ELEMENTS or ANALYTICAL MECHANICS, BY W. H. C. BARTLETT, L L. I. PROFESSOR OF NATURAL AND EXPERIMENTAL PIITLOSOPHY IN THE UNiTHD STATES MILITARY ACADEMY AT WEST POINT AND &AUTHOR OF ELEMENTS OF SYNTHETICAL MECHANICS, ACOUSTICS, OPTICS, AND SPHERItCAL ASTRONOMY. EIGHTH EDITIONI REVISED, CORRECTED, AND ENLARGED. NEW YORK: A. S. BARNES & CO., 111 & 113 WILLIAM STREET, (CORNER OF JOHN STREET.) O0LD BY BOOKSELLERS, GENERALLY, THROUGHOUT THE UNITED STAT]S 18 6 6. Entered accordine to Act of Congress, in the year One Thousand Eight Hundred and Fifty-eight, BY W. R. C. BARTLETT, m to Clerk's Office of the District Court of the United States for lthe Southern District of New-York. G. W. WOOD, Printcr, John-street, cor. Dutch, TO COLONEL SYLVANUS TIIAYER, OF THE CORPS OF ENGINEERS, AND LATE SUPERINTENDENT OF T11t UNITED STATES MIL.,ITARtY ACADEMY, IS MOST RESPE''tTi.,:UVLY AND) AFFECTIONNAfELY DEDICATED, 1.' (it A't' tl'U')E FOR Tr iE PRIVILEGES ITS AUTHOR IfAS ENJOYED UNDER A. SYSTEM OF INSTRUOTrJ'N AN[) V:R':EIINMENT WtICII' GAVE VITALITr TO THEl ACAI)E''Y,:t4 ND OF WHICH HE IS THE FATHEFR P Pt EF ACE. IT is now six years since the publication of the first edition of the present work. During this interval, it has been corrected and amended according to the suggestions of daily experience in its use as a text-book. It now appears with an additional part, under the head, MECHANICS OF iMOLECULES; and this completes —in so far as he may have succeeded in its execution-the design of the author to give to tile classes committed to his instruction, in the Military Academy, what has appeared to him a proper elementary basis for a systematic study of the laws of matter. The subject is the action of forces upon bodies,-the source of all physical phenomena-and of which the sole and sufficient foundation is the comprehensive fact, that all action is ever accompanied by an equal, contrary, and simultaneous reaction. Neither can have precedence of the other in the order of time, and from this comes that character of permanence, in tihe midst of endless variety, apparent in the order of nature. A mathematical formula which shall express the laws of this antagonism will contain the whole subject; and whatever of specialty may mark our perceptions of a particular instance, will be found to have its origin in corresponding peculiarities of physical condition, distance, place, and time, which are the elements of this formula. Its discussion constitutes the study of'Mechanics. All phenomena in which bodies have a part are its legitimate subjects, and no. form of matter under extraneous influences is exempt from its iv PREFACE. scrutiny. It embraces alike, in their reciprocal action, the gigantic and distant orbs of the Celestial regions, and the proximate atoms of the ethereal atmosphere which pervades all space and establishes an unbroken continuity upon which its, Divine Architect and Author may impress the power of HIis will at a single point and be felt everywhere. Astronomy, terrestrial physics, and chemistry are but its specialties; it classifies all of human knowledge that' relates to imlert matter into groups of phenomena, of which the rationale is in a comnmoil principle; and in the hands of those gifted with the priceless boon of a copious mathematics, it is a key to external nature. The order of treatment is indicated by the heads of ME]CeIANICS OF SOLIDS, of FLUIDms, and of MOLECULES, —an order sug'gested by differences of physical constitution. The author would acknowledge his obligation to the works of many eminent writers, and particularly to those of M. Lagrange, M. Poisson, M. Couchey, iM. Fresnel, IM. Lam6, Sir William R. Hamilton, the Rev. Baden Powell, Mr. Airy, Mr. Pratt, and ]Mr. A. Smith. WEST POINT, 1858. CONTENTS. I N TROD U CTION. PAGS Preliminary Definitions................................................. 11 Physics of Ponderable bodies....................................... 14 Primary Properties of Bodies...............................,, 15 Secondary Properties...................................... 16 Force.~~~~~~~~.. ~......................... 20 Physical Constitution of Bodies.......................................... 22 PART I. MECHANICS OF SOLIDS. Space, Time, Motion, and Force-........................................ 31 Work.................................................................. 38 Varied Motion............................................... 42 Equilibrium............................................................. 46 The Cord.............................................................. 47 The Muffle............................................................. 48 Equilibrium of a Rigid System —Virtual Velocities......................... 50 Principle of D'Alembert..................................... 55 Free Motion......................................................... 58 Composition and Resolution of Oblique Forces............................ 62 Composition and Resolution of Parallel Forces............................ 75 Work of Resultant and of Component Forces.............................. 82 Moments..... 84 Composition and Resolution of Moments................................. 88 Translation of General Equations...................................... 91 Centre of Gravity...................................................... 93 Centre of Gravity of Lines............................................ 97 Centre of Gravity of Surfaces................. 102Centre of Gravity of Volumes........................................... 109 Centrobaryc Method............................................... 114 Centre of Inertia........,.,.,.., 116 Motion of the Centre of Inertia.................................... I18 NV CONTENTS. PAM1. Motion of Translation....................................,....., 120 General Theorem of Work and Living Force...................1......... 20 Stable and Unstable Equilibrium............................,,... 123 Initial Conditions, Direct and Reverse Problem.......................... 126 Vertical Motion of Heavy Bodies....................................... 127 Projectiles......................................................... 185 Rotary Motion....................................... 165 Moment of Inertia, Centre and Radius of Gyration....................... 175 Impulsive Forces...................................................... 169 Motion under the Action of Impulsive Forces........................... 171 Motion of the Centre of Inertia.................................... 171 Motion about the Centre of Inertia.. 173 Angular Velocity............................................... 174 Motion of a System of Bodies........................... 179 Motion of Centre of Inertia of a System.............................. 180 Motion of the System about its Centre of Inertia........................ 181 Conservation of the Motion of the Centre of Inertia of a System.......... 181 Conservation of Areas................................................. 183 Invariable Plane..............,.................... 185 Principle of Living Force.............................................. 186 Planetary Motions.................................................... 188 Laws of Central Forces............................................... 190 Orbits................................................................ 196 System of the World................................................. 198 Consequences of Kepler's Laws...................................... 198 Perturbations................................................... 203 Coexistence and Superposition of Small Motions.......................... 205 Universal Gravitation.................................................. 206 Impact of Bodies................................................. 211 Constrained Motion on a Surface....................................... 218... " on a Curve.................................... 220 ".." about a Fixed Point............................. 246 t" " about a Fixed Axis................................. 247 Compound Pendulum.................................................. 249 Ballistic Pendulum........2........5............................... 25.9 Gun Pendulum..................................................... 261 PART II. MECHANICS OF FLUIDS. Introductory Remarks................................................. 263 Mariotte's Law....................................................... 265 Law of Pressure, Density, and Temperature..................... 266 Equal Transmission of Pressure.................................. 268 i4otion of Fluid Particles..................................... 270 Equilibrium of Fluids............................................ 280 Pressure of Heavy Fluids...................................... 289 Equilibrium and Stability of Floating Bodies..................... 295 CONTENTS. vii PAGE. Specific Gravity......................................... 304 Atmospheric Pressure. 316 Barometer........................................................... 317 Motion of Heavy Incompressible Fluids in V-ssels....................... 326 Motion of Elastic Fluids in Vessels..................................... 338 PART III. MECHANICS OF MOLECULES. Introductory Remarks.................................. 345 Periodicity of Molecular Condition..................................... 345 Waves.............................................................. 352 Wave Function...................................... 353 Wave Velocity......................................................... 360 Relation of Wave Velocity to Wave Length.............................. 363 Surface of Elasticity.................................................. 365 Wave Surface......................... 367 Double Wave Velocity.................. 372 Umbilic Points..................................................... 375 Molecular Orbits........................ 378 Reflexion and Refraction....................................... 381 Resolution of Living Force by Deviating Surfaces..................... 384 Polarization by Reflexion and Refraction 388 Diffusion and Decay of Living Force.................................... 394 Interference............................................ 395 Inflexion......................... 400 PART IV. APPLICATIONS TO SIMPLE MACHINES, PUMPS, &c. General Principles of all Machines..................................... 405 Friction.............................................................. 407 Stiffness of Cordage.................................................... 415 Friction on Pivots................ 420 Friction on Trunnions........................................... 425 The Cord as a Simple Machine............................. 429 The Catenary......................................................... 439 Friction between Cords and Cylindrical Solids.......................... 441 Inclined Plane.........................................443 The Lever........................................................... 446 Wheel and Axle................................................... 449 Fixed Pulley........................................................... 451 Movable Pulley....................................................... 454 The Wedge.................................................. 460 The Screw........................................................ 464 Viii CONTENTS. PAGE. P u mps.. 46........................................................... 469 The Siphon...................................................... 479 The Air-Pump...................................................... 481 TABLES. Table I.-The Tenacities of Different Substances, and the Resistances which they oppose to Direct Compression......................... 488 " II.-Of the Densities and Volumes of Water at Different Degrees of Heat (according to Stampfer), for every 24 Degrees of Fahrenheit's Scale.............................................. 490 III.-Of the Specific Gravities of some of the most Important Bodies. 491 IV.-Table for finding Altitudes with the Barometer............... 494 V.-Coefficient Values, for the Discharge of Fluids through thin Plates, the Orifices being Remote from the Lateral Faces of the Vessel. 496 " VI.-Experiments on Friction, without Unguents. By M. Morin...... 497 VII.-Experiments on Friction of Unctuous Surfaces. By M. Morin.... 500 "VIII.-Experiments on Friction with Unguents interposed. By M. Morin. 501 " IX.-Friction of Trunnions in their Boxes......................... 503 X.-Of Weights necessary to Bend different Ropes around a Wheel one Foot in Diameter.............................. 504 The Greek Alphabet is here inserted to aid those who are not already familiar with it, in reading the parts of the text in which its letters occur. letters. Names. Letters. Names A a Alpha N v Nu B A Beta g Xi y r Gamma O o Omicron.A 6 Delta I w r Pi E s Epsilon P Pg Rho Z Ad- Zeta Y: d Sigma IH ~ Eta T -7 Tau o. i: Theta T u, Upsilon I l Iota, q Phi K x Kappa X X Chi A X Lambda + Psi MPa Mu 1 Omega ELEMENTS or ALYTICAL MEtCHANI INTRODUCTION. PHYSICAL SCIENCE..-The term Nafture, is employed to signify all the bodies of the univse, collectively. Of the existence of bodies, we are rendered conscious by the impressions they make upon the mind through the senses. The condition of every body is subject to a variety of chages. These changes are brought about by agents external to the. bodies themselves; and to -investigate nature with reference to these changes and their causes' is the object of Physical Science. PHYSICAL PROPERTIES. ~2.-Physical Properties, are those external signs by which the existence of bodies are made known to us through the medium of the senses. These properties are either primary or -secondary. PRIMARY PROPERTIES. ~3. —A Primary Pro~perty is that without which the existence of the body cannot be conceived. There are two of these-'Eaten...sion and.hmpcnetrability. 14 ELEMENTS OF ANALYTICAL MECHANICS. Extension is that by which every body occupies a limited portion of space. From it the body derives its figure and volume. Impenetrability is that which prevents two bodies from occupying the same space at the same time. It determines a body's identity. A body, then, is any thing which has extension and impenetrability. SECONDARY PROPERTIES. ~ 4.-Secondary Properties are those which are not necessar to a conception of a body's existence, though all bodies may, and indeed do, possess them in a greater or less degree. They are Copressibility, Expansibility, Porosity, Divisibility, and Elasticity. 1.-Compressibility is that property bywhich made to occupy a smaller, and expansibility that by which it may be made to occupy a larger space, without, in either case, altering the quantity of its matter. 2.-7Porosity is that property by which a body does not fill all the space within its exterior bounday but leaves holes o'rpores between its elements. in'many cases the pores are visible to the -naked eye; in others they are only seen by the aid of the microscope; and when so.minute as to elude the power of this instrument, their existence may be inferred from. experiment. Sponge, cork, wood, bread, are bodies whose pores are obvious, to unassisted vision. The humanD skin appears full of them, when viewed with the magnifying glass. The pores of one body are filled with some other body, and the pores of this with a third, as in the case of a sponge containing water,.and the water containing air, and so on till we come to the most subtle of substances, ether, which pervades all bodies and all spa~ie. INTRODUCTION. 15 3.-Diviibility is that property in consequence of Which, by various mechanical means such as beating, pounding, grinding, a body may be reduced to fragments, homogeneous to each other, and to the entire mass. By the aid of mathematical processes, the mind may be led to adit the infiite divisibility of bodies, though their practical division by mechanical means, is subject to limitation. Many examples, however prove that this process may be carried to an incredible extent. Natre furnishes numerous instances of objects, whose existence can only be detected by means of the most acute senses assisted by the most powerful artificial aids. Mechanical subdivisions for purposes connected with the arts are exemplified in the grinding of corn, the pulverizing of sulphur charcoal and saltpetre, for the manufacture of gunpowder; and omopathy affords a remarkable instance of the extended application of this property of bodies. In common gold lace, a silver thread is covered with gold so attenuated, that the quantity on a foot of thread weighs -less than of a grain. An inch of such thread will therefore contain of a grain of gold; and if the'inch be divided into 100 equal parts, each of which would be distinctly visible, the quantity of the precious metal in each of such pieces would be I of a grain. One of these particles examined through a mIicroscope of a magnifying power equal to 500, will appear 500 times as long, and the' gold covering it will be visible, having been divided into. 3,600,000,'00 parts, each of which exhibits all the characteristics of this'metal. Dyes are likewise susceptible of an incredible divisibility. With 1 grain of blue carmine, 10 lbs. of water ma~y be tinged blue. These 10 lbs. of water contain about 617,000 drops. Supposing that 100 particles of carmine are required in each drop to produce a uniform tint, it follows that this one grain of carmine has been subdivided 62 millions of times. 16 ELEMENTS OF ANALYTICAL MECHANICS. According to Biot, the thread by whih a spider suspends herself is composed of more than 5000 single threads. Our blood, which appears like a uniform red mass consists of small red globules swimming in a transarent fluid called serum. The diameter of one of these globules does not exceed the 4000th part of an inch: whence it follows that one drop of blood, such as would hang from the point of a needle, contains at least one million of these globules. But more surprising than all, is the microcosm of organized nature in the Infusoria. Of these creatures, which for the most part we can see only by the aid of the microscope there exist many species so small that millions piled on each other would not equal a single grain of sand, and thousandsmight swim at once through the eye of the finest needle. The coats-of-mail and shells of these animalcules exist in such prodigious quantities, that extensive strata of rocks, as, for instance, the smooth slate near Bilin, in Bohemia, consist almost entirely of them. By microscopic mea~surements, 1 cubic line of this slate contains about 23 millions, and 1 cubic inch about 41,000 millions of these animals. As a cubic inch of this slate weighs 220 grains, 187 millions of these shells must go to a grain, each of whic would consequently weigh about the TIT m11illionth part of a grain. Conceive furher, that each of these anfimalcules, as microscopic'investigation has proved, has its limbs, entrails, &C., the possibility vanishes of our forming the most remote conception of the dimensions of these organic forms. In. cases where the finest'instrumnents are -unable to give the least aid in estimating the minuteness of bodies,-in other words, when bodies evade the perception of our sight and touch,-our olfactory nerves frequently detect the presence of matter in the atmosphere, of which no chemical analysis could afford us the slightest intimation. Thus, for instance, a single ganof muskc diffuses in a large and airy room a powerful scent, that frequently lasts for years; INTRODUCTION. and papers laid near musk will make a voyage to the East ndies and back without losing the smell. Imagine how many particles of musk mst radiate from such a body every second, in order to render the scent perceptible in all directions. 4.-Eastieity is that property by which a body resumes of itself its fgure and dimensions, when these have been changed or altered by any extraneous cause. Different bodies possess this property in very different degrees, and retain it with very unequal tenacity. Glass, tempered steel, ivory and whalebone, are among the more elastic solids. All fluids are highly elastic. REST, MOTION, FORCE. 5.-The state of a body by which it continues in the same Place is called rest; that by which it passes from one place to another is called mzotion; and whatever changes the state of a body or the elements of a body, with respect to rest or motion, is called~force. The existence of force is inferred from the changes, with, respect to rest or motion, which all bodies and their internal elements are found to be.continuallyunndergoing. Its nature, or in what it consists, is unknown. CONSTITUTION OF BODIES. ~6.-Several hypotheses have been proposed to explain the constitution of a body, and the mode of its formation. The most remarkable of these was by Boscovi'ch, about the middle of the last. century. According to this eminent philosopher: 1. All matter consists of indivisible and inextended atoms. 2. These atoms are endowed with attractive and repulsive. forces, varying' both in intensity and direction by a change of dis tance, so that at one distance two atoms attract one another, and~ at another distance they repel. 2 18 ELEMENTS OF ANALYTICAL MECHANICS. 3. This law of variation is the same in all atoms. It is, therefore, mutual; for the distance of atom a from atom b, being the same as that of b from a, if a attract b, b must attract a with precisely an equal force. 4. At all considerable or sensible distances, these mutual forces are attractive and sensibly proportional to the square of the distance inversely. It is the attraction called gravitatiom. 5. At the small and insensible distances in which sensible contact is observed, and which do not exceed the 1000th or 150Oth part of an inch, there are many alternations of attraction and repulsion, according as the distance of the atoms is changed. Consequently, there are many situations within this narrow limit, in which two atoms neither attract nor-repel. 6. The force which is exerted between two atoms when their distance is diminished without end, and is just vanishing, is an insuperable repulsion, so that no force whatever can press two atoms into mathematical contact. Such, according to Boscovich, is the constitution of a material atom and the whole of its constitution, and the immediate efficient cause of all its properties. C~~ C Boscovich represents his law of atomical action by what may be called an exponential curve. Let the distance of two atoms be estimated on the line A C' C, A being the situation of one of th4m, while the other is placed anywhere on this line. When placed at i, for example, we may suppose that it is attracted by INTRODUCTION. 19 A, with a certain intensity. We represent this intensity by the length of the line i, perpendicular to A C, and express the direction of the force, namely, from i to A, because it is attractive, by placing above the axis A C. Should the atom be at in, and be repelled by A, we express the intensity of repulsion by mn n, and its direction from n towards G by placing m n below the axis.. This maybe supposed for every point on the axis, and a curve drawn through the extremities of all the perpendicular ordinates, ill be the exponential curveor scale of force. Asthere are supposed a great many alternations of attractions and repulsions, the curve must consist of many branches lying on opposite sides of the axis, and must therefore cross it at C', D' D &c. and at G. All these are supposed to be contained within a very small fraction of an inch. Beyond this distance, which terminates at G, the force is always attractive and is called the force of grcavitat;ion, the maximium intensity of which occurs at g, and is expesdby the length of the ordinate G'g. Further on, the ordinates are sensibly proportional to -the square of their distances from A, inversely. The branch G' G" has the line A C', therefore, for its asymptote. Within the limit A C' there'is repulsion, which becomes infinite, when the distance from A is zero; whence the branch 6" D' has the perpendiculiar axis, A y, for its asymptote. An atom being placed at G, and then, disturbed so as to move it in the direction towards A, will be repelled, the ordinates of the curve being below the axis; if disturbed so as to move it from A, it will be. attracted, the corresponding'ordinates being above the axis. The point G is. therefore a position in which the atoin is neither attracted -nor repelled, and to which it will tend to return when slightly removed in either direction, and is called the If time atom be at C, or 6"', &c., and be moved ever so, little 20 ELEMENTS OF ANALYTICAL MECHANICS. G' C DAnI towards A, it will be repelled, and when the disturbing cause is removed, will fly back; if moved from A, it will be attracted and return. Hence C', C, &c., are positions similar to G, and are called limgnits of cohesion, a being called the last limit of cohesion. An atom situated at any one of these points will, with that at A, constitute a pernanent molecule of the simplest kind. On the contrary, if an atom be placed at D', or D", &c., and be then slightly disturbed in the direction either from or towards A, the action of the atom at A will cause it to recede still further from its first position, till it reaches a limit of cohesion. The points D', D", &c., are also positions of indifference, in which the atom will be neither attracted nor repelled by that at A, but they differ from C', C", G, &c., in this, that an atom being ever so little removed fi-om one of them has no disposition to return to it again; these points are called limits of dissol5ion. An atom situated in one of them cannot, therefore, constitute, with that at 4, a permanent molecule, but the slightest disturbance will destroy it. It is easy to infer, from what has been said, how three, four, &c., atoms may combine to form molecules of different orders of complexity, and how these again may be arranged so as by their action upon each other to form particles. Our limits will not permit us to dwell upon these points, but we cannot dismiss the subject without suggesting one of its most interesting consequgnces. According to the highest authority, the SUN and other heavenly bodies have been formed by the gradual subsidence of a vast INTRODUCTION. 21 neula towards its centre. Its molecules, forced by their gravi, tating action within their neutral limits, are in a state of tension, which is the more intense as the accumulation is greater; and the mmolecular agitations the sun caused by thile successive depositions at its surface make this body, in consequence of its vast size, a perpetual fountain of that incessant stream of ethereal waves which constitute the essence of light and heat. The internal heat of the earth has the same explanation. All bodies would appear self-luminous were the acuteness of our sense of sight increased beyond its present limit in the same proportion that the sun exceeds them in size. The sun far transcends all the other bodies of our systemin regard to heat and light, and is in a state of incadecence because of the mode of its formation and of its vastly greater dimensions. 2.-The molecular forces, here considered, are the effective causes which determine a body to be a. solid liquid, or gas. If the attractions'prevail over the -repulsions, the body is.solid; if these antagonistic forces be equal, it is liquid; and if the repulsions prevail over the attractions, it is a a 3.- -The molecular forces may so act upon, the elements of' dissimilar bodies as to cause a new combination or union of their atoms. This mayas prdce a separation bet~weeu the com.bined atoms or molecules, in such manner as to entirely change the individual properties of the bodies. Such efforts of the miolecular forces are called c'hemicral action; and the disposition to exert these'efforts, helmical afniy 4.-Beyond the. last limit of gravitation, atoms attract each other: hence, all the atoms of one body attract each atom of another, and vice versa: thus giving rise to attractions between bodies of sensible magnitudes through sensible, distances. Tho 22 ELEMENTS OF ANALYTICAL MECHANICS. intensities of these attractions are proportional to the number of atoms in the attracting body directly, and to the square of the distance between the bodies inversely. 5.-The term universal gravitation is applied to this force, when it is intended to express the action of the heavenly bodies on each other; and that of terrestrial gravitation or simple gravity, when we wish to express the action of the earth upon the bodies forming with itself one whole. The force is always of the same kind, however, and varies in intensity only by reason of a difference in the number of atoms and their distances. Its effeet is always to generate motion, when the bodies are free to move. Gravity, then, is a property common to all terrestrial bodies, since they constantly exhibit a tendency to approach the earth and its centre. In consequence of this tendency all bodies possess weight, and, unless supported, fall to the surface of the earth; and if prevented by any other bodies from doing so, they exert a pressure on these latter. This is one of the most important properties of terrestrial bodies, and the cause of many phenomena, of which a fuller account will be given hereafter. DENSITY. ~ 7.-Density is a term employed to express the greater or less proximity of a body's atoms. The relative densities of different bodies must, therefore, be proportional to the number of atoms they contain under equal volumes. The weights of bodies being proportional to the number of their atoms, the density of any body is measured by the quotient arising firom dividing its weight by tihe weight of an equal volume of some other body, assumed as a standard, and whose density is regarded as unity. The density of pure water, at the temperature of 38~,75 Fahren INTRODUCTION. 23 heit, is assumed as the unit, the water possessing its maximum density at that temperature. MASS. ~ 8.-The Mass of a body is the quantity of matter it contains; and this being proportional to its weight, the mass of a body may be measured by the quotient arising from dividing its weight by the weight of some other body assumed as the unit of mass. A cubic foot of distilled water, at its maximum density, may be assumed as the unit of mass. And, therefore, a body whose mass is expressed by any number, say 20, will contain twenty times the matter contained in a cubic foot of distilled water at its greatest density. If the mass of a body be denoted by X, its weight by 1V and that of a unit of mass by g, then will W.......... (1) If V denote the body's volume, and D its density; then will X= V. D..... (2) and by substitution above, W= V.D.g.....3) The masses of bodies are so constantly in view in discussing and applying physical principles, as to make it important to understand well the method of getting their numnerical values. Equation (1) lnay be written IF_ g andi in which TV and g must be expressed in terms of the same unit. But g may have two values, very different in kind. It 24 ELEMENTS OF ANALYTICAL MECHANICS. may be expressed in pounds, or any other unit of weight, or in feet, or any other unit of length. In the first case, the body assumed as the unit of mass and that whose mass is desired are simply weighed, and the ratio of the weights taken. In the second case, the body assunmed as the unit of mass is permitted to fall in vacuo, and the velocity its own weight can generate in it, in one second of time, ascertained. A cubic foot of pure water, at its maximum density, weighs 62,3791 pounds avoirdupois, and the measure of a body's mass is given by IF 62,3791 in which W must be expressed in avoirdupois pounds. The velocity which the weight of a body can impress upon itself, in one second of time, on the parallel of 45~O is 32,1801 feet; and the measure for a body's mass is given by -W 32,1801' in which WImay be expressed, as before, in pounds; but in this case the cubic foot of water ceases to be the unit of mass, and in its stead we take so much of the water, or of any other body, as will weigh 32,1801 pounds, as the unit of mass. In this latter case, any body which weighs one pound will be 3 T of the unit of mass. Ilad the pound been made greater than it is in the proportion of 62,3791 to 32,1801, or the foot less in the proportion of 32,1801 to 62,3791, then would the same number have expressed both the pounds avoirdupois in the weight of a cubic foot of pure water at the standard temperature, and the number of feet in the velocity this weight could generate in the same cubic foot in one second of time. INTRODUCTION. 25 UNORGANIZED AND ORGANIZED BODIES. 9.-All bodies connected with the earth are distributed into two classes, viz.: Unorganized and Organized. The unorganized class brace all minerals, as metals, stones, earths alkalies water air and the like. The organized class include all animals and vegetables. The unorganized bodies form the lower class, and are, so to speak, the substratum of the organized. They are acted upon olely by influences external to themselves, and have nothing that can properl be called life. They have no definite or periodical duration. Or d bodie ar more r less perfect individuals, possessing organs adapted to the performance of certain functions. They possess vitality, and are continually appropriating to themselves unorganized bodies, changing their properties, and, by this process increasing their bulk. They possess the facllty of reproduction. They retain, only for a limited time the vital principle, and, when life is extin-ct, they sink into the class of unorganized bodies. HEAVENLY BODIES. ~1O. —— The Heavenly Bodies form a distinct class. In the changes they bring about within themselves, they resemble organized bodies; and may, in one Isense, be said to possess organs. Those of our earth are its continents, oceans,' and atmosphere. The researches of' Geology furnish the most ample evidence of vast changes having taken place' in the earth. It now supports and nourishes bot h the animal and vegetable kingdoms..There was a time when neither of these existed upon it. It was once all fluid, from excessive heat; it is now incrusted with an inidurated envelope of many miles in thickness, inclosing a molten, liquid mass. In many places its continents -are being elevated, while in others they are being depressed; correspondin g changes 26 ELEMENTS OF ANALYTICAL MECHANICS. are taking place in the shores of the ocean, and the climates of the same zones are undergoing modifications. What is true of our earth, is doubtless equally-true of the other heavenly bodies. NATURAL PHILOSOPHY. 11.-Natural Philosophy is a name given to that branch of physical science which treats of the general properties of unorganizecl bodies; the changes they undergo without affecting their internal constitution; the causes of these changes, and the laws which g9vern both the causes and changes. THERMOTICS, ACOUSTICS, OPTICS, ELECTRIC$. ~ 12.-Natural Philosophy embraces the subjects of Thermoties, Acoustics, Optics, and Electrics. The first treats of heat, the second of sound, the third of light, and the fourth of electricity. The subject of mnagnetics is omitted here, because it is now merged into that of electrics. The phenomena which appertain to these different heads, all have a common source in the action of forces upon bodies-the nature of the bodies, the kind and mode of action of the forces, and the sense employed to excite the mind to a perception of the effects, constituting the main distinction. Natural Philosophy relies upon observation and experiment for its data. From these we deduce the varied information we have acquired about bodies; by the former we notice any changes that transpire in the condition or relations of any body, as they spontaneously arise without interference on our part; whereas, in the performance of an experiment, we purposely alter the natural arrangement of things, to bring about some particular condition we desire. To accomplish this, we make use of appliances called hilosophical apparatus, the proper use and application of which, it is the office of Experimental Philosophy to teach. INTRODUCTION. 27 If we notice that in winter water becomes converted into ice, e are said to make an observation; if, by means of freezing mixturesor evaporation, we cause water to freeze, we are then said to perform an experiment. These observations and experiments are next subjected to calculation, from which are deduced what are called the laws of nature or the rules that like causes will invariably produce like results. To express these laws with the greatest possible brevity, mathematical formulas are sed. When it is not practicable to represent them with mathematical precision, we are content with inferences and assumptions based on analogies, or with probable ypotheses, as the means for explanation and further deductions. A hypothess gains in probability the more nearly it accords ith the ordinary course of nature, the more numerous the olbservations and experiments on which it is founded, and the more simple the explanation it offers of the phenomena for which it is intended to account. CHEMISTRY. ~13.-Chemistry treats of the individual properties of unorganized bodies, by which, as regards their constitution, they-may be distinguished from one another. It also investiga tes the transformations that take place in the interior of these bodies, and by which their substance is altered and remodelled; andi lastly, it detects and classifies the laws that regnlate chemical changes. NATURAL HISTORY. ~14.-Natural Hfistory treats -of the organized bodies. lIt comprises three divisions, viz.: Anatomy, which is concerned with the dissections of plants and'animals; TFeyetable, and Animal~ Cheiqisry, which- investigatcs their internal, coiistitution; and Phys&iology, wh~ich explains the objects and offices of. their various organs. 28 ELEMENTS OF ANALYTICAL MECHANICS. ASTRONOMY. ~ 15.-Astronomny teaches the knowledge of the heavenly bodies. It consists of two branches-Physical and Spherical Astronomy. The foriner treats of the constitution and physical condition of the heavenly bodies, their mutual influences and actions on each other, and, generally, seks to explain the causes of celestial phenomena; the latteris concerned with the appearances, magnitudes, distances, arrangements, and motions of these bodies. All measurements upon them are made from stations on thie earth, and by instruments that give the sides and angles of spherical triangles projected upon the concave of the celestial vault; and hence the name. GEOLOGY, PHYSICAL GEOGRAPHY, METEOROLOGY. ~ 16.-Geology, Physical Geography, and Meteorology, are strictly branches of Physical Astronomy. The first teaches a knowledge of the structure and history of the earth's crust; the second treats of the nature and character of its surface; and the third is concerned with the phenomena of the atmosphere and climate. PHYSICS. 17.-Natural Philosophy, Chemistry, Natural History, and Astronomy, are but branches of the more general subject called Physies-a science so vast in its range as to embrace whatever is known and can be discovered of the nature and properties of bodies, their source, effects, affections, operations, phenomena, and laws. MECHANICS. ~18.-All phenomena of the physical world arise directly from the-action of forces- upon the various forms of bodies. That INTRODUCTION. 29 branch of science which treats of this action is called 2Jeean;cas. A careful study of a course of mechanics is, therefore, an indispensable preparation for that of any branch of physical science. Mechanics is the subject of the present volume. It will be treated under three heads, suggested by peculiarities of physical condition, viz.: Xieehanics qf Solids, ]Ieehanies of Fluids, and Afechanies of Alolecu6es; the first treating of the action of forces upon solid bodies; the second, upon fluid bodies; and the third, upon the molecules or elements of both solids and fluids. Mechanics is founded ill a single fact, viz. that all action is ever accompanied by an, equal, contrcay, a)d siinmltaneous reaction. INERTIA. ~19.-This reaction very often arises from a property, common to all bodies, by which they resist, of themselves, every change of their own state in regard to rest or motion, and with an effort equal to that which produces the change. This property, known from experience, is called Inertia. It is force, but )passive and conservative force. PART I. MECHANICS OF SOLIDS. SPACE, TIME, MOTION, AND FORCE. ~20.-Space is indefinite extension, without limit, and contains all bodies. ~21.-Time is any limited portion of duration. We may conceive of a time which is longer or shorter than a given time. Time has, therefore, magnitude, as well as lines, areas, &c. To measure a given time, it is only necessary to assume a certain interval of time as unity, and to express, by a number, how often this unit is contained in the given time. When we give to this number the particular name of the unit, as hour, minute, second, &c., we have a complete expression for time. The Instruments usually employed in measuring time are clocks, chronometers, and common watches, which are too well known to need a description in a work like this. The smallest division of tiine indicated by these time-pieces is the second, of which there are 60 in a minute, 3600 in an hour, and 86400 in a day; and chronometers, which are nothing more than a species of watch, have been brought to such perfection as not to vary in their rate a half a second in 365 days, or 31536000 seconds. Thus the number of hours, minutes, or seconds, between any two events or instants, may be' estimated with as much precision and MECHANICS OF SOLIDS. 31 ase as the number of yards, feet, or inches between the extremities of any given distance. Time may be, represented, by AndH ~~2~~~~~t the sun3 nothing ch onnected wiof thes it equal distances representingd but as a theative eunit ofprs time.he same dA second is usually taken as which we may regard a foot as thfixed.'linear unit. ~ 22.-A body is in a state of absolute rest when it continues in the' same place in space. There is perhaps no body absolutely at rest; our earth being in motion about the sun, nothing connected with it can be at rest. Rest mnust, therefore, be considered but as a relative term. A body is said to be at rest, when it'preserves the same position in respect to other bodies which we may regard as fixed. A body, for example, which continues in the, same place in a boat, is said to be at rest in relation to the boat, although the, boat itself may be in motion in relation to, the banks of a river on whose sur. face it is floating. 23.-Ak bodly'is in motion when it occupies successively different positions in space. Motion, like rest, is but relative. A body is in motion whe n it changes its place in reference to those which we may regard at rest. Motion'is essentially continuous; that is, a body cannot pass from on~e position to another without passing through a series of interme. diate positions; a point, in motion, therefore describes a continuous line. When we speak of the path described by a body, we are to understand. that of a certain, point connected with the body. Thus, the path of a ball, is that of its centre. ~24.-The m otion of a body is said to be curvilinear or rectilinear, accordin.g as thE- path described is a curve or right line. Motion is 32 ELEMENTS OF ANALYTICAL MECHANICS. either uniform or varied. A body is said to have uniform motion when it passes over equal spaces ill equal successiveportions of time: and it is said to have varied motion when it passes over unequal spaces in equal successive portions of time. The motion is said to be accelerated when the successive increments of space in equal times become greater and greater. It is retarded hen these re ments become smaller and smaller. ~ 25.- Velocity is the rate of a body's motion. Velocity is measured by the.length of path described uniformly in a unit of time. ~ 26.-The spaces described in equal successive portions of time being equal in uniform motion, it is plain that the length of path described in any time will be equal to that described in a unit of time repeated as many times as there are units in the time. Lt v denote the velocity, t the time, and s the lenth of path described, then will s8= v.t...(3) If the position of the body be referred to any assumed origin whose distance from. the point" where the motion begins, estimated in the direction of the path described, be, denoted by S, then will s =.8+ v. t........ (4) Equation (3) shows that in uniform motion, the space described is9 always equal to the product of the time into the velocity; that, the saces described by dil7'rent bodies moving with different velocities drn the same time, are to each other as the velocities; and that when the velocities are the same, the spaces,are to each other as the times. ~ 27.-Differentiating Equation (3) or. (4), we find ds V;........ ~(5) that is to say, the velocity is equals to the first differential co-efficien of the space regarded as a function of the time. Dividing both members of Equation (3) by t4 we- have S -V........(6) MECHANICS OF SOLIDIS. 33 which shows that, in uniform motion, the velocity is equal to the whole space divided by the time in which it is described. 28.-atter on the earth, in its unorganized state, is inanimate or inert. It cannot give itself motion, nor can it change of itself the motion riving at b, to move towards c in the wh ar rived at b, there is no vedeason why ~A'~body at re than another. will foreover, if the ~~~ ~ it will retain this velocity unless disturbtered d why it should be increased rather ~oby somethingall extraneous tocauses. ~~~~~ponit should deviate to one side more than another. Moreover, if the body have a certain velocity at 6, it will retain this velocity unaltered, since no reason can be assigned why it should be increased rather than diminished in the absence of all extraneous causes. If a billiard-ball, thrown upon the table, seem to'diminish its rate of motion'till it, stops, it is because its motion is resisted by the cloth and the atmosphere. If a body thrown vertically flownward seem to increase its velocity, it is because its weight is incessantly urging it onward. If the direction of the motion of a stone, thrown into the air, seem continually to change,. it is because the, weight of the stone urges it incessantly towards'the surface of the earth. Experience proves that in proportion as the obstacles to a body's motion are removed, will the motion itself remain unchanged. When a body is at rest, or moving with uniform motion, its inertia is not called into action. ~29.-A force has been. defined to'be that which changes or tends to change the state of a body in respect to rest or motion. Weight and Elasticity are examples. A body laid upon a table, or suspended fronm a fixed point by means of a thread, wvoild move under the action of. its weight, if- the, resistance of the table or that of the fixed point didi not continually prevent by an equal, simultaneous, and contrary reacti~on. A body, subjected alone to the action of a'spring, would changr,a its state by moving faster or slower. 3 ELEMENTS OF ANALYTICAL MIECHANICS. When we push or pull a body, be it free or fixed, we experience a sensation denominated pressure, traction, or. in general, effort. This effort is analogous to that which we exert in raising a weight. Forces are real pressures. Pressure may be strong or feeble; it therefore has magnitude, and may be expressed in numbers by assumin certain pressure as unity. The unit of pressure will be taken to be at exerted by the weight of h part of a cubic foot of distilled ~water, at 8,75, and is called a pound. ~30.The intensity of a force is its greater or less aacity to produce pressure. This intensity may be expressed in pounds, or in quatity of motion. Its value in pounds is called its statical ea sure in antity of motion, its dynamical measure. 3 -The point of application of a force, is the materil point to which the force may be regarded as directly applied. 32.-The linte of direction of a force is the right line which the point of application would describe; if it were perfectly free. 33. —The effect~ of a force depends upon its intensity,, point of application, and line of direction, and when these are given the force. is known. ~ 34.,-Two forces -are equal when substituted, one for the other, ihi the same circumstances, they. produce the same effect, or whendirectly opposed, they neutralize each other. ~ 35. —-There can be no action of a,, force without an equal and contrary reaction. This is a law of nature, and, -0ur knowledge of it c-omes from experience. If a force, act upon a body retained by a tixed obstacle, the latter will' opoea qal and contrary resistance. If it. act upon a free body, the. latter will change, its state, and in the. act of doing so, its'inertia, will oppose an equal and contrary resistance. Action and reaction -,,arel ever equal, contrary and simulta. neous. ~ 36.,-If a free body be drawn by a thread, the thread -N ill stretch and even break if the action, be too violent, and this will the more probably happen in proportion as the body is more massive. If a ECANICS OF SOLIDS. 35 body be suspended by means of a vertical chain, and a weighing spring be interposed in the line of traction, the graduated scale of the spring will indicate the weight of the body when the latter is at rest; but if the Upper end of the chain be suddenly elevated, the spring will immediately bend more in consequence of the resistance opposed by the inertia of the body while acquiring motion. When the motion acquired becomes uniform, the spring will resume and preserve the degree of flexure which it had at rest. If now, the motion be checed by relaxing the effort applied to the upper end of the chain, the spring will unend and indicate a pressure less than the weight of the body, in conseeeof the inertia acting in opposition to the retardation. The oscillatis of the spring may therefore serve to indicate the varia, tions. in the motion s of a body, and the energy of its force. of inertia, which acts against or with a force, according as the velo. city' is increased or diminished. ~ 37.-Forces produce vari-ous effects accoirding to circumstances. They someti'mes leave a body at rest., by balancing one another, through'its intervention; sometimes: they change its form or brIeak it; sometimes they impress upon it motion, they accelerate or retard that which it has, or change its direction; sometimes these effect~s are pr')duced gradually, sometimes abruptly, but however produced, they require. some deftiti tinw, and. are. effected by continuous degrees. If a bady is sometimes seen to \change suddenly its state, either in respect to the direction or the rate.of'its motion,'t is because the force'is so great as to produce its effect in a time, so short as:to make its duration'imperceptible to our senses, yet some definite portion of time'is necessary for the change. A ball fired from a gun wilbreak through a pane of glass, a pic f board, or a sheet of paper, when freely suspended, with a rapidity so great as to call into 36 ELEMENTS OF ANALYTICAL MECHANICS. action a force of inertia in the parts which remain, greater than the molecular forces which connect the latter with those torn away. In such cases the effects are obvious while the times in which they are accomplished are so short as to elude the senses: and yet these times have had some definite duration since the changes corres ponding to these effects, have passed in successionthrough their differ ent degrees from the beginning to the ending. ~38.-Forces which give or tend to givemotion to bodies are called motive forces. The agent, by means of which the forces exerted, is called a Motor. ~39.-The statical measure of forces may be obtained by an instrument called the Dynamometer, which in principle does not differ from the spring balance. The dynamical measure will be explained further on. ~40. —When a force acts against a point in the surface of a body, it exerts a pressure,which crowds -together the. neighboring particles;- the body yields, is compressed and its surface, indented; the crowded,particles make an effort, by their molecular forces, to regain their primitive places, and thus transmit this -crowding action even to the remotest particles of the body. If these latter.particles are fixed, or prevented.by obstacles from- moving,-the result will be a compression and ch~ange, of figure throughout the body. If, on the contrary, these extreme particles are flee, they will advance, and motion will be comnimunicated by degrees. to all the, parts of the body. This internal motion, the result of a series of compressions, proves that a certain time is necessary fo-r a. force to produce its entire effect, and the error of supposing that a finite velocity may be generated instantaneously. The same kind of action will take place when the force is employed to destroy the motion which a body has already acquired; it will first destroy the motion of the molecules at and nearest the point of action, and, then, by degrees, that. of those which are more remote in the order of distance. MECHANICS OF SOLIDS. 37 The molecular springs cannot be compressed without reacting in a ctrary diretion, and with an equal effort. The agent which presses a body will experience an equal pressure; reaclion is equal and contrary to action. In pressing the finger against a body, in pulling it with a thread or pushing it with a bar, we are pressed, drawn, or pushed in a contrary direction, and with an equal effort. Two weighin springs attached to the extremities of a chain or bar, will indicate the same degree of tension and in contrary directions when made to act upon each other through its intervention. In every cse, therefore, the action of a force is transmitted through a body to the ultimate point of resistance, by a series of equal and contrary actions and reactions which balance each other, and which the molecular springs of all bodies exert at every point of the right line, along which the force acts. It is in virtue of this property of bodies, that the action of a force, may be assumed to be exert ed at any point in its.line -of direction within the boundary of the body. 41.-Bodies being more or less extensible and compressible, when interposed between the motor and resistance, will be, stretched or compressed to a certain degree, depending upon the en ergy with which these forces act'; but as long as the force and resistance remain the same, the body having attained its, new dimensions, will cease to change. On this account, we may, in the investigations which follow) assume that the bodies employed to transmit the action of forces from one point to another, are inextensible, and rigid. 42. -To work is to overcome a resistance continually recurring along some path. Thus, to raise a body through a vertical height, its weight must be overcome at every pofht of the vertical path. If a 38 ELEMENTS OF ANALYTICAL MECHANICS. body fall through a vertical height, its weight develops its inertia at every point of the descent. To take a shaving from a board with a plane, the cohesion of the wood must be overcome at every point along the entire length of the path described by the edge of the chisel. ~43.-The resistance may be constant, o it may be variable. In the first case, the quantity of work performed is the constant resistance taken as many times as there are points at which it has acted, and is measured by the product of the resistance into the path described by its point of application, estimated in the direction of the resistance. When the resistance is variable, the qantity of work is obtained by estimating the elementary quantities of work and taking their sum. By the elementary quantity of work is meant the intensity of the variable resistance taken as many times as there ar oints in the indefinitely small path over which the resistance may be regarded as constant; and is measured by the intensity of the resistance intothe differential of the path, estimated in the direction of the resistance. ~44.-In general, let P denote any variable resistance, and s the path described by its point of application, estimated in the direction of the resistance; then will the quantity of work, den oted by Q, be given by Q =fP.d3.(7).. which integrated between erktain limits, will give the value of Q. ~ 45.-The simplest kind of work is that performed in raising a weight through a vertical height. It is taken as a standard of comparison, and suggests'at once an idea of the quantity of work expended in any particular case.. Let the weight be, denoted by W, and the vertical height by HI; then will Q= —W. H..... (8). If W become one pound, and H one foot, then will Q=I and the unit of work is, therefore, the unit of force, one pound, exerted. over the unit of distance, one foot; and is measured by a MECHANICS OF SOLIDS. 39 square f which the adjacent sides are respectively one foot and one pound, taken from the same scale of equal parts. 4.-To illustrate the use of Equation (7), let it be required t compute the quantity of work necessary to compress the spiral spring of the commo spring balance to any given degree, say from the length AB to DB. Let the resistance vary directly as the degree of compression, and denote the distance AD' by x; then will P = C. X; d in which C denotes the resistance of the spring when the balance is compressed through the distance unity. This value of P in Equation (7), gives Q fP.d=C C.x X = C. + C 22 Q Let C = 10 pounds, a=- 3 feet; then will Q 45 units of work, and the quantity of work will be equal to that required to raise 45 pounds through a vertical height of one foot, or one, pound through a height of 45 feet, or 9 pounds through 5 feet, or 5 pounds through 9 feet, &e., all of which amounts to the same thing. 47.-A mea resistance is that which, multi'lied into the entire path described inthe direction of the resistance, will give the. entire quantity of, work. Denote this by B, and the entire path by s, and from- the definition, we have =~ fJP.ds; whence, P. ds.() S 40 ELEMENTS OF ANALYTICAL MECHANICS. That is, the mean resistance is equal to the entire work, divided by the entire path. In the above example the path being 3 feet, the mean resistance would be 15 pounds. ~48.-Equation (7) shows that the quantity of work is equal to the area included between the path s, in the direction of the resistance, the curve whose ordinates are the different values of P, and the ordinates which denote the extreme resistances. Whenever, therefore, the curve which connects the resistance with the path is known, the process for finding the quantity of work is one of simple integration. Sometimes this law cannot be found and the intensity of the resistance is given only at certain points of the path. In this case we proceed as follows, viz.: At the several points of the path where the resistance is known, erect ordinates equal to the corresponding resistances, and connect their extremities by a curved line; then divide the path described into any even number of equal parts, and erect the ordinates at the points of division, and at the extremities; number the, ordinates in the order 7. r of, the natural numbers; add together the extreme ordinates, increase this sum by four times that of the even ordinates and ~ twice that of the uneven qrdi- A? d7 nates, and multiply by one-third of the distance between any two consecutive ordinates. Demons/ration: To compute the area comprised by a curve, any two of its ordinates and the axis of abscisses,7 by plane geometry, divide it into elementary areas, by drawing ordinates, as in the -last figure, and regard. each of -the elementary figuires, e, e2 r, ri, e, e,, ra r., &C., as trapezoids; it is obvious that. the error of MECHiANICS OF SOLIDS. 41 this supposition will be less, in proportion as the number of trapezoids between given limits is greater. Take the first two trape- r,. C,u zoids of the preceding figure, and divide the l distance el e3 into three equal parts, and at the points of division, erect the ordinates m n, ml nh; the area computed from the three trapezoids el m n r,, m?l n, n, ml e3 rs, n, will be more accurate than if computed from the two el e2 r2 r,, e2 e3 r3 ra. The area by the three trapezoids is elrl + mn m n + m, n, m, n, + e r elm X - 2 - + mm,l e3 But by construction, el m- = mml = ml ee el em = -el e=, and the above may be written, e: e e(e, ri ~ 2 m n + 2 ml n, + ea r3), but in the trapezoid m ml n, 2m n + -2m n, = 4 e, r2, very nearly; whence the area becomes - e, e. (el r, + 4 e2 r + er3); the area of the next two trapezoids in order, of the preceding figure, will be el e (e3 r3 + 4 e4 r4 + e5')); and similar expressions for each succeeding pair of trapezoids Taking the sum of these, and we have the whole area bounded b) the curve, its extreme ordinates, and the axis of abscisses; or, Q = e-e, [elr, + 4ear, -- 2e3r. - 4e r4 + 2esr5 + 4eor. + e,r,1. (10) whence the rule. 42 ELEMENTS OF ANALYTICAL MECHANICS. ~49.-By the processes now explained, it is easy to estimate the quantity of work of the weights of bodies, of the resistances due to the forces of affinity which hold their elements together, of their elasticity, &c. It remains to consider the rules by which the quantity of work of inertia may be computed. Inertia is exerted only during a change of state in respect to motion or rest, and this brings us to the subject of varied motion. VARIED MOTION. ~ 50.-Varied motion has been defined to be that in which unequal spaces are described in equal successive portions of time. In this kind of motion the velocity is ever varying. It is measured at any given instant by the length of path it would enable a body to describe in the first subsequent unit of time, were it to remain unchanged. Denote the space described by s, and the time of its description by t. However variable the motion, the velocity may be regarded as constant during the indefinitely small time, dt. In this time the body will describe the small space ds; and as this space is described uniformly, the space described in the unit of time would, were the velocity constant, be ds repeated as many times as the unit of time contains dA. Hence, denoting the value of the velocity at any instant by v, we have 1 v=ds Xd; or, ds v= —........ *(11) dt...... ~51.-Continual variation in a body's velocity can only be produced by the incessant action of some force. The body's inertia opposes an equal and contrary reaction. This reaction is directly proportional to the mass of the body and to the amount of change in its velocity; it is, therefore, directly proportional to the product of the mass into the increment or decremnent of the velocity. The product of a mass into a velocity, represents a quantity of motion. MEHANrCS OF SOLIDS. 43 The intensity of a motive force, at any instant, is assumed to be measured by the quantity of motion which this intensity can generate im a unit of time. The mass remainin th ae, the velocities generated in equal Successive portions of time, by a constant force, must be equal to each other. However a force may vary, it may be regarded as constant during the indefinitely short interval dt; in this time it will enerate a velocity dv, and were it to remain constant, it would generate in a unit of time, a velocity equal to dv repeated as many times as d is contained in this unit; that is, the velocity generated would be equal to dv. dv. I -dr; and denoting the intensity of the force by P, and the muss by M, we shall have pM. dv...... (12) dt Again, differentiating Equation (11), regarding t as the independent variable, we get, d v -_ dt' and this, in Equation (1 2), gives p -=M. d2s.......(13) d12 From Equation (11), we conclude that in varied motion, the velocity at any instant -is equal to the first diferential co-efficient of the space regarded as a function of the lime. From Equation (12), that the intensity of any motive force, or of the'inertia it develops, at any instant,'is measured by the, product of the mass into the first differential co-efficient of the velocity regarded as a function of the time. And from Equation (130), that the intensity of the. motive force or of inertia, is measured by the, proditct of the mas~s into the second differential co-effcient of th'?e space regarded vs a function o~f the time. 44 ELEMENTS OF ANALYTICAL MECHANICS. ~52.-To illustrate. Let there be the relation s = at3 + bt2.(14) required the space described in three seconds, the elocty at the end of the third second, and the intensity of the motive force at the same instant. Differentiating Equation (14) twice, dividing each result by d, and multiplying the last by M, we find ds -- v = 3at22bt (15) H 2 P =31f [Oat ~ 261 (16G) dt ~r. -t = P- M[4 Make a = 20 feet, b = 10 feet and t 3 seconds, we have, from Equations (14), (15), and (16), s = 20.33 + 10.32=630 feet v=3.20. 32+ 2.10.3 600 feet; P = 31(6.20.3 + 2.10) 380.AM. That is to say, the body will move over the distance 630 feet in three seconds, will have a velocity of 600 feet at the end of the third second, and the force will have at that instant an intensity capable of generating fin the mass -Ia velocity of 380 feet in one secoDid, were it to retain that intensity unchanged. ~53.-Dividirig -.Equations, (12) and (13) by M, they give P dv (17) P d2s The first member is the same in both, and it is obviously that portion of the force's intensity which'is'impressed upon the unit of mass. The second member in each is the, velocity imrpressed in the mi11t of time. and is called the acceleratiort due to the motive fo-'ce. MECHANICS OF SOLIDS. 45 ~ 54.-From Equation (11) we have, ds = v. dt..... (19) multiplying this and Equation (12) together, there will result, P. ds M. v. dv... (20) and integrating, 2][. v2 fP.ds= 2..... (21) The first member is the quantity of work of the motive force, which is equal to that of inertia; the product lV.v2, is called the living force of the body whose mass is M. Whence, we see that the work of inertia is equal to half the living force; and the living force of a body is double the quantity of work expended by its inertia while it is acquiring its velocity.' 55.-If the force become constant and equal to F, the motion will be uniformly varied, and we have, from Equation (18), F d2s Multiplying by dt and integrating, we get P ds, t = i+ C v + C ~ (22) and if the body be moved from rest, the velocity will be equal to zero when t is zero; whence C -= 0, and F va....... (23) Multiplying Equation (22) by dt, after omitting C from it, and integrating again, we find F t2. -= Is + C', and if the body start from the origin of spaces, C' will be zero, and F t2; > - -.... —8... (24) 46 ELEMENTS OF ANALYTICAL MECHAE~.US. Making t equal to one second, in Equations (24) and (23), and dividing the last by the first, we have 1 s 2v' or, v = 2s....... (W) that is to say, the velocity generated in the first unit of time is measured by double the space described in acquiring this velocity. Equations (23), (24), and (25) express the lawst (of constant forces. ~ 56.-The dynamical measure for the intensity of a force, or the pressure it is capable of producing, is assumed to be the effect this pressure can produce in a unit' of time, this effect being a quantit) of motion, measured by the product of the mass into the velocity generated. This assumed measure must not be confounded with the quantity of work of the force while producing this effect. The former is the measure of a single pressure; the latter, this pressure repeated as many times as there are points in the path over which this pressure is exerted. Thus, let the body be moved from A to B, under the action of a constant force, in one second; the velocity generated will, A Equation (25), be 2AB. Make BC = 2AB, and complete the square BCFE. BE will s be equal to v; the intensity of the fot:e will be Al. v; and the quantity of work, the product of M. v by AB, or by its equal I v; thus making the quantity of work M v2, or the mass into one half the square BF; which agrees with the result obtained from Equation (21). EQUILIBRIUM. ~57.-Equilibrium is a term employed to express the state of two or more forces which balance one another through the intervention of some body subjected to their simultaneous action. When applied to a body, it means that the body is at rest. MECANIS OF SOLIDS. 47 We must be careful to distinguish between the extraneous forces which act upon a body, and the forces of inertia which they may, or may not, develop. If a body subjected to the siultaneous action of several extraneous forces, be at rest, or have uniform motion, the extraneous forces are in equilibrio, and the force of inertia is not developed. If the body have varied motion, the extraneous forces are not in equilibrio, but develop forces of inertia which, with the extraneous forces, are in equilibrio. Forces, therefore, including the force of inertia, are ever in equilibrio; and the indication of the presence or absence of the force of inertia, in any case, shows that the body is or is not chang ing its condition in respect to rest or motion. This is but a consequence of the universal law that every action is accompanied by anll equal and contrary reaction.D. THE CORD. 58.-A cord-'is a collection of, material points, so united as. to form one continuous and flexible line. It will be considered, in what imniediately follows, as perfectly flexible, itneXtensible, a'ud with-out thickness or weight. ~ 59.-By the tension of a cord is meant, the, effort by which any two of its adjacent particles are, urged- to separate from each other. 60.-Two equal forces, P and PI, applied at, the extremities A, A' of a straight cord, and acting in opposite directions from its middle, point, will L A' A 1PJ maintain each other in equilibrio. For, all the points of the cord- being situated on the line of direction of the forces, any one of them, as 0, may be taken as the common point of application without altering their effects; but in this case., the forces being equal will, ~ 34, neutral ize each other. 4S ELEMENTS OF ANALYTICAL MECHANICS. 61.-If two equal forces, P and P', solicit in opposite directions the extremities of the cord A A', the tension of the cord will be measured by the in- _,_ A p tensity of one of the forces. For, the cord being in this case in equilibrio, if we suppose any one of its points as 0, to become fixed, the equilibrium will not be disturbed, while all communication between the forces will be intercepted, and either force may be destroyed without affecting the other, or the part of the cord on which it acts. But if the part AO of the cord be attached to a fixed point at 0, and drawn by the force P alone, this force must measure the tension. TIHE MUFFLE. 62. —Suppose A, A', B, B', &c., to be several small wheels or pulleys perfectly free to move about their centres, which, conceive for the present... ( - to be fixed points. / Let one end of a cord be fastened to a fixed point C, and be wound around the L pulleys as represented in the figure; to the other extremity, attach a weight w. The weight w will be maintained in equilibrio by the resistance of the fixed point U, through the medium of the cord. The tension of the cord will be the same throughout its entire length, and equal to the weight wv; for, the cord being perfectly flexible, and the wheels perfectly free to move about their centres, there is nothing to intercept the free transmission of tension from one end to the other. Let the points s and r of the cord be supposed for a moment, fixed; the intermediate portion s r may be removed without affecting MECHANICS OF SOLIDS. 49 the tension of the cord, or the equilibrium of the weight w. At the point r, apply in the direction from r to a, a force whose intensity is equal to the tension of the cord, and at s an equal force acting in the direction from s to b; the points r and s may now be regarded as free. Do the same at the points s', r', s", r", s/' and r'/, and the action of the weight w, upon the pulleys A and A' will be replaced by the four forces at s, s', s// and s"', all of equal intensity and acting in the same direction. Now, let the centres of the pulleys A and A' be firmly connected with each other, and with some other fixed point as m, in the direction of BA produced, and suppose the pulleys diminished indefinitely, or reduced to their centres. Each of the points A and A' will be solicited in the same direction, and along the same line, by a force equal to 2w, and therefore the point m, by a force equal to 4w. Had there been six pulleys instead of four, the point mn would have been solicited by a force equal to 6w, and so of a greater number. That is to say, the point m would have been solicited by a force equal to w, repeated as many times as there are pulleys. If the extremity C of the cord had been connected with the point Vn, after passing round a fifth pulley at C, the point mn would have been subjected to the action of a force equal to 5w; if seven pulleys had been employed, it would have been urged by a force 7w; and it is therefore apparent, that the intensity of the force which solicits the point m, is found by multiplying the tension of the cord, or weight w, by the number of pulleys. This combination of the cord with a number of wheels or pulleys, is called a muffle. ~63.-Conceive the point mn to be transferred to the position m or m", on the line AB. The centres of the pulleys A, A', &c.> being invariably connected with the point m, will describe equal paths, and each equal to m mi, or m min", so that each of the parallel portions of the cord will be shortened in the first case, or lengthened in the second, by equal quantities; and if e denote the length of the path described by m, n the number of parallel portions of 3 50 ELEMENTS OF ANALYTICAL MECHANICS. the cord, which is equal to the number of pulleys, and, the change in length of the portion u w in consequence of the otion of m, we shall have, because the entire length of the cord remains the same, n.e (26) The first member of this equation we shall refer to as the change in length of cord on the pulleys. ~ 64.-The action of any force P, upon a material point, may be replaced by that of a muffle, by making the tension of its cord equal to the intensity of the given force, divided by the mber of parallel portions of the cord, or number of pullies. VIRTUAL VELOCITIES. 65.-Let M represent a collection of material points, united in any manner whatever, forming a solid body, and subjected to the action of several forces, P, P P and suppose these forces in equilibrio. Find the, greatest force w, which will divide each of the given forces without a remainder; replace the force P by a muffle, havinig / a number of pulleys. denoted by -; the tension of the cord will MECHANICS OF SOLIDS. 51 be denoted by w. Do the same for each of the forces, and we shall have as many muffles as there are forces, and all the cords will have the same tension. Let the several cords be united into one, as represented in the figure, one end being attached at C, the other acted upon by a weight equal to the force w. The action upon the body will remain unchanged; that is, the substituted forces, including wv, will be in equilibrio. In this state of the system, let a force Q be applied to put the body in motion, and at the instant motion begins, withdraw this force and stop the motion before the equilibrium of the forces is des troyed. The points of application of the original forces will each have described an indefinitely small path, as rn n. Let m r be the projection / of this path upon the original direc- r P tion of the force, and denote the length of this projection by e. Join r - the point n with any point o, on the direction of the force and at some definite distance from mn. From the triangle onr, we have -2 -2 -2 on = or + ur; ~2 the displacement being indefinitely small, nr may be neglected in -2 comparison with or, being an indefinitely small quantity of the second order; hence, On1 = or, and, om - on = om - or = e. But the number of pulleys in the muffle which acts along the direction of the force P is, P hence, the change in the length of the cord on the pulleys of this 52 ELEMENTS OF ANALYTICAL MECHANICS. muffle, caused by the slight motion of the point of application of the force P, will, since the centre of the pulley B is fixed, be P.e w IV and denoting by e', e", e'", &c., the projections of the paths described by the points to which the forces P', P, P &C., are respectively applied, on the original directions of these forces, we shall have P'. el P —. e' P,,, W W for the corresponding changes in the length of the cord on the other muffles. In all these changes, the cord being inextensible, its entire lengt remains the same, and if the change in length which the portion uw undergoes be denoted by I, we shall have I(P.e 4- P'.e' P"e +P P'e" + &c.) +9~=0..(27) w This equation expresses the algebraic sum of all the changes -in the lengths of the several parts of the cord, between the, points of application, and the fixed points towards which the points of application- are solicited; the effect of these changes being to shorten some and lengthen others, some of the terms of Equation (27) must be. negative. Now it is one of the essential properties of a system of forces in equilibrio, to leave a body subjected to their,action as free to move as though these forces did not exist. The additional force 9, thrfrwas wholly employed in developing the inertia of the body if; it was neither assisted nor opposed by the forces reprewented by the action of the muffles, because these forces balanced each other, and the, motion was arrested before the points of application were sufficiently disturbed to break up the equilibrium;'nor, reciprocally, ~ 35, was the action, of the muffles, nor the tension of the cord which produced this action, affected by ~Q. Hence the MECHIANICS OF SOLIDS. 53 tension of the cord wa invariable during the disturbance. But an invariable tension must have kept the weight w at rest during the displacement, and we have and Equation (27) ill reduce to, PC + F P"'e"+ P'"e'' + &c. =.X (28).-t may be objected, that the given forces are incommensturable, and that therefore, a force cannot be found which will divide each without a remainder; to which it is answered, that Equation (28), being perfectly independent of the value of the weight w, or tension of the cord, this weight may be taken so small as to render the remainder after division in any particular case, perfectly inappreciable. ~7.-The idefitely small paths mn, Den', described by the ts of application of the forces, P and P', during the slight motion we have supposed, are called virtual velocities; and they are so called; because, being the actual distances passed over by the / oints to which the foces are applied, in nthe same time they measure the relative A', -' rates of motion of these points. The distances r n and r'm', represented by e and are therefore the projections of the virtual velocities upon the directions of the forces. These projections may fall on the side towards which the orces tend to urge these points, or the reverse, depending upon the direction of the motion imparted to the system. the projections are regarded as positive, and in the second, as negative. Thus, in the case taken for illustration, m r is positive, and m'r' negative. The, products P e and Fe', are called virtual mzoments. They are the elementary quantities of work of the forces P and P'. The forces, are always regarded as positive; the sign of a virtual moment will, therefore, depend upon that of the, projection of the virtual velocity ~648. —Referring to Equation (28), we conclude, therefore, that whenever several forces are in equilibria, tihe algebraic sum of their virtual 54 ELEMENTS OF ANALYTICAL MECHANICS. moments is equal to zero; and in this consists what is called the pri ciple of virtual velocities. ~ 69.-Conversely, if in any system of forces, the algebraic sum of the virtual moments be equal to zero, the forces will be in equi. librio. For, if they be not in equilibrio, some, if ot all the points of application will have a motion. Let q, &. be the pro jections of the paths which these points describe in the first instant of time, and Q, Q', Q", &c., the intensities of such forces as will, when applied to these points in a direction opposite to the actual motions, produce an equilibrium. Then by the principle of virtual velocities, we shall have Pe + P'e' + P"e" + &c. + Qq + q + Q + &C. 0 But by hypothesis, Pe + P'e' + P"e" &c. 0 and hence, Qq + Q'q' + Q" + &C. 0 Now, the forces Q, Q', Q", &c., have each been applied in a dire tion contrary to the actual motion; hence, all the virtual moments, in Equation (28)' will have the negative sign; each term must, therefore, be equal to zero, which can only be the case by making Q, Q', Q", &C,, separately equal to zero, since by supposition the quantities denoted by q, q', q", are not so. We therefore conclude, that when the algebraic sum of the virtual moments of a system. of forces is equal to zero, the forces will be in e'quilibrio. Whatever be its nature, the effect of a force will be the same if we attribute its effort to attraction between its point of application and some remote point assumed arbitrarily and as fixed upon its line of direction, the intensity of the attraction being equal to that of the force. Denote the distance from the point of application of P, to that towards which it is attracted, by p, and the corresponding dis. trances in the case of the, forces P', P", &c., by p', P" &C., respect. ively; also, let Jp, clp','Ip", &c., represent the augmentation or dimi. nution of these distances caused by the displacement, supposed indcfi. nitely small, then ~ 65, will e = &jp, e' = Jp9', err - ap" &C., MECHANICS OF SOLIDS. 55 and Equation (28) may be written Ap2 + P'&p' + Pf"p" + &c. = 0... (29) in which the Greek letter a simply denotes change in the value of the letter written immediately after it, this change arising from the small displacement. ~70.-If the extraneous forces applied to a body be not in equilibrio, they will communicate motion to it, and will develop forces of inertia in its various elementary masses with which they will be in equilibrio; and if extraneous forces equal in all respects to these forces of inertia were introduced into the system, the algebraic sum of the virtual moments would be equal to zero. But if m denote the mass of any element of the body, s the path it describes, its force of inertia will, Eq. (13), be d2S rn. d___ dt2 and denoting the projection of its virtual velocity on s by 6s, its virtual moment will be m. d2s as' dt2 and because the forces of inertia act in opposition to the extraneous forces, their virtual moments must have signs contrary to those of the latter, and Equation (29) may be written 1P. Jp- _ n. 2;... (30) in which 1 denotes the algebraic sum of the terms similar to that written immediately after it. PRINCIPLE OF D'ALEMBERT. ~ 71.