Daughter or William Stuart Smith, U.S. Navy AJ 'MO/i J^?. ELEMENTS op .'», » TO, 5 ' > :> > > ANALYTICAL MECHANICS. BY % W. H. C. BARTLETT, LL.D., COLONEL U. S. A., RETIRED, AND LATE PROFESSOR OP NATURAL AND EXPERIMENTAL PHIL- OSOPHY IN THE U. 8. MILITARY ACADEMY AT WEST POINT, AND AUTHOR OP " ELE- MENTS OF SYNTHETICAL MECHANICS, ASTRONOMY, ACOUSTICS AND OPTICS." X NINTH EDITION, EEVISED AND COREECTED. A. S. BARNES AND COMPANY, NEW YORK AND CHICAGO. Valuable Works to Lealino; Authors v TN THE HIGHER MATHEMATICS. W. H. C. BARTLETT, L.L.D., ?*rof. of jVat. & JZzp. iPhilos. in the U. S. Military Academy, West Point \ c < ( c < c <{ << IBARTLETT'S SYNTHETIC MECHANICS. ,' ,' ,r ' / c ^ Elements of M^chank-s, embracing Mathematical formulae for observing and calculating the action of Forces upon Bodies — the source of all physical phenomena. BARTLBTTS ANALYTICAL MECHANICS. Fer more advanced students than the preceding, the subjects being discussed Analytically, •by the aid of Calculus. BARTLETT'S ACOUSTICS AND OPTICS. Treating Sound and Light as disturbances of the normal Equilibrium of an analogous char- acter, and to be considered under the same general laws. BARTLETT'S ASTRONOMY. Spherical Astronomy in its relations to Celestial Mechanics, with full applications to lh« current wants of Navigation, Geography, and Chronology. A. E. CHURCH, L.L.D., 'Prof, Mathematics in the United States Military Academy , West jPoint. CHURCH'S ANALYTICAL GEOMETRY. Elements of Analytical Geometry, preserving the true spirit of Analysis, and rendering thw whole subject attractive and easily acquired. CHURCH'S CALCULUS. Elements of the Differential and Integral Calculus, with the Calculus of Variations. CHURCH'S DESCRIPTIVE GEOMETRY. Elements of Descriptive Geometry, with its applications to Spherical Projections, Shades And Shadows, Perspective and Isometric Projections. 2 vols. ; Text and Plates respectively. EDWARD M. COURTENAY, LL.D., Z,ale Pro ft Mathematics in the University of Virginia. COURTENAYS CALCULUS. A treatise on the Differential and Integral Calculus, and on the Calculus of Variations. CHAS. W. HACKLEY, S.T.D., Late iProf. of Mathematics and Astronomy in Columbia College. HACKLEY'S TRIG-OX OMETRY. A treatise on Trigonometry, Plane and Spherical, with its application to Navigation and Surveying, Nautical and Practical Astronomy and Geodesy, with Logarithmic, Trigonomet- rical, and: Nautical Tables. ^^^^^^^ DAVIES & PECK, Department of Mathematics , Columbia College. MATHEMATICAL DICTIONARY And Cyclopedia of Mathematical Science, comprising Definitions of all the terms employed in Mathematics— an analysis of each branch, and of the whole as forming a single science. C H ARLES DAVIES, L L. D., JLate of the United States Military Academy and of Columbia College. A COMPLETK COURSE IN MATHEMATICS. See A. S. Barnes & Co.'s Descriptive Catalogue. — — — — — - . B 1 — . — " Entered, according to Act of Congress, Li the year 1874, by W. H. C. BARTLETT. In the Clerk's Office of the District Court of the United States :<>r the S01.tr em District of New York. B'S AX AT. MECTI. Qjpj QJT V TO « COLONEL SYLVANUS THAYER, OF THE CORPS OF ENGINEERS, AND LATE SUPERINTENDENT OF THK UNITED STATES MILITARY ACADEMY, ftfcij Mark. 18 MOST RESPECTFULLY AND AFFECTIONATELY DEDICATED IN GRATITUDE FOR THE PRIVILEGES ITS AUTHOR HAS ENJOYED UNDER A SYSTEM OF INSTRUCTION AND GOVERNMENT WHICH GAVE VITALITY TO THE ACADEMY, AND OF WHICH HE IS THE FATHES. * 863899 PREFACE TO THE SECOND EDITION. .It is now six years since the publication of the first edi- tion 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 Molecules ; and this completes — in so far as he may have succeeded in its ex- ecution — the design of the author to give to the classes com- mitted 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 char- acter of permanence, in the 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 sub- ject; and whatever of specialty may mark our perceptions of a particular instance, will be found to have its origin in corre- sponding peculiarities of physical condition, distance, place, and time, which are the elements of this formula. Its discus- sion 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 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 His 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 inert matter into groups of phenomena, of which the rationale is in a com- mon principle; and in the hands of those gifted with the priceless boon of a copious mathematics, it is a key to exter- nal nature. The order of treatment is indicated by the heads of Me- chanics of Solids, of Fluids, 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 MM. La- grange, Poisson, Poncelet, Fresnel, Lame, Sir William R. Hamilton, the Rev. Baden Powell, Mr. Airy, Mr. Pratt, and Mr. A. Smith. West: Point, 1858 PREFACE TO THE NINTH EDITION. Twenty years ago, the course of Mechanics taught, for several previous years, to classes in the United States Military Academy, was published in the first edition of this work. In that edition the following assertion was made: "All physical phenomena are but the necessary results of a perpetual conflict of equal and opposing forces, and the mathe- matical formula expressive of the laws of this conflict must involve the whole doctrine of Mechanics. The study of Mechanics should, therefore, be made to consist simply in the discussion of this for- mula, and in it should be sought the explanation of all effects that arise from the action of forces." From the single fundamental formula thus referred to, the whole of Analytical Mechanics was then deduced. That formula was no other than the simple analytical expres- sion of what is now generally called the law of the conservation of energy, which has since revolutionized physical science in nearly all its branches, and which at that time was but little developed or accepted. It is believed that this not only was the first, but that it even still is the only treatise on Analytical Mechanics in which all the phenomena are presented as mere consequences of that single law. iv PREFACE. And in offering to the public this new edition, which has been most carefully revised and in many parts rewritten, one of the principal objects sought has been to render it more worthy of use, by making it what it ought to be in view of the great progress achieved during the last twenty-five years, in consequence chiefly of the more general recognition and acceptance of the grand law of work and energy, by Newton called that of action and reaction. To Professor EL S. McCulloch my acknowledgments are due not only for suggestions, but also for valuable aid in preparing the present edition for the press. And to Professor P. S. Michie, my former pupil and now able successor in the Military Academy at West Point, I am also much indebted. Yonkers, N. Y., 1874. CONTENTS. INTRODUCTION. PAGE Preliminary Definitions.' 13 Rest, Motion, Force 14 Constitution of Bodies 14 Inertia 16 Mass 17 Mechanics 18 PART I. Force and Motion 19 Motion and Rest 25 Work 26 Varied Motion 30 Equilibrium 34 General Laws of Work and Energy 35 Principle of D'Alembert 36 Virtual Velocities 37 Interpretation of Equation (A). 39 Reference to Co ordinate Axes. 44 Composition and Resolution of Oblique Forces 50 Parallelogram of Forces 53 Parallelopipedon of Forces 57 Parallel Forces 63 Work of Resultant and Components 72 Work of Rotation 73 Moments 76 Composition and Resolution of Moments 78 Translation of Equations B and C 81 Centre of Gravity . . 83 Centre of Gravity of Lines 87 Centre of Gravity of Surfaces 92 Centre of Gravity of Volumes 99 Centrobaryc Method 104 Centre of Inertia 106 Motion of the Centre of Inertia 107 Rotation around the Centre of Inertia 108 Motion of Translation 110 General Theorem of Work, Energy, &c 110 Stable and Unstable Equilibrium 113 Potential Function 116 vi CONTENTS. PAGE Conservation of Energy 117 Discussion of Function II. ^ 118 Initial Conditions, Direct and Inverse Problems 122 Vertical Motion of Heavy Bodies 123 Projectiles 131 Rotary Motion . . 147 Moment of Inertia, Radius and Centre of Gyration 1 59 Motion of a Body under Impulsion 171 Motion of the Centre of Inertia 171 Motion about the Centre of Inertia 173 Angular Velocity 174 Axis of Instantaneous Rotation : 175 Axis of Spontaneous Rotation 170 Stable and Unstable Rotation 177 Motion of a System of Bodies . 179 Motion of the Centre of Inertia of the System 180 Motion of the System about its Common Centre of Inertia 181 Conservation of the Motion of the Centre of Inertia 182 Conservation of Areas 183 Invariable Plane 185 Conservation of Kinetic Energy 185 Principle of Least Action 187 Planetary Motions s . 198 Laws of Central Forces 200 The Orbit 206 System of the World 208 Consequences of Kepler's Laws 208 Perturbations 213 Coexistence and Superposition of Small Motions 215 Universal Gravitation 216 Impact of Bodies 221 Constrained Motion 228 Constrained Motion on a Curve and Surfaces 230 Constrained Motion about a Fixed Point 255 Constrained Motion about a Fixed Axis 257 Compound Pendulum 259 Ballistic Pendulum 269 Gun Pendulum 271 PAET II. MECHANICS OF FLUIDS. Introductory Remarks 273 Mariotte's Law 275 Law of Pressure. Density, and Temperature 275 Equal Transmission of Pressure 278 Motion of Fluid Particles 280 Equilibrium of Fluids 290 Pressure of Heavy Fluids 299 CONTENTS. vii PAGK Equilibrium and Stability of Floating Bodies 307 Specific Gravity 316 Atmospheric Pressure 320 Barometer 321 Motion of Heavy Incompressible Fluids in Vessels 330 Steady Flow of Fluids 342 Steady Motion of Elastic Fluids 352 Digression on the Action of Heat upon Air 356 New Equations of Steady Flow 359 PART III. MECHANICS OF MOLECULES. Introductory Remarks 355 Periodicity of Molecular Condition 365 Waves 372 Wave Function 373 Wave Velocity 380 Relation of Wave Velocity to Wave Length 383 Surface of Elasticity 385 Wave Surface 387 Double Wave Velocity 392 Umbilic Points 395 Molecular Orbits 398 Reflexion and Refraction 401 Resolution of Living Force by Deviating Surfaces 404 Polarization by Reflexion and Refraction 408 Diffusion and Decay of Living Force 414 Interference 415 Inflexion 420 PAET IV. APPLICATIONS TO SIMPLE MACHINES, PUMPS, &c. General Principles of all Machines 435 Friction 427 Stiffness of Cordage 435 Friction on Pivots 440 Friction on Trunnions 445 The Cord as a Simple Machine . . . . 449 The Catenary 459 Friction between Cords and Cylindrical Solids 461 Inclined Plane. . . . 463 The Lever 466 Wheel and Axle 469 Fixed Pulley 471 Movable Pulley 474 The Wedge 480 The Screw 484 viii CONTENTS. PAGE Pumps 489 The Siphon ...» 489 The Air-Pump . 501 TABLES. Table I. — The Tenacities of Different Substances, and the Resistances which they oppose to Direct Compression 508 " II. — Of the Densities and Volumes of Water at Different Degrees of Heat (according to Stampfer), for every 2\ Degrees of Fahrenheit's Scale 510 ** III. — Of the Specific Gravities of some of the most Important Bodies »» «.».... 511 * IV. — Table for finding Altitudes with the Barometer...... 514 " V. — Coefficient Values, for the Discharge of Fluids through thin Plates, the Orifices being Remote from the Lateral Faces of the Vessel.. 510 * VI. — Experiments on Friction, without Unguents. By M. Morin. 517 kx VII. — Experiments on Friction of Unctuous Surfaces, By M. Morin. 520 "VIII. — Experiments on Friction with Unguents interposed. By M.. Morin.. 521 " IX. — Friction of Trunnions in their Boxes. ................... 528 *' X — Of Weights necessary to Bend different Ropes around a Wheel one Foot in Diameter. .......................... 524 The Greek Alphabet is here inserted to aid those who are not already famil far with it, in reading the parts of the text in which its letters occurw Letters. Names. A a Alpha b jse Beta r yf Gamma A 6 Delta E s Epsilon z a Zeta Uri Eta &d Theta I 1 Iota K x Kappa A X Lambda Mp 111 Letters. Names N v Nu « I Xi O o O micron n #*• Pi p p* Rho £ ds Sigma T rl Tau T u Upsilon $

§ 50. — Continual variation in a body's velocity can only be pro- duced 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 decrement of the velocity. The product of a mass into a velocity represents a quantity of motiov. MECHANICS OF SOLIDS. 31 i The intensity of a motive force, at any instant, is assumed to be measured by the quantity of motion which this intensity can generate in a unit of time. The mass remaining the same, the velocities generated in equal successive portions of time, by a constant force, must be equal to each other. However a force may varv, were it to remain constant, it would generate in a unit of time a velocity equal to dv repeated as many times as dt is contained in this unit; that is, the velocity generated would be equal to . 1 dv dv . — = — : dt dt ' and denoting the intensity of the force by P and the mass by M, we shall have n ir dv ■ P = M -Tt < 12 > Again, differentiating Equation (11), regarding t as the independent variable, we get, dv =di ; and this, in Equation (12), gives p = M -% < 13 > From Equation (11), we conclude that in varied motion, the velocity at any instant is equal to the first differential coefficient of the space regarded as a function of the time. 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 coefficient of the velocity regarded as a function of the time. And from Equation (13), that the intensity of the motive force, or of the inertia, is measured by the product of the mass into the necond differential coefficient of the space regarded as a function of the time,. § 51. — To illustrate. Let there be the relation s = at* + bfi (14) required the space described in three seconds, the velocity at the end 32 ELEMENTS OF ANALYTICAL MECHANICS. • of the third second, and the intensity of the motive force at the same instant. Differentiating Equation (14) twice, dividing each result by dt, and multiplying the last by M, we find — = v = Sat 2 + 2bt (15) (X v M.— —P — M[Qat-\-2b} (16) 4 Make a = 20 feet, 6 = 10 feet, and t—3 seconds; we have from Equations (14), (15), and (16), f = 20 . 3 3 -f 10 . 3 2 = 630 feet ; v = 3 . 20 . 3 2 -f 2 . 10 . 3 = 600 feet ; P = M(G.20.S + 2. 10) = 380. M. 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 in the mass M a velocity of 380 feet in one second, were it to retain that intensity unchanged. § 52. — Dividing Equations (12) and (13) by Jf, they give ^ = ^ (17) M-dfi • • * (18) 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 impressed in the unit of time, and is called the acceleration due to the motive force. § 53. — From Equation (11) we have, ds = v.dt (19) multiplying this and Equation (12) together, there will result, P.ds = M.v.dv (20) md integrating, fP.d^^ (21) MECHANICS OF SOLIDS. 33 The first member is the quantity of work of the motive force, which is equal to that of inertia ; the product M. v 2 is called the vis viva or 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 Hie quantity of work expended by iU inertia while it is acquiring its velocity. § 54, — If the force become constant and equal to F, the motion will be uniformly varied, and we have, from Equation (18), F _cfts M~dft Multiplying by dt and integrating, we get F ds M' t = dt+ 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 M' t = v < 23 ) . Multiplying Equation (22) by dt, after omitting C from it, and integrating again, we find and if the body start from the origin of spaees, C will be zero, and F ft M'2= S < 24 > Making t equal to one second, in Equations (23) and (24), and dividing the last by the first, we have 1 s ¥ ~ ~v" or, v = 2s (25) That is to say, the velocity generated in the first unit of time in measured by double the space described in acquiring this velocity Equations (23), (24), and (25) express the laws of constant forces. 34 ELEMENTS OF ANALYTICAL MECHANICS. g 55. — 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 quantity 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 y under the action of a constant force, in one second ; the velocity generated will, Equation (25), be 2AB. Make BC—2AB, and complete the square BCEE. BE will be equal to v ; the intensity of the force will be M.v; and the quantity of work, the product of M.v by AB, or by its equal -g-u; thus making the quantity of work ^Mv 2 , or the mass into one half the square BF\ which agrees with the result obtained from Equation (21). EQUILIBRIUM. § 56. — Equilibrium is a term employed to express the state of two or more forces which balance one another through the interven- tion of the body subjected to their simultaneous action. When applied to a body, it means that the state of the body may either :be rest or uniform motion. § 57. — 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 simultaneous 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, MECHANICS OF SOLIDS. 35 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 changing its condition in respect to rest or motion. This is but a consequence of the universal law that every action is accom- panied by an equal and contrary reaction. GENERAL LAW OF WORK AND ENERGY. § 58. — The extraneous forces, called impressed forces, being, there- fore, always in equilibrio, either among themselves or with the forces of inertia, the sum of the quantities of work performed in any one direction, regarded as positive, must be equal to the sum of the quantities of work performed in the contrary direction, regarded as negative. In other words, the work performed by the entire system of impressed and inertia forces, taken collectively, must be zero. To state this mathematically, the inertia forces, denoted by / t1 I iy /j, etc., are exerted in and by the elementary masses, m„ m 2i m 3 , etc., respectively, and these elementary masses describe in a definite time t ? the respective paths, »„ «* s 3 , etc. Similarly, the points of application of the impressed forces, whose intensities are denoted by P l9 P 2 > ^3> etc., describe in the definite time t paths whose lengths are denoted by p u p& p 3 , etc. Let &?„ 6s 2} 6s 3 , t etc., denote the orthographic projections of the paths described, in the element of time dt, by the points of application of the inertia forces, upon the respective directions of those forces. Then will 2 /. 6s = /j cJ$j -j- / 2 6s 2 -f- I 3 5s 3 -J- etc., be the elementary work of reaction of all the inertia forces. And if 6/>„ 6p. 2 , 6p 3j etc., denote the orthographic projections of the paths described by the points of application of the extraneous or impressed forces, on the respective directions of those forces, during the same element of time dt, then will the quantity of the elementary work of the impressed forces be lP6p= P, d Pl -f P 2 o> 2 -f P 3 o> 3 + etc. 36 ELEMENTS OF ANALYTICAL MECHANICS. Regarding the algebraic sum of the work of the impressed forces as positive, and that of the inertia forces as negative, since these latter forces oppose all changes of motion, we must always have 2Po> — 1,16s = 0. Hut Hence, _. dht T cPs* T d 2 So / 1= m, — ; /^m,^; /. = ~,^; etc. 2 Ids = 2 m — 6s; at 4 which substituted gives cPs 2P6p—2m—ds = . . . . (A) And this is the single formula referred to in the preface to this book as the one fundamental equation which embraces in its dis- cussion the whole of physical and mechanical science. § 59.— For the sake of simplicity in the demonstration, we have supposed the elementary masses, ?n u m 2 , m 3 , etc., to compose a single body. But it is evident that the same reasoning is applicable to systems of bodies, or masses, of any size, connected in any manner whatever; such as, for example, machines composed of many parts; or the solar system, in which the sun, planets and satellites, constantly pulling each other together, are kept from falling into one confused heap of ruins, and are held apart, each in its proper orbit, by that precisely-balancing resistance which the action of gravitation finds ever opposed to it in the exactly equal reaction of the inertia forces. Than which magnificent example of perpetual conservative equilibrium, noth- ing more grand is known to us in the material world. § 60. — Our Equation (A) is, therefore, perfectly general, or appli- cable to all bodies, or systems of bodies, connected by such forces as those of cohesion, gravitation, etc., or in any manner. PRINCIPLE OF D'ALEMBERT. § 61. — The forces of inertia developed by the impressed force* P, P\ P'\ &c, may or may not be equal to them, depending upon MECHANICS OF SOLIDS. 37 the manner of their application. If the impressed forces be in equi- librio, for instance, they will develop no force of inertia; but in all cases the forces of inertia developed will 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 (A) we infer that the impressed and effective forces are always in equilibrio when the directions of the latter are reversed. This is usually known as D'Alemberfs Principle, and is nothing more than a plain consequence of the law that action and reaction are ever equal and contrary. This same principle is also enunciated in another way. Since the effective forces reversed would maintain the impressed forces in equi- librio, and prevent them from producing a change of motion, it follows that whatever forces may be lost and gained must be in equilibrio ; else a motion different from that which actually takes place must occur. VIRTUAL VELOCITIES. § 62. — The indefinitely small paths mn, m'n', described by the points of application of the forces P and P' during the slight motion we have supposed, are called virtual veloci- ties ; and they are so called, because, being the actual distances passed over by the points to which the forces are applied, in «, the same time, they measure the relative r , ~m'' —**J" rates of motion of these points. The dis- tances rm and r'm', represented by dp and dp,' are, therefore, the orthographic projections of the virtual velocities upon the directions of the forces. These projections may fall on the side toward which the forces tend to urge these points, or the reverse, depending upon the direction of the motion imparted to the system. In the first case the projections are regarded as positive, and in the second as negative. Thus, in the case taken for illustration, mr is positive and m'r' negative. The products P dp and P' dp are called virtual moments. They are the elementary quantities of work of the forces P and P'. The forces are always regarded as positive ; 38 ELEMENTS OF ANALYTICAL MECHANICS. the sign of a virtual moment will, therefore, depend upon that of the projection of the virtual velocity. § 63. — Referring to Equation (A), we conclude, therefore, that whenever several forces are in equilibrio, the algebraic sum of their virtual moments is equal to zero ; and in this consists what is called the principle of virtual velocities. That this is true is evident, for if the impressed forces, P, P\ P", etc., be in equilibrio, they will develop no inertia, and Equation (^4) will reduce to lP6p = ...... (26) 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 toward which it is attracted by p, and the corresponding distances in the case of the forces P', P", &c, by p\ p", ' + P"o>" + (h 2 — h x ) be the change of the value of the potential for such a fall of that weight. Take again the descent by gravity, or fall, of a given weight of water, w, for a mill from one level, or dynamic head, h Q , to another, A„ then the same equation, §2 — Ql = W (hi ~ !h), expresses the amount of work, or change of the potential, due to the fall of the given mass. And it is so called, because it indicates the availability of the change to do other work, such as driving the pile by the pile-driver, or grinding corn, sawing wood, etc., by the mill. Level surfaces, being those for which the product wh, or potential, has the same value, are, therefore, called equipotential. § 66. — As work done has been denoted (in Equation *7) by the expression Q = fPds, in which P is the variable force exerted to overcome a resistance over the path s ; and as we use the letters P, P', P", etc., to denote the impressed forces of any system, the integral Q = ZfPdp, whether taken alone, or used as in Equation (A), may be ambiguous, or capable of two distinct significations. It may be taken to mean either the kinetic energy, which is the work done, or the potential energy, which determines the capacity of the forces to do that work, and these two energies must not be con- founded. This important distinction can be made clear only by a thorough understanding of terms which many authors too often use interchangeably, although representing different things. Then when we speak of force we mean the essence of the thing ; when we speak of intensity we mean an instantaneous effort, and when of the measure of that effort we mean its effect exerted during a unit of time, which, as we have seen, is represented by the mass multiplied by the velocity of a free body, moved from rest, due to MECHANICS OF SOLIDS. 41 that effort. "When we speak of work we mean that measure repeated as many times as the path described, while the effort acts and is supposed constant, contains the unit of path. Therefore the measure of the force's intensity is, in fact, work, hut a definite quantity of work, and is the work of that force over a unit of space. We know that the effect of a force is very different acting under different circumstances. If acting on a free body at rest it will pro- MV 2 dncc a quantity of work represented by — — . But we know that MV 2 the work embodied in may be transformed into some other representative of work, and since it may thus be transformed, this quantity of work is called kinetic energy, because it involves the idea of motion. If the force act to bend a spring, for example, or transform the shape of a body, the latter when relieved from the action of the force will restore the work communicated to it, and the whole measure of its capacity to do so is called potential energy. To illustrate, if we suppose muscular effort to be employed in bending a bow, to which an arrow is adjusted, this is an instance of potential energy being stored in the changed elasticities of the fibres of the bow, and when these are allowed to act on the arrow, through the intervention of the cord, the potential energy is very quickly being transformed into kinetic energy of the arrow, which attains its maximum when the arrow leaves the cord. The arrow carries this kinetic energy with it through space, and if the arrow in its flight meet another bow-string, it will deliver up to this bow its kinetic energy, being transformed into an equivalent amount of potential energy, and become stored up in increased elas- ticities of the fibres of the second bow. When all is thus transformed, the latter gives back its accumulated potential energy, and the arrow leaves it with its restored kinetic energy, and thus there would be forever a mutual equal interchange of these two energies. When, therefore, in the following pages the term energy is employed, it must be understood to mean accumulation of intensity, frequently, though very improperly, called quantity of force, the terms potential 42 ELEMENTS OF ANALYTICAL MECHANICS. and kinetic being simply adjectives to qualify the nature of this accumulation. § 67. — It thus appears that the modern and expressive name of kinetic energy may be applied to denote the ability of a moving mass to perform work by its inertia; and that this term may be very advantageously substituted for that of the half of the vis viva, or half of the living force, which are only unmeaning names for the algebraic quantity ^* = fp ^ 23 which we have shown (in Equation 21) measures the quantity of work of the inertia forces of a body in acquiring, or losing, its velocity. § 68. — All the various forms in which the physical forces of nature present themselves, as the causes of observed effects, may now be simply classified under the two general heads of 'potential and kinetic energy. The forces which act at sensible and measurable distances, in the phenomena of gravitation, electrical and magnetic induction, attraction and repulsion ; or those more hidden forces which act only at insensi- ble distances to combine atoms into molecules and molecules into masses, or which at times separate molecules, as in the phenomena of elasticity, or those of chemical and electrolytic decomposition ; these all are but instances of potential energy performing work by changes of position. On the other hand, the phenomena of the winds due to solar action, the radiant heat and light, forms of kinetic vibratory energy, emanating from the sun and burning bodies, the inertia of projectiles and other moving masses when their motions are changed, these are instances of what was formerly called vis viva, but is now better named kinetic energy. § 69. — It has been stated that energy, though readily transmutable from one of its forms into another, is not destructible. Thus heat is changed into mechanical work" in the steam-engine, and mechanical friction develops or rather is changed into heat. Chemical affinity, by union of zinc with oxygen, develops electrolytic action, which may MECHANICS OF SOLIDS. 43 be made to decompose oxide of zinc, or drive an electro-magnetic engine doing mechanical work. And this electrical force may be transmitted through a metallic wire to long distances ; a fact first discovered about a century and a half ago by Stephen Gray, an invalid pensioner in a charity hospital, experimenting to relieve the tedium of sad hours, and little dreaming of the future importance his discovery would acquire when used to convey swift messages of human affairs around our globe. Many and beautiful indeed are the transformations of energy; marvelous also are the changes due to the physical forces and the mutual dependence of terrestrial phenomena and of vegetable and animal life upon each other and upon solar action. Certainly, too, nothing can be better calculated to cause us to form proper conceptions of the infinite power, wisdom, and goodness of the Divine Ruler of the Universe, than the study and contempla- tion of the simple laws ordained for its continued harmonious exist- ence amid perpetual change, yet all without the least destruction of force or matter. These subjects are full of varied interest, but they do not belong to Analytical Mechanics, except in so far as their discussion may be requisite to set forth the full meaning and scope of the laws of force and motion, and to give proper conceptions of their relations to other branches of physical science. § 70. — It is now evident that energy may be given to and become stored up in dead masses, either as potential, or as kinetic, energy ; and in either case it is capable only of passing from one of these forms into the other, without loss or gain. Nothing can be created, nothing destroyed, except by God himself. The dormant potential energy of gunpowder, brought into action by a spark, is simply transferred as kinetic energy to the ball which tears apart the particles of a mass. The muscular effort which expends itself in winding up the weight or spring of a clock is only converted into potential energy of grav- itation, or elasticity, to be in turn transformed into actual, or kinetic, energy when it gives motion to the wheelwork. Those wheels also transfer their kinetic energy to adjacent particles of air and the sup 44 ELEMENTS OF ANALYTICAL MECHANICS. porting framework, to be in turn indefinitely given to other surround- ing portions of matter. Such is the conception of what is called conservation of energy. § 71. — We end this discussion by simply drawing attention to the fact that as the first term of Equation (^1) is susceptible of two different meanings, related as cause and effect, so also has the second term two distinct significations. Inertia being, by definition, resistance to any change, whether of acceleration or retardation, it is evident that, when a system is acquiring motion, the second term of Equation (A), or rather its definite integral P 2 d' z s m will indicate work of inertia done, by expenditure of potential energy converted into gain of kinetic energy. While, on the contrary, when the system is retarded, the same integral must express loss, or expend- iture of kinetic energy in work done, or potential energy stored. REFERENCE TO CO-ORDINATE AXES. § 72. — First Transformation. Equation (A) is of a form too general for easy discussion, and may be simplified by referring the forces and motions to rectangular axes. Denote by a, /3, y, the angles which the direction of the force P makes with the axes .?, y, z, respectively ; by a, b, c, the angles which its virtual velocity makes with the same axes ; and by , the angle which the virtual velocity and direction of the force make with each other, then will cos (j) = cos a . cos a -f cos b . cos (5 -\- cos c . cos y. Denote by k the virtual velocity, and multiply the above equation by Pic, and we have Pk cos ■= Pk cos a . cos a -f- Ph cos b . cos j3 -f- Pk cos c . cos y. But denoting the co-ordinates of the point of application of P by ~> y, z, we have k cos (f>=: dp; k cos a = 6x ; k cos b = dy; k cos c = 8z ; end these values substituted above, give P . dp = P cos a . 6x -\- P cos (3 . 8y + P cos y . 6z. . (31 ). Similar values mav be found for the virtual moments of other forces. MECHANICS OF SOLIDS. 45 § 73. — If P be replaced by the force of inertia, then will a, (3, and y denote the inclinations of the direction of this force to the axes xyz: k. its virtual velocity ; «, 6, and c tlie inclinations of the latter to tire* axes, and (ji its inclination to the direction of the force of inertia, and we may, Eq. (13), write d2g i <*** i d °'* a i i. du i m • — - k cos . x' = r'" cos , the triangle X'CO' give \ , t the triangle Z'CO". \ . 6 ( z ' = the triangle F'CO"', \ , , . 't ° ( 2 = r sin sr, f #' =V sin ip, -I r" cosi/?, [ (35), (36), (37). We here have two values of x\ one dependent upon 0, and the other upon tp. 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' = — /" sin $ . d(ff ; if it be turned through a like angle about the axis y\ the correspond- ing increment of x' is found by differentiating the first of Equations (36), and dx' = r" cos i/> . dtp. If these motions take place simultaneously about both axes, the above become partial differentials of x\ and we have for its total differential, dx' = r" cos ip . dip — r" sin

) 48 ELEMENTS OF ANALYTICAL MECHANICS. replacing r" cos 4* and r"' sin 9, by their values in the. above Equa- tions, and we get dx' = z'.d-^ — y'.c?(p; and in the same way, dy' = x' . dcp — z f . d &, dz' = y' .dtx — x' . d 4, which substituted in Equations (34), give (38) dx = dx 4 -\- z f .d \ — y dy = dy 4 -f- x' .d

d & — x'. d 4. .(39) and because the displacement is indefinitely small, we may write dx = 8x t + z r .8-^ — y' . #=Z; 4 &c. P' (2' cos a' — a;' cos 7') R (z cos a — x cos c) = ^ 4?" (z" cos a" — x" cos 7") f P' (y' cos 7' - z' cos /3') ft (y cos c — 2 cos b) = < +P" (y" cos 7" — z" cos ,8") =Jf; V • (42) =M 52 ELEMENTS OF ANALYTICAL MECHANICS. or, R cos a = X, > R cos 6 = y, 72 COS C =z Z. (43) i2 (a: cos b — y cos a) = Z, i2 ( z cos a — # cos c) = iV, > (44) R (y cos c — z cos b) — N. ^ Eliminating ii! cos a, R cos b and 72 cos c, from Equations (44), Dy means of Equations (43), we get, by transposing all the terms to the first member, Xy - Yx + L = 0, ] Zz - Xz -f >/ = 0, \ • (45) Yz - Zy + N = 0. J 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 tlte points of application ; and from which it is apparent, that the point of application of a force may be taken anywhere on its line of direction, .within the limits of the body, without altering the etfects 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. §7S. — To find this condition, multiply the first of these Equations by Z, the second by Y, the third by X, and add the products we obtain, ZZ+ F¥+ JJ=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, R 2 (cos 2 a 4- cos 2 b + cos 2 c) = X 2 +■ Y 2 + Z 2 . MECHANICS OF SOLIDS, 53 Extracting the square root and reducing by the relation, cos 2 a -f- cos 2 b -f cos 2 c = 1, there will result, R = V X* + Y* + Z* (47) which gives the intensity of the resultant, since X, Y and Z are known. Again, from the same Equations, cos a = — 5 cos b = cos c = — - R Y R Z R' which make known the direction of the resultant. The group of Equations (45) give, Xy - Yx + XP' (cos/S's' -cos a'/) = 0, j Zz-Xz + ZR' (cos a' 2' - cos7'x') = 0, J> . Y z - Z y + 2 />' (cos 7' / - cos /6 V) = 0. ] which are the equations of the line of the resultant. (48) (49) PARALLELOGRAM OF FORCES. §80. —If all the forces be applied to the same point, this point may be taken as the origin of co-ordinates, in which case, x' = x" = x'" &c == 0, y' = y" = y"' &c. = 0. z' = z" = z'" &c. = 0, and the last term in each of Equations (49), will reduce to zero. Hence, to determine the intensity direction and equations of th« 5 54 ELEMENTS OF ANALYTICAL MECHANICS. line of direction of the resultant, we have, Equations (47), (48) and (49), R == y/X* +P + 2 2 .... (50) X \ cos a = — j R , r cos 6 = -—•> R cos c = — > (51) Xy - Fa: = 0, Zx - Xz = 0, [ (52) •' Yz - Zy = 0.} 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, and take the plane of these forces as that of x y ; then will / = y" = y'" = &c. = 90° ; z = 0, the last Equation of group (41) reduces to, Z = 0; and the above Equations become, R = ^/X 2 + Y 2 (53) X ~}. cos a = — ? R , Y COS J-z —■> R (54) cos c = 0, Xy - Yx = (55) The last is an equation of a right line passing through the origin. The direction of the resultant wilt, therefore, pass through the point of application of the forces. The cos c being zero, c is 90°, and the direction of the resnl'ant is therefore in the plane of the forces. MECHANICS OF SOLIDS. 55 Substituting U) Equation (53), for X. and K, their values from Equations (41), we obtain. R = y/ {P f cos a' + P" cos a") 2 + (P~cos /3' + P" cos /3") 2 ; and since cos 2 a' -|- cos 2 ,6' =1, cos 2 a" -f cos 2 £" t= 1, this reduces to m R - y/P" 1 + P ' 2 + 2 P' P" (cos a' cos a"~4- cos /3' cos /3") ; denoting the angle made bv the directions of* the forces by 8, we have, cos a' cos a" -f- cos /3' cos /3" = cos 5 ; and therefore, i? = y/P'* + P" 2 4- 2 P' P" cos 5 (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 intensities of the components, which passes through th* 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' 4- P" cos a", R cos b = P' cos /3' 4- P" cos /3", and because a' = 90° - /S', a" = 90° - /3", a = 90° - 6, these Equations reduce to, i2 cos a = P' cos a' 4 P' f cos a", # sin a = P' sin a' + P" sin a" ; 66 ELEMENTS OF ANALYTICAL MECHANICS, and, by division, • Ttl ' > i Till ' II sin a P sin a -f- P sin a • eosu " P' cos a' + P" cos a' 7 ' clearing fractions and transposing, we find, P" (sin a" cos a — cos a" sin a) == ■/*' (sin a cos a' — cos a sin a') ; whence, P' sin a" cos a — cos a" sin r/. sin (a" — a) i~tt\* P" sin a cos a' — cos a sin a' sin (a — a') That is to say, the intensities of the components are inversely propor- tional to the sines of the angles which their directions make 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 which they make with the diagonal through their point of meeting. Whence, Eqs. (06) and (5«)\ The resultant of any two forces, applied to the same point, is repre- sented, in intensity and direction, by that diagonal of a parallelogram of which the adjacent sides represent the components. Making a — a' = the angle R m P' = p' and R + P > + p> — o, 2 we have, from the usual trigonometrical formula, ^'=\^HrJ^ W § 83. — In the triangle R m P', since P' R is equal and parallel to the line which represents the force P", the angle m P' R = m — -? and m»— -■> denote the dt* dt 2 at 1 components of this force \n the directions ->f the axes. MECHANICS OF SOLIDS. 59 §87. — Examples. — 1. Let the point wi, be solicited by two forces whose intensities are 9 and 5, and whose directions make an angle with each other of 57° 30'. Re- quired the intensity of the force by which the point is urged, and the direction in which it is compelled to move. Fir->t, the intensity ; make in Equation (56), P = 9, P" = 5, S = 57° 30' ; and there will result, R = V 81 + 25 + 90 x 0, 537 = 12,422. Again, substituting the values of j, P' P" and R in the first of Equations (59), we have, , 5 X sin 57° 30' sin

