F CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA 5 | ^ ~ )F CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA a * o i OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA /- it * p ?- w v8 r OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIB Y OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA Lll Y OF CUIFORNU LIBRARY OF THE UNIVERSITY OF CALIFORNIA LI Hn :; Y F CALI IF GAL AN ELEMENTARY TREATISE ON MECHANICS, COMPREHENDING THE DOCTRINE OF EQUILIBRIUM AND MOTION. AS APPLIED TO SOLIDS AND FLUIDS, CHIEFLY COMPILED, AND DESIGNED FOR THE USE OF THE STUDENTS OF THE UNIVERSITY AT * CAMBRIDGE, NEW ENGLAND. BY JOHN FARRAR, \\ PROFESSOR OF MATHEMATICS AND NATURAL PHILOSOPHY. CAMBRIDGE, N. E. PRINTED BY HILLIARD AND METCALF, At the University Press. SOLD BV W. HILLIARD, CAMBRIDGE, AND BY CFMMINGS, HILLIARD, & CO. NO. 134 WASHINGTON STREET, BOSTON. 1825. DISTRICT OF MASSACHUSETTS, TO WIT. District Clerk's Office. BE it remembered that on the twenty-ninth day of September, J825, in the fiftieth year of the Independence of the United States of America, Cum- mings, Billiard & Co. of the said district, have deposited in this office the title of a book, the right whereof they claim as proprietors, in the words following, viz : " An Elementary Treatise on Mechanics, comprehending the Doctrine of Equilibrium and Motion, as applied to Solids and Fluids, chiefly compiled, and designed for the use of the Students of the University at Cambridge, New England. By John Farrar, Professor of Mathematics and Natural Philosophy." In conformity to the act of the Congress of the United States, entitled " An act for the encouragement of learning, by securing the copies of maps, charts, and books, to the authors and proprietors of such copies, during the times therein mentioned ;" and also to an act, entitled, " An act supplementa- ry to an act, entitled, 'An act for the encouragement of learning, by securing the copies of maps, charts, and books, to the authors and proprietors of such copies, during the times therein mentioned,' and extending the benefits thereof to the arts ot designing, engraving, and etching, historical and other prints." JNO. W. DAVIS, Clerk of the District of Massachusetts. Q.A801 Fes ADVERTISEMENT. IN selecting materials for this treatise particular re- gard has been had to the practical uses of the science ; at the same time the theoretical principles are rigorously demonstrated. Where the nature of the subject ad- mitted of it, the geometrical method has been pre- ferred, as being more perspicuous and better adapted to most learners. There are many refinements in the later and more improved treatises not adopted in this, for the reasons above mentioned, and on account of the insufficient provision, as to time and preparatory studies, that is made in most of the seminaries of the United States for a text-book upon such a plan. The works principally used in preparing this treatise are those of Biot, Bezout, Poisson, Francoeur, Gregory, Whewell, and Leslie. In the portions selected it was found necessary to make considerable alterations and additions in order to give a uniform character to the whole. There has often been occasion, moreover, in appropriating the substance of a proposition or course of reasoning, to amplify or condense it, or to vary the phraseology. It became inconvenient, therefore, to dis- tinguish by quotations the respective portions taken from different authors. Bezout has been adopted, in substance, as the basis in what relates to statics, dynam- ics, and hydrostatics, although the matter is arranged iv Advertisement. according to a different system ; and Gregory, with many changes and substitutions, has been principally used in that which is comprehended under hydrodynamics. This volume constitutes the first part of a course of Natural Philosophy. The next, containing Electricity and Magnetism, will be published in a few weeks. Cambridge, Sept 24, 1825. CONTENTS. PRELIMINARY REMARKS AND DEFINITIONS 1 STATICS. Uniform Motion ... ; 13 Forces and the Quantity of Motion ; . . 16 Equilibrium between Forces directly opposite . . .18 Compound Motion ..... 20 Composition and Decomposition of Forces . . .25 Moments and their Use in the Composition and Decomposition of Forces ...... 31 Parallel Forces which act in different Planes . . .39 Forces, the Directions of which are neither in the same Plane nor parallel to each other .... 41 Centre of Gravity . . . . . .44 Application of the Principles of the Centre of Gravity to par- ticular Problems . . . . , 51 Properties of the Centre of Gravity . . . .70 General Principle of the Equilibrium of Bodies , . 76 D'Alembert's Principle, and Concluding Deductions . . 77 Application of the Principles of Equilibrium to the Machines usually denominated Mechanical Powers . . 80 Rope Machine . . . . . .80 Lever . .... 91 Pulley . . . . . . .103 Wheel and Axle . . . . . .109 Inclined Plane . . . . .117 Screw ....... 125 Wedge ....... 129 General Law of Equilibrium in Machines . . 131 vi Contents. Friction . . . . . . .136 Stiffness of Cords . . . . . . 156 Balance, Steelyard, 8fc. . . . . . 161 DYNAMICS. Motion uniformly accelerated . . . .165 Free Motion in heavy Bodies . . . . .168 Direct Collision of Bodies . . .175 Direct Collision of unelastic Bodies . . . .176 Force of Inertia . . . . . . 178 Application of the Principles of Collision of unelastic Bodies 181 Collision of elastic Bodies . . . . 185 Motion of Projectiles . . . . .190 Motion of heavy Bodies down inclined Planes . . 205 Motion along curved Sarfaces . . . .410 Motion of Oscillation . . . . .215 Pendulums . . . . . .221 Line of swiftest Descent .' . . . . 225 Moment of Inertia . . . . . 230 Centre of Percussion and Centre of Oscillation . .236 Actual Length of the Seconds Pendulum . . . 246 Measure of the Force of Gravity . . . .249 Application of the Pendulum to Time-Keepers . . 250 Rotation of Bodies unconfined . . * .257 Method of estimating the Forces applied to Machines . 265 Maximum Effect of Agents and Machines . . . 273 HYDROSTATICS. INTRODUCTORY REMARKS . . . . .289 Pressure of Fluids ...... 290 Solids immersed in Fluids .... 306 Specific Gravities . . . . . .308 Spirit Level . . . . . .321 Equilibrium of Floating Bodies . , . .322 Capillary Attraction . . . . . 336 Apparent Attraction and Repulsion observed in Bodies floating near each other on the Surface of Fluids . .349 Barometer . . . . . .351 Contents. vii HYDRODYNAMICS. Discharge of Fluids through Apertures in the Bottom and Sides of Vessels ...... 369 Motion of Gases . . . . 377 Resistance of Fluids to Bodies moving in them -. .388 Theory of the Air-Pump .... 403 Pumps for raising Water ... . 407 Syphon ....... 416 Steam Engine . . . . . .417 NOTES 431 ELEMENTARY TREATISE MECHANICS. Preliminary Remarks and Definitions. 1. MATTER has been variously defined by philosophers, and some have even doubted whether we can be morally certain of its existence. It is not our intention, nor does it belong to the nature of our subject, to enter into discussions of this kind. Re- lying solely upon experiment, we give the name of matter or body, to whatever is capable of producing, through our organs, certain determinate sensations ; and the power of exciting in us these sensations, constitutes for us so many properties, by which we recognise the presence of bodies. But among these properties, two only are absolutely essential in order to our having a per- ception of matter. These are extension and impenetrability, of which the sight and touch are the first judges. 2. The character derived from exte'nsion is self-evident; when we see or touch a body, this body, or, if you please, the power which it has of affecting us, resides in a certain portion of space. The place which it occupies is therefore determinate ; and by this very circumstance it is extended. 3. When we pass our hands over the surface of a body, we perceive that the matter of which it is composed, is without us ; moreover, two distinct portions of matter can never be made to coincide, or identify themselves, the one with the other, in such a manner that the same absolute points of space shall at the same time give us the sensation of both. In this consists the property of impenetrability. 2 Preliminary Remarks. To show how this property, together with that of extension.. is necessary to constitute a body, I will refer to familiar phenom- ena in which these properties are observed separately. If an object be placed before a concave mirror, there will be formed, at a certain distance from the mirror, an image of the object. This image, distinct from the parts of the space that surround it, is extended but not impenetrable. The hand may be thrust through it without experiencing the smallest resistance, and the parts that come in contact with the hand, vanish instead of being displaced. A piece of wood or stone does not admit of being thus penetrated. Moreover, by means of a second mirror properly disposed, an image of another object may be made to oc- cupy the same place with that of the first, without the latter being displaced, or in any way deranged. Indeed the same coincidence may be effected with a third, a fourth, or any number of images. These images are extended, but not impenetrable ; they are/orras, but not sensible matter. I say sensible matter, for we shall see here- after that light which constitutes these images, is itself probably composed of material particles of an insensible tenuity, which move with amazing velocity, and only pass by each other in this case at immense intervals, by which they are separated from each other. 4. It is here proper to speak of certain phenomena which seem, at first sight, to be opposed to what we have laid down with regard to the impenetrability of matter, but which, ex- amined more attentively, only tend to confirm it. When a solid body is suffered to fall into any fluid, as water for example, it sinks and seems to penetrate the fluid ; but it in fact only separates and displaces the parts that compose it, for if the vessel containing the fluid be formed with a narrow neck toward the top, like a bottle, the fluid will be seen to rise as the body enters, and to a greater or less height, exactly in propor- tion to the size of the immersed body. What has taken place therefore, is only a division and separation of parts, and not strictly a penetration. The same may be said when an edged tool is forced into a block of wood, only the parts of the wood are separated with more difficulty than those of water. The same may be said also when a nail is driven into clay, lead, or gold, in which cases it only makes an opening sufficient for its admis- Indeed the mass thus pierced is not entirely separated, Preliminary Remarks. 3 but the parts are nevertheless pressed and crouded together ; and if we examine those which surround the opening, caused by the nail, we shall find sensible marks of this pressure. The nail in its turn may likewise be pierced by steel, and this again by other bodies. We hence infer that bodies, even the most hard and compact, are not composed of matter absolutely continuous, but of parts aggregated together, and placed at distances, which, under the influence of external causes, may become greater or less. It is on this account that the dimensions of any given mass of mat- ter are capable of being increased by heat, or diminished by cold, that the particles of salt admit of being separated and dis- tributed, and as it were, lost among the particles of water ; that mercury attaches itself to a piece of gold immersed in it, and in- sinuates itself into the interior of this compact substance. These mixtures and dissolutions sometimes take place without any ap- parent augmentation of bulk, this bulk being estimated according to the exterior surface of the bodies in question, without regard being had to the void spaces, sensible or insensible to us, which may be found to exist among their parts. In all this there is only separation and mixture without any actual penetration of material particles. This want of material continuity in bodies is known under the general name of porosity, and we call pores the interstices or empty spaces by which these particles are separated from each other. Porosity seems to be a property common to all bodies, although it does not belong to the essence of matter, since we can conceive of sensible bodies which are entirely destitute of void space. 6. Thus admitting that bodies may be considered as com- posed of smaller parts which constitute their essence, we may be asked, what is the form and magnitude of these parts. As to the magnitude, it should seem that if is extremely minute ; for to whatever extent we carry the division, in the case of gold, for example, by the processes of wire-drawing, filing, and beating, the smallest particles preserve invariably all the properties that belong to the entire mass. Crystallized bodies reduced to an almost impalpable powder, upon being examined with a micro- scope, are found to exhibit the same forms and the same angles which characterize the whole mass of the crystal. We have 4 Preliminary Remarks. examples of a division carried to a still greater extent in odours, the sense being affected in this case by particles proceeding from the odoriferous body that are absolutely invisible and impalpable. From these - few instances, and a thousand others that might be mentioned, it is evident that a body without chang- ing its character, without ceasing to be of the same identical nature with the largest masses that surround us, may be divided into parts, the smallness of which eludes the power of the senses, and almost that of the imagination. 6* The question has been much discussed, whether matter by infinitely divisible ; but it is now pretty generally agreed that the dispute is about words. If the point in question relate to abstract geometrical divisibility, there can be no doubt of the truth of the affirmative ; for however infinitely small we suppose a particle, from the very circumstance of its being extended, we can always conceive this extent divided into two halves, and each of these into two others, and so on without end (Calc. 4). But if we mean by the question an actual physical divisibility, nothing can be decided absolutely one way or the other. It seems however by all we can learn, that we should at some stage of the division arrive at material particles which would not admit of being brok- en, or altered, or transmuted the one into the other ; for to what- ever chemical operation they are subjected, into whatever com- binations they are made to enter, however they may be brought to constitute a part of living beings, they always return to their former state, with their original properties unchanged. The infinite variety of processes of this kind through which the same material particles have been made to pass since the world was created, does not appear to have produced the smallest alteration, 7, But how can such a system of particles exist collected together in the form of solid and resisting masses, as we see they are. in a great number of Bodies, in all indeed when they are properly examined ? This state, as we shall see hereafter, is pro- duced and maintained by the natural powers with which all parts of matter are endued, and which cause them to tend to- ward each other, as it were by an attraction. But if there existed only forces of this kind, the particles would continue to approach till they came into actual contact with each other, that is, until they were arrested by the impenetrability of their parts, which would not admit of the contraction and dilatation which are con- Preliminary Remarks. $ stantly observed in bodies. We accordingly infer that there is a general cause of interior repulsion in bodies, by which the attractive forces are continually balanced. This cause which resides in all bodies, seems to be referrible to the principle of heat. The particles of each body, actuated at the same time by these two opposite forces, naturally put themselves in a state of equilibrium, resulting from a compensation of energies, and they approach and recede, according as the forces to which they are exposed from without favor the attractive or repulsive principle. It is with these minute bodies as it is with the plan- ets of our system, which are found to move and oscillate, as it were, in orbits of variable forms and dimensions, without the system being destroyed, or the general equilibrium being dis- turbed. From these different conditions of equilibrium arise, as we shall see hereafter, all the secondary and changeable forms of bodies, such, for example, as are denominated aeriform, liquid, and solid, crystallized, hard, elastic, &c. 8. In all the phenomena which present themselves, the particles of matter act, or rather are acted upon, as if they were perfectly inert, that is, deprived of all power of self-direction. They can be moved, displaced, stopped, by causes foreign to themselves, but we never have been able to discover the least trace of any thing like choice or will proper to the particles themselves. If the ball which rolls upon a billiard table in con- sequence of the impulse that is given to it, loses by little and little its velocity, and at length comes to a state of rest, it is entirely the effect of the continual resistance that it meets with from the roughness of the cloth with which it comes in contact, and from the particles of the air through which it passes. Make the cloth more smooth, or the air more rare, and the same impulse would keep it longer in motion ; substitute for the cloth a marble slab highly polished, or a band of stretched wire, the elasticity of which is still more perfect, and the ball would continue its motion for a much longer time ; from all which it is to be infer- red, that if the obstacles were completely removed, there would be no diminution of the velocity first communicated, and the motion would never cease. A stone thrown from the top of a tower, and urged at the same time by the impulse of the hand and by gravity, will come to the ground after proceeding a certain distance, losing at the same time its horizontal velocity, by imparting it to the particles of air against which it impinges-. 6 Preliminary Remarks. But let us suppose the air removed or annihilated, and the pro* jectile force (in the direction of a tangent), to be sufficient to carry the stone as far from the earth as gravity would cause it to descend each instant, and the stone would describe a circle round the earth, and if there were nothing to stop or obstruct it, it would thus continue to revolve without end. We have indeed this principle exemplified in the motion of the moon, which re- volves in a void about the earth ; we see moreover the same renewed, perpetual motion in the planets which pass in like man- ner through spaces destitute of all material resistance. We are hence lead to believe that matter is incapable of effecting any change in itself, either with respect to motion or rest, and once put into either of these states, it would continue in this state so long as it should remain undisturbed by any cause foreign to itself. This indifference to motion and rest, this want of all power of self-direction, has obtained the name of inertia. There is one class of bodies, however, that seem to form an exception to this law of matter. It comprehends those which we call animated, which put themselves in motion or stop themselves by an act of the will ; but even in these the material elements which consti- tute their parts or members, and these members themselves, are perfectly inert. It is their union or combination that possesses the quality of life. Separated, they have no longer this power, but return to the condition of ordinary matter. We* are entirely in the dark with regard to the cause of this remarkable difference in the bodies that surround us. As to what constitutes a state of life, we can pretend to no knowledge whatever. But seeing matter under all other circumstances destitute of the power of self-direction, and knowing also that in living beings it loses this faculty by death and by sleep, we are led to regard it as foreign to the essence of matter, and to consider the volition of animated beings, as the act of an immaterial principle which resides within them. We are unable to say in what part this principle is seat- ed, or in what it consi^s, and still less how, being immaterial, it is capable of acting upon matter ; but with the little attention that we have paid to ourselves and to the objects about us, these obscurities, unfortunately too common, in which our imper- fect knowledge has left us, ought not to be made the grounds of an objection against the essence of things with which we must be contented to remain unacquainted. So that we here proceed philo- sophically, according to the rule adopted in other cases, by Preliminary Remarks. 7 bringing together things that are analogous, and making the motion of animated beings to depend upon a cause foreign to matter ; matter being found inert under all other circum- stances in which we have been able to examine it. Another rea- son is given in the schools of philosophy for attributing sponta- neous motion to an immaterial principle ; namely, that the will, by the very nature of its acts, can proceed only from a simple being, and that consequently it cannot belong to a substance essentially compounded, or at least divisible and decomposable, like matter ; but this metaphysical argument would carry us too far from our subject. We content ourselves with merely sug- gesting it ; for all experimental purposes, it will be sufficient to consider the immateriality of the principle of volition as a dis- tinction founded upon analogy, and the inertia of matter as a general property in the actual state of the world. 9. We are moreover made acquainted by experiment with several other properties of matter which are also acciden- tal, that is, which seem not to be absolutely necessary in order that material bodies may manifest themselves to our stns- es, but the co- existence of which with the primitive conditions of materiality is important to be known, since it supplies the want of other evidence, in a great number of cases in which the essential properties do not admit of being recognised. Such, for example, is gravity. Among natural bodies which we can see and touch, none is to be found which is not heavy, that is, which does not tend to fall toward the centre of the earth when left to itself; and since these properties are always found to accompany each other, the presence of the one is with respect to us always a sufficient ground to infer the existence of the other. Thus, al- though we can neither see nor touch the air, as we can see and touch other bodies, still we believe it to be a material substance, because it is heavy, capable of being confined in vessels and of exhibiting other phenomena, all similar to those which belong to a heavy fluid. A careful examination of tnese properties teach- es us at length that there are airs of very different kinds, which are all so many substances differing essentially from each other in the action which they are capable of exerting on other bodies, and which is exerted in turn upon them by these bodies. 10. Moreover attraction is one of those contingent properties which supplies what is wanting in the evidence furnished by the 8 Preliminary Remarks. immediate testimony of the senses. I have said that th'e particles of all known bodies exert upon one another attractive and repulsive forces. On the other hand when we can demonstrate the exis- tence of these forces in an unknown principle, we infer that this principle is material. Thus, light is not tangible, it is not, so far as we can perceive, extended ; it has no weight, or at least none capable of being appreciated by our balances. It is so subtle as to elude all the ordinary methods by which matter manifests itself to the senses. But by causing it to pass through transparent bodies, as glass, water, &c., it deviates from a direct course in its passage, and is bent precisely as if it were repelled by a force proceeding from the surface, and attracted on the other hand within by the particles which compose the transparent body. We know also that it employs a certain time, very short indeed, but yet capable of being estimated, in passing from luminous bodies to us. In fine, by subjecting rays of light to certain tests, we find that transparent bodies attract and repel them differently on certain sides from what they do on others. From these prop- erties, taken together, we are led to conclude that light is a ma- terial substance, composed of particles extremely small, the form of which is symmetrical on certain faces, which are susceptible of particular attractions and repulsions, and which move in free space, and through transparent bodies, with a given and deter- ciinable velocity. 1 1 . There are still other principles which act upon material bodies without being either visible or tangible, or susceptible of being weighed by our balances, which even present much fewer indications of materiality than light, and which not- withstanding are believed to be material substances. Such is the unknown principle of electricity. Nothing absolutely ma- terial has yet been detected in the cause of electrical phenomena, nothing indeed which does not admit of being explained without the supposition of matter. Still in its distribution over bodies, in its passage from ont to the other through the obstacles which separate them, this principle acts in a manner so exactly con- formable to the laws of equilibrium and motion which belong to fluid substances, that we can on this hypothesis calculate with the utmost precision, and in all their details, the phenomena that are to take place under given circumstances. It seems extreme- ly probable, therefore, that the principle in question is a flu- id, and that it is accordingly material. The same reasoning Preliminary Remarks. <9 is applicable to the principle of magnetism, which manifests itself in several metals. 12. We have still less evidence of any thing material in the principle of heat. Not only does it want, like the preceding, the sensible properties by which matter is characterized, but the laws of its motion and equilibrium, not being completely known, we cannot arrive at the same probable conclusion, in this case as in the former. By following it however in our experiments, we find that it diffuses itself in bodies, passes from one to another, modifies the disposition, the distances, and attractive properties of their particles. But all this does not prove incontestibly, that the principle in question is itself a body. The strongest argument in favor of its materiality is derived perhaps from certain anal- ogies, lately discovered, between the radiant properties of heat and those of light, which lead us to believe that one of these principles may change itself gradually into the other,' that is, they may acquire and lose successively the modifications by which they are respectively distinguished. The develope- ment of these analogies furnishes a most important subject of in- vestigation. 1 3. It will be perceived from what has been said, that all bodies of a sensible magnitude, the materiality of which can be immediately determined, consist in the grouping together of a multitude of material particles of extreme minuteness, in which a difference in the mode of aggregation is the only circumstance that constitutes a body solid, liquid, or gaseous. There are moreover strong reasons for believing, as we have seen, that these particles are inert masses, incapable from any inherent power of their own, of modifying themselves, and susceptible only of of obeying causes from without ; whether this want of choice and self-direction is, in fact, as observation seems to prove, a general and essential characteristic of matter, or whether we so regard it intellectually for the purpose merely of considering by them- selves those properties which remain to matter, after it is depriv- ed of this. Now material particles being considered as in this inert state, there will hence arise, in the phenomena which their aggregation presents, certain necessary conditions, which are applicable to all bodies, independently of the chemical nature of their constituent parts, being the simple consequences of Mech. 10 Preliminary Remarks and Definitions. their materiality. Such are the general laws of equilibrium and motion, which are deduced indeed mathematically from the sin" gle property of inertia. 14. We have already ijsed the words rest, motion, and/orce, as making a part of ordinary language. It now becomes neces- sary to fix their meaning with precision. We begin with defin- ing the place in which the phenomena under consideration are supposed to occur. In order to this let us conceive of space without bounds, immaterial, immoveable, and of which all the parts, similar among themselves, are capable of being penetrated by matter without opposing the smallest resistance. Whether space in this sense exist in nature or not, is of little consequence ; the definition presents to us merely an abstract extension. Now imagine in this space the particles of which we have been speaking, the material elements of bodies, and let us first consider with- respect to them the mere circumstance of their existence. This simple fact will be capable of two distinct modifications; it may be that the same particle shall remain without change in its actual place, or that by the influence of external causes it shall leave its place to pass to some other part of space. The first of these states constitutes absolute rest, and the second motion. 15. But we can conceive further, that two or several particles are displaced at the same time, and impressed with a common motion, preserving with regard to one another their respective positions. Then if we consider them with reference to immove- able space, they will actually be in absolute motion ; but if we consider them simply in their mutual relations to one another, these will continue the same as if the whole group had remained at rest ; and if there were upon one of these particles an intelli- gent being who should observe all the others, it would be impos- sible for him to decide from this observation alone, whether the whole system were in motion or not. The permanence of these relations in the midst of a common motion, is what we understand by relative rest. This will be the condition of a number of bod- ies placed in a boat and abandoned to the course of a smooth stream. This is indeed the condition of all the bodies about us so long as they remain fixed to the same point of the terrestrial surface. They are at rest among themselves ; but the earth which turns daily on its axis, impresses upon them a common Preliminary Remarks and Definitions. 