their separation. Hence this effect of -the mutual approximation of bodies towards each other is also called -_force_. - -Force is generally defined to be “whatever produces or opposes the -production of motion in matter.” In this sense, it is a name for -the unknown cause of a known effect. It would, however, be more -philosophical to give the name, not to the _cause_, of which we are -ignorant, but to the _effect_, of which we have sensible evidence. -To observe and to classify is the whole business of the natural -philosopher. When _causes_ are referred to, it is implied, that effects -of the same class arise from the agency of the same cause. However -probable this assumption may be, it is altogether unnecessary. All -the objects of science, the enlargement of mind, the extension and -improvement of knowledge, the facility of its acquisition, are obtained -by generalisation alone, and no good can arise from tainting our -conclusions with the possible errors of hypotheses. - -It may be here, once for all, observed, that the phraseology of -causation and hypotheses has become so interwoven with the language -of science, that it is impossible to avoid the frequent use of it. -Thus, we say, “the magnet _attracts_ iron;” the expression _attract_ -intimating the cause of the observed effect. In such cases, however, -we must be understood to mean the _effect itself_, finding it less -inconvenient to continue the use of the received phrases, modifying -their signification, than to introduce new ones. - -Force, when manifested by the mutual approach or cohesion of bodies, is -also called _attraction_, and it is variously denominated, according to -the circumstances under which it is observed to act. Thus, the force -which holds together the atoms of solid bodies is called _cohesive -attraction_. The force which draws bodies to the surface of the earth, -when placed above it, is called the _attraction of gravitation_. The -force which is exhibited by the mutual approach, or adhesion, of the -loadstone and iron, is called _magnetic attraction_, and so on. - -When force is manifested by the motion of bodies from each other, it -is called _repulsion_. Thus, if a piece of glass, having been briskly -rubbed with a silk handkerchief, touch successively two feathers, these -feathers, if brought near each other, will move asunder. This effect is -called _repulsion_, and the feathers are said to _repel_ each other. - -(9.) The influence which forces have upon the form, state, arrangement, -and motions of material substances is the principal object of physical -science. In its strict sense, MECHANICS is a term of very extensive -signification. According to the more popular usage, however, it has -been generally applied to that part of physical science which includes -the investigation of the phenomena of motion and rest, pressure and -other effects developed by the mutual action of solid masses. The -consideration of similar phenomena, exhibited in bodies of the liquid -form, is consigned to HYDROSTATICS, and that of aeriform fluids to -PNEUMATICS. - - - - -CHAP. II. - -DIVISIBILITY--POROSITY--DENSITY--COMPRESSIBILITY--ELASTICITY--DILATABILITY. - - -(10.) Besides the qualities of magnitude and impenetrability, there are -several other general properties of bodies contemplated in mechanical -philosophy, and to which we shall have frequent occasion to refer. -Those which we shall notice in the present chapter are, - - 1. Divisibility. - 2. Porosity--Density. - 3. Compressibility--Elasticity. - 4. Dilatability. - -(11.) _Divisibility._--Observation and experience prove that all bodies -of sensible magnitude, even the most solid, consist of parts which are -separable. To the practical subdivision of matter there seems to be no -assignable limit. Numerous examples of the division of matter, to a -degree almost exceeding belief, may be found in experimental enquiries -instituted in physical science; the useful arts furnish many instances -not less striking; but, perhaps, the most conspicuous proofs which can -be produced, of the extreme minuteness of which the parts of matter -are susceptible, arise from the consideration of certain parts of the -organised world. - -(12.) The relative places of stars in the heavens, as seen in the -field of view of a telescope, are marked by fine lines of wire placed -before the eye-glass, and which cross each other at right angles. The -stars appearing in the telescope as mere lucid points without sensible -magnitude, it is necessary that the wires which mark their places -should have a corresponding tenuity. But these wires being magnified -by the eye-glass would have an apparent thickness, which would render -them inapplicable to this purpose, unless their real dimensions were of -a most uncommon degree of minuteness. To obtain wire for this purpose, -Dr. Wollaston invented the following process:--A piece of fine -platinum wire, _a b_, is extended along the axis of a cylindrical -mould, A B, _fig. 1._ Into this mould, at A, molten silver -is poured. Since the heat necessary for the fusion of platinum is much -greater than that which retains silver in the liquid form, the wire -_a b_ remains solid, while the mould A B is filled with -the silver. When the metal has become solid by being cooled, and has -been removed from the mould, a cylindrical bar of silver is obtained, -having a platinum wire in its axis. This bar is then wire-drawn, by -forcing it successively through holes C, D, E, F, G, H, diminishing -in magnitude, the first hole being a little less than the wire at -the beginning of the process. By these means the platinum _a b_ -is wire-drawn at the same time and in the same proportion with the -silver, so that whatever be the original proportion of the thickness -of the wire _a b_ to that of the mould A B, the same will -be the proportion of the platinum wire to the whole at the several -thicknesses C, D, &c. If we suppose the mould A B to be ten times -the thickness of the wire _a b_, then the silver wire, throughout -the whole process, will be ten times the thickness of the platinum -wire which it includes within it. The silver wire may be drawn to a -thickness not exceeding the 300th of an inch. The platinum will thus -not exceed the 3000th of an inch. The wire is then dipped in nitric -acid, which dissolves the silver, but leaves the platinum solid. By -this method Dr. Wollaston succeeded in obtaining wire, the diameter of -which did not exceed the 18000th of an inch. A quantity of this wire, -equal in bulk to a common die used in games of chance, would extend -from Paris to Rome. - -(13.) Newton succeeded in determining the thickness of very thin laminæ -of transparent substances by observing the colours which they reflect. -A soap bubble is a thin shell of water, and is observed to reflect -different colours from different parts of its surface. Immediately -before the bubble bursts, a black spot may be observed near the top. At -this part the thickness has been proved not to exceed the 2,500,000th -of an inch. - -The transparent wings of certain insects are so attenuated in their -structure that 50,000 of them placed over each other would not form a -pile a quarter of an inch in height. - -(14.) In the manufacture of embroidery it is necessary to obtain very -fine gilt silver threads. To accomplish this, a cylindrical bar of -silver, weighing 360 ounces, is covered with about two ounces of gold. -This gilt bar is then wire-drawn, as in the first example, until it -is reduced to a thread so fine that 3400 feet of it weigh less than -an ounce. The wire is then flattened by passing it between rollers -under a severe pressure, a process which increases its length, so that -about 4000 feet shall weigh one ounce. Hence, one foot will weigh the -4000th part of an ounce. The proportion of the gold to the silver in -the original bar was that of 2 to 360, or 1 to 180. Since the same -proportion is preserved after the bar has been wire-drawn, it follows -that the quantity of gold which covers one foot of the fine wire is the -180th part of the 4000th of an ounce; that is the 720,000th part of an -ounce. - -The quantity of gold which covers one inch of this wire will be twelve -times less than that which covers one foot. Hence, this quantity will -be the 8,640,000th part of an ounce. If this inch be again divided -into 100 equal parts, every part will be distinctly visible without -the aid of microscopes. The gold which covers this small but visible -portion is the 864,000,000th part of an ounce. But we may proceed even -further; this portion of the wire may be viewed by a microscope which -magnifies 500 times, so that the 500th part of it will thus become -visible. In this manner, therefore, an ounce of gold may be divided -into 432,000,000,000 visible parts, each of which will possess all the -characters and qualities found in the largest masses of the metal. -It will retain its solidity, texture, and colour; it will resist the -same agents, and enter into combination with the same substances. If -the gilt wire be dipped in nitric acid, the silver within the coating -will be dissolved, but the hollow tube of gold which surrounded it will -still cohere and remain suspended. - -(15.) The organised world offers still more remarkable examples of the -inconceivable subtilty of matter. - -The blood which flows in the veins of animals is not, as it seems, -an uniformly red liquid. It consists of flat discs of a red colour, -floating in a transparent fluid called _serum_. In different species -these discs differ both in figure and in magnitude. In man and all -animals which suckle their young, they are perfectly circular or nearly -so. In birds, reptiles, and fishes, they are of oval form. In the human -species, the diameter of these discs is about the 3500th of an inch. -Hence it follows, that in a drop of blood which would remain suspended -from the point of a fine needle, there must be about 3,000,000 of such -discs. - -Small as these discs are, the animal kingdom presents beings whose -whole bodies are still more minute. Animalcules have been discovered, -whose magnitude is such, that a million of them do not exceed the -bulk of a grain of sand; and yet each of these creatures is composed -of members as curiously organised as those of the largest species; -they have life and spontaneous motion, and are endued with sense and -instinct. In the liquids in which they live, they are observed to -move with astonishing speed and activity; nor are their motions blind -and fortuitous, but evidently governed by choice, and directed to an -end. They use food and drink, from which they derive nutrition, and -are therefore furnished with a digestive apparatus. They have great -muscular power, and are furnished with limbs and muscles of strength -and flexibility. They are susceptible of the same appetites, and -obnoxious to the same passions, the gratification of which is attended -with the same results as in our own species. Spallanzani observes, that -certain animalcules devour others so voraciously, that they fatten and -become indolent and sluggish by over-feeding. After a meal of this -kind, if they be confined in distilled water, so as to be deprived of -all food, their condition becomes reduced; they regain their spirit -and activity, and amuse themselves in the pursuit of the more minute -animals, which are supplied to them; they swallow these without -depriving them of life, for, by the aid of the microscope, the one -has been observed moving within the body of the other. These singular -appearances are not matters of idle and curious observation. They lead -us to enquire what parts are necessary to produce such results. Must we -not conclude that these creatures have heart, arteries, veins, muscles, -sinews, tendons, nerves, circulating fluids, and all the concomitant -apparatus of a living organised body? And if so, how inconceivably -minute must those parts be! If a globule of their blood bears the same -proportion to their whole bulk as a globule of our blood bears to our -magnitude, what powers of calculation can give an adequate notion of -its minuteness? - -(16.) These and many other phenomena observed in the immediate -productions of nature, or developed by mechanical and chemical -processes, prove that the materials of which bodies are formed are -susceptible of minuteness which infinitely exceeds the powers of -sensible observation, even when those powers have been extended by all -the aids of science. Shall we then conclude that matter is infinitely -divisible, and that there are no original constituent atoms of -determinate magnitude and figure at which all subdivision must cease? -Such an inference would be unwarranted, even had we no other means of -judging the question, except those of direct observation; for it would -be imposing that limit on the works of nature which she has placed -upon our powers of observing them. Aided by reason, however, and a due -consideration of certain phenomena which come within our immediate -powers of observation, we are frequently able to determine other -phenomena which are beyond those powers. The diurnal motion of the -earth is not perceived by us, because all things around us participate -in it, preserve their relative position, and appear to be at rest. But -reason tells us that such a motion must produce the alternations of -day and night, and the rising and setting of all the heavenly bodies; -appearances which are plainly observable, and which betray the cause -from which they arise. Again, we cannot place ourselves at a distance -from the earth, and behold the axis on which it revolves, and observe -its peculiar obliquity to the orbit in which the earth moves; but we -see and feel the vicissitudes of the seasons, an effect which is the -immediate consequence of that inclination, and by which we are able to -detect it. - -(17.) So it is in the present case. Although we are unable by direct -observation to prove the existence of constituent material atoms of -determinate figure, yet there are many observable phenomena which -render their existence in the highest degree probable, if not morally -certain. The most remarkable of this class of effects is observed in -the crystallisation of salts. When salt is dissolved in a sufficient -quantity of pure water, it mixes with the water in such a manner as -wholly to disappear to the sight and touch, the mixture being one -uniform transparent liquid like the water itself, before its union -with the salt. The presence of the salt in the water may, however, be -ascertained by weighing the mixture, which will be found to exceed -the original weight of the water by the exact amount of the weight -of the salt. It is a well-known fact, that a certain degree of heat -will convert water into vapour, and that the same degree of heat does -not produce the same effect upon salt. The mixture of salt and water -being exposed to this temperature, the water will gradually evaporate, -disengaging itself from the salt with which it has been combined. When -so much of the water has evaporated, that what remains is insufficient -to keep in solution the whole of the salt, a part of the latter thus -disengaged from the water will return to the solid state. The saline -constituent will not in this case collect in irregular solid molecules; -but will exhibit itself in particles of regular figure, terminated by -plane surfaces, the figure being always the same for the same species -of salt, but different for different species. These particles are -called _crystals_. There are several circumstances in the formation of -these _crystals_ which merit attention. - -If one of them be detached from the others, and the progress of its -formation observed, it will be found gradually to increase, always -preserving its original figure. Since its increase must be caused -by the continued accession of saline molecules, disengaged by the -evaporation of the water, it follows that these molecules must be so -formed, that by attaching themselves successively to the crystal, they -maintain the regularity of its bounding planes, and preserve their -mutual inclinations unvaried. - -Suppose a crystal to be taken from the liquid during the process of -crystallisation, and a piece broken from it so as to destroy the -regularity of its form: if the crystal thus broken be restored to the -liquid, it will be observed gradually to resume its regular form, the -atoms of salt successively dismissed by the vaporising water filling -up the irregular cavities produced by the fracture. Hence it follows, -that the saline particles which compose the surface of the crystal, -and those which form the interior of its mass, are similar, and exert -similar attractions on the atoms disengaged by the water. - -All these details of the process of crystallisation are very evident -indications of a determinate figure in the ultimate atoms of the -substances which are crystallised. But besides the substances which are -thus reduced by art to the form of crystals, there are larger classes -which naturally exist in that state. There are certain planes, called -_planes of cleavage_, in the directions of which natural crystals are -easily divided. These planes, in substances of the same kind, always -have the same relative position, but differ in different substances. -The surfaces of the planes of cleavage are quite invisible before the -crystal is divided; but when the parts are separated, these surfaces -exhibit a most intense polish, which no effort of art can equal. - -We may conceive crystallised substances to be regular mechanical -structures formed of atoms of a certain figure, on which the figure of -the whole structure must depend. The planes of cleavage are parallel to -the sides of the constituent atoms; and their directions, therefore, -form so many conditions for the determination of their figure. The -shape of the atoms being thus determined, it is not difficult to assign -all the various ways in which they may be arranged, so as to produce -figures which are accordingly found to correspond with the various -forms of crystals of the same substance. - -(18.) When these phenomena are duly considered and compared, little -doubt can remain that all substances susceptible of crystallisation, -consist of atoms of determinate figure. This is the case with all solid -bodies whatever, which have come under scientific observation, for -they have been severally found in or reduced to a crystallised form. -Liquids crystallise in freezing, and if aëriform fluids could by any -means be reduced to the solid form, they would probably also manifest -the same effect. Hence it appears reasonable to presume, that all -bodies are composed of atoms; that the different qualities with which -we find different substances endued, depend on the magnitude and figure -of these atoms; that these atoms are indestructible and immutable by -any natural process, for we find the qualities which depend on them -unchangeably the same under all the influences to which they have been -submitted since their creation; that these atoms are so minute in their -magnitude, that they cannot be observed by any means which human art -has yet contrived; but still that magnitudes can be assigned which they -do not exceed. - -It is proper, however, to observe here, that the various theorems -of mechanical science do not rest upon any hypothesis concerning -these atoms as a basis. These theorems are not inferred from this -or any other supposition, and therefore their truth would not be in -anywise disturbed, even though it should be established that matter is -physically divisible _in infinitum_. The basis of mechanical science -is _observed facts_, and, since the reasoning is demonstrative, the -conclusions have the same degree of certainty as the facts from which -they are deduced. - -(19.) _Porosity._--The _volume_ of a body is the quantity of space -included within its external surfaces. The _mass_ of a body, is the -collection of atoms or material particles of which it consists. Two -atoms or particles are said to be in contact, when they have approached -each other until arrested by their mutual impenetrability. If the -component particles of a body were in contact, the _volume_ would be -completely occupied by the _mass_. But this is not the case. We shall -presently prove, that the component particles of no known substance are -in absolute contact. Hence it follows that the volume consists partly -of material particles, and partly of interstitial spaces, which spaces -are either absolutely void and empty, or filled by some substance of a -different kind from the body in question. These interstitial spaces are -called _pores_. - -In bodies which are constituted uniformly throughout their entire -dimensions, the component particles and the pores are uniformly -distributed through the volume; that is, a given space in one part -of the volume will contain the same quantity of matter and the same -quantity of pores as an equal space in another part. - -(20.) The proportion of the quantity of matter to the magnitude is -called the _density_. Thus if of two substances, one contain in a -given space twice as much matter as the other, it is said to be “twice -as dense.” The density of bodies is, therefore, proportionate to the -closeness or proximity of their particles; and it is evident, that the -greater the density, the less will be the porosity. - -The pores of a body are frequently filled with another body of a more -subtle nature. If the pores of a body on the surface of the earth, and -exposed to the atmosphere, be greater than the atoms of air, then the -air may pervade the pores. This is found to be the case with many sorts -of wood which have an open grain. If a piece of such wood, or of chalk, -or of sugar, be pressed to the bottom of a vessel of water, the air -which fills the pores will be observed to escape in bubbles and to rise -to the surface, the water entering the pores, and taking its place. - -If a tall vessel or tube, having a wooden bottom, be filled with -quicksilver, the liquid metal will be forced by its own weight through -the pores of the wood, and will be seen escaping in a silver shower -from the bottom. - -(21.) The process of filtration, in the arts, depends on the presence -of pores of such a magnitude as to allow a passage to the liquid, but -to refuse it to those impurities from which it is to be disengaged. -Various substances are used as filtres; but, whatever be used, this -circumstance should always be remembered, that no substance can be -separated from a liquid by filtration, except one whose particles -are larger than those of the liquid. In general, filtres are used to -separate _solid_ impurities from a liquid. The most ordinary filtres -are soft stone, paper, and charcoal. - -(22.) All organised substances in the animal and vegetable kingdoms -are, from their very natures, porous in a high degree. Minerals are -porous in various degrees. Among the silicious stones is one called -_hydrophane_, which manifests its porosity in a very remarkable manner. -The stone, in its ordinary state, is semi-transparent. If, however, -it be plunged in water, when it is withdrawn it is as translucent -as glass. The pores, in this case, previously filled with air, are -pervaded by the water, between which and the stone there subsists -a physical relation, by which the one renders the other perfectly -transparent. - -Larger mineral masses exhibit degrees of porosity not less striking. -Water percolates through the sides and roofs of caverns and grottoes, -and being impregnated with calcareous and other earths, forms -stalactites, or pendant protuberances, which present a curious -appearance. - -(23.) _Compressibility._--That quality, in virtue of which a body -allows its volume to be diminished without diminishing its mass, is -called _compressibility_. This effect is produced by bringing the -constituent particles more close together, and thereby increasing the -density and diminishing the pores. This effect may be produced in -several ways; but the name “compressibility” is only applied to it -when it is caused by the agency of mechanical force, as by pressure or -percussion. - -All known bodies, whatever be their nature, are capable of having their -dimensions reduced without diminishing their mass; and this is one of -the most conclusive proofs that all bodies are porous, or that the -constituent atoms are not in contact; for the space by which the volume -may be diminished must, before the diminution, consist of pores. - -(24.) _Elasticity._--Some bodies, when compressed by mechanical agency, -will resume their former dimensions with a certain energy when relieved -from the operation of the force which has compressed them. This -property is called _elasticity_; and it follows, from this definition, -that all elastic bodies must be compressible, although the converse is -not true, compressibility not necessarily implying elasticity. - -(25.) _Dilatability._--This quality is the opposite of compressibility. -It is the capability observed in bodies to have their volume enlarged -without increasing their mass. This effect may be produced in several -ways. In ordinary circumstances, a body may exist under the constant -action of a pressure by which its volume and density are determined. It -may happen, that on the occasional removal of that pressure, the body -will _dilate_ by a quality inherent in its constitution. This is the -case with common air. Dilatation may also be the effect of heat, as -will presently appear. - -The several qualities of bodies which we have noticed in this chapter, -when viewed in relation to each other, present many circumstances -worthy of attention. - -(26.) It is a physical law, of high generality, that an increase -in the temperature, or degree of heat by which a body is affected, -is accompanied by an increase of volume; and that a diminution of -temperature is accompanied by a diminution of volume. The exceptions -to this law will be noticed and explained in our treatise on HEAT. -Hence it appears that the reduction of temperature is an effect which, -considered mechanically, is equivalent to compression or condensation, -since it diminishes the volume without altering the mass; and since -this is an effect of which all bodies whatever are susceptible, it -follows that all bodies whatever have _pores_. (23.) - -The fact, that the elevation of temperature produces an increase of -volume, is manifested by numerous experiments. - -(27.) If a flaccid bladder be tied at the mouth, so as to stop the -escape of air, and be then held before a fire, it will gradually swell, -and assume the appearance of being fully inflated. The small quantity -of air contained in the bladder is, in this case, so much dilated by -the heat, that it occupies a considerably increased space, and fills -the bladder, of which it before only occupied a small part. When the -bladder is removed from the fire, and allowed to resume its former -temperature, the air returns to its former dimensions, and the bladder -becomes again flaccid. - -(28.) Let A B, _fig. 2._ be a glass tube, with a bulb at the -end A; and let the bulb A, and a part of the tube, be filled with any -liquid, coloured so as to be visible. Let C be the level of the liquid -in the tube. If the bulb be now exposed to heat, by being plunged in -hot water, the level of the liquid C will rapidly rise towards B. This -effect is produced by the dilatation of the liquid in the bulb, which -filling a greater space, a part of it is forced into the tube. This -experiment may easily be made with a common glass tube and a little -port wine. - -Thermometers are constructed on this principle, the rise of the liquid -in the tube being used as an indication of the degree of heat which -causes it. A particular account of these useful instruments will be -found in our treatise on HEAT. - -(29.) The change of dimension of solids produced by changes of -temperature being much less than that of bodies in the liquid or -aeriform state, is not so easily observable. A remarkable instance -occurs in the process of shoeing the wheels of carriages. The rim of -iron with which the wheel is to be bound, is made in the first instance -of a diameter somewhat less than that of the wheel; but being raised by -the application of fire to a very high temperature, its volume receives -such an increase, that it will be sufficient to embrace and surround -the wheel. When placed upon the wheel it is cooled, and suddenly -contracting its dimensions, binds the parts of the wheel firmly -together, and becomes securely seated in its place upon the fellies. - -(30.) It frequently happens that the stopper of a glass bottle or -decanter becomes fixed in its place so firmly, that the exertion of -force sufficient to withdraw it would endanger the vessel. In this -case, if a cloth wetted with hot-water be applied to the neck of the -bottle, the glass will expand, and the neck will be enlarged, so as to -allow the stopper to be easily withdrawn. - -(31.) The contraction of metal consequent upon change of temperature -was applied some time ago in Paris to restore the walls of a tottering -building to their proper position. In the _Conservatoire des Arts -et Métiers_, the walls of a part of the building were forced out of -the perpendicular by the weight of the roof, so that each wall was -leaning outwards. M. Molard conceived the notion of applying the -irresistible force with which metals contract in cooling, to draw the -walls together. Bars of iron were placed in parallel directions across -the building, and at right-angles to the direction of the walls. Being -passed through the walls, nuts were screwed on their ends, outside the -building. Every alternate bar was then heated by lamps, and the nuts -screwed close to the walls. The bars were then cooled, and the lengths -being diminished by contraction, the nuts on their extremities were -drawn together, and with them the walls were drawn through an equal -space. The same process was repeated with the intermediate bars, and -so on alternately until the walls were brought into a perpendicular -position. - -(32.) Since there is a continual change of temperature in all bodies on -the surface of the globe, it follows, that there is also a continual -change of magnitude. The substances which surround us are constantly -swelling and contracting, under the vicissitudes of heat and cold. They -grow smaller in winter, and dilate in summer. They swell their bulk -on a warm day, and contract it on a cold one. These curious phenomena -are not noticed, only because our ordinary means of observation are -not sufficiently accurate to appreciate them. Nevertheless, in some -familiar instances the effect is very obvious. In warm weather the -flesh swells, the vessels appear filled, the hand is plump, and the -skin distended. In cold weather, when the body has been exposed to the -open air, the flesh appears to contract, the vessels shrink, and the -skin shrivels. - -(33.) The phenomena attending change of temperature are conclusive -proofs of the universal porosity of material substances, but they are -not the only proofs. Many substances admit of compression by the mere -agency of mechanical force. - -Let a small piece of cork be placed floating on the surface of water -in a basin or other vessel, and an empty glass goblet be inverted over -the cork, so that its edge just meets the water. A portion of air will -then be confined in the goblet, and detached from the remainder of -the atmosphere. If the goblet be now pressed downwards, so as to be -entirely immersed, it will be observed, that the water will not fill -it, being excluded by the _impenetrability_ of the air inclosed in it. -This experiment, therefore, is decisive of the fact, that air, one of -the most subtle and attenuated substances we know of, possesses the -quality of impenetrability. It absolutely excludes any other body from -the space which it occupies at any given moment. - -But although the water does not fill the goblet, yet if the position -of the cork which floats upon its surface be noticed, it will be -found that the level of the water within has risen above its edge or -rim. In fact, the water has partially filled the goblet, and the air -has been forced to contract its dimensions. This effect is produced by -the pressure of the incumbent water forcing the surface in the goblet -against the air, which yields until it is so far compressed that it -acquires a force able to withstand this pressure. Thus it appears -that air is capable of being reduced in its dimensions by mechanical -pressure, independently of the agency of heat. It is _compressible_. - -That this effect is the consequence of the pressure of the liquid will -be easily made manifest by showing that, as the pressure is increased, -the air is proportionally contracted in its dimensions; and as it is -diminished, the dimensions are on the other hand enlarged. If the -depth of the goblet in the water be increased, the cork will be seen -to rise in it, showing that the increased pressure, at the greater -depth, causes the air in the goblet to be more condensed. If, on the -other hand, the goblet be raised toward the surface, the cork will be -observed to descend toward the edge, showing that as it is relieved -from the pressure of the liquid, the air gradually approaches to its -primitive dimensions. - -(34.) These phenomena also prove, that air has the property of -_elasticity_. If it were simply compressible, and not elastic, it would -retain the dimensions to which it was reduced by the pressure of the -liquid; but this is not found to be the result. As the compressing -force is diminished, so in the same proportion does the air, by its -elastic virtue, exert a force by which it resumes its former dimensions. - -That it is the air alone which excludes the water from the goblet, -in the preceding experiments, can easily be proved. When the goblet -is sunk deep in the vessel of water, let it be inclined a little to -one side until its mouth is presented towards the side of the vessel; -let this inclination be so regulated, that the surface of the water -in the goblet shall just reach its edge. Upon a slight increase of -inclination, air will be observed to escape from the goblet, and to -rise in bubbles to the surface of the water. If the goblet be then -restored to its position, it will be found that the cork will rise -higher in it than before the escape of the air. The water in this case -rises and fills the space which the air allowed to escape has deserted. -The same process may be repeated until all the air has escaped, and -then the goblet will be completely filled by the water. - -(35.) Liquids are compressible by mechanical force in so slight a -degree, that they are considered in all hydrostatical treatises -as incompressible fluids. They are, however, not absolutely -incompressible, but yield slightly to very intense pressure. The -question of the compressibility of liquids was raised at a remote -period in the history of science. Nearly two centuries ago, an -experiment was instituted at the Academy _del Cimento_ in Florence, -to ascertain whether water be compressible. With this view, a hollow -ball of gold was filled with the liquid, and the aperture exactly -and firmly closed. The globe was then submitted to a very severe -pressure, by which its figure was slightly changed. Now it is proved -in geometry, that a globe has this peculiar property, that any -change whatever in its figure must necessarily diminish its volume -or contents. Hence it was inferred, that if the water did not issue -through the pores of the gold, or burst the globe, its compressibility -would be established. The result of the experiment was, that the water -_did_ ooze through the pores, and covered the surface of the globe, -presenting the appearance of dew, or of steam cooled by the metal. -But this experiment was inconclusive. It is quite true, that if the -water _had not_ escaped upon the change of figure of the globe, the -_compressibility_ of the liquid would have been established. The escape -of the water does not, however, prove its _incompressibility_. To -accomplish this, it would be necessary first to measure accurately the -volume of water which transuded by compression, and next to measure -the diminution of volume which the vessel suffered by its change of -figure. If this diminution were greater than the volume of water which -escaped, it would follow that the water remaining in the globe had -been compressed, notwithstanding the escape of the remainder. But this -could never be accomplished with the delicacy and exactitude necessary -in such an experiment; and, consequently, as far as the question of -the compressibility of water was concerned, nothing was proved. It -forms, however, a very striking illustration of the porosity of so -dense a substance as gold, and proves that its pores are larger than -the elementary particles of water, since these are capable of passing -through them. - -(36.) It has since been proved, that water, and other liquids, are -compressible. In the year 1761, Canton communicated to the Royal -Society the results of some experiments which proved this fact. He -provided a glass tube with a bulb, such as that described in (28), and -filled the bulb and a part of the tube with water well purified from -air. He then placed this in an apparatus called a condenser, by which -he was enabled to submit the surface of the liquid in the tube to very -intense pressure of condensed air. He found that the level of the -liquid in the tube fell in a perceptible degree upon the application of -the pressure. The same experiment established the fact, that liquids -are _elastic_; for upon removing the pressure, the liquid rose to its -original level, and therefore resumed its former dimensions. - -(37.) Elasticity does not always accompany compressibility. If lead or -iron be submitted to the hammer, it may be hardened and diminished in -its volume; but it will not resume its former volume after each stroke -of the hammer. - -(38.) There are some bodies which maintain the state of density in -which they are commonly found by the continual agency of mechanical -pressure; and such bodies are endued with a quality, in virtue of which -they would enlarge their dimensions without limit, if the pressure -which confines them were removed. Such bodies are called _elastic -fluids_ or _gases_, and always exist in the form of common air, in -whose mechanical properties they participate. They are hence often -called _aeriform fluids_. - -Those who are provided with an air-pump can easily establish this -property experimentally. Take a flaccid bladder, such as that already -described in (27.), and place it under the glass receiver of an -air-pump: by this instrument we shall be able to remove the air which -surrounds the bladder under the receiver, so as to relieve the small -quantity of air which is inclosed in the bladder from the pressure -of the external air: when this is accomplished, the bladder will be -observed to swell, as if it were inflated, and will be perfectly -distended. The air contained in it, therefore, has a tendency to -dilate, which takes effect when it ceases to be resisted by the -pressure of surrounding air. - -(39.) It has been stated that the increase or diminution of temperature -is accompanied by an increase or diminution of volume. Related to this, -there is another phenomenon too remarkable to pass unnoticed, although -this is not the proper place to dwell upon it: it is the converse of -the former; viz. that an increase or diminution of bulk is accompanied -by a diminution or increase of temperature. As the application of heat -from some foreign source produces an increase of dimensions, so if the -dimensions be increased from any other cause, a corresponding portion -of the heat which the body had before the enlargement, will be absorbed -in the process, and the temperature will be thereby diminished. In the -same way, since the abstraction of heat causes a diminution of volume, -so if that diminution be caused by any other means, the body will _give -out_ the heat which in the other case was abstracted, and will rise in -its temperature. - -Numerous and well-known facts illustrate these observations. A smith by -hammering a piece of bar iron, and thereby compressing it, will render -it _red hot_. When air is violently compressed, it becomes so hot as -to ignite cotton and other substances. An ingenious instrument for -producing a light for domestic uses has been constructed, consisting -of a small cylinder, in which a solid piston moves air-tight: a little -tinder, or dry sponge, is attached to the bottom of the piston, which -is then violently forced into the cylinder: the air between the bottom -of the cylinder and the piston becomes intensely compressed, and -evolves so much heat as to light the tinder. - -In all the cases where friction or percussion produces heat or fire, -it is because they are means of compression. The effects of flints, of -pieces of wood rubbed together, the warmth produced by friction on the -flesh, are all to be attributed to the same cause. - - - - -CHAP. III. - -INERTIA. - - -(40.) The quality of matter which is of all others the most important -in mechanical investigations, is that which has been called _Inertia_. - -Matter is incapable of spontaneous change. This is one of the earliest -and most universal results of human observation: it is equivalent to -stating that mere matter is deprived of life; for spontaneous action -is the only test of the presence of the living principle. If we see a -mass of matter undergo any change, we never seek for the cause of that -change in the body itself; we look for some external cause producing -it. This inability for voluntary change of state or qualities is a -more general principle than inertia. At any given moment of time a -body must be in one or other of two states, rest or motion. _Inertia_, -or _inactivity_, signifies the total absence of power to change this -state. A body endued with inertia cannot of itself, and independent of -all external influence, commence to move from a state of rest; neither -can it when moving arrest its progress and become quiescent. - -(41.) The same property by which a body is unable by any power of its -own to pass from a state of rest to one of motion, or _vice versâ_, -also renders it incapable of increasing or diminishing any motion which -it may have received from an external cause. If a body be moving in a -certain direction at the rate of ten miles per hour, it cannot by any -energy of its own change its rate of motion to eleven or nine miles an -hour. This is a direct consequence of that manifestation of inertia -which has just been explained. For the same power which would cause a -body moving at ten miles an hour to increase its rate to eleven miles, -would also cause the same body at rest to commence moving at the rate -of one mile an hour; and the same power which would cause a body moving -at the rate of ten miles an hour to move at the rate of nine miles in -the hour, would cause the same body moving at the rate of one mile -an hour to become quiescent. It therefore appears, that to increase -or diminish the motion of a body is an effect of the same kind as to -change the state of rest into that of motion, or _vice versâ_. - -(42.) The effects and phenomena which hourly fall under our observation -afford unnumbered examples of the inability of lifeless matter to put -itself into motion, or to increase any motion which may have been -communicated to it. But it does not happen that we have the same -direct and frequent evidence of its inability to destroy or diminish -any motion which it may have received. And hence it arises, that while -no one will deny to matter the former effect of inertia, few will at -first acknowledge the latter. Indeed, even so late as the time of -KEPLER, philosophers themselves held it as a maxim, that “matter is -more inclined to rest than to motion;” we ought not, therefore, to be -surprised if in the present day those who have not been conversant with -physical science are slow to believe that a body once put in motion -would continue for ever to move with the same velocity, if it were not -stopped by some external cause. - -Reason, assisted by observation, will, however, soon dispel this -illusion. Experience shows us in various ways, that the same causes -which destroy motion in one direction are capable of producing as -much motion in the opposite direction. Thus, if a wheel, spinning on -its axis with a certain velocity, be stopped by a hand seizing one of -the spokes, the effort which accomplishes this is exactly the same as, -had the wheel been previously at rest, would have put it in motion in -the opposite direction with the same velocity. If a carriage drawn -by horses be in motion, the same exertion of power in the horses is -necessary to stop it, as would be necessary to _back_ it, if it were -at rest. Now, if this be admitted as a general principle, it must be -evident that a body which can destroy or diminish its own motion must -also be capable of putting itself into motion from a state of rest, -or of increasing any motion which it has received. But this latter is -contrary to all experience, and therefore we are compelled to admit -that a body cannot diminish or destroy any motion which it has received. - -Let us enquire why we are more disposed to admit the inability of -matter to produce than to destroy motion in itself. We see most of -those motions which take place around us on the surface of the earth -subject to gradual decay, and if not renewed from time to time, at -length cease. A stone rolled along the ground, a wheel revolving on -its axis, the heaving of the deep after a storm, and all other motions -produced in bodies by external causes, decay, when the exciting -cause is suspended; and if that cause do not renew its action, they -ultimately cease. - -But is there no exciting cause, on the other hand, which thus gradually -deprives those bodies of their motion?--and if that cause were -removed, or its intensity diminished, would not the motion continue, -or be more slowly retarded? When a stone is rolled along the ground, -the inequalities of its shape as well as those of the ground are -impediments, which retard and soon destroy its motion. Render the -stone round, and the ground level, and the motion will be considerably -prolonged. But still small asperities will remain on the stone, and on -the surface over which it rolls: substitute for the stone a ball of -highly-polished steel, moving on a highly-polished steel plane, truly -level, and the motion will continue without sensible diminution for -a very long period; but even here, and in every instance of motions -produced by art, minute asperities must exist on the surfaces which -move in contact with each other, which must resist, gradually diminish, -and ultimately destroy the motion. - -Independently of the obstructions to the continuation of motion arising -from friction, there is another impediment to which all motions on the -surface of the earth are liable--the resistance of the air. How much -this may affect the continuation of motion appears by many familiar -effects. On a calm day carry an open umbrella with its concave side -presented in the direction in which you are moving, and a powerful -resistance will be opposed to your progress, which will increase with -every increase of the speed with which you move. - -(43.) We are not, however, without direct experience to prove, that -motions when unresisted will for ever continue. In the heavens we find -an apparatus, which furnishes a sublime verification of this principle. -There, removed from all casual obstructions and resistances, the vast -bodies of the universe roll on in their appointed paths with unerring -regularity, preserving without diminution all that motion which they -received at their creation from the hand which launched them into -space. This alone, unsupported by other reasons, would be sufficient -to establish the quality of inertia; but viewed in connection with the -other circumstances previously mentioned, no doubt can remain that this -is an universal law of nature. - -(44.) It has been proved, that inability to change the _quantity_ of -motion is a consequence of _inertia_. The inability to change the -_direction_ of motion is another consequence of this quality. The same -cause which increases or diminishes motion, would also give motion -to a body at rest; and therefore we infer that the same inability -which prevents a body from moving itself, will also prevent it from -increasing or diminishing any motion which it has received. In the -same manner we can show, that any cause which changes the direction of -motion would also give motion to a body at rest; and therefore if a -body change the direction of its own motion, the same body might move -itself from a state of rest; and therefore the power of changing the -direction of any motion which it may have received is inconsistent with -the quality of inertia. - -(45.) If a body, moving from A, _fig. 3._ to B, receive at -B a blow in the direction C B E, it will immediately -change its direction to that of another line B D. The cause which -produces this change of direction would have put the body in motion in -the direction B E, had it been quiescent at B when it sustained -the blow. - -(46.) Again, suppose G H to be a hard plane surface; and let the -body be supposed to be perfectly inelastic. When it strikes the surface -at B, it will commence to move along it in the direction B H. This -change of direction is produced by the resistance of the surface. If -the body, instead of meeting the surface in the direction A B, had -moved in the direction E B, perpendicular to it, all motion would -have been destroyed, and the body reduced to a state of rest. - -(47.) By the former example it appears that the deflecting cause would -have put a quiescent body in motion, and by the latter it would have -reduced a moving body to a state of rest. Hence the phenomenon of a -change of direction is to be referred to the same class as the change -from rest to motion, or from motion to rest. The quality of inertia -is, therefore, inconsistent with any change in the direction of motion -which does not arise from an external cause. - -(48.) From all that has been here stated, we may infer generally, that -an inanimate parcel of matter is incapable of changing its state of -rest or motion; that, in whatever state it be, in that state it must -for ever continue, unless disturbed by some external cause; that -if it be in motion, that motion must always be _uniform_, or must -proceed at the same rate, equal spaces being moved over in the same -time: any increase of its rate must betray some impelling cause; any -diminution must proceed from an impeding cause, and neither of these -causes can exist in the body itself; that such motion must not only be -constantly at the same uniform rate, but also must be always in the -same direction, any deflection from one uniform direction necessarily -arising from some external influence. - -The language sometimes used to explain the property of inertia in -popular works, is eminently calculated to mislead the student. The -terms resistance and stubbornness to move are faulty in this respect. -Inertia implies absolute passiveness, a perfect indifference to rest -or motion. It implies as strongly the absence of all resistance to -the reception of motion, as it does the absence of all power to move -itself. The term _vis inertiæ_ or _force of inactivity_, so frequently -used even by authors pretending to scientific accuracy, is still more -reprehensible. It is a contradiction in terms; the term _inactivity_ -implying the absence of all force. - - * * * * * - -(49.) Before we close this chapter, it may be advantageous to point out -some practical and familiar examples of the general law of inertia. -The student must, however, recollect, that the great object of -science is generalisation, and that his mind is to be elevated to the -contemplation of the _laws_ of nature, and to receive a habit the very -reverse of that which disposes us to enjoy the descent from generals to -particulars. Instances, taken from the occurrences of ordinary life, -may, however, be useful in verifying the general law, and in impressing -it upon the memory; and for this reason, we shall occasionally in the -present treatise refer to such examples; always, however, keeping -them in subservience to the general principles of which they are -manifestations, and on which the attention of the student should never -cease to be fixed. - -(50.) If a carriage, a horse, or a boat, moving with speed, be suddenly -retarded or stopped, by any cause which does not at the same time -affect passengers, riders, or any loose bodies which are carried, they -will be precipitated in the direction of the motion; because by reason -of their inertia, they persevere in the motion which they shared in -common with that which transported them, and are not deprived of that -motion by the same cause. - -(51.) If a passenger leap from a carriage in rapid motion, he will fall -in the direction in which the carriage is moving at the moment his feet -meet the ground; because his body, on quitting the vehicle, retains, by -its inertia, the motion which it had in common with it. When he reaches -the ground, this motion is destroyed by the resistance of the ground to -the feet, but is retained in the upper and heavier part of the body; so -that the same effect is produced as if the feet had been tripped. - -(52.) When a carriage is once put in motion with a determinate speed -on a level road, the only force necessary to sustain the motion is -that which is sufficient to overcome the friction of the road; but -at starting a greater expenditure of force is necessary, inasmuch as -not only the friction is to be overcome, but the force with which the -vehicle is intended to move must be communicated to it. Hence we see -that horses make a much greater exertion at starting than subsequently, -when the carriage is in motion; and we may also infer the inexpediency -of attempting to start at full speed, especially with heavy carriages. - -(53.) _Coursing_ owes all its interest to the instinctive consciousness -of the nature of inertia which seems to govern the measures of the -hare. The greyhound is a comparatively heavy body moving at the same -or greater speed in pursuit. The hare _doubles_, that is, suddenly -changes the direction of her course, and turns back at an oblique angle -with the direction in which she had been running. The greyhound, unable -to resist the tendency of its body to persevere in the rapid motion it -had acquired, is urged forward many yards before it is able to check -its speed and return to the pursuit. Meanwhile the hare is gaining -ground in the other direction, so that the animals are at a very -considerable distance asunder when the pursuit is recommenced. In this -way a hare, though much less fleet than a greyhound, will often escape -it. - -In racing, the horses shoot far beyond the winning-post before their -course can be arrested. - - - - -CHAP. IV. - -ACTION AND REACTION. - - -(54.) The effects of inertia or inactivity, considered in the last -chapter, are such as may be manifested by a single insulated body, -without reference to, or connection with, any other body whatever. They -might all be recognised if there were but one body existing in the -universe. There are, however, other important results of this law, to -the development of which two bodies at least are necessary. - -(55.) If a mass A, _fig._ 4., moving towards C, impinge upon an equal -mass, which is quiescent at B, the two masses will move together -towards C after the impact. But it will be observed, that their speed -after the impact will be only half that of A before it. Thus, after the -impact, A loses half its velocity; and B, which was before quiescent, -receives exactly this amount of motion. It appears, therefore, in this -case, that B receives exactly as much motion as A loses: so that the -real quantity of motion from B to C is the same as the quantity of -motion from A to B. - -Now, suppose that B consisted of two masses, each equal to A, it would -be found that in this case the velocity of the triple mass after impact -would be one-third of the velocity from A to B. Thus, after impact, A -loses two-thirds of its velocity and, B consisting of two masses each -equal to A, each of these two receives one-third of A’s motion; so that -the whole motion received by B is two-thirds of the motion of A before -impact. By the impact, therefore, exactly as much motion is received by -B as is lost by A. - -A similar result will be obtained, whatever proportion may subsist -between the masses A and B. Suppose B to be ten times A; then the whole -motion of A must, after the impact, be distributed among the parts of -the united masses of A and B: but these united masses are, in this -case, eleven times the mass of A. Now, as they all move with a common -motion, it follows that A’s former motion must be equally distributed -among them; so that each part shall have an eleventh part of it. -Therefore the velocity after impact will be the eleventh part of the -velocity of A before it. Thus A loses by the impact ten-eleventh parts -of its motion, which are precisely what B receives. - -Again, if the masses of A and B be 5 and 7, then the united mass after -impact will be 12. The motion of A before impact will be equally -distributed between these twelve parts, so that each part will have -a twelfth of it; but five of these parts belong to the mass A, and -seven to B. Hence B will receive seven-twelfths, while A retains -five-twelfths. - -(56.) In general, therefore, when a mass A in motion impinges on a mass -B at rest, to find the motion of the united mass after impact, “divide -the whole motion of A into as many equal parts as there are equal -component masses in A and B together, and then B will receive by the -impact as many parts of this motion as it has equal component masses.” - -This is an immediate consequence of the property of inertia, explained -in the last chapter. If we were to suppose that by their mutual impact -A were to give to B either more or less motion than that which it (A) -loses, it would necessarily follow, that either A or B must have a -power of producing or of resisting motion, which would be inconsistent -with the quality of inertia already defined. For if A give to B _more_ -motion than it loses, all the overplus or excess must be excited in B -by the _action_ of A; and, therefore, A is not inactive, but is capable -of exciting motion which it does not possess. On the other hand, B -cannot receive from A _less_ motion than A loses, because then B must -be admitted to have the power by its resistance of destroying all the -deficiency; a power essentially active, and inconsistent with the -quality of inertia. - -(57.) If we contemplate the effects of impact, which we have now -described, as facts ascertained by experiment (which they may be), -we may take them as further verification of the universality of the -quality of inertia. But, on the other hand, we may view them as -phenomena which may certainly be predicted from the previous knowledge -of that quality; and this is one of many instances of the advantage -which science possesses over knowledge _merely_ practical. Having -obtained by observation or experience a certain number of simple facts, -and thence deduced the general qualities of bodies, we are enabled, -by demonstrative reasoning, to discover _other facts_ which have -never fallen under our observation, or, if so, may have never excited -attention. In this way philosophers have discovered certain small -motions and slight changes which have taken place among the heavenly -bodies, and have directed the attention of astronomical observers to -them, instructing them with the greatest precision as to the exact -moment of time and the point of the firmament to which they should -direct the telescope, in order to witness the predicted event. - -(58.) Since by the quality of inertia a body can neither generate -nor destroy motion, it follows that when two bodies act upon each -other in any way whatever, the total quantity of motion in a given -direction, after the action takes place, must be the same as before -it, for otherwise some motion would be produced by the action of the -bodies, which would contradict the principle that they are inert. The -word “action” is here applied, perhaps improperly, but according to the -usage of mechanical writers, to express a certain phenomenon or effect. -It is, therefore, not to be understood as implying any active principle -in the bodies to which it is attributed. - -(59.) In the cases of collision of which we have spoken, one of the -masses B was supposed to be quiescent before the impact. We shall now -suppose it to be moving in the same direction as A, that is, towards C, -but with a less velocity, so that A shall overtake it, and impinge upon -it. After the impact, the two masses will move towards C with a common -velocity, the amount of which we now propose to determine. - -If the masses A and B be equal, then their motions or velocities added -together must be the motion of the united mass after impact, since no -motion can either be created or destroyed by that event. But as A and B -move with a common motion, this sum must be equally distributed between -them, and therefore each will move with a velocity equal to half the -sum of their velocities before the impact. Thus, if A have the velocity -7, and B have 5, the velocity of the united mass after impact is 6, -being the half of 12, the sum of 7 and 5. - -If A and B be not equal, suppose them divided into equal component -parts, and let A consist of 8, and B of 6, equal masses: let the -velocity of A be 17, so that the motion of each of the 8 parts being -17, the motion of the whole will be 136. In the same manner, let the -velocity of B be 10, the motion of each part being 10, the whole motion -of the 6 parts will be 60. The sum of the two motions, therefore, -towards C is 196; and since none of this can be lost by the impact, -nor any motion added to it, this must also be the whole motion of the -united masses after impact. Being equally distributed among the 14 -component parts of which these united masses consist, each part will -have a fourteenth of the whole motion. Hence, 196 being divided by 14, -we obtain the quotient 14, which is the velocity with which the whole -moves. - -(60.) In general, therefore, when two masses moving in the same -direction impinge one upon the other, and after impact move together, -their common velocity may be determined by the following rule: “Express -the masses and velocities by numbers in the usual way, and multiply -the numbers expressing the masses by the numbers which express the -velocities; the two products thus obtained being added together, and -their sum divided by the sum of the numbers expressing the masses, the -quotient will be the number expressing the required velocity.” - -(61.) From the preceding details, it appears that _motion_ is not -adequately estimated by _speed_ or _velocity_. For example, a certain -mass A, moving at a determinate rate, has a certain quantity of motion. -If another equal mass B be added to A, and a similar velocity be given -to it, as much more motion will evidently be called into existence. In -other words, the _two_ equal masses A and B united have _twice_ as much -motion as the single mass A had when moving alone, and with the same -speed. The same reasoning will show that _three_ equal masses will with -the same speed have _three times_ the motion of any one of them. In -general, therefore, the velocity being the same, the quantity of motion -will always be increased or diminished in the same proportion as the -mass moved is increased or diminished. - -(62.) On the other hand, the quantity of motion does not depend on the -mass _only_, but also on the speed. If a certain determinate mass move -with a certain determinate speed, another equal mass which moves with -twice the speed, that is, which moves over twice the space in the same -time, will have twice the quantity of motion. In this manner, the mass -being the same, the quantity of motion will increase or diminish in the -same proportion as the velocity. - -(63.) The true estimate, then, of the quantity of motion is found -by multiplying together the numbers which express the mass and the -velocity. Thus, in the example which has been last given of the impact -of masses, the quantities of motion before and after impact appear to -be as follow: - - Before Impact. | After Impact. - | - Mass of A 8 | Mass of A 8 - Velocity of A 17 | Common velocity 14 - -----------------+ -------------- - Quantity of } 8 × 17[1] or 136 | Quantity of } 8 × 14 or 112 - motion of A } | motion of A } - -----------------+ -------------- - Mass of B 6 | Mass of B 6 - Velocity of B 10 | Common velocity 14 - -----------------+ -------------- - Quantity of } 6 × 10 or 60 | Quantity of } 6 × 14 = 84 - motion of B } | motion of B } - -----------------+ -------------- - - * The sign × placed between two numbers meant that they are to be - multiplied together. - -By this calculation it appears that in the impact A has lost a quantity -of motion expressed by 24, and that B has received exactly that amount. -The effect, therefore, of the impact is a _transfer_ of motion from A -to B; but no new motion is produced in the direction A C which did -not exist before. This is obviously consistent with the property of -inertia, and indeed an inevitable result of it. - -These results may be generalised and more clearly and concisely -expressed by the aid of the symbols of arithmetic. - -Let _a_ express the velocity of A. - -Let _b_ express the velocity of B. - -Let _x_ express the velocity of the united masses of A and B after -impact, each of these velocities being expressed in feet per second, -and the masses of A and B being expressed by the weight in pounds. - -We shall then have the momenta or moving forces of A and B before -impact, expressed by A × _a_ and B × _b_, and the moving force of the -united mass after impact will be expressed by (A + B) × _x_. - -The moving force of A after impact is A × _x_, and therefore the force -it loses by the collision will be (A × _a_ - A × _x_). The force of B -after impact will be B × _x_, and therefore the force it gains will be -B × _x_ - B × _b_. But since the force lost by A must be equal to the -force gained by B, we shall have - - A × _a_ - A × _x_ = B × _x_ - B × _b_ - -from which it is easy to infer - - (A + B) × _x_ = A × _a_ + B × _b_ - -and if it be required to express the velocity of the united masses -after impact, we have - - _x_ = (A × _a_ + B × _b_)/(A + B) - -When it is said that A × _a_ and B × _b_ express the moving forces of -A and B, it must be understood that the _unit_ of momentum or moving -force is in the case here supposed, the force with which a mass of -matter weighing 1 lb. would move if its velocity were 1 foot per -second, and accordingly the forces with which A and B move before -impact are as many times this as there are units respectively in the -numbers signified by the general symbols A × _a_ and B × _b_. - -In like manner, the force of the united masses after impact is as many -times greater than that of 1 lb. moving through 1 foot per second -as there are units in the numbers expressed by (A + B) × _x_. - -(64.) These phenomena present an example of a law deduced from the -property of inertia, and generally expressed thus--“action and reaction -are equal, and in contrary directions.” The student must, however, be -cautious not to receive these terms in their ordinary acceptation. -After the full explanation of inertia given in the last chapter, it -is, perhaps, scarcely necessary here to repeat, that in the phenomena -manifested by the motion of two bodies, there can be neither “action” -nor “reaction,” properly so called. The bodies are absolutely incapable -either of action or resistance. The sense in which these words must -be received, as used in the _law_, is merely an expression of the -_transfer_ of a certain quantity of motion from one body to another, -which is called an _action_ in the body which loses the motion, and a -_reaction_ in the body which receives it. The _accession_ of motion to -the latter is said to proceed from the _action_ of the former; and the -_loss_ of the same motion in the former is ascribed to the _reaction_ -of the latter. The whole phraseology is, however, most objectionable -and unphilosophical, and is calculated to create wrong notions. - -(65.) The bodies impinging were, in the last case, supposed to move in -the same direction. We shall now consider the case in which they move -in opposite directions. - -First, let the masses A and B be supposed to be equal, and moving in -opposite directions, with the same velocity. Let C, _fig. 5._, be -the point at which they meet. The equal motions in opposite directions -will, in this case, destroy each other, and both masses will be -reduced to a state of rest. Thus, the mass A loses all its motion in -the direction A C, which it may be supposed to transfer to B at -the moment of impact. But B having previously had an equal quantity -of motion in the direction B C, will now have two equal motions -impressed upon it, in directions immediately opposite; and these -motions neutralising each other, the mass becomes quiescent. In this -case, therefore, as in all the former examples, each body transfers -to the other all the motion which it loses, consistently with the -principle of “action and reaction.” - -The masses A and B being still supposed equal, let them move towards -C with different velocities. Let A move with the velocity 10, and B -with the velocity 6. Of the 10 parts of motion with which A is endued, -6 being transferred to B, will destroy the equal velocity 6, which B -has in the direction B C. The bodies will then move together in -the direction C B, the four remaining parts of A’s motion being -equally distributed between them. Each body will, therefore, have two -parts of A’s original motion, and 2 therefore will be their common -velocity after impact. In this case, A loses 8 of the 10 parts of its -motion in the direction A C. On the other hand, B loses the entire -of its 6 parts of motion in the direction B C, and receives 2 -parts in the direction A C. This is equivalent to receiving 8 -parts of A’s motion in the direction A C. Thus, according to the -law of “action and reaction,” B receives exactly what A loses. - -Finally, suppose that both the masses and velocities of A and B are -unequal. Let the mass of A be 8, and its velocity 9: and let the mass -of B be 6, and its velocity 5. The quantity of motion of A will be 72, -and that of B, in the opposite direction, will be 30. Of the 72 parts -of motion, which A has in the direction A C, 30 being transferred -to B, will destroy all its 30 parts of motion in the direction -B C, and the two masses will move in the direction C B, with -the remaining 42 parts of motion, which will be equally distributed -among their 14 component masses. Each component part will, therefore, -receive 3 parts of motion; and accordingly 3 will be the common -velocity of the united mass after impact. - -(66.) When two masses moving in opposite directions impinge and move -together, their common velocity after impact may be found by the -following rule:--“Multiply the numbers expressing the masses by those -which express the velocities respectively, and subtract the lesser -product from the greater; divide the remainder by the sum of the -numbers expressing the masses, and the quotient will be the common -velocity; the direction will be that of the mass which has the greater -quantity of motion.” - -It may be shown without difficulty, that the example which we have -just given obeys the law of “action and reaction.” - - Before impact. | After impact. - | - Mass of A 8 | Mass of A 8 - Velocity of A 9 | Common velocity 3 - ------------+ ----------- - Quantity of motion } 8 × 9 or 72 | Quantity of motion } 8 × 3 or 24 - in direction A C } | in direction A C } - ------------+ ----------- - Mass of B 6 | Mass of B 6 - Velocity of B 5 | Common velocity 3 - ------------+ ----------- - Quantity of motion } 6 × 5 or 30 | Quantity of motion } 6 × 3 or 18 - in direction B C } | in direction A C } - ------------+ ----------- - -Hence it appears that the quantity of motion in the direction A C -of which A has been deprived by the impact is 48, the difference -between 72 and 24. On the other hand, B loses by the impact the -quantity 30 in the direction B C, which is equivalent to receiving -30 in the direction A C. But it also acquires a quantity 18 in -the direction A C, which, added to the former 30, gives a total -of 48 received by B in the direction A C. Thus the same quantity -of motion which A loses in the direction A C, is received by B in -the same direction. The law of “action and reaction” is, therefore, -fulfilled. - -This result may in like manner be generalised. Retaining the former -symbols, the moving forces of A and B before impact will be A × _a_ and -B × _b_ and their forces after impact will be A × _x_ and B × _x_. The -force lost by A will therefore be A × _a_ - A × _x_. The mass B will -have lost all the force B × _b_ which it had in its former direction, -and will have received the force B × _x_ in the opposite direction. -Therefore the actual force imparted to B by the collision will be B -× _b_ + B × _x_. But since the force lost by A must be equal to that -imparted to B, we shall have - - A × _a_ - A × _x_ = B × _b_ + B × _x_ - -and therefore - - (A + B) × _x_ = A × _a_ - B × _b_ - -and if the common velocity after impact be required, we have - - _x_ = (A × _a_ - B × _b_)/(A + B) - -As a general rule, therefore, to find the common velocity after impact. -Multiply the weights by the previous velocities and take their sum if -the bodies move in the same direction, and their difference if they -move in opposite directions, and divide the one or the other by the sum -of their weights. The greatest will be the velocity after impact. - -(67.) The examples of the equality of action and reaction in the -collision of bodies may be exhibited experimentally by a very simple -apparatus. Let A, _fig. 6._, and B be two balls of soft clay, or -any other substance which is inelastic, or nearly so, and let these -be suspended from C by equal strings, so that they may be in contact; -and let a graduated arc, of which the centre is C, be placed so that -the balls may oscillate over it. One of the balls being moved from its -place of rest along the arc, and allowed to descend upon the other -through a certain number of degrees, will strike the other with a -velocity corresponding to that number of degrees, and both balls will -then move together with a velocity which may be estimated by the number -of degrees of the arc through which they rise. - -(68.) In all these cases in which we have explained the law of “action -and reaction,” the transfer of motion from one body to the other has -been made by impact or collision. The phenomenon has been selected only -because it is the most ordinary way in which bodies are seen to affect -each other. The law is, however, universal, and will be fulfilled -in whatever manner the bodies may affect each other. Thus A may be -connected with B by a flexible string, which, at the commencement of -A’s motion, is slack. Until the string becomes stretched, that is, -until A’s distance from B becomes equal to the length of the string, -A will continue to have all the motion first impressed upon it. But -when the string is stretched, a part of that motion is transferred to -B, which is then drawn after A; and whatever motion B in this way -receives, A must lose. All that has been observed of the effect of -motion transferred by impact will be equally applicable in this case. - -Again, if B, _fig. 4._, be a magnet moving in the direction -B C with a certain quantity of motion, and while it is so moving a -mass of iron be placed at rest at A, the attraction of the magnet will -draw the iron after it towards C, and will thus communicate to the iron -a certain quantity of motion in the direction of C. All the motion thus -communicated to the iron A must be lost by the magnet B. - -If the magnet and the iron were both placed quiescent at B and A, the -attraction of the magnet would cause the iron to move from A towards B; -but the magnet in this case not having any motion, cannot be literally -said to _transfer_ a motion to the iron. At the moment, however, when -the iron begins to move from A towards B, the magnet will be observed -to begin also to move from B towards A; and if the velocities of the -two bodies be expressed by numbers, and respectively multiplied by the -numbers expressing their masses, the quantities of motion thus obtained -will be found to be exactly equal. We have already explained why a -quantity of motion received in the direction B A, is equivalent -to the same quantity lost in the direction A B. Hence it appears, -that the magnet in receiving as much motion in the direction B A, -as it gives in the direction A B, suffers an effect which is -equivalent to losing as much motion directed towards C as it has -communicated to the iron in the same direction. - -In the same manner, if the body B had any property in virtue of which -it might _repel_ A, it would itself be repelled with the same quantity -of motion. In a word, whatever be the manner in which the bodies may -affect each other, whether by collision, traction, attraction, or -repulsion, or by whatever other name the phenomenon may be designated, -still it is an inevitable consequence, that any motion, in a given -direction, which one of the bodies may receive, must be accompanied by -a loss of motion in the same direction, and to the same amount, by -the other body, or the acquisition of as much motion in the contrary -direction; or, finally, by a loss in the same direction, and an -acquisition of motion in the contrary direction, the combined amount of -which is equal to the motion received by the former. - -(69.) From the principle, that the force of a body in motion depends on -the mass and the velocity, it follows, that any body, however small, -may be made to move with the same force as any other body, however -great, by giving to the smaller body a velocity which bears to that of -the greater the same proportion as the mass of the greater bears to the -mass of the smaller. Thus a feather, ten thousand of which would have -the same weight as a cannon-ball, would move with the same force if -it had ten thousand times the velocity; and in such a case, these two -bodies encountering in opposite directions, would mutually destroy each -other’s motion. - -(70.) The consequences of the property of inertia, which have been -explained in the present and preceding chapters, have been given by -Newton, in his PRINCIPIA, and, after him, in most English treatises on -mechanics, under the form of three propositions, which are called the -“laws of motion.” They are as follow:-- - - -I. - -“Every body must persevere in its state of rest, or of uniform motion -in a straight line, unless it be compelled to change that state by -forces impressed upon it.” - - -II. - -“Every change of motion must be proportional to the impressed force, -and must be in the direction of that straight line in which the force -is impressed.” - - -III. - -“Action must always be equal and contrary to reaction; or the actions -of two bodies upon each other must be equal, and directed towards -contrary sides.” - -When _inertia_ and _force_ are defined, the first law becomes an -identical proposition. The second law cannot be rendered perfectly -intelligible until the student has read the chapter on the composition -and resolution of forces, for, in fact, it is intended as an expression -of the whole body of results in that chapter. The third law has -been explained in the present chapter, as far as it can be rendered -intelligible in the present stage of our progress. - -We have noticed these formularies more from a respect for the -authorities by which they have been proposed and adopted, than from any -persuasion of their utility. Their full import cannot be comprehended -until nearly the whole of elementary mechanics has been acquired, and -then all such summaries become useless. - - * * * * * - -(71.) The consequences deduced from the consideration of the quality -of inertia in this chapter, will account for many effects which fall -under our notice daily, and with which we have become so familiar, that -they have almost ceased to excite curiosity. One of the facts of which -we have most frequent practical illustration is, that the quantity of -motion or _moving force_, as it is sometimes called, is estimated by -the velocity of the motion, and the weight or mass of the thing moved -conjointly. - -If the same force impel two balls, one of one pound weight, and the -other of two pounds, it follows, since the balls can neither give force -to themselves, nor resist that which is impressed upon them, that they -will move with the same force. But the lighter ball will move with -twice the speed of the heavier. The impressed force which is manifested -by giving velocity to a double mass in the one, is engaged in giving a -double velocity to the other. - -If a cannon-ball were forty times the weight of a musket-ball, but the -musket-ball moved with forty times the velocity of the cannon-ball, -both would strike any obstacle with the same force, and would overcome -the same resistance; for the one would acquire from its velocity as -much force as the other derives from its weight. - -A very small velocity may be accompanied by enormous force, if the mass -which is moved with that velocity be proportionally great. A large -ship, floating near the pier wall, may approach it with so small a -velocity as to be scarcely perceptible, and yet the force will be so -great as to crush a small boat. - -A grain of shot flung from the hand, and striking the person, will -occasion no pain, and indeed will scarcely be felt, while a block of -stone having the same velocity would occasion death. - -If a body in motion strike a body at rest, the striking body must -sustain as great a shock from the collision as if it had been at rest, -and struck by the other body with the same force. For the loss of force -which it sustains in the one direction, is an effect of the same kind -as if, being at rest, it had received as much force in the opposite -direction. If a man, walking rapidly or running, encounters another -standing still, he suffers as much from the collision as the man -against whom he strikes. - -If a leaden bullet be discharged against a plank of hard wood, it will -be found that the round shape of the ball is destroyed, and that it -has itself suffered a force by the impact, which is equivalent to the -effect which it produces upon the plank. - -When two bodies moving in opposite directions meet, each body sustains -as great a shock as if, being at rest, it had been struck by the other -body with the united forces of the two. Thus, if two equal balls, -moving at the rate of ten feet in a second, meet, each will be struck -with the same force as if, being at rest, the other had moved against -it at the rate of twenty feet in a second. In this case one part of the -shock sustained arises from the loss of force in one direction, and -another from the reception of force in the opposite direction. - -For this reason, two persons walking in opposite directions receive -from their encounter a more violent shock than might be expected. If -they be of nearly equal weight, and one be walking at the rate of three -and the other four miles an hour, each sustains the same shock as if he -had been at rest, and struck by the other running at the rate of seven -miles an hour. - -This principle accounts for the destructive effects arising from ships -running foul of each other at sea. If two ships of 500 tons burden -encounter each other, sailing at ten knots an hour, each sustains the -shock which, being at rest, it would receive from a vessel of 1000 tons -burden sailing ten knots an hour. - -It is a mistake to suppose, that when a large and small body encounter, -the small body suffers a greater shock than the large one. The shock -which they sustain must be the same; but the large body may be better -able to bear it. - -When the fist of a pugilist strikes the body of his antagonist, it -sustains as great a shock as it gives; but the fist being more fitted -to endure the blow, the injury and pain are inflicted on his opponent. -This is not the case, however, when fist meets fist. Then the parts -in collision are equally sensitive and vulnerable, and the effect is -aggravated by both having approached each other with great force. The -effect of the blow is the same as if one fist, being held at rest, were -struck by the other with the combined force of both. - - - - -CHAP. V. - -THE COMPOSITION AND RESOLUTION OF FORCE. - - -(72.) Motion and pressure are terms too familiar to need explanation. -It may be observed, generally, that definitions in the first rudiments -of a science are seldom, if ever, comprehended. The force of words is -learned by their application; and it is not until a definition becomes -useless, that we are taught the meaning of the terms in which it is -expressed. Moreover, we are perhaps justified in saying, that in the -mathematical sciences the fundamental notions are of so uncompounded a -character, that definitions, when developed and enlarged upon, often -draw us into metaphysical subtleties and distinctions, which, whatever -be their merit or importance, would be here altogether misplaced. We -shall, therefore, at once take it for granted, that the words _motion_ -and _pressure_ express phenomena or effects which are the subjects -of constant experience and hourly observation; and if the scientific -use of these words be more precise than their general and popular -application, that precision will soon be learned by their frequent use -in the present treatise. - -(73.) FORCE is the name given in mechanics to whatever produces motion -or pressure. This