+ p" + R nR + n R + R (2n -f 1) H 2 - - * - 2 - 2 ; and, Equation (58), . . /(S - P') (S - P") m h* = V ]Fp* ' 60 ELEMENTS OF ANALYTICAL MECHANICS. which reduces to sin** = ±JL If n be equal to unity or the resultant be equal to either force,

n + o p' p" cos s = a \ and by condition, P' - p» = b , . . , . . (c). Squaring the second and subtracting it from the first, we get 27>'P" (1 -f cos S) as a 2 - 6 2 ; which, replacing (1 -f cos £) by 2 cos 2 ^ 5, reduces to « 2 - 6 2 4P' P" ^ cos 2 ^ 8 This added to the square of the Equation ( c ), gives V cos-* ^ o from which and Equation (c) we finally obtain, V - /, 2 (I COS 2 i COS 2 **) 8 V - b 2 (1 — cos 2 **) which are the required components. To find the angles which their directions make with the resultant, we have from Equations (59), " 62 ELEMENTS OF ANALYTICAL MECHANICS. sents the first force, draw the line P' n equal and parallel to m P A 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 ; m n' 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"), join- ing 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 appli- cation, 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 = 2 IP' (cos (3' 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 con- structed from the above equation. § 90. — To find the resultant in this case, by a graphical construc- tion, let the forces P\ P", P>" &c, be ap- j, ^ plied to the points m\ \ ^r ~P *" m", m'" , &c, respec- ^» / \ / tively. Produce the / \ _./- {,„ J & / \ Oj K directions of the forces - ^r y© V /\ P f and P" till they \/ \ / \ meet at 0, and take / J? \ this as their common |2>" \ point of application ; lay off from 0, on the ines of direction, distances S and ST. MECHANICS OF SOLIDS. 63 proportional to the intensities of the forces P' and P", and construct the parallelogram S R S', then will R 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!" = + P" + P'" + &c.' " - C ° S y '- The denominator of these expressions, being the resultant, is essen- tially 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 lohost 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, itfter replacing X y Y, and Z, by their values in Equations (41), By . cos a — Rx . co.^ b 4 cos /3' . 2 P V — cos a' . 2 P'y' = ? Bx . cos c — Bz . cos a 4- cos a' . 2 P'z' — cos y' . 2 P'x'z= 0, Pz.cos b — By .cose + cos y' . 2 P'y' - cos £'.2 PV= j aihl because, cos a = cos a', cos b = cos /3', cos c = cos y ; vre hare, (By - IPi/) . cos a'- (Bx - 2?V) . cos fPss 0, (ite - 2PV).cos y- (Bz - IP'z') . cos a.' =. (Bz - 2PY) . cos #'- (By - 2 P' ; f) . cos y' = ; raid because a', ,8' and 7', are connected only by the relatioi MECHANICS OF SOLIDS 65 cosV + cos"/3"' -}- cos'/ = 1 ; either two of the cosines of these angles are wholly arbitrary, and from the principle of indeterminate co-efficients, we have, by dispens- ing with the sign £ and writing out the terms, Rx = P'x' + P"x" -h P"'x' n + &c ' Ry = P'y' + P"y" + P"'y'" -f &c. K • i?2 = PV + P"*" + P'"z'" -f &C. , (61) 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 given to the forces, provided, they remain parallel and retain the same intensity and points of application, these latter elements being the only 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 be defined 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' + py + p'" x'" + &c p> ~ + _ p' , + p' •• • - X = y = z = "" + &c. py + P"y" 4- P'"y"' + &c - P'z' -f P"z" -f P'"z'" + &c P' 4- P" -f P'" 4- &c. (6-2) Hence, e/7Aer co-ordinate of the centre of a system of parallel force* is equal t' the algebraic sum of the products which- result from multi- plying the intensity of each force by the corresponding co-ordinate of its point of application, divided by the algebraic sum of the forces. If >Le points of application of the forces be in the same plane. 66 ELEMENTS OF ANALYTICAL MECHANIC©. the co-ordinate plane xy, may be taken parallel to this plane, in winch case and, z = z' = z" = «"' = z"" &c. ; (P' + P" + P'" -f &c.) z' p, + p „ + p,„ + &c> = •: 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, take the axis of x parallel to this line ; then in addition to the above results we have y' = y" 'j ttt &c. ; and, (P f + P" + P'" + &c) y' V - P' + P" + P'" + &c. ~ V ' 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 xy to contain their directions, in which case, Equations (60) and (61) become, R = P' + P" Rx = P'x' + P"x" z = and y = 0. Multiplying the first by x\ and subtracting the product from the second, we obtain R(x - x') = P" (x" - O . . (a) Multiplying the first by x" and sub- tracting the second from the product, we get R (x" -x) = P' (x" - x') . . . . ( b ) Denoting oy S' and S", the distances from the points of application MECHANICS OF SOLIDS. 67 of P' and P" to that of the resultant, which are x — x' and x respectively, we have x tt x' = S f + S 1 ti and from Equations (a) and (b), there will result P' : P" : R : : S" : S' : S" + & (63) If the forces act in opposite directions, then, on the supposition that P' is the greater, will R = P' - P" Rx = P'x' - P"x" z = 0, y = 0. and by a process plainly indicated by what precedes, P' : P" : R : : flf : 5" : S" - S'. . (64). From this and Proportion (63), it is 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 com- ponents, while the reverse is the case when the components act in opposite directions. In the first case, then, the resultant 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 solicit 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, accord- ing as they act in the same or opposite directions, that it always acta 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. 68 ELEMENTS OF ANALYTICAL MECHANICS. ** TmT §"96. — Examples. — 1. The length of the line m f m" 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 re- sultant, and its point of application. R = P' 4 P" = 15 + & = 20 ; R : 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 m" and m f . 2. — Required the intensity and point •of application of the resultant of two parallel forces, whose intensities are de- noted by the numbers 11 and 3, and which solicit the extremities of a right line whose length is 16 feet in opposite .directions. mf R = P' - P" = 11 3 = 8, P* - P" : P' : P' . m" m f m" m' : m" o = ~' "' "' = 22 feet. P' - P" 3. — Given the length of a line whose extremities are solicited in the same direction by two forces, the intensities of which differ by the n a 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 /, denote the length of the line. Then, by the conditions, F = P" 4 -P" = ( ! ^) P" V ft / n '2/i 4 1 (*l±A) p" P" ::2l:m'o = 2nl 2n 4 1 SO z=z I 2nl 1 2n 4 1 2n 4 1 I 1 MECHANICS OF SOLIDS 69 1 97. — The rule at the close of §05, 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 P iv , be several forces applied to the material points m, m\ m", m"\ and m iv , in parallel directions. Join the points m and m' by a straight line, and divide this line at the point o, in the inverse ratio of the intensities of the forces P and P' ; join the points o and m" by the straight line 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 m iv 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 -{- P' : P f : : m m' : m o = P' . m m' P + P t y that of o' from P + P' + P" : P" :: om" : oo' P" .om t l E>/' » P+P* + P that of o" from P + ^ + P"- P" f : - F"' : o' «'" : o' o" = P'" .o'm"' P + P'+P" -F n > and finally, that of o'" from P+P'-f P"_J>"' + i>" : P» ::o"m*: o" o Ht P iT . o" m p + p> + p"-.p>" + l 70 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 m\ at which the jjx component P' is applied, is found from the formula m"m'.P" 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 some point between the points of application. The forces in this case act in opposite directions, are equal, but not immediately opposed. To such forces the term couple is applied. The perpendicular distance between the lines of direction of the forces is called the arm of the couple and the product of the intensity of either force into the arm is called the moment of the couple. The effect of a couple is to produce, or tend to produce, a motion of rotation about a line perpendicular to the plane in which the forces act, which line is called the axis of the couple. If there is no motion of rotation, the direction of the line is given only, but if motion takes place, both the position and direction of the axis are determined. No matter where we assume the position of the axis, in case of no motion, the rotatory effect will always be measured by the moment of the couple. And a little study will show that the effect of a couple on a rigid body will not be altered, whatever be the position of the plane of the couple, provided the direction of the axis is un- altered and the arm and forces are the same, and in general it may be shown that a couple is equivalent to, and may be replaced by, any other couple whose moment is equal and the direction of its axis is the same. Now, in all these transformations, the arm and the forces may be altered in position, in length, and in magnitude, and MECHANICS OF SOLIDS. 71 the plane in which the forces act may occupy any one of its parallel positions. But the axis and the moment must remain the same, and these latter cannot be changed without altering the effect of the couple — the former has a fixed direction and the latter is a fixed quantity. It is convenient in these forces of rotation, as in forces of trans- lation, to have geometrical lengths as adequate representatives ; and such we shall obtain if along the axis we take lengths containing the same number of linear units as the moment of the couple contains units of pressure. Thus, if the force of a couple is 4* and the length of the arm is 3, the moment is represented by 12 ; and if along the axis 12 linear units be measured, this length is a full and adequate representation of the couple ; and as, moreover, couples may be right- handed or left-handed, that is, have positive or negative signs, so from the origin of the axis may the line be taken in one or the other direction and thus indicate the sign of the couple. Now if we tech- nically call this line the moment axis of the couple, in contradistinc- tion to the direction of the axis called the rotation axis, it will indi- cate three things, viz., the line of rotation, a finite length, measured from a given point on the line, and the direction in which it is measured. This axis then fully determines all the circumstances of the couple. If then by coaxal couples w r e understand those whose rotation axes are in same direction, and by coaxal and equimomental couples those that are statically equivalent, we can readily demonstrate the following theorems, viz. : 1. The resultant of many coaxal couples is a coaxal couple whose moment is equal to the algebraic sum of the moments of the com- ponent couples. 2. If two lines meeting at a point represent the moment axes of two couples, that diagonal of the parallelogram constructed on these lines, which passes through this point will represent the moment axis of a single equivalent couple. It readily follows that couples may, by means of their moment axes, which are their geometric representatives, be resolved and com- pounded according to the same laws as forces of translation, by means 72 ELEMENTS OF ANALYTICAL MECHANICS. of their equivalent lines of action, and whatever is true of forces of translation is also true of forces of rotation, as exhibited bv the moment axes of the couples which are their geometric representatives. § 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, F=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. WORK OF THE RESULTANT AND OF ITS 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 the resultant by R, and the projection of its virtual velocity by dr, we have from Equation (27), — Rdr + P . dp + P' . dp + P" . dp" + &c, = 0, or, Rdr = P.dp + P'. dp' + P" . dp" + &c, . . . (65) in which P, P', P", &c, are the components, and dp, dp', dp", = [2, P sin y . k] 6(f> . (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 compo- nents, the one parallel, the other perpendicular to that line, and the former is neglected because, in this motion, it cannot work. 76 ELEMENTS OF ANALYTICAL MECHANICS. § 105. — The product 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, and it measures the capacity of the force to produce rotation about that line as an axis. § 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 com- ponent 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 6

in Equation (74), we may write [2 P (x' cos j3 — y cos a)] d(f> = [2 P sin y . k] deb, . (74) or, As P (x'cosfi — y' cos«)] d(f>= f[l P.siny . k] def) . (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 unWs distance from the axis. g 108. — The whole quantity of work will result from the integra- tion 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 work is obtained by mul- tiplying the constant moment into the path described by a point at MECHANICS OF SOLIDS. 77 the unit's distance from the axis. In the second, the force may bo constant and the lever arm variable ; 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 remarks are equally true of the forces of inertia. The in- tensities of these depend upon the masses of the material elements and their degree of acceleration or retardation ; their points of appli- cation are on the elements themselves; the elementary arc described at the unit's distance is the same for both sets of moments, and the value of the moment of inertia depends upon the distribution of the material with reference to the axis of motion. The moments of the forces which urge a body to tarn in opposite directions about any assumed axis must have contrary signs. The sign of P siny k', or its equal P cos/3 . x — P cos« . y\ de- pends upon the angles which the direction 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 makes 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 cos/3 . x' and P cos a . y' will be positive, and if the first of these products exceeds the second, the moment will be positive ; but if the latter be the greater, the moment will be negative. The same remarks apply to the other axes. Since the effect of the moment of a force is analogous to that of a couple, and since the measure of this effect depends no less upon the lever arm than upon the intensity, we may, as in couples, repre- sent geometrically the value of the moment with reference to any moment axis, by taking as its representative a length on the axis, in the proper direction, equal to as many linear units as there are units in the product of the intensity by the lever arm. 78 ELEMENTS OF ANALYTICAL MECHANICS. If the line with reference to which the moment is taken is a component axis, then the length will be found by multiplying the intensity of the component perpendicular to the axis by the lever arm of the component. COMPOSITION AND RESOLUTION OF MOMENTS. § 109. — The forces being supposed to act in any directions what- ever, join the point of application of the resultant R 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 -jj = cos £ -^ — cos £ z -Yj. = cos e, XZ in which £, £, and e, denote the angles which the line H makes with the axes x, y, and z, respectively ; then will R . H . (cos b . cos £ — cos a . cos £) = L, R . H . (cos a . cos e — cos c . cos £) = M, r . . (75) R . H . (cos c . cos £ — cos b . cos e) = N. J Squaring each of these Equations and adding, we find f cos 2 b . cos 2 £ — 2 cos b . cos a . cos £. cos £ -f- cos 2 « . cos 2 £ R l . H* i -f- cos 2 a . cos 2 e — 2 cos a . cos c . cos e . cos £ -f cos 2 c . cos 2 J < -r-cos^c . cos 2 £ — 2 cos&.cosc .cos^.cose -f- cos 2 6.cos 2 e > = Z 2 -f if 2 + iV 2 (76) But cos 2 a -+- cos 2 b -f cos 2 c = 1, (77) cos 2 £-f cos 2 £ + cos 2 e = 1, ..... (78) cos a . cos £ + cos 6 . cos £ -J- cos c . cos e = cos 0, . (79) MECHANICS OF SOLIDS. 79 the angle being that made by the line H with the direction of the resultant. Collecting the coefficients of cos 2 a, cos 2 6, cos 2 c, and reducing by the following relations, deduced from Equation (78) ; viz.: cos 2 e -\- cos 3 £ =s 1 — cos 2 £ cos 2 £ -f- cos 2 e as 1 — cos 2 £, cos 2 £ -f- cos 2 £ = 1 — cos 8 e, we find, IP . iT 3 . [1 — (cos a . cos £+ cos 6 . cos |-J~ cos c , cos e) 2 ] =Z 2 -f 3P + N* ; from Equation (79), 1 — (cos«»cos£ -}- cos 6, cos£ -^ cose, cose) 2 = 1 — cos 2 = sin 2 ; which reduces the above to i? 2 . IP, sin 2 ss Z 2 + JP + A' 2 . But /T 2 .sin 2 ^ is the square of the perpendicular drawn from the origin to the direction of the resultant; it is, therefore, the square or the lever arm of the resultant referred to the origin as a centre of moments. Denoting this lever arm by R, we have, after taking the square root, R.X- */£> + M* + N 2 (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 com- ponent moments, taken in reference to any three rectangular axes through the point assumed as the centre of moments, § 110. — This important relation is evidently the same as that of a resultant force to its components, and it is clear that, if we geometrically represent a moment by the diagonal of a rectangular parallelopipedon, then will its sides represent the component moments. Equation (80) may, therefore, be called that of the parallelopipedon of moments. 80 ELEMENTS OF ANALYTICAL MECHANICS. § 111. — Assuming the linear representative for the moment of a force as indicated in Article 108, and combining the results that follow with Equation (80) y we derive in succession all the rules for the composition and resolution of moments, and they are perfectly analogous to the rules for the composition and resolution of forces. Thus, representing by 0„ y , and O z the angles which the resultant axis makes with any three rectangular co-ordinate axes through the centre of moments, we shall have R. K . cos0 2 =Z 1 R. K. cos0 y = Jf \ (81) R .K . cos0, = iVj 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. And since the axis z may have an infinite number of positions »nd still satisfy the condition of making equal angles with the result- ant axis, we see, Equation (81), that the sum of the moments of the forces in reference to all component axes which make equal angles with thi resultant axis will be constant. From Equations (81) we readily obtain L L cos 2 = ^ — r- = — __ . . . . (82) M M cos0 y = — — ^ = . =3 .... (83) N N COS 0, =r — -, = - — .... (84) R . K V/,2 + M2 + j\T2 whence we conclude that, the cosine of the angle which the resultant axis makes ivith any assumed line is equal to the sum of the moments of the forces in reference to this line taken uj a component axis divided by the resultant moment. MECHANICS OF SOLIDS. 81 For the same system of forces and the same centre of moments, it is obvious that R and K will be constant ; whence, Equation (80), the sum of the squares of the sums of the moments in reference to any three rectangular axes through the centre of moments, taken as com- ponent axes, is a constant quantity. § 112. — Denote by rt Q y , 0„ the angles which any component axis makes with the co-ordinate axes z, y, and #, respectively, and by 6 the angle which the component and resultant axes make with each other, then will cos 6 = cos O z . cos 6 Z 4- cos y . cos 6 y -j- cos Q x . cos B z ; multiplying both members by R . K, we have R.K.cos 6 = R. iT.cos O z . cos 6 Z -j- R.K. cos O y . cos 6 y -f R.K.cos & x . co&B^ But, Equations (81), R . K . cos O z = Z, R . K . cos Q v ss Mi R. K .cose x =zW; which substituted above, give R . K . cos 5 ■=. L . cos d z -f- M . cos B v -f N . cos X „ „ (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 mul- tiplying 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 (B) AND ( C). § 113. — Equations (B) and (C) 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 body. These conditions arc six in number; viz. : *. 82 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 2 P. cos « = 0, 2P cos/3 = 0, > (B) 2 P. cos 7 = 0;J 2P. (x f cos (3 — y' cos a) = 0, "* 2 P. (z\ cos a — x f cos y) == 0, 2 P . (y f cos 7 — z cos j3) = 0. in The above conditions, which relate to the action of a system of forces on a free 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, $x t = 0, *y, f o, H = 0; and it will only be necessary that the forces satisfy Equations ( C), these being the co-efficients of the indeterminate quantities that do not reduce to zero. Hence, in the ease 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 zero. Should the system contain two fixed points, one of the axes, as MECHANICS OF SOLIDS. 83 that of x, may be assumed to coincide with the line joining these points, in which case, there will result in Equation (40), 6x t ~ 0, i? =0, 8y t = 0, S± = 0. Sz, = 0, and it will only be necessary that the forces satisfy the last Equa- tion in group ( C) ; 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, 8 x t 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 the given forces parallel to the line joining the fixed points, also reduce to zero. If three points of the system be constrained to remain IE a fixed plane, one of the co-ordinate planes, as that of xy, may be assumed parallel to this plane; in which case, ty a 0, 8* = 0, <*4, = 0; and the forces must satisfy the first and second of Equations (B). and the first of (C); that is, the algebraic sum of the components ef the given forces parallel to each of two rectangular axes parallel to the given plane, must separately reduce to zeto, and the sum of the moments in reference to an axis perpendicular to this plane must reduce to zero. CENTRE OF GRAVITY. §115. — Gravity is the name given to that force wnich urges 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, tie same co-ordina'es as thes* part icles. 84 ELEMENTS OF ANALYTICAL MECHANICS. 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 bod v. The direction of the force of gravity is perpendicular to tin- earth's surface. The earth is an oblate spheroid, of small eccentri- city, whose mean radius is nearly four thousand miles ; hence, as tho 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 system of forces is determined by Equa- tions (62), § 94, which are _ P'x' + P"x" + P"V" + &c. 5 ] *' ~ P'.rfc P" +]P'" + &e> _ FY + P"y" + P'"y'" + &c. I , Hi - pf + p» + p»> + &c II _ P'z' + P"z" -f- P'"z'" + &c. *' ~~ f +7"T^"' + &c. ' in which x t y t z i% 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\ m", &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 teference 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 (I), P' = w' = mY; P" r= *" -r. m" g" ; P'" = w'" s* m'" q'" , &c. MECHANICS OF SOLIDS. 85 and, Equations (86), x. = y» z. = __ m'g'x' + m"g"x" + m"' g'" x' n -f &c m'/ + m" g" + m'"/" + &c. ro^y' + m" g" y" 4- m f " g'" y'" + &c. wt'/g' + m" g" z" + m'" g'" J" ± &c »»y + »» f7 p + m"'/" 4- &c. .... {sr\ g 1 16. — 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 nV 4- m" x" 4- »'"*"' 4- &c ^ x t = y t = m! 4- m" 4- i»"' ■ ■ &c m r y' 4~ m " y" 4- ni "' y'" 4- &c m t i m a 4- m"' 4- &c. z s = m' z' 4- m" z" 4- m'" z'" + &c. , m' + m" 4- »*'" 4- &c (88) 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)'. tnere vail result, *i = y t = * t = v'd'x' 4- v"d" x" 4- v"' d iti jnt „tn 4- &c. — 1 v'd' 4- v"d" 4- v"'d'" 4 &c. v'd'y' + v"d"y" + v"' d'" y"' 4- &c. v' d f 4- v" d" 4- v'" d'" 4- &c. ' v >d'g' 4- v" d"z" 4- v'" d'" z' n 4- &c v' d' 4- v" d" 4- v'" d'" 4- &c. (SQ) 86 ELEMENTS OF ANALYTICAL MECHANICS and if the elements be of homogenous density throughout, we shalJ have, tt = d" = d"' = &c. ; Mid Equations (89) become, *i = y, = */ = v'x' 4- v"x" +. v'"x'" + &c. 1 — > v' 4- v" -t- v'" 4- &c. *V 4- »*jg 4- v" f y"' 4- &c. v' 4- v" 4- *>'" 4- &c. pV 4- gig 4- p"'z'" 4-&c.. *' 4- v" 4- *'" 4- &c. ' ' > (90) 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 2, in its usual signification, Equa- tions (90), may be written, 2 (vx) " *,= y t = Z{vy) _ M>j and if the system be so united as to be continuous, ,/ p m x.dV (91) ** V 5 Jv" y> dV y/ V 5 «> Jv" Z ' dV > m V §119. — If the collection be divided symmetrically by the plane ry, then will Z(vz) = 0, MECHANICS OF SOLIDS. 87 and, therefore, 0; hence, the centre of gravity will lie in this plane. If, at he same time, the collection of elements be symmetrically divided by the plane xz, we shall have, 2 (vy) = 0, y, = ° ; 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. Equations (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-ordi- nates of the extremities are x\ y' ', z\ and x", y", z". To be applicable to this general case of a curve, included within the given limits, Equations (92) become x, = n* f , i in? hf " J xdx.yj ! + _ + — y, = t t — yda t c 'V 1 + dy 2 dx 2 dz 2 + dx* z d x S c :. N /"l + dy 2 dx 2 dz 2 + dx* j (WSj 88 ELEMENTS OF ANALYTICAL MECHANICS. ii» which = fl d*s[ 1 + dy 2 dx 2 + dz 2 dli? (94) Example 1. — Find the position of the centre of gravity of a right line. Let, y — a x + ft Z z = a! x + £', be the equations of the line. Differentiating, substi- tuting in Equations (94) and (93), integrating be- tween the proper limits, and reducing, there will result, — X x t = x' 4- x 2~ i£+£L + A z t z= a', (z' + x") 2 + (3 f , which are the co-ordinates of the middle point of the line ; x' y r z' and x" 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 ol" 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 fol- lows . 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 centn* MECHANICS Ob SOLIDS. 89 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 this process till all the sides be taken, and the lai^t 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 xy to coincide with the plane of the curve, m which case, d z d x = 0, and Equations (93) and (94) become, x. ~ V, = c xdx W 1 dy 2 dx 2 s ydx y 1 nX> L" , dy> ■ T dx 2 8 } r" d x k / 1 - dy 2 (95) • • • • • (96) Example. 3~ — Find the centre 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, y 2 = a 2 — z 2 , whence^ dr. y which substituted in Equations (95), 90 ELEMENTS OF ANALYTICAL MECHANICS. will give on reduction, *, = o, y* = a (x' + *") . s and denoting the chord of the arc by c = x' + x'\ x, = 0, ac . whence we conclude 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, dx = y -dy -y/2 ay — y 2 (a) the origin being at A, and A B 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 DA. Then will and therefore, y = 2a — x\ x ac a if — y' ; dx dy f ^ 2 a — x f w MECHANICS OF SOLIDS. 91 this, in Equations (96) and (95), gives, omitting the accent on the variables, = f» dx Vn? * = x, =. Js xdx \r\ s x y, = f," ydx \l— x * Integrating the first two equations between the limits indicated, and substituting the value of s, deduced from the first, in the second, we have, * = 2z x = — • — : - 3 yV' - y*' ' and from the third equation we have, after integrating by parts, sy, = 2y/2a (y t/ x — f-y/xdy)\ substituting the value of dy, obtained from Equation (a)', and re- ducing, there will result, s Vi = 2 V% a (y V x ~ fV% a — x.dx) 9 and taking the integral between the indicated limits, ^=2 v / ^[y(v / 7 7 - V*) + § (2a -*")*- f (2 a -*')*1: hence, replacing s by its value, and dividing, Supposing the arc *) begin at (7, we have, *' = 0, and, *, = **", 92 ELEMENTS OF ANALYTICAL MECHANICS. If the entire semi-arc from C to A be taken, these values become, *, = |«i y, = a (* - |). Taking the entire arc A € B, the curve will be symmetrical with res pect to the axis of x\ and therefore, V, = 0; hence, the centre of gravity of the arc of the cycloid, generated by one entire revolution of the generating circle, is on the line which divides the curve symmetrically, a^d at a distance from the summit of the curve equal to one-third of its height. THE CENTRE OF GRAVITY OF SURFACES. § 121. — Let L = 0, be the equation of any surface; L being a function of x y z ; then will dxdy, be the projection of an element of this surface, whose co-ordinates are xyz, upon the plane xy; and if d" 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 . dy cos 6" ' But the angle which a plane makes with the co-ordinate plane xy, is equal to the angle which the normal to the plane makes with the axis z, and,, therefore, cos 6" - ± JL dz v <£>• ♦ m * m 1 = rfc — W (97) MECHANICS OF SOLIDS. 93 and hence, in Equations (92), omitting the double sign, d V = dx-dy > w, . . . . . . . (98) nni those Equations become, »y' /»*' *, * i. f Jy'Jx'w x.dx.dy »y' />*' y, = Jy'L" W . y dx . dy .y' /»*' s , = f f Jy" J x" W . Z .dx. dy (99) in which, ,y> n xt *= V= f ,, f ,, w.dx.dy , .... (100) w being a function of ar, y, z. If the surface be plane, the plane of xy may be taken in the surface, in which case, w = J. z = o, and Equations (99), and (100), be- come, A J" »y' /»*' t/y t/x dy .xd x s .y' /»*' y, J y "Jx" dx.ydy 8 (101) »*' /»*' * = /„ [„ dx .dy, ...... (102) in which rho integral is to be taken first with respect to y, and 94 ELEMENTS OF ANALYTICAL MECHANICS. between the limits y" = P m" and y' = P m' ; then in respect to a\ between the limits x" = A P'\ and x' = A P\ lienor V x l = J*"(y" -y').xdx s Vi s _hL"(y m -y f2 )dx = fa" ~y')d> (103) (104) y' and y", denoting running co-ordinates, which may be either roots of the same equation, resulting from the same value of .t, or they may belong to two distinct functions of x, the value of x being the same in each. For instance, if F (xy) = 0, be the equation of the curve n' m" n" m\ it is obvious that between the limits x" = AP" and x' = A P\ every value of .r, as A P. must give two values for y, viz.: y" = Pm" and y' as Pm'. Or if F(xy) = 0, F'(xy) = 0, be the equations of two distmct curves m" n" and m' n\ referred to the same origin A, then will y" and y' result from these functions separately, when the same value is given to x in •each. A rt' X Example 1. — Required the position of the centre of gravity of flu area of a triangle. MECHANICS OF SO- LIDS. 95 Let A B C, be the triangle. Assume the origin of co-ordi- nates at one of the angles A, and draw the axis y parallel to the opposite side B C. Denote the distance A P by x\ and suppose, y" — <**, y' = bx, to be the equations of the sides A C and A B, respectively, then will y" — i' = (« — h ) z, and, x, = y>'2 _ y>2 _ ( a 2 _ £2) ^ / (a — b) x 2 dx Jxt * , ~~3 *' y< J (a — b) x dx if (a* - V)x*dx ■■ tfj V (a — 6) # rf # whence we conclude, ^a£ £/*e 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 J as A, of the polygon, draw lines to all the other angles except those which are adjacent on either \ W x ^ side; the polygon will thus be divided into triangles. Find by the rule just given, the centre of gravity of each of the triangles; 96 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 deter- mination 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 8. — Determine the position of the centre of gravity of a circular sector. The centre of gravity of the sec- tor will be on the radius drawn to the middle point of the arc, since this radius divides the sector symmetri- cally. Conceive the sector C A B, to be divided into an indefinite number uf elementary sectors : each one of these may be regarded as a triangle whose centre of gravity is at a dis- tance 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 t the distance of the centre >f gravity from the centre C, we have, x. l r -l c 2 r . c A'' If the figure be one of revolution about the axis of x. then, denoting by F{xy) = 0, (10») the equation of the meridian section by the plane x y, will X=«y 2 , and Equations (107) and (108), may be written, I ,, (xy) = y 2 — 2px = 0, whence, V = 2it p I xd x, J a i 2 'x and for the entire pyramid, make x" = c, and a:' = 0, which give 8 y, = ?<**; 104 ELEMENTS OF ANALYTICAL MECHANICS. wIhmu-" we conclude that the centre of gravity of a pyramid is on 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 inde- pendent of the figure of the base. The weight of a body always acting at its centre of gravity, and in a vertical direction, it follows, that if the body be freely sus- pended 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 hence it will only be necessary, in any particular case, to deter- mine 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. f'y dxJ\ + d y* nr. ▼ in which d x l y,= ■ '=//W> tHr and y, in which x _- > S /xl n(y"~y') dx ; clearing the fractions and multiplying both members by 2tt, we shall have, 2n.y,s —fl 2ny y/dx 2 +dy\ .... (112) ZTt yi s=f*' l TT{y' 2 -y' 2 )dx ... . (113) MECHANICS OF SOLIDS. 105 The 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 r and x", about the axis x. In the first member, * is the entire length of this arc, and 2 —4-' = 0, dt 2 2P.cos/3 - M> ^f = 0, dt 2 d 2 z 2 P. cos 7 — M • -^ = 0; • • • • (117) Equations which are wholly independent of the relative positions of the elementary masses m\ m" &c, since their co-ordinates x\ y\ z f , ' ("») ?(i»co8r).y,-?(J»««i/a).« J -if(5-r--^-« ( ) =©; Equations from whieh may be found the circumstances of motion of the centre of inertia about the fixed origin. 110 ELEMENTS OF ANALYTICAL MECHANICS. MOTION OF TRANSLATION". § 130. — Regarding the forces as applied directly to the centre of inertia, replace in Equations (117), the values 2 P. cos a, 2 P. cos (3 t and 2 P, cos y; by X, F, and Z, respectively, and we may write,, dv 1 ' dt ~ U ' (120) from which the accents are omitted, and in which x, y, and z, must he understood as appertaining to the centre of inertia GENERAL THEOREM OF WORK, ENERGY AND LIVING FORGE. 5 131. — Multiply the first of Equations (120) by 2 d .r, the second by 2 d y, the third by 2 d z, add and integrate, we have 2j'(Xdx + Ydy + Zdz) - M ., dx 2 + d y 2 + rfz 2 + C = t\ But, dx 2 + rf# 2 -+- dz 2 dfi dt 2 - whence. %f(Xdx + Ydy + Zdz) - M.V 2 + C = • • (121) The first term is, § 101, twice the quantity of energy expended or of work done by the extraneous forces, the second is twice the quan- tity of work of the inertia, measured by the living force, and the third is the constant of integration, or twice the quantity of work, of inertia in system before the forces began to act. If the forces X, Y, Z, be variable, they must be expressed in functions of x, y, z, before the integration can be performed. MECHANICS OF SOLIDS. Ill Supposing this latter condition fulfilled, and that the forms* of the functions are such as make the integration possible, we may write, F{xyz) - \M.V> -f C'= 0, . . ; . (122) und between the limits x 4 y 4 z 4 and x/ y' z/ , *W y/ */) - F <*. v, *,) = i *&* - 7S% : ■ (123) whence we conclude, that the quantity of energy 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 = Q, the extraneous forces will, §101, be in equilibrio, and v 2. L" that is, the velocity will be constant, and the motion, . therefore, uniform, § 132. — Again, multiply the first of Equations (118) by rf, the third by d zs ; add and reduce by the relations given in Equations (38) : we find , , . r, , ,, r, . , /j Pr> rs j dx'' + dy'* + dz'> J Rdr =J RK.ds t \ -^ = v\ &c; whence fR.K.ds, = 12 7??/+ C Adding this to Equation (121), there will result 2f(Xdx + Vdy + Zdz) + 2j*R.K.ds, = AfV*+ Iwr**+C (121/ From which it is apparent that the quantity of energy expended upon a body, or the living force with which it will move, is dependent not only upon the intensity of the force, but also upon the distance of its line of direction from the centre of inertia. v § 133. — If Equation (121) be applied to each one of a collection of elements, of which the masses are m, m\ 8 = C. This is called the conservation of living force. STABLE AND UNSTABLE EQUILIBRIUM. 134. — Resuming Equation (123), omitting the subscript accents, and bearing in mind that the co-ordinates refer to the centre ol inertia, into which we may suppose for simplification the body to be concentrated, we may write, $ AT V' 2 - \MV* = F(x'y'z') - F{xyz\ in which F(xyz) = f(Xdx + Ydy + Zdz\ and dF(xyz) = Xdx + Ydy -f Zdz. Now, if the limits x' y' z' anil xyz be taken very near to eacli other, then will x' = x + dx\ y' = y + dy\ z' = z + dz\ which substituted above, give \MV' 2 - \MV 2 = F(x + dx, y + dy, z + dz) - F(xyz), ; and developing by Taylor's theorem, i Adx + Bdy -f Cdu * * ( + A'dx* + B'dy 2 + &c. +/>, in which D denotes the sum of the terms involving the highei powers of dx, dy and dz. 114 ELEMENTS OF ANALYTICAL MECHANICS. If £ M V 2 be a maximum or minimum, then will Adx + fidy + Cdz = 0; (123)' and since Adx + Bdy + Cdz = dF{xyz) = Xdx + Ydy + Zdg t we. have. Xdx + Frfy + Zdz = 0. But when this condition is fulfilled, the forces will, Equation (6!>), 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 work* or energy will be a maximum or minimum, though this is not alwttys vtrue, since the function is not necessarily either a maximum or a minimum when its first differential coefficient is zero. § 135. — Equation (123)' , being satisfied, we have $M V' 2 - \M F 2 = ± (A'dx* + B'dy* + &c. + D) • • • (124) The upper sign answ r ers 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 verv small and less %han F, in order that the second member may be negative ; whence it appears that whenever the system arrives at a position in which the living force or quantity of work ij» 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 l)t ought it tnere. This position, which we have seen is one of equi- librium, 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 the forces, which, as soon as it overcomes the living force already existing, will cause the body to retrace its course. MECHANICS OF SOLIDS. 115 If, on the contrary, the body reach a position in which the living force 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. If the entire second member of Equation (124) be zero, then will \M V' 2 - \MV* = 0, ana 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. g 136. — 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 con- stant. Denoting these weights by W, W", W"\ &c, and assum- ing the axis of z vertical, we have from Equations (87), Rz, = W'z' + W" z" 4 WT"V" + &c, in which R, is the weight of the entire system, and z t the co-ordi- nate of its centre of gravity; and differentiating, Rdz, = W'dz + W'dz" 4- W"'dz'" + &c . . . (125) Now, if z, be a maximum or minimum, then will W'dz' 4 W'dz" + W'"dz'" 4 &c. = 0, which is the condition of equilibrium of the weights. Whence, wo conclude that when the centre of gravity of the system is at the highest or lowest point, the system will be in equilibrio. In ordei 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 t , 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. 116 ELEMENTS OF ANALYTICAL MECHANICS. Integrating- Equation (125) between the limits z=zH and z =JT, • e' = h and z' s= h\ «fcc., and we find, J2 (#, - H) = IT ft - h') + WP» ft - A") + cfcc. ; . . (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. THE POTENTIAL FUNCTION. § .137. — If, for any limited system of invariable masses, 2m, acted upon by the forces X, Y, Z, functions of the masses and of the variables of position, x, y, z, integration be possible, and we denote by the letter IT the function which expresses the quantity of action of the forces exerted during any change of position, or configuration, then will dU = iPdp = Xdx -f Ydy -f Zdz be an exact differential. This function, called by Lagrange simply the function IT, but now generally known as the potential, is one of great importance in the investigation of the forces of nature. In Articles 131 and 134, we have already discussed some of the relations of the function, F(x, y, z,) = f(Xdx + Ydy + Zdz). But as the second member of this equation is often used to signify work done, or resistance overcome, it becomes necessary that we should consider what other important relations present themselves when it is used to denote not work itself, but the quantity of action of the forces expended in doing that work. The distinction is that of action and reaction, cause and effect; and we have already remarked upon it. To indicate that amount of stored action, rather than work, is the meaning of the function II. The name potential energy is used for it by Thomson and Rankine ; and these able writers also employ the terms actual, or kinetic, energy, in place of the unmeaning half vis viva, or living force, which measures the quantity of working power of a moving body. New names, which appear to be coming rapidly MECHANICS OF SOLIDS. 117 into general use, and with advantage of greater clearness of thought and expression. For the sake of brevity, we shall use the simple name potential for the function IT, but it will always be understood to mean poten- tial energy, the power to do, instead of the work done. And it is the quantity of action of the forces, attractive or repellent, when exerted to produce changes of position, or distance, between masses, or molecules. To illustrate physically the meaning of the term, the potential of elasticity for a bent steel spring is its power or quantity of action in the recoil to a neutral state of equilibrium, from which it has been forced by bending. The stored power of an elevated weight to per- form by descent an amount of work is its potential of gravity. The amount of power exerted by the iron rim, or tire, of a carriage wheel, when chilled from a hot state, and thus made to contract powerfully upon the wooden frame of the wheel, is its heat potential. All such are simply changes of power, varying with relative position; and many like examples might easily be ~ adduced, i CONSERVATION OF ENERGY. § 138. — Power and work, action and reaction, always bear to each other the algebraic relation of positive and negative quantities. When forces work, the potential is, therefore, a decreasing and the work an increasing function. Or expenditure of power produces increase of work. Bearing this in mind, and integrating between limits, the expression gives n -n = 2^(,2-v). From which we obtain, Hence, it appears that the total amount of power, or of energy, in any limited system of masses and forces, in which the forces are functions only of the variables x, y, ar, must always be constant. 118 ELEMENTS OF ANALYTICAL MECHANICS. Or, following Lagrange, we may denote the total energy of the system by the constant H, and the kinetic energy, or half vis viva, by F, which substitution gives n + V = H, (D) an equation which may be enunciated thus: the total energy, both potential and kinetic, is constant. This principle is now usually called the law of conservation of energy ; but in the precise form here given, Equation (Z>), it is used by Lagrange, though by him called the principle of living force. It is evident that the total energy of any system can be constant only when neither the masses, nor the forces acting upon them, all other things being equal, vary with time. In other words, there must be neither growth nor decay, neither invigoration nor enfeeblement, with age or lapse of time. Any system thus invariable in energy is said to be dynamically conservative. If, however, the energy be not confined in the system itself, but be expended upon bodies foreign thereto, then clearly there will be loss, or dissipation of energy. Thus, motive power in machines is by friction transformed into heat, and then dissipated by conduction and radiation. But if dissipated and transformed, it is only transferred to other masses and not annihilated; power, therefore, appears to be indestructible. From the above it will be seen that Equation (Z)) is substantially identical with our fundamental formula (yl), they being only equiva- lent algebraic transformations for the same general law DISCUSSION OP THE FUNCTION II. § 139. — The potential II has many properties which render it of great importance in the investigation of the phenomena of gravitation, elasticity, heat, electricity, and magnetism, only a few of which can here be given. When the action of the component forces, X, J", Z, depends upon changes of position, ar, y, z, and may be expressed by the sin- gle potential function d II = Xdx + Ydy + Zdz, MECHANICS OF SOLIDS 119 or by its equivalent, ail = -— ax 4- -— rt?/ H — — dz, dx ^ dy J ^ dz ' . the requisite conditions of integrability are dX _dY dX _dZ dY _dZ dy dx 1 dz dx' dz dy' From the above equations, and from Equations (43), we obtain, A = —r- = It cos a, u t - —i 9 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, « 2 §143. — In the preceding discussion, no account is taken of the atmospheric resistance For the same body, this resistance varies m 126 ELEMENTS OF ANALYTICAL MECHANICS. the square of the velocity, so that if £, denote the velocity when the resistance becomes equal to the body's weight, then will M . g . v 2 be the resistance when the velocity is v, and in Equations (117), we shall have, v 2 2 P cos a ss X = M g cos a -\- M g - — • cos a', K * 2 2 P cos /3 = F = if a COS /3 -f if flr . . C os /3', 2 P cos 7 = Z = M g cos 7 -f- M g • — • cos 7' ; taking the co-ordinate z, vertical and positive downward, then will, cos a ss cos a! = 0, cos (3 = cos /S' = 0, cos y s= 1, cos 7' = — 1 ; and Equations (120) give, 1 c? 2 z . rfv Omitting the common factor if, and replacing — — by its value — - dv / v 2 \ whence, F.rfv & (dv dv \ integrating and supposing the initial velocity zero, !,t = ik.log £4—- ... • (140) MECHANICS OF SOLIDS. 127 which gives the time in terms of the velocity; or reciprocally, r> in which e, is the base of the Naperian system of logarithms, and. from which we find, e k — e k ) v = — - e k + t k which gives the velocity in terms of the time. Substituting for », dz its value — i integrating and supposing the initial space zero, w« (X v have k 2 , — -'—^ 2 = l.l og i^- +e A ( 143 j Multiplying Equation (139) by dz we have, 5i= •• . k 2 .v .dv adz — 1 and integrating, observing the initial conditions as above, k 2 k 2 which gives the relation between the space and velocity. _ l± As the time increases, the quantity e * becomes less and less, and the velocity, Equation (142), becomes more nearly uniform ; for, if t be infinite, then will r * = o, and, Equation (142), v = k\ making the resistance of tho air equal to the body's weight. 128 ELEMENTS OF ANALYTICAL MECHANICS. §144. — If the body had been moving upwards with a velocity v, then, taking z positive upwards, would, Equations (120), d 2 z a v 2 dv d 2 z substituting — - for -ry an ^ omitting the common factor, we find, k .dv g d t k 2 + v 2 k ' integrating, ~~ l v 9 * ^ tan T = -T +C; and supposing the initial velocity equal to a, we find _1 « V = tan — > k (145) tan - = tan — - *— (146) nid, —l tan k k k Taking tne tangent of both members and reducing, we find fft a — k . tan -— ■ t = * (147) at k 4* « • tan — k which may be put under the form, 9* j . 9* a . cos — k . sin — vz=k 1 1 .... (148) • 9* , . 9* . a . sin - — (- * • cos — & At Substituting for v its value — > integrating, and supposing the initial space zero, we have k2 i ( a • 9* 9*\ fiAQ\ f s= — • log ( — • sin V 4- cos T ) • • (I4y) g * \ k k k / MECHANICS OF SOLIDS. 129 Multiplying Equation (145), by dz and we have, k 2 .v.dv g ,dz = — k 2 + v 2 ' and integrating, with the same initial conditions of v being equal to a, when z is zero, there will result, « = -*! . log F + a2 (150) 2g & k 2 + v 2 v ; §145. — If we denote by A, the greatest height to which the body will ascend, we have z =s h, when v — 0, and hence, h = Jl..} 0g F + a2 (151) 2g ^ k 2 y ' Finding the value of £, from Equation (146), we have, -i n -i t = (tan T -tan y ) . . . . (152) from which, by making v = 0, we have, k — l a . __. * o = tan — ....... (153) 9 * which is the time required for the body to attain the greatest eleva- tion. 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, k 2 . K* h = irr ' ] °g 2g & k 2 - a' 2 and placing this value of h equal to that given by Equation (151 ^ there will result, k 2 k 2 + a 2 -* k 2 - a' 2 k 2 130 ELEMENTS OF ANALYTICAL MECHANICS whence, a* = a 2 a 2 + k 2 ' 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 f in Equation (140), we have, and, substituting for a', its value above, (154) k . %/ a 2 -f k 2 + a t . = — — • log — , • • . . a value very different from that of t a , given by Equation (158), for the ascent. Multiplying both numerator and denominator of the quantity whose logarithm is taken, by v / « 2 + ^ — a, the above becomes, • *' = f ,og 7PW^ (,55) Adding Equations (153) and (155), we have, k T - 1 a k "I t + t, = — tan — -f log n% making t = t a + t- T = tan T + log 7^T=f ' " " (156) K K y k 2 -j- a 2 — a If a ball be thrown vertically upwards, and the time of its absence from the surface of the earth be carefull v 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. i MECHANICS OF SOLIDS 131 PROJECTILES. § 146. — Any body projected or impelled forward, is called a pro- jectile, 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 (1 17), 2 P cos a = X = Mg cos a, 2 P cos £ = Y — Mg cos (3, 2 P cos y = Z = M g cos y. Assuming the origin at the point of de- parture, or the mouth of the piece, and taking the axis z vertical, and posi- tive upwards, then will cos a = ; cos (3 = 0; cos j = — 1 ; and, Equations (120), M~=0; M- d -JL=0- M>~= -Mg; dt 2 dt 2 dt 2 and integrating, omitting if, d x ~d~t = u dy dt d z = u ; -7 — = — gt -j- u y ' . dt y * (157) Integrating again, and recollecting that the initial spaces are zero, we have, x — ■= u • t ; y = u • t ; =•. — £ g - 2 -f- u • t • 132 ELEMENTS OF ANALYTICAL MECHANICS and eliminating /, from the first two, we obtain, u z wnich is the equation of a right line, and from which we see that the traje3tory is a plane curve, and that its plane is vertical. Assume the plane zx, in this plane, then will y = 0, and Equa- tions (158), become, x = u x • t ; z — — \g t 2 -|- * - 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 a?, then will, V. cos a, and V . sin a, be the lengths of the paths described in a unit of time, in the direc- tion of the axes x and z, respectively, in virtue of the velocity V ; they are, therefore, the initial velocities in the directions of these axes; and we have, , u = V cos a : u = V . sin a ; X ' X which, in Equations (159), give x = V . cos a . t ; z = - £ g t 2 -f V. sin a . t • (160) and eliminating t, we find Q X 2 z = x tan a > 2 V 2 . cos 2 a ' or substituting for V its value in Equation (135), x 2 z = x tan a — — -— • • • (161) 4 h . cos* a which is the equation of a parabola. MECHANICS OF SOLIDS 133 § 148. — The angle a is called the angle of projec- tion ; and the horizontal distance A D, from the place of departure A, to the point Z>, at which the projectile attains the same level, is called the range. To find the range, ma«ve z = 0, and Equation (161) gives x = 0, and x = 4 k sin a cos a = 2 h sin 2 a, and denoting the range by i?, i2 = 2 ^ . sin 2 a (162) the value of which becomes the greatest possible when the angle of projection is 45°. Making a = 45°, we have R = 2k (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 £, we have, v 2 = d s 1 dz 2 -+■ dx 2 dt 2 dt 2 or, replacing the values of dz and d x, obtained from Equations (160), v 2 = V 2 - 2 V.g . t. sin a + g 2 fl (164) and eliminating t, by mears of the first of Equations (160), and replacing V 2 , in ths last term by its value 2g A, v i — v 2 — 2g . tan a . x -f- g 2 h cos 2 a . • • (165) 134 elements of analytical mechanics. in which, if we make x s= 4 h . sin a cos a, we have the velocity at the point D, v 2 = F 2 , rthich shows that the velocity at the furthest extremity of the range is equal to the initial velocity. Differentiating Equation (161), we get dz x — z= tan 6 = tan a — — — .... (166) ax 2 h. cos 2 a v ' in which & is the angle which the direction of the motion at any instant makes with the axis x. Making tan & = 0, we find x = 2 h . cos a . sin a, which, in Equation (161), gives z = h. sin 2 a, the elevation of the highest point. Substituting for x, the range, 4 h cos a sin a, in Equation (166), tan 6 — — 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 coordinates are x = a and z = b. Substituting these in Equation (161), we have b — a tan a — <7 2 4 /* . cos 2 a from which to determine a. Making tan a = 4A6 + a 2 . Making, 4 k 2 — 4 h .b — a 2 = 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 6, and this curve being con- structed 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 k. § 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 tra- jectory, and gives to the subject no little complexity. Denote, as before, the velocity of the projectile when the atmos- pheric resistance equals its weight, by k, and assuming that the resistance varies as the square of the velocity, the actual resistance at any instant when the velocity is v, will be, M . g . v 2 Mcv 2 , 136 ELEMENTS OK ANALYTICAL MECHANICS. by making, *-• The forces acting apon the projectile after it leaves the piece being its weight and the atmospheric resistance, Equations (120), become, d 2 x M • -r-z- = M g . cos a -f Mc.v 2 . cos a', a t 2 M.tl. = Mg. cos + Mc.v 2 . cos /3', a r d 2 z M' •—• — Mg . cos y + Mc . v 2 . cos y* (Jb L Taking the coordinates z vertical, and positive when estimated upwards, cos a = ; cos jS = ; cos y = — 1, and because the resistance takes place in the direction of the trajec- tory, and ill opposition to the motion, if the projectile be thrown in the first angle, the angles a', /3', and y\ will be obtuse, dx nt dy dz cos a' = ; — : cos p' = r— ; cos y' = 7 — , d s as «« and the equations of motion become, after omitting the common factor M y d 2 x d x — — - = — c • v l • — : — : dt 2 ds ' &y o d v — — =• — c • v z • — — - : dt 2 ds ' d 2 z , dz dt 2 y ds From the first two we have, by division, d 2 y d 2 x dy dx * MECHANICS OF SOLIDS. 137 and by integration, log dy = log dx -f- log (7; 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, .hus 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 v 2 , by its value from the relation, we have, dt and, dp dt dp dx dx dt whence, making V 2 = 2g A, Equation (173) becomes , let dp e -i » • • • • dx 2 h . cos 2 a and multiplying this by the identical equation, (175) d x . -/I -f p 2 = ds, obtained from Equation (174), we find, 2ct e • ds and integrating, let e in which C is the constant of integration ; to determine which, make 8 = ; this gives p =z tan a ; and 1 . . C = — jr-— + tan a . y 1 -f tan 2 a + log (tana T yi+ tan 2 a) • (177) From Equation (175) w r e have, — 2 c * dx = — 2 A. cos 2 a . e • rfp from Equation (171), dz z= p .dx \ 140 ELEMENTS OF ANALYTICAL MECHANICS. from Equation (172), g d fi = — d x . dp ; and eliminating the exponential factor by means of Equation (176), we find. c.dx = £ ; . (178) p «/ 1 + P l + log (p'+ ■/! + f) - C c.dz = - P ^ - ; • (179) p V { + p 2 + ] °g (p + V l + p 2 ) - i ■ c pd p p V 1 + p 2 + ] °g (p + V 1 + p 2 ) - - c — dp V~^9 ,dt= ■ { ; • (180) J V -P V 1 + P 2 ~ lo S ( P + V J + ^ 2 ) 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 by 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, in which, dz = T.p.dp; .(182) T = l ; • (183) c p y/T~+~P 2 + log (p + V^ +P 2 ) ~ and dividing Equations (181) and (182), by dp, d x dp dz dp T; . . (184) = r.i>; (185) MECHANICS OF SOLIDS. 143 r (1.) m A 9 A c 2 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), multi- plied 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 ordi- nates x and z, separated by intervals equal to dp, may be found, and the curves traced through their extremities. At the point from — ^ E which the projectile is thrown, we have, z=0] z = ; jo=tana, and the auxiliary curves will cut the axis of p, in the same point, and at a dis- tance 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 A B = tan a, and let BzD, and BxE, be constructed as above. Draw the axes Ax and Az, through the point of departure A, Fig. (2) ; draw any ordinate c z t x d to the auxiliary curves Fig. (1); lay off ^Fig. (2) equal to Cx t Fig. (1), and draw through x t , the line x t z t parallel to the axis Az, and equal to cz t Fig. (1) ; the point z t will be a point of the trajectory. The range A D, is equal to ED, Fig. (\\. 10 142 ELEMENTS OF ANALYTICAL MECHANICS. By reference to the value of (7, Equation (177), it will be seen that the value of T r 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, w r ill 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 6 7 , and unity may be neglected in comparison with p, which will reduce Equations (178) and (179) to c .p z — C" -\ • log », c p ' c which will become, on making p very great,. x = C"; z = C" + -log jo, c which shows that the curve whose ordinates are the values of a*, will ultimately become parallel to the axis p, while the other has no limit to its retrocession from this axis. Whence we conclude, that 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°. ' d x dp c .p 2 and integrating, x = = C - 1 MECHANICS OF SOLIDS. 143 Denoting the velocity at any instant by v, we have p dx 2 -f dz 2 dx* — = (i + jp 2 ) ■ r/* 2 rf* 2 and replacing c?x 2 and d* 2 by their values in Equations (178) and (180). we find 1 9 .{l+p 2 ) v 2 _ C C -p y/\ + p 2 - log (p + -/l + J9 2 ) (180) and supposing /> 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 • = v/t=^ 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 1 k 2 in vacuo through a vertical distance equal to 2c 2<7 §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 trajec- Z tory as lies in the imme- diate neighborhood of this line may easily be found. For, the angle of projec- tion being very small, p will be small, and its second power may be neglected in comparison with unity, and we may take, d s = d x ; and * = x ; which in Equation (175), gives, 2c* 3* J) dp ~dx~ d 2 z dx 1 2 h . cos 2 a (187; 144 ELEMENTS OF ANALYTICAL MECHANICS. Integrating, 2 ex dZ ' +0: dx \cji. cos 2 a dz makirg x = 0, we have — — = tan a, whence, C = tan a + 4 c . h . cos 2 a which substituted above, gives, 2c* e?z e = tan a — - — \- dx 4 c . h . cos 2 a 4 c . h . cos 1 a ' «nd integrating again lex € X z = tan a . s — — — — + - — -— 4. C\ 8c 2 . h . cos 2 a \c.h. cos z a making # = 0, then will z = 0, and 1 (7' = 8 c 2 . /i . cos 2 a hence, z -. tanaar— — — ~- (e * — 2tx — l) . . (188) 8 c 2 . A . cos 2 a V / From Equation (172), we have, g .dt 2 = — dx . dp, and substituting the value of dp, from Equation (187), ex - e . dx a t = -y/2 ( ) in which p is the resistance in kilogrammes, v the velocity, n the ratio of the diameter to the circumference, r the radius of the ball, A the resistance on a square mP.tre when the velocity is one t» etre, and 9 the velocity which w r ould 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 rotarv 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. 147 EOTAKY MOTION. § 156. — Having discussed the motion of translation of a single body, we now come to its motion of rotation. To find the circum- stances 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 uniCs 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 d t. give dz' ~~dt djf_ dt dz' dt = z = X = y d-^ IT d 9 ~~dl d-ti ~d7 - r — z — X d