1 1 motion, and at the same time bears them all together in its orbit round the sun, which perhaps in its turn carries the earth and the whole system of planets toward some distant constellation. Relative rest, therefore, is really the only kind of rest which can actually take place among the objects to which our attention is di- rected. It is at least all that we can ever be certain of observing. 16. We are hence led to make a similar distinction with respect to motion, and to separate the absolute motions of bodies, considered with reference to immoveable space, from the relative changes of position which may happen among them. These last therefore may be called relative motions, whether that body of the system to which they are referred, be itself in motion or at rest. The changes of place, for example, among the heavenly bodies, which we observe from the surface of the earth, are not absolute but relative motions, because the earth to which we refer them, as a fixed centre, has actually a motion of rotation on its axis and a progressive motion about the sun. Even when by calculation we have inferred from these observations the ac- tual motions of the heavenly bodies as they would appear, if seen from the sun, we cannot affirm positively that these are absolute motions, since it may be that the sun and the whole planetary system have a common motion in space. 1 7. According to the idea of inertia which we derive from experience, we must regard the state of motion and that of rest, as simple accidents of matter, which it is incapable of imparting to itself, and which it can only receive from without, and which once received, it cannot alter. When therefore we see a body passing from one of these states to the other, we must re- gard this change as produced and determined by the action of external causes. These causes, whatever they may be, are denominated forces. Nature presents us with an infinite number of them which are at least in appearance of different kinds. Such are the forces produced by the muscles and organs of liv- ing animals, the exercise of which, for the most part, depends solely on the will. Such are also the forces of physical agents, as the expansion of bodies by heat, and their contraction by cold, &c. There are moreover others which seem to be inhe- rent in certain bodies, as the attraction of the magnet for iron, and that which is manifested among electrified bodies. 1 2 Preliminary Remarks and Definitiojis. From the very nature of matter as thus presented to our con- sideration, it will be seen that a body once put into a state of mo- tion or rest, by any cause whatever, must continue in that state forev- er, if no new cause is made to act upon it* If it cannot give itself motion when at rest, it cannot stop itself when in motion, for this would be equivalent to giving itself motion in the opposite direc- tion; neither can it change its velocity or direction, for this would equally imply a new force. -Thus, motion is naturally equal or uniform and rectilinear. 1 8. When several forces are applied at the same time to a body, they are mutually modified by the connexion which exists among the different parts of the body, and which prevents each from taking the motion which the force exerted upon it tends to produce. If these forces happen entirely to destroy each other, so that the body remains at rest, we say that the forces are in equilibrium, or that the body is in equilibrium, under the actioa of these forces. 19. Mechanics is the science which treats of the equilibrium and motion of bodies. That part, the object of which is to dis- cover the conditions of equilibrium, is called statics.* We give the name of dynamics^ to the other part which has for its object to determine the motion which a body takes, when the forces ap- plied to it are not in equilibrium. The general laws of statics and dynamics are applicable to fluids ; but on account of the peculiar difficulty attending the consideration of this class of bodies we are accustomed to treat them separately. That part of the mechanics of fluids which relates to their eqilibrium is called hydrostatics, J and that which comprehends their motions, hydrodynamics.^ 20. In our inquiries on these subjects, we first proceed upon the supposition that there are no other bodies, and no other forces, in nature, except those under consideration. Thus all bodies are supposed to be destitute of weight, and free from friction, resistance, and obstructions of every kind. Regard is afterwards had to these causes ; but to estimate their effects, it is necessary to begin by investigating each point separately. * From trryfii, I stand. jFrom Jyvse^/s, power I" From vap, water, and irmpi. From vfvo and STATICS Of Uniform Motion* 21. A body is said to have a uniform motion when it passes continually over the same space in the same time. In order to compare the motions of two bodies which move uniformly, it is necessary to consider the space which each des- cribes in the same determinate time, as one minute, one second, &c. This space is what is called the velocity of the body. 22. The velocity of a body therefore is, properly speaking, only the space which this body is capable of describing uniform- ly in the interval of time which we take for unity. Thus in the uniform motion of two bodies, the time being reckoned in seconds, if one passes over five feet in a second, and the other six feet in a second, we say that the velocity of the first is five feet, and that of the second six feet. 23. But if, the second being always taken as the unit of time, I am told that a body passes over 100 feet in 5 seconds, 100 feet does not express the velocity, since this space is not that which answers to the unit of time, a second ; but it will be perceived, that in each second it would pass over a fifth part of this 100 feet, or 20 feet ; that is, in order to find the velocity, I divide the number 100, the parts of the space passed over, by 5, the num- ber of units in the elapsed time. Hence universally, the velocity is equal to the space divided by the time ; for it is clear, that if we divide the whole space into as many equal parts, as there are units in the time elapsed, each part will be the space described during this unit of time, and will consequently be the velocity according to our definition. Thus calling v the velocity and s 14 Statics. the space passed over in any portion of time denoted by *, we shall have this is one of the fundamental principles of mechanics. 24. The equation i? = , gives not only the measure of the velocity, but also that of the time and space. Indeed if we con- sidered t and s as unknown quantities successively, we shall have, by the common rules of algebra, Thus, to find the time we divide the space by the velocity ; and, to find the space we multiply the velocity by the time. If, for example, it is asked what time is required to describe 200 feet, when the body in question has a uniform velocity of 5 feet in a second ; it is evident that it would require as many sec- onds as there are 5 feet in 200 feet ; that is, we should have the time sought, or the number of seconds, by dividing the space 200 by the velocity 5 ; we shall find for the answer 40 seconds ; or, in other words, a number of seconds equal to the quotient arising from dividing the space by the time. In like manner, if it is asked what space would be described in 20 seconds by a body moving with a constant velocity of 5 feet in a second ; it is manifest that it would describe 20 times 5 feet ; that is, it is necessary in this case to multiply the velocity by the time. Thus, although we have here employed algebraic characters, it is not because they are necessary to the investigation of these fundamental truths, but because, by means of them, the proposi- tions, and their dependence, the one upon the other, are more con- cisely expressed, and more easily remembered. Indeed it will be seen by the above example, that the first principle, expressed algebraically, being once fixed in the mind, the two others are rendity deduced from it by the most familiar rules. 25. It will be easy now to compare the uniform motions of two, or of a greater number of bodies. If it is asked, for exam- Uniform Motion. 1 5 pie, what is the ratio of the velocities of two bodies which des- cribe the known spaces s, /, in the times t, t respective^ ; by call- ing v, v', the velocities of these two bodies respectively, we shall have v =s -, and -' = -;-, whence s s' v : v' : : - : that is, the velocities are as the spaces divided by the times. In a word, if it is proposed to compare the velocities, the spaces, or the times, the principle above laid down, will give the express- ion for each of these particulars with respect to each body ; we have therefore only to compare together these expressions. For example, if we would compare the spaces, the fundamental prop- osition v = , gives s = v i ; we have in like manner for the second body s f =vV; whence s : s' : : v t : v' t', that is, the spaces are as the velocities multiplied by the times. 26. Of these three things, namely, the space, time, and velocity, if we would compare two together, when the third is the same for each body, we have only to deduce from the same funda? mental theorem, the expression for this third particular, with respect to each body, and to put these two expressions equal to each other. If, for example, we would know the ratio of the spaces when the velocities are the same, we should have s , , s' v = 7? and i = y ; 23. s s' whence, since by supposition v = v, we have = 7 , and ac- cordingly s : s r :: t : t'; that is, the velocities being equal, the spaces are as the times. It will be found in like manner that, the times being equal, the spaces are as the velocities ; and that the spaces being equal t the velocities must be 16 Statics. inversely as the times. Indeed we have in this last case s = v t. and s' = v t 1 ; from which we obtain, when s = *', v : v' : : t' : t. Thus the single proposition v = furnishes the means of compar- ing all the circumstances of uniform motion. Of Forces and the Quantity of Motion* 27. The sum of the material parts of which a body con- sists, is called its mass ; but in the use we shall make of the word is to be understood the number of material parts of which the body is composed. Force, as we have said, is the cause which either moves ex- tends to move a body. As forces interest us only by their effects, it is by the effects of which they are capable, that we are to measure them. Now the effect of a force is to cause in each particle of a body a certain velocity. Accordingly, if all the parts receive the same velocity, as is here supposed, the effect of the moving cause has for its measure the velocity multiplied by the number of material parts contained in the body, that is, by the mass. Therefore, a force is measured by the velocity which it is capable of impressing upon a known mass multiplied by this mass. 28. The product of the mass of a body by its velocity is call- ed the quantity of motion of this body. Forces are therefore measur- ed by the quantities of motion which they are capable of producing respectively. Thus, if we designate the above product by p, the mass by m, and the velocity by v, we shall have p = mv. This equation gives v = , and m = ; from which it will be seen that, 1. The moving force of a body and its mass being known, we shall Jind the velocity by dividing the moving force by the mass ; 2. The moving force and the velocity being known, we shall find the mass belonging to this velocity and moving force, by dividing the, moving force by the velocity ; Uniform Motion. 1 7 3. If the moping forces are equal, the velocities are inversely as (he masses. The truth of these propositions is easily shown by put- ting successively the value of m equal to m', that of v equal to T', and that of p equal to / ; the equations thus obtained, reduc- ed and converted into proportions, form the several propositions above stated. Remark. 30. The mass or number of material parts of a body, de- pends upon its bulk or volume, and what is called its density, that is, the greater or less degree of closeness or proximity among its particles. As all bodies have more or less of void space within them, their quantities of matter are not proportional to their bulks ; since, under the same bulk, the quantity of matter is greater according as the parts are more crouded and com- pressed together. A body is said to be more dense than anoth- er, when under the same bulk it has more matter ; and on the other hand to be more rare than another, when under the same bulk it has less matter. Accordingly, by means of the density of a body, we are able, when the bulk is known, to judge of the number of material parts which compose it; so that the density may be considered as representing the number of material parts in a given bulk. When we say that gold is 1 9 times as dense as water, we mean that gold contains 1 9 times as many parts in the same space. By considering density as expressing the number of material parts of a determinate bulk, taken as the unit of bulk, it is evident that in order to find the mass, or total number of material parts of a body whose bulk is known, we should simply multiply the density by the bulk. If, for example, the density of a cubic inch of gold be represented by 19, the quantity of matter con- tained in 10 cubic inches would be 10 times 19. Thus, designa- ting the mass by m, the bulk by 6, and the density by D, we shall have m b X D. It will hence be easy to compare together the masses, the bulks, and the densities of bodies. Mech. 3 -18 Statics. Moreover, as the particles of matter of whatever kind tend by the force of gravity to move with the same velocity, or exert the same power, the combined action arising from this cause will be proportional to the number of particles ; that is, the weight of a body is as its density, other things being the same. The relative weights of the different kinds of matter under equal bulks, are called the specific gravities of the bodies respectively, the weight of pure water, in a vacuum, at a particular temperature, being taken as the unit. Thus, if a cubic inch, a cubic foot, &c., of gold be 1 9 times heavier than the same bulk of water, under the same circumstances, as to temperature, &c., the specific gravity of gold is said to be 1 9. Of Equilibrium between Forces directly opposite. 31. We shall represent forces, as we have said, by their ef- fects, that is, by the quantities of motion which they are capable of producing respectively in a determinate mass. But not to embrace too many objects at once, we shall consider each mass, or body, as reduced to a single point, at which we suppose the same quantity of matter as in the body of which it takes the place. We shall see hereafter that there is in fact in every body a point through which motion is transmitted as if the whole mass were concentrated there. We shall, moreover, unless the contrary is expressly stated, consider bodies as composed of particles absolutely hard, and connected together in such a man- ner as not to admit of any change in their respective situations by the action of any force whatever. 32. This being premised, let us suppose two bodies m, n. to be put in motion, the first from A toward C, with a ve- locity w, the second from C toward A with a velocity v. When these bodies come to meet, they will be in equilibrium, if the quantity of motion in ra is equal to the quantity of motion in n ; that is, if m u is equal to n r. Indeed it is evident, that if m is equal to n, and the velocity u is equal to T;, there must be an equilibrium ; for in this case, whatever reason there may be for supposing m to prevail over Equilibrium between Forces directly opposite. 1 9 ?i, might also be given for supposing n to prevail over ra, since they are by hypothesis in all respects equal. 33. Let us suppose now, that ra is double of n, but that i>, at the same time is double of w, that is, that n passes over two feet, for example, in a second, while m passes over one foot in a second. It is clear that we may consider m as composed of two masses equal each to n ; and that at the instant of meeting, we may represent the body n as having a velocity of one foot in a second, to which is added at the same instant, another velocity of one foot in a second. We may then conceive, that in meeting the mass n expends one of its velocities against a portion of the mass m equal to itself, and its other velocity against the remain- ing portion of m of the same magnitude. If now, instead of supposing the masses m and n in the ratio of 2 to 1, and their velocities in the ratio of 1 to 2, we suppose them in any other ratio, it is evident that we may always con- ceive the greater mass as decomposed into a certain number of portions equal each to the smaller, and of which each shall des- troy in the smaller, a velocity equal to its own. We may there- fore consider the following proposition as established. Two bodies which act directly against each other in the same straight /me, are in equilibrium when their quantities of motion are equal; that is, when the product of the mass of the one into the velocity with which it moves, or tends to move, is equal to the product of the mass of the other into its actual or virtual velocity. This proposition is to be regarded as general, whether the two bodies move freely and directly the one against the other, or whether they act against each other by the intervention of a rod inflexible and without mass, or whether they are considered as pulling in opposite directions by means of a thread m n inca- pable of being extended. And reciprocally, if two bodies are in equilibrium, we may conclude that their motions are directly opposite, and that their quantities of motion are equal. 34. We infer, moreover, that if three or a greater number of bodies ra, n, o, &c., moving, or tending to move, in the same Fig. a, straight line, with velocities w, v, zy, &c., are in equilibrium, the sum 20 Statics. of the quantities of motion of those which act in one direction is equal to the sum of the quantities of motion of those which act in the opposite direction. For, they being in equilibrium, we maj always suppose that, m and n acting in the same direction, n destroys a part of the motion of o, and that m destroys the remaining part. Now if we represent by x the velocity that o loses by the action of n, we shall have o a?, for the quantity of motion destroyed by the action of n ; we have accordingly n v = o x. The body m, therefore, will have only to destroy in o the remain- ing quantity, namely, o w o x ; we have consequently m u = o w o # ; or since o x = n u, m u = o w n v^ that is, Of Compound Motion. 35. We still consider the masses to which the forces under consideration are applied, as concentrated each in a point. We call compound motion that which takes place in a body, when urged at the same time by two or more forces having any given direction with each other. Fig. 3. If a body m moving in the line C5, receive upon arriving at the point A, an impulse in the direction AD, perpendicular to CB, this impulse can produce no other effect, except that of removing the body from CB. It can neither augment nor dimin- ish the velocity with which, at the time of receiving the impulse, it was departing from AD. Indeed, since AD is perpendicular to CB, there is no reason why a force acting in the direction AD should produce an effect to the right, rather than to the left, of this line, and as it cannot act in both these directions at once, it can have no influence either way. The same reasoning will hold true, if we suppose that the body m, moving in the line AD, receives, upon arriving at A, an impulse in the direction AB. This impulse will neither add to nor take from the velocity with which the body m was departing from AB. Compound Motion. 21 36. If two forces p and q, the directions of which are right angles to each other, act at the same instant upon a body m, and the force q is such as by its sole and instantaneous action to cause the body to pass over AB in a determinate time, as one second, and the force p is such as to cause the body to pass over AD in the same time, we say that by the joint action of the two forces, q and p, the body m will in the same time pass over the diagonal AE, of the parallelogram, DABE, which has for its sides these same lines, AB, AD. Since the two forces act at the same instant upon the given body, we may suppose it moving in the line AD, and that at the instant of its arriving at the point A, it receives the force q in a direction perpendicular to AD. Now according to article 35 the force q can neither increase nor diminish the velocity with which it was at this moment departing from AB ; if, therefore, through the point Dwe drawD.E parallel to AB, the body must at the end of a second be somewhere in the line JD E, all parts of which are equally distant from AB. The same reasoning may be adopted with regard to the force q, by which it will be seen, that if through the point B, we draw BE parallel to AD, the body must at the end of a second be somewhere in BE. But there is only the point E which is at the same time in DE and BE-, therefore at the end of a second the body will be in E. It is also evident, that whatever course the body takes, by the instantaneous action of the forces, this course must be a straight line since, from the instant that the forces are ex- erted, the body is abandoned to itself, and there is no cause to incline it one way rather than another. Accordingly, as this body passes through A and E, and without any thing to change its direction, the course must be AE, that is, the diagonal of the parallelogram DABE. We will add moreover, that the body describes AE with a uniform motion, since after the joint action of the two forces, it is left equally without any cause to alter its rate of moving. 1 37. Since the two forces p and q, acting simultaneously upon the body m, have no other effect than to make it describe the 1 diagonal AE, we infer, that instead of two forces whose directions 22 Statics. are at right angles to each other, we may always substitute a single one, provided that this single one is such as to cause the body to describe the diagonal of a right-angled parallelogram, the sides of which would be described in the same time, each separately, by the action of the force of which it represents the direction. The single force AE, which results from the action of the two forces AB, AD, is called the resultant of these two forces. As the lines AB, AD, represent the effects which the forces q and p are singly capable of producing, and AE the effect which they are able to produce conjointly, we may regard AB, AD, AE, as representing these forces themselves. We may thus consider any single force AE, as being the result of two other forces AB, AD, the directions of which are at right angles to each other, provided that, the first being represented by the diagonal AE, the others are represented by the sides AB, AD, of this same right-angled parallelogram. For the single force AE, therefore we may substitute the two forces AB, AD, since these two will in fact only produce AE. 38. In general, whatever be the angle formed by the directions of Fig.5, 6. the two forces p and q which act at the same time upon a body m, this body will still describe the diagonal AE, of the parallelogram DA BE, the sides of which represent, in the directions of the forces, the effects which they are separately capable of producing; and the body mill describe this diagonal in the same time in which by the action of either of the two forces, it would have described the side which repre- sents this force. Through the point A let the line FAH be drawn perpendicular to the diagonal AE, and through the points D and B, let DF, BH, be drawn parallel, and DG, BI, perpendicular to the diagonal AE. Instead of the force p, represented by AD, the diagonal of the rectangular parallelogram FAGD, we may take the two for- 37. ces AF, AG. For the same reason, instead of the force q, represented by the diagonal AB, of the rectangular parallelogram AHBI, we may take the two forces AH, AL We may therefore, instead of the two forces p and q, substitute the four forces AF, AG,AH, AI ; and these cannot but have the same resultant as the two forces p and q. Now of these four forces, the two AH. Compound Motion. 23 AF, contribute nothing to the resultant, because they act in op- posite directions, and are equal to each other. Indeed it will be readily seen, that, from the nature of a parallelogram, the 32, two triangles DGA, EIB, are equal ; therefore DG = BI, and consequently AF AH. As to the two forces AI, AG (fig. 5.), since they are exerted ac- cordins: to the same line, and are directed the same way, the result must be the sum of the two effects AG, AI ; and in fig. 6, since AI, AG, are exerted according to the same line, and are opposed the one to the other, the result must be the difference of the two effects AG, AI. But as the triangle EIB is equal to DGA, we shall have (fis. 5.) JlI-\ AG=;AI + EI=AE; and ( fig. 6.) AI AG=AI EI = AE. We conclude, therefore, that the four forces AF, API, AG, AI, and consequently the two forces AD, AB, have no other effect, than the force AE, represented by the diagonal of the parallelogram DABE, of which the two sides AB, AD, denote the forces q, p. This proposition is known by the name of i\\e parallelogram offerees. 39. We have in what precedes, represented the two for- ces p, q, by the lines AD, AB, which they are capable Fig. 4,5, of making the body m describe in the same time, that is, 6 - by the velocities which they would communicate ; although, according to what we have said, the true measure of any 28. force is the quantity of motion that it is capable of producing. But as the quantities of motion are in the ratio of the velocities, when the mass is the same, as is the fact in the present case ; 29 we may always, as we have now done, take the velocities AD, AB, as representing the two forces. But if, instead of having immediately the velocities which the two forces p, q, are capable of giving to the body m, we had the quantities of motion which they would produce in known masses, we should take AD, AB, in the ratio of these quantities of motion. If, for example, I know the forces p, q, only by this circumstance, that the force p is capable of giving a known velocity u, to' a known mass n ; and that the force q is capable of giving a veloc- ity D to a known mass o ; I should take AD '/AB ::'nw : er. 24 Statics. For, according to what has been shown, ./ID, AB, are to be taken in the ratio of the velocities which they are capable of giving to the body m. Now the first being capable of producing the quantity of motion n u, is capable of giving to the body m the 28 velocity -- . For the same reason the second, or the force q, is capable of giving to the body m, the velocity . The lines AD, AB, are consequently to be taken according to the following pro- portion ; but nv oi) : :: nu : ov. m m We see therefore that AD, AB, must be in the ratio of the quan- tities of motion nu, ov, which are the measures respectively of the forces p, q. What has now been remarked, will be found useful in com- paring the effects of different forces applied to different bodies. 38. The general proposition above demonstrated is of the great- est importance, as almost every thing we have to offer, consists in an application of it. 40. From what has been said, it will be seen that it is imma- terial whether we regard a body as urged by the combined action Fig.5,6.of the two forces AB, AD, whioh make with each other any assumed angle, or whether we regard it as urged by the single action of a force represented by the diagonal AE. And reciprocally, it amounts to the same thing, whether we consider a body as urged by a single force AE, or by two forces represented by the two sides of a parallelogram of which the single force AE is the diagonal. Let a body, for example, be supposed to pass from A to E by a uniform motion in one second. or let it be supposed to move through AB at such a rate as to describe it in one second, while in the same time this line is car- ried parallel to itself along AE ; in this case, as in the former, the body will merely describe the line AE. 41. The two forces AB, AD, meeting at the point A, are Geom. necessarily in the same plane. Since, therefore, they have 321 ' for their resultant the diagonal AE, which is in the plane of Composition and Decomposition of Forces. 25 the parallelogram, we may infer generally that any two forces which unite in the same point are always in the same plane with their resultant. Of the Composition and Decomposition of Forces. 42. Not only is it possible, by the principle above establish- ed, to reduce two concurring forces to one, and to decompose one into two others ; but we can in general reduce to a single force, as many o.ther forces as we please, when they are in the same plane, or when they unite at the same point ; and recipro- cally, we can decompose one or several forces into as many other forces as we please. 43. But before we proceed to explain this, we must observe that when a force p acts upon a body either by pushing or by Fig. 7. drawing it, it is of no consequence at what point of the direction of this force we suppose the action to be applied. For example, let the force p be exerted upon the body m by means of a rod inflexible and without mass, or by a thread inextensible and without mass, it is the same thing, whether the force p be applied at the point .B, or at the point C, or whether it be of such a nature * as to admit of being exerted at any point Z), on the other side of the body. So long as its action is employed in the same direc- tion, the effect will be the same. Distance can have no influ- ence, except so far as the action of the power transmits itself by the aid of some instrument, as a Lever or a cord, the matter of which would partake of the action of the power, all which in- struments we at present leave out of consideration. Thus, if two forces p and , line ABC being drawn ; as the resultant p is equal to p -f 9, it is only necessary to find the point B through which this resultant 50 must pass. Now we have p : g : : BC : AC, that is, p : p + q : : BC : AC. We have therefore only to take between the two points A, C, a 237. point B such that jBC shall be equal to - . Fig. 16. If the two parallel forces are opposed to each other, the re- 50 sultant will be equal to their difference p p : ? : : BC : AC or p : p q :: BC : AC-, in other words, it is necessary to take BC equal to ? . p q If q is greater than jo, the point B will be in AC produced beyond C with respect to A. Fig. 17. 