word is also often used to express the motion or -pressure itself; and when the cause of the motion or pressure is not -known, this is the only correct use of the word. Thus, when a piece of -iron moves toward a magnet, it is usual to say that the cause of the -motion is “the attraction of the magnet;” but in effect we are ignorant -of the _cause_ of this phenomenon; and the name _attraction_ would -be better applied to the effect of which we have experience. In like -manner the _attraction_ and _repulsion_ of electrified bodies should be -understood, not as names for unknown causes, but as words expressing -observed appearances or effects. - -When a certain phraseology has, however, gotten into general use, it -is neither easy nor convenient to supersede it. We shall, therefore, -be compelled, in speaking of motion and pressure, to use the language -of causation; but must advise the student that it is effects and not -causes which will be expressed. - -(74.) If two forces act upon the same point of a body in different -directions, a single force may be assigned, which, acting on that -point, will produce the same result as the united effects of the other -two. - -Let P, _fig. 7._, be the point on which the two forces act, and -let their directions be P A and P B. From the point P, upon -the line P A, take a length P _a_, consisting of as many inches -as there are ounces in the force P A; and, in like manner, take P -_b_, in the direction P B, consisting of as many inches as there -are ounces in the force P B. Through _a_ draw a line parallel to -P B, and through _b_ draw a line parallel to P A, and suppose -that these lines meet at _c_. Then draw P C. A single force, -acting in the direction P C, and consisting of as many ounces as -the line P c consists of inches, will produce upon the point P -the same effect as the two forces P A and P B produce acting -together. - -(75.) The figure P _a c b_ is called in GEOMETRY a -_parallelogram_; the lines P _a_, P _b_, are called its _sides_, and -the line P _c_ is called its _diagonal_. Thus the method of finding an -equivalent for two forces, which we have just explained, is generally -called “the parallelogram of forces,” and is usually expressed thus: -“If two forces be represented in quantity and direction by the sides of -a parallelogram, an equivalent force will be represented in quantity -and direction by its diagonal.” - -(76.) A single force, which is thus mechanically equivalent to two or -more other forces, is called their _resultant_, and relatively to it -they are called its _components_. In any mechanical investigation, -when the resultant is used for the components, which it always may -be, the process is called “the composition of force.” It is, however, -frequently expedient to substitute for a single force two or more -forces, to which it is mechanically equivalent, or of which it is the -resultant. This process is called “the resolution of force.” - -(77.) To verify experimentally the theorem of the parallelogram -of forces is not difficult. Let two small wheels, M N, -_fig. 8._, with grooves in their edges to receive a thread, be -attached to an upright board, or to a wall. Let a thread be passed over -them, having weights A and B, hooked upon loops at its extremities. -From any part P of the thread between the wheels let a weight C be -suspended: it will draw the thread downwards, so as to form an angle -M P N, and the apparatus will settle itself at rest in some -determinate position. In this state it is evident that since the weight -C, acting in the direction P C, balances the weights A and B, -acting in the directions P M and P N, these two forces must -be mechanically equivalent to a force equal to the weight C, and acting -directly upwards from P. The weight C is therefore the quantity of the -resultant of the forces P M and P N; and the direction of the -resultant is that of a line drawn directly upwards from P. - -To ascertain how far this is consistent with the theorem of “the -parallelogram of forces,” let a line P O be drawn upon the upright -board to which the wheels are attached, from the point P upward, in the -direction of the thread C P. Also, let lines be drawn upon the -board immediately under the threads P M and P N. From the -point P, on the line P O, take as many inches as there are ounces -in the weight C. Let the part of P O thus measured be P _c_, and -from _c_ draw _c a_ parallel to P N, and _c b_ parallel -to P M. If the sides P _a_ and P _b_ of the parallelogram thus -formed be measured, it will be found that P _a_ will consist of as many -inches as there are ounces in the weight A, and P _b_ of as many inches -as there are ounces in the weight B. - -In this illustration, _ounces_ and _inches_ have been used as the -subdivisions of _weight_ and _length_. It is scarcely necessary to -state, that any other measures of these quantities would serve as well, -only observing that the same denominations must be preserved in all -parts of the same investigation. - -(78.) Among the philosophical apparatus of the University of London, -is a very simple and convenient instrument which I constructed for -the experimental illustration of this important theorem. The wheels -M N are attached to the tops of two tall stands, the heights -of which may be varied at pleasure by an adjusting screw. A jointed -parallelogram, A B C D, _fig. 9._, is formed, whose -sides are divided into inches, and the joints at A and B are moveable, -so as to vary the lengths of the sides at pleasure. The joint C is -fixed at the extremity of a ruler, also divided into inches, while -the opposite joint A is attached to a brass loop, which surrounds the -diagonal ruler loosely, so as to slide freely along it. An adjusting -screw is provided in this loop so as to clamp it in any required -position. - -In making the experiment, the sides A B and A D, C B -and C D are adjusted by the joints B and A to the same number -of inches respectively as there are ounces in the weights A and B, -_fig. 8._ Then the diagonal A C is adjusted by the loop and -screw at A, to as many inches as there are ounces in the weight C. -This done, the point A is placed behind P, _fig. 8._, and the -parallelogram is held upright, so that the diagonal A C shall be -in the direction of the vertical thread P C. The sides A B -and A D will then be found to take the direction of the threads -P M and P N. By changing the weights and the lengths of the -diagonal and sides of the parallelogram, the experiment may be easily -varied at pleasure. - -(79.) In the examples of the composition of forces which we have here -given, the effects of the forces are the production of pressures, or, -to speak more correctly, the theorem which we have illustrated, is “the -composition of pressures.” For the point P is supposed to be at rest, -and to be drawn or pressed in the directions P M and P N. -In the definition which has been given of the word force, it is -declared to include motions as well as pressures. In fact, if motion be -resisted, the effect is converted into pressure. The same cause acting -upon a body, will either produce motion or pressure, according as the -body is free or restrained. If the body be free, motion ensues; if -restrained, pressure, or both these effects together. It is therefore -consistent with analogy to expect that the same theorems which regulate -pressures, will also be applicable to motions; and we find accordingly -a most exact correspondence. - -(80.) If a body have a motion in the direction A B, and at the -point P it receive another motion, such as would carry it in the -direction P C, _fig. 10._, were it previously quiescent at -P, it is required to determine the direction which the body will take, -and the speed with which it will move, under these circumstances. - -Let the velocity with which the body is moving from A to B be such, -that it would move through a certain space, suppose P N, in one -second of time, and let the velocity of the motion impressed upon it -at P be such, that if it had no previous motion it would move from P -to M in one second. From the point M draw a line parallel to P B, -and from N draw a line parallel to P C, and suppose these lines to -meet at some point, as O. Then draw the line P O. In consequence -of the two motions, which are at the same time impressed upon the body -at P, it will move in the straight line from P to O. - -Thus the two motions, which are expressed in quantity and direction -by the sides of a parallelogram, will, when given to the same body, -produce a single motion, expressed in quantity and direction by its -diagonal; a theorem which is to motions exactly what the former was to -pressures. - -There are various methods of illustrating experimentally the -composition of motion. An ivory ball, being placed upon a perfectly -level square table, at one of the corners, and receiving two equal -impulses, in the directions of the sides of the table, will move along -the diagonal. Apparatus for this experiment differ from each other only -in the way of communicating the impulses to the ball. - -(81.) As two motions simultaneously communicated to a body are -equivalent to a single motion in an intermediate direction, so -also a single motion may be mechanically replaced, by two motions -in directions expressed by the sides of any parallelogram, whose -diagonal represents the single motion. This process is “the resolution -of motion,” and gives considerable clearness and facility to many -mechanical investigations. - -(82.) It is frequently necessary to express the portion of a given -force, which acts in some given direction different from the -immediate direction of the force itself. Thus, if a force act from -A, _fig. 11._, in the direction A C, we may require to -estimate what part of that force acts in the direction A B. If the -force be a pressure, take as many inches A P from A, on the line -A C, as there are ounces in the force, and from P draw P M -perpendicular to A B; then the part of the force which acts along -A B will be as many ounces as there are inches in A M. The -force A B is mechanically equivalent to two forces, expressed by -the sides A M and A N of the parallelogram; but A N, -being perpendicular to A B, can have no effect on a body at A, -in the direction of A B, and therefore the effective part of the -force A P in the direction A B is expressed by A M. - -(83.) Any number of forces acting on the same point of a body may -be replaced by a single force, which is mechanically equivalent to -them, and which is, therefore, their resultant. This composition may -be effected by the successive application of the parallelogram of -forces. Let the several forces be called A, B, C, D, E, &c. Draw the -parallelogram whose sides express the forces A and B, and let its -diagonal be A′. The force expressed by A′ will be equivalent to A and -B. Then draw the parallelogram whose sides express the forces A′ and -C, and let its diagonal be B′. This diagonal will express a force -mechanically equivalent to A′ and C. But A′ is mechanically equivalent -to A and B, and therefore B′ is mechanically equivalent to A, B, and -C. Next construct a parallelogram, whose sides express the forces B′ -and D, and let its diagonal be C′. The force expressed by C′ will be -mechanically equivalent to the forces B′ and D; but the force B′ is -equivalent to A, B, C, and therefore C′ is equivalent to A, B, C, and -D. By continuing this process it is evident, that a single force may be -found, which will be equivalent to, and may be always substituted for, -any number of forces which act upon the same point. - -If the forces which act upon the point neutralise each other, so that -no motion can ensue, they are said to be in equilibrium. - -(84.) Examples of the composition of motion and pressure are -continually presenting themselves. They occur in almost every instance -of motion or force which falls under our observation. The difficulty is -to find an example which, strictly speaking, is a simple motion. - -When a boat is rowed across a river, in which there is a current, it -will not move in the direction in which it is impelled by the oars. -Neither will it take the direction of the stream, but will proceed -exactly in that intermediate direction which is determined by the -composition of force. - -Let A, _fig. 12._, be the place of the boat at starting; and -suppose that the oars are so worked as to impel the boat towards B -with a force which would carry it to B in one hour, if there were no -current in the river. But, on the other hand, suppose the rapidity of -the current is such, that without any exertion of the rowers the boat -would float down the stream in one hour to C. From C draw C D -parallel to A B, and draw the straight line A D diagonally. -The combined effect of the oars and the current will be, that the boat -will be carried along A D, and will arrive at the opposite bank in -one hour, at the point D. - -If the object be, therefore, to reach the point B, starting from A, -the rowers must calculate, as nearly as possible, the velocity of the -current. They must imagine a certain point E at such a distance above -B that the boat would be floated by the stream from E to B in the time -taken in crossing the river in the direction A E, if there were no -current. If they row towards the point E, the boat will arrive at the -point B, moving in the line A B. - -In this case the boat is impelled by two forces, that of the oars -in the direction A E, and that of the current in the direction -A C. The result will be, according to the parallelogram of forces, -a motion in the diagonal A B. - -The wind and tide acting upon a vessel is a case of a similar kind. -Suppose that the wind is made to impel the vessel in the direction of -the keel; while the tide may be acting in any direction oblique to that -of the keel. The course of the vessel is determined exactly in the same -manner as that of the boat in the last example. - -The action of the oars themselves, in impelling the boat, is an example -of the composition of force. Let A, _fig. 13._, be the head, -and B the stern of the boat. The boatman presents his face towards -B, and places the oars so that their blades press against the water -in the directions C E, D F. The resistance of the water -produces forces on the side of the boat, in the directions G L and -H L, which, by the composition of force, are equivalent to die -diagonal force K L, in the direction of the keel. - -Similar observations will apply to almost every body impelled by -instruments projecting from its sides, and acting against a fluid. The -motions of fishes, the act of swimming, the flight of birds, are all -instances of the same kind. - -(85.) The action of wind upon the sails of a vessel, and the force -thereby transmitted to the keel, modified by the rudder, is a problem -which is solved by the principles of the composition and resolution -of force; but it is of too complicated and difficult a nature to be -introduced with all its necessary conditions and limitations in this -place. The question may, however, be simplified, if we consider the -canvass of the sails to be stretched so completely as to form a plane -surface. Let A B, _fig. 14._, be the position of the sail, -and let the wind blow in the direction C D. If the line C D -be taken to express the force of the wind, let D E C F -be a parallelogram, of which it is the diagonal. The force C D is -equivalent to two forces, one in the direction F D of the plane -of the canvass, and the other E D perpendicular to the sail. -The effect, therefore, is the same as if there were _two winds_, one -blowing in the direction of F D or B A, that is against the -edge of the sail, and the other, E D, blowing full against its -face. It is evident that the former will produce no effect whatever -upon the sail, and that the latter will urge the vessel in the -direction D G. - -Let us now consider this force D G as acting in the diagonal of -the parallelogram D H G I. It will be equivalent to two -forces, D H and D I, acting along the sides. One of these -forces, D H, is in the direction of the keel, and the other, -D I, at right angles to the length of the vessel, so as to urge -it _sideways_. The form of the vessel is evidently such as to offer a -great resistance to the latter force, and very little to the former. -It consequently proceeds with considerable velocity in the direction -D H of its keel, and makes way very slowly in the sideward -direction D I. The latter effect is called _lee-way_. - -From this explanation it will be easily understood, how a wind which -is nearly opposed to the course of a vessel may, nevertheless, be made -to impel it by the effect of sails. The angle B D V, formed -by the sail and the direction of the keel, may be very oblique, as may -also be the angle C D B formed by the direction of the wind -and that of the sail. Therefore the angle C D V, made up of -these two, and which is that formed by the direction of the wind and -that of the keel, may be very oblique. In _fig. 15._ the wind -is nearly contrary to the direction of the keel, and yet there is an -impelling force expressed by the line D H, the line C D -expressing, as before, the whole force of the wind. - -In this example there are two successive decompositions of force. -First, the original force of the wind C D is resolved into two, -E D and F D; and next the element E D, or its equal -D G, is resolved into D I and D H; so that the original -force is resolved into three, viz. F D, D I, D H, -which, taken together, are mechanically equivalent to it. The part -F D is entirely ineffectual; it glides off on the surface of the -canvass without producing any effect upon the vessel. The part D I -produces _lee-way_, and the part D H impels. - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -(86.) If the wind, however, be directly contrary to the course which -it is required that the vessel should take, there is no position which -can be given to the sails which will impel the vessel. In this case -the required course itself is resolved into two, in which the vessel -sails alternately, a process which is called _tacking_. Thus, suppose -the vessel is required to move from A to E, _fig. 16._, the wind -setting from E to A. The motion A B being resolved into two, by -being assumed as the diagonal of a parallelogram, the sides A _a_, _a_ -B of the parallelogram are successively sailed over, and the vessel by -this means arrives at B, instead of moving along the diagonal A B. -In the same manner she moves along B _b_, _b_ C, C _c_, _c_ D, D _d_, -_d_ E, and arrives at E. She thus sails continually at a sufficient -angle with the wind to obtain an impelling force, yet at a sufficiently -small angle to make way in her proposed course. - -The consideration of the effect of the rudder, which we have omitted in -the preceding illustration, affords another instance of the resolution -of force. We shall not, however, pursue this example further. - -(87.) A body falling from the top of the mast when the vessel is in -full sail, is an example of the composition of motion. It might be -expected, that during the descent of the body, the vessel having sailed -forward, would leave it behind, and that, therefore, it would fall -in the water behind the stern, or at least on the deck, considerably -behind the mast. On the other hand, it is found to fall at the foot -of the mast, exactly as it would if the vessel were not in motion. To -account for this, let A B, _fig. 17._, be the position of -the mast when the body at the top is disengaged. The mast is moving -onwards with the vessel in the direction A C, so that in the time -which the body would take to fall to the deck, the top of the mast -would move from A to C. But the body being on the mast at the moment it -is disengaged, has this motion A C in common with the mast; and -therefore in its descent it is affected by two motions, viz. that of -the vessel expressed by A C, and its descending motion expressed -by A B. Hence, by the composition of motion, it will be found -at the opposite angle D of the parallelogram, at the end of the fall. -During the fall, however, the mast has moved with the vessel, and has -advanced to C D, so that the body falls at the foot of the mast. - -(88.) An instance of the composition of motion, which is worthy of -some attention, as it affords a proof of the diurnal motion of the -earth, is derived from observing the descent of a body from a very high -tower. To render the explanation of this more simple, we shall suppose -the tower to be on the equator of the earth. Let E P Q, -_fig. 18._, be a section of the earth through the equator, and -let P T be the tower. Let us suppose that the earth moves on its -axis in the direction E P Q. The foot P of the tower will, -therefore, in one day move over the circle E P Q, while the -top T moves over the greater circle T T′ R. Hence it is -evident, that the top of the tower moves with greater speed than the -foot, and therefore in the same time moves through a greater space. Now -suppose a body placed at the top; it participates in the motion which -the top of the tower has in common with the earth. If it be disengaged, -it also receives the descending motion T P. Let us suppose that -the body would take five seconds to fall from T to P, and that in the -same time the top T is moved by the rotation of the earth from T to -T′, the foot being moved from P to P′. The falling body is therefore -endued with two motions, one expressed by T T′, and the other by -T P. The combined effect of these will be found in the usual way -by the parallelogram. Take T _p_ equal to T T′; the body will -move from T to _p_ in the time of the fall, and will meet the ground -at _p_. But since T T′ is greater than P P′, it follows -that the point _p_ must be at a distance from P′ equal to the excess -of T T′ above P P′. Hence the body will not fall exactly -at the foot of the tower, but at a certain distance from it, in the -direction of the earth’s motion, that is, eastward. This is found, by -experiment, to be actually the case; and the distance from the foot of -the tower, at which the body is observed to fall, agrees with that -which is computed from the motion of the earth, to as great a degree of -exactness as could be expected from the nature of the experiment. - -(89.) The properties of compounded motions cause some of the equestrian -feats exhibited at public spectacles to be performed by a kind of -exertion very different from that which the spectators generally -attribute to the performer. For example, the horseman standing on the -saddle leaps over a garter extended over the horse at right angles -to his motion; the horse passing under the garter, the rider lights -upon the saddle at the opposite side. The exertion of the performer, -in this case, is not that which he would use were he to leap from the -ground over a garter at the same height. In the latter case, he would -make an exertion to rise, and, at the same time, to project his body -forward. In the case, however, of the horseman, he merely makes that -exertion which is necessary to rise directly upwards to a sufficient -height to clear the garter. The motion which he has in common with the -horse, compounded with the elevation acquired by his muscular power, -accomplishes the leap. - -To explain this more fully, let A B C, _fig. 19._, be -the direction in which the horse moves, A being the point at which -the rider quits the saddle, and C the point at which he returns to -it. Let D be the highest point which is to be cleared in the leap. At -A the rider makes a leap towards the point E, and this must be done -at such a distance from B, that he would rise from B to E in the time -in which the horse moves from A to B. On departing from A, the rider -has, therefore, two motions, represented by the lines A E and -A B, by which he will move from the point A to the opposite angle -D of the parallelogram. At D, the exertion of the leap being overcome -by the weight of his body, he begins to return downward, and would fall -from D to B in the time in which the horse moves from B to C. But at -D he still retains the motion which he had in common with the horse; -and therefore, in leaving the point D, he has two motions, expressed -by the lines D F and D B. The compounded effects of these -motions carry him from D to C. Strictly speaking, his motion from A to -D, and from D to C, is not in straight lines, but in a curve. It is not -necessary here, however, to attend to this circumstance. - -(90.) If a billiard-ball strike the cushion of the table obliquely, -it will be reflected from it in a certain direction, forming an angle -with the direction in which it struck it. This affords an example of -the resolution and composition of motion. We shall first consider the -effect which would ensue if the ball struck the cushion perpendicularly. - -Let A B, _fig. 20._, be the cushion, and C D the -direction in which the ball moves towards it. If the ball and the -cushion were perfectly inelastic, the resistance of the cushion would -destroy the motion of the ball, and it would be reduced to a state of -rest at D. If, on the other hand, the ball were perfectly elastic, it -would be reflected from the cushion, and would receive as much motion -from D to C after the impact, as it had from C to D before it. Perfect -elasticity, however, is a quality which is never found in these bodies. -They are always elastic, but imperfectly so. Consequently the ball -after the impact will be reflected from D towards C, but with a less -motion than that with which it approached from C to D. - -Now let us suppose that the ball, instead of moving from C to D, moves -from E to D. The force with which it strikes D being expressed by -D E′, equal to E D, may be resolved into two, D F and -D C′. The resistance of the cushion destroys D C′, and the -elasticity produces a contrary force in the direction D C, but -less than D C or D C′, because that elasticity is imperfect. -The line D C expressing the force in the direction C D, let -D G (less than D C) express the reflective force in the -direction D C. The other element D F, into which the force -D E′ is resolved by the impact, is not destroyed or modified by -the cushion, and therefore, on leaving the cushion at D, the ball is -influenced by two forces, D F (which is equal to C E) and -D G. Consequently it will move in the diagonal D H. - -(91.) The angle E D C is in this case called the “angle of -incidence,” and C D H is called “the angle of reflection.” -It is evident, from what has been just inferred, that the ball, being -imperfectly elastic, the angle of incidence must always be less than -the angle of reflection, and with the same obliquity of incidence, -the more imperfect the elasticity is, the less will be the angle of -reflection. - -In the impact of a perfectly elastic body, the angle of reflection -would be equal to the angle of incidence. For then the line D G, -expressing the reflective force, would be taken equal to C D, -and the angle C D H would be equal to C D E. This -is found by experiment to be the case when light is reflected from a -polished surface of glass or metal. - -Motion is sometimes distinguished into _absolute_ and _relative_. What -“relative motion” means is easily explained. If a man walk upon the -deck of a ship from stem to stern, he has a relative motion which is -measured by the space upon the deck over which he walks in a given -time. But while he is thus walking from stem to stern, the ship and -its contents, including himself, are impelled through the deep in -the opposite direction. If it so happen that the motion of the man, -from stem to stern, be exactly equal to the motion of the ship in the -contrary way, the man will be, relatively to the surface of the sea -and that of the earth, at rest. Thus, relatively to the ship, he is in -motion, while, relatively to the surface of the earth, he is at rest. -But still this is not absolute rest. The surface itself is moving by -the diurnal rotation of the earth upon its axis, as well as by the -annual motion in its orbit round the sun. These motions, and others to -which the earth is subject, must be all compounded by the theorem of -the parallelogram of forces before we can obtain the _absolute state_ -of the body with respect to motion or rest. - - - - -CHAP. VI. - -ATTRACTION. - - -(92.) Whatever produces, or tends to produce, a change in the state -of a particle or mass of matter with respect to motion or rest, is a -force. Rest, or uniform rectilinear motion, are therefore the only -states in which any body can exist which is not subject to the present -action of some force. We are not, however, entitled to conclude, that -because a body is observed in one or other of these states, it is -therefore uninfluenced by any forces. It may be under the immediate -action of forces which neutralise each other: thus two forces may be -acting upon it which are equal, and in opposite directions. In such -a case, its state of rest, or of uniform rectilinear motion, will be -undisturbed. The state of uniform rectilinear motion declares more with -respect to the body than the state of rest; for the former betrays the -action of a force upon the body at some antecedent period; this action -having been suspended, while its effect continues to be observed in the -motion which it has produced. - -(93.) When the state of a body is changed from rest to uniform -rectilinear motion, the action of the force is only momentary, in which -case it is called an _impulse_. If a body in uniform rectilinear motion -receive an impulse in the direction in which it is moving, the effect -will be, that it will continue to move uniformly in the same direction, -but its velocity will be increased by the amount of speed which the -impulse would have given it had it been previously quiescent. Thus, if -the previous motion be at the rate of ten feet in a second, and the -impulse be such as would move it from a state of rest at five feet in -a second, the velocity, after the impulse, will be fifteen feet in a -second. - -But if the impulse be received in a direction immediately opposed to -the previous motion, then it will diminish the speed by that amount of -velocity which it would give to the body had it been previously at -rest. In the example already given, if the impulse were opposed to the -previous motion, the velocity of the body after the impulse would be -five feet in a second. If the impulse received in the direction opposed -to the motion be such as would give to the body at rest a velocity -equal to that with which it is moving, then the effect will be, that -after the impulse no motion will exist; and if the impulse would give -it a still greater velocity, the body will be moved in the opposite -direction with an uniform velocity equal to the excess of that due to -the impulse over that which the body previously had. - -When a body in a state of uniform motion receives an impulse in -a direction not coinciding with that of its motion, it will move -uniformly after the impulse in an intermediate direction, which may be -determined by the principles established for the composition of motion -in the last chapter. - -Thus it appears, that whenever the state of a body is changed either -from rest to uniform rectilinear motion or _vice versa_, or from one -state of uniform rectilinear motion to another, differing from that -either in velocity or direction, or in both, the phenomenon is produced -by that peculiar modification of force whose action continues but for a -single instant, and which has been called _an impulse_. - -(94.) In most cases, however, the mechanical state of a body is -observed to be subject to a continual change or tendency to change. We -are surrounded by innumerable examples of this. A body is placed on the -table. A continual pressure is excited on the surface of the table. -This pressure is only the consequence of the continual tendency of the -body to move downwards. If the body were excited by a force of the -nature of an impulse, the effect upon the table would be instantaneous, -and would immediately cease. It would, in fact, be _a blow_. But the -continuation of the pressure proves the continuation of the action of -the force. - -If the table be removed from beneath the body, the force which excites -it being no longer resisted, will produce motion; it is manifested, -not as before, by a tendency to produce motion, but by the actual -exhibition of that phenomenon. Now if the exciting force were an -impulse, the body would descend to the ground with an uniform velocity. -On the other hand, as will hereafter appear, every moment of its fall -increases its speed, and that speed is greatest at the instant it meets -the ground. - -A piece of iron placed at a distance from a magnet approaches it, but -not with an uniform velocity. The force of the magnet continues to act -during the approach of the iron, and each moment gives it increased -motion. - -(95.) The forces which are thus in constant operation, proceed from -secret agencies which the human mind has never been able to detect. All -the analogies of nature prove that they are not the immediate results -of the divine will, but are secondary causes, that is, effects of -some more remote principles. To ascend to these secondary causes, and -thus as it were approach one step nearer to the Creator, is the great -business of philosophy; and the most certain means for accomplishing -this, is diligently to observe, to compare, and to classify the -phenomena, and to avoid assuming the existence of any thing which -has not either been directly observed, or which cannot be inferred -demonstratively from natural phenomena. Philosophy should follow -nature, and not lead her. - -While the law of inertia, established by observation and reason, -declares the inability of matter, from any principle resident in it, to -change its state, all the phenomena of the universe prove that state -to be in constant but regular fluctuation. There is not in existence -a single instance of the phenomenon of absolute rest, or of motion -which is absolutely uniform and rectilinear. In bodies, or the parts -of bodies, there is no known instance of simple passive juxtaposition -unaccompanied by pressure or tension, or some other “tendency to -motion.” Innumerable secret powers are ever at work, compensating, -as it were, for inertia, and supplying the material world with a -substitute for the principles of action and will, which give such -immeasurable superiority to the character of life. - -(96.) The forces which are thus in continual operation, whose existence -is demonstrated by their observed effects, but whose nature, seat, and -mode of operation are unknown to us, are called by the general name -_attractions_. These forces are classified according to the analogies -which prevail among their effects, in the same manner, and according -to the same principles, as organised beings are grouped in natural -history. In that department of natural science, when individuals are -distributed in classes, the object is merely to generalise, and thereby -promote the enlargement of knowledge; but nothing is or ought to be -thus assumed respecting the essence, or real internal constitution of -the individuals. According to their external and observable characters -and qualities they are classed; and this classification should never be -adduced as an evidence of any thing except that similitude of qualities -to which it owed its origin. - -Phenomena are to the natural philosopher what organised beings are to -the naturalist. He groups and classifies them on the same principles, -and with a like object. And as the naturalist gives to each species a -name applicable to the individual beings which exhibit corresponding -qualities, so the philosopher gives to each force or attraction a name -corresponding to the phenomena of which it is the cause. The naturalist -is ignorant of the real essence or internal constitution of the thing -which he nominates, and of the manner in which it comes to possess or -exhibit those qualities which form the basis of his classification; -and the natural philosopher is equally ignorant of the nature, seat, -and mode of operation of the force which he assigns as the cause of an -observed class of effects. - -These observations respecting the true import of the term “attraction” -seem the more necessary to be premised, because the general phraseology -of physical science, taken as language is commonly received, will seem -to convey something more. The names of the several attractions which -we shall have to notice, frequently refer the seat of the cause to -specific objects, and seem to imply something respecting its mode of -operation. Thus, when we say “the magnet attracts a piece of iron,” the -true philosophical import of the words is, “that a piece of iron placed -in the vicinity of the magnet, will move towards it, or placed in -contact, will adhere to it, so that some force is necessary to separate -them.” In the ordinary sense, however, something more than this simple -fact is implied. It is insinuated that the magnet is the seat of the -force which gives motion to the iron; that in the production of the -phenomenon, the magnet is an _agent_ exerting a certain influence, of -which the iron is the _subject_. Of all this, however, there is no -proof; on the contrary, since the magnet must move towards the iron -with just as much force as the iron moves towards the magnet, there is -as much reason to place the seat of the force in the iron, and consider -it as an agent affecting the magnet. But, in fact, the influence -which produces this phenomenon may not be resident in either the one -body or the other. It may be imagined to be a property of a medium in -which both are placed, or to arise from some third body, the presence -of which is not immediately observed. However attractive these and -like speculations may be, they cannot be allowed a place in physical -investigations, nor should consequences drawn from such hypotheses be -allowed to taint our conclusions with their uncertainty. - -The student ought, therefore, to be aware, that whatever may seem -to be implied by the language used in this science in relation to -attractions, nothing is permitted to form the basis of reasoning -respecting them except _their effects_; and whatever be the common -signification of the terms used, it is to these effects, and to these -alone, they should be referred. - -(97.) Attractions may be primarily distributed into two classes; one -consisting of those which exist between the molecules or constituent -parts of bodies, and the other between bodies themselves. The former -are sometimes called, for distinction, _molecular_ or _atomic_ -attractions. - -Without the agency of molecular forces, the whole face of nature would -be deprived of variety and beauty; the universe would be a confused -heap of material atoms dispersed through space, without form, shape, -coherence, or motion. Bodies would neither have the forms of solid, -liquid, or air; heat and light would no longer produce their wonted -effects; organised beings could not exist; life itself, as connected -with body, would be extinct. Atoms of matter, whether distant or in -juxtaposition, would have no tendency to change their places, and all -would be eternal stillness and rest. If, then, we are asked for a proof -of the existence of molecular forces, we may point to the earth and -to the heavens; we may name every object which can be seen or felt. -The whole material world is one great result of the influence of these -powerful agents. - -(98.) It has been proved (11. _et seq._) that the constituent particles -of bodies are of inconceivable minuteness, and that they are not in -immediate contact (23), but separated from each other by interstitial -spaces, which, like the atoms themselves, although too small to be -directly observed, yet are incontestably proved to exist, by observable -phenomena, from which their existence demonstratively follows. The -resistance which every body opposes to compression, proves that a -repulsive influence prevails between the particles, and that this -repulsion is the cause which keeps the atoms separate, and maintains -the interstitial spaces just mentioned. Although this repulsion is -found to exist between the molecules of all substances whatever, yet -it has different degrees of energy in different bodies. This is proved -by the fact, that some substances admit of easy compression, while in -others, the exertion of considerable force is necessary to produce the -smallest diminution in bulk. - -The space around each atom of a body, through which this repulsive -influence extends, is generally limited, and immediately beyond it, a -force of the opposite kind is manifested, viz. attraction. Thus, in -solid bodies, the particles resist separation as well as compression, -and the application of force is as necessary to break the body, or -divide it into separate parts, as to force its particles into closer -aggregation. It is by virtue of this attraction that solid bodies -maintain their figure, and that their parts are not separated and -scattered like those of fluids, merely by their own weight. This force -is called the _attraction of cohesion_. - -The cohesive force acts in different substances with different degrees -of energy: in some its intensity is very great; but the sphere of its -influence apparently very limited. This is the case with all bodies -which are hard, strong, and brittle, which no force can extend or -stretch in any perceptible degree, and which require a great force to -break or tear them asunder. Such, for example, is cast iron, certain -stones, and various other substances. In some bodies the cohesive force -is weak, but the sphere of its action considerable. Bodies which are -easily extended, without being broken or torn asunder, furnish examples -of this. Such are Indian-rubber, or caoutchouc, several animal and -vegetable products, and, in general, all solids of a soft and viscid -kind. - -Between these extremes, the cohesive force may be observed in various -degrees. In lead and other soft metals, its sphere of action is -greater, and its energy less, than in the former examples; but its -sphere less, and energy greater, than in the latter ones. It is from -the influence of this force, and that of the repulsion, whose sphere of -action is still closer to the component atoms, that all the varieties -of texture which we denominate hard, soft, tough, brittle, ductile, -pliant, &c. arise. - -After having been broken, or otherwise separated, the parts of a solid -may be again united by their cohesion, provided any considerable -number of points be brought into sufficiently close contact. When this -is done by mechanical means, however, the cohesion is not so strong -as before their separation, and a comparatively small force will be -sufficient again to disunite them. Two pieces of lead freshly cut, with -smooth surfaces, will adhere when pressed together, and will require a -considerable force to separate them. In the same manner if a piece of -Indian-rubber be torn, the parts separated will again cohere, by being -brought together with a slight pressure. The union of the parts in -such instances is easy, because the sphere through which the influence -of cohesion extends is considerable; but even in bodies in which -this influence extends through a more limited space, the cohesion of -separate pieces will be manifested, provided their surfaces be highly -polished, so as to insure the near approach of a great number of their -particles. Thus, two polished surfaces of glass, metal, or stone, will -adhere when brought into contact. - -In all these cases, if the bodies be disunited by mechanical force, -they will separate at exactly the parts at which they had been united, -so that after their separation no part of the one will adhere to the -other; proving that the force of cohesion of the surfaces brought into -contact is less than that which naturally held the particles of each -together. - -(99.) When a body is in the liquid form, the weight of its particles -greatly predominates over their mutual cohesion, and consequently if -such a body be unconfined it will be scattered by its own weight; if -it be placed in any vessel, it will settle itself, by the force of its -weight, into the lowest parts, so that no space in the vessel below -the upper surface of the liquid will be unoccupied. The particles of -a solid body placed in the vessel have exactly the same tendency, by -reason of their weight; but this tendency is resisted and prevented -from taking effect by their strong cohesion. - -Although this cohesion in solids is much greater than in liquids, and -productive of more obvious effects, yet the principle is not altogether -unobserved in liquids. Water converted into vapour by heat, is divided -into inconceivably minute particles, which ascend in the atmosphere. -When it is there deprived of a part of that heat which gave it the -vaporous form, the particles, in virtue of their cohesive force, -collect into round drops, in which form they descend to the earth. - -In the same manner, if a liquid be allowed to fall gradually from the -lip of a vessel, it will not be dismissed in particles indefinitely -small, as if its mass were incoherent, like sand or powder, but will -fall in drops of considerable magnitude. In proportion as the cohesive -force is greater, these drops affect a greater size. Thus, oil and -viscid liquids fall in large drops; ether, alcohol, and others in small -ones. - -Two drops of rain trickling down a window pane will coalesce when they -approach each other; and the same phenomenon is still more remarkable, -if a few drops of quicksilver be scattered on an horizontal plate of -glass. - -It is the cohesive principle which gives rotundity to grains of shot: -the liquid metal is allowed to fall like rain from a great elevation. -In its descent the drops become truly globular, and before they reach -the end of their fall they are hardened by cooling, so that they retain -their shape. - -It is also, probably, to the cohesive attraction that we should assign -the globular forms of all the great bodies of the universe; the sun, -planets, satellites, &c., which originally may have been in the liquid -state. - -(100.) Molecular attraction is also exhibited between the particles of -liquids and solids. A drop of water will not descend freely when it -is in contact with a perpendicular glass plane: it will adhere to the -glass; its descent will be retarded; and if its weight be insufficient -to overcome the adhesive force, it will remain suspended. - -If a plate of glass be placed upon the surface of water without being -permitted to sink, it will require more force to raise it from the -water than is sufficient merely to balance the weight of the glass. -This shows the adhesion of the water and glass, and also the cohesive -force with which the particles of the water resist separation. - -If a needle be dipped in certain liquids, a drop will remain suspended -at its point when withdrawn from them: and, in general, when a solid -body has been immersed in a liquid and withdrawn, it is _wet_; that -is, some of the liquid has adhered to its surfaces. If no attraction -existed between the solid and liquid, the solid would be in the same -state after immersion as before. This is proved by liquids and solids -between which no attraction exists. If a piece of glass be immersed in -mercury, it will be in the same state when withdrawn as before it was -immersed. No mercury will adhere to it; it will not be _wet_. - -When it rains, the person and vesture are affected only because this -attraction exists between them and water. If it rained mercury, none -would adhere to them. - -(101.) When molecular attraction is exhibited by liquids pervading the -interstices of porous bodies, ascending in crevices or in the bores of -small tubes, it is called _capillary attraction_. Instances of this -are innumerable. Liquids are thus drawn into the pores of sponge, -sugar, lamp-wick, &c. The animal and vegetable kingdom furnish numerous -examples of this class of effects. - -A weight being suspended by a dry rope, will be drawn upwards through -a considerable height, if the rope be moistened with a wet sponge. The -attraction of the particles composing the rope for the water is in this -case so powerful, that the tension produced by several hundred weight -cannot expel them. - -A glass tube, of small bore, being dipped in water tinged by mixture -with a little ink, will retain a quantity of the liquid suspended when -withdrawn. The height of the liquid in the tube will be seen by looking -through it. It is found that the less the bore of the tube is, the -greater will be the height of the column sustained. A series of such -tubes fixed in the same frame, with their lower orifices at the same -level, and with bores gradually decreasing, being dipped in the liquid, -will exhibit columns gradually increasing. - -A _capillary syphon_ is formed of a hank of cotton threads, one end of -which is immersed in the vessel containing the liquid, and the other is -carried into the vessel into which the liquid is to be transferred. The -liquid may be thus drawn from the one vessel into the other. The same -effect may be produced by a glass syphon with a small bore. - -(102.) It frequently happens that a _molecular repulsion_ is exhibited -between a solid and a liquid. If a piece of wood be immersed in -quicksilver, the liquid will be depressed at that part of the surface -which is near the wood; and in like manner, if it be contained in a -glass vessel, it will be depressed at the edges. In a barometer tube, -the surface of the mercury is convex, owing partly to the repulsion -between the glass and mercury. - -All solids, however, do not repel mercury. If any golden trinket be -dipped in that liquid, or even be exposed for a moment to contact -with it, the gold will be instantly intermingled with particles of -quicksilver, the metal changes its colour, and becomes white like -silver, and the mercury can only be extricated by a difficult process. -Chains, seals, rings, &c. should always be laid aside by those engaged -in experiments or other processes in which mercury is used. - -(103.) Of all the forms under which molecular force is exhibited, -that in which it takes the name of _affinity_ is attended with the -most conspicuous effects. Affinity is in chemistry what inertia is -in mechanics, the basis of the science. The present treatise is not -the proper place for any detailed account of this important class -of natural phenomena. Those who seek such knowledge are referred -to our treatise on CHEMISTRY. Since, however, affinity sometimes -influences the mechanical state of bodies, and affects their mechanical -properties, it will be necessary here to state so much respecting it as -to render intelligible those references which we may have occasion to -make to such effects. - -When the particles of different bodies are brought into close contact, -and more especially when, being in a fluid state, they are mixed -together, their union is frequently observed to produce a compound -body, differing in its qualities from either of the component bodies. -Thus the bulk of the compound is often greater or less than the united -volumes of the component bodies. The component bodies may be of the -ordinary temperature of the atmosphere, and yet the compound may be of -a much higher or lower temperature. The components may be liquid, and -the compound solid. The colour of the compound may bear no resemblance -whatever to that of the components. The species of molecular action -between the components, which produce these and similar, effects, is -called _affinity_. - -(104.) We shall limit ourselves here to the statement of a few examples -of these phenomena. - -If a pint of water and a pint of sulphuric acid be mixed, the compound -will be considerably less than a quart. The density of the mixture is, -therefore, greater than that which would result from the mere diffusion -of the particles of the one fluid through those of the other. The -particles have assumed a greater proximity, and therefore exhibit a -mutual attraction. - -In this experiment, although the liquids before being mixed be of the -temperature of the surrounding air, the mixture will be so intensely -hot, that the vessel which contains it cannot be touched without pain. - -If the two aeriform fluids, called oxygen and hydrogen, be mixed -together in a certain proportion, the compound will be water. In this -case, the components are different from the compound, not merely in -the one being _air_ and the other _liquid_, but in other respects -not less striking. The compound water extinguishes fire, and yet of -the components, hydrogen is one of the most inflammable substances -in nature, and the presence of oxygen is indispensably necessary to -sustain the phenomenon of combustion. - -Oxygen gas, united with quicksilver, produces a compound of a black -colour, the quicksilver being white and the gas colourless. When -these substances are combined in another proportion, they give a red -compound. - -(105.) Having noticed the principal molecular forces, we shall now -proceed to the consideration of those attractions which are exhibited -between bodies existing in masses. The influence of molecular -attractions is limited to insensible distances. On the contrary, the -forces which are now to be noticed act at considerable distances, -and to the influence of some there is no limit, the effect, however, -decreasing as the distance increases. - -The effect of the loadstone on iron is well known, and is one of this -class of forces. For a detailed account of this force, and the various -phenomena of which it is the cause, the reader is referred to our -treatise on MAGNETISM. - -When glass, wax, amber, and other substances are submitted to friction -with silken or woollen cloth, they are observed to attract feathers, -and other light bodies placed near them. A like effect is produced -in several other ways, and is attended with other phenomena, the -discussion of which forms a principal part of physical science. The -force thus exhibited is called electricity. For details respecting it, -and for its connection with magnetism, the reader is referred to our -treatises on ELECTRICITY and ELECTRO-MAGNETISM. - -(106.) These attractions exist either between bodies of particular -kinds, or are developed by reducing the bodies which manifest them to -a certain state by friction, or some other means. There is, however, -an attraction, which is manifested between bodies of all species, and -under all circumstances whatever; an attraction, the intensity of which -is wholly independent of the nature of the bodies, and only depends on -their masses and mutual distances. Thus, if a mass of metal and a mass -of clay be placed in the vast abyss of space, at a mile asunder, they -will instantly commence to approach each other with certain velocities. -Again, if a mass of stone and of wood respectively equal to the former, -be placed at a like distance, they will also commence to approach -each other with the same velocities as the former. This universal -attraction, which only depends on the quantity of the masses and their -mutual distances, is called the “attraction of gravitation.” We shall -first explain the “law” of this attraction, and shall then point out -some of the principal phenomena by which its existence and its laws are -known. - -(107.) The “law of gravitation” sometimes from its universality called -the “law of nature,” may be explained as follows: - -Let us suppose two masses, A and B, placed beyond the influence or -attraction of any other bodies, in a state of rest, and at any proposed -distance from each other. By their mutual attraction they will approach -each other, but not with the same velocity. The velocity of A will be -greater than that of B, in the same proportion as its mass is less -than that of B. Thus, if the mass of B be twice that of A, while A -approaches B through a space of two feet, B will approach A through a -space of one foot. Hence it follows, that the force with which A moves -towards B is equal to the force with which B moves towards A (68). This -is only a consequence of the property of inertia, and is an example of -the equality of action and reaction, as explained in Chapter IV. The -velocity with which A and B approach each other is estimated by the -diminution of their distance, A B, by their mutual approach in a -given time. Thus, if in one second A move towards B through a space of -two feet, and in the same time B moves towards A through the space of -one foot, they will approach each other through a space of three feet -in a second, which will be their relative velocity (91). - -If the mass of B be doubled, it will attract A with double the former -force, or, what is the same, will cause A to approach B with double the -former velocity. If the mass of B be trebled, it will attract A with -treble the first force, and, in general, while the distance A B -remains the same, the attractive force of B upon A will increase or -diminish in exactly the same proportion as the mass of B is increased -or diminished. - -In the same manner, if the mass A be doubled, it will be attracted by -B with a double force, because B exerts the same degree of attraction -on every part of the mass A, and any addition which it may receive will -not diminish or otherwise affect the influence of B on its former mass. - -To express this in general arithmetical symbols let _a_ and _b_ express -the space through which A and B respectively would be moved towards -each other by their mutual attraction. We would then have - - A × _a_ = B × _b_. - -Thus, it is a general law of gravitation, that so long as the distance -between two bodies remains the same, each will attract and be attracted -by the other, in proportion to its mass; and any increase or decrease -of the mass will cause a corresponding increase or decrease in the -amount of the attraction. - -(108.) We shall now explain the law, according to which the attraction -is changed, by changing the distance between the bodies. At the -distance of one mile the body B attracts A with a certain force. At the -distance of two miles, the masses not being changed, the attraction of -B upon A will be one-fourth of its amount at the distance of one mile. -At the distance of three miles, it will be one-ninth of its original -amount; at four miles, it is reduced to a sixteenth, and so on. The -following table exhibits the diminution of the attraction corresponding -to the successive increase of distance: - - +-----------+---+----+----+----+----+----+----+----+----+ - |Distance | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | &c.| - +-----------+---+----+----+----+----+----+----+----+----+ - |Attraction | 1 | 1/4| 1/9|1/16|1/25|1/36|1/49|1/64| &c.| - +-----------+---+----+----+----+----+----+----+----+----+ - -In ARITHMETIC, that number which is found by multiplying any proposed -number by itself, is called its _square_. Thus 4, that is, 2 multiplied -by 2, is the square of 2; 9 that is, 3 times 3, is the square of 3, and -so on. On inspecting the above table, it will be apparent, therefore, -that the attraction of gravitation decreases in the same proportion as -the square of the distance from the attracting body increases, the mass -of both bodies in this case being supposed to remain the same; but if -the mass of either be increased or diminished, the attraction will be -increased or diminished in the same proportion. - -(109.) Hence the _law of gravitation_ may be thus expressed: “The -mutual attraction of two bodies increases in the same proportion -as their masses are increased, and as the square of their distance -is decreased; and it decreases in proportion as their masses are -decreased, and as the square of their distance is increased.” - -This law may be more clearly expressed by means of general symbols. -Let _f_ express the force with which a mass weighing 1 lb. will -attract another mass weighing 1 lb., at the distance of 1 foot. -The force with which they will mutually attract, when removed to the -distance expressed in feet by D, will be - - _f_/D^2 - -that is, the force _f_ divided by the square of the number D. - -If one of the bodies, instead of weighing 1 lb., weigh the number -of pounds expressed by A, their mutual attraction will be increased A -times, and will therefore be expressed by - - (A × _f_)/D^2 - -In fine, if the other be also the number of pounds expressed by B, -their mutual attraction will be - - (A × B × _f_)/D^2 - -(110.) Having explained the law of gravitation, we shall now proceed to -show how the existence of this force is proved, and its law discovered. - -The earth is known to be a globular mass of matter, incomparably -greater than any of the detached bodies which are found upon its -surface. If one of these bodies suspended at any proposed height -above the surface of the earth be disengaged, it will be observed to -descend perpendicularly to the earth, that is, in the direction of the -earth’s centre. The force with which it descends will also be found -to be in proportion to the mass, without any regard to the species of -the body. These circumstances are consistent with the account which -we have given of gravitation. But by that account we should expect, -that as the falling body is attracted with a certain force towards the -earth, the earth itself should be attracted towards it by the same -force; and instead of the falling body moving towards the earth, which -is the phenomenon observed, the earth and it should move towards each -other, and meet at some intermediate point. This, in fact, is the case, -although it is impossible to render the motion of the earth observable, -for reasons which will easily be understood. - -Since all the bodies around us participate in this motion, it would -not be directly observable, even though its quantity were sufficiently -great to be perceived under other circumstances. But setting aside -this consideration, the space through which the earth moves in such a -case is too minute to be the subject of sensible observation. It has -been stated (107), that when two bodies attract each other, the space -through which the greater approaches the lesser, bears to that through -which the lesser approaches the greater, the same proportion as the -mass of the lesser bears to the mass of the greater. Now the mass of -the earth is more than 1000,000,000,000,000 times the mass of any body -which is observed to fall on its surface; and therefore if even the -largest body which can come under observation were to fall through an -height of 500 feet, the corresponding motion of the earth would be -through a space less than the 1000,000,000,000,000th part of 500 feet, -which is less than the 100,000,000,000th part of an inch. - -The attraction between the earth and detached bodies on its surface is -not only exhibited by the descent of these bodies when unsupported, -but by their pressure when supported. This pressure is what is called -_weight_. The phenomena of weight, and the descent of heavy bodies, -will be fully investigated in the next chapter. - -(111.) It is not alone by the direct fall of bodies that the -gravitation of the earth is manifested. The curvilinear motion of -bodies projected in directions different from the perpendicular, is -a combination of the effects of the uniform velocity which has been -given to the projectile by the impulse which it has received, and the -accelerated velocity which it receives from the earth’s attraction. -Suppose a body placed at any point P, _fig. 21._, above the -surface of the earth, and let P C be the direction of the earth’s -centre. If the body were allowed to move without receiving any impulse, -it would descend to the earth in the direction P A, with an -accelerated motion. But suppose that at the moment of its departure -from P, it receives an impulse in the direction P B, which would -carry it to B in the time the body would fall from P to A, then, by -the composition of motion, the body must at the end of that time be -found in the line B D, parallel to P A. If the motion in -the direction of P A were uniform, the body P would in this case -move in the straight line from P to D. But this is not the case. The -velocity of the body in the direction P A is at first so small as -to produce very little deflection of its motion from the line P B. -As the velocity, however, increases, this deflection increases, so that -it moves from P to D in a curve, which is convex, towards P B. - -The greater the velocity of the projectile in the direction P A, -the greater sweep the curve will take. Thus it will successively take -the forms P D, P E, P F, &c., and that velocity can be -computed, which (setting aside the resistance of the air) would cause -the projectile to go completely round the earth, and return to the -point P from which it departed. In this case, the body P would continue -to revolve round the earth like the moon. Hence it is obvious, that the -phenomenon of the revolution of the moon round the earth, is nothing -more than the combined effects of the earth’s attraction, and the -impulse which it received when launched into space by the hand of its -Creator. - -(112.) This is a great step in the analysis of the phenomenon of -gravitation. We have thus reduced to the same class two effects -apparently very dissimilar, the rectilinear descent of a heavy body, -and the nearly circular revolution of the moon round the earth. Hence -we are conducted to a generalisation still more extensive. - -As the moon’s revolution round the earth, in an orbit nearly circular, -is caused by the combination of the earth’s attraction, and an original -projectile impulse, so also the singular phenomena of the planets’ -revolution round the sun in orbits nearly circular, must be considered -an effect of the same class, as well as the revolution of the -satellites of those planets which are attended by such bodies. Although -the orbits in which the comets move deviate very much from circles, yet -this does not hinder the application of the same principle to them, -their deviation from circles not depending on the sun’s attraction, but -only on the direction and force of the original impulse which put them -in motion. - -(113.) We therefore conclude that gravitation is the principle -which, as it were, animates the universe. All the great changes and -revolutions of the bodies which compose our system, can be traced -to or derived from this principle. It still remains to show how -that remarkable law, by which this force is declared to increase or -decrease in the same proportion as the square of the distance from -the attracting body is decreased or increased, may be verified and -established. - -It has been shown, that the curvilinear path of a projectile -depends on, and can be derived, by mathematical reasoning, from the -consideration of the intensity of the earth’s attraction, and the -force of the original impulse, or the velocity of projection. In the -same manner, by a reverse process, when we know the curve in which a -projectile moves, we can infer the amount of the attracting force which -gives the curvature to its path. In this way, from our knowledge of the -curvature of the moon’s orbit, and the velocity with which she moves, -the intensity of the attraction which the earth exerts upon her can be -exactly ascertained. Upon comparing this with the force of gravitation -at the earth’s surface, it is found that the latter is as many times -greater than the former, as the square of the moon’s distance is -greater than the square of the distance of a body on the surface of the -earth from its centre. - -(114.) If this were the only fact which could be brought to establish -the law of gravitation, it might be thought to be an accidental -relation, not necessarily characterising the attraction of gravitation. -Upon examining the orbits and velocities of the several planets, the -same result is, however, obtained. It is found that the forces with -which they are severally attracted by the sun are great, in exactly the -same proportion as the squares of the several numbers expressing their -distances are small. The mutual gravitation of bodies on the surface of -the earth towards each other is lost in the predominating force exerted -by the earth upon all of them. Nevertheless, in some cases, this effect -has not only been observed, but actually measured. - -A plumb-line, under ordinary circumstances, hangs in a direction truly -vertical; but if it be near a large mass of matter, as a mountain, -it has been observed to be deflected from the true vertical, towards -the mountain. This effect was observed by Dr. Maskeline near the -mountain called Skehallien, in Scotland, and by French astronomers near -Chimboraco. For particulars of these observations, see our treatise on -GEODÆSY. - -Cavendish succeeded in exhibiting the effects of the mutual gravitation -of metallic spheres. Two globes of lead A, B, each about a foot in -diameter, were placed at a certain distance asunder. A light rod, -to the ends of which were attached small metallic balls C, D, was -suspended at its centre E from a fine wire, and the rod was placed -as in _fig. 22._, so that the attractions of each of the leaden -globes had a tendency to turn the rod round the centre E in the same -direction. A manifest effect was produced upon the balls C, D, by the -gravitation of the spheres. In this experiment, care must be taken that -no magnetic substance is intermixed with the materials of the balls. - -Having so far stated the principles on which the law of gravitation is -established, we shall dismiss this subject without further details, -since it more properly belongs to the subject of PHYSICAL ASTRONOMY; to -which we refer the reader for a complete demonstration of the law, and -for the detailed development of its various and important consequences. - - - - -CHAP. VII. - -TERRESTRIAL GRAVITY. - - -(115.) GRAVITATION is the general name given to this attraction, by -whatever masses of matter it may be manifested. As exhibited in the -effects produced by the earth upon surrounding bodies, it is called -“terrestrial gravity.” - -As the attraction of the earth is directed towards its centre, it might -be expected that two plumb-lines should appear not to be parallel, but -so inclined to each other as to converge to a point under the surface -of the earth. Thus, if A B and C D, _fig. 23._, be two -plumb-lines, each will be directed to the centre O, where, if their -directions were continued, they would meet. In like manner, if two -bodies were allowed to fall from A and C, they would descend in the -directions A B and C D, which converge to O. Observation, -on the contrary, shows, that plumb-lines suspended in places not far -distant from each other are truly parallel; and that bodies allowed -to fall descend in parallel lines. This apparent parallelism of the -direction of terrestrial gravity is accounted for by the enormous -proportion which the magnitude of the earth bears to the distance -between the two plumb-lines or the two falling bodies which are -compared. If the distance between the places B, D, were 1200 feet, the -inclination of the lines A B and C D would not amount to a -quarter of a minute, or the 240th part of a degree. But the distance, -in cases where the parallelism is assumed, is never greater than, and -seldom so great as, a few yards; and hence the inclination of the -directions A B and C D is too small to be appreciated by any -practical measure. In the investigation of the phenomena of falling -bodies, we shall, therefore, assume, that all the particles of the -same body are attracted in parallel directions, perpendicular to an -horizontal plane. - -(116.) Since the intensity of terrestrial gravity increases as the -square of the distance decreases, it might be expected that, as a -falling body approaches the earth, the force which accelerates it -should be continually increasing, and, strictly speaking, it is so. But -any height through which we observe falling bodies to descend bears so -very small a proportion to the whole distance from the centre, that -the change of intensity of the force of gravity is quite beyond any -practical means of estimating it. The radius, or the distance from -the surface of the earth to its centre, is 4000 miles. Now, suppose -a body descended through the height of half a mile, a distance very -much beyond those used in experimental enquiries, the distances from -the centre, at the beginning and end of the fall, are then in the -proportion of 8000 to 8001, and therefore the proportion of the force -of attraction at the commencement to the force at the end, being that -of the squares of these numbers, is 64,000,000 to 64,016,001, which, in -the whole descent, is an increase of about one part in 4000; a quantity -practically insignificant. We shall, therefore, in explaining the laws -of falling bodies, assume that, in the entire descent, the body is -urged by a force of uniform intensity. - -Although the force which attracts all parts of the same body during -its descent in a given place is the same, yet the force of gravity, -at different parts of the earth’s surface, has different intensities. -The intensity diminishes with the latitude, so that it is greater -towards the poles, and lesser towards the equator. The causes of -this variation, its law, and the experimental proofs of it, will be -explained, when we shall treat of centrifugal force, and the motion of -pendulums. It is sufficient merely to advert to it in this place. - -(117.) Since the earth’s attraction acts separately and equally on -every particle of matter, without regard to the nature or species of -the body, it follows that all bodies, of whatever kind, or whatever -be their masses, must be moved with the same velocity. If two equal -particles of matter be placed at a certain distance above the surface -of the earth, they will fall in parallel lines, and with exactly the -same speed, because the earth attracts them equally. In the same -manner, a thousand particles would fall with equal velocities. Now, -these circumstances will in no wise be changed if those 1000 particles, -instead of existing separately, be aggregated into two solid masses, -one consisting of 990 particles, and the other of 10. We shall thus -have a heavy body and a light one, and, according to our reasoning, -they must fall to the earth with the same speed. - -Common experience, however, is not always consistent with this -doctrine. What are called light substances, as feathers, gold-leaf, -paper, &c., are observed to fall slowly and irregularly, while -heavier masses, as solid pieces of metal, stones, &c., fall rapidly. -Nay, there are not a few instances in which the earth, instead of -attracting bodies, seems to repel them, as in the case of smoke, -vapours, balloons, and other substances which actually ascend. We are -to consider that the mass of the earth is not the only agent engaged in -these phenomena. The earth is surrounded by an atmosphere composed of -an elastic or aeriform fluid. This atmosphere has certain properties, -which will be explained in our treatise on PNEUMATICS, and which are -the causes of the anomalous circumstances alluded to. Light bodies -rise in the atmosphere, for the same reason that a piece of cork rises -from the bottom of a vessel of water; and other light bodies fall more -slowly than heavy ones, for the same reason that an egg in water falls -to the bottom more slowly than a leaden bullet. This treatise is not -the place to give a direct explanation of these phenomena. It will -be sufficient for our present purpose to show, that if there were no -atmosphere, all bodies, heavy and light, would fall at the same rate. -This may easily be accomplished by the aid of an air-pump. Having -by that instrument abstracted the air from a tall glass vessel, we -are enabled, by means of a wire passing air-tight through a hole in -the top, to let fall several bodies from the top of the vessel to the -bottom. These, whether they be feathers, paper, gold-leaf, pieces of -money, &c. all descend with the same speed, and strike the bottom at -the same moment. - -(118.) Every one who has seen a heavy body fall from a height, has -witnessed the fact, that its velocity increases as it approaches -the ground. But if this were not observable by the eye, it would be -betrayed by the effects. It is well known, that the force with which a -body strikes the ground increases with the height from whence it has -fallen. This force, however, is proportional to the velocity which it -has at the moment it meets the ground, and therefore this velocity -increases with the height. - -When the observations on attraction in the last chapter are well -understood, it will be evident that the velocity which a body has -acquired in falling from any height, is the accumulated effects of the -attraction of terrestrial gravity during the whole time of the fall. -Each instant of the fall a new impulse is given to the body, from which -it receives additional velocity; and its final velocity is composed -of the aggregation of all the small increments of velocity which are -thus communicated. As we are at present to suppose the intensity of the -attraction invariable, it will follow that the velocity communicated to -the body in each instant of time will be the same, and therefore that -the whole quantity of velocity produced or accumulated at the end of -any time is proportional to the length of that time. Thus, if a certain -velocity be produced in a body having fallen for one second, twice that -velocity will be produced when it has fallen for two seconds, thrice -that velocity in three seconds, and so on. Such is the fundamental -principle or characteristic of _uniformly accelerated motion_. - -(119.) In examining the circumstances of the descent of a body, the -time of the fall and the velocity at each instant of that time are not -the only things to be attended to. The spaces through which it falls in -given intervals of time, counted either from the commencement of its -fall, or from any proposed epoch of the descent, are equally important -objects of enquiry. To estimate the space in reference to the time and -the final velocity, we must consider that this space has been moved -through with varying speed. From a state of rest at the beginning of -the fall, the speed gradually increases with the time, and the final -velocity is greater still than that which the body had at any preceding -instant during its descent. We cannot, therefore, _directly_ appreciate -the space moved through in this case by the time and final velocity. -But as the velocity increases uniformly with the time, we shall obtain -the average speed, by finding that which the body had in the middle of -the interval which elapsed between the beginning and end of the fall, -and thus the space through which the body has actually fallen is that -through which it would move in the same time with this average velocity -uniformly continued. - -But since the velocity which the body receives in any time, counted -from the beginning of its descent, is in the proportion of that time, -it follows that the velocity of the body after half the whole time of -descent is half the final velocity. From whence it appears, that the -height from which a body falls in any proposed time is equal to the -space through which a body would move in the same time with half the -final velocity, and it is therefore equal to half the space which would -be moved through in the same time with the final velocity. - -(120.) It follows from this reasoning, that between the three -quantities, the height, the time, and the final velocity, which enter -into the investigation of the phenomena of falling bodies, there are -two fixed relations: _First_, the time, counted from the beginning of -the fall and the final velocity, are proportional the one to the other; -so that as one increases, the other increases in the same proportion. -_Secondly_, the height being equal to half the space which would be -moved through in the _time_ of the fall, with the _final velocity_, -must have a fixed proportion to these two quantities, viz. the _time_ -and the _final velocity_, or must be proportional to the product of the -two numbers which express them. - -But since the time is always proportional to the final velocity, they -may be expressed by equal numbers, and the product of equal numbers -is the square of either of them. Hence, the product of the numbers -expressing the time and final velocity is equivalent to the square -of the number expressing the time, or to the square of the number -expressing the final velocity. Hence we infer, that the height is -always proportional to the square of the time of the fall, or to the -square of the final velocity. - -(121.) The use of a few mathematical characters will render these -results more distinct, even to students not conversant with -mathematical science. - -Let S = the height from which the body falls, expressed in feet. - - V = the velocity at the end of the fall in feet per second. - - T = the number of seconds in the time of the fall. - - _g_ = the number of feet through which a body would fall in one - second. - -It will therefore follow that the velocity acquired in one second will -be 2_g_, and the velocity acquired in T seconds will therefore be 2_g_ -× T; so that - - V = 2_g_ × T [1] - -Since the space which a body falls through in T seconds is found by -multiplying the space it falls through in one second by T^2, we shall -have - - S = _g_ × T^2 [2] - -from which, combined with [1] we deduce - - S = V^2/(4_g_) [3] - - S = (1/2)V × T [4] - -By these formularies, if the height through which a body falls freely -in one second be known, the height through which it will fall in any -proposed time may be computed. For since the height is proportional -to the square of the time, the height through which it will fall in -_two_ seconds will be _four_ times that which it falls through in -_one_ second. In _three_ seconds it will fall through _nine_ times -that space; in _four_ seconds, _sixteen_ times; in _five_ seconds, -_twenty-five_ times, and so on. The following, therefore, is a general -rule to find the height through which a body will fall in any given -time: “Reduce the given time to seconds, take the square of the number -of seconds in it, and multiply the height through which a body falls in -one second by that number; the result will be the height sought.” - -The following table exhibits the heights and corresponding times as far -as 10 seconds: - - +-------+---+---+---+----+----+----+----+----+----+-----+ - |Time | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | - +-------+---+---+---+----+----+----+----+----+----+-----+ - |Height | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | - +-------+---+---+---+----+----+----+----+----+----+-----+ - -Each unit in the numbers of the first row expresses a second of time, -and each unit in those of the second row expresses the height through -which a body falls freely in a second. - -(122.) If a body fall continually for several successive seconds, -the spaces which it falls through in each succeeding second have a -remarkable relation among each other, which may be easily deduced from -the preceding table. Taking the space moved through in the first second -still as our unit, four times that space will be moved through in the -first two seconds. Subtract from this 1, the space moved through in the -first second, and the remainder 3 is the space through which the body -falls in the _second_ second. In like manner if 4, the height fallen -through in the first two seconds, be subtracted from 9, the height -fallen through in the first three seconds, the remainder 5 will be the -space fallen through in the third second. To find the space fallen -through in the fourth second, subtract 9, the space fallen through in -the first three seconds, from 16, the space fallen through in the first -four seconds, and the result is 7, and so on. It thus appears that if -the space fallen through in the first second be called 1, the spaces -described in the second, third, fourth, fifth, &c. seconds, will be -expressed by the odd numbers respectively, 3, 5, 7, 9, &c. This places -in a striking point of view the accelerated motion of a falling body, -the spaces moved through in each succeeding second being continually -increased. - -(123.) If velocity be estimated by the space through which the body -would move uniformly in one second, then the final velocity of a body -falling for one second will be 2; for with that final velocity the body -would in one second move through twice the height through which it has -fallen. - -(124.) Since the final velocity increases in the same proportion as -the time, it follows that after two seconds it is twice its amount -after one, and after three seconds thrice that, and so on. Thus, the -following table exhibits the final velocities corresponding to the -times of descent: - - +---------------+---+---+---+---+----+----+----+----+----+----+ - |Time | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | - +---------------+---+---+---+---+----+----+----+----+----+----+ - |Final velocity | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | - +---------------+---+---+---+---+----+----+----+----+----+----+ - -The numbers in the second row express the spaces through which a body -with the final velocity would move in one second, the unit being, as -usual, the space through which a body falls freely in one second. - -(125.) Having thus developed theoretically the laws which characterise -the descent of bodies, falling freely by the force of gravity, or by -any other uniform force of the same kind, it is necessary that we -should show how these laws can be exhibited by actual experiment. -There are some circumstances attending the fall of heavy bodies which -would render it difficult, if not impossible, to illustrate, by the -direct observation of this phenomenon, the properties which have -been explained in this chapter. A body falling freely by the force -of gravity, as we shall hereafter prove, descends in one second of -time through a height of about 16 feet[1]; in two seconds, it