-This simple equation involves the whole doctrine of Mechanics. The extraneous forces P, P', P",, &c., are called impressed forces. The forces of inertia which they develop may or may not be equal to them, depending upon the manner of their application. If the impressed forces be in equilibrio, for instance, they will develop no force of inertia; 56 ELEMENTS OF ANALYTICAL MECHANICS. but in all cases, the forces of inertia developed i be equal and contrary to so much of the impressed forces as determines the change of motion. The portions of the impressed forces which determine a change of motion are called effective forces; and from Equation (30), we infer that the impressed and effective forces are always in equilibri wen the directions of the latter are reversed. This is usually known as D'Alebert's Principle, and is nothing more than a plain consequence of the law that action and reaction are ever equal ad contrary. This same principle is also enunciated in another way. Since the effective forces reversed would maintain the impressed forces in equilibrio, and prevent them from producing a change of motion, it follows that whatever forces may be lost and gained must be in equilibrio; els a motion, different from that which actuall takes place must occur. REFERENCE TO CO-ORDINATE AXES. ~ 72.-First Transformation. Equation (30) is of a form too general ior easy discussion, and may be simplified by referring the forces and rotions to rect.angular axes. Denote by a, g3, yv, the angles which the direction of the force P makes with the axes x, y, z, respectively; by a, b, c, the angles which its virtual velocity makes with the same axes; and by p, the angle which the -virtual velocity and direction of the force make with each other, then will 005 9p =cos a.. Cos a~ + cos b. cos f + cos c.- cos y Deno te by Ic, the -virtual velocity, and multiply the above equation by Pkc, and wre have Pke cos ip =Pkc cos a. cos a + Pk cos b.cos 16 + Pke cos c. cos7y; Bu1t denoting the, co-ordinates of the, point of application of P by x, y, z, we have k Cos p =Y p;kcos a = x;k cos b = y;k cos c -z; and these -values substituted abo-ve, give P.Jp=P -PCOS a. JX +P COSR. 6! P COS.. Z..(31). Similar values- may be, found for the virtual moments of other forces. MECHANICS OF SOLIDS. 57 73.-If P be replaced by the force of inertia, then will,, and y denot'the inclinations of the direction of this force to te axes xyz; its virtual velocity; a, b, and c the inclinations of the latter to the xes and its inclination to the direction of the forc of inertia and we may, Eq. (13), write d2 s d2 8 d s d2 s nl keosb c_ a k-cos a +in kos am cos b+m COS. k cos b+. But k cos= ds; k cos a = dx osbdy; k cos b = zy; d's.cos a =d d2x ds cos -= d'y; d2s cos Y-dz whence, d2s d2x d2y M t.;s = m M dd. 62x + M (3 2) and similar expressions may be found for the virtual moments of the forces of inertia of the other elementary masses. ~ 74.-If the intensity of the force P, be represented by a portion of- its line of direction, which is the practice, in all geometrical illustrations of Mechanics, the factors P cos a', P cos /3, and P cos 7 in Equation (31), would represent the intensities of forces equal to the projections of the intensity P, on the axes; and regarding these as acting in the directions of the axes, the factors 6x, 6y, and 6z, will represent the projections of their virtual velocities, which virtual velocities will coincide with that of the force P., Again, Equation (32), d2X d2Y d2z are forces of inertia in the directions of the axes, and ax, 6y&z, are the projections of their virtual velocities; these virtual velocities coincide with that of the inertia of m. The values of these virtual velocities depend upon the nature of the displacement. 5 ELEMENTS OF ANALYTICAL MECHANICS. FREE MOTION OF A RIGID SYSTEM..-Second Transformation. By the substitution, in Equation (30), for P dp and d s their values in Equations (31) and (32), there would result an equation containing, in general, three times as many variations of y z as there are extraneous forces and elementary masses,. Where the forces are applied to a body whose elementary masses are inLvariably connected-tat is, to a rigid solid-the number of these variations is greatly reduced, in consequence of the relations determined by this connection. ~~The~ most general otion we can attribute to a body is one of translation and of rotation combined. A motion of translation carries a body from place to place through space, and its position, at any instant, is determined by that of some one of its elements. A motion of rotation carries the elements of a body around some assumed point. In. this investi~d gation, let this point~be that which deterinines the body's place. Denote its co-ordinates by X~ ~Z and those of the element in, referred to this pointVA as an origin by x', y' z'; there will thus be two sets of axes, and supposing them parallel, ~ we have Y =,+ y'. (33), Z = Z +z' J and differentiating, dx = dx, ~ dx', dz = dz, d' MECHANICS OF SOLIDS. 59 Deit from mn, the perpendiculars mX', m Y',mZ' upon the movable axes. Denote the first by r', the / \ second by r", and the third by r"'. Let O', 0,", ~ _ be the projections of m, on the planes x y, x z, y z, res- Ie, pectively. Join the several points by right lines as indicated in the figure. Denote the angle m Z' 0" by, mX'O' by a, m Y' 0"' by 4. Then will x' rfff Cos A 7..... y35sA the triangle m Z' 0" give cos r'l sin (3 x' r'" sinos, thL-4Ariangle m Y' 0"', j'=r"cs. (30), (I rI Cos Zs) the triangle m X' 0',..- r'sin.'(37). We here have two values of x', one dependent upon (p, and the other upon 4. If the body be turned through an. indefinitely small angle about the axis z', the corresponding increment of x' is obtained by differentiating the first of Equations (35); and we have dx'-=-r... sinpydy if it be turned through a like angle about the axis y', the corresponding increm-ent of x' is found by diff-erentiating the first of Equations (306), anid If these motionDs takt- place simultaneously ab-out both axes, the above becoine partial diil~rentiails of x', and we' have, Ibr its, total diff'erential, d x' = —r" eos4~. d4,r"' sin ). d p, 60 ELEMENTS OF ANALYTICAL MECHANICS. replacing r" cos + and r"' sin l, by their values in the above Equa tions, and we get dx' = z'.d- y'.d; and in the same way, dy' = - x'. d q dz' = y'.d -x.d J which substituted in Equations (34), give dx= dx, + z'.d y'.dp( dy- dy, + x'.d Z dz d z, + y'.d x'.d and because the displacement is indefinitely small, we may write x 6X,; + z'.6.q ay = 6y, + x'. 6 6z = 6z + y'. 6'a and these in Equations (31) and (32), give r PCos a. 6x +P Cosf3. 6y, + Pcosy. az' P.6p = + P. (x. COsO - y'. Cos U).6(P{ (z'.. Cos a - z'. cosy). 6 + P.(y'. Cos y - Z'.cosP3). a. ___ d d2z dt2' dt2 / cc'. d2y - d2X6 d~~~~s ~~~dt2 dt =Z'. d2X - c'. d2Z ___ d z'. d2y ~r Y d12zdt di2~~~~~dS Similar values may be found fo r P'. p'and mn'. -t.6s, &e. III these values 6 xc, & y, and 6 z,, will be the same, as also 6 p), a +~, and 6 z, for the first relate to the movable origin, and the latter to the angular rotation which, since the body is a solid, must be, of e-qual MECHANICS OF SOLIDS. 61 ves for all the elements; so that to find the values of the virtual moments of the other forces, it will be only necessary suitably to accent P, a, 9, y, x,7 y, z, x',y', z'. These values being found ad substituted in Equation (30), we shall find, Es m -t2 ) Sx~ d2X /y dt2 ) j ++ I~~ P..=O-d2y. dx a 1-.(40) +. d2y -- y'. d2] j d2x dt2J d2.y * P. Cosf3- n. -0 (A)0'.Z. d2xY -- x.d2z oh dd f 0 2P -....Y. d2z I +;P.( Z'. Cosf3 x-y. cos*) —m dt2 Y'. d2x - Z' d2z-(B +;P. (z' Cosy - f. cosy) -1?nm*Y _ _ _ _ _ _ Now (ye dis~ placon as N)oly rbitrary tz ale odyx y 62 ELEMENTS OF ANALYTICAL MECHANICS. ~ 76.-These six equations express either all the circumstances of motion attending the action of forces, or all the circumstances of equilibrium of the forces, according as inertia is or is not brought into action; and the study of the principles of Mechanics is little else than an attentive consideration of the conclusions which follow from their discussion. Equations (A) relate to a motion of translation, and Equations (B) to a motion of rotation. They are perfectly symmetrical and may be memorized with great ease. COMPOSITION AND RESOLUTION OF FORCES. ~ 77.-When a free body is subjected to the simultaneous action of several extraneous forces which are not in equilibrio, its state will be changed; and if this change may be produced by the action o1 a single force, this force is called the resultant, and the several forces are termed components. The resultant of several forces is a single force which, acting alone, will produce the same effect as the several forces acting simultaneously; and the components of a single force, are several forces whose simultaneous action produces the same effect as the single force. If, then, several extraneous forces applied to a body, be not in equilibrio, but have a resultant, a single force, equal in intensity to this resultant, and applied so as to be immediately opposed to it, will produce an equilibrium; or, what amounts to the same thing, if in any system of extraneous forces in equilibrio, the resultant of all the forces but one be found, this resultant will be equal in intensity and irmmrediately opposed to the remaining force; otherwise the systemn could not be in equilibrio. Conceive a system of extraneous forces, not in equilibrio, and applied to a solid body, and suppose that the equilibrium may be produced by the introduction of an additional extraneous force. Denote the intensity of this force by R, the angles which its direc. tion makes with the axes x, y and z, by a, b and c, respectively, and the co-ordinates of its point of application by x, y, z. Then, because the inertia cannot act, d2x, d2y, d2z will be zero, and taking MECHiANICS OF SOLIDS. 63 the two origins to coincide, Equations (A) and (B), will give R cos a + P' cos a' + P" cos at" + P"' cos a"' + &c. = 0, R cos b + P' cos /' + P" cos /" + P"' cos /3' + &c. = 0, B cos c + P' cos y' + P"' cos 7" + P"' cos Y"' + &c. = 0; B (x cos b - y cos a) + P' (x' cos,' -y' cos a') o + P" (x" cos /" - y" cos a") + &c. (z cos a - x cos c) + P' (z' cos a' - cos') + P" (z" cos a" - x" cos'Y") + &c. (y cos c -z cos 6) + P' (y' cos y' - z' cos') + P" (y" cos 7" - z" cos /3") + &c. Now R is equal in intensity to the resultant of all the other forces of the system, or in other words, to the resultant of all the original forces; and if we give it a direction directly opposite to that in which it is supposed to act in the above equations, it becomes in all respects the same as that resultant, being equal to it in intensity and having the same point of application and line of direction. Adding, therefore, 180~ to each of the angles a, b, and c, the first terms of the foregoing equations become negative, and transposing the other terms to the second member and changing all the signs, we have, B cos a = P' cos a' + P" cos a" + P"' cos a' + &c.-X; R cos b = P' cos /' + P" os s " + P"' cos i3"' + &c. = Y;..(41) R cos c = P' cos 7' + P" cos 7" + P"' cos 7"' + &c. = Z. P' (x cos /' -'y' cos ca') R(xcosb —ycosa) = + P" (x" cos {S" —y"cos -)L; + &c. r P' (z' cos a' - X' Cos;R(zcosa.- xcosc) = +P" cos" - xcosos - y") [; (42) + &c. P' (y' cos Y' - z' cos P') R (y cosC - z co )Ts + PI" (y" cos r" - z" cos'") =N. + &c. 64 ELEMENTS OF ANALYTICAL MECHANICS. Or, Rcosa = X, Rcosb = Y,.... (43) R cos c = Z. 1R (x cos b y cos a)- L, R (z cos a- x cos c) = M,... *.. (44) R (y cos c - z cos b) N V. Eliminating R cos a, R cos b and R cos c, from Equations (44), ~y means of Equations (43), we get, by transposing all the terms to the first member, Xy - Y x+L = 0,1 Zx-Xz+M=0,.~~~~~~~(45) Zx — Xz + N = 01...... (45 Yz - Zy + N = 0. Either one of these equations is but a consequence of the other two. They are, therefore, the equations of a right line-the locus of the points of application; and from which it is apparent, that the point of application of a foree may be taken anywhere on its line of direction, within the limits of the body, without altering the effects of the force. The condition expressive of the existence of the dependence of one of these equations on the others, will, also, express the existence of a single resultant. ~78.-To find this condition, multiply the first of these Equatioans by Z, the second by Y, the third by X, and add the products; we obtain, ZL + YM + XN-= 0...(46). ~79.-Having ascertained, by the verification of this Equation, that the forces have a single resultant, its intensity, direction, and the equations of its direction may be readily found from Equations (43) and (44). Squaring each of the group (43), and adding, we obtain, R2 (cos2 a + cos2 b + cos2 c) =X2 + Y2 + Z2. MECHNICS OF SOLIDS. 65 Extracting the square root and reducing by the relation, cos'a2 b + cos2 c 1, there will result, R = + y2 Z2.... (47) which gives the intensity of the resultant, since X, Y and Z are ~~~~~kn~~OllSwn. Again, from. the samne Equations, X cos a =, F os=,7 (48) Z I cos C= J which make known the direction of the resultant. The group of Equations (45) give, X y - Yx + F' (cos /S'z' - cos a'y') =, Z x -X z + P'(cos a'z' - cosy'':0,. (49) Yz -Z y + IP' (COSy''Y' - cosf'er') =0.J nwhich are the equations of the line of the resultant. PARALLELOGRAM OF FORCES. ~ O. —-If all the forces be applied to the same point, this point may be taken as the origin of co-ordinates, in which case, 5xP = Xi/" x/f &C. = 0, I= y"1 = y"' &c. = 0, Z/ = f = Z1 &C. =0) and the last term in each of Equations (49), will reduce to zero-. Hence, to determine the intensity-, direction and equations of tho 5 66 ELEMENTS OF ANALYTICAL MECHANICS. line of direction of the resultant, we have Equations (47), (48) and (49), = X2 +Y+Z.(50) X cos a = A y cos b =- W (5 1) cos c - -, Z COS ~ -- Xy - Yx= 0, Zx -Xz = O (52) Yz - Zy = O. The last three equations show that the direction of the resultant passes through the common point of application of all the forces which might have been anticipated. ~81.-Let the forces be now reduced to two, andtake the plane of these forces as that of x y; then will r' =r" 1 = 7"1= &C. =900;z 0, the 1 st Equation of group (41) reduces to, Z = 0 and the above Equations become, - 2~+~YZ. (53) x cos a = W y....(54) cos b =, COS C = 0 XY-yYx = 0........(55) The last is an equation of a right line passing through the origin. The direction of the resultant will, therefore, pass through the point of app~lication of the forces. The cos c beingy zero. c is 900, and the direction of the resultant is therefore in the plane of' the forces. MECHANICS OF SOLIDS. 67 Substituting in Equation (53), for X and Y. their values from Equations (41), we obtain, = (P'os a'" + (P' Cos /3' + P"(+ os -,,); and since OS2 at + COS2 a 1 I cos23 a" + cos2 /3" - 1, _, ji this reduces to R = Vp2 + P"2 + 2 P' P" (co os a' " + cos' cos /"); denoting the angle made by the directions of the forces by 8, we have, Cos a' cos'a" + cos' c/s3" cos C;os and therefore, R =VP' + p"2 + 2P'Pcos.... (56) from which we conclude that the intensity of the resultant is equal to that diagonal of a parallelogram whose adjacent sides represent the directions and intensites of the components, which passes through the point of application. ~82.-Substituting in Equations (54), the values of X and Y, from Equations (41), we have, R cos a = P' cos a' + P" cos a", R cos b = P' cos /' + P" cos /", and because at = 90~go-', a" = 90 — ", a = 90~ - b these Equations reduce to, 2B cos a = P' cos a' + P" cos a", R sin a = P' sin a' + P" sin ao"; 68 ELEMENTS OF ANALYTICAL MECHANICS. and, by division, sinll a P' sin a' + P" sin a" cosa P' cos a' + P" cos a"; clearing fractions and transposing, we find, P" (sin a" cos a —cos, a" sin a) = P' (sin a cos a' - cos a sin a); whence, P' sin a" cos a - cos a" sina sin (a"t - a) P" sin a cos a'- cos a sin a' sin (a-a-) () That is to say, the intensities of the components are inversely proportionaI to thle sines of the angles which their directions makse with that of their resultant; but this is the relation that subsists between the two adjacent sides of a parallelogram and the sines of the angles whjich they make with the diagonal through their point of n meeting. Whence, Eqs. (.56) and (56)', The resultant of any two forces, applied to the same point, is tepresented, in intensity and direction, by that diagonal of a par llelogrcmn of which the adjacent sides represent the components. Making a - at'= the angle R m P' q, and.2 + PI + P"t - S 2. we have, fromn the usual trigonometrical formnaula, sin ~ S' - /(s-p')s- (S-) ~ 83.-In the triangle R mP', since P'R is equal and parallel to the line which represents the force P", the angle mn P'R =, is the supplement of the angle d, made by the directions of the'components, and there will result the following equation: C24 2 vp p" ) os ~6- sin j-.cO-V/s-P)(s P" (8 2~~~~~~~, PI, MECHANICS OF SOLIDS. 69 Equation (57), will make known the angle made by the direction of the resultant with that of either of two oblique components, pro vided, the intensities of the components and resultant be known. 84.-Also, from the two triangles mP' and B m P", we find, it~~~~~~~) three forces, Pt s", appied to ~~~~I P 1?' o. sinS 2 / from which the angles made by the direction of the result.. ant with its two components may be found. ~ 85.-Let there now be the three forces IP, P, P", applied to the material point m, in the directions m P, m P', mi pi", not in the same plane; the resultant will be repre- sented. in intensity and direc- -=. tion by the diagonal of a parallelopipedon, constructed upon the lines representingm the directions and. intensities of these components. For, A _ lay off the distances mA, m C, and m E, proportional to the intensities of the components which act in the direction of these lines, and construct the parallelopipedon.E B; the resultant of the components B' and. P will, ~ 82, be represcnted by the diagonal m B, of the parallelogram mA B C; and the resultant of this resultant and the remaining com - ponent B", will be represented by the diagonal mD of the parallelogram Em BD, which is that of the parallelopipedon. ~ 86.-If the forces act at right angles to each other, the parallelopipedon will become rectangular, and the intensity of the resultant, denoted by R, will bccoiae known fromt the formula 87) ELEMENTS OF ANALYTICAL MECHANICS. R= VP2 + PV2+Ptr2; and if the angles which the direction of the resultant makes with those of the forces P, P and P", be represented by a, b, and c, respectively, then will P~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ R cos a = P, ) B cos b =- P, B cos -- Pc/. Let three lines be drawn through the point of application m', of the force P', parallel to any three rectangular axes x, y, z; and denote by a', 3', y', the angles which the direction of this force Z mnakes with these axes respectively; then will PI cos aX P' cos A', / P' cos 7, jr/ be the components of the force P', in the direction of the axes, and they will act along the lines drawn through the point W'. These are the same as the terms composing in part Equations (A), and as the effect of the components is identical with that of the resultant, these components may always be substituted for the force _. The same d2x d2y d7z for the forces of inertia,, and m-i-, m.d, and m.-C, denot the components of this foree in the directions of the axes. MECHANICS OF SOLIDS. 71 87.-Examples. —1. Let the point m, be solicited by two forces whose intensities are 9 and 5, and whose directions make an angle with each other of 570 30'. Required the intensity of the force by which the 27 point is urged, and the direction in which it is A., compelled to move.. First, the intensity; make in Equation (56), and there will result,,R __5 81 + 25 + 90 x 0, 537 - 12,422. Again, substituting the values of a, P' P" and R? in the first f Equations (59), we have, si,' 5 X sin 570 30' 12,422 or, is = 19= 50' 35" nearly, which is the angle made by the direction of the force 9 with that of the resultant. 2.-Required the angle under which two equal components should act, in order that their resultant shall be the,N1th part of either of them separately. By condition, we have P = P" = R; hence, P' - P"? -. nR - nR +R _ (2n - 1)R end, Equation (58), S1 l ( - P' (S - P") P') P-"t 72 ELEMENTS OF ANALYTICAL MECHANICS. which reduces to sin - - If n be equal to unity, or the resultant be equal to either fore q) = 60 and, ~83, the angle of the components should be 120. 3.-Required to resolve the force 18 a into two component whose difference shall be 5 b, andirections make wit each other an angle of 38~ = 1. Also, to find the angle which th direction of each component makes with that of the resultant. Writing a for 1 in Equation (56), we have, p12 + p22 + 2 P" cos =. and by condition:?p_ Up,, Squaring the second and subtracting it from the first we et 2P'P" (1 + cos ) a2 which, replacing (1 + cos ci) by 2 cos2 -1 i reduces to a.. - b This added to the square of the Equation (c), gives Pt + PI,, - 2- (-csa i from which and Equation (c) we finally obtain, Pi a-2(1-Cos2c) 6)=709 co oswhich are the required components. To find the angles which their directions make with the resultant~ we have from Equations (59), C f= 240 - the angle which P" makes with the resultant. MECHANICS OF SOLIDS. 73 140 = angle which P' makes with the resultant. 4.-Required the angle under which two components whose inten. sities are denoted by 5 and 7 should act, to give a resultant whose intensity is represented by 9. Ans. 84~ 15' 39" 5.-From Equation (5) it appears that the resultant of two components applied to the same point, is greatest when the angle made by their directions is 00, and least when 1800~. Required the angle under which the components should act, in order that the resultant may be a mean proportional between these values; and also the angle which the resultant makes with the greater component. Callnt makes with the greater component. Ans. a = cos — P. iF'.-1.?'' = sin p 0.-Given a force whose intensity'is denoted by 17. Required the two com ponents which make with it angles of 270 and 430, SS8.-The theorem of the parallelogram of forces, just explained, enables us to determine by an easy graphical construction the in. tensity and direction of the resultant of several forces applied to the same point. Let Pi' PI, pil &C., be several forces applied to the same point m. Upon the directions of the forces, lay -~ off from the point of ap- plication distances proportional to the intensities of the forces, and let these dis. tacsrepresent the. forces. From the extremity P' of the line, rn P', which repre. T4 ELEMENTS OF ANALYTICAL MECHANICS. sents the first force, draw the line P' n equal and parallel to m PA which represents the second, then will the line joining the extremity of this line and the point of application, represent the resultant of these two forces. From the extremity n, draw the line n n' equal and parallel to m P"' which represents the third force; man' will represent the resultant of the first three forces. The construction being thus continued till a line be drawn equal and parallel to every line representing a force of the system, the resultant of the whole will be represented by the line, (in this instance m n"), joining the point of application with the last extremity of the last line drawn. Should the line which is drawn equal and parallel to that which represents the last force, terminate in the point of application, the resultant will be equal to zero. The reason for this construction is too obvious to need expla. nation. ~89.-If the forces still be supposed to act in the same plane, but upon different points of the plane, the first of Equations (49) takes the form, Yx - Xy = E [P' (cos /' x' -cos a' y') ], thus, differing from Equation (55), in giving the equation of the line of direction of the resultant an independent term, and showing that this line no longer passes through the origin. It may be constructed from the above equation. ~ 90.-To find the resultant in this case, by a graphical construe. tion, let the forces P', P", P"' &e., be ap- V plied to the points m', m", m"', &c., respec-;, f // tively. Produce the \ e- J directions of the forces.s -_... P' and P" till they x "/, meet at 0, and take this as their common p,,, — ~~~~~~~~~~~~~0 point of application lay off from 0, on the ines of direction, distances OS and OS', MECHANICS OF SOLIDS. 75 proportional to the intensities of the forces P' and P", and construct the parallelogram 0 SR S', then will OR represent the resultant of these forces. The direction of this resultant being produced till it meet the direction of the force P"', produced, a similar construction will give the resultant of the first resultant and the force P"', which will be the resultant of the three forces P', P" and P"'; and the same for the other forces. OF PARALLEL FORCES. ~91.-If the forces act in parallel directions, cos a' = cos a" - cos a'" = &c., cos' cos /" = cos f"'= &c., cos y' =os 7" = cos 7"' = &c., and Equations (41) become, X = (P' + P" + P"' + &T.) cos a', Y = (P' + F P"~ "'" + &c.) cos /', Z = (P' + P" + P"' + &c.) cos 7'; these values in Equation (47) give, R = C S/(' + F" + F"' + &c.)2 (cos2 a' + cos2 f' + cos2 7), but, cos2 a' + cos2 /' + cos2 7' = 1; hence, = Pt J+ pi J P +,&c.. ~ (60) If some of the forces as P", p"', act in directions opposite to the others, the cosines of a" and a"' will be negative while they have the same numerical value; and the last equation will become R =-t - Ft - pF"'l &c. Whence we conclude, that the resultant of a number of parallel forces is equal in intensity to the excess of the sutm of the intea. sities of those which act in one direction over the sum of the intensities of those which act in the opposite direction. 76 ELEMENTS OF ANALYTICAL MECHANICS. ~ 92.-The values of R, X, Y and Z being substituted in Equa tiolns (48) give, cos a = (P' + P" + P"' + &e.)cos a' cos a = + + & = cos a', pt + Pt - _Pt"/ + &c. cosb = p, + Pit + Pt/' + &c.) Cos =cos. (P' + P" + P"' + &c.) cos 7 cos c = Pt F p+ + PM-" + &c. COS 7. The denominator of these expressions, being the resultant, is essentially positive; the signs of the cosines of the angles a, b and c, will, therefore, depend upon the numerators; these are sums of the components parallel to the three axes. Hence, the resultant acts in the direction of those forces whose cosine coefficients are negative or positive according as the sum of the former or latter forces is the greater. ~ 93.-Equations (49), which are those of the resultant, become, after replacing X, Y, and Z, by their values in Equations (41), Ry. cos a - fx. cos b + cos /3'. 1 P'x' - cos'. v P'y' 0, ix. cos e - Rz. cos a +- cos Ca'. P'z' - cos 7. E P'x' — 0, Rz.cos b - y. Cos Co + Cos y 3' y' - eos /3'. P'z'= 0 aidI because, cos a cos a' cos b = cos )', cos c -cos'; we have, (f, - lP'y'). Cos a'- (R.f - 2P'x'). cos:3'= 0, (f;x - z P'x'). cos'- (R- P'z') cos a; 0. (Oz. "z') cos.(:'- (R: - P':/') c. y, = o; 0 and because a', a3i and y" are connected only by the reclat. MECHIAICS AND SOLIDS. 7 costc' + cos/3' -- ecos7' -= 1; either two of the cosines of these angles are wholly arbitrary, and from the principle of indeterminate co-efficients, we have, by dispenswith the sign I and writing out the terms, x - P'x' + P"x" + P"'x"' + &e. RBy - P'y' + P"y" + P"'y"' + &c. ~ (61) Rz = P'z' + P"z" + P"'z"' + &c. J The forces being given, the value of R, ~91, becomes known, and the co-ordinates x, y, z, are determined from the above equations; these co-ordinates will obviously remain the same whatever direction be iven to the forces, provided, they remain parallel and retain the same intensity and points of application, these latter elements being the ony ones upon which the values of x, y, z, depend. The point whose co-ordinates are x, y, z, which is a point of application of the resultant, is called the centre of parallel forces, and may bedefined to be, that point in a system of parallel forces through which the resultant of the system will always pass, whatever be the direction of the forces, provided, their intensities and points of appli. cation remain the same. 94.-Dividing each of the above Equations by R, we shall have P'x' + P"x" + P"'x"' + &c.' P' q- p it + p"' + &C. P'y' + P"y" + P"'y"' + &c. y- P' + P" + P"' 4- &e.X P'z' + P"z" + P'"z"' + &e. z = z = PI + PI, + P"' + &c. Hence, either co-ordinate of the centre of a system of parallel forces is equal t the algebraic sum of the products which result from multiplying the intensity of each, force by the corresponding co-ordinate of its point of application, divided by the algebraic sum of the forces. If'kle points of application of the forces be in the same plane. 78 ELEMENTS OF ANALYTIA MECHANIC. the co-ordinate plane x y, may be taken parallel to this plne, in which case Z' = Zt! = Zt. and, (P'+ P + P + &CP ZR_ UP' + P"f + P & from which it follows that the centre of parallel forces is also in this plane. If the points of application be upon the same straight line, tak the axis of x parallel to this line; then in addition tothe above results we have y' - y" - =&c.;y' and, (P' P + p " + &pc.)y':~ P =,p, + p,, + &p, whence, the centre of parallel forces is also upon this line. 95.-If we suppose, the parallel forces to be reduced to two, viz. P' and P", we may assume the axis x to pass through their points of application, and the plane x y to contain their directions, in which case,'Equations (60) and (61) become, - P'x ~ Ptfx" z =0 and y=O. Multiplying the first by x', and subtracting the product from the second, we obtain R1 (x - x') - P" (x" - x').. (a ) Multiplying'the first by x" and subtracting the second from the product, we get Denoting by S' and 8", the distances from the, poi nts of application MIECHANICS OF SOIIDfS. 79 of P' and P" to that of the resultant, which are x- x' and x".- x respectively, we have x- x'- = S'+ S"; and from Equations (a) and (b), there will result P': P": R:: S": S': S" + S'... (63) If the forces act in opposite directions, then, on the supposition that P' is the greater, will Rx = P'x' -PX z =O0, y=O. and by a process plainly indicated by what precedes, P': P":1 R:: S": S' S"-S'.. (64). From this and Proportion (63), it is A. obvious that the point of application of the resultant is always nearer that of the, greater component; and that when the components act in the same direction, the distance between the point of application of the smaller component and that of the resultant, is less than the distance between the points of application of the components, while the reverse is the case when the components act in opposite directions. In the first case, then, the isultant is between the components, and in the second, the larger component is always between the smaller component and the resultant. And we conclude, generally, that the resultant of two forces which 3olicit two points of a right line in parallel directions, is equal in inten. sity to the sum or difference of the intensities of the components, according as they act in the same or opposite directions, that it always acts in the direction of the greater component, that its line of direction is contained in the plane of the components, and that the intensity of either component is to that of the resultant, as the distance between the point of application of the other component and that of the resultant, is to the distance between the points of application of the components. 80 ELEMENTS OF ANALYTICAL MECHANICS. ~96.-Examples. —1. The length of the line'" joining the points of application of two parallel forces acting in the same direction, is 30 feet; the forces are represented by the numbers 15 and 5. Required the intensity of the resultant, and its point of application. - R P' + P" = 15 + 5 20; B P'::m" m':m" o, 20: 15:: 30: m"o = 22,5 feet. A single force, therefore, whose intensity is represented by 20, applied at a distance from the point of application of the smaller force equal to 22,5 feet, will produce the same effect as the given forces applied at mn" and in'. 2.-Required the intensity and point of application of the resultant of two parallel forces, whose intensities are denoted by the numbers 11 and 3, and which solicit the extremities of a right line whose length is 16 feet in opposite directions. BR = P' P"t I I - 3 = 8, P, -PitP, " M. " 0= P _- M = 22fet 3.-Given the i~ngth of a line whose extremities are solicited in the same direction/*%y two forces, the intensities of which differ by the n'tA part of that of the smaller. Required the distance of the.point of application of the resultant from the middle of the line. Let 2 1, denote the length of the line. Then, by the conditions, Pt,, + I =f n+ i B= ~~n_ __ 2n l~ 1 2n1 + I + p": B": 21: ino 2n+1PI 2n1 1 c co 2n I 2ni 4- 1. MECHANICS OF SOLIDS. 81 97.-The rule at the close of ~ 95, enables us to determine by a very easy graphical construction, the position and point of application of the resultant of a number of parallel forces, whose directions, intensities, and points of application are given. Let P, P',, P P"', and Prv, be several forces applied to the material points m, Mt, in", m"', and miv, in parallel directions. Join the points m and m' by a 0"o o' straight line, and divide this line at the point o, in the inverse 0 ratio of the intensities of the \ I forces P and P'; join the points o and m" by the straight line r om", and divide this line at o', in the inverse ratio of the sum of the first two forces and the force P"; and continue this construction till the last point mni' is included, then will the last point of division be the point of application of the resultant, through which its direction may be drawn parallel to that of the forces. The intensity of the resultant will be equal to the algebraic sum of the intensities of the forces. The position of the point o will result from the proportion P'. mm P P' P':: mm n=; that of o' from F". oit" P + P' + P P" o" o: 00' = P - Pt q- pit that of o" from P + P + P/ " - Pt t P":o' M"'"~ o' off" = p 0,,, - P +p, ________,,,_________ and finally, that of o"' from pt,. off mi, p~p,+p,,_p,,,+pi,. pt p t: 0,min: o" o"'= P. + f6+ P+ P"-P+-, 6 82 ELEMENTS OF ANALYTICAL MECHANICS. OF COUPLES. ~98.-When two forces P' and P" act in opposite directions, the distance of the point o, at which the resultant is applied, from the point vz', at which the f, component P' is applied, is found from the formula m'o= in'.i P"; m'i0 m' o -- pi, pit,; and if the components P' and P" become equal, the distance m'o will be infinite, and the resultant, zero. In other words, the forces will have no resultant, and their joint effect will be to turn the line m" m', about somine point between the points of application. The forces in this case act in opposite directions, are equal, but 3ot immediately opposed. To such forces the term couple is applied. A couple having no single resultant, their action cannot be compared to teat of a single force. ~99.-The analytical condition, Equation (46), expressive of the existence of a single resultant in any system of forces, will obviously be fulfilled, when X= 0, Y=0, and Z-= 0. But this may arise from the parallel groups of forces whose sums are denoted by X, Y, and Z, reducing each to a couple. These three couples may easily be reduced by composition to a single couple, beyond which, no further reduction can be made. It is, therefore, a failing case of the general analytical condition referred to. WORK OF THE RESULTANT AND OF ITrs COMPONENTS. ~ 100.-We have seen that when the resultant of several forces is introduced as an additional force with its direction reversed, it will hold its components in equilibrio. Denoting the intensity of MECHAANICS OF SOLIDS 83 the resultant by R, and the projection of its virtual velocity by 6r, we have from Equation (29), - RJr + P. p + P'. 6p' + P".6p" + &c. = 0, or, R 6 r = P. 6p + Pt 6 p' + P" a p" + &c.,.-. (65) in which P, P' P", &c. are the components, and 6p, 6p' 6p", &c. the projections of their virtual velocities. ~101.-Now, the displacement by which Equation (29) was deduced, was entirely arbitrary; it may, therefore, be made to conform in all respects to that which would be produced by the components P, P, &c., acting without the opposition of the force equal and contrary to their resultant; and writing dr for 6r, dp for 6p, &c., Equation (65) will become Rdr = Pdp + P'dp' + P"dp't + &c., (66) and integrating, fRdr = f Pdp + f P'dp'+ f pdp + &c., (67) in which R, P, P', &c. may be constant or functions of r, p, p', &c., respectively. From Equations (66) and (67), it appears that the quantity of work of the resultant of several forces is equal to the algebraic sum of the quantities of work of its components. Again, replacing P p, P' p', &c. in Equation (6k), by their values in Equation (31), and writing dr for C r, dp for r2p, &c., we find, fRdr = f2P.cos e.dx + fP. cosf.dy + fP. cos y.dz,.. (68) in which R may be constant or a function of r; P, constant or a function of x, y, z, &c. If the forces be in equilibrio, then will R = 0, and, Y P. cos a. d x +:P. cos P.dy + P. cos 7. dz 0-. (69) 84 ELEMENTS OF ANALYTICAL MECHANICS. MOMENTS. ~ 102.-It is now apparent that in the transformation of Equation (30) to Equation (40), each force of the original system was replaced by its three components in directions of three rectangular axes, arbitrarily assumed. The components parallel to either axis will, ~ 43, work during any motion which will carry their points of application in the direction of that axis, and will cease to work when the motion becomes perpendicular to the same line. Let the points of application ofthe forces move in lines parallel to the axis z; the components parallel to z alone can work, for the paths being perpendicular to the directions of the other components, the work of the latter will be nothing, because the projections of the paths upon their lines of direction will be zero. The elementary work of the extraneous forces will, in this case, be found in the third term of Equation (40), and equal to (E P cosy). d z,. Again, let the points of application turn around the axis z, parallel to the plane x y; the components parallel to the axes x and y alone can work since the paths will be perpendicular to the components in the direction of z, and their projections, therefore, zero. The elementary work in this case will be found in the fourth term of Equation (40), and equal to [ P (' cosf3- y' cos a)dfq Now let both of these motions take place simultaneously; that is, let the points of application move in the direction of the axis z, and also turn about that line; all the components will work, because the paths will be oblique to their directions, and, therefore, have projections of measurable values. The amount of elementary work of the extraneous forces will, in this case be found in the third and fourth terms of Equation (40), and equal to [( P cos )]. z Z, + [I P (x' Cos 3 - y' Cos )].. MECHANICS OF SOLIDS. 85 The same remarks apply to motion in the direction of and about ea..h of the other axes. ~ 10.-The rule for estimating the quantity of work when the motion is parallel to either axis or to a right line oblique to the three axes, is simple; that for getting the work during motion about an axis, is not so obvious. Let the motion take place around the axis z; and consider, first, the work of the force P. The two components of this force viz., P cos 3 and P cos a, which enter the fourth term of Equation (40), have for their resultant P sin y. This resultant, ~ 81, acts in a plane parallel to that of x y, and, therefore, at right angles to the axis z. Denote by a, the angle which this resultant makes with the axis x; then will P cos = P sin y. cos a}.(70) P cos = P sin y. sin a, and these values in the ter P (x Cos -y' cos a), give P(xCos3YCosa). P. =P sin y (x sin a,- y' cos o,). (71) From the point of application m of P, draw the line 7n A' perpendien- - lar to the axis z; denote "- its length by h', and its inclination to the -axis x by p' Multiply and di- vide Equation (7 1) by h' and reduce by the relntions COS sin I? then will result Pf (x' cos f3 - y' cos a) = P sin yh' (sin a,. cos 0'-cos a,. sin 0') = P sin yh'sin (a, -' Draw from A' the line A' 4' perpendicular to the direction of the line P m (produced), and denote i-ts length by 4'; then will A' sin (a, - k'l4, 86 ELEMENTS OF ANALYTICAL MECHANICS. and then will result P (x' cos -y'cosa) = P siny.k'.... (72) and the same for the forces P', P", &c.; so that we may write, omit. ting the accent from k, ZP(x'cosj3-y'cosa)= _ P.siny.k;... (73) and the measure of the elementary work due to rotation about the axis z, will be given by either member of the Equation [ P (x' cos 3-y cos a)] 6 P =[E P sin y. k].. (74) ~ 104.-So that in estimating the work due to rotation alone about the axis z, each force is, in effect, replaced by its two components, the one parallel, the other perpendicular to that line, and the former is neglected because, in this motion, it cannot work. ~ 105.-The quantity of work obtained by multiplying that one of the two components of a force which is perpendicular, while the other is parallel, to a given line, into the perpendicular distance between this line and that of the force, is called the component moment of the force in reference to the line. ~ 106.-The line in reference to which the moment is taken, is called, in general, a component axis; the perpendicular distance from the axis to the line of direction of the force, is called the lever arm of the force; and the extremity of the lever arm on the axis is called a centre of the moment. When the direction of the force is perpendicular to the axis, the latter is called the moment axis of the force. In this case the component parallel to the axis becomes zero, and the normal component the force itself. The moment of the resultant of several component forces, taken in reference to its moment axis, is called the resultant moment. The moments of the component forces are called component moments. ~ 107.-Changing d q into d p in Equation (74), we may write [ P(x' cos3-'cos a)]dz=4[P siny.ljdp.. (74) MECHANICS OF SOLIDS. 87 or fi P(X'cos3y'cosa)] dp =f[2 P. sin y.k]d p. (74)' Whence it appears, that the elementary quantity of work a force will perform during the motion of its point of application about an axis, is equal to the product of the moment of the force into the differential of the path described at the unit's distance from the axis. 108.-The whole quantity of work will result from the integration of Equation (74)' between limits. In this integration two cases may arise, viz.; either the moment may be constant, or it may be variable. In the first case, the quantity of ork is obtained by multiplying the constant moment into the path described by a point at the unit's distne from the axis. In the second, the force may be constant and the lever armvariable; the force variable and the lever arm constant; or both may be variable, and in such way as not to make their product constant. In all such cases, relations between the intensity of the force, its lever arm, and the path described at the unit's distance, must be, known in'Order to reduce, by elimination, the second member of Equation (74)' to a function of a single variable. These remiarks are equally true of the forces of inertia. The, intensities of these depend upon the, masses of the material elements and their degree of acceleration or retairdation; their points of application are on the elements themselves; the elementary arc described at the unit's distance is the same for both sets of moments, and its value depends upon the distribution of the material with reference to the axis of motion. The moments of the forces which urge a body to turn in opposite directions about any assumed axis must have contrary signs. The sign of P sin yh', or its equal P cosi. x' - P cos a.y', d epends upon the angles which the directiomn of the force makes with. the axes, and upon the signs and relative values of the co-ordinates of the point of application. Let the angles which the direction of any force ma kes with the co-ordinate axes be estimated from the positive side of the origin; then, if the angles which this direction makes with both, axes be acute, and the point of application lie in the first angle, P'mos x3 88 ELEMENTS, OF ANALYTICAL MECHANICS. and P cos a. y', will be positive, andif the first of these produtt exceed the second, the moment will be positive; but if the latte be the greater, the moment will benegative. The same remarks apply to the other axes. COMPOSITION AND RESOLUTION OF MOMENTS. ~109.-The forces being supposed to act in any directions whatever join the point of application of the resultant and the origin by a right line, and denote its length by H. Multiply and divide each of the Equations (44) by H, and reduce by the relations, x - - cos z _ = COS in which, and s, denote the angles which the line H makes with the axes x, y and z, respectively; then will B?.H. (cos. cos -cosa. cos)B. H. (cosa.cos a- cos c. cos )M. (75) B?.H. (cos c. cos- -- Cos b. cos s) N. Squaring each of these Equations and adding, we find 1?2 H2{cos2b. cos2 E-2 cosb. Cosa. cose. cos +cos2a. cos2E =L2 + HF~N.......(76) But cos2 a+cos2 b + cos2 C......(77) cos2,~ ~ cos2 E + cos2 s 1.......(78) cos a.cos~ + cos b. cos E + cos c. cos s = cos 9,. (79) MECHANICS OF SOLIDS. 89 the angle A, being that made by the line H, with the directiofi of the resultant. Collecting the coefficients of cos2 a, cos2 b, cos2 c, and reducing by the following relations, deduced from Equation (78); viz.: cos2 s + cos2 E 1 - cos2, cos2 + cos2 - 1 -cos2, cos2 + cos2' = 1 - cos2 s, we find, R2. H2. [1-(cos a.cos ~ + cos bC.OS + COS C. CoSe )2]=L2+P+N2; from Equation (79), 1-(cos a. cos + cos b. cos + cos m cos s)2 = 1 cos2 - = sin2 q; which reduces the above to R2. H2. sin2 q= L2 + M2 + N2. But H2. sin2 p is the square of the perpendicular drawn from the origin to the direction of the resultant; it is, therefore, the square of the lever arm of the resultant referred to the origin as a centre of moments. Denoting this lever arm by K, we have, after taking the square root, R. K = zM2 + L2..... (80) That is to say, the resultant moment of any system of forces is equal to the square root of the sum of the squares of the sums of the conponent moments, taken in reference to any three rectangular axes through the point assumed as the centre of moments. ~ 110.-Dividing the first of Equations (75), by Equation (80), we find, H (cos b. cos -cos a. cos ) L ~~-K G/2 + M2 + ~N2 The effect of a force is, ~ 77, independent of the position of its point of application, provided it be taken on the line of direction. Let the point of application of R, be taken at the extremity of its 90 ELEMENTS OF ANALYTICAL MECHANICS. lever armn, then will II coincide with and be equal ill length to K and g will become the angles which the lever arm makes with the axes x and y, respectively, and the well known relation obtained from the formulas for the transformation of co-ordinates from one set of rectangular axes to another, will give COs O, = cos b.cos - cos a. cos; in which 0. is the angle the resultant axis makes with the axis z; whence,* L Cos O.,. (81) jVL~' + MI + NV~ In the same way, denoting by 0r and 0o the angles which the moment axis of BR makes with the co-ordinate axes y and x respectively, will cos O = ---. +..... (82) VL2+ + MN+ CosO0 = N..... (83) VL2+M2~+N2 whence we conclude that, the cosine of the angle which the resultant axis makes with any assumed line is equal to the sum of the moments of the forces in reference to this line taken as a component axis divided by the resultant moment. ~ 111.-Multiplying Equation (81) by Equation (80), there will result, B. K. cos -L....... (84) which shows that the component moment of any system of forces in reference to any oblique axis is equal to the product of the resultant moment of the system into the cosine of the angle between the resultant and component axes. For the same system of forces and the same centre of moments, it is obvious that B and K will be constant; whence, Equation (80), the sum of the squares of the sums of the moments in reference * See Appendix, No. I. MECHANICS OF SOLIDS. 91 to any three rectangular axes through the centre of moments, taken as component axes, ts a constant quantity. Also, since the axis z, may have an infinite number of positions and still satisfy the condition of making equal angles with the resultant axis, we see, Equation (84), that the sum of the moments of the forces in reference to all component axes which make equal angles with the resultant axis will be constant. ~112.-Denote by 0,,y, 0G, the angles which any component axis makes with the co-ordinate axes z, y and x, respectively, and by 6 the angle which the component and resultant axes make with each other, then will cos d = cos 0. cos 0~ + cos 0Y. cos 0 + cos 0. cos O; multiplying both members by R. K, we have B.K. cos d =R.K. cos Oz. cos 0 +R.K.cos 0 cos 0, +B. K. cos O. cos o. But, Eouation (84), R. K. cos 0O - L R. K. cosOr = M, R. K. cos O.,= -; which substituted above, give R. K. cos d = L.cos0 + M. cos0 + X. cos0.. (85) That is to say, the component moment in reference to any assumed component axis, is equal to the sum of the products arising from multiplying the sum of the moments in reference to the co-ordinate axes, by the cosines of the angles which the direction of the component axis makes with these co-ordinate axes, respectively. TRANSLATION OF EQUATIONS (A) AND (B). ~113.-Equations (A) and (B) may now be translated. They express the conditions of equilibrium of a system of forces acting in various directions and upon different points of a solid lody. These condi. tions are six in number; viz.: 92 ELEMENTS OF ANALYTICAL MECHANICS. 1.-The algebraic sum of the components of the forces in each of any three rectangular directions must be separately equal to zero; 2.-The algebraic sum of the moments of the forces taken in refer. ence to each of three rectangular axes drawn through any assumed centre of moments, must be separately equal to zero. If the extraneous forces be in equilibrio, the terms which measure the forces of inertia will disappear, and these conditions of equilibrium will be expressed by: P. cos a = O,1 mP cost3 = 0,.**** (A)' Z P. cos _= O; z P. (x' cos - y' cos a) O, P. (z'.cos a - x'cos) =0,... (B), z P. (y' cos /- z' cos) = O. The above conditions, which relate to the action of a system of forces on a firee body, are qualified by conditions of constraint that determine the possible motion. 114.-If the body contain a fixed point, the origin of the mova. ble co-ordinates, in Equation (40), may be taken at this point; in which case we shall have, xi 0, 8y, = O, 6y, — O, 6z, =0; and it will only be necessary that the forces satisfy Equations (B), these being the co-efficients of the indeterminate quantities that do not reduce to zero. Hence, in the case of a fixed point, the sum of the moments of the forces, taken in reference to each of three rectangular axes, passing through the point, must separately reduce to cero. Should the system contain two fixed points, one of the axes, as MECHANICS OF SOLIDS. 93 that of x, may be assumed to coincide with the line joining these points, in which case, there will result in Equation (40), x, = q =0, y, S+=0. 6 zi= 0, and it will only be necessary that the forces satisfy the last Equa. tion in group (B); or that the sum of the moments of the forces in reference to the line joining the fixed points, reduce to zero. If the System be free to slide along this line, J x, will not reduce to zero and it will be necessary that its co-efficient, in Equation (40), reduce to zero or that the algebraic sum of the components of ~~~~~~th~e given forces parallel to the line oinig the fixed points, also reduce to zero. If three points of the system be constrained to remain in a fixed plane, one of the co-ordinate planes, as that of x y, may be assumed parallel to this plane; in which case, zi = 0, =0, =0; and the forces must satisfy the first and second of Equations (A) and the first of (B )'; that is, the algebraic sum of the components ofthe given forces parallel to each of two rectangular axes, parallel to the givens plane, must separately reduce to zero, and the sum of the moments in reference to an axis perpendicular to this plane must reduce to zero. CENTRE OF GRAVIT. 115.-Gravity is the name given to that force which urg es all bodies towards the centre of the earth. This force acts upon every particle of matter. Every body may, therefore, be regarded as subjected to the action of a system of forces whose number is equal to the number of its particles, and whose points of application have, with respect to any system of axes, the same co-ordinates as these particles. 94 ELEMENTS OF ANALYTICAL ME(CHANICS. The weight of a body is the resultant of this system, or the resultant of all the forces of gravity which act upon it, and is equal, in intensity, but directly opposed to the force which is just sufficient to support the body. The direction of the force of gravity is perpendicular to the earth's surface. The earth is an oblate spheroid, of small eccentricity, whose mean radius is nearly four thousand miles; hence, as.the directions of the force of gravity converge towards the centre, it is obvious that these directions, when they appertain to particles of the same body of ordinary magnitude, are sensibly parallel, since the linear dimensions of such bodies may be neglected, in compari son with any radius of curvature of the earth. The centre of such a systeiim of forces is determined by Equations (62), ~94, which are P'x' + P"x" 4- P"'x"' + &c. PX = - PI + - I+ it- &c. P'y' + P"y" + P"y" + &c. ~ ~ ~ (86) P' + P" + P"' + &c. P'z' + P"z"f + P'z"'t + &c. z P'I P"P p + P"' + &e. in which x, y, z,, are the co-ordinates of the centre; P', P", &c., the forces arising from the action of the force of gravity, that is, the weights of the elementary masses m', mn", &c., of which the co-ordinates are respectively x' y' z', x" y" z", &c. This centre is called the centre of gravity. From the values of its co-ordinates, Equations (86), it is apparent that the position of this point is independent of the direction of the force of gravity in reference to any assumed line of the body; and the centre of gravity of a body may be defined to be that point through which its weight always passes in whatever way the body may be turned in regard to the direction of the force of gravity. The values of P', P", &c., being regarded as the weights w', w", &c., of the elementary masses m', m", &c., we have, Equation (1), P' = w'= 7;P = W"g; = P"iw m" g"; w P"'wi = w" = "'g"; &c., MECHANICS OF SOLIDS. - and, Equations (86), n' g' x' + m" g" x" + m"' g"' x"' + &c. m' g' + Dn"g" - + mlg"' + &c.' = m' g'y'If m"g" y" - m"' g"' y"' + &c. Yd = mI' g' + i"g" + g n"ig"' + &c.' ( al' g' a' + n" g" z" + mi"! g"' z"' + &c.' 1 ng' ~m Ig" + m"' "' &c. ~ 116.-It will be shown by a process to be given in the proper place, that the intensity of the force of gravity varies inversely as the square of the distance from the centre of the earth. The distance from the surface to the centre of the earth is nearly four thousand miles; a change of half a mile in the distance at the surface would therefore, only cause a change of one four-thousandth part of its entire amount in the force of gravity; and hence, within the limits of bodies whose centres of gravity it may be desirable in practice to determine, the change would be inappreciable. Assuming, then, the force of gravity at the same place as constant, Equations (87), become Mn' x' I X + " x + m"'" + &c. x Im' + in" J+ — i " + &C. mi' y' "" + "y " + m' y"' + &c. Y rn' =+ in" + in"' + &c.' (88) n' z' " z" +I m"F zi"' + &c. zX In' + rn" + mi"' + &c.'J from which it appears, that when the action of the force of gravity is constant throughout any collection of particles, the position of the centre of gravity is independent of the intensity of the force. 117.-Substituting the value of the masses, given in Equation (1)', trere will result, v' d' x' + v" d" x" + v"' d"' x"' + &c. v' d' + v" d" + v' d"' + &c. v' d'y' + v" d"y" + v"'d"' "' ~ c+ &c. =e v' d' + v" d" -+ — v"' d"' + &c.r. (s ) v' d'z' + v" d" z" + v"' d"' z"' + &c. Z4 = v' d' + v" d" + v"'d"' + &c. (', ELEMENTS OF ANALYTICAL MECHANICS, and if the elements be of homogenous density throughout, we sha have, d' = d" = d "'= &c. and Equations (89) become, v'x' v x Vx" -+ v"' x"' + &c. x - xi V' + v"' + v"1 + &c.' v' y' + v" y" + v"' y"' + &c.(90) Y = V' + V + V' &c. v' z' + v" za" + v"' z"' + &c.. z: v' + v " + v+" - &c.' wnence it follows, that in all homogeneous bodies, the position of the centre of gravity is independent of the density, provided the intensity of gravity is the same throughout. ~118.-Employing the character.:, in its usual signification, Equa tions (90), may be written, x (V) X) x,= ~~ —y-,) z (V) X — (vz). (V Y) (91) and if the system be so united as to be continuous, VI fy. d V.(t.2. 119. —If the collection be divided symmetrically by the plane r y, then will (V Z) = 0, MECHANICS OF SOLIDS. 9T and, therefore, hence, the centre of gravity will lie in this plane. If, at the same time, the collection of elements be symmetrically divided by the plane xz, we shall have, vy) = 0,, 0; the collection of elements will be symmetrically disposed about the axis x, and the centre of gravity will be on that line. Although it is always true, that the centre of gravity will lie in a plane or line that divides a homogeneous collection of particles symmetrically; yet, the converse, it is obvious, is not always true, viz.: that the collection will be symmetrically divided by a plane or line that may contain the centre of gravity. 1Equations (92) are employed to determine the centres of gravity of all geometrical figures. THE CENTRE OF GRAVITY OF LINES. ~120.-Let s represent the entire length of an arc of any curve, whose centre of gravity is to be found, and of which the co-ordixiates of the extremities are x,', z', and x", y" z. To be applicable to this general case of a curve, included within the given limits, Equations (92) become XI ~ ~ d2 dZ2 fI x dx. d~ ~ XI ~ ~ d2 dx2 zdx. dy2 dzx2 S 98 ELEMENTS OF ANALYTICAL MECHANICS. in which s = If; dxw 1 +_ dy2 J d 2 * * * * (94) Example 1.-Find the position of the centre of gravity of a right line. Let, y=a x + 3, z = a'x + 3', be the equations of the line. Differentiating, substituting in Equations (94) and (93), integrating be- / tween the proper limits, I - and reducing, there will result, X + X" 1- 2'0 a. (x' + x") Y =+ - - a/.X (XI + X11)+a 2 which are the co-ordinates of the middle point of the line; a' y' z' and a" y" z", being those of its extremities; whence we conclude that the centre of gravity of a straight line is at its middle point. Example 2.-Find the centre of gravity of the'perimeter of a polygon. This may be done, according to Equations (90), by taking the sum of the products which result from multiplying the length of each side by the co-ordinate of its middle point, and dividing this sum by the length of the perimeter of the polygon. Or by construction, as follows: The weights of the several sides of the polygon constitute a system of parallel forces, whose points of application are the centres of gravity of the sides. The sides being of homogeneous density, their weights are proportional to their lengths. Hence, to find the centre MECHANICS O SOLIDS. 99 of gravity of the entire polygon, join the middle points of any two of the sides by a right line, and divide this line in the inverse ratio of the lengths of the adjacent sides, the point of division will, ~ 97, be the centre of gravity of these two sides; next, join this point with the middle of a third side by a straight line, and divide this line in the inverse ratio of the sum of first two sides, and this third side the point of division will be the centre of gravity of the three sides. Continue thisprocess till all the sides be taken, and the last point of division will be the centre of gravity of the polygon. Find the position of the centre of gravity of a plane curve. Assume the plane of x y to coincide with the plane of the curve, in which case, Z = O. dx and Equations (93) and (94) become, dx i dxy dx2, f,,ydx dx2 I'.. ~~~~(95).T f ~ dy2 s= dx 1+ dx2(96) Example 3.-Find the Icentre of gravity of a circular arc. Take'the, origin at the centre of curvature, and the axis of y passing through the middle point of the arc. The equation of the curve is, whence, dy_ dx which substituted in Equations (95), 09 100 ELEMENTS OF ANALYTICAL MECHANICSo will give on reduction, x, = O, a (x' + X"). Y.- s - and den)ting the chord of the arc by c x= at + az", xl —0~ ac y, = -; $ whence we.onclude that the centre of gravity of a circular arc ts on a line drawn through the centre of curvature and its middle point, and at a distance from the centre equal to a fourth proportional to the arc, radius and chord. Example 4.-Find the centre of gravity of the arc of a cycloid. The radius of the generating circle being a, the differential equa tion of the curve is, d=y. dy dX — y'd,~~~(a) ~2 a y — y2 the origin being at A, and AB being the axis of x. ~ - Transfer the origin to C, and denote by x', y' the new co-ordinates, the former being estimated in the direction CD, and the latter in the direction D A. Then will y = 2a - ax, x = acr -y and therefore, dx dy' 2a-' - x d y d x' I;2a X' -* * 2 MECHANICS OF SOLIDS. 101 this in Equations (96) and (95), gives, omitting the accent on the variables, s fdx, ~2a s =jA dx -- f dx S..~.t t f. ydx Integrating the first two equations between the limits indicated, and substituting the value of s, deduced from the first, in the second, we, have, = 2 V/2a (V2x"- Ix'), V/ X I 3 V/ X3 1 V /x" - V' x3 and from the third equation we have, after integrating by parts, sy 2-V/ 2a (y Va' - f Vxdy); substituting the value of d si, obtained from Equation (a)',, and reducing, there will result, Sy,2V12 a (yi/'- f V2a -x'. da), anid taking the integral between the indicated limits, henee, replacing s by its value, and dividing, 33 - 2 (a - a"') 2 -(a - xi 2 Supposing the are ~o begin at 6', we have, a" 0 and, YfJ = Y +3 2 a xr")2 a/a 3 m [(x a - - 2 102 ELEMENTS OF ANALYTCA MEHAICS. If the entire semi-arc from C to A be taken, these values become, x, l a, y,= a ( - Taking the entire arc A CB, the curve will be symmetrical with res pect to the axis of x$ and therefore, y,= 0; hence, the centre of gravity of the arc of the cycloid, generated ey one entire revolution of the generating circle, is on he linewhich divides the curve symmetrically, and at a distance from the summit the curve equal to one-third of its height. TH:E CENTRE OF GRAITY OF SURFACES. ~121.-Let L -= 0 be the equation of any surface; L being a function of x y z; then will d x d y, be the projection of an element of this surface whose co-ordinates are xyz, upon the plane xy; and'if 0 " denote the angle which a plane tangent to the surface at the'Same point makes with the plane xy, the value of the element itself will be. dx. dy1 cos 0 But the angle which a plane makes with the co-ordinate plane x y, is equal to. the angle which the normal to0 the plane makes with the axis Z, and, therefore, dJL CosO" 11 (917) (dL)2(dL)2(dL)2 W MECHANICS OF SOLIDS. 103 andhence, in Equations.(92), omitting the double sign, Vdx. dy. w,...... (98) ani those Equations become, w.x.d v.dy | 8. Z, _. y.d x. dy (99) $ J w.z.dx.dy in which, Vt XI s, V= ff w.dx,.dy,. (100) wo being/a function of x, y, z. If the surface be plane,, -the plane of x y may be taken in them surface, in which case, z-0) and Equations. (99), and (100), be- ~ - j. JI come, x jy dy.xdx = _____' ____ I (101 y, z,9 dx. dy...... (102) in which the. initegral is to be taken first witth respect to y, and 104 ELEMENTS OF ANALYTICAL MECHANICS. between the limits y" = P Pm" and y' P'; then in respect to, between the limits x" = A P", and x' A P. Hence Ay 0. s St f,, (y",- y') Y. =.8 ~'~[. (108 =.j(y, - y')dx y' and y", denoting running co-ordinates, which may be either roots of the same equation, resulting from the same value of x, or they may belong to two distinct functions of x, the value of x being the same in each. For instance, if F (zy) = 0, be the equation of the curve n' i" nn'W, it is obvious that between the limits x" = 4AP" and x' = A Pt', every value of x, as A P, must give two values for y, viz.: y"= Pm" and y' = Pmi'. Or if P (x Y) 0, be the equations of two distinct curves WI" n" and m' n', referred to the same origin A, then will y" and y' result from these N functions separately, when the' same value is given to x in AX each. Example 1I.-IRequired the porition of the centre of gravity of the trea of a triangle. MECHANICS OF SOLIDS. 105 Let A B C, be the triangle. te A Assume the origin of co-ordinates at one of the angles A, and draw the axis y parallel to~ the opposite side B C. Denote0 the distance A P by x', and suppose, y" =ax, to be the equatior of the sides A C and AB, respectively, then will = (~ - b) n, - b ) X2 dx y 12 _12 0 -) x and, f(a- ) X2dx2 x XI 0~~~ 3x (a - b) xdx Wf,( -2b2)x2 dx _2(a +b) XI f (a -b) xd x whence we conclude, that the centre of gravity of a triangle is on a line drawn from any one of the angles to the middle of the opposite side, and at a distance from this angle equal to two-thirds of the line thus drawn. Example 2.-Find the centre of gravity of the area of any polygon. From any one of the angles as A, of the polygon, draw lines to all the other angles except C, those which are adjacent on either/ side; the polygon will thus be divided into triangles. Find by the rule just given, the centre of gravity of each of the triangles; 106 ELEMENTS OF ANALYTICAL MECHANICS. join any two of these centres by a right line, and divide this line in the inverse ratio of the areas of the triangles to which these centres belong; the point of division will be the centre of gravity of these two triangles. Join, by a straight line, this centre with the centre of gravity of a third triangle, and divide this line in the inverse ratio of the sum of the areas of the first two triangles and of the third, this point of division will be the centre of gravity of the three triangles. Continue this process till all the triangles be embraced by it, and the last point of division will be the centre of gravity of the polygon, the reasons- for the rule being the same as those given for the determination of the centre of gravity of the perimeter of a polygon, it being only necessary to substitute the areas of the triangles for the lengths of the sides. Example 3.-Determine the position of the centre of gravity of a circular sector. The centre of gravity of the seetor will be on the radius drawn to the middle point of the arc, since this radius divides the sector symmetri- cally. Conceive the sector CAB, to be divided into an indefinite number i of elementary sectors; each one of these may be regarded as a triangle whose centre of gravity is at a distance from the centre C, equal to two-thirds of the radius. If, therefore, from this centre an arc be described with a radius equal to two-thirds the radius of the sector, this arc will be the locus of the centres of gravity of all the elementary sectors; and for reasons already explained, the centre of gravity of the entire sector will be the same as that of the portion of this arc which is included between the extreme radii of the sector. Hence, calling r the radius of the sector, a and c its arc and chord respectively, and x, the distance of the centre of gravity from the centre C, we have,.2 3 2 r.c a3 = a MECHANICS OF SOLIDS. 107 The centre of gravity of a circular sector is therefore on the radius drawn to the middle point of the arc of the sector, and at a distance from the centre of curvature equal to two-thirds of a fourth proportional to the arc, chord and radius of the sector. Example 4.-Find the centre of gravity of a:ircular segment. Assume the origin at the centre C, and take the axis x passing through the.7 middle point of the arc, the centre of gravity in question will be on this axis, and, therefore, y 0 = O. t h Let A B HA be the segment, and y = __ /a2 -_ x2, the equation of the circle, the origin being at the centre C, then will yr /a2 _ X2' y = a2x2, and, Equations (103) and (104), XI - f a2-xdx a2( n. 2)d x 2 (a2 _ a2-x)2 8- a te x2. dx a2 (- sin~l xf ) - a2 Xfi-, s being the area of the entire segment. Denoting the chord A B by c, we have, a2 - x2-= IC; whence, C3, 12.s; and we conclude, that the centre of gravity of a circular segment is on the radius drawn to the middle of the arc, and at a distance from the centre equal to the cube of the chord, divided by twelve times the area of the segment. 108 ELEMENTS OF ANALYTICAL MECHANICS. Replacing the value of s, and supposing x' to be zero, in which case the segment becomes a semicircle, we shall find, c = 2a, 4a x, = - 122.-If the surface be one of revolution, about the axis x fo instance, it will be symmetrical with respect to this axis hence, y, = O; z,=; and if F(xy)= O0, be the equation of a meridian section in the plane x y, then will the area of an elementary zone comprised be tween two planes perpendicular to the axis of revolution be, 2~r. y. dx2+dy2, and therefore, Equations (92), fyx 1. dx' Xi = - (105) 3 = 2 I Y. + dx2(06 Example 1.-Find the position of the cen tre of gravity of a right conical surface. The equation of the element in the plane x y, is, assumbig the origin at they vertex, y =ax; hence, 2mr, a x2d x 1_+ a2 2, xi 0 3x 2qJax dxV 1+a 2 MECHANICS OF SOLIDS. 103 e-X Xtt2 _- Xt2 wtt J}_ Xt Example 2.-Required the pos-t tion of the centre of gravity of a spherical zone. Assuming the origin at the centre, the equation of the me- / ridian curve is,a 2y =ad2-X2 whence, ydy =-xdx, dy2 x2 and, fax dx ad Hence, the centre of gravity of a spherical zone, is at the middle point of a line joining the centres of its circular bases. And in the case of one base it is only necessary to make x" -I a, which gives, XI 2So that the centre of gravity of a zone of one base is at the middle of the ver-sine of its meridian curve. THE CENTRES OF GRAVITY OF VOLUMES. ~123.-When it is the'question to determine the c-entre of gravity of the volume of any body, we have dV = dx. dy. dz, and Equations (92) become, it "1,1~ x. d y. dz. dx 110 ELEMENTS OF ANALYTICAL MECHANICS. XI X'!/I'z,,],,],,y.dy.dz.dz y_ X - Y V JrI,, z e d y. d z. d x I x iy~ zP V and, V- i,,,,,, dy.dz.dx. In which the triple integral must be extended to include the entire space embraced by the surface of the body; this surface being given by its equation. If the volume be symmetrical with respect to any line, this line may be assumed as one of the co-ordinate axes, as that of x; in which case, if X represent the area of a section perpendicular to this axis, and x, its distance from the plane y z, then will Xd x, be an elementary volume symmetrically disposed in regard to the axis, and Equations (92), become Jf,,Xxd x If(107) =, 0, =, 0, and, XI V = -Xdx. (108).Example 1.-Find the position of the centre of gravity of a semi. ellipsoid, the equation of whose surface is The senii-axes of the elliptical section parallel to the plane y z, are, y =B 1 —, A2 x2 = C MECHI ANICS OF SOCLIDS. it whence,..-X2 X-fYBC I1A2 and, Equations (107) and (108), V=f BC (1 - A ) dx fr.B C (1- )xdx 3 x2'- - $,8 =-A x2 ~~8 f B C ( 1-2)2 d 8 If the figure be one of revolution about the axis of x, theni, defiot~ng by F(xy) = 0........ (109) the equation of the meridian section by the plane x y, will X= A, and Equations (10io7) and (108), may be written, _' y2xdx. e] v =',,.....(110) -- / T, y2 dx........ V='t ry2dx,............... (111) Example 1.-Required the position of the centre of gravity of a paraboloid of revolution. In this case, Equaticn (109), F (xy) = y2 - 2p x = 0, whence, V - 2qrp xdx, ro 2 p x2 dx 2 i- — 3 a. 2%rp xdx 3 112 ELEMENTS OF ANALYTICAL MECHANICS. Exanmple 2.-Required the position of the centre of gravity of the volume of a spherical segment. F (x y) = y2 + x2 - a2 = 0, whence, V = (a2 - X2) d x zutt f (a2 -x2). x. d x I XI -J 02_ x2) d x or, 3 rx"2(2a' - x"2) -x'2 (2a2 - x'2) XI - 4 I (3a2 - x"2) -x' (3a2 -x'2)' and for a segment of one base, x" = a, 3 a4 - x'2 (2a2 - x'2) 4 2a3 - x' (3a2 - x'2) If the volume have a plane face, and be of such figure that the areas of all sections parallel to this face, are connected by any law of their distances from it, the position of the centre of gravity, may also be found by the method of single integrals. Example 1.-Find the centre of gravity of any pyramid. Find by the method explained, the centre of gravity of the base of the pyramid, and join this point with the vertex by a straight line. All sections parallel to the base are similar to it, and will be pierced by this line in homologous points and therefore in their centres of gravity. Each section being supposed indefinitely thin, and its weight acting at its centre of gravity, the centre of gravity of the entire pyramid will, ~94, be found somewhere on the same line. Take the origin at the vertex, draw the axis x perpendicular to the plane of the base, and the plane x y through its centre of MECHANICS OF SOLIDS. 113 gravity; and let X represent any section parallel to the base, then will Equations (92) become,,,Xxdx J, z 0, w= V —-,,Xydx and,~~~ V- f!Xdx. Represent by A the base of the pyramid, c its altitude, and let y ax, be the equation of the line joining the vertex and centre of gravity of the base. Then, A X and for any frustum, X3 x~d xCd2 3 1"4 - 4 Xi A P'4 -1,x2dx aAf d 3 / T a~ ~-f2dx X and for the entire pyramid, make x" c, and x? = 0 —, which give i4 8~~~~~ 114 ELEMENTS OF ANALYTIGAL MECHANICS. whence we conclude that the centre of gravity of a pyramid is onl the line drawn from the vertex to the centre of gravity of the base, and at a distance from the vertex equal to three-fourths of the length of this line. The same rule obviously applies to a cone, since the result is independent of the figure of the base. The weight of a body always actiqg at its centre of gravity, and in a vertical direction, it follows, that if the body be freely suspended in succession from any two of its points by a perfectly flexible thread, and the directions of this thread, when the body is in equilibrio, be produced, they will intersect at the centre of gravity; and hehce it will only be necessary, in any particular case, to determine this point of intersection, to find, experimentally, the centre of gravity of a body. THE CENTROBARYC METHOD. 124.-Resuming the second of Equations (95) and (103), which are,:,,ydx 1 + d x2 in which sf dx 1 dx2' and ~,, (y,,2 _y) dz $ My'~~ y'' )ds in which sJ./, (y"_a o s= (y- y')dx; clearing the fractions and multiplying both members by 2 7r, we shall have, 2.y,s =,, 2'y /dx2 + d*y2, (112) y, =, (y'12 - y12) d x.... (1138 MECHANICS OF SOLIDS. 15 he second member of Equation (112) is the area of' a surface generated by the revolution of a plane curve, whose extremities are given by the ordinates answering to the abscisses x' and x", about the axis x. In the t member, s is the entire length of this ar, and 2y, is the circumference generated by its centre of gravity. Hence, we have this simple rule for finding the area of a figure of revolution, viz.: Multiply the length of the generating curve by thee circumference described by its ctre of gravity about the axis of rotation; the product will be the required surface. The second member of Equation (113) is the volume generated by a plane area, boundedby two branches of the same curve or by two different curves, and the ordinates answering to the abscisses and x" about the axis X. s in the first member, is the generating area, and 2 y the circumference described by its centre of gravity. Hence this rule for finding the volume of any figure of revolution, viz.: Multiply the generating area by the circumference described by its centre of gravity about the axis of rotation; the product will be the volume sought. Example 1. -Required the measure of the surface of a right cone. Let thre cone be generated by the rotation of the line A B about the line A C. The centre of gravity of the generatrix is at its, middle point G, and therefore, the radius of the circle described by it will be onehalf of the radius CBA of the circular base of the cone. Hence, 2,r y4. s = 2 r. 2 C.A rBC.AB. -Exainple 2.-Find the volume of the cone. The, area of the generatrix A B C, is + B C. A C; and the radius of the circle described by its centre of gravity is ~-B C.Hence, B C.A C BC2. A C 23y =eB. 2 I 3 116 ELEMENTS OF ANALYTICAL MECHANICS CENTRE OF INERTIA. ~125.-When the elementary masses of a body exert their forces of inertia simultaneously and in parallel directions, they must experience equal accelerations or retardations in the same time, and the factor d2s dt' in the measures of these forces, as given in Equation (13), must be the same for all. Substituting these measures for P', P", &c., inr Equations (62), we find, d2s dt2 mx M mx X s =; d2s dt2. y m y! d2____ 2 |. *(114) dt2 d2s. dt2 Whence, Equations (88), the centre of inertia coincides with the centre of gravity when the force of gravity is constant, both being at the centre of mass. In strictness, however, the centre of gravity is always below the centre of inertia; for when the variation in the force of gravity, arising from change of distance, is taken into account, the lower of two equal masses will be found the heavier. And in bodies whose linear dimensions bear some appreciable proportion to their distances from the centre of attraction, the distance between these centres becomes sersible, and gives rise to some curious phenoxnena. MECHANICS OF S"OID S. 117 MOTION OF THE CENTRE OF INERTIA, 126.-Substitute in Equations (A), the values of d2 x, d2 y, and d2 z, given by Equations (34), and we have, because d t is constant, and d2 X d2y and 2z, will each be a common factor for all the elementary masses., d x, I~~P~cos a- i M f. d xn.t d2x' 0, dt2 d2d2- 1 y —-_ ~Pdos-[. d d.2 d2z, l. d -O. ed2 dt2 in which M denotes the entire mass of the body, being equal to n Denote by x y, z, the coordinates of the centre of inertia referred to the movable origin, then, Equations (114), Mf.x = lx and differentiating twice, f. d2x = v m. d2x', Mf.d2y = M. d2y'l j115 which substituted'in the, preceding Equations, give, I P.o aO -M - if.-A if. —07 d12~~dt dt2z dt2 (16:EPcosy -M.d2z, 3 d2z j 118 ELEMENTS OF ANALYTICAL MECHANICS. and if the movable origin be taken at the centre of inertia, then will, da2 0, d2y-=0, d2z = 0; and x, y,, z,, will become the coordinates of the centre of inertia referred to the fixed origin, and we have, d2x:zp.cosa~ - M. — t O,' d2 yj:P. cos s - MO. -, (117) J': P cos 7 - Ml. = 0; J Equations which are wholly independent of the relative positions of the elementary masses min', mn" &c., since their co-ordinates x', y',', &c., do not enter. It will also be observed that the resistance of inertia is the same as that of an equal mass concentrated at the body's centre of inertia. Whence we conclude, that when a body is subjected to the action of any system of extraneous forces, the motion of its centre of inertia will be the same as though the entire mass were concentrated into that point, and the forces applied without change of'intensity and direction, directly to it. This is an important fact, and shows that in discussing the motion of translation of bodies, we may confine our attention to the motion of their centres of inertia regarded as material points. ROTATION AROUND THE CENTRE OF INERTIA. ~ 127.-Now, retaining the movable origin at the centre of inertia, substitute in Equations (B), the values of d2x, d2y, and d2z, as given by Equations (34), and reduce by the relations, M.X x= m.X' % 0, =. M - n. y' = 0, _X1. z = In. z' - 0; MECHANICS OF SOLIDS. 119 and we have, (d2 d2XP) 1 P. (ds' X -Csd.y'M. -0 P.~(Cos~a.~Z~' -.(- -z'- cts.2 ) =~0 (118) \dt2 d2z' - d2Y'); m.\-W-~d - dt2. from which all traces of the position of the centre of inertia have disappeared, and from which we infer that when a free body is acted upon by any system of forces, the body will rotate about its centre of inertia exactly the same whether that centre be at rest or in motion. 12.-And we are to conclude, Equations (117) and (118), that when a body is subjected to the action of one or more forces, it will in general, take up two motions-one of transla.tion. and one of rota..6ion, each being perfectly independent of the other. 129.-Multiply the first of Equations (1 17), by y~, the second by, x,, and subtract the first product from the second; also, the first by Zd, the third by x,, and subtract the second of these products from the first; also the third by y,, and the, second by z,, and subtract the second of these products from the first, and we have,.1(P Cosf3). x,-1(P Cosca). y1-M. ~Y- - Id Y j(11 dt2 I/d - 7,(P Cos 7X).y1-Y.2(P cos/3).z1,-A _ - dz y, =0; Equations from which may be found the circumstances of motion of the centre'of inertia about the fixed origin. 120 ELEMENTS OF ANALYTICAL MECANICS. MOTION OF TRANSLATION. ~130.-Regarding the forces as applied directly to the centre of inertia, replace in Equations (117), the alues P. cos, P. cos /3, and.: P. cos T, by X, Y, and Z, respectively, and we may write, X M. d2x 0 dt2 _~'__ I ~ dt2 Y_ M. days... I 0 Z-M. dt2: from which the accents are omitted, and in which x,, and z, must be understood as appertaining to the centre of inertia. GENERAIL THEOREM OF WORK, VELOCITY AND LIVING FORCE. ~ 131.-Multiply the first of Equations (120) by 2 dx, the second by 2 d y, the third by 2 ci z, add and integrate, we have 2f(Xdx + Ydy + Zdz) -.M.x2d2~z + C o..A ~~~~~~~~~~~dt2 But, dX2 + dy- + dz2 dS2 V2 whence, 2f(Xdx~+Ydy +Zdz) M. V2 + C:.(11 The first term is, ~ 101, twice the quantity of work of the extraneous forces, the second is twice the quantity of work of the inertia, measured by the living force, and the third is the constant of integration. If the forces X, F, Z, be variable, they mnust be expressed in functions of x, y, z, before the integration cani be perfojimed MECHANICS OF SOLIDS. 