54. if we had a third force r, we should first find the resultant p' of the two forces p, q, and then seek the resultant p of the two forces Q' and r, as if there were only these two ; that is, we should proceed in precisely the same manner as we have done in the preceding article. 55. Hence, reciprocally, if we would decompose any force p *$ 16 ' into two others parallel to it, we should take arbitrarily a line Composition and Decomposition of Forces. 31 AF parallel to the direction of p; and having assumed this line as the direction of one of the components, we take arbit^ari- ly for the value of this component any quantity p smaller than p, if it is proposed that the two components should act on oppo- site sides of the force p ; the second component q must in this case be equal to p p ; and in order to find its position, it is only -necessary, having drawn any straight line CBA, to take in AB produced the part J5C, such as to give the proportion q : p :: AB : BC; then through the point C we draw /C, parallel to EB, and this will be the direction of the force q. But if the two component forces are required to be on the same side (in which case they will be directed opposite ways), then we take for p any quantity, whether greater or less than p, and if it be greater it will be directed the same way with p, and if less, it will have a contrary direction with respect to p. Hav- ing drawn a line AF parallel to EB, as the direction of p, we take upon any assumed line BAG the point C, such as will give p porp p : p :: AB : AC ; and C will be the point through which the force q must pass par- allel to the given force p; and the point C will be beyond A with respect to , when p \$ greater than p ; and it will be between A and B, when p is less than p. 56. Since what we have now said of the force p with respect to the components />, B r X A FA = tpF, (pB FB = F; that is, the whole being divided by (pF, P + q r = Q. If we examine the process now pursued, we shall see that it does not depend in any degree upon the number of forces, but that it is applicable, whatever this number may be. We must infer therefore, that the resultant of any number of parallel forces is equal to the sum of those which act in one direction minus the sum of those which act in the opposite direction. 36 Statics: If now in the equation pxFA + qxFB rxFC = gx FD, found above, we put for p its value p + q r, just obtained, we shall have p X FA + q x FB r X FC = (p + q r) X FD, from which we deduce FD = p x FA + q x FB ~ r x FC p + q r or, bearing in mind, that the process by which we have arrived at this result, does not depend upon the number of forces em- ployed, we infer, as a general conclusion, that in order to deter- mine at what distance from a given point the resultant of several par- allel forces passes, from the sum of the moments of the forces which tend to turn the system in one direction, we must subtract the sum of the moments of the forces tending to turn it in the opposite direc- tion, and divide the remainder by the sum of the forces which act in one direction, minus the sum of those which act in a contrary direc- tion.* 67. If the point F, assumed arbitrarily, should happen to be so taken as to fall in D, through which the resultant passes, the distance FD being zero, its value p x FA + g X FB r x FC p + fif r since the force p tends to turn the system about the point D in a direction opposite to that in which the force q tends to turn it, becomes P x D ^ "*" 9 X DB rx DC p + q r and is equal to zero ; we have consequently p x DA + q X DB r x DC = o, * We must take care not to confound the forces which act in op- posite directions, with those which tend to turn the system in oppo- site directions. Two forces which act in opposite directions often tend to turn the system in the same direction. This depends upon the point to which the rotation, or the moments, is referred. The Fig. 21. two forces q, r, for example, act in opposite directions, but they both tend to turn the line EC in the same direction, about a point taken between B and C; and if we consider the rotation with refer- ence to the point F, the force q tends to turn FC in a direction op- posite to that in which the force r tends to turn it. Composition and Decomposition of Forces. 37 or qXDB=pxDA + rX DC. Moreover, as the point F, taken arbitrarily, may be higher or 44 ' lower, as we please, the point D has not been supposed to be in one point of the direction of the resultant rather than in another ; it follows, therefore, that the moments of several parallel forces, taken with respect to any point whatever in the direction of the resul- tant, are such, that the sum of the moments of the forces which tend to turn the system in one direction, is always equal to the sum of the moments of those which tend to turn it in the opposite direction. 68, Therefore by taking with contrary signs the moments of the forces which tend to turn the system in opposite directions, and by taking also with contrary signs the forces which act in opposite directions, we may infer as a general conclusion j 1. That the resultant of any number whatever of parallel forces is always equal to the sum of all these forces ; 2. That this resultant, which is parallel to the component forces, passes through a series of points each of which has this property, that the sum of the moments, taken with respect to this point, is zero. The above propositions are of the greatest importance. We shall see soon with what facility they enable us to find the centre of gravity of bodies. We proceed now to the consideration of forces the directions of which are inclined to each other. 69. Let there be any number of forces p, q, r, &c., all exert- Fig. 22. ed in the same plane, let the force p, acting according to AE, be represented by AE, and let the force q, acting according to BG, be represented by BG, and the force r, acting according to CL be represented by CL. Through a point F, taken arbitrarily in the plane of these forces, suppose two straight lines FC', FB", mak- ing any angle with each other (and for the sake of greater sim- plicity, let this angle be a right angle) ; and let us imagine the forces p, q, r, or AE, BG, CL, decomposed each into two others, one of which shall be parallel to FC, and the other to FB", and which will consequently be represented, each by the correspond- ing side of the parallelogram, the diagonal of which represents 4^ the given force. It is clear from what has been said, that the forces AV, BH, 66}68t CK, being parallel, will have for their resultant a single force DN, parallel to AV, BH, &c., the value of which will be 38 Statics. AV+BH+CKf and which will pass at a distance DD from FC', equal to the expression below, namely, jy D _ AV x AA + BH x BB' + CK x CC 1 J1V + BH + CK In like manner, the forces AI, BR, CM, parallel to FB", are reduced to a single one DO, parallel to AI, &c., and equal to AI -\-BR CM, and which (by supposing that D is the point where the direction of this force meets that of the force ND) will pass at a distance D'D from FB", equal to the following express- ion, namely, AI X AA " + BR * EW CM x CC" AI + BR CM This being supposed, the forces p, 7, r, and their directions, (that is, the angles which they make with the known iixed lines FC', FB", or with their parallels,) being considered as known, we know in each of the triangles AEL BGR, CLK, the hypothenuse and the angles. It will accordingly be easy to determine the Trig. 30. lines AI, BR, KL, or CM, and the lines IE or AV, RG or BH, and CK. We shall consequently know the values of the two resultants AV + BH + CK, and AI + BR CM. Moreover, as we cannot but know the distances A A, A A", BB', BB", &c., since the position of the points A, B, &c., where the forces are applied arc supposed to be given, we are acquainted with all the quantities which enter into the expression of the distances D'D, Jy'D. It will be easy, therefore, to determine the point D, where these two resultants meet. Accordingly, taking DO - AI + BR CM, and-DJV" = AV + BH + CK, and forming the parallelogram DNTO, we shall have the diago- nal D T for the resultant g, of the two partial resultants, parallel * We must not lose sight of what was said art. 39. By the forces AE, BG, &c., we are to understand that the lines AE, BG, &c., are to each other as the quantities of motion capable of being produced by the forces p, q, &c., in the masses to which they are applied. It is to be observed, likewise, with respect to the forces AV, BH, &c., that we mean by them quantities of motion, which are to the quanti- ties of motion represented by AE, BG, &c., as AV, BH, &c. ? are to, AS, BG, &c., respectively. Parallel Forces acting in different Planes. 39 to FC' and FB", that is, for the resultant of all the proposed forces. Of parallel Forces which act in different Planes. 70. Let there be three forces/*, q, r, directed according to Fig 23. the lines Ap, B q, C r, parallel to each other, but situated in different planes. Imagine a plane XZ to which the three straight lines Ap, &c., are perpendicular, and another plane ZV to which they are par- allel, and let A, B, C, be the three points where these lines meet the plane XZ. The two forces j9, r, are in the same plane, the intersection of which with the plane XZ is the straight line AC. These two335,324> forces may therefore be reduced to a single one p', equal to 50> p :'- r, having a direction parallel to that of the components, and 67 passing through a point D, such that;) X AD r x CD. The two forces p', q, are in the same plane, the intersection of which with the plane XZ is BD. These may accordingly be reduced to a single one p, equal to p' -f q, that is, equal to P -f q -f- r, having a direction parallel to that of p' and r, and passing through a point E, such that p' x DE = q X BE. -It follows, therefore, from this and what is said above, that any number of forces, the directions of which are parallel, may be reduced to a single one, equal to the sum of those which act in one direction, minus the sum of those which act in the contrary direction, whether the given forces are in the same or in different planes. Let us now inquire more particularly how we are to deter- mine through what point the resultant passes. If from the points A, D, C, B, E, we draw the lines AA', DD, CC, BE', EE 1 , perpendicularly to the common intersection of the two planes XZ, ZV -, on account of the parallels A A, DD, f CC*, we shall have AD : CD : : AD' : C'D\ Now the equation p X AD = r X CD, found above, gives AD : CD : : r : p ; 40 Statics. hence A'D : CD : : r : p, and consequently p x A'D = r X CD'. In like manner from the parallels DD, EE f , BB, we ob- tain, DE : BE :: DE : B'E'; and from the equation p' X DE = q X BE, we have the propor- tion DE : BE : : q : Q' ; therefore DE' : B'E' :: q : p' :: q : p + r ; whence (p + r) X DE' = q x B'E. Let us now take in the intersection ZF of the two planes, a fixed point F, and seek the distance FE of this point from the point E', corresponding to E, through which the resultant passes. It is clear that A'D = FD FA 1 , CD = FC FD, DE = FE FD, BE = FB' FE. Substituting these values for their equals in the two equations, p x A'D' =rx CD, (p + r)xD'E' = qX B'E', -we shall have p x FD p x FA' = r x FC r x FD, (p + r ) x FE' (p + r)FD'=qxFB'qX FE'. The first of these two equations gives (p + r ) x FD =p X FA' + r X FC '; substituting this value for its equal in the second equation, we obtain (p + r ) X FE' p X FA' r X FC = q X FB' q x FE'- 9 or, the terms multiplied by FE' being collected into one factor, (p + q + r ) X FE' = P xFA' + qxFB' + rX FC, which gives = P X FA> + 9 X FB' + r X FC' P + r directions perpendicular to the surface of tranquil waters. If the earth, or the surface of these waters, were perfectly spherical, the directions of gravity would all meet at the centre. But although this surface is not perfectly spherical, the deviation is so inconsiderable, that, for the objects we have more immediately in view, we may, without sensible error, regard the* directions of gravity as meeting at the centre of the earth. The earth being considered as a sphere, its radius is estimat- ed at 3956 miles, or 20887680 feet. From this it may be read- ily inferred, that it would require at the surface of the earth an extent of about 100 feet to subtend an angle of one second at the centre. Thus, with respect to a machine of 100 feet in length, the directions of gravity at the extremities would want only one second of being parallel. Hence, on account of the great distance of the centre of the earth, compared with the dimensions of any ma- chine, we may regard the directions of gravity as parallel. We may also for the same reason, consider the force of gravity, exerted upon different parts of the same body, as the same in point of magnitude, and capable of giving the same velocity in the same time. Centre of Gravity. 45 75. By the centre of gravity of a body, or system of bodies (that is, any assemblage whatever of bodies), we mean that point through which passes the resultant of all the particular forces ex- erted by the gravity of the several parts of the body, or system of bodies, in whatever position the body or system is placed. If, for example, in the actual position of the triangle ABC, the Fi S- 26 - resultant force of all the actions of gravity, upon the several parts of this triangle, passed through a certain point G of its surface, and in another position a b c, it should pass through the same point G, this point is what we call the centre of gravity. We shall see here- after that the resultant in question passes through the same point in all possible positions of the given body. 76. The centre of gravity is easily determined by means of what we have said upon the use of moments in finding the resul- tant of several parallel forces. . 61. Let there be any number of bodies m, n, o, whose masses we Fi S- 27 will consider for the present as concentrated in the points A, B, C, situated in the same plane. Let u be the velocity which gravity tends to give to each in an instant ; u X m, u X n, u X o, or u m, u w, u o, will be the quantities of motion, or forces, with which the bodies tend to move according to the parallel directions A" A, B"B, C"C. In order, therefore, to find the position of the resultant, we take the sum of the moments with regard to any point F, assumed at pleas- ure, in a line perpendicular to the directions of the forces, and divide this sum by the sum of the forces ; we have therefore for the value of the distance FG", at which this resultant passes, 66. FG , f _ umx FA" + unx FB" + uo x FC" urn -f un -f- u o ' or, by suppressing the common factor w, FG = m X FA" + n x FB' + o x FC" m + n + o In like manner, if we draw the lines AA\ BB, CO, parallel to FG\ and terminating in the vertical FC ; and suppose moreover, that the point G, taken in the direction of the resultant, is the centre of gravity sought, by drawing G'G also parallel to FG\ we shall have 46 Statics. FG" = G'G. FB" = B'B, FA" = A'A, whence m x A A 4- n X B'B + o X C C Cr Cr = - - - - m + /i + o that is (by considering the masses m, w, o, as representing the for- ces, thevelocity u being common), the distance of the centre of gravity of several bodies from an assumed straight line, is found by dividing the sum of the moments of these bodies (taken with respect to this line) by the sum of the masses. Let us now conceive the system of bodies m, n, o, reversed in such a manner that FA", instead of being horizontal, shall become vertical, &c. ; it is apparent, that in order to find the distance of the resultant from the line FA' 1 , how vertical, it will be necessary to take the sum of the moments with respect to FA", and to di- vide this sum by the sum of the masses ; which gives n ,, n m x A" A 4 n x B"B + o x C"C Cr Cr = - . m -\- n -\- o Having found the distance of the pointG from two fixed known lines Top. l. FA'', FO, the position of this centre G is evidently determined. It is here taken for granted that the distances A' A, A" A, B'B, BB', &c., are known, since the point through which FA", FC\ are drawn, is assumed at pleasure. 77. If the distances A'' A, B"B, &c., are each zero ; that is, if all the bodies are in the same straight line FA," the sum of the moments with respect to this line is zero ; the distance G"G is therefore zero. Accordingly, if several bodies, considered as points, are in the same straight line, their common centre of gravity is also in this line. 78. If the lines FA' 1 , FC f , are either of them drawn in such a manner as to have bodies situated on each side of it, instead of the sum of the moments, we should say the sum of the mo- ments that are found on one side, minus the sum of the moments that are found on the other side. As to the denominator of the fraction which expresses the distance of the centre of gravity, it will always be composed of the sum of the masses, since all the forces, by the nature of gravity, act in the same direction. What is here said is applicable to any number of bodies, which, being considered as points, are situated hi the same plane. Centre of Gravity. 47 The lines FA", FO, are called the axes of the moments. 79. If now we suppose the point F, which we at first took arbitrarily, to be in G, G'G and G''G become each equal to zero. Therefore the sum of the moments with respect to FA", and the sum of the moments with respect to FC, must in this case be each equal to zero. 80. We now say, that if the sum of the moments of several bodies with respect to the straight line TS, passing through the Fig. 28. point G, is equal to zero ; and the sum of the moments with res- pect to the straight line DE, perpendicular to TS, and passing also through Gt, is in like manner equal to zero ; the sum of the moments with respect to any other straight line LH, passing through the same point G, will also be equal to zero. Indeed, having let fall upon the lines DE, TS, LH, the per- pendiculars AA', AA", AA"' ; if we suppose that the point / is that in which AA meets LH, from the right-angled triangle GA'I, we have sin GIA' : GA' : : sin DGL : A' I, or TN^ r //i// r\/^ir /itr j4A"s\n DGL cos DGL : AA" : : sin DGL : A' I = _-.. cos DGL whence cos DGL Now from the right-angled triangle IAA"', we have, radius being supposed equal to 1, 1 : AI :: sin AIA"> : A A'" : : cos DGL : A A" = AI x cos DGL; that is, substituting for AI its value above found, AA" = AA cos DGL AA' x sin DGL ; hence, if we multiply by the mass m to obtain the moment, we shall have m X AJf" = m X AA' x cos DGL m x AA" X sin DGL; in other words, the moment of the body m with respect to the axis LH, is equal to the cosine of the angle DGL, multiplied by the sum of the moments with respect to the axis DE, minus the 48 Statics. sine of the same angle Z)GL, multiplied by the sum of the mo- ments with respect to the axis TS. Now it is manifest, that with regard to any other body n, we should arrive at a similar result, with the exception only of the signs according to which the bodies are on the same or on diffe- rent sides of LH. Consequently, if we take the sum of all the moments with respect to the axis LH, we shall find that it is equal to the cosine of the angle DGL, multiplied by the sum of the moments with respect to DE, minus the sine of the angle JDGL, multiplied by the sum of the moments with respect to TS. But each of these two last sums is by supposition equal to zero ; consequently their products by the cosine and sine respectively of the angle DGL, will be each equal to zero ; therefore, also, the sum of the moments with respect to any axis whatever LH, which passes through the centre of gravity G, is equal to zero. 81. Hence we infer that the resultant action of all the partic- ular actions of gravity, which are exerted upon the several parts of a system of bodies, passes always through the same point of this system, whatever be its position ; for it is not with respect to the direction of the resultant that the sum of the moments of the 68. several parallel forces may be equal to zero. Moreover, although the inquiry hitherto has been only res- pecting bodies whose centres of gravity are in the same plane, the method is not the less applicable to the case where the parts of the system are in different planes. 82. If the bodies, still regarded as points, are not in the same Fig. 23. plane, let us imagine a horizontal plane XZ, and from each of the gravitating points/*, 9, r, let the vertical lines Ap, B q, Cr, be supposed to be drawn ; and in order to determine the point JE, through which passes the resultant p ?, in the direction of which must be the centre of gravity, we take the moments with respect to two fixed lines FX^ jPZ, assumed in the horizontal plane, perpendicular to each other ; we take, I say, the sum of the moments, as if the bodies were all in this horizontal plane ; and having divided each of the two sums of moments by the sum of the masses or forces jo, ^, r, we shall have the two distan- ces E'E, E"E. It will only remain, therefore, to find at what distance .EG, below the horizontal plane, this centre is situated. Centre of Gravity. 4.9 Now if we imagine the figure reversed, the plane XZ becom- ing vertical, and ZV horizontal, it will be seen that in order to determine the distance E'G', corresponding and equal to EG, the distance sought, it is necessary, according to the method above pursued, to take the sum of the moments with respect to Z.F, as if the bodies were all in the plane ZF, and to divide this sum by the sum of the masses ; we have then every thing that is requi- site in order to fix the position of the centre of gravity. 83. Hence, by recapitulating what we have said, this prob* lem reduces itself to the following particulars ; (1.) When the several bodies, considered as points, are situated in the same straight line, we take the sum of the moments with Fig. 29* respect to a fixed point F, assumed arbitrarily in this line, and divide this sum by the sum of the masses, and the quotient will be the distance of the centre of gravity G from the point F. (2.) When the several bodies, considered as points, are all in the same plane ; through a point -F, taken arbitrarily in this Fj g> 07. plane, we suppose two lines FA", FC', to be drawn at right an- gles to each other ; and having let fall perpendiculars upon each of these two lines from each gravitating point, we imagine that these gravitating points are applied successively to the lines FA", FC, where their perpendiculars respectively fall. We then seek, as in the case just stated, what would be the centre of gravity G" in FA", and what would be the centre of gravity G' in FO ; drawing lastly through these two points the lines G"G, G'G, parallel respectively to FO, FA", and their point of meeting G will be the centre of gravity sought. (3.) When the several bodies, considered as points, are in different planes, we imagine three planes, one horizontal, and Fi 8- 23 - the two others vertical and perpendicular to each other. From each gravitating point we suppose perpendiculars let fall upon each of these three planes ; we then take the sum of the moments with respect to each plane, and dividing each of these sums by the sum of the masses, we shall have the three distances of the centre of gravity from the three planes respectively. 84. It must be recollected, moreover, in what is above said, that when the bodies are on different sides of the line or plane with respect to which the moments are considered, it is necessa- Meck. 7 -50 Statics. ry to take with contrary signs the moments of bodies that are found on opposite sides. 85. We will here make a remark, that is suggested by what has been said, and which will enable us to abridge, in many cases, the process of finding the centre of gravity, as well as the solution of other problems. Since the distance of the centre of gravity is expressed by the sum of the moments divided by the sum of the masses, if this centre happen to be in the point, line, or plane, with respect to which the moments are considered, the distance being zero, the sum of the moments must also be zero. Therefore, the sum of the moments with respect to any such plane as may pass through the centre of gravity is zero. 86. Hitherto we have considered bodies as so many points, and we have seen how the centre of gravity of all these points may be determined, whatever be their number and position. Now a body of any size or figure whatever, being only an as- semblage of other bodies or material parts, which may be con- sidered as points, it follows that, by the method above pursued, we may determine the centre of gravity of a body of any figure whatever. Also, since the centre of gravity is simply the point through which passes the resultant of all the particular efforts made by the several parts of a body in virtue of their gravity, and since this resultant is equal to the sum of all these particular efforts ; it follows, that we may in all cases suppose the whole weight of a body united at its centre of gravity, and the weight would have the same effect upon this point, when thus united, that it would have in its actual state of distribution through all parts of the body. 87. When, therefore, it is proposed to find the common centre of gravity of several masses of whatever figure, we begin by seeking the centre of gravity of each of these masses, which is attended with no difficulty. Then, the weight of these masses being considered as united each at its centre of gravity, we seek the common centre of gravity, as if all these bodies were points situated where each has its particular centre of gravity. Centre of Gravity in particular Bodies. 51 88. Accordingly, every thing which we have said hitherto upon the common centre of gravity of several bodies, considered as points, is equally applicable to bodies of whatever figure, if we take, in estimating the moments, instead of the distance of each body, the distance of its particular centre of gravity. 89. Hence, finally, if several bodies, of whatever figure, have their particular centres of gravity in the same straight line, or in the same plane; their common centre of gravity will, in the former case, be in the given straight line, and in the latter in the given plane. Application of the Principles of the Centre of Gravity to particular Problems. 90. Let AB be a straight line uniformly heavy. It will be pig. 30. seen at once without the aid of any demonstration, that the mid- dle point P, of its length will be its centre of gravity. But in order to illustrate and confirm the theory of moments, developed in the preceding articles, let us seek the centre of gravity accor- ding to the principles of this method. We imagine this straight line divided into an infinite number of points, of which P p represents one ; and that each is multi- plied by its distance from a fixed point, as the extremity A for example. We then take the sum of these products, and divide it by the sum of the parts of which Pp is one, that is, by the line AB. Accordingly, if we call AB, a ; AP, x ; we shall have Pp = d x ; and the moment of Pp will be equal to x d x, which Cal. 7. must be integrated to obtain the sum of the moments. This sum x 2 therefore will be equal to ; and in order to have it for the Cal. 82. whole extent of the line, we must suppose x = a, which gives for the entire sum of the moments. Dividing this by the sum ft ft a of the masses, we have or for the distance of the cen- 8 g tre of gravity from the point A. Thus, the centre of gravity of a straight line, uniformly heavy, is its middle point, as was before manifest. 52 Statics. Fi. 31. ^* Hence, (1.) In order to have the centre of gravity of the perimeter of a polygon, it is necessary, from the middle of each of the sides, to let fall perpendiculars upon two fixed lines AE r AC, taken in the plane of this polygon ; and, considering the gg. weight of each side as united in the middle of this side, to seek the common centre of gravity of these weights in the manner already explained. 92. (2.) The centre of gravity of the surface of a parallelogram is the middle point of the line which joins the middle of two opposite Fig. 32. sides. For, by considering the parallelogram as composed of material lines, parallel to these two sides, each will have its cen- tre of gravity in the line which passes through the middle of these same sides. The common centre of gravity, therefore, of all the lines will be in the bisecting line. Jt will, moreover, be in its middle point, since this line, considered as sustaining the 90. weights of all the other lines, is uniformly heavy. Fig. 33. 93. (3.) To find the centre of gravity of a triangle ABC ; we draw from the vertex A to the middle D of the opposite side J5C, the straight line DA, and from the point D we take DG = Indeed, the straight line DA, which divides BC into two equal parts at the point D, divides also into two equal parts every other line ZJV, parallel to BC-, accordingly, if we consider the surface of the triangle as an assemblage of material lines parallel to BC, the line DA, which passes through the particular centres of gravity of all these lines, will also pass through their common centre of gravity, that is, through the centre of gravity of the triangle. For the same reason, the line CE, which passes through the middle of AB, will in like manner pass through the centre of gravity of the triangle. This centre is consequently at the point of intersection G of the two lines CE and DA. Now, if we join ED, it will be parallel to AC, since it divides into two Geom. equal parts the sides AB, BC. The two triangles EGD, AGC, are accordingly similar, as well as the triangles ABC, EBD ; we Geom. have, therefore, DG : AG : : DE : AC : : BD : BC : : 1 : 2 ; that is, DG is half of AG, and therefore one third of AD. Centre of Gravity in particular Bodies. 53 94. Hence, In order to find the centre of gravity G of a trape- zoid, we draw KL through the middle points of the two parallel Fig- 34. sides, and from these same points K, Z/, we draw the lines KA, LD, to the vertices of the opposite angles A, D ; then having taken we join EF, which will cut KL in G, the point sought. For, by reasoning as we have done in the case of the triangle, we shall see that the centre of gravity G must be in KL. More- over, since , F, are the centres of gravity respectively, of the triangles CAD, ADB, which compose the trapezoid ABDC, the 93. common centre of gravity of the two triangles or of the trapezoid must be in EF ; it follows, therefore, that it is at the intersec- 77. tion G. To find the distance LG, which we shall have occasion to use hereafter, we draw the lines EH, FI, parallel respectively to AE ; and since KE = \KA, and LF = -| LD, we shall have EH=^AL, and FI = -* KD, or EH = J AB, and FI = J CD. For the same reason, therefore HI = | KL. Now the similar triangles GHE, GF7, give EH : GH :: FI : G/; whence EH+FI : GH + GI or HI :: FI : G/$ that is, ^5 + | CD : | ,STL : : {. CD : G/; therefore jffLx -J CD __ $ ff L x CD - i ^jy -f A CD " AB+ CD ' and, because LG = L/ + G/, if we substitute for i/ and GI tke values above found, we shall have 54 .Statics. LG - - We remark in passing, that if the height KL of the trapezoicl were infinitely small, and the difference of the two sides AB, CD, were infinitely small, these sides must, be considered as equal, so that the distance LG would reduce itself to ^ KL ** AB or x, JiJj \ KL ; that is, the centre of gravity in this case is equally distant from the two opposite bases. Fig. 35. 