-f- g * V -vA' 2 + y' 2 + z** > < cos/, = V*' 2 + y'* -r z' 2 MECHANICS OF SOLIDS. m and eliminating .r', y' and z\ by Equations (193), cos a t = VV + V + v . 2 cos/3, = VV + \ 2 + v 2 2 :> > cos 7, = VV + %* + V z 2 (194) 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 j dx' 2 + dy' 2 -+- dz' 2 dt 2 =v = S(z\v y -y\y + (x\v-z\v z )H(/-\-*'.v y ) 2 ', Replacing v z , v y and v 2 by their values obtained by simply clearing the fractions in Equations (194), this becomes v= yv z 2 H~ v y 2 -f v 2 2 x yV 2 + y" 1 -f z' 2 — (x' cos a, -f y'cos/3, + z' cos y,) 2 , which is the velocity of any element in reference to the centre of inertia. Making x' 2 -f y' 2 + z' 2 = 1, we have the element at a unit's distance from the centre of inertia; and making x' cos a t -f y r cos fi t -\- z' cos y t = 0, (195) the point takes the position, giving the maximum velocity. In this case v becomes the angular velocity, and we have, denoting the latter by v |f h = yV + v 2 + v,» • • (196) 150 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 tb by their values given in the second ^ b d t dt J fo and third of Equations (192), . 2 m {** + z' 2 ) -»- v z . v z . 2 m (aft - *' 2 ) = M t , ~" .2 m (y' 2 + z" 1 ) -f v y . v z . 2 m (v'» - z' % ) = X> . 152 ELEMENTS OF ANALYTICAL MECHANICS. The axes *', y\ z\ which satisfy the conditions expressed ic Equations (200), are called the principal axes of figure of the body. And if we make 2 m . (y' 2 + x' 2 ) = 9 } 2 m . (x' 2 + z' 2 ) = B, 2 m . {x' 2 + z' 2 ) = B, „ . (y'a + z' 2 ) = A;J we find, by subtracting the third from the second, 2 m . {x 2 - y' 2 ) = B - A, the first from the third, 2», (*' 2 - x' 2 ) = A - C, and the second from the first, 2 m . (y' 2 - z' 2 ) = C - B\ (201) which substituted above, give, d *.£» + .,.vU- A)-. = A? C) = *M„ Jt) = :*.- (202) By means of these equations, the angular velocities v z , v v t , must be found by the operations of elimination and integration. 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 dis- tance from the principal axis z\ the second the same for the prin- cipal 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. 153 in reference to this axis, and it measures the capacity of the body to store up work in the shape of living force during a motion of rotation about that axis. A, B, and C are called principal moments of inertia. § 163. — 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 show this, assume the formulas for the transformation from one system of rectangular axes to another, also rectangular. These are x = x . cos {x' x) -f- y . cos {x' y) -f- z . cos {x' z), y y' = x.co%(yx) + y.cos(y'y) -f z . cos (y' z), I . . (203) z' = x . cos (z r x) -f- y . cos (z' y) -j- z . cos (z' z), in which (-r'#), (y'#), and (z' #), denote the angles which the new axes x, y\ z' y make with the primitive axis of x\ (x' y) y (y' y), and (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 which they make with the axis z. Assume the common origin as the centre of a 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 con- nected with the point JV y 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 = Z'AZ =z X'JYX, being the inclination of the plane x'y' to that of xy. tp = NAX, being the angular distance of the intersection of the planes xy and x'y' from the axis x. = NAX, being the angular distance of the same intersection from the axis x ', 154 ELEMENTS OF ANALYTICAL MECHANICS Then, in the spherical triangle X NX, cos (x 1 x) = cos 4 . cos 9 -f- sin y . sin (p . cos 6 ; In the triangle Y' NX, the side N Y' — -5- + • sin 9 4- sin -^ . cos 9 . cos B 9 In the triangle Z' NX, the side NZ f = —•> and SB cos (z' a;) = sin -^ . sin 0. And in the same way it will be found that cos (#' y) = — sin 4* • cos 9 -f- cos 4 . sin 9 . cos I j cos (y' y) = sin 4 . sin 9 -f- cos %}/ . cos 9 . cos 6 ; cos (z' y) = cos 4* • sin 4 ; cos (x r z) = — sin 9 . sin 6 ; cos (y f z) =. — cos 9 . sin 4 ; cos (0' z) = cos 6 ; and by substitution in Equations (203), x' = x (sin 4 . sin 9 . cos & -}- cos 4 . cos 9) -f- y (cos -^ • sin 9 . cos & — sin -^ . cos 9) — z sin 9 . sin 6 y f = x (sin 4 • cos 9 . cos & — cos 4 • sin 9) 4* y (cos 4 • cos 9 . cos & -f- sin 4 • sin 9) — z cos 9 . sin 6 % z r as # sin -^ • sin 4- y cos 4> • sin -f- cos 6 ; or making, for sake of abbreviation, D = x cos -^ — V sin 4, j£ = x sin -^ . cos & + y cos 4/ . cos 6 — 2 sin d, the above reduce to x' = E . sin 9 + & • cos 9, y f = .#. cos 9 — D . sin 9, f ' = a: . sin 4 . sin ^ +• y . cos 4 . sin 6 -f 2 . cos 6, MECHANICS OF SOLIDS. 155 Substituting these values in the equations lm.x f .1/ = 0; 2 m . ** . z' = ; 2w.y',2' = 0; we obtain from the first, sin 9 . cos 9 . 2 m (E 2 — D 2 ) + (cos 2 9 — sin 2 9) ^mE.D = 0, or, replacing sin . ^ = 0; ■ • • (204) and from the third and second, respectively, cos 9 . 2 m . E . z' — sin 9 . 2 m D . z' = 0, • • • (205) sin 9 . 2 m . E. z' + cos 9 . 2 m D . z' = 0. • • • (206) Squaring the last two and adding, we find (2mJ.z') 2 + pffl.i).2') 2 = 0. which can only be satisfied by making s «.z>.,' = o.( • < 207) These equations are independent of the angle 9, and will give the values of 4* and & ; and these being known, Equation (204) will give the angle 9. Replacing E and D by their values, we have E . z' = sin d . cos & (x 2 sin 2 -^ 4- 2 x y sin \ cos \ -\- y 2 cos 2 4> — z 2 ) 4- (cos 2 & — sin 2 &) (x z sin 4> + y z . cos 4>) , i) . 2' = sin 6 \xy (cos 2 4> — sin 2 40 + (x 2 — y 2 ) sin 4> cos 4'} -|- cos & (x z cos 4> — y 2 sin 40 • and assuming 2 w a: 2 = ^4' ; 2my 2 = 5'; 2 m z 2 = C ; 2mxy = E'; 2mxz = F' \ 2myz — H\ and replacing sin 6 . cos 0, and cos 2 — sin 2 6, by their respective values, \ sin 2 0, and cos 2 0, Equations (20V) become sin 2d {A' sin 2 4, + 2 ^' sin + cos + + £'cos 2 4, - C) + 2cos2d(/ v sin4, + #'cos40 1=., 156 ELEMENTS OF ANALYTICAL MECHANICS. sin 6 \E' . (cos 2 + - sin 2 4) + (A' - B') . sin 4, cos +} 4- cos d (F f cos 4/ — H' sin >L) f- 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* *i y, *> Dividing the first by cos 2 0, and the second by cos 0, they become tan 2 6 . (-4' sin 2 4, + 2 JF sin + cos 4, -f Jff cos 2 4- 2 (i^ sin 4, 4- H' cos 4,) f C 'H=0;(20V)' tan 4 . \E' (cos 2 4, - sin 2 4,) + (v4' - £') sin \ cos >L| ) 4- i^cos^ - #'sin4 J=-( 20 ) From the first of these we may find tan 2 &, and from the second, tan 0, in terms of sin 4', and cos 4* ; and these values in the equation 2 tan 6 . rt . tan 2 6 = - — (208) 1 — tan 2 d v ' will give an equation from which 4' may be found. In order to effect this elimination more easily, make tan 4' = w, whence . , u .. 1 sm 4^ = - ; cos 4> = yTT^ 2 yT+ u 2 ' making these substitutions above, we find 2(F'u 4- H')^/i 4- w tan20 = — A' u 2 4- % lE'u + B' — G"(l 4- m 2 ) tan 4 = (F f - H' u)^/\ 4-i is7'(l — U 2) 4- (.4' - i*') w which in Equation (208) give , B' F'-F'C'-E'H' ) 1 {J?(i-«>)HA'-S')«}\. Hctir _ A , N , +£rF> \ 1 =0 . . (209) -f (jP'w 4- H').(F' - H'uf J MECHANICS OF SOLIDS. 157 which is an equation of the third degree, and must have at least one real root, and, therefore, give one real value for ip. This value being substituted in either of the preceding equations, must give a real value for 0, and this with if), in either of the Equations (205) or (206), a real value for ; 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 xy\ 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'=zO, Equation (209) will become identical ; the problem will be indetermi- nate, 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. . § 164. — Without rotary motion, the spherical triangle XNX is invariable; with it, the point N has a motion with respect to X, X to iV, and Z' to Z. These points being on the surface of the unit sphere, their relative velocities on that surface arc angular, the first about the axis z, the second about the axis z\ and the third about the line AN. Taken in the order named they are, dxf) d t dO It' 5F ; di' 158 ELEMENTS OF ANALYTICAL MECHANICS. The components of the first about the principal axes, x\ y\ z\ are, respectively, Equation (197), dtp ~di . sin . sin 6 ; at + — .cos0; 0; 0; d( t>. di' 1 of the third, dd dO . -.cos^; -^.sm^; 0. Taking the sum of those about the same axes, there will result, dd , dib . v, = — . cos — . sin . sin ; dt Y dt Y ' dd . ^ di> . v y = — — . sin - 1 - . cos

cos«cosj3 — 2^cosacosy — 2^008 0008 7; in which D = I>mxy y E — lmxz y F = 2 myz. Upon the axis OyI, supposed to assume all directions through 0, take the length 0A> or point A, whose co-ordinates are x', y', z' } such that OA=—L=; \2nir 2 then cos a = Jf'vSflM^ cos ]3 = y'^y/Smr 2 * cosy = z'V^r 2 ? and these values give by substitution, after suppressing the common factor, 1 ss ^#' 2 + %' 2 + 6Y 2 — 2Z>*y — 2Ex'z — 2Fy'z'. This equation gives, for the locus of all such points as A, a sur- face of the second order with its centre at 0. The radius vector OA eannot be infinite, for Smr 2 cannot be zero ; the surface is therefore an ellipsoid. This ellipsoid gives a clear mental conception, or perfect geometric image, for the ratios of all the moments of inertia of any body, with reference to different axes of rotation through the same point. For every point there is such an ellipsoid, and the moments of inertia vary as the reciprocals of the squares of its semi-diameters. The ellipsoid whose axes pass through the centre of inertia of the body is called its central ellipsoid. MECHANICS OF SOLIDS. 163 If we take for co-ordinate axes those which are the principal axes of the bodv, then 2mxy =z 0, 2mxz =. 0, 2myz = 0, the ellipsoid becomes central, and its equation is 1 = Ax'* + By' 2 + Cz' 2 ; which gives, as in Equation (210), for the moment of inertia about any axis, Hmr* =s A cos 2 a + B cos 2 -f 6' cos 2 y. When A is equal to 2?, the ellipsoid becomes one of revolution; and if -? ,0- A = B = C the ellipsoid is a sphere. § 167. — Resuming Equations (33), and substituting the values of x, y y z, in the general expression, 2m (x 2 -f- y 2 ) which is the moment of inertia with reference to any axis z, parallel to the axis z\ through the centre of inertia, we have 2m {* + y 2 ) = 2m [ft + x'f + (y, + y') 2 ] = 2m (z' 2 -f y' 2 ) + ft« + y 2 ) . 2m -f 2# . 2mx' + 2y, . 2my' ; but from the principle of the centre of inertia, 2mx' ss 0, and 2my' = ; whence, denoting by c? the distance between the axes z and z', ana' ' by M the whole mass, 2m.(^+ y 2 ) == 2m (x' 2 + y' 2 ) + Md 2 . . . (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 th« 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 m from the axis z, then will r 2 = x 2 + y 2 , and 2 m (x 2 -f- y 2 ) = 2 ra r 2 . Now, denoting the whole mass by M, and assuming 2 m r 2 = Mk'\ we have (215) /2 m ) 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 concentrated 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 4 denote a principal radius of gyration, we may replace 2 m (x f2 + y' a ) in Equation (214) by Mk 2 , and we shall have Imr 2 = Mk 2 = M(k 2 + and for an axis parallel to the above at the distance d, Ma* k = -v/£a 2 + d 2 . Example 3. — The same body about an axis through its centre and in its plane. As before, dM = M. r - dr : di , 2 j) M pa /»2 w Mk 2 = / M r -^—.dr.d6 = -^— / r*(] -cos2 6) dr.dt, Mk. M P a a 2 = ^—- I r 3 . dr = M —i 1 a 2 J o 4 and k t = i a, and about an axis parallel to the above and at the distance rf. vl" 2 + rf 2 . 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 th« plate about an axis through the centre and inclined to its surface under an angle Let A' be the origin of co-ordinates, PM=y\ A'P=zx; AA' = m; A' B = n ; and V = volume of the solid. The volume of the elementary section at P will be «r y 2 .dx, and whence, V : M : : if .y 2 .dx : d M\ d M = — • if • y 2 • d x, and its moment of inertia about MM', is, Example 8, M y 2 — • * - v* • a x • -- • V y 4 and about the parallel axis, D E, M y.«. y *.dx(\y 2 + x 2 ) 168 ELEMENTS OF ANALYT.CAL MECHANICS, therefore, J m v But whence, v =f**y***i f m (i2/ 4 + * 2 r).dx k 2 = — m X The equation of the generating curve being given, y may be elimi- nated and the integration performed. Example 5. — A sphere about a line tangent to its surface. The equation of the generatrix is y 2 = 2 a x — x 2 ; in which a is the radius of the sphere. Substituting the value of y* in the last equation, recollecting that m = 0, and n = 2 a, we have f °(a 2 x 2 + ax 3 — | x 4 ) dx k 2 — — == -s a 2 - /•2« 5 J (2 ax - x 2 )dx Also Equation (216), IM = k 2 - a 2 = % a 2 , ' 5 and Thus, when the boundary of a rotating body an J 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 prop- erties of the compound pendulum, to be explained presently. MECHANICS OF SOLIDS. 169 Example 6. — Find the points in reference to which the principal mo- ments are equal. Take the origin at the centre of inertia, and the principal axe? through that point as the co-ordinate axes. Denote by x t y t z 4 the co ordinates of one of the points sought ; by A 4 , B t , and C t the principal moments with reference to this point, and by x y' z' the co-ordinates oi the element m. Then, because the moments through the point x t y i z t are to be principal, will 2m(x'-x 4 )(y'-y ( ) = Q; 2m(x'-x t ) («'-z,) = 0; 2m(/-^J(z'-2 / ) = 0. Performing tlie multiplication and reducing by the properties of the centre of inertia and principal axes, we have M . x t y t = ; Mx t z t — ; My t z t = : which can only be satisfied by making two of the co-ordinates x f y t z t separately zero. Let y t — 0; and z t — ; then, § 167 and Eq. (216). A 4 = A; B t = B + Mxy, C 4 =C+Mx 4 *; but, by the conditions, the first members are equal. Whence A = B + M x; = C + Mx 4 * ; and, therefore, B=C; and ^ = ±>/-^?; and from which it is apparent : 1st, that if all the principal momenta in reference to the centre of inertia be unequal, there is no point in reference 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 great- est, 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 they can be equal. IMPULSION. § 169. — We have thus far only been concerned with forces whose action may be likened to, and indeed represented by, the pressure arising from the weight of some defiuite body, as a cubic foot of 170 ELEMENTS OF ANALYTICAL MECHANICS. distilled water at a standard temperature. Such forces are called incessant, because thev extend their action through a definite and measurable portion of time. Such a force is assumed to be measured by the whole effect which its incessant repetition for a unit of time can produce upon a free 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), P = M.r t = M d ^.=Jf^; .... (218) ill which V t , 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 F, 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 J/", and the actual velocity generated in it when perfectly free by T, we have P = MV = M.p ( , (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 sometimes called impulsive, and their action impulsion. MOTION OF A BODY UNDER IMPULSION. § 170. — Th* components of the inertia in the direction of the axes xyz. are respectively IL ds dx w dx M-- — as M --— ; d t as d t __ ds dy dy d t ds d t _, ds dz j* dz M' — '— = M-~; dtds d t which, substituted for the corresponding components of inertia in Equations (B) and ((7), give dx 1 2 P cos a = 2 to • dt 2 Pcos/3 = 2to- -?; a £ 7 2 P cos r = 2 to • — ; 1 dt J • • • (220) 2 P(r' cos j8 — y' cos a) = 2 to (x f ~ — y' • -^J , * , . , n / / dx . dz\ 2 P {z f cos a — #' cos 7) = 2 to ( z ' • — — x • — ) , 2 P (/ cos y —z' cos /3) = 2 to (y' • -77 — 2' • ~) • y . (221) In which it will be recollected that x y z are the co-ordinates of to, referred to the fixed origin, and x' y' z r , those of the same mass referred to the centre of inertia. MOTION OF THE CENTRE OF INERTIA. § 1*71. — Substituting in Equations (220), for dx, dy, d z, their values obtained from Equations (34), and reducing by the relations 2mdx' = Q; 2*idy' — Q\ 2mdz' = 0\ • • (222) 172 ELEMENTS OF ANALYTICAL MECHANIC8 given by the principle of the centre of inertia, we find dx. 2 P cos a = -— .2m; 2Pcos/3 2 P cos y dt dt ' > dz dt and substituting M for 2 m, we have -•2m; _, , , d x . 2PcOSa ZS. M'—r^', dt 2Pcos/3 = jlf .^-'; ^ dt -.* d z. 2 P cos y = M • -t-^ ; 1 dt ' • k (223) which are wholly independent of the relative positions of the elements of the body, and from which we conclude that thr. motion of the centre of inertia will be the same as though the mass were concen- trated in it, and the forces applied immediately to that point. § 172. — Replacing 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 2 P cos a = M V cos a ; IP cos ft = M V cos b ; 2 P cos y = M V cos c ; substituting these above, we find V. cos a V . cos b F.oos c ~ dt ' - iii . dt ? dz t = 17 ' (224) MECHANICS OF SOLIDS. 173 Mid integrating, x t = V- cos a . t -f C, ' y, = F.cos&.* -f C", I . . . . . (225) z t = V. cos c.t + C", 4 and eliminating f from these equations, V will also disappear, and we find, cos c z , 1 x r z < = y,- y* x r cos a cos c cos 6 cos b cos a C" cos c — T 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 impulsive 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, theii values from Equations (34), reducing by 2mx' = 0, 2 m y' = 0, and we find, 2 P (x' cos (3 — y' cos a) = 2 m (x' • -~ — y r —^-J ; 2 P(z' cos a — x' cos y) — 2 m (z f • -^- *' • ( -^-J ; [ • . (227) 2 P (/ cos 7 — z' cos B) — 2 m (y' • -A - z' • -j- J ; 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. ANGULAB VELOCITY. § 174. — Replacing the first members of Eqs. (227) by £ , M r and N n respectively, § 162 ; and substituting in the second members fur dx\ dy' and dz\ their values in Eqs. (190), we readily find d$ dt ~dt dm ~d~t da d-L Z y -f-2 m x'z • — -f 2 m v'z' • — 2 m {x" z + y" 1 ) dt M^mz'y'.-^+Zmx'y'.^ 2 m (x' 2 + z' 2 ) ' * dt dt (228) 2 m (f* + z' 2 ) If the axes be principal, then will 2inx'y' = 0, 2 m y V = 0, 2 m z f z'=0', or if the axes be fixed in succession, then for the axis x' will dip = 0; dp = 0; for the axis y, dcp = 0; c/g) = 0; and for the axis z, d-a = 0; dip = 0, and the above become c?0 Ij t Vz ~ It ~ £t» ..(*?.+.**) : C iir. Vy ~ "^7 — 2 m . (jftfl 2 ) : B ' v„ c?cr tf7 iV y i . 2«*.(y 8 + z 2 ) A \ (220) That is, the component angular velocity about either a principal or fixed axis, is equal to the moment of the impressed forces di\ided by the .-moment of inertia with reference to that axis. The resultant angular velocity being denoted by have, (Eq. 196), ds t ^ = i- WV + rf+2 + dvfl. at a t V we also (230) MECHANICS OF SOLIDS. 175 AXIS OF INSTANTANEOUS ROTATION. § 175. — The axis of instantaneous rotation is found as in §.153, by making, in Equations (192), d x = 0, c?y'= 0, dz' — 0; and, therefore, z' . v y — y' . v t = ; x' . v z — z' . v x = () ; y' . v x — x . v y = . (23 1 ) 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), — x • * *** X ' y ~ X ' ■ ~ x ; and the inclination d of this line to the instantaneous axis, is given by . v t .Z + v u .Y+v x .X cos a = — Vv: + v; + v: . vz* + y + x* ' or, substituting for i> 2 , v y , and v x their values, Eqs. (229) and (191), l . z if ; . r ir . x -L 1 i j £__- cos = __.° B _ . (232) AW* ©> (§) '■ <*?*£»» The point in which the line of the impact pierces the plane yz is given by 9 — x » y ~ X ' dividing one by the other, we have, for the equation of the line through this point and the centre of inertia, Denote the angle which this line makes with the instantaneous axis by 6' ; then from the equations of these lines will cos a = — — • ; 2 AWW-A-W + 1 or, Eqs. (229) and (191), ^ cos0' = r= • I 238 ) 176 ELEMENTS OF ANALYTICAL MECHANICS. AXIS OF SPONTANEOUS ROTATION. § 176. — If both members of Eqs. (34) be divided by dt, we have dx dx t dx r dy dy t dt/ dz dz t dz' dt' ~di + ' ~~di " ' dt ' ~di + ~dt ' It" dt " ~*~ di " ' , .„ - , dx dy dz . . and if for any element — =0; — =z ; — = . . . . (234) ... dx t dx' dy t dy' dz dz then will -tt = — -77 ; ~§i = TV ; -r 1 = — ~r • • • ( 235 ) eft dt dt dt' dt dt v ' Substituting for the first members their values given in Equations (224), and for the second members their values given in Equations (192), we have z' . v y — y' . v z + V . cos a = x' . v, —z' . v x + V.cosb = I . . . . (236) y' . v x — x' . v y -f- 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 fine will be at rest during the body's motion. This line is called the axis of spontaneous rotation ; because, being at rest, Equations (234), while the centre of inertia is in motion, the whole body may be regarded, during impact, as rotating about this line. Its position results from the conditions of Equations (235), which are, that the velocity of each of its points and that of the centre of inertia must be equal and in contrary directions. The dis- tinction between the axes of instantaneous and of spontaneous rota- tion is, that the former is in motion with the centre of inertia, while the latter is at rest. 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 v t \ there will result, V V V cos a . — + cos &. — -{- cos c . — - = . . . (236)' v- v- v v x w i "t The first member is the cosine of the anv (A-C).(B- C) d t* A.B • ( 244 > which will make known the circumstances of motion of the common centre of inertia about the fixed origin. MOTION OF THE SYSTEM ABOUT ITS COMMON CENTRE OF INERTIA. § 182. — Substituting the values of x, y, z, d 2 x, &c, given by Equations (241), in Equations (240) and reducing by Equations (244) and (242), there will result *(/^-^-*V'-r') (245) 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 (118); 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. — If the system be subjected only to the forces arising from the mutual attractions or repulsions of its several parts, then will 2 .Y = ; 27=0; 2 Z = 0. 182 ELEMENTS OF ANALYTICAL MECHANICS. Fcr, the action of the mass J/", upon a single element of M, will vary with the number of acting elements contained in M\ 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 M' repeated as many times as there are elements in M' acted upon ; whence, the action of M upon M' will vary as the product MM. In the same way it will appear that the force required to prevent M from moving under the action of M\ will be propor- tional 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 /3, P cos y, in the values of 2 J, 2 F, 2 Z, there will be the numerically equal com- ponents, — P' cos a', — P 1 cos /3', — P' cos y\ and, Equations (243), reduce, after dividing by 2 M, to and from which we obtain, after two integrations, x 4 = C .t + D'; ^ y t = C".t + £>"; I • (247) z t = <7'".* + D'"; J in which C", C'\ C"\ D\ D" and D'" are the constants of inte- gration ; and from which, by eliminating /, we find two equations of the first degree between the variables x t , 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 2dx f , 2dy t , 2dZj, respec- tively, adding and integrating, we have ««,» + W + i;* g y» = .... (248) 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 upor. 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 (215) may be written, rv - xy + r f\ - x y n + & c . ; and considering the bodies by pairs, we have X = - X' ; Y = - V ; and eliminating X and Y' above by these values, we have y (*' _ x ») _ X(y' - y") 4- &c. But, JT= P. a' - *" ; Y =rP y' ~ y" p ' p in which p denotes the distance between the centres of inertia o( the two bodies. And substituting these above, we get p . I L. (*' _ *") - P • — (y' _ y") -= ; and the same being true of every other pair, the second members of Equations (245), will be zero, and we have M / , d 2 x' , d 2 z\ and integrating M / , d 2 z r , d 2 y'\ 2 M • ( y' Z* • — - ) — V dt 2 rf* 2 / dt y 2M. Z ' dx '- X ' dz '= C", 1M dt y f d z f — z f d y' dt =.- C". • • »* • (249) 18-1 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 A z , A y , A xi double the areas described in any interval of time, t, by the projections of the radius vector of the body J/", on the co-ordinate planes, x' y\ x' z', and y' z' , and adopting similar notations for the other bodies, we have dt ' dt . . r dt in which C", C", C", denote the sums of the 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\ %' z\ y ' z' '; and by integrating between the limits t t and t\ giving an interval equal lo I. 2M.A Z = C'.t; 2M.A y = C" f; 2M.A X = C'"t\ whence we find that when a system is in motion and is only sub jected 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. It is important to remark that the same conclusions would be true if the bodies had been subjected to forces directed toward 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 J/, will be, §185, Yx — Xy z=z ; 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. 8 1S6. — Denoting the angles which the resultant axis of rotation makes with the axes x', y', z\ by X , 6 V , Z , we have, iin _Av 1 _N i _C_ C * - K ~ K - K' _Bv 1J _M i _C^ C0S U * - K ~ K - W Cv z L C" > . . . . (250). These constant values determine the position of the resultant or invariable 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. CONSERVATION OF KINETIC ENERGY. § 187. — If, during the motion, two or more bodies of the system impinge against each other so as to produce a sudden change in their velocities, the kinetic energy of the system will undergo a change. To estimate this change, let A, B, C be the velocities of the mass m y in the direction of the axes before the impact, and a, ft, c what these velocities become at the instant of nearest approach of the centres of inertia of the impinging masses, then will A — «, B — b, C — c, be the components of the velocities lost or gained by m at the instant corresponding to this state of the impact, and m (A — «), m (B — ft), 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 186 ELEMENTS OF ANALYTICAL MECHANICS. gained by the other impinging bodies of the system. These, by the principle of D'Alembert, § 71, arc in equilibrio, whence 2 m ( A — a ) 6x + 2 m (B — b) 6y -f % m (C — c)6z = 0. The indefinitely small displacements 6x, dy, dz, (J _ a )2 + (B - by + (C- C) 2 = j or, Aa + Bb + Cc = H -f- c 2 _ 2 (^a + Bb + Cc). ^l 2 + B * + (7 2 a 2 + fc 2 + g 2 2 + 2~ (yl _ a )9 _j_ (2? _ 5)2 + ((7 _ c )2 2 which in Equation (252) gives, 2m(J 2 + £ 2 +<^)— 2m(a 2 + & 2 + c 2 ) = 2m[(.4 — a f^(B—bf+(C— e) 2 ], and making ^2 + £2 + C 2 = F 2 , «2 _j_ £2 _|_ c 2 — - w 2 ? 2wF 2 — 2mw 2 — 2m[(J — a) 2 + (£ — 6) 2 + (C — c) 2 ] ... . (253) whence we conclude, that the difference of the system's kinetic energy before the collision, and at the instant of greatest compression, is equal to the kinetic energy 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 posi- tive, it follows that at the instant of nearest approach of the impinging bodies there is a loss of kinetic energy. If the impinging masses now react upon each other in a way to (a — Ay + (b — B'Y + (e— C'f = i MECHANICS OF SOLIDS. 187 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, + * . dz); eZ* V* dt ' ^ dt r or, dMV.ds = M.dl-.dx + d l.6y + d ^. 6z\ \dt r dt J ^ dt r and by integration, tfMV. d * = M^ t .6, + d -l. S y + %*.). And for any number of bodies, <52 f MV .ds = Z m(~ .6x + (l l.6y + d 4 .6z\ J \dt dt u dt J Now MV is the body's quantity of motion and is the measure of the intensity of the force that produces it. MV.ds is the elementary quantity of work or of action ; and integrated between limits will give the quantity of action between those limits. But at these given limits 6x = ; 6y = ; dz = 0, and dJ2MV.ds = 0. That is to say, in the motion of a system of bodies, the curves they describe and their velocities are such as to make the sum of the quantities of action between given limits on their respective paths generally a maximum or minimum ; and because it is always possible to assign to each body a definite path longer than any assumed for it, the quantity of action is obviously a minimum. This is called the principle of Least Action. Because ds = Vdt, we find that 2 / MV 2 dt is a minimum, or that the quantity of kinetic energy expended during any given time is a minimum. If there be but one body and that moving upon a surface, V will be constant and / 6s will be a. minimum, and the body will describe the shortest distance between any two points arbitrarily taken on its path. . MECHANICS OF SOLIDS. 189 § 189. — To return to the rotary motion of a single body. If for the impulsion measured by Mv \vc take the moments with reference to the axes x\ y\ z', then, since ^ r dx * r dy . r clz' if—-, if-f-, M — dt ' dt ' dt are the components of Mv, it is clear that Equations (249) are ex- pressions for these moments. Designating them by Z ; , M t , N i% . respectively, for the axes z\ y\ x', Equations (229) and (249) give L t =C = Cv zy M / = C" = B Vyy N t = C" = Av £ ; or, squaring and adding, £2 + ^2 + ^2 = R C"2+ C"2+ C""2=F, A*v s * + B 2 vJ> + C*v t * = *» ; which are other expressions for the law of conservation of areas. And it is evident that the resultant area, or moment &, is a constant quantity. It has been shown by Poinsot that the principal plane coincides with that diametral plane of the central ellipsoid of inertia which is conjugate to the instantaneous axis. To prove which, as the point x' , y\ z r , is upon the instantaneous axis, Equations (193) give, x v z j • («) and, as it is also upon the ellipsoid, we have, Ax'* + By'* + Cz"2 = 1. The equation of the tangent plane to which, through the paint x\ y', z', is Ax'x -f By 'y + Cz'z = 1. 190 ELEMENTS OF ANALYTICAL MECHANICS. And, therefore, the conjugate diametral plane is Ax'x + By'y + [Cz'z = 0; which the ratios («) transform into Av x x + Bv y y + Cv z z = 0. A perpendicular to this diametral plane makes with the axes ( .x\ y\ z', angles whose cosines are Av x Bv y Cvz k ' k ! k » and, therefore, the plane coincides with that of resultant rotation around the principal axis, but generally not around the instantaneous axis ; which continually shifts its position both in the body and in space, and coincides with the principal axis only when rotation takes place about one of the natural or principal axes of the body. To find the angle made by the instantaneous with the axis of the principal plane, denote it by 0; then Equations (194) and (196) give kv t cos =r AvJ* + Bv y 2 + Cv z 2 ; the second member of which equation denotes double the kinetic energy. To prove this, Equation (210) gives for the moment of inertia, • 2 mr 2 Bs A cos 2 a -f B cos 2 /3 -f- cos 2 y ; and this, multiplied by v£, becomes, 2 mv 2 = v 2 2 mr 2 = Av* + Bv 2 -f- Cv}. , Hence, we have, Xmv 2 v lC os9 = — £-; which is constant when there are no external disturbing forces. This result shows, § 161, that under such circumstances the component. ' angular velocity with reference to the invariable axis and invariable plane is constant. Let 6 denote the semidiameter of the central ellipsoid coincident MECHANICS OF SOLIDS. 191 with the instantaneous axis, and e 2 double the kinetic energy, or simi of the living forces; then, 2 mv* = vf 2 mr 2 = e 2 ; and this gives, because of equation of tangent plane at end of semi- diameter, v t = 6e. .which shows that the angular velocity is proportional to that semi- diameter. Whenever the moving body is not acted upon by disturbing ex- ternal forces, Equations (202) give, A d -£=v v v,(C-B), ■£ d £=v z v z (A-C), Multiplying these respectively first by Av x , Bv y , Cv z , and then by r an v y> v zi we obtain by addition, A* v x dv x + B* v y dv y + C' 2 v% dK _ o, Av x dv x + Bv y dv y + Cv z dv z = 0; and bv integration, A 2 v* + E* v* + C 2 v* = k% Av£ + Bv* -f Cv? = ' and, therefore, ds(\/l — r* yl r 2^ h or, (1) . . V > = d\— —- + - r' ds rr' dt' l r \/\~?* + r' ^/T^r* which equation gives the value of r in terms of v {J s, and r'\ and, therefore, when these are known in terms of t, it determines the directing cone. Again, putting ds . , — — r(D = r a) , dt ' we have (2) <*> sin a = (*)' sin j3, and v { = 0) cos a -{- <*)' cos /3. Also, the motion of the point C on the rolling axis gives, (3) . . . . . v t sin (3 = (0 sin (a -f- (3). Hence, o) '\ 4» r . ; v t : : sin /3 : sin a : sin (a + (3) ; the geometric construction of which result is, evidently, a parallelo- gram of rotations w, v p (*>'. described on the axes A, I 7 B. MECHANICS OF SOLIDS 11*5 From the relations just found we readily get ds sin a . sin 3 . . . . — — v. - dt (4) -tl £ ' sin (« + /?)' with which equation (1) is identical. If the movable cone roll externally upon the directing cone, all the angular velocities are similar — either all positive, or all negative. Of such motion, the common top spinning around its point as a fixed centre, while its axis gyrates slowly in precessional revolution, furnishes a familiar instance. But when the movable cone rolls internally upon the surface of the directing cone, then w is in direction reverse to v f and (*)', it being posi- tive if they are negative, and nega- tive when they are positive. In this case the angle j3 is negative, and Equation (3) becomes, v t sin (3 = — g) sin (a — |3). The precessional revolution of the earth's axis, B, around A, the axis of the ecliptic, is an interesting example of this second case. The rotation of the earth about its own and about the instantaneous axis being direct (negative), that of the precession must be retro- grade, or positive. Its period is 25868 years; the obliquity of the ecliptic is 23° 27' 30"; and the length of the circle traced by the instantaneous axis on the surface of the earth is, therefore, about 52,240,000 feet. From these data it is easily calculated that the roll- ing cone goes five and a half feet per day, and that the radius of its base is 0.88 feet only. -G$ 196 ELEMENTS OF ANALYTICAL MECHANICS. § 191. — To find the cone described in the body by the instanta- neous axis when there are no external forces, the relations, v x = ex, v y = ey, v z — ez, transform the law of areas into ^2^2 + £2pfy2 + cYe 2 = 1. Also, for the central ellipsoid, Ax* -f By* -f Cz 2 = 1 ; if we put we have, A-.l b-I a- 1 a* b* c* # 2 y 2 2 ^ + T 2 + S = 1# And from these expressions we obtain, ^2 ( a 2 _ ^2) 2 2 + £2 (J2 _^2) y 2 + A: jo" & ' and these give a* + y2 _J_ 2 2 _ j0 f the equation of a sphere, with the same tangent plane as the ellipsoid. Substituting these values in the equation of living force, Av s *+ Bv* + Cv?=ze\ and reducing, we get, ofo* 4. #ty2 ^ C 2 Z 2 — pi ; MECHANICS OF SOLIDS 197 and, therefore, ( a 3 _ p 2) Z 2 + (fc2 _ j0) y 2 + (e2 _^2) s 2 _ Q, is the equation of the cone. The equations found show that the cones are elliptical, and that their axes of symmetry coincide with the principal axes of the body and of its ellipsoid of inertia. They are, therefore, right cones, whose bases are ellipses; the equations for which are found by making x or z constant. If a, b, c be unequal and denote the semiaxes of the central ellipsoid in the order of their relative length, then the cones are de- scribed about x, the greater axis, when b is less than p } and around z, the least axis of the ellipsoid, or body, when b is greater than p. But if c be not less than p, the cones are imaginary. Let secant planes cut the axes x, y, z, so as to form a cube at th centre, and assume, successively, for the constant p different increasing values, all greater than c; then the elliptical bases of the cones will each increase in size and eccentricity until p is taken equal to b, when the corresponding cone opens out into two asymptotic planes intersecting in the axis y, and whose traces in the plane xz are, for the instantaneous axis, A . /a = ±*7tV: JO 2 C p 2 — c 2 » » and for the invariable axis, =±^i — JO 2 JO 2 ,2* Beyond these limiting planes, if we give p still greater values, the body spins around x } and its cones for x give hyperbolas with a secant plane perpendicular to x. Each value of p has its particular cone, either about x, or about z; and the limiting planes divide the space around y into regions, one for rotation around x y the other for 198 ELEMENTS OF ANALYTICAL MECHANICS. rotation around z. The figure also shows that sections of the cones by planes perpendicular to y are all hyperbolas. And if we imagine a sphere, instead of the cube, at the centre, its traces with the cones will be spherical ellipses. For symmetrical solids the cones become circular and the ellipsoid one of revolution, around z if a be equal to b, around x if b equal c. And if A = B = C, the ellipsoid is .a sphere, with permanent rotation for any diam- eter. The only condition for rotation about a permanent axis has been shown, § 178, to be that the body must revolve about one of its principal axes. The rolling and the fixed cone then reduce to their axes, and the invariable, instantaneous, and rolling axes coalesce into a single line, or axis, normal to the tangent and invariable planes. That only a principal axis can be permanent is clear, for a diameter is normal to the tangent plane of an ellipsoid only at the ends of its principal axes. PLANETARY MOTIONS. § 192. — When the only forces are those arising from the mutual attractions of the several bodies of the svstem for one another, the second members of Equations (239) reduce, as we have seen, § 183, to zero, and those equations become MM