121 Supposing this latter condition fulfilled, and that the forms of the functions are such as make the integration possible, we may write, F (x y z)- M. V2 +'= O.... (122) and between the limits x, y, z, and x,' y,' z/,', F (x,' y,' z,') - F (x, y, z) - 2 M (V' 2 - 2).. (123) whence we conclude, that the quantity of work expended by the extraneous forces impressed upon a body during its passage from one position to another, is equal to half the difference of the living forces of the body at these two positions. We also see, from Equation (123), that whenever the body returns to any position it may have occupied before, its velocity will be the same as it was previously at that place. Also, that the velocity, at any point, is wholly independent of the path described. If Xdx + Ydy + Zdz -_ O, the extraneous forces will, ~ 101, be in equilibrio, and T d2.tC' that is, the velocity will be constant, and the motion, therefore, uniform. ~132.-Again, multiply the first of Equations (118) by d, the sece ond by d +b, the third by d -; add and reduce by the relations given in Equations (38): we find dax'.ad' d~'a'+,'.adz' P cosadx'+xPcos ldy'+ Pcosydz'-=zm( dt2.r+ d..+ dt ) integrating and replacing the first member by its equal in Equation (68), we have fNd? ~ (dx'2 ~ d y2 —dz'2) d i = -2- m )-{c C..fR 2 ~~~~~~d t' Denoting the lever arm of R by K, the velocity of the molecule m in reference to the centre of inertia by v, &c., and the are described by a i22 ELEMENSTS OF ANALYTICAL -MECIiANICS. point in the plane of the resultant R and of its lever arin, at the unit' distance firom the centre of inertia, by s,, we have f ~~~dx"2+dY'2+dz'22 /Rdr -=f.RK.ds; d s d; d Z =va &c.; whence fR.K.d s, = - rm v + C Adding this to Equation (121), there will result 2j'(xA x + Ydy + Z d ) + 2) fR.K.ds l, M V2+,m v2+ C (121)' From which it is apparent that the quantity ot work impressed upon a body, or the living force with which it will move, is dependent not only upon tlhe intensity of the force, but also upon the distance of its line of direction frolm the centre of inertia. ~ 133.-If E(quation (121) be applied to each one of a collection of elements, of which the masses are,i, m', &c.. there will be as Ilranv cquations as elements; and it thie velocities of these elements be denoted by v v c, we have, by addition, 2Jz/(Xdx+Ydly+ Z(,z)= —mv-C.. (121)" Let the extraneous forces be only those arising fr'om the mutual actions and reactions of the elements upon one another. If the elements it and m' be separated by the distance r, and their co-ordinates be x y z and x'y'z', respectively, then, the reciprocal action being along r, will X - xY- X; Z t x-x' y-y' z-z' COs r~ - O -; CO y —; COS == Cos/3= y; Cos Y= r r1 r cos a'=-*; cos 3'= - -_-; Cosy'= — r-; r r r,red for the element m we have Xdx+ Ydy+Zdz=P ( d x + Y dy + --- d) 7' r 7' for the element m', X'dx'-+lY'dy'+Z'dz'= -P dx-+ dy' + dz) r,F 7' -r and by addition, cr + )' d yt Z d zq+X'd.'+q Y'.y'+ Z'd z' — -[(,.-= ) d (x-')+(y-y) d (y-y')+(z-z') d (z- z')]. r MEC'H'2A'NICS OF SOLIDS. 123' But r = (x - x + (y - yT + (z - T), and differentiating, rd r = (x - x')d (x - x') + (y - y')d (y - y') + (z - z')d (z - z'); so that the second member above reduces to P dr; and Equat.3r (121)" to 2 d r = I m v2 -C.. (121)"' If the elements be invariably connected during the motion, the differentials of r will be zero, and m v2 = C. This is called the conservation of living force. STABLE AND UNSTABLE EQUILIRIUM. ~134.-Resuming Equation (123), omitting the subscript accents, and bearing in mind that the co-ordinates refer to the centre of inertia, into which we may suppose for simplification the body to be concentrated, we may write, M _ I M V2 = F (x'y'z') - F(x y z), in which F(xyz) = f(Xdx + Ydy + Zdz), and dF(xyz) = Xdx + Ydy + Zdz. Now, if the limits x'y'z' and xyz be taken very near to each other, then will x' =x + dx; y' = y + dy; z' = z + dz; which substituted above, give MV'2- i M V2 = F(x + dx, y + dy, z + dz) -F(xyz), and developing by Taylor's theorem, fMV'2 —MV2-=- Adx J- ~Bdy +- Cd -+ A'd x2 -+ B'd y2 + &c. +- D, in which D denotes the sur. of the terms involving the higher powers of dx, dy and dz. 124 ELEMENTS OF ANALYTICAL MECHANICS. If M V 2 be a maximum or minimum, then will Adx + Bdy + Cdz = 0;... (123)' and since Adx + Bdy + Cdz = dF(xyz) = Xdx + Ydy + Zdz, we have, Xdx ~ Ydy +- Zdz = 0. But when this condition is fulfilled, the forces will, Equation (69), be in equilibrio; and we therefore conclude that whenever a body whose centre of inertia is acted upon by forces not in equilibrio, reaches a position in which the living force or the quantity of work is a maximum or minimum, these forces will be in equilibrio. And, reciprocally, it may be said, in general, that when the forces are in equilibrio, the body has a position such that the quantity of action will be a maximum or minimum, though this is not always true, since the function is not necessarily either a maximum or a minimum when its first differential co-efficient is zero. ~ 135. —Equation (123)', being satisfied, we have fMV'2 -M iV2 = i+2 (A'dx2 + Bd y2 + &c. + D).. ~ (124) The upper sign answers to the case of a minimum, and the lower to a maximum. Now, if V be very small, and at the same time a maximum, V' must also be very small and less than V, in order that the second' member may be negative; whehce it appears that whenever the system arrives at a position in which the living force or quantity of work is a maximum and the system in a state bordering on rest, it cannot depart far from this position if subjected alone to the forces which brought it tnere. This position, which we have seen is one of equilibrium, is called a position of stable equilibrium. In fact, the quantity of work immediately succeeding the position in question becoming negative, shows that the projection of the virtual velocity is negative, and therefore that it is described in opposition to the resultant of tho forces, which, as soon as it overcomes the living force already existing, will cause the body to retrace its course. MECHANICS OF SOLIDS. 125 ~136.-If, on the contrary, the body reach a position in which the quantity of work is a minimum, the upper sign in Equation (124) must be taken, the second member will always be positive and there will be no limit to the increase of V'. The body may therefore depart further and further from this position, however small V may be; and hence, this is called a position of unstable equilibrium. ~ 137.-If the entire second member of Equation (124), be zero, then will, I MV'2 -- M V2 = 0, and there will be neither increase nor diminution of quantity of work, and whatever position the body occupies the forces will be in equili. brio. This is called equilibrium of indifference. ~ 138.-If the system consist of the union of several bodies acted upon only by the force of gravity, the forces become the weights of the bodies which, being proportional to their masses, will be constant. Denoting these weights by W', W", W"', &c., and assume ing the axis of z vertical, we have from Equations (87), R z, = W' z' + W"z" + W"' z"' + &c., in which R, is the weight of the entire system, and z, the co-ordinate of its centre of gravity; and differentiating, Rdz, = W'dz' + W"dz" + W"'dz"' + &c.. (125) Now, if z, be a maximum or minimum, then will W' dz' + W" dz" -+ TV"'dz"' + &c. = 0, which is the condition of equilibrium of the weights. Whence, we conclude that when the centre of gravity of the system is at the highest or lowest point, the system will be in equilibrio. In order that the virtual moment of a weight may be positive, vertical distances, when estimated downwards, must be regarded as positive. This will make the second differential of z,, positive at the limit of the highest, and negative at the limit of the lowest point. The equilibrium will, therefore, be stable when the centre of gravity is at the lowest, and unstable when at the highest point. 126 ELEMENTS OF ANALYTICAL MECHANICS Integrating Equation (125), between the limits z,= H, and z, = H', z' = h, and z' = h', &c., and we find, R (H, - H') = W' (h, - h') + W" (h,, -- ") + &c.;. (126) from which we see that the work of the entire weight of the system, acting at its centre of gravity, is equal to the sum of the quantities of work of the component weights, which descend diminished by the sum of' the quantities of work of those which ascend. INITIAL CONDITIONS, DIRECT AND INVERSE PROBLEM. ~139.-By integrating each of Equations (120) twice, we obtain three equations involving four variables, viz.: x, y, z and t. By eliminating t, there will result two equations between the variables x, y and z, which will be the equations of the path described by the centre of inertia of the body. ~ 140.-In the course of integration, six arbitrary constants will be introduced, whose values are determined by the initial circumstances of the motion. By the term initial, is meant the epoch from which t is estimated. The initial elements are, 1st. The three co-ordinates which give the position of the centre of inertia at the epoch; and 2d. The component velocities in the direction of the three axes at the same instant. The general integrals determine the nature only, and not the dimensions of the path. ~ 141.-Now two distinct propositions may arise. Either it may be required to find the path from given initial conditions, or to find the initial conditions necessary to describe a given path. In the first case, by integrating Eqs. (120) twice. we obtain six equations in x, y, z, t, the component velocities, dX d- dt and six arbitrary constants of integration. Making in these equations t = 0, and substituting for the co-ordiltates and conllponent velocities their initial MECHANICS OF SOLIDS. 127 values, the constants become known. These, in the three equations obtained from last integration, give three equations in x, y, z and t, from which, if t be eliminated, two equations in x, y and z, will result. These will be the equations of the path, and the problem will be completely solved. In the second case, the two equations of the path being differentiated twice and divided each time by dt, give only four equations involving three first, and three second differential co-efficients. The inverse problem is, therefore, indeterminate. But Equation (121) being differentiated and divided by the differential of one of the variables, say dx, gives dV2y dz M. d V _ X +- Y. + — Z d.... (127) 2 d X- dx dX which is, a fifth equation involving X, Y, Z, and V. By assuming a value for any one of these four quantities, or any condition connecting them, the other five may be found in terms of x, y and z. VERTICAL MOTION OF HEAVY BODIES. ~ 142;-When a body is abandoned to itself, it falls toward the earth's surface. To find the circumstances of motion, resume Equations (120), in which the only force acting, neglecting the resistance of the air, will be the weight = Mg; and we shall have, Equations (117), Y P cos a = X = Mg. cos a;.P cos = Y= ng. cos /; P cosy =- Z = Mg. cosy; in which M denotes the mass of the body. The force of gravity varies inversely as the square of the distance from the centre of the earth, but within moderate X limits may be considered invariable. The weight will therefore be constant during the fall. Take the co-ordinate z vertical, and positive when estimated downwards, then will cos a= O; cos =0; cos -- =1, 128 ELEMENTS OF ANALYTICAL MECHANICS. and Equations (120) become, after omitting the common factor M, d2 X d2y - d2z dt2 dt2 dt2= and integrating, dx dy dt Ux; d t =v dz dt- v =. g. + U. (128) in which v is the actual velocity in a vertical direction. Making t = 0, we have dz dt = The constants ug, u and uz, are the initial velocities in the directions of the axes x, y and z, respectively. Supposing the first two zero, and omitting the subscript z, from the third, we have, dx dy 0; Y _0; dz v = t gt - u * * * (129) Integrating again, we find x= C; y= C', = gt2 +~ ut + C'", and if when t -0, the body be on the axis z, and at a distance below the origin equal to a, then will x =0; y =O; z = - gt2 + ut + a..... (130) If the body had been moving upwards at the epoch, then would u have been negative, and, Equations (129) and (130), v = g! — u... (131) z I- y t2- ut + a.... (132) MECHANICS OF SOLIDS. 129 If the body had moved from rest at the epoch and from the origin of co-ordinates, then would v be the actual velocity generated by the body's weight; and z - h, the actual space described in the time t; and Equations (129) and (130) would become, v = g t.... (133) h gt2. (134) and elimilnating I, v= /2 h..... (135) whence, we see that the velocity varies as the time in which it is generated; that the height fallen through varies as the square of the time of fall; and that the velocity varies directly as the square root of the height. The value of h, is called the height due to the velocity v; and the value v, is called the velocity due to the height h. If, in Equation (132), we suppose a = 0, we shall have the case of a body thrown vertically upwards with a velocity u, from the origin, and we may write, v = gt-u,.... (136) z =igt2 -ut;...... (137) when the body has reached its highest point, v will be zero, and we find, gt-u = 0; or, which is the time of ascent; and this value of t, in Equation (137), will give, the greatest height, h z, to which the body will attain, h....... (138)i 2g ~143. —In the preceding discussion, no account is taken of the atmospheric resistance. For the same body, this resistance varies as 9l 130 ELEMENTS OF ANALYTICAL MECHANICS. the square of the velocity, so that if k, denote the velocity when the resistance becomes equal to the body's weight, then will M. g.v2 k2 be the resistance when the velocity is v, and in Equations (117), we shall have, P cos a- X Mg cos a +- Mg' —- ~ cos a k2 V2 y P cos - Y _ Mg cos tS + Mg ~.y. cos', P cos y Z =Mg osy + Mg. cos y'; taking the co-ordinate z, vertical and positive downward, then will, COS a = COS a' = 0 cos = cos 3' = 0, cosY- = 1, cos'= —1; and Equations (120) give, d2z 2 M.. — 2 - Mg. d tia kd Omitting the common factor M, and replacing dt2 by its value d t' dv v2) d t 9 1 k2 } whence, k2. dv k ( dv d v gdt_=- k2 _ v - 2 - k + v k - v Integrating and supposing the initial velocity zero, k + v(140) gt k ~k.log (140) k - v MECHANICS OF SOLIDS. 131 which gives the time in terms of the velocity; or reciprocally, 2g t k + v T (141) k -v in which e, is the base of the Naperian system of logarithms, and. from which we find, got g t ek+, e k which gives the velocity in terms of the time. Substituting for v, its value d integrating and supposing the initial space zero, we have Z 1 0 log (e* + e k )... (143) g Multiplying Equation (139) by dz g _= 9 we have, _ k2. v. dv gdz- -_ v2, /and integrating, observing the initial conditions as above, z 2 log 2 2 (144) which gives the relation between the space and velocity. _t As the time increases, the quantity e k becomes less and less, and- the velocity, Equation (142), becomes more nearly uniform; for, if t be infinite, then will e k = -0 and,'Equation (142), =e f the ir equl t the bdys weight making the resistance of the air equal to the body's weight. 1-32 ELEMENTS OF ANALYTICAL MECIHANICS. ~ 144.-If the body had been moving upwards with a velocity v, then, taking z positive upwards, would, Equations (120), d2 Z. t dt --.Mg - M dv d2z substituting d for d-, and omitting the common factor, we find, k.dv gdt(145) kZP + v2 k;.. (145) integrating, tan - + C; k; and supposing the initial velocity equal to a, we find a C = tan and, -~ ~j -1 a gt tan -= tan..; ~ - ~ r -(146) Taking the tangent of both members and reducing, we find a-k. tan gv k t. (147) gt k + a. tan which may be put under the form, a.cos -- k. sin k k v = k.~ (148) gt gt a. sin - +. cost * k dz Substituting for v its value dt' integrating, and supposing the initial space zero, we have z -. log ( k sin +- cos (14) (a~k MECHANICS OF SOLIDS. 133 Multiplying Equation (145), by dz V =, and we have, k2. v.d v g. dz = — + and integrating, with the same initial conditions of v being equal to a, when z is zero, there will result, k2 k2 - a2(150) z= - log + (150) ~ 145.-If we denote by A, the greatest height to'which the body will ascend, we have z = h, when v 0, and hence, k k2 + a2 h = -2' log.. (151) Finding the value of t, from Equation (146), we have, t = —- tan - -tan.... (152) from which, by making v = 0, we have, k -1 a a -.tan (153) which is the time required for the body to attain the greatest elevation. Having attained the greatest height, the body will descend, and the circumstances of the fall will be given by the Equations of ~ 143. Denoting by a', the velocity when the body returns to the point of starting, Equation (144), gives, k2 k2 h = -.log k and placing this value of h equal to that given by Equation (151), there will result, k2 k2 + a2 = - a... 134 ELEMENTS OF ANALYTICAL MECHANICS. whence, at2 = a2 + aa2 +- k2; that is, the velocity of the body when it returns to the point of departure is less than that with which it set out. Making v = a' in Equation (140), we have, k k + a' tf =2 log - a'" and, substituting for a', its value above, t~='2-.log/a + *2 a * (154) a value very different from that of t,, given by Equation (153), for the ascent. Multiplying both numerator and denominator of the quantity whose logarithm is taken, by V-0'17W - a, the above becomes, k k* V -log (155) g *2+/k2 a- a Adding Equations (153) and (155), we have, -1 a ] t + t = —tan - a log] r', making = ta + t, gt 1a k -k = tan -+ log ~ (156) If a ball be thrown vertically upwards, and the time of its absence from the surface of the earth be carefully noted, t will be known, and the value of k may be found from this' equation. This experiment being repeated with balls of different diameters, and the resulting values of k calculated, the resistance of the air, for any given velocity, will be known. MECHANICS OF SOLIDS. 135 PROJECTILES. 146.-Any body projected or impelled forward, is called a projectile, and the curve described by its centre of inertia, is called a trajectory. The projectiles of artillery, which are usually thrown with great velocity, will be here discussed. ~147.-And first, let us consider what the trajectory would be in the absence of the atmosphere. In this case, the only force which acts upon the projectile after it leaves the cannon, is its own weight; and, Equations (117), I P cos a = X = Mg cos a, P cos = Y = Mg cos/, Pcos = Z = Mg cosy. Assuming the origin Z at the point of de. parture, or the mouth of the piece, and taking the axis z vertical, and positive upwards, then will. I cos aC 0; cos - = 0; cos y — 1; and, Equations (120), d2 x d2 z d2 Z - - 0 f0 M 2 _ 0 - Mg' dt2 - dt2 t My; and integrating, omitting iM, dx dy dz d t uy dt y- d - -g t- + u (157) Integrating again, and recollecting that the initial spaces are zero, we have, x - u~ t; y = uy. t; z- = -jg t2 + u. t (158) 1836 ELEMENTS OF ANALYTICAL MECHANICS and eliminating t, from the first two, we obtain, U y -, x; U 2 which is the equation of a right line, and from which we see that the trajectory is a plane curve, and that its plane is vertical. Assume the plane z x, in this plane, then will y- = 0, and Equa. tions (158), become, x-=X.t; z = — g t2 +- u.t. ~ (159) Denote by V, the velocity with which the ball leaves the piece, that is, the initial velocity, and by a, the angle which the axis of the piece makes with the axis x, then will, V. cosa, and V. sin a, be the lengths of the paths described in a unit of time, in the direction of the axes r and z, respectively, in virtue of the velocity V; they are, therefore. the initial velocities in the directions of these axes; and we have, f = V cos a; u = V. sin a; which, in Equations (159), give = V.cosa.t; z = —gt2 +. sina.t ~ (160) and eliminating t, we find z-= x$tan a- 2 V22. COS2 a or substituting for V its value in Equation (135), z = x tan a- h. * * s (161) 4 h. cos 2a which is the equation of a parabola. MECHANICS OF SOLIDS. 137 ~ 148.-The angle a is called the angle of projection; and the horizontal distance A D), from the place of departure A, to 6 the point D, at which the projectile attains the same.A level, is called the range. To find the range, make z = 0, and Equation (161) gives x- 0, and x = 4hsin a cosa = 2hsin 2a, and denoting the range by R, R = 2 h. sin 2a. (162) the value of which becomes the greatest possible when the angle of projection is 450. Making a-_ 450, we have R =-2h ~ ~... (163) that is, the maximum range is equal to twice the height due to the velocity of projection. From the expression for its value, we also see that the same range will result from two different angles of projection, one of which is the complement of the other. ~ 149.-Denoting by v the velocity at the end of any time t, wq have, ds2 dz2 + dx2 dt2 d t2 or, replacing the values of dz and dx, obtained from Equations (160), v2 = V2-2 V.g.t. sin a + 92 (164) and eliminating t, by means of the first of Equations (160), and replacing V2, in the last term by its value 2g h, 2 V2 _ 2g. tanll a. x + g 2h o * (165) 2h.cos2a 138 ELEMENTS OF ANALYTICAL MECHANICS. in which, if we make x = 4. sin a cos a, we have the velocity at the point D, V2 = V2, which shows that the velocity at the furthest extremity of the range is equal to the initial velocity. Differentiating Equation (161), we get dz a' d = tall - tan a.-. dx 2 h. Cos2a (166) in which 8 is the angle which the direction of the motion at any instant makes with the axis x. Making tan 0 = 0, we find x = 2 h. cos a. sin a, which, in Equation (161), gives z = h. sin2 a, the elevation of the highest point. Substituting for x, the range, 4 h cos a sin a, in Equation (166), tan - tan a, which shows that the angle of fall is equal to minus the angle of projection. ~ 150.-The initial velocity V being given, let it be required to find the angle of projection which will cause the trajectory to pass through a given point whose co-ordinates are x = a and z _ b. Substituting these in Equation (161), we have as b -=a tan a -, 4 h. co$2 a from which to determine a. Making tan a- -p, we find Cos2 a - 1 +- 92 MECHANICS OF SOLIDS. 139 which in the equation above, gives 4h.b + a2 - 4h.a.q +' a2q2 = 0; whence, 2h 1 q = tan -= —-- -14h2-4hb-a2. (167) a a The double sign shows that the object is attained by two angles, and the radical shows that the solution of the problem will be possible as long as 4 h2 > 4hb + a2 Making, 4 2 - 4 h. b - a2 = 0, the question may be solved with only a single angle of projection. But the above equation is that of a parabola whose co-ordinates are a and b, and this curve being constructed and revolved about its vertical axis, will enclose the entire space within which the given point must be situated in order that it may be struck with the given initial velocity. This parabola will pass through the farthest extremity of the maximum range, and at a height above the piece equal to h. ~ 151.-Thus we see that,the theory of the motion of projectiles is a very simple matter as long as the motion takes place in vacuo. But in practice this is never the case, and where the velocity is con siderable, the atmospheric resistance changes the nature of the trajectory, and gives to the subject no little complexity. Denote, as before, the velocity of the projectile when the atmospheric resistance equals its weight, by k, and assuming that the resistance varies as the square of the velocity, the actual resistance at ally instant when the velocity is v, will be,..g.2 MC2 v ='Me v2 P 140 ELEMENTS OF ANALYTICAL MECHANICS, by making, 9 -c k2 The forces acting upon the projectile after it leaves the piece being its weight and the atmospheric resistance, Equations (120), become, d2 x Me d — - Mg. cos c + Mc. v2. cos a', 31 d2 =_ Mg. cos ji + M. v2. cos i', d t' d2 z M. d t Mg. cos y + Mc. v2. cos 7' d t. Taking the co-ordinates z vertical, and positive when estimated upwards, cos = 0; cosi = 0; cos = -1, and because the resistance takes place in the direction of the trajec. tory, and in opposition to the motion, if the projectile be thrown in the first angle, the angles a', /3', and 7', will be obtuse, cos a'; os d; cos y - dzY;' ds ds ds' and the equations of motion become, after omitting the common factor M, da C ds d2z 2 dz'W t2 ds ffrom the first two we have, by division, d2y d2 x dy -x i MECHANICS OF SOLIDS. 141 and by integration, logdy = log dx log C; and, passing to the quantities, dy = Cdx. Integrating again, we have, y = Cx + C'; in which, if the projectile be thrown from the origin, C' = 0, thus giving an equation of a right line through the origin. Whence we see that the trajectory is a plane curve, and that its plane is vertical through the point of departure. Assuming the plane z x, to coincide with that of the trajectory, and replacing v2, by its value from the relation, d s2 d t -- v2' we have, d2x ds dx d t2 d-t dt' d2z ds dz dt2- -dtg-c dtFrom the first we have, d2 x d t2 ds d x.c. di' c t and by integration, dx log dt - -c. s + C. Denoting by e, the base of the Naperian system of logarithms, and making C = log A, the above may be written, log dx log d.s XC. x loge+logA, dt 142 ELEMENTS OF ANALYTICAL MECHANICS. and passing from logarithms to the quantities, dx -cs d = A.e (169) Denoting by V, the initial velocity, and by a, the angle of pro. jection, we have, by making s = 0, dx - =A = f cos a, dt which srbstituted above, gives dx = V. cos.e...... (170) To integrate the second of Equations (168), make dz dx a! K = P. -It. - *.....(171) in which p is an additional unknown quantity. Differentiating this equation, dividing by dt, and eliminating d2 x firom the result, —, by its value in the first of equations (168), we have, d2z dp dx ds dx _P = -'-~P.C. - -' and substituting this value in the second of Equations (168), we dz have, after eliminating - by its value, obtained from Equation (171), dx dp. g..... (172) d t dt and dividing this by the square of Equation (170), dp dt 2 _ g d.V2 e CO. (173) d t MECHIANICS OF SOLIDS. 143 but regarding z and p as functions of x, we have, Equation (171), dz dt dz d d * * * * (174) dt and, dp dt dp d t whence, making V2 2gh, Equation (173) becomes dp _ 2cs dp3 e,............. $ (175) dx 2 h. os2(175) and multiplying this by the identical equation, dx. /l p2 ds, obtained from Equation (174), we find, 2c, - e ds +2.=d — = 2h.cos2a; and integrating, 2cet p./1+ v2 + log (p + ) 2 c. os; (176) in which C is the constant of integration; to determine which, make s - O; this gives p = tan a; and 2c cos2 ta tan 1. + tan2a+ log(tan a+- +/1 +tan2 a). (177) From Equation (175) we have, -2 c s dx 2 - h. cos2a. e dp from Equation (171), dz =p.dx; 144 ELEMENTS OF ANALYTICAL MECHANICS. from Equation (172), gd2 =- - dx.dp; and eliminating the exponential factor by means of Equation (176), we find, c.dx dp. (178) p V1+- p+2 _ log (p + - r-2) - C pdp c. d z = dp (179) p 1- 2 + log(p + T p2) C — dp ~rK. d t -; (180) / C-p 1 p2 - log(p+ p2) Of the double sign due to the radical of the last equation, the negative is taken because p, which is the tangent of the angle made l)y any element of the curve with the axis of x, is a decreasing function of the time t. These equations cannot be integrated under a finite form. But the trajectory may be constructed by means of auxiliary curves of which (178) and (179) are the differential equations. From the first, we have, dx = T. dp;....... (181) and from the second, dz= T.p.dp;...(... (182) in which, I 1 T =- -- () p 21 +p2+ log(p / + p2)C i (183) and dividing Equations (181) and (182), by dp, dx dx T........(184) dp dz.;....... (185) d Pt MECHANICS OF SOLIDS. 145 Now, regarding x, p, and z, p, as the variable co-ordinates of two auxiliary curves, T, and T. p, will be the tangents of the angles which the elements of these curves make with the axis of p. Any assumed value of p, being substituted in T, Equation (183), will give the tangent of this angle, and this, Equation (184), multiplied by dp, will give the difference of distances of the ends of the corresponding element of the curve from the axis of p. Beginning therefore, at the point in which the auxiliary curves cut the axis of p, and adding these successive differences together, a series of ordinates x and z, separated by intervals equal to dp, may be found, and the curves traced through their extremities. At the point from' which the projectile is thrown, we have, x —O; z=O; p=tana, and the auxiliary curves will cut the axis of p, in the same point, and at a distance from the origin equal to tan a. Let A B, be the axis of p, and A C, the axis of x and of z; take AB = tan a, and let BzD, and B x F, be constructed as above. Draw the axes Ax and Az, through the point of departure A, Fig. (i); draw any ordinate c z, xA to the Z 2 auxiliary curves Fig. (1); lay off Ax, Fig. (2) equal to Cx, Fig. (1), and draw through xA x,, the line x, z, parallel to the axis Az, and equal to c z, Fig. (1); the point z, will be a point of the trajectory. The range AD, is equal to ED, Fig. (1 ). 10 146 ELEMENTS OF ANALYTICAL MECHANICS. By reference to the value of C, Equation (177), it will be seen that the value of T, Equation (183), will always be negative, and that the auxiliary curve whose ordinates give the values of x, can, therefore, never approach the axis of p. As long as p is positive, the auxiliary curve, whose ordinates are z, will recede from the axis p; but when p becomes negative, as it will to the left of the axis.A C, Fig. (1), the tangent of the angle which the element of the curve makes with the axis p, will, Equation (185), become positive, and this curve will approach the axis p, and intersect it at some point as D. The value of p will continue to increase indefinitely to the left of the origin A, Fig. (1), and when it becomes exceedingly great, the logarithmic term as well as C, and unity may be neglected in comparison with p, which will reduce Equations (178) and (179) to dp dp dx; dz -; c.p2 c.p and integrating, 1 1 X= -C' -; z= C"+ — logp, CP C which will become, on making p very great, = C'; z = C" +- logp, which shows that the curve whose ordinates are the values of x, will ultimately become parallel to the axis p, while the other has no limit to its retrocession from this axis. Whence we conclude, th:at the descending branch of the trajectory approaches more and more to a vertical direction, which it ultimately attains; and that a line G L, Fig. (2), perpendicular to the axis x, and at a distance from the point of departure equal to C', will be an asymptote to the trajectory. This curve is not, like the parabolic trajectory, symmetrical in reference to a vertical through the highest point of the curve; the angles of falling will exceed the corresponding angles of rising, the range will be less than double the abscissa of the highest point, and the angle which gives the greatest range will be less than 45.o iECHANICS OF SOLIDS. 147 Denoting the velocity at any instant by v, we have 2 dx2 + dz2 dox and replacing dx2 and dt2 by their values in Equations (178) and (180), we find l g(1+P2 (186) c p-l+p2 log(pf+ I+p2) and supposing p to attain its greatest value, which supposes the projectile to be moving on the vertical portion of the trajectory, this equation reduces, for the reasons before stated, to = s4= k; which shows that the final motion is uniform, and that the velocity will then be the same as that of a heavy body which has fallen in vacuo through a vertical distance equal to2 = 2. ~ 152. —When the angle of projection is very small, the projectile rises but a short distance above the line of the range, and the equation of so much of the trajectory as lies in the imme- diate neighborhood of this line may easily be found. For, the angle of projection being very small, p C _C will be small, and its.second power may be neglected in comparison with unity, and we may take, ds-dz; and s —x; which in Equation (175), gives, 2 Ca dp _ d2 z e (187) dx - dx2 - 2h.cos2 c 148 ELEMENTS OF ANALYTICAL MECHANICS. Integrating, 2 cx dz e dx 4 c.h. cos2 a; dz makitg x = 0, we have d= tan o, whence, C tanl a 4ch. cosa a which substituted above, gives, 2cx dz e 1 --— = tana- + dx 4 c. h.cos2 4c. h. cos2 a and integrating again 2cz e x z - tan a. x - + + ff 8 2. h. cos2a 4c. h. Cos2 a making x = 0, then will z = 0, and =/ 1 Cl 2 8 C2. h. cos2 a hence, z = tanax- 8ch.cosa e -2cx — 1. (188) From Equation (172), we have, g.dt2= -dx.dp, and substituting the value of dp, from Equation (187), CX e.dx dt_ /2 g h. cos a and integrating, making x = 0, when t = 0, -- ~ — 1 ~ ~~C t__-1. (e -1). * * - t(189) c 2h. cos a MECHANICS OF SOLIDS. 149 which will give the time of flight to any point whose horizontal distance from the piece is equal to x. ~ 153. —Let the projectile fall to the ground at the point D, and denote the co-ordinates of this point by x = I, and z = X, and suppose the time of flight or t = r. These values in Equations (188) and (189), give — 8c2.h. cos2 a ( - 1.tana) = -e-2cl- 1 ~ (189)' cos a..c.. 2gh = - 1.(189)" When the two constants h and c, as well as a and X, are known, these equations will give the horizontal distance 1, and the time of flight. Conversely, when the quantities a, l, X and r are known, they give the co-efficient of resistance c, and the height h, due to the velocity of projection, and therefore, Equation (135), the initial velocity itself. Eliminating the height h, we find - 4 (X -. tan a) (e z 1)2 = -q*2.(e2Cl- 2cl - 1);. (189)"' from which the value of c may be found, and one of the preceding equations will give h, or the initial velocity. It may be worth while to remark that if the exponential term in Equation (188) be developed, and c be made equal to zero, which is equivalent to supposing the projectile in vacuo, we obtain Equation (161). ~ 154.-Assuming that the resistance of the air varies as the square of the velocity, some idea may be formed of its actual intensity from the fact that a twenty-four-pound ball'projected with a velocity of 2,000 feet in vacuo, and under an angle of 450, would have a range of 125,000 feet; whereas actual experimnent in the air shows it to be but 7,300 feet-about one-seventeenth of the former. Many circumstances qualify both the path and velocity of projectiles. The law of the resistance may be the samne for all figures, but it is known, from actual trial, not to be that of the square of the velocity, except for very small rates of motion. For the same velocity, the in 150 ELEMIENTS OF ANALYTICAL MECHANICS. tensity of the resistance varies with the size and figure of the ball. Much depends upon the facility with which the compressed air in front may escape latterly and make its way to the rear. The actual resistance at any instant is composed of two terms, the one due to the inertia of the displaced particles, the other to the differenet of atmospheric pressure, as such, in front and rear. If during the motion the air could close in behind and exert the same pressure as in front, the resistance would be wholly due to inertia. If the ball were at rest, and all the air removed in rear of the plane of largest section perpendicular to the trajectory, the resistance would be due entirely to the barometric pressure on the extent of this section. Both terms of the resistance must be variable and a function of the velocity, till the latter is. so great as to leave a vacuum behind, when the barometric term would become constant. From a careful and elaborate investigation of the numerous experiments upon this subject, Col. Piobert has constructed this empirical formula for spherical projectiles, viz.: in' which p is the resistance in kilogrammes, v the velocity, 7r the ratio of the diameter to the circumference, r the radius of the ball, A the resistance on a square metre when the velocity is one wdetre, and v, the velocity which would make the resistance measured by the second term equal to that measured by the first. ~ 155.-If the ball be not perfectly homogeneous in density, the centre of inertia will, in general, be removed from that of figure; the resultant of the expansive action of the powder will pass through the latter centre and communicate to the ball a rotary motion about the former. The atmospheric resistance will be greater on the side of the greatest velocity, and deflect the projectile to the opposite side. MECHANICS OF SOLIDS. 151 ROTARY MOTION. ~ 156.-Having discussed the motion of translation of a single body, we now come to its motion of rotation. To find the circumstances of a body's rotary motion, it will be convenient to transform Equations (118) from rectangular to polar co-ordinates. But before doing this, let us premise that the angular velocity of a body is the rate of its rotation about a centre. The angular velocity is measured by the absolute velocity of a point at the unit's distance from the centre, and taken in such position as to make that velocity a maximum. ~ 157.-Both members of Equations (38) being divided by dt, give d x' d t d9 dt dt dt d y d, da dt dt dt'.. (190) dz',, d4S di z' d At y d t * d t in which the first members taken in order, are the velocities of any element, as ma in the direction of the axes x, y, z, respectively, zn reference to Pt:e centre of inertia, ~ 75, while dr d~2 dp dt dt' dt are the angular velocities about the same axes respectively. Denoting the first of these by v:, the second by v,, and the third by va, we have do V d V r dp,_ - v~; = v; -d' J;. 191) 152 ELEMENTS OF ANALYTICAL MECHANICS and Equations (190) may be written d x' - dt' d y' dt x'. v, - z' v t.... (192) _ zt. y'. v, - x. Vy. ~ 158.-If an element m be so situated that its velocity shall be equal and parallel to that of the centre of inertia, then, for this element, will each of the first members of Equations (192) reduce to zero, and z'. - y'.V,- 0, x'. v — z'. v 0.. (193) y'.!v - x'.vY 0; the last being but a consequence of the two others, these equations are those of a right line passing through the centre of inertia, every point of which will have a simple motion of translation parallel and equal to that of the centre of inertia. The whole body must, for the instant, rotate about this line, and it is, there. fore, called the Axis of Instantaneous Rotation. ~ 159.-Denote by a, Z k,, r,, the angles which this axis makes with the co ordinate axes x, y, z, respectively. Then, taking any point on the instantaneous axis, will, Cos a, = V',2 + y/2 + zx' COS +;v/x'2 + y'~ + z'2 MECHANICS OF SOLIDS. 153 and eliminating x', y' and z', by Equations (193), vz, COS ad V/V2 + Vy2 + V 2 COs,... (194) CO2 + v 2 + co V 2 + V2 + v22 which will give the position of the instantaneous axis as soon as the angular velocities about the axes are known. ~160.-Squaring each of Equations (192), taking their sum and extracting square root, we find dx'2 +- dy'2 + dz'2 d 2 = V y yz )2 + (.-zv) +(y'.V,-xVy)2 Replacing v., vy and v, by their values obtained by simply clearing the fractions in Equations (194), this'becomes v-V 2 + v 2 + 2 X CX2 + l 2 + Z2-(cosa+ycos e os,)2 which is the velocity of any element in reference to the centre of inertia. Making x,2 + y,2 + z2 = 1, we have the element at a unit's distance from the centre of inertia; and making x' cos a, + y'cosi, + z'cos, 0, ~ ~ ~ (195) the point takes the positions giving the maximum velocity. In this case v becomes the angular velocity, and we have, denoting the latter by vi, vi = V/VX2 + VY + Vz2...... (196) 151 ELEMENTS OF ANALYTICAL MECHANICS. Equation (195) is that of a plane passing through the centre of inertia, and perpendicular to the instantaneous axis. The position of the co-ordinate axes being arbitrary, Equation (196) shows that the sum of the squares of the angular velocities about the three co-ordinate axes is a constant quantity, and equal to the square of the angular velocity about the instantaneous axis. ~ 161.-Multiply Equation (196), by the first of Equations (194), and there will result Vi * cos =. Co....... (197) whence the angular velocity about any axis oblique to the instantaneous axis, is equal to the angular velocity of the body multiplied by the cosine of the inclination of the two axes. ~ 162. —Equation (196) gives v, when v., vy, v, are known. To find these, resume Equations (118), and write for the moments of the extraneous forces in reference to the axes,' y z,' through the centre of inertia, i,, M,, L,, respectively; then will f t* YWet ~ $ d2 x' (d 2z' - d2 * x'> = M t'.' (198) differentiating the first of Equations (192), with respect to 1, we find ~d2 x d z' d' dv dv — v _ -.xv =r. ~ z 1V dz' dd t2 and replacing -dt — and -—, by their values given in the second and third of Equations (192), d'd t d (Vy2 I +v. +IdY dv y d d v -V- q_ V. y t y z dt 71'y~~~~~~~~~~~~~ MECHANICS OF SOLIDS. 155 in the same way d Y= (Yr?+ v2)* y'+ y. V2. 2I + Vy. Z. f + dvz., dvz ~.Z d* y dv d dt2 - 2 _ vy2).z' + vz. v.x' + v vy y + d+, dt - at2 d t and these values in the first of Equations, (198), give (vY2_VX2) *m. x' y' {tz'~ d%\ + (v.v --- )*m.z''X m (-Fx-. d5.v =2 -. V m.zy, =L,.(199) d - y. Zm. - + M t 1m + y12) Similar equations will result from the remaining two of Equations (198); then by elimination and integration, we might proceed to find the values of vx, vy and v,, but the process would be long and tedious. It will be greatly simplified, however, if the co-ordinate axes be so chosen as to make at the instant corresponding to t, mf'y' = 0; Mmz'y' = 0; Zmz'x' = 0; ~ ~ (200) which is always possible, as will be shown presently. This will reduce Equation (199) to d t*2 m (y'2 + X'2) + v. Vy..m (2 y'2) - L The other two equations which refer to the motion about the axes y' and x', may be written from this one. They are, dv.tv M2 (x'2 + Z'2) +' v. V,. Zm(z'-x'2) M dvm d z M m(y,2 + z,2) + V5. m (v, v. M) = N_,v dt' z' 156 ELEMENTS OF ANALYTICAL MECHANICS. The axes x', y', zt, which satisfy the conditions expressed in Equations (200), are called the principal axes of figure of the body. And if we make Im.(y'2~+X'2)2= C, Em. (x'2 + Z'2) = B... (201) m. (y'2 + z'2) = A; J we find, by subtracting the third from the second, y n. (X'2 - y'2) B _ -A, the first from the third, m. (z'2 - X'2) = A - C, and the second from the first, Y m. (y,'2_ -Z2) = C- B; which substituted above, give, dv C.Tt + d,-, J y (B- A) = L,, B. d Y VX..(A- C) = M,. (202) dt, dvz Adt. +,,.z.(C-B) = A. J By means of these equations, the angular velocities v,,vy, v., must be found by the operations of elimination and integration. ~ 163.-It is plain that the quantities C, B and A, are constant for the same body; the first being the sum of the products arising from multiplying each elementary mass into the square of its distance from the principal axis z', the second the same for the principal axis y', and the third for the principal axis x'. The sum of the products of the elementary masses into the square of their distances from any axis, is called the moment of inertia of the body MECHANICS OF SOLIDS 157 in reference to this axis. A, B and C are called principal moments of inertia. ~ 164. —Through any assumed point there may always be drawn one set of rectangular axes, and, in general; only one which will satisfy the conditions of Equations (200). To shlow this, assume the formulas for the transformation from one system of rectangular axes to another, also rectangular. These are'= x.cos (' x) + y cos (x' y) - z cos (X'z),7 y' = x cos (y'x) + y. cos (y'y) + z. cos (y'z),... (203) x' = C () y cos' y) + z.cos (z' z),J in which (x' x), (y' x) and (z'x), denote the angles which the new axes x', y', z', make with the primitive axis of x; (x' y), (y' y) antd (z'y), the angles which the same axes make with the primitive axis of y, and (x' z), (y'z) and (z'z), the angles they make with the axis z. Assume the common z origin as the centre of a Z; sphere of which the radius is unity; and conceive the / points in which the two sets of axes pierce its sur- -_ face to be joined by the arcs of great circles; also let these points be connected with the point N, in which the intersection of the planes xy and x'y' pierces the spherical surface nearest to that in which the positive axis x pierces the same. Also, let 8 = Z'A Z = X' NX, being the inclination of the plane x' y' to that of xy. = NAX being the angular distance of the intersection of the planes x y and x' y', from the axis x. = _ N'AX' being the angular distance of the same intersection from the axis x'. 158 ELEMENTS OF ANALYTICAL MECHANICS Then, in the spherical triangle X' NX, cos (x' x) = cos -. cos q+- sin -j. sin q,. cos O; In the triangle Y'NX, the side Y' = + p, and cos (y' x) = - cos 4. sin + sin +.cos g.cos 8, In the triangle Z'NX, the side NZ'- 2, and cos (z'x) = sin 4. sin O. And in the same way it will be found that cos (x'y) - sin i.cos g + cos sin q. co~; cos (y'y) = sin. sin qp + cos 4.cos q. cos8; cos (z' y) = cos. Sill 0; cos (x' z) = - sin q. sin O; cos (y' z) - cos q. sin O; cos (z' z) = cos 0; and by substitution in Equations (203), x = z (sin.sin q.cos O + cos. cos q) + y (cos.. sin. cos - sin 4. cosq) - z sin q. sin 8, y' = x (sin 4.. cos qc. cos - cos -4. sin q) + y (cos 4. cosgq. cos. + sin 4. sin q) - z cos p. sin 8, z' sin x sinin 8 + y cos 4,. sin 0 + z cos f; or making, for sake of abbreviation, D = cos 4 - y sin 4, E = x sin 4,. cos 8 + y cos 4. cos 8 - z sin 8, the above reduce to x$ = E. sin q~- +D. cos q, y' -. cos qg - D. sin q, z'= _x.sin d. sin +~ y. cos 4. sin 0 + z.cos. MECHANICS OF SOLIDS. 159 Substituting these values in the equations m. x'. y' = O;: m.'. z' = 0;.n. y'. z" 0; we obtain from the first, sinl. cos9. I:,n (E2 - D2) + (CS2 -- sin2 p) X m E. D = 0, or, replacing sin q. cos 9, and cos2 q - sin2 p, by their equals ~ sin 2, and cos 2, respectively, sin 2. (E2 - D2) + 2 cos 2. 2. E = 0;.~ ~. (204) and from the third and second, respectively, cos. m. E.z' - sin.? m D). z' = 0, - -. (205) sin. m. E. z'+ cos. m D. z' = 0.. * * (206) Squaring the last two and adding, we find (Xm..E.')2 + (Z *. D. z')2 - 0. which can only be satisfied by making m. E. z' 0;. 2m. D. z' = 0. These equations are independent of the angle 9, and will give the values of 4, and I; and these being known, Equation (204) will give the angle 9. PReplacing E and D by their values, we have =E. z' sin. cos 0 (x2 sin2 4, + 2 x y sin 4 cos 4 + y2 cos2 - z2) + (cos2 L - sin2 L) (x z sin 4 + y z. cos 4,), D. z' sin {x y (cos2 4 - sin2 4,) + (X2 - y2) sin 4 cos. } + cos 0 (x z os - y z sin ). and assuming'mx2 =A'; my2 = B'; Xm z2 C' mxy = E'; zmxz = F'; Zn yz - H', and replacing sill. cos I, and cos2 L - sin2 8L, by their respective values, - sin 2 L, and cos 2 L, Equations (207) become sin 2 in( 2 Esin24, + 2E'sin 4 cos + B' cos2,- C') t - + 2 cos 2 (F' sin 4- H' cos,5) 160 ELEMENTS OF ANALYTICAL MECHANICS. sin { E'l. (cos2 - sin2 +) + (A' - B'). sin, cos } O + cos d (F' cos - I' sin) ) in which A', B', C', E', F' and H', are constants, depending only upon the shape of the body and the position of the assumed axes x, y, z. Dividing the first by cos 2 0, and the second by cos 0, they become tan 2. (A' sin42 + 2 E' sin a cos + ~ B' cos 2 -2 ) + 2 (F sin 4 + I cos 4) tan. {E' (COS2 - sin2 4) + (A'- B') sin 4 cos 4 = o 0.(207)' + F' cos + - H' sin ) ( From the first of these we may find tan 2 0, and from the second, tan 0, in terms of sin +, and cos 4; and these values in the equation 2tan 0 tan 2 1 -- tan2 8 (208) will give an equation from which 4 may be found. In order to effect this elimination more easily, make tan U4 =, whence Sill = Cos'I-U2 I+u2 making these substitutions above, we find tn20 =- 2(F' u H') 1T+u-2 A'u 2 + 2 E' U +B' - C' (1 u2) (F, - zt')~~ + U2 tan = -(F'-H') u tan =-E- -I' -U2) + (A' — B') u which in Equation (208) give B' F'-! C! -E! HI E'(1 -u2) + (A'-B')u} -.-( C' H' - A''+~E'')u = 0 (209) + (F'u +'). (F' - HI, )2 MECHANICS OF SOLIDS. 161 which is an equation of the third degree, and must have at least one real root, and, therefore, give one real value for {. This value ~being substituted in either of the preceding equations, must give a real value for 0, and this with {, in either of the Equations (205) or (206), a real value for pq; whence we conclude, that it is always possible to assume the axes so as to satisfy the required conditions, and that through every point there may be drawn at least one set of principal axes at right angles to each other. The three roots of this cubic equation are necessarily real; and they represent the tangents of the angles which the axis x makes with the lines in which the three co-ordinate planes x' y', y' z', x' z', cut that of x y; for there is no reason why we should consider one of these angles as given by the equation rather than the others, and the equations of condition are satisfied when we interchange the axes x' y' z'. Hence, in general, there exists only one set of principal axes. If there were more, the degree of the equation would be higher, and would, from what we have just said, give three times as many real roots as there are systems. If E' =H'- F' = 0, Equation (209) will become identical; the problem will be indeterminate, have an infinite number of solutions, and the body consequently an infinite number of sets of principal axes. Such is obviously the case with the sphere, spheroid, &c. MOMENT OF INERTIA) CENTRE AND RADIUS OF GYRATION. ~165.-The quantities A, B and C, in Equations (201) are the moments of inertia of the body in reference to the principal axes. To find these moments" in reference to any other axes having the same origin as the principal axes, denote by x', y', z', the co-ordinates of m referred to the principal axes; by x, y, z, the co-ordinates of the same element referred to any other rectangular system having the same origin; and by C', the moment of inertia referred to the axis z; then from the definition, C' = Zm.(x2 + y2) = Zmx2 +.my2; 11 162 ELEMENTS OF ANALYTICAL MECHANICS. but by the usual formulas for transformation, x- ax' + by' + c', y = a'x' + b'y' + c'z', z = a" x' + b"y' ~- c"z', in which a, b, &c., denote the cosines of the angles which the axes of the same name as the co-ordinates into which they are respectively multiplied make with the axis corresponding to the variable in the first member. Substituting the values of x and y in that of C', and reducing by the relations, I mx'y' =0; Zmx'z' =; Zmy'z'= O; and we have, C' = a"/2 m (y'2+Z'2) + bf"2. 2 m (x'2 + 2) + c"2. Z rn (x'2+y,2); and by substituting A, B and C for their values, this reduces to C' = a"2 A + b"2 B + c"2 C. (210) That is to say, the moment of inertia with reference to any axis passing through the common point of intersection of the principal axes, is equal to the sum of the products obtained by multiplying the moment of inertia with reference to each of the principal axes, by the square of the cosine of the angle which the axis in question makes with these axes. ~ 166.-Let A, be the greatest, and C, the least of the moments of inertia, with reference to the principal axes; then, substituting for a'2, its value, 1 - b"2 - c"2, inl Equation (210), we have C' = - b"2 (A - B) -c"2 (A - C). (211) By hypothesis, A - B, and A- C6', are positive; therefore, C' is always less than A, whatever be the value of b", and c". Again, substituting for c"2 its value 1 -a"2 - b"2 in Equation (210), we get C' = C + a"2 (A- C) + b"2(B - C) ~ ~ ~ (212) and C' must always be greater thail C. MECHANICS OF SOLIDS. 163 Whence, we conclude that the principal axes give the greatest and least moments of inertia in reference to axes through the, same point. If A be equal to B, then will Equation (211) become C' = (1 - Cf2) A+ c"12 C,... (213) and this only depending upon c", we conclude that the moment of inertia will be the same for all axes making equal angles with the principal axis, z'. The moments of inertia, with reference to all axes in the plane x' y', are, therefore, equal to one another. But all the axes in the plane x' y', which are at right angles to one another, are, ~ 164, when taken with z', principal axes, and we, therefore, conclude that the body has aln indefinite number of sets of principal axes. If, at the same time, we have A = B = C, then will Equation (210) reduce to C' -- A = B. that is, the moments of inertia are all equal to one another, and al axes are principal, th;e Equation (21'0) being satisfied independently of a", b", c" 1607.-Resuming Equations, (33), and substituting the values of x, y, z, in the general expression, m WZ (X2 - y2) which is the moment of inertia with reference to any axis, z, parallel to the axis z', through the centre of inertia, we have M ("X2 + y2) = m I[(x, +'). (y' + y')2] = f ('2 + Y') + (x,2 + (x 2).+ ) m + 2x,.mx' +- 2y,. mzy'; but from the principle of the centre of inertia, mxI'- =0, -and my' = 0; whence, denoting by d the distance between the axes z and z', and by M the whole mass, M. (x2 + y2) = Z m (x'2 + y12) + Mdz... (214) 164 ELEMENTS OF ANALYTICAL MECHANICS. That is, the moment of inertia of any body in reference to a given axis, is equal to the moment of inertia with reference to a parallel axis through the centre of inertia, increased by the product of the whole mass into the square of the distance of the given axis from that centre. And we conclude that the least of all the moments of inertia is that taken with reference to a principal axis through the centre of inertia. ~ 168.-Denote by r the distance of the elementary mass mn from the axis z, then will r2 = x2 + y2 and z m (x2 + y2) =. m r2. Now, denoting the whole mass by M, and assuming Zm r2 = Mk2, we have /M (215) The distance k is called the radius of gyration, and it obviously measures the distance from the axis to that point into which if the whole mass were concenItrated the moment of inertia would not be altered. The point into which this concentration might take place and satisfy the condition above, is called the centre of gyration. When the axis passes through the centre of inertia, the radius k and the point of concentration are called principal radius and prin. cipal centre of gyration. The least radius of gyration is, Equation (215), that relating to the principal axis with reference to which the moment of inertia is the least. If k, denote a principal radius of gyration, we may replace:m (x'2 J- y,2) in Equation (214) by MC,2, and we shall have zm 2 = Mk2 = (,2 + d).... (216) MECHANICS OF SOLIDS. 165 If the linear dimensions of the body be very small as compared with d, we may write the moment of inertia equal to id2. The letter k with the subscript accent, will denote a principal radius of gyration. The determination of the moments of inertia and radii of gyration of geometrical figures, is purely an operation of the calculus. Such bodies are supposed to be continuous throughout, and of uniform density. Hence, we may write d AM for m, and the sign of integration for A, and the formula becomes Zmr2 fdiM.r2. (217) Example 1. —A physical line about an axis through its pentre and perpendicular to its length. Denote the whole length by 2a; then 2a: dr:. M: d M, whence, dM, M..d, 2a anda am rA Ma2 ifk2,= yat 2.* d r - - J2 0 -',2 3 ak If the axis be at a distance d from the centre, and parallel to that above, then, Equation (216), k = k a2 + d2..Example 2.-A circular plate of uniform density and thickless, about an axis throvgh its centre and perpendicular to its plane. 166 ELEMENTS OF ANALYTICAL MECHANICS. Denote the radius by a; the angle XA Q Y by 0; the distance of d lA from the centre by r; then, Pq m a2: r. d. dr:: M: d3; M; whence, r.dr. dO r a2 and a 2M r3. dr d f =. -MaO ar a2 a2 2 and for an axis parallel to the above at the distance d, k= _ Via +d2+ Example 3. —The same body about an axis through its centre and in its plane. As before, r. dr. d dMf —.M =, qr a2.in which r denotes the distance of d M from the centre; and taking the axis to be that from which 0 is estimated, the distance of the elementary mass from the axis will be r sin 6, and.a r2 s 1 n3 s in2 a M 2 rd Akc,2= ff M 2 -- dr.dO -- JJ r3(1 -c2 ) d r.d, o o qr a2 2s: Mk2 - a r3. d r and k, =;a, and about*an axis parallel to the above and at the distance d. k -- a2 + d2. MECHANICS OF SOLIDS. 167 It is obvious that both the axes first considered in Examples 2 and 3 are principal axes, as are also all others in the plane of the plate and through the centre, and if it were required to find the moment of inertia of the plate about an axis through the centre and inclined to its surface under an angle p, the answer would be given by the Equation (210), Mk,2 - l2 a2 sin2 + I Ma2 cos2 -Ma2 (1 + sin2 ), and for a parallel axis whose distance is d, -k2 M ( a- (1 + sin2 ) + d2) Example 4.-A solid of revolution about any'axis perpendicular to the axis of the solid. Let ZD A' X be the given axis, cutting that of the solid in A'. Let A' be the origin of co-ordinates, P 1X = y; A' P = x; A A' m; A' B = n; and V= volume of the solid. The volume of the elementary section at P will be i y2. d x, and V'::: r. y2. d d M; whence, d AI. V*. Y2 d x, and its moment of inertia about MM', is, Example 3, M y2'Y2 - dx. and about the parallel axis, D) E i *.. y2.dx (I y2 + x2) 68S ELEMENTS OF ANALYTICAL MECHANICS. therefore, 4Mk2= ~ efr. Y2( y2 x2) d x. But V -- y2 dx; whence, Im (4 y + x2 y2).d k2 = - f, y2 d x The equation of the generating curve being given, y may be elidi. nated and the integration performed. Example 5.-A sphere about a line tangent to its surface. The equation of the generatrix is y2 =_ 2ax - x2; irn which a is the radius of the sphere. Substituting the value of y2 in the last equation, recollecting that m = 0, and n = 2 a, we have r2 a (a2 x2 + a x3 X4) d 7 2 - 2 - a. J (2ax — x2)dx Also Equation (216), k,2 = k2 _ a2 = a2, and k,=a a Thus, when the boundary of a rotating body and the law of its density may be defined by equations, its moment of inertia is readily found by the ordinary operations of the calculus; but when the figure is irregular and the density discontinuous, recourse is had to the properties of the compound pendulurm, to be exp,laiuied presently. MECHANICS OF SOLIDS. 169 Example 6.-Find the points in reference to which the principal mnoments are equal. Take the origin at the centre of inertia, and the principal axes through that point as the co-ordinate axes. Denote by x, y, z, the coordinates of one of the points sought; by A4,, B,, and C, the principal moments with reference to this point, and by x' y' z' the co-ordinates of the element mn. Then, because the moments through the point x, y, z are to be principal, will I m- (- x,) (y yj) = 0; I In (X - s) (Z -Z,) = O0;? n (y' - y) (Z_ -Zi)= 0. Performing the multiplication and reducing by the properties of the centre of inertia and principal axes, we have M. x, y, =O; Mx, z, = O; My, z, -0: which can only be satisfied by making two of the co-ordinates x,y,z, separately zero. Let y, = 0, and z,- 0; then, ~ 166 and Eq. (216), A, = A; B,B + fx,2; C, C + M,2; but, by the conditions, the first members are equal. Whence A = B + x,2 = Ca+ Mx 2. and, therefore, B = C; and x, =+A-; and from which it is apparent: 1st, that if all the principal moments in ~eference to the centre of inertia be unequal, there is no point in reterge. to which they can be equal; 2d, that if two of them be equal in reference to the centre of inertia and the third be the greatest, there are two points, equally distant from the centre of inertia and on the axis of the greatest moment, with reference to which they are equal; 3d, that if all three, with reference to the centre of inertia, be equal to one another, there is no other point with respect to which th~ey can be equal. IMPULSIVE PORMes. ~ 169. —We have thus far only been concerned with forces whose action may be likened to, and indeed represented by, the pressure alrising fiom the weight of some definite body, as a cubic foot of 170 E LEMENTS OF ANALYTICAL MECHANICS. distilled water at a standard temperature. Such forces are called incessant, because they extend their action through a definite and measurable portion of time. A single and instantaneous effort of such a force, called its intensity, is assumed to be measured by the whole effect which its incessant repetition for a unit of time can produce upon a fiee body. The effect here referred to is called the quantity of motion, being the product of the mass into the velocity generated. That is, Equations (12) and (13), d V d2s P =. r dt -Md; ~. (218) in which V,, denotes the velocity generated in a unit of time. The force P, acting for one, two, or more units of time, or for any fractional portion of a unit of time, may communicate any other velocity V, and a quantity of motion measured by M! V. And if the body which has thus received its motion gradually, impinge upon another which is free to move, experience tells us that it may suddenly transfer the whole of its motion to the latter by what seems to be a single blow, and although we know that this transfer can only take place by a series of successive actions and reactions between the molecular springs of the bodies, so to speak, and the inertia of their different elements, yet the whole effect is produced in a time so short as to elude the senses, and we are, therefore, apt to assume; though erroneously, that the effect is instantaneous. Such an assumption implies that a definite velocity can be generated in an indefinitely short time, and that the measure of the force's intensity is, Equation (218), infinite. In all such cases, to avoid this difficulty, it is agreed to take the actual motion generated by these blows during the entire period of their action, as the measure of their intensity. Thus, denoting the mass impinged upon by M, and the actual velocity generated in it when perfectly free by V, we have ds P =iV ]= I. d, T'.(219) in which P, denotes the intensity of the force's action, and the second member of the equation the resistances of the body's inertia. MECHANICS OF SOLIDS. 171 Forces which act in the manner just described, by a blow, are called impulsive forces. MOTION OF A BODY UNDER THE ACTION OF IMPULSIVE FORCES. ~ 170.-The components of the inertia in the direction of the axes y z, are respectively dsdx dx Mdt ds dt,V. ds dy =M d; dt ds d-t ds d dz dt ds d t which, substituted for the corresponding components of inertia in Equations (A) and (B), give dx 2Pcosa = rm.d t; Coso M dy.;... i; (220) dz Z Pcos = m,. dt J' dy d P (x' cos - y' cos a) = m (x'.-y' ) P (z'cosa-x' cos) =j jdtx'd tx,d (221) P (y' cosy - z' cos ) = -; ('/ dzt. dyt) In which it will be recollected that x y z are the co-ordinates of mn, referred to the fixed. origin, and x' y' z', those of the same mass referred to the centre of inertia. MOTION OF THE CENTRE OF INERTIA. ~ 171.-Substituting in Equations (220), for d x, d y, d z, their values obtained from Equations (34), and reducing by the relations 1mdx' = 0; Zmdy' —; lmdz' = 0; ~ 172 ELEMENTS OF ANALYTICAL MECHANICS given by the principle of the centre of inertia, we find z p ddx 1 Pcos~y _-d t.m;:Pcosa = M. d -, d d P cos=.t M' d y,; d t zPcos 7 M. d' of the body, and from which we conclude that the motion of the centre of inertia will be the same as though the mass were concentrated in it, and the forces applied immediately to that point. ~ l72.-lReplacing the first members of the above equations by their values given in Equations (41), and denoting by V the velocity which the resultant R can impress upon the whole mass, then will Y P cos = - MVcos a; ~ P cosf= M Vcos b; Y P cos' = rVcose; substituting these above, we find dx,. V. cos a = -; dt V.cosb = d, (224) dt V.cos — dt' - d t' MECHANICS OF SOLIDS. 173 and integrating, x,= V. cos a. t + C', y, = V. cosb.t + C", (225) z, = V.cosc.t+ C"', J and eliminating t from these equations, V will also disappear, and we find, cos C C' cos c- Ci"' cos a zj =xi a - I cos a cos a = cos b cos b (226) cos b C' cos - C" cos a'cos a cos a which being of the first degree and either one but the consequence of the other two, are the equations of a straight line. This line makes with the axes x, y, z, the angles a, b, c, respectively, and is, therefore, parallel to the resultant of the impressed forces. Whence we conclude, that the centre of inertia of a body acted upon simultaneously by any number of impuisive forces, will move uniformly in a straight line parallel to their common resultant. MOTION ABOUT THE CENTRE OF INERTIA. ~ 173.-Substituting, in Equations (221), for dx, dy and dz, their values from Equations (34), reducing by E m x' = 0, Z m z - 0, and we find, 2 p (s' cosi - cos a) = Y? (x * dt dx'.- x.d ( 22z7 I-P (z' cos a - X' cos r) = Z m (z' d t d (227) P (yosy - z cosz' cos ) = n (y d z' y dt Z t'd; 174 ELEMENTS OF ANALYTICAL MECHANICS. whence, the motion of the body about its centre of inertia will be the same whether that point be at rest or in motion, its co-ordinates having disappeared entirely from the equations. ANGULAR VELOCITY. ~ 174.-Replacing the first members of Eqs. (227) by L,, M,, and 1N,, respectively, ~ 162; and substituting in the second members for dx', dy' and dz', their values in Eqs. (190), we readily find dt L,+ ny (x'2 4- y'2) 1d iY' dt r' dt! d (228) =d t y m/ (y'.2 + z'2) - If the axes be principal, then will' mz x'y =0, O. my'z' 0,?m x'z'=0; or if the axes be fixed in succession, then for the axis x' will d = O; dc = 0; for the axis y, d -= 0; d- = 0; and for the axis z, d = 0; d b = 0, and the above become d -- Z m. (x'2 + y'2) _ _ _ _ _. ~ ~ (229) dt -I rn.(X'2+ z'2) d = Z n.(y2 + z2) That is, the component angular velocity about either a principal or fixed axis, is equal to the moment of the impressed forces divided by the moment of inertia with reference to that axis. ds The resultant angular velocity being denoted by dt, we also have, (Eq. 196), dts -dp2 + d+2 + dt2.... (230) dt Idt MECHANICS OF SOLIDS. 175 AXIS OF INSTANTANEOUS ROTATION. ~ 175.-The axis of instantaneous rotation is found as in ~ 158, by making, in Equations (192), d x' = O, dy' = 0, dz' = 0; and, therefore, z'.vy-y'. v O; 0 x'. v -z/.- = O;'.v-x'. vy = O. (231) which, as the last is but a consequence of the others, are the equations of a right line through. the centre of inertia. The equations of the line of'the resultant impact are, Eqs. (45), Z Is+ X-' Y, and the inclination 0 of this line to the instantaneous axis, is gi'ven by C~os = W.* Z + vY. Y + v,. X cosO= - Vz' + f Y + v. V/'2 Y + y X+'~ or, substituting for v<, vy, and v, their values, Eqs. (229) and (191), L. Z +.Y N,. X cosO C B A (232) I ( + +Q + ( N z) + YZ + X2 The point in which the line of the impact pierces the plane y z is given by zX = -; dividing one by the other, we have, for the equation of the line through this point and the centre of inertia, Y Z'. Denote the angle which this line makes with the instantaneous- axis by 0'; then from the equations of these lines will vy L, cos 0'= V; Y) + -1. - - 1 or, Eqs. (229) and (191), - - - 1 cos' =. (233) 176 ELEMENTS -OF ANALYTICAL MECHANICS. AXIS OF SPONTANEOUS ROTATION. ~ 176.-If both members of Eqs. (34) be divided by d t, we have dx dx, dx' d- d t + dt' dy dy, + dy' ct ct d- t dz dz, dz' dt -- dt + t; and if for any element dx dy d z dt =0; t=;.. (234) then will d, d x' d y, d?/' dz, d z' dt- dt' dt d dt dt. (235) Substituting for the first members their values given in Equations (224), and for the second members their values given in Equations (192), we have X'. vy - y'. v, + V. cos a = 0';:~ v,-'. v-+- V. cos b =.... (236) yK,'v,~ ix'.;V4 + V. cos c 0= Now, if either of these equations be but a consequence of the other two, then will they be the equations of a right line parallel, Equations (231), to the instantaneous axis; and all points upon this line will be at rest during the body's motion. This line is called the axis of spontaneous rotation. To find the conditions which shall express the dependence of either of the Equations (236) upon the other two, multiply each by the angular velocity it does not already contain, add the products, and divide the sum by the resultant angular velocity vi; there will result, 2)2 22V cos a osa + cos cos... (236)' Vi V i Vi The first member is the cosine of the angle which the resultant impact makes with the instantaneous axis; this being zelo, it follows that whenever a body is struck so as to make the instantaneous axis perpendicular to the direction of the impact, the spontaneous axis will exist. MECHANICS OF SOLIDS. 177 Denote by. 1, the distance from the spontaneous axis to the line of the impact; by e, and e,, the absolute terms in the second and third of Eqs. (236), solved with respect to z and y, then will (ez -_ +vY + e + X v z_ _ __ _ - (237) (v,2 V) 2(Y 2 de I) + ( + +( + 1 Equation (232) will make known the circumstances of the impact and shape of the body which will determine the existence of the spontaneous axis. Make the impact in the plane of the principal axes x' y', and parallel to the axis x'. Then, Equation (232), will 0 90~, and the spontaneous axis will exist. Also, Equation (233), 0' = 90~. And, Equations (237) and (229), and because X=2 M. V, and V= e,. v,, L, Mk v k I=e, + e+ ey. v + whence, (l -ey). ey =k,...... (238). That is, when the line of the impact is in the plane of two of the principal axes and parallel, to one of them, there will be a spontaneous axis, and the product of its distance from the centre of inertia by that of the line of the impact from the same point, is equal to the square of the principal radius of gyration in reference to the instantaneous axis. ~ 177. —The body being free, and the axis of spontaneous rotation at rest, while the other parts of the body are acquiring motion, the forces, both extraneous and of inertia, are so balanced about that line as to impress no action upon it. The line of the impact and the points of the body on this line are called, respectively, the axis and centres ofpercussion, in reference to the spontaneous axis. A centre of percussion in reference to an axis is, therefore, any point at which a body may be struck without communicating a shock to a physical line coincident in position with that axis. STABLE AND UNSTABLE ROTATION. ~ 178.-Now suppose the rotation to have been impressed, the in 178 ELEMENTS OF ANALYTICAL MECHANICS. stantaneous axis nearly coincident with the principal axis z, and the body abandoned to itself. What will be the circumstances of the motion? The first member of the third of Equations (194) will be sensibly equal to unity, v. and vX, therefore, indefinitely small, their product an indefinitely small quantity of the second order; L,, M,, and NV will be zero, and Equations (202) may be written, Cd = 0o; B. d, +. v,. (A - C) =; A. d + v.. ( C-B) =. Integrating the first, we have C v~ — -n; in which c is the constant of integration; and this in the other equations gives B. d +n.(A-C).v 0 O; A. d +- n(C-B) v, =; dt dt differentiating and substituting in each of the derived equations the values of the first differential coefficients obtained from the primitive, we find d2 vy (A — C) (B — C) d tV + n A. B d',,, (A - C). (B - C) +d t' A. B If A-C and B-C be both positive or both negative, their product will be positive, and the integrals are v,= a. sin In. A..t + cl; v =a,.sin A - ). (B - C).t } If one of the factors A - C and B -C be positive and tl. e other negative, their product will be negative, and the integrals will be v-(A a-.,e A.B. v, - a,. e In these integrals, ay, a., cy, and c,, are constants whose values result from the initial -conditions of the rotation. Theyv are smlall at the MECHIANICS OF SOLIDS. 179 epoch, because vr and v, are small. In the first integration, v, and v, will continue small and resume periodically their initial values; in the second, they will increase with the time indefinitely. If the instantaneous axis coincide with the axis z, then will a and a, be zero; v and v, will be zero, and, hence, a princiaul axis is always a permanent axis?' rotcttion; and the rotation will be stable about the axes of greatest anld least moments of inertia, and ustable about all others MIOTION OF A SYSTEM OF BODIES.' 179 —We have seen that the Equations (117) and (119) give all the circumstances of motion of the centre of inertia of a single body in reference to any assumed point taken as an origin of co ordinates. For a second, third, and indeed any number of bodies, referred to the same origin, we would have similar f equations, the only difference being in the values of the co-ordinates, of the intensities and directions of the forces, and of the magnitudes of the masses. This difference being indicated in the usual way by accents, we should obtain by addition, x M- dt2 -X; d t2 Y. Y; (239) dt2 MY Z; ( (x t2 Y d >t2 ) (Yx-Xy);' in which it must be recollected that x, y, z, &e., denote the co ordinates of the centres of inertia of the several masses M, &C, referred to a fixed origin. 180 ELEMENTS OF ANALYTICAL MECHANICS. MOTION OF THE CENTRE OF INERTIA OF THE SYSTEM. ~180.-Taking a movable origin at the centre of inertia of the entire system, denoting the co-ordinates of this point referred to the fixed origin by x,, y,, z,, and the co-ordinates of the centres of inertia of the several masses referred to the movable origin by x', y', z', &c., we have, the axes of the same name in the two systems being parallel, X = Xi + x', Y = Yj + YI Z = Z, + Z9 and,...... (241) d2x= dZ + d2 X', d2y = d2 Y + d2y', d2 z = d2 Z, + d2 z', which substituted in Equations (239),'and reducing by the relations, Mz l.dx' 0d2; Md y = O; Md2 z' -0; ~ ~ (242) obtained from the property of the centre of inertia, we find d2x, Z. M =- Z X; d2y'. M-y Y; (243) d t2 which being wholly indepe the relative positions of the severa which being wholly independent of the relative positions of the several bodies, show that the motion of the centre of inertia of the system will be the same as though its entire mass were concentrated in that point, and the forces applied directly to it. 181. —Multiplying the first of Equations, (243), by y,, the second MECHANICS OF SOLIDS. 181 by x,, and taking the difference; also, their first by z, the third by x,, and taking the difference, and again the second by z,, the third by y,, and taking the difference, we find (. dt2 y, dt2 X.i= M -;.z Y-y,. ~.X;' (z-, d *d2 A a zX. d. Jz xi. (244) d, d t2 ( d 2 _-) — Z. d2-,. M Y,. Z- Z *; I; which will make known the circumstances of motion of the common cantre of inertia about the fixed origin. MOTION OF THE SYSTEM A13OUT ITS COMMON CENTRE OF INERTIA. 182.-Substituting the values of x, y, z, d2x, &c., given by Equations (241), in Equations (240) and reducing by Equations (244) and (242), there will result z d2yr yI d2x' i * d y-t. d t2, = (YX Xy) ( d2 Y d2 z/) ('. - dt2) (XZ' - z') (245) A( dt2 d t2= (Zy' -Yz') Equations from which all traces of the position of the centre of inertia; have disappeared, and from which we conclude that the motion of the elements of the system about that point will be the same, whether it be at rest or in motion. These equations are identical in form with Equations (11S); whence we conclude that the molecular forces disappear from the latter, and cannot, there fore, have any influence upon the motion due to the action of the extraneous forces. CONSERVATION OF THE MOTION OF THE CENTRE OF INERTIA. ~183.-Ii' the system be subjected only to the forces arising from the mutual attractions or repulsions of its several parts, then will.X = 0; ^: 0; I:Z -O 182 ELEMENTS OF ANALYTICAL MECHANICS. For, the action of the mass M, upon a single element of MP, will vary with the number of acting elements contained in Af; and the effort necessary to prevent M' from moving under this action will be equal to the whole action of M upon a single element of 1J' repeated as many times as there are elements in M' acted upon; whence, the action of M upon M' will vary as the product M Mi'. In the same way it will appear that the force required to prevent.M from moving under the action of M', will be proportional to the same product, and as these reciprocal actions are exerted at the same distance, they must be equal; and, acting in contrary directions, the cosines of the angles their directions make with the co-ordinate axes, will be equal, with contrary signs. Whence, for every set of components P cos a, P cos i/, P cos y, in the values of.: X,: Y, 1 Z, there will be the numerically equal components, - P' cos a', - P' cos 3', - P' cos y', and, Equations (243), reduce, after dividing by Y. M, to ~2X, d2, d2 zj d2, -_ d; 2Y.... (246) dt2 dt2 0; d-t and from which we obtain, after two integrations, x,= C'.t+D'; I y, = C".t + D"; (247) z- e"t.t +- D"'; in which C', C", C"', D', Df" and D"'. are the constants of integration; and from which, by eliminating t, we find two equations of the first degree between the variables x,, y,, z,, whence the path of the centre of inertia, if it have any at all, is a right line. Also multiplying Equations (246) by 2 dx,, 2 d y,, 2 d z, respectively, adding and integrating, we have dx,2 + dy,2 dz,2 _ V2 = a - (248) d t2 in which C is the constant of integration and V the velocity of the centre of inertia of the system. From all of- which we conclude, that when a system of bodies is subjected only to forces arising MECHANICS OF SOLIDS. 183 from the action of its elements upon each other, its centre of inertia will either be at rest or move uniformly in a right line. This is called the conservation of the motion of the centre of inertia. CONSERVATION OF AREAS. ~184. —The second member of the first of Equations (245) may be written, Yx' - Xy' + Y x" -X'y" + &c.; and considering the bodies by pairs, we have X=-X'; Y= — Y'; and eliminating X7 and Y' above by these values, we have (X'- x") -X(y'- y") + &c. But, _ x y' - y X= P.; Y = P in which p denotes the distance between the centres of inertia of the two bodies. And substituting these above, we get y' -- Yif X' - x'P.' (x - x") -P. (y' - y") 0; P P and the same being true of every other pair, the second members of Equations (245), will be zero, and we have 3/i. (x' d2 y, d2 x' d_-M. - / = 0; z Me (Z..Q d Z _ dxd d tz ) =0;.. (Z d'z' d2y'" _ dt2 d2; and integrating x d y' - y' d x' C z'd x' - x' d z'_ ". (249) y' d z' - z' d cfy' M. I9~~ C".J - =C" 184: ELEMENTS OF ANALYTICAL MECHANICS. But ~ 190, x' dy'- y' d x is twice the differential of the area swept over by the projection of the radius vector of the body M, on the co-ordinate plane x' y', and the same of the similar expressions in the other equations, in reference to the other co-ordinate planes; whence, denoting by Az, Ay, A,, double the areas described in any interval of time, t, by the projections of the radius vector of the body iV, on the co-ordinate planes, x' y', x' z', and y' z', and adopting similar notations for the other bodies, we have d A IM A C'r dt dA dt in which. denote the sums of the products obtained byC"' in which C', C", C"', denote the sums of th e products obtained by multiplying each mass into twice the area swept over in a unit of time by the projection of its radius vector on the planes x' y', x' z', y' z'; and by integrating between the limits t, and t', giving an interval equal to t, IM.AZ = C'.t; z M. AY- = C" t; 2. M. As = C'" t; whence we find that when a system is in motion and is only subjected to the attractions or repulsions of its several elements upon each other, the sum of the products arising from multiplying the mass of each element by the projection, on any plane, of the area swept over by the radius vector of this element, measured from the centre of inertia of the entire system, varies as the time of the motion. This is called the principle of the conservation of areas. ~ 185. —It is important to remark that the same conclusions would be true if the bodies had been subjected to forces directed towards a fixed point. For, this point being assumed as the origin of co-ordinates, the equation of the direction of any one force, say that acting upon M, will be Yx - Xy O0; MECHANICS OF SOLIDS. 185 and the second members of Equations (240) will reduce to zero; and the form of these equations being the same as Equations (245), they will give, by integration, the same consequences. INVARIABLE PLANE. ~186. —If we examine Equations (249), we shall find that.M d dt is the quantity of motion of the mass M, in the direction of the axis y', and is the measure of the component of the moving force d x' in that direction; the same may be said of I-1 d', in the direction of the axis x'; whence the expression, z'd y' - y'd x' M ~ x dt is the moment of the moving force of M, with respect to the axis z'. Designating, as before, the sum of the moments with respect to the axes z', y' and x', by L,, lf,, N,, respectively, Equations (249) become L, = C'; M, ";., C"'. Denoting by O,, Oy, and 0,, the angles which the resultant axis makes with the axes z', y' and x', we have, ~ 110, os=A. __OLi, C' c o L,2 + M2 + 2 VC'2 ++ C"'2 + CC,2 cos O =;. * (250) + 1V, / ~C/ + CX,2 c, + c C" + c, cos - V, = C'" v/L,2 + 2 N2 vC'2 + C/"2 + c,,,2 These determine the position of the resultant or principal axis. The plane at right angles to this axis is called the principal plane. The position of this plane is invariable, and it is therefore called the invariable plane, either when the only forces of the system are those arising from the mutual actions and reactions of the bodies upon each other, or when the forces are all directed towards a fixed centre. 1S6 ELEMENTS OF ANALYTICAL MECHANICS. PRINCIPLE OF LIVING FORCE. ~ 187. —If, during the motion, two or more bodies of the s3 stem impinge against each other so as to produce a sudden change in their velocities, the sum of the living forces will undergo a change. To estimate this change, let A, B, C be the velocities of the mass m,, in the direction of the axes before the impact, and a, b, c what these velocities become at the instant of nearest approach of the centres of inertia of the impinging masses, then will A —a, B -b, C' —c, be the components of the velocities lost or gained by mn at the instant corresponding to this state of the impact, and m (A - a), m (B - b), m(C- c), the components of the forces lost or gained. The same expressions, with accents, will represent the components of the forces lost or gained by the other impinging bodies of the system. These, by the principle of D'Alembert, ~ 71, are in equilibrio, whence Im(A - a) ax +.1m(B -- b) y + 2n(C- c)6z = 0. The indefinitely small displacements Sx, &y, 6z, &c., must be made consistently with the connection by virtue of which the velocities are lost or gained; but as a, b, c denote the components of the actual velocities of the body whose mass is m, at the instant of its nearest approach to that with which it collides, this condition is fulfilled if we make 6cx = a.6t; 6y = b.6t; 6z = c.6t. These values being substituted in the above equation, we have, after dividing by 6 t, m (A -a)a + m (B-b)b + Z nm(C-c)c = O (251) or, zm(Aa + Bb+ Cc)-zm (a2 + b + c2) = O.