95. To find the, centre of gravity of the surface of any polygon, we divide it into triangles, and having found the centre of gravi- 93 - ty of each triangle in the manner above shown, we determine the common centre of gravity of all the triangles, by considering them as so many masses proportional to their surfaces, and con- centrated each at its particular centre of gravity, agreeably to 86. the method already adopted. It will hence be seen how we should proceed in determining the centre of gravity of the surface of any solid figure terminated by plane surfaces. 96. In fine, it is not always necessary to have recourse to moments in finding the centre of gravity. If it were proposed for example, to determine the centre of gravity of the perimeter Tig 36. f a regular pentagon ABODE, I should draw from the vertex of one of its angles A, to the middle of its opposite side, a straight line AH; likewise from the vertex of another angle E, to the middle of its opposite side, a straight line El, and the intersection G of these lines will be the centre of gravity. Indeed the common centre of gravity of the two sides AB, AE, is evidently the middle c, of the line b a, which passes through their middle points. The common centre of gravity of the two sides J5C, DE, is for the same reason the middle of the line IK which passes through their middle points; and the side CD has its centre of gravity in JFJ. Now it is manifest that the line AH passes through the middle points c, e, and H; it accordingly passes through the common centre of gravity of the five sides. It may be shown, in like manner, that IE also passes through the Centre of Gravity in particular Bodies. 55 centre of gravity ; therefore this centre is at the intersection G of AH and IE. 97. By pursuing the same kind of reasoning which we adopted in the case of the triangle, it might be demonstrated that the point G is the centre of gravity of the surface of a regular pentagon. In general, it may be shown, by the same method, that the centre of gravity of the perimeter, as well as of the surface of any regular polygon, of an odd number of sides, is the point of in- tersection of two straight lines, each of which is drawn from the vertex of one of the angles' to the middle of the opposite side. And when the number of sides is even, the centre of gravity, Fig. 37. both of the perimeter and of the surface, is the point of intersec- tion of two straight lines drawn through the middle points of two pairs of opposite sides. We might also extend this mode of rea- soning to the circle, by regarding it as a polygon of an infinite number of sides, and we should find that the centre of gravity of the circumference, and of the surface, is the centre. When the number of lines, surfaces, bodies, &c., is not con- siderable, the centre of gravity may be found by the method of articles 53, 54. Let the three points .#, B, C, for example, be the Fig. 38 centres of gravity of three lines, or three surfaces, or three bodies, whose weights are represented by the masses m, n, o. Having joined two of these points^ as B and C, by the line BC^ we divide BC at G', in such a manner as to give n : o : : CG' : BG' or n + o : n : : CB : CG' ; and the point G' thus found, will be the centre of gravity of the two weights n, o. We now draw G'A, and supposing the two masses w, o, united in G', we divide, in the same way, G'A in the inverse ratio of the two masses m and n + o, that is, so as to give n + o : m : : JIG : GG, or n + o + m : m :: AG : GG ; and the point G will be the common centre of gravity of the three weights m, w, o. We might proceed in a similar manner with a greater number of bodies. 56 Statics. 98. It would be easy to deduce from what precedes an easy method of finding the centre of gravity of the surface and of the solidity of any cylinder or prism. Indeed it is evident that this centre must be the middle of the line that passes through the centres of gravity of the two opposite bases, since bodies of this form, being composed of laminae or material planes, perfectly equal and similar to the base, may be considered as so many equal weights uniformly distributed upon this line. 99. To find the centre of gravity G of a triangular pyramid Fig. 39. SABC, we draw from the vertex to the centre of gravity F of the base, the straight line SF, and take in this line reckoning from F, iht part FG = J FS. To show that G is the centre of gravity required, from the middle D of the side AB, we draw DC, DS, to the opposite ver- tices C, S, of the pyramid, and having taken DF = DC, and DE = | DS, the points F, E, are respectively the centres of 93 gravity of the two triangles ABC, ASB. This being supposed, if we consider the pyramid as compos- ed of material planes, parallel to ABC, the line SF, which passes through the point F, of the base, will pass through a similarly point placed in each of the parallel planes or strata. 408. ' Thus the particular centres of gravity of the several parallel planes will all be in the line SF. For the same reason the par- ticular centres of gravity of the several planes parallel to ABS, of which we may suppose the pyramid in like manner com- posed, are all in the line EC. Accordingly, the centre of gravi- ty of the pyramid is the point G, where the two lines SF, EC, situated in the plane SDC, intersect each other. Now if we draw FE, it will be parallel to CS, since DF being a third of DC, and DE a third of DS, these two lines are cut proportionally. 199. ' The two triangles FEG, GCS, are therefore similar ; and the two triangles DFE, DCS, are also similar ; whence FG : GS : : FE : CS : : DF : DC : : 1 : 3 ; that is, FG is a third of GS, and consequently a fourth of FS. 100. As any solid may be decomposed into triangular pyr- amids, knowing the centre of gravity of a triangular pyramid, it Centre of Gravity in particular Bodies. 57 will be easy, by the method of moments, to find the centre of gravity of any body whatever. 101. Such is the general manner of finding the centre of grav- ity of bodies, the parts of which are independent of each other, or rather when we have not the expression of the law by which they are connected together. But when the parts of a figure or body have a relation that can be expressed by an equation, the centre of gravity may be found much more readily. 102. Let it be required to find the centre of gravity G, of any Fig. 40 arc of a curve AM ; we imagine an infinitely small arc M m, and take for the. axis of the moments any line GAT, parallel to the ordinates, which are supposed to be perpendicular among them- selves. Suppose, moreover, that the distance of C from the ori- gin A of the abscissas = h, h being taken of any magnitude at pleasure. To obtain the distance GG' of the centre of gravity from the axis G/V, we must take the sum of the moments of the 75. arcs Mm, and divide it by the sum of the arcs M m ; that is, by the arc AM. Now the arc Mm being infinitely small, the dis- tance of its middle point n, from the straight line CJV, may be considered as equal to MN. We shall have, therefore, Mm X MN for the moment of this infinitely small arc. But, calling AP, #, c a ] 97. and PM, y, we shall have Mm ^/dx 2 + di/ a , and MN = PC = h x ; therefore (h x) \/dx* + dy* is the moment of the arc Mm, and consequently f (h x) v'dx 2 -f- ay*, or the integral of (h x) v dx* + d y 2 , is the sum of the moments of all the infi- nitely small arcs Mm, of which the arc AM is composed. We have, therefore, AM With respect to the arc AM, which is a divisor in this quantity, we have given a method of determining it exactly, when that canCal. 96 be done; and another method of determining it by approxi- Ca j llo mation. Mech. 8 58 Statics. By a course of reasoning similar to the above, we should find that the distance GG", of the centre of gravity from the axis A p is f J AM Such are the general formulas which serve to determine the centre of gravity of any arc of a curve of which we have the equation, by means of the lines designated by # and y. 103. If the arc of which we wish to find the centre of gravi- ty, is composed of two equal and similar parts AM, AM, situated Fig. 41 * on each side respectively of the axis of the abscissas, it is evident that the centre of gravity G, will be in the straight line AP ; we have, therefore, only to find its distance from the point C. Now it is plain that the moments of the two arcs Mm, M rri, with respect to the axis NN', being equal, the distance GC will be equal to 76. *) MAM' For example, let the arc MAM be an arc of a circle ; we Tri g- have y = Vax a; 2 , a being the diameter. We shall easily fincl. Cal 98. an d indeed we have already seen, that adx We shall have, therefore, h x dx = af(h x)dx (ax x 2 )~~ *' Supposing now for the sake of greater simplicity, that the point C is the centre, then JlC = h = a ; we shall have, there- fore, 2 f (hx) ^dx^+dy* = a/(irt a?) dx (axx 2 )"^ J an integral to which no constant is to be added, because when # = 0, this integral becomes zero ; as indeed it ought, since the sum of the moments is then evidently nothing. We have, therefore, finally, i x) d x Centre of Gravity in particular Bodies. 59 and consequently, Wax x* CA x MM' = a MAM * MAM.' MAM which gives this proportion, MAM' : MM' : : CA : CG. Thus we obtain the following rule ; that the, distance of the centre of a circle from the centre of gravity of any arc of this circle, is a fourth proportional to the length of the arc, its chord and radius. These formulas may be applied to any other curve. We pass now to the consideration of the centre of gravity of plane surfaces bounded by curved lines. 104. Let it be required to find the centre of gravity of the Fig. 42. surface APM; and let G represent this centre. In order to ob- tain the distance GG', it is necessary to take the sum of the moments of the small trapezoids MP pm, with respect to CJV, and to divide this sum by the sum of the trapezoids, that is, by the surface APM. Now the centre of gravity F of this small trapezoid must be in the middle point of the straight line n K, equally dis- tant from MP and nip, which point we can suppose to be in MP, on account of the infinitely small height Pp. We shall have, therefore, FL = CP ; and the moment oiPpmM will be Pp m M X CP, that is, (hx)ydx, calling always CA, h, and AP, x. There* fore the sum of the moments will be f (h x) y d x, and conse- quently the distance rr 1 APM It will be found, likewise, that the distance 105. The centre of any plane surface may be found, in the same manner, by decomposing it into infinitely small trapezoids. For example, let the surface in question be the triangle ANN 1 , and take the base NN' and the height AC for the axes of the moments ; now calling AP, x, MM', y, and AC^ h, we shall have l * 43 ' MM m' m y d x ; and the moment of this trapezoid, with res- 60 Statics. pect to JVC, will be (h x) y d x. Therefore, the distance GG' of the centre of gravity from the base, will be - ^MM ' Now calling c the base, we have AC : AP :: JVJV : MM'-, that is, ex li : x : : c : y == -r- j c x d oc therefore f(h x) y d x becomes f(h x) ^ , or t-L (hxdx x 2 dx\ it the value of which is (-^ -|- J or C ~ (3 h 2 a?). Now the surface AMM' is -^ or ^-r- ; therefore the distance Z li GG of the centre of gravity of the surface from the base is or 1(3 A 2x), which, when x = h, becomes 7i, to which this distance is there- fore equal. Now if we draw the line AGL, the similar triangles ACL, GG'L, give LG : LA :: GG' : AC : : i h : h : : 1 : 3 ; 93 therefore LG = 1 Z*#, which agrees with what has been before demonstrated. 106. Let us now apply the formulas to curved lines. Sup- pose that APM is a portion of a circle whose diameter is a, and Fig. 44. whose centre is C ; we have then h = AC = | . Now y = \/ax x 2 ; the quantity /( h x) y dx, becomes therefore Cal. 88. f(a x)dx Vaxx 2 , or /(i x) dx (ax a? 2 ) *, which is an integrable quantity, and being integrated, gives 1 (a x # 2 ) * ; a quantity to which no constant is to be added, because it becomes zero, when #=0 ; as it evidently ought. We have, therefore, Centre of Gravity in particular Bodies. 61 With respect to GG", since y = \/aa: x 2 , we shall have, or f (ax dx x 2 dx ~~ or * 33 but we have, therefore, GG" = A If the question related to the entire segment, since it is evident that the centre of gravity , must be in the radius C^, which bi-^'g* sects the arc, and that it must be at the same distance from JVW as the centre of gravity of each of the two semi-segments APM, APM, we shall have CE = ^ 3 - A X 8 X PM 3 A MM' 3 ~ APM APM APM that is, the distance of the centre of a circle from the centre ofgravi* ty f an y one f it s segments, is equal to the twelfth part of the cube of the chord, divided by the surface of this segment. 107. The centre of gravity of a sector CMAM' may be easily Fig ' 45< found, by observing that E, the centre of gravity of the segment MdM', G that of the sector, and F that of the triangle, are all in the radius G/2; and that, according to the principle of mo- ments, the moment of the sector must be equal to the moment of the segment plus that of the triangle. We have then CMAM x CG = MAM x CE -f CM, M X CF. Now we have just found CE = %Tnj, which may be chang- We know, moreover, that CMM = PM X CP, and that CF = | CP, so that G/Jf-M' X CFis reduced to | PM X CP 2 . Substituting, therefore, these values, we have CMAM x CG = | plr 3 4- | PJIf x CP 2 = | PM(PM 2 4- GP 2 ) 62 Staties. = | PM X G^ 2 , on account of the right-angled triangle CPM 5 consequently, CG = But the surface of the sector CMAM', is equal to the arc MAM CM Geom. multiplied by , therefore 290. rr - PM x ^** ~ X CM Jtf jJJW' MAM 2 That is, //ie distance of the centre of a circle from the centre of gravity of any one of its sectors, is a fourth proportional to the arc, the radius, and two thirds of the chord of the arc. The formulas above found may be applied to any other curve, as the parabola, &c. 108. We now proceed to the consideration of curved surfa- ces, confining ourselves to those of solids of revolution. Reason- ing, then, as in the preceding articles, it will be perceived that the centre of gravity of each elementary zone, is in the axis of Fig. 46. revolution CA, and that it must be regarded as at the centre P of one of the bases of this zone, considered as having an infinite- ly small breadth. But we have seen that the expression for this Cal. 97. zone j s 2 a y ^/d-x 2 + dy*, TI representing the ratio of the diam- eter to the circumference. We shall have, therefore, (denoting always by ft, the distance AC of A, the origin of the abscissas, from JVW, the axis of the moments) 2 n (ft x) y \/d x 2 for the moment of this zone ; from which it follows that the dis- tance of G, the centre of gravity of the surface, from the point C, designating this surface by tf, will be /2 71 (fe a?) y\/dx* 109. Let us suppose, in order to apply this formula, that it is proposed to find the centre of gravity of the convex surface of 47> the right cone ANN' ; we denote AP by or, PM by We have also, on account of the similar triangles, ACN and APM, AC : CAT : : AP : PM, that is, ax h : a : : x : y = r- $ therefore 2 TT (/i a?) a/ -v/dz* +dy* becomes f 2 w x to art x a * x e dar - j ^ 7f x {ft #; x --X -- , -sr of which the integral is r-^-> or '-^(3 ft- 2*; Now, the surface of the portion AM'LMA, or tf, is equal to Geom. X circum PM, and we have n-p AC : AP :: AN : AM, = therefore, AP X AJV . a; x therefore the distance of the centre of gravity of the surface AMLMA, from the point C is ' or Therefore, when a? = ft, or AP = ^C, we shall have the distance CG, of the centre of gravity of the whole curved surface of the cone that is, the centre of gravity is found in the same manner as that of the surface of the triangle ANN 9 . 64 Statics. *ig. 48. JJQ^ p or a seconc [ example, we take the sphere. We now have y = y/q x z 2 ,(a being the diameter] andvra tf or is = - ' ; the distance of the centre of gravity therefore, is c x or A (4 h 3 a?) ; this quantity, when x = h = AC, is reduced to \ ft, and we have the height CG" of the centre of gravity G, above the base = 1 h. Now let G' represent the centre of gravity of the base ; the line AG' will pass through G, the centre of gravity of the pyra- mid ; and the parallels G"G, G X C, will give G"C or i h :. AC or h : : GG' : AG' ; whence GG 7 = 1 AG' which confirms what we have before said, 99 and shows that the centre of gravity of every pyramid, is one fourth of the distance from th centre of gravity of the base, to the vertex. Mech. 9 66 Statics. Geom. 113. With respect to solids of revolution, the general value * ' of jF + TH that of the trapezoid TKMH from the same point , will be, for the same reason, taken in connexion with the equality of the lines IE, 7L, &c., TH + KM TH + JSTJtf In like manner, the distance of the centre of gravity of the trapezoid NKMP, will be , * ,p , + AP J5CJH + AP and so on. Now if we multiply each distance by the surface of the cor- responding trapezoid, that is, by half the sum of the two parallel Ge om. sides, multiplied by their common height IE, we shall have for 178 - the series of these moments, J IE 3 X (DF + 2 TH\ J l 3 X (4 TH + 5 ^Jf), J Ifi 3 X (7 KM + 8 A"P), and so on ; the sum of which will be J IE* X (JDF -f 6 TH -f 12 JO/ + 18 JVP -f 24 QS 4- 14 It may be observed, that if there were a greater number of divisions, the multiplier of the last term, which is here 14, would be in general 2 -f- 3 (n* 2) or 3 n 4, n represent- ing the whole number of the perpendiculars DjP, TH, &c., in- cluding AB, which may be zero. So that the general expression for the sum of the moments, is reduced to IE 2 (J DF + TH + 2 KM -f 3 JVP -f 4 QS + &c. . . But it is evident that the surface ANDFPB has for its ex- pression, IE X (J DF -f TH + ITJIf + JVP 4- & c . . . + \ AB) ; and hence the distance of the centre of gravity G, namely, 68 Statics. EG = / x( + T# 4- KM + ./VT -|- &c. . . + | .tffl This formula, expressed in common language, furnishes the following rule ; To find the distance of the centre of gravity G, from one of the extreme ordinates DF, (l). Take a sixth of the first ordinate DF ; a sixth of the last ordinate AB, multiplied by triple the number of ordinates less 4 ; then the second ordinate, double the third, triple the fourth, and so on ; which may be called thejirst sum. (2). To half the entire sum of the two extreme ordinates, add all the intermediate ordinates, for a second sum ; ' (3). Divide the first sum by the second, and multiply the quotient by the common interval between two adjacent ordinates* For example, if there were 7 perpendiculars, whose values were 18,23,28, 30,30,21, 0, feet; and each interval were 20 feet; I should take a sixth of 18, which is 3; and since the last term is 0, I should add to 3, the second ordinate 23, double of 28, triple of 30, 4 times 30, and so on, which would give 397. To the half of 18, 1 should next add, 23, 28, &c. the result of which would be 141; now dividing 397 by 141, and multiplying by 20, 1 should have * 397 * 20 H or = 56 feet 4 inches nearly.* When we once know how to determine the centre of gravity of any section of a solid, that of the solid itself is easily found. Hence, by means of what is above laid down, we can determine the centre of gravity of the hold of a ship, or of the space em- braced by the outer surface of a ship's bottom. Let it be pro- posed to find the distance of the centre of gravity of this space from the keel. We imagine it composed of several laminae par- *" 61, allel to the section at the water's edge. The bulk of each lam- ina will be equal to half the sum of the two opposite surfaces of >ra - this lamina, multiplied by their perpendicular distance, and the *See Bouguer, Trail* du Navire, p. 213. Centre of Gravity in particular Bodies. 69 centre of gravity will be at the same height in this lamina, as in the trapezoid ABCD, which is a section of this lamina, made by a vertical plane passing through the keel. We see, therefore, that the reasoning to be made use of here, in order to find the height GE, of the centre of gravity, is precisely the same as that in the last case, substituting only for perpendicular or ordinate, the word section ; we have, therefore, this rule ; (l). Take a sixth of the lowest section ; a sixth of the highest, multiplied by triple the number of sections less 4; the second section from the lowest, double the third, triple the fourth, and so on; and call the result thejirst sum. (2). Take half the sum of the lowest and highest sections, and all the sections between them, for the second sum. (3). Divide the first sum by the second, and multiply the quotient 83. by the common distance between two adjacent sections. We may make use of the same method in finding the distance of the centre of gravity from the vertical line XZ, drawn through Fig. 51. a determinate point B of the stern-post, by imagining the bot- tom cut by planes parallel to the midship frame ; but as it would be necessary to measure the surfaces of these sections, it is better to make use of those which have been already measured, in the last operation ; accordingly we determine by the above method the centres of gravity G', G', of the several sections parallel to'the keel. Their distances from the vertical XZ wi 1 ! be each the same as that of the centre of gravity G / 01 the corresponding lamina, We now multiply each section by the distance of its centre of gravity from the line XZ, and regarding the several products as the ordinates of a curved line, like those in figure 50, we add the half sum of the two extreme products, to the sum of all the mean products, and divide the entire sum by the sum of all the mean sections, plus half the sum of the two extreme sections, the common thickness of the laminae being suppressed sa a com- 88, mon factor to the dividend and divisor. With respect to the centre of gravity of the vessel itself, wheth- er laden or not, the investigation cannot be reduced to so simple a process. We must take into particular consideration, the differ- ent parts which compose both the vessel and its lading. Having 70 Statics. found the moments of these different parts with respect to a hori- zontal plane, supposed to pass through the keel ; and the moments with respect to a vertical plane taken at pleasure perpendicular to the keel ; we divide each of these two sums by the whole weigh! of the vessel, and we obtain the height of the centre of gravity, and its distance from the vertical plane with respect to which the moments were considered ; and as it must also be in the vertical plane which passes through the keel,- we shall have its position. But it may be remarked, that in the calculation of these moments, we must multiply, not the bulk of each part, but its weight, by the distance of the centre of gravity of this part ; which centre is easily determined after all that has been said upon this subject. Properties of the Centre of Gravity. 115. It is evident from what we have said upon the subject of the centre of gravity, and upon the resultant of parallel forces, that if the parts of a body or system of bodies have each the same velocity, or tend to move with the same velocity, it is evi- dent, I say, that the resultant of all these motions or tendencies would pass through the centre of gravity of the body or system, and that consequently the system would move, or tend to move, as if the several masses were all concentrated at the. centre of gravity ; and were together urged with a velocity equal to that which urges each of the parts. 116. We must infer reciprocally, that if any force be applied at the centre of gravity of a system of bodies ; all the equal parts of this system will partake equally of this motion, and will all proceed with an equal velocity, obtained by dividing the quantity 2 ' of motion applied at this centre by the entire mass of the system, and this velocity will have for its direction that of the force ap- plied at the centre of gravity. Indeed, whatever be the motions distributed among the parts of the system, we see clearly that they must have for a resultant " the very force applied at the centre of gravity, since it is sup- posed that the system is free, and that there is consequently nothing to destroy any part of the force thus applied. Properties of the Centre of Gravity. 71 117. Also, since several forces applied at the same point, reduce themselves, by the preceding principles, to a single one, we infer generally, that whatever be the number, direction, and mag- nitude of the forces which are applied at the centre of gravity of a system of bodies ; (1). All parts of this system will have the same velocity ; (2). This velocity will be in the direction of the resultant of all the applied forces ; (3). It will be equal to the quantity of motion, which this result- ant represents, divided by the entire mass of the system. 118. Whence we conclude, that while the forces which act upon a body, are capable of being reduced to a single one, the direction of which passes through the centre of gravity, this body will not turn about the centre of gravity. 119. But if the forces which act upon a body cannot be re- duced to a single one, or on the supposition that they admit of being so reduced, if the direction does not pass through the cen- tre of gravity, all the parts of the system will not have a common motion. Nevertheless the centre of gravity will move in the same manner as if all the forces were applied directly at this point, as we now propose at show. 120. Let us in the first place suppose three bodies m, n, o, Fi the tension will communicate itself equally to every part, so that the two powers must be equal. * * &3Hsz / HMCss sin 13H', V *i*L CM~ // M C1*~ t 86 Statics. 151. When the number of cords united by the same knot exceeds three, being in the same plane, or when,'being in differ- ent planes, the number exceeds four, the directions being given, the ratios of the powers and of the tensions of the cords are not absolutely determinable ; that is, if a certain number of powers (not less than those just stated) be in equilibrium according to known directions, we can substitute instead of them a like number of other powers directed in the same manner, but which, having very different ratios among themselves, are notwithstanding in Fig. 60. equilibrium. If, for example, the four cords A p, A q, A r, A w, are all in the same plane, having taken AB to represent the force p, and having produced the cord ?/ A to C, we suppose the effort AB composed of two others AC, AD, the first of which is qual and directly opposite to the power vr, nothing can be infer- red from the direction AD of the action that is to oppose itself to the effort of the two powers q, r ; nothing, I say, can be inferred from this direction, except that, produced, it must pass into the angle q A r ; a condition which may evidently be satisfied in an infinite number of ways. Accordingly, if AD be drawn in any manner whatever, within the angle formed *by A q and A r pro- duced, and we construct upon AB as a diagonal, and upon the directions AC, AD, as sides, the parallelogram ACBD, and then upon AD as a diagonal, and upon q A, r A, produced, as sides^ we construct also the parallelogram AEDF, AB being taken to represent the value of p, AC may be taken to represent that of -, AF that of r, and AE that of q. This is evident, because the force AB is equivalent to the two forces AC, AD, the first of which, in order to be in equilibrium with &, must be equal to w, and the second AD is equivalent to the two forces AF, AE, which to be in equilibrium with r and q, must be equal to r and q res- pectively. But it will be seen at the same time, that by giving io AD a different direction, AC, AF, AE, will have different val- ues, but such notwithstanding that being taken to represent the powers acting in these directions, an equilibrium would be pro- duced ; so that in this case, the directions remaining the same, there is an infinite variety of ways in which an equilibrium can be effected among the powers in question. 152. The problem is of a similar character \vhen the cords proceeding from the same knot, are in different planes, and amount . >. * - . .* Rope Machine. 87 to more than four. But if the number does not exceed four, the directions being given, the ratios that must exist among the forces applied to these cords respectively, are determinate. For through any two of these cords, as Ap, A w, a plane may be supposed to Fig. 6U pass, which produced would meet the plane r A q of the two other cords in some line DAE, the position of which is determin- ed by the directions of the four powers. Then, the direction &A being produced, and JIB being taken to represent the power p, if upon AB as a diagonal, and upon the directions AD, AC, as sides, we construct the parallelogram DACB, AC will represent the value of the power , and AD the effort made by the power p against the two powers q and r acting conjointly. Accordingly, having produced qA and rA (which are ift the same plane with AD) to F and G, if upon AD as a diagonal, and upon AF, AG, as sides, we construct the parallelogram AFDG, AF, AG, wiH represent the values belonging to the two powers q and r. 1 53. Finally, whatever the case may be, whether the cords are in the same plane or not, as a state of equilibrium requires that each knot should remain immoveable, if the force or tension of each cord, applied to the same knot, be decomposed into three other forces parallel to three rectangular co-ordinates, it is nec- essary with respect to each- knot that the sum of the forces paral- lel to each of these lines should be equal to zero; (it being well un- derstood that by the word sum, as here used, is meant the sum of 129 - the forces that act in one direction, minus the sum of those which act in the opposite direction). If the cords united by the same knot, were in the same plane, it would be sufficient to decompose the tensions respectively into two forces parallel to two lines perpendicular to each other, and drawn in the same plane. This method would give in every case all the conditions of equilibri- um, the cords being supposed to be firmly connected among themselves. To give a simple example of this method, let it be proposed to find the ratio of three powers in equilibrium by means of three cords united by the same knot. Let us suppose for a moment that these three powers admit F 'g- 6a > of being represented by the three lines AG, AB, AF, and in or- der to abridge the decomposition, let the two powers p, q* be 88 Statics. decomposed in the manner indicated by the figure, that is, each into two parts, one in the direction of p, and the other perpen- dicular to this direction. Then in the right-angled triangles BAC, FAI, we shall have, radius being unity, Trig. 30. BC = AD = AB sin q AC, FI = AE = AF s'mrAC, AC = AB cos qAC, AI = AF cosr AC. Therefore, according to the principle above referred to, we shall 129. obtain, 4D -- ^E = 0; fr also AC+JII JG^O*, that i's, AB sin^ AC AF sin r AC = 0, and AB cos q AC -f AF cos r AC AG 0. The first of these equations gives AB sin q AC = AF sin r AC, and consequently, AB : AF :: sin r AC : sin q AC, that is, q : r :: sin r AC : sin q AC, 143. which agrees with what was before demonstrated. A If the value of AF be deduced from the first of the above equations, and sub- stituted in the second, we shall have AB sin qAC cos r AC AB cos q AC H --- /- jrfT- -- AG = 0, sin r J1C or jiBsinr.0Ccosg.tfC + AB sin qACcos rAC = AG sin r AC. But Trig. 11. sin r -AC cos 9 'ftC + sin 9 ^ C cos r ^ C = sin ( r A ^ + = sin 7*# r ; therefore, AE sin 9^ r = JIG sin r ^C ; that is, : : sin r AC : sin q A r. Rope Machine. 