. Z d'z ,x = 0, MECHANICS OF SOLIDS. 203 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 ve- locity of the body. The angular velocity therefore varies iniersely as the square of the radius vector. Multiply Equation (26(5) by d s, and it may be put undei the *brm, d s 2 c d t rda. 1 d s but — '- — , 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. — — , d s and V=— (267) P whence, the actual velocity of the body varies inversely as the distance of the tangent to the orbit at the body's place, from the centre. § 19 4. — Denoting the intensity of the acceleration on M t by F\ sub- stituting M \ . F .dr for Xdx + Ydy + Z dz, writing M t for M in the coefficient of V 2 in Equation (121), and differentiating, we find VdV= -Fdr\ and taking the logarithms of both members of Equation (267), log V = log 1c — log p ; differentiating, dV dp ~V~p 1 and dividing the equation above by this, n->:,.g.i ! r.» J ,.jr . . . (2 «8) 13 204 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 from rest at that point over the distance ME, equal to one-fourth of the chord of cur- vature M G, through the fixed cen- tre — the force retaining unchanged its intensity at M. § 195. — Resuming Equations (120), we have d x d 2 x dt X= M'-— — M — , dt 1 dt and performing the operation indicated, regarding the arc of the orbit as the independent variable, we 'have, after dividing both numerator and denominator by d s z , dt 6} x dx d? t X=M. ds d a 2 d s ds* ds 2 d* x dx da? d 2 I DUt whence, dt' -'[ .dt' ds* ds 3 ds* d 2 t de'd**~ d 2 s t ~ d~i 2 ; ~ d 2 x dx d 2 s~i In like manner, ™>[*#*-£l' d* d z d* s' 8qnanng and adding, MECHANICS OF SOLIDS. 205 - 1 &!* &h SS ! ■ *■ X'+F a + Z 2 =:^ T ™ d 2 * Idx d % x dy d}y dz d? z\ l|_ d d d I (/ s 2 /<*«■ dy' dz\ '&•<* bnt, denoting the radius of curvature by p, we have (£f\* (d'yV l 8e* Appendix No. 2. 206 ELEMENTS OF ANALYTICAL MECHANICS. The 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 de- fined to be the resistance which the inertia of a body in motion oppose* to whatever deflect* it from its rectilinear path. It is measured, Equa- tion (269), by the living force of the body divided by the radius ot curvature. The direction of its action is from the centre of curvature, and it this differs from 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 and denoting the centrifugal force by F n we have F < = ~y- ( 27 °) and integrating the next to the last equation, we have 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 in- dependent of the intensity of the extraneous force and of the 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 inte- grate ; we find, omitting the accents, d* + dy> _M+M, f {MM . 2xdx + 2ydy ^ d? ~ M.M, 'J\ mm <>' r MECHANICS OF SOLIDS. 207 but r 2 = # s -f- y 2 , and rdr = xdx-\-ydy\ also x = r cos a ; y s£ r . sin a ; d x = — r sin a rf a -f- cos a c? r ; dy = r cos a rf a -f- sin a 6? r ; and, Equation (266), 12c d t r 1 d a These substituted above, give Make H+^*<%£ ■***>*« u, and therefore — = — d w, r r a substitute above, differentiate and reduce, there will result \r/V ' J/.Jf V '' L M T if- J 1 ; . • and making ^ =* I i^ + ^ ] = relative acceleration on M t v (271) V ■■ From which the equation of the orbit may be found fry "integration, when the law of the force is known ; or the law of the force Seduced, when 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, £ exjst- iri£ in the differential equation. These are determined ty the initial or other circumstances of the motion, viz. : the body's velocity, its dis tance from 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 notion at any given time determine the species and dimensions of the orbit. 208 ELEMENTS OF ANALYTICAL MECHANICS. In the second case, find the second differential coefficient of u in 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 Riiy knowledge, and to which the preceding principles have a direct application, is that called the solar system. It consists of the Su?i, 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 the common centre of inertia of the whole within the boundary of its own volume. These bodies revolve about their respective centres of inertia, are over shifting their relative positions, and our knowledge of them is the f suit 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 de- scribing, 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 tun, are proportional to the cubes of their ?nean distances from that body. ^ These are called the laws of Kepler, and lead directly to a knowl- edge of the nature of the forces which uphold the solar system. CONSEQUENCES OF KEPLF.r's LAWS. § 198. — The first law shows, § 193, that the centripetal forces which MECHANICS OF SOLJDS. 209 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. § 199. — What law of the force will 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 r = \ + e cos a ' whence, 1 1 -h e cos a r = U= a (1-7)'' and,

tan e y ' and this in Equation (274) gives c- x 2 h, . m U 2 2 r* sin* e , 4c*r, ' but, Equation (267), 1 v? V % r 4 4 r, 2 sin* e, 4 c 2 4 c a .r. • • • (275)' in which V t is the velocity corresponding to r t ; hence, 4cV y which, substituted in the equation of the curve, gives 4 c 2 k J . m r = hV^^-(^-^~<-+»> . (276> '/ ' 4 and comparing this with the general polar equation of a com; section referred to the focus as a pole, viz. : ., (i - *) r = I s + e cos (a + ■■■■ (-) And this last value will be greater or less than unity, according as V* 2 K 171 is greater or less than r , Multiplying and dividing the last factor by M, r„ and replacing wi by its value, the orbit will be an ellipse, parabola, or hyperbola, ac- cording as 212 ELEMENTS OF ANALYTICAL MECHANICS. K v * < — --i ' ' M > ' r < J r/ M, . V; > **-= - -M t .r t . r? That is, according as the living force of the primary at any point ot its orbit is less than, equal to, or greater than twice the work its rela- tive weight, at that point, would perform over a distance equal to its radius vector. So that a primary may describe any of the conic sec- tions ; as well as the ellipse, the only condition for this purpose being an adequate value for its velocity. Substituting the value of e % in Equation (277), we find k. .m.r. . . , and denoting the semi-parameter by p, the equation of the curve gives, by making a -f-

" dX ■ *" dX rf *'~ r // 3 >*„/ M,.dx'~ " rj M t .dx" whieh, substituted above, give, after treating the other two of Equations ^258) in the same way, MECHANICS OF SOLIDS. 215 + 9 m (*' + 9,/V ; jr* + e J + b ih & + ej) ; «' + 9 J + 9 tt ,{d + •,/) , or, performing the multiplication and omitting the terms containing *' + *' (*„ + BJ y' + y' (*„ + t „) ; •' +. < (*„ + $„ t ) j y 216 ELEMENTS OF ANALYTICAL MECHANICS. in the same way, when also disturbed by M llit , ^+*;(0„+0,„+0,L,); y'+y'(0„+0„.+0„„); *+<&,+*m+',,J and for the simultaneous disturbance of all the bodies of the system, x' + x'zd^ y' + y'ZOy, z' + z'zo,,- in which x'.Ed^, y' . 2 /y , z',16^ are the increments of x' y' z f re- spectively, due to the joint action of all the disturbing bodies. Now let u = V T 4c a and integration, 2 (a + + *) + tJ x? ' sin '( a + 9) . . (285) which is the equation of an ellipse referred to its centre as a pole, the semi-axes being r, and — \ • 4 r t k,. m § 2C7. — The time required to describe the entire ellipse being deno- ted by T, we have, Equation (264), tt . r, . 2 c \ - m k..m n a / 1 T = '- — = 2 7T X ; r,.c k t .m and replacing m by its value, Equation (273)', T=2nX (M + M,) k t (286) 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 F, denote the angular velocity of a body about a centre, then wiU V=pV l , and Equation (270) becomes The earth revolves about its axis A A' once in twenty-four hours, and the circumferences of the parallels of latitude have velocities 220 ELEMENTS OF ANALYTICAL MECHANICS, which diminish from the eqiator to the poles. The law of this diminu- tion, on the supposition that the & planet is a sphere, is given by in which M is the body's mass, V y the earth's angular velocity, and R' the radius of one of its parallels of latitude. Denoting the equatorial radius C E = C P, by R, and the angle C P C = P C E, which is the latitude of the place, by - w), and W ( V cos 9' - u') • • • (288) 224 ELEMENTS OF ANALYTICAL MECHANICS. be the forces lost and gained at the instant of greatest compression ; and hence, M ( V cos

+ AT V cos 'cosd' = (l+c) U W^ ~ C P cos *i^ +*J «"* -cV^tf+ F»ri.» 9 • • • (303, ^^[(l+ ^^^/^ -^-c^cos^+^si,^':. (304) V sin

and passing to the limits, non-elasticity on the one hand and perfect elasticity On the other, we have in the first case, c = 0, and M V + M V /n ^ V = -WTM- W M V + AT V ■ ' = -lrTJr- (310) and in the second, c = 1, consequently, ft M V + iT F' v ="- 2 -IrX-177--^ (311) M + M' MV + M>V> v ~ z ~WTm> v • • • (312) CONSTRAINED MOTION. §211. — Thus far we have only discussed the subject of free motion. We now come to constrained motion. 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 con- tinue on a given surface, by causing it to slide or roll upon some other rigid surface. § 213. — We have seen, § 128, that the motion ol translation of ♦.ho centre of inertia, and of rotation about that ooint, are whollv MKCLIANTCS OF SOLTDS. 229 independent of one another, arid the generality of any discussion relating to the former will not, therefore, be affected by making, in Equation (40), o> = 0; 8+ = 0; 8& as 0; which will reduce that equation to d 2 x (2 P cos a — -— - • 2 m) x t + (2Pos/3- j^>Xm)8y t V==o. rf 2 ^ 4- (2 P cos / — j-j , Xm) £ *j Making 2 m sa M ; 2 P cos a = X; 2 Pcos as F; 2 /» cos/ = Z; and omitting the subscript accents, we may write 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 8 x, 8 y, 8 z, respectively, we have, from the principles of the calculus, dL dL dL — — o x -\ =— • o y -\- -=— •02 = 0. dx dy dz (315) Multiplying \y an indeterminate intensity X, and adding the product to Equation (313), there will result d 2 x dL d?y • dL\ „ \ = o. + (*- M • d 2 z dL dt l + X — ) dzJ dz 230 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 8x. d y, 6 z, say tliat of $x, to zero; and there will result X d 2 x d L dt 2 d x o, (316) and (" 7>4v) »' + <£*-g + >■£) * = °- < 317 > Now in Equation (315), 5y and #2 may be assumed arbitrarily, and 8x will result; hence 8y and 8 z in Equation (317) may be regarded as independent of each other, and by the principle of indeterminate coefficients, Y - M. d 2 y dt 2 + X- dL dy Z - M- d 2 z + X- dL o, o, (318) and eliminating X by means of Equation (316), we find, M- - M- d 2 y dt 2 d 2 z >• d 2 z\ dL T&) ~dy dL d x dL y (J - M- - M- d 2 x \ dL ~d~t?) '-d d 2 y\ d ~~dt 2 ) ' ~d y IL = 0, = ► --(319) which, with the equation of the surface, will determine the place of the centre of inertia at the end of a given time. MOTION 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 from their intersection, take L = F(xyz) =-. 0, > H=F'(xyz) =0tl (320) MECHANICS OF SOLIDS. 231 from which, by the process of differentiating and replacing dx, dy. d z % by the projections of the virtual velocity, d L ~ 4 L t d L „ a a: rfy dg , dH r tf# , • -— + X . -— - + X' • — — = 0; ► • • • (327) &ud Equation (325) to /_ ., c/ 2 ^\ dL dH KX — M > — — I . — — V dt z ' dz dy V dt 2 / dx d dL_ dH dx d z dL dH >= 0. • • • • (328) This, with the equations cf 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 vaiia ble reducing to zero, and d 2 y = 0, and -,— = 0: 9 dy MECHANICS OF SOLIDS. 233 hence Equations (327), bcome X ■ d 2 x dL r= 0; d*z dL dH d P dz dz • • » = 0; (329) and because the factor r-^ = o, Equation (328) becomes, on dividing out the common factor dH dy 1 t*r „ d 2 x\ dL /_ ,_ d 2 z\ dL „ ,„ rtrtV § 217. — By transposing the terms involving X, in Equations (316) and (318) and squaring we have ( r, „ d 2 Z\ 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 JV, we may write V<£)'+ 0"+ <&>'-'• • • • (331) «nd dividing each of the equations dL d*x rf# V d&J dz \ dt 2 /' 234 ELEMENTS OF ANALYTICAL MECHANICS. obtained by the transposition just referred to, by Equation (331). we find, d x X - M- J? x ~d& I SdL y /d L \ 2 /, dLy d L dy N Y - M- d 2 y I ML \» (d L \ 2 (d L V V (77) + W?) + (-37) c/2 JV if. >> (332) ^2 rf* 2 /7^"Z V 2 " rdL\ l /d LV V t© + (37) * Kti) N 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 ^©f©*(^ss are equal, but that the}' are both normal to the surface, and act m opposite directions. The second is the direct action upon 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 imposed upon the body's path. If the body be constrained to move on a rigid surface 01 line, this resistance will arise from its reaction. • § 218,— If Equations (332) be multiplied by N, and the angles which the u«ii*n.ul re&Utance <>!' the surface makes with MECHANICS OF SOLIDS. 235 the ixes :r, y, 2, respectively, be denoted by 8 X , Q pi and 0„ those equations will take the form X - M • %?- + N- cos 6 = ; rf 2 ?/ F- Jlf -yf + iV-cos*, =0; rf 2 Z Z - M • — + iV • cos d f = 0. at 1 (333) §219. — To impose the condition, therefore, that a body in motion shall remain on a rigid surface, is equivalent tc introducing into the system an additional force, which shall be equal and directly opposed to the pressure jpon 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 appli- cable to a rigid curve of anv curvature, as to a surface; the nor- mal reaction of the curve being denoted by iV 7 , and the angles which If makes with the axes x, y, ar, by 0„ Q y , and &,. § 220. — To find the value of JV, eliminate d t from Equations (333), by the relation 1 V dt ~ ds' in which V and s are the velocity and the space ; t ten oy transpo- sition these equations may be written d' z x iT.cos*. = M- V*.-— - X- 9 (X 6* jV-cosd =lf. 7 2 --tI -: Y\ tf.cos*, = M. V*.^4 - Z. as 2 15 236 Elements of analytical mechanics. Squaring, adding and reducing by the relations R 2 = X 2 -f Y 2 + Z 2 , cos 2 ^ -f- cos 2 dj, -f cos 2 6, = I. and we find J/ 2 - N* = J ^[( rf2 *) 2 + (^y) 2 + (<* 2 z) 2 ] + i? 2 Resolving i? 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 (p the inclination of R to the radius of curvature, we have Rsm y — M- — = M- V 2 - or, dt 2 = /?. sin cp -M . V 2 ds 2 d 2 s I7 2 ' Squaring and subtracting' from the equation above, there will result V* N 2 =< but M'< — • ((d 2 x) 2 + {d 2 y) 2 + (d 2 z) 2 - { multiplying the second member by p -.- p, substituting above, and reducing by the relations, dx dy d 2 x dx d 2 s ds d 2 y dy cPs ds ds 2 dsc/s 2 ~ d*' d.s 2 ds ds 2 ~ ds dz a cos

*)* in which p denotes the radius of curvature, we have. r* m v 2 N 2 = M 2 - — — 2 Rcostp -f i*2 C os 2 ■ 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 de- scribed. Example I. — 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 = x 2 ■ + y 2 + z 2 - a 2 = ; . . . . (33t>) dL dL dL -j— = 2 x ; — - = 2 y ; -r— = 2 z : dx dy * ' dz ' MECHANICS OF SOLIDS 239 and the axis of z being vertical and positive downwards, which values in Equations (319). give d 2 x d?y . "| gy -y (Pz dt 2 + * d 2 y dt 2 ^•(337) 0; and differentiating the equation of the sphere twice, we have xcPx 4* yd 2 y + z . d 2 z = — (dx 2 -f d y 2 + d z 2 ) \ dividing by d t 2 , and replacing the second member by its value F 2 , (he velocity, we find, dt 2 But, Equation (335), (P-x d 2 y d 2 z T _, dt 2 dt 2 » V 2 = 2gz + C (ms) and denoting by F' and k t the initial values of V and 2, respectively, we have F 2 = F' 2 + 2ff (z - £), wh : ch substituted above, gives ^# d?V d?z ~ /, v »t + y~ + *•— --=2<7(A:-2) - F df* 2 dt 2 dP (339) Eliminate ar, y, c? 2 #, c? 2 y, from this equation by means of Equa- tions (336) and (337). From the latter we find, ' ; jection in a horizontal direction, (3 being a very small quantity ; and a the initial value of 6. Then, because z = a . cos & = a (1 — 2 sin 2 \ff)\ k = a, cos a = a (1 — 2 sin 9 £ a) ; d z — — a . sin 6 . d d ; V t ' = ; G t — (i ■ 4 sin* \ a . cos' i a . /3* . a // = a . ji . t = 1 ; whence 2 \J- - 1 = 0, or = 2 *r, orir4ir, arid so on; and for a single interval between two consecutive maxi- ma,- without respect to sign, A" t — f\/-; • • • (344) the maximum being a. The least value occurs when cos 2 v/- • t = — 1, or 2 \ /- • £ = dt t d& dt d& suostituting for —=- , its value obtained from the relation y — ztanip, we find rf

«« ^(a 2 — d 2 )(0 2 — j8») dividing this by Equation (341), a. /3 • • • • (346) d(p _ /£ <*-$ fi_ dt ~\ a' d* "V a ' i (a 2 + /3 2 ) + £ (a 2 - 0*) . cos2 i/- • ' but cos 2 \I— • t = cos 2 '\/— • < — sin 2 \I- • * ; whence g = ^£ —^ — - ; • • • (34^ a 2 • cos 2 \ /— • / 4- /3 2 . sin 2 \ /— • t a 2 .cos 2 \/— ./ + /3 2 .sin 2 a . v a from which we find 9 >dt V a a ' . fg cos 2 \ / — • t , V a d

equal to the time of one entire oscillation. From Equation (348) we have, after substituting for tanp its * alue in the relation y = x tan dfi ' dfi ~ U> 246 ELEMENTS OF ANALYTICAL MECHANICS. and assuming the axis z vertical, positive upwards, and the origin at the lowest point A, L = x 2 + y* + # - 2az = 0, . . . . (351) d L « dL <> dL 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 F P -^> r = V* 2 + V 2 + * 2 ; • • • • (352) '7* ii *9 for the force P. cos a = — ; cos ft = — ; cos y = — ; for the r r ' r weight Mg, cos a' = 0; cos ft' = ; cos y' = — 1 ; and Fx Fy Fz These several values being substituted in Equations (319), give Fyx Fyx r 3 r 3 ~ u > /Fz \ Fv (73- -M 9 )- V - 1 f.(z-a) = 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 circumferenco is at the proper height from the bottom. From the second equation we deduce, Fa M -73- = ** (Fa\\ - \~Mg) ' F r 3 • • • • • Mg a ( 353 > MECHANICS OF SOLIDb. 247 from which r oecomes known ; and to determine the position of the circle upon which the body must be placed, we have, by makin« x = in Equations (352) and (351), y^ 2 + y 3 = r, f _j_ Z 2 _ 2 a z _ Equation (353) makes known the relation b* tween the weight of the body and the repulsive force at the unit's distance; the in- tensity 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 Mg ~ a ' hei.ce, F : F : : r 3 : r' 3 ; 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 iV, in Equation (334). g 224. — Example 3. Let it be required to find the circumstances 248 ELEMENTS OF ANALYTICAL MECHANICS. of motion of a body acted upon by its own weight while on the arc of a cycloid, of which the plane is vertical, and directrix horizontal. Taking the axis of 2, vertical; the plane zx, in the plane of the curve; and the origin at the low- est point, then will / -1 z L = x —\/2az — z 2 — a versin — =0; • • (354) in which z is taken positive upwards. dL , dL = i; d x dz -V- (355) X = ; Z = — Mg, and Equation (330) becomes d 2 x fa a —z d 2 z tW-T- +* + — dt 2 (350) and by transposition and division, d 2 x dt 2 9 d 2 z 12 a — z dt 2 ,2a — z (357) From the equation of the curve we find, 2dx = 2dz- it. 2a — z • • • (353) multiplying by Equation (357), there wi 1 result 2d x . d 2 x Ti 2 == - 2gdz — 2dz.d 2 z dt 2 MECHANICS OF SOLIDS. 219 and by integration, <*• d x 2 -f dz* and supposing the velocity zero, when z =. k- y = C—*Zgh; which subtracted from the above gives dx 2 -\- dz 2 = 2g(k-s)',. . . .359) dt 2 and eliminating dx 2 by means of Equation (358), dz2 9 /I 2\ — — = - • ih z — z l ) dt 2 a y ' whence, dz dt -Vi- y/h Z — 2 a the negative sign being taken because z is a decreasing function of t. By integration, /a f dz fa~ . -i 2* t ~ — 1/ — * / — . = — \/ versm *-— + V. V 9 J y/hz - z 2 * 9 * Making z = A, we have = — \J — * versin~ ' % -+• C ', 250 ELEMENTS OF ANALYTICAL MECHANICS, whence, and «M . -1 2z N versin • -~) ( 3(5 °) When the body has reached the bottom, then will e ~ 0, and t = * \/— i which is wholly independent of k, or the point of departure, and we hence infer that the time of descent to the lowest point will r>e the same in the same cycloid, no matter from what point the body starts. Whenever t = &, the body will, Equation (350), stop, and we shall have the times arranged in order before and after the epoch, ~ 4 *V7 5 ~ 2 *V7 ; 0; 2 Vt* ; 4 *V7' &c " the difference between any two consecutive values being 2 nt v/— • V y 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 W* be supported by the action of a constant foree upon the branch EH of an hyperbola, of which the transverse axis is vertical, the force being directed to the centre of the curve. Required the position of equilibrium. MECHANICS OF SOLIDS, 251 Denote the constant fcrce 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. De- note the distance CM by r, and the axes of the curve by A and B. Take the axis z vertical, and the curve in the plane xz. Make P ' = W, P" = W tnen will cosy' = 1, cos a' = 0, H % It < * cos y" = » cos a = , ' r r X = F cos a' + P" cos a" = r Z = P f cos 7' + P" cos 7" ss W - W . — i and as the question relates to the state of rest, The Equation of tho curve is ' L = A 2 x* - B 2 z 2 + A 2 B 2 = ; whence, dL dx dL dz sb 2^ 2 ar, = -2B 2 z; these values substituted in Equation (330), give whence, W'B* — - WA 2 x + W'A 2 — ss 0- r r (4 2 -f £ 2 ) JT •* - f*M 2 r = 16 (361) 252 ELEMENTS OF ANALYTICAL MECHANICS. But i«2 — - -r2 X 2 -f z 2 = z 2 + B 2 A 2 A 2 4- B 2 whence, denoting the eccentricity by c, r = ^/e 2 z 2 - B 2 and this, in Equation (361), gives after reduction, B . W z = e(W 2 - W' 2 erf which, with the equation of the curve, will give the positirii o^ equilibrium. If We be greater than W, the equilibrium will be mi^ja&ibie- If We = W, 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 line, and the origin at the point of departure, and let z be reckoned positive down- wards. Then will L = z — a x = 0, d L dz . dL = 1; -y- = — a; d x which in Equation (330) give, after omitting the sommon factor M s d 2 x d 2 z --dfi +a ^~ a T^ = Q - ( 362 ) From the equation of the line we have d*x m (Pz MECHANICS OF SOLIDS. 253 which in Equation (362), after slight reduction, d 2 z _ a 2 dt 2 ~ Ifo 2 Multiplying by 2dz, and integrating, '9' dz 2 a 2 ~dJ = 9 \ + a the constant of integration being zero. Whence 2 " /2(1 + a 2 ) dz V # • a 2 2 -/ z and ' , = ^3E37, = ^+^ ; . . . (3C3) V ga 2 y g a z z 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 ^4 B, we have . 2 . 2 ^.g 1 4- a z 2 . « 2 ' which substituted above, (364) in which of 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 a-> a diameter, we see that the time down any one of its chords, ter- minating 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 circumstances of the motion. 254 ELEMENTS 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 steepest descent Take the plane x z, to contain this element, the axis z vertical and positive upwards. The equation of the path will be, t L = z 4* * tan a — h ■=. ; whence, dL dz = i; dL dx = tan a. The extraneous forces are the weight of the sphere and the fric- tion. Denote the first by W, 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 con- stant force acting up the plane and opposed to the motion. We shall therefore have Z = — Mg -f Fsm a ; X = — Fcosa, which values, and those above substituted in Equation (330), give ■■'■''■ cPx / d 2 z\ — Fc.osa — M • -— - -f ( M a — .Fsin a + M* — - I ian a = 0. dt 2 \ J dt 2 / ; r! But from the equation of the path, we have h d?z = — d 2 x • tan a ; »nd eliminating vanish together, F r * * M-k* Also, because the length of path described in the direction of the $)lane is r.-vLj we have, in addition, h — z = r . 4> • sin a ; and eliminating 4- from this and the above equation, there will result t ■ « /_,** *£ (A - «), (J) V -r . r 2 .sma v ' w Dividing Equation (a) by Equation (/>), and solving with respect to /; ^= ^r .^ /if/ _____; ....... .(c) and this in Equation (&), gives f = Ji(h-z) t* + r» V */ • sin 2 \ *. x .d 2 y — y d l x _ "l 2 P (x cos ft — y cos a) — 2 m AA = ; 2 P (z cos a — x cos 7) —2m- — —— = dt 2 z . d 2 x — 2 'd 2 z dt 2 y ,d 2 z — z . d 2 y ► . . (365 2 P (y cos 7 — z cos /3) — 2 m • £- — £ = ; (365) the accents being omitted because the elements ?/<, m/, &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 dtecux sion being in all respects similar to that relating to the motion about the centre of inertia, § 127 and § 173, we pass to CONSTRAINED MOTION ABOUT A FIXED AXIS. S22S. — If the bodv be constrained to turn about a fixed axl*, both origins may be taken upon, and the co-ordinate axis y t<» coincide with this axis; in which case 8x^ Sy^ $p s1 '$$ and 8u, in Equation (40), will be zero, and to satisfy the conditions of equilibrium, it will only be necessary for the forces to fulfil thi? condition, z d x x ' d 2 z 2 P (z cos a — x COS7) —2m —f^ = <> ' ' (366) the accents being omitted for reasons just stated. §229. — The only possible motion being that of rotation, let us transform the above equation so as to contain angular co-ordinates. For this purpose we have, Equations (36), x' = r" sin -\, ; z' = r"cos-^ ..... (367) in which r" denotes the distance of the element m from the a„xis »*. Omitting the accents, differentiating and dividing by d t, we -have' dx d\ dz c?+ ftN __ = rC0 s + _; _ = _- sm + ._. • .(368, 258 ELEMENTS OF ANALYTICAL MECHANICS. Now, Z'd 2 x x> d 2 z 1 , / dx dz\ it \ dt dt/ ' dfl dt> whence by substitution, Equations (367) and (368), (Px d 2 z 1 . /. rf + \ d*± - - x . tf <2 tf (3 £/ ^ V dt/ - dt 2 ' d?-l> and since —~ must be the same for every element, we have, Equa dt* tion (366), 2 m r 2 • -yj? = 2 P (z cos a — ar cos y), and rf 2 4/ 2 P • (z cos a — or cos 7) rf t 2 2 m r 2 (369) 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 cnoment of the impressed forces divided by the moment _>f inertia with, reference to this axis. Denoting the angular velocity by V x , and the moment of inertia by /, we find, by multiplying Equation (369) by 2 c? 4* an( ^ integrating, IV X 2 = 2^2 P(z cos a — xoosy)d-^ -f C, and supposing the initial angular velocity to be F/, we have I(V 2 - F/s) = 2fzP(zcosa - xcos 7 )d^. But the second member is, § 107, twice the quantity of work about the fixed axis ; whence the quantity of work performed be- tween the two instants at which the b< 5y has any two angular velocities, is equal to half the difference of the squares of these velocities into the moment ' of inertia, 01 to half the lVng force gained or lost in the interval. MECHANICS OF SOLIDS. 259 Now, /= Mk 2 = M t . (I) 2 = M t ; so that, the moment of inertia measures that mass which would, if concentrated on the arc \, have a living force equal to that of the body which actually rotates. COMPOUND PENDULUM. § 230. — Any body suspended from a horizontal axis A B, 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 ; F* "= m f g, &c. ; the axis of z being taken vertical and positive downwards, cos a = cos a' = &c. = ; cos * cos/' == &c. and Equation (369) becomes d 2 \ 2mx dt 2 = - 9' Lmr 2 (370) Denote by e, the distance A G, of the centre of gravity from the axis; by -^, the angle HAG, which A G makes with the plane yz\ by x n the distance of the centre of gravity from this plane ; then will x 4 = e . sin -^ ; and from the principles of the centre of gravity, 2 m x = Mx t = M. e . sin ^ ; which substitute! above, gives d 2 \ = ~ 9 M . e . sin <\t 2mr 2 (371) 260 ELEMENTS OF ANALYTICAL MECHANICS. Multiplying by 2d-],, and integrating, d\ 2 M.e. d t 2 2mr* Denoting the initial value of >L by a, we have _. Me , _, = 2 9'~ r-coset -f C; whence, d-l 2 n M.e . . but C ° S+ = 1 -0- f 172^4-^ a 2 a* cos a = 1 4- &c. 1.2 x 1.2-3.4 and taking the value of 4* ? so small that its fourth power may dp neglected in comparison with radius, we have cos 4/ — cos a ss -— — ; 2 which substituted above, gives, after a slight reduction, and replacing 2mr 2 by its value given in Equation (216), d -vf- V e.g I v/-S ihe negative sign being taken because \ is a decreasing function of the time. Integrating, we have /Jc 2 _L e 2 _j r t = \/ ' — -cos — (373) V e.^ a v The constant of integration is zero, because when -^ = a, we have i = 0. MECHANICS OF SOLIDS. 261 Making -\. = — a, we have Ik 2 + e 2 which gives the time of one entire oscillat'on, and from which we conclude that the oscillations of the same pendulum will be isochro- nal, 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 6 denote the time of observation, and JV the number of oscillations, then will =i=*v k 2 + e 2 e.g and if the same pendulum be made to oscillate at some other location during the same interval &, 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, 6 _ A 2 + L — cos a) ; . - (375) 262 ELEMENTS OF ANALYTICAL MECHANICS, and making -~1 = F, : 2 m ** 2 = /: e (cos -L — cos a) = H ; we have I.V* = 2M.g.H; (376) in which H, denotes the vertical height passed over by the centre of gravity, and from which it appears that the pendulum will come to rest whenever -^ becomes equal to a, on either side of the ver- tical plant 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. Deno- ting the distance of the point of concentration from the axis by /, we have k t = ; e = Z, which reduces Equation (374) to t = «-\P- (377) If the point be so chosen that I Ik? +j? a -\"~e~:Y~ ; 9 or. • • • (378) the simple and compound pendulum will perform their oscillations in the same time. The former is then called the equivalent simple pen- dulum ; and the point of the compound pendulum into which the mass may bf concentrated tc 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. 26o £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 V , and the distance of this latter axis from the centre of gravity by e[ then will / = e 4- «' or e' = / — e\ and, Equation (378), 9 _ *, 2 ± e' 2 _ *, 2 + (/ - «)» 1 ~ ? ~ l-e and replacing /, by its value in Equation (378), we find h 2 4- e 2 e That is, if the old axis of oscillation be taken as a new axis of su;» pension, the old axis of suspension becomes the new axis of oscilla- tion. This furnishes an easy method for finding the length of an equivalent simple pendulum. Differentiating Equation (378), regarding / and e as variable, we have ii ~ g2 - k ? de e 2 and if / be a minimum, d - = o = e2 ~ k < 2 - de e 2 whence, e •=. k t . But when / is a minimum, then will t be a minimum, Equa- tion (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 ' e longer in proportion as the axis recedes from that centre. 264 ELEMENTS OF ANALYTICAL MECHANICS. u m i 1 f i i I - * M ] — i L i i Let A and a£* be twc acute parallel prismatic axes firmly con nected with the pendulum, the aaute 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 assumed position of M, let the principal radius of gyration be GC; with G as a centre, \° ) G C as radius, describe the circumference CSS'. 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 M, the centre of gravity, and therefore the gyratory circle of which it is the centre, may be thrown towards oither 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 cir- cumference, keeping the centre of this circumference between the axes, as indicated in the figure. In this position, it is obvious that anv 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 /, of the equiva- lent 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 /, we may easily find that of the simple tecotiiTs pendulum that is to say, the sirrple pendulum which vvilj MECHANICS OF SOLIDS. 265 perform its vibration in one second. Let iV, be the number of vibrations performed in one hour by the compound pendulum whose equivalent simple pendulum is /; the number performed in the same time by the second's pendulum, whose length we will denote by V, is of course 3600, being the number of seconds in 1 hour, and hence, N V g 1* fl 3600' V g * 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) 2 v ' Such is, in outline, the beautiful process by which Kater determined the length of the simple second's pendulum at the Tower of London 10 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-,2lTT59 th 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 YT2'S Xh °f a CUD,C f° ot > aR d 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 oblateness of the earth. The time of oscillation 266 ELEMENTS OF ANALYTICAL MECHANICS. being one second, and the length of the simple pendulum 3,26159 feet, Equation (377) gives 3,26159 — *\/ ; 9 whence" g = ** (3^6159) = (3,1416) 2 . (3,26159) = 32,1908 feet. From Equation (377), we also find, by making t one second, and assuming we have I = x -f- y cos 2 4s JL as # 4- y cos 2 4 (380) Now starting with the value for g at London, and causing the sume 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 un- known quantities, which may easily be found by the method of least squares. In this way it has been ascertained that «r 2 .* = 32,1808 and «r 2 .y = - 0,0821 ; whence, generally, t f g = 32,1808 - 0,0821 cos 2 4 ; .... (381) and substituting this value in Equation (377), and making t = 1, we find / I = 3,26058 -t 0,008318 cos 2 4 • . . . (382) JSuch is the length of the simple second's pendulum at any place of which the latitude is 4*. MECHANICS OF SOLIDS, 267 If we make + = 40° 42' 40", the latitude of the City Hall of Np.w York, we shall find I ft- in. 3,25938 = 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 from its position of equi- librium, oscillate, and become, in fact, a compound pendulum ; and denoting the length of its equivalent simple pendulum by /, we have, after multiplying Equation (378) by M, M.l.e = M (k* + e 2 ) = 2 m r* ; . . . . (383) or since W W M = — , 9 • I . e = Z,mr 2 , (384) '.n which W denotes the weight of the body. Knowing the latitude of the place, the length /' of the simple second's pendulum is known from Equation (382) ; and counting the number N of oscillations performed by the body in one hoar Equation (379) gives . V • (3600) 2 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 max- imum reading of the balance. De- noting by a, the distance from the axis C to the po'nt of support A', 17 268 ELEMENTS OF ANALYTICAL MECHANICS. and by b, the maximum indication of the balance, we have, frcm 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.l'.(3600) 2 9-N 2 = I. (385) If the axis pass through the centre of gravity, as, for example, in the flu-wheel, it will not oscillate; in which case, take Equation (383), from which we have Mk 2 = M.l.e - Me 2 . Mount the body upon a parallel axis A, not passing through the cen~ tre of gravity, and cause it to vibrate tor an hour as before ; from the num her of these vibrations and the length of the simple second's pendulum, the value of I may be found; M is known, bc'ing the weight W divided by g ; and e may be found by direct measure- ment, or by the aid of the spring balance, as already indicated; "whence k t becomes known. MOTION OF A BODY ABOUT AN AXIS UNDER THE ACTION OK IMPUL- SIVE FORCES. § 237. — If the forces be impulsive, we may, § 170, rep) 'ice in Equation (366) the second differential co-efficients of f, y, 2, by the first differential co-efficients of the same variables, which will reduce it to ~ tw \ ~ Z( lx — xdz 2 Piz cos a — x cos y) — 2 m • -= = : at MECHANICS OF SOLIDS. 269 and replacing dx and rfz, by their values in Equations (368), we find d\ 2 P (z cos a — x cos y) ~dT Zmr 2 (386) 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 sum 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 suspended from a horizontal axis in the shape of a knife-edge, after the manner of the compoimd pen- dulum. 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 the shape impressed upon it by the blow Denote by V and m, the initial velocity and mass of the ball ; V l the angular velocity of the ballistic pendulum the instant after the blow, / and M its moment of inertia and mass. Also let I represent the distance of the centre of oscillation of the pendulum 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 0. With this condition, Eq (386) give? 4* dt V x m.V.l mV TLmr 1 iJft4-«NJi**Hh**J i Jf +»»).« 270 ELEMENTS OF ANALYTICAL MECHANICS, whence m and supposing the angular velocity communicated to the pendulum tc be equal to that acquired by falling from rest through the initial arc a, in Equation (372), we have, from that equation and Equation (216), by writing e for d, and Eq. (374), - = V 9 e which substituted above ogives V i = 2 - • sin £ a ; and this in the value for V gives, afier substituting for the ratio of the masses that of their weights, W -+- w The quantity of motion in ball and wad, on leaving the gun, will be Y\ the corresponding pressure on the bottom of the gun is tc AS that which generates this motion, as the area of a cross-section of th«. 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 9 9 «* 9 9 tX\ which n\ like rc, is a constant to be determined by experiment: and from which we find IV .V.l.e r= * W.-t- - + nW .* + n'W m .€ 272 ELEMENTS OF ANALYTICAL MECHANICS. ^n 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 yi f . / . e — 2 W . e . versine a = 4 W , . e . sin 2 i a, 9 f g fn which a denotes the greatest inclination of e to the vertical. Whence which substituted above gives, Wt'^ + nW^n'W * £ V= j- sin ^a .... ifc88). p The methods for finding e and a are the same as in the ballistic pendulum. To find n and w', 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. — The physical condition of every body depends upon the rela- tions subsisting among its molecular forces. In the vast range of relations, from those which distinguish a solid to those which determine a gas or vapor, bodies are found in all possible conditions — solids run impercepti- bly into liquids, and liquids into vapors or 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 among each other is called a fluid — such as water, wine, mercury, the air, and, in general, liquids and gases; all of which are distinguished from solids by the great mobility of their particles among themselves. This distin- guishing 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. Sueh fluids arc 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, sealing wax, and the like. 27i ELEMENTS OF ANALYTICAL MECHANICS. § 241. — Fluids arc divided in mechanics into two classes, viz. : com- pressible and incompressible. The term incompressible cannot, in strict- ness of propriety, be applied to any body in nature, all being more or less compressible ; but the enormous power required to change, in any sensi- ble 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 gases as com- pressible. g 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 mechanical power. The existence of gases is proved by a multitude of facts. Contained in an inflexible and impermeable envelope, they resist pressure like solid bodies. Gas, in an inverted glass vessel plunged into water, will not yield its place to the liquid, unless some avenue of escape be provided for it. Tornadoes which uproot trees, overturn houses, and devastate entire dis- tricts, 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. g 243. — In the discussions which are to follow, fluids will be consid- ered as without viscosity ; that is to say, the particles will be supposed to have the utmost freedom of motion among each other. Sucli fluids are said to be 'perfect. The results deduced upon the hypothesis of per- fect fluidity will, of course, require modification when applied to fluids possessing sensible viscosity. The nature and extent of these modifica- tions can be known only from expeiiments. MECHANICS OF FLUIDS. 275 mariotte's law. § 244. — Gases readily contract into smaller volumes when pressed ex- ternally ; they as readily expand and regain their former dimensions when the pressure is removed. They arc therefore both compressible and elastic. It is found by experiment that the change in volume is, for a constant temperature, very nearly proportional to the change of pressure. The density of the same body is inversely proportional to the volume it occu- pies. If, therefore, P denote the pressure upon a unit of surface which will produce, at a given temperature, say 0° 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 experiments above referred to, will • p = P . D . . (389) This law was investigated by Boyle and Mariotte, and is known as Mariotte's Law. It has been found to be very nearly true for all gases which are not liquefied when subjected to great pressure and cold, and which are therefore called permanent gases. LAW OF THE PRESSURE, DENSITY, AND TEMPERATURE. § 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 changes for perma- nent gases may be expressed by p = P.B.(l + «0); (390) in which p denotes the pressure upon a unit of surface, D the density of the gas, the difference between the actual and some standard tempera- ture, and a a constant which is equal to y\s — 0,003665 when the standard is 0° centr., and 6 is expressed in units of that scale. First supposing D and 6 variable and p constant; then p and variable and D constant, Equation (390) gives dD a.D dp ap ii id z ~rr^ ; de~ i + ao w 276 ELEMENTS OF ANALYTICAL MECHANICS. The quantity of heat, denoted by q, necessary to change the temperature degrees from the assumed standard, will be a function of $>, D, 6 ; but because of Equation (390) we may write q=f(V,p) (b) The increment of heat which will raise a body's temperature one degree, is called its specific 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 t that when the density is constant, then will, dq dq dD C = dd~dD'~dd' or, Equations (a), dq dq dp C, = db = dp'~dd" dq a.D dD'l + ad' dq a.p and by division, making c = y . c , in which y denotes the ratio of the specific heat of the gas at a constant pressure to that at a constant density. This ratio is known from experi- ment to be constant for atmospheric air and other permanent gases. The experiments of Cazin make its value 1.41 for all permanent gases, and those of Dulong on perfectly dry air 1,417. Regarding y as constant, the integration of the foregoing equation gives Jii I / (See Appendix No. S.) D in which / denotes any arbitrary function of the quantity within the parenthesis, and from which, denoting the inverse functions by F, we may write, p = D r .F(q) (c) MECHANICS OF FLUIDS. 277 From Equation (390) we have, o= I -- = --.i)y-KF(q)^-. . . (d) a.P.D a a.P yif a v ' Sudden compression increases and a sudden expansion decreases the temperature of bodies, and if q remain the same, while suddenly p, D, 0, become p\ I) ', 0', we have, p' = D* .F(q) .(e) 1 „,-. - 1 «. P a e MECHANICS OF FL7IDS 279 C, will be measured by a' s'. But the jressuie upon the pistons and the temperature remaining the same, the entire volume of the fluid in th« vessel and tubes will be unchanged. Hence, as — a' s' \ dividing the equation above by this one, we have P P' - = - 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 prin- ciple 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 sur- face, and, Equation (396), P f = 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 transmis- sion is made. § 248 — Since the elements of the fluid are supposed in equilibrio, the pressure transmitted to the surface through t'le elements in con- 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 always in the direction of the normal. 280 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 ap plicable to a single particle, whose co-ordinates are x, y, z, we shall have and supposing the particle to have simply a motion of translation, we also have <$

tion due to the transmitted pressures, we have _ ± dp 2 P cos a = wJl • — • dx .d y . dz • dx 2 P cos P =miY — dp dy dy . dx . dz ; dp 2 P cos y = m Z — — - dz . dx . dy. Danote by D the density of the mass ;w, then will, Equation (2), m — D . dx .dy . dz, and by substitution, E< fixations (398) become 1 ii D dx 1 ■ ■ D dp dy 1 D dp dz dp' = "r - ^i W dp ' _ 7 d*z - L ~ JW ; (399) Denote by «, v and w, the velocities of the molecule whose co- ordinates 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 co- ordinates of the molecule's place; and, reciprocally, each coordinate wil be a function of t, u, v and w\ whence, Equations (12) and (13), d 2 x d u i 2 x du /du\ dt du dx du dy du dz as as I — ) 1 . -Z J . — • dt 2 dt \dt/ dt dx dt dy dt dz dt y dx dy dz and replacing — ->> — -? — > by their values u, v, w. respectively, w« (XL CL Z %M> £ have (Px x /du\ du • du du MECHANICS OF FLUIDS. 283 in the same way, d dv . Ht dz ' 1 if / dv\ d v dv 7T = (-77) + I- ' » + T- * v + ^ 2 \ dl / dx dy d 2 z / dw\ dw dw dw _^_ 3^ I I -j- . 1/ -J- . |» -1_ • d( l \ d t / dx dy dz w which, substituted in Equations (399), give 1 u dp dx 1 a dp dy 1 D ' dp d z X Y Z / du\ \dt) (—) \dt / /dio\ \dl) d u dx dv dx dw dx u u — u — d u dy V dv dy V dw dy V ■ five un du ~dz~ dv dz dw dz w ; w ; w. (400) Here are three equations involving five unknown quantities, viz. «, v, ic, p and D, which are to be found in terms of x, y, z and /. 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 y z, is v ; 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 u H — —'dx: dx and hence the relative velocity of the two molecules is du . du . u -\- - — dx — u = - — -d x. dx dx At the time t, the length of the edge joining these molecules ** dx, and at the end of the time t + d t, this length will be 1 du . . ,/, du d x -|- -, — • dx . d t = d x ( 1 -f- -r— ' at): dx v dx the second term being the distance by which the molecules in question approach toward or recede from one another in the lime dt. 18 284: ELEMENTS OF ANALYTICAL MECHANICS. In the same way the edges of the parallelopipedon which at the fcime t, were dy and dz, become respectively, , d v . . , /, dv dW 7 7 7 /I . ^ W 7 N d z -\ — - — • d z . d t = dz [l -j — - — • dt)\ d z d z and the volume of the parallelopipedon, which at the time t, waa dx .dy .dz, becomes at the time t -f- dt, dx . d „. dz(x+ ^. d() . (1 + ±L. dt) . (1+ i£. dt) . The density, which was D, at the time t, being a function of xyz and t, becomes at the time t -\- dt, dD . dD , d.D , dD . D + -r-r'dt + -J— -dx +-T— -dy -f —— *dz\ dt dx dy dz which may be put under the form, {dD dD dx dD dy dD dz\ , D 4- 1 1 4- • -— 4- I d t ; ^ Vtf* T dx dt ^ dy dt T dz rf#/ ' and replacing dx dy dz dt dt dt by their values u, v, w, respectively, 'rfi> rfi) dD . dD y Multiplying this by the volume above, we have for the mass of th* parallelopipedon, which was D .dx .dy .dz, at the time t, the value, /dD dD dD dD \ , D 4- (-77 + -7— •« + -j— -v -f -j--w) dt. \ dt dx dy dz J r (dD dD dD dD \ , 1 X dx.dy.dz (l 4- -J" d <) * ( ] + ~d~' dt> ) ' + ~T~' dt ) at the time * -}- rf & MECHANICS OF FLUIDS. 285 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 „ /dn dv dw\ dD dD dD dD D (__ + -— + -_) +_.+ _.«+_ .»+ — »== 0.(401) \dx dy dz ' dt dx dy dz This is called the Equation of continuity of the fluid. It expres ses 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) =0, • • • (402) as is illustrated in the particular instance of Mariotte's law, Equa- tion (389). The form of the function designated by the letter F will depend upon the nature of the fluid. §251. — If the fluid be incompressible, the total differential of D will be zero, and dD dD dD dD a t dx ay dz s and consequently, the equation of continuity, Equation (401), becomes, d u dv dw ^ v -I7 + 7j + -J7= > < 40 *> aid we have for the determination of ?/, t», w, D and />, the five Equations (400), (403), (404). § 252. — These equations admit of great simplification in the case of ai. incompressible homogeneous fluid when u-dx -f- v.dy -f- w.dz, is a perfect differential. For if we make ud x -f- vdy + wdz = d (412) and the above may be written, after dividing by d t, t_ t d_D d\ogD d^ dt D { dt dt' v ' which, in Equation (410), gives ^1 = a * ( d I± + ** + ^L) . . . . (414) dt 2 \dx 3 T dy* T dz*/ \ ' '} From this Equation the function