(252) MECHANICS OF SOLIDS 187 But we have the identical equation, { A2 + B2 + C2 +~ 2 + b2 (A —a)2 + (B-b)2 + (C — c)2-= + c2 - 2(Aa + Bb + Cc), or, A2 + B2+ C2 a2 + 62 + c2 Aa + Bb + Cc 1 + (A - a)2 J- (B - b)2 + (C - C)2 which in Equation (252) gives, I n(A2+B2+ C2)- m(a2+b2+c2) = m [(A -a)2+ (B-b)2+(C-c)'], and making A2 + B2 + C2 = V2 a2 + b2 + C2 = U2, m v- m, -2 m [ (A - a)2 + (B - b)2 +( (253) whence we conclude, that the difference of the sums of the living forces before the collision, and at the instant of greatest compression, is equal to the sum of the living forces which the system would have, if the masses moved with the velocities lost and gained at this stage of. the collision. Since all the terms of the preceding equation are essentially positive, it follows that at the instant of nearest approach of the impinging bodies, there is a loss of living force. If the impinging masses now react upon each other in a way to cause them to be thrown asunder, and A', B', C', &c., denote the components of the actual velocities, in the direction of the axes, at the instant of separation, then will the components of the velocities lost and gained while the separation is taking place, be a- A', b -B', c - C', &c., &c.; and Equation (251) will become m (a — A') a + Zm (b - B') b + m (C - C')C = 0, or, m (a2+ b2 + C2) - mn (A' a 4- LB'b + C'c) = —. 0; 188 ELEMENTS OF ANALYTICAL MECHANICS. and eliminating A'a + B'b + C'c, by means of the identical equa. tion, (a- A')'2 -+ (b — /)2 - +q_' (a b2 )2 c= + a0 + An2 - B'2 + C'2 - 2 (A'a+/B'b + C'c), we obtain, (a -A)2 A 2t n(a2 + b2+c2) — m(A'2+B'2 +C'2)- - m + (b - B')2, + (c - C')2 and making At2 + B'2 + C'2 = vT2, z mU -2,r V'2 =- - m [(a - A')2 + (b - B')2 + (c - C')2].. (254) All the terms of this equation being essentially positive, it follows, from the sign of the second member, that during the reaction of the bodies by which they are separated, there is a gain of living force. If the loss and gain of velocities after, be the same as before the instant of greatest compression, then will there be no loss or gain of living force by the collision. PLANETARY MOTIONS. ~ 188. —When the only forces are those arising from the mutual attractions of the several bodies of the system for one another, the second members of Equations (239) reduce, as we have seen, ~ 183, to zero, and those equations become dt x:g J~. W7= 0, d M =. d-2 = i25) * d t 0' Jet us now find the motion of any one body of the system in reference to any other, taken at pleasure. This latter body will be called the central, the former the primary, and the others, collectively, the MECIANICS OF' SOLIDS. 1'; pertlurbatinzy bodies. Let the centyal and ptimary bodies be those whose masses are A1l and Al, respectively; the perturbating bodies those whose masses are M,,, M,,,, &c. The first of the above equations may be written d2 x d'x xx I. 7- x + V z, d+ = o.. *i. (256) dt1 dt2 "P cd t' If the perturbating bodies alone acted upon one another, the last term would be zero; and when the action of the central and primary are included, the numerical value of' this term will result from the action of these latter bodies. Denote the reciprocal action of any two bodies upon one another by writing their masses within the parenthetic sign, and use the subscript x to denote the component of this action parallel to the axis x. Then will d" xj~ Z (MMI,,), + z(M, P-,, ),-zr d' t2-o; adding this to the next equation above, we get d'x d' x, M. dt2 d t + (fM,,) + (A,,,),.. (257) Taking the movable origin at the centre of the body M, we have x, = - x', and d2 x, = d2 X - d2, which, substituted above, gives ( d2 i d2X 2 x' (-, _M (, d,) +' t d +(LM,,)Z + Y (I M,,,),= o; dividing by M + A_, and multiplying by M, there will result d'x il. A, dx' -.2 A+Al, dt + (if.I(M, MI j)=O. * d t2 M+ M,' d ta if M+' *, M -) + M+-j. z (MM,, ),,= O. The value of the first term results from the component action of the primary and perturbating bodies upon M; whence d'2 x M. E-[(Mi,)Z - Z (M I,,),] = 0; from which subtracting the equation above, there will result AI M, d x' Ml M 190 ELEMENTS OF ANALYTICAL MECHANICS. Dividing by the coefficient of the first term, and treating the othel' two of Equations (255) in the saume way, we finally get d' x' M+M,)1 1 dt y M+f A, d l. i.l(l 1M)+{. (M - l.(M,,,)-o,= i0, (258) dt2 - (M Which, by integration, will give all the circumstances of the primary's motion in reference to the central body. LAWS OF CENTRAL FORCES. ~ 189.-A central force is one which is directed towards a centre, movable or fixed, and of which the intensity is a function of the distance firom the centre. T1he forces of nature are of this description. If the perturbating bodies did not exist, then would the action on the primary be directed to the central body as a centre, the Equations (258) would reduce to their first two terms, and, denoting the distance from the central to the primary by r', they would be written, d'x' A+L~A, M+M 12 y-v + ffr, ( -- + m d Yy' kf+2~~ ~~l[lr/~(.M,).?); +. *(259) d t' M' f M m' M).? I,. d2 Z' M+:~, M+M, z' ~t-w = (ifV,.(. -f, (M M.) Multiply the first by y', the second by x', and take the difference of the products; also multiply the first by z', the third by x', and take the difference of the products; and again the second by z', the third by y', and take the difference of the products: there will result, omitting the accents, d' d2X d2x d' z dt2.z- x=O, d'z d& y y -. z = 0; MECIIANICS OF SOLIDS. 191 which, being integrated, give dy dx.27-x. yy= C', dx dz d..2 d 6'x C, ff... (260) dz dy d-. Y - J = CC'; in which C/, C/", and C"' are the constants of integration. Multiplying each by the first power of the variable which it does not contain, and adding, we have C/z + C"y + C///x = O; which is the equation of an invariable plane passing through the centre, and of which the position depends upon the constants C/, C"/, C"". Whence we conclude that the primary deflected by the central body alone, will describe a plane curve of which the plane'will contain the centres of both.. 190.-Take the co-ordinate plane xy to coincide with this plane, and the Equations (260) will reduce to dy dx -' y....... 261) Transform to polar co-ordinates; for this purpose we have x=-r.cosa; y=r.sin a; differentiating, d-x = d r cos a - r sin a d a, dy = dr sin a +- r cos a da. Substituting in Equation (261), we find dy dx dc, dt dt dt (262) integrating again, we have fr. d = C' t + C", and taking between the limits r,, a, and r,,, a,,, corresponding to the time t, and t,,,'. da, = C' (t,,- t,)..... (263) a,, a,, 1 EiI2 EEMENTS OF ANALYTICAL MECHIANICS. BlutJit. ci is (ouble the area described by the motion of the radius vector; whence we see, Equation (263), that the areas described by the radius vector of a body revolving about a centre, are proportional to the intervals of time required to describe them. MIalking, in Equation (263), t,, - t, equal to unity, the first member becomes double the area described in a unit of time. Denoting this by 2 c, that equation gives C'- 2 c. Placing this in Equation (263), we find f r,'a, r2 d -.... (264) 2 c That is to say, any interval of time is equal to the area described in that interval, divided by the area described in the unit of time. ~ 191.-The converse is also true; for, differentiating Equation (262), we find d2 y d2x d t2 X t2 Y; Multiplying by M, and replacing Ml. d t and H. d - by their values in Equations (120), there will result Yx-Xy= O... (265) which is the Equation of the line of direction of the force; and having no independent term, this line passes through the centre. Whence we conclude, that a body whose radius vector describes about any point areas proportional to the times, is acted upon by a force of which the line of direction passes through that point as a centre. The force will be attractive or repulsive according as the orbit turns its concave or convex side towards the centre. ~ 192.-Replacing C' by its value 2c, in Equation (262), and divi ding by r2, we have dM 2c dt r2 (266) MECHANICS OF SOLIDS. 193 The first member being the actual velocity of a point on the radius vector at the distance unity from the centre, is called the angular velocity of the body. The angular velocity therefore varies inversely as the square of the radius vector. ~ 193.-Multiply Equation (266) by ds, and it may be put undel the form, ds 2c dt rda' r.ds but d, is equal to the sine of the angle which the element of the orbit makes with the radius vector, and denoting by p the length of the perpendicular from the centre on the tangent to the orbit at the place of the body, we have r.d a p=r. ds' and =. (267) whence, the actual velocity of the body varies inversely as the distance of the ta'ngent to the orbit at the body's place, from the centre. ~ 194.-Denoting the intensity of the acceleration on M, by F; substituting M,. F. d r for Xd x + Yd y + Z d z, writing JM, for M in the coefficient of V2 in Equation (121), and differentiating, we find VdV= - Fdr; and taking the' logarithms of both members of Equation (267), log V = log 2 c- log p; differentiating, d V dp -V- P and dividing the equation above by this, dr 1 dr V2 =.p F. 2P. d.. (268) 13 19)4 ELEMENTS OF ANALYTICAL MECHANICS Whence we conclude that, the velocity of a body at any point of its orbit is the same as that which it would have acquired had it fallen freely firom rest at that / point over the distance ME, equal / to one-fourth of the chord of cur- G 4 _ vature M G, through the fixed centre-the force retaining unchanged its intensity at I[. ~ 195. —Resuming Equations (120), we have dx d2x d t X-' = M. -- d t' dt' and performing the operation indicated, regarding the are of the orbit qas the independent variable, we have, after dividing both numerator and denominator by d s3, d t d'x dx d' t ds ds2 ds ds' d [s' d'xz dx ds3 d't. X'Af=M.d s d t3 Ld t' d s' d ~ds d ts' d s3 Put ds' d2t d2 s ds d t3 d s2 d t2; dt ='; whence, X= M. [ V d S2 ds t2 In like manner,:y'=_M' [IV2* d s2 y dy d 2 s] Z=M [V2 *d-z2 dz d's] quaring and ds dtding, Squaring and adding, MBECHIANICS OF SOLIDS. 195 f { (22 (2 d2 )2 (d2g)2 } 22,~2+ y2+ V2. ds d +x dy dmy dz mz\, y~2+ y2+Z2 = + 2 [.d t (ds'$.d2? d Y d Y, + d - d sV) *2 but, denoting the radius of curvature by p, we have (d2) + ds( 1 + (d s2- p2 and nmultiplying the second termn of thle second member of the preceding equation by -, it may be put under the form, MV~ Jard'2s~dx (Z 2 dy d2?, dz d'z 2 kV t2 2d-s 2P d- + dds' P dss2 + d s; I V2 M- d's 2 c. os 6; p d t2 in which d denotes the angle made by the element of the curve and radius of curvature; also d 2 d?2f dz 1' dd+ + + ds2 s ds ds 5 whence, substituting for X 2.p2 - Z2 is value R2, we have 2 — +2. cs o+ s~. -I - p2.p d t2 d t and comparing this with Equation (pt) we fnd that R is equal to the resugant of the two component forces and 1f. - which make with each other the angle d. But d is equal to 90~, and therefore3J2 174 Md sd' + G.. A. (269) See Appendix No. 2, 19i ELEMENTS OF ANALYTICAL MECOHANICS. Thle second of these components is, Equation (13), the intensity of the reaction of inertia in the direction of the tangent, and the first is therefore its reaction in the direction of the radius of curvature. This first component is called the centrifugal force, and may be defined to be the resistance which the inertia of a body in motion ojpposes to whatever deflects it from its rectilinear path. It is measured, Equation (269), by the living force of the body divided by the radius of curvature. The direction of its action is from the centre of curvature, and it thus differs firom the force which acts towards a centre, and which is called centripetal force. The two are called central forces. If the component in the direction of the orbit be zero, then will Me d ts-~7 dts and denoting the centrifugal force by F,, we have =,- -.. [.... (270) p and integrating the next to the last equation, we have ds = V= C; dt in which C is the constant of integration. Whence, the velocity will be constant, and we conclude that a body in motion and acted upon by a force whose direction is always normal to the path described, will preserve its velocity unchanged. These laws, except that expressed by Equation (268), are wholly independent of the intensity of the extraneous force and of th'e law of its variation. Not so, however, of THE ORBIT. ~ 196.-To find the differential equation of the orbit, multiply the first of Equations (259) by 2 d x, the second by 2 d y, add and integrate; we find, omitting the accents, d x'+ f.1 M) ( xax + 2ydya dt2 - M. M, r MZECHANICS OF SOLIDS. 197 but r2 = XZ+y -, and r d - d x- ydy; also x = r cos a; y=r. sin a; dx= -r sin a d a +- cos a dr; dy = r cos ada + sin adr; and, Equation (266), 1 2c dt r2da' These substituted above, give 4 cl( -- 4 ) = 2 - uhf f (MM,)d Make 1 dr -=u, and therefore - -— d u r r substitute above, differentiate and reduce, there will result 4c'u(d'2 + 83 = i) r $r' * (MM,) = (AM,) j~ (M1]; and making mF = [(if) + =fJ relative acceleration on,. (271) F=4 C2 2 d i+ )....... (272) From which the equation of the orbit may be found by integration, when the law of the force i:s known; or the law of the force deduced, whlen the equation of the orbit is given. In the first case, the integral will contain three arbitrary constants — two introduced in the process of integration, and the third, c, existing in the differential equation. These are determined by the initial or other circumstances of the motion, viz.: the body's velocity, its dis tance firom the centre, and direction of the motion at a given instant. The general integral only determines the nature of the orbit described: the circumstances of the mition at any given time determine the species and dimensions of the orbit. 198 ELEMENTS OF ANALYTICAL MECHANICS. In the second case, find the second differential coefficient of u il regard to a, from the polar equation of the curve; substitute this in the above equation, eliminating a, if it occur, by means of the relation between u and a, and the result will be F, in terms of u alone. SYSTEM OF THE WORLD. ~ 197. —The most remarkable system of bodies of which we have any knowledge, and to which the preceding principles have a direct application, is that called the solar system. It consists of the Sun, the Planets, of which the earth we inhabit is one, the Satellites of the planets, and the Comets. These bodies are of great dimensions, are spheroidal in figure, are separated by distances compared to which their diameters are almost insignificant, and the mass of the sun is so much greater than that of the sum of all the others, as to bring tlie common centre of inertia of the whole within the boundary of its own volume. These bodies revolve about their respective centres of inertia, are ever shifting their relative positions, and our knowledge of them is the result of computations based upon data derived from actual observation. Kepler found; I. That the areas swept over by the radius vector of each planet about the sun, in the same orbit, are proportional to the times of describing them. II. That the planets move in ellipses, each having one of its foci in the sun's centre. III. That the squares of the periodic times of the planets about the sun, are proportional to the cubes of their mean distances from that body. These are called the laws of Kepler, and lead directly to a knowledge of the nature of the forces which uphold the solar system. CONSEQUENCES OF KEPLER'S LAWS. ~ 198.-The first law shows, ~ 191, that the centripetal forces which MECIIANICS OF SOLTDS. 199 keep the planets in their orbits, are all directed to the sun's centre; and that the sun is, therefore, the centre of the system., ~ 1'99.-What law of the force 4vill cause a primary to describe about a central body an ellipse having one of its foci at the centre of the latter? The equation of the ellipse referred to its focus as a pole is = a(I -e2) 1 + e cos a' whence, I 1 + e cos cc r U a (1 - e)' and, de - e cos a da - a (1 - e')' which, substituted in Equation (272), give F4 Cu2 I- ecosa 1~ e cos c) (-e)-+ (-e)) reducing and replacing u by its value -, we have 4 ca 1 a (l-e) r (273) and from which we conclude, that the only law for the relative acceleration, is that of the inverse square of the distance. ~ 200.-Conversely, let the force vary inversely as the square of the distance; required the orbit. Denote by k, the reciprocal attraction of one unit of mass upon another at the unit's distance; then will (M AI) = M. M k; and, Equation (271), F = k}. (i + M). u = k,,. O.m; in which m = + M,. (73), 200 ELEMENTS OF ANALYTICAL MECHANICS. and, Equation (272), d2' k,. m d,+U- — 4d (2 4 C2 multiplying by 2 d u and integrating, dut 2k.m do = 4.c2~ - - u 2 + C.... (274) whence — du C+ 2 k,.m. ff _ 2 the negative sign being taken, because du \d dci dct= - r~dc. (275) da d a r'd( Place under the radical ( 4 9 )-( 4( ), and we may write, I — du a a \/(km) (km) 2 T 4 c(krn 4c! k, mynz and integrating, k,.m 1 4 c2 a ) =C=os-' 4 in which q) is the constant of integration. Replacing u by its value, taking cosine of both members and solving with respect to r, there will result 4 c2 r C,. _ c which+ is+ k eqation of a conic section, hvin its pole at te central which is the equation of a conic section, having its pole at the central MECHANICS: OF SOLIDS. 201 body. To find the precise curve, we must find C. To do this, denote by r, the initial value of the radius vector, and by s, the angle which the orbit makes with r, at the point of intersection therewith. Then, Equation (275), duz 1 da — r, tan e,and this in Equation (274) gives 1 2 ki,.n C = r2 sin2 El 4 c2 r, but, Equation (267), 1 _ 72 V I2 r,,2t sin' e' 4c2 4.r, *, (275)' in which V, is the velocity corresponding to r,; hence, C=2 V. r, -2 k,. m 4 C2 r which, substituted in the equation of the curve, gives 4 c2 ~r =.kI. m ). (278) 1+ 1 2 2. (V - Cr ).os ( + q) and comparing this with the general polar equation of a congz section referred to the focus as a pole, viz.: 1 + e cos (a + )' a, (1-e') =k....... 277) 4C 2 k,. m).... 28) e I - VI (278) and this last value will be greater or less than unity, according as V,/ is greater or less than d t. Multiplying and dividing the lanAt ei: s Jrb rl, r,, and replacing m by its value, the orbit will be an ellipse, parabola, or hyperbola, according as 202 ELEMENTS OF ANALYTICAL MECHANICS. 1i 2 < 2 4e.(f + M,). M,. 2 k. ( _ MI,)'2. f I V12 > --- M. r,. That is, according as the living force of the primary at any point of its orbit is less than, equal to, or greater than twice the work its relative weight, at that point, would perform over a distance equal to its radius vector. So that a primary may describe any of the conic sections as well as the ellipse, the only condition for this purpose being an adequate value for its velocity. Substituting the value of e2 in Equation (277), we find k. n. r, a, 2;...... (279) and denoting the semi-parameter by p, the equation of the curve gives, by making ca + = 90~, 4 c2 V,2. sin2 E,.r,2 - k:.i -,.m and denoting the semi-conjugate axis by b,, b, = -Va, p = V,. sin,. r, \//,. m (279)' Whence it appears that the nature of the orbit and its transverse axis are independent of the direction of the primary's motion, while the conjugate axis is dependent upon this element. ~ 201.-The consequence of Kepler's third law is not less important. Denote the periodic time of the primary by T,; then, Equation (264), or. a, b, C and substituting the values of b,, m, and c, Equations (279), (273)', and (275)', T, = 2r.a,/.V(M ) M, k MECIANICS OF SOLIDS. 203 and for another body whose mass is Al,;, about the same central body, 3 1 Ta, = 2 X- a,, ( and by division, T,2 a,3 +,. * (20) 2 aI +, k.(280) If the difference of the masses M, and Mf,, be so small in comparison with M as to make its omission insensible to ordinary observation, which is the case in the solar system, the above may be written, T2_ a3 k,, But by Kepler's third law, T2 a 3 T - a 3' whence That is, the central body M would act equally on the unit of mass of each of the primaries Mi and M,,, were they at the same distance; so that not only is the law of the central force the same, but the absolute force at the same distance is the same, and it is one and the same force that keeps the planets in their orbits about the sun. ~ 202. —The observations of Dr. Maskelyne on the fixed stars, show that a neighboring mountain, Schehallien, drew the plumb-line of his instrument sensibly from the vertical; and those of Cavendish and Baily upon leaden and other balls, demonstrate this power of attraction to reside in every particle of matter wherever found; and that it is exerted under all circumstances, without the possibility of being inter cepted. It is, therefore, concluded that matter is endowed with a general gravitating principle by which every particle attracts every other particle, and according to the law before given. PERTURBATIONS. ~ 203.-Granting, for the present, that universal gravitation is a principle of nature, and denoting the distances of the several bodies of 204 ELEMENTS OF ANALYTICAL MECIIANICS. the system from the central by r with subscript accents corresponding to those of the bodies to which they belong, and employing the same notation in regard to the co-ordinates, we shall have (MM,),k * " = *MMJr (1 —- =.-; m M' z (MM,,)k = ck M.'I " -=,,=c. Mz I Ar 2". r di (x" -x')2+(y, _y"-')+(z' -z') V(X_x)+(y, _yl)2+(ZI_z)2 which, substituted in first of Equations (258), give d P r 7. x.,., X' ~ [(" -x'"{'+(Y" —Y')~.%(z" —z') ]=-] but _ _ _.I 1 1. [(x"- X')2 + (y - y'l)2 + (z" —'Z)21] - dx' d (x" — x'T +(y" — y')2 + (I "- Z) and making'(x" - X')2 + (y" - y)2 (z"').. (281) the last term of the equation above becomes 1 d, M, d x" and d' x' x'H, M',,rA x"I/ 1 d d~d x ( M 3- 3 +M, d'] Make BR= M,,'"'" ("+ xM,(Y"+y+z + &c.- (282) r,13 P1 then will dR M 31 " 3I' M x"' d t x" d x = +-&ce — A = - M _... W-P 3 r III3 M,. dX' I r'3 M,. dx which, substituted above, give, after treating the other two of Equations (258) in the same way, MIECHANICS OF SOLIDS. 205 d k' [(M+M,) _ dR- ~ d - _k [(M+ t,). Y3/ (283) d t2 [) r/ d (Z The curve which would be described by the primary about the central, under the reciprocal action of these two bodies alone, and which we have seen is a conic section, is called the uIndisturbed orbit of the primary. That which it actually describes under the joint action of all the bodies of the system, is called the disturbed orbit. The undisturbed orbit is given by the first two terms of Equations (283); the disturbed by all three. The departures of the disturbed from the undisturbed orbit are called perturbations, and the last terms of Equations (283), which determine them, are called perturbating functions. The constructions of the perturbating functions are given in Equations (281) and (282), and the methods of computing their values are greatly facilitated by the principle of the COEXISTENCE AND SUPERPOSITION OF SMALL MOTIONS. ~ 204.-Denote by 0,,, 0,,,, &c., numerical quantities which depend upon the perturbating actions of the bodies whose masses are M,,, M,.,, &c., and of which the values are, so small as to justify the omission of all terms into which their products enter as factors, in comparison with such as contain them singly. The co-ordinates of M,, at the time t, when undisturbed, being x' y' z', become, when the body M, is disturbed by M,, at the same time, x'- +0,,.x'; y'.y'; + z' I0,,.z'; and for the same reason, when also disturbed by M,,7, x + o,x' + a,, (' +,,x');' + o,,y' + o,,, (y' + o,,y'); z' + 0,Z' + 0,,(Z' + 01,), or, performing the multiplication and omitting the terms containing 0 X,.,); x' + x' (,, + 0,,,); y' + y' (0,+,,,);' + Z' (0,, +,,,); 206 ELEMENTS OF ANALYTICAL MECHANICS. in the same way, when also disturbed by M,.., x' x' (0,, +0,,,,,,,); y' y' (,,+O,,,+O,,,,); z' +'(O,,- +,,,+0,,,,) and for the simultaneous disturbance of all the bodies of the system, x' +x'0,,; y'f y'YO,,; z'+z'fO,,; in which x'. Z0,,, y'. 0j,,, z'. 0O,, are the increments of x'y'z' respectively, due to the joint action of all the disturbing bodies. Now let u = p (x' y' z'), in which qp denotes any function of x' y''. Differentiating, we have dze duz d u 0 W ~ x' Z O. qt- ~ y'Y. 0,, q- ~' 0. and performing the multiplications indicated, we I'ave d ud d du d u1y d u + l d u d 0t, 0 Cl (I 0 |+ x' * + 0,,O + at d"' 0,,,, + -x k&c. + &c. s+ &c. + &c. + &C. Whence it appears that the perturbation in u or g (x' y' z'), is equal to the sum of the separate perturbations due to each of the perturbating bodies, supposing the others not to exist. The practical effect of this principle is to reduce the problem of the perturbations from one of several to one of a single perturbating body, and to give rise to What is known as the problem of the three bodies, viz.: the central, primary, and perturbating. UNIVERSAL GRAVITATION. ~ 205.-From all of which it is manifest that either Kepler's laws cannot be rigorously true, or universal gravitation is not a Principle of Nature. Now, in point of fact, observations of far greater nicety than MECHANICS OF SOLIDS. 207 those or Kepler prove that his laws are not accurately true, though they differ but slightly from the truth; a circumstance arising entirely from the fact of the great mass of the sun as compared with the sum of the masses of all the planets. Were there but a single body in existence besides the sun, it would describe accurately an elliptical, paraboliq, Or hyperbolic orbit about the centre of the sun, depending upon its living force and the sun's attraction. A third body would derange this motion and cause a departure fi'om this simple path, and the degree -f the disturbance would depend upon the mass, distance, and direction of the disturbing body as compared with those of the sun. The same remark would apply to a fourth, fifth, and to any number of additional bodies. The disturbed orbits in the solar system have been computed by Equations (283), and the complete harmony which is found to subsist between the numerical results deduced from theory and observation, is the strongest possible evidence in support of the Law of Universal Gravitation. If the principal plane of the solar system, as determined at different and remote periods, be found to have undergone no change, this will show that the system is uninfluenced by the action of the fixed stars and other distant bodies, and its cenltre of inertia will, ~ 193, either be at rest or be moving uniformly through space in a right line; but if the principal plane be found to have changed its place, it will be a sign that the system is in motion, and that its centre of inertia is describing a curvilinear path about some distant centre. ~ 206.-Thus much for the larger bodies of nature. But these are themselves built up of innumerable molecules which are ever on the move about their respective places of relative rest. The molecular forces within the range of their natural action vary directly as the distance from their respective centres. Let it be required to determine the nature of the orbits under this law. Then will F=in r nk,. r = which, in Equiation (272), gives d'u ik. m d2 + U = u 20S ELEMENTS OF ANALYTICAL MECHANICS. multiplying by 2 d u, and integrating, we find d 2k,. d +2 C_ -..., (284) from which we get udu drC u, U4 4 C2 the negative sign being taken, because du d _ dr da d - - r=_ 2da..* (284)' Placing 1 02 - I C' under the radical, we may write 1 — 2uduq C_ k, i (42 _ C y2 2 4 4 c / C~2 k,. m V 4 4 c and integration, 2 (a + Q) = cos -; 4 4C2 in which q is the constant of integrating. Taking cosine of both members, replacing u by its value and solving with respect to r, we find r-^ Is C2 +r 2 iC c. cos2 (a + g) Denote by r, the radius vector which is normal to the orbit; corre sponding to this value we have du = 0, da and, by Equation (284), 1 k,. m. r, c= ~ 4 c'; MECHANICS OF SOLIDS. 209 and because cos 2 (a + $q) = cos' (a + q) - ain (a + p), the above reduces to 1 1. =.. (285) //2 cos, (a + 9) + X4 sin'(a~ + I ) which is the equation of an ellipse referred to its centre as a pole, the semi-axes being r, and 2c. 1 r, k,. m ~ 207.-The time required to describe the entire ellipse being denoted by T, we have, Equation (264), ~r.r,.2ca 1 7r. r,. 2 c j T_= _ - -2 7r; r,. C k,. m and replacing m by its value, Equation (273)', T= 2 rr (286) (M + O,) k, Thus the time is wholly independent of the dimensions of the orbit, and will be the same in all orbits, great and small. This result finds its application in the subject of acoustics, thermotics, optics, &c. ~ 208.-Let us conclude the planetary motions with the centrifugal force on its surface, arising from the rotation of one of these bodies, say the earth, about its axis. If V1 denote the angular velocity of a body about a centre, then will V= p V,, and Equation (270) becomes F, =M VY p. The earth revolves about its axis A A' once in twenty-four hours, and the circumferences of the parallels of latitude have velocities 14 210 ELEMENTS OF ANALYTICAL MECHANICS. which diminish from the eqmlator to -A _ the poles. The law of this diminution, on the supposition that the - planet is a sphere, is given by F. = M V1 RI; in which M is the body's mass, V/ the earth's angular velocity, and R' the radius of one of its parallels of latitude. Denoting the equatorial, radius CE= CP, by R, and the angle CP C' = P CE, which is the latitude of the place, by p, we have R' - R cos pq; which substituted for R' above, gives F = M V,2R cos...... (286)' The only variable quantity in this expression, when the same maw,3 is taken from one latitude to another, is q; whence we conclude that tie centrifugal force varies as the cosine of the latitude. The centrifugal force is exerted in the direction of the radius R' of the parallel of latitude, and therefore in a direction oblique to the horizon T T'. The normal and tangential components are, respectively, F,. cos p = AI V12 R cos2 qp, F,. sin qp = M V,12 R. sin pq cos = X M V12 R sin 2 q; whence we conclude, that the diminution of the weights of bodies arising fronm the centrifugal force at the earth's surface, varies as the square of the cosine of the latitude; and that all bodies are, in consequence of the centrifugal force, urged towards the equator by a force which varies as the sine of twice the latitude. At the equator the diminution of the force of gravity is a maximlrm, and equal to the entire centrifugal force; at the poles it is zero. The earth is not perfectly spherical, and. all observations agree in demonstrating that it is protuberant at the equator and flattened at the poles, the difference between the equatorial and polar diameters being about twenty-six English miles. If we suppose the earth to have been MECHIANICS OF SOLIDS. 211 at'one time in a state of fluidity, or even approaching to it, its )present figure is readily accounted for by the foregoing considerations. To find the value of the centrifugal force at the eqluator, nIll. in Equation (286)', M= - and cos q 1, which is equivalent to'osingl a unit of mass on the equator, and we have F,.= VIP,, in which, if the known radius of the equator and ang.tlalr ve\'l..:', et substituted, we shall find f F= V,2. _R= 0, 1112.'To find the angular velocity with which the earth should rotate, to mnake the centrifugal force of a body at the equator equal to its weight, make g = 32, 1937 = V'2 R; in which 32, 1937 is the force of gravity at the equator. Dividing the second by the first, we find 32,1937 V1,'2 0,l111 2- -, -289, nearly; whence,,' = 17 V,; that is to say, if the earth were to revolve seventeen times as fast as it does, bodies would possess no weight at the equator. IMPACT OF BODIES. ~ 209.-When a body in motion comes into collision with another, either at rest or in motion, an impact is said to arise. The action and reaction which take place between two bodies, when pressed together, are exerted along the same right line, perpendicular to the surfaces of both, at their common point of contact. Th-fis arises fi'om the symmetrical disposition of the molecular springs about this line. When the motions of the centres of inertia of the two bodies are parallel to this normal before collision, the imnpact is said to be direct. When this normal passes through the centres of inertia of both 212 ELEMENTS OF ANALYTICAL MECHANICS. bodies, and the motions of these centres are along that line, the impact is said to be direct and central. When the motion of the centre of inertia of one of the bodies is along the common normal, and the normal does not pass through the centre of inertia of the other, the impact is said to be direct and eccentric. When the path described by the - centre of inertia of one of the bodies, makes an angle with this normal, the impact is said to be oblique. When two bodies come into collision, each will experience a pressure from the reaction of the other; and as all bodies are more or less compressible, this pressure will produce a change in the figure of both; the change of figure will increase till the instant the bodies cease to approach each other, when it will have attained its maximum. The molecular spring of each will now act to restore the former figures, the bodies will repel each other, and finally separate. Three periods must, therefore, be distinguished, viz.: 1st., that occupied by the. process of compression; 2d., that during which the greatest compression exists; 3d., that occupied by the, process, as far as it extends, of restoring the figures. The force of restitution must also be distinguished from the force of distortion; the latter denoting the reciprocal action exerted between the bodies in the first, and the former in the third period. The greater or less capacity of the molecular springs of a body to restore to it the figure of which it has been deprived by the application of soIne extraneous force when the latter ceases to act, is called its elasticity. The ratio of the force of restitution to that of distortion, is the measure of a body's elasticity. This ratio is sometimes called the co-efficient of elasticity. When these two forces are equal, the ratio MECHANICS OF SOLIDS.'213 is unity, and the body is said to be perfectly elastic; when the ratio is zero, the body is said to be non-elastic. There are no bodies that satisfy these extreme conditions, all being more or less elastic, but none perfectly so. Let the two bodies AB and A'B', the former moving along the line H I', and the latter along / II' T', come into collision at the point 0. Through 0, draw the common normal NL. De-, 0 \ L note the angle HG N by ~, and HI' EN by' —these being'lj the angles which the directions of the two motions make with the normal. Also denote the velocity and mass of the body A B by V and X respectively, and the velocity and mass of A'B' by V' and M'. The components of the quantity of motion of the two bodies in the direction of the normal and of the perpendicular to the normal, will be M V cos I, M7 V' cos0' and M V sin p, M' V' sin q'.,The former of these components will alone be involved in the impact; for if the bodies were only animated by the latter, they would not collide, but would simply move the one by the other. For simplicity, let the body A B be spherical; the normal will pass through its centre of inertia. Denote by u, the velocity of the body AB in the direction of the normal at the instant of greatest compression, and by u' the velocity of the body A'B' at the same instant in the same direction. Then will Vcos uI, and V' cos' -I u' ~ ~ ~ (287) be the velocities lost and gained in the direction of the normal, and M(Vcos 9 - u), and d' (V'cosp' - u') ~ ~ ~ (288) 214 ELEMENTS OF ANALYTICAL MECHANICS. be the forces lost and gained at the instant of greatest compression; and hence, M(V fcos g, -) +J M'(V cosq' - C') = 0; ~ ~ (289) and denoting the angular velocity of the body A'B' by V,', the distance G' D from the centre of inertia of A' B' to the normal by e, and the principal radius of gyration of A'B', with reference to the instantaneous axis by k,, then will, AM(Vcoos P - u). e K, - 2 (290) and since the velocity u must be equal to that of the point D at the end of the lever arm e, we have U= Q' + e.V,'. *. (291) Substituting the values of u and u' from this equation successively in Equation (289), we find M V cos p ~- M' V' cos I' + M' e V,'22 ~.. (292) Mb = M + M'. M Vcos q + MA' V' cos q' - Me V,' (293) = 3/4 M' After the instant of greatest compression, the molecular springs of the bodies will be exerted to restore the original figures, and if c denote the co-efficient of elasticity, then will the velocities lost by AB and gained by A'B' during the process of restitution be, respectively, c ( V cos g-u) and c ( V' cos' -'); and the entire loss of AB, and gain of A' B', will be, respectively, Vcos - u + c(Vcos - u), and V'cos' -' + c(V'cos q'-,'). Also the gain of angular velocity of the body A' B', during thA process of restitution, will be cV,' - c(Vcosq - u).e.M C, _k,2 A' r ~~~r" M'~~~ MECHANICS OF SOLIDS. 215 and the whole angular velocity produced by the impact and denoted by V,, will be given by the equation, (Vcos q9 -u) e M v, = (1 + C) (V. -). (294) Denoting the velocities of A B and A' B', after the collision by v and v', and the angles which the directions of these velocities make with the normal by 6 and 0', respectively, then will v cos 0 = Vcosq - Vcosp + u-c (V cos- ) =(1 + c) u - c Vcos,( v'cos I'_= V'cos -- V'cos q~'+u'-c( V' cosp'-')-(1 +c)u'-cV' cos q', and replacing the values of u and u', as given by Equations (292) and (293), M V cos p+M' V' cos'Me V/ c'Vcos (295) v cos =(l +c) -c cs (295 M + M' M Vcos p+M' V' cosp' —Me V' c (296) v' cos0'=(l1c) ++ H') +M N' Moreover, because the effects of the impact arising from the compo. nents of the quantities of motion in' the direction of the normal will be wholly in that direction, the components of the quantities of motion before and after the impact at right angles to the normal will be the same, and hence v sin 8= Vsin q, ~..... (297) v' sin 0' =' sin Q'...... (298) Squaring Equations (295) and (297) and adding; also Equations (296) and (298) and adding, we find after taking square root, and reducing by the relations cos2 8 + sin2 - 1=; Cos2 4- sin28' 1; ) oI Gosh + M' V'COSt + M'e V,' * 2 V=-/1 [(1 c~ ) M V'cos'+ -c Vcosq]z2+ V2sin2q (299)'=/[(1+ Vif ~fC-os + M' V'cos'-MeV, 2(300) C [(1~c)i 3+ - -c V'cosq']2 + V'2sin2q'.(300) M + M' 216 ELEMENTS OF ANALYTICAL MECHANICS. Dividing Equation (297) by Equation (295), and Equation (298) by Equation (296), we have, V. sin ip tan V,(301) + M Vcos q' V' os' M' V'e Vco _ - c (OS eV (1+ c) M -cV'COSp' tan = V~02) M Vcos 0 + M'V' Cosq~' -- Me V,' M q- M' Equations (290) and (292), will give the values of u and V,', in known terms, and these in Equations (294), (295) and (296) will give the values of V, v, v, and v', and all the circumstances of the collision will be known. 210.-If the bodies be both spherical, then will e = 0, and Equation (294) gives V, = 0; and Equations (299) and (300), (301) and (302), become /[(1 +c) fVcosq+M'V' C -c VCOS ]2_ V2sin2 (303) taMV c os + +'V' c' cos VP',' (l1 q-c). 4- MV' COS q']c2- Vt2'sinq.. (304) V sin 9 tan' = (305) 1+ c) MVcos M' V' cos' - V Cos q VI sin q)' tan 8(306) CM V cos qP q MI V' cos q)' - ~ V' Cos ip' (+ c) M+M The Equations (303) and (304) will make known the velocities, and (305) and (306) the directions in, which the bodies will move, after the impact. Now, suppose the body A'B' at rest, and its mass so great that the mass of AB is insignificant in comparison, then will V' be zero, M' may be written for M - AM' and will be a fraction so MECHANICS OF SOLIDS. 217 small that all the terms into which it enters as a factor may be neglected, and Equation (303) becomes v = V /c2cos2q + sin29; -and Equation (305), tan d tan.. (307) The tangent of 0 being negative, shows that the angle NHK, which the direction of A B's motion makes with the normal NN' after the A' impact, is greater than 90 degrees; in \ / other words, that the body A B is driven back or reflected from A' B'. This explains why it is that a cannonball, stone, or other body thrown ob- liquely against the surface of the earth, will rebound several times before it ~ comes to rest. If the bodies be non-elastic, or, which is the same thing, if c be zero, the tangent of 8 becomes infinite; that is to say, the body AB will move along the tangent plane, or if the body.A'B' were reduced at the place of impact to a smooth plane, the body A B would move along this plane. If the body were perfectly elastic, or if c were equal to unity, which expresses this condition, then would Equation (307) become tan8 = - tanp...... (308) which means that the angle -NHF- EFuN' becomes equal to KTHN'. The angle EIFN' is called the angle of incidence, the angle _KHN', commonly, the angle of reflection. Whence we see, that when a perfectly elastic body is thrown against a smooth, hard, and fixed plane, the angle of incidence will be equal to the angle of reflection. If the angles q) and o' be zero, then- will cos p = 1, cos 9'.- 1, 218 ELEMENTS OF ANALYTICAL MECHANICS. sin p = 0, sin q' - 0; the impact will be direct and central, and Equations (303) and (304) become v=(1 + C) M + M'' M+At - I V' v' =(1 +) v+ c'VI and passing to the limits, nof~-elasticity on the one hand and perfect elasticity on the other, we have in the first case, c = 0, and V = V+M'... (309) M V + J M'.e (310 V" = r + (316.) and in the second, c 1, consequently, M V + If' V' v = 2 M-' V..... (311) M V + M'V' v'=2 + - v'..... (312) CONSTRAINED MOTION. ~ 211. —Thus far we have only discussed the subject of free nmotion. We now come to constrained motions Motion is said to be constrained when by the interposition of some rigid surface or curve, or by connection with some one or more fixed points, a body is compelled to pursue a path different from that indicated by the forces which impart motion. ~212.-The centre of inertia of a body may be made to continue on a given surface, by causing it to slide or roll upon some other rigid surface. ~ 213.-We have seen, ~ 128, that the motion of translation of the. centre of inertia, and of rotation about that point, are wholly MECHATtICS Ot SOLIDS. 219 independent of one another, and the generality of any discussion relating to the- former will not, therefore, be affected by making, in Equation (40), 6q = 0; 6+ = 0; J-Sz O; which will reduce that equation to d2 X (3 Pcos a - d t m)6 x, d2 y + (P cos/ - d z t m) y, =o. d2 Z d tz Making Im = M; ZPcosa = X; I Pcos /- Y; P cos7 = Z; and omitting the subscript accents, we may write dt2 d UfdyN/ / d22 \ Now, assuming the movable origin at the centre of inertia, and supposing this latter point constrained to move on the surface of which the equation is L = F(xyz) -- 0,. (314) the virtual velocity must lie in this surface, and the generality of Equation (313), is restricted to the conditions imposed by this cir cumstance. Supposing-the variables x y z, in the above equations, to receive the increments or decrements 6 x, 6 y, 6 z, respectively; we have, from the principles of the calculus, dL. d L dL d axy + d y.-1 = O. ~ ~ ~ (315) Multiplying by an indeterminate intensity X, and adding the product to Equation (313), there will result (d2 d L (x-M. +X. )6x= + (y.Z ~ ~ ~d t2 Xd + - ~6~*.d2y + dL d y 220 ELEMENTS OF ANALYTICAL MECHANICS. The quantity X, being entirely arbitrary, let its value be such as to reduce the coefficient of one of the variables 6 x, y, 6 z, say that of x, to zero; and there will result d2 x dL x -:~'W ~ + d Lx = o,..... (316) dt2 + d -(316) and r d2 dLY /y+ Z Id2z dL\d (Y-M.dt2 dy) y + - + _ a' d z = 0. (317) Now in Equation (315), 6y and 6z may be assumed arbitrarily, and 6x will result; hence 6y and 6 z in Equation (317) may be regarded as independent of each other, and by the principle of indeterminate coefficients, d2M y dL = 0 d -- (318) and eliminating X by means of Equation (316), we find, (Y Mx d2* d- -(X - M. d2dL ) dLx 0;.. (319) (Z -- - M. ~F / d (Zd Md d2zd d_ ~i' d z =O-j which, with the equation of the surface, will determine the place of the centre of inertia at the end of a given time. MIOTION ON A CURVE OF DOUBLE CURVATURE. ~ 214.-If the centre of inertia be constrained to move upon two surfaces at the same time, or, which is the same thing, upon a curve of double curvature resulting firom their intersection, take L= F(xyz)-0, o(320) H= hlxys t) O; _ r32. MECHANICS OF SOLIDS. 221 from which, by the process of differentiating and replacing dx, dy, d z, by the projections of the virtual velocity, dL dL _dL d 6x + d*y + d Z = ~; d* (321) dH dH dil 6 * + 6z =- O. (322) d x d-y dz Multiplying the first of these by X, and the second by X', adding the products to Equation (313), and collecting the coefficients of 6 x, 6y, and 6 z, we have X ~I d2$ X dL d ] t + x' ( d t2 d + d X J i d +h dL dH + Y -.31. d y 2 + + y v -o.(323), z - ~.~ -F+ x'- + X'-76z Now the coefficients of two of the three variables 6 x, 6 y and 6 z, say those of 6x and 6y, may be made equal to zero by assigning proper values for that purpose to the indeterminate intensities X and X', in which case, since 6 z is not equal to zero, its coefficient must also be equal to zero; whence d2 x dL +. d_ X- Mr ~.'- - o, d t2 d x dx d2y dL dH Y- M.-Y+ x. +. = ov d....(324) dt2 dy dy d2 z d +'J d= ~j z- d.T2 dz dz+.. and eliminating X and X', there will result d2xX /dL dH dL dH x dt2 dz dy dy dz d2y..... dH dL d 0. (35) + (Y-Med- (d-x -dz - dz -d x) =O. (325) (Z d2 Z\d L d tI dL dl 1 +~ dM dt2,'dy dx dz dy ) 222 ELEMENTS OF ANALYTICAL MECHANICS. which, with the equations of the surfaces, is sufficient to determine the co-ordinates of the centre of inertia when the time is given. ~215.-If the given surfaces be the projecting cylinders of a curve of double curvature, then will Equations (320) become L = F (x) —;} (326) H =F' (y z) = 0. And because L is now independent of y, and H is independent of x, we have d L dH d —=0; d -0 d? dx which reduce Equations (324) to X-M'd 2 + X *-0d Y-ed dYJra*d -; i''(327) + ddI. d. o d z d. d2 Z d- +' +' — d tX dz d0 ali, Equation (325) to (X-M. dd2x dL dI d t2/ dz y + (- d2 M ) * d-' do -~ = * * (328) d d dz -(Z-.d-z d2.z dL dfH d'.1 dx dy This, with the equations of the curve, will give the place of the centre of inertia at the end of a given time. ~ 216.-If the curve be plane, the co-ordinate plane x z, may be assumed to coincide with that of the curve; in which case the second of Equations (327), becomes independent of y, that variable reducing to zero, and d df d2y- = 0, and -- 0; dy MECHANICS OF SOLIDS. 223 hence Equations (327), bcome d2 X dL X-M - + x. = o; dt2 d I Y= 0o; (329) Z -. j dt2 dz d -~ and because the factor d2 y d H Equation (328) becomes, on dividing out the common factor (1E, --- z M - = 0-. 0330) d 2 dz dt2 * dz ~ 217.-By transposing the terms involving X, in Equations (316) and (318) and squaring we have d2 x2 2[ +d L)2 (d L)2 + (,), (. dY ) 2 +f (Z M d2z' 2 The second member of this equation is, Equation (50), the square of the intensity of the resultant of the extraneous forces and the forces of inertia. Denoting this resultant by N, we may write /d 2 dz 2 (aL 2 Xh + (d) + d = N. ( 331) and dividirg each of the equations dL d2X'd- d- (X — * d2 dy d t2/ dl / d2 N dz dt2/ 2.24 ELEMENTS OF ANALYTICAL MECHANICS. obtained by the transposition just referred to, by Equation (331), we find, dL d2 x d L dt2 V/d.L 2 /d L 2 7 d L'2 N (/ ~. (dz ~d L 2 d L 2. ('2) dL d2 z -i z_______-,tdx)~ + y + The second members are the cosines of the angles which the resultant of all the forces including those of inertia, makes with the axes; the first members are the cosines of the angles which the normal to the surface at the body's place makes with the same axes. These being equal, with contrary signs, it follows not only that the forces whose intensities are / (dL) (d< + a ( dL\ N, are equal, but that they are both normal to the surface, and act in opposite directions. The second is the direct action upon the diret action the surface; the first is the reaction of the surface. Equation (331), will, therefore, give the value of a passive resistance sufficient to neutralize all action in the system which is inconsistent with the arbitrary condition inposed upon the body's path. If the body be constrained to move on a rigid surface or line, this resistance will arise friom its reaction. ~ 218.-If Equations (332) be multiplied by and the angles which the, nlortma resistance of the surface makes with RMECHANIC S OF SOLIDS. 225 the axes x, y, z, respectively, be denoted by 0,, e,y and 8,, 8ose equations will take the form X-M- dt2 cos _ —0; d2 Y Y-M *d-t'+ N. cos: - 0; ~..... (333) d2 z Z- M df. + N. cos' — 0. ~219.-To impose the condition, therefore, that a body in motion shall remain on a rigid surface, is equivalent to introducing into the system an additional force, which shall be equal and directly opposed to the pressure upon the surface. The motion may then be regarded as perfectly free, and treated accordingly. The same might be shown from Equations (324) to be equally true of a rigid curve, but the principle is too obvious to require further elucidation. Equations (333), may, therefore, be regarded as equally applicable to a rigid curve of any curvature, as to a surface; the -normal reaction of the curve being denoted by N, and the angles which N makes with the axes x, y, z, by, O, and 0. ~ 220. —To find the value of N, eliminate d t from Equations (333), by the relation 1 V d- 7 d-7; in which V and s are the velocity and the space; then by transposition these equations may be written d2 x N. cos 6 = M.V2. — - X d s2 d2 V cos6O = M 2. Y;Z. N. cosks, —. V2. d- Z. d s2 226 ELEMENTS OF ANALYTICAL MECHEANICS. Squaring, adding and reducing by the relations R2 = X2 + 2 + Z2, cos28, + Cos2Oy + cos2 O, = I, and we find [*24 (d2)2 + (d2y)2 + (d2Z )2] + J2 AN2= — _ 2 Mx V2 X d2 + d22 z]J Resolving R into two components, one parallel and the other per. pendicular to the path, the former will be_ in equilibrio with the inertia it develops in the direction of the curve; and denoting by qg the inclination of R to the radius of curvature, we have d2 s d2 S Rsin c M --.2 =. V2Y'5. or, 0 =R. sinq -J. V2d. d s2, Squaring and subtracting friom the equation above, there will result. M2 d. ((d2 X)2 + (d2 y)2 + (d2 )2 _ (d2 )2,) + R2 cos2 V2I V2 X d2 x Y d2y Z d2z d2 s 2.. ( ds2 R ds2 R ds2 d s2 J but X dx Y dy Z dz 81 =R ds+ Rds R ds; multiplying the second member by p- p, substituting above, and reducing by the relations, dx:dy dz d d ad2x d2s ds d2y dy d2s ds d2z dz d2s ads d s2ds d ds' ds2 ds ds2 ds ds2 s ds2 ds' ddx ddy d X ds +Y ds Z ds CosP = *.p-s +. p-~s +p N. P —s; *See Appendix 1No. 2. MECHANICS OF SOLIDS. 22T and d s2 (d2z )2 + - (d2Y)Z + (d2 )2 - (d2 8)2 in which p denotes the radius of curvature, we have, V4 23 V2 2 = M2. - 2 V cos q 4- B2 cos2 9; P2 P and taking square root, ffV2 X= _R cosq... (334) The first term of the second member is, 195, the centrifugal force arising from the deflecting action of the curve, and the last term is the normal component of the B resultant B. As the equation stands, its signs apply to the case in which the body is on the concave side of the curve, and the resultant acts from the curve. The angle p, must be measured from the radius of curvature, or that radius produced, according as the body is on the concave or convex side of the curve. When the body is moving on the convex side of the curve, the first term of the second member must change its sign and become negative. ~221. — Writing Equations (333) under the form d2 X M.* = X + N cos 8., d2y Y + NcosO *d t2 -: COSB, d2 z M. dt -Z + NcosO8; multiplying the first by 2 dx, the second by 2d y, the third by 2 dz, adding and reducing by the relation ldx dy dz ds - cos,+ -. cos -.Cos0 ds " /ds 228 ELEMENTS OF ANALYTICAL MECHANICS. the second factor being the cosine of the angle made by the nor mal and tangent to the curve, we have (2 dX. d2X + 2 dy o d2y + 2dz d 2z) g @ t~ - =2(:d+ Ydy+Zdz)); integrating and reducing by V2 dx2 + dy2 +- dz2 d t2 we find if V2 -= 2 f(Xd x + Yd y + Zd z) + Ce (335) This being independent of the reaction of the curve, it can have no effect upon the velocity. If the incessant forces be zero, then will X = 0 Y=O; and Z =0; and C that is, a body moving upon a rigid surface or curve, and not acted upon by incessant forces, will preserve its velocity constant, and the motion will be uniform. We also recognize, in Equation (335), the general theorem of the living force and quantity of work; and from which, as before, it appears that the velocity is wholly independent of the path described. Ex2am le 1.-Let the body be required to move upon the interior surface of a spherical bowl, under the action of its own weight. In this case, L = x2 + y2 + z2 - 0;. (336) dL dL dL 2x; dy 2y; -d-= 2 dx' dy dr MECHANIC'S O'P SOOLIDS 229 and the, axis: of z. being vertical and positive downwards, Z X = 0; Y 0;' Z = Mg; which values in Equations (319), give d2x d-y 0 yd 2a d2' gy —yd- - d.- (337) Z and differentiating the equation of the sphere twice, we have x d2x + yd2y + z.d2z = - (d2 + dy2: + dZ2); dividing by d t2, and replacing the second member by its value V.z the velocity, we find,..:P Z. dP y d Z z, d~x dY y, d2z _ d d Y.- d t2 - But, Equation (335), V2 = 2gz + C.... (338) and denoting by VI and k, the initial values of V and z, respectively, we have V2 = V'2 + 2g (z -), wh.ich substituted above, gives 2-X + Y d2 + Z' d t = 2g (k -z) V ~ ~(339) Eliminate x, y, d2 x, d2y, from this equation by means of Equations (336) and (337). From the latter we find, d2y y'd2z - -- gj --- d2X x d(2z;~,d -- -2-x~-~~~Z -. 230 ELEMENTS OF ANALYTICAL MECHANICS. which substituted in Equation (339), and reducing by means of Equation (336), we get 2 d2 z a2 d - = g (a2 - 3z2 + 2kz) - ~2 z; multiplying by 2dz, and integrating, we find d z2 a2 d- 2 = 2g (a2z - z3 + kz2) - V'2 z2 + C; in which C is the constant of integration, and to determine which, we denote the component of the velocity V', in the direction of the axis z, by yV', and make z = k. This being done, we get = a2.V,'2 +'2k2 - 2ga2k; whence, d z2 a0* - 2g (a2 z -z3 + k z2) - V'2 2 + a2 V 2 + V2 k2 _ 2g a k, adding and subtracting a2 y'2 in the second member, this reduces to dz2. d2 =_ (a2 - Z2) [ V2 - 2g (k - )] -,, in which C, = (a2 k2) V'2-a2 V12. Finding the value of d t, and integrating, we have t= adz... 340 2- ) [V2 _ 2g (k -z)]- C, Could this equation be integrated in finite terms, then would z become known for a given value of t; and this value of z in Equation (336), and the first of Equations (337), after integration, would make known the values of x and y, and hence the position of the body; its velocity would be known from Equation (335). But this integration is not possible. MECHANICS OF SOLIDS. 231 ~ 222. -We may, however, approximate to the result when the initial impulse is small and in a horizontal direction, and the point of departure is near the bottom of the bowl. Let 6 be the angle which the radius drawn to the variable position of the body makes with the axis of z;?, the angle which the plane of the angle 6 makes with the plane through the axis z and initial place of the body, supposed in the plane x z; zV'- = P a, the velocity of pro. jection in a horizontal direction, 1 being a very small quantity; and a the initial value of 6. Then, because = a. cos 6 a (1 - 2 sin.' 1); k = a cos a = a (l - 2 sin' e); dz= — a.sin. d6; V'- 0; C, a'. 4 sin ~a. cos2 a.'. /. = ag. _. =a. sin' a; Equation (340) becomes \/af _ sin 6. d6 Yg V/3a(sin2 -- sin' a)- 4 sin' 0 (sin'2 Siln' ~ a) and making 6 and a very small, their arcs may be taken for their sines, and the above becomes, after differentiating, drt _ a/ ~. 2 **(341) d - g V /(2 - 62) ( - (3)) which may be put under the form = f(a - -48.d6 Vg=J /32( ) _ ) [22 _ (a2 + 32)]2 whence, by integration, 2 t 28 3)] + C; * * (342) making t - 0, and = -c, we have C =- cos 1. /a — i'g, or C = 0O; and solving the equation with reference to 6, we get 0 (U2 ~ + /2) + ~ (2f -032).ccs2 X t. *.(343) it" ~ 0iL+ P) +& (a _ 2)~CCS Va 282 ELEMENTS OF AINALYTICAL MECHANICS. From which it appears that the greatest- and least values of 0 will occur periodically, and: at equal intervals of time. The formert of these values is found by making cos 2.t= 1; whence 2 - t O or= 2-, or - 4;, a a and so on-; and for a, single interval between two consecutive maxi. ma, without respect to sign, ~t =.see ** ** *(344) the maximum being a. The least value occurs when cos 2 \ t- -1, or 2 t -', or = 3, &c. a a whence for a single interval between any maximum and the succeed. ing minimum, = -;...... (345) the minimum being J3. The movement by which these recurring values are brought about, is called oscillatory motion; that- between any two equal values is called an oscillation; and when the oscillations are performed in equal times, they are said to be Isochronous. Again, dp d p dt. d) - dt de' suostituting for d- ILs value obtained from the relation y = ztan q we find dq 1 dy d dy IY dx ) dt d- - x +2 y X dt -- tJ' Integrating the first of Equations (337), we get dy dx Y d- =2c=: V'aoa=a/3a /g-a; ME-CH A.NICS OF SOLIDS. 233 substituting this above, and also the value of di, given by Equa. tion (341), we find dv_;..... (346) d 8 3/ (X2 _ 82) (82, _2) dividing this by Equation (341), dt V a d2 V j(M2+ I2) + (M2 - g2).cos2 a /I.t but cos2 t = cos2 t-sin2 t; Va a a whence dcp _ g (347) ad t da 2COS2' t +- ~ 2.sin2't a a from which we find dt a Va cos2 d t dq = a 1+ - tan2 t jX2 a integrating, and taking tangents of both members, tan q -- tan tt... (348) from which the azimuth of the plane of oscillation may be found at the end of any time. Making tan g = GiA, we have 1 3 5 t =-2; or = -r; or a 2 2 284 ELEMENTS OF ANALYTICAL MECHANICS. and the interval from the epoch to the first azimuth of 900, is 1 a t, =.-. -, and to the first azimuth of 270~, 3 o and the interval from the azimuth of 900 to the next azimuth of 2700 td -- t, = t = ~. v g equal to the time of one entire oscillation. From Equation (348) we have, after substituting folr tan 9 its value in the relation y = x tan d, a2 y2 i2 x2 = tan2 ta adding unity to both members, 3 22xY2 = 1 + tan2.t; also from y = x.tanp, 2 = 1 + tal2 q; dividing the last equation by this one, and replacing x2 + y2 by its value a2 - z2, from the equation of the surface, we get I + tan2 t a2Y2 y+ f2x2 = 2. (a2 _ z2). + tan2q but, neglecting the term involving 84, a2 -_ 2 = a2 82; substituting this above, replacing tan2q by its value in Equation (348), and 82 by its value in Equation (343), after making cos 2 /.t = 1 - 2 sin2 1.t a a MECHANICS OF SOLIDS. 235 and reducing by the relation, tan2 H- t sin2 _t = a we have a I +tan2 ~.t x2 Y= a +;....... (349) which shows that the projection of the path of the body on the plane x y, is an ellipse whose centre is on the vertical radius of the sphere, and that the line connecting the body with the centre of the sphere, describes a conical surface. If a- =, then will, Equations (343) and (348), 82 = p2- -_ gt; and, Equation (349), X2 + y2 = a2a2;..... (350) hence, the body will describe a horizontal circle with a uniform motion. The pressure upon the surface, at any point of the body's path, is given by the value of N in Equation (334). ~ 223.-Example 2.-Let the body, still reduced to its centre of inertia and acted upon by its own weight, be also repelled Z from the bottom point A of the bowl, by a force which varies inversely as the square of the distance; required the position of the body in which it would remain at rest. >As the body is to be at rest, there will be no inertia exerted, and we have d2 X d2 y d2 z d t2 d 2; d 12 236 ELEMENTS" OF ANALYTICAL MECHANICS. and assuming the axis z vertical, positive upwards, and the origin at the lowest point A, L x2 + y2 + z2_ 2 az = 0,... (351) d.L dL dL = 2x; = 2y; -- 2(z-a) dx dy dz (-a); and denoting the distance of the body from the lowest point by r, the intensity of the repelling force at -the unit's distance by F, and the force at any distance by; P, then will P =; r = v2 +-y + z2;. (352) for the force P, cos -; cos; cosy -; for the weight Mg, cos a' = 0; cos 1i' 0;, cos y' = - 1; and Fx y y Fz X — 3 Z - g —.These several values being substituted in Equations (319), give ( 3 y -— I*-~ r (3 - ~7a)= 0. The first equation establishes no relation between x and y, since the equilibrium, which depends upon the distance of the particle from the source of repulsion, would obviously exist at any point of a horizontal circle whose circumference is at the proper height from the bottom. From the second equation we deduce, Fa 3 = Mg, F r3 _ —_,........ ~(353).q a MECHANICS OF SOLIDS. 237 from which r becomes known; and to determine the.position of the circle upon which the body must be placed, we have, by making x 0 in Equations (352) and (351), z2 + y2 =r, y2 _+ 72 2 2az = 0. Equation (353) makes known the relation between the weight of the body and the repulsive force at the unit's distance; the intensity of the force at any other distance may therefore be deter. mined. If there be substituted a repulsive force of different intensity, but whose law of variation is the same, we should have, in like manner, F' r'3 2Xg - a' henice, F: F' 3 r3 r: that is, the forces are as the cubes of the distances at which the body is brought to rest. If, instead of being supported on the surface of a sphere, the body had been connected by a perfectly light and inflexible line with the centre of the sphere and the surface removed, the result would have been the same. In this form of the proposition, we nave the common Electroscope. The differential co-efficients of the second order, or the terms which measure the force of inertia, being equal to zero, Equations (332), show that the resultant of the extraneous forces, in this case the weight and repulsion, is normal to the surface, which should be the case; for then there is no reason why the body should move in one direction rather than another. The pressure upon the surface is given by the value of N, in Equation (334). ~ 224. —Example 3. Let it be required to find the circumstances 238 ELEMENTS OF ANALYTICAL MECHANICS. of motion of a body acted upon by its own weight while on the are of a cycloid, of which the plane is vertical, and Z directrix horizontal. Taking the axis of z, vertical; the plane zx, in the plane of the curve; and the origin at the lowest point, then will L L = x - 2az - 2 - a versin -= =.. (354) a in which z is taken positive upwards. dL _;d-* (355)L dx dz z' X= 0;- Z= - Mg, and Equation (330) becomes d2 x /2 a- z d2 Z 2~-*'/; + g+dt-=~0.. (356) and by transposition and division, d2 g d2 z 1(357) dt 2/ z ~ (357) - From the equation of the curve we find, 2dx = 2 d z.V~-;..... (358) multiplying by Equation (357), there will result 2dx. d2 - 2dz. d2z = t -2gd-z - d j2 d 12 MECHAANICS OF SOLIDS. 239 and by integration, dSC dz2 = ( — g z -- 2-2 dtt2 or,.d x2 4- dZ2 dz+2d= V2 = C - 2gz; d t2 and supposing the velocity zero,:when z =h; 0 = C- 2gh; w.'ch subtracted from the above gives d X2 - dzd + = 2 g(h- z);... (359) ar,d eliminating dx2 by means of Equation (358), dz _9 (hz- Z2) d t2 -a 2) whence, /a dz dt = — ~ / -VIX g* -; the negative sign being taken because z is a decreasing function of t. By integration, dz -I 2z t' —/ -- — * versin + C. II hz-v2 g rsin Making z = h, we have 0 ---- versin 2 + C; -240 ELEMENTS OF ANALYTICAL:MECHANICS. whence, C=~-, and t= ~QX -versiri' *)(300) When the body has reached the bottom, then will z -0, and a which is wholly independent of h, or the point of departure, and we hence infer that the time of descent to the lowest point will oe the same in the same cycloid, no matter from what point the body qtarts. Whenever z = h, the body will, Equation (359), stop, and we shall have the times arranged in order before and after the epoch, -4r -; -2- 2; 0; 2 -; 40, &c., the difference between any two consecutive values being The body will, therefore, oscillate back and forth, in equal times. The cycloid is a Tautochrone. The pressure upon the curve is given by Equation (334). The time being given and substituted in Equation (360), the value of z becomes known, and this, in Equations (359) and (354), will give the body's velocity and place. ~225.-Example 4.-Let a body reduced to its centre of inertia, and whose weight is denoted by WV, be supported by the action of a constant force upon the branch EH of an hyperbola, of which the transverse axis is vertical, the force being directed to the centro of the curve. Required the position of equilibrium. MECHANICS OF SOLIDS. 241 Denote the constant force by W', which may be a weight at the end of a cord passing over a small wheel at C, and attached to the body M. Denote the distance C M by r, and the axes X of the curve by A and B. Take the axis z vertical, and the curve in the plane xz. Make P'=W, P" = W' =W then will i z x cos?" - -, cos a" = —, r r X = P' cos a' + P" cos a"' W -'r Z = P' cos y' + P" cos' W - W' -- and as the question relates to the state of rest, d2x d2 Z -0; dt- -. d Pdtz The Equation of tho curve is L = A2x2 - B2z2'+ A2B2 = 0; whence, 2 A2x, /d x dL d _ 2/B2z; dz these values substituted in Equation (330), give W'B2 xz -WA2X W'+ A =0 r r whence, (A2 + B2)'f.z - WA2r = O (361 16 242 ELEMENTS OF ANALYTICAL. MECHANICS. But 1.2 = x2 + Z2 = z2 + 22 - B2 = 2 ~B. B2 A2 A2 whence, denoting the eccentricity by e, r _ /e2z2 - _B2 and this, in Equation (361), gives after reduction, B. W e (W2 - W'12 e2)2 which, with the equation of the curve, will give the position of equilibrium. If W'e be greater than WV, the equilibrium will be impossibleIf W'e = WV, the body will be supported upon the asymptote. The pressure upon the curve is given by Equation (334). ~ 226.-Example 5. —Required the circumstances of motion of a body moving from rest under the action of its own weight upon an inclined right line. Take the axis of z vertical, the plane z x to contain the A line, and the origin at the point of departure, and -let z be reckoned positive downwards. Then will L -- z - ax =0, d.L d.L — 1; ~ d —— a; dz dx X 0=; Z Mg; which in Equation (330) give, after omitting the common factor M, d2 x d2 z d-2 + ug -- ~ * * ~..t (362) From the equation of the line we have -d2x = d2.. MECHANICS OF SOLIDS. 243 which in Equation (362), after slight reduction, d2 z a2 d 2 I +a2g Multiplying by 2dz, and integrating, d z2 a2 dhe constant of integration being zero. Whence d=( + a2) dz and g2 q= z (.a) the constant of integration being again zero. The body being supposed at B, then will z = AD; and if we draw from B the perpendicular B C to A B, we have AB 2 1+ a Z- a2 X which substituted above, _ _. =.-. *. *(364) z.*: g in which d denotes the -distance A C. But the second member is the time of falling freely through the vertical distance. d; if, therefore, a circle be described upon A C as a diameter, we see that the time down any one of its chords, terminating at the upper or lower point of this diameter, will be the same as that through the vertical diameter itself. This is called the mechanical property of the circle. Example 6.-A spherical body placed on a plane inclined to the horizon, would, in the absence of friction, slide under the action of its'own weight; but, owing to friction, it will roll. Required the cireumstances of the motion. 244 E LEMENTS OF ANALYTICAL MECHANICS. If the sphere move from rest with no initial impulse, the centre will describe a straight line parallel to the element of z steepest descent. Take the plane x z, to contain this element, the axis z vertical and positive upwards. A X The equation of the path will be, L = z + x tana - h = 0; whence, dL dL — = 1; * =tan a. dz' d The extraneous forces are the weight of the sphere and the frick tion. Denote the first by WI, and the second by F. The nature of friction and its mode of action will be explained in the proper place, ~ 354; it will be sufficient here to say that for the same weight of the sphere and inclination of the plane, it will be a constant force acting up the plane and opposed to the motion. We shall therefore have Z- -Mg + Fsina; X =-Fcosa, which values, and those above substituted in Equation (330), give -Fcosa-. dd t + (Mg - Fsina+ M.d2) an = O. d — 0. But from the equation of the path, we have d2z= -d2 x. tan a; and eliminating d2x by means of this relation, there will result d2 = s-n a -g sin a). MECHANICS OF SOLIDS. 245 Multiplying by 2 d z, integrating and making the velocity zero when z = A, we have d 2 2= 2 sin a - sin a)(z This gives dt1 dz 2 sin a ( -gsina) and by integration, the time being zero when z = h, a -- z = sin a (g i sin a- ). t2..pn (a). Again, all axes in the sphere through its centre, are principal axes;.the sphere will only rotate about the movable axis y, in which ease v, and v, will each be zero, and Equations (202) will give. dv'B. = At",; wherei B = Mk,2; dv _ d2~ t — k; -t = dt2, -Fr; r being the radius of the sphere. Whence, dq Fr d tz -- Fk2 Multiplying by 2 d, integrating, and making the angular velocity and the are 4, vanish together, d+2 2 Fr whence,.2 fFr 4 246 ELEMENTS OF ANALYTICAL MECHANICS. and by integration, making t and.; vanish together, F. rt2 Also, because the length of path described in the direction of the plane is r..+, we have, in addition, h - z = r.~.sina; and eliminating ~ from this and the above equation,, there will.result 2. (h4- sin, (b Dividing, Equation (a) by Equation (b), aind solving with respect to F,' M - k 2 and this- in Equation (6), gives/(h,- z) k.,4 ~r2 g = /sin2-*;, r~ (d) If the sphere be homogeneous, then will k - 2r2 and d= g.jsin2a 5 if the matter be all concentrated into the surface, then will wich times are to o another as tsn2a whiich times are to one another as - to Z25. CONSTRAINED MOTION ABOUT A FIXED POINTo ~ 227.-If a body be retained by a fixed point, the fixed and what has been thus far regarded as a movable origin may both be taken at this point; in which case, xz,, Sy,, &z,, in Equation (40), will bhe zero, the first three terms of that general equation cXf equ MECHANIOS; OF SOLIDS. 247 librium will reduce, to zero independently of the forces, and the equilibrium will be satisfied by simply making x.d2y - yd'x Y P (x cos - y cos a) -x- m- dt = 0; z. d x -- x d2 3 Z P (z cos a - xcosy) - m d; (365) dt0:^P (y cos - z cos y) - m. d z z. d 0; the accents being omitted because the elements m, t', &c., being referred to the same origin, x', y', z' will become x, y, z. The motion of the body about the fixed point might be discussed both for the cases of incessant and of impulsive forces, but the discussion being in all respects similar to that relating to the motion about the centre of inertia, ~ 127 and ~ 173, we pass to CONSTRAINED 3MOTION ABOUT A FIXED AXIS. ~ 228.-If the body be constrained to turn about a fixed axis, both origins may be taken upon, and the co-ordinate axis y. to coincide with this axis; in which case 6x,, 6y,, 6z,1, 6 and 6 d, in Equation (40), will be zero, and to satisfy the conditions of equilibrium, it will only be necessary for the forces to fulfil the condition, Z P (z cosa - xcosy) -- *m d-tz - 0 3(66) the accents being omitted for reasons just stated. ~229. —The only possible motion being that of rotation, let -s transform the above equation so as to contain angular co-ordinates. For this purpose we have, Equations (36), X* = r" sin 4; z' = r" cosJ.... (367) in which r" denotes the distance of the element nm from the axis y. Omitting the accents, differentiating and dividing by d t, we have dx d+ dz d_ (368 r-cos -, = - r sin-,.. (6s I dt ~dt dt dt 248 ELEMENTS OF ANALYTICAL MECHANICS. Now, z.d2 x.d2Z I. d Z d d t2 - d t2 =dt' d- Z - -t; whence by substitution, Equations (367) and (368), d2 X d2z 1 d d2 Z' — d t2 x-' d -- d'-t Zdt2 -x dt2 d'= Td 2r dt d2; and since d must be the same for every element, we have, Equation (366), Z m r2 = L P (z cos a - x cos ), 2 dt2 and d2~ P. (z cos -xoy)... (369) d t2 X2 m r2 That is to say, the angular acceleration of a body retained by a fixed axis, and acted upon by incessant forces, is equal to the moment of the impressed forces divided by the moment of inertia with reference to.cthis axis. Denoting the angular velocity by kV, and the moment of inertia by I, we find, by multiplying Equation (369) by 2 d;- and integrating, IV 2 = 2f P(z cosa - xcosy) d + C, and supposing the initial angular velocity to be V,', we have I(V1 - V2) = 2 f P (zcosa - cosy) d4. But the second member is, ~107, twice the quantity of work about the fixed axis; whence the quantity of work performed between the two instants at which the body has any two angular velocities, is equal to half the difference of the squares of these velocities into the moment of inertia, or to half the l ving force gained or lost in the interval. MEYCH-ANICS OF SOLIDS. 249 Now, I — Mk,2 =,. (1)2 = M,; so that, the moment of inertia measures that mass which would, if concentrated on the arc 4A, have a living force equal to that of the body which actually rotates. COMPOUND PENDULUM. ~ 230.-Any body suspended from a horizontal axis AB, about which it may swing with freedom under the action of its own weight, is called a compound pendulum. The elements of the pendulum being acted upon only by their own weights, we have P= mg; P' ='g, &c.; the axis of z being taken vertical and positive A downwards, cosa C= cos at &c.= 0 cos y cos T =&c. = 1, and Equation (369) becomes d2 _ m.... (370) Denote by e, the distance A G, of the centre of gravity from the axis; by 4, the angle HA G, which A G makes with the plane y z; by x,, L the distance of the centre of gravity from this plane; then will x, e. sin4;a and from the principles of the centre i of gravity, m mx = Mxz = M.e.sin~; Z which substitutea above, gives d2z M. e. sin. 2T2 sn. (371) 250 ELEMENTS OF ANALYTICAL MECHANICS. Multiplying by 2 d, and integrating, d-2 Me. 2 2.M ecos + C. d r2 mr2 Denoting the initial value of { by a, we have Me 0 = 2g. f.cos cc + C; whence, d{2 M2. o e dt2 tg Z (Co s -co a); (372) but cos I. 3 c 1.2 1.2.3.4 a2 a4 cos a = 1 -- &c. 1.2 1.2.3.4 and taking the value of 4, so small that its fourth power may be neglected in comparison with radius, we have a,2 _ 42 cos a - Cos 2 which substituted above, gives, after a slight reduction, and replacing m Mr2 by its value given in Equation (216), d; dt= 2 + e2 a e.g 2 a2 the negative sign being taken because h is a decreasing function of the time. Integrating, we have k/2 + e2 -C1 ( ) t --.' *cos -C O.... (373) e.g a The constant of integration is zero, because when 4 a, we have t = 0. MECHANICS OF SOLIDS. 251 Making =- a, we have +/k + e' e.g which gives the time of one entire oscillation, and from which we conclude that the oscillations of the same pendulum will be isochronal, no matter what the lengths of the arcs of vibration, provided they be small. If the number of oscillations performed in a given interval, say ten or twenty minutes, be counted, the duration of a single oscillation will be found by dividing the whole interval by this number. Thus, let 0 denote the time of observation, and N the number of oscillations, then will t - +V= so ie; and if the same pendulum be made to oscillate at some other location during the same interval 0, the force of gravity being different, the number N' of oscillations will be different; -but we shall have, as before, g' being the new force of gravity, 8 /k 2 + e2 =NV' e. g' Squaring and dividing the first by the second, we find N'2 g' _....... 9 (374)' N2 - g that is to say, the intensities of the force of gravity, at different places, are to each other as the squares of the number of oscillations performed in the same time, by the same pendulum. Hence, if the intensity of gravity at one station be known, it will be easy to find it at others. ~ 231.-From Equation (372), we have d 2 m Xmr2 = 2 1.g. e(cos - cosa); o (375) dt2 252 ELEMENTS OF ANALYTICAL MECHANICS. and making d V,; Zmr2 =I; e(cos{-cosa) =H; dt we have I. 12 = 2M.g.H; ~ ~. (376) in which 1, denotes the vertical height passed over by the centre of gravity, and from which it appears that the pendulum will come to rest whenever 4, becomes equal to a, on either side of the ver. tical plane through the axis. ~232.-If the whole mass of the pendulum be conceived to be concentrated into a single point, the centre of gravity must go there also, and if this point be connected with the axis by a medium without weight and inertia, it becomes a simple pendulum. Denoting the distance of the point of concentration from the axis by 1, we have k, =O; e= l, which reduces Equation (374) to..-t (37r) If the point be so chosen that k,2 + e2 ~; * (378) the simple and compound pendulum will perform their oscillations in the same time. The former is then called the equivalent simple pendulum; and the point of the compound pendulum into which the mass may be concentrated to satisfy this condition of equal duration, is called the centre of oscillation. A line through the centre of oscillation and parallel to the axis of suspension, is called an axis of oscillation. MECHANICS OF SOLIDS. 253 ~233. —The axes of oscillation and of suspension are reciprocal. Denote the length of the equivalent simple pendulum when the com, pound pendulum is inverted and suspended from its axis of oscillation, by 1', and the distance of this latter axis from the centre of gravity by e, then will I e +- e' or e' = l-e; and, Equation (378), Ilt k2 + e'2 k,2 + (I -e)2 e' - I-e' and replacing 1, by its value in Equation (378), we find' k,2 + e2 That is, if the old axis of oscillation be taken as a new axis of sus pension, the old axis of suspension becomes the new axis of oscillation. This furnishes an easy method for finding the length of an equivalent simple pendulum. Differentiating Equation (378), regarding I and e as variable, we have dl e2- k,2 de - e2 and if I be a minimum, dl e2 - k 2 de = ~= —'e2; whence, e- = k,. But when I is a minimum, then will t be a minimum, Equation (377). That is to say, the time of oscillation will be a minimum when the axis of suspension passes through the principal centre of.gyration, and the time will be longer in proportion as the axis recedes from that centre. 254 ELEMENTS OF ANALYTICAL MECHANICS. Let A and A' be two acute parallel prismatic axes firmly con, nected with the pendulum, the acute edges being turned towards each other. The oscillation may be made to take place about either axis by simply inverting the pendulum.. Also, let if be a sliding mass capable of being retained in any position by the clamp-screw H. For any assumedI 7 position of M, let the principal radius of / gyration be G C; with G as a centre, G G C as radius, describe the circumference \ / C S S'. From what has been explained, the time of oscillation about either axis will be shortened as it approaches, and lengthened as it recedes from this circumference, being a minimum, or least possible, when on it. By moving the mass H, the centre of gravity, and therefore the gyratory circle of which it is the centre, may be thrown towards either axis. The pendulum bob being made heavy, the centre of gravity may be brought so near one of the axes, say A', as to place the latter within the gyratory circumference, keeping the centre of this circumference between the axes, as indicated in the figure. In this position, it is obvious that any motion in the mass M would at the same time either shorten or lengthen the duration of the oscillation about both axes, but unequally, in consequence of their unequal distances from the gyratory circumference. The pendulum thus arranged, is made to vibrate about each axis in succession during equal intervals, say an hour or a day, and the number of oscillations carefully noted; if these numbers be the same, the distance between the axes is the length 1, of.the equivalent simple pendulum; if not, then the weight M must be moved towards that axis whose number is the least, and the trial repeated till the numbers are made equal. The distance between the axes may be measured by a scale of equal parts. ~234.-From this value of 1, we may easily find that of the simple second's pendulum; that is to say, the simple pendulum which \Mill MECHANICS OF SOLIDS. 255 perform its vibration in one second. Let N, be the number of vibrations performed in one hour by the compound pendulum whose equivalent simple pendulum is 1; the number performed in the same time by the second's pendulum, whose length we will denote by I', is of course 3600, being the number of seconds in 1 hour, and hence, m1' Y=T = 3600s = T'=r and because the force of gravity at the same station is constant, we find, after squaring and dividing the second equation by the first, ~' ~.... (379) (3600) (379) Such is, in outline, the beautiful process by which KATER determined the length of the simple second's pendulum at the Tower of London to be 39,13908 inches, or 3,26159 feet. As the force of gravity at the same, place is not supposed to change its intensity, this length of the simple second's pendulum must remain forever invariable; and, on this account, the English have adopted it as the basis of their system of weights and measures. For this purpose, it was simply necessary to say that the 3,2 159th part of the simple second's pendulum at the Tower of London shall be one English foot, and all linear dimensions at once result from the relation they bear to the foot; that the gallon shall contain ~2-sth of a cubic foot, and all measures of volume are fixed by the relations which other volumes bear to the gallon; and finally, that a cubic foot of distilled water at the temperature of sixty degrees Fahr. shall weigh one thousand ounces, and all weights are fixed by the relation they bear to the ounce. ~235.-It is now easy to'find the apparent force of gravity at London; that is to say, the force of gravity as affected by, the cen. trifugal force and the oblaten-ress of the earth. The time of oscillation 256 ELEMENTS OF ANALYTICAL MECHANICS. being one second, and the length of the simple pendulum 3,26159 feet, Equation (377) gives /ft. 3,26159 g whence, g = r2 (3,26159) = (3,1416)2. (3,26159) = 32,1908 feet. From Equation (377), we also find, by making t one second, g = 2 1, and assuming = x + y cos 2, we have = x + y cos 2.....