89 or, q : p : : sin r AC ; sin q A r, which agrees also with the proposition above referred to. 143. 154. We shall now inquire into the changes that take place in the communication of the action exerted by the powers in consequence of the gravity of the cords. Let there be any number of powers applied to the same cord gABC-sr drawn at its two extremities by the two powers p, w,jpi gt ($3. and retained at two fixed points p and *. If we produce the two extreme cords Q A^ C, until they meet in F, it is evident that the resultant of the particular ten- sions of these two cords must pass through the point F. And since an equilibrium is supposed, the resultant of the three pow- ers j, q, r, and of the tensions of the two intermediate cords AB, BC, must also pass through the point F; since, in order to an equilibrium, this resultant must be equal and directly opposite to the resultant of the tensions of the two cords QA,vrC. But the resultant of the three powers, and of the tensions of the two in- termediate cords, is nothing but the resultant of the three powers simply, because each of the two cords AB, C, has by itself no action whatever, and consequently no effect upon any part of the system. Therefore the resultant of all the powers jo, , as the arm to which it is applied is longer than the arm to which p is appli- ed. As to the force p exerted against the fulcrum, it is equal to the sum of the two powers 7, p ; since these being represented by FD) FB) the former, or p, is represented by BD. 162. If we make a distinction in the forces or powers, by J^L regarding one, q for instance, as giving motion, p as receiving it, and F as a pivot or point of support, we may, in the manner of the ancients, make three sorts of levers, according to the three different situations in which the agent q can be placed with re- gard to p and F. Figure 71 represents what is called a lever of the first kind, in which the agent and the resistance are on op- posite sides of the fulcrum, and the agent will have so much the more advantage according as its distance from the fulcrum is greater than that of the resistance. Figure 72 represents a lever of the second kind, in which the resistance is between the agent and the fulcrum, and which consequently is always favourable to the agent. Figure 73 represents a lever of the third kind, in which the agent is between the resistance and the fulcrum ; in this case, therefore, the power of the agent is always employed to disadvantage ; and such a lever is never to be used, where the object is to augment the effect of the agent ; that is, where -it is proposed to overcome a greater force. But as the purpose to be fulfiled is not always to increase the power of the agent, this circumstance d< es not prevent this third kind of lever being very usefully employed in machinery, where we would avail ourselves of every species of motion that we can dispose of. Thus in turning, in weaving, in spinning, and in various kind of manufacture, where great velocity, and not great force is requir- ed, and where the hands of the labourer are occupied with the more important parts of the work, this species of lever is adopted with obvious advantage, the feet being employed to give motion to the machinery. 163. Before proceeding farther, we will observe, that setting aside friction, the fulcrum is not to be considered as simply a pivot, or support. Indeed, if the fulcrum F, instead of penetrat- Fig ing into the interior of th,e lever, as represented in the figure, only touched the surface, it is evident that, although the two powers 9, p, were in the inverse ratio of the distances of the perpendiculars FM, FJL, they would still not be in equilibrium, except in the single case, where the direction AF is perpendicu- lar to BD (or to the tangent at F in figure 68) ; for, if JLF were oblique, it would clearly tend to communicate motion to the lev- er in the direction BD. Thus, we should err in supposing, for % Statics. example, that (friction and the gravity of the lever being left out Fig. 74. of the question), the two weights p and q, would remain in equi- librium in the inclined position represented, if, p being to q as Fq to Fp, the surface of the lever merely rested upon the point F. The fulcrum, in order that there may be an equilibrium in all posi- tions of the lever, must have the effect of a pin passing through it. In short, when we say, it is sufficient, that the resultant AF of the two powers should pass through the fulcrum F, it is taken for granted that the corresponding point F of the lever does not admit of any motion ; for otherwise this condition is not sufficient. Fig. 75. For example, if the lever BD were drawn by three forces jo, r/, r , applied to the three chords Bp, D q, F r, there would not be an equilibrium, if AF were the direction of the resultant of p and q, although it should pass through the point F ; it would be further 64. necessary that the point of meeting A should be in r F. 164. Since the two forces /?,, q, is to be understood the product of a cer- tain mass by the velocities, that these forces would respectively communicate to this mass, if it were free. Thus let m be a cer- tain mass, and u the velocity that the force p, acting freely, is capable of giving it ; also, let n be another mass, and v the veloc- ity that the force q is capable of giving it ; in order that there may be an equilibrium, the following proportion is necessary ; namely, m X u : n X v :: FM : FL. 166. Let 10 be the velocity produced by the force of gravity _. ^ in an instant ; and let m, n, be two heavy bodies attached to two cords BI m, DKn, which passing over two round bodies 7, K, transmit entirely to the lever BFD, according to any proposed directions BI, DK, the action of gravity of these bodies ; we shall have ic m, w n, as the measures of the forces with which these Lever. ' 97 bodies act respectively ; we must have, therefore^ in order that 27 - there may be an equilibrium, w m : wn : : FE : FC, that is, m : n : : FE : FC ; therefore, in order that there may be an equilibrium between two bodies which are urged only with the force of gravity, or between any two bodies that tend to move with equal velocities ; it is sufficient that the masses of these bodies be in the inverse ratio of the distances of their directions from the fulcrum. 167. But if the velocities with which the bodies tend to move, be unequal, it is not the masses, but the products of the masses into the velocities, which must be in the inverse ratio of the dis- tances of their directions from the fulcrum. 168. If two finite and heavy masses m, n, are urged by finite and unequal velocities, according to the directions Im, Kn; as the velocity which gravity is capable of giving in an instant (or infinitely small portion of time) is infinitely small ; in order that the two finite velocities may mutually destroy each other, it is sufficient that the quantities of motion which the two bodies would have in virtue of these velocities should be in the inverse ratio of FE, FC. But this equilibrium would not exist except for an instant ; for when these velocities are mutually destroyed, the bodies m, n, subjected to the action of gravity, would receive quantities of motion, which would be in the simple ratio of the masses, and which consequently would no longer be in the in- verse ratio of the distances FE, FC. We hence see the difference between an equilibrium among bodies urged by gravity only, and an equilibrium among bodies urged by unequal finite velocities. It may be remarked, moreover, that it is impossible to put in equilibrium a body urged by gravity only with a body urged by a finite velocity ; and we may hence conclude, that if the weight p is in equilibrium with a force q exerted by an animal, Fig- 7] this last does not tend to move the point D, except with a velo- city infinitely small. If on the contrary the force q, applied at Mcch. 1 3 98 Statics. D, acted by means of a blow or finite impression, it would raise the weight p, however great it might be, at least during a certain time, which, when p is very large, may be such that the eye can- not distinguish it ; but the motion would not be the less real. This subject will be placed in a clearer light hereafter, when we come to treat of Collision. 157. 169. By means of the ratio which we have established be- tween the two powers/?, q, and the force exerted against the ful- Fig. 68, crum F, we shall be able to solve this general question. Three of these six things, namely, the two powers, the force exerted against the fulcrum, and the three directions, being given, to find the three others. When, however, the directions only are given, we 144. can merely find the ratio of the forces p, q, p. The solution in this case is evident from what has been said. It may be easily obtained also by geometrical construction, upon which we will only observe, that when the directions are parallel, the question is solved by articles 53, 160; and that, in general, if it is pro- posed to determine the position of the fulcrum, when the powers p and q, and their position are known, the question reduces itself to finding, by article 38, the resultant of these two powers. 170. The problem is different when more than two powers are applied to the lever; in this case, as in that of the cords in article 151, we can vary without end the ratio or the directions of some of the powers, the others remaining the same, and yet not destroy the equilibrium. There is, however, this difference, between the lever and cords, that the condition of equilibrium in the former is single, whereas in the latter there are as many 153. conditions of equilibrium as there are knots. It will suffice to point out the condition of equilibrium in the lever, when three powers are employed, to make it evident that the proposition will hold true, for any greater number of powers. Fig. 77. 171. Let the three powers p, q, r, directed according to Bp, E q, Dr, be in equilibrium by means of the lever BFD. The power q may be considered as exerted in part against each of the powers jo and ?,* and in part against the fulcrum F. Having * The power q cannot, strictly speaking, be considered as exerted against r, since they both tend to turn the lever in the same direction Leter. 99 produced the directions Bp, E q, and taken from the point of meeting A, the line AH to represent the power q, we decompose this power into two others, one AG equal and directly opposite to the power/?, and the other AC such as will admit of being in equilibrium with the power r, by means of the fulcrum F, Ac- cordingly, if the direction D r meet AC at the point /, we may suppose the force AC applied at 7, according to the direction ACIL ; the force AC or IL must therefore be capable of being 33. decomposed into two others, one IK equal and directly opposite to the power r, and the other IM directed against the fulcrum F. Thus the force q produces the three effects AG, IK, IM, of which the two first, being equal and directly opposite to the forces/) and r, are destroyed, and the last, being directed against the fixed point F, cannot but be destroyed also. Now, since all the forces which act upon the lever, are p, q, r, or AG, IK, IM, and AG, IK are destroyed, we conclude that IM is the resul- tant of the three powers p, q, r, and that consequently the only condition necessary to an equilibrium is, that this resultant should pass through the fulcrum F. We see, therefore, that the powers p, q, r, act upon the fulcrum as if they were immediately applied to it according to directions parallel to those which they actually have ; and this c6nclusion would hold true for any num- ber whatever of powers, for we may always suppose one of the powers to be in equilibrium with all the others by means of the resistance of the fulcrum. 1 72. Since F must be in one of the points of the resultant, it must have the properties of which mention has already been made ; that is, when several powers, exerted in the same plane, are 63 in equilibrium by means of a. lever of whatever figure, if from the fulcrum we let fall perpendiculars upon the directions of these forces, and multiply each force by the corresponding perpendicular, in other words, if we take the moments of the forces with respect to the fulcrum, the sum of the moments of the forces zvhich tend to turn the lever in about the fulcrum. The two powers p, q, being represented by AH, and GA or AG' respectively, and being exerted in these directions, are equivalent to A C or IL, the direction of which is opposed to r, since it tends to turn the lever in the contrary direction ; and the resultant of IL and IK', that is, of p, q, r, is IM. AH 100 Statics. one direction, must be equal to the sum of the moments of those which tend to turn it in the opposite direction ; which may be expressed generally, by taking with contrary signs the moments of the forces which tend to turn the lever in opposite directions, and saying, that the sum of the moments must be zero. 173. According] jr, all that we have said with respect tq the 63 value and direction of the resultant, is applicable here to the determination of the force exerted against the fulcrum, and the position of this point, whatever be the number of powers. Fig 78. 174. Knowing, for example, the twoweightsp and ^together with the length and weight BD of the lever, if we would determine the fulcrum JP, upon which the whole would remain in equilibri- um, we should consider the weight of the lever as a new force r, applied at the centre of gravity G of the lever, and it would be necessary that the moment of p with respect to the unknown point F should be equal to the sum of the moments of r and ^, taken with respect to the same unknown point F. Let the lever BD be straight, and of a uniform magnitude and specific gravity ; and, bearing in mind, that on account of the directions of the forces being parallel, instead of the per- pendiculars jPL, FK) FM, we may employ the parts BF, FG, FD, which have the same ratio to each other, we shall have p x BF = r X FG + q X FD. Let a be the length of the lever, a; the distance BF-, we shall have, BG = I a, FG = -J a x, FD = a x. Let s be the specific gravity of the lever, or, in other v.ords, the weight of each inch in length of this lever; a and a; being also counted in inchts ; s a will be the whole weight r. We have accordingly, p x = s a ( J a x) + q (a x), = -I sa 2 sax -f- q a q x, from which we obtain x = ii^!_JL a p + s a -f q Leder. 101 Let a = 24 inches, p = 20 pounds, q = 4 pounds, s = -^ of a pound ; we shall have, 2 +' 4.24 24 + 96 - that is, the fulcrum F, in order that there may be an equilibrium, must be placed 4 T 8 inches from the extremity B ; whereas by neglecting the weight of the lever, we should have q a 96 A . . x = = .-. = 4 inches. p + q 24 If, on the other hand, the point B and the point F were given, and it were proposed to find the point D where the force q, sup- posed to be known as well as />, must be applied to produce an equilibrium ; designating BF by 6, and BD by y, the equation of the moments, becomes ' p b = s y (-|- y 6) -f- q (y 6) = \ sy 2 syb + qy qb; (2 q 2 s 6) y _. 2^ * s and _ s 6 9 V(q s b) 2 + (2 p b + 2 q 6) s s The positive value of t/ in this result gives the distance BD in figure 78, and the negative value gives the distance BD in figure 79, the distance BF being supposed without gravity. If we would have the distance or length y at which the weight of the part FD would of itself be sufficient to counterbal- ance the weight p, we should put q = 0, which reduces the above result to = y = 102 Statics: Fig so ' knowing p, 7, &F, and the specific gravity of the lever DF, we would determine the distance FD at which the power q must be placed ; designating FD by ?/, BF by fe, we shall have sy for the weight r\ and, accordingly, j> 6 -f I s # 2 = q y, from which y is easily obtained. In figure 78, it is evident that the longer the lever is, the more the power q is to be diminished, till it becomes zero, after which it must act in a contrary direction to produce an equili- brium. In figure 80, as the lever is increased in length, the power q becomes at first less and less to a certain point bej^ond which it begins to augment. This may be easily shown in several ways, and among others by the equation pl> + isi/ 2 = gry, which gives by which it will be seen, that when y = 0, q must be infinite ; and that when y is infinite, q must also be infinite. Accordingly, between these extremes the values of q must be finite, and there must be some point where it will be the smallest possible. In Cal 44 or( ^ er to determine this point, we have merely to put equal to zero the differential of the value of directions that are tangents to the circumference of the pulley. In the moveable pulley, the weight or resistance is applied Fig. 84, at the centre, or in a direction passing through the centre or 85 ' 8 * axis of the pulley. This machine, considered in a general point of view, is sus- ceptible of two sorts of motion ; one by which the rope passing through the groove of the pulley, changes its place without alter- ing the position of the body of the pulley, the other is such that the body of the pulley changes its situation at the same time. Thus a state of equilibrium requires two different conditions. The first is that the two parts of the rope which embraces the 104 Statics. pulley should be equally stretched, and thus mutually destroy each other. The second condition is derived from the first in the following manner ; . 176. From the tension of the two parts of the rope which passes over the pulley, there results an effort upon the body of the machine which may be determined by taking in the direc- Fig. 83, ti ns f tne r P es ? beginning at their point of meeting, I A, IB, equal 83, 84. to each other, and forming the parallelogram IADB, in which the diagonal ID represents the force exerted upon the body of the machine, I A being considered as representing the tension of the rope Op or OG. Now since IT, 70, are tangents, and IB = IA, it will be seen that ID produced would pass through the centre F of the pulley. Therefore, if the body of the pulley is not firmly fixed, ID cannot be destroyed, except the obstacle, what- ever it be, which is to prevent the motion of the body of the pulley, is situated in some point of the line IF, extending from the centre F to the point of meeting of the two ropes. Thus, if the pulley is destined to turn in a block FG, fixed to some point G without, and admitting of a motion about G, an equilibrium will Flg ' 8 ' not take place except when the block has the direction FL In like manner, if the body of the pulley, being embraced by a rope fixed to the point G, is moveable, there will not be an Fig. 84. equilibrium, except the effort applied at the centre F, or to the fixed block at this centre, is exerted in such a direction as to bisect the angle formed by the two parts of the rope OG, Tq and is at the same time to the tension of OG, T q, :: ID : IA : IB. 177. It is now easy to find the ratio of the tension of each part of the rope that passes round the pulley, to the force exert- ed upon the body of the pulley, and consequently to the force of which the moveable pulley is capable. The tension of each part ' of the rope being represented by 1A or its equal IB, the effort which is exerted upon the body of the pulley, will be expressed by ID. But in the triangle IAD Pulley. 105 IA : ID : : sin IDA : sin IAD, sin Flq : sin OAD, sin FI q : sin G7 q. We may say, therefore, universally, that when there is an equili- brium by means of the simple pulley, freed or moveable ; (1.) The tensions of the two parts of the rope which passes round the polity, or the powers applied to them, are equal', (2.) That each of these powers is to the force exerted at the centre of the pulley, as the sine of half the angle formed by the two parts of the ropes in question, is to to the sine of the whole of this angle. Thus in the fixed pulley, there is no other advantage gained Fi g- 82, by the agent q, except that of being able to change at pleasure the direction in which the action shall be employed. But in the moveable pulley, there is possessed by the agent 9, the double pig. 84. advantage of a change of direction and an augmention of the 85j 86- effect of the action. But it is to be remarked, that according as the direction is changed, the force exerted upon the centre va- ries, so that there is a direction in which the effect produced by a given power is the greatest possible ; and this is when the two parts of the rope GO, Tq, are parallel, as will be readily per- ceived. 178. If we draw the radii OF, FT, and the chord OT, the Fig. 84. the triangle OFT, having its sides perpendicular respectively to those of the triangle BID, will be similar to BID ; whence Geom. IB : ID : : FT : OT, or q : g :: FT : OT '; that is, the tension of either part of the rope is to the force exerted against the centre F, as the radius of the pulley is to the chord of the arc embraced by the rope. Now it is evident that this last ratio is the greatest possible when the two parts of the rope are parallel ; hence in the move- able pulley the power is the least possible, or is exerted to the greatest advantage, when the two parts of the rope are parallel ; and it is then half of the force exerted against the centre of the pulley. This second kind of pulley is made use of in tightening Mech. 14 106 Statics. the sails of a vessel, by attaching it to one of the corners as rep- resented in figure 86. Fig. 87. 179. If, therefore, the weight p is sustained by the power 5, by means of several moveable pulleys, embraced each by a rope, one extremity of which is attached to a fixed point, and the other to the block of a pulley, the ratio of the power to the weight will be that of the product of the radii of all the moveable pulleys to the product of the chords of the arcs embraced by the ropes. Indeed if we call , p, the forces exerted at the centres of the pulleys JV, JW, which are at the same time the tensions respective- ly of the two ropes attached to the centres of N and M\ R, R/, R", being the radii, and c, c', c", the chords of the arcs embraced 178. by the ropes in the several pulleys .AT, .A/, L, we shall have q : s : : R : c, -a : (> : : R' : c', 9 : p : : R" : c" ; Alg.226. whence, by taking the product of the corresponding terms, (}&(> : -srQp :: RR'R" : c c'c", or q : p : : RR'R" : c c f c" ; that is, when the cords are parallel, which gives c = 2 R, c' == 2 R 7 , c" * 2 R", q : p : : RR'R" : 2 R X 2 R' X 2 R", : : 1 : 2 X 2 X 2; in other words, the power is to the weight as unity to the num- ber 2 raised to the power denoted by the number of moveable pulleys. With three pulleys, for example, the power would sus- tain a force eight times as great. 130. But this arrangement of pulleys is not the most conven- ient. It is more common to employ one of the forms represented in figures 88, 89, 90, 91, 92, in which all the pulleys, both fixed and moveable, are embraced by the same rope. Moreover all the fixed pulleys are attached to one block, and all the moveable pulleys to another. Sometimes the centres are distributed upon different points of the same block as in figures 88, 89, 90, 91. Pulley. 107 Sometimes they are united upon the same axis as in figures 92, 93. The latter arrangement has the advantage of being more compact ; but when a large number of pulleys are thus disposed in the same block, the power being applied on one side instead of being directed through the middle, the system is drawn awry, and part of the force employed is lost by the oblique manner in which it is exerted. This inconvenience does not belong to the pulleys represented in figures 88, 89, 90, 91 5 and in that repre- sented by figure 94, the peculiar advantages of the two systems are united. Here two sets of pulleys having a common axis are attached to the moveable block, and two to the fixed block, the inner set in each case being of a less diameter than the outer so as to allow a free motion to the rope. Then the rope com- mencing at the middle of the upper block after being made to pass over all the pulleys will terminate also in the middle. This arrangement was invented by Smeaton. 181. But whatever difference there may be in this respect in the particular disposition of the pulleys, the ratio of the power to the weight may always be found by the following rule. The power is to the weight as radius, or sine of 90 , is to the sum of the sines of the angles made by the several ropes (meeting at the moveable pulley) with the Iwrizon. Indeed, if upon each of the ropes we take the equal parts /,/, JVP, &c., to represent the tension, and upon each of these J^S- 88 ' lines, as a diagonal, we form a parallelogram, having one pair of its opposite sides vertical, and the other pair horizontal ; instead of considering the weight p as sustained by the immediate ten- sion of the rop^.s, we may regard it as supported by the hori- zontal forces IK, JVO, &c., and the vertical forces /L, JVQ, &c. Now the first being perpendicular to the action of the weight contribute nothing to counterbalance this action; and in the case of an equilibrium these horizontal forces mutually destroy each other. The weight />, therefore, is wholly sustained by the resultant, that is, by the sum of the vertical forces /L, JVQ, & c j and the ropes being all equally stretched, it is evident that q is to p as the tension of one of these ropes is to the entire sum of the vertical forces. But in the right-angled triangles IML, JVQP, &c., we have IM : IL : : 1 : sin IML ; NP or IM : NQ : : 1 : sin 108 Statics. the same may be said of the other ropes ; whence IL = IM sin IML ; NQ = IM sin NPQ ; accordingly, q i p n IM : IM sin ML -f /./V sin JVPQ + &c., : : 1 : sin IML + sin NPQ + &c. If the ropes are parallel and consequently vertical, the angles IML, NPQ, &c., will be right angles, and their sines will be each equal to radius, or 1. Therefore the power in this case will be to the weight as 1 is to the sum of so many units as there are ropes meeting at the moveable pulley. Hence it will be seen, that if one of the extremities of the rope is attached to the fixed Fig. 88, pulley, the power will be to the weight as unity is to double the num.' ber of moveable pulleys ; and if the extremity of the rope is attached g.J s< 89 ' to Hit moveable pulley, the power will be to the weight as unity is t& double the number of moveable pulleys, plus 1. 182. The general proposition above demonstrated, holds true, whether the ropes are in the same plane or not ; and if the obstacle to be overcome be not a weight, that is, if the direction of the whole power of the pulley be not vertical, we have only to substitute for the angles which the ropes are supposed to make with the horizon, those which they would make with a plane perpen- dicular to the whole action of the pulley. In figure 93, for exam- ple, the power q is to the force exerted at G, as radius is to the sum of the sines of the angles made by the several ropes (meet- ing in CF) with a plane perpendicular to FG. 183. If several sets of pulleys are employed, it will be easy after what has been said to assign the ratio of die power to the weight. In figure 93, for example, the ropes being supposed parallel, the power q will be to the force exerted in the direction is*. CB, as 1 is to 5. Now this last force performs the office of a power with respect to the system of pulleys BA, and accordingly is to the weighty, as 1 to 4. Therefore the power q is to the weight p, as 1 X 1 is to 4 X 5, that is, as 1 to 20. 184. In all that precedes, we have- supposed the system of pulleys destitute of gravity and friction, and the ropes perfectly flexible. We shall see hereafter what allowance is to be made for friction and the stiffness of the ropes. With respect to the Wheel and Axle. 109 gravity of the parts of the system which the power has to sustain, allowance is made for it, in the case of an equilibri- um by adding it to the weight, when its action coincides with Fig. that of the weight. But if, as in figure 93, the gravity of the system CF is not exerted in the same direction with the power q. BC, instead of being in the same direction with the power , q : 'p : : BI : AE 5 Wheel and Axle. Ill that is, in the wheel and axle the power is to the weight, as the radius of the cylinder to the radius of the wheel. 187. If the weight/) be attached at some point B in the plane of the wheel, in such a manner, that the perpendicular IB upon its direction shall be equal to the radius of the cylinder, we may consider AIB as an angular lever the fulcrum of which is at the centre /; and in order to an equilibrium, we must have 159 q : p : : El : AI, that is, the ratio between the power and the weight would be the same as the above. Therefore the action of the power is trans- mitted to the weight by means of the wheel and axle, in ihe same manner as if the power and the weight were in the same plane. 