= £.-. /♦ r r MECHANICS OF FLUIDS. 289 Differentiating the first of the above equations, we have xdx-\-ydy-\-zdz — r . dr. Substituting the values of a-, y, and z from the second, third, and fourth, there will result udx -\- vd y + w dz = C, . dr\ go that this satisfies the condition of the first member beinjr an exact differential ; and, therefore, d $ = £. d r ; or *"" dr' And hence d cp d «p x d (p d

z d x dr r ' t/ y dr >r' dz dr r ' differentiating, o? 2

' of which the integral is, Appendix No. IV n r ? = ^r + a/)+/(r-fl<); and in which ^ and / denote any arbitrary functions whatever. From this we have 290 ELEMENTS OF ANALYTICAL MECHANICS. 9 = -l F ( r + at )+f( r - at )] ••.. (415) Taking the first differential coefficient of

), (424) in which F denotes any function whatever, the above equation be- comes —j^ = Xdx f Ydy + Zdz-, . . . (425) but for a level surface or stratum, the second member reduces to tero ; whence, dF(D) = 0-, and by integration, F(D) = 0; 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. 293 simply requires that the potential function Xdx -f Ydy + Z dz, Equa- tion (419), shall be an exact differential of the three independent variables #, 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 Mere contained in a closed vessel. But the function above 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 equi- librium, 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, § 136, and hence the denser strata should be the lowest. When thf> fluid is incompressible, the density may be any function whatever of the co-ordinates of place. It may be continuous or dis- continuous. When it is given, the value of the pressure is found from Equation (419). § 2G3. — 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, PXdx + Ydy + Zdz . „ , A - 294 ELEMENTS OF ANALYTICAL MECHANICS, denoting the base of the Naperian system by e, we have fXdx + Ydy+Zdi . ( \?> 2 .xdx +

(431) which is the equation of a paraboloid whose axis is that of rotation. 296 ELEMENTS OF ANALYTICAL MECHANICS. To find the constant 6', 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 and « 2

— r » ~- r > — i - r f and r 3 kg r3 ' which in Equation (430), give k — {xdx + ydy + zdz) — 2 • cos 2 & L . and this in the equation rsl+u, gives . • . r r= 1 + -?- . cos 2 H3m 2 ; 2k and replacing w 2 by its approximate value - — >i above, by neg- lecting 3 m 2 , we have , , 9 2 o * , 3

, the specific gravity, which is also the tangent of the angle en 0. That the term — represents a line may readily be seen, for p, the (1) pressure upon the unit of surface, is a weight w divided by an area n'P ; and the specific gravity u is another weight w" divided by a volume n"P. So that p hi' to P\ 1 . .0) \n w Pf in which the coefficient », within brackets, is only a product of abstract numbers, but / is linear. Equation (435'), or its construction, shows the law that in heavy liquids pressiwes are pi'oportional to depths. No portion of the liquid can be above the level AA\ where z is equal to H and p is zero ; for if z be supposed to exceed H, then either p or l (j must become negative, which is impossible. If the atmosphere rests upon the liquid, then its surface sustains the pressure p due to the weight of the air; and drawing BB' below AA at the distance *>,. — P° . nr = — ; MECHANICS OF FLUIDS. 301 this plane BB' will be the top of the liquid ; above wjiich none of it can exist except in the form of vapor mixed with the air. The atmospheric pressure may be eliminated by transferring the origin of co-ordinates and axis of z to the new position O'n', for which p is zero at n. And generally, whenever in hydraulic questions only differ- ences of liquid pressure need to be considered, the atmospheric -pressure may be eliminated. For gases, the value of D given by the law of Gay Lussac, Equation (390), may be substituted in Formula (435) ; and thus, after dividing* we find 9 P If in this be constant, we may write A for the factor of the second term, and integration gives z -f- A \ogp = c ; for any other height z 9 and pressure p g , we have, z e + A \ogp = c; and, therefore, z — z„ = A log — ; (436) an equation which gives the altitude z — z^ when p, p , and A have been determined by observation. § 267. — Let now the liquid, acted upon by its weight only, be con- tained in any vessel ; and let the axis z be taken vertical and positive downwards, then X=0, F=0; Z=f/; and Equation (418) becomes, after integrating, p = £ffz+ 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 Furfacc. Whence, p — p'=zD.n.z (437) "302 ELEMENTS OF ANALYTICAL MECHANICS The first member is the pressure exerted upon a unit of surface, every point of which unit sustains a pressure equal to that upon the element whose co-ordinate is z. If p' = k the pressure on the free surface, then will P = Dgz\ • . (437) which shows that 2wesswcs are 'proportional to depths. Denoting by b the area of the surface pressed, and by db the element of this surface, whose co-ordinate is z, we have, Equation (307), for the •pressure p t upon this element, p t == Dg.z. db, and the same for any other element of the surface ; whence, denoting the entire pressure by P, we shall have P = lp / = Bg.lz.db (437") But if z denote the co-ordinate of the centre of gravity of the entire surface b, then will, Equations (91), Zz.db = bz t , and P=zDg.b.z i (438) Now bz t is the volume of a right cylinder or prism, whose base is b and altitude Z f \ Dg.b.z t is the weight of this volume of the pressing fluid. Whence we conclude, that the pressure exerted upon any surface by a heavy fluid is equal to the weight of a cylindrical or prismatic column of the fluid whose base is equal to the surface pressed, and whose altitude is equal to the distance of the centre of gravity of the surface below the upper surface of the fluid. When 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 ves- sel filled with water, one of its faces being horizontal. Call the edge of the cube a, the area of each face will be a 2 , the dis- tance of the centre of gravity of each vertical face below the upper surface will be -J-rt, and that of the f\ \ — \ MECHANICS OF FLUIDS 303 lower face a ; whence, the principle of the centre of gravity gives. 4 a 2 X I a + a 2 X a 5« 2 8 a Again, b — 5 a 2 ; and these, substituted in Equation (438). give P - D .g-b.z, = B.g.Sa 3 . Now D g x I 3 = Dg, is the weight of a cubic foot of water ^62,5 lbs., whence, lbs. 62,5 X 3a 3 . 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 a 3 , yet the pressuw is 62,5 X 3a 3 , whence we see that the outward pressure to break the vessel, is. three times 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 R. The surface of the sphere being equal to that of four of its great .circles, we have b = 4«R 2 ', whence. and, Equation (438), b.t t = 4*i2 3 ; P = 4*.£.g.R 3 . The quantity Dg X V = D g, is the weight of a cubic foot of mercury =843,75 lbs., and therefore, substituting the value ( of r = 3,1416, lbs. P = 4 x 3,1416 x 843,75 . R 3 . 304 ELEMENTS OF ANALYTICAL MECHANICS. Now suppose the radius of the sphere to be two feet, then will R 3 = 8, and lbs. lbs. P = 4 x 3,1416 X 843,75 X 8 = 84822,4. The volume of the sphere is £ * R 3 ; and the weight of the con- tained mercury will therefore be ±*R 3 gD — W % Dividing the whole pressure by this, we find 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 I, 6 feet. Then will b.z, = «rl{r + I) = 3,1416 X 2 X 6 X 8; which, substituted in Equation (438), P = 301,5936 X Dg, *nd W - 3,1416 x 2 2 x 6 X Dg = 75,398 x Dg\ whence, P _ 301,5936 X Dg W ~ 75,3984 . D g ~~ * lhat is, the pressure against this particular vessel is four times the Aeight of the fluid. § 268.— The point through which the resultant of the pressure upon all the elements of the surface passes, is called the centre of pressure. i' sin 9 db\ its moment with reference to the line MN, D gr' 2 sin 9 . db\ and for the entire surface, the moment becomes D g . sin 9 . 2 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 the fluid will be r . sin 9 ; and the pressure upon this surface will be ' D g . r sin 9 . b ; and if / denote the distance of the centre of pressure from the line M N, then will Dg .ramp.b.l = Dg . sin 9 . 2 r' 2 . db, from which we have, Irt ^_7±=.t_±I.i (439) r . o r , • . whence, Equation (238), the centre of pressure is found at the centre cf percussion of the surface pressed. The principles which have just been explained are of grtirit practical importance. It is often necessary to know the precise 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 col- lected and retained till needed for purposes of irrigation, the supply of cities and towns, or to drive machinery ; dykes to keep the sea 306 ELEMENTS OF ANALYTICAL MECHANICS. and lakes from inundating low districts ; artificial embankments con- structed 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 pres- sure. Let A B C be a section of pipe perpen- dicular to the axis, the inner surface of which is subjected to a pressure of p pounds on each superficial unit. Denote by R the radius of the interior circle, and by / the length of the pipe parallel to the axis ; then will the surface pressed be measured by % f R . / ; and the whole pressure by 2*R.l.p. By virtue of the pressure, the pipe will stretch ; its radius will become R -f d R, the path described by the pressure will be d R, and its quantity of work 2« R.l.pdR. The interior circumference before the pressure was 2tfi£, afterwards 2 and there can be no vertical motion of translation fi:m the fluid pressure and the body's weight. When D' > 2), then will d 2 z ■Zm. J¥ ={D'-D)V, 9i and the body will sink with an accelerated motion. When D < D, then will 2m~=-(D'-D) V'.g, and the body will rise with an accelerated motion till 2m.°^ ^ V D'g - VDv = 0', • • fcl42) 310 ELEMENTS OF ANALYTICAL MECHANICS. in which V denotes the volume ABC, of the iluid displaced. At this instant we have V'D'g = VLg- (443) and if the body be brought to rest, it will remain so. That is, the body will float at the surface when the weight of the fluid it dis- places 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 effort. 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 .2 c . d b . cos y . x' = D g . V. x ; D g . 2 c .d b . cos y . y' = D g . V . y ; (444) 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 x' . d 2 y' — y' . d 2 x f * . „ = ; 2 m 2 m d& 6 -d 2 x' - x' • d 2 z' dt 2 y'' d 2 z' - z' d 2 y' dt 2 — Dg* V'X\ — — Dg-V-y. (445) 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 = 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 ha\e no tendency to rotate. To recapitulate, w r e rind, MECHANICS OF FLUIDS. 311 1st. That the pressures upon the surface of a body immersed in a heavy fluid have a sinyle resultant, called the buoyant effort of the ffuid, 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. 4 th. 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 A BCD 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 It that of the fluid dis- placed. 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 ^JJ will be vertical, and denoting the density of the fluid by D, we shall have M = D. V. (446) Suppose the section ABCD 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, «', &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 from it a variable section. Let A' B' 6" D' be one of these sections at any g'ven instant of 312 ELEMENTS OF ANALYTICAL MECHANICS. time; A B" C D", another variable section of the body by a hori zontal plane through the centre of gravity of the primitive section A BCD. and A C the intersection of the two Denote bv the inclination of these two sections, and by £ the vertical distance of A B" CD", from the plane of floatation, which now coincides with A' B' C D\ this distance being regarded as negative or positive, ac- cording as A B" C D" is below or above the plane of floatation. The variable quantities and £ 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), fu 2 .dM = 2f(Xdx -f Ydy 4- Zdz) + C. The forces acting are the weights of the elements dM and the verti- cal pressures, the horizontal pressures destroying one another ; whence, X = 0, Y = 0, and JutdM = %fz dz + C=22Zz -f C . . (447) The force which acts upon the body downwards is its own weight, and the force which acts upon it upwards is the difference between its own weight and that of the fluid it displaces; the first will be the integral of g.dM, and the second that of g.D.dV, whence, ^Zz — Jg.z.dM — JgD.z.dV. . . . (448) But, drawing from the centre of gravity G, of the body, the perpen- dicular G E, to the plane of floatation A' B' C D' , and denoting G E by z t , we have ■ / g . z .d M = g Mz r The integral J gD. z. d V, will be divided into two parts, viz: one relating to the volume of the body below A B CD, or the volume immersed in a state of rest, and the other that comprised between MECHANICS OF FLUIDS. 313 A BCD and the plane of floatation A' B' C D', when the body is in motion. Denote by g D V z\ the value of the first, in which z f denotes the variable distance H F, of the centre of gravity H, of the volume V, from the plane of floatation A' B' C D f . And repre- senting for the instant by h tne value of the integral / zdV, com- prehended between the planes A BCD and A' B' C f D f , gDh will be the second part; and Equation (447) becomes f utdM = 2g.Mz t - 2gDVz' - 2gDh + C. • • (449) The line G //, being perpendicular to the plane A B C D, the angle which it makes with the line G E is equal to d, and denoting the dis- tance G H by a y we have z t — z r ± a cos 8 ; 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 fu 2 dM = ±2gD Vacosd — 2gDk + C. • • • (450) Let us now find the integral //. For this purpose, conceive the area ABCD to be divided into indefinitely small elements denoted by d\ and let these be projected upon the plane of floatation. A' B' C D'. The projecting surfaces will divide the volume com prised between these two sections into an indefinite number of vertical elementary prisms, and these being cut by a series of hori- zontal planes indefinitely near each other, will give a series of ele- mentary volumes r each of whicji will be denoted by d V } and we shall have d V = dz . d\. cosd ; whence, for a single elementary vertical prism JzdV — Jzdz.dX.cosQ = J (z) 2 . cts . d\ ; in which (z) denotes the mean altitude of the prism, and consequently h = I cosd. f{zf.d\ which must be extended to embrace the entire sur ace A B CD, 314 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 AB"CD", and which has been denoted by £; the other comprised between the base d\ and the second of these planes, and which is equal to / , sin , y, ' and making the volumes equal, W w, m J) (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 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 condi- tion—gaseous, liquid and solid; and in every kind of shape and of all sizes. The determination of their specific gravity, in every in- stance, 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 fluid are equal. Hence the 1 - 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 "20 318 ELEMENTS OF ANALYTICAL MECHANICS. standard, and the loss of weight be made to occupy the denomi nator, this ratio becomes the measure of the specific gravity of the bcdv 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 temperature, 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 D 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 r 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 stan- dard, and corresponding to which it has a density D in then will D S' = A, Dividing the first of these equations by the second, we have S' D/ whence, S= S'-^-, (455) and if the density D t , be taken as unity, ,, S = S'D„. - . (450) MECHANICS OF FLUIDS. 319 That is to say, the specific gravity of a body as determined at tht standard temperature of the water, is equal to its specific gravity deter- mined at any 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 pur- pose, take any pure 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 32° Fahr. be denoted by /, and the radius of its base by nd\ its volume at this temperature will be, 77 m 2 P x I = 77 m? I s . Let nl be the mean expansion in length for each degree of the ther- mometer above 32°. Then, for a temperature denoted by t, will the whole expansion in length be nl x (t - 32°), and the entire length of the cylin- der will become l+nl(t-32°) = l[\+n (*-32°)]; which, substituted for I in the first expression, will give the volume for the temperature t, equal to «m 2 P[l + n(t — 32°)] 3 . The cylinder is now weighed in vacuo and in the w r ater, at differ- ent 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 320 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 38°. 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, 38°.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 /, exceed s 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. ATMOSPHERIC PRESSURE. § 277. — The atmosphere encases, as it were, the whole earth. It has weight, else the expansive action among its own particles would cause it to extend itself through space. The weight of the upper stratum of the atmosphere is in equilibrio with the expansive 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. 321 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. The force thus exerted. is called the atmospheric pressure. As the absolute amount of atmospheric pressure was first discovered by Torricelli, the tubes employed iu such experiments arc called Torricellian tubjs, and the vacant space above the mercury in the tube is called the Torricellian vacuum. The pressure of the atmosphere at the level of the sea, support- ing 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. BAROMETER. §278. — The atmosphere being a heavy and elastic fluid, is com- pressed by its own weight. Its density cannot be the same through- out, 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 decompositions, throw large quantities of aqueous vapor, carbonic acid, and other foreign ingredients temporarily into the permanent 322 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 incessant change. These changes arc 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 changes and those of the weather. The barometer consists of a glass tube about thirty-four or thirty- five inches long, open at one end, partly filled with distilled mer- cury, 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 corresponding increase of weight; if, on the contrary, the density of the air diminish, the column will fall till its diminished weight is sufficient to restore the equilibrium. In the Common Ba?ometer, the tube and its cis- tern are partly inclosed in a metallic case, upon which the scale is cut, the cistern, in this case, hav- ing a flexible bottom, 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 par- allel legs : the longer is hermetically sealed, and constitutes the Torricellian tube ; the shorter is open, and on the surface of the quicksilver the pressure of the atmosphere is exerted. The difference be- tween the levels in the longer and shorter legs is the barometric A \ 31 .— 30 SS£ 29 ^1 m *=f f r^ 5 i b MECHANICS OF FLUIDS 323 r 30 1=1 height. The most convenient and practicable way of measuring thi* 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 shorter leg. Different contrivances have been adopted to ren- der the minute variations in the atmospheric pres- sure, and consequently in the height of the barome- ter, more readily perceptible by enlarging the di- visions 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 Diago- nal, and Hook's Wheel-Barometer, but especially Huygcns 1 Double-Barometer. The essential properties of a good barometer are perfection of vacuum, width of tube, purity of the mercury, accurate graduation of the scale, and a good vernier. § 279. — 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 be indi- cated ; the height of the mercury in the tube, expressed in inches, reduced to a standard tempera- ture, 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 tilled 7 with mercury, a is a tube communicating with the boiler, and through which the steam flows and presses upon the mercury ; the barometer tube be, op«n at top, reaches nearly to the bottom of the vessel A. 60 45 30 15 324 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 Ma- riotte's manometer is considered preferable. The tube being filled with air and the upper end closed, the surface of the mercury in both branches will stand at the same level as long as no steam is admitted. The steam being admitted through d, presses on the surface of the mercury a and forces it up the branch b c, and the scale from J to c marks the force of compression in atmospheres. The greater width of 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. § 280. — Another very important 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 instru- ment 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 known that the surface of the earth is not uniform, and does not. in consequence, sustain an equal atmospheric pressure MECHANICS OF FLUIDS. 325 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 instru- ment of great value in the hands of the traveller in search of geographical information. § 281. — To find the relation which subsists between the altitudes of two barometric columns, and the difference of level of the points where 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 = 0; Y = 0; Z= - g. and hence, p = Ce~- S T (462) Making 2 = 0, and denoting the corresponding pressure by p { , we find p,= C; and dividing the last equation by this one, P - e />, whence, denoting the reciprocal of the common modulus by M, MP , P, z = log — (463) 9 P V ' Denote by h { and A, the barometric heights at the lower and upper stations, respectively, then will 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 2i h p h [1 + (T - T) .0,0001001]' in which T denotes the temperature of the mercury at '.he lowrr and T' that at the upper station. 21 326 ELEMENTS OF ANAL7TICAL MECHANICS. Moreover, Equation (381), a = g' (1 - 0,002551 cos 2 +) ; in which. f* = 32,1808 = force of gravity at the latitude of 45°. P Substituting the value of — -■, of ,A, l4-|(/ / + ''-G4°)0,00204 ("A, 1 xlog -^-x ~ D t 1 -0,002551 eos 2 + The factor — — --•> we have seen, is constant, and it only rc- D t mains to determine its value. For this purpose, measure with aecuracv the difference of level between two stations, one at the base and the other on the summit of some lofty mountain, bv 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 t ; 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 t and V . 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 M D„ h m ' a - 60345,51 English feet ; D, which will finally give for z, ft. i-f-i(; / + ;'_64°)0. 00204 Vh 1 "1 *=60345,oL l _ o,002551 cos 2+ X ,0g U~ X l +( ^-7^)0,000 1001 J To find the difference of level between any two stations, the lati- tude 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. 328 ELEMENTS OF ANALYTICAL MECHANICS. Make 60345,51 [1 + (t 4 + t' -64°)0,00102] = A, , = B. Then will 1 — 0,002551 cos 2-1 1 1 + (T - T) 0,0001 = C. z — A B > log h z = AB> [log C + log h t — log h] ; and taking the logarithms of both members, log 3 = log A + log B + log [log C + log h t — log A] . . (404) Making t t + t' to vary from 40° to 162°, 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 + t', as an argument. Causing the latitude -^ to vary from 0° to 90°, 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 — 30° to + 30°, 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 Howlet, 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 ob- served value of T — T ', and subtract from this sum the logarithm of the barometric reading at the upper 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 + t' and -^ ; the siim will be the logarithm of the height in English feet. > MECHANICS OF FLUIDS 329 Example. — At the mountain of Guana xuato, in Mexico, Von Hum boldt observed at the Upper Station. flower Station. Detached thermometer, t' = 70 c ,4 ; t t = 77°,6 Attached " T' = 70,4 ; T =r 77,6. Barometric column, h = 23,66 ; h t — 30,05. What was the difference of level 1 Here t g + f = 148° ; T — T = 7°,2 ; Latitude 21°. in. To log 30,05 = 1,4778445 Add C for 7°,2 = 9,9996814 1,4775259 in. Sub. log 23,66 = 1,3740147 Log of - - - 0,1035112 = - 1,0149873 Add A for 148° - - - - = 4,8193975 Add B for 21° - - - - = 0,0008689 ft. - 6843.1 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 equilibrium, 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 accu- racy, 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 j 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 tempera- tures of the air at this station, and use these means as a single 330 ELEMENTS OF ANALYTICAL MECHANICS set of observations made simultaneously with those at the highei station. Example. — The following observations were made to determine the height of a hill near West Point, N. Y. Upper Station. Lower Station. (1) (2) Detached thermometer, V = 57° ; t t = 56° and 61°. Attached " T = 57,5 ; T = 56,5 and 63. in. in. in. Barometric column, h = 28,94 ; h t = 29,62 and 29,63. First, to reduce 29,63 inches at 63°, to what it would have been at 56°, 5. For this purpose, Equation (394) gives in. k (I + T - T X 0,0001) = 29,63 (1 - 6,5 x 0,0001) = 29,611 Then \ = °±+«l - - = 58o,5, ■ l t -f t' = 5S°,5 + 57°- - = 115°,5, T— T = 56°,5 - 57°,5- = - 1°. in. To log 29,6155 = 1,4715191 Add C for — 1° = 0,0000434 1,4715625 Sub. log of 28,94 = 1,4614985 Log of - - - - 0,0100640 = - 2,0027706 Add A for 115°,5 - - - = 4,8048112 Add B for 41 a ,4 - - - - - 0,0001465 /* 642,28 2,8077283; whence the height of the hill is 642,28 English feet. MOTION OF HEAVY INCOMPRESSIBLE FLUIDS IN VESSEia. § 282. — Let it now be the question to investigate the flow of a heavy, brmogeneous and incompressible fluid through an opening in a vessel which contains it. And for this purpose, resume Eq. (407), which is MECHANICS OF FLUIDS. 331 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 = ; Y = ; and Z = — g. Any lateral or horizontal motions will have no vertical components and may, therefore, be disregarded, and we shall have, Eq. (405), which will reduce the general Eq. (407) to and integrating, &» d(p p^-Dgz-D-jj 4 D . «* 4 C • (465) Next, find the function , at the time t The continuity of the fluid requires that w • s = Wj • s t , because the same quantity must flow through every horizontal section it the same time : whence w = w. — : which, in Eq. (466), gives the integration being taken with respect to the variable 2, 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 mav be well to remark here, that z r J z + h 9 dz * will be constant for the same vessel and same value of h; and if the figure of the vessel be that of revolution about a vertical axis, it will only be necessary to have the equation of this vertical section to find the value of the integral. The quantity h is called the head of fluid. Differentiating

-v^/ /T -^|>- x J- (468) Also, if P' denote the pressure at the upper surface corresponding to which z = z\ we have Now z' — z, ■=. h = height of the fluid surface above the section CD) whence, by substitution and transposition, The quantity of fluid flowing through every section in. the same time being equal, we also have — Sdk = s t ,w t .dt. • * •-••■• * • (4T1); By means of this equation, t may be eliminated from Equation' (470) ; then knowing the quantity of the liquid, the size and figure y n%t d z r^dz — == / — ♦ in which s is a function of z. *t * rl in § 283. — The value of —r— - being found from Equation (470), and (X C 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. § 284. — 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 334 ELEMENTS OF ANALYTICAL MECHANICS. as it flows out at the bottom, the head h must be constant, and Equation (471) will not be used. Making, in Equation (4V0), A = 2t J.T> £ = 2g ( h + IL*L) ; I>9 s? n _ zi 1 • and solving with respect to d t, we have ■ « = £&? V < 472 > Now, three cases may occur. 1 st. S may be less than s, , and C will be positive. 2d. S may be equal to s n 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 t = 0, t = —== • tan w t K/rg ; (473) whence, W/=v /|. tan ^. t . . . . . (474) from which we see that the velocity of egress increases rapidly with *he time; it becomes infinite when a > ' t ~ : IT w t = -==- • • (475) When (7=0, then will the integration of Equation (472) give A ~B ***•**> • ' ' < 4 ™) MECHANICS OF FLUIDS. 335 or replacing A and B by their values, and finding the value of w t , "< = =QT ' ; < 4r » 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 A . VB + w. *JC t = , • log -*—= * ; 2-y/lfC ^B-W t y/U whence, • t "< = '*U*7 l '\/i! 1 (478) e A +1 in Which e is the base of the Naperian system of logarithms = 2,718282. If the section S exceeds s t considerably, the exponent of e will soon become very great, and unity may be neglected in comparison with the corresponding power of e ; whence, w - = \/J = \/ : t^ — ;• • • («») 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 t = 0, and (480) and if the opening below become a mere orifice, the fraction s 2 — = 0; and u> t = V27A; (481) 336 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, which is called the law of Torricelli, who discovered it experimentally. The velocities given by Equations (479), (480), (481), are inde- pendent of the figure of the vessel. If the velocity w t be multiplied by the area s t of the orifice, the product will be the volume of fluid discharged in the unit of time. This is called the expense. The expense multiplied by the time of flow will give the whole volume discharged. § 285. — The velocity w 4 being constant in the case referred to in Equation (479), we shall have dt * fcnd Equation (468) becomes f = P, - Dg (, - o - D. \ (5l _ i) ; or, substituting the value of w 4 , given by Equation (470), £ - P t ) .£_ ; . . (482) 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 * of the stratum, its distance s 2 from the upper surface of the fluid, and upon the ratio -^. § 286. — 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, = y/^gff, (483) MECHANICS OF FLUIDS. 337 in which H denotes the height due to the velocity of discharge : we have dw t = *—=. (484) and, Equation (471), , S • d h . ^ dt= * (485) and by integration. 1 rS>dh t= C =• I —;= (486) To effect the integration, S and H must be found in terms of A. 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 w t \ d w 4 , and d t from Equation (470), by the values above, and we have or, dividing by s* r h dz ~s'Jo V * ■ (^-' + g '•('-£ ^-= dh + dE i-f B dh =0.(487) /* dz n P h dz v ' 'Jog 'Jos and making 'Job ' J o 8 Qdh + dlf + RJIdk = 0. . . . (488) fRdh Multiplying by e , fRdk f^dh fRdh dh>Qe +dH>e +- H . e xRdh=0; 338 ELEMENTS OF ANALYTICAL MECHANICS. or J Rdh , ftldh. dh- Q-e + d (He ) = 0; and integrating //Rdh /Rdh dh-Q-e + He = C\ . . . . (489) whence, —/Rdh . /Rdh. H= e • (C - fdh-Q-e )' ' ' (490) The constant must result from the condition, that when H = 0, h. must be h t , the initial height of the fluid in the vessel. Thus H becomes known in terms of A, 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) must be determined, so that when t = 0, h = h t . § 28*7. — Equation (490) gives a direct relation between S, A, and H\ the figure and dimensions of the vessel give another between S and h. From these, two of the three variables mav be eliminated from the Equation (486) and the integration performed. Take, for example, the case of a right cylinder or prism. Here S will be constant, and equal to s. L h ilz h 7 ~~s' Moreover, let us suppose P' — P t = 0, which would be sensibly true were the fluid to flow into the atmosphere that surrounds the vessel. Also, for the sake of abbreviation, make — = &, ther will K and J Rdh = (1 - &)fj = (1 -* 8 )1ogfc MECHANICS OF FLUIDS. 339 and Eq. (490) becomes -*»)logA • [C-JWdh-e -*s)le t 1 Multiplying the last term by 2 -k 2 h 2 i -k 2 K we may write -(l-ka)log k H=e .[»- -.2 -**/ rf ('" (l-k»> log A — (1— l2)log h = e .[,. * 2 , •»- -it*) log 1. when H •=. 0, then will /i = A/ and C =2 * 2 - k 2 (1-42) lor ft, h r e ; which substituted above, gives. after reduction, * = 2 **.* -F -]' but, 1 ll-Jfcs e ) lo sr /AA 1- * 2 ; >] and therefore, -^[(ir-]-^D-o"n---« which substituted in Equation (486), gives Ik 2 — 2 P dh ltMK in which the only variable is h. § 288. — The particular case in which k 2 = 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 2kdh + hdff- Hdh = 0: 340 ELEMENTS OF ANALYTICAL MECHANICS — 2 multiplying by h , 2h~ l dk + h~ l .dH -H.hT 2 dh = 0, 2 • h d — = 0, h v . * 21ogA + y = (7; and because IT = when h = h, , 21ogA, = C; whence, # = 2 A • log y> and this, in Equation (486), gives ' 1 /» G?A vT* \/ 2A - Io «T Making ~ = — ri this becomes between ine limits .r = 0, a: = 1 X' t = C § 289. — If the orifice be very small in comparison with a cross section of the prismatic or cylindrical vessel, then will H = A, and Equation (486) gives < <=<7--^L--/I Making t = when h = J), we have « = -M= . (y>T, _ -/*). • • • • • (494) and for the time required for the vessel to empty itself, h = 0, and MECHANICS OF FLUIDS. 311 Now, with the same relation of the orifice to the cross section of the cylindrical vessel, we have, Equation (481), w t — y/2yh, and for the volume of fluid discharged in the time t, when the vessel is kept full, w t . s t .t = s 4 ,t . -\Z2gh, and if this be equal to the contents of the vessel, * t .t. y/2gh l = S . h t ; whence, S fh. s 4 V 2g That is, Equation (495), the time required for a prismatic or cylin- drical 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. § 290. — The orifice being still small, we obtain, from Equa- tion (485), d 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 whence, s? .2g h & = a* but supposing the horizontal sections circular, ^ . »?'2gh 22 a* 342 ELEMENTS OF ANALYTICAL MECHANICS, and, therefore, whence the radii of the sections must vary as the fourth root of their distances from the bottom. These considerations apply to the con- struction of Clepsydras or Water Clocks. STEADY FLOW OF FLUIDS. § 291. — When with lapse of time the form, quantity, density, pressure and velocity of a stream, at and for any given point therein, do not change, though differing from one point, or cross section, to another, then the flow is steady or permanent; and under such cir- cumstances the general equations for fluid motion, (400) and (401), become greatly simplified. Moreover, as the flow of water in rivers, channels, and pipes may generally be considered steady for short periods of time, or its mean value for longer durations, very many of the most important appli- cations of fluid motion are only problems of steady flow. From the definition given it is clear that the steady motion of fluids will be expressed by Equations (399) and (400), if the five quantities, u, v y w, p, *nd D are assumed not to be functions of time, but only of x, y, z y the co-ordinates of position ; or, analyti- cally, if du dv dw dp dD It ~~ di "" ~dt ~ It ~ ~di ~~ * which conditions give d?x d*y d?z Jv2 1a "* dt* ' ' di 2 ' dt* and cause Equations (399) to become, whence also, dx dy dz dU = Xdx -f Ydy -f Zdz = — dp. MECHANICS OF FLUIDS. 343 These are the same as Equations (417) and (423); which proves that, in the uniform or steady flow of fluids, pressure acts as it does when they are in a state of equilibrium of rest. Multiplying now the first of Equations (399) by e/.r, the second by dy, and the third by dz, then adding, we obtain, 1 , ,„ , Jdx* dy* d*\ 3 d *= dn ^* d \dfi+M+m But the quantity within brackets is the square of the velocity ; and we have, therefore, for the general law of steady motion of any fluid, ll l + vdv — dll = 0. . . . . . , f (496) For gases obeying Mariotte's law (§ 244), if the temperature is constant, D-P-- ■ P' and substituting, we have, by integration, p \ ogp _+_ _ _ n = a (496') Also, for liquids, D being constant, | + J-n = C. (497) If now the motion of the liquid be due only to its weight, then dU. = Zdz = — gdz, and D or integrating, dividing by g, and reducing, J? 4- v dv + gdz = ; (497') z + 2 l + - = H, (497") T to T 2g ' v ' a very important formula, which is called the theorem of Bernouilli. 344 ELEMENTS OF ANALYTICAL MECHANICS. § 292. — To interpret which, let z and z' be the vertical ordinate*, or heights above the co- ordinate horizontal plane xy, for the successive posi- tions of the same particle in a steadily flowing stream. Then it has been shown in § (266) that, if q from z to z', we get w (^-0? + (p-p')q, the total amount of work done by the pressure and by tke weight, which are the only forces supposed to work. For the equivalent change of living force, we have (D - („* _ V 2) q ; and, therefore, 346 ELEMENTS OF ANALYTICAL MECHANICS. or transposing, p v 2 , p v' 2 which is Equation (498), or the theorem of Bernouilli as fouud ' before. § 294. — By comparing this theorem with its analogous formula, found in § 266, for the head of a heavy liquid at rest, it appears that the only difference of head, in passing from problems of rest to those of steady motion, is the addition of the term v 2 — - = 071, or the height due to the velocity of flow. And conversely, if the velocity v be zero, then there is no motion, and the theorem of Ber- nouilli, Equation (497"), reduces to Formula (435"), that of the total head of a heavy liquid at rest. § 295. — Multiplying Equation (497") by 2>mg, the weight of the mass 2 m, we obtain, 2 (p\ V* z -f — ) + 2 m — == 2 mg H, an expression stating the total potential energy, or stored work, due to the elevation H, to be equal always to the sum of the variable potential energy s*f (*+!), added to its supplementary kinetic energy, or to the half sum of its living force. And this is, evidently, only the reverse of the demonstration by which Equation (498) has just been obtained from our general fun- damental law. MECHANICS OF FLUIDS. 347 § 296. — If we suppose a jet to spout from a small orifice a, in the bottom or side of a vessel, kept constantly filled to the same level z, where the sectional area a is so large that its velocity i a , v = — v a can be neglected; and if the atmospheric pressure be then Equation (498) becomes, j£=(«-V)=A'. ....... (499) and we have, v' = \/2ffh/, (499') the law of Torrecelli for the velocity of a spouting jet. § 297. — It should be borne in mind that these laws are deduced under the hypothesis that there is no friction, viscosity, or other obstruction preventing particles from gliding without loss of velocity upon each other and upon enveloping surfaces. This is far from being true ; and, consequently, to calculate their real flow corrections must be applied. We shall give but one example, interesting histor- ically, and in its uses important, that of the contracted vein. 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 con- tinuous 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 ful- filled 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 find it to a certain degree viscous, and exhibiting a tendency to adhere to surfaces 348 ELEMENTS OF ANALYTICAL MECHANICS. with which it may be brought in contact. When water flows through an opening, the 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 at 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. Lagrange gives the following account (Mec. Anal., 2 e partie, sect. X) of the discovery of the contracted vein : " Newton tried to demonstrate the law of Torrecelli in the second book of his Principia, which appeared in 1684; but that passage is, it must be avowed, the least satisfactory of any in his great book. Having measured the flow of water from an orifice, during a given time, he thence concluded, in the first edition, that its velocity of escape is only that due to half the height. This error arose from his not having then observed the contraction; but in the second edition, which appeared in 1714, he corrected it, by stating that the smallest section of the vein is to the orifice nearly as 1 to V2; so that, taking this section for the true area, the velocity must be increased in the ratio of y2 to 1, and thus it becomes that due to the height, as found by Torrecelli." § 298. — For steady flow from a vessel kept filled to a constant height, the contraction is calculable in the particular case of discharge through an adjutage of Borda, which is only a re-entrant cylindrical MECHANICS OF FLUIDS 349 tube, so short that the liquid escapes without touching its inner .sur- face. Such a tube causes the pressure within the vessel to distribute itself as in case of equilibrium : so that the pressure on the area a of its orifice will always be the same and equal to that on s, its pro- jection on the opposite side of the vessel.. Omitting the atmospheric pressure, which acts equally and oppositely, the pressure on the area « of the orifice, or mouth of the tube, will be p a •=. 0) h! a, which produces during the lapse of time dt, the quan- tity of impulsion, padt = G)k' adt. For the contracted vein a! the equivalent impulsion is that of the mass escaping in the same time dt, multiplied by its velocity, or 1 •' a s • (G) . . ■ \ . a) a v 2 _ -a'v'dt) x *' = dt. And these equal values give a) a h! — oa v 9 but, by the law of Torrecelli, and, therefore, v' 2 = 2gh r , 2a' = a; or the contracted vein has an area equal only to 0.5 of that of the tube. The velocity through the orifice a will, therefore, be only one- half of that through the contracted neck, or v = 0.5 ^2gh'. Experience shows that the greatc st contraction takes place , at a 350 ELEMENTS OF ANALYTICAL MECHANICS. distance from tho 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, 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 by simply multiplying the theoretical discharge into a coefficient whose numerical value depends upon the size of the aperture and head of the fluid. So tha. a v = C a \2gJi, in which C is the coefficient and a the sectional area of the orifice. Moreover, all other circumstances being the same, it is ascertained that this coefficient 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 small- est dimension of the former with the diameter of the latter. The value of this coefficient depends, therefore, when 6ther 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 coefficient also vary. When the flow takes place through thin plates, or through orifices whose lips are bevelled externally, the coefficient 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. MECHANICS OF FLUIDS. 51 As the orifice approaches one of the lateral faces of the reservoir, the contraction on that side becomes less and less, and will ultimately become nothing, and the coefficient will be greater than those of the 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 coefficient corresponding 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" = 1,06C; if nothing on three sides, C" = l,12 C; 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. 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 0,81.5 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 adhesion which will cause ihe former to spread themselves over the latter and stick with considerable pertinacity. 352 ELEMENTS OF ANALYTICAL MECHANICS. This adhesion 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, 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 contraction of the vein will be less, and the discharge consequently greater. STEADY MOTION OF ELASTIC FLUIDS. § 299. — As in the case of incompressible, so also in that of elastic fluids, if we suppose the motion to have been established and become permanent, the velocity of a stratum as it passes any partic- ular cross section of the vessel will always be constant, and the quan- tity of fluid which flows through every cross section will be the same ; while its density and volume may vary from one position of the section to another. All lateral velocity will be disregarded. And the motion will be supposed to be due only to the weight of the ele- ments and to the elastic force arising from some external force of compression. Our fundamental formula, ■ ' cPs 2 i Pdp — 2?n—ds = 0, o-ives 2 P cos a — 2 m — = (a) at Therefore, we need only find the forces and their equivalent accel- eration. Let z and z — dz be the vertical co-ordinates of the same particle hi the two consecutive cross sections a and a\ at the distance ds from each other. Then, the same mass, — ads = —avdL 9 9 flows in the same time through each of them. MECHANICS OF FLUIDS. 353 The normal component of its weight is — . a) a ds = <>) a dz, as which is negative because it acts downwards and z is taken positive upwards. The positive pressure on a is p a, and the negative pressure or resistance on a' is p a -}- d (p a) ; their difference, — a dp — p da, acting in the direction ds, reduces to — a dp; for pda, being lateral, has no effective component in the direction ds, and tends only to increase, or diminish, the area of the cross section without affecting the quantity of the flow. Substituting these values in Equation («), we obtain, o)adz + a dp -f- i—av dt\ — = ; or reducing, . dp vdv ♦ /,*„#» dz + J-^ — 0; (49V) (D g v and from this, if we suppose o) constant, integration gives, P v 2 which is, evidently, the same as Equation (497"), or the theorem of Bernouilli. If we eliminate the constant H, we get v 2 — v« 2 L = <*— > + g-3 < 498 > 2-7 If v be very small, and the velocity due to weight from z to z can 354 ELEMENTS OF ANALYTICAL MECHANICS. be neglected, which is always the case when the pressure p is the principal moving force, then £=(2-9.- •••••• m a formula applicable both to airs and liquids. And which, if we put h in place of the second member, gives v* = 2gh, ........ (499) or the law of Torrecelli. But as g> is not constant in the flow of air, integration in the manner supposed cannot be performed. Seeking its value, therefore, from the law of Mariotte and Gay Lussac, Equation (390), we have and, consequently, P p = A a) = — (1 + a 6) 0), if vdv . .dp , \-dz + A— = (501) g p v ' If in this the temperature be supposed coustant, integration will Bfive v 2 -f z + .4 log;? = const, (501') which is the theorem of Navier. It may be put under the form, ?^= (*„_*)+ ,4 log J. .... (501") And if in this equation we make v equal to v 0i or the velocity be uniform for different cross sections, as it would be in steady flow through a long and smooth cylindrical pipe, then z — z = A log — , P which is the same as Equation (436) for the equilibrium of airs at rest. MECHANICS OF FLUIDS. 355 As the expense, or quantity, is constant in steady flow for any two sections a and «', we have, apv = ccoPoVq, and this gives Generally the compressing force p cc greatly exceeds the resistance pa, and we may, therefore, make g=( 2o -z) + ^logj; .... (501'") and if the effect of weight from z to z can be neglected, this sim- plifies and becomes — = A\oA (501 w ) 2g P If p and p do not differ much, then integration is practicable without logarithms, for we may put and we shall have A f*dp_ /"°dp _/ Po — Pi \. which gives ^=fc->+fe-S); . . . («., And this, when v and (z — z) can be neglected, reduces to V = 2^-5^1), . (500") an approximative formula, much used and the same as Equation (500). The identity of these equations is due to the fact that in one th» specific gravity w being supposed constant, while in the other, though considered variable, it is replaced by its mean value, assumed to be constant, these two hypotheses do thus substantially become one. 356 ELEMENTS OF ANALYTICAL MECHANICS. Moreover, the same formula was first obtained for liquids by Ber- nouilli; and it is clear that the hypothesis of constant density must give always the same equations, both for liquids and gases. The preceding equations and discussion, due to Navier, are those given in most of the treatises on Mechanics. But they are very- defective, for the reason that the density of air in its flow cannot be supposed either to remain constant or to vary according to the law of Gay Lussac, Equation (390), as assumed by Navier. The expansion, in fact, generally takes place according to Equation (391), or the law of Laplace. The mechanical theory of heat has furnished more exact equations for the flow of airs, but, before deducing them, some facts to which reference is needed should be given. DIGRESSION ON THE ACTION OF HEAT UPON AIRS. i § 300. — The work of expansion done by any substance, as by steam on a piston, is expressed by the integral of the pressure p on the unit of surface, multiplied by the area a, and by the distance moved, ds\ this gives for it, / pads = I p dv, in which dv is the elementary variation of volume. The total work, or energy, of any system may be expressed by the formula, JPdp = U+S, in which U is internal and S external work. If S denote work of expansion, this becomes ^Jpdp = U + Jpdv. When air flows by its expansion into a partially exhausted re- ceiver, out of another into which it has been previously compressed, the mean temperature of the whole system does not vary. Hence, • MECHANICS OF FLUIDS. 357 if the internal work U be regarded as a function of the volume v and temperature 0, its variation for 6 is independent of v, or »» f >»w>»»»„jj*s»js,w;s;ss;;;,- a ra V a' N For the work of gravity on the unit of weight from z to z„ the heights of the centres of gravity of a and a\ we have simply (z — z x ) = h. And denoting the internal energy by U, we have, for the total work of all the forces, Uj 2 — Mq 8 *9 = (*o - *i) + (Po^o —Pi »,) + (*7o — CT m ), . . . (502) a general equation for all fluids, whether liquid or gaseous; but for which we may find a simpler form. The internal work is only that of expansion from the volume v 9 under the pressure p to the volume v x under the pressure p x . It takes place for a particle along a line of flow mo ; and if rapidly, 360 ELEMENTS OF ANALYTICAL MECHANICS. then sufficient time is not allowed for heat to be either absorbed or emitted. We have, therefore, dv or integrating by parts, p iVl — p v + / °vdp; which, by substitution, gives u t 2 — u *9 2 /JO °- = (z — z 1 )+J vdp (503) Three distinct cases now present themselves, and we shall discuss them successively : Case I. — The density of the fluid, liquid or gaseous, is constant. Its internal work will be zero, and our general Equation (502) reduces to - u2 + z jl. P - - U * 4. z + P « (±a«\ 2^ + * + u-27~ Mo+ u' * * * * (498) or the theorem of Bernouilli. Which, if u is neglected, as is always the case if a is much greater than a\ reduces to the law of Torricelli, u*=2g(z -z), (499) when external pressure is neglected; and becomes when weight is neglected, u 2 Po—Pi 2g ' a (500) a true formula for liquids; but for gases merely approximative, and to be used only when p differs little from jo„ and when the mean value of a> may be assumed to be constant. For airs there must be loss of temperature, for the equation pv = Rt gives, if v be supposed to remain constant, v dp = R dr ; MECHANICS OF FLUIDS. 361 • and therefore, R Po—Pi = — (~o — t,) ; or the temperature varies proportionally to the pressure. This is sim- ply verified by feeling the breath blown upon the hand to be warm or cold proportionally to the amount of force employed. Case II. — The fluid is supposed to be an air obeying the laws of Mariotte and Gay Lussac, but receiving heat during expansion, so as to keep its temperature constant. Then we have p v = p v = A = const. ; and this gives p dv = v dp = A — . P which, by substitution and integration, transforms Equation (503) into U " ~ U l = (*. - Zl )+A log* 2 from which r l is easily calculated when r , p and jo, are given by observation. As an example, let gas escape from a vessel at 30° C. and a pressure of one and a half atmospheres into the air, and let u be neglected, then we have g = 9.81 m. ; p s 1.5 ; j»i = 1 *, r =:303 o ; c = 0.2375; c, = 0.1684; and these data give for r, the value, r, = r (f)o-2909 =269°.16. So that the temperature will descend to nearly 4° below the freezing of water. And this result causes Equation (504') to become, — = Ec (303° — 269°.16) : which gives for the velocity of escape, u m 250 m. nearly. MECHANICS OF FLUIDS. 363 We have, therefore, three formulas for the steady flow of airs — one that of Bernoulli, Equation (500), which can only be used when p and p x differ but slightly ; another that of Navier, rarely applicable ; and the third, Equation (504), which is given by the thermoelastic properties of air, and which should generally be employed. Analogous equations have been determined for steam, which does not obey the law of Mariotte, but for those we must refer to treatises on heat. PART III. MECHANICS OF MOLECULES. 8 302. — The more general circumstances attending the action of ft 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 Me- chanics of Molecules ; which comprehends the whole theory of Electrics, Thermotics, Acoustics, and Optics. It is assumed that all bodies are built up of elementary mole- cules 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 dis- tance between the acting molecules. A displacement of a single mole- cule from its position of relative rest T will break up the equilibrium of the surrounding forces, and give rise to a general and progressive dis- turbance 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 connections which "nite the molecules into a system 366 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 Solid?, by L = ; If = ; L" = ; &c (506) Z, L\ L", &c, being functions of the co-ordinates of the molecules Denote by X, T,Z; X, T, Z'; Ac, the accelerations impressed upon the molecules whose masses are ft, m', &c, in the directions of the axes. Equation (313) will obtain for each molecule. There will be as many equations as molecules, and by addi« tion, we find, by inverting the terms, There will be three co-ordinates for each molecule. Denote tin- 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 in- &tant determined. Denote the m co-ordinates by x y z, x' y' z\ ; v (a j9 y a', &c ) = P..; &c. = ,+ * = />,+ d P* e+ d Py w , da *^ tfj3 '^ a( y of y dp, y £ + &c, &c, d y % -f- &c, &c, ► . . (508) ■ ■ ■ • J aa dp ay >. . (509) Z=P,+ d a f + rfP. 1 + rfP. rfy 4* + &c, ), S - "~ ... [ (514) MECHANICS OF MOLECULES. 369. :n which R and r are arbitrary constants, and p, N ' N , N ' , &c, are constants to be determined. For, after two differentiations, regarding £, 77, £, &c, and t variable, we have d 2 £ — t = - B.N r sm(tVp — r)p, . dl = ~ H.N n .sm{tVi-r)p, d*? - jj = -£.iV^sm(tVp-r)p, ^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 £, 77, £, £', » MECHANICS OF MOLECULES. 371 When these intervals are commensurable, then will £, 77, £, &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 dis- turbed 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 derange- ment 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 £„ 77,, £„ n = R. ,N . n t\p- e tVp- r • r S = :R w r e 1 372 ELEMENTS OF ANALYTICAL MECHANICS. which give d*£ tVp-r dt % Ti=P-*'K-e d*S Ar t.yfi-r J¥ =p.B.tf.e and these substituted in Equations (513), give Equations (515), wiUi 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 £, J], £, £', •).- = 0, 2 /«'V = °' I (519) As V« • - = 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 «• by S i\, £ m' " | + A£ ?PHM, ?+Af, m" " ?+Af, 7/ + A77', £+A^, &c, (r) (Ax . Af + Ay . At) + Az . A£) . Asj !»,-J J=S{q)(r).Aiy+V»(»")(Aaf.A| + Ay.Aiy + A2.A5).Ay| m.jji =Z\y(r).Ar+i}>(?-)(Ax.AZ + A ! /.A7 ) + Az.AZ).Azl (523) (. (524) Pe>*forming the multiplication as indicated in the last term of the sec- ond members, there will result terms of the form, 2 ij) (r) . A 7) . A x . A y ; Zip (r) .A^.Az.Az; lip (r) . A % . A x . A y ; 2^(r).A»|.Ay.Az; 2 i/> (r) . A £ . A z . A y ; 2i/>(r).Ag.A:r.Az; and it may be shown by the process of § 163, 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 sa'isfy these condition*, Equations (524) become d*l m . jj ? = 2 {