would, -therefore, fall through four times that space, or 64 feet; in three -seconds, through 9 times the height, or 144 feet; and in four seconds, -through 256 feet. In order, therefore, to be enabled to observe the -phenomena for only four seconds, we should command an height of at -least 256 feet. But further; the velocity at the end of the first -second would be at the rate of 32 feet per second; at the end of the -second second, it would be 64 feet per second; and towards the end of -the fall it would be about 120 feet per second. It is evident that this -great degree of rapidity would be a serious impediment to accurate -observation, even though we should be able to command the requisite -height. It appears therefore that the number expressed by _g_ in the -preceding formulæ is 16·083. - -[1] More exactly through 16-1/12 feet, or 193 inches. - -It occurred to Mr. George Attwood, a mathematician and natural -philosopher of the last century, that all the phenomena of falling -bodies might be experimentally exhibited and accurately observed, if -a force of the same kind as gravity, viz. an uniformly accelerating -force, be used, but of a much less intensity; so that while the motion -continues to be governed by the same laws, its quantity may be so much -diminished, that the final velocity, even after a descent of many -seconds, shall be so moderated as to admit of most deliberate and exact -observation. This being once accomplished, nothing more would remain -but to find the height through which a body would fall in one second, -or, what is the same, the proportion of the force of gravity to the -mitigated but uniform accelerating force thus substituted for it. - -(126.) To realise this notion, Attwood constructed a wheel turning on -its axle with very little friction, and having a groove on its edge -to receive a string. Over this wheel, and in the groove, he placed a -fine silken cord, to the ends of which were attached equal cylindrical -weights. Thus placed, the weights perfectly balance each other, and no -motion ensues. To one of the weights he then added a small quantity, so -as to give it a slight preponderance. The loaded weight now began to -descend, drawing up on the other side the unloaded weight. The descent -of the loaded weight, under these circumstances, is a motion exactly -of the _same kind_ as the descent of a heavy body falling freely by -the force of gravity; that is, it increases according to the same -laws, though at a very diminished rate. To explain this, suppose that -the loaded weight descends from a state of rest through one inch in -a second, it will descend through 4 inches in two seconds, through 9 -in three, through 16 in four, and so on. Thus in 20 seconds, it would -descend through 400 inches, or 33 feet 4 inches, a height which, if it -were necessary, could easily be commanded. - -It might, perhaps, be thought, that since the weights suspended at the -ends of the thread are in equilibrium, and therefore have no tendency -either to move or to resist motion, the additional weight placed upon -one of them ought to descend as rapidly as it would if it were allowed -to fall freely and unconnected with them. It is very true that this -weight will receive from the attraction of the earth the same force -when placed upon one of the suspended weights, as it would if it were -disengaged from them; but in the consequences which ensue, there is -this difference. If it were unconnected with the suspended weights, -the whole force impressed upon it would be expended in accelerating -its descent; but being connected with the equal weights which sustain -each other in equilibrium, by the silken cord passing over the wheel, -the force which is impressed upon the added weight is expended, not -as before, in giving velocity to the added weight alone, but to it -together with the two equal weights appended to the string, one of -which descends with the added weight, and the other rises on the -opposite side of the wheel. Hence, setting aside any effect which the -wheel itself produces, the velocity of the descent must be lessened -just in proportion as the mass among which the impressed force is to -be distributed is increased; and therefore the _rate_ of the fall -bears to that of a body falling freely the same proportion as the added -weight bears to the sum of the masses of the equal suspended weights -and the added weight. Thus the smaller the added weight is, and the -greater the equal suspended weights are, the slower will the rate of -descent be. - -To render the circumstances of the fall conveniently observable, a -vertical shaft (see _fig. 24._) is usually provided, which is -placed behind the descending weight. This pillar is divided to inches -and halves, and of course may be still more minutely graduated, if -necessary. A stage to receive the falling weight is moveable on this -pillar, and capable of being fixed in any proposed position by an -adjusting screw. A pendulum vibrating seconds, the beat of which ought -to be very audible, is placed near the observer. The loaded weight -being thus allowed to descend for any proposed time, or from any -required height, all the circumstances of the descent may be accurately -observed, and the several laws already explained in this chapter may be -experimentally verified. - -(127.) The laws which govern the descent of bodies by gravity, being -reversed, will be applicable to the ascent of bodies projected upwards. -If a body be projected directly upwards with any given velocity, it -will rise to the height from which it should have fallen to acquire -that velocity. The earth’s attraction will, in this case, gradually -deprive the body of the velocity which is communicated to it at the -moment at which it is projected. Consequently, the phenomenon will be -that of _retarded motion_. At each part of its ascent it will have the -same velocity which it would have if it descended to the same place -from the highest point to which it rises. Hence it is clear, that all -the particulars relative to the ascent of bodies may be immediately -inferred from those of their descent, and therefore this subject -demands no further notice. - -To complete the investigation of the phenomena of falling bodies, it -would now only remain to explain the method of ascertaining the exact -height through which a body would descend in one second, if unresisted -by the atmosphere, or any other disturbing cause. As the solution -of this problem, however, requires the aid of principles not yet -explained, it must for the present be postponed. - - - - -CHAP. VIII. - -OF THE MOTION OF BODIES ON INCLINED PLANES AND CURVES. - - -(128.) In the last chapter, we investigated the phenomena of bodies -descending freely in the vertical direction, and determined the laws -which govern, not their motion alone, but that of bodies urged by any -uniformly accelerating force whatever. We shall now consider some of -the most ordinary cases in which the free descent of bodies is impeded, -and the effects of their gravitation modified. - -(129.) If a body, urged by any forces whatever, be placed upon a -hard unyielding surface, it will evidently remain at rest, if the -resultant (76) of all the forces which are applied to it be directed -perpendicularly against the surface. In this case, the effect produced -is pressure, but no motion ensues. If only one force act upon the -body, it will remain at rest, provided the direction of that force be -perpendicular to the surface. - -But the effect will be different, if the resultant of the forces which -are applied to the body be oblique to the surface. In that case this -resultant, which, for simplicity, may be taken as a single force, may -be considered as mechanically equivalent to two forces (76), one in the -direction of the surface, and the other perpendicular to it. The latter -element will be resisted, and will produce a pressure; the former will -cause the body to move. This will perhaps be more clearly apprehended -by the aid of a diagram. - -Let A B, _fig. 25._, be the surface, and let P be a particle -of matter placed upon it, and urged by a force in the direction -P D, perpendicular to A B. It is manifest, that this force -can only press the particle P against A B, but cannot give it any -motion. - -But let us suppose, that the force which urges P is in a direction -P F, oblique to A B. Taking P F as the diagonal of a -parallelogram, whose sides are P D and P C (74), the force -P F is mechanically equivalent to two forces, expressed by the -lines P D and P C. But P D, being perpendicular to -A B, produces pressure without motion, and P C, being in -the direction of A B, produces motion without pressure. Thus -the effect of the force P F is distributed between motion and -pressure in a certain proportion, which depends on the obliquity of -its direction to that of the surface. The two extreme cases are, 1. -When it is in the direction of the surface; it then produces motion -without pressure: and, 2. When it is perpendicular to the surface; it -then produces pressure without motion. In all intermediate directions, -however, it will produce both these effects. - -(130.) It will be very apparent, that the more oblique the direction -of the force P F is to A B, the greater will be that part -of it which produces motion, and the less will be that which produces -pressure. This will be evident by inspecting _fig. 26._ In this -figure the line P F, which represents the force, is equal to -P F in _fig. 25._ But P D, which expresses the pressure, -is less in _fig. 26._ than in _fig. 25._, while P C, -which expresses the motion, is greater. So long, then, as the obliquity -of the directions of the surface and the force remain unchanged, so -long will the distribution of the force between motion and pressure -remain the same; and therefore, if the force itself remain the same, -the parts of it which produce motion and pressure will be respectively -equal. - -(131.) These general principles being understood, no difficulty can -arise in applying them to the motion of bodies urged on inclined planes -or curves by the force of gravity. If a body be placed on an unyielding -horizontal plane, it will remain at rest, producing a pressure on the -plane equal to the total amount of its weight. For in this case the -force which urges the body, being that of terrestrial gravity, its -direction is vertical, and therefore perpendicular to the horizontal -plane. - -But if the body P, _fig. 25._, be placed upon a plane A B, -oblique to the direction of the force of gravity, then, according to -what has been proved (129), the weight of the body will be distributed -into two parts, P C and P D; one, P D, producing a -pressure on the plane A B, and the other, P C, producing -motion down the plane. Since the obliquity of the perpendicular -direction P F of the weight to that of the plane A B must be -the same on whatever part of the plane the weight may be placed, it -follows (130), that the proportion P C of the weight which urges -the body down the plane must be the same throughout its whole descent. - -(132.) Hence it may easily be inferred, that the force down the -plane is uniform; for since the weight of the body P is always the -same, and since its proportion to that part which urges it down the -plane is the same, it follows that the quantity of this part cannot -vary. The motion of a heavy body down an inclined plane is therefore -an uniformly-accelerated motion, and is characterised by all the -properties of uniformly-accelerated motion, explained in the last -chapter. - -Since P F represents the force of gravity, that is, the force -with which the body would descend freely in the vertical direction, -and P C the force with which it moves down the plane, it follows -that a body would fall freely in the vertical direction from P to F -in the same time as on the plane it would move from P to C. In this -manner, therefore, when the height through which a body would fall -vertically is known, the space through which it would descend in the -same time down any given inclined plane may be immediately determined. -For let A B, _fig. 25._, be the given inclined plane, and let -P F be the space through which the body would fall in one second. -From F draw F C perpendicular to the plane, and the space P C -is that through which the body P will fall in one second on the plane. - -(133.) As the angle B A H, which measures the elevation -of the plane, is increased, the obliquity of the vertical direction -P F with the plane is also increased. Consequently, according to -what has been proved (130), it follows, that as the elevation of the -plane is increased, the force which urges the body down the plane is -also increased, and as the elevation is diminished, the force suffers a -corresponding diminution. The two extreme cases are, 1. When the plane -is raised until it becomes perpendicular, in which case the weight is -permitted to fall freely, without exerting any pressure upon the plane; -and, 2. When the plane is depressed until it becomes horizontal, in -which case the whole weight is supported, and there is no motion. - -From these circumstances it follows, that by means of an inclined plane -we can obtain an uniformly-accelerating force of any magnitude less -than that of gravity. - -We have here omitted, and shall for the present in every instance -omit, the effects of _friction_, by which the motion down the plane -is retarded. Having first investigated the mechanical properties of -bodies supposed to be free from friction, we shall consider friction -separately, and show how the present results are modified by it. - -(134.) The accelerating forces on different inclined planes may be -compared by the principle explained in (131). Let _figs. 25._ -and _26._ be two inclined planes, and take the lines P F in each -figure equal, both expressing the force of gravity, then P C will -be the force which in each case urges the body down the plane. - -As the force down an inclined plane is less than that which urges a -body falling freely in the vertical direction, the space through which -the body must fall to attain a certain final velocity must be just so -much greater as the accelerating force is less. On this principle we -shall be able to determine the final velocity in descending through any -space on a plane, compared with the final velocity attained in falling -freely in the vertical direction. Suppose the body P, _fig. 27._, -placed at the top of the plane, and from H draw the perpendicular -H C. If B H represent the force of gravity, B C will -represent the force down the plane (131). In order that the body -moving down the plane shall have a final velocity equal to that of -one which has fallen freely from B to H, it will be necessary that it -should move from B down the plane, through a space which bears the same -proportion to B H as B H does to B C. But since the -triangle A B H is in all respects similar to H B C, -only made upon a larger scale, the line A B bears the same -proportion to B H as B H bears to B C. Hence, in falling -on the inclined plane from B to A, the final velocity is the same as in -falling freely from B to H. - -It is evident that the same will be true at whatever level an -horizontal line be drawn. Thus, if I K be horizontal, the final -velocity in falling on the plane from B to I will be the same as the -final velocity in falling freely from B to K. - -(135.) The motion of a heavy body down a curve differs in an important -respect from the motion down an inclined plane. Every part of the -plane being equally inclined to the vertical direction, the effect of -gravity in the direction of the plane is uniform; and, consequently, -the phenomena obey all the established laws of uniformly-accelerated -motion. If, however, we suppose the line B A, on which the -body P descends, to be curved as in _fig. 28._, the obliquity -of its direction at different parts, to the direction P F of -gravity, will evidently vary. In the present instance, this obliquity -is greater towards B and less towards A, and hence the part of the -force of gravity which gives motion to the body is greater towards -B than towards A (130). The force, therefore, which urges the body, -instead of being uniform as in the inclined plane, is here gradually -diminished. The rate of this diminution depends entirely on the nature -of the curve, and can be deduced from the properties of the curve by -mathematical reasoning. The details of such an investigation are not, -however, of a sufficiently elementary character to allow of being -introduced with advantage into this treatise. We must therefore limit -ourselves to explain such of the results as may be necessary for the -development of the other parts of the science. - -(136.) When a heavy body is moved down an inclined plane by the force -of gravity, the plane has been proved to sustain a pressure, arising -from a certain part of the weight P D, _fig. 25._, which -acts perpendicularly to the plane. This is also the case in moving -down a curve such as B A, _fig. 28._ In this case, also, the -whole weight is distributed between that part which is directed down -the curve, and that which, being perpendicular to the curve, produces -a pressure upon it. There is, however, another cause which produces -pressure upon the curve, and which has no operation in the case of -the inclined plane. By the property of inertia, when a body is put in -motion in any direction, it must persevere in that direction, unless -it be deflected from it by an efficient force. In the motion down an -inclined plane the direction is never changed, and therefore by its -inertia the falling body retains all the motion impressed upon it -continually in the same direction; but when it descends upon a curve, -its direction is constantly varying, and the resistance of the curve -being the deflecting cause, the curve must sustain a pressure equal to -that force, which would thus be capable of continually deflecting the -body from the rectilinear path in which it would move in virtue of its -inertia. This pressure entirely depends on the curvature of the path -in which the body is constrained to move, and on its inertia, and is -therefore altogether independent of the weight, and would, in fact, -exist if the weight were without effect. - -(137.) This pressure has been denominated _centrifugal force_, because -it evinces a tendency of the moving body to _fly from_ the centre of -the curve in which it is moved. Its quantity depends conjointly on the -velocity of the motion and the curvature of the path through which -the body is moved. As circles may be described with every degree of -curvature, according to the length of the radius, or the distance from -their circumference to their centre, it follows that, whatever be the -curve in which the body moves, a circle can always be assigned which -has the same curvature as is found at any proposed point of the given -curve. Such a circle is called “the circle of curvature” at that point -of the curve; and as all curves, except the circle, vary their degrees -of curvature at different points, it follows that different parts of -the same curve will have different circles of curvature. It is evident -that the greater the radius of a circle is, the less is its curvature: -thus the circle with the radius A B, _fig. 29._, is more -curved than that whose radius is C D, and that in the exact -proportion of the radius C D to the radius A B. The radius of -the circle of curvature for any part of a curve is called “the radius -of curvature” of that part. - -(138.) The centrifugal pressure increases as the radius of curvature -increases; but it also has a dependence on the velocity with which the -moving body swings round the centre of the circle of curvature. This -velocity is estimated either by the actual space through which the body -moves, or by the _angular velocity_ of a line drawn from the centre of -the circle to the moving body. That body carries one end of this line -with it, while the other remains fixed at the centre. As this angular -swing round the centre increases, the centrifugal pressure increases. -To estimate the rate at which this pressure in general varies, it is -necessary to multiply the square of the number expressing the angular -velocity by that which expresses the radius of curvature, and the force -increases in the same proportion as the product thus obtained. - -(139.) We have observed that the same causes which produce pressure -on a body restrained, will produce motion if the body be free. -Accordingly, if a body be moved by any efficient cause in a curve, it -will, by reason of the centrifugal force, _fly off_, and the moving -force with which it will thus retreat from the centre round which -it is whirled will be a measure of the centrifugal force. Upon this -principle an apparatus called a _whirling table_ has been constructed, -for the purpose of exhibiting experimental illustrations of the laws -of centrifugal force. By this machine we are enabled to place any -proposed weights at any given distances from centres round which they -are whirled, either with the same angular velocity, or with velocities -having a certain proportion. Threads attached to the whirling weights -are carried to the centres round which they respectively revolve, and -there, passing over pulleys, are connected with weights which may be -varied at pleasure. When the whirling weights fly from their respective -centres, by reason of the centrifugal force, they draw up the weights -attached to the other ends of the threads, and the amount of the -centrifugal force is estimated by the weight which it is capable of -raising. - -With this instrument the following experiments may be exhibited:-- - -Exp. 1. Equal weights whirled with the same velocity at equal distances -from the centre raise the same weight, and therefore have the same -centrifugal force. - -Exp. 2. Equal weights whirled with the same angular velocity at -distances from the centre in the proportion of one to two, will raise -weights in the same proportion. Therefore the centrifugal forces are in -that proportion. - -Exp. 3. Equal weights whirled at equal distances with angular -velocities which are as one to two, will raise weights as one to four, -that is, as the squares of the angular velocities. Therefore the -centrifugal forces are in that proportion. - -Exp. 4. Equal weights whirled at distances which are as two to three, -with angular velocities which are as one to two, will raise weights -which are as two to twelve; that is, as the products of the distances -two and three, and the squares one and four, of the angular velocities. -Hence, the centrifugal forces are in this proportion. - -The centrifugal force must also increase as the mass of the body moved -increases; for, like attraction, each particle of the moving body is -separately and equally affected by it. Hence a double mass, moving -at the same distance, and with the same velocity, will have a double -force. The following experiment verifies this:-- - -Exp. 5. If weights, which are as one to two, be whirled at equal -distances with the same velocity, they will raise weights which are as -one to two. - -The law which governs centrifugal force may then be expressed in -general symbols briefly thus:-- - -Let _c_ = the centrifugal force with which a weight of one lb. -revolving in a circle in one second, the radius of which is one foot, -would act on a string connecting it with the centre. The force with -which it would act on a string, the length of which is R feet, would -be _c_ × R; and if instead of revolving in one second it revolved in T -seconds, the force would be - - (_c_ × R)/T^2; - -and if the revolving mass were W lbs. the force would be - - C = (_c_ × W × R)/T^2. - -This formula includes the entire theory of centrifugal force. - -But it can be shown that the number expressed by _c_ is 1·226, and -consequently - - C = (1·226 × W × R)/T^2. - -It is often more convenient to use the number of revolutions made in -a given time than the time of one revolution. Let N then express the -number of revolutions, or fraction of a revolution, made in one second, -and we shall have - - T = 1/N. - -Therefore - - C = 1·226 × W × R × N^2. - -(140.) The consideration of centrifugal force proves, that if a body -be observed to move in a curvilinear path, some efficient cause must -exist which prevents it from flying off, and which compels it to -revolve round the centre. If the body be connected with the centre by -a thread, cord, or rod, then the effect of the centrifugal force is -to give tension to the thread, cord, or rod. If an unyielding curved -surface be placed on the convex side of the path, then the force will -produce pressure on this surface. But if a body is observed to move -in a curve without any visible material connection with its centre, -and without any obstruction on the convex side of its path to resist -its retreat, as is the case with the motions of the planets round -the sun, and the satellites round the planets, it is usual to assign -the cause to the attraction of the body which occupies the centre: -in the present instance the sun is that body, and it is customary to -say that the _attraction_ of the sun, neutralising the effects of the -centrifugal force of the planets, _retains them_ in their orbits. We -have elsewhere animadverted on the inaccurate and unphilosophical style -of this phraseology, in which terms are admitted which intimate not -only an unknown cause, but assign its seat, and intimate something of -its nature. All that we are entitled to declare in this case is, that -a motion is continually impressed upon the planet; that this motion is -directed towards the sun; that it counteracts the centrifugal force; -but from whence this motion proceeds, whether it be a virtue resident -in the sun, or a property of the medium or space in which both sun and -planets are placed, or whatever other influence may be its proximate -cause, we are altogether ignorant. - - * * * * * - -(141.) Numerous examples of the effects of centrifugal force may be -produced. - -If a stone or other weight be placed in a sling, which is whirled round -by the hand in a direction perpendicular to the ground, the stone will -not fall out of the sling, even when it is at the top of its circuit, -and, consequently, has no support beneath it. The centrifugal force, in -this case, acting from the hand, which is the centre of rotation, is -greater than the weight of the body, and therefore prevents its fall. - -In like manner, a glass of water may be whirled so rapidly that even -when the mouth of the glass is presented downwards, the water will -still be retained in it by the centrifugal force. - -If a bucket of water be suspended by a number of threads, and these -threads be twisted by turning round the bucket many times in the same -direction, on allowing the cords to untwist, the bucket will be whirled -rapidly round, and the water will be observed to rise on its sides and -sink at its centre, owing to the centrifugal force with which it is -driven from the centre. This effect might be carried so far, that all -the water would flow over and leave the bucket nearly empty. - -(142.) A carriage, or horseman, or pedestrian, passing a corner moves -in a curve, and suffers a centrifugal force, which increases with -the velocity, and which impresses on the body a force directed from -the corner. An animal causes its weight to resist this force, by -voluntarily inclining its body towards the corner. In this case, let -A B, _fig. 30._, be the body; C D is the direction of -the weight perpendicular to the ground, and C F is the direction -of the centrifugal force parallel to the ground and _from_ the corner. -The body A B is inclined to the corner, so that the diagonal force -(74), which is mechanically equivalent to the weight and centrifugal -force, shall be in the direction C A, and shall therefore produce -the pressure of the feet upon the ground. - -As the velocity is increased, the centrifugal force is also increased, -and therefore a greater inclination of the body is necessary to resist -it. We accordingly find that the more rapidly a corner is turned, the -more the animal inclines his body towards it. - -A carriage, however, not having voluntary motion, cannot make this -compensation for the disturbing force which is called into existence -by the gradual change of direction of the motion; consequently it -will, under certain circumstances, be overturned, falling of course -outwards, or _from_ the corner. If A B be the carriage, and -C, _fig. 31._, the place at which the weight is principally -collected, this point C will be under the influence of two forces: the -weight, which may be represented by the perpendicular C D, and the -centrifugal force, which will be represented by a line C F, which -shall have the same proportion to C D as the centrifugal force -has to the weight. Now the combined effect of these two forces will be -the same as the effect of a single force, represented by C G. -Thus, the pressure of the carriage on the road is brought nearer to the -outer wheel B. If the centrifugal force bear the same proportion to the -weight as C F (or D B), _fig. 32._, bears to C D, -the whole pressure is thrown upon the wheel B. - -If the centrifugal force bear to the weight a greater proportion than -D B has to C D, then the line C F, which represents it, -_fig. 33._, will be greater than D B. The diagonal C G, -which represents the combined effects of the weight and centrifugal -force, will in this case pass outside the wheel B, and therefore this -resultant will be unresisted. To perceive how far it will tend to -overturn the carriage, let the force C G be resolved into two, one -in the direction of C B, and the other C K, perpendicular to -C B. The former C B will be resisted by the road, but the -latter C K will tend to lift the carriage over the external wheel. -If the velocity and the curvature of the course be continued for a -sufficient time to enable this force C K to elevate the weight, -so that the line of direction shall fall on B, the carriage will be -overthrown. - -It is evident from what has been now stated, that the chances of -overthrow under these circumstances depend on the proportion of -B D to C D, or what is to the same purpose, of the distance -between the wheels to the height of the principal seat of the load. -It will be shown in the next chapter, that there is a certain point, -called the centre of gravity, at which the entire weight of the vehicle -and its load may be conceived to be concentrated. This is the point -which in the present investigation we have marked C. The security of -the carriage, therefore, depends on the greatness of the distance -between the wheels and the smallness of the elevation of the centre of -gravity above the road; for either or both of these circumstances will -increase the proportion of B D to C D. - -(143.) In the equestrian feat exhibited in the ring at the -amphitheatre, when the horse moves round with the performer standing on -the saddle, both the horse and rider incline continually towards the -centre of the ring, and the inclination increases with the velocity of -the motion: by this inclination their weights counteract the effect -of the centrifugal force, exactly as in the case already mentioned -(142.) - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -(144.) If a body be allowed to fall by its weight down a convex -surface, such as A B, _fig. 34._, it would continue upon the -surface until it arrive at B but for the effect of the centrifugal -force: this, giving it a motion from the centre of the curve, will -cause it to quit the curve at a certain point C, which can be easily -found by mathematical computation. - -(145.) The most remarkable and important manifestation of centrifugal -force is observed in the effects produced by the rotation of the earth -upon its axis. Let the circle in _fig. 35._ represent a section -of the earth, A B being the axis on which it revolves. This -rotation causes the matter which composes the mass of the earth to -revolve in circles round the different points of the axis as centres -at the various distances at which the component parts of this mass -are placed. As they all revolve with the same angular velocity, they -will be affected by centrifugal forces, which will be greater or less -in proportion as their distances from the centre are greater or less. -Consequently the parts of the earth which are situated about the -equator, D, will be more strongly affected by centrifugal force than -those about the poles, A B. The effect of this difference has been -that the component matter about the equator has actually been driven -farther from the centre than that about the poles, so that the figure -of the earth has swelled out at the sides, and appears proportionally -depressed at the top and bottom, resembling the shape of an orange. An -exaggerated representation of this figure is given in _fig. 36._; -the real difference between the distances of the poles and equator -from the centre being too small to be perceptible in a diagram. The -exact proportion of C A to C D has never yet been certainly -ascertained. Some observations make C D exceed C A by 1/277, -and others by only 1/333. The latter, however, seems the more probable. -It may be considered to be included between these limits. - -The same cause operates more powerfully in other planets which revolve -more rapidly on their axes. Jupiter and Saturn have forms which are -considerably more elliptical. - -(146.) The centrifugal force of the earth’s rotation also affects -detached bodies on its surface. If such bodies were not held upon the -surface by the earth’s attraction, they would be immediately flung -off by the whirling motion in which they participate. The centrifugal -force, however, really diminishes the effects of the earth’s attraction -on those bodies, or, what is the same, diminishes their weights. If -the earth did not revolve on its axis, the weight of bodies in all -places equally distant from the centre would be the same; but this is -not so when the bodies, as they do, move round with the earth. They -acquire from the centrifugal force a tendency to fly from the axis, -which increases with their distance from that axis, and is therefore -greater the nearer they are to the equator, and less as they approach -the pole. But there is another reason why the centrifugal force is more -efficient, in the opposition which it gives to gravity near the equator -than near the poles. This force does not act from the centre of the -earth, but is directed from the earth’s axis. It is, therefore, not -directly opposed to gravity, except on the equator itself. On leaving -the equator, and proceeding towards the poles, it is less and less -opposed to gravity, as will be plain on inspecting _fig. 35._, -where the lines P C all represent the direction of gravity, and -the lines P F represent the direction of the centrifugal force. - -Since, then, as we proceed from the equator towards the poles, not only -the amount of the centrifugal force is continually diminished, but also -it acts less and less in opposition to gravity, it follows that the -weights of bodies are most diminished by it at the equator, and less so -towards the poles. - -Since bodies are commonly weighed by balancing them against other -bodies of known weight, it may be asked, how the phenomena we have been -just describing can be ascertained as a matter of fact? for whatever -be the body against which it may be balanced, that body must suffer -just as much diminution of weight as every other, and consequently, all -being diminished in the same proportion, the balance will be preserved -though the weights be changed. - -To render this effect observable, it will be necessary to compare the -effects of gravity with some phenomenon which is not affected by the -centrifugal force of the earth’s rotation, and which will be the same -at every part of the earth. The means of accomplishing this will be -explained in a subsequent chapter. - - - - -CHAP. IX. - -THE CENTRE OF GRAVITY. - - -(147.) By the earth’s attraction, all the particles which compose -the mass of a body are solicited by equal forces in parallel -directions downwards. If these component particles were placed in mere -juxtaposition, without any mechanical connection, the force impressed -on any one of them could in nowise affect the others, and the mass -would, in such a case, be contemplated as an aggregation of small -particles of matter, each urged by an independent force. But the bodies -which are the subjects of investigation in mechanical science are not -found in this state. Solid bodies are coherent masses, the particles -of which are firmly bound together, so that any force which affects -one, being modified according to circumstances, will be transmitted -through the whole body. Liquids accommodate themselves to the shape of -the surfaces on which they rest, and forces affecting any one part are -transmitted to others, in a manner depending on the peculiar properties -of this class of bodies. - -As all bodies, which are subjects of mechanical enquiry, on the surface -of the earth, must be continually influenced by terrestrial gravity, -it is desirable to obtain some easy and summary method of estimating -the effect of this force. To consider it, as is unavoidable in the -first instance, the combined action of an infinite number of equal and -parallel forces soliciting the elementary molecules downwards, would -be attended with manifest inconvenience. An infinite number of forces, -and an infinite subdivision of the mass, would form parts of every -mechanical problem. - -To overcome this difficulty, and to obtain all the ease and simplicity -which can be desired in elementary investigations, it is only necessary -to determine some force, whose single effect shall be equivalent to the -combined effects of the gravitation of all the molecules of the body. -If this can be accomplished, that single force might be introduced into -all problems to represent the whole effect of the earth’s attraction, -and no regard need be had to any particles of the body, except that on -which this force acts. - -(148.) To discover such a force, if it exist, we shall first enquire -what properties must necessarily characterise it. Let A B, -_fig. 37._, be a solid body placed near the surface of the -earth. Its particles are all solicited downwards, in the directions -represented by the arrows. Now, if there be any single force equivalent -to these combined effects, two properties may be at once assigned to -it: 1. It must be presented downwards, in the common direction of -those forces to which it is mechanically equivalent; and, 2. it must -be equal in intensity to their sum, or, what is the same, to the force -with which the whole mass would descend. We shall then suppose it to -have this intensity, and to have the direction of the arrow D E. -Now, if the single force, in the direction D E, be equivalent -to all the separate attractions which affect the particles, we may -suppose all these attractions removed, and the body A B influenced -only by a single attraction, acting in the direction D E. This -being admitted, it follows that if the body be placed upon a prop, -immediately under the direction of the line D E, or be suspended -from a fixed point immediately above its direction, it will remain -motionless. For the whole attracting force in the direction D E -will, in the one case, press the body on the prop, and, in the other -case, will give tension to the cord, rod, or whatever other means of -suspension be used. - -(149.) But suppose the body were suspended from some point P, not in -the direction of the line D E. Let P C be the direction of -the thread by which the body is suspended. Its whole weight, according -to the supposition which we have adopted, must then act in the -direction C E. Taking C F to represent the weight; it may be -considered as mechanically equivalent to two forces (74), C I and -C H. Of these C H, acting directly from the point P, merely -produces pressure upon it, and gives tension to the cord P C; but -C I, acting at right angles to C P, produces motion round P -as a centre, and in the direction C I, towards a vertical line -P G, drawn through the point P. If the body A B had been on -the other side of the line P G, it would have moved in like manner -towards it, and therefore in the direction contrary to its present -motion. - -Hence we must infer, that when the body is suspended from a fixed -point, it cannot remain at rest, if that fixed point be not placed in -the direction of the line D E; and, on the other hand, that if -the fixed point _be_ in the direction of that line, it cannot move. A -practical test is thus suggested, by which the line D E may be at -once discovered. Let a thread be attached to any point of the body, and -let it be suspended by this thread from a hook or other fixed point. -The direction of the thread, when the body becomes quiescent, will -be that of a single force equivalent to the gravitation of all the -component parts of the mass. - -(150.) An enquiry is here suggested: does the direction of the -equivalent force thus determined depend on the position of the body -with respect to the surface of the earth, and how is the direction -of the equivalent force affected by a change in that position? This -question may be at once solved if the body be suspended by different -points, and the directions which the suspending thread takes in each -case relatively to the figure and dimensions of the body examined. - -The body being suspended in this manner from any point, let a small -hole be bored through it, in the exact direction of the thread, so that -if the thread were continued below the point where it is attached to -the body, it would pass through this hole. The body being successively -suspended by several different points on its surface, let as many small -holes be bored through it in the same manner. If the body be then cut -through, so as to discover the directions which the several holes have -taken, they will be all found to cross each other at one point within -the body; or the same fact may be discovered thus: a thin wire, which -nearly fills the holes being passed through any one of them, it will be -found to intercept the passage of a similar wire through any other. - -This singular fact teaches us, what indeed can be proved by -mathematical reasoning without experiment, that there is _one_ point in -every body through which the single force, which is equivalent to the -gravitation of all its particles, must pass, in whatever position the -body be placed. This point is called _the centre of gravity_. - -(151.) In whatever situation a body may be placed, the centre of -gravity will have a tendency to descend in the direction of a -line perpendicular to the horizon, and which is called the _line -of direction_ of the weight. If the body be altogether free and -unrestricted by any resistance or impediment, the centre of gravity -will actually descend in this direction, and all the other points of -the body will move with the same velocity in parallel directions, -so that during its fall the position of the parts of the body, with -respect to the ground, will be unaltered. But if the body, as is most -usual, be subject to some resistance or restraint, it will either -remain unmoved, its weight being expended in exciting pressure on the -restraining points or surfaces, or it will move in a direction and -with a velocity depending on the circumstances which restrain it. - -In order to determine these effects, to predict the pressure produced -by the weight if the body be quiescent, or the mixed effects of motion -and pressure, if it be not so, it is necessary in all cases to be -able to assign the place of the centre of gravity. When the magnitude -and figure of the body, and the density of the matter which occupies -its dimensions, are known, the place of the centre of gravity can be -determined with the greatest precision by mathematical calculation. -The process by which this is accomplished, however, is not of a -sufficiently elementary nature to be properly introduced into this -treatise. To render it intelligible would require the aid of some -of the most advanced analytical principles; and even to express the -position of the point in question, except in very particular instances, -would be impossible, without the aid of peculiar symbols. - -(152.) There are certain particular forms of body in which, when they -are uniformly dense, the place of the centre of gravity can be easily -assigned, and proved by reasoning, which is generally intelligible; -but in all cases whatever, this point may be easily determined by -experiment. - -(153.) If a body uniformly dense have such a shape that a point may be -found on either side of which in all directions around it the materials -of the body are similarly distributed, that point will obviously be -the centre of gravity. For if it be supported, the gravitation of the -particles on one side drawing them downwards, is resisted by an effect -of exactly the same kind and of equal amount on the opposite side, and -so the body remains balanced on the point. - -The most remarkable body of this kind is a globe, the centre of which -is evidently its centre of gravity. - -A figure, such as _fig. 38._, called an _oblate spheroid_, has its -centre of gravity at its centre, C. Such is the figure of the earth. -The same may be observed of the elliptical solid, _fig. 39._, -which is called a prolate spheroid. - -A cube, and some other regular solids, bounded by plane surfaces, have -a point within them, such as above described, and which is, therefore, -their centre of gravity. Such are _fig. 40._ - -A straight wand of uniform thickness has its centre of gravity at the -centre of its length; and a cylindrical body has its centre of gravity -in its centre, at the middle of its length or axis. Such is the point -C, _fig. 41._ - -A flat plate of any uniform substance, and which has in every part -an equal thickness, has its centre of gravity at the middle of its -thickness, and under a point of its surface, which is to be determined -by its shape. If it be circular or elliptical, this point is its -centre. If it have any regular form, bounded by straight edges, it is -that point which is equally distant from its several angles, as C in -_fig. 42._ - -(154.) There are some cases in which, although the place of the centre -of gravity is not so obvious as in the examples just given, still -it may be discovered without any mathematical process, which is not -easily understood. Suppose A B C, _fig. 43._, to be -a flat triangular plate of uniform thickness and density. Let it be -imagined to be divided into narrow bars, by lines parallel to the side -A C, as represented in the figure. Draw B D from the angle -B to the middle point D of the side A C. It is not difficult to -perceive, that B D will divide equally all the bars into which the -triangle is conceived to be divided. Now if the flat triangular plate -A B C be placed in a horizontal position on a straight edge -coinciding with the line B D, it will be balanced: for the bars -parallel to A C will be severally balanced by the edge immediately -under their middle point; since that middle point is the centre of -gravity of each bar. Since, then, the triangle is balanced on the edge, -the centre of gravity must be somewhere immediately over it, and must, -therefore, be within the plate at some point under the line B D. - -The same reasoning will prove that the centre of gravity of the plate -is under the line A E, drawn from the angle A to the middle -point E of the side B C. To perceive this, it is only necessary -to consider the triangle divided into bars parallel to B C, -and thence to show that it will be balanced on an edge placed under -A E. Since then the centre of gravity of the plate is under the -line B D, and also under A E, it must be under the point G, -at which these lines cross each other; and it is accordingly at a depth -beneath G, equal to half the thickness of the plate. - -This may be experimentally verified by taking a piece of tin or card, -and cutting it into a triangular form. The point G being found by -drawing B D and A E, which divide two sides equally, it will -be balanced if placed upon the point of a pin at G. - -The centre of gravity of a triangle being thus determined, we shall -be able to find the position of the centre of gravity of any plate of -uniform thickness and density which is bounded by straight edges, as -will be shown hereafter. (173.) - -(155.) The centre of gravity is not always included within the volume -of the body, that is, it is not enclosed by its surfaces. Numerous -examples of this can be produced. If a piece of wire be bent into any -form, the centre of gravity will rarely be in the wire. Suppose it be -brought to the form of a ring. In that case, the centre of gravity of -the wire will be the centre of the circle, a point not forming any -part of the wire itself: nevertheless this point may be proved to have -the characteristic property of the centre of gravity; for if the ring -be suspended by any point, the centre of the ring must always settle -itself under the point of suspension. If this centre could be supposed -to be connected with the ring by very fine threads, whose weight would -be insignificant, and which might be united by a knot or otherwise at -the centre, the ring would be balanced upon a point placed under the -knot. - -In like manner, if the wire be formed into an ellipse, or any other -curve similarly arranged round a centre point, that point will be its -centre of gravity. - -(156.) To find the centre of gravity experimentally, the method -described in (149, 150) may be used. In this case two points of -suspension will be sufficient to determine it; for the directions of -the suspending cord being continued through the body, will cross each -other at the centre of gravity. These directions may also be found -by placing the body on a sharp point, and adjusting it so as to be -balanced upon it. In this case a line drawn through the body directly -upwards from the point will pass through the centre of gravity, and -therefore two such lines must cross at that point. - -(157.) If the body have two flat parallel surfaces like sheet metal, -stiff paper, card, board, &c., the centre of gravity may be found by -balancing the body in two positions on an horizontal straight edge. -The point where the lines marked by the edge cross each other will -be immediately under the centre of gravity. This may be verified by -showing that the body will be balanced on a point thus placed, or that -if it be suspended, the point thus determined will always come under -the point of suspension. - -The position of the centre of gravity of such bodies may also be found -by placing the body on an horizontal table having a straight edge. -The body being moved beyond the edge until it is in that position in -which the slightest disturbance will cause it to fall, the centre of -gravity will then be immediately over the edge. This being done in two -positions, the centre of gravity will be determined as before. - -(158.) It has been already stated, that when the body is perfectly -free, the centre of gravity must necessarily move downwards, in a -direction perpendicular to an horizontal plane. When the body is not -free, the circumstances which restrain it generally permit the centre -of gravity to move in certain directions, but obstruct its motion in -others. Thus if a body be suspended from a fixed point by a flexible -cord, the centre of gravity is free to move in every direction except -those which would carry it farther from the point of suspension than -the length of the cord. Hence if we conceive a globe or sphere to -surround the point of suspension on every side to a distance equal to -that of the centre of gravity from the point of suspension, when the -cord is fully stretched, the centre of gravity will be at liberty to -move in every direction within this sphere. - -There are an infinite variety of circumstances under which the motion -of a body may be restrained, and in which a most important and useful -class of mechanical problems originate. Before we notice others, -we shall, however, examine that which has just been described more -particularly. - -Let P, _fig. 44._, be the point of suspension, and C the centre -of gravity, and suppose the body so placed that C shall be within the -sphere already described. The cord will therefore be slackened, and in -this state the body will be free. The centre of gravity will therefore -descend in the perpendicular direction until the cord becomes fully -extended; the tension will then prevent its further motion in the -perpendicular direction. The downward force must now be considered as -the diagonal of a parallelogram, and equivalent to two forces C D -and C E, in the directions of the sides, as already explained in -(149). The force C D will bring the centre of gravity into the -direction P F, perpendicularly under the point of suspension. -Since the force of gravity acts continually on C in its approach to -P F, it will move towards that line with accelerated speed, and -when it has arrived there it will have acquired a force to which no -obstruction is immediately opposed, and consequently by its inertia -it retains this force, and moves beyond P F on the other side. -But when the point C gets into the line P F, it is in the lowest -possible position; for it is at the lowest point of the sphere which -limits its motion. When it passes to the other side of P F, it -must therefore begin to ascend, and the force of gravity, which, in the -former case, accelerated its descent, will now for the same reason, and -with equal energy, oppose its ascent. This will be easily understood. -Let C′ be any point which it may have attained in ascending; -C′ G′, the force of gravity, is now equivalent to C′ D′ and -C′ E′. The latter as before produces tension; but the former -C′ D′ is in a direction immediately opposed to the motion, and -therefore retards it. This retardation will continue until all the -motion acquired by the body in its descent from the first position -has been destroyed, and then it will begin to return to P F, and -so it will continue to vibrate from the one side to the other until -the friction on the point P, and the resistance of the air, gradually -deprive it of its motion, and bring it to a state of rest in the -direction P F. - -But for the effects of friction and atmospheric resistance, the body -would continue for ever to oscillate equally from side to side of the -line P F. - -(159.) The phenomenon just developed is only an example of an extensive -class. Whenever the circumstances which restrain the body are of such -a nature that the centre of gravity is prevented from descending below -a certain level, but not, on the other hand, restrained from rising -above it, the body will remain at rest if the centre of gravity be -placed at the lowest limit of its level; any disturbance will cause -it to oscillate around this state, and it cannot return to a state of -rest until friction or some other cause have deprived it of the motion -communicated by the disturbing force. - -(160.) Under the circumstances which we have just described, the body -could not maintain itself in a state of rest in any position except -that in which the centre of gravity is, at the lowest point of the -space in which it is free to move. This, however, is not always the -case. Suppose it were suspended by an inflexible rod instead of a -flexible string; the centre of gravity would then not only be prevented -from receding from the point of suspension, but also from approaching -it; in fact, it would be always kept at the same distance from it. -Thus, instead of being capable of moving anywhere within the sphere, -it is now capable of moving on its surface only. The reasoning used in -the last case may also be applied here, to prove that when the centre -of gravity is on either side of the perpendicular P F, it will -fall towards P F and oscillate, and that if it be placed in the -line P F, it will remain in equilibrium. But in this case there is -another position, in which the centre of gravity may be placed so as to -produce equilibrium. If it be placed at the highest point of the sphere -in which it moves, the whole force acting on it will then be directed -on the point of suspension, perpendicularly downwards, and will be -entirely expended in producing pressure on that point; consequently, -the body will in this case be in equilibrium. But this state of -equilibrium is of a character very different from that in which the -centre of gravity was at the lowest part of the sphere. In the present -case any displacement, however slight, of the centre of gravity, will -carry it to a lower level, and the force of gravity will then prevent -its return to its former state, and will impel it downwards until it -attain the lowest point of the sphere, and round that point it will -oscillate. - -(161.) The two states of equilibrium which have been just noticed, are -called stable and instable equilibrium. The character of the former is, -that any disturbance of the state produces oscillation about it; but -any disturbance of the latter state produces a total overthrow, and -finally causes oscillation around the state of stable equilibrium. - -Let A B, _fig. 45._, be an elliptical board resting on -its edge on an horizontal plane. In the position here represented, -the extremity P of the lesser axis being the point of support, the -board is in stable equilibrium; for any motion on either side must -cause the centre of gravity C to ascend in the directions C O, -and oscillation will ensue. If, however, it rest upon the smaller -end, as in _fig. 46._, the position would still be a state of -equilibrium, because the centre of gravity is directly above the point -of support; but it would be instable equilibrium, because the slightest -displacement of the centre of gravity would cause it to descend. - -Thus an egg or a lemon may be balanced on the end, but the least -disturbance will overthrow it. On the contrary, it will easily rest on -the side, and any disturbance will produce oscillation. - -(162.) When the circumstances under which the body is placed allow the -centre of gravity to move only in an horizontal line, the body is in a -state which may be called _neutral equilibrium_. The slightest force -will move the centre of gravity, but will neither produce oscillation -nor overthrow the body, as in the last two cases. - -An example of this state is furnished by a cylinder placed upon an -horizontal plane. As the cylinder is rolled upon the plane, the centre -of gravity C, _fig. 47._, moves in a line parallel to the plane -A B, and distant from it by the radius of the cylinder. The body -will thus rest indifferently in any position, because the line of -direction always falls upon a point P at which the body rests upon the -plane. - -If the plane were inclined, as in _fig. 48._, a body might be -so shaped, that while it would roll the centre of gravity would move -horizontally. In this case the body would rest indifferently on any -part of the plane, as if it were horizontal, provided the friction be -sufficient to prevent the body from sliding down the plane. - -If the centre of gravity of a cylinder happen not to coincide with -its centre by reason of the want of uniformity in the materials of -which it is composed, it will not be in a state of neutral equilibrium -on an horizontal plane, as in _fig. 47._ In this case let G, -_fig. 49._, be the centre of gravity. In the position here -represented, where the centre of gravity is immediately _below_ the -centre C, the state will be stable equilibrium, because a motion -on either side would cause the centre of gravity to ascend; but in -_fig. 50._, where G is immediately above C, the state is instable -equilibrium, because a motion on either side would cause G to descend, -and the body would turn into the position _fig. 49._ - -(163.) A cylinder of this kind will, under certain circumstances, roll -up an inclined plane. Let A B, _fig. 51._, be the inclined -plane, and let the cylinder be so placed that the line of direction -from G shall be _above_ the point P at which the cylinder rests upon -the plane. The whole weight of the body acting in the direction -G D will obviously cause the cylinder to roll towards A, provided -the friction be sufficient to prevent sliding; but although the -cylinder in this case ascends, the centre of gravity G really descends. - -When G is so placed that the line of direction G D shall fall on -the point P, the cylinder will be in equilibrium, because its weight -acts upon the point on which it rests. There are two cases represented -in _fig. 52._ and _fig. 53._, in which G takes this position. -_Fig. 52._ represents the state of stable, and _fig. 53._ of -instable equilibrium. - -(164.) When a body is placed upon a base, its stability depends upon -the position of the line of direction and the height of the centre of -gravity above the base. If the line of direction fall within the base, -the body will stand firm; if it fall on the edge of the base, it will -be in a state in which the slightest force will overthrow it on that -side at which the line of direction falls; and if the line of direction -fall without the base, the body must turn over that edge which is -nearest to the line of direction. - -In _fig. 54._ and _fig. 55._, the line of direction G P -falls within the base, and it is obvious that the body will stand firm; -for any attempt to turn it over either edge would cause the centre of -gravity to ascend. But in _fig. 56._ the line of direction falls -upon the edge, and if the body be turned over, the centre of gravity -immediately commences to descend. Until it be turned over, however, the -centre of gravity is supported by the edge. - -In _fig. 57._ the line of direction falls outside the base, the -centre of gravity has a tendency to descend from G towards A, and the -body will accordingly fall in that direction. - -(165.) When the line of direction falls within the base, bodies will -always stand firm, but not with the same degree of stability. In -general, the stability depends on the height through which the centre -of gravity must be elevated before the body can be overthrown. The -greater this height is, the greater in the same proportion will be the -stability. - -Let B A C, _fig. 58._, be a pyramid, the centre of -gravity being at G. To turn this over the edge B, the centre of -gravity; must be carried over the arch G E, and must therefore -be raised through the height H E. If, however, the pyramid were -taller relatively to its base, as in _fig. 59._, the height -H E would be proportionally less; and if the base were very small -in reference to the height, as in _fig. 60._, the height H E -would be very small, and a slight force would throw it over the edge B. - -It is obvious that the same observations may be applied to all figures -whatever, the conclusions just deduced depending only on the distance -of the line of direction from the edge of the base, and the height of -the centre of gravity above it. - -(166.) Hence we may perceive the principle on which the stability of -loaded carriages depends. When the load is placed at a considerable -elevation above the wheels, the centre of gravity is elevated, and the -carriage becomes proportionally insecure. In coaches for the conveyance -of passengers, the luggage is therefore sometimes placed below the body -of the coach; light parcels of large bulk may be placed on the top with -impunity. - -When the centre of gravity of a carriage is much elevated, there is -considerable danger of overthrow, if a corner be turned sharply and -with a rapid pace; for the centrifugal force then acting on the centre -of gravity will easily raise it through the small height which is -necessary to turn the carriage over the external wheels (142). - -(167.) The same waggon will have greater stability when loaded with -a heavy substance which occupies a small space, such as metal, than -when it carries the same weight of a lighter substance, such as hay; -because the centre of gravity in the latter case will be much more -elevated. - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -If a large table be placed upon a single leg in its centre, it will -be impracticable to make it stand firm; but if the pillar on which it -rests terminate in a tripod, it will have the same stability as if it -had three legs attached to the points directly over the places where -the feet of the tripod rest. - -(168.) When a solid body is supported by more points than one, it is -not necessary for its stability that the line of direction should fall -on one of those points. If there be only two points of support, the -line of direction must fall between them. The body is in this case -supported as effectually as if it rested on an edge coinciding with a -straight line drawn from one point of support to the other. If there -be three points of support, which are not ranged in the same straight -line, the body will be supported in the same manner as it would be by -a base coinciding with the triangle formed by straight lines joining -the three points of support. In the same manner, whatever be the -number of points on which the body may rest, its virtual base will be -found by supposing straight lines drawn, joining the several points -successively. When the line of direction falls within this base, the -body will always stand firm, and otherwise not. The degree of stability -is determined in the same manner as if the base were a continued -surface. - -(169.) Necessity and experience teach an animal to adapt its postures -and motions to the position of the centre of gravity of his body. When -a man stands, the line of direction of his weight must fall within the -base formed by his feet. If A B, C D, _fig. 61._, be the -feet, this base is the space A B D C. It is evident, -that the more his toes are turned outwards, the more contracted the -base will be in the direction E F, and the more liable he will be -to fall backwards or forwards. Also, the closer his feet are together, -the more contracted the base will be in the direction G H, and the -more liable he will be to fall towards either side. - -When a man walks, the legs are alternately lifted from the ground, and -the centre of gravity is either unsupported or thrown from the one -side to the other. The body is also thrown a little forward, in order -that the tendency of the centre of gravity to fall in the direction of -the toes may assist the muscular action in propelling the body. This -forward inclination of the body increases with the speed of the motion. - -But for the flexibility of the knee-joint the labour of walking would -be much greater than it is; for the centre of gravity would be more -elevated by each step. The line of motion of the centre of gravity in -walking is represented by _fig. 62._, and deviates but little -from a regular horizontal line, so that the elevation of the centre -of gravity is subject to very slight variation. But if there were no -knee-joint, as when a man has wooden legs, the centre of gravity would -move as in _fig. 63._, so that at each step the weight of the body -would be lifted through a considerable height, and therefore the labour -of walking would be much increased. - -If a man stand on one leg, the line of direction of his weight must -fall within the space on which his foot treads. The smallness of this -space, compared with the height of the centre of gravity, accounts for -the difficulty of this feat. - -The position of the centre of gravity of the body changes with the -posture and position of the limbs. If the arm be extended from one -side, the centre of gravity is brought nearer to that side than it was -when the arm hung perpendicularly. When dancers, standing on one leg, -extend the other at right angles to it, they must incline the body in -the direction opposite to that in which the leg is extended, in order -to bring the centre of gravity over the foot which supports them. - -When a porter carries a load, his position must be regulated by the -centre of gravity of his body and the load taken together. If he bore -the load on his back, the line of direction would pass beyond his -heels, and he would fall backwards. To bring the centre of gravity -over his feet he accordingly leans forward, _fig. 64._ - -If a nurse carry a child in her arms, she leans back for a like reason. - -When a load is carried on the head, the bearer stands upright, that the -centre of gravity may be over his feet. - -In ascending a hill, we appear to incline forward; and in descending, -to lean backward, but in truth, we are standing upright with respect to -a level plane. This is necessary to keep the line of direction between -the feet, as is evident from _fig. 65._ - -A person sitting on a chair which has no back cannot rise from it -without either stooping forward to bring the centre of gravity over -the feet, or drawing back the feet to bring them under the centre of -gravity. - -A quadruped never raises both feet on the same side simultaneously, -for the centre of gravity would then be unsupported. Let -A B C D, _fig. 66._, be the feet. The base on -which it stands is A B C D, and the centre of gravity -is nearly over the point O, where the diagonals cross each other. The -legs A and C being raised together, the centre of gravity is supported -by the legs B and D, since it falls between them; and when B and D -are raised it is, in like manner, supported by the feet A and C. The -centre of gravity, however, is often unsupported for a moment; for the -leg B is raised from the ground before A comes to it, as is plain from -observing the track of a horse’s feet, the mark of A being upon or -before that of B. In the more rapid paces of all animals the centre of -gravity is at intervals unsupported. - -The feats of rope-dancers are experiments on the management of the -centre of gravity. The evolutions of the performer are found to be -facilitated by holding in his hand a heavy pole. His security in -this case depends, not on the centre of gravity of his body, but on -that of his body and the pole taken together. This point is near the -centre of the pole, so that, in fact, he may be said to hold in his -hands the point on the position of which the facility of his feats -depends. Without the aid of the pole the centre of gravity would be -within the trunk of the body, and its position could not be adapted to -circumstances with the same ease and rapidity. - -(170.) The centre of gravity of a mass of fluid is that point which -would have the properties which have been proved to belong to the -centre of gravity of a solid, if the fluid were solidified without -changing in any respect the quantity or arrangement of its parts. This -point also possesses other properties, in reference to fluids, which -will be investigated in HYDROSTATICS and PNEUMATICS. - -(171.) The centre of gravity of two bodies separated from one another, -is that point which would possess the properties ascribed to the centre -of gravity, if the two bodies were united by an inflexible line, the -weight of which might be neglected. To find this point mathematically -is a very simple problem. Let A and B, _fig. 67._, be the two -bodies, and _a_ and _b_ their centres of gravity. Draw the right line -_a b_, and divide it at C, in such a manner that _a_ C shall have -the same proportion to _b_ C as the mass of the body B has to the mass -of the body A. - -This may easily be verified experimentally. Let A and B be two bodies, -whose weight is considerable, in comparison with that of the rod -_a b_, which joins them. Let a fine silken string, with its -ends attached to them, be hung upon a pin; and on the same pin let a -plumb-line be suspended. In whatever position the bodies may be hung, -it will be observed that the plumb-line will cross the rod _a b_ -at the same point, and that point will divide the line _a b_ into -parts _a_ C and _b_ C, which are in the proportion of the mass of B to -the mass of A. - -(172.) The centre of gravity of three separate bodies is defined in the -same manner as that of two, and may be found by first determining the -centre of gravity of two; and then supposing their masses concentrated -at that point, so as to form one body, and finding the centre of -gravity of that and the third. - -In the same manner the centre of gravity of any number of bodies may be -determined. - -(173.) If a plate of uniform thickness be bounded by straight edges, -its centre of gravity may be found by dividing it into triangles by -diagonal lines, as in _fig. 68._, and having determined by (154) -the centres of gravity of the several triangles, the centre of gravity -of the whole plate will be their common centre of gravity, found as -above. - -(174.) Although the centre of gravity takes its name from the -familiar properties which it has in reference to detached bodies of -inconsiderable magnitude, placed on or near the surface of the earth, -yet it possesses properties of a much more general and not less -important nature. One of the most remarkable of these is, that the -centre of gravity of any number of separate bodies is never affected -by the mutual attraction, impact, or other influence which the bodies -may transmit from one to another. This is a necessary consequence of -the equality of action and reaction explained in Chapter IV. For if A -and B, _fig. 67._, attract each other, and change their places -to A′ and B′, the space a a′ will have to _b b′_ the same -proportion as B has to A, and therefore by what has just been proved -(171) the same proportion as _a_ C has to _b_ C. It follows, that the -remainders _a′_ C and _b′_ C will be in the proportion of B to A, and -that C will continue to be the centre of gravity of the bodies after -they have approached by their mutual attraction. - -Suppose, for example, that A and B were 12lbs. and 8lbs. respectively, -and that _a b_ were 40 feet. The point C must (171) divide -_a b_ into two parts, in the proportion of 8 to 12, or of 2 to -3. Hence it is obvious that _a_ C will be 16 feet, and _b_ C 24 feet. -Now, suppose that A and B attract each other, and that A approaches -B through two feet. Then B must approach A through three feet. Their -distances from C will now be 14 feet and 21 feet, which, being in the -proportion of B to A, the point C will still be their centre of gravity. - -Hence it follows, that if a system of bodies, placed at rest, be -permitted to obey their mutual attractions, although the bodies will -thereby be severally moved, yet their common centre of gravity must -remain quiescent. - -(175.) When one of two bodies is moving in a straight line, the other -being at rest, their common centre of gravity must move in a parallel -straight line. Let A and B, _fig. 69._, be the centres of gravity -of the bodies, and let A move from A to _a_, B remaining at rest. -Draw the lines A B and _a_ B. In every position which the body B -assumes during its motion, the centre of gravity C divides the line -joining them into parts A C, B C, which are in the proportion -of the mass B to the mass A. Now, suppose any number of lines drawn -from B to the line A _a_; a parallel C _c_ to A _a_ through C divides -all these lines in the same proportion; and therefore, while the body A -moves from A to _a_, the common centre of gravity moves from C to _c_. - -If both the bodies A and B moved uniformly in straight lines, the -centre of gravity would have a motion compounded (74) of the two -motions with which it would be affected, if each moved while the other -remained at rest. In the same manner, if there were three bodies, each -moving uniformly in a straight line, their common centre of gravity -would have a motion compounded of that motion which it would have if -one remained at rest while the other two moved, and that which the -motion of the first would give it if the last two remained at rest; and -in the same manner it may be proved, that when any number of bodies -move each in a straight line, their common centre of gravity will have -a motion compounded of the motions which it receives from the bodies -severally. - -It may happen that the several motions which the centre of gravity -receives from the bodies of the system will neutralise each other; and -this does, in fact, take place for such motions as are the consequences -of the mutual action of the bodies upon one another. - -(176.) If a system of bodies be not under the immediate influence of -any forces, and their mutual attraction be conceived to be suspended, -they must severally be either at rest or in uniform rectilinear -motion in virtue of their inertia. Hence, their common centre of -gravity must also be either at rest or in uniform rectilinear motion. -Now, if we suppose their mutual attractions to take effect, the -state of their common centre of gravity will not be changed, but the -bodies will severally receive motions compounded of their previous -uniform rectilinear motions and those which result from their mutual -attractions. The combined effects will cause each body to revolve in -an orbit round the common centre of gravity, or will precipitate it -towards that point. But still that point will maintain its former state -undisturbed. - -This constitutes one of the general laws of mechanical science, and is -of great importance in physical astronomy. It is known by the title -“the conservation of the motion of the centre of gravity.” - -(177.) The solar system is an instance of the class of phenomena to -which we have just referred. All the motions of the bodies which -compose it can be traced to certain uniform rectilinear motions, -received from some former impulse, or from a force whose action has -been suspended, and those motions which necessarily result from the -principle of gravitation. But we shall not here insist further on this -subject, which more properly belongs to another department of the -science. - -(178.) If a solid body suffer an impact in the direction of a line -passing through its centre of gravity, all the particles of the body -will be driven forward with the same velocity in lines parallel to the -direction of the impact, and the whole force of the motion will be -equal to that of the impact. The common velocity of the parts of the -body will in this case be determined by the principles explained in -Chapter IV. The impelling force being equally distributed among all the -parts, the velocity will be found by dividing the numerical value of -that force by the number expressing the mass. - -If any number of impacts be given simultaneously to different points -of a body, a certain complex motion will generally ensue. The mass -will have a relative motion round the centre of gravity as if it were -fixed, while that point will move forward uniformly in a straight line, -carrying the body with it. The relative motion of the mass round the -centre of gravity may be found by considering the centre of gravity -as a fixed point, round which the mass is free to move, and then -determining the motion which the applied forces would produce. This -motion being supposed to continue uninterrupted, let all the forces be -imagined to be applied in their proper directions and quantities to the -centre of gravity. By the principles for the composition of force they -will be mechanically equivalent to a single force through that point. -In the direction of this single force the centre of gravity will move -and have the same velocity as if the whole mass were there concentrated -and received the impelling forces. - -(179.) These general properties, which are entirely independent of -gravity, render the “centre of gravity” an inadequate title for this -important point. Some physical writers have, consequently, called it -the “centre of inertia.” The “centre of gravity,” however, is the name -by which it is still generally designated. - - - - -CHAP. X. - -THE MECHANICAL PROPERTIES OF AN AXIS. - - -(180.) When a body has a motion of rotation, the line round which it -revolves is called an _axis_. Every point of the body must in this -case move in a circle, whose centre lies in the axis, and whose radius -is the distance of the point from the axis. Sometimes while the body -revolves, the axis itself is moveable, and not unfrequently in a state -of actual motion. The motions of the earth and planets, or that of -a common spinning-top, are examples of this. The cases, however, which -will be considered in the present chapter, are chiefly those in which -the axis is immovable, or at least where its motion has no relation to -the phenomena under investigation. Instances of this are so frequent -and obvious, that it seems scarcely necessary to particularise them. -Wheel-work of every description, the moving parts of watches and -clocks, turning lathes, mill-work, doors and lids on hinges, are all -obvious examples. In tools or other instruments which work on joints or -pivots, such as scissors, shears, pincers, although the joint or pivot -be not absolutely fixed, it is to be considered so in reference to the -mechanical effect. - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -In some cases, as in most of the wheels of watches and clocks, -fly-wheels and chucks of the turning lathe, and the arms of wind-mills, -the body turns continually in the same direction, and each of its -points traverses a complete circle during every revolution of the -body round its axis. In other instances the motion is alternate or -reciprocating, its direction being at intervals reversed. Such is -the case in pendulums of clocks, balance-wheels of chronometers, the -treddle of the lathe, doors and lids on hinges, scissors, shears, -pincers, &c. When the alternation is constant and regular, it is called -_oscillation_ or _vibration_, as in pendulums and balance-wheels. - -(181.) To explain the properties of an axis of rotation it will be -necessary to consider the different kinds of forces to the action of -which a body moveable on such an axis may be submitted, to show how -this action depends on their several quantities and directions, to -distinguish the cases in which the forces neutralise each other and -mutually equilibrate from those in which motion ensues, to determine -the effect which the axis suffers, and, in the cases where motion is -produced, to estimate the effects of those centrifugal forces (137.) -which are created by the mass of the body whirling round the axis. - -Forces in general have been distinguished by the duration of their -action into instantaneous and continued forces. The effect of an -instantaneous force is produced in an infinitely short time. If -the body which sustains such an action be previously quiescent and -free, it will move with a uniform velocity in the direction of the -impressed force. (93.) If, on the other hand, the body be not free, -but so restrained that the impulse cannot put it in motion, then the -fixed points or lines which resist the motion sustain a corresponding -shock at the moment of the impulse. This effect, which is called -_percussion_, is, like the force which causes it, instantaneous. - -A continued force produces a continued effect. If the body be free and -previously quiescent, this effect is a continual increase of velocity. -If the body be so restrained that the applied force cannot put it in -motion, the effect is a continued pressure on the points or lines which -sustain it. (94.) - -It may happen, however, that although the body be not absolutely free -to move in obedience to the force applied to it, yet still it may not -be altogether so restrained as to resist the effect of that force and -remain at rest. If the point at which a force is applied be free to -move in a certain direction not coinciding with that of the applied -force, that force will be resolved into two elements; one of which is -in the direction in which the point is free to move, and the other at -right angles to that direction. The point will move in obedience to -the former element, and the latter will produce percussion or pressure -on the points or lines which restrain the body. In fact, in such cases -the resistance offered by the circumstances which confine the motion -of the body modifies the motion which it receives, and as every change -of motion must be the consequence of a force applied (44.), the fixed -points or lines which offer the resistance must suffer a corresponding -effect. - -It may happen that the forces impressed on the body, whether they -be continued or instantaneous, are such as, were it free, would -communicate to it a motion which the circumstances which restrain it -do not forbid it to receive. In such a case the fixed points or lines -which restrain the body sustain no force, and the phenomena will be -the same in all respects as if these points or lines were not fixed. - -It will be easy to apply these general reflections to the case in which -a solid body is moveable on a fixed axis. Such a body is susceptible -of no motion except one of rotation on that axis. If it be submitted -to the action of instantaneous forces, one or other of the following -effects must ensue. 1. The axis may resist the forces, and prevent any -motion. 2. The axis may modify the effect of the forces sustaining a -corresponding percussion, and the body receiving a motion of rotation. -3. The forces applied may be such as would cause the body to spin round -the axis even were it not fixed, in which case the body will receive a -motion of rotation, but the axis will suffer no percussion. - -What has been just observed of the effect of instantaneous forces is -likewise applicable to continued ones. 1. The axis may entirely resist -the effect of such forces, in which case it will suffer a pressure -which may be estimated by the rules for the composition of force. 2. -It may modify the effect of the applied forces, in which case it must -also sustain a pressure, and the body must receive a motion of rotation -which is subject to constant variation, owing to the incessant action -of the forces. 