188. It is not the same, however, with respect to the force exerted against the supports. This varies according to the dis- tance of the plane BDL from the plane of the wheel. In order Fig.lOL to determine what it is, we decompose the power q, considered as applied at parallel to A q, into two forces parallel to A q, and passing through F and F. We decompose likewise the S5 power /?, considered as applied at /, into two forces parallel to p D, and passing through F and F. By this means each support will be urged by two forces, the magnitude and direc- tions of which will be known. It will be easy, therefore, to re- duce these forces, in the case of each support, to a single one of a known magnitude and direction. This method of finding the forces exerted against the two sup- ports, is founded upon the fact, that the two forces & and p reduce themselves to one which acts at /. If we conceive this decompos- ed into two forces parallel to the direction & and/?, and applied in /, they will have simply the values of & and jo. Accordingly, (1.) we may regard/? as applied at 7; (2.) The force ar, considered as applied at /, and the force r applied at G, cannot but have for a resultant the force q, by which they are produced, as we have seen above ; moreover this resultant passes through E, since G7 : GE : : GB : GA : : q : *. 189. If the power, instead of being applied in the direction of a tangent to the wheel, acted by means of the arms EE, and i' 112 Statics. at right angles to their length, the ratio of the power to the weight would always be found to be the same as above stated, by substituting for radius of the wheel the words length of the arm; this length being reckoned from the axis of the cylinder. But Fig. 99. if the power acted in a direction not perpendicular to the arm IE, instead of the length of the arm we should take that of the per- pendicular IR let fall upon the direction of the power ; so that in this case the power will be to the weight as the radius of the cylinder to the perpendicular IR. Fig.101. 19a g ince ^ . p .. gf . ^ we Jj ave ^ x __ p x JT. that is, the moment of the power is equal to the moment of the weight, these moments being taken with respect to the axis FF. If, therefore, several powers are employed at the same time, ap- plied to different arms, the sum of the moments of these powers must be equal to the moment of the weight. 191. If the cord which supports the weight or which transmits the action of the power to the weight, were wound round a conical surface, or a surface of a variable diameter, instead of that of a cylinder, the ratio of the power to the weight, would also vary continually ; and reciprocally, if the power, whose action is to be communicated through the medium of such a machine as that under consideration, varies continually, and is intended, notwith- standing, to produce the same effect, we arrive at the end pro- posed, by causing the action to be applied successively to radii that increase in length according as the power diminishes. We have an example of this adaptation of the machine to a varying power in watches and chronometers, in which the moving or maintaining power is a spring fixed at one of its extremities to Fig.103. the axis or arbor of a barrel Z, and which, after several revolu- tions or coils, is attached to the interior of this barrel. A chain with one of its extremities fixed to the convex surface of the bar- rel is wound round the conical axis or fusee F, to which the other extremity of the chain is attached. As the spring uncoils, the barrel turns, and drawing the chain, causes the fusee to turn ; but since the force of the spring diminishes as it uncoils, a com- pensation is made for this reduction of the power, by giving a greater diameter to those parts of the fusee on which the last coils of the spring are exerted. By this contrivance, the ma- chinery receives nearly equal impulses in equal times. Wheel and Jlxle. 113 192. It would seem, therefore, by having regard only to an equilibrium, that we might diminish at pleasure the ratio of the power to the weight, and make a force, however small, counter- balance one however large, by means of the wheel and axle, and such machines as depend upon the same principle. But if we take into consideration their motion, and have respect also, as we must, to the nature of the agents to be employed, we cannot aug- ment the effect at pleasure. The ratio of the radius of the cyl- inder to that of the wheel, is not arbitrary. It requires a par- ticular adaptation to the purpose proposed, in order to produce the greatest possible effect. Suppose, for example, that the agent applied to the arm E, Fig 99. tends to move with a velocity w, a id that the force of which it is capable, is raw, that is, equal to a known mass ra urged with a velocity u. Let v be the velocity with which the point E would be moved in virtue of the resistance of p ; then, if we call jD the perpendicular distance of E from the axis, and tf that of p from the axis, we shall obtain the velocity thatp would have, by the proportion, since it is evident, that the point E and the point where the cord touches the cylinder, would have velocities proportional to tLeir distances from the axis. We must suppose, therefore, that at the instant when the power comes to exert itself, the velocity u is composed of the velocity i>, which actually takes place, and the velocity u i>, which is destroyed ; and that at the same instant the weight p has the velocity -^-, which actually takes place, and the velocity -jr- in the contrary direction which is destroyed ; that is, the moving force m (u v) must be in equilibrium with the mass p, urged with the force ? f) . Accordingly, whence muD* V m D* Meeh. 15 114 Statics. and the velocity of p, namely -^- will be mud D m D* -f p d a* Therefore, in order to know what ratio there must be between -D and d, in order thatp may have the greatest velocity possible, it is necessary to put equal to zero the differential of this express- ion, taken by regarding <5 only as variable ; thus muDdS (?nD 2 + pd*} muDS X Zpddd = 0,' (mD* +p$ 2 ) 2 or muDdd (mD 2 + p J 2 ) m uDd X 2p$ dd = 0, whence or m which gives If, for example, the weight /> be lOOOOOlb, and the mass m or moving force be equivalent to a weight of lOlb, we shall have = D 1-12 \ 1000 !2- = J> 1 00000 that is, the radius of the cylinder must be a hundredth part of the arm /E, in order that the effect may be the greatest possible. 193. There are many machines which are referrible, either wholly or in part, to the wheel and axle, and consequently to the lever ; such as rack-work, machinery in which wheels are con- nected by bands, tooth and pinion work, and instruments intended for drilling, boring, and screwing, although these last operations Fig.105. often depend in part upon another machine that remains to be described, namely, the inclined plane. In rack-work, the axis FE having a winch FR q, carries a pinion the teeth or leaves of which act upon the toothed bar AB. The leaves of the pin- ion, in turning, raise the bar JIB with a force which is to the force q applied to the winch, as the radius of the winch is to that of the pinion ; and as the radius of the pinion is for the most part small compared with that of the winch, by the aid of such a Wheel and Axle. 115 machine, we are able to raise a very considerable w< ight with a moderate force. 194. Toothed wheels serve several purposes. Sometimes we employ them to augment a force, at others to increase a velocity, often to change the direction of a motion, and still more frequently to adapt motion to certain periods of time, or to render sensible certain motions or spaces, that the eye cannot distinguish. Several toothed wheels W, X, Y, Z, being connected together Fig.lOS. by the pinions w, a?, ?/, z, it is proposed to find the ratio of the power , D", D"', be the greater distances or radii of the wheels, d, d', d", d'", the less distances or radii of the pinions. We shall consider the effort made by the leaf of any one of the pinions upon the tooth of the neigh- bouring wheel, as a power applied to this last ; then E, E', E", being these efforts, we shall have 186 q : E : : d : D, E : E' : : 6' : &, E' : E" : d" : D", E" : p : : d'" ' D"' ; whence, by taking the products of the corresponding terms, q : p : : d d' d" d'" : D- D' D" D"' ; that is, the power is to the weight as the product of the radii of all the pinions to the product of the radii of all the wheels. If, for example, the radius of each pinion is one tenth of that of the corresponding wheel, we should have q : p :: 1 : 10 X 10 X 10 X 10, 1 : 10000; that is, a power of one pound would counterbalance a weight of 10000 pounds. What is gained, however, in point of force by the use of wheels and pinions, is lost in respect to velocity. Indeed while the wheel W turns once, the pinion w, turning in the same time, causes to pass only as many teeth of the wheel A 7 , as it has leaves ri S- 106 - in its own circumference, so that if the wheel X has 48 teeth, and the pinion w six leaves, the wheel X would make only T 8 or | part of a revolution, while W turns once round ; it will hence be H6 Statics. seen that the wheel X goes just so much slower than W, Y so much slower than X, and so on. 195. From what is above said, it will be perceived how, bj means of toothed wheels, the velocity may be augmented in any Fig.107. given ratio. Let there be, for example, the toothed wheel W, acting upon the pinion w ; it is clear, that during one revolution of W, the pinion zv will turn as many times as the number of leaves in the pinion is contained in the number of teeth of the wheel ; that is, during one revolution of the wheel, the pinion will turn times, JV denoting the number of teeth in the wheel, and v the number of leaves in the pinion. If therefore the axis of the pinion w carries a wheel, which acts also on a pinion a:, we shall see that during one revolution ,7V* of the wheel X^ or of the pinion w, the pinion x will turn T times, JV denoting the number of teeth in the wheel X, and V* the number of leaves in the pinion x. Therefore while the wheel X N makes a number of turns expressed by , that is, during one revolution of the wheel W, the pinion x revolves a number of JV' jV N' 'V times expressed by X or -f . And by reasoning in this manner for a greater number of wheels and pinions, it will be perceived that the number of times that the last pinion turns, during one revolution of the first wheel, is expressed by a frac- tion having for its numerator the product of the number of teeth in the several wheels, and for a denominator the product of the number of leaves in the several pinions. When it is asked, therefore, what must be the number of teeth and leaves for a proposed number of wheels and pinions, in order that the velocity of the last piece shall be to that of the first in a given ratio, the question is indeterminate, that is, one which admits of several answers. Two examples will suffice to show how we ought to proceed in questions of this kind. We will suppose that it is required to find how many teeth must be given to the two wheels W and Jf, and how many leaves to the pinions w and , in order that the pinion x may make 50 revolutions while the wheel W makes one. We shall have Inclined Plane. 117 . = 50. vv We know in this case only the quotient obtained by dividing NN' by r i' ; we do not know either the dividend or the divisor. Let us take, therefore, arbitrarily for the divisor v V' a number composed of two factors which shall be neither too small nor too great for the number of leaves to be allowed to the pinions. Sup- pose, for example, v v f 1 X 8 = 56, v being 7, and i>' 8. We shall then have ~, = 50, or JVJV' = 50X56. Now 50 and 56 56 not exceeding the number of teeth that can be given to the wheels W and X, I will suppose JV to have 50 ; and consequent- ly those of N' will be 56. If these two factors, or one of them, should happen to be too great, I should decompose them into their prime factors, and see if from the combination of these fac- tors there would not result two smaller factors ; or another num- ber might be taken for v v'. Suppose, for a second example, that it is proposed to find the number of teeth and leaves to be given to three wheels and their pinions, in order that while the last pinion turns once in twelve hours, the first wheel shall require a year to make one revolution. The common year consisting of 365,25 X 24 X 60 or 525949 minutes, and 12 hours being equal to 12 X 60 or 720 minutes, it is evident that during one revolution of the first wheel the last pinion will make a number of revolutions expressed by JVYW" 5 Wo 49 5 we have, therefore, , = 52 T Yo 49 . Let us take .AOV./V" arbitrarily v = 7, v' = 8 ; and we shall have - -zss or JVWJV" = 5 9 X 7 X 8 v" = X 3681643 v" 90 ' Of the Inclined Plane. 196. Jfabody^of any figure whatever, touching a plane Fig 108. XZ in any point C, is urged by a single force, it can remain at rest on this plane only when the direction of this force is perpen- dicular to the plane, and is such at the same time as to pass 118 Statics. through the point C. The necessity of the first condition is evi- 88. dent. As to the second, it will be seen, with a moment's atten- tion, that this is not the less necessary ; since, if the direction AT) of the body p', for example, although perpendicular to the plane, does not pass through the point of contact C', the resist- ance of the plane, which cannot be exerted except according to the perpendicular at C, would not be directly opposed to the force AD, and consequently would not destroy it, even when it 137. is supposed equal to this force. 197. If the body, instead of touching the plane only in one Fig. 109, point, touches it in several points, it is not indispensable that the 10 ' single force AD, which acts upon it, should pass through any one of these points ; but it is necessary that it should be perpendic- ular to the plane, and that it should be capable of being decom- posed into as many forces perpendicular to the plane, as there are points which rest upon it, and that they should be such as to pass through these points. Thus if the bodyp, for example, were Fig.109. in contact with the plane at the points C, C', and the force AD were not in the plane which passes through the two perpendicu- lars raised at the points C, C', an equilibrium would not take place, because the force AD could not be decomposed into forces passing through C and C', without a third arising which would not be counterbalanced. 198. Hence, if a body which touches a plane in one or in several points, be urged by several forces directed at pleasure, it is necessary, (1.) That these forces should admit of being reduc- ed to a single one perpendicular to the plane ; (2.) That this, in the case where it does not pass through one of the points of con- tact, should be capable of being decomposed into as many forces parallel to it, as there are points of contact, and that these should pass each through one of the points of contact. 1 99. If the single force which urges a body be gravity, it is necessary that the plane should be horizontal ; and if the vertical plane, drawn through the centre of gravity of the body, do not pass through one of the points of contact, it is necessary, at least, that it should not leave all the touching points on the same side. Inclined Plane. 119 200. If therefore, the body be urged only by two forces, it is necessary; (1.) That the two forces should be in the same plane ; (2.) That this plane should be perpendicular to that on which the body rests ; (3.) That the resultant (which must be always perpendicular to this last plane), should not leave all the points of contact on the same side ; and if one of these forces be gravity, it is necessary, moreover, that this plane should be verti- cal and pass through the centre of gravity of the body. 201. Let us now see what ratio must exist between two forces which hold a body in equilibrium upon a plane. Let F q, Fp, be the directions of these two forces, and AE the intersection ofFig.m the plane of these forces with that upon which the body rests ; having drawn the perpendicular FH upon AE, let us suppose that on this line, as a diagonal, and upon F q, Fp, as sides, that the parallelogram FEDC is constructed. In order that the resultant of the two forces q and p may be directed according to FD orFH, it is necessary that the two forces q and p should be to each other as FC to FE ; and then the two forces p and q, and the pressure which they exert upon the plane, and which I shall represent by p, will be such as to give the proportion q : p : p : : FC : FE : FD. 38. 202. According to article 40, we have likewise q : p : p : : sin EFD : sin CFD : sin EFC. 203. From the two points A, E, taken arbitrarily in AE, we let fall upon the directions of the two forces q,p, the perpendicu- lars AG, EG. The triangle AEG having its sides perpendicular respectively to those of the triangle FDE, the two triangles will be similar ; hence AG : EG : AE : : DE or FC : FE : FD : : q : p : g accordingly AG : EG : AE : : q : p : g . But Trig. 32. AG : EG : AE : : sin AEG : sin EAG : sin AGE, therefore 120 Statics* q :/:?:: sin ABG : sin BAG : sin AGB ; that is, when two forces only act upon a body to retain it in equilibrium upon a plane ; if we imagine two other planes to which the forces are perpendicular, these two forces and the pressure upon the given plane, are represented each by the sine of the angle comprehended between the planes to which the two other forces are perpendicular. 204. Since the ratios which we have established, take place Fig.l 12. whatever be the nature of the two forces^) and q, they will hold true when one of the forces p for example is gravity ; in this case the plane BG is horizontal, and the intersection BG is called the base, and AL, perpendicular to BG, the height of the plane. 205. Since by article 202, q : p : p : : sin EFD : sin CFD : sin EFC, we have q : p : : sin EFD : sin CFD, : : sin HFp : sin HF q ; if, therefore, knowing the weight p, the power q, and the angle HFp, which the direction of the weight p makes with the per- pendicular to the plane, we would determine the angle which the direction of the power q must make with the same perpen- dicular, we shall obtain it by the above proportion, which gives But, when an angle is determined by its sine, there is no reason for taking as the value of this angle, the angle itself found in the Trig. 13. tables, rather than its supplement. Accordingly, the same weight may be supported upon the same plane, by the same power, di- rected in two different ways. These two directions must there- fore be such that the two angles HF q, HF q, which they form with the perpendicular FH, may be supplements to each other. Now if we produce the perpendicular HF, toward /, the greater of these two angles HF q is the supplement of q FI ; therefore, since it must also be the supplement of the smaller angle HF q^ it follows that q FJ is equal to the smaller angle HF q. Hence: Inclined Plant. 121 the two directions according to which the same power will sus- tain a given weight upon the same plane, are equally inclined with respect to a perpendicular to this plane, and consequently with respect to this plane itself; and they both fall on the side of a perpendicular to this plane, opposite to that in which the gravity of the body is directed. 206. In the same proportion, q : p : : sin HFp : sin HF q, if, instead of the angle HF 'p, we put the inclination ABG of the plane, which is equal to this angle, and instead of sin HFq, its Geom. equal cos A'F ' q, FA' being drawn parallel to BA, we shall 209 ' have q : p : : sin ABG : cos A'Fq, and hence x 9in Therefore, the inclination of the plane and the weight remaining the same, the power q must be so much the smaller, as the cosine of its inclination to the plane is greater ; accordingly, as the greatest of all the cosines is that of 0, we say that the direction's- 1 **- in which a power acts to the greatest advantage, in sustaining a weight upon an inclined plane, is that which is parallel to this plane. 207. In this case the proportion q : p :: sin ABG : cos A'Fq becomes q : p : : sin ABG : 1 or radius. Now if, from the point A, we let fall the perpendicular AL upon Fig113 the horizontal line BG, we shall have in the right-angled triangle ALB, therefore q ' p :: AL : AB-, that is, when the power acts in a direction parallel to the plane ; it to the weight as the height of the plane is to its length. Mech. 16 122 Statics. Fig.U4. 208. If the direction of the power be horizontal, the angle A'F q, being the complement of BAL, the proportion becomes q : p : : sin ABG : cos A'Fq, q : p : : sin AEG : sin BAL, Trig. 32. :: ^ : BL; that is, when the direction of the power is parallel to the base of the inclined plane, the power is to the weight as the height of the plane to its base. From the proportion q : p :: sin ABG : cosd'Fq, we infer, as a general conclusion, that so much less power is re- quired according as the inclination of the plane is less, and according also as the inclination of the power to the plane is less. We have said nothing of the point where the direction of the power is to be applied to the body. This point is determined only by the condition, that the direction of the power meet the ver- tical drawn through the centre of gravity of the body in a point from which a perpendicular let fall upon the plane has the con- ditions mentioned, article 1 96, &c. We hence see that a homogeneous sphere cannot be sustained upon an inclined plane, except when the direction of the sustain- ing force passes through the centre of the figure, which is at the same time the centre of gravity. 209. If several powers, instead of one, are opposed to the action of the weight, what we have said respecting the power q, is to be understood of the resultant of these several powers. If the Fig.115.body p, for example, is supported upon an inclined plane by the cojnbined action of a power 9, and of the resistance of a fixed point jB, to which is attached the cord HD q, passing round the body ; through the point of meeting S, of the two cords BH, q D, suppose a line SF drawn so as to bisect the angle formed by the cords. If this line cut a vertical line passing through the centre of gravity in a point F, from which a perpendicular can be let fall upon the plane that shall pass through the point of contact Inclined Plane. 123 If, the equilibrium will be possible ; and the ratio of the weight p to the effort in the direction SF will be determined by the fore- going rules. The ratio of the effort, in the direction F, to the power , to be perpen- dicular to the back of the wedge, since if it is not, it may always be decomposed into two others, one perpendicular, and the 45. other parallel to the back, of which the latter is incapable of 35. urging the wedge backward or forward. This perpendicular force, therefore, may be considered as keeping the wedge ABC in equilibrium, while pressed at 7, K, by the parts of a body that tend to unite. The theory of the inclined plane is accordingly 211. applicable to this case, and the resistance exerted at 7, K, can- not destroy the action of the power p, except while this power admits of being decomposed into two others q, r, passing through these points, and directed perpendicularly to the faces BC, AC, of the wedge. Therefore, the forces/?, q, r, must meet in the same point E, be in the same plane ABC, and have the following pro- portion to each other, namely, p : q : r :: sin q E r : sin p Er : sin p E q, reom or ' s ^ nce ^6 sines of the angles q E r, p E r, p E q, are equal res- so pectively to the sines of their supplements C, A, B, Trig. 13. p : q : r : : sin C : sin A : sin B, :: AB : BC : AC, that is, the three forces p, q, r, are to each other as the three sides of the triangle to which their directions are perpendicular. Principle of Virtual Velocities* 131 224. The three straight lines AB, EC, AC, are to each other as the faces of the wedge to which they respectively belong, for these faces are parallelograms of the same base, and whose al- titudes are JiB, BC, AC; it follows, therefore, that the power pGeom. and its two components, are to each other as the back and two 175t sides of the wedge, or in other words, that the power being repre- sented by the back of the wedge, the force exerted against the two sides will be represented by these sides respectively. A very acute wedge, therefore, or one whose sides are very long compared with the back, possesses an advantage in the same proportion, and may be made to exert a great power by means of a very moderate blow on the back. 225. There has not been a perfect agreement among mechan- ical writers as to the theory of the wedge. The direction of the resistance has sometimes not been sufficiently attended to, and the circumstance of one of the resistances proceeding from an immoveable obstacle in certain cases, has sometimes been over- looked. 226. To the wedge are referred all cutting instruments, as knives, scissors, the teeth of animals, &c. A saw is a series of wedges on which the motion impressed is oblique to the resist- ance. A wimble is a combination of the screw and the wedge. To the wedge of the pyramidal form, are reduced all piercing instruments, as nails, bayonets, stakes, piles, &c. General Law of Equilibrium in Machines. 227. By combining together, in different ways, the machines above considered, we can form others, the number of which may be multiplied without end. With respect to compound machines, we determine the ratio of the power to the resistance necessary to an equilibrium, by having regard to the tensions of the cords that connect the different parts, this ratio being supposed to be known for each of the component parts. But however complicated the machine, there is a simple rule by which the ratio of the power to the resistance is obtained 132 Statics. directly from the ratio of the spaces which the points of appli- cation of the two forces tend to describe in the same time. This is a particular case of what is called the principle of virtual velocities. 228. Let us suppose a very small motion given to the ma- chine, and that the points of application of the two forces, describe curves to which the directions of these forces are tangents. Let u denote the velocity of the force />, or the space described in any given time by />, and v the corresponding velocity of , nec- essary to an equilibrium, is required, we shall obtain it very near- ly by the proportion q : p :: u : i>; and this will approach so much the more nearly to the exact ratio, according as the motion impressed upon ihe machine is less, Trig, 16. so that by taking the limit of the ratio of u to i>, we shall have exactly the ratio sought of q top. If the directions of the forces q,p, are not tangents to the curves described by the points of application of these forces, we take, instead of the spaces described by these points, the projec- tions of these spaces upon the directions of the forces, and the inverse ratio of these projections will be the ratio of the forces, in the case of an equilibrium. We are at liberty to take for the point of application of each force, any point we may choose in its direction, provided we regard it as firmly attached to the machine. 229. We shall now apply this rule to a few examples in order to show its truth and utility. In the lever, I take for the points of application of the forces rig<124 >, q, the feet L, M, of the perpendiculars PL, FM, let fall from the fulcrum F, upon the directions of the forces. When the lev- er turns about the point F, the points L, M, will describe the similar arcs LZ/, MM', to which the directions^) L, qM, of the forces, are tangents. The lengths of these arcs are to each other Geom. as their radii FL, FM ; so that we have the proportion, OQD LU : MM' :: FL : FM, that is, u : v :: FL : FM', Principle of Virtual Velocities. 133 and as this ratio remains the same, however small the motion impressed upon the lever, it holds true, when u : v : : q : p, which gives exactly q : p : : FL : FM. In the case of an equilibrium, therefore, the two forces are to each other in the inverse ratio of the perpendiculars let fall upon their directions, as already determined by a different method. 230. If we had taken the extremities B, D, of the lever for the points of application of the forces p, q, the directions of the forces would no longer be tangents to the arcs described by these points. We should therefore have to project these arcs upon the straight lines Bp, D q, then to take the ratio of these projec- tions, and seek the limit of this ratio. On the supposition of motion, the angles DFD', BFB', describ- ed by the arms FB,FD, are equal; and the arcs BB', DD', des- cribed by B, D, about the point F, as a centre, are to each other as the radii FB, FD. This ratio continues the same when the arcs become infinitely small, so that we have constantly FB : BB / :: FD : DD / . From the points B', D, let fall the perpendiculars B'A, DC, upon the directions of the forces p, q ; and we shall have u = BA, and -o = DC. Also from F, let fall the perpendiculars FM, FL, upon the direc- tions of the forces. By considering the infinitely small arcs BB', DD', as straight lines, perpendicular respectively to the Geom. radii FB, FD, the triangles DD'C, FMD, are similar, as also the 209 - triangles BB'A, FLB-, whence FB : FL : : BB' : BA = J^L x FL, and FD : FM : : DD' : DC = ^ x FM; accordingly we have, by substitution, D f)f T-) TV u = -JjfL- x FL, and v = -~L x FM, 134 Static** and hence ** X FL : X FM, that is, since -- = u : v :: FL : FM; from which we obtain as before, q : p : : FL : FM. . 10 _ 231. In the wheel and axle, when motion commences, the points of application of the power and resistance describe similar arcs, or arcs of the same number of degrees, the one upon the circumference of the wheel, and the other upon that of the axle. The directions of the forces are tangents to these arcs, whose lengths are to each other as the radii IB, /^, of the axle and wheel ; in this machine, therefore, we have : v : : IB : 1A, and accordingly q : p : : IB : IA< 232. In the pulley, if, while the weight p describes in rising ^ l> 95 ' a space equal to w, each of the two cords which meet at the moveable pulley, is shortened by the same quantity w, the cord to which is suspended the power 9, will be lengthened by a quantity equal to 2tf, which will consequently be the space passed through by q in its descent. Taking, therefore, v = 2 w, w.e shall have, in the case of an equilibrium, q : p : : u : v, :: 1 : 2; or generally, the cords being parallel, q :p :: 1 ; n, n denoting the number of cords that meet at the moveable pulley. I will take, as the last example, the assemblage of pulleys, represented in figure 87. If the power ; calling this T', the point M will be elevated by a quantity equal to \ v' or \ v ; this being designated by i/', the point L will be elevated by a quantity j- D" or | r ; and this is the space through which the weight p passes in rising ; calling this w, therefore, we shall have, whatever v may be, u = \ T, which gives in the case of an equilibrium q : p : : u : z>, :: 1 : 8; or generally q : p : : 1 : 2, n denoting the number of moveable pulleys in the system, which agrees with what has been before shown. 