(380) Now starting with the value for g at London, and causing the same pendulum to vibrate at places whose latitudes are known, we obtain, from the relation given in Equation (374)', the corresponding values of g, or the force of gravity at these places; and these values and the corresponding latitudes being substituted successively in Equation (380), give a series of Equations involving but two unknown quantities, which may easily be found by the method of least squares. In this way it has been ascertained that q2,. = 32,1808 and 2.y = - 0,0821; whence, generally, f f g = 32,1808 - 0,0821 cos 2; ~ ~ ~ ~ (381) and substituting this value in Equation (377), and making t = 1, we find f I = 3,26058:- 0,008318 cos 2 ~ ~ ~ ~ (382) Such is the length of the simple second's pendulum at any place of which the latitude is,~. MECHANICS OF SOLIDS. 257 If we make + = 400 42' 40", the latitude of the City Hall of New York, we shall find ft. tn. 1 = 3725938.39,11256. ~236.-The principles which have just been explained, enable us to find the moment of inertia of any body turning about a fixed axis, with great accuracy, no matter what its figure, density, or the distribution of its matter. If the axis do not pass through its centre of gravity, the body will, when deflected front its position of equilibrium, oscillate, and become, in fact, a compound pendulum; and denoting the length of its equivalent simple pendulum by 1, we have, after multiplying Equation (378) by M, M.l.e = M(k,2 + e2) =:mr2;... (383) or since W g -*. ie = lmr2, ~...... (384) in which TW denotes the weight of the body. Knowing the latitude of the place, the length 1' of the simple second's pendulum is known from Equation (382); and counting the number N of oscillations performed by the body in one hour Equation (379) gives 1'. (3600)2 JN2 To find the value of e, which is the distance of the centre of gravity from the axis, attach a spring or other balance to any point of the body, say its lower end, and bring the centre of gravity to a horizontal plane through the axis, which posi- - tion will be indicated by the maximum reading of the balance. Denoting by a, the distance from the axis C to the point of support t, 17 258 ELEMENTS OF ANALYTICAL MECHANICS. and by b, the maximum indication of the balance, we have, from the principle of moments, -b a We. The distance a, may be measured by a scale of equal parts. Sub. stituting the values of W, e and I in the expression for the moment of inertia, Equation (384), we get b. a. 1'. (3600)2 g. N2 If the axis pass through the centre of gravity, as, for example, in the fly-wheel, it will not oscillate; in which case, take Equation (383), from which we have Mk,2 =.3. l.e - Me2. Mount the body upon a parallel axis A, not passing through the centre of gravity, and cause it to vibrate for an hour as before; from the number of these vibrations and the length of the simple second's pendulum, the value of 1 maybe found; M is known, being the weight W divided by g; and e may be found by direct measurement, or by the aid of the spring balance, as already indicated; whence k, becomes known. MOTION OF A BODY ABOUT AN AXIS UNDER THE ACTION OF IMPULSIVE FORCES. ~ 237.-If the forces be impulsive, we may, ~ 170, replace in Equation (366) the second differential co-efficients of x, y, z, by the first differehntial co-efficients of the same variables, which will reduce it to zdx - xdz XP(z cos a - xcosy) -m. = 0; MECHANICS OF SOLIDS. 259 and replacing dx and dz, by their values in Equations (368), we find d4A, P (z cos a - X cosy).6 dt =- zmr2 That is, the angular velocity of a body retained by a fixed axis, and subjected to the simultaneous action of impulsive forces, is equal to the sun,m of the moments of the impressed forces divided by the moment of inertia with reference to this axis. BALLISTIC PENDULUM. ~ 238.-In artillery, the initial velocity of projectiles is ascertained by means of the ballistic pendulum, which consists of a mass of matter A suspended from a horizontal axis in the shape of a knife-edge, after the manner of the compound pendulum. The bob is either made of some unelastic substance, as wood, or of metal provided with / a large cavity filled with some -"/ - soft matter, as dirt, which re- - ceives the projectile and retains iH the shape impressed upon it by the blow Denote by V and m, the initial velocity and mass of the ball; V, the angular velocity of the ballistic pendulum the instant after the blow, I and M its moment of inertia and mass. Also let I represent the distance of the centre of oscillation of the, pendulunm from the axis A. That no motion may be lost by the resistance of the axis arising from a shock, the ball must be received in the direction of a line passing through this centre and perpendicular to the plane of the axis and line A O. With this condition, Eq. (386) gives C2 +- e2 -d m. V.l 1 V/e Tn dt z,, -' ) t,+ ). e 2IL0 ELEMENTS OF ANALYTICAL MECHANICS. whence M+m and supposing the angular velocity communicated to the pendulum tc be equal to that acquired by falling from rest through the initial are a, in Equation (372), we have, from that equation and Equation (216), by writing e for d, (M ) (,2 + e2) ) = A sin aye and Eq. (374), r / g e t k, + e' which substituted above gives V1=2- sin lta; t 2 and this in the value for V gives, after substituting for the ratio of the masses that of their weights, + w r V = 2 -- e *sin ~....... (l387) w t (387). From this equation we may find the initial velocity V; and for this purpose, it will only be necessary to have the duration of a single oscillation, and the amplitude of the are described by the centre of gravity of the pendulum. The process for finding the time has been explained. To find the arc, it will be sufficient to attach to the lower extremity of the pendulum a pointer, and to fix, on a permanent stand below, a circular graduated groove, whose centre of curvature is at A; the groove being filled with some soft substance, as tallow, the pointer will mark on it the extent of the oscillation. Knowing thus the arc a, and the value of e, found as already described, ~ 236, we have eV. MECHANICS OF SOLIDS. 261 THE GUN PENDULUM. This consists of a gun suspended from a horizontal axis. The shot is fired from the gun, and its velocity is inferred from the recoil, as in the Ballistic Pendulum. The forces measured by the quantities of motion developed by the expansive action of the exploded powder, must be in equilibrio. Make = velocity of the ball on leaving the gun, n F = average velocity of the inflamed powder, V = angular velocity of pendulum on parting from shot, % = weight of gun pendulum, VWb = 44 ball and wad, We = 4" the charge of powder and bag, Wvl = 4" "4 "4 of powder alone, d = diameter of bore, d = diameter of ball, E = distance of axis of bore from axis of suspension. The quantity of motion in ball and wad, on leaving the gun, will be E V; the corresponding pressure on the bottom of the gun is to g that which generates this motion, as the area of a cross-section of the bore is to that of a great circle of the ball. Again, the blast of the powder will continue its action on the gun after the ball leaves it. Let this action be proportional to the charge of powder. The moment of the force impressed upon the pendulum, in reference to the axis of suspension, will be given by Eqs. (384) and (229); and taking the moments of the other forces in reference to the same axis, we have Tv vF 12 Wt tV W,.. e- W V..E...-..,-.2'.T,=o0; an which n', like n, is a constant to be determined by experiment; and from which we find W. v,.. el. Wt.6 c +W nW.c + n' Wp.s Zi(:1 ELEMENTS OF ANALYTICAL MECHANICS.'The living force with which the pendulum separates from the ball must equal twice the work performed by the weight while the centre of gravity is moving to the highest point; whence W 2-.l. e = 2 Wg. e. versine a = 4 Wg. e. sin2. a, inl which a denotes the greatest inclination of e to the vertical. Whence V, = 2 9. sin Ia; which substituted above gives, 9. y. ~ s (88). T= r sin is..a. (3882 W6, - + rWn 1+ n' W The methods for finding e and a, are the same as in the ballistic pendulum. To find n and n', fire the ball from the gun into the ballistic pendulum; the effect upon the latter will give the initial velocity V. Repeat as often as may be thought desirable, and with different charges. The corresponding initial velocities substituted in Eq. (388), will give as many equations as trials. These equations will contain only n and n' as unknown quantities, which may be found by the method of least squares. For full and valuable information on this subject, consult Mordecai's "Experiments on Gunpowder." PART II. {MECHANICS OF FLUIDS. INTRODUCTORY REMARKS. ~239. —TE physical condition of every body depends upon the relation subsisting among its molecular forces. When the attrac. tions prevail greatly over the repulsions, the particles are held firmly together, and the body is solid. In proportion as the difference between these two sets of forces becomes less, the body is softer, and its figure yields more readily to external pressure. When these forces are equal, the particles will yield to the slightest force, the body will, under the action of its own weight, and the resistance of the sides of a vessel into which it is placed, readily take the figure of the latter, and is liquid. Finllally, when the repulsive exceed the attractive forces, the' elements of the body tend to separate from each other, and require either the application of some extraneous for'ce or to be confined in a closed vessel to keep them together; the body is then a gas. In the vast range of relation among the molecular forces, from that which distinguishes a solid to that which determines a gas or vapor, bodies are found in all possible conditions-solids run imperceptibly into liquids, and liquids into gases. Hence all classification of bodies founded on their physical properties alone, must, of necessity, be arbitrary. ~240.-Any body whose elementary particles admit of motion 264 ELEMENTS OF ANALYTICAL MECHANICS. among each other, is called a Jluid-such as water, wine, mercury, the:air, and, in general, liquids and gases; all of which are distin. guished from solids by the great mobility of their particles among themselves. This distinguishing property exists in different degrees in different liquids-it is greatest in the ethers and alcohol; it is less in water and wine; it is still less in the oils, the sirups, greases, and, melted metals, that flow with difficulty, and rope when poured into the air. Such fluids are said to be viscous, or to possess viscosity. Finally, a body may approach so closely both a solid and liquid, as to make it difficult to assign it a place among either class, as paste, putty, and the like. ~ 241.- Fluids are divided in mechanics into two classes, viz.: comnzpressible and incompressible. The term incompressible cannot, in strictness of propriety, be applied to any body in nature, all being more or less compressible; but the enormous power required to change, in any sensible degree, the volumes of liquids, seems to justify the term, when applied to them in,a restricted sense. The gases are highly compressible. All liquids will, therefore, be regarded as incompressible; the yases as compressible. ~242. —The most important and remarkable of the gaseous bodies is the atmosphere. It envelops the entire earth, reaches far beyond the tops of our highest mountains, and pervades every depth from which it is not excluded by the presence of solids or liquids. It is even found in the pores of these latter bodies. It plays a most important part in all natural phenomena, and is ever at work to influence the motions within it. It is essentially composed of oxygen and nitrogen, in a state of mechanical mixture. The former is a supporter of combustion, and, with the various forms of carbon, is one of the principal agents employed in the development of mechan-. ical power. The existence of gases is proved by a multitude of facts. Con. tained in an inflexible and impermeable envelope, they resist pressure like solid lodies. Gas, in an inverted glass vessel plunged into water, will not yield tts place to the liquid, unless some avenue of escape be provided for it. Tornadoes which uproot trees, overtuin MECHANICS OF FLUIDS. 265 houses, and devastate entire districts, are but air in motion. Air opposes, by its inertia, the. motion of other bodies through it, and this opposition is called its resistance. Finally, we know that wind is employed as a motor to turn mills and to give motion to ships of the largest kind. ~243.-In the discussions which are to follow, fluids will be considered as without viscosity; that is to say, the particles will be supposed to have the utmost freedom of motion among each other. Such fluids are said to be perfect. The results deduced upon the hypothesis of perfect fluidity will, of course, require modification when applied to fluids possessing sensible viscosity. The nature and extent of these modifications can be known only from experiments. MARIOTTE' S LAW. ~244.-Gases readily contract into smaller volumes when pressed externally; they as readily expand and regain their former dimensions, when the pressure is removed. They are therefore both cornpressible and elastic. It is found by experiment, that the change in volume is, for a constant temperature, always directly proportional to the change of pressure. The density. of the same body is inversely proportional to the volume it occupies. If, therefore, P denote the pressure upon a unit of surface which will produce, at a given temperature, say O~ Centr., a density equal to unity, and D any other density, and p the pressure upon a unit of surface which will, at the same temperature of the gas, produce this density, then, according to the ex. periments above'referred to, will p = P.D...... (389) This law was investigated by Boyle and Mariotte, and is known as Mariotte's Law. By experiments made at Paris, it was found that this law obtains, when air, in its ordinary condition, is condensed 27 an&d rarefied 112 times. 266 ELEMENTS OF ANALYTICAL IECHANICS. LTAW OF THE PR.ESSURE, DE NSITY, AND TE:MPERATURE. ~ 245.-Under a constant pressure, all bodies are expanded by heat; under a constant volume, their elastic force is increased by the same agent. Experiment has shown that the laws of these changres for gases are expressed by p=P.I.(l Jrct); * * (390) in which p denotes the pressure upon a unit of surface, D the density of the gas, 0 the difference between the actual and some standard temperature, and oc a constant which is equal to 213-_0,003665 when the standard is 0~ centr., and 0 is expressed in units of that scale. First supposing D and 0 variable and p constant; then p and 0 variable and D constant, Equation (390) gives dD a. D dp ap d - 1 +a; d8 - cI ad The quantity of heat, denoted by q, necessary to change the temperature 0 degrees from the assumed standard, will be a function of p, D), 0; btlt because of Equation (390,) we may write q =f (D, p). * * * *(b) The increment of heat which will raise a body's temperature one degree, is called its specife heat. The specific heat being the increment of q for each unit of 0, if c denote the specific heat when the pressure is constant, and c, that when the density is constant, then will d q dq d D d q dq dp c = d-d D d; = ddp d' i or, Equations (a), dq a.D dq a.p dl) D I-oO' a - dp 1 +- a' and by division, making c = y.c,, (I q d q D - d D 1Y.pe = 0, in which T, denotes the ratio of the specific heat of the gas at a constant pressure to that at a co(nstant density. This ratio is known from experiment to l)e constanlt fir atmospheric air, and is piobably so for all gsses.'Th11e xlperlimrenlts of I)esormes and MECHANICS OF FLUIDS. 267 Clements make its value 1,3482; those of Gay-Lussac and Walter 1,3748; and those of Dulon)g on perfectly dry air 1,421. Regarding r as constant, the integration of the foregoing equation gives q =/ (5) (See Appendix No. 3.) in which f, denotes any arbitrary function of the quantity witlhln the parenthesis, and from which, denoting the inverse functions by F, we may write.F(q)........ (q)(c) From Equation (390), we have 0 __-=p 1 1..(q)!-..(d) B_=.1v (d) a.. - a =a~ —7~ Sudden compression increases, and a sudden expansion decreases the temperature of bodies, and if q remain the same, wbhile suddenly p, D, 8, become p', D', 0', we have p' =D'7.F(q), ~ ~ (e) 8' _ /'7 -'.F(q). (g) Eliminating F (q) first from Equations (c) and (e), and then fiom Equations (d) and (g), we have, replacing 7 and oa by their numerical values, (t\D 1421 _ ).\0,421 0'e (273 + 0) (.D - ) 273 *(392) These equations give the relation between the densities, elastic forces, and the tenmperatures of a gas suddenly compressed or dilated, and retaining the quantity of its heat unchanged. The pressure being constant, mlake,in Eq. (390), 0 -- 0, 1) = D,, and divide same equation by the result; we find D= I), - (- cc0). MBake p=-D,,, hj ~ q' -- weight of a columln of mereli'y at stlad!card telnperature T, and resting oi a base unity, in Lat. 45~, -ohi re gravity is g'. Thes( in Eq. (39) give. after lwrit in 0.0' 204 fr a,;dl t~ — 3'0 tr 0, p_ [1 + (t~ - 2) o. 0 4] (' If -he te lell- tlre ol f }- t tE - ni, r.lil ry v-.1y fir,)m the st-, t 268 ELEMENTS OF ANALYTICAL MECHANICS. and become T' then will D, also vary and become D2,', and to exert the same pressure A,, must have a new value A, and such that D)m..jj g' = D-,. h. 9:. RMercury expands or contracts 0,001001th part of its entire vol umne for each degree of Fahr. by which it increases or diminishes its temperature. And as the density of the same body varies inversely as its volume, we have D) = D, [1 + (T — T') 0,0001001] which substituted above gives h,,= h [1 + (T- T').0,0001001]..(394) EQUAL TRANSMISSION OF PRESSURE. ~ 246.-Let EHL, represent a closed vessel of any shape, with which two piston tubes A B' and D C' communicate, each tube being provided with a piston that fits it accurately and which may' move within it with the utmost fieedom. The vessel being filled with any fluid, let forces P and P', be applied, the former perpendicularly to the piston A B, and the latter in like direction to the piston CD, and suppose these forces in equilibrio, which they may be, since the fluid cannot escape. Now let the piston A B be moved to the position A' B'; the piston CD will take some new position, as C('D'. And denbting by s and s', the dis. tances A A' and C C', respectively, we have, from the principle of virtual velocities, Ps = P' s'. Denote the area of the piston AB by a, and that of the piston C D by a', then will the volume of the fluid which was thrust from the tube A B', be measured by a. s, and that which entered the tube MECHANICS OF FL.UIDS 269 ) C', will be measured by a' s'. But the pressure upon the pistons and the temperature remaining the same, the entire volume of the fluid in the vessel and tubes will be unchanged. Hence, as = a' s'; dividing the equation above by this one, we have P P' a = -...... (396) a a' That is to say, two forces applied to pistons which communicate freely with each other through the intervention of some confined fluid, will be in equilibrio when their intensities are directly proportional to the areas of the pistons upon which they act. This result is wholly independent of the relative dimensions and positions of the pistons; and hence we conclude that any pressure communicated to one or more elements of a fluid mass in equilibrio, is equally transmitted throughout the whole fluid in every direction. This law which is fully confirmed by experiment, is known as the principle of equal transmission of pressure. ~247.-Let a become the superficial unit, say a square inch or square foot, then will P be the pressure applied to a unit of surface, and, Equation (396), P' = P a'. ~. * * (397) That is, the pressure transmitted to any portion of the surface of the containing vessel, will be equal to that applied to the unit of surface multiplied by the area of the surface to which the transmission is made. ~ 248.-Since the elements of the fluid are supposed in equilibrio, the pressure transmitted to the surface through the elements incon. tact with it, must, ~ 217 and Equations (332), be normal to the sur face. That is, the pressure of a fluid against any surface, acts alwayi in the direction of the normal. 270 ELEMENTS OF ANALYTICAL MECHANICS. MOTION OF THE FLUID PARTICLES. ~ 249.-The particles of a fluid having the utmost freedom of motion among one another, all the forces applied at each particle must be in equilibrio. Regarding the general Equation (40) as applicable to a single particle, whose co-ordinates are x, y, z, we shall have Z = Xi, y = Y, z = I,, and supposing the particle to have simply a motion of translation, we also have 6a =0; 64 =0; 6S=0; and that equation becomes (P cosa - m. -- x + QPcos8- m.5->y. =0; + (HPcosy-m< dt2 j whence, upon the principle of indeterminate co-efficients, d2 X ZP cos a - nZ - = 0; d tP d2y Y.Peos -M-.-12 - (398) d2 z Pcosy-? dt2 =0. Now the terms: P cos a, Z P cos /3 and z P cos y, are each composed of two distinct parts, viz.: 1st., the component of the resultant of the forces applied directly to the particle; and 2d., the component of the pressure transmitted to it from a distance, arising from the forces impressed upon other particles. Denote by X, Y and Z, the accelerations, in the directions of the axes x, y, z, respectively, due to the forces applied directly to the MECHANICS OF FLUIDS. 271 particle; then m, being the mass of the particle, the components of the forces directly impressed will be mX; m Y; mZ. The pressure transmitted will depend upon the particle's place, and will' be a function of its co-ordinates of position. Denote by p, the pressure upon a unit of surface, on the supposition that every point of the unit sustains a pressure equal to that communicated to the particle from a distance; then, for a given time, will p = F(x,y,z). Conceive each particle of the fluid to consist of a small rectan. gular parallelopipedon whose faces are parallel to the co- Z1 ordinate planes, and whose contiguous edges at the time t, are dxr, dy and dz; and let x, y, z, be the co-ordinates of the molecule in the solid angle nearest the origin of coordinates. Then would the difference of pressure on the / opposite faces, which are parallel to the plane z y, were these faces equal to unity, be dp F(x + dx, yz,)-F(x, y, z,)=P. dx; and upon the actual faces whose dimensions are each dz.dy, this difference becomes, Equation (397), dp dp.dxdy.dz. dx In like manner will the difference of the pressures transmitted to the opposite faces parallel to the planes zx and xy, be, respeo tively, dp d y da dx, and d.dz dx dy. dy dz 272 ELEMENTS OF ANALYTICAL MECHANICS. These pressures being normal to the surfaces to which they are respectively applied, they will act, the. first in the direction of x, the second in the direction of y, and the third in the direction of z. And as these differences alone determine that portion of the motion due to the transmitted pressures, we have, EY Pcos-a rmX-'- d x. dy.dz; dx IP cos -= mY- d dy. dx. dz; dy dp Z P cos y M Z- dz. d x. dy. dz Denote by D the density of the mass m, then will, Equation (1)', m =D.dx.dy.dz, anld by substitution, Eluations (398) become 1 d1r d2 X D dx dt2 z I dp d2 y!j dy/ Y= d12 (399) 1 dp _ d2 z dz dt2 Denote by u, v and w, the velocities of the molecule whose coordinates are xyz, parallel to the axes x, y, z, respectively, at the time t. Each of these will be a function of the time and the coordinates of the molecule's place; and, reciprocally, each co-ordinate will be a function of t, u, v and w; whence, Equations (12) and (13), d2x du (du d t du dx du,dy du dz dt2 —7 dt' d t' dtd dt dy dt dz dtt dx dy dz and replacing -dt d-, c —, by their values u, v, w, respectively, we have d2 x d u dU du du d-dt2dxU d u d u.dt2 \ d y d MECHANICS OF FLUIDS. 2T3 in the same way, d2 y dv dv dv dv d d(.t)+''j +' *'+ -d., dt d/ ddt dy dz d2z (d \ dw dw d w dt2 tdtJ dr t d * dw which, substituted in Equations (399), give 1 dp d u d u d d t d I _ -.....V D d dt dX 7 dy dzl 1 dp dv\ dv dv dv - I —t 0 u -- ~.w; - ~.(400).d y d dt dy dz 1 dp z /dwv dw dw dw —.z ='~ -- - ZU - - - -W. D dz ( dt d w Here are three equations involving five unknown quantities. viz. t, v, w, p and D, which are to be found in terms of x, y, z and t. Two other equations may be found from these considerations, viz: the velocity in the direction of x, of the molecule whose co-ordinates are x yz, is u; the velocity of the molecule in the angle of the parallelopipedon at the opposite end of the side dx, at the same time, is du du + and hence the relative velocity df the two molecules is du du u +.dx - it dx. At the time t, the length of the edge joining these molecules is dx, and at the end of the time t + dt, this length will be dx + d. dx. dt = dx(l +d * dt); the second term being the distance by which the molecules in question approach toward or recede from one another in'tho time dt. 18 274- ELEMENTS OF ANALYTICAL MECHANICS. In the same way the edges of the parallelopipedon which at the time t, were dy and dz, become respectively, dv dv dy +.dy.dt = dy (1 + dt); dw dw d z dd d -z + -*.dz.dt =dz (l + -7*.dt); and the volume of the parallelopipedon, which at the time t, was dx.dy.dz, becomes at the time t + dt, du dv dw d.y. dz (l + dx.t).(1+ d(1+ d- dd t). The density, which was D, at the time t, being a function of xyz and t, becomes at the time t + dt, d D dD d D dD D + -d t + -dx +. dy + - dz; dt d dydz which may be put under the form, IdI d D dx d dy dD dzd D+ +..Y- d ~ + d +d dt d+ y dt. dz dt and replacing dx dy. dz dt dt dt' by their values u, v, w, respectively, dd D dD EdD D D+ (d D u + DV+ W) d i. -~ ~-E. dx d+ y dz Multiplying this by the volume above, we have for the mass of the parallelopipedon, which was D. d. dy. dz, at the time t, the value, [D + (dD + dDo+ dD d d] at th tm +W X dzx.dy.d2(1+d t).*1 " di d I t) * d * di) at the tdi-d dy dz at the time t - dr. MECHANICS OF FLUIDS. 275 But these masses must be equal, since the quantity of matter is unchanged. Equating them., striking out the common factors, per. forming the multiplication,'and neglecting the second powers of' the differentials, we have;) + + +- + ++ -vy+ + WQ' + *+-= 0.(401) dz d dz dt dz dy dz This is called the Equatioc of co tinuity of the Jfuid. It expresses the relation between the velocity of the molecules and the den. sity of the fluid, which are necessarily dependent upon each other. This is a fourth equation. ~250. —If the fluid be compressible, then will the fifth equation be given by the relation, F (D, p) =,. ~ ~ ~ (402) as is illustrated in the particular instance of Mariotte's law, Equa. tion (389). The'form of the function designated by thq letterPF will depend upon the nature of the fluid. ~251.-If the fluid be incompressible, the total differential of D will be zero, and d:D d D dD dDI a+dD. 4x - -- D Oy ~ ~d (403) d t Xr a + d * = and consequently, the equation of continuity, Equation (401), becomes, du dv dw dj x+ +d g; d 0.. ~. (404) ar:d we have for the determination of u. v, w, D and,. the five Equations (400), (403), (404). ~ 252.-These equations admit of great simplification in the case of ant incom3ressible homogeneous fluid, when u- d x + v. dy + w. d z, is a perfect differential. For if we make u'dx + v dy + wdz = d 276 ELEtMEN TS OF ANALYTICAL MECHEANICS. then from the partial differentidls will d ~ d ~ d ~ u r-V7- d w;*. (405) dx; v- dy; W dz which, in Equation (404), gives for the equation of continuity, d2 9 d2 9 d2 q0 d2 - ~ 2 + 2 (406) by the integration of which the function cq may be found. Differentiating the values of u, v and w above, we have d2 q9 d2 - d2 q d u dx; dv -; dw — dx dy dz Eliminating u, v, w, d u, d v and d w, from Equation (400), by means of the values of these quantities above, we have 1 dp (2'p d d9 d2 p d p d2 dip d2 cp Judx dxdt dx dx d dx. dx dy dz dx.dz' I dp d2 q9 d q d2 i d d2a) d q d2cp D dy dy.dt dx dy.dx dy dy2 dz dy.dz' 1 dp d2c9 dcp d29 d d29c dcp d29 ) dz dz.dt dx dz.dx dy dz.dy dz dz2 Multiplying the first by dx, the second by d y, the third by d z, and adding, we, find, -.dp=Xdx+ Ydy+Zdz- d -Id[(d y+ ()2 (d9] (407) From which, by integration, may be found the pressure at any point of an incompressible fluid mass in motion, when Equation (406) is the equation of continuity. ~253. —When the excursions of the molecules are small, the second powers of the velocities may be neglected, which will reduce Equation (407) to' *dp = Xdx + Ydy + Zdz - d d9 (408) MECHANI CS OF F4%UI DS. 277 ~ 254.-If the condition expressed by Equation (406) be not fulfilled, then we must have recourse to Equation (404) to find the pressure..~255.-Resuming Equation (401), which appertains to a compres. sible fluid, retaining the condition that udx + vdy'+ wdz = dcq is a perfect differential, and from which, therefore, d p d (p dq = d; - d-; = d —;. (409) we obtain by substitution, dfi d v dw dD dDvd ddw dD d Ddq dx dy. dz dt dx dx dy dy dz dz If the excursions of the molecules from their places of rest be very small, both the change of density and velocity will be so small that the products which constitute the last three terms of this equation may be neglected, and the equation of continuity becomes'D (- + z + t = and replacing -du, d v and d w, by their values from Equations (409), and dividing by D, we find dlogD d~ d2p d2~ (410) dt d+ d+y + d - = 0. e (410) from which, and Eq. (408), the equation connecting the extraneous forces with the co-ordinates xyz, and that expressive of AMariotte's law, the funetion p may be found, then the value of D, and finally that of p. The excursions being small, if we impose the additional condition that the molecules of the fluid are not acted up(n by extra 278 ELEMENTS OF ANALYTICAL MECHANICS. neoas forces, in which case the motions can only arise from some arbitrary initial disturbance; then, Equation (408), 1 - d9 dqo.dp_ -- = - d-t and by Mariotte's law, p-P. D a2. D (411) from which dp = a2 dD............ (412) and the above may be written, after dividing by d t, a' d 2 d logD dd-p - D = a - - - - I? 0 0 0 0 (413),which, in Equation (410), gives d2 q d2 c d2s d p' = t a.\c2 + y ~ (414) From this Equation the function q is to be determined, then the value of D, from Equation (410), and that of p, from either of the Equations (411) or (413). 0 256.-Conceive a homogeneous elastic fuidu to he disturbed at one of its points by the swlden expansion or contraction of the element there situated. This will break up the equilibrilui of the surrounding molecular forces at that point, the par-ticles adcjacent will nmlove to restore t1he unifoiknlty of (densitv, and ai expandinil' disturb-tlwne will proceed ottward firom this as a ccntre.'T'ke the origiin at tl;e point, andll denote the distance of anyii particle involve(ld in the cdisturbance, at ainy time t, slubsequient to the disturb'ance by r, then will Denote the -elocity of tIhe par:ticle, supposed in tlhe direction of r, by S; then will =_-. - V = *-; w --..1'.' P- MECHANICS OF FLUII)S. 27W Differentiating the first of the above equations, we have xdx + y dy + zdZ -- r. d r. Substituting the values of x, y, and z firom the second, third, and fourth, there will result udx + vd y wdz = d. dr; so that this satisfies the condition of the first ixember being an exact differential; and, therefore, d,. d d; or dp And hence dc dyp x do dp~ y dp d c z differentiating, 4d"p d'p x'~ d y'~-z2 d 4x =~ d,2',q - t a. *,; dp d2cp'!/' dc z'. +x~ dy d,'- r's+8 dr r~; 4 q). 1 X + y d Z' d r' 2?' d?' rs t and these values, substituted in Equation (414), give which may be written, dt d 2 _'. (414) which dte a is Appendi N. IV. of which the integral is, Appendix No. IV., r =F(r +. t) +f(r - a t); an(l in which F and f denote any arbitrary fun,tions whatever. From this we 11ave 280 ELEMENTS OF ANALYTICAL MECHANICS. (p = _[F(r+a +at)f(r — t)] (415) Taking the first differential coefficient of q with respect to r, and replacing its value by', 1 I = [F' (r + a +fl( at) - [F( a ) f (r - a t)t)] r r For any considerable distance from the origin, the second terni may be omitted in comparison with the first; and there will result, after squaring and multiplying by mn, the mass of the moving particle, -m -=g[F' (r + at)+f' (pr -a... (410) The first member is the living force of the moving particle, or double the quantity of work it may impress upon the organs of sense exposed to its action. The effect it may produce will, therefore, all other things being equal, vary inversely as the square of its distance from the place of primitive disturbance. Equations (415) and (416) are employed in discussing the theory of sound. EQUILIBRIUM OF FLUIDS. ~ 257.-If the fluid be at rest, then will d2x X. d2 y d2 z and Equations (399) become dx y= D. Y;...... 417) dy dp _ D.Z. dz ~ 258.-Multiplying the first by dx, the second by d y, the third by dz, and adding we find, dp = D (Xdx + Yd4/ 4 Zdz); ~ ~ ~ (418) MECHANICS OF FLUIDS. 281 and by integration, p =JD. (Xdx + Ydy + Zdz);. (419) whence, in order that the value of p may be possible for any point of the fluid mass, the product of the density by the function Xdx + Ydy + Zdz, must be an exact differential of a function of the three independent variables x,y, z. Reciprocally, when this condition is fulfilled, not only will the pressure at any point become known by substituting its co-ordinates, but the Equations (417), will be satisfied, and the fluid will be in equilibrio. ~ 259.-Conceiving those points of the fluid which experience equal pressures to be connected by, indeed to form a surface, then in passing from one point to another of this surface, we shall have dp = 0, and Xdx r+ Ydy + Zdz = 0,.. (420) which is obviously the differential equation of the surface. Dividing this by R ds, in.which mi, denotes the resultant of the forces which act upon any particle, and ds, the element of any curve upon the surface passing through the particle, we have Xdx Y ly Z dz d + R ds ds ds(421) whence the resultant of the forces acting upon any one of the elements of a surface of equal pressure, is normal to that surface. This is the characteristic of what is called a level surface, which may be defined to be. any surface which cuts at right angles the direction of the resultant of the forces which act upon its particles. 260.-If Equation (420) be integrated, we have (Xd + Ydy + Zdz) =,. (422) in which C is the constant of integration. The magnitudes of this constant must result from the dimensions of the surface, or from the volume of the fluid it envelops. By giving it different and 282 ELEMENTS OF AN'ALYTICAL MECHANICS. suitable values, we may start from a single particle and proceed outwards to the boundary of the fluid, and if the successive values:differ by a small.quantity, we shall have a series of level concentric strata. The last possible value for C will determine the exterior or bounding surface of the fluid; because this surface being free, the pressure upon it will be zero; the differential of the pressure from one point to another will, therefore, be zero, and the differential equation will be that numbered (420), or that of equal pressure. Every free surface of a fluid in equilibrio is, therefore, a level surface. ~261.-Putting Equation (418) under the form dp = Xdx + Ydy + Zdz, ~ (423) we see that whenever the second member is an exact differential, p must be a function of D, since the first member must also be an exact differential. Making, therefore, p = F(D),. (424) in which F denotes any function whatever, the above equation becomes dF(D) = Xdx + Ydy + Zdz;... (425) but for a level surface or stratum, the second member reduces to zero; whence, dF(D)= o; and by integration, F(D) C; whence, not only will each level stratum be subjected to an equal pressure over its entire surface, but it will also have the same density throughout. ~ 262.-If the fluid be homogeneous and of the same temperature throughout, then will D be constant, and the condition of equilibrium MECHANICS OF FLUIDS. 283 simply requires that the function Xdx + Ydy + Zdz, Equation (419), shall be an exact differential of the three independent variables x, y, z, and when this is not the case, the equilibrium will be impossible, no matter what the shape of the fluid mass, and though it were contained in a closed vessel. But the function abod;e referred to is, ~'133, always an exact differential for the forces of nature, which are either attractions or repulsions, whose intensities are functions of the distances from the centres through which they are exerted. And to insure the equilibrium, it will only be necessary to give the exterior surface such shape as to cut perpendicularly the resultants of the forces which act upon the surface particles. This is illustrated in the simple example of a tumbler of water, or, on a larger scale, by ponds and lakes which only come to rest when their upper surfaces are normal to the resultant of the force of gravity and the centrifugal force arising from the earth's rotation on its axis. In the case of a heterogeneous fluid subjected to the action of a central force, its equilibrium requires that it be arranged in concentric level strata, each stratum having the same density throughout. * And the equilibrium will be stable when the centre of gravity of the whole is the lowest possible, ~ 138, and hence the denser strata should be the lowest. When the fluid is incompressible, the density may be any function whatever of the co-ordinates of place. It may be continuous or discontinuous. When it is given, the value of the pressure is found from Equation (419). ~ 263.-In compressible fluids the density and pressure are con. nected by'law, and the former is no longer arbitrary. Dividing Equation (418) by Equation (389), we have dp _ Xdx + Ydy + Zdz. (425)' P P Integrating, -ogp fXdx ~r Ydy + Zdz'log P p + o'; ~ ~ (426) 284 ELEMENTS OF ANALYTICAL MECHANICS. denoting the base of the Naperian system by e, we have fXdZ+ Ydy+ Zd;... (427) p=C.e" P and this substituted in Equation (389), gives fXdx + Ydy+Zdz C.e P xD' * (428) These equations determine the pressure and density. For any surface of constant pressure, the exponent of e, in Equation (427), must be constant, its differential must, therefore, be zero, and all the consequences deduced from Equation (420) will follow; that is, when the fluid is at rest, it must be arranged in level strata, each stratum having the same density throughout, with the addition that the law of the varying density must be continuous by the requirements of Mariotte's law. If the temperature vary, then will P vary, and in order that Equation (425)' may be an exact differential, P must be a function of x y z, and hence, Equations (427) and (428), when p is constant, D will be constant; that is, each level stratum must be of uniform temperature throughout. It is obvious that the atmosphere can never be in equilibrio; for the sun heating unequally its different portions as the earth turns upon its axis, the layers of equal pressure, density and temperature can never coincide. Hence, those perpetual currents of air known as the trade winds, and the periodical monsoons; also, the sea and land breezes, variable winds, &c., &c. ~264.-Rest is a relative term; when applied to a particle of a fluid mass, it means that that particle preserves unaltered its place in regard to the other particles; a condition consistent with a bodily movement of the entire mass. If a liquid mass turn uniformly about an axis, the preceding equations will make known its permanent figure. For this purpose it will be sufficient to join to the forces X, Y, Z, the centrifuigal force MECIIANICS OF FLUIDS. 285 Take the axis z as the axis of rotation; denote the angular velocity by p, and the distance of the particle M11 from the z axis z by r; then will 2'2 = x2 + y2; the centrifugal force of M regarded as a unit of mass, wvill be r p2, / and its components in the direction of x and y, respectively, 2 r. 92. * = - q9f; r. q2 L= Qy2. and these in Equation (418), give dp -=).(Xdz + Ydy + Zdz,+ p2.xdx + q2 y.dy).. (429) When the second member is an exact differential, the permanent form will be possible. ior the free surface dp = O, and we have Xdx + Ydy + Zdz + (p2.x.dx + q2ydy = O. ~ ~ (430) Example 1.-Let it be required to find the figure assumed by the free surface of a heavy and homogeneous fluid contained in an open vessel and rotating about a vertical axis. Here, X= 0; Y= 0; Z=- g; and Equation (430) becomes gdz -= 2(xdx + ydy). Integrating, z _ -(x2 + y2) + C;. (431) vwhich is the equation of a paraboloid whose axis is that of rotation. 286 ELEMENTS OF ANALYTICAL MECHANICS. To find the constant C, let the vessel be a right cylinder, with circular base, whose radius is a, and denote by h the height due to the velocity of the fluid at the circumference, then a2 2=- 2gh, and h. r2 z + C....... (432) Denote by b the height of the liquid before the rotation; its volume will be r a2. b. Conceive the whole body of the liquid to be divided into concentric cylindrical layers, having for a common axis the axis of rotation. The base of any one of these layers will have for its area, neglecting d r2, 2 r r. dr, and for its volume, taking the origin of co-ordinates in the bottom of the vessel, 2qrr. dr.z, which being integrated between the I limits r 0= and r = a, will give the whole volume of the fluid, and hence, a2b = 2 zr.dr; replaning r. d r by its value from Equation (432), and integrating between the limits z = C and z = h + C, which are the values given by Equation (432) for r 0 and r = a, we find C = b - h, and the equation of the upper surface becomes h r2 z = - + b - ~h. The least and greatest values for z, are b - -- h and b + h, obtained by making r = 0 and r = a, so that the depression of the MECHANICS OF FLUIDS. 287 liquid at the axis is equal to its- elevation at the surface of She cylindrical vessel, and is equal to half She height due to the velocity of the latter. ~ 265.-Example 2.-Let the fluid elements be attracted to the centre of the mass by a force varying inversely as the square of the distance. Take the origin at the centre; denote the distance to the particle m from that pointby r, and the intensity of the / attractive force at the unit's distance by k. Then will k x z P = m; COS Cosf = --—; COS -- r2 ~ r r r and kx y koz = r3 r3 r3; which in Equation (430), give -3 (xdx + ydy + zdz) - q2(xdx + ydy) - O or kldr d2 r2 f2 d (x2 + y2) = 0, and by integration, k + (X2 + y2) C; r making x2 + y2 = r2 cos2, in which 8 denotes the angle made by r, with the plane x y, k q.2 -- r - ~2 COS2 r 2, 288 ELEMENTS OF ANALYTICAL MECHANICS. and denoting the distance from the origin to the point in which.he free surface cuts the axisr z by unity, we have, by making 0 = 900~, k C -=; which substituted above, and solving with respect to cos2, gives ~p2. COS2 - - (r 1) (434) r3 and mnaking r = 1 + u, we have cu ~ q02. COS2 0 = ~i 2. cos20 (1 + U)3 If the angular velocity be small, then will u be very small. Developing the second member with this supposition, and limiting the terms to the first power of u, we find 2. os2 08 - k (u - 3U2)...... (434)' Neglecting 3u2, and replacing u by its value, viz.: r- 1, we have for a first approximation, 2 r= 1 + 2kCos28. From Equation (434)', we find 02. c os2 U 2k - 3U2, and this in the equation r = 1 ~ u, gives r=1 + 2.cos2 + 3ut2; q4. Cos4 and replacing u2 by its approximate value 4k2, above, by neg. lecting 3 u2, we have 20~ 3" q4. cos4 0 r = 1 — +.kcos2C + 4k2 for the polar equation of the meridian section. MECHANICS OF FLUIDS 289 Comparing this with the equation r = - - = 1 + ~e2cos2 +. e4.cos40 + &c., a/17 — e2 cos2 they become identical by neglecting the higher powers and making The fiee surface of the fluid approximates therefore very closely to an ellipsoid of revolution of which the eccentricity of its meridian section is equal to the square root of the quotient arising from dividing the centrifugal force at the unit's distance from the axis of rotation, by the force of attraction at an equal distance from the centre. PRESSURE OF HEAVY FLUIDS. ~266.-When a fluid contained in any vessel is acted upon by its own weight, if the axis z be taken vertical and positive downwards, then will X=0; Y= 0; Z= g; and Equation (418) becomes, after integrating, p = Dgz + C; and assuming the plane xy to coincide with the upper surface of the fluid, which must, when in equilibrio, be horizontal, we have, by making, z= 0, P C; in which p' denotes the pressure exerted upon the unit of the free surface. Whence, p1-p-2 =D.g.z. ~.... *.. (435) The first member is the pressure exerted upon a unit of surface, every point of which unit having a pressure equal to that sustained by the element whose co-ordinate is z. 19 290 ELEMENTS OF ANALYTICAL MECHANICS. If p' - 0, then will p = gz;............ (436) and denoting by b the area of the surface pressed, and by d b, the element of this surface, whose co-ordinate is z, we have, Equation (3'97), for the pressure upon this element denoted by p,, p, = Dg.z.db, and the same for any other element of the surface; whence, denoting the entire pressure by P, we shall have P = zp, = Dg. z.d b. ~.. (437) But if z, denote the co-ordinate of the centre of gravity of the entire surface b, then will, Equations (91), Y z.d b = bz,, and P = Dg.,. z. ~..... (438) Now bz, is the volume of a right cylinder or prism, whose base is b, and altitude z,; Dg.b.z, is the weight of this volume of the pr6ssing fluid. Whence we conclude, that the pressure exerted apon any sumfclce by a heavy fluid is equal to the wveight of a cylindrical or prismatic column of the.flic whose base is equal to the surfice pressed, and whose altitude is equal to the distance of the centre of gravity of the suTzzface below the eupper surface of the fluid. W~hen the surface pressed is horizontal, its centre of gravity will be at a distance from the upper surface equal to the depth of the fluid. This result is wholly independent of the quantity of the pressing fluid, and depends solely upon the density of the fluid, its height, and the extent of the surface pressed. Example 1. - Required the pressure against the inner surface of a cubical vessel filled with water, one of its faces being horizontal. Call the edge of the cube a,, the area of each face will be a2, the distance of the centre of gravity of each "- face below the upper surface will be 1a, and that of tho MECHANICS OF FLUIDS. 291 lower face a;; whence, the principle of the centre of gravlty gives, 4a2 X -a +- a2 X a 5 a2 Again, b - 5a2; and these, substituted in Equation (43S), give P = D.g b. z -D. g.3a3. Now D g x 13 = D g, is the weight of a cubic foot of water = 62,5 lbs., whence, Ibs. P- 62,5 X 3a3. Make a = 7 feet, then will lbs. P= 62,5:X 3 x (7)3 - 64312,5. The weight of the water in the vessel is 62,5 a3, yet the pressure is 62,5 x 38a3, whence we see that the outward pressure to break the vessel, is three tinmes the weight of the fluid. Example 2.-Let the vessel be a sphere filled with mercury, and let its radius be R. Its centre of gravity is at the centre, and therefore below the upper surface at the distance B. The surface of the sphere being equal to that of four of its great circles, we have b 4;R2; whence, b.z 4 sr R3; and, Equation (438), P = 4~r.D.g.R3. The quantity Dg X 13= Dg, is the weight of a cubic foot of mercury = 843,75 lbs., and therefore, substituting the value of = 3,1416, P = 4 X 3,1416 x 843,T75.X3. 292 ELEMENTS OF ANALYTICAL MECHANICS. Now suppose the radius of the sphere to be two feet, then will R3 = 8, and Ibs. lbs. P = 4 x 3,1416 X 843,75 X 8 = 84822,4. The volume, of the sphere is 3'R3; and the weight of the con. tained mercury will therefore be 4 R3g gD= D Wf. Dividing the whole pressure by this, we find P _ 3 W 3; whence the outward pressure is three times the weight of the fluid. Example 3.-Let the vessel be a cylinder, of which the radius r of the base is 2, and altitude 1, 6 feet. Then will b.z, = r l(r + 1) = 3,1416 x 2 X 6 x 8; which, substituted in Equation (438), P = 301,5936 x D)g,.and - 3,1416 x 22 x 6 x Dg = 75,398 x Dg; xwhence, P 301,5936 X Dg _ 4 W - 75,3984. Dg that is, the pressure against this particular vessel is four times the weight of the fluid. ~ 267.-The point through which the resultant of the pressure upon all the elements of the surface passes, is called the centre of pressure. Let E IF be any plane, and A N the intersection of this plane produced vwith the upper surface of the fluid,which presses against it. Denote the / area of any elementary portion n of the plane ElIF by db; and let m be the projection of its place upon the I upper surface of the fluid; draw m 1f perpendicular to MAN~, and join n with Mby the right line n M; the MECHANICS OF FLTTIDS. 299 latter will also be perpendicular to MA N, and the angle n Mm will measure the inclination of the plane EIF to the surface of the fluid. Denote this angle by pq, the distance m pa by A', and Mn by r' then will h' - r' sin; the pressure upon the element d b, Dg.rtsinqp db; its moment with reference to the line MN, D g r'2 sin q. d6b; and for the entire surface, the moment becomes D g. sin 9.. r.'2 d b. Denote by r the distance of the centre of gravity of the surface pressed from the line M N, its distance below the upper surface of te fluid will be r. sin q; and the pressure upon this surface will be D g. r sin q h.; and if I denotee the distance of the centre of pressure from the line M~2N, then will Dg. sin qo. b. I =- /)1. sin q-p. Z r'2. d b, from which we have, 2r'. d b k,2 + r2; t= = y' t2' d b (439) whence, Equation (238), the centre of pressure is found at the centre of percussion of the surface pressed. ~268.-The principles Which have just been explained, are of great practical importance. It is often necessary to know the preeIise. amount of pressure exerted by fluids against the sides of vessels and obstacles exposed to their action, to enable us so to adjust the dimensions of the latter as to give them sufficient strength to resist. Reservoirs in which considerable quantities of water are collected and retained till needed for purposes of irrigation, the supply of cities and towns, or to drive machinery; dykes to keep the sea 294 ELEMENTS OF ANALYTICAL MECH:ANICS. and lakes from inundating low districts; artificial embankments constructed along the shores of rivers to protect the adjacent country in times of freshets; boilers in which elastic vapors are pent up in a high state of tension to propel boats and cars, and to give motion to machinery, are examples. ~ 269.-As a single instance, let it be required to find the thick ness of a pipe of any material necessary to resist a given pressure. Let AB C be a section of pipe perpendicular to the axis, the inner surface of which is subjected to a pressure of p pounds on each superficial unit. Denote by B the radius of the interior circle, and by I the length of the pipe parallel to the axis; then will the surface pressed be measured 0 by 2a R. I; and the whole pressure by t Il.p. By virtue of the pressure, the pipe will stretch; its radius will become R + dR, the path described by the pressure will be dB, and its quantity of work 2.R..p d R. The interior circumference before the pressure was 2 7r R, afterwards 2 nr(R + d -), and the path described by resistance, 2 d R. And if B denote the resistance which the material of the pipe is capable of opposing, to a stretching force, without losing its elasticity over each unit of section, t the thickness of the pipe, then, by the prin. ciple of the transmission of work, must 2cr. B. I. d R. t = 2R. l..dR; whence, Bp The value of p is estimated in the case of water pressure by the rules just given. That in the case of steam or condensed gases, MECHANICS OF FLUIDS. 295 by rules to be given presently. The value of B is readily obtained from Taole I, giving the results of experiments on the strength of materials. EQUILIBRIUM AND STABILITY OF FLOATING BODIES. ~ 270.-When a body is immersed in a fluid it is not only acted upon by its own weight, but also by the pressure arising from the weight of the fluid, and the circumstances of its rest or motion will be made known by Equations (A) and (B). Let ED be the body; take the plane x y in the plane of the upper surface of the fluid, supposed at rest, and the axis of z therefore Xi vertical. Denote by h the entire surface of the body, and by d b, one of its elements, whose co-ordinates of position are x y z. The pressure upon this element will be D.g.z.db, in which D is the density of the fluid, and g the force of gravity. This pressure is, ~ 248, normal to the surface, and denoting by a, / and H,, the angles which this normal makes with the axes xyz, respectively, the components of the pressure in the direction of these axes will be D g.z.db.cosa; D.y.z.db.cos[/; D.g.z.db.cosy. Similar expressions being found for the components of the pressure on other elements, we have, by takingr their sum, ) g. I z.d. cos cc; Dg.2'z.db. cos; D g. z. db. cos y. But db.cCosg, db.cos,8, and db.co0sy, are the prqjecticns of the iarea d b on t'ih. co-ordinate planes z y, z x and( x y, respectively; and 296 ELEMENTS OF ANALYTICAL MECHANICS. 1 z.db.cos a, z.db.cosf, z. d b.cosy, are volumes of righ prisms whose bases are projections of the entire surface pressed upon the same co-ordinate planes, and of which the altitude of each is the depth of the common centre of gravity of the elements of its base submerged to the depths of their corresponding surface elements. Whence we conclude, that the component of the pressure on any surface, estimated in any direction, is equal tb the pressure on so much of that surfajce as is equal to its projection on a plane at right angles to the given direction. The cylinder or prism which projects an element on one side of the body will also project an element situated on the opposite side; these projections will, therefore, be equal in extent, but will have contrary signs, for the normal to the one will make an acute, and to the other an obtuse angle with the axis of the plane of projection. When these projections are made upon any vertical plane, the value of z will be the same in both, and hence, for each positive product, z. d b. cos a and z. d b. cos /, there will be an equal negative product; therefore, D g. y z. d b. cOS CC= =P COS a = 0; D g.X z. d b. eos 3 =- P cos = O. That is, the sum of the horizontal pressures in the directions of x and y, and therefore in all horizontal directions, will be zero; and the first and second of Equations (120), give d2 x d2 y d tP dt5 or, which is the same thing, there can be no horizontal motion of translation from the fluid pressure. When the projections of opposite elements are made upon a horizontal plane, they will still be equal with contrary signs, the normal to the elements on the lower side making obtuse, while the normals to the elements above make acute angles with the axis z; but the corresponding values of z will differ, and by a length equal to that of the vertical filament of the body of which these elements form the opposite bases, and hence Dg. I z.db. cosy = Dg. (z'-z,) d6 cosy - c db cosy (440) MECHANICS OF FLUIDS. 297 in which z' denotes the ordinate for the upper, and z, that for the lower element in the same vertical line, and c the distance between the elements; and the third of Equations (120) becomes d2z d2z. (Pcos y- d = My- ). g c.db.cosY - r. —- O. d P j = Y1Y c UY ~ Y C;~ U V ~ C;VY'Y ~- L~ ~l~r =0dt2 But c. d b. cos yis the volume of the immersed body which is obviously equal to that of the displaced fluid; also D g. Z c. d b. cosy is the weight of the displaced fluid; and.Mg that of the body. Denoting the volume of the body by V', its density by D', the above may be written d2 z VD'g - V' Dg-Ems d2 0.. (441) Now, when V'D'g V'Dg = 0, or D = D', then will d2 z.m O; dt2 and there can be no vertical motion of translation fi m the fluid pressure and the body's weight. When D' > D, then will m dt2 = (DI D) V'.; and the body will sink with an accelerated motion. When D' < D, then will d2 z z. d t — = - (I)' —) ) V'. g, and the body will rise with an accelerated motion till d2 z m. t2 V'D'g — V.Dgg=O; (142) 298 ELEMENTS OF ANALYTICAL MECHANICS. in wNhich V denotes the volume AB C, of the fluid displaced. At this instant we have V'y7Dg = VDg; ~ ~ ~ (443) and if the body be brought to rest, it will remain sb. That is, the body will float at the surface when the weight of the fluid it displaces is equal to its own weight. The action of a heavy fluid to support a body wholly or partly immersed in it, is called the buoyant eg ort. The intensity of the buoyant effort is equal to the weight of the fluid displaced. Substituting the values of the horizontal and vertical components of the pressures in Equations (118), and reducing by the relations, Dg. I c. d b. cosy. x' = g. V. x; Dg. c. d b. cos y.' = D g. V. y; 44) in which x and y are the co-ordinates of the centre of gravity of the displaced fluid referred to the centre of gravity of the body, we find'. d2 y'- y'. d2 x'' Ea.'z =0; dt2 zt. d2 t _ xt. d2 zt d t2 P = Dy. V-X; (445) y'. d2 z' - z'. d2 y x d ~ Dg - V.Y. d t2' Equations (444) show that the line of direction of the buoyant effort passes through the centre of gravity of the displaced fluid. This point is called the centre of buoyancy. And from Equations (445), we see that as long as x and y are not zero, there will be an angular acceleration about the centre of gravity. At the instant x -0 and y-0, that is to say, when the centres of gravity of the body and displaced fluid are on the same vertical line, this acceleration will cease, and if the'body were brought to rest, it would has e ho tendency to rotate. To recapitulate, we find, MECHANICS OF FLUIDS. 299 1st. That the pressures eupon the surface of a body immersed in a heavy fluid have a single resultant, called the buoyant effort of the fluid, and that this resultant is directed vertically upwards. 2d. That the buoyant effort is equal in intensity to the weight of the fluid displaced. 3d. That the line of direction of the buoyant effort passes through the centre of gravity of the displaced fluid. 4th. That the horizontal pressures destroy one another. ~271. —Having discussed the equilibrium, consider next the sta bility of a floating body. The density of the body may be homo. geneous or heterogeneous. Let AB CD be a section of the body by the upper surface of the fluid when the body is at rest, (G _ its centre of gravity, and -__ H, that of the fluid displaced. Denote by V the, volume of the displaced _ fluid, and by M the mass -- of the entire body. The body being in equilibrio, the line CGH will be vertical, and denoting the density of the fluid by D, we shall have M= = D. V... (446) Suppose the section A B CD either raised above or depressed below the surface of the fluid, and at the same time slightly careened; also suppose, when the body is abandoned, that the elements have a slight velocity denoted by u,; u', &c. Now the question of sta. bility will consist in ascertaining whether the body will return to its former position, or will depart more and more from it. The free surface of the fluid is called the plane of floatation, and during the motion of the body this plane will cut fioom it a variable section. [Let A' B' (/' D)' be one of these sections at any given instant of 300 ELEMENTS OF ANALYTICAL MECHANICS. time; A B" CD", another variable section of the body by a hori. zontal plane through the centre of gravity of the primitive section A B CD, and A C the intersection of the two. Denote bvy the inclination of these two sections, and by C the vertical distance of A _B" CD)", from the plane of floatation, which now coincides with A' B' C' )', this distance being regarded as negative or positive, ac. cording as AB" C D" is below or above the plane of floatation. The variable quantities 8 and Y will be supposed very small'at the instant the body is abandoned. Will they continue so during the whole time of motion! From the principles of living force and quantity of work, we have, Equation (121), U2. d M=2 (Xdx + Ydy - Zdz) + /,. The forces acting are the weights of the elements d M and the verti. cal pressures, the horizontal pressures destroying one another; whence, X-= O, Y= 0, and u2d M 2 Zdz+ C= 2 Zz+ C. ~ ~ (447) The force which ats upon an element above the plane of floatation is its own weight, and the force which acts upon any element below that plane is the difference between its own weight and that of the fluid it displaces; the first of these latter will be g. d M, and the second, g. D. d V, in which d V is the volume of d M; whence, fZz Sg.z.dM-fg.z. ~ ~. (448) But, drawing from the centre of gravity G, of the body, the perpendicular G E, to the plane of floatation A' B' C' D', and denoting G / by z,, we have fg.z.dM= gMz,. The integral D..z d V, will be divided into two parts, viz: onm relating to the volume of the body below A B CD, or the volum( immersed in a state of' rest, and the other that comprised betweei MECHANICS OF FLUIDS. 301 A B CD and the plane of floatation A' B' C' D', when the body is in nmotion. Denote by g D Vz', the value of the first, in which z' denotes the variable distance HF, of the centre of gravity H., of the volume V, from the plane of floatation A' B' C' D'. And representing for the instant by h tie value of the integral Jzd V, comprehended between the planes A B CD and A' B' C' D', g D h will be the second part; and Equation (447) becomes fu2 di = 2q. Az, - 2g D Vz' - 2gDh + C. ~ ~ (449) The line G H, being perpendicular to the plane AB CD, the angle which it makes with the line G E is equal to 0, and denoting the distance G H by a, we have z, -- Z' _ acos0; the upper sign being taken when the point G is below the point II, and the lower when it is above. This value reduces Equation (449) to ju2d f= + 2gD Vacos - 2gD)h +. ~ ~ ~ (450) Let us now find the integral h. For this purpose, conceive the area AB CD to be divided into indefinitely small elements denoted by dX, and let these be projected upon the plane of floatation, A' B' C' D'. The projecting surfaces will divide the volume comprised between these two sections into an indefinite number of vertical elementary prisms, and these being cut by a series of horizontal planes indefinitely near each other, will give a series of elementary volumes, each of which will be denoted by d V, and we shall have d V = dz. dX. cos; whence, for a single elementary vertical prism, fzdV,fzdr.dX.cos =z (Z)2. cos 0.dX; in which (z) denotes the mean altitude of the prism, and consequently h = I cos.f(z)d. d X, which must be extended to embrace the entire surface A B CD. 302 ELEMENTS OF ANALYTICAL MECHANICS. The value of (z) is composed (f two parts, viz.: one comprised between the parallel sections A'B' C' D' and A B"CD", and which has been denoted by A; the other comprised between the base dX and the second of these planes, and which is equal to 1. sin0, denoting by I the distance of dX from the intersection A C; whence, (z) - + 1. sin6, in which 1 will be positive or negative according as dX happens to be below or above the plane AB" CD". Substituting this in the value of h, and recollecting that C and 0 are constant in the integration, we find h, 2. os.d X + Sill C dX + I sin2'.cos s l2 d X. Denote by 6 the area of AB C(D, or the value of Ed x. The line A C passing through the centre of gravity of A B CD), we have jid = 0, And denoting by k, the principal radius of gyration of the surface b, in reference to the axis A C, j12dzX = b,2, in which the value of k, is dependent upon the figure and extent of the surface AB CD), and upon the position of the line A C. Whence, h = 4.cos (~2 + lc2 sin2 0).... (451) Taking 02 sin 0 = - + &c; cos = 1 1 +.2 & Neglecting all the terms of the third and higher orders, substitutingin the value of h, and then in Equation (450) we find, after trans. posing and including the term 4- 2gD V a, in the constant C; fu2.d + g bC 2 + (b6,2 _ va) — j= C. ~ ~ (452) Now the value of the constant (C depends upon the initial values of u, 0 and A; but these by hypothesis are very small; hence C, must also be very small. As long as the second term of the first MECHANICS OF FLUIDS. 303 member is positive,fu2 d AI must remain very small, since it is essentially positive itself, and being increased, by a positive quantity, the sum is very small. Hence C and 8 must remain very small. But when the second term is negative, which can only be when ik,2+- VFa, is negative and greater than b6, the value of u2dM may increase indefinitely; for, being diminished by a quantity that increases as fast as itself, the difference may be constant and very small. HIence, C and 8 may increase more and more after the body is abandoned to itself, and finally it may overturn. The stability of the equilibrium depends, therefore, upon the sign of b Ic2 4- Va; the equilibrium is always stable when this quantity is positive; it is unstable when it is negative and greater than b ~2 The value of b li, =- 2dX, must always be positive, since all its elements are positive; the value of 4- Va becomes negative when the centre of gravity of the body is above that of the displaced fluid, in which case the stability requires that b k,2 > Va, or, kj2 > V When the centre of gravity of the body is below that of the displaced fluid, the sign of Va is positive. Whence we conclude that the equilibrium of a body floating at the surface of a heavy fluid, will be stable as long as the centre of gravity of the body is below that of the displaced fluid; that it will also be stable about all lines A C, with reference to which the principal radius of gyration of the section of the body by the plane of floatation squared, is greater than the. volume of the displaced fluid multiplied by the distance between the centres of gravity of the displaced fluid and that of the body, when the latter is in equilibrio, divided by the area of the section of the body by the plane of floatation. When this condition is not fulfilled, the equilibrium may be unstable. A ship whose centre of gravity is above that of the water she displaces, may overturn about her longer, but not about her shorter axis. ~272.-A line B]K through the centre of gravity G of the body 304 ELEMENTS OF ANALYTICAL MECHANICS. and which is vertical when the body is in equilibrio, is called a line of rest. A vertical line H'M through the centre of gravity _' of the displaced fluid, is called a line of support. The point MV, in which the line of support cuts the line of rest, __ is called the mnetacentre. The body will be in equilibrio when the line of rest and of support coincide. The equilibrium will be stable if the metacentre fall above the centre of gravity; it may be unstable if below. ~ 273.-When the equilibrium is stable, and the body is disturbed and then abandoned to the action of its own weight and that of the fluid pressure, it will, in its efforts to regain its place of rest, oscillate about this position, and finally come to rest. The circumstances of those oscillations about the centre of gravity of the body will readily result from Equations (445). SPECIFIC GRAVITY. ~ 274.-The specific gravity of a body, is the weight of so much of the body, as would be contained under a unit of volume. It is measured by the quotient arising from dividing the weight of the body by the weight of an equal volume of some other substance, assumed as a standard; for the ratio of the weights of equal volumes of two bodies being always the same, if the unit of volume of each be taken, and one of the bodies become the standard, its weight will become the unit of weight. The term density denotes the degree of proximity among the particles of a body. Thus, of two bodies, that will have the greater density which contains, under an equal volume, the greater number of particles. The force of gravity acts, within moderate limits, equally upon all elements of matter. The weight of a substance MECHANICS OF FLUIDS. 305 is, therefore, directly proportional to its density, and the ratio of the weights of equal volumes of two bodies is equal to the ratio of their densities. Denote the weight of the first by IW, its density by D, its volume by V, and the force of gravity by g, then will WV= g.D. V; and denoting the like elements of the other body by WT, D, and V,, we have WI=y. Xi, V,. Dividing the first by the second, TW gD V D V W gDiV, -D,V and making the volumes equal, w )......... ii.(453) Now suppose the body whose weight is W. to be assumed as the standard both for specific gravity and density, then will D, be unity, and WIV S- ~:< = Z).............. (454) in which S denotes the specific gravity of the body whose density is D; and from which we see, that when specific gravities and densities are referred to the same substance as a standard,. the numbers which express the one will also express the other. ~275.-Bodies present themselves under every variety of condition-gaseous, liquid, and solid; and in every kind of shape and of all sizes. The determination of their specific gravity, in every instance, depends upon our ability to find the weight of an equal volume of the standard. When a solid is immersed in a fluid, it loses a portion of its weight equal to that of the displaced fluid. The volume of the body and that of the displaced Fuid are equal. Hence the weight of the body in vacuo, divided by its loss of weight when immersed, will give the ratio of the weights of equal volumes of the body and fluid; and if the latter be taken as the 306 ELEMENTS OF ANALYTICAL MECHANICS; standard, and the loss of weight be made to occupy the denorni. nator, this ratio becomes the meastire of the specific gravity of the body immersed. For this reason, and in view of the consideration that it may be obtained pure at all times and places, water is assumed as the general standard of specific gravities and densities for all bodies. Sometimes the gases and vapors are referred to atmospheric air, but the specific gravity of the latter being known as referred to water, it is very easy, as we shall presently see, to pass from the numbers which relate to one standard to those that refer to the other. 276.-But water, like all other substances, changes its density with its tempelature, and, in consequence, is not an invariable standard. It is hence necessary either to employ it at a constant temperature, or to have the means of reducing the apparent specific gravities, as determined by means of it at different temperatures, to what they would have been if the water had been at the standard temperature. The former is generally impracticable; the latter is easy. Let n denote the density of any solid, and S its specific gravity, as determined at a standard temperature corresponding to which the density of the water is D,. Then, Equation (453), Again, if S' denote the specific gravity of the same body, as indi. cated by the water when at a temperature different from the standard, and corresponding to which it has a density -D,,, then will, D Dividing the first of these equations by the second, we have S D_.D' whence, S= S' D,,. (455) and if the density D,, be taken as unity, S = SS'..D,* ~... (456) MECHANICS OF FLUIDS. 307 That is to say, the specific gravity of a body as determined at the standard temperature of the water, is equal to its specific gravity determined at aly other temperature, multiplied by the density of the water corresponding to this temperature, the density at the standard temperature being regarded as unity. To make this rule practicable, it becomes necessary to find tho relative densities of water at different temperatures. For this purpose, take any metal; say silver, that easily resists the chemical action of water, and whose rate of expansion for each degree of Fahr. thermometer is accurately known from experiment; give it the form of'a slender cylinder, that it may readily conform to the temperature of the water when immersed. Let the length of the cylinder at the temperature of 320 Fahr. be denoted by 1, and the radius of its base by m 1; its volume at this temperature will be, qrm 2 12 X I- = qr213. Let n 1 be the amount of expansion in length for each degree of the thermometer above 32~0. Then, for a temperature denoted by t, will the whole expansion in length be nl X (t - 320), and the entire length of the cylinder will become l+ n I (t- 320)-= [1+n (t- 32~)]; which, substituted for I in the first expression, will give the volume for the temperature t, equal to rm 2l [ 1 J- nI (t - 320)13 The cylinder is now weighed in vacuo and in the water, at different temperatures, varying from 32 ~ upward, through any desirable range, say to one hundred degrees. The temperature at each process being substituted above, gives the volume of the displaced fluid; the weight of the displaced fluid is known 308 ELEMENTS OF ANALYTICAL MECHANICS. from the loss of weight of the cylinder. Dividing this weight by the volume, gives the weight of the unit of volume of the water at the temperature t. It was found by Stampfer, that the weight of the unit of volume is greatest when the temperature is 380.75 Fah. renheit's scale. Taking the density of the water at this temperature as unity, and dividing the weight of the unit of volume at each of' the other temperatures by the weight of the unit of volume at this, 30~.75, Table II will result. The column under the head V, will enable us to determine how much the volume of any mass of water, at a temperature t, exceeds that,of the same mass at its maximum density. For this purpose, we have but to multiply the volume at the maximum density by the tabular number corresponding to the given temperature. ~ 277.-Before proceedin-g to the practical methods of finding the specific gravity of bodies, and to the variations in the processes rendered necessary by the peculiarities of the different substances, it will be necessary to give some idea of the best instruments employed for this purpose. These are the Htydrostatic Balance and NVicholson's Hlydrometer. The first is similar in principle and form to the common balance. It is provided with numerous weights, extending through a wide range, from a small fraction of a grain to several ounces. Attached to the under surface of one of the / basins is a small hook, from which may be suspended any body by means of a thin platinum wire, horsehair', or any other delicate 4,thread that will neither absorb: ___ nor yield to the chemical action of the fluid in which it may be desirable to immerse it. EicWholson's tHydrometer consists of a hollow metalic ball A, through MECHANICS OF FLUIDS. 309 the centre of which passes a metallic wire, prolonged in both di. rections beyond the surface, and supporting at either end a basin B and B'. The concavities of these basins are turned in the same direction, and the basin B' is made so heavy that when the instrument is placed in water the stem CC' shall be ) vertical, and a weight of 500 grains being placed in the basin B, the whole instrument A will sink till the upper surface of distilled water, at the standard temperature,. comes to a point C marked on the upper stem near its middle. This instrument is provided with weights similar to those of the Hydrostatic Balance. ~ 278. —(l). If the body be solid, insoluble int water, and will sink in that fluid, attach it, by means of a hair, to the hook of the basin of the hydrostatic balance; counterpoise it by placing weights in the opposite seale; now immerse the body in water, and restore the equilibrium by placing weights in the basin above the body, and note the temperature of the water. Divide the weights in the basin, to which the body is not attached by those in the basin to which it is, and multiply the quotient by the density corresponding to the temperature of the water, as given by the table; the result will be the specific gravity. Thus denote the specific gravity by S, the density of the water by D,, the weight in the first case by W, and that in the scale albove the solid by w, then will S=, Di -- W (2). If the body be insoluble, but will not sink in water, as would be the case with most varieties of wood, wrax, and the like, attach to it some body, as a metal, whose weight in the air and loss of weight in the water are previously found. Then proceed, as in the case before, to find the weights which will counterpoise the compound in air and restore the equilibrium of the balance when it is 310 ELEMENTS OF ANALYTICAL MECHANICS. immersed in the water. From the weight of the compound in air, subtract that of the denser body in air; front the loss of weight of the compound in water, subtract that. of the denser body; divide the first difference by the second, and multiply by the density of the water answering to its temperature, and the result will be the specific gravity sought. Example. grs. A piece of wax and copper in air = 438 = + Wr, Lost on immersion in water - - = 95,8 = w + we, Copper in air = 388 = W', Loss of copper in water - - = 44,2 = w'. Then [W+ W'- W' = 438- 388 = 50, =, wu + wt - w' = 95,8 - 44,2 = 51,6 = w. Temperature of water 430,25,'=,, = 0,999952, w 516 (3). If the body readily dissolve in water, as many of the salts, sugar, &c., find its apparent specific gravity in some liquid in which it is insoluble, and multiply this apparent specific gravity by the density or specific gravity of the liquid referred to water at its maximum density as a standard; the product will be the true specific gravity. If it be inconvenient to provide a liquid in which the solid is insoluble, saturate the water with the substance, and find the appa. rent specific gravity with the water thus saturated. Multiply this apparent specific gravity by the density of the saturated fluid, and the product will be the specific gravity referred to the standard. This is a common method of finding the specific gravity of gunpowder, the water being saturated with nitre. (4). If Wte body be a liquid, select some solid that will resist its chemical action, as a massive piece of glass suspended from fine MECHANICS OF FLUIDS. 311 platinum wire; weigh it in air, then in water,'and finally in the liquid; the differences between the first weight and each of the latter, will give the weights of equal volumes of water and the liquid. Divide the weight of the liquid by that of the water, and the quotient will be the specific gravity of the liquid, provided the temperature of water be at the standard. If the water have not the standard temperature, multiply this apparent specific gravity by the tabular density of the water corresponding to the actual temperature. Example. grs. Loss of glass in water at 410, 150 = w' " " sulphuric acid, 277,5 -= w, 277,5 S 150 0,999988 =- 1,85. (5). If the body be a gas or vapor, provide a large glass flask. shaped vessel, weigh it when filled with the gas; withdraw the gas, which may be done by means to be explained presently, fill with water, and weigh again; finally, withdraw the water and exclude the air, and weigh again. This last weight subtracted from the first, will give the weight of the gas that filled the vessel, and subtracted from the second will give the weight of an equal volume of water; divide the weight of the gas by that of the water, and multiply by the tabular density of the water answering to the. actual temperature of the latter; the result will be the specific gravity of the gas. The atmosphere in which all these operations must be performed, varies at different times, even during the same day, in respect to temperature, the weight of its column. which presses upon the earth, and the quantity of moisture or aqueous vapor it contains. That is to say, its density depends upon the state of the thermometer, barometer, and hygrometer. On all these accounts corrections must be made, before the specific gravity of atmospheric air, or that of any gas exposed to its pressure, caln be accurately determined. The principles according to which these corrections are made, will be discussed when we come. to treat of the properties of elastic fluids. 312 ELEMERTS OF ANALYTICAL MECHANICS. To find the specific gravity of a solid by means of Nicholson's Hydrometer,place the instrument in water, and,add weights to the upper basin until it sinks to the mark on the upper stem; remove the weights and place the solid in the upper basin, and add weights till the hydrometer sinks to the same point; the difference between the first weights and those added with the body, will give the weight of the latter in air. Take the body from the upper basil, leasing the weights behind, and place it in the lower basin; add weights to the upper basin till the instrument sinks to the same point as before, the last added weights will be the weight of the water displaced by the body; divide the weight in air by the weight of the displaced water, and multiply the quotient by'the tabular density of the water answering to its actual temperature; the result will be the specific gravity of the solid. To find the specific gravity of a fluid by this instrument, immerse it in water as before, and by weights in the upper basin sink it to the mark on the upper stem; add the weights in the basin to the weight of the instrument, the sum will be the weight of the displaced water. Place the instrument in the fluid whose specific gravity is to be found, and add weights in the upper basin till it sinks to the mark as before; add these weights to the weight of the instrument, the sum will be the weight of an equal volume of the fluid;. divide this weight by the weight of the water, and multiply by the tabular density corresponding to the temperature of the water, the result will be the specific gravity. } 279.-Besides the hydrometer of Nicholson, which requires the use of weights, there is another form of this instrument which is employed solely in the determination of the specific gravities of liquids, and its indications are given by means of a scale of equal parts. It is called the Scale-Areometer. It consists, generally, of a glass vial-shaped vessel A, terminating at one end in a long slender neck C, -__ to receive the scale, and at the other in a MECHANICS OF FLUIDS. 