(r) A *» j A & \. • • • (525) 07/' ELEMENTS OF ANALYTICAL MECHANICS, Making m p' =

T] dt % ~ 2p .AT], dt* ' 2p'".AZ. (527) An initial and arbitrary displacement of a molecule at on« 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 7 confirmed bv manv 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 xy. The integrations of Equations (527) are given by 2 7T /Tr v 1 | = a e . sin — ( . 2 77 , __ x fj= a,, sin -j- (P, t — rX Ay £= a,, sin -rr (V t .t — r f ), (528) MECHANICS OF MOLECULES. 377 in which V x , V f , and V t are the velocities with which the disturbance is propagated in directions perpendicular to the axes a*, y, and z, re- spectively; A a , X 9 , and X, the shortest distances, in the same directions, between the places of rest of anv two molecules that mav have at the same instant the same phase; r,, r fJ and r z the distances of any mole- cule's place of rest from that of primitive disturbance, estimated in the sime direct! /ns. This being understood, we have the relations, r m =t ^y* + z* ; r ¥ = v^ + z 2 ; r 2 -— %V _j- y 2 . 2tt 2tt 2tt t- p, = ». ; -3- ^ = » f ; y- ^ = *. ; r A, A„ A, 27T 27T 2tt Aj A,y A., and the above become J = «. . sin (n x .t — h x . r x ), 77 = a y . sin (ra, A — k y . r y ), )■ £ = a 2 . sin (w z . * — & 2 . r 2 ). _ (529) (53(>) § 312. — To show that these are the solutions of Equations (52*7), it will be sufficient to prove that they will satisfy those equations with real values for n z , n y , and n z . Differentiate twice with respect tc t, and we have d7 de = — n* 2 • £ = - nj . r\, (531) Give to r m , r r , and r t the increments Ar Jf A r y , and Ar 1( respectively; the corresponding increments of £, 7/, and £ are A |, A 77, and A £ and Equations (530) become 378 ELEMENTS OF ANALYTICAL MECHANICS. $ + *$ a m . sin (n x . t — k, . r x + Je m . A r.), a y . sin («, . * — k y . r y -f k y . A rj, a,, . sin {n t . t — k t . i\ -\- k t . A r t ). Develoj ing the second members, regarding n z . t — k, . r», n y . t — k y . r f and n t ,t — k t ,r t as single arcs; subtracting Equations (530) in order replacing 1 — cos k m Ar„ 1 — cos k y A r r , and 1 — cos k z A r z by tlieii respective values, we find *. . r m ), (k x A r m ) A % = — 2 £ . sin 2 ~ - 4* sin {K & r m) • a m cos (n, . t m h (£ A r ) A tj = — 2 ?| . sin 2 - + sin (& r A r y ) . a y cos («, . t — £ y . r y ), (& A r ) A £ = —24". sin 2 — i — — - + sin (& e A rj . « r cos (n z .t — k z . r E ). ► (532) Substituting these in the second members of Equations (527), we have dp dP (A A*' ) -2g.Sp'.sm» * * + S/.sin(* B Ar,).« .cos(» a .<-* x .r # ), - 2 if . SJ9" . sin 2 — ^ S- -f- %p" . sin (£ A rj . « . cos (n . t — k . r ), jj " ™ " » y ™ = - 2 { . Z jpT . sin 3 -f- £ /?"' . sin (* A r t ) . « t . cos (» g . * - * . r f ). K588) Tn 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 m. Indeed, this is the most general way in which we may conceive the equilibrium to exist. Then, since for every positive arc k m . A r x there will be an pqual negative one, we must have 2 p' . sin (fc t . A 7\) . a. . cos (n e . t — h m . r a ) as 0, 2 p" . sin {k y . A r y ) . a y . cos (n y . t — k y . r y ) as 0, Sy ' . sin (k t . A r,) . a, . cos (w, . t — &, . r t ) =» 0, „ ► . . (534) and therefore, MECHANICS OF MOLECULES. 379 dt 2 d¥ d¥ 2 £ . 2 p . sirr , r 2 2^2/\sin 2 ^— *, 2 £.2;? .sin' , v . . . . (535) whence, Equations (531) and (535), w, 2 = 2 2//. sin 2 — -, n* = 22/. sin '// _i_i n 2 =r 2 2;/". sin 2 ' (536) which are, Equations (526) and (522), real values for v„ » fJ and ft,. § 313. — Substituting the values of w,, w^, n z , and f^ & y , & M Equa- tions (529), there will result, after multiplying the first, second, and third by 1 = A r 2 -f- A rj ; 1 = A r 2 -^ A r 2 ; 1 = A r 2 -f- A r£ re- spectively, sin J n A r a 1 F. ! = I2/ .Ar, 2 . /7r A r s y sin t tt A r y F 2 ^".Ar, 2 . r. f = j2y.Ar, 2 . . 2 7T A r f sin 1 * — r — (7T. Ar,v» > . . . . (537) 380 ELEMENTS OF ANALYTICAL MECHANICS. WAVE SECTION. §314. — Resuming either of Equations (528), say the first, viz.: v . 2 7T £ — a x sin — ( v x . t A_ it is apparent that if t be made constant and r x variable, so as to reach in succession all the molecules in its direction between the limits V m . t — X„ and V x . t, the displacement £ will also vary, and from zero to zero, passing between these limits' through the maximum values a a and minimum value — ol, ; thus deter- as i mining the curved 'line C D, of the annexed figure, to be the locus of the corresponding dis- placed molecules, of which the places of rest are on the straight line A B, coincident in direction with the line r x 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 t be made to vary between the limits V a .t' — a., and V x .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 dis- turbance 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 depcr js upon the ratio between MECHANICS OF MOLECULES. 381 7T . A T the arc — -= — - and its sine. If the distance A r # , between the mole- cules, in the direction of r„, have any appreciable value as compared with the wave length A,, 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 9 is insignificant in compari- son with the wave length A x , 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'" ; A r„ = A r y = A r, ; and, therefore, V =v =v That is, the velocity will be the same in all directions. Denote this velocity by V\ we may write sin .- V s -IT. k /7T . A r\2 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 in- terval 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, Ar becomes Aar, and, first of Equations (537), replacing p' by its value, Equations (526) and (523), 2 m L r Vrfrr* r / J The distances between the molecules being very small, the term of which A a; 4 is a factor may be neglected in comparison with that con- taining A .t 2 , and the above may be written 382 ELEMENTS OF ANALYTICAL MECHANICS. P* = -L.2/(r).— .Ax. i ... A x . Now, /(r). — is the component of the elastic force exerted betweer two molecules whose distance is r, in the direction of the axis x; and f(r). — - .Ax is the quantity of work of this component acting through a distance Ax. Making we may, by the principle of parallel forces, write */ v) • ~~r • A x — 2 e * x i » in which e { is the sum of the component molecular forces which act on one side of the molecule ra, in the direction of the axis #, or, which is the same thing, the elastic force limited to a single molecule; and r t the path over which this force would perform an amount of work equal to that measured by the first member. Substituting this above, F 2 = ^. m .Denote by i the number of molecules in a unit of length, and multiply both numerator and denominator by f\ we have f *2 * TT2 _ l ' e ,' lX , . — •* ? V . m but i 2 . e t 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 i x t is the number of molecules in the distance x t ; call this k. The denominator &m is the quantity of matter in a unit of volume, which is the density; call this A, and the above becomes V=\J-.k (539) Denote by c the ratio which the contraction produced in a given vol MECHANICS OF MOLECULES. 383 ume of the medium by the pressure of a standard atmosphere A, bears to the volume without any external pressure; then will e = A = <7.i>„. 30 c c tneKe*, . . . . (540) in which g is the force of gravity and D ti 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 in- definitely, thus making their increments of volumes sensibly equal \£ the volumes they would ultimately attain. RELATION OF WAVE VELOCITY TO WAVE LENGTH. § 317. — Denote the resultant displacement, of which £, r\, and £ are the components, by tf; and the angles which , then will cos if) = cos a . cos a y -f- cos j3 . cos (3 4 -f- cos y . cos y t , e . cos a { = e x = — = - tf . g . F 2 = - tf . $• ( F, 2 . cos 8 a -f ^' . cos* j3 + F, 2 . cos* y) ; in which F is the velocity perpendicular to the displacement. Making we have V=V; V x = a; V-b; V, = c; V= Va 2 . cos 8 a4-^ 2 . cos 2 j3 + c 8 .cos 8 y . . . (549) The quantities a, b, and c are called definite axes of elasticity, in con- tradistinction 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 F will measure the velocity of any point on the wave surface in a direction normal to the displacement, and being squared and multiplied by a . g will give the elasticity de\ eloped in the direction of the displacement itself. MECHANICS OF MOLECULES. 387 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, simulta neously, 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 disturb- ance 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 /, m, and n, will be the cosines of the angles which the normal to this plane makes with the axes xyz, respectively, and its length will measure the velocity F, of wave propagation in its own direction. This plane must be parallel to the displacement and its normal perpendicular thereto; hence I cos a -f m cos ]3 + n cos y = . . . . (551) ; also cos 9 a + cos* (3 -f cos 9 y = 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 V* • cos 9 a, it becomes 388 ELEMENTS OF ANALYTICAL MECHANICS. , COS 2 , COS* y COS" 1 a COS 2 a (558) V* COS 2 a divide Eq. (551) by cos a, we have eogij , cos y m /e *,x l + m + n = (554); COS a COS a and divide Eq. (552) by cos 2 a, the result is COS 2 /? COS«y 1 1 H H = . (555). COS 2 a COS 2 a COS 2 a Equations (553) and (555) give , .. cos 2 B cos 2 > a 2 -|- b* + c' - cos 2 B cos 2 y cos 2 rt cos 2 a "cos 2 a ~*~ cos 2 a ~~ F 2 " ' whence r*i-« a +(F 2 -& 2 ). < ^-^ + (F 2 -c 2 ).^-? = .... (556). ' ' cos 2 a COS 2 a V From Equation (554) we have cos/3 f -f- rn cos y cos a cos a *t ' which in Equation (556) gives r(r 2 -6 2 )n 2 +(F 2 -c')m 2 ].^— +2(F 2 -c 2 )./-OT.^-^ = -(F 2 -a 2 )n 2 -(F 2 -c»)/ 2 r 1 Cos 2 a COS a or • I eos 2 ( F 2 — c 2 ) . / . m cosfl _ ( F » — a 2 ) n 2 -f( F 2 — c 2 ) P ^r; + 2 ( V 1 - ¥) n? + {V* - c*)m* ' cosl " " ( F 2 — 6 2 ) n 2 + ( F 2 - c 2 )ro 2 * cos j3 and solving with respect to , there will result, • ° r cos a cos _ (r»-c»).m.lTw^-[ ( F»-a »)( F 2 -6a)»«+(r» -a a )( F a -c*)m*+( F 2 - c 2 )(F 2 - & 2 )i »] iosl ~ ~ < F*-6») » a + ( F» -c«) «i» (557,; MECHANICS OF MOLECULES. 389 and this in Equation (554) gives cos y _ (J™-b*).nJ±my/^{V*-a*)(V*-b*)?i*+(V*-a*){F*-c*)m*+(V*--c*){r*--b*)l*] e^~~ (P»-&*)n» + (F»-c«)m* (558). For any assumed displacement, the value of F, Eq. (549), becomes known, and the values of the first members of Eqs. (55 7) and (558) must be real; whence /, m, and n, must, in addition to Eq. (549), also satisfv the condition ( V' — a 2 )( F* - 6 2 )- 2 -f { V* - «•)( F 2 - c 2 )wi 2 + ( F 2 - 6 2 )( K» - c 2 )/ 2 = 0. Dividing- by ( F 2 - a 2 ) ( 7 2 - 6 2 ) ( V % - c 2 ), and inverting the order of the terms, P m* n 1 _ |Tt a i "T" jrs _ £2 + jT2 __ c « — 0- • • • • (559) From this equation, together with Equation (550), and the relation P + m 2 + n 2 = 1, (560) we have all the conditions necessary to find the equation of the wave surface ; this is done by eliminating V, m, /, 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) ldl-\-mdm + ndn = 0' 1 and from Equation (559), Idl ' mdm , ndn (3) Idl , mdm ndn _ _. . v / P m* w» \ V*~^a? "*" V* - #> "*" V* - c 2 =: \(V*-a*)* ""*" (P»-*)J "*" ( F« _ <<*")i/' Multiply the first by A, the second by — A', tlie third by — 1, and add members to members, and collect the coefficient? of l 4 ke differ- entials ; there will result, 24 390 ELEMENTS OF ANALYTICAL MECHANICS, (xx-X'l -^l—.^dl + \Xy-X'm- v ™_ b i }d m n H x - r \w^ = 0. / TTO 19.9 (V*-b 2 f ' (V- Taking X and X' of such values as to make the coefficients of d V auci in each zero, the equation will reduce to the first two terras; and as dm and dl are wholly arbitrary, Equation (560),. as long as dn is undetermined, we may, from the principle of indeterminate coefficients, write, <*) (5) (6) I • '-~ "• 7* -a 2 '7 1 u_, Vm — o • A y A m yi _ p — u » JLs - x'n — — . A * AW J^2 _ c 2 — U » 2 ri. Z 2 m 8 n* i I. Multiply (4) by I, (5) by m, (6) by w, add and reduce by Equation* (550), (560), and (559); we have (8) aF-a'=0; Multiply (4) by x, (5) by y, and (6) by z; add and reduce by Equa- tion (550) and the relation a* 8 + y 2 + z 2 = r* ; we have Xr {XV+—— + —— + nz \ yr-i .) = o; substituting for a' its value, (8), and transposing, (9) . . a (r»- F 5 ) = /# my "*" V* A3 "T" 172 V> _ tf » ' JTi _ a ,8 » MECHANICS OF MOLECULES. 391 transposing in (4), (5), and (6), squaring and adding, we have ^ H - 1" 4- —1 + 5l_ 4- n —— • T j v 2 - a y \ (v* - by T (F 2 - c>f ' substituting for X' 1 its value, (8), and reducing by (7), we have jy (r ._ H = ^; and, therefore, ( 10 ) • • • • ^» — TTTIs lzn i ^' — "3 V (r 8 - V 2 ) ' "" H - F 8 ' Substituting these in (4), we find _x_ / 1 1 \ TZT\nj ~ \r r ZTv> + V" - a) ' whence similarly, F(r 8 FZ r * - o} ~ V 3 - a 2 ' r*'l'^ = F 8 - 6 2 ; z V n r 2 -c*~ F 8 - c* ' multiply the first by #, the second by y, the third by 2, add and re- duce by (9) and (10) ; we have x* y* z % ~1 i" + "1 Ta + 1 2 = 1 ( 56 ^) r 8 — - ar r 2 — b 2 r 8 — - r From this, which is one form of the equation of the wave surface, sub- tract and we have x % 4- y % + s' , a 8 * 8 &V cV /-B _„ 3 i + -r-hi + j r = o . . . . (562 r 1 — a 8 r 8 — 6" r* — r which is a second form of the equation of the wave surface. Clearing the fractions, it becomes, after substituting for r* its value ** + y f + •*, 392 ELEMENTS OF ANALYTICAL MECHANICS (*« + f + z 2 ) (« 2 x* + i 2 f + c 2 z 2 ) ^| - a 8 (6 2 + c 2 ) *" - & 9 (a 2 + c*)y 2 - c 2 (a 2 + 6 2 ) s l + a 2 b 2 c* , = . . . (563) DOUBLE WAVE VELOCITY. § 320. — The radius vector r measures the velocity of the point ol the wave to which it belongs; and denoting by l t , m,, and n t the sosines of the angles which r makes with x, y, and 0, respectively, we have x == r . l t ; y r= r m i ; 2 se f *, ; and writing V r for r, we have, by substituting in Equation (563), and dividing by V r A . a 2 . 6 2 . c 2 , a trinomial equation, of which the second powers of the equal roots are i =i (? + ?) + * (? ~ i) W ■ A " ± VTZrir ' x ^F^ (565) and in which, A' =1 a' V ? + », a' ^1"=/,. 6 2 a 2 — n. . 1 1 c 2 6 2 1 1 ?~ a 2 1 1 ? 6 2 . . (566) / i i ' / i i V c 2 a 2 V c 2 a 1 ; . , . (567) If a > b > c, the values of -4' and .4" will be real, and there will, in general, be two real values for -j=- 2 ; and with this condition, Equation (565) will give two pairs of real and equal roots with contrary signs. MECHANICS OF MOLECULES. 393 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 t) ; with the axis 6, an- gles whose cosines are j3 ; and j3 /y ; with the axis c, angles whose co- sines are y t and y u ; and such that J (568) and denote the angle which r makes with the first of these lines by « , and that which it makes with the second by u lt ; then will A' = l t a t + n t . y i = cos u t , A" = l,**-, — n t y J = cos u t) . Vl — A' 1 = sin u t ; Vl — A! n = sin u u . These, in Equation (565), give for the two values of -=- 4 , |Ti = i [f + J) + i (-> ~ ^) • ( cos u i • cos u » + sin ** ■ sin M ><) • • ( 569 ) T"» = * (? * a" 8 ) + * (? ~ ) = 0, "fj) \ . (572) * = 0; (a' + y'-c 2 ) (aV +6 2 y 2 - a 2 6 2 ) = 0, If a be greater, and c less than 6, 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, intersecting one another in four points. This last is the most important. It is the section parallel to the axes of greatest and least elasticities, 9 § 322.— If b = c, then, Equations (568), a , = 1 ; 7, = : MECHANICS OF MOLECULES. 395 the axes will coincide with one another and with the axis a, that is, with x\ u t will equal u tl , and, Equation (571), Also, Equation (563), (x 2 + y* + z 2 - c 8 ) [a 5 g + c 2 (y 2 + z 1 ) - a 2 c 2 ] = . . (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 anv other value for u. since u. = u , cos u 4 cos u u + sin u t sin u u = 1, Equations (569) and (570) become ^ = 7-; f^. = ?■-(■?■- -yj • «« f «/.- • (^) and it hence appears, that the relocity 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 or- dinary, 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 V r 2 = V r 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. 9 § 323. — Let L = represent Equation (563), and take cos A = — ; cos B = r— ; cos C = — -7— ; (5 ! 6) w dz w ay w dx in which J, B, and C are the angles ivhich a tangent plane to the sur 396 ELEMENTS OF ANALYTICAL MECHANICS. face makes with the co-ordinate planes xy, xz 1 and y z, respectively and, l_ 1 w J idLV 1 , /dL\* (dL\ l M*)+fo)+U7) . . . . (577) Performing the operation here indicated on Equation (563), we have d_L dz Z-^ =2 z(a i x t + b*y' + c 2 z 2 ) + 2 c* z (x> + y* + z* - a* - b% dL dy d_L dx fL = 2 y (a 8 x* + 6* y* + c 2 z ? ) + 2 6 s y (x* + f + z 8 - a 8 - c ! ) ; = 2 z (a 8 a; 8 + 6 2 y 9 + c 2 z 2 ) + 2 a 2 z (z 2 + y 8 + z 2 - 6 2 - c') . Making y = 0, brings the tangential point in the plane a c, and the above become dL dz dL dy dL dx = 2 z (a 8 x> + c 2 2 s ) + 2 c 2 2 (z 8 + z 2 - a 2 - fr 2 ), 0, = 2 * (a 8 ^ + *' z 2 ) +2o l a;(^ + 2 , -J f -4 f. . . (578) the second of which shows the tangent plane to be normal to the plane a c. But y = gives, Equations (572), - x* + z 8 - &* = o ; a 2 z 8 + c 8 z 2 — a 8 c 2 = 0, whence we have z = a r — i | I -*Vf? -ft 2 a' — c (579) for the co-ordinates of the points in which the circle and ellipse inter MECHANICS OF MOLECULES 397 sect, and which are real as long as a > b > c. Substituting these in Equations (576), (577), and (578), we have cos A = - ; cos B = - ; cos C ■ = - ; hence the points of intersection of the ellipse and circle in the plane of the axis a c, are the vertices of couoidal cusps, each having a tan- gent cone. If a line be drawn tangent both to the ellipse and the circle in the plane a c, the tangential points will belong to the cir- cumference 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 grad- ual increase of the inclination of its elements, as the tangential cir- cumference recedes from the cusp point. A narrow annular plane wave, starting from this circle, will contract to a point in one direc- tion ; and, conversely, an element of a plane wave starting in the op- posite 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 snow two of the umbilic points, as well as the general course of the nappes, by the re- moval of a pair of the resulting diedral quadrantal fragments. MOLECULAR VELOCITY. § 324,— Multiply the first of Equations (531) by 2 d |, the second by 2drj, the third by 2 d £ and integrate ; there will result, recollect- ing that the molecule is moved from its place of rest 398 ELEMENTS OF ANALYTICAL MECHANICS. dp df "~ - < . ?, drf df ~ - K • n\ d? df ' -n\.?. (580) whence it appears that the velocity of a molecule in the direction ol either axis is proportional to its displacement in that direction, from its place of rest. The place of rest is only relative. When a mole- cule 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 f to be superposed upon that due to 77, and take a molecule of which the place of rest is on the axis r. 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, ajfer writing z for r m and r r , (1) . . . . ^.(V x .t-z) = sirr 1 ^-, At a, (2). . . . ^.(P^-^sin- 1 -! (3). . . . ^ . {V m . t - z) = cos-'l/l - £, a. (4). . . . y^.(V y .t-z) = cor 1 Vl-£. a f MECHANICS OF MOLECULES. 399 Subtracting (2) from (1), rr / V > A - Z V y 1 - Z \ V,.t — z V„ . t — z\ . _i £ . _, r t 2 7r I - -7- — — -1 = sin sin — ; a, a in which V m . t «— t, is the distance of the wave front due to g from the molecule's place of rest, and V y . t — z, that of the wave front due to r\ from the same point. Make t, = time required for the wave front due to £-to travel over V m . t — z\ u »f •> . A,, ?/ " V y .t-z; A y , r.= u M M ^ = u u M r y ~ u u U then will F. .t — 1 2. i which substituted above gives, after taking cosine of both members, Clearing the radical and reducing, — 4 + -^ - 2 cos 2 7T — £ . -L . Jl — sir. 2 2 ?r — ^ = . . (582) «. V 1-, «. « y T, 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 (p — 7]' sin /* , a ff = *'; v^ . ^ -= u ' ; there will result, tan (