233. In the screw, when the nut passes over a space equal to the distance between two adjacent threads, or when it is elevated through a height equal to DE, the point to which the power is applied describes a spiral about the axis AB, and rises through a space equal to Z)E, the projection of which spiral upon a horizontal plane is a circle, of which BG is the radius. More- over, these motions of the nut, of the point of application, and its projection are such that if the nut describes one half, one third, or any other part of the distance between two threads, the point of application will describe a similar part of the length of the spiral, and its projection a similar part of a circumference of which BG is the radius ; accordingly, if we call u the height through which the nut rises, and v the arc of a circle described in the same time by the horizontal projection in question, we shall have u : v : : DE : circum. BG. Now the, direction of the force 9, not being a tangent to the spiral, it is necessary, in order to apply to this case the general law of equilibrium, to consider the projection of a very small arc of the spiral upon the direction of the force q ; but this direction being supposed to be a tangent to the circumference above mentioned, the projection upon this tangent will be very nearly equal to the projec- tion upon the circumference, and one may be taken for the other, when we consider only an infinitely small motion of the nut ; consequently, the ratio of the force q, to that of p, will be given,. 136 Statics. by taking the limit of the ratio of u to v. Now as this ratio is constantly the same, and equal to the ratio of DE to circum. EG; according to the preceding proportion, we shall have, in the case of an equilibrium, q : p : : DE : circum. EG, as before shown by a different process. Of Friction,, 234. The surfaces of bodies, even the most smooth, are cov- ered with elevations and depressions ; and when two bodies are brought in contact with each other, the prominent parts of the one enter the cavities of the other, and they cannot be moved the one over the other, without employing a certain force. The resistance arising from this cause is called friction. There are two sorts of friction, one which takes place when the bodies in contact have simply a sliding motion ; the other when one or both the bodies move by turning upon an axis. We have an instance of the former in the motion of skates and sledg- es, and of the latter in the action that exists between the wheels of wheel-carriages and the ground. The resistance arising from the second kind of friction, is much less than that of the first, since a rotatory motion serves to disengage the parts in contact without breaking down the eminences or lifting them out of the cavities. When the rising up of one body over the other is complete- ly prevented, the friction becomes very intense, as appears in what may be accounted an extreme case, the drawing of wire, and the rolling of bars of iron, copper, &c., into plates, where the extension of the metal is the effect of the friction acting all round or on two sides. If the asperities with which the surfaces of bodies are cover- ed, were perfectly hard, and immoveably attached to these sur- faces, it would be necessary, in overcoming the friction, to raise the incumbent mass. If these asperities, on the other hand, were perfectly flexible, there would be no resistance and no friction.. Frictim. 137 But as neither supposition is true in any case, it follows, (1.) That the resistance of friction arises in part from the difficulty of bending the asperities, and in part from the necessity of rais- ing in a degree the body or incumbent mass. (2.) That the asperities having only a limited degree of adhesion, when the force necessary to cause the body to move exceeds this, the as- perities yield, or are broken down, and the surfaces are grad- ually worn. Thus the effect of friction in machines is not only to consume a part of the force employed, but also to destroy the machines themselves. It would seem difficult, if not impossible, to establish general rules, sufficiently exact for determining the force of friction. In- deed it will be readily seen, that this resistance must vary ac- cording to the nature and texture of the surfaces in contact, their flexibility, and the adaptation in size and figure of the prominent parts and cavities to each other, and according as the force is greater or less by which the surfaces are pressed together ; moreover, on account of the flexible nature of surfaces, the prom- inent parts are found to penetrate to a greater depth when more time is allowed for enlarging the openings which they tend to enter. It belongs to experiment alone to enlighten us upon these points, and to teach us the proportional effect due to each. The information, however, derived from this source, is not yet so perfect and complete as could be wished, though it is such as may be useful on many occasions. We proceed now to make known some of the results of experiments, as also the method of applying them in calculating the effect of friction in the different kinds of machines, and the different kinds of motion. 235. (1.) When the surfaces which are to rub the one upon the other, are of the same kind of matter, the resistance of fric- tion, other things being the same, is greater than when the sur- faces are of different kinds. Thus, two pieces of wood of differ- ent kinds slide upon each other with less difficulty than two of the same kind. Iron rubbing on copper has less friction than iron on iron or copper on copper. This is supposed to be owing to the prominent parts and cavities being more nearly fitted to each other in the latter case than in the former. Mech. 18 138 Statics. * 2, The more rough the surfaces, or the less they are planed or polished, the greater is the friction. This resistance, there- fore, may be diminished, by smoothing the surfaces, or by filling the openings and pores with other matter, as oil, soap, grease y black-lead, &c., with any substance indeed, which, while it fills the cavities, does not give rise to a new adhesion. (3.) It would seem that the extent of surface ought to contrib- ute sensibly to the friction; it appears, however, by a great va- riety of experiments, that this circumstance makes but little difference ; we find in fact the same difficulty for the most part in drawing a body upon one of its surfaces as upon another, though very different in extent, provided they are equally smoothed. Thus, oak rubbing on oak, is found to have a fric- tion of about 44 per cent; and on diminishing the surface as much as possible, this is reduced only to 4H per cent. We must except, however, the case of bodies resting upon a point, when the friction is more considerable, than when the contact takes place in several points. (4.) It is principally from pressure that friction arises, and this resistance is found to increase in proportion to the pressure ; that is, we require twice the force to overcome the friction when, the weight is doubled, other things being the same. (5.) Still the time, during which the two surfaces are acting upon each other, either by their own gravity, or by any other force, is to be taken into the account, although its effect has not yet. been accurately determined ; it is found that the augmentation depending on this cause, has its limits, and that these limits vary according to the nature of the rubbing surfaces. Coulomb found that in wood sliding on wood, without grease, the friction at first increased, but in a minute or two came to a limit, which it did not afterward exceed. Oak, for example, sliding on oak, though the pressure was varied from 74 lb * to 2474 lb - had a friction after- a minute always nearly 44 parts in the hundred. In the case, however, of iron rubbing on iron, or iron on brass, the friction is the same whether the bodies are just beginning to move from rest, or have acquired any given velocity. Friction. 139 When heterogeneous bodies are made to slide upon one another, as wood on metal, the friction increases slowly with the time, and docs not arrive at its maximum in less than four or five days. Iron on oak after ten seconds, is found to have a friction of 74 per cent, and after four days it amounts to nearly 20 per cent. In heterogeneous substances, too, the friction increases sensibly with the velocity, nnd follows nearly an arithmetical, while the velocity follows a geometrical progression. When the surfaces are smeared with some unctuous substance, although the friction is diminished, a certain time is required in order that the friction may attain its maximum. Oak rubbing on oak, the surfaces being covered with tallow, has a friction that continues to increase for five or six days, and becomes sta- tionary at about 42 or 43 per cent. In the case of brass on iron with fresh tallow between the surfaces, the friction is four days in coming to a maximum, when it is 10 or 12 per cent, it being 9 per cent at the commencement. The increase where metals are used, is much less considerable than in experiments with substances more porous and yielding. 236. The quantity of friction being determined for a partic- ular kind of matter, ;et us now see if the effect upon a given ma- chine, or given motion, may be thence deduced, friction being considered as simply proportional to the pressure. Let us take, as the first example, the body p, situated upon Fi S- 126 - a horizontal plane AB, and drawn by the weight of the body 9, parallel to JiB. Suppose that the body q has a weight just suf- ficient to put the body p in motion. The ratio of the weight q to the friction is thus found. From the centre of gravity G of the body p, let fall the per- pendicular GH upon the plane AB. The body p is urged by gravity in the direction G//, and by the weight q in the direction KM which meets GH in K. From the joint action of these two forces, there will result an effort according to some line KI, meeting in /, the horizontal plane AB ; and this effort must be just counterbalanced, since we have supposed that the body p is only upon the point of moving. Suppose the effort according to KI or KIZ applied at the point 7, and decomposed into two 140 Statics. others, the one perpendicular to the plane AB, and the other in the direction of this plane. These efforts will evidently be the same as those which were directed according to KH and KL. Moreover the first will be destroyed, especially if it meet the plane AB in some point / common to this plane and the surface of the body. As to the second, since it is in the direction of friction, it will not be destroyed unless it happen to be exactly equal to the force of friction. We hence see how the value of friction is to be determined 5 we take successively for q different weights until we find one that is just sufficient to cause a motion in the body p. But not to comprehend, in estimating the friction of the body/), effects for- eign to that which is sought, .it is necessary to attend to several particulars; (1.) The pulley M should move with the greatest ease, and the cord KM q should be as flexible as it can be made. (2.) The cord CM should be attached to some point C as near as possible to the plane AB. The necessity of this precaution arises from the circumstance, that other things being the same, the point /, where the effort in the direction KI meets the plane AB, will approach so much the nearer to the extremity E of the base of the body, or will fall without the base so much the far- ther from the extremity , according as the point C is the more elevated above the plane AB. Now in the case where the point / falls without the base, the effort perpendicular to the plane, not being entirely destroyed, there will hence result a tendency in the body to rotate ; and the friction thence arising would be somewhat more considerable than the proper friction in question. Fig 127. 237. Let us now consider a weight p, put upon an inclined plane, and retained by the effect of friction alone. The action of gravity directed according to the vertical GZ passing through the centre of gravity G of the body, and meeting in / some point of the plane AB, may be decomposed into two parts, one in the direction of the plane, and the other perpendicular to it. The second will be destroyed, if the point / does not fall without the base DE, and the first in order to be destroyed must be equal to the force of friction. Now it is evident that by forming the parallelogram IHZL, IZ will represent the weight of the body. Friction. 141 IH the pressure, and HZ or IL the force of friction ; hence the triangles ///Z, ABC being similar, we have Geom HZ or IL : IH : : BC : O4, * 02 ' from which it will be seen, that the force of friction is to the pressure as the height of the plane to its base. It will be perceived, in like manner, that HZ : IZ :: BC : ./IB; that is, the force of friction is to the weight of the body as the height of the plane is to its length. In order, therefore, to determine the friction in different sub- stances, we have only to raise the plane JIB till the body p is upon the point of moving ; then measuring the height and the base, we shall have the ratio of the force of friction to the pres- sure. 238. By these two examples, it will be seen, that regard being had to friction, the condition required in order that a body may remain in equilibrium upon a proposed surface, and be in a state approaching the nearest to motion, is, that the single force which acts upon it, if there be but one, or the resultant of all the forces, if there be several, have with respect to the sur- face upon which it is to move, an inclination GIE or ZJZ, such that IL shall be"///, as the force of friction is to the pressure. v But IL is to LZ, as one is to the tang. LIZ, I being the radius of the tables. Consequently the inclination LIZ must be such that the radius shall be to the tangent of this inclination, as the force of friction is to the pressure ; therefore, the ratio of the force of friction to the pressure being once ascertained, it will always be easy to determine the inclination belonging to the resultant of all the forces which act upon the body, this body being in a state of equilibrium approaching as near as possible to motion. Hereafter we shall call the angle LIZ the angle of friction. It is different for different substances, and for different degrees of smoothness of the same substance. If the friction is 331 per cent, or one third of the pressure, which is nearly the case with respect to a great many kinds of matter, the rubbing surfaces being tolerably smooth, the tangent of LIZ will be triple of the radius, that is, LZ will be equal to 3 IL. Now the angle 142 Statics. of which the tangent is triple the radius is 71* 34'. This will accordingly be the angle of friction in these cases. 239. By means of this result, it will be easy to determine in each machine what ratio ought to exist between the power and the. weight in order that motion may be upon the point of taking place, allowance being made for friction. In the lever, for example, let us suppose that the fulcrum is a simple support, as represented in figure 1 28. We have seen, that with respect to this machine, there cannot be an equilibri- um, unless the resultant DF of the two forces;?, 7, be perpendic- ular at F to the common tangent to the surface of the lever and 163> that of the fulcrum. On the supposition of friction, the case is different ; it is still necessary that the resultant should be direct- ed from the point D to the fulcrum F; but it is sufficient in order to an equilibrium, that one of the two inclinations DFA, DFB, according as we wish q or p to prevail, should be greater than the angle of friction, and for the state of equilibrium approach- ing the nearest to motion on the part of the power q, it suffices that the inclination DFA should be precisely equal to the angle of friction ; since, if we imagine the force according to DF de- composed into two others, the one perpendicular to AB^ and the other in the direction A.F, the force in the direction AF will be less than the friction in the first case, and exactly equal to it in the second. With respect to the two forces , they will still be in the inverse ratio of the perpendiculars FK, FL. 240. But if the fulcrum is such that the lever can have no other motion except that of a rotation, that is, if it turn on an axis or pin, we adopt the following method which is common to the lever, the pulley, and the wheel and axle, especially when in this last machine the power and weight are in the same plane. We shall consider first the wheel and axle ; the manner of treat- ing the lever and pulley will afterwards appear. Fi 129 Let HCI be the plane of the wheel, GKL a section of the cylinder, and NDM the axis about which the machine is to turn. On the supposition that there is no friction, the resultant of the two powers p, , which would give, 2p cos/ = Although in the wheel and axle we have supposed the direc- tions in the same plane, the solution is not the less adapted to the ordinary construction of this machine in which the weight and power are exerted in planes but little distant from each other. 242. To ascertain the effect of friction in the moveable pulley we proceed thus. In order that the power q may be upon the Fig.130. point of causing the pulley to turn about its axis F, it is necessa- ry that it should receive an augmentation sufficient to overcome the friction. Now by this augmentation, the power causes the weight p to depart from its position to such a degree, that a Mech. 19 146 Statics. vertical drawn through the centre of gravity shall make at the point D, with the surface of the axle, an angle equal to that of friction. Then any increase of the power will cause the pul- ley to turn. By the position which the weight has taken, it will tend to turn the axle upon its centre J 1 , with a force, the moment of which will be p X FE, FE being perpendicular to a vertical passing through D. Now we have already seen, that this moment, therefore, will bepd cosy*. But in order that the power, by its augmentation, which I shall call 2', may be upon the point of overcoming this effort, it is necessary that the moment Z* X FG of the force z' ', with which it tends to cause a rotation 172. about F, should be equal to the moment pd cos/ ; we have, therefore, by calling the radius FG, Z), = pd cos/, and consequently, / _ p cos/ ~D~~' from which it will be seen, that the effect of friction will be less according as the radius of the axle is less than that of the pulley, and in the same ratio. It will hence be easy to determine the effect of friction in the different systems and combinations of pulleys. Suppose that the question related to one similar to that represented in figure 92, the weight p being equal to 400 lb , and the radius d of the axle as well for the fixed as for the moveable pulleys being a fifth part of the radius D of the pulley. The two cords 1, 2, sustain half of the weight or 200 lb j thus, if in the value of z, we put for p 200u>, j D for d, and on the supposition that the friction is one fourth of the pressure, which would give the angle of friction equal to 75 58', and the os/or cos 75 58, equal to 0,24249,* or simply 0,24, we shall have, * See table of natural sines and cosines, Topography. Friction. 147 : g ,_/> very rigorously, it would be necessary to consider the subject a little differently. In fact, by determining in the way we have done, the particular tensions of each of the cords, it is taken for granted that each pair of cords acts as it would do in the case of a simple pulley, which is not strictly true, perhaps. But this approximation will suffice for the present. 244. In order to determine upon the inclined plane, the ratio of the power to the weight, when the former is upon the point of causing the body to move, we proceed thus. Through the point ^ 13J of meeting F, of the directions of the power 9, and weight />, we suppose the line FI drawn, making with the plane JIB an angle FIA equal to that of friction. In order that the power q may be upon the point of causing motion, it is necessary, (\.) That the Friction. 1 49 resultant of the power and weight should be directed according to FI. (2.) That the point /, where the line FI meets the plane, should belong to some point of the base DE ; otherwise the body would tend to turn. This being premised, we have p : q : : sin q FI : sin p FI, or, letting fall upon the plane the perpendicular FH, p : q :: sin (qFH HFI} : sin ( P FH + HFI). Now the angle HFI is the complement of the angle of friction; and the angles q FH, p FH, are supposed to be known, since the the direction of the power is considered as known, together with the inclination of the plane, which is equal to the angle p FH ; Geom. we have accordingly the ratio of p to q. If we would determine this ratio in lines, we have only to draw through any point B of the inclined plane, the line BT, making with AB the angle ABT = HF q, and the line BV, mak- ing with AB the angle ABV = HFI, the complement of the angle of friction. Then drawing the horizontal line AT, we shall have p : q : : VT i BT, since the angle VBT - ABT ABV = HF q HFI, and the Ge om. angle BVT = BAY + ABV = p FH + HFI. Now in the 78 - triangle BVT VT : BT :: sin VBT : sin BVT. Trig<32v Instead of making the angle ABT = HF q, and the angle ABV = HFI, we may draw BT perpendicular to the direction of the power q, and BV perpendicular to FI\ this amounts to the same thing, and is moreover analogous to the course pursued in article 203. 245. The second condition shown to be necessary in order that the power q may be upon the point of moving the body, renders it evident, that when the body does not rest upon a point, the direction of the power produced must meet the vertical, drawn through the centre of gravity, at the point F, where thisFig.132. last line is met by the line IF, proceeding from some point of con- 1 50 Statics. tact /, and making with the plane an angle equal to the angle of friction. 246. We proceed in the same manner in determining the second kind of friction, or that which is to be overcome in giving a rolling motion to bodies terminated by curved surfaces ; I say curved surfaces, since with respect to bodies terminated by plane surfaces, as they cannot roll, except by turning on a point or some angular part, we shall not treat of these, the laws and value of friction m such cases not being sufficiently known. But as to the friction of bodies terminated by curved surfaces, the method is precisely the same as that above pursued. We have only to suppose the angle of friction to approach more nearly to 90 ; and it is to experiment that we are to look for the determination of this angle in all cases. 247. Friction may be the occasion of motions very different from those which take place without this cause, some of which it may be worth while to notice. 186, &c. \y e nave already mentioned more than once, what must hap- Fig.133. pen to a free body BO g, which receives an impulse in a direc- tion not passing through the centre of gravity. But if the body were struck externally according to any direction AB, it would not receive the whole of this impulse. The impelling force is to be decomposed into two others, one in the direction of a tan- gent to the surface, and the other perpendicular to this surface. When there is no friction, the impelling force would have no effect in the direction of a tangent. It is only the force in the direction BF, therefore, which would be transmitted to the body, and this would not cause the body to turn, except when it hap- pened not to pass through the centre of gravity G. It will hence be seen that if the body were of a spherical form and homogene- ous, it would never be made to turn in virtue of an external force unaccompanied with friction, since in this case a perpendicular to the surface would always pass through the centre of the figure, which would, at the same time, be the centre of gravity. On the supposition of friction, the case is different. The force in the direction of a tangent would transmit itself by means of the as- perities of the surface, and to a greater or less degree according to the amount of friction ; so that in addition to the motion aris- Friction* 161 ing from the perpendicular force BF, the body would turn, and the centre of gravity G would advance in a line parallel to the tangent, as if the point B were drawn in that direction by means of a thread attached at this point, and with a power equal to the force of friction. 248. Let us suppose that a hard, spherical body ABC, falls Fig.134. freely upon a horizontal plane HR, and that it receives from some cause or other, a motion of rotation about its centre of gravity ; if there were no friction, this body, after meeting the plane would preserve only its rotatory motion, and its centre of gravity would be at rest. But in the case of friction, when the body has reached the plane, it will roll from / toward R, or from /toward //, accord- ing as the rotatory motion is in the direction CAB, or in that of BAC ; since the resistance of friction, which is exerted in the direc- tion of the plane,is equivalent to a force acting upon this body in a direction opposite to its motion ; and as it does not pass through the centre of gravity of the body, it must give it a motion paral- lel to the plane, and a rotatory motion, both in the direction contrary to the actual motion of rotation. Now of these two motions, the latter diminishes continually the original motion; and on the other hand, the motion of the centre will be acceler- ated to a certain point, after which it will be diminished till it is destroyed with the motion of rotation. 249. We are hence enabled to explain several phenomena ; as (1.) Why a spherical body ABC, struck in the direction DB, Fig. 135 after having advanced in the direction IE, returns afterward from E toward 7, and may even pass beyond / toward H. The impulse in the direction DB causes it to turn (on account of fric- tion at B), according to ABC, and to advance in the line IE-, but the friction upon the plane, being now a friction of the first kind, the motion of the centre of gravity is soon destroyed, and the motion of rotation gives rise to another in the opposite direction, as in the preceding case. 24g (2.) We are moreover furnished, upon the same principles, with the reason why a cannon ball, which had apparently lost nearly all its force, seems on striking to recover it again, and often with violence. When it is impelled by the force of the 1 52 Statics. powder, it acquires, at the same time in consequence of the friction against the bottom of the bore, a rotatory motion, which is but little affected while in the air ; but when the ball comes to reach the ground, as the rotatory motion on the part toward the ground takes place in a direction opposite to the progressive mo- tion, the consequence must be an acceleration in the motion of 248. the centre, that is, of the progressive motion. 250. Finally, if friction is disadvantageous in many cases, it is still not without its utility. Were it not for this, the least in- clination would be continually subjecting us to a fall. No man or other animal could turn while in rapid motion without falling, F Jg-136 wna tever position he might take ; whereas, on account of friction, an animal may incline himself toward the point .F, for example, about which he is moving, in such a manner that his weight, di- rected according to the vertical GK, passing through the centre of gravity G, and the tendency to fly off GC, acquired by turn- ing, and which is directed from F toward C, will conspire to pro- duce a single force according to the line G7, passing through a point / between the legs of the animal ; this force, although ob- lique, is still destroyed by friction, provided the inclination be within the limits required by the laws of friction. It is moreover to friction that we are indebted for the power of diminishing friction, when injurious ; since it is only by means of this resistance that we are able to work and polish the surfa- ces of bodies. It is to friction that we owe the facility with which the parts of certain machines are rendered sometimes fixed, and sometimes moveable. It is by friction that scissors and the like instruments, pincers, forceps, files, &c., produce their effect. If the parts of scissors, for example, were not saws, armed with small teeth, which take into the cavities of the bodies to be cut, these bodies would slip from between the two edges. Friction is also very often of service in moving bodies in certain directions ; thus if we would raise the body p by means of the lever AB, it is very easily done, by making the body bear Fig 137. on the edge CD. The friction in a case like this, being very considerable, renders CD fixed, and prevents the body from slip- ing. The same cause keeps the extremity A of the lever in its Friction, 158 place. In this ease, if we would know the ratio of the weight/) to the power q, we imagine the weight of p. directed according to the vertical GK, passing through the centre of gravity G, decomposed into two parallel forces, the one passing through the point / in which the body rests upon the lever, arid the other through a point of CD, situated in the plane of the two parallels GK, 1M\ then the resulting force in / will be to p as EK to EM ; 52 - and if from A we let fall upon IM the perpendicular AL, the force q will be to the force at /, as AL to AB ; whence H ^ q : p : : AL X EK : AB x EM. In short, it is only on account of the friction at the point / that we regard the force according to IM, as transmitted entirely to the lever. The lever would otherwise receive only a part of this force which would exert itself in the direction of a perpen- dicular to AB. 251. It is to friction and friction only, that we are to refer the singular motion by which certain bodies in a state of rotation are seen to elevate themselves contrary to the tendency of grav- ity ; I speak of the phenomena of the top. It is well known that when a body of this description, that is, one which is symmetri- cal with respect to one of its axes, as A r Z), has received a rotato- Fig.lSa. ry motion about this axis, and moves upon its point JV, over a horizontal plane XZ ; it is well known, I say, that the smaller the point JV, and the greater the divergency of the sides from it, the greater is the tendency of this body to rise, and thus to place the axis ND in a vertical position. We proceed now to show that this phenomenon could not take place without friction ; we shall speak, moreover, of the nature of this friction. To simplify the subject, let us consider only the axis JVD of Fig.139. the top, and let us suppose the point N, and the horizontal plane XZ, to be perfectly smooth. The only cause which opposes the motion of the centre of gravity G, being the plane XZ, the resis- tance which the centre of gravity meets with can have no other direction than the line NK, perpendicular to XZ, whatever be the rotatory motion about the axis JVD. Now it is evident that this resistance takes place only because gravity urges the body Mech. 20 Statics. towards the plane, for the rotatory motion about the axis ND cannot cause any pressure upon this plane ; the resistance in question, therefore, can never be equivalent to the force of grav- ity, and there will always remain in the centre of gravity G, a force tending to bring it to the plane. Hence, when there is no friction, and the top has received at first no motion of rotation, except about the axis of its figure, it must of necessity fall. 252, It is not the same on the supposition of friction. In this case, the resistance which takes place at .AT, acts not accord- ing to the perpendicular JVA 7 , but according to the line NK'^ which makes with the plane XZ an angle equal to the angle of fric- tion, and passes through JV, one of the points of friction. What- ever may be this angle and this point, the resistance which takes place along the line NK', is equivalent to a force acting upon the body in a contrary direction ; now as this direction does not pass through the centre of gravity, it must produce in the body 137. a rotatory motion, that is, a variation in its actual motion of ro- tation ; but it must also transmit itself entirely to the centre of gravity. Let us suppose, therefore, that GL parallel to NK' is this force ; if the vertical line GI represent the force of gravity, and the parallelogram GLEI be completed, GE will be the actu- al force which belongs to the centre of gravity G. Now the angle LGI and the force G7 remaining the same ; the greater the force acting in the direction K'N, and consequent- ly the greater the force GL, the more nearly will the line GE approach the line GL; that is, the greater will be the tendency of the point E to rise above G. It remains therefore to be seen, whether from the nature of friction, together with the figure of the body, and its motion of rotation, the ratio of the force in the \, the diameter of the cord which bears 158 Statics. the weight being d, and the radius of the surface R. We wish to know what this addition must be, when the weight is j/, the diameter of the cord tf', and the radius of the surface R'. It will be observed, after what has been said, that if there were no dif- ference except with respect to the entire weight by which the cord is stretched, we should arrive at a solution by the propor- tion p : p' : : k : - = the addition required. But if, beside the difference in the weights, there is also a differ- ence in the curvature of the surfaces ; then, by the second of the above remarks ; namely, that the additions arising from this cause are in the inverse ratio of the radii of the surfaces, we should obtain the addition in question, together with that due to a change of weight, by the following proportion^ Rl . R . . Zt .ft kR P R' Regard being had to the third remark, we shall obtain the addi- tion to be made on account of the three causes united, by the proportion , ., kRp' d'kRp' o : o' :; - _ : x = - _. R p dR'p The resistance in the first case, therefore, will be to the resist- ance in the second that is, the resistances arising from the stiffness of the cords are as the weights which stretch the cords, multiplied by the diame- ters of these cords, and divided by the radii of the surfaces over which they pass. These conclusions, it may be observed, are not perfectly rig- orous ; but they may be regarded as sufficiently exact for prac- tice, till experiment has thrown new light upon the subject. In- deed, experiment shows that the resistance arising from the stiff- Stiffness of Cords. 