3. The forces may be such as would communicate to the -body the same rotatory motion if the axis were not fixed. In this case -the forces will produce no pressure on the axis. - -The impressed forces are not the only causes which affect the axis of -a body during the phenomenon of rotation. This species of motion calls -into action other forces depending on the inertia of the mass, which -produce effects upon the axis, and which play a prominent part in the -theory of rotation. While the body revolves on its axis, the component -particles of its mass move in circles, the centres of which are placed -in the axis. The radius of the circle in which each particle moves is -the line drawn from that particle perpendicular to the axis. It has -been already proved that a particle of matter, moving round a centre, -is attended with a centrifugal force proportionate to the radius of the -circle in which it moves and to the square of its angular velocity. -When a solid body revolves on its axis, all its parts are whirled round -together, each performing a complete revolution in the same time. The -angular velocity is consequently the same for all, and the difference -of the centrifugal forces of different particles must entirely depend -upon their distances from the axis. The tendency of each particle to -fly from the axis, arising from the centrifugal force, is resisted by -the cohesion of the parts of the mass, and in general this tendency is -expended in exciting a pressure or strain upon the axis. It ought to -be recollected, however, that this pressure or strain is altogether -different from that already mentioned, and produced by the forces which -give motion to the body. The latter depends entirely upon the quantity -and directions of the applied forces in relation to the axis: the -former depends on the figure and density of the body, and the velocity -of its motion. - -These very complex effects render a simple and elementary exposition -of the mechanical properties of a fixed axis a matter of considerable -difficulty. Indeed, the complete mathematical development of this -theory long eluded the skill of the most acute geometers, and it was -only at a comparatively late period that it yielded to the searching -analysis of modern science. - -(182.) To commence with the most simple case, we shall consider the -body as submitted to the action of a single force. The effect of this -force will vary according to the relation of its direction to that of -the axis. There are two ways in which a body may be conceived to be -moveable around an axis. 1. By having pivots at two points which rest -in sockets, so that when the body is moved it must revolve round the -right line joining the pivots as an axis. 2. A thin cylindrical rod may -pass through the body, on which it may turn in the same manner as a -wheel upon its axle. - -If the force be applied to the body in the direction of the axis, it -is evident that no motion can ensue, and the effect produced will be a -pressure on that pivot towards which the force is directed. If in this -case the body revolved on a cylindrical rod, the tendency of the force -would be to make it slide along the rod without revolving round it. - -Let us next suppose the force to be applied not in the direction of -the axis itself, but parallel to it. Let A B, _fig. 70._, -be the axis, and let C D be the direction of the force applied. -The pivots being supposed to be at A and B, draw A G and B F -perpendicular to A B. The force C D will be equivalent to -three forces, one acting from B towards A, equal in quantity to the -force C D. This force will evidently produce a corresponding -pressure on the pivot A. The other two forces will act in the -directions A G and B F, and will have respectively to the -force C D the same proportion as A E has to A B. Such -will be the mechanical effect of a force C D parallel to the axis. -And as these effects are all directed on the pivots, no motion can -ensue. - -If the body revolve on a cylindrical rod, the forces A G and -B F would produce a strain upon the axis, while the third force -in the direction B A would have a tendency to make the body slide -along it. - -(183.) If the force applied to the body be directed upon the axis, -and at right angles to it, no motion can be produced. In this case, -if the body be supported by pivots at A and B, the force K L, -perpendicular to the line A B, will be distributed between the -pivots, producing a pressure on each proportional to its distance from -the other. The pressure on A having to the pressure on B the same -proportion as L B has to L A. - -If the force K H be directed obliquely to the axis, it will be -equivalent to two forces (76.), one K L perpendicular to the axis, -and the other K M parallel to it. The effect of each of these may -be investigated as in the preceding cases. - -In all these observations the body has been supposed to be submitted -to the action of one force only. If several forces act upon it, the -direction of each of them crossing the axis either perpendicularly -or obliquely, or taking the direction of the axis or any parallel -direction, their effects may be similarly investigated. In the same -manner we may determine the effects of any number of forces whose -combined results are mechanically equivalent to forces which either -intersect the axis or are parallel to it. - -(184.) If any force be applied whose direction lies in a plane oblique -to the axis, it can always be resolved into two elements (76.), one of -which is parallel to the axis, and the other in a plane perpendicular -to it. The effect of the former has been already determined, and -therefore we shall at present confine our attention to the latter. - -Suppose the axis to be perpendicular to the paper, and to pass through -the point G, _fig. 71._ and let A B C be a section of -the body. It will be convenient to consider the section vertical and -the axis horizontal, omitting, however, any notice of the effect of the -weight of the body. - -Let a weight W be suspended by a cord Q W from any point Q. This -weight will evidently have a tendency to turn the body round in the -direction A B C. Let another cord be attached to any other -point P, and, being carried over a wheel R, let a dish S be attached to -it, and let fine sand be poured into this dish until the tendency of -S to turn the body round the axis in the direction of C B A -balances the opposite tendency of W. Let the weights of W and S be -then exactly ascertained, and also let the distances G I and -G H of the cords from the axis be exactly measured. It will be -found that, if the number of ounces in the weight S be multiplied by -the number of inches in G H, and also the number of ounces in W -by the number of inches in G I, equal products will be obtained. -This experiment may be varied by varying the position of the wheel R, -and thereby changing the direction of the string P R, in which -cases it will be always found necessary to vary the weight of S in -such a manner, that when the number of ounces in it is multiplied by -the number of inches in the distance of the string from the axis, the -product obtained shall be equal to that of the weight W by the distance -G I. We have here used ounces and inches as the measures of weight -and distance; but it is obvious that any other measures would be -equally applicable. - -From what has been just stated it follows, that the energy of the -weight of S to move the body on its axis, does not depend alone upon -the actual amount of that weight, but also upon the distance of the -string from the axis. If, while the position of the string remains -unaltered, the weight of S be increased or diminished, the resisting -weight W must be increased or diminished in the same proportion. But -if, while the weight of S remains unaltered, the distance of the string -P R from the axis G be increased or diminished, it will be found -necessary to increase or diminish the resisting weight W in exactly the -same proportion. It therefore appears that the increase or diminution -of the distance of the direction of a force from the axis has the -same effect upon its power to give rotation as a similar increase -or diminution of the force itself. The power of a force to produce -rotation is, therefore, accurately estimated, not by the force alone, -but by the product found by multiplying the force by the distance of -its direction from the axis. It is frequently necessary in mechanical -science to refer to this power of a force, and, accordingly, the -product just mentioned has received a particular denomination. It is -called the _moment_ of the force round the axis. - -(185.) The distance of the direction of a force from the axis is -sometimes called the _leverage_ of the force. The _moment_ of a force -is therefore found by multiplying the force by its leverage, and the -energy of a given force to turn a body round an axis is proportional to -the leverage of that force. - -From all that has been observed it may easily be inferred that, if -several forces affect a body moveable on an axis, having tendencies -to turn it in different directions, they will mutually neutralise -each other and produce equilibrium, if the sum of the moments of those -forces which tend to turn the body in one direction be equal to the -sum of the moments of those which tend to turn it in the opposite -direction. Thus, if the forces A, B, C, ... tend to turn the body from -right to left, and the distances of their directions from the axis be -_a_, _b_, _c_, ... and the forces A′, B′, C′, ... tend to move it from -left to right, and the distances of their directions from the axis be -_a′_, _b′_, _c′_, ...; then these forces will produce equilibrium, -if the products found by multiplying the ounces in A, B, C, ... -respectively by the inches in _a_, _b_, _c_, ... when added together -be equal to the products found by multiplying the ounces in A′, B′, -C′, ... by the inches in _a′_, _b′_, _c′_, ... respectively when added -together. But if either of these sets of products when added together -exceed the other, the corresponding set of forces will prevail, and the -body will revolve on its axis. - -(186.) When a body receives an impulse in a direction perpendicular to -the axis, but not crossing it, a uniform rotatory motion is produced. -The velocity of this motion depends on the force of the impulse, the -distance of the direction of the impulse from the axis, and the manner -in which the mass of the body is distributed round the axis. It is to -be considered that the whole force of the impulse is shared amongst the -various parts of the mass, and is transmitted to them from the point -where the impulse is applied by reason of the cohesion and tenacity -of the parts, and the impossibility of one part yielding to a force -without carrying all the other parts with it. The force applied acts -upon those particles nearer to the axis than its own direction under -advantageous circumstances; for, according to what has been already -explained, their power to resist the effect of the applied force is -small in the same proportion with their distance. On the other hand, -the applied force acts upon particles of the mass, at a greater -distance than its own direction, under circumstances proportionably -disadvantageous; for their resistance to the applied force is great in -proportion to their distances from the axis. - -Let C D, _fig. 72._, be a section of the body made by a plane -passing through the axis A B. Suppose the impulse to be applied at -P, perpendicular to this plane, and at the distance P O from the -axis. The effect of the impulse being distributed through the mass will -cause the body to revolve on A B, with a uniform velocity. There -is a certain point G, at which, if the whole mass were concentrated, -it would receive from the impulse the same velocity round the axis. -The distance O G is called the _radius of gyration_ of the axis -A B, and the point G is called the _centre of gyration_ relatively -to that axis. The effect of the impulse upon the mass concentrated at -G is great in exactly the same proportion as O G is small. This -easily follows from the property of moments which has been already -explained; from whence it may be inferred, that the greater the radius -of gyration is, the less will be the velocity which the body will -receive from a given impulse. - -(187.) Since the radius of gyration depends on the manner in which the -mass is arranged round the axis, it follows that for different axes -in the same body there will be different radii of gyration. Of all -axes taken in the same body parallel to each other, that which passes -through the centre of gravity has the least radius of gyration. If the -radius of gyration of any axis passing through the centre of gravity be -given, that of any parallel axis can be found; for the square of the -radius of gyration of any axis is equal to the square of the distance -of that axis from the centre of gravity added to the square of the -radius of gyration of the parallel axis through the centre of gravity. - -(188.) The product of the numerical expressions for the mass of the -body and the square of the radius of gyration is a quantity much used -in mechanical science, and has been called the _moment of inertia_. The -moments of inertia, therefore, for different axes in the same body are -proportional to the squares of the corresponding radii of gyration; and -consequently increase as the distances of the axes from the centre of -gravity increase. (187.) - -(189.) From what has been explained in (187.), it follows, that the -moment of inertia of any axis may be computed by common arithmetic, if -the moment of inertia of a parallel axis through the centre of gravity -be previously known. To determine this last, however, would require -analytical processes altogether unsuitable to the nature and objects of -the present treatise. - -The velocity of rotation which a body receives from a given impulse -is great in exactly the same proportion as the moment of inertia is -small. Thus the moment of inertia may be considered in rotatory motion -analogous to the mass of the body in rectilinear motion. - -From what has been explained in (187.) it follows that a given impulse -at a given distance from the axis will communicate the greatest -angular velocity when the axis passes through the centre of gravity, -and that the velocity which it will communicate round other axes -will be diminished in the same proportion as the squares of their -distances from the centre of gravity added to the square of the radius -of gyration for a parallel axis through the centre of gravity are -augmented. - -(190.) If any point whatever be assumed in a body, and right lines -be conceived to diverge in all directions from that point, there are -generally two of these lines, which being taken as axes of rotation, -one has a greater and the other a less moment of inertia than any of -the others. It is a remarkable circumstance, that, whatever be the -nature of the body, whatever be its shape, and whatever be the position -of the point assumed, these two axes of greatest and least moment will -always be at right angles to each other. - -These axes and a third through the same point, and at right angles to -both of them, are called the _principal axes_ of that point from which -they diverge. To form a distinct notion of their relative position, -let the axis of greatest moment be imagined to lie horizontally from -north to south, and the axis of least moment from east to west; then -the third principal axis will be presented perpendicularly upwards and -downwards. The first two being called the principal axes of greatest -and least moment, the third may be called the _intermediate principal -axis_. - -(191.) Although the moments of the three principal axes be in general -unequal, yet bodies may be found having certain axes for which these -moments may be equal. In some cases the moment of the intermediate axis -is equal to that of the principal axis of greatest moment: in others it -is equal to that of the principal axis of least moment, and in others -the moments of all the three principal axes are equal to each other. - -If the moments of any two of three principal axes be equal, the moments -of all axes through the same point and in their plane will also be -equal; and if the moments of the three principal axes through a point -be equal, the moments of all axes whatever, through the same point, -will be equal. - -(192.) If the moments of the principal axes through the centre of -gravity be known, the moments for all other axes through that point may -be easily computed. To effect this it is only necessary to multiply -the moments of the principal axes by the squares of the co-sines of -the angles formed by them respectively with the axis whose moment is -sought. The products being added together will give the required moment. - -(193.) By combining this result with that of (189.), it will be evident -that the moment of all axes whatever may be determined, if those of the -principal axes through the centre of gravity be known. - -(194.) It is obvious that the principal axis of least moment through -the centre of gravity has a less moment of inertia than any other axis -whatever. For it has, by its definition (190.) a less moment of inertia -than any other axis through the centre of gravity, and every other -axis through the centre of gravity has a less moment of inertia than a -parallel axis through any other point (187.) and (189.) - -(195.) If two of the principal axes through the centre of gravity have -equal moments of inertia, all axes in any plane parallel to the plane -of these axes, and passing through the point where a perpendicular from -the centre of gravity meets that plane, must have equal moments of -inertia. For by (191.) all axes in the plane of those two have equal -moments, and by (189.) the axes in the parallel plane have moments -which exceed these by the same quantity, being equally distant from -them. (187.) - -Hence it is obvious that if the three principal axes through the centre -of gravity have equal moments, all axes situated in any given plane, -and passing through the point where the perpendicular from the centre -of gravity meets that plane, will have equal moments, being equally -distant from parallel axes through the centre of gravity. - -(196.) If the three principal axes through the centre of gravity have -unequal moments, there is no point whatever for which all axes will -have equal moments; but if the principal axis of least moment and -the intermediate principal axis through the centre of gravity have -equal moments, then there will be two points on the principal axis -of greatest moment, equally distant at opposite sides of the centre -of gravity, at which all axes will have equal moments. If the three -principal axes through the centre of gravity have equal moments, no -other point of the body can have principal axes of equal moment. - -(197.) When a body revolves on a fixed axis, the parts of its mass are -whirled in circles round the axis; and since they move with a common -angular velocity, they will have centrifugal forces proportional to -their distances from the axis. If the component parts of the mass were -not united together by cohesive forces of energies greater than these -centrifugal forces, they would be separated, and would fly off from -the axis; but their cohesion prevents this, and causes the effects of -the different centrifugal forces, which affect the different parts of -the mass, to be transmitted so as to modify each other, and finally -to produce one or more forces mechanically equivalent to the whole, -and which are exerted upon the axis and resisted by it. We propose -now to explain these effects, as far as it is possible to render them -intelligible without the aid of mathematical language. - -It is obvious that any number of equal parts of the mass, which are -uniformly arranged in a circle round the axis, have equal centrifugal -forces acting from the centre of the circle in every direction. These -mutually neutralise each other, and therefore exert no force on the -axis. The same may be said of all parts of the mass which are regularly -and equally distributed on every side of the axis. - -Also if equal masses be placed at equal distances on opposite sides -of the axis, their centrifugal forces will destroy each other. Hence -it appears that the pressure which the axis of rotation sustains from -the centrifugal forces of the revolving mass, arises from the unequal -distribution of the matter around it. - -From this reasoning it will be easily perceived that in the following -examples the axis of rotation will sustain no pressure. - -A globe revolving on any of its diameters, the density being the same -at equal distances from the centre. - -A spheroid or a cylinder revolving on its axis, the density being equal -at equal distances from the axis. - -A cube revolving on an axis which passes through the centre of two -opposite bases, being of uniform density. - -A circular plate of uniform thickness and density revolving on one of -its diameters as an axis. - -(198.) In all these examples it will be observed that the axis of -rotation passes through the centre of gravity. The general theorem, of -which they are only particular instances, is, “if a body revolve on a -principal axis, passing through the centre of gravity, the axis will -sustain no pressure from the centrifugal force of the revolving mass.” -This is a property in which the principal axes through the centre -of gravity are unique. There is no other axis on which a body could -revolve without pressure. - -If two of the principal axes through the centre of gravity have equal -moments, every axis in their plane has the same moment, and is to be -considered equally as a principal axis. In this case the body would -revolve on any of these axes without pressure. - -A homogeneous spheroid furnishes an example of this. If any of the -diameters of the earth’s equator were a fixed axis, the earth would -revolve on it without producing pressure. - -If the three principal axes through the centre of gravity have equal -moments, all axes through the centre of gravity are to be considered as -principal axes. In this case the body would revolve without pressure on -any axis through the centre of gravity. - -A globe, in which the density of the mass at equal distances from the -centre is the same, is an example of this. Such a body would revolve -without pressure on any axis through its centre. - -(199.) Since no pressure is excited on the axis in these cases, the -state of the body will not be changed, if during its rotation the axis -cease to be fixed. The body will notwithstanding continue to revolve -round the axis, and the axis will maintain its position. - -Thus a spinning-top of homogeneous material and symmetrical form will -revolve steadily in the same position, until the friction of its point -with the surface on which it rests deprives it of motion. This is a -phenomenon which can only be exhibited when the axis of rotation is a -principal axis through the centre of gravity. - -(200.) If the body revolve round any axis through the centre of -gravity, which is not a principal axis, the centrifugal pressure is -represented by two forces, which are equal and parallel, but which act -in opposite directions on different points of the axis. The effect of -these forces is to produce a strain upon the axis, and give the body a -tendency to move round another axis at right angles to the former. - -(201.) If the fixed axis on which a body revolves be a principal axis -through any point different from the centre of gravity, then a pressure -will be produced by the centrifugal force of the revolving mass, and -this pressure will act at right angles to the axis on the point to -which it is a principal axis, and in the plane through that axis and -the centre of gravity. The amount of the pressure will be proportional -to the mass of the body, the distance of the centre of gravity from the -axis, and the square of the velocity of rotation. - -(202.) Since the whole pressure is in this case excited on a single -point, the stability of the axis will not be disturbed, provided that -point alone be fixed. So that even though the axis should be free to -turn on that point, no motion will ensue as long as no external forces -act upon the body. - -(203.) If the axis of rotation be not a principal axis, the centrifugal -forces will produce an effect which cannot be represented by a single -force. The effect may be understood by conceiving two forces to act -on _different points_ of the axis at right angles to it and to each -other. The quantities of these pressures and their directions depend -on the figure and density of the mass and the position of the axis, -in a manner which cannot be explained without the aid of mathematical -language and principles. - -(204.) The effects upon the axis which have been now explained are -those which arise from the motion of rotation, from whatever cause that -motion may have arisen. The forces which produce that motion, however, -are attended with effects on the axis which still remain to be noticed. -When these forces, whether they be of the nature of instantaneous -actions or continued forces, are entirely resisted by the axis, their -directions must severally be in a plane passing through the axis, or -they must, by the principles of the composition of force [(74.) et -seq.], be mechanically equivalent to forces in that plane. In every -other case the impressed forces _must_ produce motion, and, except in -certain cases, must also produce effects upon the axis. - -By the rules for the composition of force it is possible in all cases -to resolve the impressed forces into others which are either in planes -through the axis, or in planes perpendicular to it, or, finally, some -in planes through it, and others in planes perpendicular to it. The -effect of those which are in planes through the axis has been already -explained; and we shall now confine our attention to those impelling -forces which act at right angles to the axis, and which produce motion. - -It will be sufficient to consider the effect of a single force at right -angles to the axis; for whatever be the number of forces which act -either simultaneously or successively, the effect of the whole will -be decided by combining their separate effects. The effect which a -single force produces depends on two circumstances, 1. The position of -the axis with respect to the figure and mass of the body, and 2. The -quantity and direction of the force itself. - -In general the shock which the axis sustains from the impact may be -represented by two impacts applied to it at different points, one -parallel to the impressed force, and the other perpendicular to it, -but both perpendicular to the axis. There are certain circumstances, -however, under which this effect will be modified. - -If the impulse which the body receives be in a direction perpendicular -to a plane through the axis and the centre of gravity, and at a -distance from the axis which bears to the radius of gyration (186.) -the same proportion as that line bears to the distance of the centre -of gravity from the axis, there are certain cases in which the impulse -will produce no percussion. To characterise these cases generally would -require analytical formulæ which cannot conveniently be translated -into ordinary language. That point of the plane, however, where the -direction of the impressed force meets it, when no percussion on the -axis is produced, is called the _centre of percussion_. - -If the axis of rotation be a principal axis, the centre of percussion -must be in the right line drawn through the centre of gravity, -intersecting the axis at right angles, and at the distance from the -axis already explained. - -If the axis of rotation be parallel to a principal axis through the -centre of gravity, the centre of percussion will be determined in the -same manner. - -(205.) There are many positions which the axis may have in which there -will be no centre of percussion; that is, there will be no direction in -which an impulse could be applied without producing a shock upon the -axis. One of these positions is when it is a principal axis through -the centre of gravity. This is the only case of rotation round an axis -in which no effect arises from the centrifugal force; and therefore it -follows that the only case in which the axis sustains no effect from -the motion produced, is one in which it must necessarily suffer an -effect from that which produces the motion. - -If the body be acted upon by continued forces, their effect is at each -instant determined by the general principles for the composition of -force. - - - - -CHAP. XI. - -ON THE PENDULUM. - - -(206.) When a body is placed on a horizontal axis which does not -pass through its centre of gravity, it will remain in permanent -equilibrium only when the centre of gravity is immediately below the -axis. If this point be placed in any other situation, the body will -oscillate from side to side, until the atmospherical resistance and the -friction of the axis destroy its motion. (159, 160.) Such a body is -called a _pendulum_. The swinging motion which it receives is called -_oscillation_ or _vibration_. - -(207.) The use of the pendulum, not only for philosophical purposes, -but in the ordinary economy of life, renders it a subject of -considerable importance. It furnishes the most exact means of measuring -time, and of determining with precision various natural phenomena. By -its means the variation of the force of gravity in different latitudes -is discovered, and the law of that variation experimentally exhibited. -In the present chapter, we propose to explain the general principles -which regulate the oscillation of pendulums. Minute details concerning -their construction will be given in the twenty-first chapter of this -volume. - -(208.) A simple pendulum is composed of a heavy molecule attached -to the end of a flexible thread, and suspended by a fixed point O, -_fig. 73._ When the pendulum is placed in the position O C, -the molecule being vertically below the point of suspension, it will -remain in equilibrium; but if it be drawn into the position O A -and there liberated, it will descend towards C, moving through the arc -A C with accelerated motion. Having arrived at C and acquired -a certain velocity, it will, by reason of its inertia, continue to -move in the same direction. It will therefore commence to ascend the -arc C A′ with the velocity so acquired. During its ascent, the -weight of the molecule retards its motion in exactly the same manner -as it had accelerated it in descending from A to C; and when the -molecule has ascended through the arc C A′ equal to C A, -its entire velocity will be destroyed, and it will cease to move in -that direction. It will thus be placed at A′ in the same manner as in -the first instance it had been placed at A, and consequently it will -descend from A′ to C with accelerated motion, in the same manner as -it first moved from A to C. It will then ascend from C to A, and so -on, continually. In this case the thread, by which the molecule is -suspended, is supposed to be perfectly flexible, inextensible, and -of inconsiderable weight. The point of suspension is supposed to be -without friction, and the atmosphere to offer no resistance to the -motion. - -It is evident from what has been stated, that the times of moving from -A to A′ and from A′ to A are equal, and will continue to be equal so -long as the pendulum continues to vibrate. If the number of vibrations -performed by the pendulum were registered, and the time of each -vibration known, this instrument would become a chronometer. - -The rate at which the motion of the pendulum is accelerated in its -descent towards its lowest position is not uniform, because the force -which impels it is continually decreasing, and altogether disappears -at the point C. The impelling force arises from the effect of gravity -on the suspended molecule, and this effect is always produced in the -vertical direction A V. The greater the angle O A V is, -the less efficient the force of gravity will be in accelerating the -molecule: this angle evidently increases as the molecule approaches -C, which will appear by inspecting _fig. 73._ At C, the force of -gravity acting in the direction C B is totally expended in giving -tension to the thread, and is inefficient in moving the molecule. It -follows, therefore, that the impelling force is greatest at A, and -continually diminishes from A to C, where it altogether vanishes. The -same observations will be applicable to the retarding force from C to -A′, and to the accelerating force from A′ to C, and so on. - -When the length of the thread and the intensity of the force of -gravity are given, the time of vibration depends on the length of the -arc A C, or on the magnitude of the angle A O C. If, -however, this angle do not exceed a certain limit of magnitude, the -time of vibration will be subject to no sensible variation, however -that angle may vary. Thus the time of oscillation will be the same, -whether the angle A O C be 2°, or 1° 30′, or 1°, or any -lesser magnitude. This property of a pendulum is expressed by the word -_isochronism_. The strict demonstration of this property depends on -mathematical principles, the details of which would not be suitable -to the present treatise. It is not difficult, however, to explain -generally how it happens that the same pendulum will swing through -greater and smaller arcs of vibration in the same time. If it swing -from A, the force of gravity at the commencement of its motion impels -it with an effect depending on the obliquity of the lines O A and -A V. If it commence its motion from _a_, the impelling effect from -the force of gravity will be considerably less than at A; consequently, -the pendulum begins to move at a slower rate, when it swings from -_a_ than when it moves from A: the greater magnitude of the swing is -therefore compensated by the increased velocity, so that the greater -and the smaller arcs of vibration are moved through in the same time. - -(209.) To establish this property experimentally, it is only necessary -to suspend a small ball of metal, or other heavy substance, by a -flexible thread, and to put it in a state of vibration, the entire -arc of vibration not exceeding 4° or 5°, the friction on the point -of suspension and other causes will gradually diminish the arc of -vibration, so that after the lapse of some hours it will be so small, -that the motion will scarcely be discerned without microscopic aid. If -the vibration of this pendulum be observed in reference to a correct -timekeeper, at the commencement, at the middle, and towards the end of -its motion, the rate will be found to suffer no sensible change. - -This remarkable law of isochronism was one of the earliest discoveries -of Galileo. It is said, that when very young, he observed a chandelier -suspended from the roof of a church in Pisa swinging with a pendulous -motion, and was struck with the uniformity of the rate even when the -extent of the swing was subject to evident variation. - -(210.) It has been stated in (117.) that the attraction of gravity -affects all bodies equally, and moves them with the same velocity, -whatever be the nature or quantity of the materials of which they are -composed. Since it is the force of gravity which moves the pendulum, we -should therefore expect that the circumstances of that motion should -not be affected either by the quantity or quality of the pendulous -body. And we find this, in fact, to be the case; for if small pieces -of different heavy substances such as lead, brass, ivory, &c., be -suspended by fine threads of equal length, they will vibrate in the -same time, provided their weights bear a considerable proportion to the -atmospherical resistance, or that they be suspended _in vacuo_. - -(211.) Since the time of vibration of a pendulum, which oscillates in -small arcs, depends neither on the magnitude of the arc of vibration -nor on the quality or weight of the pendulous body, it will be -necessary to explain the circumstances on which the variation of this -time depends. - -The first and most striking of these circumstances is the length of -the suspending thread. The rudest experiments will demonstrate the -fact, that every increase in the length of this thread will produce a -corresponding increase in the time of vibration; but according to what -law does this increase proceed? If the length of the thread be doubled -or trebled, will the time of vibration also be increased in a double -or treble proportion? This problem is capable of exact mathematical -solution, and the result shows that the time of vibration increases not -in the proportion of the increased length of the thread, but as the -square root of that length; that is to say, if the length of the thread -be increased in a four-fold proportion, the time of vibration will be -augmented in a two-fold proportion. If the thread be increased to nine -times its length, the time of vibration will be trebled, and so on. -This relation is exactly the same as that which was proved to subsist -between the spaces through which a body falls freely, and the times -of fall. In the table, page 89, if the figures representing the -height be understood to express the length of different pendulums, the -figures immediately above them will express the corresponding times of -vibration. - -This law of the proportion of the lengths of pendulums to the squares -of the time of vibration may be experimentally established in the -following manner:-- - -Let A, B, C, _fig. 74._, be three small pieces of metal each -attached by threads to two points of suspension, and let them be placed -in the same vertical line under the point O; suppose them so adjusted -that the distances O A, O B, and O C shall be in the -proportion of the numbers 1, 4, and 9. Let them be removed from the -vertical in a direction at right angles to the plane of the paper, so -that the threads shall be in the same plane, and therefore the three -pendulums will have the same angle of vibration. Being now liberated, -the pendulum A will immediately gain upon B, and B upon C, so that A -will have completed one vibration before B or C. At the end of the -second vibration of A, the pendulum B will have arrived at the end of -its first vibration, so that the suspending threads of A and B will -then be separated by the whole angle of vibration; at the end of the -fourth vibration of A the suspending threads of A and B will return -to their first position, B having completed two vibrations; thus the -proportion of the times of vibration of B and A will be 2 to 1, the -proportion of their lengths being 4 to 1. At the end of the third -vibration of A, C will have completed one vibration, and the suspending -strings will coincide in the position distant by the whole angle of -vibration from their first position. So that three vibrations of A are -performed in the same time as one of C: the proportion of the time of -vibration of C and A are, therefore, 3 to 1, the proportion of their -lengths being 9 to 1, conformably to the law already explained. - -(212.) In all the preceding observations we have assumed that the -material of the pendulous body is of inconsiderable magnitude, its -whole weight being conceived to be collected in a physical point. -This is generally called a simple pendulum; but since the conditions -of a suspending thread without weight, and a heavy molecule without -magnitude, cannot have practical existence, the simple pendulum must -be considered as imaginary, and merely used to establish hypothetical -theorems, which, though inapplicable in practice, are nevertheless the -means of investigating the laws which govern the real phenomena of -pendulous bodies. - -A pendulous body being of determinate magnitude, its several parts -will be situated at different distances from the axis of suspension. -If each component part of such a body were separately connected with -the axis of suspension by a fine thread, it would, being unconnected -with the other particles, be an independent simple pendulum, and -would oscillate according to the laws already explained. It therefore -follows that those particles of the body which are nearest to the -axis of suspension would, if liberated from their connection with the -others, vibrate more rapidly than those which are more remote. The -connection, however, which the particles of the body have, by reason -of their solidity, compels them all to vibrate in the same time. -Consequently, those particles which are nearer the axis are retarded -by the slower motion of those which are more remote; while the more -remote particles, on the other hand, are urged forward by the greater -tendency of the nearer particles to rapid vibration. This will be more -readily comprehended, if we conceive two particles of matter A and B, -_fig. 75._, to be connected with the same axis O by an inflexible -wire O C, the weight of which may be neglected. If B were removed, -A would vibrate in a certain time depending upon the distance O A. -If A were removed, and B placed upon the wire at a distance B O -equal to four times A O, B would vibrate in twice the former time. -Now if both be placed on the wire at the distances just mentioned, -the tendency of A to vibrate more rapidly will be transmitted to B -by means of the wire, and will urge B forward more quickly than if -A were not present: on the other hand, the tendency of B to vibrate -more slowly will be transmitted by the wire to A, and will cause it to -move more slowly than if B were not present. The inflexible quality -of the connecting wire will in this case compel A and B to vibrate -simultaneously, the time of vibration being greater than that of A, and -less than that of B, if each vibrated unconnected with the other. - -If, instead of supposing two particles of matter placed on the wire, -a greater number were supposed to be placed at various distances from -O, it is evident the same reasoning would be applicable. They would -mutually affect each other’s motion; those placed nearest to point -O accelerating the motion of those more remote, and being themselves -retarded by the latter. Among these particles one would be found -in which all these effects would be mutually neutralised, all the -particles nearer O being retarded in reference to that motion which -they would have if unconnected with the rest, and those more remote -being in the same respect accelerated. The point at which such a -particle is placed is called _the centre of oscillation_. - -What has been here observed of the effects of particles of matter -placed upon rigid wire will be equally applicable to the particles of -a solid body. Those which are nearer to the axis are urged forward by -those which are more remote, and are in their turn retarded by them; -and as with the particles placed upon the wire, there is a certain -particle of the body at which the effects are mutually neutralised, and -which vibrates in the same time as it would if it were unconnected with -the other parts of the body, and simply connected by a fine thread to -the axis. By this centre of oscillation the calculations respecting the -vibration of a solid body are rendered as simple as those of a molecule -of inconsiderable magnitude. All the properties which have been -explained as belonging to a simple pendulum may thus be transferred -to a vibrating body of any magnitude and figure, by considering it as -equivalent to a single particle of matter vibrating at its centre of -oscillation. - -(213.) It follows from this reasoning, that the virtual length of -a pendulum is to be estimated by the distance of its centre of -oscillation from the axis of suspension, and therefore that the times -of vibration of different pendulums are in the same proportion as the -square roots of the distances of their centres of oscillation from -their axes. - -The investigation of the position of the centre of oscillation is, in -most cases, a subject of intricate mathematical calculation. It depends -on the magnitude and figure of the pendulous body, the manner in which -the mass is distributed through its volume, or the density of its -several parts, and the position of the axis on which it swings. - -The place of the centre of oscillation may be determined when the -position of the centre of gravity and the centre of gyration are known; -for the distance of the centre of oscillation from the axis will always -be obtained by dividing the square of the radius of gyration (186.) -by the distance of the centre of gravity from the axis. Thus if 6 be -the radius of gyration, and 9 the distance of gravity from the axis, -36 divided by 9, which is 4, will be the distance of the centre of -oscillation from the axis. Hence it may be inferred generally, that -the greater the proportion which the radius of gyration bears to the -distance of the centre of gravity from the axis, the greater will be -the distance of the centre of oscillation. - -It follows from this reasoning, that the length of a pendulum is not -limited by the dimensions of its volume. If the axis be so placed -that the centre of gravity is near it, and the centre of gyration -comparatively removed from it, the centre of oscillation may be placed -far beyond the limits of the pendulous body. Suppose the centre of -gravity is at a distance of one inch from the axis, and the centre -of gyration 12 inches, the centre of oscillation will then be at the -distance of 144 inches, or 12 feet. Such a pendulum may not in its -greatest dimensions exceed one foot, and yet its time of vibration -would be equal to that of a simple pendulum whose length is 12 feet. - -By these means pendulums of small dimensions may be made to vibrate as -slowly as may be desired. The instruments called _metronomes_, used -for marking the time of musical performances, are constructed on this -principle. - -(214.) The centre of oscillation is distinguished by a very remarkable -property in relation to the axis of suspension. If A, _fig. 76._, -be the point of suspension, and O the corresponding centre of -oscillation, the time of vibration of the pendulum will not be -changed if it be raised from its support, inverted, and suspended from -the point O. It follows, therefore, that if O be taken as the point -of suspension, A will be the corresponding centre of oscillation. -These two points are, therefore, convertible. This property may be -verified experimentally in the following manner. A pendulum being put -into a state of vibration, let a small heavy body be suspended by -a fine thread, the length of which is so adjusted that it vibrates -simultaneously with the pendulum. Let the distance from the point of -suspension to the centre of the vibrating body be measured, and take -this distance on the pendulum from the axis of suspension downwards; -the place of the centre of oscillation will thus be obtained, since -the distance so measured from the axis is the length of the equivalent -simple pendulum. If the pendulum be now raised from its support, -inverted, and suspended from the centre of oscillation thus obtained, -it will be found to vibrate simultaneously with the body suspended by -the thread. - -(215.) This property of the interchangeable nature of the centres -of oscillation and suspension has been, at a late period, adopted -by Captain Kater, as an accurate means of determining the length of -a pendulum. Having ascertained with great accuracy two points of -suspension at which the same body will vibrate in the same time, the -distance between these points being accurately measured, is the length -of the equivalent simple pendulum. See Chapter XXI. - -(216.) The manner in which the time of vibration of a pendulum -depends on its length being explained, we are next to consider how -this time is affected by the attraction of gravity. It is obvious -that, since the pendulum is moved by this attraction, the rapidity -of its motion will be increased, if the impelling force receive any -augmentation; but it still is to be decided, in what exact proportion -the time of oscillation will be diminished by any proposed increase -in the intensity of the earth’s attraction. It can be demonstrated -mathematically, that the time of one vibration of a pendulum has the -same proportion to the time of falling freely in the perpendicular -direction, through a height equal to half the length of the pendulum, -as the circumference of a circle has to its diameter. Since, therefore, -the times of vibration of pendulums are in a fixed proportion to the -times of falling freely through spaces equal to the halves of their -lengths, it follows that these times have the same relation to the -force of attraction as the times of falling freely through their -lengths have to that force. If the intensity of the force of gravity -were increased in a four-fold proportion, the time of falling through -a given height would be diminished in a two-fold proportion; if the -intensity were increased to a nine-fold proportion, the time of falling -through a given space would be diminished in a three-fold proportion, -and so on; the rate of diminution of the time being always as the -square root of the increased force. By what has been just stated this -law will also be applicable to the vibration of pendulums. Any increase -in the intensity of the force of gravity would cause a given pendulum -to vibrate more rapidly, and the increased rapidity of the vibration -would be in the same proportion as the square root of the increased -intensity of the force of gravity. - -(217.) The laws which regulate the times of vibration of pendulums in -relation to one another being well understood, the whole theory of -these instruments will be completed, when the method of ascertaining -the actual time of vibration of any pendulum, in reference to its -length, has been explained. In such an investigation, the two elements -to be determined are, 1. the exact time of a single vibration, and, -2. the exact distance of the centre of oscillation from the point of -suspension. - -The former is ascertained by putting a pendulum in motion in the -presence of a good chronometer, and observing precisely the number of -oscillations which are made in any proposed number of hours. The entire -time during which the pendulum swings, being divided by the number of -oscillations made during that time, the exact time of one oscillation -will be obtained. - -The distance of the centre of oscillation from the point of suspension -may be rendered a matter of easy calculation, by giving a certain -uniform figure and material to the pendulous body. - -(218.) The time of vibration of one pendulum of known length being -thus obtained, we shall be enabled immediately to solve either of the -following problems. - -“To find the length of a pendulum which shall vibrate in a given time.” - -“To find the time of vibration of a pendulum of a given length.” - -The former is solved as follows: the time of vibration of the known -pendulum is to the time of vibration of the required pendulum, as the -square root of the length of the known pendulum is to the square root -of the length of the required pendulum. This length is therefore found -by the ordinary rules of arithmetic. - -The latter may be solved as follows: the length of the known pendulum -is to the length of the proposed pendulum, as the square of the time -of vibration of the known pendulum is to the square of the time of -vibration of the proposed pendulum. The latter time may therefore be -found by arithmetic. - -(219.) Since the rate of a pendulum has a known relation to the -intensity of the earth’s attraction, we are enabled, by this -instrument, not only to detect certain variations in that attraction in -various parts of the earth, but also to discover the actual amount of -the attraction at any given place. - -The actual amount of the earth’s attraction at any given place is -estimated by the height through which a body would fall freely at that -place in any given time, as in one second. To determine this, let the -length of a pendulum which would vibrate in one second at that place -be found. As the circumference of a circle is to its diameter[2] (a -known proportion), so will one second be to the time of falling through -a height equal to half the length of this pendulum. This time is -therefore a matter of arithmetical calculation. It has been proved in -(120.), that the heights, through which a body falls freely, are in the -same proportion as the squares of the times; from whence it follows, -that the square of the time of falling through a height equal to half -the length of the pendulum is to one second as half the length of -that pendulum is to the height through which a body would fall in one -second. This height, therefore, may be immediately computed, and thus -the actual amount of the force of gravity at any given place may be -ascertained. - -[2] This ratio is that of 31,416 to 10,000 very nearly. - -(220.) To compare the force of gravity in different parts of the earth, -it is only necessary to swing the same pendulum in the places under -consideration, and to observe the rapidity of its vibrations. The -proportion of the force of gravity in the several places will be that -of the squares of the velocity of the vibration. Observations to this -effect have been made at several places, by Biot, Kater, Sabine, and -others. - -The earth being a mass of matter of a form nearly spherical, revolving -with considerable velocity on an axis, its component parts are affected -by a centrifugal force; in virtue of which, they have a tendency to fly -off in a direction perpendicular to the axis. This tendency increases -in the same proportion as the distance of any part from the axis -increases, and consequently those parts of the earth which are near the -equator, are more strongly affected by this influence than those near -the pole. It has been already explained (145.) that the figure of the -earth is affected by this cause, and that it has acquired a spheroidal -form. The centrifugal force, acting in opposition to the earth’s -attraction, diminishes its effects; and consequently, where this force -is more efficient, a pendulum will vibrate more slowly. By these means -the rate of vibration of a pendulum becomes an indication of the amount -of the centrifugal force. But this latter varies in proportion to the -distance of the place from the earth’s axis; and thus the rate of a -pendulum indicates the relation of the distances of different parts of -the earth’s surface from its axis. The figure of the earth may be thus -ascertained, and that which theory assigns to it, it may be practically -proved to have. - -This, however, is not the only method by which the figure of the earth -may be determined. The meridians being sections of the earth through -its axis, if their figure were exactly determined, that of the earth -would be known. Measurements of arcs of meridians on a large scale have -been executed, and are still being made in various parts of the earth, -with a view to determine the curvature of a meridian at different -latitudes. This method is independent of every hypothesis concerning -the density and internal structure of the earth, and is considered by -some to be susceptible of more accuracy than that which depends on the -observations of pendulums. - -(221.) It has been stated that, when the arc of vibration of a pendulum -is not very small, a variation in its length will produce a sensible -effect on the time of vibration. To construct a pendulum such that the -time of vibration may be independent of the extent of the swing, was a -favourite speculation of geometers. This problem was solved by Huygens, -who showed that the curve called a _cycloid_, previously discovered and -described by Galileo, possessed the isochronal property; that is, that -a body moving in it by the force of gravity, would vibrate in the same -time, whatever be the length of the arc described. - -Let O A, _fig. 77._, be a horizontal line, and let O B -be a circle placed below this line, and in contact with it. If this -circle be rolled upon the line from O towards A, a point upon its -circumference, which at the beginning of the motion is placed at O, -will during the motion trace the curve O C A. This curve is -called a _cycloid_. If the circle be supposed to roll in the opposite -direction towards A′, the same point will trace another cycloid -O C′ A′. The points C and C′ being the lowest points of the -curves, if the perpendiculars C D and C′ D′ be drawn, they -will respectively be equal to the diameter of the circle. By a known -property of this curve, the arcs O C and O C′ are equal to -twice the diameter of the circle. From the point O suppose a flexible -thread to be suspended, whose length is twice the diameter of the -circle, and which sustains a pendulous body P at its extremity. If -the curves O C and O C′, from the plane of the paper, be -raised so as to form surfaces to which the thread may be applied, the -extremity P will extend to the points C and C′, when the entire thread -has been applied to either of the curves. As the thread is deflected -on either side of its vertical position, it is applied to a greater -or lesser portion of either curve, according to the quantity of its -deflection from the vertical. If it be deflected on each side until -the point P reaches the points C and C′, the extremity would trace a -cycloid C P C′ precisely equal and similar to those already -mentioned. Availing himself of this property of the curve, Huygens -constructed his cycloidal pendulum. The time of vibration was subject -to no variation, however the arc of vibration might change, provided -only that the length of the string O P continued the same. If -small arcs of the cycloid be taken on either side of the point P, they -will not sensibly differ from arcs of a circle described with the -centre O and the radius O P; for, in slight deflections from the -vertical position, the effect of the curves O C and O C′ on -the thread O P is altogether inconsiderable. It is for this reason -that when the arcs of vibration of a circular pendulum are small, they -partake of the property of isochronism peculiar to those of a cycloid. -But when the deflection of P from the vertical is great, the effect of -the curves O C and O C′ on the thread produces a considerable -deviation of the point P from the arc of the circle whose centre is -O and whose radius is O P, and consequently the property of -isochronism will no longer be observed in the circular pendulum. - - - - -CHAP. XII. - -OF SIMPLE MACHINES. - - -(222.) A MACHINE is an instrument by which force or motion may be -transmitted and modified as to its quantity and direction. There are -two ways in which a machine may be applied, and which give rise to -a division of mechanical science into parts denominated STATICS and -DYNAMICS; the one including the theory of equilibrium, and the other -the theory of motion. When a machine is considered statically, it is -viewed as an instrument by which forces of determinate quantities -and direction are made to balance other forces of other quantities -and other directions. If it be viewed dynamically, it is considered -as a means by which certain motions of determinate quantity and -direction may be made to produce other motions in other directions -and quantities. It will not be convenient, however, in the present -treatise, to follow this division of the subject. We shall, on the -other hand, as hitherto, consider the phenomena of equilibrium and -motion together. - -The effects of machinery are too frequently described in such a manner -as to invest them with the appearance of paradox, and to excite -astonishment at what appears to contradict the results of the most -common experience. It will be our object here to take a different -course, and to attempt to show that those effects which have been held -up as matters of astonishment are the necessary, natural, and obvious -results of causes adapted to produce them in a manner analogous to the -objects of most familiar experience. - -(223.) In the application of a machine there are three things to -be considered. 1. The force or resistance which is required to be -sustained, opposed, or overcome. 2. The force which is used to sustain, -support, or overcome that resistance. 3. The machine itself by which -the effect of this latter force is transmitted to the former. Of -whatever nature be the force or the resistance which is to be sustained -or overcome, it is technically called the _weight_, since, whatever it -be, a weight of equivalent effect may always be found. The force which -is employed to sustain or overcome it is technically called the _power_. - -(224.) In expressing the effect of machinery it is usual to say that -the power sustains the weight; but this, in fact, is not the case, and -hence arises that appearance of paradox which has already been alluded -to. If, for example, it is said that a power of one ounce sustains the -weight of one ton, astonishment is not unnaturally excited, because -the fact, as thus stated, if the terms be literally interpreted, is -physically impossible. No power less than a ton can, in the ordinary -acceptation of the word, support the weight of a ton. It will, however, -be asked how it happens that a machine _appears_ to do this? how it -happens that by holding a silken thread, which an ounce weight would -snap, many hundred weight may be sustained? To explain this it will -only be necessary to consider the effect of a machine, when the power -and weight are in equilibrium. - -(225.) In every machine there are some fixed points or props; and the -arrangement of the parts is always such, that the pressure, excited by -the power or weight, or both, is distributed among these props. If the -weight amount to twenty hundred, it is possible so to distribute it, -that any proportion, however great, of it may be thrown on the fixed -points or props of the machine; the remaining part only can properly be -said to be supported by the power, and this part can never be greater -than the power. Considering the effect in this way, it appears that -the power supports just so much of the weight and no more as is equal -to its own force, and that all the remaining part of the weight is -sustained by the machine. The force of these observations will be more -apparent when the nature and properties of the mechanic powers and -other machines have been explained. - -(226.) When a machine is considered dynamically, its effects are -explained on different principles. It is true that, in this case, a -very small power may elevate a very great weight; but nevertheless, -in so doing, whatever be the machine used, the total expenditure of -power, in raising the weight through any height, is never less than -that which would be expended if the power were immediately applied to -the weight without the intervention of any machine. This circumstance -arises from an universal property of machines by which the velocity of -the weight is always less than that of the power, in exactly the same -proportion as the power itself is less than the weight; so that when -a certain power is applied to elevate a weight, the rate at which the -elevation is effected is always slow in the same proportion as the -weight is great. From a due consideration of this remarkable law, it -will easily be understood, that a machine can never diminish the total -expenditure of power necessary to raise any weight or to overcome any -resistance. In such cases, all that a machine ever does or ever can -do, is to enable the power to be expended at a slow rate, and in a -more advantageous direction than if it were immediately applied to the -weight or the resistance. - -Let us suppose that P is a power amounting to an ounce, and that W is -a weight amounting to 50 ounces, and that P elevates W by means of a -machine. In virtue of the property already stated, it follows, that -while P moves through 50 feet, W will be moved through 1 foot; but -in moving P through 50 feet, 50 distinct efforts are made, by each -of which 1 ounce is moved through 1 foot, and by which collectively -50 distinct ounces might be successively raised through 1 foot. But -the weight W is 50 ounces, and has been raised through 1 foot; from -whence it appears, that the expenditure of power is equal to that which -would be necessary to raise the weight without the intervention of any -machine. - -This important principle may be presented under another aspect, which -will perhaps render it more apparent. Suppose the weight W were -actually divided into 50 equal parts, or suppose it were a vessel of -liquid weighing 50 ounces, and containing 50 equal measures; if these -50 measures were successively lifted through a height of 1 foot; the -efforts necessary to accomplish this would be the same as those used -to move the power P through 50 feet, and it is obvious, that the total -expenditure of force would be the same as that which would be necessary -to lift the entire contents of the vessel through 1 foot. - -When the nature and properties of the mechanic powers and other -machines have been explained, the force of these observations will be -more distinctly perceived. The effects of props and fixed points in -sustaining a part of the weight, and sometimes the whole, both of the -weight and power, will then be manifest, and every machine will furnish -a verification of the remarkable proportion between the velocities -of the weight and power, which has enabled us to explain what might -otherwise be paradoxical and difficult of comprehension. - -(227.) The most simple species of machines are those which are commonly -denominated the MECHANIC POWERS. These have been differently enumerated -by different writers. If, however, the object be to arrange in distinct -classes, and in the smallest possible number of them, those machines -which are alike in principle, the mechanic powers may be reduced to -three. - - 1. The lever. - 2. The cord. - 3. The inclined plane. - -To one or other of these classes all simple machines whatever may be -reduced, and all complex machines may be resolved into simple elements -which come under them. - -(228.) The first class includes every machine which is composed of -a solid body revolving on a fixed axis, although the name lever has -been commonly confined to cases where the machine affects certain -particular forms. This is by far the most useful class of machines, and -will require in subsequent chapters very detailed development. The -general principle, upon which equilibrium is established between the -power and weight in machines of this class has been already explained -in (183.) The power and weight are always supposed to be applied in -directions at right angles to the axis. If lines be drawn from the axis -perpendicular to the directions of power and weight, equilibrium will -subsist, provided the power multiplied by the perpendicular distance -of its direction from the axis, be equal to the weight multiplied by -the perpendicular distance of its direction from the axis. This is a -principle to which we shall have occasion to refer in explaining the -various machines of this class. - -(229.) If the moment of the power (184.) be greater than that of the -weight, the effect of the power will prevail over that of the weight, -and elevate it; but if, on the other hand, the moment of the power be -less than that of the weight, the power will be insufficient to support -the weight, and will allow it to fall. - -(230.) The second class of simple machines includes all those cases -in which force is transmitted by means of flexible threads, ropes, -or chains. The principle, by which the effects of these machines are -estimated, is, that the tension throughout the whole length of the same -cord, provided it be perfectly flexible, and free from the effects -of friction, must be the same. Thus, if a force acting at one end be -balanced by a force acting at the other end, however the cord may be -bent, or whatever course it may be compelled to take, by any causes -which may affect it between its ends, these forces must be equal, -provided the cord be free to move over any obstacles which may deflect -it. - -Within this class of machines are included all the various forms of -_pulleys_. - -(231.) The third class of simple machines includes all those cases in -which the weight or resistance is supported or moved on a hard surface -inclined to the vertical direction. - -The effects of such machines are estimated by resolving the whole -weight of the body into two elements by the parallelogram of forces. -One of these elements is perpendicular to the surface, and supported -by its resistance; the other is parallel to the surface, and supported -by the power. The proportion, therefore, of the power to the weight -will always depend on the obliquity of the surface to the direction of -the weight. This will be easily understood by referring to what has -been already explained in Chapter VIII. - -Under this class of machines come the inclined plane, commonly so -called, the wedge, the screw, and various others. - -(232.) In order to simplify the development of the elementary theory -of machines, it is expedient to omit the consideration of many -circumstances, of which, however, a strict account must be taken before -any practically useful application of that theory can be attempted. -A machine, as we must for the present contemplate it, is a thing -which can have no real or practical existence. Its various parts are -considered to be free from friction: all surfaces which move in contact -are supposed to be infinitely smooth and polished. The solid parts are -conceived to be absolutely inflexible. The weight and inertia of the -machine itself are wholly neglected, and we reason upon it as if it -were divested of these qualities. Cords and ropes are supposed to have -no stiffness, to be infinitely flexible. The machine, when it moves, is -supposed to suffer no resistance from the atmosphere, and to be in all -respects circumstanced as if it were _in vacuo_. - -It is scarcely necessary to state, that, all these suppositions -being false, none of the consequences deduced from them can be true. -Nevertheless, as it is the business of art to bring machines as near -to this state of ideal perfection as possible, the conclusions which -are thus obtained, though false in a strict sense, yet deviate from -the truth in but a small degree. Like the first outline of a picture, -they resemble in their general features that truth to which, after many -subsequent corrections, they must finally approximate. - -After a first approximation has been made on the several false -suppositions which have been mentioned, various effects, which have -been previously neglected, are successively taken into account. -Roughness, rigidity, imperfect flexibility, the resistance of air and -other fluids, the effects of the weight and inertia of the machine, -are severally examined, and their laws and properties detected. The -modifications and corrections, thus suggested as necessary to be -introduced into our former conclusions, are applied, and a second -approximation, but still _only_ an approximation, to truth is made. -For, in investigating the laws which regulate the several effects -just mentioned, we are compelled to proceed upon a new group of false -suppositions. To determine the laws which regulate the friction of -surfaces, it is necessary to assume that every part of the surfaces of -contact are uniformly rough; that the solid parts which are imperfectly -rigid, and the cords which are imperfectly flexible, are constituted -throughout their entire dimensions of a uniform material; so that the -imperfection does not prevail more in one part than another. Thus, -all irregularity is left out of account, and a general average of the -effects taken. It is obvious, therefore, that by these means we have -still failed in obtaining a result exactly conformable to the real -state of things; but it is equally obvious, that we have obtained -one much more conformable to that state than had been previously -accomplished, and sufficiently near it for most practical purposes. - -This apparent imperfection in our instruments and powers of -investigation is not peculiar to mechanics: it pervades all departments -of natural science. In astronomy, the motions of the celestial bodies, -and their various changes and appearances as developed by theory, -assisted by observation and experience, are only approximations to the -real motions and appearances which take place in nature. It is true -that these approximations are susceptible of almost unlimited accuracy; -but still they are, and ever will continue to be, only approximations. -Optics and all other branches of natural science are liable to the same -observations. - - - - -CHAP. XIII. - -OF THE LEVER. - - -(233.) An inflexible, straight bar, turning on an axis, is commonly -called a _lever_. The _arms_ of the lever are those parts of the bar -which extend on each side of the axis. - -The axis is called the _fulcrum_ or _prop_. - -(234.) Levers are commonly divided into three kinds, according to the -relative positions of the power, the weight, and the fulcrum. - -In a lever of the first kind, as in _fig. 78._, the fulcrum is -between the power and weight. - -In a lever of the second kind, as in _fig. 79._, the weight is -between the fulcrum and power. - -In a lever of the third kind, as in _fig. 80._, the power is -between the fulcrum and weight. - -(235.) In all these cases, the power will sustain the weight in -equilibrium, provided its moment be equal to that of the weight. (184.) -But the moment of the power is, in this case, equal to the product -obtained by multiplying the power by its distance from the fulcrum; and -the moment of the weight by multiplying the weight by its distance from -the fulcrum. Thus, if the number of ounces in P, being multiplied by -the number of inches in P F, be equal to the number of ounces in -W, multiplied by the number of inches in W F, equilibrium will be -established. It is evident from this, that as the distance of the power -from the fulcrum increases in comparison to the distance of the weight -from the fulcrum, in the same degree exactly will the proportion of the -power to the weight diminish. In other words, the proportion of the -power to the weight will be always the same as that of their distances -from the fulcrum taken in a reverse order. - -In cases where a small power is required to sustain or elevate a great -weight, it will therefore be necessary either to remove the power to a -great distance from the fulcrum, or to bring the weight very near it. - -(236.) Numerous examples of levers of the first kind may be given. A -crow-bar, applied to elevate a stone or other weight, is an instance. -The fulcrum is another stone placed near that which is to be raised, -and the power is the hand placed at the other end of the bar. - -A handspike is a similar example. - -A poker applied to raise fuel is a lever of the first kind, the fulcrum -being the bar of the grate. - -Scissors, shears, nippers, pincers, and other similar instruments are -composed of two levers of the first kind; the fulcrum being the joint -or pivot, and the weight the resistance of the substance to be cut or -seized; the power being the fingers applied at the other end of the -levers. - -The brake of a pump is a lever of the first kind; the pump-rods and -piston being the weight to be raised. - -(237.) Examples of levers of the second kind, though not so frequent as -those just mentioned, are not uncommon. - -An oar is a lever of the second kind. The reaction of the water against -the blade is the fulcrum. The boat is the weight, and the hand of the -boatman the power. - -The rudder of a ship or boat is an example of this kind of lever, and -explained in a similar way. - -The chipping knife is a lever of the second kind. The end attached -to the bench is the fulcrum, and the weight the resistance of the -substance to be cut, placed beneath it. - -A door moved upon its hinges is another example. - -Nut-crackers are two levers of the second kind; the hinge which unites -them being the fulcrum, the resistance of the shell placed between -them being the weight, and the hand applied to the extremity being the -power. - -A wheelbarrow is a lever of the second kind; the fulcrum being the -point at which the wheel presses on the ground, and the weight being -that of the barrow and its load, collected at their centre of gravity. - -The same observation may be applied to all two-wheeled carriages, which -are partly sustained by the animal which draws them. - -(238.) In a lever of the third kind, the weight, being more distant -from the fulcrum than the power, must be proportionably less than -it. In this instrument, therefore, the power acts upon the weight to -a mechanical disadvantage, inasmuch as a greater power is necessary -to support or move the weight than would be required if the power -were immediately applied to the weight, without the intervention of a -machine. We shall, however, hereafter show that the advantage which -is lost in force is gained in despatch, and that in proportion as the -weight is less than the power which moves it, so will the speed of its -motion be greater than that of the power. - -Hence a lever of the third kind is only used in cases where the -exertion of great power is a consideration subordinate to those of -rapidity and despatch. - -The most striking example of levers of the third kind is found in the -animal economy. The limbs of animals are generally levers of this -description. The socket of the bone is the fulcrum; a strong muscle -attached to the bone near the socket is the power; and the weight -of the limb, together with whatever resistance is opposed to its -motion, is the weight. A slight contraction of the muscle in this case -gives a considerable motion to the limb: this effect is particularly -conspicuous in the motion of the arms and legs in the human body; a -very inconsiderable contraction of the muscles at the shoulders and -hips giving the sweep to the limbs from which the body derives so much -activity. - -The treddle of the turning lathe is a lever of the third kind. The -hinge which attaches it to the floor is the fulcrum, the foot applied -to it near the hinge is the power, and the crank upon the axis of the -fly-wheel, with which its extremity is connected, is the weight. - -Tongs are levers of this kind, as also the shears used in shearing -sheep. In these cases the power is the hand placed immediately below -the fulcrum or point where the two levers are connected. - -(239.) When the power is said to support the weight by means of a lever -or any other machine, it is only meant that the power keeps the machine -in equilibrium, and thereby enables it to sustain the weight. It is -necessary to attend to this distinction, to remove the difficulty which -may arise from the paradox of a small power sustaining a great weight. - -In a lever of the first kind, the fulcrum F, _fig. 78._, or axis, -sustains the united forces of the power and weight. - -In a lever of the second kind, if the power be supposed to act over -a wheel R, _fig. 79._, the fulcrum F sustains a pressure equal -to the difference between the power and weight, and the axis of the -wheel R sustains a pressure equal to twice the power; so that the total -pressures on F and R are equivalent to the united forces of the power -and weight. - -In a lever of the third kind similar observations are applicable. The -wheel R, _fig. 80._, sustains a pressure equal to twice the power, -and the fulcrum F sustains a pressure equal to the difference between -the power and weight. - -These facts may be experimentally established by attaching a string -to the lever immediately over the fulcrum, and suspending the lever -by that string from the arm of a balance. The counterpoising weight, -when the fulcrum is removed, will, in the first case, be equal to the -sum of the weight and power, and in the last two cases equal to their -difference. - -(240.) We have hitherto omitted the consideration of the effect of the -weight of the lever itself. If the centre of gravity of the lever be -in the vertical line through the axis, the weight of the instrument -will have no other effect than to increase the pressure on the axis by -its own amount. But if the centre of gravity be on the same side of -the axis with the weight, as at G, it will oppose the effect of the -power, a certain part of which must therefore be allowed to support -it. To ascertain what part of the power is thus expended, it is to -be considered that the moment of the weight of the lever collected -at G, is found by multiplying that weight by the distance G F. -The moment of that part of the power which supports this must be -equal to it; therefore, it is only necessary to find how much of the -power multiplied by P F will be equal to the weight of the lever -multiplied by G F. This is a question in common arithmetic. - -If the centre of gravity of the lever be at a different side of the -axis from the weight, as at G′, the weight of the instrument will -co-operate with the power in sustaining the weight W. To determine what -portion of the weight W is thus sustained by the weight of the lever, -it is only necessary to find how much of W, multiplied by the distance -W F, is equal to the weight of the lever multiplied by G′ F. - -In these cases the pressure on the fulcrum, as already estimated, will -always be increased by the weight of the lever. - -(241.) The sense in which a small power is said to sustain a great -weight, and the manner of accomplishing this, being explained, we -shall now consider how the power is applied in moving the weight. Let -P W, _fig. 81._, be the places of the power and weight, and -F that of the fulcrum, and let the power be depressed to P′ while the -weight is raised to W′. The space P P′ evidently bears the same -proportion to W W′, as the arm P F to W F. Thus if -P F be ten times W F, P P′ will be ten times W W′. -A power of one pound at P being moved from P to P′, will carry a weight -of ten pounds from W to W′. But in this case it ought not to be said, -that a lesser weight moves a greater, for it is not difficult to show, -that the total expenditure of force in the motion of one pound from P -to P′ is exactly the same as in the motion of ten pounds from W to W′. -If the space P P′ be ten inches, the space W W′ will be one -inch. A weight of one pound is therefore moved through ten successive -inches, and in each inch the force expended is that which would be -sufficient to move one pound through one inch. The total expenditure -of force from P to P′ is ten times the force necessary to move one -pound through one inch, or what is the same, it is that which would be -necessary to move ten pounds through one inch. But this is exactly what -is accomplished by the opposite end W of the lever; for the weight W is -ten pounds, and the space W W′ is one inch. - -If the weight W of ten pounds could be conveniently divided into ten -equal parts of one pound each, each part might be separately raised -through one inch, without the intervention of the lever or any other -machine. In this case, the same quantity of power would be expended, -and expended in the same manner as in the case just mentioned. - -It is evident, therefore, that when a machine is applied to raise a -weight or to overcome resistance, as much force must be really used as -if the power were immediately applied to the weight or resistance. All -that is accomplished by the machine is to enable the power to do that -by a succession of distinct efforts which should be otherwise performed -by a single effort. These observations will be found to be applicable -to all machines whatever. - -(242.) Weighing machines of almost every kind, whether used for -commercial or philosophical purposes, are varieties of the lever. The -common balance, which, of all weighing machines, is the most perfect -and best adapted for ordinary use, whether in commerce or experimental -philosophy, is a lever with equal arms. In the steel-yard one weight -serves as a counterpoise and measure of others of different amount, by -receiving a leverage variable according to the varying amount of the -weight against which it acts. A detailed account of such instruments -will be found in Chapter XXI. - -(243.) We have hitherto considered the power and weight as acting on -the lever, in directions perpendicular to its length and parallel to -each other. This does not always happen. Let A B, _fig. 83._, -be a lever whose fulcrum is F, and let A R be the direction of the -power, and B S the direction of the weight. If the lines R A -and S B be continued, and perpendiculars F C and F D -drawn from the fulcrum to those lines, the moment of the power will be -found by multiplying the power by the line F C, and the moment of -the weight by multiplying the weight by F D. If these moments be -equal, the power will sustain the weight in equilibrium. (185). - -It is evident, that the same reasoning will be applicable when the -arms of the lever are not in the same direction. These arms may be of -any figure or shape, and may be placed relatively to each other in any -position. - -(244.) In the rectangular lever the arms are perpendicular to each -other, and the fulcrum F, _fig. 84._, is at the right angle. The -moment of the power, in this case, is P multiplied by A F, and -that of the weight W multiplied by B F. When the instrument is in -equilibrium these moments must be equal. - -When the hammer is used for drawing a nail, it is a lever of this kind: -the claw of the hammer is the shorter arm; the resistance of the nail -is the weight; and the hand applied to the handle the power. - -(245.) When a beam rests on two props A B, _fig. 85._, and -supports, at some intermediate place C, a weight W, this weight is -distributed between the props in a manner which may be determined by -the principles already explained. If the pressure on the prop B be -considered as a power sustaining the weight W, by means of the lever of -the second kind B A, then this power multiplied by B A must -be equal to the weight multiplied by C A. Hence the pressure on -B will be the same fraction of the weight as the part A C is of -A B. In the same manner it may be proved, that the pressure on A -is the same fraction of the