313 small globe B, filled with some heavy substance, as lead or mercury to keep it upright when immersed in a fluid. The application and use of the scale depend upon this, that a body floating on the surface of different liquids, will sink deeper and deeper, in proportion as the density of the fluid approaches that of the body; for when the body is at rest its weight and that of the displaced fluid must be equal. Denoting the volume of the instrument by V, that of the displaced fluid by V', the density of the instrument by D, and that of the fluid by ZD', we must always have g V) - g V' D'; in which g denotes the force of gravity, the first member the weight of the instrument, and the second that of the displaced fluid. Divi-.ding both members by D' V and omitting the common factor g we have D V' In which, if the densities be equal, the volumes must be equal; if the density D' of the fluid be greater than D, or that of the solid, the volume V of the solid must be greater than V', or that of the displaced fluid; and in proportion as D' increases in respect to D, will V' diminish in respect to V; that is, the solid will rise higher and higher out of the fluid in proportion as the density of the latter is increased, and the reverse. The neck C of the vessel should be of the same diameter throughout. To establish the scale, the instrument is placed in distilled water at the standard temperature, and when at rest the place of the surface of the water on the neck is marked and numbered 1; the instrument is then placed in some heavy solution of salt, whose specific gravity is accurately known by means of the Hydrostatic Balance, and when at rest the place on the neck of the fluid surface is again marked and characterized by its appropriate number. The same process being repeated for rectified alcohol, will give another point towards the opposite extreme of the scale, which may be completed by graduation. 31i ELEMENTS OF ANALYTICAL MECHANICS. To use this instrument, it will be sufficient to immerse it in a fluid and take the number on the scale which coincides with the surface. To ascertain the circumstances which determine the sensibility both of the Scale-Areometer and Nicholson's Hydrometer, let s denote the specific gravity of the fluid, c the volume of the vial, I the length of the immersed portion of the narrow neck, r its semi-diameter, and tw the total weight of the instrument. Then will ir r2, denote the area of a section of the neck, and Ir r2 I, the volume of fluid displaced by the immersed part of the neck. The weight, therefore, of the whole fluid displaced by the vial and neck will be sc + sir 221; but this must be equal to the weight of the instrument, whence, w = s(c + f r21), from which we deduce, c + sr?21' W - SC =-....... (457) Now, immersing the instrument in a second fluid whose specific gravity is s', the neck will sink through a distance 1', and from the last equation we have W -. S'I o7r q2 St i subtracting this equation from that above and reducing, we find qrr2 ( 8 St The difference I - I' is the distance between two points on the scale which indicates the difference s' -s of specific gravities, and this we see becomes longer, and the instrument more sensible, therefore, in proportion as wo is made greater and r less. Whence we conclude that the Areometer is the more valuable in proportion as the vial portion is made larger and the neck smaller. MECHANICS OF FLUIDS. 315 Jf the specific gravity of the fluid remain the same, which is the case with Nicholson's Hydrometer, and it becomes a question to know the effect of a small weight added to the instrument, denote this weight by w', then will Equation (457) become' - w~ -sc I - Trqr2S C subtracting from this Equation (457), we find qr r2 s From which we see that the narrower the upper stem of Nicholson's instrument, the greater its sensibility. The knowledge of the specific gravities or densities of different substances, Table III, is of great importance, not only for scientific purposes, but also for its application to many of the useful arts. This knowledge enables us to solve such problems as the follow ing, viz.:1st. The weight of any substance may be calculated, if its volume and specific gravity be known. 2d. The volume of any body may be deduced from its specific gravity and weight. Thus we have always W= gD V; in which g is the force of gravity, D the density, V the volume, and W the weight, of which the unit of measure is the weight of a unit of volume of water at its maximum density. Making) Dand V equal to unity, this equation becomes V, = g; but if the density be one, the substance must be water at 380,75 Fahr. The weight of a cubic foot of water at 600.~ is 62,5 lbs., and, therefore, at 38~0,75, it is Ibs. 62,5 lbs. 0,99914 - 6 whence, if the volume be expressed in cubit feet, lbs. V-= 6_,556 x D V. (458) 316 ELEMENTS OF ANALYTICAL MECHANICS. in which W is expressed in pounds; and if the unit of volume be a cubic inch, w, 1728 D V = 0,036201D V, ~.. (459) Also, V- s........ *.. (460) 62,556..D IV'. V= bs.... (461) 0,036201. D Example 1.-Required the weight of a block of dry fir, containing 50 cubic inches. The specific gravity or density of dry fir is 0,555, and V = 50; substituting these values in Equation (459), lbs. - = 0,036201 X 0,555 X 50 = 1,00457. Example 2.-How many cubic inches are there in a 12-pound cannon-ball? Here TV is 12 pounds, the mean specific gravity of cast iron is 7,251, which, in Equation (461), give 12 inl. V, = 0,036201 X 7,251 = 45 ATMIOSPHIERIC PRESSURE. ~ 2S0.-The atmosphere encases, as it were, the whole earth. It has weight, else the repulsive action among its. own particles would cause it to expand and extend itself through space. The weight of the upper stratum of the atmosphere is in equilibrio with the repulsive action of the strata below it, and this condition determines the exterior limit. Since the atmosphere has weight, it must exert a pressure upon all bodies within it. To illustrate, fill with mercury a glass tube, about 32 or 33 inches long, and closed at one end by an iron stop-cock. Close the open end by pressing the finger against it, and invert the tube in a basin of mercury; remove the finger, the mercury will not _ escape, but remain apparently suspended, at MECHANICS OF FLUIDS. 31T the level of the ocean, nearly 30 inches above the surface cf the mercury in the basin. The atmospheric air presses on the mercury with a force sufficient, to maintain the quicksilver in the tube at a height of nearly 30 inches; whence, the intensity of its pressure must be equal to the weight of a column of mercury whose base is equal to that of the surface pressed and whose altitude is about 30 inches. T'ie Jbrce thus exerted, is called the atmospheric pressure. The absolute amnount of atmospheric pressure was first discovered by Torricelli, and the tubes employed in such experiments are called, on this account, Torricellian tubes, and the vacant space above the mercury in the tube, is called the Torricellian vacuunzm. The pressure of the atmosphere at the level of the sea, supporting as it does a column of mercury 30 inches high, if we suppose the bore of the tube to have a cross-section of one square inch the atmospheric pressure up the tube will be exerted upon this extent of surface, and will support 30 cubic inches of mercury. Each cubic inch of mercury weighs 0,49 of a pound-say half a pound-from which it is apparent that the surfaces of all bodies, at the level of the sea, are subjected to an atmospheric pressure of fifteen pounds to each'square inch.:JAROMETER. ~281.-The atmosphere being a heavy and elastic fluid, is compressed by its own weight. Its density cannot be the same throughout, but diminishes as we approach its upper limit where it is least, being greatest at the surface of the earth. If a vessel filled with air be closed at the base of a high mountain and afterwards opened on its summit, the air will rush out; and the vessel being closed again on the summit and opened at the base of the mountain, the air will rush in. The evaporation which takes place from large bodies of water. the activity of vegetable and animal life, as well as vegetable decom positions, throw considerable quantities of aqueous vapor, carboni( acid, and other foreign ingredients temporarily into the permanent 318 ELEMENTS OF ANALYTICAL MECHANICS. portions of the atmosphere. These, together with its ever-varying temperature, keep the density and elastic force of the air in a state of almost incessant change. These changes are indicated by the Barometer, an instrument employed to measure the intensity of atmospheric pressure, and frequently called a weather-glass, because of certain agreements found to exist between its indications and the state of the weather. The barometer consists of a glass tube about thirty-four or thirtyfive inches long, open at one end, partly filled with distilled mercury, and inverted in a small cistern also containing mercury. A scale of equal parts is cut upon a slip of metal, and placed against the tube to measure the height of the mercurial column, the zero being on a level with the surface of the mercury in the cistern. The elastic force of the air acting freely upon the mercury in the cistern, its pressure is transmitted to the interior of the tube, and sustains a column of mercury whose weight it is just sufficient to counterbalance. If the density and consequent elastic force of the air be increased, the column of mercury will rise till it attain a c6rresponding increase of weight; if, on the contrary, the density of the air diminish, the columnn will fall till its diminished weight is sufficient to restore the equilibrium. In the Common Barometer, the tube and its cistern are partly inclosed in a metallic case, upon which the scale is cut, the cistern, in this case, having a flexible bottom of leather, against which a plate a at the end of a screw b is made to press, in order to elevate or depress the mercury in the cistern to the zero of the scale. De Luc's Siphon Barometer consists of a glass tube bent upward so as to form two unequal parallel legs: the longer is hermetically sealed, and constitutes the Torricellian tube; the shorter is open, and on the surface of the quicksilver the pressure I1 4b of the atmosphere is exerted. The difference between the levels in the longer and shorter legs is the baror,; MECHANICS OF FLUIDS. 319 height. The most convenient and practicable way of measuring this difference, is to adjust a movable scale between the two legs, so that its zero may be made to coincide with the level of the mercury in the 31shorter leg. 20 Different contrivances have been adopted to ren-:er the minute variations in the atmospheric pressure, and consequently in the height of the barometer, more readily perceptible by enlarging the divisions on the scale, all of which devices tend to hinder the exact measurement of the length of the column. Of these we may name?Morland's Diagonal, and Hook's Wheel-Barometer, but especially EHuygen's Double-Barometer. ~ The essential properties of a good barometer are width of tube; purity of the mercury; accurate graduation of the scale; and a good vernier. ~ 282.-The barometer may be used not only to measure the pressure of the external air, but also to determine the density and elasticity of pent-up gases and vapors. When thus employed, it is called the barometer-gauge. In every case it will only be necessary to establish a free connection between the cistern of the barometer and the vessel containing the fluid whose elasticity is to te indi- 4 cated; the height of the mercury in the tube, - 5 expressed in inches, reduced to a standard temperature, and multiplied by the known weight of a cubic inch of mercury at that temperature, will give the pressure in pounds on each square inch. In the case of the steam in the boiler of an en-. gine, the upper end of the tube is sometimes left open. The cistern A is a steam-tight vessel, partly filled with mercury, a is a tube communicating with the boiler, and through which the steam flows and presses upon the mercury; the barometer tube b c, open at top, reaches nearly to the bottom of the vessel A. 320 ELEMENTS OF ANALYTICAL MECHANICS. having attached to it a scale whose zero coincides with the level of the quicksilver. On the right is marked a scale of inches, and on the left a scale of atmospheres. If a very high pressure were exerted, one of several atmospheres for example, an apparatus thus constructed would require a tube of great length, in which case HIla- e riotle's mnanometer is considered preferable. The tube.. Zz i, being filled with air and the upper end closed, the 2 surface of the mercury in both branches will stand at the same level as long as no steam is admitted. by1 _ The steam being admitted through d, presses on the r surface of the mercury a and forces it up the branch be c, and the scale from b to c marks the force of L compression in atmospheres. The greater width of I tube is given at a, in order that the level of the mercury at this point may not be materially affected by its ascent up the branch be, the point a being the zero of the scale. ~ 283.-Another very irnportant use of the barometer, is to find the difference of level between two places on the earth's surface, as the foot and top of a hill or mountain. Since the altitude of the barometer depends on the pressure of the atmosphere, and as this force depends upon the height of the pressing column, a shorter column will exert a less pressure than a longer one. The quicksilver in the barometer falls when the instruinent is carried from the foot to the top of a mountain, and rises again when restored to its first position: if taken down the shaft of a mine, the barometric column rises to a still greater height. At the foot of the mountain the whole column of the atmosphere, from its utmost limits,'presses with its entire weight on the mercury; at the top of the mountain this weight is diminished by that of the intervening stratum between the two stations, and a shorter column of mercury will be sustained by it. It is well kiown that the surface of the earth is not uniform, and does not. in consequence. sustain an equal atmospheric pressure MECHANICS OF FLVIIDS. 321 at its different points; whence the mean altitude of the barometric column will vary at different places. This furnishes one of the best and most expeditious means of getting a profile of an extended section of the earth's surface, and makes the barometer an instrument of great value in the hands of the traveller in search of geographical information. ~284.-To find the relation which. subsists between the altitudes of two barometric columns, and the difference of level of the points whtere they exist, resume Equation (427). The only extraneous force acting being that of gravity, we have, taking the axis z vertical, and counting z positive upwards, X= O; Y= O; Z=-g. and hence, p = Ce........ (462) Making z - 0, and denoting the corresponding pressure by p,, we find pI= C; and dividing the last equation by this one, p ffgz = —e p P, whence, denoting the reciprocal of the common modu0is'by -M, MP p, z= - log.... (463) g P Denote by h, and h, the barometric heights at the lower and upper stations, respectively, then will p, _ h, P- h' and reducing the barometric column h to what it would have been had the temperature of the mercury at the upper not differed from that at the lower station, by Equation (394), we have P, _ h, 2p - h [1 + (T - T').0,0001001]' in which T denotes the temperature of the mercury at the lower and T' that at the upper station. 21 G22 ELEMENTS OF ANALYTICAL MECHANICS. Moreover, Equation (381), g = g' (1 -0,002551 cos 2 ); in which, f g = 32,1808 = force of gravity at the latitude of 450. Substituting the value of Pi_, of g, and that of P, as given by Equation (393), in Equation (463), we find, _D, 1 -_0,002551cos 2 - Lh 1 + (T- T')0,0001001 In this it will be remembered that t denotes the temperature of the air; but this may not be, indeed scarcely ever is, the same at both stations, and thence arises a difficulty in applying the formula. But if we represent, for a moment, the entire factor of the second member. into which the factor involving t is multiplied, by X, then we may write z = [1 + (t - 320)0,00204] X. If the temperature of the lower station be denoted by t,, and this temperature be the same throughout to the upper station, then will z, = [1 + (t, - 320) 0,00204] X. And if the actual temperature of the upper station be denoted by t', and this be supposed to extend to the lower station, then would' [1 + (t' - 320) 0,00204] X. Now if t, be greater than t', which is usually the case, then will the barometric column, or h, at the upper station, be greater than would result from the temperature t', since the air being more expanded, a portion which is actually below would pass above the upper station and press upon the mercury in the cistern; and because h enters the denominator of the value X, z, would be too small. Again, by supposing the temperature the same as that at the upper station throughout, then would the air be more condensed at the lower station, a portion of the air would sink below the upper station that before was above it, and would cease to -act upon the mercurial column A, which would, in consequence, become too small, MECHANICS OF FLUIDS. 323 and this would make z' too great. Taking a, mean between z, and z' as the true value, we find 2 = [1 +- (t, + t' - 64~) 0,00204] X. Replacing X by its value, MDhi. 1+~~(t,+t' —640)0,00204 o 1 - Dd — 1-0,002551 cos2 log + (T-Tf)0,001001I MD, h,'IAhe factor Y- ", we have seen, is constant, and it only remains to determine its value. For this purpose, measure with accuracy the difference of level between two stations, one at the base and the other on the summit of some lofty mountain, by means of a Theodolite, or levelling instrument-this,will give the value of z; observe the barometric column at both stations-this will give h and h,; take also the temperature of the mercury at the two stations-this will give T and T'; and by a detached thermometer in the shade, at both stations, find the values of t, and t'. These, and the latitude of the place, being substituted in the formula, every thing will be known except the co-efficient in question, which may, therefore, be found by the solution of a simple equation. In this way, it is found that MD' h,, = 60345,51 English feet; -which will finally give for z, ft. 1 ~(tt'-640)0,00204 l [h, 1 z=60345,5. 1- 0,002551 cos 2 g 1 + (T-T')0,000100 To find the difference of level between any two stations, the Iatitude of the locality must be known; it will then only be necessary to note the barometric columns, the temperature of the mercury, and that of the air at the two stations, and to substitute these observed elements in this formula. Much labor is, however, saved by the use of a table for the computation of these results, and we now proceed to explain how it may be formed and used. 324 ELEMENTS OF ANALYTICAL MECHANICS. Make 60345,51 [1 + (t, + t' - 640)0,00102] = A, -B, 1 - 0,002551 cos 2' i + (T- T',) 0,0001 Then will z = AB -log - -, z = AB. [log C + log h, - log h]; and taking the logarithms of both members, log- = log A + logB + log [log C + log h, - log h] (464) Making t, + t' to vary from 400 to 1620, which will be sufficient for all practical purposes, the logarithms of the corresponding values of A are entered in a column, under the head A, opposite the values t, + t', as an.argmenu-m. t, e Causing the latitude?- to vary from 00 to 900~, the logarithms of the corresponding values of B are entered in a column headed B, opposite the values of.4. The value of T — T' being made, in like manner, to vary from - 300 to + 300, the logarithms of the corresponding values of C are entered under the head of C, and opposite the values of T - T'. In this way a table is easily constructed. Table IV was computed by Samuel Hlowlet,' Esq., from the formula of Mr. Francis Baily, which is very nearly the same as that just described, there being but a trifling difference in the co-efficients. Taking Equation (464) in connection with Table IV, we have this rule for finding the altitude of one station above another, viz.:Take the logarithm of the barometric reading at the lower station, to which add the number in the column headed C, opposite the observted value of T - T', and subtract from this sum the logarithm of the barometric reading at the utpper station; take the logarithm of this difference, to which- add the numbers in the columns headed A and B, corresponding to the observed values of t, + t' and, 9 the sum will be the logarithm of the height in English feet. MECHANICS OF FLUIDS. 325 Example.-At the mountain of Guanaxuato, in Mexico, M. Ilum. boldt observed at the Upper Station. Lower Station. Detached thermometer, t' = 70,4; t, = 770,6. Attached " T' = 70,4; - = 77,6. Barometric column, h = 23,66; h, = 30,05. What was the difference of level Here +t t' = 148"; T- T = 70,2; Latitude 21~. in. To log 30,05 = 1,4778445 Add C for 70,2 = 9,9996814 1,4775259 Sub. log 23,66 = 1,3740147 Log of o - - 0,1035112 = - 1,0149873 Add A for 1480 -. - = 4,8193975 Add B for 210 - - - - = 0,0008689 ft. 843,1 - - - - 3,8352537; whence the mountain is 6843,1 feet high. It will be remembered that the final Equation (464) was deduced on the supposition that the air is in equilibrio-that is to say, when there is no wind. The barometer can, therefore, only be used for levelling purposes in calm weather. Moreover, to insure accuracy, the observations at the two stations whose difference of level is to be found, should be made simultaneously, else the temperature of the air may change during the interval between them; but with a single instrument this is impracticable, and we proceed thus, viz.: Take the barometric column, the reading of the attached and detached thermometers, and time of day at one of the stations, say the lower; then proceed to the upper station, and take the same elements there; and at an equal interval' of time afterward, observe these elements at the lower station again; reduce the mercurial columns at the lower station to the same temperature by Equation (394), take a mean of these columns, and a mean of the temperatures of the air at this station, and use these means as a,ingle 326 ELEMENTS OF ANALYTICAL MECHANICS. set of observations made simultaneously with those at the higher station. Example.-The fi)llowing observations were made to determine the height of a hill near West Point, N. Y. Upper Station. Lower Station. (1) (2) Detached thermometer, t' = 57~; t, = 56~ and 610. Attached T' 57,5; T - 56,5 and 63. in. in. in. Barometric column, h - 28,94; h, = 29,62 and 29,63. First, to reduce 29,63 inches at 630, to what it would have been at 560,5. For this purpose, Equation (394) gives in. h (1 + T - T' x 0,0001) = 29,63 (1 - 6,5 x 0,0001) _ 29,611 Then 29,62 - 29,611 in. hi2 29,6155. 56~ +- 61~ 560~0- - - =580,5,, +- t' = 580,5 5 - - = 1150,5, T- T' 560,5 - 570,5 - -- 1 ". To log'29,6155 = 1,4715191 Add C for - 10 - 0,0000434 1,4715625 in. Sub. log of 28,94 = 1,4614985 Log of - - - - 0,0100640 = - 2,0027706 Add A for 1150,5 - - - = 4,8048112 Add B for 410~,4 - - - - = 0,0001465 ft. 642,28 - - - - 2,8077283; whence the height of the hill is 642,28 English feet. MOTION OF HEAVY INCOMPRESSIBLE FLUIDS IN VESSELS. ~ 285. —Let it now be the question to investigate the flow of a heavy, hol'loceneo,)',s n,(I incr)nlnplFcesib7e fluid througlh an, pening in,lny parit of a wvs.se! i\}:i } I (' - it. A ii foir thiis.:)U'Ipose, I-resutIme Eq. (407), whlich is MECHANICS OF FLUIDS. 327 directly applicable to the case. The only incessant force being the weight of the fluid, take the axis z vertical and positive upwards; then will X = 0; Y=-0; andZ =-g. The lateral or horizontal velocities will be insignificant in comparison with the vertical, and may be disregarded or neglected, and we have, Eq. (405), IdI\2 /d(P\s a,2 {d q2 -d ) = ~; dy ) =~0; andd z which will reduce the general Eq. (407) to dp= - gdz d- - i ad (w)2; and integrating, dcp p=-Dgz-Dt -y-nd2 + C. (465) Next, find the function p, and its differential co-efficient of the first order with respect to t. Equations (405) give d r w.dz, =fw d z........ (466) Let A B C D be a vertical section of the vessel, and take the following notation, viz.: Z s = the variable area of the stratum A whose velocity is w. s, = the constant area of any determinate horizontal section of the vessel, as C D. S = the area of the section of the vessel by the upper surface of the liquid; this may be constant or variable, according as the upper surftce is stationary or movable. 328 ELEMENTS OF ANALYTICAL MECHANICS. 2a - velocity of the stratum passing the section s, at C D, at the time t. The continuity of the fluid requires that W * = WI Si $ because the same quantity must flow through every horizontal section in the same time: whence W Ws; which, in Eq. (466), gives a, dz 1 - W. *, JS.+h $ i the integration being taken with respect to the variable z, of which s is a function. This function will be given by the figure of the vessel, h being the height of the upper surface of the fluid above the opening. It may be well to remark here, that j" dz will be constant for the same vessel and same value for h; and if the figure of the vessel be that of revolution about a vertical axis, it will only be necessary to havae the equation of this vertical section to find the value of the integral. The quantity h is called the head of fluid. Differentiating qp with respect to t, and recollecting that the velocity downward is negative, we find d dw?u dz d _' d t JfZ +A, T - -t and this, with the value of w, in Eq. (465), gives Dgz + D.- T. dD 2 C -. (4 7) a~~~fJ{ t: -F' L4~r1 MECHANICS OF FLUIDS. 329 To find the value of C, let p = P., when z = z,, which corre. sponds to the section C(D of the liquid; then will dw, f dz w2 s'2 P, =.D —Dgz, + D.s,. * -t.J -D 2. + s-, which, subtracted from the equation above, gives p — -[g- (z-z,) + 5. ts, J — 1D 2 I- 1 ~(468) Also, if P' denote the pressure at the upper surface corresponding to which zx z', we have dw -~z'd z q-.D P' -P, =- Dg (' -,) + ) s,'-t-' l D s — 2 -S. (469) Now z' - z = h = height of the fluid surface above the section C' D; whence, by substitution and transposition, P'-P, + Dgh- D s, * t' — 2) O- o(470) The quantity of fluid flowing through every section in the same time being equal, we also have - Sdh =s, w,.d...... (471) By means of this equation, t may be eliminated from Equation (470); then knowing the quantity of the liquid, the size and figure of the vessel, we will know h, S and the integral -f - f -, ~S S in which s is a function of z. d w, ~2s7.-The value of dt being found from Equation (470), and substituted in Equation (468), this latter equation will give the value of the pressure p at any point of the fluid mass as soon as w, be. comes known. Two cases may arise. Either the vessel may be kept constantly full while the liquid is flowing out at the bottom, or it may be suffered to empty itself. ~288. —To discuss the case in which the vessel is always full, or the fluid retains the same level by being supplied at the top as fast 830 ELEMENTS OF ANALYTICAL MECHANICS. as it flows out at the bottom, the quantity h must be constant, and Equation (471) will not be used. And making, in Equation (470), r dz A =2s,; B=( = h+ PI i); Dg S 2 S2-1; and solving with respect to d.t, we have dt = A.d, (472) B~ Cw2 Now, three cases may occur. 1st. S may be less than s,, and C will be positive. 2d. S may be equal to s,, in which case C will be zero. 3d. S may be greater than s,, when C will be negative, and this is usually the case in practice. In the first case, when C is positive, we have, by integrating Equa. tion (472), and supposing t = 0, when w, = 0, A ta-1 1=..tan WI. (473) whence, W -- = /. tan A t.... (474) from which we see that the velocity of egress increases rapidly with the time; it becomes infinite when A 2 or ~r.A t.......(475) 2V/BC When C = 0, then will the integration of Equation (472) give A t -W-. (476) MECHANICS OF FLUIDS. 331 or replacing A and B by their values, and finding the value of w,, g (h + P Pr ) * W.- ~ t;..... (477) sohd - whence, the velocity varies directly as the time, as it should, since the whole fluid mass would fall like a solid body under the action of its own weight. When C is negative, the integration gives t= A.log - ~W+ w, 2 BC -v/B - w, V7 whence, X —..... (478) e - +1 in which e is the base of the Naperian system of logarithms = 2,7182822. If the section S exceeds s, considerably, the exponent of e will soon become very great, and unity may be neglected in comparison with the corresponding power of e; whence, 2g (h + P -gP) W V. ~ (479) S2 that is to say, the velocity will soon become constant. If the pressure at the upper surface be equal to that at the place of egress, which would be sensibly the case in the atmosphere, P -P, = 0, and;....... (480) and if the opening below become a mere orifice, the fraction 0; S2 and w, = /2#g h;... * (481) 332 ELEMENTS OF ANALYTICAL MECHANICS. that is to say, the velocity with which a heavy liquid will issue from a small orifice in the bottom of a vessel, when subjected to the pressure of the superincumbent mass, is equal to that acquired by a heavy body in falling through a height equal to the depth of the orifice below the upper surface of the liquid. The velocities given by Equations (479), (480), (481), are independent of the figure of the vessel. If the velocity w, be multiplied by the area s, of the orifice, the product will be the volume of fluid discharged in a unit of time. This is called the expense. The expense multiplied by the time of flow will give the whole volume discharged. ~ 289. —The velocity w, being constant in the case referred to in Equation (479), we shall have dw, _ dt and Equhtion (468) becomes p = P - Dg (Z - z,)- D - * -1) or, substituting the value of w,, given by Equation (470), 82 p, - Dg (z -,) + (Dgh + P' - P,) 2 -;.. (482) S~ whence, it appears, that when the flow has become uniform, the pres. sure upon any stratum is wholly independent of the figure of the vessel, and depends only upon the area s of the stratum, its distance S 2 from the upper surface of the fluid, and upon the ratios. ~ 290.-If the vessel be not replenished, but be allowed to empty itself, h will be variable, as will also S except in the particular cases of the prism and cylinder. Making w. =.1~tI....... (483) MECHANICS OF FLUIDS. 333 in which H denotes the height due to the velocity of discharge: we have g-dH dw d (484) and, Equation (471), S.d h d t... (485) and by integration, 1 rS.__dh s= *. (486) To effect the integration, S And H must be found in terms of h. The relation between S and h will be given by the figure of the vessel. Then to find the relation between H and h, eliminate w1, d w,, and d t from Equation (470), by the values above, and we have P' - b- I)' + t) dh d ff dz _ (1 s'2- dh 0; or, dividing by s82 h d z S P (t D P hp S ( Si2) and making R; _ Sfh 2 dz QdIt + dH - Ed. dh = o. (488) SR d h fR dh h, dh R d h dh. Q. e + d. e -. e Rdh =O; 334 ELEMENTS OF ANALYTICAL MECHANICS. or JRdh fR dh dh. Q.e +d(Ze )=O; and integrating fRd h fRdh dh. Q e + e = C;... (489) whence, -fR dhRd (490) H = e (C- fdh. Q.e ) The constant must result from the condition, that when Hf = 0, A must be h,, the initial height of the fluid in the vessel. Thus H becomes known in terms of h, and its value substituted in Equation (486) Will make known the time required for the fluid to reach any altitude h. The constant in Equation (486) rrmust be determined, so that when t = 0, h = h,. ~ 291.-The Equation (490) gives a direct relation between S, h, and Hi; the figure and dimensions of the vessel give another between S and h.'From these, two of the three variables may be eliminated from the lqlatioll (486) and the integration perfornmed. Take, for exaii lp)l, tlhe case of a right cylinder ori prl'ib;n. liere S will be tonstant, and' e,:dil, Jo. Moreover, let us suppOse P'- P,- 0, which would be sensibly true were the fluid to flow into tlhe atmosphere that surrounds the vessel. Also, for the sake of abbreviation, make - k, then will k2 - I I - k2 h h Q = — and R]dh = (1- k2)fdh (1- k2)log h. MECHANICS OF FLUIDS. 335 and Eq. (490) becomes — (1-k2)log h (l-k:) Icg k H = e *C -[C fk2dh e ] Mlultiplying the last term by 2 - k2 h 2 —k2 Jh we may write -(-k2)log 4 k[ (I-) log h(] H (-e ~- 2 - T2 -of d —1cz -(I-kH2)10g [ C2 h1-ik) log h = e C-.h e when HI- 0 then will h - h, and ~2 (1-k2) lo ha, = --- -2 h, ~ e which substituted above, gives. after reduction, k2h -h. [-h, ) hi og~- ] but, h, (l —k2) log ~e _ ( /X and therefore,.2. h (h, - k - if —Ikk2[ h- I h24k2h92) 2_ - ]... I which substituted in Equation (486), gives t=-'9'S (193) in which the only variable is h. ~292.-The particular case in which k2 = 2, gives to this value for t the form of indetermination. When this occurs, we must have recourse to the form assumed by Equation (488), which, under this supposition, becomes 2 ddh +,~ dH- - Hdh 0: 336 ELEMENTS OF ANALYTICAL MECHANICS multiplying by h-2 2h -'dh + h1-.d.H -H.h 2dh 0, 2.- h + d = o h T 2logh + = C; and because H = 0 when h = h,, 2 log h, = C; whence, H = 2h log, and this, in Equation (486), gives 1_ dh t =C H 2 A.log hi_ 1, Making h = 2 this becomes, between the limits x = x, = 1, t = Cane -z = /7ih' /log x ~293.-If the orifice be very small in comparison with a cross section of the prismatic or cylindrical vessel, then will H= h, and Equation (486) gives =C_ 2S Making t = 0 when h = h,, we have 2S t= h -- ),.(494) and for the time required for the vessel to empty itself, h = 0, and, 2S h_ t.=.... (495) 2g MECHANICS OF FLUIDS. 337 Now, with the same relation of the orifice to the cross section of the cylindrical vessel, we have, Equation (481), w y-h, and for the volume of fluid discharged in the time t, when the yessel is kept full, w..s,.t = s,.t. 52h, and if this be equal to the contents of the vessel, S,. 2., S. hi,; whence, t -. That is, Equation (495), the time required for a prismatic or.cylindrical vessel to discharge itself through a small orifice at the bottom is double that required to discharge an equal volume, if the vessel were kept full. ~ 294.-The orifice being still small, we obtain, from Equation (485), dh V gh; dt S h whence it appears that, for a cylindrical or prismatic vessel, the motion of the upper surface of the fluid is uniformly retarded. It will be easy to cause S so to vary, in other words, to give the vessel such figure as to cause the motion of the upper surface to follow any law. If, for example, it were.required to give such figure as to cause the motion of the upper surface to be uniform, then would the first member of the above equation be constant; and, denoting the rate of motion by a, we should have s'./2gh whence, s 2. 2 g h S2 -i12g a2 but supposing the horizontal sections circular,, S2 2 U2 r S = s2.2gh O0,~2~~a 338 ELEMENTS OF ANALYTICAL MECHANICS. and, therefore, r = 4 42; whence the radii of the sections must vary as the fourth root of their distances from the bottom. These considerations apply to the construction of Clepsydras or Water Clocks. MOTION OF ELASTIC FLUIDS IN VESSELS. ~295.-As in the case of incompressible, so also in that of elastic fluids, it is assumed that in their movement through vessels, they arrange themselves into parallel strata at right angles to the direction of the motion. The quantity of matter in each stratum is supposed to remain the same, while its density, which is always uniform throughout, may vary from one position of the stratum to another; hence, the volume of each stratum may vary. All lateral velocity among the particles will be supposed zero; and as the weight of the elements of elastic fluids is insignificant in comparison to their elasticity, the former will be disregarded. The motion will, therefore, be due only to the elastic force arising from some force of compression; and as the fluid will be supposed to communicate freely with the air, or with a vessel partly filled with some other elastic fluid, this force within may be greater or less than it is on the exterior of the vessel. ~ 296. —Assumning the axis of the vessel horizontal, take that line as the axis of. x. Then, by the supposi- tion above, will x-o; _a _ P_ Z'-, L ~X, w =0; MECHANICS OF FLUIDS. 339 and Equations (400) give 1 dp du du D dx = - d- - (d. u. * * * (496) Moreover, if we suppose the motion to have been established and become permanent, the velocity of a stratum as it passes any particular cross section of the vessel will always be constant, and the quantity of fluid which flows through every cross section will be the same. Hence the partial differential of u in regard to the timlle, that is, supposing x, y, z, to be constant, must be zero, and tne above equation reduces to dp= - D..du. From Mariotte's law, Equation (389), p= 2.D, and by division, dp P - - I, U d ul and by integration, log p = C - 2 *u2...... (497) To determine the constant, let p, be the pressure at the opening C D, that is, the pressure of the atmosphere, and denote by %u the velocity of the fluid at this point, then will I log, = C- - 2. and by subtraction, log,- -. (2 u2).... (498) Denote by s the area of any section of the vessel A' B', at which the pressure is p and velocity u, by D the density of the fluid at this section, and by D, that at the section C() equal to s,. Then, since the quantities of fluid flowing through these sections in a unit of time must be equal, we have D.s. s =,. s,. U; 340 ELEMENTS OF ANALYTICAL MECHANICS. but, ~244, D _ whence, or p, 8, U, P.s which, in Equation (498), gives Po p 2[1- -, s)2 -. (499) log If p' denote the pressure exerted by the piston AB, and S denote its area, we have log -t (- _2 (1- (500) whence. 2 P. log - ~ ~ r (501) (p, Si,2 This is the velocity with which the fluid will issue into the atmosphere or other fluid whose pressure on the unit of surface is p,. 297.-The volume discharged in a unit of time is 1O' 2 P1 log VUI S" S/ p while under the pressure p,; and under a pressure equal to that on the unit of surface of the piston, or top of a gasometer, and which would be indicated by a gauge, since the volumes are inversely as the pressures, P. log Ud- Si. (502)L ( pi Si2 MECHANICS OF FLUIDS. 34=1 ~9298.-Dividing Equation (499) by Equation (500), we have log9 1 -i (P' S'I log __ ___.e e e ^ *,(503) log 1 which will give the pressure p at any section of the vessel. -299.-If the opening CD is very small in reference to A B, the,velocity u, will become, Equation (501), a h u a/2 u -ni oti anoa and the volume of fluid discharged in a unit of time and of a density equal to that pressing upon the gauge,' 8. 2P.log;.... (505) and Equation (503) becomes log log - pS P4 -300.-A stream flowing through an orifice is called a vein. In estimating the quantity of fluid discharged, it is supposed that there are neither within nor without the vessel any causes to obstruct the free and continuous flow; that the fluid has no viscosity, and does not adhere to the sides of the vessel and orifice; that the particles of the fluid reach the upper surface with a common velocity, and also leave the orifice with equal and parallel velocities. None of these conditions are fulfilled in practice, and the theoretical discharge must, therefore, differ from the actual. Experience teaches that the former always exceeds the latter. If we take water, for example, which is far the most important of the liquids in a practical point of view, we shall find it to.a certain degree viscous, and always exhibiting a tendency to adhere to ununctuous surfaces with which it may be brought in contact. When water flows through an opening, the 342 ELEMENTS OF ANALYTICAL MECHANICS. adhesion of its particles to the surface will check their motion, and the viscosity of the fluid will transmit this effect towards the interior of the vein; the velocity will, therefore, be greatest a. the axis of the latter, and least on and near its surface; the inner particles thus flowing away from those without, the vein will increase in length and diminish in thickness, till, at a certain distance from the orifice, the velocity becomes the same throughout the same cross-section, which usually takes place at a short distance from the aperture. This effect will be increased by the crowding of the particles, arising from the convergence of the paths along which they approach the aperture, every particle, which enters near the edge, tending to pass obliquely across to the opposite side. This diminution of the fluid vein is called the veinal contraction. The quantity of fluid discharged must depend upon the degree of veinal contraction, and the velocity of the particles at the section of greatest diminution; and any cause that will diminish the viscosity and cohesion, and draw the particles in the direction of the axis of the vein as they enter the aperture, will increase the discharge. Experience shows that the greatest contraction takes place at a distance from the vessel varying from a half to once the greatest dimension of the aperture, and that the amount of contraction depends somewhat upon the shape of the vessel about the orifice and the head of fluid. It is further found by experiment, that if a tube of the same shape and size as the vein, from the side of the vessel to the place of greatest contraction, be inserted into the aperture, the actual discharge of fluid may be accurately computed by Equation (478), provided the smaller base of the tube be substituted for the area of the aperture; and that, generally, without the use of the tube, the actual may be deduced from the theoretical discharge, as given by that equation, by simply multiplying the theoretical discharge into a co-efficient whose numerical value depends upon the size of the aperture and head of the fluid. Moreover, all other circumstances being the same, it is ascertained that this co-efficient remains constant, whether the aperture be circular, square, or oblong, which embrace all cases of practice, provided that in comparing rectangular with circular orifices, we compare the smallest MECHANICS OF FLUIDS. 343 dimens.on of the former with the diameter of the latter. The value of this co-efficient depends, therefore, when other circumstances are the same, upon the smallest dimension of the rectangular orifice, and upon the diameter of the circle, in the case of circular orifices. But should other circumstances, such as the head of fluid, and the place of the orifice, in respect to the sides and bottom of the vessel, vary, then will the co-efficient also vary. When the flow takes -place through thin plates, or through.'> orifices whose lips are bevelled externally, _. the co-efficient corresponding to given heads and orifices, may be found in Table V, provided the orifices be remote from the lateral faces of the vessel. This table is deduced from the experiments of Captain Lesbros, of the French engineers, and agrees with the previous experiments of Bossut, Michelotti, and others. As the orifice approaches one of the lateral faces of the reservoir, the contraction on that side becomes less and less, Il tJ l and will ultimately become nothing, and the \ co-efficient will be greater than those of the \\\, i table. If the orifice be near two of these faces, the contraction becomes nothing on two sides, and the co-efficient will be still greater. Under these circumstances, we have the following rules: —Denote by C the tabular, and by C' the true co-efficient corresponding 11 to a given aperture and head; then, if the contraction be nothing on one side, will C' - 1,03 C; if nothing on two sides, C' = 1,06 C; if nothing on three sides, (' - 112 C; 344 ELEMENTS OF ANALYTIC'AL MECHANICS. and it must be borne in mind, that these results and those of the table are applicable only when the fluid issues through holes in thin plates, or through apertures so bevelled externally that the particles may not be drawn aside by molecular action along their tubular contour. 301.-When the discharge is through thick plates without bevel, or through cylindrical tubes whose lengths are from two to three times the smaller dimension of the orifice, the expense is increased, the mean coefficient, in such cases, augmenting, according to experiment, to about 0,815 for orifices of which the smaller dimension varies from 0,33 to 0,66 of a foot, under heads which give a coefficient 0,619 in the case of thin plates., The cause of this increase is obvious. It is within the observation of every one, that water will wet most surfaces not highly polished or covered with an unctuous coating-in other words, that there exists between the particles of the fluid and those of solids an affinity which will cause the former to spread themselves over the latter and adhere with considerable pertinacity. This affinity becoming effective between the inner surface of the tube and those particles of the fluid which enter the orifice near its edge, the latter will not only be drawn aside from their converging directions, but will take with them, by the force of viscosity, the other particles, with which they are in sensible contact. The fluid filaments leading through the tube will, therefore, be more nearly parallel than in the case of orifices through thin plates, the con. traction of the vein will be less, and the discharge consequently greater. PART III. MECHANICS OF MOLECULES. ~ 302. —THE more general circumstances attending the action of f(rces upon bodies of sensible magnitudes have been discussed. They constitute the subjects of Mechanics of Solids and of Fluids. Those which result from the action of forces upon the elements of both solids and fluids remain to be considered. They form the subject of AMechanics of Molecules; which comprehends the whole theory of Electrics, Thermotics, Acoustics, and Optics. It has been seen, that all bodies are built up of elementary molecules in sensible, though not in actual, contact; that the relative places of equilibrium of these molecules are determined by the molecular forces, and that the intensities of these forces are some function of the distance between the acting molecules. A displacement of a single molecule friom its position of relative rest, will breakup. the equilibrium of the surrounding forces, and give rise to a general and progressive disturbance throughout the body. It is proposed to investigate the nature of this disturbance, the circumstances of its progress, and the conduct of the molecules as they become involved in it. PERIODICITY OF MOLECULAR CONDITION. ~ 303.-Molecular motions cannot, like the initial disturbances which produce them, be arbitrary; but must fulfil certain conditions imposed by the physical connectiens wlvich unite' the inolecules into a system. 346 ELEMENTS OF ANALYTICAL MECHANICS. These motions are, so to speak, constrained by this connection. Let the conditions of constraint be expressed, as in ~ 213, Mech. of Solids, by L-=; L'=o; L" =O; &c. (506) L, L', L", &c., being functions of the co-ordinates of the molecules. Denote by X, Y, Z; X', Y', Z'; &c., the accelerations impressed upon the molecules whose masses are m, in', &c., in the directions of the axes. Equation (313) will obtain for each molecule. There will be as many equations as molecules, and by addition, we find, by inverting the terms, I2t2 (ddt X d d Y-Bbtd + Y (dt z) a 1= 0, (50 ) There will be three co-ordinates for each. molecule. Denote the number of molecules by i; the number of Equations (506) of condition by m; then will 3 i - m = n,'be the number of co-ordinates which, being given, will reduce the number of unknown co-ordinates to the number of equations. These unknown co-ordinates may, hence, be found in functions of the known, and -the places of the molecules at any instant determined. Denote the m co-ordinates by x y z, x'y' z', &c., and the n co-ordli nates by r3 y, a'P'y', &c.: then we may write, x = p, (c 3y a', &c.) =,; y gy (a 3 y a', &c.) Py; z - z, (c y c', &c.) = p,; x'= P, (3 y a', &c.) = p.,; &e. = &e. = &c.; also, X=, (a Py ac', &c.) = P,; Y=- 9CI (p3 c', &c.) =P,; Z = I, (aya', &c.) P,,; X'= fbl(cz3y ca',&c.)= P.; &c. - &c. = &e.; in -,hich pq, P%, qz, &c., izb,'ly, %, &c., denote any functions of the co. MECIIANICS OF MOLECULES. 347 ordinates a 3 y, a' &c., which result from the conditions of Equations (506) and the process of elimination. At any time t, suppose a 3 and y to become a + —, y +; a' 3' and Y' to become a'+ +', /3'+i/', Y'+ e'; and suppose the increments ~ /,''/' ~', &c., to continue so small during the entire motion as to justify the omission of all termtn into which their second powers and products enter; then will x = p+.f+ 3- f 3+- ~ ~+ &c., +c., d dpy d p, d p YY J d, * 3 vS+- r d0 + &c., &c., d p d p + d, (508) =,+ —. -7)~- -. + -+ &c, &c., ~'pf, d'+ dp d + &., &c, + dZ d dP dP, xz = P $-. d + r' +. d + &c., &c., dP, dP& dP& Y =P,Y + ~ ~+ - v + &o., &o.,. dP dP dPP z =P.d+, d T.d d + -. +2 + &C., &o., da da dy From Equations (508) we have d d pp dpd, d= x - d2 -d+ *d 1+- d2+ &c., &c., d p d/ dy p( Y d = 7-1~ + -' ~ vd + -. &c.. &c., c., d a d dy d2 x'-= &C &c., &c. 348 ELuEMENTS OF ANALYTICAL MECHANICS. also from the same d p dp, d p. d p, d p, d p, da d dy 6z 32 - li~z.~=-~.- 6 n + d d + d.d; + &c. y The Equation (507) contains three times as many terms as there are molecules, each term consisting of a variation with its coefficient. Eliminate from this equation X, Y, Z, X', &c., d' x, d'y, d'z, d2', &c., and 6x, dy, 6z, dx', &c., by means of Equations (509), (510), and (511); collect the coefficients of, 6, d6;,6 d', &c.; the number of terms will reduce to n, this being the number of the co-ordinates a, 13, y, a', &c. These variations are independent of one another, since the coordinates a, 3, y, a', &c., are so. The coefficients of these variations must, therefore, be separately equal to zero. Performing the operation, omitting all the terms containing products and powers of f, b, A', &c., higher than the first, there will result n equations of the form, D. +E. da; +F G, + 2EH, q+1K. - +A=O; (512) in which D, El F, C, H, &c., are functions of the differential co-efficients in Equations (509), (510), and (511); and -A consists of a. series of terms each composed of two factors, one of which is either Pa, Py, P,, or some other P with subscript co-ordinate accented. If a/3 y, a', &c., give the places of rest of the molecules, then will P. = P = P, = &c. = 0, and Equations (512) become d2 2 d2 r/ d2 Z' dt.. + d' + G. f + + i. K. =. (513) These equations are satisfied by making:= R. N,,.sin (t./p- r), z/= ]R. N_. sin (t /p - r),.. ~ T',./(514) MECHIANICS OF MOLECULES. 349:n which R and r are arbitrary constants, and p, N, N2V,, &c., are constants to be determined. For, after two differentiations, regarding 4, r, A, &c., and t variable, we have d-t -R t. sin (t p ) p d R2 _ N.sin (t Vp - r) p, d t' d t2 = R. N. sin (t Vp-r) p, &c., &c. which, substituted in Equations (513), give, after dividing out the common factor R. sin (t /p - r), ( D. Nf +E. N=+ 0EFNNF)p-l GN- 11N - IKAn _ O. (515) Now, there being n of these equations; n- 1 of them will give the values of NV, 2N, NN', &c., in terms of No; and these being substituted in the nth equation, must, from the form of the equations, give a resulting equation having NF as a common factor and of the nth degree in p. The common factor NV will divide out. The quantity p will have n values. The values of N, N,,, &c., will be rational fractions of the (n-i1)th degree in p, having a common denominator, and each multiplied by NV, which is, as yet, arbitrary. Make N equal to the common denominator, and IV, N, NV,, &c., will be expressed in symmetrical functions of p, of the (n - I)th degree. Each of the quantities NV, N, N, &c., will have as many values as p; and each of the increments [, i, /,',, &c., will also have n values, each set of which will satisfy Equations (513). But Equations (513) are linear; not only, therefore, will each of the values of ~, r, (, A', &c., satisfy them, but their respective sums substituted for;:, /,',:', &c., will also satisfy them. Denoting the roots of the nth equation in p by p, pa, p,, &c., and using the subscript figures to designate the corresponding values for the other letters, the general solution of Equations (513) will be 350 ELEMENTS OF ANALYTICAL MECHAINICS. f =R1?.N. snt- R.i.P ).sin (t.. p-r)L. sil(t. VP-p12-?2)-lt&C. q=R.N. sin(t. V/p —r)f -Ri.N.sin(t. /tp -vr,)q-I2. 2.sin(t. V /-r2)+&C. 1 s - *sin(t*.Vp-rl)-1.1V;. sin (t. /Vg-r )+tIJ2Y.1.sin (t. /p 2 -2)+&C. R, R,, &c., and r, r1, &c., are arbitrary constants, in these complete integrals. They must be found in terms of the initial values of 5, 91,; &c., and their differential coefficients. They are small, because the original'disturbance is supposed small. ~ 304.-If all the quantities R, N, R,, &c., except the first, vanish, Equations (516) lose all their terms in the second members except the first, and Equations (508) become = ( d a X.rd( + d?: * + ~ -- R. ) R. ln (t.'p — r), ___, _ _ dp d p xPZ, + (.. d, f..) R. sin (t.pV-r), ~ 305.-If the roots p, pl, P2, &c., be real, the different terms in the values of, &c., as given in Equations (516), will disappear periodically, and the precise times of disappearance of each will be found by making tv/p —rarr; t /p,-r,-=arr; t — r =a7r; &c., or, a 7rr + r a t+ qtr, a r -+ra; t ~_ t=; &c., 1/P Pi V/p in which a is any whole number whatever. The intervals of disappearance will be 7r+r rr r, rr +.2 &c. v/-; /V- p2 AiECIIANICS OF MOLECULES. 351 \Nhen these intervals are commensurable, then will ~, i, l, &c., resume the values they had at some previous time, the molecules -will return to their former simultaneous places, the movement will become periodical, and the period will be equal to the least common multiple of the above intervals. This phenomenon of periodical returns of molecules to their initial places, is called the periodicity of molecular condition. ~ 306.-From Equations (516) it is apparent that each and every individual of a system of molecules in which the connection is such as to leave n of their co-ordinates independent, may, when slightly disturbed from rest in positions of stable equilibrium, assume a number n of oscillatory movements, and that all or any number of these may take place simultaneously. And conversely, whatever be the initial derangement of such a system, the resulting motions of each molecule may be resolved into n or less than n simple components parallel to each of any three rectangular axes. Here we have, under a different form, the principle of the coexistence of small motions. ~ 307.-Again, let ~,, 71j, e,, &c., be the values of 4, y7,', &c., when the system is in motion by the action of one set of forces; 2, 7?l, a2, &c., when under the action of another set, and so on-the initial condition being determined for each set of movements-then, Equation (516) being linear, will the resultant values of A, Ay,', &c., be given by -= + 2 + 3 + &c.,:- = 91 + 71Y2 + 713 + &c.,'=,+ ~ + g; + &c.; and here we also have again the superposition of small motions. That is, each molecule may take up simultaneously the motions due to each disturbing cause acting separately and alone. ~ 308.-Equations (513) may also be satisfied by making = I.?.ee l = R. V. e t' Vp- r C R.N e.e 352 ELEMENTS OF ANALYTICAL MECHANICS. which give d2y P t Vp-r dr- = p. R. n. e* ~~~d t2~~ 7 d t~ - P. -R.-S. e; and these substituted; in Equations (513), give Equations (515), with the exceptions of the signs of the terms which are independent of p. But with this solution there would be no limit to the increase of 7, 1,,', &c., which is contrary to the conditions that the disturbances are to continue small. In fact, this last solution supposes the molecules to be moved from positions of ucnstable equilibrium; the other, which is the case of nature, from stable equilibrium. WAVES. ~ 309.-It thus appears that every molecule, subjected to certain conditions of aggregation, may, when disturbed from its place of relative rest, describe, under the action of surrounding molecules, a closed orbit. The disturbed molecule being acted upon by its neighbors, will react upon the latter, and cause them, in turn, to take up their appropriate paths; and the same being true of the next molecules in order of distance, the disturbance will be progressive and in all directions. That is, an initial disturbance of a molecule at one time and place, becomes a cause of disturbance of another molecule at another time and place. While, therefore, any molecule A, is travelling over its orbit, the disturbance is being propagated on all sides, and at the instant the former completes its circuit, the latter will have reached a molecule A2, in the distance, which will then, for the first time, begin to move; and the molecules A, and A2 will, thereafter, always be at the same relative distance from their respective starting points. In the same way, a molecule A3, still further in the distance, will begin its first circuit when -A begins its second and A, its third, and so on. MECHANICS OF MOLECULES. 353 Between the molecules Al and A,, as also between A, and A,, &c., molecules will be found at all distances from their starting points and moving in all directions, consistently with the dimensions and shapes of their respective orbits. The term phase is used to express the condition of a molecule with respect to its displacement and the direction of its motion. Molecules are said to be in similar phases, when moving in parallel orbital elements and in the same direction; and in opposite phases, when moving in parallel orbital elements and in opposite directions. The particular form of aggregation assumed by the molecules between the nearest two concentric surfaces in which the same phases simultaneously exist throughout, is called a wave. A surface which contains molecules only in similar phases, is called a wave front. This latter term is generally, though not always, applied to the surface upon which the molecules are just beginning to move. The velocity of a wave front will always be that of disturbance propagation. A wave length is the interval, measured in the direction of wave propagation, between two consecutive surfaces upon which the molecules have similar phases. WAVE FUNCTION. ~ 310.-Denote the masses of the molecules by m, m', &c.; the coordinates of m by x,, z, m' " x+ Ax, y + y, z+ Az, m" " x.+Ax', y+Ay', z+Az', &c., &c., &c., &c., and the distance between any two molecules, as m and n', by r; then will r = /aAx+ z.... (518) Let f (r) be the intensity of the reciprocal action between m and m';. in which f denotes any function whatever. This reciprocal action will. determine the elastic force of the body. 23 354 ELEMENTS OF ANALYTICAL MECHANICS. Before the system is disturbed, there will be no inertia developed, the inertia terms in Equations (A ) will disappear, and we shall have for the action on any molecule as m, 0, Zf (r)' =0. If (r) * y = 0, (519) If (r) - -- o. Now suppose the system slightly disturbed, and denote the displacement at the time t in the direction of the axes x, y, z, respectively, of m by +, I, +,.",, ~+a/:', e+ a', +A? a', &c., &c., &c., &c. Then, denoting the change in r by Ar, Equation (518) becomes r~+ ~Axr -(ax+ i\n $)"+ (xyv + a v)P +3 (aiz + v )a. (520) and by the principle of the superposition of small motions, Equations (A) give for the action on m, m. d2 If (r+A r) A-++' d2. -- a r+A (521) m =f (rA. Ar). (521) m.- f(r'+A) r+Ar''But 1 1 Ar Ar2 = —7 +- &c., r+ar r r' r f (r + a r) =f (r) + df r + &c., whence, neglecting the powers of A r higher than the first, MECGlANICS OF MOLECULES. 35S (r+ r- * x) = lJ rf+ (r df(?,) f(r) Az /__A._ r + Aw r )= -9+(~ ~)r I.(r +) Squaring Equation (520), neglecting the squares of A. r6 L, A 9, and AL, and subtracting the square of Equation (518) fiom the resuit1 we find aX.zE + y. 1+ Az. A Ar==..... (522) r Substituting this above, and making f(r) r' I __ efd f(~). <> i (523) (r) = 2~.dr- r) );.. Equations (521) become..d2. d t2 =- Z~(r) a t+ f (,) (ax.a + a q az. a ) ay!}! (524) ndt = {p(r) q ++(r)(.a +Ay. * +AZag).ha} i(524) Performing the multiplication as indicated in the last term of the second members, there will result terms of the form, zn6(r).a y.a.z; p(r).a.az.z; zpO.). A.ax.Az; and it may be shown by the process of ~ 164, to prove the existence of principal axes, that the co-ordinate axes may be so taken as to cause these terms to vanish. Assuming the axes to satisfy these conditions, Equations (524) become'n Z? = z I, (r) (+ t (') A } I, m. = z I (r) + b (r) A y2.... (525) d 2 = () + ) a t2 -1aq(r 356 ELEMENTS OF ANALYTICAL MECHANICS. Making mp' =9 (r) + i (r). a x, mp" = q7 (r) Jr+ - (r) * 4 32, * (526) m p"'== b(r) Jr + (r).Az2. Equations (525) take the form, j=Kp'.z 4, d tzd P" - X,...... (527) An initial and arbitrary displacement of a molecule at one time and place, becomes, through a series of actions and reactions of the molecular forces alone, the cause of displacement of another molecule, at another time and place. In this latter displacement, which results alone from the molecular forces, the molecular motions must take place in the direction of least molecular resistance. This direction is at right angles to that of wave propagation; for, the force which resists the approach of any two strata of molecules will be much greater than that which opposes their sliding the one by the other. Indeed, this view is abundantly confirmed by many of the phenomena that result from wave transmission; and it will be taken for granted, without further remark, that the molecular orbits are in planes at right angles to the direction of wave propagation. 311. —The first of Equations (527) appertains, therefore, to wave propagation in the plane y z, the second in the plane x z, and the third in the plane x y. The integrations of Equations (527) are given by =. 2w =ar. sin (v.t-r,), 7 — %y.sin (2.t —r' 2..... (528) c= a..sin2 ( v'. t -rz, MECHANICS OF MOLECULES. 357 in which V~, Vy, and V~ are the velocities with which the disturbance is propagated in directions perpendicular to the axes x, y, and z, respectively; X, R, and 2, the shortest distances, in the same directions, between the places of rest of any two molecules that may have at the same instant the same phase; r%, ry, and r, the distances of any molecule's place of rest from that of primitive disturbance, estimated in the same directions. This being understood, we have the relations, r_ = By2 + Z';, = -'Vi2 + z2; r, = vx2 q- y2. Make 2r V; 27rV 2 r (529) 2 ay Y gt a, XZ 2 7r 27r 27r a, = k - A k and the above become a= a. sin (n,. t-. X), 7= a,. sin (n,/. t- kX r-),. e... (53.Q) 4= a. sin (n,. t- k-,. rI,). ~ 312. —To show that these are the solutions of Equations (527), it will be sufficient to prove that they will satisfy those equations with real values for n,, n, and n,. Differentiate twice with respect to t, and we have d2~ f. d- = - n~'. i,...... (531) d2 =' d t -,.'. Give to r,, ry, and r, the increments Ar,, Arys, and A r,, respectively; the corresponding increments of ~, I, and; are A, IA rj, and A', and Equations (530 become 358 ELEMENTS OF ANALYTICAL MECHIANICS. s + - =. sin (n. t - k,. r, + k,. A ), 1 +{ Aq X= a. sin (n,. t - k. ry - k. A r),'+ Aq' = ac. sin (n,. t - kg. r, + k.A r,). Developing the second members, regarding n,. t - k,. r7, nv. t - k,. r, and n,. t -k,.re as single arcs; subtracting Equations (530) in order replacing 1 -- cos ko A r,, - cos k A r, and I - cos k, A r by theih respective values, we find F = - 2 (.sing A )+sin(k;ar(). acos(.t-k).r,)) a =-2-. sin2 (k2 + sin (k1 A r). a CoS (n. t - k y) ( A= -2,.sin2 (k A r) + si (kA r,). cos (n. t-k, 2 ~.sin + sin (k, a r,). aL, Cos (na,. t - ~.. Substituting these in the second members of Equations (527), we have d2 (kar) -2 p sinl' 2 d -2. z: p'. sin~ z 2 p'. sin(kj, a r'. a9. cos * t-k') dt-= —2.'".-sin Y + Xp'.sin(,Ab ).a. es(i.g - a.).jr In the state of equilibrium of the molecules, we may suppose their masses equal, two and two, and symmetrically disposed on either side of that whose mass is Mn. Indeed, this is the most general way in which we may conceive the equilibrium to exist. Then, since for every positive arc k,. A r; there will be an equal negative one, we must have Mp'. sin (k,. A r). a, cos (n,. t - k. r) = 0,: pt, sin (k.. A r,). a, cos (n..t (534) zp"'. sin (k,. h rz). a,. cos (n,. t -. r,) = 0, and therefore, MECOHANICS OF MOLECULES. 359 dt' - d t2 - -2. Xp. sina' dt-' — 2'q.' p sin~ 2 (535) d =_2.. 1; p" 2 sin' k d t' whence, Equations (531) and (535), n _ 2 p'. sin' k Ar = p", sin''2. (536) n2 - 2 Z p"'. sin k A r, which are, Equations (526) and (522), real values for n., n., and n,. ~ 313.-Substtituting the values of n.,, n., n,, and k,, k, k,, Equations (529), there will result, after multiplying the first, second, and third by 1 =a r.?2 AQ; 1 = A-r*; 1 =Ar- A -r,2, respectively, sin', si.n 7r A r VZ2= 2 p'lxt. I r _ 4 22 7 A 2'i?=~ vp". _'.,... (537) 7rr ai ra. sin A r v 2, g~~PIIMI(rid135) — ~~~A I 360 ELEMENTS OF ANALYTICAL MECHANICS. WAVE SECTION. ~ 314.-Resuming either of Equations (528), say the first, viz.: = a, sin (V. t - it is apparent that if t be made constant and r. variable, so as to reach in succession all the molecules in its direction between the limits V. t - z, and V,,. t, the displacement 5 will also vary, and from zero to zero, passing between these limits through the maximum values a, and minimum value -a c; thus determining the curved line A - B CD), of the annexed C c' figure, to be the locus of the corresponding displaced molecules, of which the places of rest are on the straight line A B, coincident in direction with the line r. in the plane y z. And it is also apparent that if the above value of t receive an increment, making the time equal to t', and, with this new value for the time, r, be made to vary between the limits Va.t'- x, and V'.t', the locus of the corresponding displaced molecules will be found to have shifted its place to C' D)', in the direction towards which the disturbance is propagated. This peculiar arrangement of a series of consecutive molecules, by which the latter are made to occupy the various positions, arranged in the order of continuity about their places of rest, is, as we have seen, ~ 305, called a wave, and the functions, Equations (528), from which a section of the waves may be constructed, are called wave functions. WAVE VELOCITY. ~ 315.-From either of Equations (537), say the first, it appears that the velocity of wave propagation depends upon the ratio between MECHANICS OF MOLECULES. 361 the arc and its sine. If the distance A r,, between the molecules, in the direction of r,, have any appreciable value as compared with the wave length X,, this ratio will be less than unity; and in proportion as the wave length increases, in the same medium, will the velocity increase. When the distance A r is insignificant in comparison with the wave length?a, the ratio of the sine to the arc will be unity, and that factor will cease to appear. ~ 316.-If the medium be homogeneous, then will p'= p"= p"';. = ry = -ArAr; and, therefore, V, =v,, = v,. That is, the velocity will be the same in all directions. Denote this velocity by V; we may write 7r.A r sin "= r H...... (538) 7T. Ar2 in which the two factors that compose the second member have such average values as to give a product equal to the sum of the products which make up the second members of either of Equations (537). Supposing, in addition to the existence of homogeneity, that the interval between the molecules is insignificant in regard to the wave length, the last factor of Equations (537) reduces to unity, and taking the axis x in the direction of the velocity to be estimated, A r becomes A x, and, first of Equations (537), V2 =~ p' (A X)2; replacing p' by its value, Equations (526) and (523), v 1. d [ () Ax + (d-(r) I _ (r))*4] dr r2 r3 The distances between the molecules being very small, the term of which A x is a factor may be neglected in, comparison with that containing A x2, and the above may be written 362 ELEMENTS OF ANALYTICAL MECHANICS. 1 A x V 2 nz r Now, f (r) * is the component of the elastic force exerted betweer two molecules whose distance is r, in the direction of the axis:; and f (r).-. A x is the quantity of work of this component acting through a distance A x. Making zif(r). - - 2 ei we may, by the principle of parallel forces, write Zf(r)..Ax = 2e, x,; in which e, is the sum of the component molecular forces which act on one side of the molecule m, in the direction of the axis x, or, which is the same thing, the elastic force limited to a single molecule; and x, the path over which this force would perform an amount of work equal to that measured by the first member. Substituting this above, V2 _ e, x, Denote by i the number of molecules in a unit of length, and multiply both numerator and denominator by iV; we have m2 _. e,. x, but i2.e, is the elastic force extended to a unit of surface, and is the measure of the elastic force of the medium; call this e. The factor mx, is the number of molecules in the distance x,; call this k. The denominator i3m is the quantity of matter in a unit of volume, which is the density; call this A, and the above becomes A*.. (539) Denote by c the ratio which the contraction produced in a given vol. MECHIANICS OF MOLECULES. 363 ume of the medium by the pressure of a standard atmosphere A, bears to:the volume without any external pressure; then will A g D 30 0", g=-=.30..(540) ~ C in which g is the force of gravity and D,, the density of mercury at a standard temperature. In the case of gases, c is sensibly equal to unity; for if such bodies were relieved from their atmospheric pressures they would expand indefinitely, thus making their increments of volumes sensibly equal to the volumes they would ultimately attain. RELATION OF WAVE VELOCITY TO WAVE LENGTH. 317. —Denote the resultant displacement, of which I, qr, and; are the components, by a; and the angles which a' makes with the axes x, y, and z, by a, I, and y, respectively; then will =d'. cos a; ='. cos 3; ='. cosy; which, substituted in the second members of Equations (531), give d2 _ tj = -. -os P,,...... (C41) d ta d t —-- ~'. nz. COS 7. Squaring, adding, taking square root, and denoting the resultant by E,m we have 62 (d)a (d2 2o,) d2 a2 (n4 ca 4y COS2 P C no 4. Cos2 y). (542) The first member is the square of the resultant acceleration due to the molecular action developed by the displacement ~. Denote by oc,,,, and y, the angles which the direction of this '364 ELEMENTS OF ANALYTICAL MECHANICS. resultant makes with the axes. x, y, and z, respectively; and by b the inclination of this direction to that of displacement. Then will cos = cos a. cos a, + cos p cos 3, + cos y. cos y,. (543) The components of the acceleration, in the directions of the axes x, y, and z, are, respectively, E cosa,; E C os; e Cos 5y,; and, therefore, Equations (541), sm cos 1 =- - ~n. n. cs a, em cOS 3, -= - y2. OS, e,, cos = --. n,a. COs y. Whence, o - cr.n.2.cosa a COS g, = _. Co0 13.cos - (544) COS -, n'. coss, These, in Equation (54i3), give, Equations (531), cm nC b. - a. (n= -. cos' a + ~. cos' + n. cos); and replacing n, n,, ny, and n., by their values, Equations (529), V a V2 V2 V2 va: = - *cos2 a~ Cos2 + COS y. But, because the number of waves, in a unit of time, arising from the components of a common initial disturbance must be the same, the coefficients of the circular functions above must be equal, and hence, V2 V2 V2 a _'a (cos2 a + cs2+os3 y)- 2 (545) Whence the wave velocity is proportional to the wave length. MECHANICS OF MOLECULES. 365 SURFACE OF IELASTICITY. ~ 318.-Replacing, in Equations (541), n,, ny, n,, by their values in Equations (529), multiplying the first by c. r it. nm, the second by c. r'2. m, jnd the third by c. Tr 2. em, we have C.. t,.m. d t- ~c.4 X3. m 2. cos a, c.r.2. d =. c. 43. m VY2. cos, (546) C'TrX 2. CM. = a.c. 4rr'.m V2I. cos y. Now, r. X,.rr 12, and Xr. Z are the projections of the waves arising from the component displacements F, q/, and', on the planes yz, xz, and x y, respectively; and if every molecule in each of these waves had the same acceleration, the first members would measure the elastic forces exerted over these projections by making c equal to unity. These are, however, not equal; but if c denote a proper fractional coefficient, and,X, ey, and e, the actual elastic forces in the three waves, we may write, E =- -. ~.. Vy2. cos...... (547) 83 =-so SgjTz *tCos Y. in which = 4 c.,r'. m. Squaring, adding, taking square root of sum, and denoting the resultant by e%, X= v'E E+ y 2= +' /=. V. COS2 ~. opf V os. o+ v. osy; from which it is apparent that if the displacement be made in the direction of either axis, the elastic force will be wholly in the direction of that axis-a property possessed by these particular axes in consequence of the fact that they were assumed in directions to satisfy the conditions of symmetry in molecular arrangement, which caused Equations (524) to reduce to Equations (525). The directions of these 3,66 ELEMENTS OF ANALYTICAL MECHANICS. special axes are called axes of elasticity. The resultant elastic force will not, in general, act in the direction of the displacement. Denote the angles which se makes with the axes of elasticity by a,, B,, and y,, and the angle which it makes with the displacement by ib, then will cos = cos a. cos c, + cos. cos, + cos y. cos y,, e. COs a, = E, - a' V-~... cos a. Cos i = ey = - r. c.. s op, e.Cos y e = -... cos. Whence.. v.2. COS a cos a, - 2 COS72 COS COS, -- -. which substituted above, give, en. COS 9 = -~ d. f. Z2 (V_ -. ( V2. COS2 a+y2, OS2 a 2 1;z2. 7 OSO 3); in which V is the velocity perpendicular to the displacement. Making if= V 2= a; Vy - b; T7=c; we have V= Va2.cos2a+b.os2 3+c2. cos2y.. (549) The quantities a, b, and c are called definite axes of elasticity, in contradistinction to axes of elasticity which merely give direction. The surface of which the above is the equation, is called the surface of elasticity. The value of V will measure the velocity of any point on the wave surface in a direction normal to the displacement, and being squared and multiplied by. will give the elasticity developed in the. direction of the displacement itself. MECHANICS OF MOLECULES. 36T The definite axes of elasticity are the geometrical axes of figure of the surface of elasticity; the general axes of elasticity are directions parallel to these, and drawn from any point in the medium taken at pleasure. WAVE SURFACE. ~ 319.-This is the locus of those molecules which have, simultaneously, the same phase, ~ 309; and whatever this phase may be, the particular surface characterized by it will be concentric with that which marks, at any epoch, the exterior limits of the disturbance, or upon which the molecules are beginning to participate in the disturbance propagation. It is now the question to determine the equation of this latter surface; for this purpose, assume the origin of co-ordinates at the point of primitive disturbance, and let Ix + my + nz = V...... (550) be the equation of a plane tangent to the wave front at any point, and at the end of a unit of time. The coefficients 1, m, and n, will be the cosines of the angles which the normal to this plane makes with the axes x yz, respectively, and its length will measure the velocity V, of wave propagation in its own direction. This plane must be parallel to the displacement and its normal perpendicular thereto; hence cos a + mco ncos = 0... (551); also cos2 c + cos2j + cosy = 1.... (552). Equations (549), (550), (551), and (552) must exist simultaneously for real values of the cosines of a, 3, and y. To find an equation which shall express this condition, square Eq. (549), and divide it by Va. cos2 a, it becomes 368 ELEMENTS OF ANALYTICAL MECIIANICS. COS2 ( COs' y a2 + 62. CS + c2 s 1 - ost a + COS a. (553) V2 Cos2 a divide Eq. (551) by cos a, we have cos Cosy I + m.-+n. =0......... (554); COs a COS a and divide Eq. (552) by cos' a, the result is ~ cos7 p COs2 y 1 +3~ -. (555). COS a COS2-a COS2 a Equations (553) and (555) give a2 + 6b2 c82ol + 2. cos2cos,2 j cos y COS2 a COS2 a 1+ $- + = Cos2 a cos2 a V2 whence ~C~ - 2 cf-(COSl.. + (.V2 C2). -- o.. (556). V2 _.2 + (V2-b2). c- a = ~ COS2 a C0OS a From Equation (554) we have cos 3 I + ma. cos y COS a COS a n which in Equation (556) gives [(I2- b2) n2 +- ( V2 - c2) 2]. cos" +2_ o (V2- C2).. m. o - (Va-a2) n2-(a —C2) l, COSO a COs a or cos2 - ( y2 - c2).. n coS Co ( 2 -- a2) n7 + (V2 - c2) 1 cosa- ( V - b2) na + (V2 - C) m cos a ( V2 - 62) n2 + ( V2- c2)m2' and solving with respect to Cos thlere will result, cos 1B (V2 -c').m..':/ - [(V2 -a2)(V2 -ba2)l-2+(V2 -a2)(V2 -c2)m2+( V2 - c2)(V'-b2)12] cOS a (V-2) n2 ( Vr2 -C2) m (557); MECHANICS OF MOLECULES. 369 and this in Equation (554) gives cos y _ _(V2-b2).n.z m~ —[(V2 -au)(V'-b2) n+(V-au)(V-c2)m'-+(V'2 — )(-ba) z2] cos a ( V2 —bl).n2 + ( 2-ca) ym (558). For any assumed displacement, the value of V, Eq. (549), becomes known, and the values of the first members of Eqs. (557) and (558) nluit be real; whence 1, m, and n, must, in addition to Eq. (549), also satisfy the condition ( -a2)( i V - 2b a2' ( + - a2)( F2 - c2)j2 + ( f - b)( V2 - C2)12 = 0. Dividing by (VI a') (V2 - b) (V' - C), and inverting the order of the terms, 12 i2 n 2 _ b2 + 0._ -.. (559) From this equation, together with Equation (550), and the relation 12+m2+nff = 1...... (560) we have all the conditions necessary to find the equation of the wave surface; this is done by eliminating F, m, 1, and n. For this purpose, differentiate each of these equations with respect to the quantities to be eliminated. We have, from Equation (550), (1)...... xdl + ydm zdn- d V; from Equation (560), (2)..... l + mdm + ndn 0; and from Equation (559), ldl mdm ndnr -/2 1d 2 (3) V2-b c= V d V 2-' +' VI-b)2"- -- +(' 172a V l( 2- a (Vb0)2 (Vc2)t Multiply the first by A, the second by - -', the third by - 1, and add members to members, and collect the coefficients of like differentials; there will result, 24 370 ELEMENTS OF ANALYTICAL MECHANICS. V2 a-2 )dl i +- _ - — _ -+ (V A)d TJ Taking X and X' of such values as to make the coefficients of d V and d n each zero, the equation will reduce to the first two terms; and as Jdm and dl are wholly arbitrary, Equation (560), as long as dn is undetermined, we may, from the principle of indeterminate coefficients, write, (4)..... - X —.- ~-x -, (5)......;. y-'mm-v - b = 0, (6)...... z -Xn - (7). Azi - v + - 7 + ( lo (V2 _ a2)2 (V2 _ b2)2 (V2 _ C2)2 MIulttply (4) by 1, (5) by m, (6) by n, add and reduce by Equations,(550), (560), and (559); we have (s).... Multiply (4) by x, (5) by y, and (6) by z; add and reduce by Equa.,tion (550) and the relation x' + y2 + z' = r2; we have lx my r n Xr _ -- _' V +V- L _ b- a V =o C2 0; substituting for X' its value, (8), and transposing,. Ix my nz (9). 0 ~' —- Vt 2 + V2 =VV. C fV 27 MECHANICS OF MOLECULES. 371 transposing in (4), (5), and (6), squaring and adding, we have 2(V2 = 2 ( V2 )+)( C)2(V substituting for A'" its value, (8), and reducing by (7), we have A'.- 2) = y;) and, therefore, (10).A... 2= - r-V; x Substituting these in (4), we find V(r'2 V') r 72 V2. 2 whence?-_- a_ e V2-a 2; similarly, _y _ Vm sr_,b2- 72 _ b2; z n r2 -- Va - c"; multiply the first by x, the second by y, the third by z, add and re duce by (9) and (10); we have | t — -- 1..... (561} _,a2 b2 2 2 2+ a 2 _ b2+ r2 c2 From this, which is one form of the equation of the wave surface, subtract x2 + y2 q- z2 and we have a' x b2 y2 CZ2 Q ~x + 2~ y2 _ b =O.,.. (562) ~ - co+ r2 - b' r" cwhich is a second form of the equation of the wave surface. Clearing the fractions, it becomes, after substituting for rt its value x2 + y2 + zI, 372 ELEMENTS OF ANALYTICAL MECHANICS,. (,' + y' +'2) (a2 + 62 2 + e z2) 2 _ a,2 (ba 3+ C) x - b2 (a + e) y, =~=0... (563) - c2 (a2 + b2) z2 + aa b2 C2 DOUBLE WAVE VELO-CITY. ~ 320. —The radius vector r measures the velocity of the point of the wave to which it belongs; and denoting by /,, m,, and n, the cosines of the angles which r makes with x, y, and z, respectively, we have x = r. 1,; y = rm,; z = rn,; and writing V. for r, we have, by substituting in Equation (563), and dividing by V.4. a2. b6. c, [(1 1) +I 1) (1 l l1 1,2, 2 ZII4 [(L12) 2+( )+ m2+ ( + 3i * c2~ 22c2 a2 b2 = 0, (564) a trinomial equation, of which the second powers of the equal roots are 2 A" + 2 [A.x A"1 /-' Al "2], (565) and in which, 1 1. /1 1 4 = ac b2~ A'=1 a + n, b;. (566) 1b _ __ V a2 A"=, a;.... (567) V c2 a ca aa If a > b > c, the values of A' and A" will be real, and there will, in general, be two real values for and with this condition, Equation (565) will give two pairs of real and equal roots with contrary signs. MECHANICS OF MOLECULES. 373 The positive roots give two velocities in any one direction, and the negative in a direction contrary to this. Through the origin, conceive two lines to be drawn, making with the axis a, angles whose cosines are a, and a,,; with the axis b, angles whose cosines are f3 and 3,,; with the axis c, angles whose cosines are yA and 7,,; and such that 1 1 1 1 av = /~, = ~; P, = 3 -,, = o; 7 = c, b; (568) \/ c a_ V c2 a2 and denote the angle which r makes with the first of these- lines by u,, and that which it makes with the second by u,,; then will A' = 1, a, - n,. 7, = cos u,, A" = 1, a, -n, y = cos u,,. ~/1- A2" = sin u,; V/1A"2 = sin u,,. These, in Equation (565), give for the two values of -i, 3 (1 ) (2 (cos os,, + sin it,. sinu,..(569) -~ =, ) (sO ~ C -in,. sin u,). (50) + - 2 (COS -. Cos.. and by subtraction, 2 2 = C-).sin,. sin u,. Now1 1 1 - and - are the retardations of wave velocity. As long as a and c differ, the second member can only reduce to zero, when u, or u,, is zero; whence it appears that, as a general rule, every direction except two is distin 374 ELEMENTS OF ANALYTICAL MECHANICS. guished by transmitting two waves, one in advance of the other. The two directions which form the exceptions are in the plane of the axes of greatest and least elasticity, and make with these axes the angles of which the cosines are a, and y,, a,, and y,,, Equations (568). In these directions the waves will travel with equal velocities. Any direction along which the component waves travel with equal velocities is called an axis of equal wave velocity. All bodies in which the elasticities in three rectangular directions differ, possess, Equation (571), two of these axes, and are called biaxial bodies. "The retardation of one component wave over that of the other, will vary with the inclination of the -direction of its motion to the axis of equal wave velocity; and Equation (571) shows that the loci of equal retardations will be arranged in the form of spherical lemniscates about the poles of the axes. ~ 321.-The form of the wave surface and its properties become better known from its principal sections and singular points. Its sections by the planes yz, xz, and xy give, respectively, x=; (yt + z'- a') (b y + 2 z' - c) o, y=-; (z, +,-be) (d ze +.ax-c a) = o, (572) 0=; (X'+ y_ )(..+'cx- V 2 - b2) If a be greater, and c less than b, then will the first give a circle and an ellipse, the latter lying wholly within the former; the third will give the same kind of curves, but the ellipse will wholly envelop the circle; the second will give the same kind of curves, ifitersecting one another in four points. This last is the most important. It is' the section parallel to the axes of greatest and least elasticities. ~ 322.