se parallel to the other; w$ have, cos = sin 2 v — ; (614) « sin 6 = sin 2 " (^ ± ^) • ' ( 615 ) Differentiating, regarding — as constant, we find, dd 2n -=— = . cos 2 at, r . cos v and, developing the last factor, » d0 2tt 4 r . _ £, . i t' dt r r . cos 6 cos 2 77 • — • cos 2 rr • — ^ sin 2 77 • — • sin 2 77 — I ; and making — = |, n d$ 2r . L ■ , cosd.-r- = ^f — .sin 2 77.— (616) a L r r Differentiating (614), we find, sinfl.— - «fc _.cos2 7r— (617) (It- T T Squaring, adding to the square of Equation (616), and taking square root, dd 277 17 = *- < 618) 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 as, and these equations show that the upper sign must be taken in Equation (618) when if 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 <" stance equal to \ of a wave MECHANICS OF MOLECULES. 415 length ; and the motion will be from right to left, or the converse, ac- cording to wave precedence. Two waves distinguished by these peculiarities are said to be oppo- sitely 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 wave* is called the plane of crossing. § 342.— Let ( 1 ) a y cos Q = £ = a, sin 2 n — , * / * t'\ (2) a^ sin = tj = a, sin I 2 7r . — -\ 1, (3) . . . . . . a 4 cos# = | = a y sin |2ip . -*- + --V i (4) a, sin = 7/ = a, sin 2 7T — > T be the displacements in two oppositely circularly polarized waves, Tlk* 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 s = v in the plane of crossing. It thus appears that the union of two circu- larly polarized waves, polarized in opposite directions, gives a plane- polarized wave, of which the intensity is double of either. Conversely* a w r ave 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 time, we have, denoting the velocity of wave propagation by V, X = Vt, whence X 414 ELEMENTS OF ANALYTICAL MECHANICS. which, in Equation (618), gives, after multiplying b} 7 t x and dividing by 2tt, <[[*'_^_Yi* ( Gl9 ) 2tT ~~ X ,The first member is the arc, expressed in circumferences, described by the molecule while the wave is moving through a thickness V . t a of the medium. So that a wave, compounded of many components hav ing 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, 44° 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 ve- locitv v, . is m v* ; and denoting by n the number of molecules on a superficial unit of tf.he wave front, the living force on this unit will be n . m . vj ; and on the surface of a spheie of which the radius is r f , 4 7r . rf . n . m v* ; nnd for another sphere, of which the radius is r (t , and molecular velo* 4 n . r ii t . n m vj. If these spherical surfaces occupy the same relative positions in a di- verging wave, in any two of its positions, their molecular living foioes must be equal ; whence, suppressing the common factors, t} . m v 2 = r 2 m v * (620) MECHANICS DF MOLECULES. 415 The molecules describe elliptical c rbits, and under the action of molec- ular forces directed to the centres of these curves. The periodic time will, therefore, § 207, Equation (286), be constant, however the dimen- sions of these orbits may vary ; and the average velocities of the mole- cules will be proportional to the lengths of their respective orbits, or, in similar orbits, to anv homologous dimensions of the same — as their transverse axes or greatest molecular displacements. Denoting the latter by c' and c" in the two waves, then will which, with Equation (620), gives c" r u =r.c'r t (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., -, + -h j - sin 2 2 n - = ; denote the radius vector of the molecular orbit by p', and the angle it makes with the axis of g by 0', then will £ = p' . cos 0' ; tj = p\ sin 0' ; which, in the above, give i . * P = — - " — . sin 2 ft . - ; l/a,, 1 cos' 0' + a/ sin* 0' T and making a / • a /y t > v/a,, 1 . cos» 0' + a, f . Bin* 0' i\Q ELEMENTS OF ANALYTICAL MECHANICS. we have I p' = c' . sin 2 rr . - (622) § In this equation, p' is the actual displacement of the molecule from ita t place of rest, and becomes a maximum when - is any odd multiple of t i. If, however, there be added to the arc 2 rr -, an arbitrary arc a, tin's latter may be so taken as to make the maximum or any other displacement occur at such time and place as we please, aud, there- fore, to give to the molecule any particular phase at pleasure, at the time t. We may write, then, generally, p' = c' . sin ^2 rr . - -f «') ? . . . . (623) and for a second resultant wave, p" = c" . sin (2 7T . - + a") ; (624) and if these waves act simultaneously upon the same molecules, the re- sultant displacement, denoted by p, will, § 306, be given by p — p' + p" = c'\ sin (2 n . - + a'\ + c" . sin (2 n . - + a"\. Developing the circular functions aid collecting the coefficients of like factors, ( t' ...... . t p = (c r cos a' +'c" cos a") . sin 2 n- -f (c ' sin a' + V sin a") . cos 2 n - ; T 7" and making c cos a = c' . cos a' -} c" cos a", . (625) c sin a = c' sin a' -j- e" sin a", we have t t o ss f . cos a . sin 2 tt . - -f c sin a . cos 2 n . - ; t T MECHANICS OF MOLECULES. 417 p = c sin y2 7T . — h a). (626) Squaring Equations (625), and adding, c* = c' 2 + c" 2 + 2 c' c" cos (a' and dividing the second by the first, c . sin a -f- c . sin a a"), . . . (627) tan a = c' cos a' + c" cos a .// (628) 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 t; 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 = vV* + c" 2 -{-2 c' c" . cos (a' - a") ; . . . (629) which will be the greatest possible when a' — a" == 0, and least pos- sible when a — a" — 180° ; the maximum in the former case being given by c = e' + c" and the minimum, by e = c' - c". In the first case, Equation (628), (c f 4- c") . sin a' tan a = )—. ~^ = tan a . (c -|- c ) . cos a' Whence a = a = a", and the maximum displacement will occur al the same place and time in the resultant and c mponent waves. 418 ELEMENTS OF ANALYTICAL MECHANICS. In the second case, Equation (628), if we make a' = 180° -\- a" t (c f — c") • sin a" ., . , tan a — )- Tr { - = tan a" = tan (a' — 180°) = tan a : (c — c ) . cos a v ' that is, a will be equal to one at least of the arcs a' ancf 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\\ -f- cos (a' — a")] ; and because 2 a — a 1 + cos (a' — a") = 2 cos 2 t _ _ ft c = 2 c . cos — -, . . . . . . (630) and, Equation (628), sin a -f- sin a" a' + a" , tana = - == tan ... . (631) cos a + cos a 2 v ' If, while c' and c" continue equal, we also have a' — a" — 180% then, Equation (630), c = 0. Thus it appears that 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 ->ause this mutual destruction, make, in Equation (623), a = a rt7r = a ± — , and that equation becomes, p = c .sin ( n i t, 2 7T.r\ I27T- + a"db 1, \ r 2r /' 2tt l-.:. a "y } .... (632) MECHANICS OF MOLECULES. 419 which becomes identical with Equation (624) by making c' = c", anc t = t 3: ^- t. . . . . . (633) Now, the same value for t, in Equations (623) and (624), will, tbi equal values of the arbitrary arcs a' and a", determine the component 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 pro- gress 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 simi- lar, 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 r is the same in all ; that is, wherein the t t wave lengths are the same; for, in that case, sin 2 tt . ~ and cos2 7r..- being common factors, after developing each function in the sum, the resultant displacement p becomes, p = sin 2 77 . - . 2 c' cos a' + cos 2 7r . — . 2 c' sin a\ T T and assuming c . cos a '■ = 2 c' cos a', c . sin a = 2 c' sin a' ; p = c . sin (2 7T - -f a), . . (634) T thus making the resultant wave of the same length as that of either of its components. 420 ELEMENTS OF ANALYTICAL MECHANICS. But, if the component waves be not of equal lengths, the sum of the corresponding functions cannot reduce to the form of Equation (634) fl because of the absence of common factors, arising from a A -J^ change in the valuo of r from one com- ponent to another. Such components can never destroy one anothei. INFLEXION. § 348. — Make, in Equation (621), r" = 1, and that equation be- comes , c" and this value being substituted for c', in Equation (622J, gives, and making c " . n t p' = — . sin 2 7r . -; r r t Vt- ■ r , r X » we have, omitting all the accents, c . Vt — r p = - . sin 2 7r r — , (635) Y A which is of the same form as Equations (528), and in which V is the velocity of wave propagation ; /, the time of its motion from primitive disturbance ; A, the wave length ; -, the maximum displace- ment 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 caiwe of MECHANICS OF MOLECULES. 421 jisturbar.ee 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 f, its displacement 2 p. Suppose a wave, whose centre of disturbance is (7, to t \ __— -^ss^D 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 AB, of a section of this wave front? Draw the normal C D iV, through the middle of P Q ; denote the variable distance D Q by z, and Q by r. The displacement of the molecule 0, by the secondary waves from the arc AB = 2 6, will, Eq. (635), be given by /+ 6 r +b cdz Vt — r pdz — l .sin2rr. = . . (636) — b J —b r A Here r and z are variable. To eliminate the former, join with the middle of AB by the line D 0, and denote its length by /, and the angle Q D 0, which it makes with the wave front, by 0. Then wHI r = Vl* + z>— 2/zcos0; and by Maclaurin's formula, r = I — cos B . z + 5^-j- . z % — _ . . c X 2 rr . b . cos 2 c b ( sp >' = indole x — = — ; - • ' ( ti4I > • ■ 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 Ihe front. This lateral action of secondlirv waves proceeding from a small portion of a primitive wave, is called wave inflection. When 6 approaches nearly to 90°, cos 6 will be exceedingly small, and the arc 2 n . b . cos 6 may again be substituted for its sine ; again Equation (641) suits the case, and determines the maximum displacement immediately about the normal. The maximum of the maxima displacements will occur when, in Equation (640), 2 n . b . cos 6 . sin . r = ± 1 ; and which would reduce that equation to c X (*rt„ IT . I . cos 6 ' and as the living forces are proportional to the squares of the greatest displacements, we have 4c*6 J and we have ZPSp — SQSq == £2j».rff». .... (64J) Integrating, J % 2PSp-f2Q6q = $2?nv*+ C; and denoting by v t the initial velocity, and taking the integral so as to vanish when t sz 0, J 2 PSp — J 2 QSq = Jlmt; 2 - jlw^.. . . (645) The products P Sp and $ £ q are the elementary quantities of work performed by a power and a resistance respectively, it the element of time d t ; the product %mdv 2 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 to 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 fzPSp— j*2 Q8q = ±2mv 2 , • - - - (646) and as the second member is essentially positive, the work of the motors must exceed that of the resistances embraced in the term flQfiq; 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 tha 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, und this will continue till the machine com*** APPLICATIONS. 427 to rest, and the excess whence, v«;: This constant ratio / is called the co-ejicimt of friction, because, 'when multiplied by the total normal pressure, the product gives the entire friction. , Assuming the first law of fric- tion, the co-efficient of friction may easily be obtained by means of the inclined plane. Let W denote the weight of any body placed upon the inclined plane A B. Resolve this, weight G G' into two compo- nents, one GM perpendicular to the plane, and the other G JV par APPLICATIONS. 431 allel to it. Because the angles G' G M and BAC are equal, the . first of these comporents will be GM = W.cosA, and the second, GN = W.sinA, in which A denotes the ansjle B A C. The first of these components determines the total pressure ujk>d 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, ff 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, /. W . cos A = W . sin A ; vhence, . sin A ■ f — = tan A. cos 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 ordei 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 M. Morin ; it has suggested to him what appears to be the true explanation of the Jifference between his results and those of Coulomb. He conceive*, that in the ex- 432 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 nands of the workman who hn>-) 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 | ds 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, ind 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 con- tinuous 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 thai co-efficient being in all cases included between 0,07 and 0,08, and the limiting angle of resistance therefore between 4° and 4° 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 coefficient 0,10, and for its limiting angle of re- sistance 5° 43 APPLICATIONS. 433 356. — 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 num- ber 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 be- tween 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 area 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 otv 4:34 ELEMENTS OF ANALYTICAL MECHANICS. viously expresses the value of adhesion on each unit of surface, for making S=l, 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 be- tween 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 adhe- sion 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 globu- lar 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 x A PPLIC AT10NS. 435 from any species of water-fowl, remains dry the lgh 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 containing the quicksilver in contact with water, and the metal will also flov through. It is difficult to ascertain the precise value of the force of ad he 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 unguents 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 W, being suspended from one end of the rope, while a force F, 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 436 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 K, and the ratio of the variable part to the weight W by /, the measure will be given by the expression K+ I. W; in which K represents the stiffness arising from the natural torsion or tension of the threads, and / the stiffness of the same cord due to a tension resulting from one unit of weight; for, making W == 1, the above becomes K + /. 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 2E 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 sub- jected 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, Couloml) also ascertained the law which connects the stiffness with the diame- ter 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 drv *»r moist j 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. 437 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 & 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 numbei of yarns in any tarred rope to that of the table, we shall have for AVjo white rope, dry or moist. Half worn white rope, dry or moist. Tarred rope. K+ I.W Packthread. b = a - — — • (651) li -ti For packthread, it will always be sufficient to use the tabular values given, corresponding to the least tabular diameters, and substi- tute 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, '' = S = 1 ' 5nearly: and from the table at the side, d? = 2,25. From No. 1, opposite 0,79, we find K = 1,6097, / = 0,03195; 27 438 ELEMENTS OF ANALYTICAL MECHANICS. ft. which, together with the weight W = 882 lbs., and 2 R = 1,64, substituted in Equation (648), give lb. lb. 8 = 2,25 - 1 ' 6097 + Off" *_ggg = 4 $ 17 , 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 diam- eter 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, d = -~ = 1,3 nearly; the square root of the cube of which is, by the table at the side, di = 1,482. In No. 4 we find, opposite 1,57, K = 6,4324, . / = 0,06387; ft. which values, together with W = 2205 lbs., and 2 R = 0,82, in Equation (649), give lbs . lbs. s = M 82 x 6 ' 4324 + %gg x 2205 --= aeffog, 0,0^ which is *,he required resistance. Example Sd. 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 running foot of the rope being about 0,6 of a pound I By referring to No. 5, Table X, we find the tabular number of yarns next less than 22 to be 15, and hence, 22 n = — = 1,466 nearly. 15 J APPLICATIONS 439 In the same table, opposite 15, we find K = 0,7664, / = 0,019879; ft which, together with W = 4212, and 2 Ji = 1,3, in Equation (G50), give S = ,,40« °1 7664 + W/fTC >< 4al2 = oft* l,o Example 4tk. Required the resistance due to the stiffness of a new white packthread, whose diameter is 0,196 inches, when moist- ened 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 0.196 0,390 = 0,5 nearly. In No. 2, Table X, we find, opposite 0,39, K = 0,8048, I = 0,00798 ; which, with W = 275, and 2 R — 0,5, we find, after substituting in Equation (651), 8 = 05 0,8048 + 0,00708 X^75 = ^ . 0,5 § 358. — The resistance just found is expressed in pounds, and is the amount of w r eight which would be necessary to bend any given rope around a vertical wheel, so that the portion A E, between the first point of contact ^4,-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 noint A ; for, if motion be com- 440 ELEMENTS OF ANALYTICAL MECHANICS. inunieated to the wheel in the direction indicated by the airow head, the rope, supposed not to slide, will, at this point, take ana 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 £ will have described in its own direction the distance A D. Denoting the arc described by a point at the unit's distance from the centre of the wheel by 8 4 , and the radius of the wheel by R, we shall have AD m Rs d ; and representing the quantity of work of the force S by L, we get L = S.Bs,; replacing S by its value in Equations (648) to (651), r „ . K+ I. W L = Rs r d t — (652) 3 in which d 4 represents the quantity d 2 , rf*, 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 por- tion of the rope, equal to 20 feet in length, V) have been brought •n 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. — All 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 tc be different according as the rotating pieces are kept APPLICATIONS. Ul in place by trunnions or by pivots. By trunnions are meant cylindrical projections a a from the ends of the arbor A B of a wheel. The trunnions rest on the concave surfaces of cylindrical boxes CD, with which they usu- 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 sock- ets FGHL w PIVOTS. Let N denote the force, in the direction of the axis, by whict 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 pres- sures exerted upon all the ele- mentary surfaces of which this circle is composed. Denote by A the area of the entire circle, then will the pressure sustained by each unit of surface be N A ' and the pressure on any small portion of the surface denoted by a, will obviously be a.N 442 ELEMENTS OF ANALYTICAL MECHANICS, and the friction on th<. same will be f.a.N This friction may be regarded as applied to the centre of the ele- nentary 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 *, then will the mo- ment of the friction be Now. if t denote the length of any variable portion of the circumfer enee at the unit's distance from the centre (7, then will also, a = x . d s . d x\ A = « E 2 ; which substituted above give f.N x 2 . dx . d s « .R 2 ' end by integration, f.N x 2 d x I d s « R 2 = /-A r -j A; (653) whence we conclude, that, in the fric- tion of a pivot, we may regard the whole friction due to the pressure as acting in a single point, and at a dis- tance from the centre of motion equal to two-thirds of the radius of the base of the pivot. This distance is called' the wean lever of friction. § 360. — If the extremity of the pivot, instead of rubbing upon an entire circle, is only in contact with a ring or sur- face comprised between two concentric APPLICATIONS. 443 circles, as when the irbor of a wheel is urged in the direction of its length by the force N against a shoulder d c b a ; then will A = « (R 2 - It' 2 ) ; and the integration will give pR /»2 t / x 2 dx I d _ _ 2 f. y. R * ~ ^ . J « (R 2 - R' 2 ) f/ R 2 — R' 2 ' in which R denotes the radius of the larger, and R' that of the smaller circle. Finally, denote by I the breadth of the ring, that is, the dis- tance A f A\ by r, its mean radius or distance from C to a point half way between A' and A, and we shall have R' = r-il; substituting these values above and reducing, we have r P~\ /. n x \r + iV • 7 1 » ( 654 ^ and making 12r r + T7T~ " T , 1 we obtain, for the moment of the friction on the entire ring, f-N.T i (655) The quantity r t is called the mean lever of friction for a ring. Since the whole friction fN may be considered as applied at a point whose distance from the centre is § R, or r t = r -j- tjt— > according l «* /* 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 di- rection in which the friction acts, the quantity of work consumed by it will be equal t3 the product of its intensity fN into this path. Designating the length of the arc described at the unit's distance from C by s t , the path in q lestion will be either %Rs t , or r, v, 444 ELEMENTS OF ANALYTICAI MECHANICS. and the quantity of work either %R.s t .f.N for an entire circle, or 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 * X n ; and we shall have Q = %«.R.f. N.n (656) for the circle, and Q = 2*-f'N- (r + — ) . n .... (657) for a ring; in which * == 3,1416. The co-efficient of friction /, when employed in either of the fore- going 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. R = « = 3 = 0,25, lbs. N = 1784, / = 0,147 ; which, substituted in Equation (653), gives lbs. 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 «r = 3,1416 in Equation (656), and find Q = i X 3,1416 X 0,25 x 0,147 x 1784 x 20 = 5402,80} APPLICATIONS. 445 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' sup- ports 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. / = — - — = 1 inch, r = 2 + 0,5 .= 2,5 inches, (1)2 in. ft. T * = 2 ' 5 + 12x2,5 = 2 ' 5333 = °' 2111 ' N = 2046 pounds, J = 0,314; jrhich, substituted in Expression (655), gives for the moment of friction. 0.314 x 2046 *'x 0,2111 = 135,62. The quantity of work consumed in one minute, there being sup- posed 10 revolutions in that unit, will be found by making 4n Equation (657), * = 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 dis- tance of one foot. The quantity of work consumed in any given time would result from multiplying the work above found, by tho time reduced to minutes. • TRUNNIONS. §361. — The friction on trunnions and axles, which we now pio- ceed 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- m ELEMENTS OF ANALYTICAL MECHANICS. merit, common to the surfaces of both. A section perpendiculai 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 it3 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- 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 inclina- tion of the tangent urn v is such, that the friction is just sufficient to pre- vent the trunnion from sliding. Here let the trunnion be in equili- brio. But the equilibrium requires that the resultant of all the forces which act, friction 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 iV 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 othe r , 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 ad bm, of which the side m b shall coincide with the tangent, then, denoting the angle d (1 + Z 3 ) ; whence, iVcos 9 = y x VTT?' and multiplying both members by /, / . i\ r . cos 9 = N ' f (659) but the first member is the total friction ; whence we conclude that to find the friction upon a trunnion, we have but to multiply thr 448 ELEMENTS OF ANALYTICAL MECHANICS. 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 trim nion and in the direction of the tangent, its moment will be 7 if. -/i +/ 2 x R. (660) Arid 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 . s, x / /M 7 / 2 ' (661) in which s, denotes the path described by a point at the unit's dis- tance 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 t ; and making t == 3,1416, we have and s, = 2 at . n ; Q = 2« .R.n.N. f VT+/ 2 (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 fric- tion 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 ithan that of the axle, the trunnion has the advantage, in this respect over the axle. APPLICATIONS 449 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 paiL It is now proposed to discuss its action under the operation of fom-s applied to it in any manner whatever. Let the points A\ A", A"\ be connected with each other by means of two perfectly flex- ible and inextensible cords A' A", A" A'", the first point being acted upon by the forces P\ P", &c. ; the second by the forces Q\ Q'\ &c. ; and the third by the forces S\ £", &c. ; and sup- pose these forces to be in equilibrio. Denote the co- ordinates of A' by x'y'z', A" by *" y" z", and A'" by *"' y'" z'". Also, the alge- braic sum of the components of the forces acting at A' in the direo tion of xyz, by X' T Z ', at A" by X" T' Z", and at A m by X" T" Z" f . Then will, § 101, X' $x' + Y' By' + Z' Sz' + X" S x" + Y" S y" + Z" 8 z" }= 0. + X"8x'" + Y'"8y'" + Z"'Sz'" ) Denote the length A' A" by /, and A" A"' by g; then will (663} L =/-VP -x'f + (y" -y') 2 + ( 2 " -*') 2 = 0; I- . (664) H = g- (66C>) *"' - *" ■> 0; y» + x.JL—JL _ fcw.L. — y_ = . / Z" + X' •" - 2' / 2'" — 2" _ x'" . — = (667) X"' + X ttt x'" - x" = 0; y'" — y" Y»' + x'" . y - y — - 0; \ Z'" + X /// *'" - z" 0; (668) Taking from each group its first equation ar d adding, and doing the same for the second and third, we have X' + X" + X"l = ; Y' + Y" + Y'" = ; Z' + Z" 4- Z'" = 0. (669) AFPI ICATIONS. 451 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 X'", 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"' + y (x" - x') = ; T' + Y" + —'V' - jfO = 0; / Z" + Z'" + h. ( Z " - Z ') = 0. irom which we find by elimination, *LJ L?L (X" + X'") = ; *" - x- Z" + 2 nt z" - z' x" - X - (X" + X"') = 0. \ (670) From group (666), by eliminating X', r-^, ^^' = 0; X — X Z' - z" - z' x" - X 7^ = 0; (671) and finally from group (668) we obtain, by eliminating X"', \rttr ^"^ y"' — y" l Z'" x'" z'" x" z" X"' — x" . X'" = . X'" = 0; 0. • • • • (672) 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 oil 462 ELEMENTS OF ANALYTICAL MECHANICS. the displacement will be made known by Equation (331), which because f dL V r dL V r dL -i* ld{x"-x')\ " 1 "L rf (y"-y')J U(*"-*')J - 1; U(x'" - x")\ T ly(y'" - y")\ \d (*'" - *")J ' becomes for the cord A' A" % V = JP ; and for the cord A" A'", X'" = 2V" ; from which we conclude, that X' and X'" are respectively the ten sions 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'f + (y" - y') 2 ± {?" - z'f P * we have = 1, g 2 X' = y/X'i + Y' 2 + Z' 2 = i?', (673) x/" = yx'" 2 + y" 2 + z"' 2 = .#'", in which i?' 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 £! - *" ~ *' IL - y " ~" y ' E. - z " r *' . ^ ~ 7 ; IB ~ 7 ; ]sP ~ 7~ ' X'" x' n — x" T" _ y'" — y" Z"' z'" — z" ill 777 ~ a ; ~W* = a ' R ttt whence the resultants of the forces applied at the points A' and A"' t 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 453 § 364. — From Equations (669), we have by transposition, X" = - (X'" + X') ; Y" = - ( Y"' + Y') ; Z" = - (Z" f 4- £')• Squaring, adding and denoting the resultant of the forces applied at A" by R", we have R" = ^(X"' + X') 2 + ( Y"' + F') 2 + (Z" f -h Z') 2 ' * ( 6 ? 4 ) and dividing each of the above equations by this one X" R" Y" R" Z" ~RT X"' + X t ^ R" > Y" + r 7 • R" ? Z"' 4- Z' • i? it y (675) 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", distances A" m and A" n be taken proportional to R' and R'" respectively, and a paral- lelogram A" m Cn be constructed, A" C will represent the value of R". If A' A" A"' be a contin- uous cord, and the point A" capable of sliding thereon, the tension of the cord would be the same throughout, in which case R' would be equal to R'", and the direction of R" would bisect the angle A' A" A"'. The same result is shown if, instead of making Sf = and 8 g = separately, we make 28 454 ELEMENTS OF ANALYTICAL MECHANICS. ^ (/ 1" 9) — 0> multiply by a single indeterminate quantity X, and proceed as before. § 365. — Had there been four points, A', A", A'" and A", connected by the same means, the general equation of equili- brium would become, by call- ing h the distance between the points, A'" and A iy , X' 8 x' + X" 8 x" + X" 8 x'" 4- ^ iv * *** 1 • 4- Y' I y' 4- Y" 8 y" 4- Y" 8 y"' + F iv 8 y iv + Z' 6 z' 4- Z" 8 z" 4- £'" * z'" + Z" S z" 4- X' 8/ 4- X" 8y 4- X'" or'" _ x » X • Y" + X' • 2- -r-^- - X Z" 4- X' . X"' 4- X // Z"' 4- X" / f - y' f z" - <_ / x'" - x" 9 y"' - y" 9 z ,„ _ z" y>» _ y» Jit Jf — X // = 0, = 0, = 0, — X Itt a: IT — x tit = 0, ,nt Y'" 4. X" • Z 1 X'" . ylT t y = 0, h X f/> 2 iv — 2'" A = 0, (677) > ' (678) APPLICATIONS IK x y * — x tt> X" + V" — h - ' y\r + X'" yiv _ y f " Z ,r + X'" ■ — I. — u > (679) Eliminating the indeterminate quantities X^ X" and X'", we obtain eight equations, from which, and the three equations of conditions expressive of the lengths of /, g, and A, the position of the points A', A", A"', and A iY may be determined. If there be n points, connected in the same way and acted upon by any forces, the law which is manifest in the formation of Equa- tions (676), (677), (678), and (679), plainly indicates the following r equations of equilibrium : Y 7 x" — x' >' ■ — A.\ A. • j. — v/, T - X' . fLjzX = o, Z' z" — z' V • - . - 0, /' (680* X" + X' *" - *' / — X ft x'" - x" = 0, r + vJ-LzlL-vJ"-* 9 ^ f Z" + x' z" - *' n 9 — / = 0, (f«M) X'" + X tit ~n - X •> III x — x .III .11 y ' y \ttt y y Y"' -f X' — X 1 Z'" + x mill Z " tt . * f_ y^ilt h z" — z' h 0, 0, 0, = 0. V (682) 450 ELEMENTS OF ANALYTICAL MECHANICS. X m _, + X._ 8 . «-' 7 *— - X„_, «■ ' *~* r.i, + \._ 8 A y. -i — y- -8 k *— -i — «»- -1 - V-. i y. — y- -l / 2. — *.- • 1 = o, = 0, Z.-i + *.-. • ""• , ""' ■ X.., . 2--^=i = o, x. + x.,,. *--,**-' ^,] (683) r. + x*-, £„ +X... y» — r— ! = 6, > » • z._ m-1 / = 0. (684) In which X, with its particular accent, denotes the tension of the cord into the difference of whose extreme co-ordinates it is multi- plied. Adding together the equations containing the components of the forces parallel to the same axis, there will result X 9 + Jg" + X" + I 1T . . . X n = 0, } Y> + F" f- Y'" + T ir • • • Y m = 0, I • • Z' + Z" + 3'" + £ iT • • - z % = 0, J (685) from which we infer, that the conditions of equilibrium are the same as though the forces were all applied to a single point. From group (680), we find by transposing, squaring, adding and extracting square root, f y" - y' £ _ *" - z ' B!~ f APPLICATIONS. 457 Treating the equations of group (684) in the 3ame way, we have R. R. R. *» — #•- -1 / Vn — Uk- -l l *» — *m- -I / whence, the resultants of the forces applied to the extreme points A' and A n , act in the direction of the extreme cords. And from Equations (685) it appears that the resultant of these two resultants is equal and contrary to that of all the forces applied to the other points. §366.— If the extreme points be fixed, X\ Y\ Z' and X n , Y n , Z n , will be the components of the resistances of these points in the directions of the axes ; these resistances will be equal to the ten- sions \' and X n of the cords which terminate in them. Taking the sum of the equations in groups (680) to (684), stopping at the point whose co-ordinates are #,»_», y^_ m , s»_m> we have X> + U-^ r + u-u.,. Z' + SZ-X^,.,. X n— m X n—r*—\ Vn—m Vn-m-\ Z Z 0; 0; 0; (686) in which 2 X y 2F, 2 Z, denote the algebraic sums of the components in the directions of the axes of the active forces; X w _ lir _ 1 the tension on the side of which the extreme co-ordinates are ar,^., #»_,», *»_», and s„_*_i, y»_ ^_i, K-m-\\ a n( i *»— tne length of this side. §367. — Now, suppose the length of the sides diminished and 458 ELEMENTS OF ANALYTICAL MECHANICS. their number increased indefinitely ; the polygon will become a curve; also, making '^ t ^ m - l = t 1 we have y-m — y»-«-i == dy, K~m — *»-m-l = dz, Cm = dS, i being any length of the curve ; and Equations (686) become dx I' + U-/.-=0; d s as a*2 a s (687) which will give the curved locus of a rope or chain, fastened at its ends, and acted upon by any forces whatever, as its own weight, the weight of other materials, the pressure of winds, curreuts of water, &c, &c. This arrangement of several points, connected by means of flexi- ble cords, and subjected to the action of forces, is called a Funi- cular Machine. • §368. — If the only forces acting be pressure from weights, we have, by taking the axis of z vertical, X" = X" = X T &c, = ; Y" = Y'" &c = ; and from Equations (680) to (684), X sz X' x" - x 1 f = X n x'" - *" • • • • X n — £„_, x - 1 TT "*' whence, the tensions on all the cords, estimated in a horizontal direction, are equal to one another. Moreover, we obtain from the *ame equations, by division, y" - y' x" - x- y'" - y" x m - x" y. - y»-i — • APPLICATIONS. 450 These are the tangents of the angles which the projections of the sides on the plane xy make with the axis x. The polygon is therefore contained in a vertical plane. THE CATENARY. §369. — If a single rope or chain cable be taken, and subjected only to the action of its own weight, it will assume a curvilinear shape called the Catenary curve. It will lie in a vertical plane. Take the axes z and x in this plane, and z positive upwards, then will U=0; 2F=0; F' = 0; 2Z=-W; in which W denotes the weight of the cable, and Equations (687) become dx X'-l-- =0, as d z z f -w -t.— =0. d s (688) Z These are the differential equations of the curve. The origin may be taken at any point. Let it be at the bottom point of the curve. The curve being at rest, will not be disturbed by taking any one of its points fixed at pleas- ure. Suppose the lowest point for a moment to be- come fixed. As the curve is here horizontal, Z' = 0, § 366, and from the second of Equations (688), we have dz W = ds' (689) whence, the vertical component of the tension at any point as of the curve, is equal to the weight of that part of the cable between this point and the lowest point. The first of Equations (688) shows 460 ELEMENTS OF ANALYTICAL MECHANICS. that the horizontal component <»f the tension at is equal to the tension at the lowest point, as it should be, since the horizontal tensions are equal throughout. Taking the unit of length of the cable to give a unit of weight, * which would give the common catenary, we have W = s ; and, de- noting' the tension at the lowest point by c, we have t = =fc -y/s 2 + c 2 , and from Equation (689), s • ds dz = q= — • y/c 2 + •* Taking the positive sign, because z and s increase together, inte- grating, and finding the constant of integration such that when z — 0, we have s = 0, z -\- c =. y/ c 2 -j- 6' 2 ; whence, ,? 2 a z 2 -f 2 c 2. Also, dividing the first of Equations (688) by Equation (689), dx c c dz s y^ 2 + 2cz ' and integrating, and taking the constant such that x and z vanish together, * = ,. log i±c+y* + 2c ± ; ; ; (690) ■ which is the equation of the catenary. This equation may be put under another form. For we mav write the above, • c e^ = z + c + y/{? + c) 2 — c 2 ; transposing z 4- c and squaring, c 2 > e e — Z c e e (z -f c) = — c 2 ; whence, ? + « = |c >£ -j- cause the area of the cross section to be proportional to the tension at the point where the section is made. The general Equa- tions (638) will give the solution for every possible ease. FRICTION BETWEEN CORDS AND CYLINDRICAL SOLIDS. § 371. — When a cord is wrapped around a solid cylinder, and motion is communicated by applying the power F at one end while a resistance W acts at the other, a pressure is exerted by the cord upon the cylinder ; this pressure produces friction, and this acts as a resistance. To estimate its amount, denote the radius of the cylinder by is?, the arc of contact by », the tension of ths cord at any point by t. The tension t being the same throughout the length d s — a t t of the cord, this element will be pressed against the cylinder by two forces each equal to /, and applied at its extremities a and t t , the first acting from a towards W, the second from t t towards b'. Denoting by & the angle a b t t , and by p the resultant brn of these forces, which is obviously the pressure of ds against the cylinder, we have, Equation (56), p = y/P + t* + 2 t . : cos 6 = t y^2(l + cos 6) ; but 1 -J- cos 6 == 2 cos 2 £ 6 ; (t - ») = -g ; 462 ELEMENTS OF ANALYTICAL MECHANICS. and taking the arc fur its sine, because n — 6 is very small, we have ds and hence, 8 355. the friction on ds will be > • , ds f'P =f' t '-ft' The element t t £ 2 of the cord which next succeeds a t t 7 will have its tension increased by this friction before the latter can be over come ; this friction is therefore the differential of the tension, being the difference of the tensions of two consecutive elements ; whence, dt=f.t--- i dividing by t and integrating, or, log t = f.± + log C, t = Ce R (693) making s = 0, we have t = W = C; whence, t = W*tT\ (694) and making « = & = al, ^ f 8 , we have t = F; and F= W-e R (695) Suppose, for example, the cord to be wound around the .ylinder three times, and / = ^ ; then will S = 3«.2B =-. 6 . 3,1416. E s 18,849 B, and ^ = Trx^ X,884S =^X(2 ? 71825) 63838 ; or, I = W. 535,3, that is to say, one man at the end W could resist the combined effort of 535 men, of the same strength as himself, to put the cord in motion when wound three times round the cylindei. A PPLtCATIONS 463 THE INCLINED PLANE. $ 372. — The inclined plane is used to support, in part, the weight of a body while at rest or in motion upon its surface. Suppose a hody to rest with one of its faces on an inclined plane ni which the Equation is L = cos a x -f cos b y -+- cos c z — d = ; • • • • (a) in which d denotes the distance of the plane from the origin of co- ordinates, and a, b, c, the angles which a normal to the plane makes with the axes x, y, z, respectively. .Denote the weight of the body by W t the power by F ; the nor- mal pressure by N ; the angles which the power makes with the axes ./•, //, i, by a,, /3,, y t , respectively; and the path described by the point of application of" the resultant friction by s. Then, taking ihe axis Z vertical and positive upwards, and supposing the force to produce a uniform motion of simple translation, will, Eq. (645), dx^ (Vcosa, +/JV -^) Sx + (F< KM & 4 +fNp?) d s dif d a r dz y \ -0; + {Fcos y 4 +/N ~ - W ) Sz d s and. Equation (a), cos a d x -f cos b d y -f- cos c o 2 = Multiplying this last by X, adding and proceeding as in § 213, dx F cos a -\- f N h X cos a = 0, d s F cos ft, + fff %2 -fc X.oob* = 0, a s F cos y, + f N -^ + X cos c — W ~ ; W and, Eq. (331), XT ■> / f^yfe) + W r (rfl) =x - r/ /A" ■dL\* (d) Substituting the vai le of X in Equations ('„), the tirst two give by eliminating iV", 464 ELEMENTS OF ANALYTICAL MECHANICS. d x f (- cos a ds cos p, - — — ... - * — — dy , cos a / — - -h cos b ds + 1 = {*) I and th3 first and third, bj eliminating N. I I// cos v,— — cos a,— ) + cos 7. cos a — cos a cos c \—W( f — -f cos « ) . i_ \ "ds 'dsl : J V efc / / a If there be no friction, then will fss 0, and, Eq. (e), cos a cos # i + 1=0; cos cos a t whence, Eqs. (45) and (a), the power must he applied in a jiaiir iioimnl hoth to the inclined plane and to the horizon. If without disregarding friction, the power be applied in a plane fulfilling - the above condition, and also con- taining the centre of gravity, the resultant friction may be regarded as acting in this plane, and we may take it as the co- ordinate plane z x, in which case cos b = ; cos (3 = ; -- — ; d t and denoting the inclination of the plane to the horizon by a, and that 01 thi power to the inclined plane by

=s 0, and F— JF(sina ±/cos a) * . (698) the upper sign answering to the case of motion up, and the lower, down the plane; the difference of the two values being 2/ JFcosa. If /= 0, then will F . £ C that is, the power is to the weight as the height of the plane is to its length; and there will be a gain of power. § 375. — If the power be applied horizontally, then will

'- ^==£2 as O.ttOl] Omitting the common factor rf«, , and making = /; m = r; w = we have, P - m Q - ^P 2 + § 2 +2/ ) £ . cos • /'» = 0. Transposing, squaring, and solving, with respect to P, we find, *= « = " l-f>2 T ? " L i 70 V If the fraction n be so small as to justify the omission of every term into which it enters as a„factor, or if the co-efficient of friction be sensibly zero, then would i= m =i (703) That is, the power and the resistance are to each other inversely as the lengths of their respective lever arms. If the power or the resistance, or both, be applied in a plane oblique to the axis of the trunnion, each oblique action must be replaced by its components, one of which is perpendicular, and the. other parallel to the axis of the trunnion. The perpendicular com- ponents must be treated as above. The parallel components will, if APPLICATIONS. 469 the friction arising from the resultant of the norma] components be not too great, give motion to the whole body of the lever along the trunnion : and if this be prevented by a shoulder, the friction upon this shoulder becomes an additional resistance, whose elementary quantity of work may be computed by means of Eq. (657) and made another term in Equation (701). WHEEL AND AXLE. §379. — This machine consists of a wheel , mounted upon an arbor, supported at either end by a trun- nion resting in a box or trunnion bed. The plane of the wheel is at right angles to the arbor ; the pow- er P is applied to a rope wound round the wheel, the resistance to another rope wound in the opposite direction about the arbor, and both act in planes at right angles to the axis of motion. Let us suppose the arbor to be horizontal and the re- sistance Q to be a weight. Make N and N' = pressures upon the trunnion boxes at A and B ; R = radius of the wheel ; r = radius of the arbor ; p and p' = radii of the trunnions at A and B ; f /' = -/i +p s, = arc described at unit's distance from axis of motion Ihen, the system being retained by a fixed axis, we have P Sp = PRd s x \ Q 8 q = Q r d 8 V The elementary work of the friction will, Eq. (661), be f(2tf+ N' P ')ds ri 29 470 ELEMENTS OF ANALYTICAL MECHANICS. and the elementary work of the stiffness of cordage. Equation (652), , K+I. Q d 4 • — r.d Si ; and when the machine is moving uniformly, PRds x -Qrds x -f{N ? + N' ? , )ds l -d r : ^ r ^-^..r'ds l = 0', . (704) & v The pressures N and N' arise from the action of the power P. the weight of the machine, and the reaction of the resistance Q, in creased bv the stiffness of cordage. To find their values, resolve each of these forces into two parallel components acting in planes which are perpendicular to the axis of the arbor at the trunnion beds* then resolve each of these components which are oblique to the components of Q into two others, one parallel and the other perpendicular to the direction of Q. Make w = weight of the wheel and axle, g = the distance of its centre of gravity from A, p = the distance m A, q — the distance n A, I ■=: length of the arbor A B. 9 =s the angle which the direction of P makes with the vertical or direction of the resistance Q. Then the force applied in the plane perpendicular to the trunnion A. and acting parallel to the resistance Q, will, § 95, be, and the force applied in this plane and acting at right angles to the direction of Q, will be P j-^- • sin (^-/»)cos(pia + i w (/- i >)».8iii»(p; (705) JST= l r J[w.g + £.? 4- P.jo.coscp] 2 + P* . p* . sin* 9 ; . -(706) If d and &' be the angles which the directions of JV and A Tf make with that of the resistance Q, we have . P(l-P) . . At Pp . sin = — — .— - • sin 9 ; sin r = — — • sin 9. iV\ / N'l r Equations (704), (705), and (706) are sufficient to determine the rela- tion between P and Q to preserve the motion uniform, or an equili- brium without the aid of inertia. The values of iV and J\ Tf being substituted in Equation (704), and that equation solved with refer- ence to P, will give the relation in question. §380. — If the power P act in the direction of the resistance Q, then will cos 9 sac. I, sin 9 = 0, and Equation (704) would, after substituting the corresponding values of iV and iV 7 , transposing omitting the common factor d Sj , and supposing p = p', become PB= Q r +f ? (w+ Q + P) 4- d r K + IQ -r. • ■ (707) And omitting the terms involving the friction and stiffness of cordage, P_ _ r_ Q ~ ~R % that is, the power is to the resistance as the radius of the arbor is to that of the wheel ; which relation is exactly the same as that of the common lever. FIXED PULLEY. §381.— -The pulley is a small wheel having a groove in its cir- cumference for the reception of a rope, to one end of which the 472 ELEMENTfc OF ANALYTICAL MECHANICS. power P is applied, and to the other the resistance Q. The pulley may turn either upon trunnions or aoout an axle, supported in what is called a block. This is usually a solid piece of wood, through which is cut an opening large enough to receive the pulley, and allow it to turn freely between its cheeks. Sometimes the block is a simple framework of metal. When the block is stationary, the pulley is said to be fixed. The principle of this machine is obvi- ously the same as that of the wheel and axle. The friction between the rope and pulley will be sufficient to give the latter motion. Making, in Equations (705) and (706), we have g = q =p = $ I, N = i y/{w + Q + P cos n PR- QR-f'ty/{w+ # + Peos=V(1 +2/'i.cosi9) + rf,.^j^ • • • (710) (711) in which