weight as B C is of B A. Thus, if -A C be one third, and therefore B C two thirds of B A, -the pressure on B will be one third of the weight, and the pressure on -A two thirds of the weight. - -It follows from this reasoning, that if the weight be in the middle, -equally distant from B and A, each prop will sustain half the weight. -The effect of the weight of the beam itself may be determined by -considering it to be collected at its centre of gravity. If this point, -therefore, be equally distant from the props, the weight of the beam -will be equally distributed between them. - -According to these principles, the manner in which a load borne -on poles between two bearers is distributed between them may be -ascertained. As the efforts of the bearers and the direction of the -weight are always parallel; the position of the poles relatively to the -horizon makes no difference in the distribution of the weights between -the bearers. Whether they ascend or descend, or move on a level plane, -the weight will be similarly shared between them. - -If the beam extend beyond the prop, as in _fig. 86._, and the -weight be suspended at a point not placed between them, the props must -be applied at different sides of the beam. The pressures which they -sustain may be calculated in the same manner as in the former case. -The pressure of the prop B may be considered as a power sustaining the -weight W by means of the lever B C. Hence, the pressure of B, -multiplied by B A, must be equal to the weight W multiplied by -A C. Therefore, the pressure on B bears the same proportion to the -weight as A C does to A B. In the same manner, considering B -as a fulcrum, and the pressure of the prop A as the power, it may be -proved that the pressure of A bears the same proportion to the weight -as the line B C does to A B. It therefore appears, that the -pressure on the prop A is greater than the weight. - -(246.) When great power is required, and it is inconvenient to -construct a long lever, a combination of levers may be used. In -_fig. 87._ such a system of levers is represented, consisting of -three levers of the first kind. The manner in which the effect of the -power is transmitted to the weight may be investigated by considering -the effect of each lever successively. The power at P produces an -upward force at P′, which bears to P the same proportion as P′ F -to P F. Therefore, the effect at P′ is as many times the power -as the line P F is of P′ F. Thus, if P F be ten times -P′ F, the upward force at P′ is ten times the power. The arm -P′ F′ of the second lever is pressed upwards by a force equal -to ten times the power at P. In the same manner this may be shown to -produce an effect at P″ as many times greater than P′ as P′ F′ -is greater than P″ F′. Thus, if P′ F′ be twelve times P″ F′, the -effect at P″ will be twelve times that of P′. But this last was ten -times the power, and therefore the P″ will be one hundred and twenty -times the power. In the same manner it may be shown that the weight is -as many times greater than the effect at P″ as P″ F″ is greater than -W F″. If P″ F″ be five times W F″, the weight will be five -times the effect at P″. But this effect is one hundred and twenty times -the power, and therefore the weight would be six hundred times the -power. - -In the same manner the effect of any compound system of levers may be -ascertained by taking the proportion of the weight to the power in -each lever separately, and multiplying these numbers together. In the -example given, these proportions are 10, 12, and 5, which multiplied -together give 600. In _fig. 87._ the levers composing the system -are of the first kind; but the principles of the calculation will not -be altered if they be of the second or third kind, or some of one kind -and some of another. - -(247.) That number which expresses the proportion of the weight to the -equilibrating power in any machine, we shall call the _power of the -machine_. Thus, if, in a lever, a power of one pound support a weight -of ten pounds, the power of the machine is _ten_. If a power of 2lbs. -support a weight of 11lbs., the power of the machine is 5-1/2, 2 being -contained in 11 5-1/2 times. - -(248.) As the distances of the power and weight from the fulcrum of -a lever may be varied at pleasure, and any assigned proportion given -to them, a lever may always be conceived having a power equal to that -of any given machine. Such a lever may be called, in relation to that -machine, the _equivalent lever_. - -As every complex machine consists of a number of simple machines acting -one upon another, and as each simple machine may be represented by an -equivalent lever, the complex machine will be represented by a compound -system of equivalent levers. From what has been proved in (246.), it -therefore follows that the power of a complex machine may be calculated -by multiplying together the powers of the several simple machines of -which it is composed. - - - - -CHAP. XIV. - -OF WHEEL-WORK. - - -(249.) When a lever is applied to raise a weight, or overcome a -resistance, the space through which it acts at any one time is small, -and the work must be accomplished by a succession of short and -intermitting efforts. In _fig. 81._, after the weight has been -raised from W to W′, the lever must again return to its first position, -to repeat the action. During this return the motion of the weight is -suspended, and it will fall downwards unless some provision be made to -sustain it. The common lever is, therefore, only used in cases where -weights are required to be raised through small spaces, and under these -circumstances its great simplicity strongly recommends it. But where -a continuous motion is to be produced, as in raising ore from the -mine, or in weighing the anchor of a vessel, some contrivance must be -adopted to remove the intermitting action of the lever, and render -it continual. The various forms given to the lever, with a view to -accomplish this, are generally denominated the _wheel and axle_. - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -In _fig. 88._, A B is a horizontal axle, which rests in -pivots at its extremities, or is supported in gudgeons, and capable of -revolving. Round this axis a rope is coiled, which sustains the weight -W. On the same axis a wheel C is fixed, round which a rope is coiled -in a contrary direction, to which is appended the power P. The moment -of the power is found by multiplying it by the radius of a wheel, and -the moment of the weight, by multiplying it by the radius of its axle. -If these moments be equal (185.), the machine will be in equilibrium. -Whence it appears that the power of the machine (247.) is expressed by -the proportion which the radius of the wheel bears to the radius of -the axle; or, what is the same, of the diameter of the wheel to the -diameter of the axle. This principle is applicable to the wheel and -axle in every variety of form under which it can be presented. - -(250.) It is evident that as the power descends continually, and the -rope is uncoiled from the wheel, the weight will be raised continually, -the rope by which it is suspended being at the same time coiled upon -the axle. - -When the machine is in equilibrium, the forces of both the weight and -power are sustained by the axle, and distributed between its props, in -the manner explained in (245.) - -When the machine is applied to raise a weight, the velocity with which -the power moves is as many times greater than that with which the -weight rises, as the weight itself is greater than the power. This is -a principle which has already been noticed, and which is common to all -machines whatsoever. It may hence be proved, that in the elevation of -the weight a quantity of power is expended equal to that which would be -necessary to elevate the weight if the power were immediately applied -to it, without the intervention of any machine. This has been explained -in the case of the lever in (241.), and may be explained in the -present instance in nearly the same words. - -In one revolution of the machine the length of rope uncoiled from -the wheel is equal to the circumference of the wheel, and through -this space the power must therefore move. At the same time the length -of rope coiled upon the axle is equal to the circumference of the -axle, and through this space the weight must be raised. The spaces, -therefore, through which the power and weight move in the same time, -are in the proportion of the circumferences of the wheel and axle; but -these circumferences are in the same proportion as their diameters. -Therefore the velocity of the power will bear to the velocity of the -weight the same proportion as the diameter of the wheel bears to the -diameter of the axle, or, what is the same, as the weight bears to the -power (249). - -(251.) We have here omitted the consideration of the thickness of the -rope. When this is considered, the force must be conceived as acting in -the direction of the centre of the rope, and therefore the thickness -of the rope which supports the power ought to be added to the diameter -of the wheel, and the thickness of the rope which supports the weight -to the diameter of the axle. It is the more necessary to attend to -this circumstance, as the strength of the rope necessary to support -the weight causes its thickness to bear a considerable proportion to -the diameter of the axle; while the rope which sustains the power not -requiring the same strength, and being applied to a larger circle, -bears a very inconsiderable proportion to its diameter. - -(252.) In numerous forms of the wheel and axle, the weight or -resistance is applied by a rope coiled upon the axle; but the manner in -which the power is applied is very various, and not often by means of a -rope. The circumference of a wheel sometimes carries projecting pins, -as represented in _fig. 88._, to which the hand is applied to -turn the machine. An instance of this occurs in the wheel used in the -steerage of a vessel. - -In the common _windlass_, the power is applied by means of a _winch_, -which is a rectangular lever, as represented in _fig. 89._ The arm -B C of the winch represents the radius of the wheel, and the power -is applied to C D at right angles to B C. - -In some cases no wheel is attached to the axle; but it is pierced with -holes directed towards its centre, in which long levers are incessantly -inserted, and a continuous action produced by several men working at -the same time; so that while some are transferring the levers from hole -to hole, others are working the windlass. - -The axle is sometimes placed in a vertical position, the wheel or -levers being moved horizontally. The _capstan_ is an example of this: -a vertical axis is fixed in the deck of the ship; the circumference is -pierced with holes presented towards its centre. These holes receive -long levers, as represented in _fig. 90._ The men who work the -capstan walk continually round the axle, pressing forward the levers -near their extremities. - -In some cases the wheel is turned by the weight of animals placed at -its circumference, who move forward as fast as the wheel descends, -so as to maintain their position continually at the extremity of the -horizontal diameter. The _treadmill_, _fig. 91._, and certain -_cranes_, such as _fig. 92._, are examples of this. - -In water-wheels, the power is the weight of water contained in -buckets at the circumference, as in _fig. 93._, which is called -an over-shot wheel: and sometimes by the impulse of water against -float-boards at the circumference, as in the under-shot wheel, -_fig. 94._ Both these principles act in the breast-wheel, -_fig. 95._ - -In the paddle-wheel of a steam-boat, the power is the resistance which -the water offers to the motion of the paddle-boards. - -In windmills, the power is the force of the wind acting on various -parts of the arms, and may be considered as different powers -simultaneously acting on different wheels having the same axle. - -(253.) In most cases in which the wheel and axle is used, the action of -the power is liable to occasional suspension or intermission, in which -case some contrivance is necessary to prevent the recoil of the weight. -A ratchet wheel R, _fig. 88._, is provided for this purpose, which -is a contrivance which permits the wheel to turn in one direction; -but a catch which falls between the teeth of a fixed wheel prevents -its motion in the other direction. The effect of the power or weight -is sometimes transmitted to the wheel or axle by means of a straight -bar, on the edge of which teeth are raised, which engage themselves in -corresponding teeth on the wheel or axle. Such a bar is called a rack; -and an instance of its use may be observed in the manner of working the -pistons of an air-pump. - -(254.) The power of the wheel and axle being expressed by the number -of times the diameter of the axle is contained in that of the wheel, -there are obviously only two ways by which this power may be increased; -viz. either by increasing the diameter of the wheel, or diminishing -that of the axle. In cases where great power is required, each of these -methods is attended with practical inconvenience and difficulty. If the -diameter of the wheel be considerably enlarged, the machine will become -unwieldy, and the power will work through an unmanageable space. If, -on the other hand, the power of the machine be increased by reducing -the thickness of the axle, the strength of the axle will become -insufficient for the support of that weight, the magnitude of which had -rendered the increase of the power of the machine necessary. To combine -the requisite strength with moderate dimensions and great mechanical -power is, therefore, impracticable in the ordinary form of the wheel -and axle. This has, however, been accomplished by giving different -thicknesses to different parts of the axle, and carrying a rope, which -is coiled on the thinner part, through a wheel attached to the weight, -and coiling it in the opposite direction on the thicker part, as in -_fig. 96._ To investigate the proportion of the power to the -weight in this case, let _fig. 97._ represent a section of the -apparatus at right angles to the axis. The weight is equally suspended -by the two parts of the rope, S and S′, and therefore each part is -stretched by a force equal to half the weight. The moment of the force, -which stretches the rope S, is half the weight multiplied by the radius -of the thinner part of the axle. This force being at the same side of -the centre with the power, co-operates with it in supporting the force -which stretches S′, and which acts at the other side of the centre. By -the principle established in (185.), the moments of P and S must be -equal to that of S′; and therefore if P be multiplied by the radius of -the wheel, and added to half the weight multiplied by the radius of the -thinner part of the axle, we must obtain a sum equal to half the weight -multiplied by the radius of the thicker part of the axle. Hence it is -easy to perceive, that the power multiplied by the radius of the wheel -is equal to half the weight multiplied by the difference of the radii -of the thicker and thinner parts of the axle; or, what is the same, the -power multiplied by the diameter of the wheel, is equal to the weight -multiplied by half the difference of the diameters of the thinner and -thicker parts of the axle. - -A wheel and axle constructed in this manner is equivalent to an -ordinary one, in which the wheel has the same diameter, and whose axle -has a diameter equal to half the difference of the diameters of the -thicker and thinner parts. The power of the machine is expressed by the -proportion which the diameter of the wheel bears to half the difference -of these diameters; and therefore this power, when the diameter of the -wheel is given, does not, as in the ordinary wheel and axle, depend -on the smallness of the axle, but on the smallness of the difference -of the thinner and thicker parts of it. The axle may, therefore, -be constructed of such a thickness as to give it all the requisite -strength, and yet the difference of the diameters of its different -parts may be so small as to give it all the requisite power. - -(255.) It often happens that a varying weight is to be raised, or -resistance overcome by a uniform power. If, in such a case, the weight -be raised by a rope coiled upon a uniform axle, the action of the -power would not be uniform, but would vary with the weight. It is, -however, in most cases desirable or necessary that the weight or -resistance, even though it vary, shall be moved uniformly. This will -be accomplished if by any means the leverage of the weight is made -to increase in the same proportion as the weight diminishes, and to -diminish in the same proportion as the weight increases: for in that -case the moment of the weight will never vary, whatever it gains by the -increase of weight being lost by the diminished leverage, and whatever -it loses by the diminished weight being gained by the increased -leverage. An axle, the surface of which is curved in such a manner, -that the thickness on which the rope is coiled continually increased -or diminishes in the same proportion as the weight or resistance -diminishes or increases, will produce this effect. - -It is obvious that all that has been said respecting a variable -weight or resistance, is also applicable to a variable power, which, -therefore, may, by the same means, be made to produce a uniform effect. -An instance of this occurs in a watch, which is moved by a spiral -spring. When the watch has been wound up, this spring acts with its -greatest intensity, and as the watch goes down, the elastic force of -the spring gradually loses its energy. This spring is connected by a -chain with an axle of varying thickness, called a _fusee_. When the -spring is at its greatest intensity, the chain acts upon the thinnest -part of the fusee, and as it is uncoiled it acts upon a part of the -fusee which is continually increasing in thickness, the spring at the -same time losing its elastic power in exactly the same proportion. A -representation of the fusee, and the cylindrical box which contains -the spring, is given in _fig. 98._, and of the spring itself in -_fig. 99._ - -(256.) When great power is required, wheels and axles may be combined -in a manner analogous to a compound system of levers, explained -in (246.) In this case the power acts on the circumference of the -first wheel, and its effect is transmitted to the circumference -of the first axle. That circumference is placed in connection with -the circumference of the second wheel, and the effect is thereby -transmitted to the circumference of the second axle, and so on. It -is obvious from what was proved in (248.), that the power of such a -combination of wheels and axles will be found by multiplying together -the powers of the several wheels of which it is composed. It is -sometimes convenient to compute this power by numbers expressing the -proportions of the circumferences or diameters of the several wheels, -to the circumferences or diameters of the several axles respectively. -This computation is made by first multiplying the numbers together -which express the circumferences or diameters of the wheels, and then -multiplying together the numbers which express the circumferences or -diameters of the several axles. The proportion of the two products -will express the power of the machine. Thus, if the circumferences or -diameters be as the numbers 10, 14, and 15, their product will be 2100; -and if the circumferences or diameters of the axles be expressed by the -numbers 3, 4, and 5, their product will be 60, and the power of the -machine will be expressed by the proportion of 2100 and 60, or 35 to 1. - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -(257.) The manner in which the circumferences of the axles act upon -the circumferences of the wheels in compound wheel-work is various. -Sometimes a strap or cord is applied to a groove in the circumference -of the axle, and carried round a similar groove in the circumference -of the succeeding wheel. The friction of this cord or strap with the -groove is sufficient to prevent its sliding and to communicate the -force from the axle to the wheel, or _vice versa_. This method of -connecting wheel-work is represented in _fig. 100._ - -Numerous examples of wheels and axles driven by straps or cords occur -in machinery applied to almost every department of the arts and -manufactures. In the turning lathe, the wheel worked by the treddle is -connected with the mandrel by a catgut cord passing through grooves -in the wheel and axle. In all great factories, revolving shafts are -carried along the apartments, on which, at certain intervals, straps -are attached passing round their circumferences and carried round -the wheels which give motion to the several machines. If the wheels, -connected by straps or cords, are required to revolve in the same -direction, these cords are arranged as in _fig. 100._; but if they -are required to revolve in contrary directions, they are applied as in -_fig. 101._ - -One of the chief advantages of the method of transmitting motion -between wheels and axles by straps or cords, is that the wheel and -axle may be placed at any distance from each other which may be found -convenient, and may be made to turn either in the same or contrary -directions. - -(258.) When the circumference of the wheel acts immediately on the -circumference of the succeeding axle, some means must necessarily be -adopted to prevent the wheel from moving in contact with the axle -without compelling the latter to turn. If the surfaces of both were -perfectly smooth, so that all friction were removed, it is obvious that -either would slide over the surface of the other, without communicating -motion to it. But, on the other hand, if there were any asperities, -however small, upon these surfaces, they would become mutually inserted -among each other, and neither the wheel nor axle could move without -causing the asperities with which its edge is studded to encounter -those asperities which project from the surface of the other; and -thus, until these projections should be broken off, both wheel and -axle must be moved at the same time. It is on this account that if the -surfaces of the wheels and axles are by any means rendered rough, and -pressed together with sufficient force, the motion of either will turn -the other, provided the load or resistance be not greater than the -force necessary to break off these small projections which produce the -friction. - -In cases where great power is not required, motion is communicated in -this way through a train of wheel-work, by rendering the surface of the -wheel and axle rough, either by facing them with buff leather, or with -wood cut across the grain. This method is sometimes used in spinning -machinery, where one large buffed wheel, placed in a horizontal -position, revolves in contact with several small buffed rollers, each -roller communicating motion to a spindle. The position of the wheel W, -and the rollers R R, &c., are represented in _fig. 102._ Each -roller can be thrown out of contact with the wheel, and restored to it -at pleasure. - -The communication of motion between wheels and axles by friction has -the advantage of great smoothness and evenness, and of proceeding with -little noise; but this method can only be used in cases where the -resistance is not very considerable, and therefore is seldom adopted in -works on a large scale. Dr. Gregory mentions an instance of a saw mill -at Southampton, where the wheels act upon each other by the contact of -the end grain of wood. The machinery makes very little noise, and wears -very well, having been used not less than 20 years. - -(259.) The most usual method of transmitting motion through a train of -wheel-work is by the formation of teeth upon their circumferences, so -that these indentures of each wheel fall between the corresponding ones -of that in which it works, and ensure the action so long as the strain -is not so great as to fracture the tooth. - -In the formation of teeth very minute attention must be given to their -figure, in order that the motion may be communicated from wheel to -wheel with smoothness and uniformity. This can only be accomplished -by shaping the teeth according to curves of a peculiar kind, which -mathematicians have invented, and assigned rules for drawing. The ill -consequences of neglecting this will be very apparent, by considering -the nature of the action which would be produced if the teeth were -formed of square projecting pins, as in _fig. 103._ When the -tooth A comes into contact with B, it acts obliquely upon it, and, -as it moves, the corner of B slides upon the plane surface of A in -such a manner as to produce much friction, and to grind away the side -of A and the end of B. As they approach the position C D, they -sustain a jolt the moment their surfaces come into full contact; and -after passing the position of C D, the same scraping and grinding -effect is produced in the opposite direction, until by the revolution -of the wheels the teeth become disengaged. These effects are avoided by -giving to the teeth the curved forms represented in _fig. 104._ -By such means the surfaces of the teeth roll upon each other with very -inconsiderable friction, and the direction in which the pressure is -excited is always that of a line M N, touching the two wheels, and -at right angles to the radii. Thus the pressure being always the same, -and acting with the same leverage, produces a uniform effect. - -(260.) When wheels work together, their teeth must necessarily be of -the same size, and therefore the proportion of their circumferences may -always be estimated by the number of teeth which they carry. Hence it -follows, that in computing the power of compound wheel-work, the number -of teeth may always be used to express the circumferences respectively, -or the diameters which are proportional to these circumferences. When -teeth are raised upon an axle, it is generally called a _pinion_, and -in that case the teeth are called _leaves_. The rule for computing the -train of wheel-work given in (256.) will be expressed as follows: when -the wheel and axle carry teeth, multiply together the number of teeth -in each of the wheels, and next the number of leaves in each of the -pinions; the proportion of the two products will express the power of -the machine. If some of the wheels and axles carry teeth, and others -not, this computation may be made by using for those circumferences -which do not bear teeth the number of teeth which would fill them. -_Fig. 105._ represents a train of three wheels and pinions. The -wheel F which bears the power, and the axle which bears the weight, -have no teeth; but it is easy to find the number of teeth which they -would carry. - -(261.) It is evident that each pinion revolves much more frequently in -a given time than the wheel which it drives. Thus, if the pinion C be -furnished with ten teeth, and the wheel E, which it drives, have sixty -teeth, the pinion C must turn six times, in order to turn the wheel -E once round. The velocities of revolution of every wheel and pinion -which work in one another will therefore have the same proportion as -their number of teeth taken in a reverse order, and by this means the -relative velocity of wheels and pinions may be determined according to -any proposed rate. - -Wheel-work, like all other machinery, is used to transmit and modify -force in every department of the arts and manufactures; but it is also -used in cases where motion alone, and not force, is the object to be -attained. The most remarkable example of this occurs in watch and -clock-work, where the object is merely to produce uniform motions of -rotation, having certain proportions, and without any regard to the -elevation of weights, or the overcoming of resistances. - -(262.) A _crane_ is an example of combination of wheel-work used for -the purpose of raising or lowering great weights. _Fig. 106._ -represents a machine of this kind. A B is a strong vertical beam, -resting on a pivot, and secured in its position by beams in the floor. -It is capable, however, of turning on its axis, being confined between -rollers attached to the beams and fixed in the floor. C D is a -projecting arm called a _gib_, formed of beams which are mortised into -A B. The wheel-work is mounted in two cast-iron crosses, bolted on -each side of the beams, one of which appears at E F G H. -The winch at which the power is applied is at I. This carries a pinion -immediately behind H. This pinion works in a wheel K, which carries -another pinion upon its axle. This last pinion works in a larger wheel -L, which carries upon its axis a barrel M, on which a chain or rope -is coiled. The chain passes over a pulley D at the top of the gib. At -the end of the chain a hook O is attached, to support the weight W. -During the elevation of the weight it is convenient that its recoil -should be hindered in case of any occasional suspension of the power. -This is accomplished by a ratchet wheel attached to the barrel M, as -explained in (253.); but when the weight W is to be lowered, the catch -must be removed from this ratchet wheel. In this case the too rapid -descent of the weight is in some cases checked by pressure excited on -some part of the wheel-work, so as to produce sufficient friction to -retard the descent in any required degree, or even to suspend it, if -necessary. The vertical beam at B resting on a pivot, and being fixed -between rollers, allows the gib to be turned round in any direction; so -that a weight raised from one side of the crane may be carried round, -and deposited on another side, at any distance within the range of the -gib. Thus, if a crane be placed upon a wharf near a vessel, weights may -be raised, and when elevated, the gib may be turned round so as to let -them descend into the hold. - -The power of this machine may be computed upon the principles already -explained. The magnitude of the circle, in which the power at I moves, -may be determined by the radius of the winch, and therefore the number -of teeth which a wheel of that size would carry may be found. In -like manner we may determine the number of leaves in a pinion whose -magnitude would be equal to the barrel M. Let the first number be -multiplied by the number of teeth in the wheel K, and that product -by the number of teeth in the wheel L. Next let the number of leaves -in the pinion H be multiplied by the number of leaves in the pinion -attached to the axle of the wheel K, and let that product be multiplied -by the number of leaves in a pinion, whose diameter is equal to that of -the barrel M. These two products will express the power of the machine. - -(263.) Toothed wheels are of three kinds, distinguished by the position -which the teeth bear with respect to the axis of the wheel. When they -are raised upon the edge of the wheel as in _fig. 105._, they are -called _spur wheels_, or _spur gear_. When they are raised parallel to -the axis, as in _fig. 107._, it is called a _crown wheel_. When -the teeth are raised on a surface inclined to the plane of the wheel, -as in _fig. 108._, they are called _bevelled wheels_. - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -If a motion round one axis is to be communicated to another axis -parallel to it, spur gear is generally used. Thus, in _fig. 105._, -the three axes are parallel to each other. If a motion round one -axis is to be communicated to another at right angles to it, a crown -wheel, working in a spur pinion, as in _fig. 107._, will serve. -Or the same object may be obtained by two bevelled wheels, as in -_fig. 108._ - -If a motion round one axis is required to be communicated to another -inclined to it at any proposed angle, two bevelled wheels can always be -used. In _fig. 109._ let A B and A C be the two axles; -two bevelled wheels, such as D E and E F, on these axles will -transmit the motion or rotation from one to the other, and the relative -velocity may, as usual, be regulated by the proportional magnitude of -the wheels. - -(264.) In order to equalise the wear of the teeth of a wheel and -pinion, which work in one another, it is necessary that every leaf -of the pinion should work in succession through every tooth of the -wheel, and not continually act upon the same set of teeth. If the -teeth could be accurately shaped according to mathematical principles, -and the materials of which they are formed be perfectly uniform, this -precaution would be less necessary; but as slight inequalities, both -of material and form, must necessarily exist, the effects of these -should be as far as possible equalised, by distributing them through -every part of the wheel. For this purpose it is usual, especially -in mill-work, where considerable force is used, so to regulate the -proportion of the number of teeth in the wheel and pinion, that the -same leaf of the pinion shall not be engaged twice with any one tooth -of the wheel, until after the action of a number of teeth, expressed -by the product of the number of teeth in the wheel and pinion. Let us -suppose that the pinion contains ten leaves, which we shall denominate -by the numbers 1, 2, 3, &c., and that the wheel contains 60 teeth -similarly denominated. At the commencement of the motion suppose the -leaf 1 of the pinion engages the tooth 1 of the wheel; then after -one revolution the leaf 1 of the pinion will engage the tooth 11 of -the wheel, and after two revolutions the leaf 1 of the pinion will -engage the tooth 21 of the wheel; and in like manner, after 3, 4, and -5 revolutions of the pinion, the leaf 1 will engage successively the -teeth 31, 41, and 51 of the wheel. After the sixth revolution, the -leaf 1 of the pinion will again engage the tooth 1 of the wheel. Thus -it is evident, that in the case here supposed the leaf 1 of the pinion -will continually be engaged with the teeth 1, 11, 21, 31, 41, and 51 -of the wheel, and no others. The like may be said of every leaf of the -pinion. Thus the leaf 2 of the pinion will be successively engaged with -the teeth 2, 12, 22, 32, 42, and 52 of the wheel, and no others. Any -accidental inequalities of these teeth will therefore continually act -upon each other, until the circumference of the wheel be divided into -parts of ten teeth each, unequally worn. This effect would be avoided -by giving either the wheel or pinion one tooth more or one tooth less. -Thus, suppose the wheel, instead of having sixty teeth, had sixty-one, -then after six revolutions of the pinion the leaf 1 of the pinion would -be engaged with the tooth 61 of the wheel; and after one revolution of -the wheel, the leaf 2 of the pinion would be engaged with the tooth 1 -of the wheel. Thus, during the first revolution of the wheel the leaf -1 of the pinion would be successively engaged with the teeth 1, 11, -21, 31, 41, 51, and 61 of the wheel: at the commencement of the second -revolution of the wheel the leaf 2 of the pinion would be engaged with -the tooth 1 of the wheel; and during the second revolution of the wheel -the leaf 1 of the pinion would be successively engaged with the teeth -10, 20, 30, 40, 50, and 60 of the wheel. In the same manner it may be -shown, that in the third revolution of the wheel the leaf 1 of the -pinion would be successively engaged with the teeth 9, 19, 29, 39, 49, -and 59 of the wheel: during the fourth revolution of the wheel the -leaf 1 of the pinion would be successively engaged with the teeth 8, -18, 28, 38, 48, and 58 of the wheel. By continuing this reasoning it -will appear, that during the tenth revolution of the wheel the leaf -1 of the pinion will be engaged successively with the teeth 2, 12, -22, 32, 42, and 52 of the wheel. At the commencement of the eleventh -revolution of the wheel the leaf 1 of the pinion will be engaged with -the tooth 1 of the wheel, as at the beginning of the motion. It is -evident, therefore, that during the first ten revolutions of the wheel -each leaf of the pinion has been successively engaged with every tooth -of the wheel, and that during these ten revolutions the pinion has -revolved sixty-one times. Thus the leaves of the pinion have acted six -hundred and ten times upon the teeth of the wheel, before two teeth can -have acted twice upon each other. - -The odd tooth which produces this effect is called by millwrights the -_hunting cog_. - -(265.) The most familiar case in which wheel-work is used to produce -and regulate motion merely, without any reference to weights to be -raised or resistances to be overcome, is that of chronometers. In watch -and clock work the object is to cause a wheel to revolve with a uniform -velocity, and at a certain rate. The motion of this wheel is indicated -by an index or hand placed upon its axis, and carried round with it. -In proportion to the length of the hand the circle over which its -extremity plays is enlarged, and its motion becomes more perceptible. -This circle is divided, so that very small fractions of a revolution -of the hand may be accurately observed. In most chronometers it is -required to give motion to two hands, and sometimes to three. These -motions proceed at different rates, according to the subdivisions of -time generally adopted. One wheel revolves in a minute, bearing a -hand which plays round a circle divided into sixty equal parts; the -motion of the hand over each part indicating one second, and a complete -revolution of the hand being performed in one minute. Another wheel -revolves once, while the former revolves sixty times; consequently the -hand carried by this wheel revolves once in sixty minutes, or one hour. -The circle on which it plays is, like the former, divided into sixty -equal parts, and the motion of the hand over each division is performed -in one minute. This is generally called the _minute hand_, and the -former the _second hand_. - -A third wheel revolves once, while that which carries the minute hand -revolves twelve times; consequently this last wheel, which carries -the _hour hand_, revolves at a rate twelve times less than that of -the minute hand, and therefore seven hundred and twenty times less -than the second hand. We shall now endeavour to explain the manner in -which these motions are produced and regulated. Let A, B, C, D, E, -_fig. 110._, represent a train of wheels, and _a_, _b_, _c_, _d_ -represent their pinions, _e_ being a cylinder on the axis of the wheel -E, round which a rope is coiled, sustaining a weight W. Let the effect -of this weight transmitted through the train of wheels be opposed by -a power P acting upon the wheel A, and let this power be supposed -to be of such a nature as to cause the weight W to descend with a -uniform velocity, and at any proposed rate. The wheel E carries on its -circumference eighty-four teeth. The wheel D carries eighty teeth; -the wheel C is also furnished with eighty teeth, and the wheel B with -seventy-five. The pinions _d_ and _c_ are each furnished with twelve -leaves, and the pinions _b_ and _a_ with ten. - -If the power at P be so regulated as to allow the wheel A to revolve -once in a minute, with a uniform velocity, a hand attached to the axis -of this wheel will serve as the _second hand_. The pinion _a_ carrying -ten teeth must revolve seven times and a half to produce one revolution -of B, consequently fifteen revolutions of the wheel A will produce two -revolutions of the wheel B; the wheel B, therefore, revolves twice in -fifteen minutes. The pinion _b_ must revolve eight times to produce -one revolution of the wheel C, and therefore the wheel C must revolve -once in four quarters of an hour, or in one hour. If a hand be attached -to the axis of this wheel, it will have the motion necessary for the -minute hand. The pinion _c_ must revolve six times and two thirds to -produce one revolution of the wheel D, and therefore this wheel must -revolve once in six hours and two thirds. The pinion _d_ revolves seven -times for one revolution of the wheel E, and therefore the wheel E will -revolve once in forty-six hours and two thirds. - -On the axis of the wheel C a second pinion may be placed, furnished -with seven leaves, which may lead a wheel of eighty-four teeth, so -that this wheel shall turn once during twelve turns of the wheel C. If -a hand be fixed upon the axis, this hand will revolve once for twelve -revolutions of the minute hand fixed upon the axis of the wheel C; -that is, it will revolve once in twelve hours. If it play upon a dial -divided into twelve equal parts, it will move over each part in an -hour, and will serve the purpose of the hour hand of the chronometer. - -We have here supposed that the second hand, the minute hand, and the -hour hand move on separate dials. This, however, is not necessary. The -axis of the hour hand is commonly a tube, inclosing within it that of -the minute hand, so that the same dial serves for both. The second -hand, however, is generally furnished with a separate dial. - -(266.) We shall now explain the manner in which a power is applied -to the wheel A, so as to regulate and equalise the effect of the -weight W. Suppose the wheel A furnished with thirty teeth, as in -_fig. 111._; if nothing check the motion, the weight W would -descend with an accelerated velocity, and would communicate an -accelerated motion to the wheel A. This effect, however, is interrupted -by the following contrivance:--L M is a pendulum vibrating on the -centre L, and so regulated that the time of its oscillation is one -second. The pallets I and K are connected with the pendulum, so as to -oscillate with it. In the position of the pendulum represented in the -figure, the pallet I stops the motion of the wheel A, and entirely -suspends the action of the weight W, _fig. 110._, so that for -a moment the entire machine is motionless. The weight M, however, -falls by its gravity towards the lowest position, and disengages the -pallet I from the tooth of the wheel. The weight W begins then to take -effect, and the wheel A turns from A towards B. Meanwhile the pendulum -M oscillates to the other side, and the pallet K falls under a tooth -of the wheel A, and checks for a moment its further motion. On the -returning vibration the pallet K becomes again disengaged, and allows -the tooth of the wheel to escape, and by the influence of the weight W -another tooth passes before the motion of the wheel A is again checked -by the interposition of the pallet I. - -From this explanation it will appear that, in two vibrations of the -pendulum, one tooth of the wheel A passes the pallet I, and therefore, -if the wheel A be furnished with 30 teeth, it will be allowed to make -one revolution during 60 vibrations of the pendulum. If, therefore, the -pendulum be regulated so as to vibrate seconds, this wheel will revolve -once in a minute. From the action of the pallets in checking the motion -of the wheel A, and allowing its teeth alternately to _escape_, this -has been called the _escapement_ wheel; and the wheel and pallets -together are generally called the _escapement_, or _’scapement_. - -We have already explained, that by reason of the friction on the -points of support, and other causes, the swing of the pendulum would -gradually diminish, and its vibration at length cease. This, however, -is prevented by the action of the teeth of the scapement wheel upon the -pallets, which is just sufficient to communicate that quantity of force -to the pendulum which is necessary to counteract the retarding effects, -and to maintain its motion. It thus appears, that although the effect -of the gravity of the weight W in giving motion to the machine is at -intervals suspended, yet this part of the force is not lost, being, -during these intervals, employed in giving to the pendulum all that -motion which it would lose by the resistances to which it is inevitably -exposed. - -In stationary clocks, and in other cases in which the bulk of the -machine is not an objection, a descending weight is used as the -moving power. But in watches and portable chronometers, this would be -attended with evident inconvenience. In such cases, a spiral spring, -called the _mainspring_, is the moving power. The manner in which this -spring communicates rotation to an axis, and the ingenious method of -equalising the effect of its variable elasticity by giving to it a -leverage, which increases as the elastic force diminishes, have been -already explained. (255.) - -A similar objection lies against the use of a pendulum in portable -chronometers. A spiral spring of a similar kind, but infinitely -more delicate, called a _hair spring_, is substituted in its place. -This spring is connected with a nicely-balanced wheel, called _the -balance wheel_, which plays in pivots. When this wheel is turned to -a certain extent in one direction, the hair spring is coiled up, and -its elasticity causes the wheel to recoil, and return to a position -in which the energy of the spring acts in the opposite direction. -The balance wheel then returns, and continually vibrates in the same -manner. The axis of this wheel is furnished with pallets similar to -those of the pendulum, which are alternately engaged with the teeth of -a crown wheel, which takes the place of the scapement wheel already -described. - -A general view of the work of a common watch is represented in -_fig. 111._ _bis._ A is the balance wheel bearing pallets _p_ -_p_ upon its axis; C is the crown wheel, whose teeth are suffered to -escape alternately by those pallets in the manner already described -in the scapement of a clock. On the axis of the crown wheel is placed -a pinion _d_, which drives another crown wheel K. On the axis of this -is placed the pinion _c_, which plays in the teeth of the third wheel -L. The pinion _b_ on the axis of L is engaged with the wheel M, called -the centre wheel. The axle of this wheel is carried up through the -centre of the dial. A pinion _a_ is placed upon it, which works in -the great wheel N. On this wheel the mainspring immediately acts. -O P is the mainspring stripped of its barrel. The axis of the -wheel M passing through the centre of the dial is squared at the end -to receive the minute hand. A second pinion Q is placed upon this -axle which drives a wheel T. On the axle of this wheel a pinion _g_ -is placed, which drives the hour wheel V. This wheel is placed upon a -tubular axis, which incloses within it the axis of the wheel M. This -tubular axis passing through the centre of the dial, carries the hour -hand. The wheels A, B, C, D, E, _fig. 110._, correspond to the wheels -C, K, L, M, N, _fig. 112._; and the pinions _a_, _b_, _c_, _d_, -_e_, _fig. 109._, correspond to the pinions _d_, _c_, _b_, _a_, -_fig. 111_. From what has already been explained of these wheels, -it will be obvious that the wheel M, _fig. 111._, revolves once -in an hour, causing the minute hand to move round the dial once in -that time. This wheel at the same time turns the pinion Q which leads -the wheel T. This wheel again turns the pinion _g_ which leads the -hour wheel V. The leaves and teeth of these pinions and wheels are -proportioned, as already explained, so that the wheel V revolves once -during twelve revolutions of the wheel M. The hour hand, therefore, -which is carried by the tubular axle of the wheel V, moves once round -the dial in twelve hours. - -Our object here has not been to give a detailed account of watch and -clock work, a subject for which we must refer the reader to the proper -department of this work. Such a general account has only been attempted -as may explain how tooth and pinion work may be applied to regulate -motion. - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - - - - -CHAP. XV. - -OF THE PULLEY. - - -(267.) The next class of simple machines, which present themselves -to our attention, is that which we have called the _cord_. If a rope -were perfectly flexible, and were capable of being bent over a sharp -edge, and of moving upon it without friction, we should be enabled by -its means to make a force in any one direction overcome resistance, or -communicate motion in any other direction. Thus if P, _fig. 112._, -be such an edge, a perfectly flexible rope passing over it would be -capable of transmitting a force S F to a resistance Q R, -so as to support or overcome R, or by a motion in the direction of -S F to produce another motion in the direction R Q. But as -no materials of which ropes can be constructed can give them perfect -flexibility, and as in proportion to the strength by which they are -enabled to transmit force their rigidity increases, it is necessary, -in practice, to adopt means to remove or mitigate those effects which -attend imperfect flexibility, and which would otherwise render cords -practically inapplicable as machines. - -When a cord is used to transmit a force from one direction to another, -its stiffness renders some force necessary in bending it over the -angle P, which the two directions form; and if the angle be sharp, -the exertion of such a force may be attended with the rupture of the -cord. If, instead of bending the rope at one point over a single angle, -the change of direction were produced by successively deflecting it -over several angles, each of which would be less sharp than a single -one could be, the force requisite for the deflection, as well as the -liability of rupturing the cord, would be considerably diminished. But -this end will be still more perfectly attained if the deflection of the -cord be produced by bending it over the surface of a curve. - -If a rope were applied only to sustain, and not to move a weight, -this would be sufficient to remove the inconveniences arising from -its rigidity. But when motion is to be produced, the rope, in passing -over the curved surface, would be subject to excessive friction, and -consequently to rapid wear. This inconvenience is removed by causing -the surface on which the rope runs to move with it, so that no more -friction is produced than would arise from the curved surface rolling -upon the rope. - -(268.) All these ends are attained by the common pulley, which consists -of a wheel called a _sheave_, fixed in a block and turning on a pivot. -A groove is formed in the edge of the wheel in which the rope runs, -the wheel revolving with it. Such an apparatus is represented in -_fig. 113._ - -We shall, for the present, omit the consideration of that part of the -effects of the stiffness and friction of the machine which is not -removed by the contrivance just explained, and shall consider the rope -as perfectly flexible and moving without friction. - -From the definition of a flexible cord, it follows, that its tension, -or the force by which it is stretched throughout its entire length, -must be uniform. From this principle, and this alone, all the -mechanical properties of pulleys may be derived. - -Although, as already explained, the whole mechanical efficacy of this -machine depends on the qualities of the cord, and not on those of the -block and sheave, which are only introduced to remove the accidental -effects of stiffness and friction; yet it has been usual to give the -name pulley to the block and sheave, and a combination of blocks, -sheaves, and ropes is called a _tackle_. - -(269.) When the rope passes over a single wheel, which is fixed in -its position, as in _fig. 113._, the machine is called a _fixed -pulley_. Since the tension of the cord is uniform throughout its -length, it follows, that in this machine the power and weight are -equal. For the weight stretches that part of the cord which is between -the weight and pulley, and the power stretches that part between the -power and the pulley. And since the tension throughout the whole length -is the same, the weight must be equal to the power. - -Hence it appears that no mechanical advantage is gained by this -machine. Nevertheless, there is scarcely any engine, simple or complex, -attended with more convenience. In the application of power, whether -of men or animals, or arising from natural forces, there are always -some directions in which it may be exerted to much greater convenience -and advantage than others, and in many cases the exertion of these -powers is limited to a single direction. A machine, therefore, which -enables us to give the most advantageous direction to the moving power, -whatever be the direction of the resistance opposed to it, contributes -as much practical convenience as one which enables a small power to -balance or overcome a great weight. In directing the power against the -resistance, it is often necessary to use two fixed pulleys. Thus, in -elevating a weight A, _fig. 114._, to the summit of a building, -by the strength of a horse moving below, two fixed pulleys B and C may -be used. The rope is carried from A over the pulley B; and, passing -downwards, is brought under C, and finally drawn by the animal on -the horizontal plane. In the same manner sails are spread, and flags -hoisted on the yards and masts of a ship, by sailors pulling a rope on -the deck. - -By means of the fixed pulley a man may raise himself to a considerable -height, or descend to any proposed depth. If he be placed in a chair -or bucket attached to one end of a rope which is carried over a fixed -pulley, by laying hold of this rope on the other side, as represented -in _fig. 115._, he may, at will, descend to a depth equal to half -of the entire length of the rope, by continually yielding rope on the -one side, and depressing the bucket or chair by his weight on the -other. Fire-escapes have been constructed on this principle, the fixed -pulley being attached to some part of the building. - -(270.) A _single moveable pulley_ is represented in _fig. 116._ -A cord is carried from a fixed point F, and passing through a block -B, attached to a weight W, passes over a fixed pulley C, the power -being applied at P. We shall first suppose the parts of the cord on -each side the wheel B to be parallel; in this case, the whole weight W -being sustained by the parts of the cords B C and B F, and -these parts being equally stretched (268.), each must sustain half the -weight, which is therefore the tension of the cord. This tension is -resisted by the power at P, which must, therefore, be equal to half the -weight. In this machine, therefore, the weight is twice the power. - -(271.) If the parts of the cord B C and B F be not parallel, -as in _fig. 117._, a greater power than half the weight is -therefore necessary to sustain it. To determine the power necessary -to support a given weight, in this case take the line B A in the -vertical direction, consisting of as many inches as the weight consists -of ounces; from A draw A D parallel to B C, and A E -parallel to B F; the force of the weight represented by A B -will be equivalent to two forces represented by B D and B E. -(74.) The number of inches in these lines respectively will represent -the number of ounces which are equivalent to the tensions of the parts -B F and B C of the cord. But as these tensions are equal, -B D and B E must be equal, and each will express the amount -of the power P, which stretches the cord at P C. - -It is evident that the four lines, A E, E B, B D, and -D A, are equal. And as each of them represents the power, the -weight which is represented by A B must be less than twice the -power which is represented by A E and E B taken together. It -follows, therefore, that as parts of the ropes which support the weight -depart from parallelism the machine becomes less and less efficacious; -and there are certain obliquities at which the equilibrating power -would be much greater than the weight. - -(272.) The mechanical power of pulleys admits of being almost -indefinitely increased by combination. Systems of pulleys may be -divided into two classes; those in which a single rope is used, and -those which consist of several distinct ropes. _Fig. 118._ and -_119._ represent two systems of pulleys, each having a single rope. -The weight is in each case attached to a moveable block, B, in which -are fixed two or more wheels; A is a fixed block, and the rope is -successively passed over the wheels above and below, and, after passing -over the last wheel above, is attached to the power. The tension of -that part of the cord to which the power is attached is produced by -the power, and therefore equivalent to it, and the same tension must -extend throughout its whole length. The weight is sustained by all -those parts of the cord which pass from the lower block, and as the -force which stretches them all is the same, viz. that of the power, -the effect of the weight must be equally distributed among them, their -directions being supposed to be parallel. It will be evident, from -this reasoning, that the weight will be as many times greater than the -power as the number of cords which support the lower block. Thus, if -there be six cords, each cord will support a sixth part of the weight, -that is, the weight will be six times the tension of the cord, or six -times the power. In _fig. 118._ the cord is represented as being -finally attached to a hook on the upper block. But it may be carried -over an additional wheel fixed in that block, and finally attached -to a hook in the lower block, as in _fig. 119._, by which one -will be added to the power of the machine, the number of cords at -the lower block being increased by one. In the system represented in -_fig. 118._ the wheels are placed in the blocks one above the -other; in _fig. 119._ they are placed side by side. In all systems -of pulleys of this class, the weight of the lower block is to be -considered as a part of the weight to be raised, and in estimating the -power of the machine, this should always be attended to. - -(273.) When the power of the machine, and therefore the number of -wheels, is considerable, some difficulty arises in the arrangement of -the wheels and cords. The celebrated Smeaton contrived a tackle, which -takes its name from him, in which there are ten wheels in each block: -five large wheels placed side by side, and five smaller ones similarly -placed above them in the lower block, and below them in the upper. -_Fig. 120._ represents Smeaton’s blocks without the rope. The -wheels are marked with the numbers 1, 2, 3, &c., in the order in which -the rope is to be passed over them. As in this pulley 20 distinct parts -of the rope support the lower block, the weight, including the lower -block, will be 20 times the equilibrating power. - -(274.) In all these systems of pulleys, every wheel has a separate -axle, and there is a distinct wheel for every turn of the rope at each -block. Each wheel is attended with friction on its axle, and also with -friction between the sheave and block. The machine is by this means -robbed of a great part of its efficacy, since, to overcome the friction -alone, a considerable power is in most cases necessary. - -An ingenious contrivance has been suggested, by which all the advantage -of a large number of wheels may be obtained without the multiplied -friction of distinct sheaves and axles. To comprehend the excellence -of this contrivance, it will be necessary to consider the rate at -which the rope passes over the several wheels of such a system, as -_fig. 118._ If one foot of the rope G F pass over the -pulley F, two feet must pass over the pulley E, because the distance -between F and E being shortened one foot, the total length of the rope -G F E must be shortened two feet. These two feet of rope -must pass in the direction E D, and the wheel D, rising one foot, -three feet of rope must consequently pass over it. These three feet of -rope passing in the direction D C, and the rope D C being -also shortened one foot by the ascent of the lower block, four feet of -rope must pass over the wheel C. In the same way it may be shown that -five feet must pass over B, and six feet over A. Thus, whatever be -the number of wheels in the upper and lower blocks, the parts of the -rope which pass in the same time over the wheels in the lower block -are in the proportion of the odd numbers 1, 3, 5, &c.; and those which -pass over the wheels in the upper block in the same time, are as the -even numbers 2, 4, 6, &c. If the wheels were all of equal size, as in -_fig. 119._, they would revolve with velocities proportional to -the rate at which the rope passes over them. So that, while the first -wheel below revolves once, the first wheel above will revolve twice; -the second wheel below three times; the second wheel above, four times, -and so on. If, however, the wheels differed in size in proportion to -the quantity of rope which must pass over them, they would evidently -revolve in the same time. Thus, if the first wheel above were twice the -size of the first wheel below, one revolution would throw off twice the -quantity of rope. Again, if the second wheel below were thrice the size -of the first wheel below, it would throw off in one revolution thrice -the quantity of rope, and so on. Wheels thus proportioned, revolving -in exactly the same time, might be all placed on one axle, and would -partake of one common motion, or, what is to the same effect, several -grooves might be cut upon the face of one solid wheel, with diameters -in the proportion of the odd numbers 1, 3, and 5, &c., for the lower -pulley, and corresponding grooves on the face of another solid wheel -represented by the even numbers 2, 4, 6, &c., for the upper pulley. The -rope being passed successively over the grooves of such wheels, would -be thrown off exactly in the same manner as if every groove were upon a -separate wheel, and every wheel revolved independently of the others. -Such is White’s pulley, represented in _fig. 121._ - -The advantage of this machine, when accurately constructed, is very -considerable. The friction, even when great resistances are to be -opposed, is very trifling; but, on the other hand, it has corresponding -disadvantages which greatly circumscribe its practical utility. In the -workmanship of the grooves great difficulty is found in giving them -the exact proportions. In doing which, the thickness of the rope must -be accurately allowed for; and consequently it follows, that the same -pulley can never act except with a rope of a particular diameter. A -very slight deviation from the true proportion of the grooves will -cause the rope to be unequally stretched, and will throw on some parts -of it an undue proportion of the weight, while other parts become -nearly, and sometimes altogether slack. Besides these defects, the rope -is so liable to derangement by being thrown out of the grooves, that -the pulley can scarcely be considered portable. - -For these and other reasons, this machine, ingenious as it -unquestionably is, has never been extensively used. - -(275.) In the several systems of pulleys just explained, the hook to -which the fixed block is attached supports the entire of both the power -and weight. When the machine is in equilibrium, the power only supports -so much of the weight as is equal to the tension of the cord, all the -remainder of the weight being thrown on the fixed point, according to -what was observed in (225.) - -If the power be moved so as to raise the weight, it will move with a -velocity as many times greater than that of the weight as the weight -itself is greater than the power. Thus in _fig. 118._ if the -weight attached to the lower block ascend one foot, six feet of line -will pass over the pulley A, according to what has been already proved. -Thus, the power will descend through six feet, while the weight rises -one foot. But, in this case, the weight is six times the power. All the -observations in (226.) will therefore be applicable to the cases of -great weights raised by small powers by means of the system of pulleys -just described. - -(276.) When two or more ropes are used, pulleys may be combined in -various ways so as to produce any degree of mechanical effect. If -to any of the systems already described a single moveable pulley -be added, the power of the machine would be doubled. In this case, -the second rope is attached to the hook of the lower block, as in -_fig. 122._, and being carried through a moveable pulley -attached to the weight, it is finally brought up to a fixed point. The -tension of the second cord is equal to half the weight (270.); and -therefore the power P, by means of the first cord, will have only half -the tension which it would have if the weight were attached to the -lower block. A moveable pulley thus applied is called a _runner_. - -[Illustration: _C. Varley, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -(277.) Two systems of pulleys, called _Spanish bartons_, having -each two ropes, are represented in _fig. 123._ The tension of -the rope P A B C in the first system is equal to -the power; and therefore the parts B A and B C support a -portion of the weight equal to twice the power. The rope E A -supports the tensions of A P and A B; and therefore the -tension of A E D is twice the power. Thus, the united -tensions of the ropes which support the pulley B is four times the -power, which is therefore the amount of the weight. In the second -system, the rope P A D is stretched by the power. The rope -A E B C acts against the united tensions A P and -A D; and therefore the tension of A E or E B is twice -the power. Thus, the weight acts against three tensions; two of which -are equal to twice the power, and the remaining one is equal to the -power. The weight is therefore equal to five times the power. - -A single rope may be so arranged with one moveable pulley as to support -a weight equal to three times the power. In _fig. 124._ this -arrangement is represented, where the numbers sufficiently indicate the -tension of the rope, and the proportion of the weight and power. In -_fig. 125._ another method of producing the same effect with two -ropes is represented. - -(278.) If several single moveable pulleys be made successively to act -upon each other, the effect is doubled by every additional pulley: -such a system as this is represented in _fig. 126._ The tension -of the first rope is equal to the power; the second rope acts against -twice the tension of the first, and therefore it is stretched with -a force equal to twice the power: the third rope acts against twice -this tension, and therefore it is stretched with a force equal -to four times the power, and so on. In the system represented in -_fig. 126._ there are three ropes, and the weight is eight times -the power. Another rope would render it sixteen times the power, and so -on. - -In this system, it is obvious that the ropes will require to have -different degrees of strength, since the tension to which they are -subject increases in a double proportion from the power to the weight. - -(279.) If each of the ropes, instead of being attached to fixed points -at the top, are carried over fixed pulleys, and attached to the several -moveable pulleys respectively, as in _fig. 127._, the power of -the machine will be greatly increased; for in that case the forces -which stretch the successive ropes increase in a treble instead of a -double proportion, as will be evident by attending to the numbers which -express the tensions in the figure. One rope would render the weight -three times the power, two ropes nine times, three ropes twenty-seven -times, and so on. An arrangement of pulleys is represented in _fig. -128._, by which each rope, instead of being finally attached to a fixed -point, as in _fig. 126._, is attached to the weight. The weight -is in this case supported by three ropes; one stretched with a force -equal to the power; another with a force equal to twice the power; -and a third with a force equal to four times the power. The weight is -therefore, in this case, seven times the power. - -(280.) If the ropes, instead of being attached to the weight, pass -through wheels, as in _fig. 129._, and are finally attached to the -pulleys above, the power of the machine will be considerably increased. -In the system here represented the weight is twenty-six times the power. - -(281.) In considering these several combinations of pulleys, we have -omitted to estimate the effects produced by the weights of the sheaves -and blocks. Without entering into the details of this computation, -it may be observed generally, that in the systems represented in -_figs. 126._, _127._ the weight of the wheel and blocks acts -against the power; but that in _figs. 128._ and _129._ they -assist the powers in supporting the weight. In the systems represented -in _fig. 123._ the weight of the pulleys, to a certain extent, -neutralise each other. - -(282.) It will in all cases be found, that that quantity by which the -weight exceeds the power is supported by fixed points; and therefore, -although it be commonly stated that a small power supports a great -weight, yet in the pulley, as in all other machines, the power supports -no more of the weight than is exactly equal to its own amount. It -will not be necessary to establish this in each of the examples which -have been given: having explained it in one instance, the student -will find no difficulty in applying the same reasoning to others. In -_fig. 126._, the fixed pulley sustains a force equal to twice the -power, and by it the power giving tension to the first rope sustains a -part of the weight equal to itself. The first hook sustains a portion -of the weight equal to the tension of the first string, or to the -power. The second hook sustains a force equal to twice the power; and -the third hook sustains a force equal to four times the power. The -three hooks therefore sustain a portion of the weight equal to seven -times the power; and the weight itself being eight times the power, it -is evident that the part of the weight which remains to be supported by -the power is equal to the power itself. - -(283.) When a weight is raised by any of the systems of pulleys which -have been last described, the proportion between the velocity of -the weight and the velocity of the power, so frequently noticed in -other machines, will always be observed. In the system of pulleys -represented in _fig. 126._ the weight being eight times the power, -the velocity of the power will be eight times that of the weight. If -the power be moved through eight feet, that part of the rope between -the fixed pulley and the first moveable pulley will be shortened by -eight feet. And since the two parts which lie above the first moveable -pulley must be equally shortened, each will be diminished by four feet; -therefore the first pulley will rise through four feet while the power -moves through eight feet. In the same way it may be shown, that while -the first pulley moves through four feet, the second moves through two; -and while the second moves through two, the third, to which the weight -is attached, is raised through one foot. While the power, therefore, is -carried through eight feet, the weight is moved through one foot. - -By reasoning similar to this, it may be shown that the space through -which the power is moved in every case is as many times greater than -the height through which the weight is raised, as the weight is greater -than the power. - -(284.) From its portable form, cheapness of construction, and the -facility with which it may be applied in almost every situation, -the pulley is one of the most useful of the simple machines. The -mechanical advantage, however, which it appears in theory to possess -is considerably diminished in practice, owing to the stiffness of the -cordage, and the friction of the wheels and blocks. By this means it -is computed that in most cases so great a proportion as two thirds of -the power is lost. The pulley is much used in building, where weights -are to be elevated to great heights. But its most extensive application -is found in the rigging of ships, where almost every motion is -accomplished by its means. - -(285.) In all the examples of pulleys, we have supposed the parts of -the rope sustaining the weight and each of the moveable pulleys to be -parallel to each other. If they be subject to considerable obliquity, -the relative tensions of the different ropes must be estimated -according to the principle applied in (271.) - - - - -CHAP. XVI. - -ON THE INCLINED PLANE, WEDGE, AND SCREW. - - -(286.) The inclined plane is the most simple of all machines. It is -a hard plane surface forming some angle with a horizontal plane, -that angle not being a right angle. When a weight is placed on such -a plane, a two-fold effect is produced. A part of the effect of the -weight is resisted by the plane, and produces a pressure upon it; and -the remainder urges the weight down the plane, and would produce a -pressure against any surface resisting its motion placed in a direction -perpendicular to the plane (131.) - -Let A B, _fig. 130._, be such a plane, B C its -horizontal base, A C its height, and A B C its angle -of elevation. Let W be a weight placed upon it. This weight acts in -the vertical direction W D, and is equivalent to two forces, -W F perpendicular to the plane, and W E directed down the -plane (74.) If a plane be placed at right angles to the inclined -plane below W, it will resist the descent of the weight, and sustain -a pressure expressed by W E. Thus, the weight W resting in the -corner, instead of producing one pressure in the direction W D, -will produce two pressures, one expressed by W F upon the inclined -plane, and the other expressed by W E upon the resisting plane. -These pressures respectively have the same proportion to the entire -weight as W F and W E have to W D, or as D E and -W E have to W D, because D E is equal to W F. Now -the triangle W E D is in all respects similar to the triangle -A B C, the one differing from the other only in the scale on -which it is constructed. Therefore, the three lines A C, C B, -and B A, are in the same proportion to each other as the lines -W E, E D, and W D. Hence, A B has to A C the -same proportion as the whole weight has to the pressure directed toward -B, and A B has to B C the same proportion as the whole -weight has to the pressure on the inclined plane. - -We have here supposed the weight to be sustained upon the inclined -plane by a hard plane fixed at right angles to it. But the power -necessary to sustain the weight will be the same in whatever way it is -applied, provided it act in the direction of the plane. Thus, a cord -may be attached to the weight, and stretched towards A, or the hands of -men may be applied to the weight below it, so as to resist its descent -towards B. But in whatever way it be applied, the amount of the power -will be determined in the same manner. Suppose the weight to consist -of as many pounds as there are inches in A B, then the power -requisite to sustain it upon the plane will consist of as many pounds -as there are inches in A C, and the pressure on the plane will -amount to as many pounds as there are inches in B C. - -From what has been stated it may easily be inferred that the less the -elevation of the plane is, the less will be the power requisite to -sustain a given weight upon it, and the greater will be the pressure -upon it. Suppose the inclined plane A B to turn upon a hinge -at B, and to be depressed so that its angle of elevation shall be -diminished, it is evident that as this angle decreases the height of -the plane decreases, and its base increases. Thus, when it takes the -position B A′, the height A′ C′ is less than the former -height A C, while the base B C′ is greater than the former -base B C. The power requisite to support the weight upon the plane -in the position B A′ is represented by A′ C′, and is as much -less than the power requisite to sustain it upon the plane A B, -as the height A′ C′ is less than the height A C. On the -other hand, the pressure upon the plane in the position B A′ is -as much greater than the pressure upon the plane B A, as the base -B C′ is greater than the base B C. - -(287.) The power of an inclined plane, considered as a machine, is -therefore estimated by the proportion which its length bears to its -height. This power is always increased by diminishing the elevation of -the plane. - -Roads which are not level may be regarded as inclined planes, and -loads drawn upon them in carriages, considered in reference to the -powers which impel them, are subject to all the conditions which have -been established for inclined planes. The inclination of the road is -estimated by the height corresponding to some proposed length. Thus it -is said to rise one foot in fifteen, one foot in twenty, &c., meaning -that if fifteen or twenty feet of the road be taken as the length of -an inclined plane, such as A B, the corresponding height will be -one foot. Or the same may be expressed thus: that if fifteen or twenty -feet be measured upon the road, the difference of the levels of the two -extremities of the distance measured is one foot. According to this -method of estimating the inclination of roads, the power requisite to -sustain a load upon them (setting aside the effect of friction), is -always proportional to that elevation. Thus, if a road rise one foot in -twenty, a power of one ton will be sufficient to sustain twenty tons, -and so on. - -On a horizontal plane the only resistance which the power has -to overcome is the friction of the load with the plane, and the -consideration of this being for the present omitted, a weight once put -in motion would continue moving for ever, without any further action of -the power. But if the plane be inclined, the power will be expended in -raising the weight through the perpendicular height of the plane. Thus, -in a road which rises one foot in ten, the power is expended in raising -the weight through one perpendicular foot for every ten feet of the -road over which it is moved. As the expenditure of power depends upon -the rate at which the weight is raised perpendicularly, it is evident -that the greater the inclination of the road is, the slower the motion -must be with the same force. If the energy of the power be such as to -raise the weight at the rate of one foot per minute, the weight may be -moved in each minute through that length of the road which corresponds -to a rise of one foot. Thus, if two roads rise one at the rate of a -foot in fifteen feet, and the other at the rate of one foot in twenty -feet, the same expenditure of power will move the weight through -fifteen feet of the one, and twenty feet of the other at the same rate. - -From such considerations as these, it will readily appear that it may -often be more expedient to carry a road through a circuitous route -than to continue it in the most direct course; for though the measured -length of road may be considerably greater than in the former case, yet -more may be gained in speed with the same expenditure of power than is -lost by the increase of distance. By attending to these circumstances, -modern road-makers have greatly facilitated and expedited the -intercourse between distant places. - -(288.) If the power act obliquely to the plane, it will have a twofold -effect; a part being expended in supporting or drawing the weight, -and a part in diminishing or increasing the pressure upon the plane. -Let W P, _fig. 130._, be the power. This will be equivalent -to two forces, W F′, perpendicular to the plane, and W E′ -in the direction of the plane. (74.) In order that the power should -sustain the weight, it is necessary that that part W E′ of the -power which acts in the direction of the plane should be equal to that -part W E, _fig. 130._, of the weight which acts down the -plane. The other part W F′ of the power acting perpendicular to -the plane is immediately opposed to that part W F of the weight -which produces pressure. The pressure upon the plane will therefore -be diminished by the amount of W F′. The amount of the power -which will equilibrate with the weight may, in this case, be found -as follows. Take W E′ equal to W E, and draw E′ P -perpendicular to the plane, and meeting the direction of the power. -The proportion of the power to the weight will be that of W P to -W D. And the proportion of the pressure to the weight will be that -of the difference between W F and W F′ to W D. If the -amount of the power have a less proportion to the weight than W P -has to W D, it will not support the body on the plane, but will -allow it to descend. And if it have a greater proportion, it will draw -the weight up the plane towards A. - -(289.) It sometimes happens that a weight upon one inclined plane is -raised or supported by another weight upon another inclined plane. -Thus, if A B and A B′, _fig. 131._, be two inclined -planes forming an angle at A, and W W′ be two weights placed -upon these planes, and connected by a cord passing over a pulley at -A, the one weight will either sustain the other, or one will descend, -drawing the other up. To determine the circumstances under which these -effects will ensue, draw the lines W D and W′ D′ in the -vertical direction, and take upon them as many inches as there are -ounces in the weights respectively. W D and W′ D′ being the -lengths thus taken, and therefore representing the weights, the lines -W E and W′ E′ will represent the effects of these weights -respectively down the planes. If W E and W′ E′ be equal, the -weights will sustain each other without motion. But if W E be -greater than W′ E′, the weight W will descend, drawing the weight -W′ up. And if W′ E′ be greater than W E, the weight W′ will -descend, drawing the weight W up. In every case the lines W F and -W′ F′ will represent the pressures upon the planes respectively. - -It is not necessary, for the effect just described, that the inclined -planes should, as represented in the figure, form an angle with each -other. They may be parallel, or in any other position, the rope being -carried over a sufficient number of wheels placed so as to give it the -necessary deflection. This method of moving loads is frequently applied -in great public works where rail-roads are used. Loaded waggons descend -one inclined plane, while other waggons, either empty or so loaded as -to permit the descent of those with which they are connected, are drawn -up the other. - -(290.) In the application of the inclined plane which we have hitherto -noticed, the machine itself is supposed to be fixed in its position, -while the weight or load is moved upon it. But it frequently happens -that resistances are to be overcome which do not admit of being thus -moved. In such cases, instead of moving the load upon the planes, -the plane is to be moved under or against the load. Let D E, -_fig. 132._, be a heavy beam secured in a vertical position -between guides F G and H I, so that it is free to move -upwards and downwards, but not laterally. Let A B C be an -inclined plane, the extremity of which is placed beneath the end of -the beam. A force applied to the back of this plane A C, in the -direction C B, will urge the plane under the beam so as to raise -the beam to the position represented in _fig. 133._ Thus, while -the inclined plane is moved through the distance C B, the beam is -raised through the height C A. - -(291.) When the inclined plane is applied in this manner, it is called -a _wedge_. And if the power applied to the back were a continued -pressure, its proportion to the weight would be that of A C to -C B. It follows, therefore, that the more acute the angle B is, -the more powerful will be the wedge. - -In some cases, the wedge is formed of two inclined planes, placed base -to base, as represented in _fig. 134._ The theoretical estimation -of the power of this machine is not applicable in practice with any -degree of accuracy. This is in part owing to the enormous proportion -which the friction in most cases bears to the theoretical value of -the power, but still more to the nature of the power generally used. -The force of a blow is of a nature so wholly different from continued -forces, such as the pressure of weights, or the resistance offered by -the cohesion of bodies, that it admits of no numerical comparison with -them. Hence we cannot properly state the proportion which the force -of a blow bears to the amount of a weight or resistance. The wedge is -almost invariably urged by percussion; while the resistances which it -has to overcome are as constantly forces of the other kind. Although, -however, no exact numerical comparison can be made, yet it may be -stated in a general way that the wedge is more and more powerful as its -angle is more acute. - -[Illustration: _C. Varley, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -In the arts and manufactures, wedges are used where enormous force -is to be exerted through a very small space. Thus it is resorted to -for splitting masses of timber or stone. Ships are raised in docks -by wedges driven under their keels. The wedge is the principal agent -in the oil-mill. The seeds from which the oil is to be extracted are -introduced into hair bags, and placed between planes of hard wood. -Wedges inserted between the bags are driven by allowing heavy beams to -fall on them. The pressure thus excited is so intense, that the seeds -in the bags are formed into a mass nearly as solid as wood. Instances -have occurred in which the wedge has been used to restore a tottering -edifice to its perpendicular position. - -All cutting and piercing instruments, such as knives, razors, scissors, -chisels, &c., nails, pins, needles, awls, &c. are wedges. The angle -of the wedge, in these cases, is more or less acute, according to -the purpose to which it is to be applied. In determining this, two -things are to be considered--the mechanical power, which is increased -by diminishing the angle of the wedge; and the strength of the tool, -which is always diminished by the same cause. There is, therefore, -a practical limit to the increase of the power, and that degree of -sharpness only is to be given to the tool which is consistent with the -strength requisite for the purpose to which it is to be applied. In -tools intended for cutting wood, the angle is generally about 30°. For -iron it is from 50° to 60°; and for brass, from 80° to 90°. Tools which -act by pressure may be made more acute than those which are driven by a -blow; and in general the softer and more yielding the substance to be -divided is, and