-If b = c, then, Equations (568), a, =1; y, =O; MECHANICS OF MOLECULES. 375 the axes will coincide with one another and with the axis a, that is, with x; U, will equal U,, and, Equation (571), V V2 v~ Vd-( ~-X)*sin u..... (573) Also, Equation (563), (x2 + y' + z2- c) [a2 x2 + C' (yg + z') - a c] =O.. (574) and the wave surface will be resolved into the surface of a sphere, and that of an ellipsoid of revolution. Making u, = 0, it will be seen from Equation (571) that these waves travel with equal velocities in the direction of the axis a. For any other value for u, since u, =,, cos u, cos Z,,- +sin u, sin u,, = 1, Equations (469) and (570) become Tr: a = cw; a =Tz c- (1 a - s) *snu,;..(575) and it hence appears, that the velocity of one of the component waves will be constant throughout its entire extent, while that of the other will be variable from one point to another. The first is called the ordinary, the second the extra-ordinary wave. If c be greater than a, then will the ellipsoid be prolate; if less than a, it will be oblate. There is but one direction which will make VI -= -V,2, and that is coincident with the axis a. Bodies in which this is true have but one axis of equal wave velocity, and are called Uniaxial bodies. From Equation (571) it appears that the loci of equal retardations are concentric surfaces, of which the common axis is on the axis of equal wave velocity, and common vertex at the origin. UMBILIC POINTS. ~ 323.-Let L = 0 represent Equation (563), and take 1 dl 1 dL I dL cco B=-.; cosC -- (576) w dz w dy w dx in which A, B, and C are the angles which a tangent plane to the sur 376 ELEMENTS OF ANALYTICAL MECHANICS. face makes with the co-ordinate planes x y, x z, and y z, respectively, and, 1 1 W(dL)+ \( d Ly.. (5'7) Performing the operation here indicated on Equation (563), we have dL' = 2z (a x2 + b y2 + c2 2) + 22 z (x2 + y +z z2 - b'), dz d L d= 2 y (a + x' + ba y2 + c2 z2) + 2 bV y (x2 + y + _ 2 - c); ay d L d =2 z (a b y- + ) + 2 a X (z + yl + -- b2 _ C2). Making y = 0, brings the tangential point in the plane a c, and the above become dL dz= 2 z (a' 2 + c2' ) + 2 cz (x + z2 - b), dL d=,.. (578) dy dL d = 2x(a' +c z) + 2 a (x' +z' - vb- ). the second of which shows the tangent plane to be normal to the plane a c. But y = 0 gives, Equations (572), x' +z' - b2 = 0; a22 x+ c2 - a' c = 0, whence we have 2 _ C2 z=b [ -.S * *(579) a2 - c2 for the co-ordinates of the points in which the circle and ellipse inter MECHANICS OF MOLECULES. 377 sect, and which are real as long as a > b > c. Substituting these in Equations (576), (577), and (578), we have cosA-=; cosB=-; cos C-=; 0 0' hence the points of intersection of the ellipse and circle in the plane of the axis a c, are the vertices of conoidal cusps, each having a tangent coney If a line be drawn tangent both to the ellipse and the circle in the plane ac, the tangential points will belong to the circumference of a circle along which a plane through this line may be drawn tangent to the wave surface. This circumference is in fact the margin of the conoidal or umbilic cusp, determined by the surface of the tangent cone reaching its limit by becoming a plane in the gradual increase of the inclination of its elements, as the tangential circumference recedes from the cusp point. A narrow annular plane wave, starting from this circle, will contract to a point in one direction; and, conversely, an element of a plane wave starting in the opposite direction will expand into a ring. It thus appears that the general wave surface, and of which (563) is the equation, consists of two nappes, the one wholly within the other, except at four points, where they unite, and at each of which they form a double umbilic, somewhat after the manner of the opposite nappes of a very obtuse cone. The figure represents a model of the wave surface, so cut, by three rectangular planes, as to show two of the umbilic points, as well as the general course of the nappes, by the removal of a pair of the resulting diedral quadrantal fragments, MOLECULAR VELOCITY. ~ 324.-Multiply the first of Equations (531) by 2 d the second by 2 d y, the third by 2 d S, and integrate; there will result, recollecting that the molecule is moved from its place of rest, 378 ELEMENTS OF ANALYTICAL MECHANICS. d 1= 2 *2 d t7 d 1= _ (580) d 2 - 2. whence it appears that the velocity of a molecule in the direction of either axis is proportional to its displacement in that direction, from its place of rest. The place of rest is only relative. When a molecule is in a position such that its neighbors are symmetrically disposed around it, it is in its place of rest, and its displacement therefrom will be directly proportional to the excess of condensation on one side over that on the other. This excess and the molecule's motion will reduce to zero simultaneously, and a single displacement, not repeated, can only give rise to what is called a pulse. These equations also show that the living force of the molecule is proportional to the square of the displacement. MOLECULAR ORBITS. ~ 325.-The molecular orbits are on the wave front. Suppose the wave due to the displacement g to be superposed upon that due to q7, and take a molecule of which the place of rest is on the axis z. The first and second of Equations (528), will be sufficient to find the orbit of this molecule under the simultaneous action of both waves. From these two equations we find, after writing z for ro and rp, (1)'.. l, (V,. t- z) = sin, (2).,. (V t- z) in ().., (Tr,. t-,Z) = CoS I - V~, all MECHANICS OF MOLECULES. 379 Subtracting (2) from (1), _.( t__-)z'v _.* 2rX Vi t _) sin -- sin - in which V.. t - z, is the distance of the wave front due to X from the molecule's place of rest, and V.. t - z, that of the wave front due to v7 from the same point. Make t, = time required for the wave front due to 5 to travel over V. t - z; ty = " V,.t - z; ty ~ " " " " " 2ym;t-Z; then will V.t- -z v,..t-z _t ty {t *.- ty t. - t ) ix t 3Vy - r r7. TOy T which substituted above gives, after taking cosine of both members, cos2 rr = 41- 1. aX a.' a.cat Clearing the radical and reducing, - — 2 cos 2'..-sin22'-A =.. (582),. ay T. a, yY2 7 which is the equation of an ellipse referred to its centre. ~ 326.-To find the position of the transverse axis, take the usual formulas for the transformation of co-ordinates from one set, which are rectangular, to another, also rectangular. They are, - ='cos q? — o' sin p, r1 = ~' sin q +- 7' cos p; in which qp is the angle which the axis 5' makes with that of 5. Substituting these values of f and q7 in Equaticx (582), collecting 380 ELEMENTS OF ANALYTICAL MECHANICS. the coefficients, and placing that of the rectangle I' y', equal to zero, we have 2 sin p. cos p (a.' - as) - 2 (sin' p - Cos2 )) a. a. cos 2 rr = 0; and because sin2 - cos2 p = cos 2 p, 2 sin p. cos p = sin 2 p, the above becomes, (a,. a, t tan2 q —2. = 2.cos2 c.'.... (583) C-c 7! ~ 327.-Now, if the successive pairs of component waves which disturb the molecule, reach it with a variable difference of phase, then t will cos 2 7r- be variable,, and the transverse axis of the elliptical orbit be continually shifting its place. A wave in which the molecular motions fulfil this condition is called a common wave; being far the most frequent in nature. When the successive pairs of component waves are such as to make the second member of Equation (583) constant, the transverse axes of the molecular orbits will retain the same direction, and the wave is said to be elliptically polarized. ~ 328.-If -'- equal I, or any odd multiple of 1, and a.= a., then will, Equation (582), 2 + -,2 = 0,.... (584) and the orbit becomes a circle. When this happens, the wave is said to be circularly polarized. ~ 329.-If t' be equal to any even multiple of {, then will tT t 4 cos 2 r.-; sin 2 rr.- -0; 7 T and, Equation (582), -— =1 0.1 i.,. (585) a, ay MECHANICS OF MOLECULES. 381 and the orbit is a straight line through the molecule's place of rest. The motion of the molecule will take place in a plane normal to the wave front, and the wave is said to be plane polarized; and a plane normal to the wave front and in the molecular paths, is called the plane of polarization. ~ 330. —Referring the curve to the new axes, and omitting the accents from F' and y', Equation (58'2) may be written, -+ sin2 2 rr. - = 0,.... (586) In which a, and ac will take new values. REFLEXION AND REFRACTION OF WAVES. ~ 331.-The elastic force which the molecules in the surface of one body exert upon those in the surface of another, in sensible contact, must, when the molecules are at relative rest, be equal to that exerted by the molecules in the interior of either body; else these surface molecules would be urged in opposite directions by unequal forces, and relative repose would be impossible. But, for equal displacements, the elastic forces developed in different bodies are in general unequal, and this is one of the most common of the causes that produce a resolution of primitive into secondary or component waves. The velocity of a wave molecule varies, Equations (580), directly as the molecule's distance from its place of rest. If, therefore, a wave, in its progress through any medium, meet with a constitutional change of elasticity or density, the elastic force developed at the place of change will either be greater or less than that which determined the places of rest in the interior of either body. In the first case, the condensation in front cannot, by the forward movement, reduce to an equality with that behind; the surface molecules will first be checked, and then partly driven back upon those behind, and a return and an onward pulse will proceed in opposite directions from the surface which marks the change of structure, as from a primitive disturbance. In the second case, the molecules, meeting with less opposition, will go beyond their neutral 382 ELEMENTS OF ANALYTICAL MECHANICS. limits with reference to those behind, the latter will close up in sue cession, and thus a- return and transmitted pulse will arise as before, but with this difference, viz.: in the latter case, the molecular motions in the return pulse will continue in the same direction as before, whereas, in the former case, those motions will be reversed. The return pulse is said to be reflected; that transmitted, refracted. The primitive pulse, and of which these are the components, is called the incident pulse. A change of density or of elasticity will, Equation (537),'produce a change in the velocity of wave propagation. A surface which is the locus of a change of density or of elasticity, is called a deviating surface. Two planes which are tangent, the one to the deviating surface, the other to the wave front, at a point common to both, will intersect in a line parallel to that of the nodes of the molecular orbits, which are in the deviating surface and near the common tangential point. This line of intersection is called the line of nodes. A plane through the tangential point and perpendicular to the line of nodes, is called the plane of incidence. The medium through which the wave moves before it meets the deviating surface, is called the medium of incidence; that into which it enters on passing this surface, the medium of intromittance. ~ 332.-Let A be a point common both to the wave and deviating surface. A C a lincar element of the former, and A B a like element of the latter, both lying in the plane - of incidence. Denote by V and X the velocity and length of the wave in the medium of incidence; by V, and X, the same in that of intromittance; and by t the time. Now, supposing the wave to proceed in the direction CB, and taking A B = d s, we have CB = V. d t. But while the poifit C, in the incident wave front, is moving from C to B, the reflected pulse, proceeding from A as a centre of disturbance, will move over a distance equal to V d t in the medium of incidence; the refracted pulse over a distance equal to V,. d t in that of MECHANICS1 OF MOLECULES. 383 intromittance. With A as a centre, and radius V. d t, describe the arc a c, and with the radius V, d t, the arc a' c'; and from B draw the tangents B D and B D'; the first will be the fiont of the new wave element in the medium of incidence, the second in that of intromittance. ~ 333.-Denote the angle CA B = A B D by p; the angle A B D' by p'; then will ds.sinp=Vdt; ds.sin'= V, d t.... (587) and by division, denoting the ratio of the velocities by ni, sill. V ='.. *..... (5)88) sin' -(588) whence sin m sin'....... (589) The angle p measures the inclination of the incident, and a' that of the refracted wave to the deviating surface. These are equal, respectively, to the angles which the normals to the incident and refracted waves make with the normal to the deviating surface, at the point of incidence. The first is called the angle of incidence, the second the angle of refraction. The inclination of the reflected wave to the deviating surface, is called the angle of reflexion. The normals to the incident and reflected waves fall on opposite sides of the normal to the deviating surface; and because the velocity of the reflected wave is equal to that of the incident, with contrary sign, Equation (589) becomes applicable to the reflected wave, by making m —— 1. LIVING FORCE AND QUANTITY OF MOTION IN A PLANE POLARIZED WAVE. ~ 334.-Take either of Equations (528), say the first, and which relates to a wave plane polarized, the plane of polarization being perpendicular to the co-ordinate plane yz, differentiate with respect to g and t, dropping the subscripts-we get dt =a. Cos-(Vt-r)2 Denote the density of the medium by A, and the area of any portion of the wave-front by a, then will the mass between two consecutive positions of this area be a. A. d r, and the living force within a quarter of a wave-length be rd i A d 2 Vt-rO 2, 2 r 2H V (590) 38 ELEMENTS OF ANALYTICAL MECHANICS. Dividing by the volume a.V, and recalling that r and - are constant, we shall find that the quantity of living force in a unit of volume of the medium will vary directly as the product of the density and square of the greatest displacement; and the relation of these products, in the case of any two waves, will determine the relation of the effects of these waves upon the organs of sense upon which they act. Again, the quantity of motion in this quarter of wave-length will be +A a. d r.- Vt a. a; V. Cos - (Yt - r)- dr a. a.a. V. (591) r+ y d t AVt-r-1 ) A RESOLUTION OF LIVING FORCE AND OF MOTION, BY DEVIATING SURFACES. ~ 335.-Take the co-ordinate plane xz in the plane of incidence, and the axis z in the direction of the normal to the incident wave, the axis y will be parallel to the line of the nodes of the molecular orbit in the deviating surface, at the place of incidence. Then, preserving the notation of ~ 332, will the element of the deviating surface at the place of incidence be d s. d y, and its projections upon the incident, reflected and refracted wave-fronts, respectively, be d s. d y. cos, d s. d y cos p, and d s. d y. cos q'. These will take the place of a in Equations (590) and (591), in computing the living force and quantity of motion in the incident, reflected and refracted waves. The living force in the incident must be equal to the sum of the living forces of its reflected and refracted components. First take the wave in which the molecular motions are parallel to the axis x, and employ the subscripts i, rand t to denote the incident, reflected and refracted or transmitted waves, respectively. The living force in a quarter of each of these waves will, omitting the common factors, Equations (529), (545) and (590), give A. Cos q. V. a2+,.cosp'. a2-. c.Cos p.V.a2, = 0; or, Equations (588) and (589), a, cos p sin. c o ip...*99) A cos q sinp' 2 9 in which a and a, are the densities of the medium of incidence and of intromittance. The molecular motions are all parallel to the plane of incidence, and at the same time normal to the directions of their respective wave motions; MECHANICS OF MOLECULES. 385 they, therefore, make with one another angles equal to those made by the directions of these latter motions, and we obtain two more equations from the relations of Equations (59) for the resolution and composition of oblique forces. The angles made by the direction of the motion in the incident with the directions of the motions in the, reflected and refracted waves, are 180~ - 2p and 360~ - (p - y'), respectively; and the angles under which the directions of the motions in the latter waves are inclined to one another, is 180~ - (Pq+-'). Whence A. cos. V. a — A. cos p. V.a. sin (p - q')' sin 2 q) a, cos q)'. a.ax -'. COS A.. cosSin ( + ); sin (p + (pi') or, sin (p - cpq') (93 -r =- " (sin ( + )....593) xA cosQ sinQ, sin 2 (594 t -A, cos', sin p' sin (9 + 9q') Substituting these in Equation (592), we readily find, A 4 cos' qp'. sin' p' cosg Ap'. sin' p' Awhn sin 22 cos q9. sinq;' whence, sin2p 2 / cos.sin (p55) 2 cos q'.sin q. cos qp'. sin qp' Substituting the above ratio of the densities in the equation just preceding, we get 2 cos qp'. sin'. a sin (p+p') ( ) multiplying this by Equation (595), member by member, and the equation giving the value of Ca, by V/A, and taking _A-agi= 1; 7tazr=V; l;.AZt=) U we find sin (p-q') = -- +.... (597) sin 2 9. Sin ( + ( p'). To which may be added the relations, Equation (589), isin__. os P - - 1 sin' sing = A Cpos?=-25 25 386 ELEMENTS OF ANALYTICAL MECHANICS. Transposing the term of which a, is a factor to the second member in Equation (592), subtracting Equation (593) from ac,, = a,, dividing the first result by the second, and multiplying the quotient by Equation (593)', we readily find aZi+ a,, 25t h 9 9) cos q) cos q9 That is, the projection in the direction of wave propagation and on the deviating surface, of the greatest displacement in the incident, increased by that in the reflected wave, is equal to like projection of the greatest displacement in the refracted wave. Next, take the wave in which the molecular motions are parallel to the axis y; these are parallel to the deviating surface. The motions in the incident, reflected and refracted waves are parallel to one another, and, by the principles of parallel forces, the sum of the motions in the reflected and refracted waves must be equal to that in the incident. The equation obr the living force will be the same as before. Whence Equations (529), (545) and (590), omitting the common factors, A. cos p. V. a2,, -,. cos q'. V,.,,- a. cos q. V. ayi - 0; A. ces g.. ay, + A,. cos a. V,. ay - a. cos q. V.,- i =. (600) In -which A and A, are, as before, the densities of the medium of incidence and of intromittance, respectively; or, Equations (588) and (589), 2 +a, sinP c' cos p' 2 2 r!, a -a Sn y a sinp cos p - A' sin q' cosp' acy.+- +ir A oL'r',o... (601) Transposing the terms containing a,, and ay to the second members, and dividing the first by the second, we find ay, + %yi-ay......... (602) That is, the greatest displacement in the refracted is equal to the sum of the greatest displacements in the incident and reflected waves. Substituting the value of A, as given by Equation (597), in Equation (601), we have sin'.'cos I*' + sinp'.cosp''' 0... (603) singp'. Cos p1 MECHANICS OF MOLECULES. 387 Substituting in this, first the value of ay, and then of c,, deduced from Equation (602), we readily get tan (q~ - q')( -a,.;...... (604) tan (p + q') 4 cos qp'. sin qp' t"sin s.....np. (605) sin 2 q + sin 2'' Multiplying the first of these by ~/, and the second by Equation (595), and inaking ---- V a v. V'; V=. - yu'; there will result, tan (P - P') V- - tail (99 + q).'.... (606) sin 2 cp Sin (p + p').) C... (6o07 ~ 336.- Divide Equation (598) by Equation (597), and Equation (607) by Equation (606), replace v, u, v' and u' by their values, and substitute fbr the ratio of the square roots of the densities its value as given in Equation (595), we find az55s. Cos cosos' sin 2 p' a,,. cos qp cosp sin(qP -- qp')', COi sin 2 (p' a- - ii (~ -'). cos (P +')'' ( ) But a,. cos q' and a,. cos are the components parallel to the deviating surface of the displacements which are in the plane of incidence; ca, and a%, are already parallel to the deviating surface; whence, as long as >,p', that is, as long as the velocity of wave-motion in the medium of incidence exceeds that in the medium of intromittance, the molecular phases in the refracted and reflected waves will be opposite, and conversely. ~ 337.-Denote the living force in the original incident wave, supposed common, by unity; that in each of its two original components will be denoted by one half of unity, and the total living force of the reflected wave will, Equations (597), (606), be -2 + V I j tan,( ). (609) sin' (9 -- Y') tan' (q - 9') ~ + ~'~- ~'~i~ (~ +') + ~2'tan2 (p +'p). (609) and that of the refracted,. — s ( tan' (- -- ~ (610) sinl' tan2 i (9 + (')! 3iS8 ELEMENTS OF ANALYT CAL MECHANICSo POLARIZATION BY REFLEXION AND REFRACTION. ~ 338.-The first term in the second member of Equation (609), measures the living force in that portion of the reflected wave which is due to vibrations parallel to the plane of incidence; the second, that duae to vibrations perpendicular to this plane. The former exceeds the }atter. These living forces being proportional to the squares of the greatest displacements, the former may be represented by a,2, and the latter by a,, in Equation (582). The factor l, in this equation, de, termines the difference of phase simultaneously impressed by both waves upon the same molecule, and when the waves have passed fioom one medium to another, its value will depend not only upon tie nature of' both media, but also upon the action to which the waves may have been subjected while arossing the space wherein the physical changes occur that constitute the transition from one medium to another. The amount of this action, in any particular case, can only be known from experience. The resultant waves, both in the medium of incidence and of intromittance, will be elliptically polarized. When p + -' = 90~, then,'Equation (589), will sin p' = cos, and sin q.p. m — tanu; b (61i1) cos Qp the second, term of Equation (609) will disappear, and the reflected wave will be wholly polarized in the plane of incidence. This angle, of which the tangent is equal to the index of refraction, is called the polarizing angle. The index of refraction varies with the wave Iength, Eqs. (588), (545), and it will, therefore, be impossible wholly to polarize, by a single reflexion, a wave compounded of several components, having different wave lengths. Of the terms of the second member of Equation (610), the last is the greater, because sin' (p- p') tan' (, -') cos' (, -'). sin2 (p + p') -tan2 (, + q) cos' ( + p')' MECHEANICS OF 3MOLECULES. 389 and the excess will measure the preponderance of that part of the refracted wave due to vibrations perpendicular over that due to vibrations paralle to the plane of incidence. This excess is exactly equal to the excess in tile reflected wave which arises from vibrations parallel over those perpendicular to the plane of incidence. ~ 338'.-If the wave velocity in the medium of incidence be less than in that of intromittance, then will Gmi be less than unity, and the values of.v and v' become imaginary for all angles of incidence greater than that whose sine is equal to nm, and at this limit the problem changes its nature. In fact, this is the limit of refraction, according to the law of the sines, Equation (589), and for.any increase of the angle of incidence beyond this, the wave will be wholly reflected. ~ 339.-If the wave be plane polarized, and its plane of polarization inclined to that of incidence, under any ange denoted by a, then will the reflected component displacements parallel and perpendicular to the plane of incidence be, respectively, Equations (597) and'(06), sin (I - if) tan (9 - V') -cos a and.sin a. sin (I + 3') tan (c + a )n The component waves due to these displacements will proceed onwards, and may satisfy the condition of being an even multiple of A; in which case the resultant will, Equation (585), be a plane polarized wave. Denote the inclination of its plane of polarization' to that of reflexion by a', then will tan (p -- q') sin a VI tan (op +=- ) = s (P + q)I) V sin (cp - ) cos (P-9 a') - ac a sin ( + p-') Uf Q + P' = 90, then will a' = -0d whatever be a; also if a 0%o then will a' = 00; finally, if q = 0~, then will cp' = 0, and a' = a. That is, when a plane polarized wave is incident under the polarizing angle, it is reflected polarized in the plane of reflexion. Where an incident wave is polarized in the plane of incidence, the reflected wave 390 ELEMENTS OF ANALYTICAL MECHANICS. preserves its plane of polarization unchanged under all angles of incidence. Finally, under a perpendicular incidence, the plane of polarization of the incident and that of the reflected wave coincide. Equation (612) shows that a' is always less than a, and that the plane of polarization approaches that of incidence at each, reflexion, and may be made, by a sufficient number of reflexions, ultimately to coineide with it. ~ 340.-Still, supposing the velocity of the wave less in the medium of incidence than in that of intromittance, or q' > q, let the wave be plane polarized, and its plane of polarization inclined to that of incidence. The vibrations will be resolved into their components, respectively parallel and perpendicular to this latter plane; and as long as sin q < m, two components will be reflected and two refracted. If' be any even multiple of ~, in both sets of components, the reflected and intromitted resultant waves will be plane polarized. The inclinations, denoted by a' and a,, of the planes of polarization of the reflected and refracted waves, respectively, to the plane of;ncidence, will be given by V U tan a'=-. tan a; tan a, =-.tan a; in which v, v', u and u', are to be found by Equations (597), (606, (598), and (607). If a = 45~ and sin p = -n, then wilI tan a' = -1, and tan a, - At this limit, the refracted wave takes the direction of the deviating surface. An infinitesimal increment to p will cause this wave to be regected and make = - I, tan a = - 1, and give to tan a' the form of indetermination. But, retaining the limiting value of this function above, we have, 1 + tan a'. tan a, = 1 - 1 - 0; MECHANICS OF MOLECULES. 391. and since the planes of polarization pass through the same line, viz., a normal to the wave front, they will make with one another an angle of 90~, and the whole reflected wave will be compounded of two equal components polarized in planes at right angles to each other. If these waves reach the molecules in their common path, so as to satisfy the condition that t shall be an even multiple of 4, the resultant wave will be plane polarized; if an odd multiple, then circularly polarized; and if between these limits, then elliptically polarized. ~ 341.-If the polarization be. circular, then will a, = a, = a,, be equal to the radius vector of the circular orbit. Denote the angle which this radius makes with the axis x, at any instant, by 0; then will v~.t-z a. cos 0 = a=,c. sin 2 r - a,. sin 0 = j = a,. sin 2 r V. t Denote the time, required for the first wave to describe VF,. t - z, by t., that for the second to describe Vr,. t- z by t;, and the periodic time of a molecule in both waves by r; then, because the wave velocity is constant, and the wave length and orbit are described in the same time, V. t- z t V.- ty which, in the above, give, cos 0 = sill 2 v.-, sin 0 = sin 2 r.-; and making ty = to +',........ (613) in which t' denotes the time the wave due to vibrations parallel to 392 ELEMENTS OF ANALYTICAL MECHANICS. one axis is in advance of that due to th)se pxrallel to the other; we have, cos 0 = sin 2 -;... (614) sin =sin 2 tr + )... (615) Differentiating, regarding- as constant, we find, d_ 2 7r.. tf cos 2 osr -4d t —'. cos 0 \oT 7 and, developing the last factor, d t, -r. CoT s T TI 4! and making -- =, dO 2r t7 cos 0. - -=: —.sin 2r...... (616) Differentiating (614), we find, sin 0.. cos 2 r..... (617) Squaring, adding to the square of Equation (616), and taking square root, dO (618) d 4 7' whence the velocity is constant. The first member of Equation (616) is the velocity in the direction of the axis y, and Equation (617) in the direction of the axis x, and these equations show that the upper sign must be taken in Equation (618) when t' is positive in Equation (613), and the lower when t' is negative. Whence it appears, that two waves plane polarized will, by their simultaneous action upon a molecule, cause it to move uniformly in a circle, provided they be of the same length, and one wave lag as it were, behind the other, bv a distance equal, to { of a wave MECHANICS OF MOLECULES. 393 length; and the motion will be from right to left, or the converse, according to wave precedence. Two waves distinguished by these peculiarities are said to be oppositely polarized. The plane perpendicular to the wave front, and through that diameter of the orbit into which the molecule would be brought at the same instant by the separate action of the two waves, is called the plane of crossing. ~ 342.-Let (1)..... ad, cos 0 = a = a, sin 2r, (2)..s.., sin= a, sin(2r. ), (3)......,cos - = a, sin (2 r. + (4)..... a, sin 0 =- a, sin 2 r, 7. be the displacements in two oppositely circularly polarized waves. The union of (1) and (4) gives a resultant wave plane polarized; that of (2) and (3) also a wave plane polarized, the equation of the path being in the plane of crossing. It thus appears that the union of two circularly polarized waves, polarized in opposite directions, gives a plane polarized wave, of which the intensity is double of either. Conversely, a wave plane polarized may be resolved into two components of equal intensity, circularly polarized in opposite directions. ~ 343.-Because the time of describing the wave length is equal to the molecular periodic timre, we have, denoting the velocity of wave propagation by V, A -V7, whence T 394 ELEMENTS OF ANALYTICAL MECHANICS. which, in Equation (618), gives, after multip ying by t, and dividing by 2 7r, dO d t, Vt,...... (19) 27 A The first member is the are, expressed in circumferences, described by the molecule while the wave is moving through a thickness V. t, of the medium. So that a wave, compounded of many components having different wave lengths, but all polarized, on entering a medium, may emerge with the planes of polarization of its several components so twisted through different angles as to diverge from a common line perpendicular to the wave front. The department of optics furnishes some fine examples of this. A piece of quartz, of a peculiar kind, is known to twist the extreme red wave through an angle of 17~ 29' 47", and the extreme violet, 440 04' 58", for each millimetre of thickness DIFFUSION AND DECAY OF LIVING FORCE. ~ 344. —The living force of any molecule whose mass is m and velocity v,, is and denoting by n the number of molecules on a superficial unit of the wave front,' the living force on this unit will be n. in., 2; and on the surface of a sphere of which the radius is rj, 4 r. r2. n. m v; and for another sphere, of which the radius is r,,, and molecular velocity v,,, 4 7'. r,2. n m v,,2. If these spherical surfaces occupy the same relative positions in a diverging wave, in any two of its positions, their molecular living forces must be equal; whence, suppressing the common factors, r,'.?m v, =.,,2 n v,,2'...... (620) MECHANICS DF MOLECULES. 395 The molecules describe elliptical crbits, and under the,,action of molecular forces directed to the centres of these curves. The periodic time will, therefore, ~ 207, Equation (286), be constant, however the dillmensions of these orbits may vary; and the average velocities of the molecules will be proportional to the lengths of their respective orbits, or, in similar orbits, to any homologous dimensions of the same-as their transverse axes or greatest molecular displacements. Denoting the latter by c' and cl' in the two waves, then will V, c' V2 C which, with Equation (620), gives c" l,-',...(621) Whence it appears, that the living force of the molecules of any wave varies inversely as the second, and the greatest displacement inversely as the first power of the distance to which the wave has been propagated from its place of primitive disturbance..INTERFERENCE. ~ 345. —Resuming Equation (586), viz., -- sin2 2r- - 0; 5~ I denote the radius vector of the molecular orbit by p', and the angle it makes with the axis of k by 0', then will -= p'. cos O'; p'. sin 0'; which, in the above, give Vpa= c -. + a sin 2 r.-; Vao-2 cos2 0' q+ a2 sin2 0' 7 and making V/a,2. cos2 O' + a2. sin' O' 396 ELEMENTS OF ANALYTICAL MECHANICS. we have t p' = c'. sin 2r.-...... (622) In this equation, p' is the actual displacement of the molecule from its place of rest, and becomes a maximum when - is any odd multiple of t 4. If, however, there be added to the arc 2,r -, an arbitrary arc a', this latter may be so taken as to make the maximum or any other displacement occur at such time and place as we please, and, therefore, to give to.the molecule any particular phase at pleasure, at the time t. We may write, then, generally, p =c'.sin (2 r.-+a');... (623) and for a second resultant wave, p" c". sin 2 r. - + a");.... (624) and if these waves act simultaneously upon the same molecules, the resultant displacement, denoted by p, will, ~ 306, be given by p =p' + p" ='.sin (2 r. - a' )+c". sin (2 r. - a). Developing the circular functions artd collecting the coefficients of like factors, t t p = (c'cos a'+ c" cos a"). sin 2 r- -+ (:' sin a' - c" sin a"). cos 2 r-; and making c cos a = c'. cos a' cos a",) c sin a c' sin a' + c" sin a", we have t t p = c. cos a. sin 2 r.- - ~ c sin a. cos 2 r.-;' TT MECHANICS OF MOLECULES. 397 or, t a). p = c sin (2 rr.-+a). (626) Squaring Equations (625), and adding, C2 = C'2 + C-"2 + 2 c' c" cos (a' - a"),.. (627) and dividing the second by the first, C'. sin a' + c'". sin a" tan a =. (628) tan cos a' + c" cos a"' From Equation (626) we see that the resultant wave is of the same length as that of the component waves to which Equations (623) and (624) appertain; the length being determined by the molecular periodic time r; but the value of a in that equation differing from a' and a" in Equations (623) and (624), shows that the maximum displacement of a given molecule does not take place in the resultant wave at the same time as in either of its components. ~ 346.-The maximum displacement in the resultant wave is given by C = V/C2 + C'"2 + 2 c'c". cos (a'- a");..'(629) which will be the greatest possible when a' - a" = 0, and least possible when a' - a" = 180~; the maximum in the former case being given by c = c' + c" t + and the minimum, by C - C'-C". A B In the first case, Equation (628), (c' +- c"). sin a' tanaa-, tan a'. (c' + c'). cos at Whence a = a' = a", and the maximum displacement will occur at the same place and time in the resultant and component waves. 39S ELEMENTS OF ANALfTTICAL MECHANICS. In the second case, Equation (628), if we make a' = 180~ + a", tan a - -"). sin a" tan a (' c") tan a" = tan (a' - 180~) = tan a'; (C' - c"). COS a" that is, a will be equal to one at least of the arcs a' and a", and the greatest displacement will occur at the same time and place in the resultant wave as in one of its components. If c' = c" then, Equation (629), C= C' 2 [1 + cos a' -a")]; and because 1 + cos (a' - a") = 2 cos2 2 c = 2 c'.C. 2.(630) and, Equation (628), sin a' + sin a" a' + a" tan a = -= tan (631) cosa'+cos a" 2 If, while c' and c" continue equal, we also have a' - a" = 180~, then, Equation (630), c = 0. Thus it appears th'at two equal waves may reach the same molecules in such relative condition as to keep them in their places of rest; in other words, two equal waves may destroy one another. ~ 347.-To ascertain the precise relation of two waves which will cause this mutual destruction, make, in Equation (623), a' = a - 7r = a 2' 27.~ and that equation becomes, p= c' sin (2 w- + ai 2- ) p= t " Si 2 2 a );.... (632) p'- c'.sil 27r. 2 - +; (632) MECHANICS OF MOLECULES. 399 which becomes identical with Equation (624) by making C = C, and t = ti:-....... (633) Now, the same value for t, in Equations (623) and (624), will, for equal values of the arbitrary arcs a' and a", determine the cc'nponent waves to give to a molecule subjected to their simultaneous action, similar phases; and a value for t, in the one, which differs from that in the other, by one-half, or any odd multiple of one-half, of the molecular periodic time, opposite phases. And, because the waves progress by a wave length during each molecular revolution, the above result shows that, when two waves meet, after having travelled over routes, estimated from points at' which their molecular phases are similar, and which routes differ by half, or any odd multiple of half a wave length, they will destroy one another, provided the waves have the same length and equal maximum molecular displacements. This act, by which one wave destroys another, is called wave interference. The same process of combination will equally apply to three or more wave functions in which ir is the same in all; that is, wherein the t t wave lengths are the same; for, in that case, sin 2 r.- and cos 2 7r. - being common factors, after developing each functitn in the sum, the resultant displacement p becomes, t t p-sin 2 Tr.-.Z c' cos a'- cos 2 rr.-.I c' sin a', and assuming c.cos a _ I c' cos a', c. sin a = X c' sin a'; p=c.sin(2-7r+a),.. (634) thus making the resultant wave of the same length as that of either of its components. 4(X0 - ELEMENTS OF ANALYTICAL MECHIANICS. But, if the component waves be not. of equal lengths, the sum of the corresponding functions cannot reduce to the form of Equation (634), because of the absence of common. /, —- -- factors, arising from a change in the value -- of r from one component to another. Such components can never destroy one another.'INFLEXION. ~ 348. —Make, in Equation (621), r" -1, and that equation becomes C" and this'value being substituted for c', in Equation (622), gives, C. t pi -.sin 2r.-, r 1 and, making t Vt —r, we have, omitting all the accents, c Vt-r p.sin 2.. (635) which is of the same form as Equations (528), and in which V is the velocity of wave propagation; t, the time'of its motion from primitive disturbance; Z, the wave length; c, the maximum displacement of a molecule of which the distance of the place of, rest from the point of primitive disturbance is r; and p the actual displacement, at the time t, of this same molecule. And from which it is apparent that the displacements will always be the same for equal distances, Vt- r, behind the wave front. Every disturbance of a-molecule, at one time, becomes a cause of MECHANICS OF MOLECULES'. 401 disturbance to. another molecule at some subsequent time. All the molecules in a wave front, when they first begin to move, become, therefore, centres of disturbance for every molecule in advance; and if the primitive disturbance be kept up, secondary waves proceeding from these centres will reach a molecule in advance simultaneously, and determine, ~ 307, at any instant t, its displacement I p. Suppose a wave, whose centre of disturbance is C, to A have reached the position AB,..... so remote from C that a small portion, A B, may be regarded as sensibly plane: What is the displacement of a molecule at 0, produced by the simultaneous action of the secondary waves proceeding from the molecules in any portion, as A B, of a section of this wave front? Draw the normal CD N, through the middle of P Q; denote the variable distance D Q by z, and Q 0 by r. The displacement of the molecule 0, by the secondary waves from the are AB = 2 b, will, Eq. (635), be given by + b S+ bcdz snV t - r E p-f pdz= sin 2z,. s X - (636) Here r and z are variable. To eliminate the former; join 0 with the middle of AB by the line D 0, and denote its length by 1, and the angle Q D O, which it makes with the wave front, by 0. Then will r = V2/+ z2 - 2lzcos0O; and by Maclaurin's formula, sin2 0. r I -cos0.. z —&c. (637)i If the greatest value of z be small as compared to i, we may take r =- - cos0...... (6388) and regard the, displacements of the molecule 0, by the partial waves 26 402 ELEMENTS OF ANALYTICAL MECHANICS. from z to be equal. Whence, substituting the value of r, with this restriction, in Equation (636), we have, -t+b c fb' 2 n zp d = p.J sin (Vt- +cosO. z) dz, and, performing the integration without regard to limits, 1P =-2 1 co0 cos;X (Vt -tI + cos 0. z), and between the limits - b and + b, ~P= [ c.. c os c (Vt —-b cos O)- cos (Vt — +b. cos0), or, c 2rr.b.cosO Vt - I - Sill. sin2.;.... (639) r. I. cos 0 so that the function whose value gives the resultant displacement, is of the same form as that of the function which determines either of the partial displacements. The maximum value of the resultant displacement is given by c 2rr.b.cosO 2p=r. I cos' sin;... (640) and this will become zero for such values of 0 as make b. cos 0 equal to either of the following values, viz., 2T A 2 2 2 Conceiving the figure to be revolved about the normal CN, and all the wave except the circular portion whose diameter is 2 b =- A B, to be intercepted, the space in advance of the wave will, when the above values obtain, find itself divided by the secondary waves into a series of concentric cone-like zones around the normal CNJ, as an axis, and of which the alternate ones, beginning with that immediately about the axis, will be filled with molecules in motion, wrlile the molecules in the MECHANICS OF MOLECULES. 403 otheis will be at rest. A section in advance of the primitive wave will cut from these zones a series of concentric circular rings distinguisled by the same peculiarities. But if A be very great as compared with b, then will the are 2 vr. b. cos 0 be so small as to justify the substitution of the are for its sine and for the maximum value of resultant displacement, c / 2r.b. cos0 2cb P)..cos =;''' (641) and this result being independent of 0, the conic zones cannot exist, and the effect of the secondary waves will be diffused in all directions to the front. This lateral action of secondary waves proceeding fromn a small portion of a primitive wave, is called Wave inflection. When 0 approaches nearly to 90~, cos 0 will be exceedingly small, and the arc 2 r. b. oss may again be substituted for its sine; again Equation (641) suits the case, and determines the maxiimum displacement immediately about the normal. rThe maximum of the masximn displacements will occur when, in Equation (640), 2 r. b.ecos 0 sin. ~ = i 1; and which would reduce that equation to rr/. I..cos 0 and as the living forces are proportional to the squares of the greatest displacements, we have rn2 * 2 4 C2 b C2X2 a v,2. m -. v -,2.' s2:"' 404: ELEMENTS OF ANALYTICAL MECHANICS. Whence m 2, -m(2 42) MI. V,, m.v, 4 Yr b. cos (4 in which v, is the velocity of the molecule on the normal, and v,, that at the angular distance 0 from it. When the waves are very short, as compared with b, it is obvious that the living force of the molecules would be sensibly nothing, except immediately about the normal. When the waves are long, as compared with b, the living force will be appreciable far every value of 0, and, therefore, in every direction in front of the primitive wave. The importance of this discussion will be apparent in the subjects of sound and light. PART IV. APPLICATION OF THE PRECEDING PRINCIPLES TO SIMPLE MACHINES, PUMPS, ETC. ~ 349. —Any device by which the action of a force may be received at one place and transmitted to another is called a Machine. There are usually seven elementary machines discussed jn Zfechanics; viz., the Cord, Lever, Inclined Plane, Pulley, Screw, Wheel and Axle, and Wedge. The Cord, Lever, and Inclined Plane are called Simple Machines; the others, being combinations of these, are called Compound Machines. ~ 350.-In Machines, as in all other bodies; every action is accompanied by an equal and contrary reaction. A force which acts upon a Machine to impress or preserve motion is called a Power. A force which reacts to prevent or destroy motion, is called a Resistance. The Agent which is the source of power, is, ~38, callede a Motor. ~ 351. —Pesuming Equation (30), and supposing the displacement, which in that equation was wholly arbitrary, to conform in every respect to that caused by the powers and resistances, we shall have s = ds, s being the path described by the elementary mass m; and hence, s P6p - I. di. ds. 0; but d2 s ds a6 s dt2 d- dt d V f whence, z P p - m.d (v) = 0.... (643) 406 ELEMENTS OF ANALYTICAL MECHANICS. Denoting by Q, Q', &c. the resistances, by /B, P', &c. the pow ers, a q, &c. and &p, &c. the projections of their respective virtual velocities; the first term, which embraces all the forces except inertia in action on the machine, may be replaced by Y Pp - Q 6q, and we have zPp -- I QSq - Zm.dv2.. (644) Integrating, fpap -f Qaq = 2mvZ + C; and denoting by v, the initial velocity, and taking the integral se as to vanish when t - 0,.jP.p-. IQq —a zm _,2.. (645) The products P ap and Q 6 q are the elementary quantities of work performed by a power and a resistance respectively, in the element of time, dt; the product Imdv2 is the elementary quantity of work performed by the inertia, or one half the incre ment of living force of the mass m in this time. And Equation (645) shows that in any machine, in motion, the increment of the half sum of the living forces of all its parts is always equal te the excess of the work of the powers or motors over that of the resistances ~352.-If the machine start from rest, Equation (645) becomes S ap Qa -7- 1 Inz, * * (646) and as the second member is essentially positive, the work of the motors must exceed that of the resistances embraced in the term Q Qsq; in other words, the inertia will oppose the motor and act as a resistance. When the motion becomes uniform, the second member will be constant; from that instant inertia will cease to act, and the subsequent work of the motor will be equal to that of the resistances as long as this motion continues. If the motion be now retarded, the second member will decrease, the inertia will act with the power, and this will continue till the Mnichine conmes APPLICATIONS. 407 to rest, aad the excess (f work of the Resistance during retardation will be exactly equal toothat of the Power during acceleration. Generally, then, when a machine is at rest or is moving uniformly, inertia does not act; when the motion is variable, it does, and opposes or aids the motor according as the motion is accelerated or retarded. ~353.-The essential parts of every machine are those which receive i'xectly the action of the motor, those which act directly upon the body to be moved or transformed, and those which serve to transmit the action. The arrangement of the latter is often a source of resistance, arising from Friction, Adhesion, Stiffness of Cordage, &c., whose work enters largely into the general term Q J q. FRICTION. ~ 354.-When two bodies are pressed together, experience shows that a certain effort is always required to cause one to roll or slide along the other. This arises almost entirely from the inequalities in the surfaces of contact interlocking with each other, thus rendering it necessary, when motion takes place, either to break them off, compress them, or force the bodies to separate far enough to allow them to pass each other. This cause of resistance to motion is called friction, of which we distinguish two kinds, according as it accompanies a sliding or rolling motion. The first is denominated sliding, and the second rolling friction. They are governed by the same laws; the former is much greater in amount than the latter under given circumstances, and being of more importance in machines, will principally occupy our attention. The intensity of friction, in any given case, is measured by the force exerted in the direction of the surface of contact, which will place the bodies in a condition to resist, during a change of state, in respect to motion or rest, only by their inertia. 3:55. —The friction between two bodies may be measured directly by means of the spring balance. For this purpose, let the surface 408 ELEMENTS OF ANALYTICAL MECHANICS. CD of one of the bodies M be made perfectly level, so that the other body M', when laid upon it, may press with its entire weight. To some point, as E, of the body M', attach a cord with a C/ //// E0/p spring balance in the manner indicated in the figure, and apply to the latter a force F of such intensity as to produce in the body M' a uniform motion. The motion being uniform, the accelerating and retarding forces must be equal and'contrary; that is to say, the friction must be equal and contrary to the force F. of which the intensity is indicated by the balance. The experiments on friction which seem most entitled to conf: dence are those performed at Metz by M. Morin, under the order of the French government, in the years 1831, 1832, and 1833. They were made by the aid of a contrivance, first suggested by M. Poncelet, which is one of the most beautiful and valuable contributions that theory has ever made to practical mechanics. Its details are given in a work by M. Morin, entitled ".Touvelles Experiences.sur le Frottement." Paris, 1833. The following conclusions have been drawn from these experiments, viz.: The friction of two surfaces which have been for a considerable time in contact and at rest is not only different in amount, but also in nature, from the friction of surfaces in continuous motion; especially in this, that the friction of quiescence is subjected to causes of variation and uncertainty from which the friction during motion is exempt. This variation does not appear to depend upon the extent of the surface of contact; for, with different pressures, the ratio of the friction to the pressure varied greatly, although the surfaces of contact were the same. The slightest jar or shock, producing the most imperceptible movement of the surfaces of contact, causes the friction of quiescence to pass to that which accompanies motion. As every machine may be regarded as being subject to slight shocks, producing imper APPLICATIONS. 409 ceptible motions in the surfaces of contact, the kind of friction to be employed in all questions of equilibrium, as well as of motions of machines, should obviously be this last mentioned, or that which accompanies continuous motion. The LAWS of friction which accompanies continuous motion are remarkably uniform and de'finite. These laws are: 1st. Friction accompanying continuous motion of two surfaces, between which no unguent is interposed, bears a constant proportion to the force by which those surfaces are pressed together, whatever be the intensity of the force. 2d. Friction is wholly independent of the extent of the surfaces in contact. 3d. Where zunguents are interposed, a distinction is to be made between the case in which the surfaces are simply unctuous and in intimate contact with each other, and that in which the surfaces are wholly separated from one another by an interposed stratum of the unguent. The friction in these two cases is not the same in amount under the same pressure, although the law of the independence of extent of surface obtains in each. When the pressure is increased sufficiently to press out the unguent so as to bring the unctuous surfaces in contact, the latter of these cases passes into the first; and this fact nmay give rise to an apparent exception to the law of the independence of the extent of surface, since a diminution of the surface of contact may so concentrate a given pressure as to remove the unguent from between the surfaces. The exception is, however, but apparent, and occurs at the passage from one of the cases abovenamed to the other. To this extent, the law of independence of the extent of surface is, therefore, to be received with restriction. There are, then, three conditions in respect to friction, under which the surfaces of bodies in contact may be considered to exist, viz.: 1st, that in which no unguent is present; 2d, that in which the surfaces are simply unctuous; 3d, that in which there is an interposed stratum of the unguent. Throughout each of these states the friction which accompanies motion is always proportional to the pressure, but for the same pressure in each, very different in amount. 410 E LEMENTS OF ANALYTICAL MECHANICS 4th. The friction which accompanies motion is always independ ent of the velocity with which the bodies move; and this, whether the surfaces be without unguents or lubricated with water, oils, grease, glutinous liquids, syrups, pitch, &c., &c. The variety of the circumstances under which these laws obtain, and the accuracy with which the phenomena of motion accord with them, may be inferred from a single example taken from the first set of Morin's experiments upon the friction of surfaces qf oak, whose fibres were parallel to the direction of the motion. The surfaces of contact were made to vary in extent from 1 to 84; the forces which pressed them together from 88 to 2205 pounds; and the velocities from the slowest perceptible motion to 9,8 feet a second, causing them to be at one time accelerated, at another uniformrr, and at another retarded; yet, throughout all this wide range of variation, in no instance did the ratio of the friction to the pressure differ from its mean value of 0,478 by mnore than Jof this same fraction. Denote the constant ratio of the entire friction F, to the normal pressure P, by f; then will the first law of friction be expressed by the following equation, F..f;.. (647) whence, F = f. P. This constant ratio f is called the co-efficisnt of friction, because, when multiplied by the total normal pressure, the product gives the entire friction. Assuming the first law of friction, the co-efficient of friction may easily be obtained by means or the B inclined plane. Let W denote the weight of any body placed upon \ the inclined plane A B. Resolve this weight G G' into two components, one GA M perpendicular to ":: the plane, and the other G N par APPLICATIONS. 411 allel to it. Because the angles G' GM and BA C are equal, the first of these comporents will be GM= W.cosA, and the second, GN= W.sinA, in which A denotes the angle BA C. The first of these components determines the total pressure upon the plane, and the friction due to this pressure will be _ = f. W cos A. The second component urges the body to move down the plane. If the inclination of the plane be gradually increased till the body move with uniform motion, the total friction and this component must be equal and opposed; hence, f.. cosA = W. sinA; whence, sin A - A = tan A. We, therefore, conclude, that the unit or co-efficient of friction between any two surfaces, is equal to the tangent of the angle which one of the surfaces must make with the horizon in order that the other may slide over it with a uniform motion, the body to which the moving surface belongs being acted upon by its own weight alone. This angle is called the angle of friction or limiting angle of resistance. The values of the unit of friction and of the limiting angles for many of the various substances employed in the art of construction, are given in Tables VI, VII and VIII. The distinction between the friction of surfaces to which no un guent is applied, those which are merely unctuous, and those between which a uniform stratum of the unguent is interposed, appears first to have been remarked by Mi. Morin; it has suggested to him what appears to be the true explanation of the difference between his results and those of Coulomb. He conceives, that in the ex 412 ELEMENTS OF ANALYTICAL MECHANICS. periments of this celebrated Engineer, the requisite precautions had not been taken to exclude unguents from the surfaces of contact. The slightest unctuosity, such as might present itself accidentally, unless expressly guarded against-such, for instance, as might have been left by the hands of the workman who had given the, last polish to the surfaces of contact-is sufficient materially to affect the co-efficient of friction. Thus, for instance, surfaces of oak having been rubbed with hard dry soap, and then thoroughly wiped, so as to show no traces whatever of the unguent, were found by its presence to have lost 2ds of their friction, the co-efficient having passed from 0,478 to 0,164. This effect of the unguent upon the friction of the surfaces may be traced to the fact, that their motion upon one another without unguents was always found to be attended by a wearing of both the surfaces; small particles of a dark color continually separated from them, which it was found from time to time necessary to remove, and which manifestly influenced the friction: now, with the presence of an unguent the formation of these particles, and the consequent wear of the surfaces, completely ceased. Instead of a new surface of contact being continually presented by the wear, the same surface remained, receiving by the motion continually a more perfect polish. A comparison of the results enumerated in Table VIII, leads to the following remarkable conclusion, easily fixing itself in the memory, that with the unguents, hogs' lard and olive oil interposed in a continuous stratum between them, surfaces of wood on metal, wood on wood, metal on wood, and metal on metal, when in motion, have all of them very nearly the same co-efficient of friction, the value of that co-efficient being in all cases included between 0,07 and 0,08, and the limiting angle of resistance therefore between 40 and 40 35'. For the unguent tallow the co-efficient is the same as the above in every case, except in that of metals upon metals; this unguent seems less suited to metallic surfaces than the others, and gives for the mean value of its co-efficient 0,10, and for its limiting angle of re. sistance 50 43'. APPLICATIONS. 413 356.-Besides friction, there is another cause of resistance to the motion of bodies when moving over one another. The same forces which hold the elements of bodies together, also tend to keep the bodies themselves together, when brought into sensible contact. The effort by which two bodies are thus united, is called the force of Adhesion. Familiar illustrations of the existence of this force are furnished by the pertinacity with which sealing-wax, wafers, ink, chalk and black-lead cleave to paper, dust to articles of dress, paint to the surface of'wood, whitewash to the walls of buildings, and the like. The intensity of this force, arising as it does from the affinity of the elements of matter for each other, must vary with the number of attracting elements, and therefore with the extent of the sur. face of contact. This law is best verified, and the actual amount of adhesion between different substances determined, by means of a delicate spring-balance. For this purpose, the surfaces of solids are reduced to polished planes, and pressed together to exclude the air, and the efforts necessary to separate them noted by means of this instrument. The experiment being often repeated with the same substances,.laving different extent of surfaces in contact, it is found that the effort necessary to produce the separation divided by the area of the surface gives a constant ratio. Thus, let S denote the area2 of the surfaces of contact expressed in square feet, square -inches, or any other superficial unit; A the effort required to separate them, and a the constant ratio in question, then will A = a, or, A = a.S. The constant a is called the unit or co-efficient of adhesion, and ob. 414 ELEMENTS OF ANALYTICAL MECHANICS. viously expresses the value of adhesion on each unit of surface, for making S= 1, we have A =a. To find the adhesion between solids and liquids, suspend the solid from the balance, with its polished surface downward and in a hori. zontal position; note the weight of the solid, then bring it in contact with the horizontal surface of the fluid and note the indication of the balance when the separation takes place, on drawing the balance up; the difference between this indication and that of the weight will' give the adhesion; and this divided by the extent of surface, will give, as before, the co-efficient a. But in this experiment two opposite conditions must be carefully noted, else the cohesion of the elements of the liquid __ for each other may be mistaken for the adhesion of the solid for the fluid. If the solid on being removed take with it a layer of the fluid; in other words, if the solid has been wet by the fluid, then the attraction of the elements of the solid for those of the liquid is stronger than that of the elements of the liquid for each other, and a will be the unit of adhesion of two surfaces of the fluid. If, on the contrary, the solid on leaving the fluid be perfectly dry, the elements of the fluid will attract each other more powerfully than they will those of the solid, and a will denote the unit of adhesion of the solid for the liquid. It is easy to multiply instances of this diversity in the action of solids and fluids upon each other. A drop of water or spirits of wine, placed upon a wooden table or piece of glass, loses its globular form and spreads itself over the surface of the solid; a drop of mercury will not do so. Immerse the finger in water, it becomes wet; in quicksilver, it remains dry. A tallow candle, or a feather APPLICATIONS. 415 from any species of water-fowl, remains dry thoe gh dipped in water. Gold, silver, tin, lead, &c., become moist on being immersed in quicksilver, but iron and platinum do not. Quicksilver when poured into a gauze bag will not run through; water will: place the gauze conitaining the quicksilver in contact with water, and the metal will also flow through. It is difficult to ascertain the precise value of the force of adhe sion between the rubbing surfaces of machinery, apart from that of friction.'But this is attended with little practical inconvenience, as long as a machine is in motion. The experiments of which the results are given in Tables VI, VII and VIII, and which are applicable to machinery, were made under considerable pressures, such as those with which the parts of the larger machines are accustomed to move upon one another. Under such pressures, the adhesion of upguents to the surfaces of contact, and the opposition to motion presented by their viscosity, are causes whose influence may be safely disre garded as compared with that of friction. In. the cases of lighter machinery, however, such as watches, clocks, and the like, these considerations rise into importance, and cannot be neglected. STIFFNESS OF CORDAGE.' 357.-Conceive a wheel turning freely about an axle or trunnion, and having in its circumference a groove to receive a cord or rope. A weight WV, being suspended from one end of the rope, while a force 1, is applied to the other extremity to draw it up, the latter will experience a resistance in consequence of the rigidity of the rope, which opposes every effort to bend it around the wheel. This resistance must, of necessity, consume a portion of the work of the force F. The measure of the resistance due to the rigidity of cordage has been made the 416 ELEMENTS OF ANALYTICAL MECHANICS. subject of experiment by Coulomb; and, according to him, it results that for the same cord and same wheel, this measure is composed of two parts, of which one remains constant, while the other varies with the weight W, and is directly proportional to it; so that, designating the constant part by If, and the ratio of the variable part to the weight vf by I, the measure will be given by the expression K+I. Tf; in which K represents the stiffness arising from the natural torsion or tension of the threads, and I the stiffness of the same cord due to a tension resulting from one unit of weight; for, making IV = 1, the above becomes K+ I. Coulomb also found that on changing the wheel, the. stiffness varied in the inverse ratio of its diameter; so that if K+ I. W be the measure of the stiffness for a wheel of one foot diameter, then will K + I. W 2R be the measure when the wheel has a diameter of 2 R. A table giving the values of K'and I for all ropes and cords employed in practice, when wound around a wheel of one foot diameter, and subjected to a tension arising from a unit of weight, would, therefore, enable us to find the stiffness answering to any other wheel and weight whatever. But as, it would be impossible to anticipate all the different sizes of ropes used under the various circumstances of practice, Coulomb also ascertained the law which connects the stiffness with the diameter of the cross-section of the rope. To express this law in all cases, he found it necessary to distinguish, 1st, new white rope, either dry or moist; 2d, white ropes partly worn, either dry or moist; 3d, tarred ropes; 4th, packthread. The stiffness of the first class he found nearly proportional to the square of the diameter of the cross-section; that APPLICATIONS. 417 of the second, to the square root of the cube of this diameter, nearly; that of the third, to the number of yarns in the rope; and that of the fourth, to the diameter of the cross-section, So that, if S denote the resistance due to the stiffness of any given rope; d the ratio of its diameter to that of the table; and n the ratio of the number of yarns in any tarred rope to that of the table, we shall have for,New white rope, dry or moist. K~+I. W S= d2... (648) Half worn white rope, dry or moist. S- d2.,... (649) Tarred rope. K+ I. S =.. n,...... (650) Packthread.:d eI. 2k S d. +. W..... (651) For packthread, it will always be sufficient to use the tabular values given, corresponding to the least tabular diameters, and substitute them in Equation (651). An example or two will be sufficient to illustrate the use of these tables. Example 1st. Required the resistance due to the stiffness of a new dry white rope, whose diameter is 1,18 inches, when loaded with a weight of 882 pounds, and wound about a wheel 1,64 feet in diameter. Seek in No. 1, Table X, the diameter nearest that of the, given rope; it is 0,79; hence, t 1,18 _ 1,5 nearly; 0,79 and from the table at the side, d2 = 2,25. From No. 1, opposite 0,79, we find K - 1,6097, I = 0,03195; 27 418 ELEMENTS OF ANALYTICAL MECHANICS. Pt. which, together with the weight W = 882 lbs., and 2 R = 1,64, substituted in Equation (648), give lb. lb. 1, 6097 + 0,03195 X 882. lbs. S- 2,25. 164 = 40,817, which is the true resistance due to the stiffness of the rope in question. Example 2d. What is the resistance due to the stiffness of a white rope, half worn and moistened with water, having a diameter equal to 1,97 inches, wound about a wheel 0,82 of a foot in diameter, and loaded with a weight of 2205 pounds? The tabular diameter in No. 4, Table X, next less than 1,97, is 1,57, and hence, I.97 d = 1,3 nearly; 1,57 the square root of the cube of which is, by the table at the side, 3 d2 = 1,482. In No. 4 we find, opposite 1,57, K = 6,4324, I = 0,06387; ft. which values, together with W = 2205 lbs., and 2 R = 0,82, in Equation (649), give lbs. lbs. 6;4324 -+- 0,06387 x 2205 lbs. S=1,482 082 =266 109, 0,82 which is the required resistance. Example 3d. What is the resistance due to the stiffness of a tarred rope of 22 yarns, when subjected to the action of a weight equal to 4212 pounds, and wound about a wheel 1,3 feet diameter, the weight of one runnirg foot of the rope being about 0,6 of a pound? By referring to No. 5, Table X, we find the tabular number of yarns next less than 22 to be 15, and hence, n = =5 - 1,466 nearly. 15 APPLICATIONS 419 In the same table, opposite 15, we find K_ = 0,7664, I -0,019879; ft. N7which, together with WV = 4212, and 2 R - 13, in Equation (650), give 1,66 7664 + 0,019879 x 4212 lbs. S = 1,466' 195,188. 1,3 Example 4th. Required the resistance due to the stiffness of a new white packthread, whose diameter is 0,196 inches, when moistened or wet with water, wound about a wheel 0,5 of a foot in diameter, and loaded with a weight of 275 pounds. The lowest tabular diameter is 0,39 of an inch, and hence d = 0,196 0,5 nearly. 0,390 In No. 2, Table X, we find, opposite 0,39, lb. K 0,8048, I = 0,00798; which, with V = 275, and 2 R = 05, we find, after substituting in Equation (651), 0,8048 + 0,00798 x 275 lbs. S = 05 05 2,999. ~ 358.-The resistance just found is expressed in pounds, and is the ~t amount of weight which would be necessary to bend any given rope around a vertical wheel, so thati the portion AE, between the first point of contact A, and the point E, where the rope is attached to the weight, shall be perfectly straight. The entire process of bending takes place at this first or tangential point A; for, if motion be com 420 ELEMENTS OF ANALYTICAL MECHANICS. municated to the wheel in the direction indicated by the airow. head, the rope, supposed not to slide, will, at this point, take and retain the constant curvature of the wheel, till it passes- from the latter on the side of the power F, When, therefore, by the motion of the wheel, the point m of the rope, now at the tangential point, passes to m', the working point of the force S will have described in its own direction the distance AD. Denoting the arc described by a point at the unit's distance from the centre of the wheel by s,, and the radius of the wheel by R, we shall have AD = Rs,; and representing the quantity of work of the force S by L, we get L = S. Rs; replacing S by its value in Equations (648) to (651), L- R. K+ I. (6-2) 3 in which d, represents the quantity d2, d>, n, or d, in Equations (648) to (651), according to the nature of the rope. Example.-Taking the 2d example of ~ 357, and supposing a portion of the rope, equal to 20 feet in length, to have been brought in contact with the wheel, after the motion begins, we shall have L = 20 X 266,109 = 5322,18 units of work; that is, the quantity of work consumed by the resistance due to the stiffness of the rope, while the latter is moving over a distance of 20 feet, would be sufficient to raise a weight of 5322,18 pounds through a vertical height of one foot. FRICTION ON PIVOTS, AND TRUNNIONS. ~ 359.-A11 rotating pieces, such as wheels supported upon other pieces, give rise by their motion to friction. This is an important element in all computations relating to the performance of machinery. It seems to be different according as the rotating pieces are kept A P PLIC AT I O N S. 421 in place by trunnions or by pivots. By trunnions are meant cylindrical projections a a from the ends of the arbor AB of a wheel. The trunnions rest on the concave surfaces of cylindrical boxes CD, with which they usu- c ally have a small surface of contact m, the linear elements of both being parallel. Pivots are shaped like the trunnions, but support the weight of the wheel and its arbor upon their circular end, which rests against the bottom of cylindrical sockets F6G HI. PIVOTS. Let N denote the force, in the direction of the axis, by which the pivot is pressed against the bottom of the socket. This force may be regarded as passing through the centre of the cir. cular end of the pivot, and as the resultant of the partial pressures exerted upon all the elementary surfaces of which this AQ U circle is composed. Denote by 0 - A the area of the entire circle, then will the pressure sustained by each unit of surface be N and. the pressure on any small portion of the surface denoted by a, will obviously be a. * 422 ELEMENTS OF ANALYTICAL MECHANICS. and the friction on the same will be f a. a A This friction may be regarded as applied to the centre of the elementary surface a; it is opposed to the motion, and the direction of its action is tangent to the circle described by the centre of the element. Denote the radius of this circle by x, then will the moment of the friction be a. N f'A.X Now, if s denote the length of any variable portion of the circumference at the unit's distance from the centre C, then will a= x.ds. dx; also, A = R2; which substituted above give x2. dx. d f. 3 R... R2 and by integration, f X2 dx ds /f f * ~ - f. N. Jo R;.. - 53) whence we conclude, that, in the friction of a pivot, we may regard the whole friction due to the pressure as T acting in a single point, and at a distance from the centre of motion equal to two-thirds of the radius of the base of the pivot. This distance is called i the mean lever of friction. ~ 360.-If the extremity of the pivot, instead of rubbing upon an entire circle, 3 is only in contact with a ring or surface comprised between two concentric APPLICA T IONS. 423 circles, as when the arbor of a wheel is urged in the direction of its length by the force N against a shoulder d cb a; then will A = -r (R2 - _R2); and the integration will give xr2 dx d s3-'3 I s2 d~t~n~r/ /?P'db R3 -:R13 f. N 2 f.R. N',(R0 -- R'2) 3. R2 - _Rf2 in which B denotes the radius of the larger, and B' that of the smaller circle. Finally, denote by I the breadth of the ring, that is, the dis. tance A'A; by r, its mean radius or distance from C to a point half way between A' and A, and we shall have X r + 12 B' = r- I; substituting these values above and reducing, we have fyVX [r +~ *];. (654) and making r 1 -- r=, we obtain, for the moment of the friction on the entire ring, f. Nf. r,........ (655) The quantity r, is called the mean lever of friction for a ring. Since the whole friction fN may be considered as applied at a point 12 whose distance from the centre is RB, or r'= r + -— r according as the friction is exerted over an entire circle or over a ring, and since the path described by this point lies always in the direction in which the friction acts, the quantity of work consumed by it will be equal t the product of its intensity fN into this path. Designating the length of the are described at the unit's distance from C by s,, the path in question will be either 2Rs,, or ris,; T I d 424 ELEMENTS OF ANALYTICAL MECHANICS. and the quantity of work either R. s,.f. N for an entire circle, or f N (r+ 12r) S for a ring. Let Q denote the quantity of work consumed by fric. tion in the unit of time, and n the number of revolutions performed by the pivot in the same time; then will s, = -'2 Xn; and we shall have Q =.. R.f..... (656) for the circle, and Q = 2.f7N (rf + 12 n e.*.. (657) for a ring; in which r — = 3j1416. The co-efficient of friction f, when employed in either of the foregoing cases, must be taken from Table VI, VII, or VIII. Example. —Required the moment of the friction on a pivot of cast iron, working into a socket of brass, and which supports a weight of 1784 pounds, the diameter of the circular end of the pivot being 6 inches. Here in. ft. = = 3 = 0,25, Ibs. N = 1784, f = 0,147; which, substituted in Equation (653), gives Ibs. ft. 0,147 X 1784 x - x 0,25 = 43,708. And to obtain the quantity of work in one unit of time, say a minute, there being 20 revolutions in this unit, we make n = 20, and ir = 3,1416 in Equation (656), and find Q = 4 x 3,1416 x 0,25 x 0,147 x 1784 x 20 = 5492,80; APPLICATIONS. 425 that is to say, during each unit of time, there is a quantity of work lost which would be sufficient to raise a weight of 5492,80 pounds through a vertical distance of one foot. Example.-Required the moment of friction, when the pivot supports a weight of 2046 pounds, and works upon a shoulder whose exterior and interior diameters are respectively 6 and 4 inches; the pivot and socket being of cast iron, with water interposed. 6-4 I= - - =1 inch, r = 2 + 0,5._ 2,5 inches, (1)2 in. ft. r, = 2,5 + 12 ( 2 2,5333 = 02111, N = 2046 pounds, f= 0,314; which, substituted in Expression (655), gives for the moment of friction, lbs. ft. 0,314 X 2046 X 0,2111 = 135,62. The quantity of work consumed in one minute, there being supposed 10 revolutions in that unit, will be found by making in Equation (657), -r = 3:1416 and n- 10, Q =2 x 3,1416 x 0,314 x 2046 x 0,211 x 10 = 8517,24; that is to say, friction will, in one unit of time, consume a quantity of work which would raise 8517,24 pounds through a vertical distance of one foot. The quantity of work consumed in any given time would result from multiplying the work above found, by the time reduced to minutes. TRUNNIONS. ~361.-The friction on trunnions and axles, which we now proceed to consider, gives a considerably less co-efficient than that which accompanies the kinds of motion referred to in ~ 355. This will appear from Table IX, which is the result of careful experiment. The contact of the trunnion with its box is along a linear ele 426 ELEMENTS OF ANALYTICAL MECHANICS. ment, common to the surfaces of both. A section perpendicular to its length would cut from the trunnion and its box, two circles tan. gent to each other internally. The trunnion being acted on only by its weight, would, when at rest, give this tangential point at o, the lowest point of the section p o q of the box. If the trunnion be put in motion by the application of a force, it would turn around the point of contact and roll indefinitely along the sur- N face of the box, if the latter were level; but this not being the case, it will ascend along the inclined \ surface op to some point as m, where the inclination of the tangent u m v is such, that the friction d is just sufficient to prevent the trunnion from sliding. Here let the trunnion be in equilibrio. But the equilibrium requires that the resultant of all the forces which act, friiction included, shall'pass through the point m and be normal to the surface of the trunnion at that point. The friction is applied at the point m; hence the resultant N'of all the other forces must pass through m in some direction as m d; the friction acts in the direction of the tangent; and hence, in order that the resultant of the friction and the force N shall be normal to the surface, the tangential component of the latter must, when the other component is normal, be equal and directly opposed to the friction. Take upon the direction of the force N the distance m d to represent its intensity, and form the rectangle a d b nz, of which the side m b shall coincide with the tangent, then, denoting the angle dma by q, will the component of N perpendicular to the tan. gent be N. cos q; and the friction due to this pressure will be f. N. cos q. APPLICATIONS. 427 The component of N, in the direction of the tangent, will be N. sin pq; and as this must be equal to the friction, we have f. N. os = N. sin;.... (658) whence, f = tan qp; that is to say, the ratio of the friction to the pressure on the trunnion is equal to the tangent of the angle which the direction of the resultant N, of all the forces except the friction, makes with the normal to the surface of the trunnion at the point of contact. This gives an easy method of finding the point of contact. For this purpose, we have but to draw through the centre A a line A Z, parallel to the direction of N, and through A the line Am, making with A Z an angle of which the tan- Hi gent is f; the point m, in which this / line cuts the circular section of the / trunnion, will be the point of contact. Because m ad b, last figure, is a rectangle, we have N2 = N2 cos2 9 +- 1~2 sin2 a; and, substituting for N2 2sin2 q its equal f2 N2cos2 p, we have N2 = N2COS2 p + f2 N2 COS2 N = 1y2 cos2 9 (1 + f2); whence, Ncosp = N x and multiplying both members by f, f.. cosp= N..... (659) but the first member is the total friction; whence we conclude that to find the friction qupon a trunnion, we have but to multiply the 428 E LEMENTS OF ANALYTICAL MECIHANICS. resultant of the forces which act upon it by the unit of friction, found in Table IX, and divide this product by the square root of the square of this same unit increased by unity. This friction acting at the extremity of the radius R of the trunnion and in the direction of the tangent, its moment will be N f x R....... (660) And the path described by the point of application of the friction being denoted by Rs,, the quantity of work of the friction will be N. R. fS X +;.. (661) in which s, denotes the path described by a point at the unit's distance from the centre of the trunnion. Denoting, as in the case of the pivot, the number of revolutions performed by the trunnion in a unit of time, say a minute, by n; the quantity of work performed by friction in this time by Q.; and making -r 3,1416, we have s, — 2. n; and Q, = 2,..n. N.. (662) When the trunnion remains fixed and does not form part of the rotating body, the latter will turn about the trunnion, which now becomes an axle, having the centre of motion at A, the centre of the eye of the wheel; in this case, the lever of friction becomes the radius of the eye of the wheel. As the quantity of work consumed by friction is the greater, Equation (662), in proportion as this radius is greater, and as the radius of the eye of the wheel must be greater than that of the axle, the trunnion has the advantage, in this respect over the axle. APPLICATIONS. 429 The value of the quantity of work consumed by friction is wholly independent of the length of the trunnion or axle, and no advantage is therefore gained by making it shorter or longer. THE CORD. ~ 362. —The cord and its properties have been considered in part at ~58. It is now proposed to discuss its action under the opera. tion of forces applied to it in any manner whatever. Let the points A', All, A"', be connected with each other by means of two perfectly flexible and inextensible cords A' A", A" A"', the first point being acted upon by' the forces P', P", &c.; the j I i second by the forces Q', Q",! &c.; and the third by the i 3 forces S', S", &c.; and sup-., pose these forces to be in equilibrio. Denote the coordinates of A' by x'y' z', / A" by x" y" z", and A"' by x"' y"' z"'. Also, the algebraic sum of the components of the forces acting at A' in the direction of x yz, by X' Y' Z', at A" by X" Y"Z", and at A'" by X"' Y"' Z"'. Then will, ~ 101, X' 6xf' + Y' 6y' + Z' Sz' +- X" 6x" + Y"' y" +- Z" 6 z"' 0... (663) + X"'71 x"' + Y'" y y'" + Z'"f z"' Denote the length A' A" by f, and A" A"' by g; then will L =Z f-aV(' - x' )2 + (y" -- y )2 + (z" -z' )2 -; } =1 g - -1x'7" - X")2 + (y"' -_ y1")2 + (z"' -. ") o The displacement by which we obtain the. virtual velocities whose 430 ELEMENTS OF ANALYTICAL MECHANICS. projections are 6 x', &y', 6z', &c., is not wholly arbitrary; but must be made so as to satisfy the condition 6f= 0 and lg = 0. ~.. (665) Differentiating Equations (664), and writing for dx', d y', d z', 6x', 6y', 6z', &c., we find (x" - x')(x" - 6x') + (y" - y')(ay" - y') + (z" - z')(z" -z 6z') = 0; (x"'-x")(6x"'-6x")+(y"'-y")(6y"'-Sy")+(z"'- z")(6z"'-6z") OThese being multiplied respectively by -X' and X"', and added to Equation (663), we obtain by reduction, and by the principle of indeterminate co-efficients, exactly as in ~ 213, Z - 1 Y'-x. -Y = 0o;.. ~ ~ (666) X" + A'. XI. 0; f gf X,,,Z' ~?'