-i)>.*' iT+ /. IT 2 sin 4- 4 2i2 . (710) The ouantity of work is found by multiplying both members by R $i , in which s x is the arc described at the unit's distance. If the arc enveloped by the rope be 1 80°, then will £ Q = 90°, sin \ & = 1, and P= W (l +/>■■£) + <*,- 2B (»«) If the friction and stiffness of cordage be neglected, then will. Equation (716), W = 2 P sin J 0, and multiplying by P, R W = P . 2 R . sin | 6 ; but xr hence, 2 P sin £ d = A B ; P. TT= P . AB; that is, $e power is to the resistance as the radius of the pulley is to the chord of the arc enveloped by the rope. 476 ELEMENTS OF ANALYTICAL MECHANICS § 383. — The Muffle is a collection of pulleys in two separate « blocks or frames. One of these blocks is attached to a fixed point A\ by which all of its pulleys become Jixed, while the other block is attached to the resist- ance W 7 , and its pulleys thereby made mov- able. A rope is" attached at one end to a hook h at the extremity of the fixed block, and is passed around one of the movable pulleys, then about one of the fixed pulleys, and so on, in order, till the rope is made to act upon each pulley of the combination. The power P is applied to the other end of the rope, and the pulleys are so proportioned that the parts of the rope between them, when stretched, are parallel. Now, suppose the power P to main- tain in uniform motion the point of applica- tion of the resistance W; denote the tension of the rope between the hook of the fixed block and the point where it comes in con- tact with the first movable pulley by t x ; the radius of this pulley by R x ; that of its eye by r, ; the co-efficient of friction on the axle by f\ the constant and co-efficient of the stiff- ness of cordage by K and /, as before; then, denoting the tension of the rope between the last point of contact with the first movable, and first point of contact with the first fixed pulley, by £ 2 , the quan- tity of work of the tension t t will, Equation (652). be t x R x s, = t x R, ^ + d 4 — iL-l 1 jfc, *, + f (t, + t,) r, s, ; V 2R, in which r = f VTT7' dividing by *,, t H /v, ft *i Va, • y gffi • * + / ft + *) r * - 118) applications. 477 Again, denoting the tension of that part of the rope which passes from the first fixed to the second movable pulley by / 3 , the radius of the first fixed pulley by i? 2 , and that of its eye by r, , we shall, in like manner, have t.R, = m + d t *£J* 22, + f («, -f k) r,. . (719) And denoting the tensions, in order, by t A and t b , this last being equal to P, we shall have t 4 R 3 = t z R 3 -f d, ***** 'Ri+f (4 + h) r,. . (720) PP 4 = * 4 P 4 + d t ^~ &* + /' (U + P) r+ . (721) so that we finally arrive at the power P, through the tensions which are as yet unknown. The parts of the rope being parallel, and the resistance W being supported by their tensions, the latter may ob- viously be regarded as equal in intensity to the components of W\ hence, UUU^^; • • • • • (722) which, with the preceding, gives us five equations for the determi- nation of the four tensions and power P. This would involve a tedious process of elimination, which may be avoided by contenting ourselves with an approximation which is found, in practice, to be sufficient! v accurate. If the friction and stiffness be supposed zero, for the moment* Equations (718) to (721) become ^R % = hR*, t 4 R z = /,i?,, PR< = t K R^ from which it is apparent, dividing out the radii P, , ff 8 , R t1 4tu, 478 ELEMENTS OF ANALYTICAL MECHANICS. that t % = t Xi t % = tf 8 , t A = / 3 , P = t 4 -, and hence, Equation (7221 becomes 4 t x = W; whence, W the denominator 4 being the whole number of pulleys, movable and fixed. Had there been n pulleys, then would W t x = n With this approximate value of t l1 we resort to Equations (718) to (721), and find the values of £ 2 , t s . t 4 , &c. Adding all these tensions together, we shall find their sum to be greater than W, and hence we infer each of them to be too large. If we now suppose the true tensions to be proportional to those just found, and whose sum is W x > W, we may find the true tension corre- sponding to any erroneous tension, as t x , by the following propor- tion, viz. : W rr j . rr . . t x . i, , or, which is the same thing, multiply each of the tensions found by W the constant ratio — > the product will be the true tensions, very nearly. The value of t 4 thus found, substituted in Equation (721), will give that of P. Example. — Let the radii R x , /? 9 , R z and i2 4 , be respectively 0.26, 0,39, 0,52, 0,65 feet ; the radii r, = r 2 = r 3 = r 4 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 / = 0,0319501 ; and suppose the pulley of brass, and its axle of wrought iron, of which the co-efficient / = 0,09, and the resistance W a weight of 2400 pounds. Without friction and stiffness of cordage, 2400 '*»• t x = —— = 600. APPLICATIONS 479 Dividing Equation (718) by 22,, it becomes, since d t = 1, Substituting the value of 22, , and the above value of Jj , and regard, ng in the last term t 2 as equal to £, , which we may do, because of the small co-efficient -^- /', we find 22, U= \ + 600 1,6097 + 0,0319501 x 600 2 x (0,26) f = 623,39. + jjljj- X 0,09 x (600 + 600) Again, dividing Equation (719) by H.^ and substituting this value of t. 2 and that of 22,, we find lbs. / t 3 = 673,59. Dividing Equation (720) by fi z , and substituting this value and 2400 = 0,919 ; 2611,80 which will give for the true values of t x = 0,919 x 600 - 551,400 t 2 =z 0,919 x 628,39 = 577,490 t 3 = 0,919 X 673,59 as 619,029 f 4 = 0,919 x 709,82 = 652,324 2400,243 480 ELEMENTS OF ANALYTICAL MECHANICS. The above value for t 4 = 05*2,324, in Equation (^21), will give, aflei dividing by i? 4 , and substituting its numerical value. P = < + 652,324 1,0097 + 0.03195 x 652,324 2 x 0,05 0,06 f ~g X 0,09 x (652,324 + P) \ tnd making in the last factor P = l 4 = '652,324, we find lbs. lbs. lbs. lbs. P = 652,324 -f 17.270 + 10,831 == 680,425. Thus, without friction or stiffness of cordage, the intensity of P would be 600 lbs. ; with both oi these causes of resistance, which cannot be ..voided in practice, it becomes 680,425 lbs., making a difference of S0,425 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 man if these sources ot 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. The acute dihedral angle A Cb is called the edge ; the opposite plane face A b the lack-, and the planes Ac and Cb, 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 such mannei that after each advance it shall be supported against the faces of ♦he opening till the work is accomplished. APPLICATIONS. 481 § 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 ACB being inserted in the opening a hb, and in col. tact with its jaws at a and &, we know that the resistance of the latter will be perpendicular to the faces of the wedge. Through the points a and 6 draw the lines a q and b p normal to the faces A C 'and B C ; from their point of intersection lay off the distances Oq and Op equal, respec- tively, to the resistances at a and b. Denote the first by Q, and the second Dy P. Completing the parallelogram Oqmp, m will represent the re- sultant of the resistances Q and P. Denote this resultant by R', and the angle A C B of the wedge by $, which, in the quadrilateral a b C, will be tance CD of this helix from the axis z ; a = the- angle which this helix makes with the plane xy\ £ = the angle C BD which the generatrix of the helicoidal surface makes with the axis z ; y = the co-ordinate AB of the point in which the genera- trix, 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 -f BD = y + r. cotan €, and for any subsequent position, as C B\ z = y -f- r . cotan § + r . 9 . tan a, • • . • (726) APPLICATIONS. 487 which is the equation sought, and in which a and r are constant for the same helix, and variable from one helix to anothei The power P acts in a direction perpendici lar to the axis uf the newel. Denote by / its lever arm ; its virtual moment will be ', Pldy. The resistance Q acts in the direction of the axis of the newel ; its virtual moment will be Qdz. The friction acts in the direction of the helicoidal surface and paral- lel 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 JV, and co-efficient of friction by f. The virtual moment of friction will be , * f.N.d*\ and Equation (645), Pldcp - Qdz -j\N.ds = (727) But the displacement must satisfy Equation (726), or, as in § 213, the condition, L = z — r . (p . tan a — r . cotan € — y = 0; . (728) and also, r = constant (729) Differentiating, we have, dz — cotan £ . d r — r tan a dcp = 0, dr = 0. Multiplying the first by X, the second by X', adding to Equation (727), and eliminating d s by the relation d s = r . dcp . cos a + dz . sin a, . . , (730) we find, tPl— f.N.ooaa .r - \tena.r)df + (X - Q - /. JV*. sin a) dz +(A'- Xcotan€)(/r - 488 ELEMENTS OF ANALYTICAL MECHANICS and, from the principle of indeterminate co -efficients, PI — f . N . cos a . r — X . tan ct . r = 0; . . (731) Q +fN. sin a - X = 0; (732) X' — X cotan * - (732)' The variables d z, d r, andrd , ^ • cotan § V = G ; -= ; . (734) 1 — /. sin a y^l -f tan 2 a -{- cotan 2 § in whicrt X' is, § 217, the value of the force acting in the direction of r. § 387.— If the fillet be nectangular, § — 90°, cotan § = 0, and _ _ r tan a + /. cos a . -i/l 4- tan 2 a ,. v P = Q . ^ . V " . . (735) * 1 — / . sin a . yl -|- tan 2 a ftni X' r= 0. § 388. — If we neglect the friction, / = ; and PI = $ . r . tan a, multiplying both members by 2 at, P . 2 ^r / = # . 2 o . tan a (736) That is, the power is to the resistance as tlie helical interval is to the circumference described by the end of the level arm of the power. APPLICATIONS. 489 TUMI'S. § 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 bucking, forcing, and lifting pumps. § 390. — The Sacking-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 ca- pable of moving up and down freely. The piston is perforated in the direction of the bore of the barrel, and the orifice is covered by a valve F called the piston-valve, which opens up- ward ; a similar valve E, called the sleeping-valve, at the bottom of the barrel, covers the upper end of the sucking-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 of a lever H, or other contriv- ance, attached to the piston-rod G. The distance A A', between the highest and lowest points of the piston, is called the play. 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 same level L L both within and without the sucking-pipe. Now suppose the piston raised to its highest point A', the air contained in the barrel and sucking-pipe wijl tend by ita D 490 ELEMENTS-OF ANALYTICAL MECHANICS elastic force lo 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 i 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, wall 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 Vie again elevated, the water will rise still higher, for the same reason as before. This operation of raising and depressing the piston being repeated a few times, the water will at length entei the barrel, through the valve F, and be delivered from the dis- charge-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 to which the piston may reach, should never have a greater altitude above the water in the reser voir than that of the column of this fluid which the atmospheric pressure may support, ir 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 w r hen the air below it has a greater density and therefore greater elasticity than the external air ; am 1 APPLICATIONS, 491 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 ib*r 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 otjier dimensions, in order to make the pump effec- tive, suppose the water to have reached a sta- tionary level X, at some one ascent of the 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 in- definitely, within these limits for the play, without moving, the water. Denote the play of tne 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 mav also be regarded as the altitude of the discharge pipe ; 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 tc A', pro- vided the water does not move, will be B (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 measure! by Bh.ff. D; 492 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 3 same «*ir will support, when expanded between the limits X and A\ will ue Bh'.g.D\ in which /*' denotes the height of this new column. But, fr. m Ms* riotte's law, we have ♦ B (b - ■ x - a) : B(b — x) : : B h' g D : Bhg D\ whence, b — x — a h' = h> b — x But there is an equilibrium between the pressure of the external air and thr«t of the rarefied air between the limits X and A f , 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 ; 6 — x or, clearing the fraction and solving the equation in reference to #, we find • x - £6 ± $ ^b 2 - 4 ah. (737) When x has a real value, the water will cease to rise, but x will be real as long as b 2 is greater than 4 a h. If, on the con- trary, 4 a h is greater than b 2 , 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. : — 4 a h > 6 2 , or, b 2 a > 4h that is to say, the play of the -piston must be greater titan 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 ti'ues the height to which the atmospheric pressure at tlie place, where the pump APPLICATIONS. 493 is used, will 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 13J, this being the specific gravity of mercury referred to water as a standard, and the product will give the value of .A in feet. Example. — Required the least play of the piston in a sucking- pump intended to raise water through a height of 13 feet, at a place where the barometer stands at . 28 inches. j Here Barometer, b = 13, and b 2 = 169. *n. 2S — = 2,333 feet. 12 . ft. h = 2,333 X 13,5 = 31,5 feet. Play b 2 = « >77 169 ft 4A 4 x 31,5 = 1,341 + ; I 7* V :, M that is, the play of the piston must be greater than one and on third of a foot. i § 392. — The quantity of work performed by the motor during the delivery of water through the discharge-pipe, is easily computed. Sup- pose the piston to have .any position, as M, and to be moving upward, the water being at the level LL 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, increased bv the weight of the column of water MP', whose 1 height is e', and whose base is the area B of the piston \ that is, the pressure upon the top of the piston will be A + Bc'gD, in which g and D are the force of gravity and density of the water, respectively Again, the pressure upon the undei surface of the x JV 494 ELEMENTS Of 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 JVM below the piston, and jf 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 + B c' g B — (A - B eg D) = B g D (c + c') ; but, employing the notation of the sucking-pump just described, C + C' = ft; whence, the foregoing expression becomes Bb.g.B-, 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 an} useful effect ; that is, F = BbgD + we shall find, by exactly the same process, w Bp z t , for the quantity of work of the motor during the descent of "he piston ; and hence the quantity of work during an entire double stroke will be the sum of these, or w Bp (z' 4- z t ). But z' -f- z t is the height of the point of delivery P above the surface of the water in the reservoir ; denoting this, as before, by 6, we have w Bpb ; and calling the number of double strokes n, and the whole quantity of work Q, we finally have Q = nw Bpb. (730) b If we make z t = z\ or 6 = 2^, which will give z t == — » 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. 4913 the water in the reservoir and point of delivery, and being wholly relieved during the descent, when tlje l>ad is thrown upon the sleeping-valve and its box, the work becomes variable, and the motion irregular. THE SIPHON. g 396. — The Siphon is a bent tube of unequal branches, open ai both ends, and is used to convey a liquid from a higher to a lower level, over an in- termediate point higher than either. Its parallel branches being in a vertical plane and plunged into two liquids whose upper surfaces are at L M and L' J\f, 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 0. 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' M' 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 atmospheric pressures upon the surfaces L M and U 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 coun- teracted 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 longer column will therefore be more counter- acted than that opposed to the weight of the shorter, thus leaving 500 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 at- mospheric 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 M and the bend ; by A/ the difference of level between the same point 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 M, be A — ah D g\ and that which tends to force the fluid up the branch immersed in the other reservoir, be A — ah' D g ; and subtracting the first from the second, we find aDg(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, aiid 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 alD\ and if we denote the velocity by V, we shall have, for the living force of the moving mass, alD. F 2 ; whence, aDg{h' — h)lz= ; and, V = -/20(A'-A); 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. 501 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 furnished with internal hoops to prevent its collapsing by the pres- sure 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 atmos- phere 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 0, over which the water is to be conveyed, should never exceed the height to which water may 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 pro- perty, it may be rarefied and brought to almost any degree of teru- ity. 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 1st. A Receiver R, or chamber from which the exterior air is ex- cluded, that the air within may be rarefied. This is commonly a bell-shaped glass vessel, with ground edge, over which a small quaj tity of grease is smeared, that no air may pass through any remain 31 502 ELEMENTS OF ANALYTICAL MECHANICS. fif»M^^ P **»i<»^iAtt!)AMa ing inequalities on its surface, and a ground glass plate m n imbedded in a metallic table, on which it stands. 2d. K Barrel B, or chamber into which the air in the reservoir is to expand itself. It is a hollow cylin- der of metal or glass, connected with the receiver R by the commu- nication 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 re- ceiver is estab- lished ; the second MMtesfo ■-* y\\\'\\»\^m%%^^to^- 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. 503 direction indicated by the arrow-head, through the 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 occu- pied the receiver alone. Turn the cock a quarter round, the com- munication between the receiver and barrel is cut off, and that be- tween 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 :he 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 bar- rel, 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 , '"„ J) . r+ P . r -\- p -\- b The cock being turned a quarter round, the piston pushed back to 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 : : /> . — 1±£^ : *'; in which x' is the density after the second backward motion -of fk« piston, or after the second double stroke ; and we find 504 ELEMENTS OF ANALYTICAL MECHANICS. x -»-c r 4- p *■ + p + fc and if n denote the number of double strokes of the piston, and g a the corresponding density of the remaining air, then will V x. V 4- » + fr/ + /> + 6> 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-pump. 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 — 2 . 30 "" 3 * T -f- p + b and suppose 4 double strokes of the piston; then will n — 4, and t^y m W = Jf.=..Mw. ***** 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 gauges. These (&JR not only indicate the condition of the air in the receiver, but also warn the operator of any leakage that may take place either at the edge of the receiver or ill 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 suffi- cient 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 ew\ is £S^ APPLICATIONS. 505 filled with mercury. A scale of equal parts is placed between the branches, having its zero at a point midway from 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 communicales 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 in- stead 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 if LB=£Q tras that by which a portion of the air was retained between the piston and the bottom of the barrel. This intervening space is filled with air of the ordinary density at each descent of the piston ; 506 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 rarefac- tion 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 can 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 im- provements has been to diminish this space as much as possible. A B is a highly polished cylinder of glass, which serves as the bar- rel 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 (76, which is toothed, is elevated and depressed by means* of a cog-wheel turned by the handle M. If a thin film of oil be poured upon 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 abso- lute contact with the cock E\ this space is therefore rendered indefi nitely small, the oo?ing of the oil down the barrel contributing still further to lessen it. The exchange-cock E has the double bore already described, and is turned by a short lever, to which motion is communicated by a -rod c d. The communication G H is carried to the two plates / and K. on one or both of which receivers ma) be placed ; the two cocks N and 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 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 I. THE TENACITIES OF DIFFERENT SUBSTANCES, AND THE RESISTANCES WHICH THEY OPPOSE To DIRECT COMPRESSION.— See §269. SUBSTANCES EXPERIMENTED ON. Wrought-iron, in wire from l-20t!i i to 1--301 li of au inch in diame- > tor ) in wire, l-10th of iin inch • • • in bars, Russian (mean) • English (mean) • hammered rolled in >lieets, and cut length- » wise J ditto, cut crosswise • • • in chains, oval links 6 in. clear, * iron Is in. diameter • • ditto, Bruntou's, with stay across I link j Cast lion, quality No. 1 2 . . . • 3 • • • • Steel, cast cast and tilted blistered and hammered • shear raw Damascus • ditto, once refined .... ditto, twice refined .... Copper, cast hammered slicct wire Platinum wire ........ Silver, cast wnc Gold, cast • •••••••• wire Brass, yellow (tine) Gun metal (hard) Tin, ca>t wire • • • Lead, cast • miiled sheet ' wire c - 2 X fl s C = 3 >J5 9 6o to 91 36 to 43 27 3o 14 id 2U 6 6 6 25 to 7J to 8 to qJ 44 60 5oi 57 ' 5o 3i 36 44 Sh i5 21 27J- 17 id 9 a 8 16 2 3 4-5ths 14 I|I w ■: c s S.= Lame Telford Lame Brunei Mitis Bro vn Barlow Hodgkinson Mitis Rennie Mitis Rennie Kingston Guyton Rennie Tredgold Guyton ■ C' . g — t £ ? fcl ■ s" - ■g 6" >= sou = - ■- t&- 2. * 2 «< a. 38 to 41 37 to 48 5i to 65 52 46 73 7 3* Hodgkinson Rennie •The stronger quality 01 cast iron, isji Scotch iron known as the Devon Hot Blast, No. 3: its tenaci- ty Is 9J tons per square inch, and its resistance to compression 65 tons. The experiments of Major Wade on the gun Iron at VVesi Point Foundry, and at Boston, give results as high as 10 to 16 tons, and on small cast bars, as high as 17 tuns. — See Ordnance Manual, 1850, p. 402 >AULE I. 509 TABLE I — continued. SUBSTANCES EXPERIMENTED ON. ■) Stone, shite (Welsh) • Marble (white) • • Givry .... Portland .... Craigleith freestone Bra m ley Fall sandston Cornish granite Peterhead ditto Limestone (compact blk) Pui beck .... Aberdeen granite • Brick, pale red . • • red Hammersmith (pavior ditto (burnt) • Chilk • • • « [Plaster of Paris • • • 61ftg<& nlute .... Bone (ox) .... Hemp fibres glued aether Strips of paper gluea together Wood, Box, spec, gravity Ash Teak Beech .... Oak Ditto Fir Pear Mahogany • • • Elm ... • • Pine, American Deal, white • • • ,862 ,6 ,9 ,7 ,92 f ,646 ,63 7 a -8 C r. — CCS 5,7 4 1 ,13 ,o3 4 2,2 41 i3 5 4 5 4* 3* 6 6 6 X W n c - ■! z 8. Barlow ■ c . SI - — • £ * - o£2L M 1,6 2,4 2,8 3,7 4 4 5 ,56 ,8 I M ,22 «,7 ' 5 ? .7* e W v.' c s £* Kcnnie 510 TABLE II. TABLE H. 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. (Jahrbiu des Polytechnischen Institutes in (Vein, Bd. 16, S. 70). t Temperature. Density. Din". V Volume. Diff. o 32. oo 0,999887 < i,oooii3 34,25 0,999950 63 1 ,oooo5o 63 36,5o 0,999988 38 1,000012 38 38,75 1 ,000000 12 1 ,000000 12 4i,oo 0,999988 12 1,000012 12 43,25 0,999952 0,999894 35 1 ,000047 35 45, 5o 58 1,000106 £ 47,75 0,999813 81 1,000187 81 5o.oo 0,999711 102 1.000289 102 52,25 0,999587 124 1. ooo4i3 124 54, 5o 0,999442 i45 i,ooo558 145 56,75 0,999278 164 1,000723 1 65 59,00 0,999095 1 83 1 ,000906 1 83 61, 25 0,998893 202 1,001108 202 63.5o 0,998673 220 1, 001329 221 65, 7 5 0,998435 238 1, 00 1 567 238 68,00 0,998180 255 • 1,001822 255 70,25 0,997909 271 1,002095 . 2 7 3 72,5o 0.997622 287 1,002384 289 74,75 0,997320 3o2 1,002687 3o3 77,00 0,997003 3i 7 1 .oo3oo5 3i8 79,25 0,996673 33o i,oo3338 333 81, 5o 0,996329 344 i,oo3685 347 83,75 0,993971 358 1 ,004045 36o 86,00 0,995601 370 1,004418 373 88,25 0,995219 382 1 ,004804 386 90,50 0,994825 394 I,O0D2O2 3 9 8 92,75 0,994420 4o5 i,oo56i2 410 95,00 0,994004 416 i,oo6o32 420 97,25 0,993575 425 1,006462 43o 99, 5o 1 0,993143 434 1,006902 440 With this table it is easy to find the specific gravity by means of water at any temperature Suppose, for example, the specific gravity S' in Equation (456), had been found at the tempera- lure of 59°, then would D,i 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' X 0,999095 TABLE III. 511 TABLE in. F THE SPECIFIC GRAVITIES OF SOME OF THE MOST IMPORTANT BODIES. (The density of distilled water is reckoned in this Table at its maximum 38J° F. = 1 ,000]. Name of the Body, Specific Gravity. I. SOLID BODIES, (1) Metals. Antimony (of the laboratory) Brass .... Bronze for cannon, according to Lieut. Mttzka Ditto, mean • Copper, melted Ditto, hammered • Ditto, wiYe-drawn • Gold, melted • Ditto, hammered Iron, wrouglit Ditto, cast, a mean • Ditto, gray • • Ditto, white • Ditto for cannon, a mean Lead, pure melted • Ditto, flattened Platinum, native Ditto, melted • Ditto, hammered and wire-drawn Quicksilver, at 32° Fahr Silver, pure melted Ditto, hammered • Steel, cast • • Ditto, wrought Ditto, much hardened Ditto, slightly Tin, chemically pure Ditto, hammered Ditto, Bohemian and Saxon Ditto, English Zinc, melted • Ditto, rolled • (2) Bou.din» Stones Alabaster Basalt • • . Dole rite • Gneiss • Granite • Hornblende Limestone, various Phonolite • Porphyry Quartz • • Sandstone, various Stones for building Syenite • Tract iy te Brick kinds kinds, a mean 4,2 — 4,7 7,6 — 8,8 8,4U — 8,974 8,758 7,788 8.878 - 8,726 - 8.9 8,78 ig.238 — 19,253 io,36i — 19>° 7.207 - 7,7^ 7,25i 7,2 7,5 7,21 — 7;3o n,33o3 u,388 16,0 — 18,94 2o,855 21,25 I 3, 568 — i3,5 9 8 10,474 io,5i — 10,622 7,? ! 9 7,840 7,818 7,833 7,291 7,299 - 7,475 7,3 1 J 7,291 6,861 — 7,2l5 7>«9« 2 »2 — 3,0 M - 3,i 2,72 - 2, 9 3 2,5 — 2,9 2,5 — 2,66 2,9 - 3,i 2,64 — 2.72 »,5i — 2,69 2,4 — 2,6 2,56 = \f 2,2 1,66 — 2,62 =,5 — 3. 2,4 - 1,6 i,4> — i,86 512 TABLE III. TABLE III— Continued Name of the Body.'' ]' Specific Gravity. I. SOLID BODIES. (3) Woods. ' - v * .... Fresh-fH'.ecl. Pry. ; AMer • • < 0,8371 o,5ooi Ash • • < o 9036 6440 Aspen • • < * ■ ' *■" t 0.7654 0.4302 Birch • • < 0,9012 0,6274 Box • • < 0,9822 0,5907 Elm • • < 9476 0.5474 o,555o Fir > • • 0,8941 Hornbeam • « 0,9452 0,7695 , Horse-chestnut < 0,8614 0,5749 : Larch • • < 0,0206 0,4735 Lime • • « 0,8170 0,4390 Maple • • ■ 0.9036 0.6592 Oak 1,0494 0,6777 Ditto, another specimen < 1.0754 0.7075 Pine, Pinus Abies Picea < 0,8699 0,4716 Ditto, Pinvs Sylvestris « 0,9121 o,55o2 Poplar (Italian) • < 0,7634 0.3931 Willow < 0,71 55 0,5289 0,4873 Ditto, white • • < . ' 0,9859 (4) Various Solid Bodiis. ' Charcoal, of cork 0,1 Ditto, soft wood • 0,28 — o,44 Ditto, oak .... 1,573 1 ! Coal • • « 1.232 — I,5l0 Coke • 1,865 Earth, common • • « 1,48 rough sand • • • 1,92 rough earth, with gravel • 2,02 moist sand • . 2,o5 gravelly soil 2,07 2.l5 clay ..... • clay or loam, with gravel < 2,48 Flint, dark • • 2,542 Ditto, white .... 2,741 > Gunpowder, loosely filled in coarse powder » • < 0,886 nmsket ditto • • < 0.992 ' Ditto, sliffhtly shaken down musket-powder 1,069 Ditto, solid .... 2,248 — 2,563 Ice .... ► • < 0,916 — 0,9268 Lime, unslacked » • 1.842 Resin, common > • 1,089 Rock-salt • < ft • 2.257 Saltpetre, melted > • 2,743 Ditto, crystallized i • 1,900 Slate-pencil i • 1,8 — 2,24 Sulphur « • 1.92 — 1,99 Tallow • I • 0,942 Turpentine • • 0.991 Wax, white • l • 0.969 Ditto, yellow ♦ • o.o65 Ditto, shoemaker's *••••••• 0,897 TABLE III. 513 TABLE III— Continued. Name of the Body. Specific Gravity. II. LIQUIDS. Acid, acetic l,o63 Ditto, muriatic 1,211 Diito, nitric, concentrated 1,521 — 1,522 Ditto, sulphuric, Enirlish 1.845 Ditto, concentrated (Nordh.) I,86o Alcohol, free from water • • . O.792 Ditto, common O.824 — 0,79 Ammoniac, liquid 0,875 Aquafortis, double i,3oo Ditto, single • 1,200 Beer • • 1 023 — i,o3i Eiher, acetic • 0.866 Ditto, muriatic 0,845 — 0,674 Ditto, nitric • < 0,886 Ditto, sulphuric 0,715 Oil, linseed , > 0.928 — 0,953 Ditto, olive • « 0.915 Ditto, turpentine ■ 0,792 — 0,891 Ditto, whale • « 1 0.92J Quicksilver • < i3.568 — 13,098 Water, distilled < 1.000 Ditto, rain • « 1 .00 1 3 Ditto, sea • « i,0265 — 1,028 Wine • • < * . • 0,992 — i,o38 III. GASES. Kriroinetei W;.ter = 1. 3D !•.. , . . i Temp. 38JO F. rrtyjsJQp Atmospheric air = . i Q — • ' » • 0,00 i3o 1 ,0000 Carbonic acid eras .... » • 0.00198 1,524c Carbonic oxide gas .... > - , » . 0.00126 0,9069 Carbureted hydrogen, a maximum < . 0,00127 00784 Ditto, from Coals • • • : i o,ooo3o 0,0008a 0.J000 0.5596 Chlorine •"•■••« 0,00321 2,4700 Hydrio.lic gttfl ••"«•« 0.00577 4.4430 Hydrogen •••••« 0.0000895 0,0688 Hydrosiilphuric acid gas • • < , . o.ooi55 I. 1912 Muriatic acid gas .... 00162 1.2474 Nitrogen ..... • 0,00127 0,9760 ; Oxygen •••••« 0,00143 1,1026 Phospliureted hydrogen gas • < 0.001 13 0,8700 Steam at 212° Fahr. 0.00082 0.6235 Snlrhurous acid gas • < » . 0,00292 2,2470 p" 1 - TABLE IV. TABLE IY. TABLE FOE FINDING ALTITUDES.-See § 284. Detached Thermometer. • t t +t> A t, + t' A A 40 4,7689067 75 4,7859208 no 4,8022936 145 4,8180714 4i ,7694021 76 ,7853973 in ,8027525 146 ,8i85i4o 42 ,7698971 77 ,7868733 112 ,8032109 147 ,8189559 43 ,770391 1 78 ,7873487 u3 ,8036687 148 ,8193973 44 ,77o885 I 79 ,7878236 114 ,8041261 149 ,8198387 45 , 77 i3 7 85 80 ,7882979 n5 ,8o4583o i5o .8202794 46 ,7718711 81 ,7887719 116 ,8000393 i5i ,8207196 47 ,7723633 82 ,7892451 U7 ,8o54953 l52 ,b2II394 48 ,7728548 83 ,7897180 118 ,8039309 i53 ,8213958 ,8220377 49 ,7733457 84 ,7901903 119 ,8o64o58 i54 5o ,7738363 85 ,7906621 120 ,8068604 1 55 ,8224761 5i ,7743261 86 ,7911335 121 ,8073144 • 106 ,8229141 5a ,7748i53 87 ,7916042 122 ,8077680 1 5 7 .8233317 53 ,7753o42 88 ,7920745 123 ,8082211 1 58 ,6237888 54 ,7757925 89 ,7925441 124 ,8086737 139 .6242236 55 ,7762802 90 ,793oi 35 125 ,8091238 160 ,8246618 56 ,7767674 9 1 ,7934822 126 ,8095776 161 ,8230976 57 ,7772540 92 ,7939504 127 ,8100287 162 ,b25533i 58 ,7777400 93 ,7944182 128 ,8104795 i63 ,6239680 5 9 ,7782256 94 ,7948854 129 ,8109298 164 ,6264024 6o ,7787105 95 ,7953521 i3o ,8113796 163 ,8268365 6i ,7791949 96 ,7958184 i3i ,8118290 166 ,6272701 6a ,7796788 97 ,7962841 132 ,8122778 167 ,8277034 63 ,7801622 98 ,7967493 i33 ,8127263 168 ,82Si362 64 ,7806430 99 ,7972141 i34 ,8131742 169 ,8285685 65 ,7811272 100 ,7976784 i35 ,8i362i6 170 .8290005 66 ,7816090 101 ,7981421 i36 ,8140688 Hi ,8294319 u ,7820902 102 ,7986034 137 ,8i45i53 172 ,829^-629 ,7825709 io3 ,799068 1 i38. ,8149614 173 ,8302937 69 ,783o5u 104 ,79953o3 • i3 9 ,8154070 174 ,8307238 70 ,78353o6 io5 ,7999921 ■ 140 ,81 58523 175 ,83 11 536 71 ,7840098 106 ,8oo4333 141 ,8162970 176 .83i583o Ti ,7844883 107 ,8009142 142 ,8167413 177 ,8320119 73 ,7849664 108 ,8013744 143 ,8171852 178 ,8324404 74 4,7854438 109 4,8oi8343 144 4,8176285 179 4,8328686 TABLE IV. 515 TABLE YV— continued. WITH THE BAROMETER.— See § 284. Latitude. Attached Thermometer. ¥ B T— T' c c# 0° 0,0011689 — + 3 ,0011624 0° 0,0000000 0,0000000 6 ,001 1433 I ,0000434 9,9999566 9 ,0011 1 17 2 ,0000869 ,99991 3 1 12 ,0010679 3 ,oooi3o3 ,9998697 i5 ,0010124 4 ,0001738 ,9998263 18 ,0009439 5 ,0002172 ,9097829 ,9997395 21 ,0008689 6 ,0002607 24 ,0007820 7 ,ooo3o4r ,9996961 27 ,0006874 8 ,0003476 ,9096527 3o ,0000848 9 ,0003910 ,9996093 33 ,0004738 10 ,0004345 ,9995659 36 ,ooo36i5 11 ,0004780 ,9995225 39 ,ooo2433 12 ,ooo52i5 ,9994792 42 ,0001223 i3 ,ooo565o ,9994358 45 ,0000000 14 ,0006084 ,9993924 48 . 9'999 8 77 5 i5 ,0006319 ,9993490 49 ,9998372 16 ,ooo6q54 ,0993057 5o ,9997967 17 ,0007389 ,9992623 • Si ,9997366 18 ,0007824 ,9992190 52 53 ,9997167 ,9996772 19 20 9 ,ooo82 5o ,0008696 ,9991756 ,9991 323 54 ,9996381 21 ,0009 1 3c ,9990889 55 ,9990993 Ti ,0009565 ,9990456 56 ,9995613 23 ,00 1 0000 ,9990023 57 ,9995237 24. ,ooio436 ,9989589 58 ,9994866 25 ,0010871 ,9989 1 56 59 ,9994502 26 ,001 i3o6 ,9988723 6o ,9994144 27 ,00 i 1742 ,9988290 63 ,999.3 1 1 5 28 ,0012177 ,9987867 66 ,9992161 29 ,ooi26i3 ,9987424 2 ,9991293 ,9989832 3o 3i ,00 1 3o48 0,00 1 3484 ,9986991 9,9986558 ^9988834 * 90 9,9988300 I 516 TABLE V. TABLE V. COEFFICIENT VALUES, FOR THE DISCHARGE OF FLUIDS THROUGH THIN PLATES, THE ORIFICES BEING REMOTE FROM THE LATERAL FACES OF THE VESSEL.— See § 300. • Head of fluid above the centre of the orifice, in feet. Values of the coefficients for orifices whose smallest dimensions or diameters are — ft- 0,66 ft- o,33 ft- 0,16 ft. 0,08 ft- 0,07 ft o,o3 o,o5 0,07 o,i3 0,20 0,26 o,33 0,66 1,00 1,64 3,28 5,oo 1 6,65 32,75 0,593 0,596 0,601 0,602 o,6o5 o,6o3 0,602 0,600 0,592 0,602 0,608 o,6i3 0,617 0.617 o,6i5 0,612 0,610 0,600 0,618 0,620 0.625 o,63o o,63 r o,63o 0,628 0,626 0,620 0,61 5 0,600 0,627 o,632 0,640 o,638 o,63i o634 o,632 o,63o 0,628 0.620 o,6i5 0,600 0,660 0,657 o.656 o,655 o,655 0.604 0.644 0.640 o,633 621 0,610 0,600 0,700 0.696 0.6S5 0.677 0.672 0.667 o,655 o,65o 0,644 o.632 0.618 0,610 0,600 In the instance of gas, the generating head is always greater than 6,63 fu, and the coefficient 0,6 «r 0,61, is taken in all cases. For orifices larger than 0,66 ft., the coefficients are taken as for this dimension ; for orifices smallei *han 0,03 ft., the coefficients are the same as for this latter; finally, for orifices between those of tht table, we lake coefficients whose values are a mean between the latter, corresponding to the given head TABLE VI. 517 TABLE VI. EXPER MENTS ON FRICTION, WITHOUT UNGUENTS. BY If. MORIN. The surHtcea of friction were varied from o,o3336 to 2,7987 square feet, the pressures from 88 lbs. to 22 >5 lbs., and the velocities from a scarcely perceptible motion to 9,84 feet per second. Tne surfaces of wood were planed, and those of metal filed and polished with the greate>t care, and carefully wiped after every experiment. The presence of unguents was •specially guarded agaiust.— See § 855. SURFACES OF CONTACT. Oak upon oak, the direction of the fibres beiiiir parallel t<> the motion • • Oak upon oak, the directions of the fibres of tne moving surface being perpen- dioular to those uf the quiescent sur- face and to the direction of the motion.}: Oak upon oak, the fibres of the both sur- faces being perpendicular to the direc- tion of the motion Oak upon oak, the fibres of the moving surface being perpendicular t»> the sur- face of contact, and those of the surface at rest pantile! to the direction of the motion Oak upon oak. tliefinresof both surfaces being perpendicular to the surface of contact, or the pieces end to end • Elm upon oak, the direction of the fibres behi in mat tct. , , t Th- diunn in. is of I be surfaces of contact were In this experiment .947 square feet, ard the results were nearlv uniform. When thedimensions were iliminisheil to .043 a tearing of tie fit re Im 1 ine appa- rent in thee -e of mntin:i, and there were symptoms of the e m w l i— I IW of the wood: Ironi IIM cir eumst nee- there re-nlt-d «a irregularity in the friction indicative of BSCSM Iw pTWMWB. $ It is worthy of rem-. rk that tie friction of oak ii|M»n elm i* Inn five-i ir.tb* ol ih.tof elm ii| H, n oak. || In the experiments in which me of the surfaces w.is of Met I. Mini I particles of the metal began, after a time, 10 be appoeni ii|mmi the wood, giving it a polished m-iallic appaaraar* these were at every experimen \\ip of the motion ) Hornbeam upon cast iron — fibres paral- i lei to motion j Wild pear-tree upon cast iron — fibres j parallel to the motion • • • • • \ Steel upon cast iron Steel upon brass ■ • Yellow copper upon cast iron .... Ditto oak .... Brass upon cast iron Brass upon wrought iron, the fibres of) the iron being parallel to the motion • j Wrought iron upon brass Brass upon brass Black leather (curried) upon oak* • Ox hide (such as that used for soles and J for the stuffing of pistons) upon oak, V rou 12 i5 9 9 9 46 1 1 22 14 5i 27 29 18 3i 16 3o 27 29 17 45 27 29 Friction or Quiescknck. c s c 3 0.649 0,194 0,137 0,64^) 0,162 0,617 0,74 o,6o5 o,43 • • 0,64 o,5o o,79 ■ ttTT a C "* 23 2'S * = W.J .2 = 0> 33° o' 10 59 7 49 32 52 9 i3 3i 41 36 3i 3i 11 23 17 • • 32 38 26 34 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 ap- parent in the rough and the smooth skins. t All the above ex|ieriments, except that with curried black leather, presented the phenomenon of <» change in the polish of the surfaces of friction — a state of their surfaces necessary tn, and dependent apon. their motion upon one another. TABLE VI. 519 TABLE Yl—contii ued. SURFACES OF CONTACT. Calcareous oolitic stone, used in building, of a moderately hard quality, called stone of Jaumont — upon the same stone Hard calcareous stone of Brouck, of a light gray color, susceptible of taking a tine polish, (the mu>chelkalk,) mov- ing upon the same stone The soft stone mentioned above, upon the hard The hard stone mentioned above upon the soft Common brick upon the stone of Jaumont Oak upon ditto, the fibres of the wood being perpendicular to the surface of the stone • Wrought iron upon ditto, ditto • Common brick upon the stone of Brouck Oak as before (endwise) upou ditto • lion, ditto ditto • • Friction of Motion. — c c S 6*N .= = V 3a° 38' 20 49 33 2 33 5o 33 2 20 49 34 37 3o 58 20 49 i3 3o Friction of Ul'ltMlt.M t. 2 J 0* o,74 0,70 0.75 0,75 o,65 o,63 0,49 0,67 0,64 0,42 ■ B -1 -- s - -i /. ~ - o> •— ^. — 36° 3i' 35 36 53 36 53 33 2 32 i3 26 7 33 5o 3a 38 22 47 520 TABLE VIL TABLE VII. EXPERIMENTS ON THE FRICTION OF UNCTUOUS SURFACES. BY M. MORIN.— See §£55. In these experiments the surfaces, after having been smeared with an unguent, \ver« wiped, so that no interposing layer of the unguent prevented their intimate contact. Friction of Friction or SURFACES OF CONTACT. Motion. Q.11KSCKNCK. *j — u •« s ■ £ -5 Z ■- •5 : US. i 2* t^ = »"3 X ■■ = 8 "t r* *z r ^ '~ *? ae w 3 w w Oak upon oak, the fibres being parallel t< the motion .... ?( 0,108 6° 10' 0,3gO 21° 19' Ditto, the fibres of the moving body be intr perpendicular to the motion* " £ °>'43 8 9 0,3 14 17 26 Oak upon elm, fibres parallel' o.i 36 7 45 Kun upon oak, ditto 0,119 6 48 0,420 22 47 beech upon oak. ditto • • • o,33o 18 16 Elm upon elm, ditto 0, 1 40 7 5 9 Wrought irou upon elm, ditto • o.i38 7 52 Ditto upon wrought iron, ditto 0.177 10 3 Ditto upon cast iron, ditto • • • 0,Il8 6 44 Cast iron upon wrought iron, ditto 0.143 8 9 Wrought iron upon brass, ditto • 0.160 9 6 Brass upon wrought irou 0.166 9 26 Cast iron upon oak, ditto 107 6 7 0,100 5 43 Ditto upon elm, diuo, the unguent beiiu tallow ..... T ( f O.I 2D 7 8 Ditto, ditto, the unguent being hog': j' °> ,3 7 lard and black lead • 7 49 Elm upon cast iron, fibres parallel • o.i35 7 42 O,093 5 36 Cast iron upon cast iron • • 0,144 8 12 Ditto upon brass . 0.1 3a 7 32 Brass upon cast iron • 0.107 6 7 Ditto upon bra>s • • 0.1 34 7 3S 0,l64 9 '9 Copper upo i oak • • 0.100 5 43 Yellow copper upon cast iron 0.1 15 6 34 Leather (ox hi le) well tanned upon cast iron, wetted • ' ' 0,229 12 54 0,267 U 57 Ditto upon brass, wetted • • 0,244 i3 43 TABLE VIII. 521 TABLE VIII. EXPERIMENTS ON FRICTION WITH UNGUENTS INTERPOSED. BY M. MORIN. Tlie extent of the surfaces in these experiments bore such a relation to tlie pressure, a* to cau-e tlicin to be separated from one another throughout by an interposed stratum of the unguent.— See § 855. SURFACES OF CONTACT. Oak upon oak, fibres parallel Ditto ditto Ditto ditto Ditto, fibres perpendicular Ditto ditto Ditto ditto Ditto upon dm, fibres parallel Ditto ditto Ditto ditto Ditto upon cast iron, ditto Ditto upon wi ought iron, ditto Beech upon oak, ditto Elm upon oak, ditto • Ditto ditto • ♦ Ditto ditto Ditto upon elm, ditto • Ditto upon cast iron, ditto Wrought iron upon oak, ditto Ditto ditto ditto • Ditto ditto ditto • Ditto upon elm, ditto • Ditto ditto ditto ■ Ditto ditto ditto • Ditto upon cast iron, ditto Ditto ditto ditto • Ditto ditto ditto • Ditto upon wrought iron, ditto Ditto ditto ditto • Ditto ditto ditto - Wrought iron upon brass, fibre Kiraliel •'•"!•• itto ditto ditto • Ditto ditto ditto • Cast iron upon oak, ditto • Ditto ditto ditto • Ditto ditto ditto • Ditto ditto ditto • Ditto ditto ditto • Ditto upon elm, ditto • Dit'o ditto ditto • Ditt5 ditto ditto • Ditto, ditto upon wrought iron Cast iron upon cast iron • Ditto ditto Friction or Motion. - ■ 0,164 0,073 0,067 o,o83 0,072 0.230 o. i36 o 073 0.066 0.080 0.098 o,o55 0.137 0.070 0.060 0.139 0,066 o,256 0.214 o.o85 0,078 0.076 o.o55 o,i o3 0.076 0,066 0,082 o,©8i 0,070 o,io3 0.075 0.078 0,189 0,218 0,078 0,075 0,075 0.077 o 061 0,091 o,3 1 4 0.197 Friction OF QulESCKNCK. e c 5 c fc. 0,440 0,164 • • 0,254 0,178 0,411 0,142 0,217 • • 0,649 0,108 0,100 • • o,n5 • • 0,646 0,100 0,IOO • • O,I0O • • UNGUENTS. Drv soap. Tallow. lh'g"s lard. Tallow. Hog's lard. 'W ater. Drv soap. Tallow. Hog's laid. Tallow. Tallow. Tallow. Dry soap. Tallow. Hog's lard. Drv soap. Tallow. (Greased, nnd saturated with water. Drv soap. Tallow. Tallow. Hog'.- lard. Olive oil. Tallow. Hog's lard. Olive oil. Tallow. 1 loir's lard. Olive oil. Tallow. Hotr's lard. Olive oil. Drv soap. (Greased, and saturated with water. Tallow. Ilo74 posed, after from 10 to 15 min- utes' cortact. TABLE IX 523 TABLE IX. FRICTION OF TRUNNIONS IN THEIR BOXES.-See § 361. KINDS OF MATERIALS. Trunnions of cast iron and boxes of cast iron. STATE OF SURFACES. Trunnions of cast iron and boxes of brass. Trunnions of castiron and boxes of liguum-vitee. Trunnions of wrought iron and boxes of cast iron. Trunnions of wrought iron and boxes of brass. Trunnions of wrought iron ( and boxes of lignum-vi- -J tee. { Tr-i unions of brass and ( boxes of brass. 1 Trunnions of brass and ) boxes of cast iron. J Trunnions of lignum-vitse j and boxes of cast iron. ) Trunnions of lignum-vitse i and boxes of lignum- > vita. J Unguents of olive oil, hogs' lard, and tallow .... The same unguents moistened with water ..... Unguent of asphaltum Unctuous ..... Unctuous and moistened with wa- ter ..... Unguents of olive oil, hogs' lard, and tallow .... Unctuous ..... Unctuous and moistened with wa- ter ..... Very slightly unctuous Without unguents • Unguents of olive oil and hogs' | lard • • • . , Unctuous with oil and hogs' lard Unctuous with a niixture of hours' lard and plumbago Unguents of olive oil, tallow, and hogs' lard .... Unguents of olive oil, hogs' lard, and tallow .... Old umruents hardened • Unctuous and moistened with wa- ter ..... Very slightly unctuous Unguents of oil or hogs' lard • Unctuous • . • . • Unguent of oil- Unguent of hogs' lard Unguents of tallow or of olive oil Ungrucnts of hogs' lard Unctuous. .... Unguent of hogs' lard Ratio of friction to pressure when the unguent is renewed. By the onlinnry method. i r i I 0,08 ) 0,08 o.o54 0,14 o.ii ( 0,07 I 0,07 i to \ 0.08 ) 0,16 0.16 0,10 0,l8 O.I9 0,23 0,11 O.I9 o. 10 0.09 0,12 0,1 5 Or. con- tinuously o,o54 o,o54 0,034 • • o,o54 • • 0,090 o,o54 o,o54 o.o.'»5 to o,o5a ) ! o,c7 524 TABLE X. TABLE X. OF WEIGUTS NECESSARY TO BEND DIFFERENT ROPES AROUND A WJ EEL ONE FOOT IN DIAMETER.-- See § =357. No. 1. White Ropes — new and dry. Stiffness proportional to the square of the diameter. Diameter of rope in inch s. Natural stiffness, or value of K. 1 Stiffness for load of] 1 lb., or value of /. 0.39 79 1,57 3,i5 lbs. 0,4024 1,6097 6,4389 25,7553 lbs. 0.0079877 o,o3i95oi 0,1278019 o,5i 12019 No. 2. White Ropes- -new and moistened with water. Stiffness proportional to square of diameter. Diameter of rope in inch.es. Natural stiff ess, or value of K. Stiffness for load ol 1 lb., or value of /. 0,39 0,79 1,57 3,i5 lbs. O.8048 3,2194 12.8772 5i,5iii lbs. 0,0079877 o,o3i95oi 0.1278019 o,5i 12019 1 No. 3. White Ropes — half worn and dry. Stiffness proportional to the square root of the cube of the diameter. Diameter of ro|ie in inches. Natural Stiffness, or value of K. Stiffness for load of j 1 lb., or value of /. O.39 0,79 1.5 7 3,i5 lbs. 0.4O243 i,i38oi 3,21844 9 ioi5o lbs. 0.0079877 0,o525889 0,o638794 0,l8o6573 No. 4. White Ropes— half worn and moistened WITH WATER. Stiffness proportional to the square root of the cul/e of the diameter. Diameter of rope in inches. Natural Stiffness, or value of K. Stiffness for load of 1 lb.. or value of I. o,39 0.79 1, 37 3,i5 1 lbs. 0,8048 2 2761 6,4324 18,2037 lbs. 0,0079877 o,o525889 0,0638794 0,1806573 1 Squ 1 res of 1 he ratios ol diameter, or Tal ties of (/%. Squ res Ratios (J. d*. 1,00 1. 00 1,10 1. 21 1.20 1.44 i.3o 1,69 1, 4o 1,96 I.3o 2,25 l,6o 2.56 I.70 2 89 1,80 3.24 I.90 36i 2,00 4,00 Square roots of the cubes of the r t o« of diameter, or v.il . 3 ues ol 2 m Ratios or d. l»o\ver ? or ... (IS .... u> S . , cos 8' = p • — r- ; cos 6" = p- — — ; cosd = p — • • • (1) ds as Is v ' Squaring, adding and reducing by the relation, cos 2 8' + cos 2 6" + cos 2 8"' = 1, we have performing the operations indicated under the radical sign, and redu- cing by the relations d s 2 = d x 2 4- d y 2 -f- d z 2 , d 2 sd s as d 2 x dx -f d 2 y d y + d 2 z d z, we find P ~~ V(^ z) 2 + (d 2 y) 2 +"(# z) 2 - (aP 'is) 2 ' * * ' * ( 2 ) If 5 be taken as the independent variable, then will ^5 = 0, and Eqs. (1) and (2) become Af d 2 x At , d 2 y .... d 2 z ... cos &' = p • — - ; cos 8" = • -5-f ; cosd"' 7= p ■ ^— ; • • (3) r ds 2 r ds 2 ds 1 ds 2 p = . - ; . . . . . (4) ^/{d 2 x) 2 4- (d 2 y) 2 + id 2 z) 2 V ' 528 APPENDIX. No. 1 1 I To integrate the partial differential equation •^ dq da transpose and divide by Z>, and we have dD~ 7 'D'dp' and because q is a function of p and D, we hav& da , ^ da , j and substituting the value of -7-=, a J J . dq D-dp — y'V'dD d i = r P 3 ; multiplying and dividing by y • D • p7 , 1 --' - D'-'p7 'dp — p7-dD dq = *l.ll9__ -1 but dpi £>* p7 m D — p7 -dp—p7'dD M ry D 2 and making 1 dp 1 1 ,:»/' • p7 we may write 1 1 dq: = F f (£>•< (©■ > in which F, denotes any arh itrary function. APPENDIX. 529 No. IV. To integrate Equation (414)' of the text, add t«> both members d*r (p dt.dr' and we have 1 ,fdiy d r y~i a ,rdra> dry~» — .d\ — — + a -j— I = — . d I -— " + a — -? I ; rf* Vdt d r A dr Ldt dr V and making dry d r

> in which /' denotes any arbitrary function. Whence, by addition, 1 630 AFPExm* »u4 by subtraction, But Whence, ■ dr

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