the less the power required to act upon it, the more -acute the wedge may be constructed. - -In many cases the utility of the wedge depends on that which is -entirely omitted in its theory, viz. the friction which arises between -its surface and the substance which it divides. This is the case when -pins, bolts, or nails are used for binding the parts of structures -together; in which case, were it not for the friction, they would -recoil from their places, and fail to produce the desired effect. Even -when the wedge is used as a mechanical engine, the presence of friction -is absolutely indispensable to its practical utility. The power, as -has already been stated, generally acts by successive blows, and is -therefore subject to constant intermission, and but for the friction -the wedge would recoil between the intervals of the blows with as much -force as it had been driven forward. Thus the object of the labour -would be continually frustrated. The friction in this case is of the -same use as a ratchet wheel, but is much more necessary, as the power -applied to the wedge is more liable to intermission than in the cases -where ratchet wheels are generally used. - -(292.) When a road directly ascends the side of a hill, it is to be -considered as an inclined plane; but it will not lose its mechanical -character, if, instead of directly ascending towards the top of the -hill, it winds successively round it, and gradually ascends so as after -several revolutions to reach the top. In the same manner a path may be -conceived to surround a pillar by which the ascent may be facilitated -upon the principle of the inclined plane. Winding stairs constructed in -the interior of great columns partake of this character; for although -the ascent be produced by successive steps, yet if a floor could -be made sufficiently rough to prevent the feet from slipping, the -ascent would be accomplished with equal facility. In such a case the -winding path would be equivalent to an inclined plane, bent into such -a form as to accommodate it to the peculiar circumstances in which it -would be required to be used. It will not be difficult to trace the -resemblance between such an adaptation of the inclined plane and the -appearances presented by the thread of a _screw_: and it may hence be -easily understood that a screw is nothing more than an inclined plane -constructed upon the surface of a cylinder. - -This will, perhaps, be more apparent by the following contrivance: -Let A B, _fig. 135._, be a common round ruler, and let -C D E be a piece of white paper cut in the form of an -inclined plane, whose height C D is equal to the length of the -ruler A B, and let the edge C E of the paper be marked with -a broad black line: let the edge C D be applied to the ruler -A B, and being attached thereto, let the paper be rolled round -the ruler; the ruler will then present the appearance of a screw, -_fig. 136._ the thread of the screw being marked by the black -line C E, winding continually round the ruler. Let D F, -_fig. 135._, be equal to the circumference of the ruler, and draw -F G parallel to D C, and G H parallel to D E, the -part C G F D of the paper will exactly surround the -ruler once: the part C G will form one convolution of the thread, -and may be considered as the length of one inclined plane surrounding -the cylinder, C H being the corresponding height, and G H -the base. The power of the screw does not, as in the ordinary cases of -the inclined plane, act parallel to the plane or thread, but at right -angles to the length of the cylinder A B, or, what is to the same -effect, parallel to the base H G; therefore the proportion of the -power to the weight will be, according to principles already explained, -the same as that of C H to the space through which the power -moves parallel to H G in one revolution of the screw. H C is -evidently the distance between the successive positions of the thread -as it winds round the cylinder; and it appears from what has been just -stated, that the less this distance is, or, in other words, the finer -the thread is, the more powerful the machine will be. - -(293.) In the application of the screw the weight or resistance is -not, as in the inclined plane and wedge, placed upon the surface of -the plane or thread. The power is usually transmitted by causing the -screw to move in a concave cylinder, on the interior surface of which -a spiral cavity is cut, corresponding exactly to the thread of the -screw, and in which the thread will move by turning round the screw -continually in the same direction. This hollow cylinder is usually -called the _nut_ or _concave screw_. The screw surrounded by its -spiral thread is represented in _fig. 137._; and a section of the -same playing in the nut is represented in _fig. 138._ - -There are several ways in which the effect of the power may be conveyed -to the resistance by this apparatus. - -First, let us suppose that the nut A B is fixed. If the screw be -continually turned on its axis, by a lever E F inserted in one -end of it, it will be moved in the direction C D, advancing every -revolution through a space equal to the distance between two contiguous -threads. By turning the lever in an opposite direction, the screw will -be moved in the direction D C. - -If the screw be fixed, so as to be incapable either of moving -longitudinally or revolving on its axis, the nut A B may be turned -upon the screw by a lever, and will move on the screw towards C or -towards D, according to the direction in which the lever is turned. - -In the former case we have supposed the nut to be absolutely -immoveable, and in the latter case the screw to be absolutely -immoveable. It may happen, however, that the nut, though capable of -revolving, is incapable of moving longitudinally; and that the screw, -though incapable of revolving, is capable of moving longitudinally. In -that case, by turning the nut A B upon the screw by the lever, the -screw will be urged in the direction C D or D C, according to -the way in which the nut is turned. - -The apparatus may, on the contrary, be so arranged, that the nut, -though incapable of revolving, is capable of moving longitudinally; -and the screw, though capable of revolving, is incapable of moving -longitudinally. In this case, by turning the screw in the one direction -or in the other, the nut A B will be urged in the direction -C D or D C. - -All these various arrangements may be observed in different -applications to the machine. - -(294.) A screw may be cut upon a cylinder by placing the cylinder in -a turning lathe, and giving it a rotatory motion upon its axis. The -cutting point is then presented to the cylinder, and moved in the -direction of its length, at such a rate as to be carried through the -distance between the intended thread, while the cylinder revolves -once. The relative motions of the cutting point and the cylinder being -preserved with perfect uniformity, the thread will be cut from one end -to the other. The shape of the threads may be either square, as in -_fig. 137._, or triangular, as in _fig. 139._ - -(295.) The screw is generally used in cases where severe pressure -is to be excited through small spaces; it is therefore the agent in -most presses. In _fig. 140._, the nut is fixed, and by turning -the lever, which passes through the head of the screw, a pressure is -excited upon any substance placed upon the plate immediately under -the end of the screw. In _fig. 141._, the screw is incapable of -revolving, but is capable of advancing in the direction of its length. -On the other hand, the nut is capable of revolving, but does not -advance in the direction of the screw. When the nut is turned by means -of the screw inserted in it, the screw advances in the direction of its -length, and urges the board which is attached to it upwards, so as to -press any substance placed between it and the fixed board above. - -In cases where liquids or juices are to be expressed from solid bodies, -the screw is the agent generally employed. It is also used in coining, -where the impression of a die is to be made upon a piece of metal, and -in the same way in producing the impression of a seal upon wax or other -substance adapted to receive it. When soft and light materials, such -as cotton, are to be reduced to a convenient bulk for transportation, -the screw is used to compress them, and they are thus reduced into hard -dense masses. In printing, the paper is urged by a severe and sudden -pressure upon the types, by means of a screw. - -(296.) As the mechanical power of the screw depends upon the relative -magnitude of the circumference through which the power revolves, and -the distance between the threads, it is evident, that, to increase -the efficacy of the machine, we must either increase the length -of the lever by which the power acts, or diminish the magnitude of -the thread. Although there is no limit in theory to the increase of -the mechanical efficacy by these means, yet practical inconvenience -arises which effectually prevents that increase being carried beyond -a certain extent. If the lever by which the power acts be increased, -the same difficulty arises as was already explained in the wheel and -axle (254.); the space through which the power should act would be so -unwieldy, that its application would become impracticable. If, on the -other hand, the power of the machine be increased by diminishing the -size of the thread, the strength of the thread will be so diminished, -that a slight resistance will tear it from the cylinder. The cases -in which it is necessary to increase the power of the machine, being -those in which the greatest resistances are to be overcome, the object -will evidently be defeated, if the means chosen to increase that power -deprive the machine of the strength which is necessary to sustain the -force to which it is to be submitted. - -(297.) These inconveniences are removed by a contrivance of Mr. Hunter, -which, while it gives to the machine all the requisite strength and -compactness, allows it to have an almost unlimited degree of mechanical -efficacy. - -This contrivance consists in the use of two screws, the threads of -which may have any strength and magnitude, but which have a very small -difference of breadth. While the working point is urged forward by -that which has the greater thread, it is drawn back by that which has -the less; so that during each revolution of the screw, instead of -being advanced through a space equal to the magnitude of either of -the threads, it moves through a space equal to their difference. The -mechanical power of such a machine will be the same as that of a single -screw having a thread, whose magnitude is equal to the difference of -the magnitudes of the two threads just mentioned. - -Thus, without inconveniently increasing the sweep of the power, on the -one hand, or, on the other, diminishing the thread until the necessary -strength is lost, the machine will acquire an efficacy limited by -nothing but the smallness of the difference between the two threads. - -This principle was first applied in the manner represented in -_fig. 142._ A is the greater thread, playing in the fixed nut; B -is the lesser thread, cut upon a smaller cylinder, and playing in a -concave screw, cut within the greater cylinder. During every revolution -of the screw, the cylinder A descends through a space equal to the -distance between its threads. At the same time the smaller cylinder -B ascends through a space equal to the distance between the threads -cut upon it: the effect is, that the board D descends through a space -equal to the difference between the threads upon A and the threads upon -B, and the machine has a power proportionate to the smallness of this -difference. - -Thus, suppose the screw A has twenty threads in an inch, while the -screw B has twenty-one; during one revolution, the screw A will -descend through a space equal to the 20th part of an inch. If, during -this motion, the screw B did not turn within A, the board D would be -advanced through the 20th of an inch; but because the hollow screw -within A turns upon B, the screw B will, relatively to A, be raised in -one revolution through a space equal to the 21st part of an inch. Thus, -while the board D is depressed through the 20th of an inch by the screw -A, it is raised through the 21st of an inch by the screw B. It is, -therefore, on the whole, depressed through a space equal to the excess -of the 20th of an inch above the 21st of an inch, that is, through the -420th of an inch. - -The power of this machine will, therefore, be expressed by the number -of times the 420th of an inch is contained in the circumference through -which the power moves. - -(298.) In the practical application of this principle at present the -arrangement is somewhat different. The two threads are usually cut -on different parts of the same cylinder. If nuts be supposed to be -placed upon these, which are capable of moving in the direction of -the length, but not of revolving, it is evident that by turning the -screw once round, each nut will be advanced through a space equal to -the breadth of the respective threads. By this means the two nuts -will either approach each other, or mutually recede, according to the -direction in which the screw is turned, through a space equal to the -difference of the breadth of the threads, and they will exert a force -either in compressing or extending any substance placed between them, -proportionate to the smallness of that difference. - -(299.) A toothed wheel is sometimes used instead of a nut, so that -the same quality by which the revolution of the screw urges the nut -forward is applied to make the wheel revolve. The screw is in this -case called an endless screw, because its action upon the wheel may be -continued without limit. This application of the screw is represented -in _fig. 143._ P is the winch to which the power is applied; and -its effect at the circumference of the wheel is estimated in the same -manner as the effect of the screw upon the nut. This effect is to be -considered as a power acting upon the circumference of the wheel; and -its proportion to the weight or resistance is to be calculated in the -same manner as the proportion of the power to the weight in the wheel -and axle. - -(300.) We have hitherto considered the screw as an engine used to -overcome great resistances. It is also eminently useful in several -departments of experimental science, for the measurement of very -minute motions and spaces, the magnitude of which could scarcely be -ascertained by any other means. The very slow motion which may be -imparted to the end of a screw, by a very considerable motion in the -power, renders it peculiarly well adapted for this purpose. To explain -the manner in which it is applied--suppose a screw to be so cut as -to have fifty threads in an inch, each revolution of the screw will -advance its point through the fiftieth part of an inch. Now, suppose -the head of the screw to be a circle, whose diameter is an inch, the -circumference of the head will be something more than three inches: -this may be easily divided into a hundred equal parts distinctly -visible. If a fixed index be presented to this graduated circumference, -the hundredth part of a revolution of the screw may be observed, by -noting the passage of one division of the head under the index. Since -one entire revolution of the head moves the point through the fiftieth -of an inch, one division will correspond to the five thousandth of an -inch. In order to observe the motion of the point of the screw in this -case, a fine wire is attached to it, which is carried across the field -of view of a powerful microscope, by which the motion is so magnified -as to be distinctly perceptible. - -A screw used for such purposes is called a _micrometer screw_. Such an -apparatus is usually attached to the limbs of graduated instruments, -for the purposes of astronomical and other observation. Without the -aid of this apparatus, no observation could be taken with greater -accuracy than the amount of the smallest division upon the limb. Thus, -if an instrument for measuring angles were divided into small arcs of -one minute, and an angle were observed which brought the index of the -instrument to some point between two divisions, we could only conclude -that the observed angle must consist of a certain number of degrees and -minutes, together with an additional number of seconds, which would -be unknown, inasmuch as there would be no means of ascertaining the -fraction of a minute between the index and the adjacent division of -the instrument. But if a screw be provided, the point of which moves -through a space equal to one division of the instrument, with sixty -revolutions of the head, and that the head itself be divided into -one hundred equal parts, each complete revolution of the screw will -correspond to the sixtieth part of a minute, or to one second, and each -division on the head of the screw will correspond to the hundredth part -of a second. The index being attached to this screw, let the head -be turned until the index be moved from its observed position to the -adjacent division of the limb. The number of complete revolutions of -the screw necessary to accomplish this will be the number of seconds; -and the number of parts of a revolution over the complete number of -revolutions will be the hundredth parts of a second necessary to be -added to the degrees and minutes primarily observed. - -It is not, however, only to such instruments that the micrometer screw -is applicable; any spaces whatever may be measured by it. An instance -of its mechanical application may be mentioned in a steel-yard, -an instrument for ascertaining the amount of weights by a given -weight, sliding on a long graduated arm of a lever. The distance from -the fulcrum, at which this weight counterpoises the weight to be -ascertained, serves as a measure to the amount of that weight. When the -sliding weight happens to be placed between two divisions of the arm, a -micrometer screw is used to ascertain the fraction of the division. - -Hunter’s screw, already described, seems to be well adapted to -micrometrical purposes; since the motion of the point may be rendered -indefinitely slow, without requiring an exquisitely fine thread, such -as in the single screw would be necessary. - - - - -CHAP. XVII. - -ON THE REGULATION AND ACCUMULATION OF FORCE. - - -(301.) It is frequently indispensable, and always desirable, that the -operation of a machine should be regular and uniform. Sudden changes -in its velocity, and desultory variations in the effective energy -of its power, are often injurious or destructive to the apparatus -itself, and when applied to manufactures never fail to produce -unevenness in the work. To invent methods for insuring the regular -motion of machinery, by removing those causes of inequality which may -be avoided, and by compensating others, has therefore been a problem to -which much attention and ingenuity have been directed. This is chiefly -accomplished by controlling, and, as it were, measuring out the power -according to the exigencies of the machine, and causing its effective -energy to be always commensurate with the resistance which it has to -overcome. - -[Illustration: _C. Varley, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -Irregularity in the motion of machinery may proceed from one or more -of the following causes:--1. irregularity in the prime mover; 2. -occasional variation in the amount of the load or resistance; and, 3. -because, in the various positions which the parts of the machine assume -during its motion, the power may not be transmitted with equal effect -to the working point. - -The energy of the prime mover is seldom if ever regular. The force of -water varies with the copiousness of the stream. The power which impels -the windmill is proverbially capricious. The pressure of steam varies -with the intensity of the furnace. Animal power, the result of will, -temper, and health is difficult of control. Human labour is most of -all unmanageable; hence no machine works so irregularly as one which -is manipulated. In some cases the moving force is subject, by the very -conditions of its existence, to constant variation, as in the example -of a spring, which gradually loses its energy as it recoils. (255.) In -many instances the prime mover is liable to regular intermission, and -is actually suspended for certain intervals of time. This is the case -in the single acting steam-engine, where the pressure of the steam -urges the descent of the piston, but is suspended during its ascent. - -The load or resistance to which the machine is applied is not less -fluctuating. In mills there are a multiplicity of parts which are -severally liable to be occasionally disengaged, and to have their -operation suspended. In large factories for spinning, weaving, -printing, &c. a great number of separate spinning machines, looms, -presses, or other engines, are usually worked by one common mover, such -as a water-wheel or steam-engine. In these cases the number of machines -employed from time to time necessarily varies with the fluctuating -demand for the articles produced, and from other causes. Under such -circumstances the velocity with which every part of the machinery is -moved would suffer corresponding changes, increasing its rapidity with -every augmentation of the moving power or diminution of the resistance, -or being retarded in its speed by the contrary circumstances. - -But even when the prime mover and the resistance are both regular, or -rendered so by proper contrivances, still it will rarely happen that -the machine by which the energy of the one is transmitted to the other -conveys this with unimpaired effect in all the phases of its operation. -To give a general notion of this cause of inequality to those who have -not been familiar with machinery would not be easy, without having -recourse to an example. For the present we shall merely state, that the -several moving parts of every machine assume in succession a variety of -positions; that at regular periods they return to their first position, -and again undergo the same succession of changes. In the different -positions through which they are carried in every period of motion, -the efficacy of the machine to transmit the power to the resistance -is different, and thus the effective energy of the machine in acting -upon the resistance would be subject to continual fluctuation. This -will be more clearly understood when we come to explain the methods of -counteracting the defect or equalising the action of the power upon the -resistance. - -Such are the chief causes of the inequalities incidental to the -motion of machinery, and we now propose to describe a few of the many -ingenious contrivances which the skill of engineers has produced to -remove the consequent inconveniences. - -(302.) Setting aside, for the present, the last cause of inequality, -and considering the machinery, whatever it be, to transmit the power to -the resistance without irregular interruption, it is evident that every -contrivance, having for its object to render the velocity uniform, -can only accomplish this by causing the variations of the power and -resistance to be proportionate to each other. This may be done either -by increasing or diminishing the power as the resistance increases or -diminishes; or by increasing or diminishing the resistance as the power -increases or diminishes. - -According to the facilities or convenience presented by the peculiar -circumstances of the case either of these methods is adopted. - -The contrivances for effecting this are called _regulators_. Most -regulators act upon that part of the machine which commands the supply -of the power by means of levers, or some other mechanical contrivance, -so as to check the quantity of the moving principle conveyed to the -machine when the velocity has a tendency to increase; and, on the -other hand, to increase that supply upon any undue abatement of its -speed. In a water-mill this is done by acting upon the shuttle; in a -wind-mill, by an adjustment of the sail-cloth; and in a steam-engine, -by opening or closing, in a greater or less degree, the valve by which -the cylinder is supplied with steam. - -(303.) Of all the contrivances for regulating machinery, that which is -best known and most commonly used is the _governor_. This regulator, -which had been long in use in mill-work and other machinery, has of -late years attracted more general notice by its beautiful adaptation -in the steam-engines of Watt. It consists of heavy balls B B, -_fig. 144._, attached to the extremities of rods B F. -These rods play upon a joint at E, passing through a mortise in the -vertical stem D D′. At F they are united by joints to the short -rods F H, which are again connected by joints at H to a ring -which slides upon the vertical shaft D D′. From this description -it will be apparent that when the balls B are drawn from the axis, -their upper arms E F are caused to increase their divergence in -the same manner as the blades of a scissors are opened by separating -the handles. These, acting upon the ring by means of the short links -F H, draw it down the vertical axis from D towards E. A contrary -effect is produced when the balls B are brought closer to the axis, and -the divergence of the rods B E diminished. A horizontal wheel W is -attached to the vertical axis D D′, having a groove to receive a -rope or strap upon its rim. This strap passes round the wheel or axis -by which motion is transmitted to the machinery to be regulated, so -that the spindle or shaft D D′ will always be made to revolve with -a speed proportionate to that of the machinery. - -As the shaft D D′ revolves, the balls B are carried round it -with a circular motion, and consequently acquire a centrifugal force -which causes them to recede from the axle, and therefore to depress -the ring H. On the edge or rim of this ring is formed a groove, which -is embraced by the prongs of a fork I, at the extremity of one arm of -a lever whose fulcrum is at G. The extremity K of the other arm is -connected by some means with the part of the machine which supplies -the power. In the present instance we shall suppose it a steam-engine, -in which case the rod K I communicates with a flat circular valve -V, placed in the principal steam-pipe, and so arranged that, when K is -elevated as far as by their divergence the balls B have power over it, -the passage of the pipe will be closed by the valve V, and the passage -of steam entirely stopped; and, on the other hand, when the balls -subside to their lowest position, the valve will be presented with its -edge in the direction of the tube, so as to intercept no part of the -steam. - -The property which renders this instrument so admirably adapted to -the purpose to which it is applied is, that when the divergence of -the balls is not very considerable, they must always revolve with the -same velocity, whether they move at a greater or lesser distance from -the vertical axis. If any circumstance increases that velocity, the -balls instantly recede from the axis, and closing the valve V, check -the supply of steam, and thereby diminishing the speed of the motion, -restore the machine to its former rate. If, on the contrary, that -fixed velocity be diminished, the centrifugal force being no longer -sufficient to support the balls, they descend towards the axle, open -the valve V, and, increasing the supply of steam, restore the proper -velocity of the machine. - -When the governor is applied to a water-wheel it is made to act upon -the shuttle through which the water flows, and controls its quantity as -effectually, and upon the same principle, as has just been explained in -reference to the steam-engine. When applied to a windmill it regulates -the sail-cloth so as to diminish the efficacy of the power upon the -arms as the force of the wind increases, or _vice versâ_. - -In cases where the resistance admits of easy and convenient change, the -governor may act so as to accommodate it to the varying energy of the -power. This is often done in corn-mills, where it acts upon the shuttle -which metes out the corn to the millstones. When the power which drives -the mill increases, a proportionally increased feed of corn is given -to the stones, so that the resistance being varied in the ratio of the -power, the same velocity will be maintained. - -(304.) In some cases the centrifugal force of the revolving balls is -not sufficiently great to control the power or the resistance, and -regulators of a different kind must be resorted to. The following -contrivance is called the _water-regulator_:-- - -A common pump is worked by the machine, whose motion is to be -regulated, and water is thus raised and discharged into a cistern. -It is allowed to flow from this cistern through a pipe of a given -magnitude. When the water is pumped up with the same velocity as it is -discharged by this pipe, it is evident that the level of the water in -the cistern will be stationary, since it receives from the pump the -exact quantity which it discharges from the pipe. But if the pump -throw in more water in a given time than is discharged by the pipe, -the cistern will begin to be filled, and the level of the water will -rise. If, on the other hand, the supply from the pump be less than -the discharge from the pipe, the level of the water in the cistern -will subside. Since the rate at which water is supplied from the -pump will always be proportional to the velocity of the machine, it -follows that every fluctuation in this velocity will be indicated by -the rising or subsiding of the level of the water in the cistern, and -that level never can remain stationary, except at that exact velocity -which supplies the quantity of water discharged by the pipe. This pipe -may be constructed so as by an adjustment to discharge the water at -any required rate; and thus the cistern may be adapted to indicate a -constant velocity of any proposed amount. - -If the cistern were constantly watched by an attendant, the velocity of -the machine might be abated by regulating the power when the level of -the water is observed to rise, or increased when it falls; but this is -much more effectually and regularly performed by causing the surface -of the water itself to perform the duty. A float or large hollow metal -ball is placed upon the surface of the water in the cistern. This ball -is connected with a lever acting upon some part of the machinery, which -controls the power or regulates the amount of resistance, as already -explained in the case of the governor. When the level of the water -rises, the buoyancy of the ball causes it to rise also with a force -equal to the difference between its own weight and the weight of as -much water as it displaces. By enlarging the floating ball, a force may -be obtained sufficiently great to move those parts of the machinery -which act upon the power or resistance, and thus either to diminish -the supply of the moving principle or to increase the amount of the -resistance, and thereby retard the motion and reduce the velocity to -its proper limit. When the level of the water in the cistern falls, -the floating ball being no longer supported on the liquid surface, -descends with the force of its own weight, and producing an effect upon -the power or resistance contrary to the former, increases the effective -energy of the one, or diminishes that of the other, until the velocity -proper to the machine be restored. - -The sensibility of these regulators is increased by making the surface -of water in the cistern as small as possible; for then a small change -in the rate at which the water is supplied by the pump will produce a -considerable change in the level of the water in the cistern. - -Instead of using a float, the cistern itself may be suspended from -the lever which controls the supply of the power, and in this case a -sliding weight may be placed on the other arm, so that it will balance -the cistern when it contains that quantity of water which corresponds -to the fixed level already explained. If the quantity of water in -the cistern be increased by an undue velocity of the machine, the -weight of the cistern will preponderate, draw down the arm of the -lever, and check the supply of the power. If, on the other hand, the -supply of water be too small, the cistern will no longer balance the -counterpoise, the arm by which it is suspended will be raised, and the -energy of the power will be increased. - -(305.) In the steam-engine the self-regulating principle is carried -to an astonishing pitch of perfection. The machine itself raises in -due quantity the cold water necessary to condense the steam. It pumps -off the hot water produced by the steam, which has been cooled, and -lodges it in a reservoir for the supply of the boiler. It carries from -this reservoir exactly that quantity of water which is necessary to -supply the wants of the boiler, and lodges it therein according as it -is required. It breathes the boiler of redundant steam, and preserves -that which remains fit, both in quantity and quality, for the use of -the engine. It blows its own fire, maintaining its intensity, and -increasing or diminishing it according to the quantity of steam which -it is necessary to raise; so that when much work is expected from the -engine, the fire is proportionally brisk and vivid. It breaks and -prepares its own fuel, and scatters it upon the bars at proper times -and in due quantity. It opens and closes its several valves at the -proper moments, works its own pumps, turns its own wheels, and is only -not alive. Among so many beautiful examples of the self-regulating -principle, it is difficult to select. We shall, however, mention one or -two, and for others refer the reader to our treatise on this subject.[3] - -[3] Lardner on the Steam-Engine, Steam-Navigation, Roads, and Railways. -8th edition. 1851. - -It is necessary in this machine that the water in the boiler be -maintained constantly at the same level, and, therefore, that as -much be supplied, from time to time, as is consumed by evaporation. -A pump which is wrought by the engine itself supplies a cistern C, -_fig. 145._, with hot water. At the bottom of this cistern is a -valve V opening into a tube which descends into the boiler. This valve -is connected by a wire with the arm of a lever on the fulcrum D, the -other arm E of which is also connected by a wire with a stone float F, -which is partially immersed in the water of the boiler, and is balanced -by a sliding weight A. The weight A only counterpoises the stone float -F by the aid of its buoyance in the water; for if the water be removed, -the stone F will preponderate, and raise the weight A. When the water -in the boiler is at its proper level, the length of the wire connecting -the valve V with the lever is so adjusted that this valve shall be -closed, the wire at the same time being fully extended. When, by -evaporation, the water in the boiler begins to be diminished, the level -falls, and the stone weight F, being no longer supported, overcomes -the counterpoise A, raises the arm of the lever, and, pulling the -wire, opens the valve V. The water in the cistern C then flows through -the tube into the boiler, and continues to flow until the level be so -raised that the stone weight F is again elevated, the valve V closed, -and the further supply of water from the cistern C suspended. - -In order to render the operation of this apparatus easily -intelligible, we have here supposed an imperfection which does not -exist. According to what has just been stated, the level of the water -in the boiler descends from its proper height, and subsequently returns -to it. But, in fact, this does not happen. The float F and valve V -adjust themselves, so that a constant supply of water passes through -the valve, which proceeds exactly at the same rate as that at which the -water in the boiler is consumed. - -(306.) In the same machine there occurs a singularly happy example of -self-adjustment, in the method by which the strength of the fire is -regulated. The governor regulates the supply of steam to the engine, -and proportions it to the work to be done. With this work, therefore, -the demands upon the boiler increase or diminish, and with these -demands the production of steam in the boiler ought to vary. In fact, -the rate at which steam is generated in the boiler, ought to be equal -to that at which it is consumed in the engine, otherwise one of two -effects must ensue: either the boiler will fail to supply the engine -with steam, or steam will accumulate in the boiler, being produced in -undue quantity, and, escaping at the safety valve, will thus be wasted. -It is, therefore, necessary to control the agent which generates the -steam, namely, the fire, and to vary its intensity from time to time, -proportioning it to the demands of the engine. To accomplish this, -the following contrivance has been adopted:--Let T, _fig. 146._, -be a tube inserted in the top of the boiler, and descending nearly -to the bottom. The pressure of the steam confined in the boiler, -acting upon the surface of the water, forces it to a certain height -in the tube T. A weight F, half immersed in the water in the tube, is -suspended by a chain, which passes over the wheels P P′, and is -balanced by a metal plate D, in the same manner as the stone float, -_fig. 145._, is balanced by the weight A. The plate D passes -through the mouth of the flue E as it issues finally from the boiler; -so that when the plate D falls it stops the flue, suspending thereby -the draught of air through the furnace, mitigating the intensity of -the fire, and checking the production of steam. If, on the contrary, -the plate D be drawn up, the draught is increased, the fire is rendered -more active, and the production of steam in the boiler is stimulated. -Now, suppose that the boiler produces steam faster than the engine -consumes it, either because the load on the engine has been diminished, -and, therefore, its consumption of steam reduced, or because the -fire has become too intense; the consequence is, that the steam, -beginning to accumulate in the boiler, will press upon the surface of -the water with increased force, and the water will be raised in the -tube T. The weight F will, therefore, be lifted, and the plate D will -descend, diminish, or stop the draught, mitigate the fire, and retard -the production of steam, and will continue to do so until the rate -at which steam is produced shall be commensurate to the wants of the -engine. If, on the other hand, the production of steam be inadequate -to the exigency of the machine, either because of an increased load, -or of the insufficient force of the fire, the steam in the boiler will -lose its elasticity, and the surface of the water not sustaining its -wonted pressure, the water in the tube T will fall; consequently the -weight F will descend, and the plate D will be raised. The flue being -thus opened, the draught will be increased, and the fire rendered -more intense. Thus the production of steam becomes more rapid, and is -rendered sufficiently abundant for the purposes of the engine. This -apparatus is called the _self-acting damper_. - -(307.) When a perfectly uniform rate of motion has not been attained, -it is often necessary to indicate small variations of velocity. The -following contrivance, called a _tachometer_[4], has been invented -to accomplish this. A cup, _fig. 147._, is filled to the level -C D with quicksilver, and is attached to a spindle, which is -whirled by the machine in the same manner as the governor already -described. It is well known that the centrifugal force produced by this -whirling motion will cause the mercury to recede from the centre and -rise upon the sides of the cup, so that its surface will assume the -concave appearance represented in _fig. 148._ In this case the -centre of the surface will obviously have fallen below its original -level, _fig. 147._, and the edges will have risen above that -level. As this effect is produced by the velocity of the machine, so -it is proportionate to that velocity, and subject to corresponding -variations. Any method of rendering visible small changes in the -central level of the surface of the quicksilver will indicate minute -variations in the velocity of the machine. - -[4] From the Greek words _tachos_ speed, and _metron_ measure. - -A glass tube A, open at both ends, and expanding at one extremity into -a bell B, is immersed with its wider end in the mercury, the surface -of which will stand at the same level in the bell B, and in the cup -C D. The tube is so suspended as to be unconnected with the cup. -This tube is then filled to a certain height A, with spirits tinged -with some colouring matter, to render it easily observable. When the -cup is whirled by the machine to which it is attached, the level of -the quicksilver in the bell falls, leaving more space for the spirits, -which, therefore, descends in the tube. As the motion is continued, -every change of velocity causes a corresponding change in the level -of the mercury, and, therefore, also in the level A of the spirits. -It will be observed, that, in consequence of the capacity of the bell -B being much greater than that of the tube A, a very small change in -the level of the quicksilver in the bell will produce a considerable -change in the height of the spirits in the tube. Thus this ingenious -instrument becomes a very delicate indicator of variations in the -motion of machinery. - -(308.) The governor, and other methods of regulating the motion of -machinery which have been just described, are adapted principally to -cases in which the proportion of the resistance to the load is subject -to certain fluctuations or gradual changes, or at least to cases in -which the resistance is not at any time entirely withdrawn, nor the -energy of the power actually suspended. Circumstances, however, -frequently occur in which, while the power remains in full activity, -the resistance is at intervals suddenly removed and as suddenly again -returns. On the other hand, cases also present themselves, in which, -while the resistance is continued, the impelling power is subject to -intermission at regular periods. In the former case, the machine would -be driven with a ruinous rapidity during those periods at which it is -relieved from its load, and on the return of the load every part would -suffer a violent strain, from its endeavour to retain the velocity -which it had acquired, and the speedy destruction of the engine could -not fail to ensue. In the latter case, the motion would be greatly -retarded or entirely suspended during those periods at which the moving -power is deprived of its activity, and, consequently, the motion which -it would communicate would be so irregular as to be useless for the -purposes of manufactures. - -It is also frequently desirable, by means of a weak but continued -power, to produce a severe but instantaneous effect. Thus a blow may -be required to be given by the muscular action of a man’s arm with a -force to which, unaided by mechanical contrivance, its strength would -be entirely inadequate. - -In all these cases, it is evident that the object to be attained is, -an effectual method of accumulating the energy of the power so as -to make it available after the action by which it has been produced -has ceased. Thus, in the case in which the load is at periodical -intervals withdrawn from the machine, if the force of the power could -be imparted to something by which it would be preserved, so as to be -brought against the load when it again returned, the inconvenience -would be removed. In like manner, in the case where the power itself -is subject to intermission, if a part of the force which it exerts in -its intervals of action could be accumulated and preserved, it might -be brought to bear upon the machine during its periods of suspension. -By the same means of accumulating force, the strength of an infant, -by repeated efforts, might produce effects which would be vainly -attempted by the single and momentary action of the strongest man. - -(309.) The property of inertia, explained and illustrated in the third -and fourth chapters of this volume furnishes an easy and effectual -method of accomplishing this. A mass of matter retains, by virtue of -its inertia, the whole of any force which may have been given to it, -except that part of which friction and the atmospheric resistance -deprives it. By contrivances which are well known and present no -difficulty, the part of the moving force thus lost may be rendered -comparatively small, and the moving mass may be regarded as retaining -nearly the whole of the force impressed upon it. To render this method -of accumulating force fully intelligible, let us first imagine a -polished level plane on which a heavy globe of metal, also polished, is -placed. It is evident that the globe will remain at rest on any part of -the plane without a tendency to move in any direction. As the friction -is nearly removed by the polish of the surfaces, the globe will be -easily moved by the least force applied to it. Suppose a slight impulse -given to it, which will cause it to move at the rate of one foot in -a second. Setting aside the effects of friction, it will continue to -move at this rate for any length of time. The same impulse repeated -will increase its speed to two feet per second. A third impulse to -three feet, and so on. Thus 10,000 repetitions of the impulse will -cause it to move at the rate of 10,000 feet per second. If the body to -which these impulses were communicated were a cannon ball, it might, -by a constant repetition of the impelling force, be at length made to -move with as much force as if it were projected from the most powerful -piece of ordnance. The force with which the ball in such a case would -strike a building might be sufficient to reduce it to ruins, and yet -such force would be nothing more than the accumulation of a number -of weak efforts not beyond the power of a child to exert, which are -stored up, and preserved, as it were, by the moving mass, and thereby -brought to bear, at the same moment, upon the point to which the force -is directed. It is the sum of a number of actions exerted successively, -and, during a long interval, brought into operation at one and the same -moment. - -But the case which is here supposed cannot actually occur; because -we have not usually any practical means of moving a body for any -considerable time in the same direction without much friction, and -without encountering numerous obstacles which would impede its -progress. It is not, however, essential to the effect which is to be -produced, that the motion should be in a straight line. If a leaden -weight be attached to the end of a light rod or cord, and be whirled by -the force of the arm in a circle, it will gradually acquire increased -speed and force, and at length may receive an impetus which would -cause it to penetrate a piece of board as effectually as if it were -discharged from a musket. - -The force of a hammer or sledge depends partly on its weight, but much -more on the principle just explained. Were it allowed merely to fall -by the force of its weight upon the head of a nail, or upon a bar of -heated iron which is to be flattened, an inconsiderable effect would be -produced. But when it is wielded by the arm of a man, it receives at -every moment of its motion increased force, which is finally expended -in a single instant on the head of the nail, or on the bar of iron. - -The effects of flails in threshing, of clubs, whips, canes, and -instruments for striking, axes, hatchets, cleavers, and all instruments -which cut by a blow, depend on the same principle, and are similarly -explained. - -The bow-string which impels the arrow does not produce its effect at -once. It continues to act upon the shaft until it resumes its straight -position, and then the arrow takes flight with the force accumulated -during the continuance of the action of the string, from the moment it -was disengaged from the finger of the bow-man. - -Fire-arms themselves act upon a similar principle, as also the air-gun -and steam-gun. In these instruments the ball is placed in a tube, and -suddenly exposed to the pressure of a highly elastic fluid, either -produced by explosion as in fire-arms, by previous condensation as in -the air-gun, or by the evaporation of highly heated liquids as in the -steam-gun. But in every case this pressure continues to act upon it -until it leaves the mouth of the tube, and then it departs with the -whole force communicated to it during its passage along the tube. - -(310.) From all these considerations it will easily be perceived that -a mass of inert matter may be regarded as a magazine in which force -may be deposited and accumulated, to be used in any way which may be -necessary. For many reasons, which will be sufficiently obvious, the -form commonly given to the mass of matter used for this purpose in -machinery is that of a wheel, in the rim of which it is principally -collected. Conceive a massive ring of metal, _fig. 149._, -connected with a central box or nave by light spokes, and turning on -an axis with little friction. Such an apparatus is called a fly-wheel. -If any force be applied to it, with that force (making some slight -deduction for friction) it will move, and will continue to move until -some obstacle be opposed to its motion, which will receive from it a -part of the force it has acquired. The uses of this apparatus will be -easily understood by examples of its application. - -Suppose that a heavy stamper or hammer is to be raised to a certain -height, and thence to be allowed to fall, and that the power used -for this purpose is a water-wheel. While the stamper ascends, the -power of the wheel is nearly balanced by its weight, and the motion -of the machine is slow. But the moment the stamper is disengaged and -allowed to fall, the power of the wheel, having no resistance, nor any -object on which to expend itself, suddenly accelerates the machine, -which moves with a speed proportioned to the amount of the power, -until it again engages the stamper, when its velocity is as suddenly -checked. Every part suffers a strain, and the machine moves again -slowly until it discharges its load, when it is again accelerated, -and so on. In this case, besides the certainty of injury and wear, and -the probability of fracture from the sudden and frequent changes of -velocity, nearly the whole force exerted by the power in the intervals -between the commencement of each descent of the stamper and the next -ascent is lost. These defects are removed by a fly-wheel. When the -stamper is discharged, the energy of the power is expended in moving -the wheel, which, by reason of its great mass, will not receive an -undue velocity. In the interval between the descent and ascent of the -stamper, the force of the power is lodged in the heavy rim of the -fly-wheel. When the stamper is again taken up by the machine, this -force is brought to bear upon it, combined with the immediate power -of the water-wheel, and the stamper is elevated with nearly the same -velocity as that with which the machine moved in the interval of its -descent. - -(311.) In many cases, when the moving power is not subject to -variation, the efficacy of the machine to transmit it to the working -point is subject to continual change. The several parts of every -machine have certain periods of motion, in which they pass through a -variety of positions, to which they continually return after stated -intervals. In these different positions the effect of the power -transmitted to the working point is different; and cases even occur -in which this effect is altogether annihilated, and the machine is -brought into a predicament in which the power loses all influence -over the weight. In such cases the aid of a fly-wheel is effectual -and indispensable. In those phases of the machine, which are most -favourable to the transmission of force, the fly-wheel shares the -effect of the power with the load, and retaining the force thus -received directs it upon the load at the moments when the transmission -of power by the machine is either feeble or altogether suspended. These -general observations will, perhaps, be more clearly apprehended by an -example of an application of the fly-wheel, in a case such as those now -alluded to. - -Let A B C D E F, _fig. 150._, be a -_crank_, which is a double winch ((252.) and _fig. 89._), by -which an axle, A B E F, is to be turned. Attached -to the middle of C D by a joint is a rod, which is connected -with a beam, worked with an alternate motion on a centre, like -the brake of a pump, and driven by any constant power, such as a -steam-engine. The bar C D is to be carried with a circular motion -round the axis A E. Let the machine, viewed in the direction -A B E F of the axis, be conceived to be represented -in _fig. 151._, where A represents the centre round which the -motion is to be produced, and G the point where the connecting rod -G H is attached to the arm of the crank. The circle through which -G is to be urged by the rod is represented by the dotted line. In -the position represented in _fig. 151._, the rod acting in the -direction H G has its full power to turn the crank G A round -the centre A. As the crank comes into the position represented in -_fig. 152._, this power is diminished, and when the point G comes -immediately below A, as in _fig. 153._, the force in the direction -H G has no effect in turning the crank round A, but, on the -contrary, is entirely expended in pulling the crank in the direction -A G, and, therefore, only acts upon the pivots or gudgeons which -support the axle. At this crisis of the motion, therefore, the whole -effective energy of the power is annihilated. - -After the crank has passed to the position represented in -_fig. 154._, the direction of the force which acts upon the -connecting rod is changed, and now the crank is drawn upward in the -direction G H. In this position the moving force has some efficacy -to produce rotation round A, which efficacy continually increases -until the crank attains the position shown in _fig. 155._, when -its power is greatest. Passing from this position its efficacy is -continually diminished, until the point G comes immediately above the -axis A, _fig. 156._ Here again the power loses all its efficacy -to turn the axle. The force in the direction G H or H G can -obviously produce no other effect than a strain upon the pivots or -gudgeons. - -In the critical situations represented in _fig. 153._, and -_fig. 156._, the machine would be incapable of moving, were -the immediate force of the power the only impelling principle. But -having been previously in motion by virtue of the inertia of its -various parts, it has a tendency to continue in motion; and if the -resistance of the load and the effects of friction be not too great, -this disposition to preserve its state of motion will extricate the -machine from the dilemma in which it is involved in the cases just -mentioned, by the peculiar arrangement of its parts. In many cases, -however, the force thus acquired during the phases of the machine, in -which the power is active, is insufficient to carry it through the -dead points (_fig. 153._ and _fig. 156._); and in all cases -the motion would be very unequal, being continually retarded as it -approached these points, and continually accelerated after it passed -them. A fly-wheel attached to the axis A, or to some other part of the -machinery, will effectually remove this defect. When the crank assumes -the positions in _fig. 151._ and _fig. 155._, the power is -in full play upon it, and a share of the effect is imparted to the -massive rim of the fly-wheel. When the crank gets into the predicament -exhibited in _fig. 153._ and _fig. 156._, the momentum which -the fly-wheel received when the crank acted with most advantage, -immediately extricates the machine, and, carrying the crank beyond the -dead point, brings the power again to bear upon it. - -The astonishing effects of a fly-wheel, as an accumulator of force, -have led some into the error of supposing that such an apparatus -increases the actual power of a machine. It is hoped, however, that -after what has been explained respecting the inertia of matter and the -true effects of machines, the reader will not be liable to a similar -mistake. On the contrary, as a fly cannot act without friction, and as -the amount of the friction, like that of inertia, is in proportion to -the weight, a portion of the actual moving force must unavoidably be -lost by the use of a fly. In cases, however, where a fly is properly -applied this loss of power is inconsiderable, compared with the -advantageous distribution of what remains. - -[Illustration: _C. Varley, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -As an accumulator of force, a fly can never have more force than has -been applied to put it in motion. In this respect it is analogous to -an elastic spring, or the force of condensed air, or any other power -which derives its existence from causes purely mechanical. In bending -a spring a gradual expenditure of power is necessary. On the recoil -this power is exerted in a much shorter time than that consumed in its -production, but its total amount is not altered. Air is condensed by a -succession of manual efforts, one of which alone would be incapable of -projecting a leaden ball with any considerable force, and all of which -could not be immediately applied to the ball at the same instant. But -the reservoir of condensed air is a magazine in which a great number of -such efforts are stored up, so as to be brought at once into action. -If a ball be exposed to their effect, it may be projected with a -destructive force. - -In mills for rolling metal the fly-wheel is used in this way. The -water-wheel or other moving power is allowed for some time to act upon -the fly-wheel alone, no load being placed upon the machine. A force -is thus gained which is sufficient to roll a large piece of metal, to -which without such means the mill would be quite inadequate. In the -same manner a force may be gained by the arm of a man acting on a fly -for a few seconds, sufficient to impress an image on a piece of metal -by an instantaneous stroke. The fly is, therefore, the principal agent -in coining presses. - -(312.) The power of a fly is often transmitted to the working point -by means of a screw. At the extremities of the cross arm A B, -_fig. 157._, which works the screw, two heavy balls of metal are -placed. When the arm A B is whirled round, those masses of metal -acquire a momentum, by which the screw, being driven downward, urges -the die with an immense force against the substance destined to receive -the impression. - -Some engines used in coining have flies with arms four feet long, -bearing one hundred weight at each of their extremities. By turning -such an arm at the rate of one entire circumference in a second, the -die will be driven against the metal with the same force as that with -which 7500 pounds weight would fall from the height of 16 feet; an -enormous power, if the simplicity and compactness of the machine be -considered. - -The place to be assigned to a fly-wheel relatively to the other parts -of the machinery is determined by the purpose for which it is used. If -it be intended to equalise the action, it should be near the working -point. Thus, in a steam-engine, it is placed on the crank which turns -the axle by which the power of the engine is transmitted to the object -it is finally designed to affect. On the contrary, in handmills, such -as those commonly used for grinding coffee, &c., it is placed upon the -axis of the winch by which the machine is worked. - -The open work of fenders, fire-grates, and similar ornamental articles -constructed in metal, is produced by the action of a fly, in the -manner already described. The cutting tool, shaped according to the -pattern to be executed, is attached to the end of the screw; and the -metal being held in a proper position beneath it, the fly is made to -urge the tool downwards with such force as to stamp out pieces of the -required figure. When the pattern is complicated, and it is necessary -to preserve with exactness the relative situation of its different -parts, a number of punches are impelled together, so as to strike the -entire piece of metal at the same instant, and in this manner the most -elaborate open work is executed by a single stroke. - - - - -CHAP. XVIII. - -MECHANICAL CONTRIVANCES FOR MODIFYING MOTION. - - -(313.) The classes of simple machines denominated mechanic powers, -have relation chiefly to the peculiar principle which determines the -action of the power on the weight or resistance. In explaining this -arrangement various other reflections have been incidentally mixed up -with our investigations; yet still much remains to be unfolded before -the student can form a just notion of those means by which the complex -machinery used in the arts and manufactures so effectually attains the -ends, to the accomplishment of which it is directed. - -By a power of a given energy to oppose a resistance of a different -energy, or by a moving principle having a given velocity to generate -another velocity of a different amount, is only one of the many objects -to be effected by a machine. In the arts and manufactures the _kind_ -of motion produced is generally of greater importance than its _rate_. -The latter may affect the quantity of work done in a given time, but -the former is essential to the performance of the work in any quantity -whatever. In the practical application of machines, the object to be -attained is generally to communicate to the working point some peculiar -sort of motion suitable to the uses for which the machine is intended; -but it rarely happens that the moving power has this sort of motion. -Hence, the machine must be so contrived that, while that part on which -this power acts is capable of moving in obedience to it, its connection -with the other parts shall be such that the working point may receive -that motion which is necessary for the purposes to which the machine is -applied. - -To give a perfect solution of this problem it would be necessary to -explain, first, all the varieties of moving powers which are at -our disposal; secondly, all the variety of motions which it may be -necessary to produce; and, thirdly, to show all the methods by which -each variety of prime mover may be made to produce the several species -of motion in the working point. It is obvious that such an enumeration -would be impracticable, and even an approximation to it would be -unsuitable to the present treatise. Nevertheless, so much ingenuity has -been displayed in many of the contrivances for modifying motion, and an -acquaintance with some of them is so essential to a clear comprehension -of the nature and operation of complex machines, that it would be -improper to omit some account of those at least which most frequently -occur in machinery, or which are most conspicuous for elegance and -simplicity. - -The varieties of motion which most commonly present themselves in the -practical application of mechanics may be divided into _rectilinear_ -and _rotatory_. In rectilinear motion the several parts of the -moving body proceed in parallel straight lines with the same speed. -In rotatory motion the several points revolve round an axis, each -performing a complete circle, or similar parts of a circle, in the same -time. - -Each of these may again be resolved into continued and reciprocating. -In a continued motion, whether rectilinear or rotatory, the parts move -constantly in the same direction, whether that be in parallel straight -lines, or in rotation on an axis. In reciprocating motion the several -parts move alternately in opposite directions, tracing the same spaces -from end to end continually. Thus, there are four principal species of -motion which more frequently than any others act upon, or are required -to be transmitted by, machines:-- - -1. _Continued rectilinear motion._ - -2. _Reciprocating rectilinear motion._ - -3. _Continued circular motion._ - -4. _Reciprocating circular motion._ - -These will be more clearly understood by examples of each kind. - -Continued rectilinear motion is observed in the flowing of a river, in -a fall of water, in the blowing of the wind, in the motion of an animal -upon a straight road, in the perpendicular fall of a heavy body, in the -motion of a body down an inclined plane. - -Reciprocating rectilinear motion is seen in the piston of a common -syringe, in the rod of a common pump, in the hammer of a pavier, the -piston of a steam-engine, the stampers of a fulling mill. - -Continued circular motion is exhibited in all kinds of wheel-work, and -is so common, that to particularise it is needless. - -Reciprocating circular motion is seen in the pendulum of a clock, and -in the balance-wheel of a watch. - -We shall now explain some of the contrivances by which a power having -one of these motions may be made to communicate either the same species -of motion changed in its velocity or direction, or any of the other -three kinds of motion. - -(314.) By a continued rectilinear motion another continued rectilinear -motion in a different direction may be produced, by one or more fixed -pulleys. A cord passed over these, one end of it being moved by the -power, will transmit the same motion unchanged to the other end. If the -directions of the two motions cross each other, one fixed pulley will -be sufficient; see _fig. 113._, where the hand takes the direction -of the one motion, and the weight that of the other. In this case the -pulley must be placed in the angle at which the directions of the two -motions cross each other. If this angle be distant from the places at -which the objects in motion are situate, an inconvenient length of rope -may be necessary. In this case the same may be effected by the use of -two pulleys, as in _fig. 158._ - -If the directions of the two motions be parallel, two pulleys must -be used as in _fig. 158._, where P′ A′ is one motion, and -B W the other. In these cases the axles of the two wheels are -parallel. - -It may so happen that the directions of the two motions neither cross -each other nor are parallel. This would happen, for example, if the -direction of one were upon the paper in the line P A, while the -other were perpendicular to the paper from the point O. In this case -two pulleys should be used, the axle of one O′ being perpendicular to -the paper, while the axle of the other O should be on the paper. This -will be evident by a little reflection. - -In general, the axle of each pulley must be perpendicular to the -two directions in which the rope passes from its groove; and by due -attention to this condition it will be perceived, that a continued -rectilinear motion may be transferred from any one direction to any -other direction, by means of a cord and two pulleys, without changing -its velocity. - -If it be necessary to change the velocity, any of the systems of -pulleys described in chap. XV. may be used in addition to the fixed -pulleys. - -By the wheel and axle any one continued rectilinear motion may be -made to produce another in any other direction, and with any other -velocity. It has been already explained (250.) that the proportion of -the velocity of the power to that of the weight is as the diameter of -the wheel to the diameter of the axle. The thickness of the axle being -therefore regulated in relation to the size of the wheel, so that -their diameters shall have that proportion which subsists between the -proposed velocities, one condition of the problem will be fulfilled. -The rope coiled upon the axle may be carried, by means of one or more -fixed pulleys, into the direction of one of the proposed motions, while -that which surrounds the wheel is carried into the direction of the -other by similar means. - -(315.) By the wheel and axle a continued rectilinear motion may be made -to produce a continued rotatory motion, or _vice versâ_. If the power -be applied by a rope coiled upon the wheel, the continued motion of -the power in a straight line will cause the machine to have a rotatory -motion. Again, if the weight be applied by a rope coiled upon the -axle, a power having a rotatory motion applied to the wheel will cause -the continued ascent of the weight in a straight line. - -Continued rectilinear and rotatory motions may be made to produce each -other, by causing a toothed wheel to work in a straight bar, called a -_rack_, carrying teeth upon its edge. Such an apparatus is represented -in _fig. 159._ - -In some cases the teeth of the wheel work in the links of a chain. The -wheel is then called a _rag-wheel_, _fig. 160._ - -Straps, bands, or ropes, may communicate rotation to a wheel, by their -friction in a groove upon its edge. - -A continued rectilinear motion is produced by a continued circular -motion in the case of a screw. The lever which turns the screw has -a continued circular motion, while the screw itself advances with a -continued rectilinear motion. - -The continued rectilinear motion of a stream of water acting upon a -wheel produces continued circular motion in the wheel, _fig. 93_, -_94_, _95_. In like manner the continued rectilinear motion of the wind -produces a continued circular motion in the arms of a windmill. - -Cranes for raising and lowering heavy weights convert a circular motion -of the power into a continued rectilinear motion of the weight. - -(316.) Continued circular motion may produce reciprocating rectilinear -motion, by a great variety of ingenious contrivances. - -Reciprocating rectilinear motion is used when heavy stampers are to -be raised to a certain height, and allowed to fall upon some object -placed beneath them. This may be accomplished by a wheel bearing on -its edge curved teeth, called _wipers_. The stamper is furnished with -a projecting arm or peg, beneath which the wipers are successively -brought by the revolution of the wheel. As the wheel revolves the -wiper raises the stamper, until its extremity passes the extremity of -the projecting arm of the stamper, when the latter immediately falls -by its own weight. It is then taken up by the next wiper, and so the -process is continued. - -A similar effect is produced if the wheel be partially furnished with -teeth, and the stamper carry a rack in which these teeth work. Such an -apparatus is represented in _fig. 161._ - -It is sometimes necessary that the reciprocating rectilinear motion -shall be performed at a certain varying rate in both directions. This -may be accomplished by the machine represented in _fig. 162._ -A wheel upon the axle C turns uniformly in the direction -A B D E. A rod _mn_ moves in guides, which only permit -it to ascend and descend perpendicularly. Its extremity _m_ rests -upon a path or groove raised from the face of the wheel, and shaped -into such a curve that as the wheel revolves the rod _mn_ shall be -moved alternately in opposite directions through the guides, with the -required velocity. The manner in which the velocity varies will depend -on the form given to the groove or channel raised upon the face of the -wheel, and this may be shaped so as to give any variation to the motion -of the rod _mn_ which may be required for the purpose to which it is to -be applied. - -The _rose-engine_ in the turning-lathe is constructed on this -principle. It is also used in spinning machinery. - -It is often necessary that the rod to which reciprocating motion is -communicated shall be urged by the same force in both directions. A -wheel partially furnished with teeth, acting on two racks placed on -different sides of it, and both connected with the bar or rod to which -the reciprocating motion is to be communicated, will accomplish this. -Such an apparatus is represented in _fig. 163._, and needs no -further explanation. - -Another contrivance for the same purpose is shown in _fig. 164._, -where A is a wheel turned by a winch H, and connected with a rod or -beam moving in guides by the joint _ab_. As the wheel A is turned -by the winch H the beam is moved between the guides alternately -in opposite directions, the extent of its range being governed by -the length of the diameter of the wheel. Such an apparatus is used -for grinding and polishing plane surfaces, and also occurs in silk -machinery. - -An apparatus applied by M. Zureda in a machine for pricking holes in -leather is represented in _fig. 165._ The wheel A B has its -circumference formed into teeth, the shape of which may be varied -according to the circumstances under which it is to be applied. One -extremity of the rod _ab_ rests upon the teeth of the wheel upon which -it is pressed by a spring at the other extremity. When the wheel -revolves, it communicates to this rod a reciprocating rectilinear -motion. - -Leupold has applied this mechanism to move the pistons of pumps.[5] -Upon the vertical axis of a horizontal hydraulic wheel is fixed another -horizontal wheel, which is furnished with seven teeth in the manner -of a crown wheel (263.). These teeth are shaped like inclined planes, -the intervals between them being equal to the length of the planes. -Projecting arms attached to the piston rods rest upon the crown of this -wheel; and, as it revolves, the inclined surfaces of the teeth, being -forced under the arm, raise the rod upon the principle of the wedge. -To diminish the obstruction arising from friction, the projecting arms -of the piston rods are provided with rollers, which run upon the teeth -of the wheel. In one revolution of the wheel each piston makes as many -ascents and descents as there are teeth. - -[5] Theatrum Machinarum, tom. ii. pl. 36. fig. 3. - -(317.) Wheel-work furnishes numerous examples of continued circular -motion round one axis, producing continued circular motion round -another. If the axles be in parallel directions, and not too distant, -rotation may be transmitted from one to the other by two spur-wheels -(263.); and the relative velocities may be determined by giving a -corresponding proportion to the diameter of the wheels. - -If a rotary motion is to be communicated from one axis to another -parallel to it, and at any considerable distance, it cannot in -practice be accomplished by wheels alone, for their diameters would -be too large. In this case a strap or chain is carried round the -circumferences of both wheels. If they are intended to turn in the same -direction, the strap is arranged as in _fig. 100._; but if in -contrary directions it is crossed, as in _fig. 101._ In this case, -as with toothed wheels, the relative velocities are determined by the -proportion of the diameters of the wheels. - -If the axles be distant and not parallel, the cord, by which the motion -is transmitted, must be passed over grooved wheels, or fixed pulleys, -properly placed between the two axles. - -It may happen that the strain upon the wheel, to which the motion is to -be transmitted, is too great to allow of a strap or cord being used. In -this case, a shaft extending from the one axis to another, and carrying -two bevelled wheels (263.), will accomplish the object. One of these -bevelled wheels is placed upon the shaft near to, and in connection -with, the wheel from which the motion is to be taken, and the other at -a part of it near to, and in connection with, that wheel to which the -motion is to be conveyed, _fig. 166._ - -The methods of transmitting rotation from one axis to another -perpendicular to it, by crown and by bevelled wheels, have been -explained in (263.). - -The endless screw (299.) is a machine by which a rotatory motion round -one axis may communicate a rotatory motion round another perpendicular -to it. The power revolves round an axis coinciding with the length of -the screw, and the axis of the wheel driven by the screw is at right -angles to this. - -The axis to which rotation is to be given, or from which it is to -be taken, is sometimes variable in its position. In such cases, an -ingenious contrivance, called a _universal joint_, invented by the -celebrated Dr. Hooke, may be used. The two shafts or axles A B, -_fig. 167._, between which the motion is to be communicated, -terminate in semicircles, the diameters of which, C D and -E F, are fixed in the form of a cross, their extremities moving -freely in bushes placed in the extremities of the semicircles. -Thus, while the central cross remains unmoved, the shaft A and its -semicircular end may revolve round C D as an axis; and the shaft -B and its semicircular end may revolve round E F as an axis. If -the shaft A be made to revolve without changing its direction, the -points C D will move in a circle whose centre is at the middle of -the cross. The motion thus given to the cross will cause the points -E F to move in another circle round the same centre, and hence the -shaft B will be made to revolve. - -[Illustration: _C. Varley, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -This instrument will not transmit the motion if the angle under the -directions of the shafts be less than 140°. In this case a double -joint, as represented in _fig. 168._, will answer the purpose. -This consists of four semicircles united by two crosses, and its -principle and operation is the same as in the last case. - -Universal joints are of great use in adjusting the position of large -telescopes, where, while the observer continues to look through the -tube, it is necessary to turn endless screws or wheels whose axes are -not in an accessible position. - -The cross is not indispensably necessary in the universal joint. A -hoop, with four pins projecting from it at four points equally distant -from each other, or dividing the circle of the hoop into four equal -arcs, will answer the purpose. These pins play in the bushes of the -semicircles in the same manner as those of the cross. - -The universal joint is much used in cotton-mills, where shafts are -carried to a considerable distance from the prime mover, and great -advantage is gained by dividing them into convenient lengths, connected -by a joint of this kind. - -(318.) In the practical application of machinery, it is often necessary -to connect a part having a continued circular motion with another which -has a reciprocating or alternate motion, so that either may move the -other. There are many contrivances by which this may be effected. - -One of the most remarkable examples of it is presented in the -scapements of watches and clocks. In this case, however, it can -scarcely be said with strict propriety that it is the rotation of -the scapement-wheel (266.) which _communicates_ the vibration to the -balance-wheel or pendulum. That vibration is produced in the one case -by the peculiar nature of the spiral spring fixed upon the axis of the -balance-wheel, and in the other case by the gravity of the pendulum. -The force of the scapement-wheel only _maintains_ the vibration, -and prevents its decay by friction and atmospheric resistance. -Nevertheless, between the two parts thus moving there exists a -mechanical connection, which is generally brought within the class of -contrivances now under consideration. - -A beam vibrating on an axis, and driven by the piston of a -steam-engine, or any other power, may communicate rotary motion to -an axis by a connector and a crank. This apparatus has been already -described in (311.). Every steam-engine which works by a beam affords -an example of this. The working beam is generally placed over the -engine, the piston rod being attached to one end of it, while the -connecting rod unites the other end with the crank. In boat-engines, -however, this position would be inconvenient, requiring more room than -could easily be spared. The piston rod, in these cases, is, therefore, -connected with the end of the beam by long rods, and the beam is placed -beside and below the engine. The use of a fly-wheel here would also -be objectionable. The effect of the dead points explained in (311.) -is avoided without the aid of a fly, by placing two cranks upon the -revolving axle, and working them by two pistons. The cranks are so -placed that when either is at its dead point, the other is in its most -favourable position. - -A wheel A, _fig. 169._, armed with wipers, acting upon a -sledge-hammer B, fixed upon a centre or axle C, will, by a continued -rotatory motion, give the hammer the reciprocating motion necessary -for the purposes to which it is applied. The manner in which this acts -must be evident on inspecting the figure. - -The treddle of the lathe furnishes an obvious example of a vibrating -circular motion producing a continued circular one. The treddle acts -upon a crank, which gives motion to the principal wheel, in the same -manner as already described in reference to the working beam and crank -in the steam-engine. - -By the following ingenious mechanism an alternate or vibrating force -may be made to communicate a circular motion continually in the same -direction. Let A B, _fig. 170._, be an axis receiving an -alternate motion from some force applied to it, such as a swinging -weight. Two ratchet wheels (253.) _m_ and _n_ are fixed on this axle, -their teeth being inclined in opposite directions. Two toothed wheels C -and D are likewise placed upon it, but so arranged that they turn upon -the axle with a little friction. These wheels carry two catches _p_, -_q_, which fall into the teeth of the ratchet wheels _m_, _n_, but fall -on opposite sides conformably to the inclination of the teeth already -mentioned. The effect of these catches is, that if the axis be made -to revolve in one direction, one of the two toothed wheels is always -compelled (by the catch _against_ which the motion is directed) to -revolve with it, while the other is permitted to remain stationary in -obedience to any force sufficiently great to overcome its friction with -the axle on which it is placed. The wheels C and D are both engaged by -bevelled teeth (263.) with the wheel E. - -According to this arrangement, in whichever direction the axis -A B is made to revolve, the wheel E will continually turn in -the same direction, and, therefore, if the axle A B be made to -turn alternately in the one direction and the other, the wheel E will -not change the direction of its motion. Let us suppose that the axle -A B is turned against the catch _p_. The wheel C will then be -made to turn with the axle. This will drive the wheel E in the same -direction. The teeth on the opposite side of the wheel E being engaged -with those of the wheel D, the latter will be turned upon the axle, -the friction, which alone resists its motion in that direction, being -overcome. Let the motion of the axle A B be now reversed. Since -the teeth of the ratchet wheel _n_ are moved against the catch _q_, the -wheel D will be compelled to revolve with the axle. The wheel E will be -driven in the same direction as before, and the wheel C will be moved -on the axle A B, and in a contrary direction to the motion of the -axle, the friction being overcome by the force of the wheel E. Thus, -while the axle A B is turned alternately in the one direction and -the other, the wheel E is constantly moved in the same direction. - -It is evident that the direction in which the wheel E moves may be -reversed by changing the position of the ratchet wheels and catches. - -(319.) It is often necessary to communicate an alternate circular -motion, like that of a pendulum, by means of an alternate motion in -a straight line. A remarkable instance of this occurs in the steam -engine. The moving force in this machine is the pressure of steam, -which impels a piston from end to end alternately in a cylinder. The -force of this piston is communicated to the working beam by a strong -rod, which passes through a collar in one end of the piston. Since it -is necessary that the steam included in the cylinder should not escape -between the piston rod and the collar through which it moves, and yet, -that it should move as freely and be subject to as little resistance as -possible, the rod is turned so as to be truly cylindrical, and is well -polished. It is evident that, under these circumstances, it must not be -subject to any lateral or cross strain, which would bend it towards one -side or the other of the cylinder. But the end of the beam to which it -communicates motion, if connected immediately with the rod by a joint, -would draw it alternately to the one side and the other, since it moves -in the arc of a circle, the centre of which is at the centre of the -beam. It is necessary, therefore, to contrive some method of connecting -the rod and the end of the beam, so that while the one shall ascend -and descend in a straight line, the other may move in the circular arc. - -The method which first suggests itself to accomplish this is, to -construct an arch-head upon the end of the beam, as in _fig. 171._ -Let C be the centre on which the beam works, and let B D be an -arch attached to the end of the beam, being a part of a circle having C -for its centre. To the highest point B of the arch a chain is attached, -which is carried upon the face of the arch B A, and the other end -of which is attached to the piston rod. Under these circumstances it is -evident, that when the force of the steam impels the piston downwards, -the chain P A B will draw the end of the beam down, and will, -therefore, elevate the other end. - -When the steam-engine is used for certain purposes, such as pumping, -this arrangement is sufficient. The piston in that case is not forced -upwards by the pressure of steam. During its ascent it is not subject -to the action of any force of steam, and the other end of the beam -falls by the weight of the pump-rods drawing the piston, at the -opposite end A, to the top of the cylinder. Thus the machine is in fact -passive during the ascent of the piston, and exerts its power only -during the descent. - -If the machine, however, be applied to purposes in which a constant -action of the moving force is necessary, as is always the case in -manufactures, the force of the piston must drive the beam in its ascent -as well as in its descent. The arrangement just described cannot effect -this; for although a chain is capable of transmitting any force, by -which its extremities are drawn in opposite directions, yet it is, from -its flexibility, incapable of communicating a force which drives one -extremity of it towards the other. In the one case the piston first -_pulls_ down the beam, and then the beam _pulls_ up the piston. The -chain, because it is inextensible, is perfectly capable of both these -actions; and being flexible, it applies itself to the arch-head of the -beam, so as to maintain the direction of its force upon the piston -continually in the same straight line. But when the piston acts upon -the beam in both ways, in pulling it down and pushing it up, the chain -becomes inefficient, being from its flexibility incapable of the latter -action. - -The problem might be solved by extending the length of the piston -rod, so that its extremity shall be above the beam, and using two -chains; one connecting the highest point of the rod with the lowest -point of the arch-head, and the other connecting the highest point -of the arch-head with a point on the rod below the point which -meets the arch-head when the piston is at the top of the cylinder, -_fig. 172._ - -The connection required may also be made by arming the arch-head with -teeth, _fig. 173._, and causing the piston rod to terminate in a -rack. In cases where, as in the steam-engine, smoothness of motion is -essential, this method is objectionable; and under any circumstances -such an apparatus is liable to rapid wear. - -The method contrived by Watt, for connecting the motion of the piston -with that of the beam, is one of the most ingenious and elegant -solutions ever proposed for a mechanical problem. He conceived the -motion of two straight rods A B, C D, _fig. 174._, -moving on centres or pivots A and C, so that the extremities B and D -would move in the arcs of circles having their centres at A and C. The -extremities B and D of these rods he conceived to be connected with a -third rod B D united with them by pivots on which it could turn -freely. To the system of rods thus connected let an alternate motion -on the centres A and C be communicated: the points B and D will move -upwards and downwards in the arcs expressed by the dotted lines, but -the middle point P of the connecting rod B D will move upwards and -downwards without any sensible deviation from a straight line. - -To prove this demonstratively would require some abstruse mathematical -investigation. It may, however, be rendered in some degree apparent -by reasoning of a looser and more popular nature. As the point B is -raised to E, it is also drawn aside towards the right. At the same -time the other extremity D of the rod B D is raised to E′, and -is drawn aside towards the left. The ends of the rod B D being -thus at the same time drawn equally towards opposite sides, its middle -point P will suffer no lateral derangement, and will move directly -upwards. On the other hand, if B be moved downwards to F, it will be -drawn laterally to the right, while D being moved to F′ will be drawn -to the left. Hence, as before, the middle point P sustains no lateral -derangement, but merely descends. Thus, as the extremities B and D move -upwards and downwards in circles, the middle point P moves upwards and -downwards in a straight line.