.', =0; fg g'" -- y; X" f g X' ~ X" + X"' =; Y+ f "=o.; Y * *(669) zY' + X" "'= 0; Zl, +; zll- - " —f = 0; Z/ + Z" + Zf I I = O.b AFPI ICATIONS. 431 That is, the conditions of equilibrium of the forces are, ~ 80, the same as though they had been applied to a single point. To find the position of the points, eliminate the factors X' and."', and for this purpose add the first, second and third equations of group (667) to the corresponding equations of group (668), and there will result X" + X"' + X'" i -x') 0; y' + Y"' + x (y" - y') = 0; Z"- + Z"' + - (z" - z') = 0. from wlich we find by elimination, "+ Y"' - (X"+ X"') =; z" z z"-Z; *. (670) z" + z'"-?,'~ _' zII + Z",, (X"+ ")= O. j From group (666), by eliminating X', y; x y- X T... (671) XI', Xl and finally fiom group (668) we obtain, by eliminating X"', it Y,, Y. X"' =O; Z -x"''-" X" - --- xr_ xt - O;... I~. (672) Z" -'' - Z. X"' O= 0. Equations (669), (670), (671) and 672), involve all the conditions necessary to the equilibrium, and the last three groups, in connection with group (664), determine the positions of the points A', A" and A"', in space. ~363. —The reactions in the system which impose conditions on 432 ELEMENTS OF ANALYTICAL MECHANICS. the displacement will be made known by Equation (331), which, because [d (x ] [+ ] (yi ) + d (z'" -' 1; r dlI 12 r d -12 r d i" 1; Ki~ ~ ~-; l I -, _ ~,_ - _ -1 d("' )j Ld(y"' - y" (Z"'- z") becomes for the cord A' A",' -V'; and for the cord A" A"', X, = Ns; from which we conclude, that X' and X"' are respectively the tensions of the cords A' A" and A" A"'. This is also manifest from Equations (666) and (668); for, by transposing, squaring, adding and reducing by the relations, (x" - x')2 + (y" - y')2 + (Z" - z')2 f2 (X' - )2 + (y" - y")2 + (Z"' - Z"-)2 Y2 we have X"'= VX'2 + y'2 + Z"'- "', t. in which R' and R,"' are the resultants of the forces acting upon the points A' and A"' respectively. Substituting these values in Equations (666) and (668), we have.X " - -' Y' y" - y' Z' " -'' f'. — f':R- f X"' x"' -Z" Y"' yWm._ yt Z"' zi" - Z't ~" = -- g R"- g; R"' = g; whence the resultants of the forces applied at the points A' and A"', act in the directions of the cords connecting these points with the point A", and will be equal to, indeed determine the tensions of these cords. APPLICATIONS. 433 ~ 364.-From Equations (669), we have by transposition, " = - (X",', + X'); Y" = - (Y"' + yr); Z" = - (z', + Z'). Squaring, adding and denoting the resultant of the forces applied at A" by R", we have " = -/(X"' + X')2 + (Y"' + Y')2 + (Z"' + Zf').2 (674) and dividing eadh of the above equations by this one Xi/ X"' + X' R' R"; y,, y,,, + Y':R", =-; * ***(675) Z" Z"' + Z' R"=- 2"'J whence, Equation (674), the resultant of the forces applied at A" is equal and immediately opposed to the resultant of all the forces applied both at A' and A"' If, therefore, from the point A' A", distances A" m and A"n be taken proportional to B' and', B"' respectively, and a parallelogram A" m Cn be constructed, A" C will represent the value of B". If A' A"A"' be a continuous cord, and the point A" capable of sliding thereon, the tension of the cord would be,' 0 A" the same throughout, in which case B' would be equal to B"', and the direction of R" would bisect the angle A' A" A"'. / The same result is shown if, 9 instead of making &f = 0 and 6g = 0 separately, we make 28 434 ELEMENTS OF ANALYTICAL MECHANICS. (f - -) = 0, multiply by a single indeterminate quantity ), and proceed as before. ~ 365.-Had there been four points, A', A", A"' and Aiv, connected by the same means, the general equation of equilibrium would become, by calling h the distance between the points, A"' and Aiv, X' 6 x' +- X" 6x" + X' 6 x"' + Xiv 6S iv 1 + y' 6yt + yt"6 y"I + rt 6 y"' + yiv 6yiv; 0* + Z' z' Z" 6 z Z" T 6Z"' Zzi - "iV + X' Sf + X"ig + X"'6h J and from which, by substituting the values of 6f, 6 g and 6 h, the follcwing equations will result, viz.: Y' x'." - y'=o, ( f7 = o, (...) fZ' -- X Z= -o z1 z +X zX X z= O j X:' -- X Xl" -- Xi/ f X' Z >,, Z - g Z"~x y'" -- y" Y" + Xi. Y'; _ y = k. (677) f g Z" + X z' z X z. 0O, f g Y'" 4- X"'t " -- y" -' h Z"' -~ "!"V - z'_ _ x",,g'' hA 9 h APPLICATIONS. 435.XiV + "'. O p1 i_? W, we may find the true tension corresponding to any erroneous tension, as t,, by the following proportion, viz.: W,: W:: tl: t'; or, which is the same thing, multiply each of the tensions found by the constant ratio W, the product will be the true tensions, very nearly. The value of t4 thus found, substituted in Equation (721), will give that of P. Example.-Let the radii R,, R,, R, and R4, be respectively 0,26, 0,39, 0,52, 0,65 feet; the radii r, = r= = r = r4 of the eyes = 0,06 feet; the diameter of the rope, which is white and dry, 0,79 inches, of which the constant and co-efficient of rigidity are, respectively, K = 1,6097 and I- 0,0319501; and suppose the pulley of brass, and its axle of wrought iron, of which the co-efficient f = 0,09, and the resistance W a weight of 2400 pounds. Without friction and stiffness of cordage, 2400 Ms. tl - = 600. APPLICATIONS. 459 Dividing Equation (718) by R., it becomes, since d,= 1, t2 = t, J ]q - f, (tl - t+). Substituting the yalue of R,, and the above value of tl, and regarding in the last term t. as equal to t,, which we may do, because of the small co-efficient l f', we find F 600 1,6097 + 0,0319501 x 600 t2= I 2 X (0,26) = 628,39. o 0o26 x 0,09 x (600 + 600) Again, dividing Equation (719) by N2, and substituting this value of t1 and that of R,, we find Ibs. t = 673,59. Dividing Equation (720) by R,, and substituting this value (f t,, as well as that of R3, there will result Ibs. t4= 709,82; whence, 600 +- 628,39 W —- tl + t2 Jr t3 +- t4 = 673,59 =2611.~ 673,59 + 709,82 J andl W 2400 WI- 261180 = 0919; which will give for the true values of t, = 0,919 x 600 = 551,400 2 = 0,919 x 628,39 -- 577,490 t = — 0,919 x 673,59 - 619,029 t4 = 0,919 x 709,82 - 652,324 2400,243 460 ELEMENTS OF ANALYTICAL MECHANICS. The above value for t4 = 652,324, in Equation (721), will give, after dividing by R4, and substituting its numerical value, 652,324 + 1,6097 + 0,03195 x 652,324 P- - 2 x 0,65 0- 065 x 0,09 x (652,324 + P); tnd making in the last factor P = t4= 652,324, we find lbs. Ibs. lbs. lbs. P = 652,324 + 17,270 + 10,831 = 680,425. Thus, without friction or stiffness of cordage, the intensity of P would be 600 lbs.; with both of these causes of resistance, which cannot be avoided in practice, it becomes 680,425 lbs., making a difference of 80,425 lbs., or nearly one-seventh; and as the quantity of work of the power is proportional to its intensity, we see that to overcome friction and stiffness of rope, in the example before us, the motor must expend nearly a seventh more work ihan if these sources of resistance did not exist. THE WEDGE. ~ 384.-The wedge is usually employed in the operation of cut. ting, splitting, or separating. It consists of an acute right triangular prism A B C. A B The acute dihedral angle A Cb is called ".. the edge; the opposite plane face A b the back; and the planes A c and C7b, which terminate in the edge, the faces. The more common application of the wedge consists in driving it, by a blow upon its back, into any substance which we wish to split or divide into parts, in C such manner that after each advance it shall be supported against the faces of the opening till the work is accomplished. APPLI CATIONS. 461 ~ 385.-The blow by which the wedge is driven forward will be supposed perpendicular to its back, for if it were oblique, it would only tend to impart a rotary motion, and give rise to complications which it would be unprofitable to consider: and to make the case conform still further to practice, we will suppose the wedge to be isosceles. The wedge A CB being inserted in the opening a h b, and in contact with its jaws at a and b, we know that the resistance of the latter will A - B be perpendicular to the faces of the wedge. Through the points a and b draw the lines a q and bp normal to the faces A C and B C; fiom their 0 point of intersection 0 lay off the distances O q and Op equal, respectively, to the resistances at a and b. Denote the first by Q, and the second by P. Completing the parallelogram O q nmy,' 0 will represent the resultant of the resistances Q and P. Denote this resultant by B', and the angle A CB of the wedge by 8, which, in the quadrilateral a 0 b C, will be equal to the supplement of the angle a 0 b p 0 q, the angle made by the directions of Q and P. From the parallelogram cf forces, we have, B'2 P P2 J Q2+ 2P QCOSp Oq-=p2+ Q2- 2P Q cc:s 0; or, R' - v/- + Q2 - 2 P Q cos d. The resistance Q will produce a friction on the face A C equal to f Q, and the resistance P will produce on the face B C the fric. tion fP: these act in the directions of the faces of the wedge. Produce them till they nmeet in Ct and lay off the distances Cq' and Cp' to represent their intensities, and complete the parallelogram 462 ELEMENTS OF ANALYTICAL MECHANICS. Cq' O' p'; C O' will represent the resultant of the frictions. Ienote this by lR", and we have, from the parallelogram of forces, R"2 - f2 Q2 + f2 P2 + 2f2 P Q cos 8; or, R" f p2 + Q2 + 2 P Q os. The wedge being isosceles, the resistances P and Q will be equal, their directions being normal to the faces will intersect on the line CD, which bisects the angle C - 0, and their resultant will coincide with this line. In like manner the frictions will be equal, and their resultant will coincide with the same line. Making Q and P equal, we have, from the above equations, R' = P /2 (1 -cos 0), R" =fP 2(i + cs0I). But, 1 - cos = 2 sin2 0, 1 + cos = 2os2; whence we obtain, by substituting and reducing, R' = 2P. sin0 8, i" = 2 2f. P. cos I 0; and further, A B sin2 8 _ C' CD C052- AC'; therefore, I- P.' A A A C' R"= 2f P. CD A C Denote by F the intensity of the blow on the back of the wedge. If this blovt be just sufficient to produce an equilibrium bordering APPLICATIONS. 463 on motion forward, call it F'; the friction will oppose it, and we must have. AB CD F =R' + R" = P. A + 2f. P. A.. (723) If, on the contrary, the blow be just sufficient to prevert the wedge from flying back, call it F"; the friction will aid it, and we must have, AB CD F" = P -a 2f. P. AC.. (724) The wedge will not move under the action of any force whose intensity is between F' and F". Any force less than F", will allow it to fly back; any force greater than F', will drive it forward. The range through which the force may vary without producing motion, is obviously, F! - "= 4fP. AP-CD... (725) which becomes greater and greater, in proportion as CD) and A C become more nearly equal; that is to say, in proportion as the wedges becomes more and more acute. The ordinary mode of employing the wedge requires that it shall retain of itself whatever position it may be driven to. This makes it necessary that F" should be zero or negative, Eq. (724), whence AB CD A B CD P = 2f P or.PA < 2f ~ P * A-C -A C' oAPC <. AC' or, omitting the common factors and dividing both members of the equation and inequality by 2 C)D, A B AB CD =f' or < f; but A A is the tangent of the angle A C(7; hence we conclude, that the wedge will retain its place when its semi-angle does not exceed that whose tangent is the co-efficient of fricti(n between the surface of the wedge and the surface of the opening which it is intended to enlarge. 464 ELEMENTS OF ANALYTICAL MECHANICS. Resuming Eq. (724), and supposing the last term of the second member greater than the first term, F" becomes negative, and will represent the intensity of the force necessary to withdraw the wedge; which will obviously be the greatest possible when A B is the least possible. This explains why it is that nails retain with such perti nacity their places when driven into wood, &c. THE SCREW. ~ 386.-The Screw, regarded as a mechanical power, is a device by which the principles of the inclined plane are so applied as to produue considerable pressures with great steadiness and regularity of motion. To form an idea of the figure of a screw and its mode of action. conceive a right cylinder, a k, with circular base, and a rectangle, or other plane figure, a bcm, having one of its sides ab coincident with a surface element, while its plane passes through the axis of this cylinder. Next, suppose the plane of the, generatrix to rotate uniformly about the axis, and the generatrix itself to move also uniformly in the direc- " tion of that line; and let this twofold motion of rotation and of translation be so regulated, wz that in one entire revolution of the plane, the generatrix shall progress in the direction of the axis over a distance greater than the side a b, which is in the surface of the cylinder. The generatrix will thus generate a projecting and winding solid called a fillet, leaving between its turns a groove called the channel. Each point as in in the perimeter of the generatrix, will generate a curve called a helix, and it is obvious, from what has been said, that every helix will enjoy this property, viz.: any one of its points as m, being taken as an origin of reference, as well for the curve itself as for its projection on a plane through this point and at right angles to the axis, the distances d' m', d" mi", &c., of the several points of the helix from this plane, APPLiCATIONS. 465 are respectively proportioned to the circular arcs m d', md", &c., into which the portions mmn', mmn", &c., of the helix, between the origin and these points, are projected. The solid cylinder about which the fillet is wound, is called the newel of the screw; the distance m m"', between the consecutive turns of the same helix, estimated in the direction of the axis, is called the helical interval. The fillet is often generated by the motion of a triangle with one of its sides coincident with ab; and as the discussion will be more general by considering this mode of generation, we shall adopt it. The surfaces of the fillet, which are generated by the inclined faces of the triangle, are each made up of an infinite number of helices, all of which have the same interval, though the helices themselves are at different distances from the axis, and have different inclinations to that line. The inclination of the different helices to the axis of the screw, increases from the newel to the exterior surface of the fillet, the same helix preserving its inclination unchanged throughout. The screw is received into a hole in a solid piece B of metal or wood, called a nut or burr. The surface of the hole through the nut is furnished with a winding op fillet of the same shape and size as the channel of the screw, so that the surfaces of the screw and nut are brought into accurate contact. From this arrangement it is obvious that when the nut is stationary, and a rotary motion is communicated to the screw, the - latter will move in the direction of its axis; also, when the screw is stationary and the nut is turned, the nut must also move in the direction of the axis. Inr 30 466 ELEMENTS OF ANALYTICAL MECHANICS. thc first case, one entire revolution of the screw will carry it longitudinally through a distance equal to the helical interval, and any fractional portion of an entire revolution will carry it through a proportional distance; the same of the nut, when the latter is movable and the screw stationary. The resistance Q is applied either to the head of the screw, or to the nut, depending upon which is the movable element; in either case it acts in the direction D C of the axis. The power P is applied at the extremity of a bar G H connected with the screw or nut, and acts in a plane at right angles to the axis of the screw. From the description of the screw and its mode of generation, we may find the equation of its fillet or helicoidal surface. For this purpose, take the axis z to coincide with the axis of the newel, and the initial position of the generatrix in the plane yz. Make s = any definite portion CC' of an assumed helix; -= the angle YA t, through which the rotating plane -/ has turned during the generation of s; r = the distance CD of this I helix from the axis z; i a = the angle which this helix makes with the plane xy; I = the angle CBD which the i i 4 I generatrix of the helicoidal i i surface makes with the axis z; y the co-ordinate A B of the point in which the generatrix, in its initial position, intersects the axis z. Then, for any point as C of the generatrix in its initial position, we have z - AD = AB + B D = y + r. cotan C, and for any subsequent position, as C'B', z = y + r. cotan C + r. 9. tan a,. (726) APPLICATIONS. 467 which is the equation sought, and in which a and r are;onstant for the same helix, and variable from one helix to another The power P acts in a direction perpendicular to the axis of the newel. Denote by I its lever arm; its virtual moment will be P ld p. The resistance Q acts in the direction of the axis of the newel; its virtual moment will be Qd z. The friction acts in the direction of the helicoidal surfaee and parallel to the helices. Conceive it to be concentrated upon a mean helix, of which the distance from the newel axis is r, and length s: denote the normal pressure by N. and co-efficient of friction by f. The virtual moment of fiiction will be f. N. ds; and Equation (645), P ldp - Qdz -f.N. ds = 0.. d s (727) But the displacement must satisfy Equation (726), or, as in ~213, the condition, L = z - r..tan - r.cotan C-y 0;. (728) and also, r = constant....... (729) Differeentiating, we have, d - cotan. d r - r tan d =0, dr_- 0. Multiplying tie first by X, the second by X', adding to Equation (727), and eliminating d s by the relation ds = r..dp.cosa + dz. sina,... (730) we find, (P l-f.N.osa.r - Xtana.r)d + (X - Q-f. sin a) dz +(\- X(otan)dr 0 468 ELEMENTS OF ANALYTICAL MECHANICS and, from the principle of indeterminate co-efficients, P -f N. cos a. r -. tan a. r = 0;.. (731) Q + f.sin. - x- Q;..... (732)'- X cotan C = 0...... (732)' The variables dz, dr, andrdq, are rectangular; whence, Equation (331), N = / (\dL)2+ (-d-Lj+ (dL)2= \ -1 + tan2 2a + cotan2. Substituting this in Equations (731) and (732), and eliminating X, there will result r tan a + f. cosa. t tan2a cotan2 (733) I 1-,f. sin 1. tan2a+cotan2 Substituting the value of X from Equation (732), in Equation (732)', we find, cotan k' Q Q. (734) 1 -f. sin a /I- + tan2 a + cotan2 (73 ) in which X' is, ~217, the value of the force acting in the direction of r. ~ 387.-If the fillet be rectangular, = 90~, cotan C = 0, and r tan a f. cos a. t/1 +-tan2 a P= Q (735) l -f. sin a. 1- tan2 a and I= 0. ~388.-If we neglect the friction, f = 0; and PI = Q.r.tana, multiplying both members by 2 c, P.2rl= Q.2r.r.tana. 7.O. t736) That is, the power is to the resistance as the helical interval is to the circumference described by the end of the lever arm of the power. APPLICATIONS. 469 PUMPS. ~389. —Any machine used for raising liquids from one level to a higher, in which the agency of atmospheric pressure is employed, is called a Pump. There are various kinds of pumps; the more common are the sucking, forcing, and lifting pumps. ~ 390. —The Sucking-Pump consists of a cylindrical body or barrel B, from the lower end of which a tube D, called the sucking-pipe, descends into the water contained in a reservoir or well. In the interior of the barrel is a movable piston C, surrounded with leather to make it water-tight, yet capable of moving up and down freely. The piston is perforated in the direction of the bore of the barrel, and the orifice is i/ covered by a valve F called the piston-valve, which opens up- A ward; a similar valve X, called the slbeping-valv, at the bottom of the barrel, covers the upper end of the sueking-pipe. Above the highest point ever occupied by the piston, a discharge-pipe P is inserted into the barrel; the piston is worked by means z of a lever 11 or other contrivance, attached to the piston-rod G.' The distance A A', between the highest and lowest points of the piston, is called the platy. To explain the action of' this pump, let the piston be at its lowest point A, the valves E and F closed by their own weight, and the air within the pump of the same density and elastic force as that on the exterior. The water of the reservoir will stand at the sarme level L L both within nad without the sucking-pipe. Now suppose the piston raised to its highest point A', the air contained in the barrel and sucking-pipe will tend by its 470) ELEMENTS OF ANALYTICAL MECHANICS. elastic force to occupy the space which the piston leaves void, the valve E will, therefore, be forced open, and air will pass from the pipe to the barrel, its elasticity diminishing in proportion as it fills a larger space. It will, therefore, exert a less pressure on the water below it in'the sucking-pipe than the exterior air does on that in the reservoir, and the excess of pressure on the part of the exterior air, will force the water up the pipe till the weight of the suspended column, increased by the elastic force of the internal air, becomes equal to the pressure of the exterior air. When this takes place, the valve E will close of its own weight; and if the piston be depressed, the air contained between it and this -valve, having its density augmented as the piston is lowered, will at length have its elasticity greater than that of the exterior air; this excess of elasticity will force open the valve F, and air enough will escape to reduce what is left to the same density as that of the exterior air. The valve F will then fall of its own weight; and if the piston be again elevated, the water will rise still hligher, for the same reason as before. This operation of raising and depressing the piston being repeated a few times, the water will at length enter the barrel, through the valve. F, and be delivered from the' discharge-pipe P. The valves E and F, closing after the water has passed them, the latter is prevented from returning, and a cylinder. of water equal to that through which the piston is raised, will, at each upward motion, be forced out, provided the discharge-pipe is large enough. As the ascent of the water to the piston is pro'duced by the difference of pressure of the internal and external air, it is plain that the lowest point teo which the piston may reach, should never have a greater altitude above the water in the resert voir than that of the column of this fluid which the atmospheric pressure may support, in vacuo, at the place. ~391.-It.will readily appear that the rise of water, during each ascent of the piston after the first, depends upon the expulsion of air through the piston-valve in its previous descent. But air can only issue through this valve when the air below it has a greater density and therefore greater elasticity than the external air; ar~nd APPLICATIONS. 471 if the piston may not descend low enough, for want of sufficient play, to produce this degree of compression, the water must cease to rise, and the working of the piston can have no other effect than alternately to compress and dilate the same air between it and the surface of the water. To ascertain, therefore, the relation which the play of the piston should bear to the other i' dimensions, in order to malike the pump effec- ---- - tive, suppose the water to have reached a stationary level X, at some one ascent of the Z - piston to its highest point A', and that, in its subsequent descent, the piston-valve will not open, but the air below it will be compressed only to the same density with the external air when the piston reaches its lowest point A. The piston may be worked up and down indefinitely, within these limits for the play, without moving the water. Denote the play of the piston by a; the greatest height to which the piston may be raised above the level of the water in the reservoir, by b, which may also be regarded as the altitude of the dischargepipe; the elevation of the point X, at which the water stops, above the water in the reservoir, by x; the cross-section of the interior of the barrel by B. The volume of the air between the level X and A will be B x (b - x - a); the volume of this same air, when the piston is raised to A', provided the water does not move, will be (b - x). Represent by h the greatest height to which water may be supported in vacuo at the place. The weight of the column of water which the elastic force of the air, when occupying the space between the limits X and A, will support in a tube, with a bore equal to that of the barrel is measureo by Bh.g. D; 472 ELEMENTS OF ANALYTICAL MECHANICS. in which D is the density of the water, and g the fcrce of gravity. The weight of the column which the elastic force of th:s same air will support, when expanded between the limits X and A', will be Bh'. g. D; in which h' denotes the height of this new column. But, frt m Mariotte's law, we have B(b - x - a) B(b - x):: Bht'gD BhgD whence, b -x -a b- x But there is an equilibrium between the pressure of the external air and that of the rarefied air between the limits X and A', when the latter is increased by the weight of the column of water whose altitude is x. Whence, omitting the common factors B, D and g, b - x - a x + h' - z + h - -= h; b -x or, clearing the fraction and solving the equation in reference to x, we find X b _ -- 4a h..~.... (737) When x has a real value, the water will cease to rise, but x will be real as long as b2 is greater than 4 a h. If, on the contrary, 4a h is greater than b2, the value of x will be imaginary, and the water cannot cease to rise, and the pump will always be effective when its dimensions satisfy this condition, viz.:4ah > b2, or, b2 a > 4h' that is to say, the play of the piston must be greater than the square of the altitude of the upper limit of the play of the piston above the surface of the water in the reservoir, divided by four timnes the height to which the atmospheric pressure at the place, where the pump APPLICATIONS. 473 is used, vwill support water in, vacuo. This last height is easily found by means of the barometer. We have but to notice the altitude of the barometer at the place, and multiply its column, reduced to feet, by 13a-, this being the specific gravity of mercury referred to water as a standard, and the product will give the value of k in feet. Examnple.-Required the least play of the piston in a suckingpump intended to raise water through a height of 13 feet, at a place where the barometer stands at 28 inches. Here b - 13, and b2 = 169. 28 Barometer, 2,333 feet. ft. h - 2,333 x 13,5 31,5 feet. b2 169 ft. Play a> > -- 1,341 ~; A >_ 4 x 31,5 I that is, the play of the piston must be greater than one and on third of a foot. ~ 392. —The quantity of work performed by the motor during the delivery of water through a the discharge-pipe, is easily computed. Suppose the piston to have any position, as M, and to be moving upward, the water being at the level L L in the reservoir, and at P in the pump. The pressure upon the upper surface of the piston will be equal to the entire atmospheric pressure denoted by A, c increased by the weight of the column of water MP', whose height is c', and whose Z. _ base is the area B of the piston; that is, the pressure upon the top of the piston will be A -+ Be' gD, in which g and D are the force of gravity and density of the water, respectively. Again, the pressure upon the under surface of the 474 ELEMENTS O1' ANALYTICAL MECHANICS. piston is equal to the atmospheric pressure A, transmitted through the water in the reservoir and up the suspended column, diminished by the weight of the column of water NV[ below the piston, and of which the base is B and altitude c; that is. the pressure from below will be A - BcgD, and the difference of these pressures will be A + Bc'g D - (A - Bcg D) = BgD(c + c'); but, employing the notation of the sucking-pump just described, c + c' -= b; whence, the foregoing expression becomes Bb. g. D; which is obviously the weight of a column of the fluid whose base is the area of the piston and altitude the height of the discharge-pipe above the level of the water in the reservoir. And adding to this the effort necessary to overcome the friction of the parts of the pump when in motion, denoted by p, we shall have the resistance which the force F, applied to the piston-rod, must overcome to produce any useful effect; that is, F = BbgD + q. Denote the play of the piston by p, and the number of its double strokes, from the beginning of the flow through the discharge-pipe till any quantity Q is delivered, by n; the quantity of work will, by omitting the effort necessary to depress the piston, be Fnp = np [Bb. gD + 1p]; or estimating the volume in cubic feet, in which case p and b must be expressed in linear feet and B in square feet, and substituting for g D its value 62,5 pounds, we finally have for the quantity of work necessary to deliver a number of cubic feet of water Q = B np Fnp -- np [62,5. Bb + qp];.. (738) in which 9 must be expressed in pounds, and may be determined APPLI CATIONS. 475 either by experiment in each particular pump, or computed by the rules already given. It is apparent that the action 6f the sucking-pump must be very irregular, and that it is only during the ascent of the piston that it produces any useful effect; during the descent of the piston, the force is scarcely exerted at all, not more than is necessary to overcofne the friction. ~ 393.-The Lifting-Pump does not differ much from /the suckingpump just described, except that the barrel and sleeping-valve E are-L placed at the bottom of the pipe, and some distance below the surface of the water L L in the reservoir; the piston may or may not'be below this same surface when at the lowest point of its play. The piston and sleeping-valves open upward. Supposing the piston at its lowest point, it will, when raised, lift the column of water above it, and the pressure of the external air, together with the head of fluid in the reservoir above the level of the sleeping-valve, will force the latter open; the water will flow into the H__ barrel and follow the piston. When the piston: reaches the upper limit of its play, the sleeping-valve will close and prevent the return of the water above it. The piston being depressed, its valves F will open and the water will flow through them till the piston reaches its lowest point. The same operation being repeated a few times, a column of water will be lifted to the mouth of the discharge-pipe P, after which every elevation of the piston will deliver a volume of the fluid equal to that of a cylinder whose base is the area of the piston and whose altitude is equal to its play. As the water on the same level within ohnd without the pump will be in equilibrio, it is plain that the resistance to be overcome by the power will bi the friction of the rub)bing surfaces of the pump, 476( ELEME'NTS Ov ANALYTICAL MECHANICS. augmented by the weight of a column of fluid whose base is the area of the piston, and altitude the difference of level between the surface of the water in the reservoir and the discharge-pipe. Hence the quantity of work is estimated by the same rule, Equation (738). If we omit for a moment the consideration of friction, and take but a single elevation of the piston after the water has reached the discharge-pipe, n will equal one, p will be zero, and that equation reduces to Fp = 62,5 Bp x b; but 62,5 X Bp is the quantity of fluid discharged at each double stroke of the piston, and b being the elevation of the discharge-pipe above the water in the reservoir, we see that the work will be the same as though that amount of fluid had actually been lifted through this vertical height, which, indeed, is the useful effect of the pump for every double stroke. ~ 394.-The Forcing-Pump is a further modification of, the simple sucking-pump. The g P barrel B and sleeping-valve 3 are placed upon the top of the sucking-pipe M. The piston F is without perforation and valve, and the water, after being forced into the barrel by the atmospheric pressure without, as in the suck- N _O ing-pump, is driven by the depression of the piston through a lateral pipe H into an airvessel MV, at the bottom of which is a second sleeping-valve E', opening, like the first, up. ward. Through the top of the ___ air-vessel a discharge-pipe K passes. air-tight, nearly to the APPLICATIONS. 4'77 bottom. The water, when forced into the air-vessel by the descent of the piston, rises above the lower end of this pipe, confines and compresses the air, which, reacting by its elasticity, forces the water up the pipe, while the valve E' is closed by its own weight and the pressure from above, as soon as the piston reaches the.lower limit of its play. A few strokes of the piston will, in general, be sufficient to raise water in the pipe K to any desired height, the only limit being that determined by the power at command and the strength of the pump. ~395. —During the ascent of the piston, the valve E' is closed and E is open; the pressure upon the upper surface of the piston is that exerted by the entire atmosphere; the pressure upon the lower surface is that of the entire atmosphere transmitted from the surface of the reservoir through the fluid up the pump, diminished by the weight of the column of water whose base is the area of the piston and altitude the height of the piston above the surface of the water in the reservoir; hence, the resistance to be overcome by the power will be the difference of these pressures, which is obviously the weight of this column of water. Denote the area of the piston by B, its height above the water of the reservoir at one instant by y, and the weight of a unit of volume of the fluid by w, then will the resistance to be overcome at this point of the ascent be w.. y; and the elementary quantity of work will be w. B.ydy; and the whole work during the ascent will be nyY?' y - y'; wB..B ydy- w B- + Y, in which y' and y, are the distances of the upper and lower limits of the play of the piston from the water in the reservoir. But B. (y' - y,) is the volume of the barrel within the limits of the play of the piston, and ~ (y' + y,) is the height of its centre of gravity above the level of the fluid in the reservoir. 478 ELEMENTS OF ANALYTICAL MECHANICS. Denoting the play by p, and making z2 =', we have for the qu.antity of work during the ascent, w. B.p.'. During the descent of the piston, the valve E is closed, and.E oper, and as the columns of the fluid in the barrel and discharge. pipe, below the horizontal plane of the lower surface'of the piston, will maintain each other in equilibrio, the resistance to be overcome by the power will be the weight of a column of fluid whose base is the area of the piston and altitude the difference of level between the piston and point of delivery P; and denoting by z. the distance of the central point of the play below the point P, we shall find, by exactly the same process, w Bp z, for the quantity of work of the motor during the descent of the piston; and hence the quantity of work during an entire double stroke will be the sum of these, or w Bp (z' + z,). But z' + a, is the height of the point of delivery P above the surface of the water in the reservoir; denoting this, as before, by b, we have wBpb; and calling the number of double strokes n, and the whole quantity of work Q, we finally have Q = nwBpb..~.... (739) If we make z, = z', or b- 2z,, which will give z, = -, the quantity of work during the ascent will be equal to that during the descent, and thus, in the forcing-pump, the work may be equalized and the motion made in some degree regular. In the lifting and sucking-pumps the motor has, during the ascent of the piston, to overcome the weight of the entire column whose base is equal to the area of the piston and altitude the difference of level between APPLICATIONS. 470 the water in the reservoir and point of delivery, and being wholly relieved during the descent, when the load is thrown upon the sleeping-valve' and its box, the work becomes variable, and the motion irregular. THE SIPIION. ~ 396. —The Siphon is a bent tube of unequal branches, open at both ends, and is used to convey a liquid from a higher to a lower level, over an intermediate point higher than either. Its A - parallel branches being in a vertical plane f. and plunged into two liquids whose upper AI surftces are at L M and L' iv', the fluid will stand at the same level both within and without each branch of the tube when a vent or small opening is made at O. if the air be withdrawn from the siphon through this vent, the water will rise in the branches by the atmospheric pressure without, and when the two columns unite and the vent is closed, the liquid will flow from the reservoir A to A', as long as the level L' iM' is below L M, and the end of the shorter branch of the siphon is below the surface of the liquid in the reservoir A. The atlmospheric pressures upon the surfaces L X and L' M', tend to force the liquid up the two branches of the tube. When the siphon is filled with the liquid, each of these pressures is counteracted in part by the pressure of the fluid column in the branch of the siphon that dips into the fluid upon which the pressure is exerted. The atmospheric pressures are very nearly the same for a difference of level of several feet, by reason of the slight density of air. The pressures of the suspended columns of water will, for the same difference of level, differ considerably, in consequence of the greater density of the liquid. The atmospheric pressure opposed to the weight of the loager column \will therefore be more counteracted than that opposed to the weight of the shorter, thus leaving 480 ELEMENTS OF ANALYTICAL MECHANICS. an excess of pressure at the end of the shorter branch, which will produce the motion. Thus, denote by A the intensity of the atmospheric pressure upon a surface a equal to that of a cross-section of the tube; by h the difference of level between the surface L If and the bend 0; by h' the difference of level between the same point 0 and the level L' M'; by D the density of the liquid; and by g the force of gravity: then will the pressure, which tends to force the fluid up the branch which dips below L.t, be A - ahDg; and that which tends t, force the fluid up the branch immersed in the other reservoir, be A - ahtLDg; and subtracting the first from the second, we find a Dg(h' - h), for the intensity of the force which urges the fluid within the siphon, from the upper to the lower reservoir. Denote by I the length of the siphon from one level to the other. This will be the distance over which the above force will be instantly transmitted, and the quantity of its work will be measured by aDg(h' - h) l. The mass moved will be the fluid in the siphon which is measured by a l D; and if we denote the velocity by V, we shall have, for the living force of the moving mass, ali)D. r2; whence, a.D V2 a Dg(h' —h)l= i and, V='2g(h'-h); from which it appears, that the velocity with which the liquid will flow through the siphon, is equal to the square root of twice the force of gravity, into the difference of level of the fluid in the two reser APPLICATIONS. 481 voirs. When the fluid in the reservoirs comes to the same level, the flow will cease, since, in that case, h' - h = 0. ~397. —The siphon may be employed to great advantage to drain canals, ponds, marshes, and the like. For this purpose, it may be made flexible by constructing it of leather, well saturated with grease, like the common hose, and 0 furnished with internal hoops to A prevent its collapsing by the pressure of the external air. It is thrown into the water to be drained, and filled; when, the ends being plugged up, it is placed across the ridge or bank over which the water is to be conveyed; the plugs are then removed, the flow will take place, and thus the atmosphere will be made literally to press the water from one basin to another, over an intermediate ridge. It is obvious that the difference of level between the bottom of the basin to be drained and the highest point O, over which the water is to be conveyed, should never exceed the height to which water nmay be supported in vacuo by the atmospheric pressure at the place. THE AIR-PUMP. ~ 398.-Air expands and tends to diffuse itself in all directions when the surrounding pressure is lessened. By means of this property, it may be rarefied and brought to almost any degree of tenuity. This is accomplished by an instrument called the Air-Pump or Exhausting Syringe. It will be best understood by describing one of the simplest kind. It consists, essentially, of lst.. A Receiver A, or chamber from which the exterior air is excluded, that the air within may be rarefied. This is commonly a bell-shaped glass vessel, with ground edge, over which a small quaa tity of grease is smeared, that no air may pass through any remain31 482 ELEMENTS OF ANALYTICAL MECHANICS. ing inequalities on its surface, and a ground glass plate m n imbedded in a -metallic table, on which it stands. 2d. A Barrel B, or chamber into which the air in the reservoir is to expand itself. It is a hollow cylinder of metal or glass, connected L with the receiver R by the communication ofg. An air-tight piston P is made to move back and forth in the barrel by means of the handle a. 3d. A Stop-cock h, by means of which the communication between the barrel and receiver is established or cut off at pleasure. This cock is a conical piece of metal fitting air-tight into an aperture just at the lower end of the barrel, and is pierced in two directions; one of the perforations runs transversely through, as shown in the first figure, and when in this position the communication between the barrel and receiver is established; the second perforation passes in the direction of the axis from the smaller end, and as it approaches the first, inclines sideways, and runs out at right angles to it, as indicated in the second figure. In this position of the cock, the, communication between the receiver and barrel is cut off, whilst that with the external air is opened. Now, suppose the piston at the bottom of the barrel, and the communication between the barrel and the receiver established; draw the piston back, the air in the receiver will rush out in the APPLICATIONS. 483 direction indicated by the arrow-head, through tle communication ofg, into the vacant space within the barrel. The air which now occupies both the barrel and receiver is less dense than when it occupied the receiver alone. Turn the cock a quarter round, the communication between the receiver and barrel is cut off, and that between the latter and the open air is established; push the piston to the bottom of the barrel again, the air within the barrel will be delivered into the external air. Turn the cock a quarter back, the communication between the barrel and receiver is restored; and the same operation as before being repeated, a certain quantity of air will be transferred from the receiver to the exterior space at each double stroke of the piston. To find the degree of exhaustion after any number of double. strokes of the piston, denote by D the density of the air in the re-, ceiver before the operation begins, being the same as that of the external air; by r the capacity of the receiver, by b that of the barrel, and by p that of the pipe. At the beginning of the operation, the piston is at the bottom of the barrel, and the internal air occu pies the receiver and pipe; when the piston is withdrawn to the opposite end of the'barrel, this same air expands and occupies the receiver, pipe, and barrel; and as the density of the same body is inversely proportional to the space it occupies, we shall have r + p + b ~ r -p:' D x; in which x denotes the density of the air after the piston is drawn back the first time. From this proportion, we find X = _D - D. + P X —D r + p+ The cock being turned a quarter round, the piston pushed back tu the bottom of the barrel, and the cock again turned to open the. communication with the receiver, the operation is repeated upon the air whose density is x, and we have r+p+b r+p D r + p ~; r — p + b in which x' is the density after the second backward motion -of the piston, or after the second double stroke; and w\e find 484 ELEMENTS OF ANALYTICAL MECHANICS. x'-= D. ++ b and if ns denote the number of double strokes of the piston, and x, the corresponding density of the remaining air, then will From which it is obvious, that although the density of the air will become less and less at every double stroke, yet it can never be reduced to nothing, however great n may be; in other words, the air cannot be wholly removed from the receiver by the air-pumlp. The exhaustion will go on rapidly in proportion as the barrel is large as compared with the receiver and pipe, and after a few double strokes, the rarefaction will be sufficient for all practical purposes. Suppose, for example, the receiver to contain 19 units of volume, the pipe 1, and the barrel 10; then will r +p'20 rtp + b 30 3 and suppose 4 double strokes of the piston; then will n = 4, and r - _p 16 ( - -) = (2)4 = 1 = 0,197, nearly; that is, after 4 double strokes, the density of the remaining air will be but about two tenths of the original density. With the best machines, the air may be rarefied from four to six hundred times. The degree of rarefaction is indicated in a very simple manner by what are called yauges. These not only indicate the condition of the air in the receiver, but also wari the operator of any leakage that may take place either at the edge of the receiver or in the joints of the instrument. The mode in which the gauge acts, will be readily understood from the discussion of the barometer; it will be sufficient here simply to indicate its construction. In its more perfect form, it consists of a glass tube, about 60 inches long, bent in the middle till the straight portions are parallel to each other; one end is closed, and the branch terminating, in this end is APPLICATIONS. 485 filled with mercury. A scale of equal parts is placed between the branches, having its zero at a point midway firom the top to the bottom, the numbers of the scale increasing in both directions. It is placed so that the branches of the tube shall be vertical, with its ends upward, and inclosed in an inverted glass vessel, which communicates with the receiver of the air-pump. Repeated attempts have been made to bring the air pump to still higher degrees of perfection since its first invention. Self-acting valves, opening and shutting by the elastic force of the air, have been used instead of cocks. Two barrels have been employed instead of one, so that an' uninterrupted and more rapid rarefaction of the air is brought about, the piston in one barrel being made to ascend while that of the other descends. The most serious defect was that by which a portion of the air was retained between the piston and the bottom of the barrel. This inter:vening space is filled with air of the ordinary density at each del1scent of the piston; 486 ELEMENTS OF ANALYTICAL MECHANICS. when the cock is turned, and the communication re-established with the receiver, this air forces its way in and diminishes the rarefaction already attained. If the air in the receiver is so far rarefied, that one stroke of the piston will only raise such a quantity as equals the air contained in this space, it is plain that no further exhaustion call be effected by continuing to pump. This -limit to ~rarefaction will be arrived at the sooner, in proportion as the space below the piston is larger; and one chief point in the improvements has been to diminish this space as much as possible. AB is a highly polished cylinder of glass, which serves as the barrel of the pump; within it the piston works perfectly air-tight. The piston consists of washers of leather soaked in oil, or of cork covered with a leather cap, and tied together about the lower end C of the piston-rod by means of two parallel metal plates. The piston-rod Cb, which is toothed, is elevated and depressed by means of a cog-wheel turned by the handle J2I. If a thin film of oil be poured uponl the upper surface of the piston the friction will be lessened; and the whole will be rendered more air-tight. To diminish to the utmost the space between the bottom of the barrel and the piston-rod, the form of a truncated cone is given to the latter, so that its extremity may be brought as nearly as possible into absolute contact with the cock E; this space is therefore rendered indefinitely small, the oozing of the oil down the barrel contributing still further to lessen it. The exchange-cock' has the double bore already described, and is turned by a short lever, to which imotion is communicated by a rod c d. The communication G H is carried to the two plates I and K, on one or both of which receivers may be placed; the two cocks 1V and O below these plates, serve to cut off the rarefied air within the receivers when it is desired to leave them for any length of time. The cock 0 is also an exchange-cock, so as to admit the external air into the receivers. Pumps thus constructed have advantages over such as work with valves, in that they last longer, exhaust better, and may be employed as condensers when' suitable receivers are provided, by merely reversing the operations of the exchange valve during the motion of the piston. TABLES. TABLE 1. TI-E TENACITIES OF DIFFERENT SUBSTANCES, AND THE R$SISTANCES WHICH THEY OPPOSE TO DIRECT COMPRESSION.-See ~ 269. SUBSTANCES EXPERIMENTED ON. aQ o _ a. Wrought-iron, in wire from 1-20th to 1-30th of an inch in diame- 60 to 9I Lame ter...... in wire, 1-10th of an inch..~ 36 to 43 Telford in bars, Russian (mean) 27 Lam6 English (mean) 251 - hammered ~ 30 Brunel rolled in sheets, and cut length- 4 Mitis wise...... ditto, cut crosswise ~ ~ 18 in chains, oval links 6 in. clear, 2l i Brown iron 11 in. diameter ~ diltto, Brunton's, with stay across 25 Barlow link......... Cast Iron, quality No. 1.... 6 to 7V Hodgkinson 38 to 4i Hodgkinson 2.. 6 to 8 - 37 to 48 - 3*.. 6 to 9 5 to 65 Steel, cast.......... 44 Mitis east and tilted... 6o Rennie blistered and hammered.. 591 - shear......... 57 raw......... 50 Mitis Damascus....... 3 ditto, once refined ~ ~ ~.. 36 ditto, twice refined.... 44 Copper, cast.8 Rennie 52 Ronnie hammered....... I5 - 46 sheet........ 2 1 Kingston wire........... 271 Platinum wire....... 17 Guyton Silver, cast....... 18 wire......... 17 Gold, cast......... 9 - wire......... 14 Brass, yellow (fine) 8 Rennie 73 Gun metal (hard)..... *6 - Tin, cast ~...... 2 - 7 wire......... 3 - Lead, cast........ 4-5ths 3 milled sheet... Tredgold wire. ~ ~ ~ ~ I Guyton *The strongest quality oi cast iron, is a Scotch iron known as the Devon Hot Blast, No. 3: its tenacity is 9{ tons per square inch, and its resistance to compression 65 tons. The experiments of Major Wade on the gun iron at West Point Foundry, and at Boston, give results as high as 10 to 16 tons, and on small cast bars, as high as 17 tons.-See Ordnance Manual, 1850, p. 4t02 TABLE I. 489 TABLE I —continued. SUBSTANCXS EXPERIMENTED ON. t. Q) g C Z C[ r. Stone, slate (Welsh)... 5,7 Marble (white)... 4 ~ ~ 1,4 Ronnie Givry........ I Portland...... ~. 1,6 - Craigleith freestone...... 24 Bramley Fall sandstone 2,7 Cornish granite....... 2,8 Peterhead ditto... 3,7 Limestone (compact blk)..... 4 Purbeck.......... 4 Aberdeen granite....... 5 Brick, pale red.....,3,56 red....,8 Hammersmith (pavior's)..... I ditto (burnt)......1,4 Chalk........,22 Plaster of Paris.......,o3 Glass, plate........ 4 Bone (ox)....... 2,2 Hemp fibres glued together. 41 Strips of paper glued together I3 Wood, Box, spec. gravity...,862 Barlow Ash.......,6 Teak. 9 7 Beech... 7 7 Oak......,92 5 1,7 - Ditto.... 77 4 Fir...,6 5 Pear.....,646 4 - Mahogany.....,637 3 - Elm. ~ ~,57 Pine, American 6 ~,73 Deal, white 6.,86 490 TABLE II. TABLE II. OF THE DENSITIES AND VOLUMES OF WATER AT DIFFERENT DEGREES OF HEAT, (ACCORDING TO STAMPFER), FOR EVERY 2* DEGREES OF FAHRENHEIT'S SCALE.-See ~ 276. (Jahrbuc des Polytechnischen Institutes in WYein, Bd. 16, S. 70). t ifD,f V Df Temperature. Density. Volume. 32,00 0,999887 I,oooI 13 34,25 o,999950 63 I,oooo5o 63 36,50 0,999988 38 1,000o02 38 38,75 1,000000 12 1,000000 12 4I,00oo 0,999988 12 1,000012 12 43,25 0,999952 35 1,oo000047 35 45,50 0,999894 58 I,ooI06 59 47,75 0o,999813 8 1,0001ooo87 81 50o00 0,999711 102 1,000289 102 52,25 0,999587 124 I,000o413 124 54,50 0,999442 145 1,000558 145 56,75 0,999278 164 I,o000723 i65 59,00 0,999095 i83 1,ooo00090o6 i83 61,25 0,998893 202 I,0oo0Io8 202 63,50 0,998673 220 I,OOI329 221 65,75 0,998435 238 I,oo567 238 68,00 o0,998180 255 I 001822 255 70,25 0,997909 271 1,o002095 273 72,50 0,997622 287 1,002384 289 74,75 0,997320 302 1,oo002687 303 77,00 0,997003 317 x,oo3oo5 318 79,25 0,996673 330 i,oo3338 333 8,o50 0,996329 344 I,oo3685 347 83,75 0,995971 358 1,00oo4045 360 86,00 o,99560I 370 1,004418 373 88,25 0,995219 382 1,oo4804 386 90,50 0,994825 394 I,005202 398 92,75 0,994420 405 1,005612 410 95,00 0,994004 416 Ioo6o32 420 97,25 0,993579 425 1,00oo6462 430 99,50 o0,99345 434 1,00oo6902 440 With this table it is easy to find the specijc gravity by means of water at any temperature. Suppose, for example, the specific gravity S' in Equation (456), had been found at the temperature of 590, then would Dl, in that equation be 0,999095, and the specific gravity of the body referred to water at its greatest density, would be given by S= St X 0,999095. TABLE III. 491 TABLE III. IF THE SPECIFIC GRAVITIES OF SOME OF THE MOST IMPORTANT BODIES. [The density of distilled water is reckoned in this Table at its maximum 38o0 F. = 1,000]. Nasme of the Body. Specific Gravity. I. SOLID BODIES. (1) METALS. i Antimony (of the laboratory)..4,2 - 4,7 Bralss....... 7,6 - 8,8 Bronze for cannon, according to Lieut. Mftzka 8,414 - 8,974 Ditto, mean... 8,758 Copper, melted..7,788 - 8,726 Ditto, hammered. 8,878 - 8,9 Ditto, wi~'e-drawn... 8,78 Gold, melted.... I9,238 -19,253 Ditto, hammered.... 19,36 - 19,6 Iron, wrouglt 7,207 - 7,788 Ditto, cast, a mean.7,251 Ditto, gray.7,2 Ditto, white. 7,5 Ditto for caanon, a mean 7,21 - 7,30 Lead, pure melted. I1,3303 Ditto, flattened. 11,388 Platinum, native.6, o - 18,94 I)itto, melted. 20,855 Ditto, hammered and wire-drawn.... 2,25 Quicksilver, at 320 Fahr... 3,568 - 13,598 Silver, pure melted.10,474 Ditto, hammered 10,5 - 10,622 Steel, cast 7,9'9 Ditto, wrought.7,840 Ditto, much hardened. 7,818 Ditto, slightly.7,833 Tin, chemically pure. 7,291 Ditto, hammered 7,299 - 7,475 Ditto, Bohemian and Saxon 7,3I2 Ditto, English.7,291 Zinc, melted.. 6,861 - 7.215 Ditto, rolled.. 7,19 (2) BUILDING STONES. Alabaster 2,7 - 3,o Basalt.2,8 - 3,I Dolerite.. 2,72 - 2,93 Gneiss.. 2,5 - 2,9 Granite.. 2,5 - 2,66 Hornblende.2,9 - 3,I Limestone, various kinds.2,64 - 2,72 Phonolite..,5 - 2,69 Porphyry.2,4 - 2,6 IQuartz 2,56 - 2,75 Sandstone, various kinds, a mean.2,2 - 2,5 Stones for building.. 1,66 - 2,62 Syenite......... 2,5 - 3, Tracityte. 2,4 - 2,6 Brick..,41 - 1,86 L ~ ~.. 492 TABLE III. TABLE III —Contrnued Name of the Body. Specific Gravity. I. SOLID BODIES. (3) WOODS. Fresh-felled. Dry. Alder.......... 0,8571 o,5oo Ash..0,9036 o,644o Aspen.. o,7654 0,4302 Birch.. o,9012 0,6274 Box 0,9822 O,5907 Elm.. o,9476 0,5474 Fir.. 0,94I 0,555o Hornbeam...... 0,9452 0,7695 Horse-chestnut. o,8614 0 574 Larch.... 206 0,2473 Lime..o, 170 0,4390o Maple ~........ 0o,go36 o,6592 Oak. 1,0494 0,6777 Ditto, unother specimen..1,o734 0,7075 Pine, Pinus Abies Picea. 0,8699 0,4716 Ditto, Pinus Sylvestris.o,912 0,5502 Poplar (Italian) 0,7634 0,393 Willow.. 0,7155 0,5282 Ditto, white.0,9859 0,4873 (4) VARIOus SOLID BODIES. Charcoal, of cork........ o, I Ditto, soft wood 0,28 - 0,44 Ditto, oak. 1,573 Coal. 1,232. - 1,5I10 Coke.,865 Earth, common.,... 1,48 rough sand.1..... 1,92 rough earth, with gravel. 2,02 moist sand.2,05 gravelly soil..,o7 clay.. 2,15 clay or loam, with gravel 2,48 Flint, dark.2,542 Ditto, white..... 2,741 Gunpowder, loosely filled in coarse powder..... 0,886 musket ditto.... o 0,992 Ditto, slightly shaken down musket-powder.,o69 Ditto, solid.2.,248 2,563 Ice. 0,916 - 0,9268 Lime, unslacked. i,842 Resin, common.,0 o89 Rock-salt.2. ~ ~ 2257 Saltpetre, melted....... 2,'745 Ditto, crystallized.9.,oo Slate-penci.. 1,8 - 2,24 Sulphur....,92 1,99 Tallow. o,942 Turpentine.0,991 Wax, white.. 0,969 Ditto, yellow. o,965 Ditto, shoemaker's.. 0,897 TABLE III. 493 TABLE III-Continued. Name of the Body. Specific Gravity. II. LIQUIDS. Acid, acetic ~....... x,o63 Ditto, inmuriatic ~...... 1,211 Ditto, litric, concentrated.1,51. 1,522 Ditto, stlphnlric, Enrlih.. 845 Ditto, concentrated (Nordhl.).,86o Alcohol, free fiom water...... o0,792 Ditto, common.. 0,824 - 0,79 Ammoniac, liquid.0,875 Aquafioltis, double....... 1,300 I)itto, single..... 1,200 Beer 1..0..... I.023 - I,o34 Ether, acetic.o....,866 Ditto, nlluiati..,845 - 0,874 Ditto, nitric......... 0,886 l)itto, sulphuric.... 0,7 1 5 Oil, linseed.. o0,928 - o0,953 Ditto, olive. o,915 I)tto, turpentine. o0,792 - o,89 Ditto, whale...0,923 Quicksilver ~..... I3,568 - I3,598 Water, distilled. I,000 Ditto, rain......... I,0013 Ditto, sea.1....,o0265 - 1,o028 Wine. o,992 - i,o38 III. GASES. t Ilriioletei Water = 1. 3u In. = 1 Temp. 38o 0 F. r'em.=32o Atmosphericair 7 7 - 001ooi3o i,0000 Carbonic acid gas..0,00198 1,5240 Carbonic oxide gas.. 0,00126 0,9569 Carbileted hydrogen, a maximum.. 0o,oo00127 o 9784 Ditto, from Coals.....0,00039,3 o,ooo85 0,5596 Chlorine.. ~ ~ ~0,0032 2,4700 Hydriodic gas. 0,00577 4,4430 Hydrogen. 0,0000895 o,o688 Hydrosullphluric acid gas... o.oo55 1,1912 Muriatic acid gas.... oo6 1,2474 Nitrogen...0,..... o,00127 0,9760 Oxygen...... o,oo043 1.1026 Phosphureted hydrogen gas..... o,ooi 13 0o8700 Steam at 2120 Fahlir..0,00082 o06235 Sulphurous acid gas ~ ~.... 0,00292 2,2470 L~~~~~ 4.93~4 TABLE IV. TABLE IV. TABLE FOR FINDING ALTITUDES.-See ~ 284. Detached Thermllometer. t1+tl A t,+t' A t + t' A t,+t' A 40 4,7689067 75 4,7859208 110 4,8022936 145 4,8180714 41,7694021 76.7863973 III,8027~25 I46,818514o 42,7698971 77,7868733 112 58032109 147 )8189559 43,7703911 78,7873487 1O3 8,36687 148,8193973 44,7708851 79,7878236 114,8041261 I49 8198387 45,7713785 80o,7882979 11 5,8045830 10,8202794 46,7718711 81,7887719 i 6,8050395 151,820796 47,7723633 82,7892451 11 7,8054953 1| 2 821t194 48,7728548 83,7897180 ii8 80359509 I53,821i988 49, 7733457 84,7901903 119,8o60o58 } 54,822o377 50,7738363 85 7906621 120 80o68604 155,822476I 51 774326I 86 7911335 121,8073144 i 56,822914 52 7748t53 87 7916042 1 22,8o77680 I57,8233517 53 7753042 88 792o0745 123 8082211 I58,8237888 54,7757925 89 77925441 124 8086737 159,8242256 55 7762802 90o,7930135 125,8091258 i6o,18246618 56,7767674 91,7934822 126,8095776 161t 8250976 57,7772540 92,7939504 127,8100287 162,825533I 58,7777400 93,7944182 128,8104795 i63,8259680 59 7782256 94,7948854 129 8 109298 I64,8264024 60 7787 105 95,7953521 13,8113 3796 I65,8268365 61 77791949 96,7958184 131 8118290 i66 18272701 62,7796788 97,7962841 I3',8122778 16,8277034 63.7801622 98,7967493 133,8127263 168,8281362 64,7806450 99,7972141 134,8131742 169,8285685 65,7811272 100 7976784 I 35,8136216 170.8290005 66,7816o090 101,7981421 i36,840o688 171,8294319 67,7820902 102,7986054 137,8145153 172,8298629 68:7825709 io3 7990681 I38,8149614 173,8302937 69,7830511 104 7995303 139,8154070 i74,8307238 70,78353o6 105,7999921 140,8158523 i75,83ii536 71 27840098 Io6,8004533 141 8162970 176,83x583o 72,7844883 107'8009142 142,8i67413 177,8320119 73,7849664 Io8,8o03744 143,817I852 178,8324404 74 4,7854438 109 4,8o I 8343 144 4,8176285 179 4,8328686 TABLE IV. 495 TABLE IV-continued. WITH THE BAROMETER.-Se ~ 284. Latitude. Attached Thermnometer.'t~ I B T-Tt C C oO 0,00oo689 T3,OOI 1624 00 0,0000000 0,0000000 6,ooI I433 I,oooo0000434 9,9999566 9,ool I I I 7 2 00o00869,9999 13 I 2,oo001o679 3,ooo13o3,9998697 15,OOIO124 4.,0oo001738 9998263 18 0oo09459 5,0002172 9997829 21,ooo8689 6,ooo000267,9997395 24,o00782 7,ooo3o041 999696I 27,ooo6874 8,ooo3476' 9996527 30,ooo5848 9,ooo391o 9996093 33,ooo4758 10,ooo4345,9995659' 36,ooo365 I I,0004780,9995225 39,0002433 12,ooo52 5,9994792 42,0001223 13,ooo565o,9994358 45,o00o000ooooo 14,ooo6o84,9993924 48 9,9998775 15,ooo6519,9993490 49,9998372 16,ooo6954,9993057 5o,9997967 17,0007389,9992623 51,9997566 18 oo0007824,9992190 52 ~9997167 19 0ooo8259,9991756 53'9996772 20,oooo8695 99913 23 54,9996381 21,0009130 9990889 55,9995995 22, ooo9565,9990456 56,9995613 23 0 OOIOoo0,9990023 57,9995237 24,oo0o436,9989589 58 999486b 25 oo00187I,9989156 59,9994502 26 ooI 1306,9988723 60,9994144 27,0011742,9988290 63 9993 I5 28,0012177,9987857 66,9992161 29 0012613 9987424 69,9991 293 30,oo 3048,9986991 73,9989852 31 0,00o 3484 9,9986558 81,9988854 90 9,9988300 496t TABLE V. TABLE V. COEFFICIENT VALUES, FOR THIE DISCHARGE OF FLUIDS THROUGH TIIN PLATES, TIHE ORIFICES BEING REMOTE FROS THE LATERAL FACES OF THE VESSEL.-See ~ 800. Values of the coefficients for orifices whose smallest (dillensions or lead of fluid diameters areabove the centre of the orifice, in feet. ft. ft. ft. ft. ft. ft. 0,66 0,33 0, 16 0,0o 0,07 0,o3 0,05 0,700 0,07 0,627 o,66o 0,696 0, I3 o,68 0o,632 0,657 0,685 0,20 0,592 0,620 0,640 0,656 0,677 0,26 0,602 o0,625 0,638 0,655 o,672 0,33 0,593 o,6o8 o0,63o 0,637 o,655 0,667 0,66 0,596 0,613 0,63I 0,634 o,654 o,655 1,00 o0,6o 0,617 o,630 o,632 o,644 o,650 1,64 0,602 0,617 0o,628 o0,630 0,640 o,644 3,28 o.605 0,615 0,626 0,628 0,633 0,632 5,00 o,603 0,612 0o,620 o,620 0,6a2 o.6i8 6,65 0,602 o,61o o,615 o,615 o,61o o,6io 32,75 o0,6 00 0,600 0,6oo00 0,600 0,600 In the instance of gats, the generating head is always greater than 6,65 ft., and the coefficient 0,6, or 0,61, is taken in all cases. For orifices larger than 0,66 ft., the coefficients are taken as for this dimension; for orifices smaller than 0,03 ft., the coefficients are the same as for this latter; finally, for orifices between those of the table, we take coefficients whose values are a mean between the latter, corresponding to the given head. TABLE VI. 49T7 TABLE VI. EXPER:MENTS ON FRICTION, WITHOUT UNGUENTS. BY M. MORIN. The surfaces of friction were varied from o,o3336 to 2,7987 square feet, the pressures from 88 lbs. to 2205 lbs., and the velocities from a scarcely perceptible motion to 9,84 feet per second. The surfaces of wood were planed, and those of metal filed and polished with the greatest care, and carefully wiped after every experiment. The presence of unguents was especially guarded against.-See ~ 855. FRICTION OF FRICTION OF MOTION.* QUIESCENCE.t SURFACES OF CONTACT. C C X, |.,.....-.. Oak upon oak, the direction of the fibres' 0,478 250 33' o,625 320 It being parallel to the motion... Oak upon oak, the directions of the fibres of the moving surface being perpen- dicular -to those of the quiescent sur- 324 7 58 0540 2 23 face and to the direction of the motions j Oak upon oak, the fibres of the both surfaces being perpendicular to the direc- 0,336 I8 35 tion of the motion.. Oak upon oak, the fibres of the moving surface being perpendicular to the surface of contact, and those of the surface 0o,192 o 52 0,271 15 Io at rest parallel to the direction of the motion........ J Oak upon oak, the fibres of both surfaces i being perpendicular to the surface of... 0,43 23 17 contact, or the pieces end to end ) [Elm upon oak, the direction of the fibres being parallel to tile motion,432 3 2 0694 34 46 Oak upon elm, ditto~..... 0,246 I3 5: 0,376 20 37 Elm upon oak, the fibres of the moving surface (the elm) being perpendicular to o,450 24 s6 0570 29 41 those of the quiescent surlace (the oak) and to the direction of the motion. J Ash upon oak, the fibres of both surfaces 29 41 being parallel to the direction of the 0,400 21 49 0,570 29 41 motion.......... Fir upon oak, the fibres of both surfaces being parallel to the direction of the o,355 i9 33 o,520 27 29 motion....... Beach upon oak, ditto...... o,36 48 o,53 27 56 Wild pear-tree upon oak, ditto 0,370 20 19 0o.440 23 45 Service-tree upon oak, ditto.... 0,400 21 49 0,570 2a 41 Wrought iron upon oak, ditto ~ ~ ~ o,6ig 3i 47 0,619 31 47 * The friction in this case varies but very slightly from the mean. t The friction in this case varies considerably from the mean. In all the experiments the surfacts. had been 15 minutes in contact.: The dimensions of the surfaces of contact were in this experiment,947 square feet, and the results were nearly uniform. When the dimensions were diminished to,043, a tearing of the fibre became apparent in the case of motion, and there were symptoms of the combustion of the wood; from these circumstances there resulted an irregularity in the friction indicatiws of excessive pressure. It is worthy of remark that the friction of oak upon eln is but five-ninths of that of elm upon oak. I In the experiments in which one of the surfaces was of metal, small particles of the metal began,, after a time, to be apparent upon the wood, giving it a polished metallic appearance; these were at every experimen,' wiped off; they indicated a wearing of the metal. The friction of motion and that of quioscence, in these experiments, coincided. The results were remarkably 4:Mrtn. 32 4i98 TABLE VI. TABLE VI —-continued. FRICTION OF FRICTION OF MOTION. QUIESCENCE. SURFACES OF CONTACT. ca Qv)oh O E 0 Wrcught iron upon oak, the surfaces 56 14 being greased and well wetted Wrought iron upon elm.0,252 14 9 Wrought iron upon cast iron, the fibres o0I94 IO 59 O0I94 IO 59 of the iron being parallel to the motion,94 Wrought iron upon wrought iron, the ) fibres of both surfaces being parallel 0 o,138 7 52 o,137 7 49 to the motion....... Cast iron upon oak, ditto... 0,49~ 26 7 Ditto, the surfaces being greased and 0,646 32 52 wetted.......... Cast iron upon elm...... o, 95 3 Cast iron upon cast iron o. 152 8 39 o,162 9 13 Ditto, water being interposed between 0,314 17 26 the surfaces........ 3I4 Cast iron upon brass.. o47 8 22 Oak upon cast iron, the fibres of the wood being perpendicular to the direction 0o,372 20 25 of the motion ~ Hornbeam upon cast iron-fibres paral- 0,394 21 3i lel to motion....... Wild pear-tree upon cast iron —fibres 0,436 23 34 parallel to the motion..... Steel upon cast iron.. 0,202 11 26 Steel upon brass ~ 0, 52 8 39 Yellow copper upon cast iron.. 0,189 o10 49 Ditto oak.... o,6I7 31 41 0,617 31 41 Brass upon cast iron..0,27 12 15 Brass upon wrought iron, the fibres of the iron being parallel to the motion 0,61 9 9 Wrought iron upon brass.....,172 9 46 Brass upon brass....... 201 11 22 Black leather (curried) upon oak* 0,265 14 51 0,74 36 3i Ox hide (such as that used for soles and for the stuffing of pistons) upon oak, 0,52 27 29 0,605 31 II rough.......... Ditto ditto ditto smooth o 0,335 i8 31 0,43 23 17 Leather as above, polished and hardened 0,296 i6 3o by hammering....... Hempen girth, or pulley-band, (sangle) de chanvre,) upon oak, the fibres of the wood and the direction of the cord 052 27 29 0,64 32 38 being paralleg to the motion... Iempen matting, woven with small 0,32 17 45 o,50 26 34 cords, ditto........ Old cordage, 1 inch in diameter, dittot 0,52 27 29 0,79 38 19 * The friction of motion was very nearly the same whether the surface of contact was the inside,)r the outside of the skin.-The constancy of the coefficient of the friction of motion was equally apI parent in the rough and the smooth skins. A-11 the above experiments, except that with curried black leather, presented the phenomenon of a change in the polish of the surfaces of friction-a state of their surfaces necessary to, and dependent upon, their motion upon one another. TABLE VI. 499 TABLE VI-conti;ued. FRICTION OF FRICTION OF MOTION. QUIESCENCE. SURFACES OF CONTACT. a'; 5' _...o __ Calcareous oolitic stone, used in building, l of a moderately hard quality, called 0,64 320 38' O74 360 3I stone of Jaumont-upon the same stone........ Hard calcareous stone of Brouck, of a light gray color, susceptible of taking C0,38 20 49 0,70 35 o a fine polish, (the muschelkalk,) moving upon the same stone.. The soft stone mentioned above, upon o,65 33 2 0,75 36 53 the hard.... The hard stone mentioned above upon 0 67 33 50 0)75 36 53 the soft b........ Common brick upon the stone of Jaumont 0,65 33 2 0,65 33 2 Oak upon ditto, the fibres of the wood being perpendicular to the surface of o0,38 20 49 o,63 32 I3 the stone........ Wrought iron upon ditto, ditto O. 0,69 34 37 0,49 26 7 Common brick upon the stone of Brouck 0,60 30 58 0,67 33 5o Oak as before (endwise) upon ditto ~ ~ o:38 20 49 0,64 32 38 Iron, ditto ditto. o0,24 I3 30 0,42 22 47 500 TABLE VIL TABLE VII. EXPERIMENTS ON THE FRICTION OF UNCTUOUS SURFACES. BY M. MORIN.-See 3855. In these experiments the surfaces, after having been smeared with an unguent, were wiped, so that no interposing layer of the unguent prevented their intimate contact. FRICTION OF FRICTION OF MOTION. QUIESCENCE. SURFACES OF CONTACT.!.'._..-. Oak upon oak, the fibres being parallel to t 0,08 60 IO 0,390 210 19' the motion. Ditto, the fibres of the moving body be-,4 ing perpendicular to the motion o,43 8 9 Oak'upon eltl, fibres parallel. o,136 7 45 Elm upon oak, ditto.. 0,119 6 48 0,420 22 47 Beech upon oak, ditto.. o0,330 ]8 16 Elm upon elm, ditto.. 0o, 140 7 59 Wrought iron upon elm, ditto 0, 38 7 52 Ditto upon wrought iron, ditto 0,177 Io 3 Ditto upon cast iron, ditto.....,118 6 44 Cast iron upon wrought iron, ditto 0,143 8 9 Wrought iron upon brass, ditto o, 6o 9 6 Brass upon wrought iron ~ * ~ o,i66 9 26 Cast iron upon oak, ditto 0,107 6 7 o,loo 5 43 Ditto upon elm, ditto, the unguent being 0,125 7 8 tallow Ditto, ditto, the unguent being hog's 49 lard and black lead.. 37 7 49 Elm upon cast iron, fibres parallel ~. o,I35 7 42 0,098 5 36 Cast iron upon cast iron 0o,. 144 8 12 Ditto upon brass... o, 32 7 32 Brass upon cast iron * ~ ~ 0,107 6 7 Ditto upon brass.. o, 34 7 38 0,164 9 19 Copper upon oak. o,100 5 43 Yellow copper upon cast iron ~ o,115 6 34 Leather (ox hide) well tanned upon cast oo229 12 54 0,267 14 57 iron, wetted.0,2 12 4 Ditto upon brass, wetted. ~ ~ 0,244 i3 43 TABLE VIII. 5i01 TABLE VIII. EXPERIMENTS ON FRICTION WITH UNGUENTS INTERPOSED. BY M. MORIN. The extent of the surfaces in these experiments bore such a relation to the pressure, as to cause them to.be separated from one another throughout by an interposed! stratum of the unguent.-See ~ 355. FRICTION FRICTION OF I OF MOTION. QUIESCENCE. SURFACES OF CONTACT.. UNGUENTS. j.- C Oak upon oak, fibres parallel 0, o64 0,440 Dry soap. Ditto ditto 0,075 o, I64 Tallow. Ditto ditto o,o67.. Hog's lard. Ditto, fibres perpendicular 0,083 0,254 Tallow. Ditto ditto 0,072.. Hog's lard. Ditto ditto ~ ~ 0,250.. Water. Ditto upon elm, fibres parallel 0,136. Dry soap. Ditto ditto * * o,o73 o, I78 Tallow. Ditto ditto ~ ~ ~ o,o66.. Hog's lard. Ditto upon cast iron, ditto ~ o,o80.. Tallow. Ditto upon wrought iron, ditto o,og08. Tallow. Beech upon oak ditto ~' o,o55.. Tallow. Elm upon oak, ditto. o, 0I37 0,411 Dry soap. Ditto ditto 0,070 o0 142 Tallow. Ditto ditto ~.. o,o60.. Hog's lard. Ditto upon elm, ditto ~ o, 139 0,217 Dry soap. Ditto upon cast iron, ditto o66 Tallow. Greased and Wrought iron upon oak, ditto 0,256 0o,649 saturated with water. Ditto ditto ditto ~ 0,214.. Dry soap. Ditto ditto ditto. o,o85 o 108 Tallow. Ditto upon elm, ditto ~ 0,078.. Tallow. Ditto ditto ditto ~ 0,076.. Hog's lard. Ditto ditto ditto ~ 0,055.. Olive oil. Ditto upon cast iron, ditto o, io3.. Tallow. Ditto ditto ditto ~ o0,076 Hog's lard. Ditto ditto ditto ~, o066,o100 Olive oil. Ditto upon wrought iron, ditto 0o,o082. Tallow. Ditto ditto ditto ~ * o,o8I.. Iog's lard. Ditto ditto ditto. o,o70 0oII5 Olive oil. Wrought iron upon brass, fibres Tllow. Ditto ditto ditto.~ 0,075.. Hog's lard. Ditto ditto ditto. 0,078. Olive oil. Cast iron upon oak, ditto.. 0189 Dry soap. Greased, and Ditto 4itto ditto. 0,2I8 0,646 satuatted with water. Ditto ditto ditto ~ 0,078 0, 00 Tallow. Ditto ditto ditto ~. 0,075.. Hog's lard. Ditto ditto ditto ~ 0 o75 0,loo00 Olive oil. Ditto upon elm, ditto * 0,077 Tallow. Ditto ditto ditto 0,061 Olive oil. Hog-'s lard and Ditto ditto ditto.. 091.'s lard and }plumbago. Ditto, ditto upon wrought iron. o,oo Tallow. Cast iron upon cast iron 0,314 I. Water. Ditto ditto ~ ~ o0,197 Soap. 502 TABLE VIII. TABLE VI. —continued. FRICTION FRICTION 0r OF MOTION. QUIESCENCE. SURFACES OF CONTACT.. UNGUENTS. v v Cast iron upon cast iron 0,100 0 o00 Tallow. Ditto ditto 0,070 0oIoo Hogs' lard. Ditto ditto. o0,o64.. Olive oil. Ditto ditto... o,o55 pluard and Ditto upon brass. 003 Tallow. Ditto ditto,o75. ogs' lard. Ditto ditto o,o78 Olive oil. Copper upon oak, fibres parallel 0,069 0,100 Tallow. Yellow copper upon east iron 0,072 o,103 Tallow. Ditto ditto o,68.. Hoes' lard. Ditto ditto o,o66.. Olive oil. Brass upon cast iron o,o86 o, 06 Tallow. Ditto ditto... 0,077. Olive oil. Ditto upon wrought iron 0,081.. Tallow. Ditto ditto... 89 plard and Ditto ditto.. 0,072.. Olive oil. Ditto upon brass. o,o58.. Olive oil. Steel upon cast iron 0,105 0, 108 Tallow. Ditto ditto 0,08.. Hogs' lard. Ditto ditto' o,079.. Olive oil. Ditto upon wrought iron 0 o,o3.. Tallow. Ditto ditto 0 * o,076.. Hogs' lard. Ditto upon brass,o56.. Tallow. Ditto ditto o,o53 Olive oil. Ditto ditto v.. *,0o67 Lard anbago. Greased, and Tanned ox hide upon cast iron o,365.. saturated with water. Ditto ditto 0,59.. Tallow. Ditto ditto ~ o, 133 O,122 Olive oil. Ditto upon brass,241. Tallow. Ditto ditto 0op. 0191.l. Olive oil. Ditto upon oak, 0,2 0,70 Water. Hempen fibres not twisted, moving upon oak, the fibres of the Greased, and hemp being placed in a direc- G8rated wita tion perpendicular to the direc-,86 water tion of the motion, and those of the oak parallel to it t The same as above, moving upon t o 194.. Tallow. cast iron... Ditto.. o, I53.. Olive oil. Soft calcareous stone of Jaumont A upon the same, with a layer of I mortat, of sand, and lime inter-.. 0,74 posed, after from 10 to 15 minutes' cortact. J TABLE IX. 503 TABLE IX. FRICTION OF TRUNNIONS IN THEIR BOXES.-See ~ 861. Ratio of friction to pressure when the unguent is renewed. KINDS OF MATERLALS. STATE OF SURFACES. By the Or, conordinary tinuously. ( Unguents of olive oil, hogs' lard, 0,07o o,o54 and tallow o 0o8 Trunions of cast iron d The same unguents moistened with Trunnions of cast iron and 0water. t.... o7o8 o054 boxes of cast iron. tUnguent of asphaltum 0,054 0,054 Unctuous.. I0,14 Unctuous and moistened with water.0 o, 14 Unguents of olive oil, hogs' lard, 0,07 and tallow..' to 0,0o54 Trunnions of cast iron and Unctuous..o I 6 boxes of brass. Unctuous and moistened with water..... 16 Very slightly unctuous * O;I Without unguents o, Unguents of olive oil and hogs' 090 Trunons of cast iron and lard..... boxes of lignum-vitoe. Unctuous with oil and hogs' lard o, Io Unctuous with a mixture of hogs' lard and plumbago 0,14 Trunn ions of wrouLght iron Unguents of olive oil, tallow, and o0,07 5 and boxes of cast iron. hogs' lard to o,o54 and tallow.. TrunnUons of wrought iron Trrnions of wrought iron Old unguents hardened 0 o09 and boxes of brass. es of br Unctuous and moistened with water.0..o,15 Very slightly unctuous,2 Trunln-lons ofwroughtiron Unguents of oil or hogs' lard o,,I and boxes of lignum-vi- Utuous0,19 Unctuous.. 0. Trunnions of brass and I Unguent of oil. o,IO boxes of brass. I Unguent of hogs' lard 0,0oog Trunnions of brass and } Unguents of tallow or of olive oil.,45 boxes of cast iron. 52 Trunnions of lignnm-vite{ Unguents of hogs' lard ~ o02 and boxes of cast iron. Unctuous. 0. 5 Trunnions oft'licnunm-vite and boxes of lignum- Unguent of hogs' lard.. 0oC7 vitto. 504 TABLE X. TABLE X. OF WEIGHTS NECESSARY TO BEND DIFFERENT ROPES AROUND A WHEEL ONE FOOT IN DIAMETER. —See ~ 857. No. 1. WHITE ROPES-NEW AND DRY. Stiffness proportional to the square of the diameter. Diameter of rope Natural stiffness, Stiffnless for load of in inches. or value of K. 1 lb., or value of I. Squares of the ratios lbs. lbs. of diameter, or val0,39 0,4024 0,0079877 ues of d2. 0,79 1,6097 o,031950oi 1,57 6,4389 o,12178oi9 3,I5 25,7553 0,5112019 Ratios d. quares Iatios d. NO. 2. WHITE ROPES-NEW AND MOISTENED WITH WATER. 1,00 1,00 0 I0 1200 Stiffness proportional to square of diameter. 1,10 1,21 1,20 1,44 i,3o 1,69 Diameter of rope Natural stiffness, Stiffness for load of 1,40 1,96 in inches. or value of K. 1 lb., or value of L 1,50 2,25 1,60 2,56 1,70 2,89 lbs. lbs.,8o 3,24 | 0,~39 | 0,8048 i o0,0079877 1,90 3,6x 0,79 3,2194 0,0319501 2,o 4,00 1,57 12,8772 o,127809 O 3,15 51,5111 0,5112019 NO. 3. WHITE IROPES-HALF WORN AND DRY. Stifness paroportional to the square root of the cube of the diameter. Diameter of rope- Natural Stiffness, Stiffness for load of in inches. or value of K. I lb., or value of I. Square roots of the cubes of the ratios lbs. lbs. of diameter, or val0,39 0,40243 0,0079877 3 0,79 1,13801 0,0525889 ues of di i,57 3,21844 0,o638794 3,15 9,IO150 O,1806573.Raios or Power 3 d. or d2 No. 4. WHITE ROPES-HALF WORN AND MOISTENED WITH WATER. Stiffness proportional to the square root of the cube of 1,00,000 the diameter. 1,10 1,154 1,20 1,315,3o 1,482 Diameter of rope Natural Stiffness, Stiffness for load of 1,40 1,657 in inches. or value of K. lb., or value of 1. 1,5 11,837 1,60 2,024 1,70 2,217 lbs. lbs.',8o 2,415 0,39 o,8o48 0o0079877 1,90 2,61 0~79 2,276 0,052 5889 00oo 2828 I,57 6,4324 0oo638794 3I15 18,2037 o, 1806573 APPENDIX. 50.5 TABLE X —-continued. No. 5. TARRED ROrEs. Stiftness proportional to the number of yarns. [These ropes are usually made of three strands twisted around each other, each strand being com. lsued of a certain number of yarns, also twisted abou, each other in the same manner.] No. of yarns. Weight of 1 foot in Natural stiffness, or Stiffness for load of No. of yarns. length of rope. value of K. 1 lb., or value of I. Ibs. lbs. lbs. 6 0,0211 0o, I534 0,oo85198 15.o,o47 0,7664 0,0198796 30 I,0137 2,5297 0,041 I799 A P PEN D I X. No. I. Take the usual formulas for the transformation of co-ordinates from one system to another, both being rectangular, viz: x = ax' + b y' + cz', ) y = a'x' +6' y' + c' z',.. (1) z = a'Y' +b"y' + c" z'; J in which a, b, &c., denote the cosines of the angles which the axes of the same name as the co-ordinates into which they are respectively multiplied make with the axis of the variable in the first member. And hence, X' =ax + a' y + a"fz, y' = b x + b' b",.. (2) z' = cx c' y+ c"z; Multiply the first of (2) by b, the second by a, and take the difference of the products; we get 6x'-' ay y(a'b - a b') + z(a"b -ab"); ~ ~ ~ (3) again, multiply the first by c, the third by a, and take the difference of products; we have x' - a z' -y (a' c- ac') + z(a" c-a c") ~ ~ (4) Find the value of y in (4), substitute in (3), and reduce, we find - = (bc' - b' c)x' + (a' c - a c')y' + (ab' - a' b) z', 506 APPENDIX. in which A = c (a' b" -a" b')+c' (a" b - a b") + c" (a b'- a' b), dividing by A, and subtracting the result from the third of Eqs. (I) we have (,, c' - b'c)~ ( a'c" -ac)y+ (, ab' -a'b=o A and since x', y' and z' are wholly arbitrary, we have b c' - b' c a' c -a c' b' - a' b a"- - o; b" - o. e"- =o-;.(5) A - A A transposing, clearing the fraction, squaring, adding, collecting the coefficients of c'2, b'2, a'2, and reducing by the relations a2 + 12 + C2 1; a'2 + b'2 + C!2; a2 + b2 1 -C2; C2+ 12 = 1-; a2;a+C — = -b2, there will result A2 = 1- (aa' + bb' + cC')2. But a a' + b b' + cc' = 0, whence A 1, and, Eqs. (5), a" =c' - b'c; b" =a'c-ac'; c" = — — a' b. No. II. To find the radius of curvature of any curve, and its inclination to the co-ordinate axes. Take the centre of curvature as the centre of a sphere of which the radius is unity. Through the same point draw the line 0 X, parallel to the axis x, and another 0 T, parallel to the C tangent to the are Oif iV, of osculation. R The planes of these lines and of the ra- x dius of curvature will cut from the sphere the spherical triangle A B C, of which the /r side B C is 90~, A C the angle which the radius of curvature makes with the axis x, and A B the angle which the tangent to the curve makes with the same axis. Make p = 0 R = radius of curvature, O'= A C; c = AB; C=- A CB. APPENDIX. 50T Then will dx cos c _-= sin 6'. cos C; differentiating, and regarding C as constant, dx d - = cos 6'. d 6'. cos C; but d'. cos C is the projection of the are d' on the osculatory plane, whence d 6'. cos C d -s P Substituting this above, we find dx ds COS' p d s and denoting by 8" and 6"', the angles which tlie radius makes with the axes y and z, respectively, we may write d x d~l dz' d dd d dCOS 6' d cos =.; c cos d; cos'"... (1) Squaring, adding and reducing by the relation, cos2 6' + cos2 6" + cos2 8"' 1, we hav6 ds V d X ( d r d ~ dS) + (d ) performing the operations indicated under the radical sign, and reducing by the relations ds2 = dx2 + dy2 + d z2, d2 s d s = d2x d x + d2ydy d2zdz, we find V(d2 x)2 2 + (d )2 (2 Z)2 - (d2 s)2' () If s be taken as the independent variable, then will d2 s = 0, and Eqs. (1) and (2) become CJ2 X d2 Y. d2 Z cos6P' CoX cCos" d y d2z. (3) dir S2 x d 2 d s2 P (-l2 (X);2 + (dc y)? + (di z)2... ( 508 APPENDIX. No. III To integrate the partial differential equation dq d q d). -' +r dp transpose and divide by D, and we have dq p dq dD " D dp' and because q is a function of p and D, we have dq dq dq.dD +;i dp; dD dp dq dq D.dp —y.p.dD d=dp =D 1 multiplying and dividing by 7y D. pY 1 1 1 ~ — d1 -p7. d D d1 —p 1 1-1 P7 but 1 1 D -. pY d p - pY. d.D; dq2 -=d (n and makintegrating.p P we may write dq yir(PD) in; kn which F, denotes any arbitrary function. APPEN DIX. 509 No. IV. To integrate Equation (414)' of the text, add to both members a. and we have dt L dt + a r d. _1 cT'a d +a d r]; and making drp + dr dt d re the above imay be written. dV d V - -. r; and V being a function of r and t, we have d V = d r +.d t; or, by substitution for dt its value above, dV dt dV cd V- d'(d = a.d = d (r + a t), and by integration, V= = F'dt +,(r +at); in which F' is any arbitrary function. In like manner, by subtracting, &c' r qp (. cl' from both members of Equation (414)', we find V' = — a -f'. (' -at); dt dr - in which f' denotes any arbitrary function. Whence, by addition, dr o V; I-+ V' - I dV+ V' (r + ( a t) + - f' (r -a ); - dt2 510 APPENDIX. and by subtraction, -- V' a- F'(r + a t)- f' (r —a t). 2 a 2 a But dr q d r q) d q) = dc.d + dp.dr. d't dr Whence, dr1p 1.r F' (r + a at d (- a t)- f'. (r - a t) d ( - at); and, by intecra-tiun, r. F(,. + t) +f (,- - a t), in wLhich.P nat3 f denote the primitive functions of which F' and f' are the derived.