[6] - -[6] In a strictly mathematical sense, the path of the point P is a -curve, and not a straight line; but in the play given to it in its -application to the steam-engine, it moves through a part only of its -entire locus, and this part extending equally on each side of a point -of inflection, the radius of curvature is infinite, so that in practice -the deviation from a straight line, when proper proportions are -observed in the rods, is imperceptible. - -The application of this geometrical principle in the steam-engine -evinces much ingenuity. The same arm of the beam usually works two -pistons, that of the cylinder and that of the _air-pump_. The apparatus -is represented on the arm of the beam in _fig. 175._ The beam -moves alternately upwards and downwards on its axis A. Every point of -it, therefore, describes a part of a circle of which A is the centre. -Let B be the point which divides the arm A G into two equal parts -A B and B G; and let C D be a straight rod, equal in -length to G B, and fixed on a centre or pivot C on which it is at -liberty to play. The end D of this rod is connected by a straight bar -with the point B, by pivots on which the rod B D turns freely. If -the beam be now supposed to rise and fall alternately, the points B and -D will move upwards and downwards in circular arcs, and, as already -explained with respect to the points B D, _fig. 174._, the -middle point P of the connecting rod B D will move upwards and -downwards without lateral deflection. To this point one of the piston -rods which are to be worked is attached. - -To comprehend the method of working the other piston, conceive a rod -G P′, equal in length to B D, to be attached to the end G of -the beam by a pivot on which it moves freely; and let its extremity -P′ be connected with D by another rod P′ D, equal in length to -G B, and playing on points at P′ and D. The piston rod of the -cylinder is attached to the point P′, and this point has a motion -precisely similar to that of P, without any lateral derangement, but -with a range in the perpendicular direction twice as great. This will -be apparent by conceiving a straight line drawn from the centre A -of the beam to P′, which will also pass through P. Since G P′ -is always parallel to B P, it is evident that the triangle -P′ G A is always similar to P B A, and has its -sides and angles similarly placed, but those sides are each twice the -magnitude of the corresponding sides of the other triangle. Hence -the point P′ must be subject to the same changes of position as the -point P, with this difference only, that in the same time it moves -over a space of twice the magnitude. In fact, the line traced by P′ -is the same as that traced by P, but on a scale twice as large. This -contrivance is usually called the _parallel motion_, but the same name -is generally applied to all contrivances by which a circular motion is -made to produce a rectilinear one. - - - - -CHAP. XIX. - -OF FRICTION AND THE RIGIDITY OF CORDAGE. - - -(320.) With a view to the simplification of the elementary theory of -machines, the consideration of several mechanical effects of great -practical importance has been postponed, and the attention of the -student has been directed exclusively to the way in which the moving -power is modified in being transmitted to the resistance independently -of such effects. A machine has been regarded as an instrument by -which a moving principle, inapplicable in its existing state to the -purpose for which it is required, may be changed either in its velocity -or direction, or in some other character, so as to be adapted to that -purpose. But in accomplishing this, the several parts of the machine -have been considered as possessing in a perfect degree qualities which -they enjoy only in an imperfect degree; and accordingly the conclusions -to which by such reasoning we are conducted are infected with errors, -the amount of which will depend on the degree in which the machinery -falls short of perfection in those qualities which theoretically are -imputed to it. - -[Illustration: _C. Varley, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -Of the several parts of a machine, some are designed to move, while -others are fixed; and of those which move, some have motions differing -in quantity and direction from those of others. The several parts, -whether fixed or movable, are subject to various strains and pressures, -which they are intended to resist. These forces not only vary according -to the load which the machine has to overcome, but also according to -the peculiar form and structure of the machine itself. During the -operation the surfaces of the movable parts move in immediate contact -with the surfaces either of fixed parts or of parts having other -motions. If these surfaces were endued with perfect smoothness or -polish, and the several parts subject to strains possessed perfect -inflexibility and infinite strength, then the effects of machinery -might be practically investigated by the principles already explained. -But the materials of which every machine is formed are endued with -limited strength, and therefore the load which is placed upon it must -be restricted accordingly, else it will be liable to be distorted by -the flexure, or even to be destroyed by the fracture of those parts -which are submitted to an undue strain. The surfaces of the movable -parts, and those surfaces with which they move in contact, cannot in -practice be rendered so smooth but that such roughness and inequality -will remain as sensibly to impede the motion. To overcome such an -impediment requires no inconsiderable part of the moving power. This -part is, therefore, intercepted before its arrival at the working -point, and the resistance to be finally overcome is deprived of it. The -property thus depending on the imperfect smoothness of surfaces, and -impeding the motion of bodies whose surfaces are in immediate contact, -is called _friction_. Before we can form a just estimate of the effects -of machinery, it is necessary to determine the force lost by this -impediment, and the laws which under different circumstances regulate -that loss. - -When cordage is engaged in the formation of any part of a machine, it -has hitherto been considered as possessing perfect flexibility. This is -not the case in practice; and the want of perfect flexibility, which -is called _rigidity_, renders a certain quantity of force necessary -to bend a cord or rope over the surface of an axle or the groove of -a wheel. During the motion of the rope a different part of it must -thus be continually bent, and the force which is expended in producing -the necessary flexure must be derived from the moving power, and is -thus intercepted on its way to the working point. In calculating the -effects of cordage, due regard must be had to this waste of power; -and therefore it is necessary to enquire into the laws which govern -the flexure of imperfectly flexible ropes, and the way in which these -affect the machines in which ropes are commonly used. - -To complete, therefore, the elementary theory of machinery, we propose -in the present and following chapter to explain the principal laws -which determine the effects of friction, the rigidity of cordage, and -the strength of materials. - -(321.) If a horizontal plane surface were perfectly smooth, and free -from the smallest inequalities, and a body having a flat surface also -perfectly smooth were placed upon it, any force applied to the latter -would put it in motion, and that motion would continue undiminished -as long as the body would remain upon the smooth horizontal surface. -But if this surface, instead of being every where perfectly even, had -in particular places small projecting eminences, a certain quantity -of force would be necessary to carry the moving body over these, and -a proportional diminution in its rate of motion would ensue. Thus, if -such eminences were of frequent occurrence, each would deprive the -body of a part of its speed, so that between that and the next it -would move with a less velocity than it had between the same and the -preceding one. This decrease being continued by a sufficient number of -such eminences encountering the body in succession, the velocity would -at last be so much diminished that the body would not have sufficient -force to carry it over the next eminence, and its motion would thus -altogether cease. - -Now, instead of the eminences being at a considerable distance asunder, -suppose them to be contiguous, and to be spread in every direction -over the horizontal plane, and also suppose corresponding eminences to -be upon the surface of the moving body; these projections incessantly -encountering one another will continually obstruct the motion of the -body, and will gradually diminish its velocity, until it be reduced to -a state of rest. - -Such is the cause of friction. The amount of this resisting force -increases with the magnitude of these asperities, or with the roughness -of the surfaces; but it does not solely depend on this. The surfaces -remaining the same, a little reflection on the method of illustration -just adopted, will show that the amount of friction ought also to -depend upon the force with which the surfaces moving one upon the -other are pressed together. It is evident, that as the weight of -the body supposed to move upon the horizontal plane is increased, a -proportionally greater force will be necessary to carry it over the -obstacles which it encounters, and therefore it will the more speedily -be deprived of its velocity and reduced to a state of rest. - -(322.) Thus we might predict with probability, that which accurate -experimental enquiry proves to be true, that the resistance from -friction depends conjointly on the roughness of the surfaces and the -force of the pressure. When the surfaces are the same, a double -pressure will produce a double amount of friction, a treble pressure a -treble amount of friction, and so on. - -Experiment also, however, gives a result which, at least at first view, -might not have been anticipated from the mode of illustration we have -adopted. It is found that the resistance arising from friction does not -at all depend on the magnitude of the surface of contact; but provided -the nature of the surfaces and the amount of pressure remain the same, -this resistance will be equal, whether the surfaces which move one upon -the other be great or small. Thus, if the moving body be a flat block -of wood, the face of which is equal to a square foot in magnitude, and -the edge of which does not exceed a square inch, it will be subject to -the same amount of friction, whether it move upon its broad face or -upon its narrow edge. If we consider the effect of the pressure in each -case, we shall be able to perceive why this must be the case. Let us -suppose the weight of the block to be 144 ounces. When it rests upon -its face, a pressure to this amount acts upon a surface of 144 square -inches, so that a pressure of one ounce acts upon each square inch. The -total resistance arising from friction will, therefore, be 144 times -that resistance which would be produced by a surface of one square inch -under a pressure of one ounce. Now, suppose the block placed upon its -edge, there is then a pressure of 144 ounces upon a surface equal to -one square inch. But it has been already shown, that when the surface -is the same, the friction must increase in proportion to the pressure. -Hence we infer that the friction produced in the present case is 144 -times the friction which would be produced by a pressure of one ounce -acting on one square inch of surface, which is the same resistance as -that which the body was proved to be subject to when resting on its -face. - -These two laws, that friction is independent of the magnitude of the -surface, and is proportional to the pressure when the quality of the -surfaces is the same, are useful in practice, and _generally_ true. In -very extreme cases they are, however, in error. When the pressure is -very intense, in proportion to the surface, the friction is somewhat -_less_ than it would be by these laws; and when it is very small in -proportion to the surface, it is somewhat _greater_. - -(323.) There are two methods of establishing by experiment the laws of -friction, which have been just explained. - -First. The surfaces between which the friction is to be determined -being rendered perfectly flat, let one be fixed in the horizontal -position on a table T T′, _fig. 176._; and let the other be -attached to the bottom of a box B C, adapted to receive weights, -so as to vary the pressure. Let a silken cord S P, attached to the -box, be carried parallel to the table over a wheel at P, and let a dish -D be suspended from it. If no friction existed between the surfaces, -the smallest weight appended to the cord would draw the box towards -P with a continually increasing speed. But the friction which always -exists interrupts this effect, and a small weight may act upon the -string without moving the box at all. Let weights be put in the dish D, -until a sufficient force is obtained to overcome the friction without -giving the box an accelerated motion. Such a weight is equivalent to -the amount of the friction. - -The amount of the weight of the box being previously ascertained, let -this weight be now doubled by placing additional weights in the box. -The pressure will thus be doubled, and it will be found that the weight -of the dish D and its load, which before was able to overcome the -friction, is now altogether inadequate to it. Let additional weights -be placed in the dish until the friction be counteracted as before, -and it will be observed, that the whole weight necessary to produce -this effect is exactly twice the weight which produced it in the former -case. Thus it appears that a double amount of pressure produces a -double amount of friction; and in a similar way it may be proved, that -any proposed increase or decrease of the pressure will be attended with -a proportionate variation in the amount of the friction. - -Second. Let one of the surfaces be attached to a flat plane A B, -_fig. 177._, which can be placed at any inclination with an -horizontal plane B C, the other surface being, as before, attached -to the box adapted to receive weights. The box being placed upon the -plane, let the latter be slightly elevated. The tendency of the box -to descend upon A B, will bear the same proportion to its entire -weight as the perpendicular A E bears to the length of the plane -A B (286.). Thus if the length A B be 36 inches, and the -height A E be three inches, that is a twelfth part of the length, -then the tendency of the weight to move down the plane is equal to a -twelfth part of its whole amount. If the weight were twelve ounces, and -the surfaces perfectly smooth, a force of one ounce acting up the plane -would be necessary to prevent the descent of the weight. - -In this case also the pressure on the plane will be represented by -the length of the base B E (286.), that is, it will bear the -same proportion to the whole weight as B E bears to B A. -The relative amounts of the weight, the tendency to descend, and the -pressure, will always be exhibited by the relative lengths of A B, -A E, and B E. - -This being premised, let the elevation of the plane A B be -gradually increased until the tendency of the weight to descend just -overcomes the friction, but not so much as to allow the box to descend -with accelerated speed. The proportion of the whole weight, which then -acts down the plane, will be found by measuring the height A E, -and the pressure will be determined by measuring the base B E. Now -let the weight in the box be increased, and it will be found that the -same elevation is necessary to overcome the friction; nor will this -elevation suffer any change, however the pressure or the magnitude of -the surfaces which move in contact may be varied. - -Since, therefore, in all these cases, the height A E and the base -B E remain the same, it follows that the proportion between the -friction and pressure is undisturbed. - -(324.) The law that friction is proportional to the pressure, has been -questioned by the late professor Vince of Cambridge, who deduced from -a series of experiments, that although the friction increases with the -pressure, yet that it increases in a somewhat less ratio; and from -this it would follow, that the variation of the surface of contact -must produce some effect upon the amount of friction. The law, as we -have explained it, however, is sufficiently near the truth for most -practical purposes. - -(325.) There are several circumstances regarding the quality of the -surfaces which produce important effects on the quantity of friction, -and which ought to be noticed here. - -This resistance is different in the surfaces of different substances. -When the surfaces are those of wood newly planed, it amounts to about -half the pressure, but is different in different kinds of wood. The -friction of metallic surfaces is about one fourth of the pressure. - -In general the friction between the surfaces of bodies of different -kinds is less than between those of the same kind. Thus, between wood -and metal the friction is about one fifth of the pressure. - -It is evident that the smoother the surfaces are the less will be the -friction. On this account, the friction of surfaces, when first brought -into contact, is often greater than after their attrition has been -continued for a certain time, because that process has a tendency to -remove and rub off those minute asperities and projections on which the -friction depends. But this has a limit, and after a certain quantity -of attrition the friction ceases to decrease. Newly planed surfaces -of wood have at first a degree of friction which is equal to half the -entire pressure, but after they are worn by attrition it is reduced to -a third. - -If the surfaces in contact be placed with their grains in the same -direction, the friction will be greater than if the grains cross each -other. - -Smearing the surfaces with unctuous matter diminishes the friction, -probably by filling the cavities between the minute projections which -produce the friction. - -When the surfaces are first placed in contact, the friction is less -than when they are suffered to rest so for some time; this is proved -by observing the force which in each case is necessary to move the one -upon the other, that force being less if applied at the first moment -of contact than when the contact has continued. This, however, has a -limit. There is a certain time, different in different substances, -within which this resistance attains its greatest amount. In surfaces -of wood this takes place in about two minutes; in metals the time -is imperceptibly short; and when a surface of wood is placed upon a -surface of metal, it continues to increase for several days. The limit -is larger when the surfaces are great, and belong to substances of -different kinds. - -The velocity with which the surfaces move upon one another produces but -little effect upon the friction. - -(326.) There are several ways in which bodies may move one upon the -other, in which friction will produce different effects. The principal -of these are, first, the case where one body _slides_ over another; the -second, where a body having a round form _rolls_ upon another; and, -_thirdly_, where an axis revolves within a hollow cylinder, or the -hollow cylinder revolves upon the axis. - -With the same amount of pressure and a like quality of surface, the -quantity of friction is greatest in the first case and least in the -second. The friction in the second case also depends on the diameter -of the body which rolls, and is small in proportion as that diameter -is great. Thus a carriage with large wheels is less impeded by the -friction of the road than one with small wheels. - -In the third case, the leverage of the wheel aids the power in -overcoming the friction. Let _fig. 178._ represent a section of -the wheel and axle; let C be the centre of the axle, and let B E -be the hollow cylinder in the nave of the wheel in which the axle is -inserted. If B be the part on which the axle presses, and the wheel -turn in the direction N D M, the friction will act at B in -the direction B F, and with the leverage B C. The power acts -against this at D in the direction D A, and with the leverage -D C. It is therefore evident, that as D C is greater than -B C, in the same proportion does the power act with mechanical -advantage on the friction. - -(327.) Contrivances for diminishing the effects of friction depend on -the properties just explained, the motion of rolling being as much -as possible substituted for that of sliding; and where the motion of -rolling cannot be applied, that of a wheel upon its axle is used. In -some cases both these motions are combined. - -If a heavy load be drawn upon a plane in the manner of a sledge, the -motion will be that of sliding, the species which is attended with -the greatest quantity of friction; but if the load be placed upon -cylindrical rollers, the nature of the motion is changed, and becomes -that in which there is the least quantity of friction. Thus large -blocks of stone, or heavy beams of timber, which would require an -enormous power to move them on a level road, are easily advanced when -rollers are put under them. - -When very heavy weights are to be moved through small spaces, this -method is used with advantage; but when loads ore to be transported to -considerable distances, the process is inconvenient and slow, owing to -the necessity of continually replacing the rollers in front of the load -as they are left behind by its progressive advancement. - -The wheels of carriages may be regarded as rollers which are -continually carried forward with the load. In addition to the friction -of the rolling motion on the road, they have, it is true, the friction -of the axle in the nave; but, on the other hand, they are free from -the friction of the rollers with the under surface of the load, or -the carriage in which the load is transported. The advantages of -wheel carriages in diminishing the effects of friction is sometimes -attributed to the slowness with which that axle moves within the box, -compared with the rate at which the wheel moves over the road; but -this is erroneous. The quantity of friction does not in any case vary -considerably with the velocity of the motion, but least of all does it -in that particular kind of motion here considered. - -In certain cases, where it is of great importance to remove the -effects of friction, a contrivance called _friction-wheels_, or -friction-rollers, is used. The axle of a friction-wheel, instead of -revolving within a hollow cylinder, which is fixed, rests upon the -edges of wheels which revolve with it; the species of motion thus -becomes that in which the friction is of least amount. - -Let A B and D C, _fig. 179._, be two wheels revolving -on pivots P Q with as little friction as possible, and so placed -that the axle O of a third wheel E F may rest between their edges. -As the wheel E F revolves, the axle O, instead of grinding its -surface on the surface on which it presses, carries that surface with -it, causing the wheels A B, C D, to revolve. - -In wheel carriages, the roughness of the road is more easily overcome -by large wheels than by small ones. The cause of this arises partly -from the large wheels not being so liable to sink into holes as small -ones, but more because, in surmounting obstacles, the load is elevated -less abruptly. This will be easily understood by observing the curves -in _fig. 180._, which represent the elevation of the axle in each -case. - -(328.) If a carriage were capable of moving on a road without friction, -the most advantageous direction in which a force could be applied to -draw it would be parallel to the road. When the motion is impeded by -friction, it is better, however, that the line of draught should be -inclined to the road, so that the drawing force may be expended partly -in lessening the pressure on the road, and partly in advancing the load. - -Let W, _fig. 181._, be a load which is to be moved upon the plane -surface A B. If the drawing force be applied in the direction -C D, parallel to the plane A B, it will have to overcome the -friction produced by the pressure of the whole weight of the load upon -the plane; but if it be inclined upwards in the direction C E, -it will be equivalent to two forces expressed (74.) by C G and -C F. The part C G has the effect of lightening the pressure -of the carriage upon the road, and therefore of diminishing the -friction in the same proportion. The part C F draws the load along -the plane. Since C F is less than C E or C D the whole -moving force, it is evident that a part of the force of draught is -lost by this obliquity; but, on the other hand, a part of the opposing -resistance is also removed. If the latter exceed the former, an -advantage will be gained by the obliquity; but if the former exceed the -latter, force will be lost. - -By mathematical reasoning, founded on these considerations, it is -proved that the best angle of draught is exactly that obliquity which -should be given to the road in order to enable the carriage to move of -itself. This obliquity is sometimes called the _angle of repose_, and -is that angle which determines the proportion of the friction to the -pressure in the second method, explained in (323.). The more rough the -road is, the greater will this angle be; and therefore it follows, that -on bad roads the obliquity of the traces to the road should be greater -than on good ones. On a smooth Macadamised way, a very slight declivity -would cause a carriage to roll by its own weight: hence, in this case, -the traces should be nearly parallel to the road. - -In rail roads, for like reasons, the line of draught should be parallel -to the road, or nearly so. - -(329.) When ropes or cords form a part of machinery, the effects of -their imperfect flexibility are in a certain degree counteracted by -bending them over the grooves of wheels. But although this so far -diminishes these effects as to render ropes practically useful, yet -still, in calculating the powers of machinery, it is necessary to take -into account some consequences of the rigidity of cordage which even -by these means are not removed. - -To explain the way in which the stiffness of a rope modifies the -operation of a machine, we shall suppose it bent over a wheel and -stretched by weights A B, _fig. 182._, at its extremities. -The weights A and B being equal, and acting at C and D in opposite -ways, balance the wheel. If the weight A receive an addition, it will -overcome the resistance of B, and turn the wheel in the direction -D E C. Now, for the present, let us suppose that the rope -is perfectly inflexible; the wheel and weights will be turned into -the position represented in _fig. 183._ The leverage by which -A acts will be diminished, and will become O F, having been -before O C; and the leverage by which B acts will be increased to -O G, having been before O D. - -But the rope not being inflexible will yield partially to the effects -of the weights A and B, and the parts A C and B D will -be bent into the forms represented in _fig. 184._ The form of -the curvature which the rope on each side of the wheel receives is -still such that the descending weight A works with a diminished -leverage F O, while the ascending weight resists it with an -increased leverage G O. Thus so much of the moving power is lost, -by the stiffness of the rope, as is necessary to compensate this -disadvantageous change in the power of the machine. - - - - -CHAP. XX. - -ON THE STRENGTH OF MATERIALS. - - -(330.) Experimental enquiries into the laws which regulate the strength -of solid bodies, or their power to resist forces variously applied -to tear or break them, are obstructed by practical difficulties, the -nature and extent of which are so discouraging that few have ventured -to encounter them at all, and still fewer have had the steadiness to -persevere until any result showing a general law has been obtained. -These difficulties arise, partly from the great forces which must be -applied, but more from the peculiar nature of the objects of those -experiments. The end to which such an enquiry must be directed is -the development of a _general law_; that is, such a rule as would be -rigidly observed if the materials, the strength of which is the object -of enquiry, were perfectly uniform in their texture, and subject to no -casual inequalities. In proportion as these inequalities are frequent, -experiments must be multiplied, that a long average may embrace cases -varying in both extremes, so as to eliminate each other’s effects in -the final result. - -The materials of which structures and works of art are composed are -liable to so many and so considerable inequalities of texture, that -any rule which can be deduced, even by the most extensive series of -experiments, must be regarded as a mean result, from which individual -examples will be found to vary in so great a degree, that more than -usual caution must be observed in its practical application. The -details of this subject belong to engineering, more properly than -to the elements of mechanics. Nevertheless, a general view of the -most important principles which have been established respecting the -strength of materials will not be misplaced in this treatise. - -A piece of solid matter may be submitted to the action of a force -tending to separate its parts in several ways; the principal of which -are,-- - -1. To a _direct pull_,--as when a rope or wire is stretched by a -weight. When a tie-beam resists the separation of the sides of a -structure, &c. - -2. To a direct pressure or thrust,--as when a weight rests upon a -pillar. - -3. To a transverse strain,--as when weights on the ends of a lever -press it on the fulcrum. - -(331.) If a solid be submitted to a force which draws it in the -direction of its length, having a tendency to pull its ends in -opposite directions, its strength or power to resist such a force is -proportional to the magnitude of its transverse section. Thus, suppose -a square rod of metal A B, _fig. 185._, of the breadth and -thickness of one inch, be pulled by a force in the direction A B, -and that a certain force is found sufficient to tear it; a rod of the -same metal of twice the breadth and the same thickness will require -double the force to break it; one of treble the breadth and the same -thickness will require treble the force to break it, and so on. - -The reason of this is evident. A rod of double or treble the thickness, -in this case, is equivalent to two or three equal and similar rods -which equally and separately resist the drawing force, and therefore -possess a degree of strength proportionate to their number. - -It will easily be perceived, that whatever be the section, the same -reasoning will be applicable, and the power of resistance will, in -general, be proportional to its magnitude or area. - -If the material were perfectly uniform throughout its dimensions, the -resistance to a direct pull would not be affected by the length of the -rod. In practice, however, the increase of length is found to lessen -the strength. This is to be attributed to the increased chance of -inequality. - -(332.) No satisfactory results have been obtained either by theory or -experiment respecting the laws by which solids resist compression. -The power of a perpendicular pillar to support a weight placed upon -it evidently depends on its thickness, or the magnitude of its base, -and on its height. It is certain that when the height is the same, -the strength increases with every increase of the base, but it seems -doubtful whether the strength be exactly proportional to the base. That -is, if two columns of the same material have equal heights, and the -base of one be double the base of the other, the strength of one will -be greater, but it is not certain whether it will exactly double that -of the other. According to the theory of Euler, which is in a certain -degree verified by the experiments of Musschenbrock, the strength will -be increased in a greater proportion than the base, so that, if the -base be doubled, the strength will be more than doubled. - -When the base is the same, the strength is diminished by increasing the -height, and this decrease of strength is proportionally greater than -the increase of height. According to Euler’s theory, the decrease of -strength is proportional to the square of the height; that is, when -the height is increased in a two-fold proportion, the strength is -diminished in a four-fold proportion. - -(333.) The strain to which solids forming the parts of structures of -every kind are most commonly exposed is the lateral or transverse -strain, or that which acts at right angles to their lengths. If any -strain act obliquely to the direction of their length it may be -resolved into two forces (76.), one in the direction of the length, and -the other at right angles to the length. That part which acts in the -direction of the length will produce either compression or a direct -pull, and its effect must be investigated accordingly. - -Although the results of theory, as well as those of experimental -investigations, present great discordances respecting the transverse -strength of solids, yet there are some particulars, in which they, for -the most part, agree; to this it is our object here to confine our -observations, declining all details relating to disputed points. - -Let A B C D, _fig. 186._, be a beam, supported -at its ends A and B. Its strength to support a weight at E pressing -downwards at right angles to its length is evidently proportional to -its breadth, the other things being the same. For a beam of double or -treble breadth, and of the same thickness, is equivalent to two or -three equal and similar beams placed side by side. Since each of these -would possess the same strength, the whole taken together would possess -double or treble the strength of any one of them. - -When the breadth and length are the same the strength obviously -increases with the depth, but not in the same proportion. The -increase of strength is found to be much greater in proportion than -the increase of depth. By the theory of Galileo, a double or treble -thickness ought to increase the strength in a four-fold or nine-fold -proportion, and experiments in most cases do not materially vary from -this rule. - -If while the breadth and depth remain the same, the length of the -beam, or rather, the distance between the points of support, vary, the -strength will vary accordingly, decreasing in the same proportion as -the length increases. - -From these observations it appears, that the transverse strength of -a beam depends more on its thickness than its breadth. Hence we find -that a broad thin board is much stronger when its edge is presented -upwards. On this principle the joists or rafters of floors and roofs -are constructed. - -If two beams be in all respects similar, their strengths will be in the -proportion of the squares of their lengths. Let the length, breadth, -and depth of the one be respectively double the length, breadth, -and depth of the other. By the double breadth the beam doubles its -strength, but by doubling the length half this strength is lost. Thus -the increase of length and breadth counteract each other’s effects, and -as far as they are concerned the strength of the beam is not changed. -But by doubling the thickness the strength is increased in a four-fold -proportion, that is, as the square of the length. In the same manner it -may be shown, that when all the dimensions are trebled, the strength is -increased in a nine-fold proportion, and so on. - -(334.) In all structures the materials have to support their own -weight, and therefore their available strength is to be estimated -by the excess of their absolute strength above that degree of -strength which is just sufficient to support their own weight. This -consideration leads to some conclusions, of which numerous and striking -illustrations are presented in the works of nature and art. - -We have seen that the absolute strength with which a lateral strain is -resisted is in the proportion of the square of the linear dimensions of -similar parts of a structure, and therefore the amount of this strength -increases rapidly with every increase of the dimensions of a body. But -at the same time the weight of the body increases in a still more rapid -proportion. Thus, if the several dimensions be doubled, the strength -will be increased in a four-fold but the weight in an eight-fold -proportion. If the dimensions be trebled, the strength will be -multiplied nine times, but the weight twenty-seven times. Again, if the -dimensions be multiplied four times, the strength will be multiplied -sixteen times, and the weight sixty-four times, and so on. - -Hence it is obvious, that although the strength of a body of small -dimensions may greatly exceed its weight, and, therefore, it may be -able to support a load many times its own weight; yet by a great -increase in the dimensions the weight increasing in a much greater -degree the available strength may be much diminished, and such a -magnitude may be assigned, that the weight of the body must exceed its -strength, and it not only would be unable to support any load, but -would actually fall to pieces by its own weight. - -The strength of a structure of any kind is not, therefore, to be -determined by that of its model, which will always be much stronger in -proportion to its size. All works natural and artificial have limits -of magnitude which, while their materials remain the same, they cannot -surpass. - -In conformity with what has just been explained, it has been observed, -that small animals are stronger in proportion than large ones; that the -young plant has more available strength in proportion than the large -forest tree; that children are less liable to injury from accident -than men, &c. But although to a certain extent these observations are -just, yet it ought not to be forgotten, that the mechanical conclusions -which they are brought to illustrate are founded on the supposition, -that the smaller and greater bodies which are compared are composed -of precisely similar materials. This is not the case in any of the -examples here adduced. - - - - -CHAP. XXI. - -ON BALANCES AND PENDULUMS. - - -(335.) The preceding chapters have been confined almost wholly to -the consideration of the laws of mechanics, without entering into a -particular description of the machinery and instruments dependant upon -those laws. Such descriptions would have interfered too much with the -regular progress of the subject, and it therefore appeared preferable -to devote a chapter exclusively to this portion of the work. - -Perhaps there are no ideas which man receives through the medium of -sense which may not be referred ultimately to matter and motion. In -proportion, therefore, as he becomes acquainted with the properties -of the one and the laws of the other, his knowledge is extended, his -comforts are multiplied; he is enabled to bend the powers of nature to -his will, and to construct machinery which effects with ease that which -the united labour of thousands would in vain be exerted to accomplish. - -Of the properties of matter, one of the most important is its weight, -and the element which mingles inseparably with the laws of motion is -time. - -In the present chapter it is our intention to describe such instruments -as are usually employed for determining the weight of bodies. To -attempt a description of the various machines which are used for the -measurement of time, would lead us into too wide a field for the -present occasion, and we shall, therefore, confine ourselves to an -account of the methods which have been practised to perfect, to perfect -that instrument which affords the most correct means of measuring time, -the pendulum. - -The instrument by which we are enabled to determine, with greater -accuracy than by any other means, the relative weight of a body, -compared with the weight of another body assumed as a standard, is the -balance. - -[Illustration: _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - - -_Of the Balance._ - -The balance may be described as consisting of an inflexible rod or -lever, called the beam, furnished with three axes; one, the fulcrum or -centre of motion situated in the middle, upon which the beam turns, -and the other two near the extremities, and at equal distances from -the middle. These last are called the points of support, and serve to -sustain the pans or scales. - -The points of support and the fulcrum are in the same right line, and -the centre of gravity of the whole should be a little below the fulcrum -when the position of the beam is horizontal. - -The arms of the lever being equal, it follows that if equal weights be -put into the scales no effect will be produced on the position of the -balance, and the beam will remain horizontal. - -If a small addition be made to the weight in one of the scales, the -horizontality of the beam will be disturbed; and after oscillating -for some time, it will, on attaining a state of rest, form an angle -with the horizon, the extent of which is a measure of the delicacy or -sensibility of the balance. - -As the sensibility of a balance is of the utmost importance in -nice scientific enquiries, we shall enter somewhat at large into a -consideration of the circumstances by which this property is influenced. - -In _fig. 187._ let A B represent the beam drawn from the -horizontal position by a very small weight placed in the scale -suspended from the point of support B; then the force tending to draw -the beam from the horizontal position may be expressed by P B, -multiplied by such very small weight acting upon the point B. - -Let the centre of gravity of the whole be at G; then the force acting -against the former will be G P multiplied into the weight of the -beam and scales, and when these forces are equal, the beam will rest -in an inclined position. Hence we may perceive that as the centre of -gravity is nearer to or further from the fulcrum S, (every thing else -remaining the same) the sensibility of the balance will be increased or -diminished. - -For, suppose the centre of gravity were removed to _g_, then to produce -an opposing force equal to that acting upon the extremity of the beam, -the distance _g p_ from the perpendicular line must be increased -until it becomes nearly equal to G P; but for this purpose -the end of the beam B must descend, which will increase the angle -H S B. - -As all weights placed in the scales are referred to the line joining -the points of support, and as this line is above the centre of gravity -of the beam when not loaded, such weights will raise the centre of -gravity; but it will be seen that the sensibility of the balance, as -far as it depends upon this cause, will remain unaltered. - -For, calling the distance S G unity, the distance of the centre of -gravity from the point S (to which the weight which has been added is -referred) will be expressed by the reciprocal of the weight of the beam -so increased; that is, if the weight of the beam be doubled by weights -placed in the scales, S _g_ will be one half of S G; and if the -weight of the beam be in like manner trebled, S _g_ will be one third -of S G, and so on. And as G P varies as S G, _g p_ -will be inversely proportionate to the increased weight of the beam, -and consequently, the product obtained by multiplying _g p_ by the -weight of the beam and its load will be a constant quantity, and the -sensibility of the balance, as before stated, will suffer no alteration. - -We will now suppose that the fulcrum S, _fig. 188._, is situated -below the line joining the points of support, and that the centre of -gravity of the beam when not loaded is at G. Also that when a very -small weight is placed in the scale suspended from the point B, the -beam is drawn from its horizontal position, the deviation being a -measure of the sensibility of the balance. Then, as before stated, -G P multiplied by the weight of the beam will be equal to -P′ B multiplied by the very small additional weight acting on the -point B. - -Now if we place equal weights in both scales, such additional weights -will be referred to the point W, and the resulting distance of the -centre of gravity from the point W, calling W G unity, will be -expressed as before by the reciprocal of the increased weight of the -loaded beam. But G P will decrease in a greater proportion than -W G: thus, supposing the weight of the beam to be doubled, W _g_ -would be one half of W G; but _g p_, as will be evident on -an inspection of the figure, will be less than half of G P; and -the same small weight which was before applied to the point B, if -now added, would depress the point B, until the distance _g p_ -became such as that, when multiplied by the weight of the whole, the -product would be as before equal to P′ B, multiplied by the before -mentioned very small added weight. The sensibility of the balance, -therefore, in this case would be increased. - -If the beam be sufficiently loaded, the centre of gravity will at -length be raised to the fulcrum S, and the beam will rest indifferently -in any position. If more weight be then added, the centre of gravity -will be raised above the fulcrum, and the beam will turn over. - -Lastly, if the fulcrum S, _fig. 189._, is above the line joining -the two points of support, as any additional weights placed in the -scales will be referred to the point W, in the line joining A and B, -if the weight of the beam be doubled by such added weights, and the -centre of gravity be consequently raised to _g_, W _g_ will become -equal to half of W G. But _g p_, being greater than one half -of G P, the end of the beam B will rise until _g p_ becomes -such as to be equal, when multiplied by the whole increased weight of -the beam, to P B, multiplied by the small weight, which we suppose -to have been placed as in the preceding examples, in the scale. - -From what has been said it will be seen that there are three positions -of the fulcrum which influence the sensibility of the balance: first, -when the fulcrum and the points of support are in a right line, when -the sensibility of the balance will remain the same, though the weight -with which the beam is loaded should be varied: secondly, when the -fulcrum is below the line joining the two points of support, in which -case the sensibility of the balance will be increased by additional -weights, until at length the centre of gravity is raised above the -fulcrum, when the beam will turn over; and, thirdly, when the fulcrum -is above the line joining the two points of support, in which case the -sensibility of the balance will be diminished as the weight with which -the beam is loaded is increased. - -The sensibility of a balance, as here defined, is the angular deviation -of the beam occasioned by placing an additional constant small weight -in one of the scales; but it is frequently expressed by the proportion -which such small additional weight bears to the weight of the beam -and its load, and sometimes to the weight the value of which is to be -determined. - -This proportion, however, will evidently vary with different weights, -except in the case where the centre of gravity of the beam is in the -line joining the points supporting the scales, the fulcrum being -above this line, and it is therefore necessary, in every other case, -when speaking of the sensibility of the balance, to designate the -weight with which it is loaded: thus, if a balance has a troy pound in -each scale, and the horizontality of the beam varies a certain small -quantity, just perceptible on the addition of one hundredth of a grain, -we say that the balance is sensible to 1/1152000 part of its load with -a pound in each scale, or that it will determine the weight of a troy -pound within 1/576000 part of the whole. - -The nearer the centre of gravity of a balance is to its fulcrum -the slower will be the oscillations of the beam. The number of -oscillations, therefore, made by the beam in a given time (a minute -for example), affords the most accurate method of judging of the -sensibility of the balance, which will be the greater as the -oscillations are fewer. - -Balances of the most perfect kind, and of such only it is our present -object to treat, are usually furnished with adjustments, by means of -which the length of the arms, or the distances of the fulcrum from the -points of support, may be equalised, and the fulcrum and the two points -of support be placed in a right line; but these adjustments, as will -hereafter be seen, are not absolutely necessary. - -The beam is variously constructed, according to the purposes to -which the balance is to be applied. Sometimes it is made of a rod of -solid steel; sometimes of two hollow cones joined at their bases; -and, in some balances, the beam is a frame in the form of a rhombus: -the principal object in all, however, is to combine strength and -inflexibility with lightness. - -A balance of the best kind, made by Troughton, is so contrived as to -be contained, when not in use, in a drawer below the case; and when -in use, it is protected from any disturbance from currents of air, by -being enclosed in the case above the drawer, the back and front of -which are of plate glass. There are doors in the sides, through which -the scale-pans are loaded, and there is a door at the top through which -the beam may be taken out. - -A strong brass pillar, in the centre of the box, supports a square -piece, on the front and back of which rise two arches, nearly -semicircular, on which are fixed two horizontal planes of agate, -intended to support the fulcrum. Within the pillar is a cylindrical -tube, which slides up and down by means of a handle on the outside -of the case. To the top of this interior tube is fixed an arch, the -terminations of which pass beneath and outside of the two arches before -described. These terminations are formed into Y _s_, destined to -receive the ends of the fulcrum, which are made cylindrical for this -purpose, when the interior tube is elevated in order to relieve the -axis when the balance is not in use. On depressing the interior tube, -the Y _s_ quit the axis, and leave it in its proper position on the -agate planes. The beam is about eighteen inches long, and is formed of -two hollow cones of brass, joined at their bases. The thickness of the -brass does not exceed 0·02 of an inch, but by means of circular rings -driven into the cones at intervals they are rendered almost inflexible. -Across the middle of the beam passes a cylinder of steel, the lower -side of which is formed into an edge, having an angle of about thirty -degrees, which, being hardened and well polished, constitutes the -fulcrum, and rests upon the agate planes for the length of about 0·05 -of an inch. - -Each point of suspension is formed of an axis having two sharp concave -edges, upon which rest at right angles two other sharp concave -edges formed in the spur-shaped piece to which the strings carrying -the scale-pan are attached. The two points are adjustable, the one -horizontally, for the purpose of equalising the arms of the beam, and -the other vertically, for bringing the points of suspension and the -fulcrum into a right line. - -Such is the form of Troughton’s balance: we shall now give the -description of a balance as constructed by Mr. Robinson of Devonshire -Street, Portland Place:-- - -The beam of this balance is only ten inches long. It is a frame of -bell-metal in the form of a rhombus. The fulcrum is an equilateral -triangular prism of steel one inch in length; but the edge on which -the beam vibrates is formed to an angle of 120°, in order to prevent -any injury from the weight with which it may be loaded. The chief -peculiarity in this balance consists in the knife-edge which forms -the fulcrum bearing upon an agate plane throughout its whole length, -whereas we have seen in the balance before described that the whole -weight is supported by portions only of the knife-edge, amounting -together to one tenth of an inch. The supports for the scales are -knife-edges each six tenths of an inch long. These are each furnished -with two pressing screws, by means of which they may be made parallel -to the central knife-edge. - -Each end of the beam is sprung obliquely upwards and towards the -middle, so as to form a spring through which a pushing screw passes, -which serves to vary the distance of the point of support from the -fulcrum, and, at the same time, by its oblique action to raise or -depress it, so as to furnish a means of bringing the points of support -and the fulcrum into a right line. - -A piece of wire, four inches long, on which a screw is cut, proceeds -from the middle of the beam downwards. This is pointed to serve as -an index, and a small brass ball moves on the screw, by changing the -situation of which the place of the centre of gravity may be varied at -pleasure. - -The fulcrum, as before remarked, rests upon an agate plane throughout -its whole length, and the scale-pans are attached to planes of agate -which rest upon the knife-edges forming the points of support. This -method of supporting the scale-pans, we have reason to believe, is -due to Mr. Cavendish. Upon the lower half of the pillar to which the -agate plane is fixed, a tube slides up and down by means of a lever -which passes to the outside of the case. From the top of this tube -arms proceed obliquely towards the ends of the balance, serving to -support a horizontal piece, carrying at each extremity two sets of Y -_s_, one a little above the other. The upper Y _s_ are destined to -receive the agate planes to which the scale-pans are attached, and thus -to relieve the knife-edges from their pressure; the lower to receive -the knife-edges which, form the points of support, consequently these -latter Y _s_, when in action, sustain the whole beam. - -When the lever is freed from a notch in which it is lodged, a spring -is allowed to act upon the tube we have mentioned, and to elevate it. -The upper Y _s_ first meet the agate planes carrying the scale-pans -and free them from the knife-edges. The lower Y _s_ then come into -action and raise the whole beam, elevating the central knife-edge above -the agate plane. This is the usual state of the balance when not in -use: when it is to be brought into action, the reverse of what we -have described takes place. On pressing down the lever, the central -knife-edge first meets the agate plane, and afterwards the two agate -planes carrying the scale-pans are deposited upon their supporting -knife-edges. - -A balance of this construction was employed by the writer of this -article in adjusting the national standard pound. With a pound troy in -each scale, the addition of one hundredth of a grain caused the index -to vary one division, equal to one tenth of an inch, and Mr. Robinson -adjusts these balances so that with one thousand grains in each scale, -the index varies perceptibly on the addition of one thousandth of a -grain, or of one-millionth part of the weight to be determined. - -It may not be uninteresting to subjoin, from the Philosophical -Transactions for 1826, the description of a balance perhaps the most -sensible that has yet been made, constructed for verifying the national -standard bushel. The author says,-- - -“The weight of the bushel measure, together with the 80 lbs. of -water it should contain, was about 250 lbs.; and as I could find -no balance capable of determining so large a weight with sufficient -accuracy, I was under the necessity of constructing one for this -express purpose. - -“I first tried cast iron; but though the beam was made as light as was -consistent with the requisite degree of strength, the inertia of such -a mass appeared to be so considerable, that much time must have been -lost before the balance would have answered to the small differences I -wished to ascertain. Lightness was a property essentially necessary, -and bulk was very desirable, in order to preclude such errors as might -arise from the beam being partially affected by sudden alterations of -temperature. I therefore determined to employ wood, a material in which -the requisites I sought were combined. The beam was made of a plank -of mahogany, about 7O inches long, 22 inches wide, and 2-1/4 thick, -tapering from the middle to the extremities. An opening was cut in the -centre, and strong blocks screwed to each side of the plank, to form a -bearing for the back of a knife-edge which passed through the centre. -Blocks were also screwed to each side at the extremities of the beam on -which rested the backs of the knife-edges for supporting the pans. The -opening in the centre was made sufficiently large to admit the support -hereafter to be described, upon which the knife-edge rested. - -“In all beams which I have seen, with the exception of those made by -Mr. Robinson, the whole weight is sustained by short portions at the -extremities of the knife-edge; and the weight being thus thrown upon a -few points, the knife-edge becomes more liable to change its figure and -to suffer injury. - -“To remedy this defect, the central knife-edge of the beam I am -describing was made 6 inches, and the two others 5 inches long. They -were triangular prisms with equal sides of three fourths of an inch, -very carefully finished, and the edges ultimately formed to an angle of -120°. - -“Each knife-edge was screwed to a thick plate of brass, the surfaces in -contact having been previously ground together; and these plates were -screwed to the beam, the knife-edges being placed in the same plane, -and as nearly equidistant and parallel to each other as could be done -by construction. - -“The support upon which the central knife-edge rested throughout its -whole length was formed of a plate of polished hard steel, screwed to -a block of cast iron. This block was passed through the opening before -mentioned in the centre of the beam, and properly attached to a frame -of cast iron. - -“The stirrups to which the scales were hooked rested upon plates of -polished steel to which they were attached, and the under surfaces of -which were formed by careful grinding into cylindrical segments. These -were in contact with the knife-edges their whole length, and were -known to be in their proper position by the correspondence of their -extremities with those of the knife-edges. A well imagined contrivance -was applied by Mr. Bate for raising the beam when loaded, in order -to prevent unnecessary wear of the knife-edge, and for the purpose of -adjusting the place of the centre of gravity, when the beam was loaded -with the weight required to be determined, a screw carrying a movable -ball projected vertically from the middle of die beam. - -“The performance of this balance fully equalled my expectations. With -two hundred and fifty pounds in each scale, the addition of a single -grain occasioned an immediate variation in the index of one twentieth -of an inch, the radius being fifty inches.” - -From the preceding account it appears that this balance is sensible to -1/1750000 part of the weight which was to be determined. - -We shall now describe the method to be pursued in adjusting a balance. - -1. To bring the points of suspension and the fulcrum into a right line. - -Make the vibrations of the balance very slow by moving the weight which -influences the centre of gravity, and bring the beam into a horizontal -position, by means of small bits of paper thrown into the scales. Then -load the scales with nearly the greatest weight the beam is fitted to -carry. If the vibrations are performed in the same time as before, no -further adjustment is necessary; but if the beam vibrates quicker, -or if it oversets, cause it to vibrate in the same time as at first, -by moving the adjusting weight, and note the distance through which -the weight has passed. Move the weight then in the contrary direction -through double this distance, and then produce the former slow motion -by means of the screw acting vertically on the point of support. Repeat -this operation until the adjustment is perfect. - -2. To make the arms of the beam of an equal length. - -Put weights in the scales as before; bring the beam as nearly as -possible to a horizontal position, and note the division at which the -index stands; unhook the scales, and transfer them with their weights -to the other ends of the beam, when, if the index points to the same -division, the arms are of an equal length; but if not, bring the index -to the division which had been noted, by placing small weights in one -or the other scale. Take away half these weights, and bring the index -again to the observed division by the adjusting screw, which acts -horizontally on the point of support. If the scale-pans are known to be -of the same weight, it will not be necessary to change the scales, but -merely to transfer the weights from one scale-pan to the other. - - -_Of the Use of the Balance._ - -Though we have described the method of adjusting the balance, these -adjustments, as we have before remarked, may be dispensed with. -Indeed, in all delicate scientific operations, it is advisable never -to rely upon adjustments, which, after every care has been employed in -effecting them, can only be considered as approximations to the truth. -We shall, therefore, now describe the best method of ascertaining the -weight of a body, and which does not depend on the accuracy of these -adjustments. - -Having levelled the case which contains the balance, and thrown the -beam out of action, place a weight in each scale-pan nearly equal to -the weight which is to be determined. Lower the beam very gently till -it is in action, and by means of the adjustment for raising or lowering -the centre of gravity, cause the beam to vibrate very slowly. Remove -these weights, and place the substance, the weight of which is to be -determined in one of the scale-pans; carefully counterpoise it by means -of any convenient substances put into the other scale-pan, and observe -the division at which the index stands; remove the body, the weight of -which is to be ascertained, and substitute standard weights for it so -as to bring the index to the same division as before. These weights -will be equal to the weight of the body. - -If it be required to compare two weights together which are intended -to be equal, and to ascertain their difference, if any, the method -of proceeding will be nearly the same. The standard weight is to be -carefully counterpoised, and the division at which the index stands, -noted. And now it will be convenient to add in either of the scales -some small weight, such as one or two hundredths of a grain, and mark -the number of divisions passed over in consequence by the index, by -which the value of one division of the scale will be known. This should -be repeated a few times, and the mean taken for greater certainty. - -Having noted the division at which the index rests, the standard -weight is to be removed, and the weight which is to be compared with -it substituted for it. The index is then again to be noted, and the -difference between this and the former indication will give the -difference between the weights in parts of a grain. - -If the balance is adjusted so as to be very sensible, it will be long -before it comes to a state of rest. It may, therefore, sometimes be -advisable to take the mean of the extent of the vibrations of the index -as the point where it would rest, and this may be repeated several -times for greater accuracy. It must, however, be remembered, that it is -not safe to do this when the extent of the vibrations is beyond one or -two divisions of the scale; but with this limitation it is, perhaps, as -good a method as can be pursued. - -Many precautions are necessary to ensure a satisfactory result. The -weights should never be touched by the hand; for not only would -this oxydate the weight, but by raising its temperature it would -appear lighter, when placed in the scale-pan, than it should do, in -consequence of the ascent of the heated air. For the larger weights a -wooden fork or tongs, according to the form of the weight, should be -employed; and for the smaller, a pair of forceps made of copper will be -found the most convenient. This metal possessing sufficient elasticity -to open the forceps on their being released from pressure, and yet not -opposing a resistance sufficient to interfere with that delicacy of -touch which is desirable in such operations. - - -_Of Weights._ - -It must be obvious, that the excellence of the balance would be of -little use, unless the weights employed were equally to be depended -upon. The weights may either be accurately adjusted, or the difference -between each weight and the standard may be determined, and, -consequently, its true value ascertained. It has been already shown how -the latter may be effected, in the instructions which have been given -for comparing two weights together; and we shall now show the readiest -mode of adjusting weights to an exact equality with a given standard. - -The material of the weight may be either brass or platina, and its form -may be cylindrical: the diameter being nearly twice the height. A small -spherical knob is screwed into the centre, a space being left under -the screw to receive the portions of fine wire used in the adjustment. -It will be convenient to form a cavity in the bottom of each weight to -receive the knob of the weight upon which it may be placed. - -Each weight is now to be compared with the standard, and should it -be too heavy, it is to be reduced till it becomes in a very small -degree too light, when the amount of the deficiency is to be carefully -determined. - -Some very fine silver wire is now to be taken, and the weight of three -or four feet of it ascertained. From this it will be known what length -of the wire is equal to the error of the weight to be adjusted; and -this length being cut off is to be enclosed under the screw. To guard -against any possible error, it will be advisable before the screw -is firmly fixed in its place, again to compare the weight with the -standard. - -The most approved method of making weights expressing the decimal parts -of a grain, is to determine, as before, with great care, the weight of -a certain length of fine wire, and then to cut off such portions as are -equal to the weights required. - -Before we conclude this article we shall give a description, from the -Annals of Philosophy for 1825, of “a very sensible balance,” used by -the late Dr. Black:-- - -“A thin piece of fir wood, not thicker than a shilling, and a foot -long, three tenths of an inch broad in the middle, and one tenth and -a half at each end, is divided by transverse lines into twenty parts; -that is, ten parts on each side of the middle. These are the principal -divisions, and each of them is subdivided into halves and quarters. -Across the middle is fixed one of the smallest needles I could procure, -to serve as an axis, and it is fixed in its place by means of a little -sealing wax. The numeration of the divisions is from the middle to each -end of the beam. The fulcrum is a bit of plate brass, the middle of -which lies flat on my table when I use the balance, and the two ends -are bent up to a right angle so as to stand upright. These two ends -are ground at the same time on a flat hone, that the extreme surfaces -of them may be in the same plane; and their distance is such that the -needle, when laid across them, rests on them at a small distance from -the sides of the beam. They rise above the surface of the table only -one tenth and a half or two tenths of an inch, so that the beam is very -limited in its play. See _fig. 190._ - -“The weights I use are one globule of gold, which weighs one grain, and -two or three others which weigh one tenth of a grain each; and also -a number of small rings of fine brass wire, made in the manner first -mentioned by Mr. Lewis, by appending a weight to the wire, and coiling -it with the tension of that weight round a thicker brass wire in a -close spiral, after which, the extremity of the spiral being tied hard -with waxed thread, I put the covered wire into a vice, and applying -a sharp knife, which is struck with a hammer, I cut through a great -number of the coils at one stroke, and find them as exactly equal to -one another as can be desired. Those I use happen to be the 1/30 part -of a grain each, or 300 of them weigh ten grains; but I have others -much lighter. - -“You will perceive that by means of these weights placed on different -parts of the beam, I can learn the weight of any little mass from one -grain, or a little more, to the 1/1200 of a grain. For if the thing to -be weighed weighs one grain, it will, when placed on one extremity of -the beam, counterpoise the large gold weight at the other extremity. -If it weighs half a grain it will counterpoise the heavy gold weight -placed at 5. If it weigh 6/10 of a grain, you must place the heavy -gold weight at 5, and one of the lighter ones at the extremity to -counterpoise it, and if it weighs only one or two, or three or four -hundredths of a grain, it will be counterpoised by one of the small -gold weights placed at the first or second, or third or fourth -division. If, on the contrary, it weighs one grain and a fraction, it -will be counterpoised by the heavy gold weight at the extremity, and -one or more of the lighter ones placed in some other part of the beam. - -“This beam has served me hitherto for every purpose; but had I occasion -for a more delicate one, I could make it easily by taking a much -thinner and lighter slip of wood, and grinding the needle to give it an -edge. It would also be easy to make it carry small scales of paper for -particular purposes.” - -The writer of this article has used a balance of this kind, and -finds that it is sensible to 1/1000 of a grain when loaded with ten -grains. It is necessary, however, where accuracy is required, to -employ a scale-pan. This may be made of thin card paper, shaped as in -_fig. 191._ - -A thread is to be passed through the two ends, by tightening which they -may be brought near each other. - -The most convenient weights for this beam appear to be two of one grain -each, and one of one tenth of a grain. They should be made of straight -wire; and if the beam be notched at the divisions, they may be lodged -in these notches very conveniently. Ten divisions on each side of the -middle will be sufficient. The weight of the scale-pan must first be -carefully ascertained, in order that it may be deducted from the -weight, afterwards determined, of the scale-pan and the substance it -may contain. - -If the scale-pan be placed at the tenth division of the beam, it is -evident that by means of the two grain weights, a greater weight cannot -be determined than one grain and nine tenths; but if the scale-pan be -placed at any other division of the beam, the resulting apparent weight -must be increased by multiplying it by ten, and dividing by the number -of the division at which the scale-pan is placed; and in this manner it -is evident that if the scale-pan be placed at the division numbered 1, -a weight amounting to nineteen grains may be determined. - -We have been tempted to describe this little apparatus, because it is -extremely simple in its construction, may be easily made, and may be -very usefully employed on many occasions where extreme accuracy is not -necessary. - - -_Description of the Steelyard._ - -The steelyard is a lever, having unequal arms; and in its most simple -form it is so arranged, that one weight alone serves to determine a -great variety of others, by sliding it along the longer arm of the -lever, and thus varying its distance from the fulcrum. - -It has been demonstrated, chapter xiii., that in the lever the -proportion of the power to the weight will be always the same as -that of their distances from the fulcrum, taken in a reverse order; -consequently, when a constant weight is used, and an equilibrium -established by sliding this weight on the longer arm of the lever, the -relative weight of the substance weighed, to the constant weight, will -be in the same proportion as the distance of the constant weight from -the fulcrum is to the length of the shorter arm. - -Thus, suppose the length of the shorter arm, or the distance of -the fulcrum from the point from which the weight to be determined -is suspended, to be one inch; let the longer arm of the lever be -divided into parts of one inch each, beginning at the fulcrum. Now -let the constant weight be equal to one pound, and let the steelyard -be so constructed that the shorter arm shall be sufficiently heavy -to counterpoise the longer when the bar is unloaded. Then suppose a -substance, the weight of which is five pounds, to be suspended from the -shorter arm. It will be found that when the constant weight is placed -at the distance of five inches from the fulcrum, the weights will be in -equilibrium, and the bar consequently horizontal. In this steelyard, -therefore, the distance of each inch from the fulcrum indicates a -weight of one pound. An instrument of this form was used by the Romans, -and it is usually described as the Roman statera or steelyard. A -representation of it is given at _fig. 192._ - -The steelyard is in very general use for the coarser purposes of -commerce, but constructed differently from that which we have -described. The beam with the scales or hooks is seldom in equilibrium -upon the point F, when the weight P is removed; but the longer arm -usually preponderates, and the commencement of the graduations, -therefore, is not at F, but at some point between B and F. The common -steelyard, which we have represented at _fig. 193._, is usually -furnished with two points, from either of which the substance, the -weight of which is to be determined, may be suspended. The value of -the divisions is in this case increased in proportion as the length -of the shorter arm is decreased. Thus, in the steelyard which we have -described, if there be a second point of suspension at the distance of -half an inch from the fulcrum, each division of the longer arm will -indicate two pounds instead of one, and these divisions are usually -marked upon the opposite edge of the steelyard, which is made to turn -over. - -This instrument is very convenient, because it requires but one weight; -and the pressure on the fulcrum is less than in the balance, when the -substance to be weighed is heavier than the constant weight. But, -on the contrary, when the constant weight exceeds the substance to -be weighed, the pressure on the fulcrum is greater in the steelyard -than in the balance, and the balance is, therefore, preferable in -determining small weights. There is also an advantage in the balance, -because the subdivision of weights can be effected with a greater -degree of precision than the subdivision of the arm of the steelyard. - - -_C. Paul’s Steelyard._ - -A steelyard has been constructed by Mr. C. Paul, inspector of weights -and measures at Geneva, which is much to be preferred to that in -common use. Mr. C. Paul states, that steelyards have two advantages -over balances: 1. That their axis of suspension is not loaded with -any other weight than that of the merchandise, the constant weight of -the apparatus itself excepted; while the axis of the balance, besides -the weight of the instrument, sustains a weight double to that of -the merchandise. 2. The use of the balance requires a considerable -assortment of weights, which causes a proportional increase in the -price of the apparatus, independently of the chances of error which it -multiplies, and of the time employed in producing an equilibrium. - -1. In C. Paul’s steelyard the centres of the movement of suspension, or -the two constant centres, are placed on the exact line of the divisions -of the beam; an elevation almost imperceptible in the axis of the beam, -destined to compensate for the very slight flexion of the bar, alone -excepted. - -2. The apparatus, by the construction of the beam, is balanced below -its centre of motion, so that when no weight is suspended the beam -naturally remains horizontal, and resumes that position when removed -from it, as also when the steelyard is loaded, and the weight is at -the division which ought to show how much the merchandise weighs. The -horizontal situation in this steelyard, as well as in the others, is -known by means of the tongue which rises vertically above the axis of -suspension. - -3. It may be discovered, that the steelyard is deranged if, when not -loaded, the beam does not remain horizontal. - -4. The advantage of a great and a small side (which in the other -augments the extent of their power of weighing) is supplied by a very -simple process, which accomplishes the same end with some additional -advantages. This process is to employ on the same division different -weights. The numbers of the divisions on the bar, point out the degree -of heaviness expressed by the corresponding weights. For example, when -the large weight of the large steelyard weighs 16 lbs., each -division it passes over on the bar is equivalent to a pound; the small -weight, weighing sixteen times less than the large one, will represent -on each of these divisions the sixteenth part of a pound, or one ounce; -and the opposite face of the bar is marked by pounds at each sixteenth -division. In this construction, therefore, we have the advantage -of being able, by employing both weights at once, to ascertain, -for example, almost within an ounce, the weight of 500 pounds of -merchandise. It will be sufficient to add what is indicated by the -small weight in ounces, to that of the large one in pounds, after an -equilibrium has been obtained by the position of the two weights, viz. -the large one placed at the next pound below its real weight, and the -small one at the division which determines the number of ounces to be -added. - -5. As the beam is graduated only on one edge, it may have the form of -a thin bar, which renders it much less susceptible of being bent by -the action of the weight, and affords room for making the figures more -visible on both the faces. - -6. In these steelyards the disposition of the axes is not only such -that the beam represents a mathematical lever without weight, but in -the principle of its division, the interval between every two divisions -is a determined and aliquot part of the distance between the two fixed -points of suspension; and each of the two weights employed has for its -absolute weight the unity of the weight it represents, multiplied by -the number of the divisions contained in the interval between the two -centres of motion. - -Thus, supposing the arms of the steelyard divided in such a manner -that ten divisions are exactly contained in the distance between the -two constant centres of motion, a weight to express the pounds on each -division of the beam must really weigh ten pounds; that to point out -the ounces on the same divisions must weigh ten ounces, &c. So that the -same steelyard may be adapted to any system of measures whatever, and -in particular to the decimal system, by varying the absolute heaviness -of the weights, and their relation with each other. - -But to trace out, in a few words, the advantages of the steelyards -constructed by C. Paul for commercial purposes, we shall only observe,-- - -1. That the buyer and seller are certain of the correctness of the -instrument, if the beam remains horizontal when it is unloaded and in -its usual position. 2. That these steelyards have one suspension less -than the old ones, and are so much more simple. 3. That by these means -we obtain, with the greatest facility, by employing two weights, the -exact weight of merchandise, with all the approximation that can be -desired, and even with a greater precision than that given by common -balances. There are few of these which, when loaded with 500 pounds -at each end, give decided indication of an ounce variation; and the -steelyards of C. Paul possess that advantage, and cost one half less -than balances of equal dominion. 4. In the last place, we may verify at -pleasure the justness of the weights, by the transposition which their -ratio to each other will permit; for example, by observing whether, -when the weight of one pound is brought back one division, and the -weight of one ounce carried forward sixteen divisions, the equilibrium -still remains. - -It is on this simple and advantageous principle that C. Paul has -constructed his universal steelyard. It serves for weighing in the -usual manner, and according to any system of weights, all ponderable -bodies to the precision of half a grain in the weight of a hundred -ounces; that is to say, of a ten-thousandth part. It is employed, -besides, for ascertaining the specific gravity of solids, of liquids, -and of air, by processes extremely simple, and which do not require -many subdivisions in the weights. - -We think the description above given will be sufficiently intelligible -without a representation of this instrument. An account of its -application to the determination of specific gravities will be found in -vol. iii. of the Philosophical Magazine. - - -_The Chinese Steelyard._ - -This instrument is used in China and the East Indies for weighing -gems, precious metals, &c. The beam is a small rod of ivory, about -a foot in length. Upon this are three lines of divisions, marked by -fine silver studs, all beginning from the end of the beam, whence the -first is extended 8 inches, the second 6-1/2, and the third 8-1/2. The -first is European weight, and the other two Chinese. At the other end -of the beam hangs a round scale, and at three several distances from -this end are holes, through which pass so many fine strings, serving as -different points of suspension. The first distance makes 1-3/5 inches, -the second 3-1/5, or double the former, and the third 4-4/5, or triple -the same. The instrument, when used, is held by one of the strings, -and a sealed weight of about 1-1/4 oz. troy, is slid upon the -beam until an equilibrium is produced; the weight of the body is then -indicated by the graduated scale above mentioned. - - -_The Danish Balance._ - -The Danish balance is a straight bar or lever, having a heavy weight -fixed to one end, and a hook or scale-pan to receive the substance, -the weight of which is to be determined, suspended from the other -end. The fulcrum is moveable, and is made to slide upon the bar, till -the beam rests in a horizontal position, when the place of the fulcrum -indicates the weight required. In order to construct a balance of this -kind, let the distance of the centre of gravity from that point to -which the substance to be weighed is suspended be found by experiment, -when the beam is unloaded. Multiply this distance by the weight of the -whole apparatus, and divide the product by the weight of the apparatus -increased by the weight of the body. This will give the distance from -the point of suspension, at which the fulcrum being placed, the whole -will be in equilibrio: for example, supposing the distance of the -centre of gravity from the point of suspension to be 10 inches, and -the weight of the whole apparatus to be ten pounds; suppose, also, it -were required to mark the divisions which should indicate weights of -one, two, or three pounds, &c. First, for the place of the division -indicating one pound we have (10 × 10)/(10 + 1) = 100/(10 + 1) = 9-1/11 -inches, the place of the division marking one pound. For two pounds we -have 100/(10 + 2) = 8-1/3 inches, the place of the division indicating -two pounds; and for three pounds 100/(10 + 3) = 7-9/13 inches for the -place of the division indicating three pounds, and so on. - -This balance is subject to the inconvenience of the divisions becoming -much shorter as the weight increases. The distance between the -divisions indicating one and two pounds being, in the example we have -given, about seven tenths of an inch, whilst that between 20 and 21 -pounds is only one tenth of an inch; consequently a very small error -in the place of the divisions indicating the larger weights would -occasion very inaccurate results. The Danish balance is represented at -_fig. 194._ - - -_The Bent Lever Balance._ - -This instrument is represented at _fig. 195._ The weight at C, is -fixed at the end of the bent lever A B C, which is supported -by its axis B on the pillar I H. A scale-pan E, is suspended from -the other end of the lever at A. Through the centre of motion B draw -the horizontal line K B G, upon which, from A and C let fall -the perpendiculars A K and C D. Then if B K and B D -are reciprocally proportional to the weights at A and C, they will be -in equilibrio, but if not, the weight C will move upwards or downwards -along the arc F G till that ratio is obtained. If the lever be so -bent that when A coincides with the line G K, C coincides with -the vertical B H, then as C moves from F to G, its momentum will -increase while that of the weight in the scale-pan E will decrease. -Hence the weight in E, corresponding to different positions of the -balance, may be expressed on the graduated arc F G. - - -_Brady’s Balance, or Weighing Apparatus._ - -This partakes of the properties both of the bent lever balance and of -the steelyard. It is represented, at _fig. 196._ A B C -is a frame of cast iron having a great part of its weight towards A. F -is a fulcrum, and E a moveable suspender, having a scale and hook -at its lower extremity. E K G are three distinct places, to -which the suspender E may be applied, and to which belong respectively -the three graduated scales of division expressing weights, _f_ C, -_c d_, and _a b_. When the scale and suspender are applied at -G, the apparatus is in equilibrio, with the edge A B horizontal, -and the suspender cuts the zero on the scale _a b_. Now, any -substance, the weight of which is to be ascertained, being put into -the scale, the whole apparatus turns about F, and the part towards B -descends till the equilibrium is again established, when the weight -of the body is read off from the scale _a b_, which registers to -ounces and extends to two pounds. If the weight of the body exceed two -pounds, and be less than eleven pounds, the suspender is placed at K; -and when the scale is empty, the number 2 is found to the right of the -index of the suspender. If now weights exceeding two pounds be placed -in the scale, the whole again turns about F, and the weight of the -body is shown on the graduated arc _c d_, which extends to eleven -pounds, and registers to every two ounces. - -If the weight of the body exceed eleven pounds, the suspender is hung -on at E, and the weights are ascertained in the same manner on the -scale _f_ C to thirty pounds, the subdivisions being on this scale -quarters of pounds. The same principles would obviously apply to -weights greater or less than the above. To prevent mistake, the three -points of support G, K, E, are numbered 1, 2, 3; and the corresponding -arcs are respectively numbered in the same manner. When the hook is -used instead of the scale, the latter is turned upwards, there being a -joint at _m_ for that purpose. - - -_The Weighing Machine for Turnpike Roads._ - -This machine is for the purpose of ascertaining the weight of heavy -bodies, such as wheel carriages. It consists of a wooden platform -placed over a pit made in the line of the road, and which contains the -machinery. The pit is walled withinside, and the platform is fitted to -the walls of the pit, but without touching them, and it is therefore -at liberty to move freely up and down. The platform is supported by -levers placed beneath it, and is exactly level with the surface of the -road, so that a carriage is easily drawn on it, the wheels being upon -the platform whilst the horses are upon the solid ground beyond it. The -construction of this machine will be readily understood by reference to -_fig. 197._, in which the platform is supposed to be transparent -so as to allow of the levers being seen below it. - -A, B, C, D, represent four levers tending towards the centre of the -platform, and each moveable on its fulcrum at A, B, C, D; the fulcrum -of each rests upon a piece securely fixed in the corner of the pit. -The platform is supported upon the cross pins _a_, _b_, _c_, _d_, -by means of pieces of iron which project from it near its corners, and -which are represented in the plate by the short dark lines crossing the -pins _a_, _b_, _c_, _d_. The four levers are connected under the centre -of the platform, but not so as to prevent their free motion, and are -supported by a long lever at the point F, the fulcrum of which rests -upon a piece of masonry at E: the end of this last lever passes below -the surface of the road into the turnpike house, and is there attached -to one arm of a balance, or, as in Salmon’s patent weighing machine, to -a strap passing round a cylinder which winds up a small weight round a -spiral, and indicates, by means of an index, the weight placed upon the -platform. - -[Illustration: _Captn. Kater, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -Suppose the distance from A to F to be ten times as great as that from -A to _a_, then a force of one pound applied beneath F would balance -ten pounds applied at _a_, or upon the platform. Again: let the -distance from E to G be also ten times greater than the distance from -the fulcrum E to F; then a force of one pound applied to raise up the -end of the lever G would counterpoise a weight of ten pounds placed -upon F. Now, as we gain ten times the power by the first levers, and -ten times more by the lever E G, it follows, that a force of one -pound tending to elevate G, would balance 100 lbs. placed on the -platform; so that if the end of the lever G be attached to one arm of -a balance, a weight of 10 lbs. placed in a scale suspended from -the other arm, will express the value of 1000 lbs. placed upon -the platform. The levers are counterpoised, when the platform is not -loaded, by a weight H applied to the end of the last lever, continued -beyond the fulcrum for that purpose. - - -_Of Instruments for weighing by means of a Spring._ - -The spring is well adapted to the construction of a weighing machine, -from the property it possesses of yielding in proportion to the force -impressed, and consequently giving a scale of equal parts for equal -additions of weight. It is liable, however, to suffer injury, unless -the steel of which it is composed be very well tempered, from a want -of perfect elasticity, and, consequently, from not returning to its -original place after it has been forcibly compressed. This, however, -must be considered to arise, in a great measure, from imperfection -of workmanship, or of the material employed, or from its having been -subjected to too great a force. - - -_The Spring Steelyard._ - -The little instrument known by this name is in very general use, and -is particularly convenient where great accuracy is not necessary, as -a spring which will ascertain weights from one pound to fifty, is -contained in a cylinder only 4 inches long and 3/4 inch diameter. - -This instrument is represented at _fig. 198._ It consists of a -tube of iron, of the dimensions just stated, closed at the bottom, -to which is attached an iron hook for supporting the substance to be -weighed; a rod of iron _a b_, four tenths of an inch wide and one -tenth thick, is firmly fixed in the circular plate _c d_, which -slides smoothly in the iron tube. - -A strong steel spring is also fastened to this plate, and passed round -the rod _a b_ without touching it, and without coming in contact -with the interior of the cylindrical tube. The tube is closed at the -top by a circular piece of iron through which the piece _a b_ -passes. - -Upon the face of _a b_ the weight is expressed by divisions, -each of which indicates one pound, and five of such divisions in the -instrument now before us occupy two tenths of an inch. The divisions, -notwithstanding, are of sufficient size to enable them to be subdivided -by the eye. - -To use this instrument, the substance to be weighed is suspended by the -hook, the instrument being held by a ring passing through the rod at -the other end. The spring then suffers a compression proportionate to -the weight, and the number of pounds is indicated by the division on -the rod which is cut by the top of the cylindrical tube. - - -_Salter’s improved Spring Balance._ - -A very neat form of the instrument last described has been recently -brought before the public by Mr. Salter, under the name of the Improved -Spring Balance. It is represented at _fig. 199._ The spring is -contained in the upper half of a cylinder behind the brass plate -forming the face of the instrument; and the rod is fixed to the lower -extremity of the spring, which is consequently extended, instead of -being compressed, by the application of the weight. The divisions, each -indicating half a pound, are engraved upon the face of the brass plate, -and are pointed out by an index attached to the rod. - - -_Marriott’s Patent Dial Weighing Machine._ - -The exterior of this instrument is represented at _fig. 200._, and -the interior at _fig. 201._ A B C is a shallow brass -box, having a solid piece as represented at A, to which the spring -D E F is firmly fixed by a nut at D. The other end of the -spring at F is pinned to the brass piece G H, to the part of which -at G is also fixed the iron racked plate I. A screw L serves as a stop -to keep this rack in its place. The teeth of the rack fit into those -of the pinion M, the axis of which passes through the centre of the -dial-plate, and carries an index which points out the weight. The brass -piece G H is merely a plate where it passes over the spring, and -the tail piece H, to which the weight is suspended, passes through an -opening in the side of the box. - - -_Of the Dynamometer._ - -This is an important instrument in mechanics, calculated to measure -the muscular strength exerted by men and animals. It consists -essentially of a spring steelyard, such as that we first described. -This is sometimes employed alone, and sometimes in combination with -various levers, which allow of the spring being made more delicate, -and consequently increase the extent of the divisions indicating the -weight. - -The first instrument of this kind appears to have been invented by Mr. -Graham, but it was too bulky and inconvenient for use. M. le Roy made -one of a more simple construction. It consisted of a metal tube, about -a foot long, placed vertically upon a stand, and containing in the -inside a spiral spring, having above it a graduated rod terminating in -a globe. This rod entered the tube more or less in proportion to the -force applied to the globe, and the divisions indicated the quantity of -this force. Therefore, when a man pressed upon the globe with all his -strength, the divisions upon the rod showed the number of pounds weight -to which it was equal. - -An instrument of this kind for determining the force of a blow struck -by a man with his fist was lately exhibited at the National Repository. -It was fixed to a wall, from which it projected horizontally. In -place of the globe there was a cushion to receive the blow, and as -the suddenness with which the spring returned rendered it impossible -to read the division upon the rod, another rod similarly divided was -forced in by the plate forming the basis of the cushion, and remained -stationary when the spring returned. The common spring steelyard, -however, which we first described, is in principle the same as M. le -Roy’s dynamometer, and is much more conveniently constructed for the -purpose we are considering. The ring at one end may be fixed to an -immovable object, and the hook at the other attached to a man, or to -an animal, and the extent to which the graduated rod is drawn out of -the cylinder shows at once the force which is applied. Though this is -perhaps the best, and certainly the most simple dynamometer, others -have been contrived, which are, however, but modifications of the -spring steelyard. One of these is represented at _fig. 202._ The -spiral spring acts in the manner before described, but its divisions -are increased in size, and therefore rendered more perceptible by means -of a rack fixed to the plate, acting against the spiral spring, the -teeth of which move a pinion upon which the arm I is fixed, pointing to -the graduated arc K. - -Another dynamometer has been invented by Mr. Salmon; it is represented -at _fig. 203._ and is a combination of levers with the spring. -By means of these levers a much more delicate spring, and which is -therefore more sensible, may be employed than in the dynamometer last -described. - -The manner in which these levers and spring act will be readily -understood by an inspection of the figure. Like the weighing machine -for carriages, the fulcrum of each lever is at one end, and the force -is diminished in passing to the spring, in the ratio of the length of -its arms. The spring moves a pinion by means of a rack, upon which -pinion a hand is placed, indicating by divisions upon a circular -dial-plate, the amount of the force employed. - -The spring used in this machine is calculated to weigh only about -50 lbs. instead of about 5 cwt., as in the last described; -but by means of the levers which intervene between it and the force -applied, it will serve to estimate a force equal to 6 cwt., and -might obviously be made to go to a much greater extent, by varying the -ratio of the length of the arms of the levers. - - -ON COMPENSATION PENDULUMS. - -(336.) It is said of Galileo that, when very young, he observed a -lamp suspended from the roof of a church at Pisa, swinging backwards -and forwards with a pendulous motion. This, if it had been remarked -at all by an uneducated mind, would, most probably, have been passed -by as a common occurrence, unworthy of the slightest notice; but to -the mind imbued with science no incident is insignificant; and a -circumstance apparently the most trivial, when subjected to the giant -force of expanded intellect, may become of immense importance to the -improvement and to the well-being of man. The fall of an apple, it is -said, suggested to Newton the theory of gravitation, and his powerful -mind speedily extended to all creation that great law which brings an -apple to the ground. The swinging of a lamp in a church at Pisa, viewed -by the piercing intellect of Galileo, gave rise to an instrument which -affords the most perfect measure of time, which serves to determine the -figure of the earth, and which is inseparably connected with all the -refinements of modern astronomy. - -The properties of the pendulum, and the manner in which it serves -to measure time, have been fully explained in chapter xi.; and if -a substance could be found not susceptible of any change in its -dimensions from a change of temperature, nothing more would be -necessary, as the centre of oscillation would always remain at the -same distance from the point of suspension. As every known substance, -however, expands with heat, and contracts with cold, the length of the -pendulum will vary with every alteration of temperature, and thus the -time of its vibration will suffer a corresponding change. The effect -of a difference of temperature of 25°, or that which usually occurs -between winter and summer, would occasion a clock furnished with a -pendulum having an iron rod to gain or lose six seconds in twenty-four -hours. - -It became, then, highly important to discover some means of -counteracting this variation to which the length of the pendulum was -liable, or, in other words, to devise a method by which the centre of -oscillation should, under every change of temperature, remain at the -same distance from the point of suspension: happily, the difference in -the rate of expansion of different metals presented a ready means of -effecting this. - -Graham, in the year 1715, made several experiments to ascertain the -relative expansions of various metals, with a view of availing himself -of the difference of the expansions of two or more of them when opposed -to each other, to construct a compensating pendulum. But the difference -he found was so small, that he gave up all hope of being able to -accomplish his object in that way. Knowing, however, that mercury was -much more affected by a given change of temperature than any other -substance, he saw that if the mercury could be made to ascend while -the rod of the pendulum became longer, and _vice versâ_, the centre of -oscillation might always be kept at the same distance from the point -of suspension. This idea happily gave birth to the mercurial pendulum, -which is now in very general use. - -[Illustration: _Captn. Kater, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -In the mean time, Graham’s suggestion excited the ingenuity of -Harrison, originally a carpenter at Barton in Lincolnshire, who, in -1726, produced a pendulum formed of parallel brass and steel rods, -known by the name of the gridiron pendulum. - -In the mercurial pendulum, the bob or weight is the material affording -the compensation; but in the gridiron pendulum the object is attained -by the greater expansion of the brass rods, which raise the bob upwards -towards the point of suspension as much as the steel rods elongate -downwards. - -In the present article, we shall describe such compensation pendulums -as appear to us likely to answer best in practice; and we trust we -shall be able to simplify the subject so as to render a knowledge -of mathematics in the construction of this important instrument -unnecessary. - -The following table contains the linear expansion of various substances -in parts of their length, occasioned by a change of temperature -amounting to one degree. We have taken the liberty of extracting -it from a very valuable paper by F. Bailey, Esq., on the mercurial -compensation pendulum, published in the Memoirs of the Astronomical -Society of London for 1824. - - -TABLE I. - -_Linear Expansion of various Substances for One Degree of Fahrenheit’s -Thermometer._ - - +----------------------+-------------+-----------------------+ - | Substances. | Expansions. | Authors. | - +----------------------+-------------+-----------------------+ - |White Deal, { | ·0000022685 | Captain Kater. | - | { | ·0000028444 | Dr. Struve. | - |English Flint Glass, | ·0000047887 | Dulong and Petit. | - |Iron (cast), { | ·0000061700 | General Roy. | - | { | ·0000065668 | Dulong and Petit. | - |Iron (wire), | ·0000068613 | Lavoisier and L. | - |Iron (bar), | ·0000069844 | Hasslar. | - |Steel (rod), | ·0000063596 | General Roy. | - | | | {Commissioners of | - |Brass, | ·0000104400 | {Weights and Measures | - | | | {--mean of several | - | | | {experiments. | - |Lead, | ·0000159259 | Smeaton. | - |Zinc, | ·0000163426 | Ditto. | - |Zinc (hammered), | ·0000172685 | Ditto. | - |Mercury _in bulk_, | ·00010010 | Dulong and Petit. | - +----------------------+-------------+-----------------------+ - -From this table it is easy to determine the length of a rod of any -substance the expansion of which shall be equal to that of a rod of -given length of any other substance. - -The lengths of such rods will be inversely proportionate to their -expansions. If, therefore, we divide the lesser expansion by the -greater (supposing the rod the length of which is given to be made of -the lesser expansible material), and multiply the given length by this -quotient, we shall have the required length of a rod, the expansion -of which will be equal to that of the rod given. For example:--The -expansion of a rod of steel being, from the above table, ·0000063596, -and that of brass, ·0000104400; if it were required to determine the -length of a rod of brass which should expand as much as a rod of steel -of 39 inches in length, we have ·0000063596/·0000104400 = ·6091, which, -multiplied by 39, gives 23·75 inches for the length of brass required. - -We shall here, in order to facilitate calculation, give the ratio of -the lengths of such substances as may be employed in the construction -of compensation pendulums. - - -TABLE II. - - +---------------------------------------------------+ - | Steel rod and brass compensation, as 1: ·6091 | - | Iron wire rod and lead compensation, ·4308 | - | Steel rod and lead compensation, ·3993 | - | Iron wire rod and zinc compensation, ·3973 | - | Steel rod and zinc compensation, ·3682 | - | Glass rod and lead compensation, ·3007 | - | Glass rod and zinc compensation, ·2773 | - | Deal rod and lead compensation, ·1427 | - | Deal rod and zinc compensation, ·1313 | - | Steel rod and mercury in a steel cylinder, ·0728 | - | Steel rod and mercury in a glass cylinder, ·0703 | - | Glass rod and mercury in a glass cylinder, ·0529 | - +---------------------------------------------------+ - -It is evident that in this table the decimals express the length of a -rod of the compensating material, the expansion of which is equal to -that of a pendulum rod whose length is unity. - -As we are not aware of the existence of any work which contains -instructions that might enable an artist or an amateur to make a -compensation pendulum, we shall endeavour to give such detailed -information as may free the subject from every difficulty. - -The pendulum of a clock is generally suspended by a spring, fixed -to its upper extremity, and passing through a slit made in a piece -which is called the cock of the pendulum. The point of suspension is, -therefore, that part of the spring which meets the lower surface of the -cock. Now the distance of the centre of oscillation of the pendulum -from this point may be varied in two ways; the one by drawing up the -spring through this slit, and the other by raising the bob of the -pendulum. Either of these methods may be practised in the compensation -pendulum, but the former is subject to objections from which the latter -is exempt. - -Suppose it were required to compensate a pendulum of 39 inches in -length, of steel, by means of the expansion of a brass rod. Here, -referring to _fig. 204._, we have S C 39 inches (which is -to remain constant) of steel; the pendulum spring, passing through -the cock at S, is attached to another rod of steel, which is fixed to -the cross piece R A at A. The other end of the cross piece at R -is fastened to a brass rod, the lower extremity of which is fixed to -the cock of the pendulum at B. Now the brass rod B R must expand -upwards, as much as the steel rod A C expands downwards; and the -length of the brass must be such as to effect this, leaving 39 inches -of the steel rod below the cock of the pendulum. - -Let us first try 80 inches of steel. Multiplying this by ·6091, we have -48·73 inches for the length of brass, which compensates 80 inches of -steel. But as 48·73 inches of the steel, equal in length to the brass, -would in this case be above the cock of the pendulum, it would leave -only 31·27 inches below it, instead of 39 inches. - -Let us now try 100 inches of steel. This, multiplied as before by -·6091, gives 60·91 inches, according to the expansions which we have -used, for the length of the brass rod, and leaves 39·09 inches below -the cock of the pendulum, which is sufficiently near for our present -purpose. - -From what has been said we may perceive that the total length of the -material of which the pendulum rod is composed must be always equal to -the length of the pendulum added to the length of the compensation. - -In this instance we have effected our object, by drawing the -pendulum-spring through the slit; but we will now show how the -same thing may be done by moving the bob of the pendulum. At -_fig. 205._, let S C, as before, be equal to 39 inches. Let -the steel rod S D turn off at right angles at D, and let a rod -of brass B R, of 61 inches in length, ascend perpendicularly from -this cross piece to R. To the upper part of the brass rod fix another -cross piece R A, and from the extremity A let a steel rod descend -to E, bending it as in the figure till it reaches C. Now the total -length of the pieces of steel expanding downwards is equal to S D, -D F, and F C (amounting together to 39 inches), to which must -be added a length of steel equal to that of the brass rod B R, (61 -inches), making together 100 inches of steel as before, the expansion -of which downwards is compensated by that of the brass rod, of 61 -inches in length, expanding upwards. - -This form, however, is evidently inconvenient, from the great length -of brass and steel which is carried above the cock of the pendulum; -but it is the same thing whether the brass and steel be each in one -piece, or divided into several, provided the pieces of steel be all -so arranged as to expand downwards, and those of brass upwards. Thus, -at _fig. 206._, the portions of steel expanding downwards are -together equal, as before, to 100 inches, and the two brass pieces -expanding upwards are together equal to 61 inches. So that, in fact, -the two last forms of compensation which we have described differ in -no respect from each other in principle, but only in the arrangement -of the materials. The last is the half of the gridiron pendulum, the -remaining bars being merely duplicates of those we have described, and -serving no other purpose but to form a secure frame-work. - - -_Harrison’s Gridiron Pendulum._ - -After what has been said, little more is necessary than to give a -representation of this pendulum. This is done at _fig. 207._, in -which the darker lines represent the steel rods, and the lighter those -of brass. The central rod is fixed at its lower extremity to the middle -of the third cross piece from the bottom, and passes freely through -holes in the cross pieces which are above, whilst the other rods are -secured near their extremities to the cross pieces by pins passing -through them. In order to render the whole more secure, the bars pass -freely through holes made in two other cross pieces, the extremities of -which are fixed to the exterior steel wires. As different kinds of the -same metal vary in their rate of expansion, the pendulum when finished -may be found upon trial to be not duly compensated. In this case one or -more of the cross pieces is shifted higher or lower upon the bars, and -secured by pins passed through fresh holes. - - -_Troughton’s Tubular Pendulum._ - -This is an admirable modification of Harrison’s gridiron pendulum. -It is represented at _fig. 208._, where it may be seen that it -has the appearance of a simple pendulum, as the whole compensation is -concealed within a tube six tenths of an inch in diameter. - -A steel wire, about one tenth of an inch in diameter, is fixed in the -usual manner to the spring by which the pendulum is suspended. This -wire passes to the bottom of an interior brass tube, in the centre of -which it is firmly screwed. The top of this tube is closed, the steel -rod passing freely through a hole in the centre. Into the top of this -interior tube two steel wires, of one tenth of an inch in diameter, -are screwed into holes made in that diameter, which is at right angles -to the motion of the pendulum. These wires pass down the tube without -touching either it or the central rod, through holes made in the piece -which closes the bottom of the interior tube. The lower extremities of -these wires, which project a little beyond the inner tube, are securely -fixed in a piece which closes the bottom of an exterior brass tube, -which is of such a diameter as just to allow the interior tube to pass -freely through it, and of a sufficient length to extend a little above -it. The top of the exterior tube is closed like that of the interior, -having also a hole in its centre, to allow the first steel rod to pass -freely through it. Into the top of the exterior tube, in that diameter -which coincides with the motion of the pendulum, a second pair of -steel wires of the same diameter as the former are screwed, their -distance from the central rod being equal to the distance of each from -the first pair. They consequently pass down within the interior tube, -and through holes made in the pieces closing the lower ends of both the -interior and exterior tubes. The lower ends of these wires are fastened -to a short cylindrical piece of brass of the same diameter as the -exterior tube, to which the bob is suspended by its centre. - -[Illustration: _Captn. Kater, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -_Fig. 209._ is a full sized section of the rod; the three -concentric circles represent the two tubes, and the rectangular -position of the two pair of wires round the middle one is shown by the -five small circles. - -_Fig. 210._ is the part which closes the upper end of the interior -tube. The two small circles are the two wires which proceed from it, -and the three large circles show the holes through which the middle -wire and the other pair of wires pass. - -_Fig. 211._ is the bottom of the interior tube. The small circle -in the centre is where the central rod is fastened to it, the others -the holes for the other four wires to pass through. - -_Fig. 212._ is the part which closes the top of the external tube. -In the large circle in the centre a small brass tube is fixed, which -serves as a covering for the upper part of the middle wire, and the two -small circles are to receive the wires of the last expansion. - -_Fig. 213._ represents the bottom of the exterior tube, in which -the small circles show the places where the wires of the second -expansion are fastened, and the larger ones the holes for the other -pair of wires to pass through. - -_Fig. 214._ is a cylindrical piece of brass, showing the manner in -which the lower ends of the wires of the last expansion are fastened -to it, and the hole in the middle is that by which it is pinned to -the centre of the bob. The upper ends of the two pair of wires are, -as we have observed, fastened by screwing them into the pieces which -stop up the ends of the tubes, but at the lower ends they are all -fixed as represented in _fig. 214._ The pieces represented by -_figs. 213._ and _214._ have each a jointed motion, by means -of which the fellow wires of each pair would be equally stretched, -although they were not exactly of the same length. - -The action of this pendulum is evidently the same as that of the -gridiron pendulum, as we have three lengths of steel expanding -downwards, and two of brass expanding upwards. The weight of the -pendulum has a tendency to straighten the steel rods, and the tubular -form of the brass compensation effectually precludes the fear of its -bending; an advantage not possessed by the gridiron pendulum, in which -brass rods are employed. - -Mr. Troughton, to the account he has given of this pendulum in -Nicholson’s Journal, for December, 1804, has added the lengths of -the different parts of which it was composed, and the expansions of -brass and steel from which these lengths were computed. The length of -the interior tube was 31·9 inches, and that of the exterior one 32·8 -inches, to which must be added 0·4, the quantity by which in this -pendulum the centre of oscillation is higher than the centre of the -bob. These are all of brass. The parts which are of steel are,--the -middle wire, which, including 0·6, the length of the suspension spring, -is 39·3 inches. The first pair of wires 32·5 inches; and the second -pair, 33·2 inches. The expansions used were, for brass ·00001666, -and for steel ·00000661, in parts of their length for one degree of -temperature. - - -_Benzenberg’s Pendulum._ - -This pendulum is mentioned in Nicholson’s Journal for April, 1804, and -is taken from Voigt’s Magazin für den Neuesten Zustande der Naturkunde, -vol. iv. p. 787. The compensation appears to have been -effected by a single rod of lead in the centre, of about half an inch -thick; the descending rods were made of the best thick iron wire. - -As this pendulum deserves attention from the ease with which it may -be made, and as others which have since been produced resemble it in -principle, we have given a representation of it at _fig. 215._, -where A B C D are two rods of iron wire riveted into the -cross pieces A C B D. E F is a rod of lead pinned -to the middle of the piece B D, and also at its upper extremity -to the cross piece G H, into which the second pair of iron wires -are fixed, which pass downwards freely through holes made in the cross -piece B D. The lower extremities of these last iron wires are -fastened into the piece K L, which carries the bob of the pendulum. - -To determine the length of lead necessary for the compensation, we must -recollect, as before, that the distance from the point of suspension -to the centre of the bob (speaking always of a pendulum intended to -vibrate seconds) must be 39 inches. Let us suppose the total length -of the iron wire to be 60 inches; then, from the table which we have -given, we have ·4308 for the length of a rod of lead, the expansion -of which is equivalent to that of an iron rod whose length is unity. -Multiplying 60 inches by ·4308, we have 25·84 inches of lead, which -would compensate 60 inches of iron; but this, taken from 60 inches, -leaves only 34·16 instead of 39 inches. Trying again, in like manner, -68·5 inches of iron, we find 29·5 inches of lead for the length, -affording an equivalent compensation, and which, taken from 68·5 -inches, leaves 39 inches. - -The length of the rod of lead then required as a compensation in this -pendulum is about 29-1/2 inches. - -The writer of this article would suggest another form for this -pendulum, which has the advantage of greater simplicity of construction. - -S A, _fig. 216._, is a rod of iron wire, to which the -pendulum spring is attached. Upon this passes a cylindrical tube of -lead, 29-1/2 inches long, which is either pinned at its lower extremity -to the end of the iron rod S A, or rests upon a nut firmly screwed -upon the extremity of this rod. - -A tube of sheet iron passes over the tube of lead, and is furnished at -top with a flanche, by which it is supported upon the leaden tube; or -it may be fastened to the top of this tube in any manner that may be -thought convenient. - -The bob of the pendulum may be either passed upon the iron tube -(continued to a sufficient length) and secured by a pin passing through -the centre of the bob, or the iron tube may be terminated by an iron -wire serving the same purpose. - -Here we have evidently the same expansions upwards and downwards as in -the gridiron form, given to this pendulum by Mr. Benzenberg, joined to -the compactness of Troughton’s tubular pendulum. - - -_Ward’s Compensation Pendulum._ - -In the year 1806, Mr. Henry Ward, of Blandford in Dorsetshire, received -the silver medal of the Society of Arts for the compensation pendulum -which we are about to describe. - -_Fig. 217._ is a side view of the pendulum rod when together. -H H and I I are two flat rods of iron about an eighth of -an inch thick. K K is a bar of zinc placed between them, and is -nearly a quarter of an inch thick. The corners of the iron bars are -bevelled off, which gives them a much lighter appearance. These bars -are kept together by means of three screws, O O O, which -pass through oblong holes in the bars H H and K K, and screw -into the rod I I. The bar H H is fastened to the bar of zinc -K K, by the screw _m_, which is called the adjusting screw. This -screw is tapped into H H, and passes just through K K; but -that part of the screw which passes K K has its threads turned -off. The iron bar I I has a shoulder at its upper end, and rests -on the top of the zinc bar K K and is wholly supported by it. -There are several holes for the screw _m_, in order to adjust the -compensation. - -The action of this pendulum is similar to that last described, the -zinc expanding upwards as much as the iron rods expand downwards, and -consequently the instance from the point of suspension to the centre -of oscillation remains the same. - -[Illustration: _Captn. Kater, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -Mr. Ward states that the expansion of the zinc he used (hammered zinc) -was greater than that given in the tables. He found that the true -length of the zinc bar should be about 23 inches; our computation would -make it nearly 26. - - -_The Compensation Tube of Julien le Roy._ - -We mention this merely to state that it is similar in principal to the -apparatus represented at _fig. 204._, with merely this difference, -that, instead of the steel rod being fixed to a cross piece proceeding -from the brass bar B R, it is attached to a cap fixed upon a brass -tube (through which it passes) of the same length as that of the brass -rod B R. Cassini spoke well of this pendulum, and it was used in -the observatory of Cluny about the year 1748. - - -_Deparcieux’s Compensation._ - -This was contrived in the same year as that invented by Julien le Roy. -It is represented at _fig. 218._, where A B D F -is a steel bar, the ends of which are to be fixed to the lower -sides of pieces forming a part of the cock of the pendulum. -G E I H is of brass, and stands with its extremities -resting on the horizontal part B D of the steel frame. The upper -part E I of the brass frame passes above the cock of the pendulum, -and admits the tapped wire K, to which the pendulum spring is fixed -through a squared hole in the middle. A nut upon this tapped wire gives -the adjustment for time. The spring passes through the slit in the cock -in the usual manner. - -It may be easily perceived that this pendulum is in principle the -same as that of Le Roy; the expansion of the total length of steel -A B S C downwards being compensated by the equivalent -expansion of the brass bar G E upwards. It is, however, preferable -to Le Roy’s, because the compensation is contained in the clock case. - -Deparcieux had previously published, in the year 1739, an improvement -of an imperfectly compensating pendulum, proposed in the year 1733 by -Regnauld, a clockmaker of Chalons. In this pendulum Deparcieux employed -a lever with unequal arms to increase the effect of the expansion of -the brass rod, which was too short. - -We may here remark, that all fixed compensations are liable to the same -objection, namely, that of not moving with the pendulum, and therefore -not taking precisely the same temperature. - - -_Captain Kater’s Compensation Pendulum._ - -In Nicholson’s Journal, for July, 1808, is the description of a -compensation pendulum by the writer of this article. In this pendulum -the rod is of white deal, three quarters of an inch wide, and a quarter -of an inch thick. It was placed in an oven, and suffered to remain -there for a long time until it became a little charred. The ends were -then soaked in melted sealing-wax; and the rod, being cleaned, was -coated several times with copal varnish. To the lower extremity of the -rod a cap of brass was firmly fixed, from which a strong steel screw -proceeded for the purpose of regulating the pendulum for time in the -usual manner. - -A square tube of zinc was cast, seven inches long and three quarters -of an inch square; the internal dimensions being four tenths of an -inch. The lower part of the pendulum rod was cut away on the two sides, -so as to slide with perfect freedom within the tube of zinc. To the -bottom of this zinc tube a piece of brass a quarter of an inch thick -was soldered, in which a circular hole was made nearly four tenths of -an inch in diameter, having a screw on the inside. A cylinder of zinc, -furnished with a corresponding screw on its surface, fitted into this -aperture, and a thin plate of brass screwed upon the cylinder, served -as a clamp to prevent any shake after the length of zinc necessary for -compensation should have been determined. A hole was made through the -axis of the cylinder, through which passed the steel screw terminating -the pendulum rod. - -An opening was made through the bob of the pendulum, extending to its -centre, to admit the square tube of zinc which was fixed at its upper -extremity to the centre of the bob. The pendulum rod passed through the -bob in the usual manner, and the whole was supported by a nut on the -steel screw at the extremity. - -In this form the compensation acts immediately upon the centre of the -bob, elevating it along the rod as much as the rod elongates downwards: -the method of calculating the length of the required compensation is -precisely the same as that we have before given. - -Assuming the length of the deal rod to be 43 inches, and multiplying -this by ·1313 from Table II., we have 5·64 inches for the length of -the zinc necessary to counteract the expansion of the deal. The length -of the steel screw between the termination of the pendulum rod and the -nut was two inches, and that of the suspension spring one inch. Now, -3 inches of steel multiplied by ·3682 would give 1·10 inches for the -length of zinc which would compensate the steel, and, adding this to -5·64 inches, we have 6·74 inches for the whole length of zinc required. - -In this pendulum, the length of the compensating part may be varied by -means of the zinc cylinder furnished with a screw for that purpose. -The bob of this pendulum and its compensation are represented at -_fig. 219._ - -It has been objected to the use of wooden pendulum rods, that it -is difficult, if not impossible, to secure them from the action of -moisture, which would at once be fatal to their correct performance. -The pendulum now before us has, however, been going with but little -intermission since it was first constructed: it is attached to a -sidereal clock, not of a superior description, and exposed to very -considerable variations of moisture and dryness; yet the change in its -rate has been so very trifling as to authorize the belief that moisture -has little or no effect upon a wooden rod prepared in the manner we -have described. Its rate, under different temperatures, shows that it -is over-compensated; the length of the zinc remaining, as stated in -Nicholson’s Journal 7·42 inches, instead of which it appears, by our -present compensation, that it should be 6·78 inches. - - -_Reid’s Compensation Pendulum._ - -Mr. Adam Reid of Woolwich presented to the Society of Arts, in 1809, a -compensation pendulum, for which he was rewarded with fifteen guineas. -This pendulum is the same in principle with that last described; -the rod, however, is of steel instead of wood, and the compensation -possesses no means of adjustment. This pendulum is represented at -_fig. 220._, where S B is the steel rod, a little thicker -where it enters the bob C, and of a lozenge shape to prevent the bob -turning, but above and below it is cylindrical. - -A tube of zinc D passes to the centre of the bob from below, and the -bob is supported upon it by a piece which crosses its centre, and which -meets the upper end of the tube. - -The rod being passed through the bob and zinc tube, a nut is applied -upon a screw at the lower extremity of the rod in the usual manner. If -the compensation should be too much, the zinc tube is to be shortened -until it is correct. - -The length of the zinc tube will be the same in this pendulum as in -that of Mr. Ward--about 23 inches, if his experiments are to be relied -upon. - -The objection to this pendulum appears to be its great length, which -amounts to 62 inches. We conceive it would be preferable to place the -zinc above the bob, as in the modification which we have suggested of -Benzenberg’s pendulum. - - -_Ellicott’s Pendulum._ - -It appears that the idea of combining the expansions of different -metals with a lever, so as to form a compensation pendulum, -originated with Mr. Graham; for Mr. Short, in the Philosophical -Transactions for 1752, states that he was informed by Mr. Shelton, that -Mr. Graham, in the year 1737, made a pendulum, consisting of three -bars, one of steel between two of brass; and that the steel bar acted -upon a lever so as to raise the pendulum when lengthened by heat, and -to let it down when shortened by cold. - -[Illustration: _Captn. Kater, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -This pendulum, however, was found upon trial to move by jerks, and was -therefore laid aside by the inventor to make way for the mercurial -pendulum. - -Mr. Short also says that Mr. Fotheringham, a quaker of Lincolnshire, -caused a pendulum to be made, in the year 1738 or 1739, consisting of -two bars, one of brass and the other of steel, fastened together by -screws with levers to raise or let down the bob, and that these levers -were placed above the bob. - -Mr. John Ellicott of London had made very, accurate experiments on the -relative expansions of seven different metals, which, however, will be -found to differ more or less from the results of the experiments of -others. It is not, however, from this to be concluded that Ellicott’s -determinations were erroneous; for the expansion of a metal will suffer -considerable change even by the processes to which it is necessarily -subjected in the construction of a pendulum. It is therefore desirable, -whenever a compensation pendulum is to be made, that the expansions -of the materials employed should be determined after the processes of -drilling, filing, and hammering have been gone through. - -It had been objected to Harrison’s gridiron pendulum, that the -adjustments of the rods was inconvenient, and that the expansion of the -bob supported at its lower edge would, unless taken into the account, -vitiate the compensation. These considerations, it is supposed, gave -rise to Ellicott’s pendulum, which is nearly similar to those we have -just mentioned. - -Ellicott’s pendulum is thus constructed:--A bar of brass and a bar of -iron are firmly fixed together at their upper ends, the bar of brass -lying upon the bar of iron, which is the rod of the pendulum. These -bars are held near each other by screws passing through oblong holes in -the brass, and tapped into the iron, and thus the brass is allowed to -expand or contract freely upon the iron with any change of temperature. -The brass bar passes to the centre of the bob of the pendulum, a little -above and below which the iron is left broader for the purpose of -attaching the levers to it, and the iron is made of a sufficient length -to pass quite through the bob of the pendulum. - -The pivots of two strong steel levers turn in two holes drilled in -the broad part of the iron bar. The short arms of these levers are in -contact with the lower extremity of the brass bar, and their longer -arms support the bob of the pendulum by meeting the heads of two -screws which pass horizontally from each side of the bob towards its -centre. By advancing these screws towards the centre of the bob, the -longer arms of the lever are shortened, and thus the compensation may -be readily adjusted. At the lower end of the iron rod, under the bob, -a strong double spring is fixed, to support the greater part of the -weight of the bob by its pressure upwards against two points at equal -distances from the pendulum rod. Mr. Ellicott gave a description of -this pendulum to the Royal Society in 1752, but he says the thought was -executed in 1738. As this pendulum is very seldom met with, we think it -unnecessary to give a representation of it. - - -_Compensation by means of a Compound Bar of Steel and Brass._ - -Several compensations for pendulums have been proposed, by means of -a compound bar formed of steel and brass soldered together. In a -bar of this description, the brass expanding more than the steel, -the bar becomes curved by a change of temperature, the brass side -becoming convex and the steel concave with heat. Now, if a bar of -this description have its ends resting on supports on each side the -cock of the pendulum, the bar passing above the cock with the brass -uppermost, if the pendulum spring be attached to the middle of the bar, -and it pass in the usual manner through the slit of the cock, it is -evident that, by an increase of temperature, the bar will become curved -upwards, and the pendulum spring be drawn upwards through the slit, and -thus the elongation of the pendulum downwards will be compensated. The -compensation may be adjusted by varying the distance of the points of -support from the middle of the bar. - -Such was one of the modes of compensation proposed by Nicholson. Others -of the same description (that is, with compound bars) have been brought -before the public by Mr. Thomas Doughty and Mr. David Ritchie; but as -they are supposed to be liable to many practical objections, we do not -think it requisite to describe them more particularly. - -There is, however, a mode of compensation by means of a compound bar, -described by M. Biot in the first volume of his Traité de Physique, -which appears to possess considerable merit, of which he mentions -having first witnessed the successful employment by the inventor, a -clockmaker named Martin. At _fig. 221._, S C, is the rod of -the pendulum, made, in the usual manner, of iron or steel; this rod -passes through the middle of a compound bar of brass and steel (the -brass being undermost), which should be furnished with a short tube and -screws, by means of which, or by passing a pin through the tube and -rod, it may be securely fixed at any part of the pendulum rod. - -Two small equal weights W W slide along the compound bar, and, -when their proper position has been determined, may be securely clamped. - -The manner in which this compensation acts is thus:--Suppose the -temperature to increase, the brass expanding more than the steel, the -bar becomes curved, and its extremities carrying the weights W and W -are elevated, and thus the place of the centre of oscillation is made -to approach the point of suspension as much, when the compensation is -properly adjusted, as it had receded from it by the elongation of the -pendulum rod. - -There are three methods of adjusting this compensation: the first, by -increasing or diminishing the weights W and W; the second, by varying -the distance of the weights W and W from the middle of the bar; and -the third, by varying the distance of the bar from the bob of the -pendulum, taking care not to pass the middle of the rod. The effect of -the compensation is greater as the weights W and W are greater or more -distant from the centre of the bar, and also as the bar is nearer to -the bob of the pendulum. - -M. Biot says that he and M. Matthieu employed a pendulum of this kind -for a long time in making astronomical observations in which they were -desirous of attaining an extreme degree of precision, and that they -found its rate to be always perfectly regular. - -In all the pendulums which we have described, the bob is supposed -to be fixed to the rod by a pin passing through its centre, and the -adjustment for time is to be made by means of a small weight sliding -upon the rod. - - -_Of the Mercurial Pendulum._ - -We have been guided, in our arrangement of the pendulums which we have -described, by the similarity in the mode of compensation employed; and -we have now to treat of that method of compensation which is effected -by the expansion of the material of which the bob itself of the -pendulum is composed. - -On this subject, as we have before observed, an admirable paper, from -the pen of Mr. Francis Baily, may be found in the Memoirs of the -Astronomical Society of London, which leaves nothing to be desired -by the mathematical reader. But as our object is to simplify, and -to render our subjects as popular as may be, we must endeavour to -substitute for the perfect accuracy which Mr. Baily’s paper presents, -such rules as may be found not only readily intelligible, but -practically applicable, within the limits of those inevitable errors -which arise from a want of knowledge of the exact expansion of the -materials employed. - -At _fig. 222._, let S B represent the rod of a pendulum, and -F C B a metallic tube or cylinder, supported by a nut -at the extremity of the pendulum rod, in the usual manner, and having -a greater expansibility than that of the rod. Now C, the centre of -gravity, supposing the rod to be without weight, will be in the middle -of the cylinder; and if C B, or half the cylinder, be of such -a length as to expand upwards as much as the pendulum rod S B -expands downwards, it is evident that the centre of gravity C will -remain, under any change of temperature, at the same distance from the -point of suspension S. M. Biot imagined that, in effecting this, a -compensation sufficiently accurate would be obtained; but Mr. Baily has -shown that this is by no means the fact. - -Let us suppose the place of the centre of oscillation to be at O, -about three or four tenths of an inch, in a pendulum of the usual -construction, below the centre of gravity. Now, the object of the -compensation is to preserve the distance from S to O invariable, and -not the distance from S to C. - -The distance of the centre of oscillation varies with the length of the -cylinder F B, and hence suffers an alteration in its distance from -the point of suspension by the elongation of the cylinder, although -the distance of the centre of gravity C from the point of suspension -remains unaltered. - -We shall endeavour to render this perfectly familiar. Suppose a -metallic cylinder, 6 inches long, to be suspended by a thread 36 inches -long, thus forming a pendulum in which the distance of the centre -of gravity from the point of suspension is 39 inches: the centre of -oscillation in such a pendulum will be nearly one tenth of an inch -below the centre of gravity. Now let us imagine cylindrical portions of -equal lengths to be added to each end of the cylinder, until it reaches -the point of suspension; we shall then have a cylinder of 78 inches in -length, the centre of gravity of which will still be at the distance of -39 inches from the point of suspension. But it is well known that the -centre of oscillation of such a cylinder is at the distance of about -two thirds of its length from the point of suspension. The centre of -oscillation, therefore, has been removed, by the elongation of the -cylinder, about 13 inches below the centre of gravity, whilst the -centre of gravity has remained stationary. - -Now the same thing as that which we have just described takes place, -though in a very minor degree, with our former cylinder, employed as a -compensating bob to a pendulum. The rod expands downwards, the centre -of gravity remains at the same distance from the point of suspension, -and the cylinder elongates both above and below this point; the -consequence of which is, that though the centre of gravity has remained -stationary, the distance of the centre of oscillation from the point -of suspension has increased. It is, therefore, evident that the length -of the compensation must be such as to carry the centre of gravity -a little nearer to the point of suspension than it was before the -expansion took place; by which means the centre of oscillation will be -restored to its former distance from the point of suspension. - -Let us suppose the expansions to have taken place, and that the -centre of gravity, remaining at the same distance from the point of -suspension, the centre of oscillation is removed to a greater distance, -as we have before explained. It is well known that the product obtained -by multiplying the distance from the point of suspension to the centre -of gravity, by the distance from the centre of gravity to the centre -of oscillation, is a constant quantity; if, therefore, the distance -from the centre of gravity to the point of suspension be lessened, the -distance from the centre of gravity to the centre of oscillation will -be proportionally, though not equally, increased, and the centre of -oscillation will, therefore, be elevated. We see, then, if we elevate -the centre of gravity precisely the requisite quantity, by employing -a sufficient length of the compensating material, that although the -distance from the centre of gravity to the point of suspension is -lessened, yet the distance from the point of suspension to the centre -of oscillation will suffer no change. - -The following rule for finding the length of the compensating material -in a pendulum of the kind we have been considering will be found -sufficiently accurate for all practical purposes:-- - -_Find in the manner before directed the length of the compensating -material, the expansion of which will be equal to that of the rod of -the pendulum. Double this length, and increase the product by its -one-tenth part, which will give the total length required._ We shall -give examples of this as we proceed. - - -_Graham’s Mercurial Pendulum._ - -It was in the year 1721 that Graham first put up a pendulum of this -description, and subjected it to the test of experiment; but it appears -to have been afterwards set aside to make way for Harrison’s gridiron -pendulum, or for others of a similar description. For some years past, -however, its merits have been more generally known, and it is not -surprising that it should be considered as preferable to others, both -from the simplicity of its construction, and the perfect ease with -which the compensation may be adjusted. - -We have already alluded to Mr. Baily’s very able paper on this -pendulum, and we shall take the liberty of extracting from it the -following description:-- - -At _fig. 223._ is a drawing of the mercurial pendulum, as -constructed in the manner proposed by Mr. Baily. - -“The rod S F is made of steel, and perfectly straight; its form -may be either cylindrical, of about a quarter of an inch in diameter, -or a flat bar, three eighths of an inch wide, and one eighth of an inch -thick: its length from S to F, that is, from the bottom of the spring -to the bottom of the rod at F, should be 34 inches. The lower part of -this rod, which passes through the top of the stirrup, and about half -an inch above and below the same, must be formed into a _coarse_ and -_deep_ screw, about two tenths of an inch in diameter, and having about -thirty turns in an inch. A steel nut with a milled head must be placed -at the end of the rod, in order to support the stirrup; and a similar -nut must also be placed on the rod _above_ the head of the stirrup, -in order to screw firmly down on the same, and thus secure it in its -position, after it has been adjusted _nearly_ to the required rate. -These nuts are represented at B and C. A small slit is cut in the rod, -where it passes through the head of the stirrup, through which a steel -pin E is screwed, in order to keep the stirrup from turning round on -the rod. The stirrup itself is also made of steel, and the side pieces -should be of the same form as the rod, in order that they may readily -acquire the same temperature. The top of the stirrup consists of a flat -piece of steel, shaped as in the drawing, somewhat more than three -eighths of an inch thick. Through the middle of the top (which at this -part is about one inch deep) a hole must be drilled sufficiently large -to enable the screw of the rod to pass _freely_, but without _shaking_. -The inside height of the stirrup from A to D may be 8-1/2 inches, and -the inside width between the bars about three inches. The bottom piece -should be about three eighths of an inch thick, and hollowed out nearly -a quarter of an inch deep, so as to admit the glass cylinder freely. -This glass cylinder should have a brass or iron cover G, which should -fit the mouth of it freely, with a shoulder projecting on each side, by -means of which it should be screwed to the side bars of the stirrup, -and thus be secured always in the same position. This cap should not -_press_ on the glass cylinder, so as to prevent its expansion. The -measures above given may require a slight modification, according to -the weight of the mercury employed, and the magnitude of the cylinder: -the final adjustment, however, may be safely left to the artist. Some -persons have recommended that a circular piece of thick plate glass -should float on the mercury, in order to preserve its surface uniformly -level.[7] The part at the bottom marked H is a piece of brass fastened -with screws to the front of the bottom of the stirrup, through a small -hole, in which a steel wire or common needle is passed, in order to -indicate (on a scale affixed to the case of the clock) the arc of -vibration. This wire should merely rest in the hole, whereby it may -be easily removed when it is required to detach the pendulum from the -clock, in order that the stirrup might then stand securely on its base. -One of the screw holes should be rather larger than the body of the -screw, in order to admit of a small adjustment, in case the steel wire -should not stand exactly perpendicular to the axis of motion. The scale -should be divided into _degrees_, and not _inches_, observing that with -a radius of 44 inches (the estimated distance from the bend of the -spring to the end of the steel wire) the length of each degree on the -scale must be 0·768 inch.” - -[7] The variation produced in the height of the column of mercury -(supposed to be 6-1/2 inches high) by an alteration of ± 16° in the -temperature will be only ± 1/100 of an inch, or in other words, 1/100 -of an inch will be the total variation from its _mean_ state, by an -alteration of 32° in the temperature. It is therefore probable that, in -most cases of moderate alteration in the temperature, the _centre_ only -of the column of mercury is subject to elevation and depression, whilst -the exterior parts remain attached to the sides of the glass vessel. It -was with a view to obviate this inconvenience that Henry Browne, Esq. -of Portland Place (I believe) first suggested the piece of floating -glass. - -In order to determine the length of the mercurial column necessary -to form the compensation for this pendulum, we must proceed in the -following manner:-- - -Let us suppose the length of the steel rod and stirrup together to be -42 inches. The absolute expansion of the mercury is ·00010010; but it -is not the absolute expansion, but the vertical expansion in a glass -cylinder, which is required, and this will evidently be influenced by -the expansion of the base of this cylinder. It is easily demonstrable -that, if we multiply the linear expansion of any substance (always -supposed to be a very small part of its length) by 3, we may in all -cases take the result for the cubical or absolute expansion of such -substance. In like manner, if we multiply the linear expansion by 2, we -shall have the superficial expansion. - -If we want the apparent expansion of mercury, the absolute or cubical -expansion of the glass vessel must be deducted from the absolute -expansion of the mercury, which will leave its excess or apparent -expansion. In like manner, deducting the superficial expansion of glass -from the absolute expansion of mercury, we shall have its relative -vertical expansion. Now, taking the rate of expansion of glass to be -·00000479, and multiplying it by 2, the relative vertical expansion -of the mercury in the glass cylinder will be ·00010010 - ·00000958 = -·00009052. - -The expansion of a steel rod, according to our table, is ·0000063596; -which, divided by ·00009052, gives ·0703 for the length of a column of -mercury, the expansion of which is equal to that of a steel rod whose -length is unity. - -We have now to multiply 42 inches by ·0703, which gives 2·95 inches; -and this, deducted from 42, leaves 39·1 inches; so that the length -of rod we have chosen is sufficiently near the truth. Now, double -2·95 inches, and add one tenth of its product, and we shall have 6·49 -inches for the length of the mercurial column forming the requisite -compensation. Mr. Baily’s more accurate calculation gives 6·31 inches. - -A mercurial compensation pendulum may be formed, having a cylinder -of steel or iron, with its top constructed in the same manner as the -top of the stirrup, so as to receive the screw of the rod. To find -the length of the mercurial column necessary in a pendulum of this -description (that is, with a cylinder made of steel), we must double -the linear expansion of steel, and take it from the absolute expansion -of mercury to obtain the relative vertical expansion of the mercury. -This will be ·00010010 - ·00001272 = ·00008738; and, proceeding as -before, we have ·0000063596/·00008738 = ·07279. - -Let the length of the steel rod be, as before, 42 inches. Multiplying -this by ·07279, we have 3·057, which being doubled, and one tenth -of the product added, we obtain 6·72 inches for the length of the -compensating mercurial column; which Mr. Baily states to be 6·59. - -A mercurial compensation pendulum having a rod of glass has been -employed by the writer of this article, who has had reason to think -well of its performance. Its cheapness and simplicity much recommend -it. It is merely a cylinder of glass of about 7 inches in depth, and -2-1/2 inches diameter, terminated by a long neck, which forms the -rod of the pendulum, the whole blown in one piece. A cap of brass is -clamped by means of screws to the top of the rod, and to this the -pendulum spring is pinned. - -We have unquestionable authority for saying, that the mercurial -pendulum of the usual construction, that is, with a steel rod and glass -cylinder, is not affected by a change of temperature simultaneously in -all its parts. Now, the pendulum of which we are treating being formed -throughout of the same material in a single piece, and in every part -of the same thickness, it is presumed it cannot expand in a linear -direction, until the temperature has penetrated to the whole interior -surface of the glass, when it is rapidly diffused through the mass of -mercury. M. Biot mentions that a pendulum of this kind was formerly -used in France, and expresses his surprise that it was no longer -employed, as he had heard it very highly spoken of. The writer of this -article has also used a pendulum with a glass rod, which differs from -that we have just mentioned, in having the lower end of the rod firmly -fixed in a socket attached to the centre of a circular iron plate, on -the circumference of which a screw is cut, which fits into a collar of -iron, supporting the cylinder (to which it is cemented) by means of a -circular lip. - -This arrangement, though perhaps less perfect than that we have just -described, the pendulum not being in one piece, has the advantage of -allowing a circular plate of glass to be placed upon the surface of -the mercury, as practised by Mr. Browne. To determine the length of -a column of mercury for a glass pendulum, let us suppose the glass, -including the cylinder, to be 41 inches in length. Multiplying this -by ·0529, the number taken from Table II. for a glass rod and mercury -in a glass cylinder, we have 2·17 inches for the uncorrected length -of mercury, which compensates 41 inches of glass. Suppose the steel -spring to be one inch and a half long: multiplying this by ·0703, the -appropriate decimal taken from Table II., we have 0·1, the length of -mercury due to the steel, making with the former 2·27 inches, which, -being doubled, and the product increased by its one-tenth part, we -obtain five inches for the length of the required column of mercury. - - -_Compensation Pendulum of Wood and Lead, on the Principle of the -Mercurial Pendulum._ - -If by any contrivance wood could be rendered impervious to moisture, -it would afford one of the most convenient substances known for a -compensation pendulum. It does not appear that sufficient experiments -have been made upon this subject to decide the question. Mr. Browne -of Portland Place, who has devoted much of his time and attention to -the most delicate enquiries of this kind, has, we believe, found that -if a teak rod is well gilded, it will not afterwards be affected by -moisture. At all events, it makes a far superior pendulum, when thus -prepared, to what it does when such preparation is omitted. - -Mr. Baily, in the paper we have before alluded to, proposes an -economical pendulum to be constructed by means of a leaden cylinder and -a deal rod. He prefers lead to zinc, on account of its inferior price, -and the ease with which it may be formed into the required shape; and -as there is no considerable difference in their rates of expansion, it -is equally applicable to the purpose. - -Let the length of the deal rod be taken at 46 inches. Then, to find the -length of the cylinder of lead to compensate this, we have, in Table -II., ·1427 for such a pendulum; which, being multiplied by 46, the -product doubled, and one tenth of the result added to it, gives 14·44 -inches for the length of the leaden cylinder. Mr. Baily’s compensation -gives 14·3 inches. - -[Illustration: _Captn. Kater, del._ _H. Adlard, sc._ - -_London, Pubd. by Longman & Co._] - -The rod is recommended to be made of about three eighths of an inch -in diameter: the leaden cylinder is to be cast with a hole through -its centre, which will admit with perfect freedom the cylindrical end -of the rod. The cylinder is supported upon a nut, which screws on the -end of the rod in the usual manner. This pendulum is represented at -_fig. 224._ - -Mr. Baily proposes that the pendulum should be adjusted nearly to the -given rate by means of the screw at the bottom, and that the final -adjustment be made by means of a slider moving along the rod. Indeed, -this is a means of adjustment which we would recommend to be employed -in every pendulum. - - -_Smeaton’s Pendulum._ - -We shall conclude our account of compensation pendulums with a -description of that invented by Mr. Smeaton. The compensation for -temperature in this pendulum is effected by combining the two modes, -which have been so fully described in the preceding part of this -article. - -The pendulum rod is of solid glass, and is furnished with a steel screw -and nut at the bottom in the usual manner. Upon the glass rod a hollow -cylinder of zinc, about the eighth of an inch thick, and about 12 -inches long, passes freely, and rests upon the nut at the bottom of the -pendulum rod. - -Over the zinc cylinder passes a tube made of sheet-iron. The edge of -this tube at the top is turned inwards, and is notched so as to allow -of this being effected. A flanche is thus formed, by which the iron -tube is supported, upon the zinc cylinder. The lower edge of the iron -tube is turned outwards, so as to form a base destined to support a -leaden cylinder, which we are about to describe. - -A cylinder of lead, rather more than 12 inches long, is cast with a -hole through its axis, of such a diameter as to allow of its sliding -freely, but without shake, upon the iron tube over which it passes, and -by the lower extremity of which it is supported. - -Now the zinc, resting upon the nut and expanding upwards, will raise -the whole of the remaining part of the compensation. This expansion -upwards will be slightly counteracted by the lesser expansion downwards -of the iron tube, which carries with it the leaden cylinder. The -cylinder of lead now acts upon the principle of the mercurial pendulum, -and, expanding upwards, contributes that which was wanting to restore -the centre of oscillation to its proper distance from the point of -suspension. - -This pendulum, we have been informed, does well in practice, and we are -not aware that any description of it has been before published. - -The method of calculating the length of the tubes required to form the -compensation is very simple; nothing more is necessary than to find the -length of zinc, the expansion of which is equal to that of the pendulum -rod. - -Let the pendulum rod be composed of 43 inches of glass, the spring -being an inch and a half long, and the screw between the end of the -glass rod and the nut half an inch, making in the whole two inches of -steel and 43 inches of glass. - -Now to find the length of zinc that will compensate the glass, we have, -from Table II., for glass and zinc ·2773, which, multiplied by 43, -gives 11·92 inches. In like manner we obtain as a compensation for two -inches of steel 0·74 of zinc, which, added to 11·92, gives 12·66 inches -for the total length of the zinc cylinder. - -Now if the iron tube and the lead cylinder be each made of the same -length as the zinc, and arranged as we have described, the compensation -will be perfect. - -To prove this, find, by means of the expansions given in Table I., the -actual expansion of each of the substances employed in the pendulum, -and we shall have the following results:-- - - The expansion of 12·66 inches of zinc expanding - upwards is ·0002186 - - Deduct that of 12·66 inches of iron expanding - downwards ·0000869 - -------- - Remaining effect of expansion upwards, referred - to the lower extremity of the iron tube ·0001317 - - Now, for the lead.--On the principle of the - mercurial compensation, subtract one tenth part - of the length of the cylinder, and take half - the remainder, and we shall have six inches of - lead, the expansion of which upwards is ·0000955 - -------- - Total expansion of the compensation upwards ·0002272 - -------- - To find the expansion of the rod, we have - the expansion of 43 inches of glass ·0002059 - - Of two inches of steel ·0000127 - -------- - Total expansion of the pendulum rod ·0002186 - -Agreeing near enough with that of the compensation before found. - -As we conceive we have been sufficiently explicit in our description -of this pendulum, in the construction of which no difficulty presents -itself, we think an engraved representation of it would be superfluous. - -We have hitherto treated only of compensations for temperature; but -there is another kind of error, which has been sometimes insisted upon, -arising from a variation in the density of the atmosphere. If the -density of the atmosphere be increased, the pendulum will experience -a greater resistance, the arc of vibration will in consequence be -diminished, and the pendulum will vibrate faster. This, however, is in -some measure counteracted by the increased buoyancy of the atmosphere, -which, acting in opposition to gravity, occasions the pendulum to -vibrate slower. If the one effect exactly equalled the other, it is -evident no error would arise; and in a paper by Mr. Davies Gilbert, -President of the Royal Society of London, published in the Quarterly -Journal for 1826, he has proved that, by a happy chance, the arc in -which pendulums of clocks are usually made to vibrate is the arc at -which this compensation of error takes place. This arc, for a pendulum -having a brass bob, is 1° 56′ 30″ on each side of the perpendicular; -and for a mercurial pendulum, 1° 31′ 44″, or about one degree and a -half. - -It is well known that, if a pendulum vibrates in a circular arc, the -times of vibration will vary nearly as the squares of the arcs; but -if the pendulum could be made to vibrate in a cycloid, the time of -its vibration in arcs of different extent would then remain the same. -Huygens and others, therefore, endeavoured to effect this by placing -the spring of the pendulum between cheeks of a cycloidal form. - -When escapements are employed which do not insure an unvarying impulse -to the pendulum, the force may be unequally transmitted through the -train of the clock in consequence of unavoidable imperfections of -workmanship, and the arc of vibration may suffer some increase or -diminution from this cause. To discover a remedy for this is certainly -desirable. - -The writer of this article some years ago imagined a mode, which he -believes has also been suggested by others, by which he conceived a -pendulum might be made to describe an arc approaching in form to that -of a cycloid. The pendulum spring was of a triangular form, and the -point or vertex was pinned into the top of the pendulum rod, the base -of the triangle forming the axis of suspension. Now it is evident that -when the pendulum is in motion, the spring will resist bending at the -axis of suspension, with a force in some sort proportionate to the base -of the triangle. - -Suppose the pendulum to have arrived at the extent of its vibrations; -the spring will present a curved appearance; and if the distance from -the point of suspension to the centre of oscillation be then measured, -it will evidently, in consequence of the curvature of the spring, be -shorter than the distance from the point of suspension to the centre of -oscillation, measured when the pendulum is in a perpendicular position, -and consequently when the spring is perfectly straight. - -The base of the triangle may be diminished, or the spring be made -thinner; either of which will lessen its effect. We cannot say how this -plan might answer upon further trial, as sufficient experiments were -not made at the time to authorize a decisive conclusion. - -We have thus completed our account of compensation pendulums; but -before we conclude, it may not be unacceptable if we offer a few -remarks on some points which may be found of practical utility. - -The cock of the pendulum should be firmly fixed either to the wall or -to the case of the clock, and not to the clock itself, as is sometimes -done, and which has occasioned much irregularity in its rate, from the -motion communicated to the point of suspension. We prefer a bracket or -shelf of cast iron or brass, upon which the clock may be fixed, and the -cock carrying the pendulum attached to its perpendicular back. This -bracket may either be screwed to the back of the clock-case, or, which -is the better mode, securely fixed to the wall; and if the latter be -adopted, the whole may be defended from the atmosphere, or from dust, -by the clock-case, which thus has no connection either with the clock -or with the pendulum. - -The point of suspension should be distinctly defined and immovable. -This may be readily effected, after the pendulum shall have taken the -direction of gravity, by means of a strong screw entering the cock -(which should be very stout) on one side, and pressing a flat piece of -brass into firm contact with the spring. - -The impulse should be given in that plane of the rod which coincides -with the plane of vibration passing through the axis of the rod. If the -impulse be given at any point either before or behind this plane, the -probable result will be a tremulous unsteady motion of the pendulum. - -A few rough trials, and moving the weight, will bring the pendulum near -its intended time of vibration, which should be left a little too slow; -when the bob should be firmly fixed to the rod, if the form of the -pendulum will admit of it, by a pin or screw passing through its centre. - -The more delicate adjustment may be completed by shifting the place of -the slider with which the pendulum is supposed to be furnished on the -rod. - -Mr. Browne (of whom we have before spoken) practises the following very -delicate mode of adjustment for rate, which will be found extremely -convenient, as it is not necessary to stop the pendulum in order to -make the required alteration. Having ascertained, by experiment, the -effect produced on the rate of the clock, by placing a weight upon the -bob equal to a given number of grains, he prepares certain smaller -weights of sheet-lead, which are turned up at the corners, that they -may be conveniently laid hold of by a pair of forceps, and the effect -of these small weights on the rate of the clock will be, of course, -known by proportion. The rate being supposed to be in defect, the -weights necessary to correct this may be deposited, without difficulty, -upon the bob of the pendulum, or upon some convenient plane surface, -placed in order to receive them: and should it be necessary to remove -any one of the weights, this may readily be done by employing a -delicate pair of forceps, without producing the slightest disturbance -in the motion of the pendulum. - - - - -INDEX. - - - A. - - Action and reaction, 34. - - Aeriform fluids, 26. - - Animalcules, 12. - - Atmosphere, impenetrability of, 22. - Compressibility and elasticity of, 23. - - Atoms, 6. - Coherence of, 7. - - Attraction, magnetic, of gravitation, 8, 50, 64. - Molecular or atomic, 69. - Cohesion, 70. - - Attwood, machine of, 92. - - Axes, principal, 138. - - Axis, mechanical properties of, 128. - - - B. - - Balance, 279. - Of Bates, 288. - Use of, 289. - Danish, 299. - Bent-lever of Brady, 301. - - Bodies, 2. - Lines, surfaces, edges, area, length of, 4. - Figure, volume, shape of, 5. - Porosity of, 17. - Compressibility of, 18. - Elasticity, dilatibility of, 19. - Inertia of, 27. - Rule for determining velocity of; motion of two bodies after - impact, 38. - - - C. - - Capillary attraction, 73. - - Capstan, 179. - - Cause and effect, 7. - - Circle of curvature, 99. - - Cog, hunting, 191. - - Components, 51. - - Cord, 163. - - Cordage, friction and rigidity of, 260. - - Crank, 241. - - Crystallisation, 14. - - Cycloid, 158. - - - D. - - Damper, self-acting, 234. - - Deparcieux’s compensation pendulum, 319. - - Diagonal, 51. - - Dynamics, 160. - - Dynamometer, 305. - - - E. - - Electricity, 76. - - Electro-magnetism, 76. - - Equilibrium, neutral, instable, and stable, 118. - - - F. - - Figure, 5. - - Fly-wheel, 239. - - Force, 6. - Composition and resolution of, 49. - Centrifugal, 98. - Moment of; leverage of, 135. - Regulation and accumulation of, 224. - - Friction, effects of, 96. - Laws of, 264. - - - G. - - Governor, 227. - - Gravitation, attraction of, 77. - Terrestrial, 84. - - Gravity, centre of, 107. - - Gyration, radius of, centre of, 137. - - - H. - - Hooke’s universal joint, 252. - - Hydrophane, porosity of, 18. - - - I. - - Impact, 39. - - Impulse, 65. - - Inclined plane, 163–209. - - Inclined roads, 211. - - Inertia, 27. - Laws of, 32. - Moment of, 137. - - - J. - - Julien le Roy, compensation tube of, 319. - - - L. - - Lever, 163. - Fulcrum of; three kinds of, 167. - Equivalent, 176. - - Line of direction, 110. - - Liquids, compressibility of, 24. - - Loadstone, 68. - - - M. - - Machines, simple, 160. - Power of, 175. - Regulation of, 225. - - Magnet, 68. - - Magnetic attraction, 8. - - Magnetism, 76. - - Magnitude, 4. - - Marriott’s patent weighing machine, 305. - - Materials, strength of, 272. - - Matter, properties of, 2. - Impenetrability of, 4. - Atoms of; molecules of, 6. - Divisibility of, 9. - Examples of the subtilty of, 12. - Limit to the divisibility of, 13. - Porosity of; density of, 17. - Compressibility of, 18. - Elasticity and dilatability of, 19. - Impenetrability of, 22. - Inertia of, 27. - - Mechanical science, foundation of, 16. - - Metronomes, principles of, 153. - - Molecules, 6. - - Motion, laws of, 46. - Uniformly accelerated, 87. - Table illustrative of, 90. - Retarded; of bodies on inclined planes and curves, 94. - Rotary and progressive, 127. - Mechanical contrivances for the modification of, 245. - Continued rectilinear; reciprocatory rectilinear; continued - circular; reciprocating circular, 246. - - - N. - - Newton, method of, for determining the thickness of transparent - substances, 10. - Laws of motion of, 46. - - - O. - - Oscillation, 129. - Of the pendulum, 145. - Centre of, 152. - - - P. - - Parallelogram, 51. - - Particle, 6. - - Pendulum, oscillation or vibration of, 145. - Isochronism of, 147. - Centre of oscillation of, 152. - Of Troughton, 284. - Compensation, 307. - Of Harrison, 313. - Tubular, of Troughton, 314. - Of Benzenberg, 316. - Ward’s compensation, 318. - Captain Kater’s compensation, 320. - Reid’s; Ellicott’s compensation, 322. - Steel and brass compensation, 324. - Mercurial, 326. - Graham’s mercurial, 329. - Wood and lead, 334. - Smeaton’s, 335. - - Percussion, 130. - Centre of, 144. - - Planes of cleavage, 15. - - Porosity, 17. - - Power, 161. - - Properties, 2. - - Projectiles, curvilinear path of, 82. - - Pulley, 164. - Tackle; fixed, 198. - Single moveable, 200. - Called a runner; Spanish bartons, 205. - - - R. - - Rail-roads, 213. - - Regulating damper, 233. - - Regulators, 227. - - Repulsion, 8. - Molecular, 74. - - Resultant, 51. - - Rose-engine, 250. - - - S. - - Salters, spring balance of, 305. - - Screw, 209. - Concave, 217. - Micrometer, 223. - - Shape, 5. - - Siphon, capillary, 73. - - Spring, 304. - - Statics, 160. - - Steelyard, 294. - C. Paul’s, 296. - Chinese, 299. - - - T. - - Table, whirling, 99. - - Tachometer, 234. - - Tread-mill, 179. - - - V. - - Velocity, angular, 99. - - Vibration, 129. - Of the pendulum, 145. - Centre of, 152. - - Volume, 5–17. - - - W. - - Watch, mainspring of; balance wheel of, 195. - - Water regulator, 229. - - Wedge, 209. - Use of, 215. - - Weight, 161–291. - - Weighing machines, 278. - For turnpike roads, 302. - By means of a spring, 303. - - Wheels, spur, crown, bevelled, 189. - Escapement, 194. - - Wheel and axle, 177. - - Wheel-work, 176. - - Winch, 179. - - Windlass, 178. - - Wollaston’s wire, 10. - - - Z. - - Zureda, apparatus of; Leupold’s application of, 251. - - - END OF MECHANICS. - - LONDON: - SPOTTISWOODES and SHAW - New-street-Square. - -*** END OF THE PROJECT GUTENBERG EBOOK A TREATISE ON MECHANICS *** - -Updated editions will replace the previous one--the old editions will -be renamed. - -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the -United States without permission and without paying copyright -royalties. 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