1 59 ness of cords, agrees nearly with this law ; but all the experi- ments that have been made upon this subject have not hitherto agreed so perfectly with the theory as might be wished. 255. We shall now illustrate the foregoing principles by an example. For this purpose, let us suppose the common radius of the pulley to be 2 inches, that of the axle j of the same quanti-f<; g4 go ty, and the diameter of the cords to be | of an inch. Let us take, moreover, the result of experiments on this subject, namely, that a cord of half an inch diameter, loaded with 120 lb , and pass- ing over a pulley of f of an' inch, occasions, by its stiffness, a resistance of 8 lb . This being established, the weighty being 400 lb , as in article 242, the branches 1 and 2 will be loaded, both on account of this weight and the force added to q to overcome the friction, by a force equal to 209 lb , 6. Multiplying, therefore, this weight by % of an inch, the diameter of the cord, and dividing by 2 inches, the radius common to all the pulleys ; multiplying also the weight 120 lb , used in our experiment, by , the diameter of the cord, and dividing by f , the radius of the pulley, in the experiment in question ; we shall have for the results, 34, 93, and 40. Now in order to obtain the force required by the stiffness of the cord 1, 2, which passes beneath the moveable pulley, we must use this proportion, 40 : 34, 93 : : 8 lb : 6, 99 or 7 lb nearly ; this fourth term is the quantity which it would be necessary to add to the tension of the cord 2, if the lower pulley were fixed ; but since it is moveable, which diminishes the tension by |, as we shall see below, we have only 3 lb , 5 to be added to 109 lb , 6, by which this cord is already stretched ; the whole tension will thus be 113 lb , 1. We have seen that on account of friction, the tension of the 242i cord 3 ought to be equal to 1 20 lb , 7 ; therefore the whole force by which the cord 2, 3, which passes over the fixed block, is extended, is 1 13, 1 -f- 120, 7 on 233 lb , 8. Multiplying this force as above, by , and dividing by 2, we shall have 38,97. Seeking as before the value of the stiffness of the cord 2, 3, which passes ever the fixed block, we shall have the proportion, 160 Statics. 40 : 38, 97 : : 8 lb : 7 lb 79. Adding now these 7 lb , 79 to 120 lb , 7, the quantity before found for the tension of the cord 3, we shall have 1 28 lb , 5 for the force of tension, both on account of friction and the stiffness of the cords. The cord 4, on account of friction is stretched by a force = 130 lb , 3 ; the whole force of tension upon the cord 3, 4, which embraces the moveable block, is therefore 258 lb , 8 ; with this quantity, the dimensions of the pulleys and cords remaining the same, we shall find, as above, that the allowance for the stiffness of the cords, would be 8 lb , 62, if this cord did not embrace the moveable block ; but proceeding as we have done above, and for the same reason, we must take but half of this quantity ; thus the tension of the cord 4, all things considered, will be 134 lb , 6 nearly. The cord 5, on account of friction, is stretched with a force = 143 lb , 5 ; the whole force by which the cord 4, 5, embracing the fixed pulley, is stretched, is therefore 278 lb ,l . With this value, the dimensions remaining the same, we shall find the force required for the stiffness of the cords to be 9 lb , 3 ; the whole tension of the eord, therefore, is 152 lb , 8. Thus friction and the stiffness of the cords together require the weight 5, which would otherwise be but 100 lb , to be 153 lb nearly, a quantity greater by more than one half. We have taken only half of what the calculation furnished as the quantity to be added to cords 2 and 4, on account of the stiffness. The reason of this is, that the cord by which the moveable pulley is made to revolve, may be considered as turn- ing it on a centre placed at the point where this pulley touches the other cord ; and consequently the case is similar to that of a fixed pulley, having a radius equal to the diameter of the move- able pulley ; and since the forces required on account of the stiffness of the cords, are in the inverse ratio of the radii of the pulleys, the force in this case must be diminished one half. Balance, Steelyard, fa. 161 Of the Balance, Steelyard, #c. 256. A balance is a lever of the first kind in which the fulcrum Fig.142. or axis is in the middle between the points of application of the forces or weights. It is apparent that with such an instru- ment, a state of equilibrium must indicate an equality in the weights to be compared. There are several particulars to be attended to in the con- struction of a good balance, (1.) The axis should be above and not too far above the centre of gravity of the two arms or beam ; for if it pass through this centre of gravity, the beam, when load- ed with equal weights, or not loaded at all, will remain at rest in any position whatever; and it is intended that a horizontal position shall indicate equal weights. If the axis be below the centre of gravity of the beam, the slightest deviation of this centre from a vertical position over the axis would be followed with a motion that would tend to reverse the position of the beam. Moreover, if the axis be at a considerable distance above the centre of gravity, the beam would have too great a tendency to a horizontal position, independently of an equality in the weights, and would accordingly want the requisite sensi- bility. (2.) The axis and points of application of the weights, or points of suspension, should be in the same straight line ; otherwise the beam when loaded is liable to the defects just mentioned, by having its centre of gravity shifted above the axis or too far be- low it. It is accordingly of great importance that the arms should be so constructed as not to bend or yield in any degree in consequence of the weights attached to them, while at the same time they should be as light as possible. To secure both these objects, in the best balances instrument makers have given to the arms the form of hollow cones. Care is taken also to preserve the equality in point of length of the two arms, and to diminish friction, by making the axis and points of suspension of hardened steel, and in the shape of a very acute, wedge, or knife? Meek. 21 162 $taties. edge, the plane surfaces with which they come in contact being likewise of the hardest substances. 257. Nevertheless, where extreme accuracy is required, the best method is to place the thing to be weighed in one of the scales, and to balance it by any convenient substances in the other scale, till the beam takes an exact horizontal position, as shown by the index FL* Then carefully remove the thing whose weight is sought, and replace it by accurate weights, as grains, &c., till the beam takes precisely the same position. The number of grains, &c., used will be the weight required. In this way, all error arising from the want of perfect equality in the two arms of the balance are completely obviated. 258. With all these precautions, the process of weighing is but an approximation, strictly speaking, to perfect accuracy. (I.) The best balance tends of itself with some degree of force to a hori- zontal position, since the slightest inclination must cause the cen- tre of gravity to rise somewhat, and consequently, if the differ- ence in the two weights does not exceed this force, the balance will not enable us to detect the inequality. (2.) The friction both of the axis and of the points of suspension, is an obstacle to motion, and the difference in the weights must be sufficient to overcome this, in addition to the force just mentioned, in order that the balance may show them to be unequal. 259. Knowing, however, what weight is sufficient to produce motion in any case, we know what degree of accuracy we have been able to attain. This is different for different weights. A small weight can be determined more exactly, other things being the same, than a large one, since the friction increases with the pressure. A balance constructed by Ramsden for the Royal Society of London, when loaded with a weight of 10 lb , would turn with a grain, or a ten-millionth part of the weight ; and one made by Fortin of Paris, when charged with a weight of 4 lb , would turn with a 70th part of a grain. * Instead of an index rising perpendicularly, a pin is sometimes attached to the end of the beam, and made to move over a nicely graduated arc provided with a microscope ; and the whole apparatus enclosed' under glass to prevent any agitation from the air. Balance, Steelyard, &c. 163 259. The steelyard also is a lever of the first kind ; but the Fi.> ces that the body would describe in the corresponding intervals of time respectively ; the whole space described during the time t will be the sum of the terms u + 2 u -f &c., of this progress- Alg.229. ion, that is, it will be expressed by (u -f- v) X | t. Therefore, if we call s this whole space passed over from the commence- of the motion, we shall have s = (u + v) X \ t. 265. Let us suppose now, that the accelerating force acts without interruption, or, which amounts to the same thing, that the time is divided into an infinite number of infinitely small parts, which we shall call instants ; and that at the beginning of each instant, the accelerating force exerts a new impulse upon the body. Let us suppose, also, that it acts by degrees infinitely decelerated Motion. 16? Small. Then, u being infinitely small with respect to T, which is the velocity acquired during the infinite number of instants de- noted by t) u is to be neglected in the equation s = (u -f v) X 1 1 i . r Cal. 6. which gives 266. This being established, let us suppose that at the end of the time *, the accelerating force ceases to act ; the body will continue its motion with the velocity i>, that it had acquired ; that is, in each unit of time, it will describe a space equal to V ; 22 accordingly, if it were to continue with this velocity during the time <, it would describe a space equal to v X /, that is, double the space s or v t, described in the same time by the successive 266. action of the accelerating force. Therefore, in motion uniformly and continually accelerated, the space described during a certain time is half of that which the body would describe in an equal time with the last acquired velocity continued uniformly. 267. Since the acquired velocity increases with the times elapsed, if we call g the velocity acquired at the end of a second, the velocity acquired after a number t of seconds, will be g t ; that is, we have, and accordingly the equation s = \ v /, found above, becomes If, therefore, we represent by 5' another space described in the same manner during a time /', we shall have, according to the above reasoning, f.j* ig*' 2 , from which we deduce the proportion, s : s' :: %gt 2 : %gt' 2 : : t* : t? 2 ; we hence learn that with respect to motion uniformly accelerated. the spaces described are as the squares of the times. 268. Moreover, since the velocities are as the times, we con- 264 elude also, that the spaces described are a* the. squares of the -odor.i- 168 Dynamics. 269. Therefore the velocities and the times are each as the square roots of the spaces described from the commencement of the motion. 270. The principles here established are equally applicable to the case of motion uniformly retarded, provided that by the times we understand those which are to elapse, and by the spa- ces those which are to be described, from the instant in question till the velocity is destroyed. 271. From the equation s = | gt 2 , the quantity g by which we have understood the velocity that the accelerating force is capable of producing by its action, exerted successively during a second of time, is what we call the accelerating force, since we must judge of this force by the effect which it is capable of pro- ducing in a body in a determinate time, an effect which is noth- ing else but the communication of a certain velocity. Of free Motion in heavy Bodies. 272. It is the kind of motion we have been considering, to which the motion of heavy bodies is to be referred. But before applying to this subject the theory above developed, it will be proper to make known a few facts concerning gravity, in addition 74. to those heretofore given. As to the magnitude of the force of gravity, it is different, strictly speaking, in different latitudes, and at different dis- tances from the centre of the earth in the same latitude. But the quantities by which it varies, as we depart from the equa- tor, are very small, and do not in any manner concern us at present. The same may be said of the variations it under- goes, according as we rise above, or descend below, the mean surface of the earth; they cannot become sensible, except by changes of distance much more considerable than any to which we are accustomed ; so that for the present we may regard gravity as a force every where the same, or one which urges bodies downward by the same quantity in the same time. This force is to f-e considered also as acting, and acting e.qually at each instant, upon every particle of the matter about Motion of heavy Bodies. 169 ufe. Now it is evident, that if each of the parts of a body receive the same velocity, the whole will move only with the velocity that any detached portion would have received ; so that the velocity which gravity impresses upon any mass whatever, does not depend upon the magnitude of this mass ; it is the same for a large body as for a small one. It is true, however, that all bodies are not observed to fall from the same height in the same time ; but this difference is the effect of the resistance of the air, as we shall see hereafter ; and when bodies are made to fall in close vessels, from which the air has been withdrawn, though of very different masses, they are found to descend through the same space in the same time. It may be well to notice here the distinction between the effect of gravity and that of weight. The effect of gravity is to cause, or tend to cause in each part of matter, a certain velocity, which is absolutely independent of the number of material particles. But weight is equal to the effort necessary to be exerted, in order to prevent a given mass from obeying its gravity. Now this effort depends upon two things; namely, the velocity that gravity tends to cause in each part, and the number of parts on which this force is exerted. But as the velocity which gravity tends to give, is the same for each part of matter, the effort to be exerted in or- der to prevent a given mass from obeying its gravity, is propor- tional to the number of parts, that is, to its mass. Thus weight depends upon the mass, whereas gravity has no relation to it. 273. Having made known these particulars with regard to gravity, we proceed to the laws of motion of heavy or gravitating bodies. Since gravity acts equally and without interruption, at what- ever distance the body is from the surface of the earth (at least, so far as our experience extends), gravity is a uniformly accele- rating force, which at each instant causes in a body a new degree of velocity, that is always the same for each equal instant ; so that the velocities acquired, increase as the times elapsed ; the spaces passed over are as the squares of the times, or as the squares of the velocities ; the velocities are as the square roots of the spaces described ; the times are also as the square roots of the spaces described ; in short, all that we have said respect- Mech. 22 1 70 Dynamics. ing a uniformly accelerating force, is strictly applicable to gravity, it being well understood at the same time, that the resistance of the air and obstructions of every kind are oat of the question. In order to determine, therefore, with respect to the motion of heavy bodies, the spaces described and the velocities acquired, we require only one single effect of gravity for a determinate time. For the equations v gt, s = |g 2 , enable us to calcu- late each of the particulars above enumerated, when the value of g is known. It must be recollected that by g we have understood the velocity which a body acquires by gravity in one second of 26<7 - time. Now we know by actual observation, that a body, not impeded by the resistance of the air or other obstacle, falls at the surface of the earth through 16,1 feet in one second. We shall see hereafter how this is determined. But we have shown that with the velocity acquired by a series of accelerations, the body would describe with a uniform 266. motion double the space in the same time. Hence the veloc- ity acquired by a heavy body at the end of the first second of its fall is such, that if gravity ceased to act, it would describe twice 16,1 feet, or 32,2 in each succeeding second. Therefore g = 32,2 feet. 274. Now of the two equations v = g t, and s = g t 2 , the first teaches us, that in order to find the velocity acquired by a heavy body, falling freely, during a number t of seconds, it is necessary to multiply the velocity acquired at the end of the first second by the time f, or number of seconds. Hence, when a heavy body has fallen during a certain number of seconds , the velocity acquired is such, that if gravity ceased to act, the body would describe in each second as many times 32,2y*ee, as there were seconds elapsed. Thus a body that has fallen during 7 seconds, will move at the end of the 7 seconds, with a velocity equal to 7 times 32,2 or 225,4 feet in a second without any new acceleration. 275. From the second of the above equations, namely. Motion of heavy Bodies* 171 we learn that in order to find the space or height s through which a heavy body falls in a number t of seconds, we have only to multiply the square of this number of seconds by ^ g, that is, by the space described in the first second. Hence, the height or number of feet through which a heavy body falls during a number t of seconds is so many times 16,1 feet as there are units in the square of this number of seconds. Thus, when a body has been suffered to fall freely during 7 seconds, we may be assured that it has passed through a space equal to 49 times 16,1 feet, or 788,9 feet. We see, therefore, that when, in the case of falling bodies, the time elapsed is known, nothing is more easy than to determine the velocity ac- quired, and the space described. 276. If the question were to find the time employed by a body in falling from a known height, the equation s = ?gt 3 9 gives t 2 = j , and consequently, t = that is, we seek how many times the height s contains 1 g, or 16,1 feet, the space described by a body in the first second of its fall, and take the square root of this number. 277. If we would know from what height a heavy body must fall to acquire a given velocity, that is, a velocity by which a certain number of feet is uniformly described in a second ; from the equation v = g J, I deduce the value of , namely, t = ; sub- stituting this value in the equation s = \gt 2 , I have T? a 7>2 s = 1 g X = , by which I learn, that in order to find the height s from which a heavy body must fall to acquire a velocity T, of a certain num- ber of feet in a second, the square of this number of feet is to be divided by double the velocity acquired by a heavy body in one second, that is, by 64,4, 172 Dynamics. Thus, if I would know, for example, from what height a heavy body must fall, to acquire a velocity of 100 feet in a second, I divide the square of 100, namely, 10000, by 64,4 ; and the quo- tient VT,T = 155,2&c., is the height through which a body must fall to acquire a velocity of 100 feet in a second. We might evidently make use of the same formula in deter- mining to what height a body would rise, when projected verti- cally upward with a known velocity. Moreover, from the above equation, s = we obtain v 2 = 2g 5, or v = \/2 g s 1= 8,024 v*r that is, the velocity acquired in falling through any space s, is equal to \-'2~F* or equal to eight times the square root of s nearly, u, g, and s, being estimated in feet. Thus the velocity acquired in falling through 1 mile or 5280 feet, is equal to 8,024 v*<*o = 583 feet very nearly. 278. By these examples it Avill be seen that all the circum- stances of the motion of heavy bodies may be easily determined ; and it is accordingly to these motions, that we commonly refer all others ; so that instead of giving immediately the velocity of a body, we often give the height from which it must fall to ac- quire this velocity. Occasions will be furnished for examples hereafter. We will merely observe, therefore, by way of recapitulation, that all the circumstances of accelerated motion, and consequent- ly of the motion of heavy bodies, are comprehended in the two equations v = g J, s = ^ g t 2 ; so that, g being known, and one of the three things, f, s, i>, or the time, space, and velocity, the two others may always be found, either immediately by one or the other of the above equations, or by means of both combined after the manner of article 277. 279. When a body is subjected to the action of a force that is exerted upon it without interruption, but in a different manner at each successive instant, we give to the motion the general de- nomination of varied. We have examples of varied motion in the unbending of springs ; although in this case the velocity goes Motion of heavy Bodies. 173 on increasing, still the degrees by which it increases go oa ishing. The same may be observed with respect to the degrees by which the motion of a ship arrives at uniformity ; the action of the wind upon the sails diminishes according as the ship ac- quires motion, because it is withdrawn so much the more from this action, according as it has more velocity. 280. The principles necessary for determining the circum- stances of this kind of motion are easily deduced from the prin- ciples that we have laid down with regard to uniform motion, and motion uniformly accelerated. (1.) In whatever manner motion is varied, if we consider it with respect to instants infinitely small, we may suppose that the velocity does not change during the lapse of one of these instants. Now, when the motion is uniform, the velocity has for its express- ion the space described during any time , divided by this time. Accordingly, when the motion is uniform only for an instant the velocity must have for its expression the infinitely small space described during this instant divided by this instant. Hence, if s represents the space described, in the case of a variable motion, du 'ing any time , d s will represent the space uniformly describ- ed during the instant d t ; we have, therefore, ds , v = or d s = v d t, as the first fundamental equation of varied motion. 281. In the equation v = g f, we have understood by g the velocity which the accelerating force is capable of giving to a body in a determinate time, as one second, by an action that is supposed to continue constantly the same. In the equation d v = g d /, the same thing is to be understood. But we must observe that the accelerating force being supposed to be variable, the quanti- ty g which represents the velocity that the accelerating force is capable of producing, if it were constant for one second, this quantity g, I say, is different for all the different instants of the motion. Indeed, it will be readily conceived, that when the accelerating force becomes less, the velocity that it is capable 1 74. Dynamics. of generating in a second by its action repeated equally during each instant of this second, must be less, and vice versa. 282. From the two equations ds =v dt, dv = g dt, we can obtain a third that may be employed with advantage. Thus from the equation d s = t> d t, we deduce d t = ; substituting this value instead of d t in the equation dv = g d t, we have, ds or, g d s = v d v. 283. We remark, that in the process by which we have just arrived at the equation d v = g d t, we regarded the velocity as increasing. If it had gone on diminishing, it would have been necessary, instead of d v to put d v ; so that the two equations dv = g d t, and g ds = v dv, to become general, must be written -h dv = gdty d=igds=zvdv, the upper sign being used when the motion is accelerated, and the lower when the motion is retarded. 284. There is a fourth equation that may be deduced from the two fundamental equations, and which should not be omitted. Thus, the equation d s = v d t gives v = -j- ; whence we obtain substituting this value for d v in the equation gdt= d i>, we have If we suppose, as we are authorized to do, that d t is constant, we shall have, gdt = -j^-or gdt* = dds. Collision. 175 But it must be recollected that, in the equation g dt* = d ds, it is supposed that d t is constant. When d t is variable, we make use of the equation Occasions will occur in which these formulas will be of great use. But we must not forget that the quantity g which they contain, represents, for each instant, the velocity which the ac- celerating force is capable of giving to the moving body in a known interval of time, as one second, if during the second, it were to act with a uniformly accelerating force ; so that as each quantity g measures for each instant, the effect of which the ac- celerating force is capable, we shall give it, for brevity's sake, the name of accelerating force. Of the direct Collision of Bodies. 285. We suppose, in what follows, that no account is taken of the gravity of bodies, of friction, or other resistance. We suppose also that the bodies, whose collision is the sub- ject of consideration, act the one upon the other according to the same straight line, passing through their centres of gravity, and that this straight line, is perpendicular to the plane touching their surfaces at the point where they meet. We shall consider bodies as divided into two classes, denom* inated unelastic and elastic ; the former are supposed to be such that no force can change their figure ; the latter are regarded as capable of having their figure changed, that is, of being com- pressed, but as endued at the same time with a property by which this figure is resumed after the compressing force is re- moved. Although there are not in nature bodies of a sensible mass 3 that answer perfectly to either of these descriptions, yet it is only by proceeding upon such suppositions, that we are able to de- termine the action of such bodies as are actually presented to our observation. 1 7S Dynamics. Of the direct Collision ofundastic Bodies. 286. Two unelastic bodies which meet, (or one of which falls upon the other at rest) communicate, or lose, a part of their motion ; and in whatever manner this takes place, we may al- ways, at the instant of the collision, represent each body, accord- ing to the principle of D'Alembert, as urged with two velocities, one of which remains after the collision, while the other is des- 133. troyed. 287. Let us, in the first place, suppose ,the two bodies to move in the same direction. That which goes the faster, will evidently lose a part of its velocity by the collision, and the other will gain by it. Let ra be the mass of the impinging body, and u its velocity before collision ; n the mass of the impinged body (which may be less or greater than m), and v its velocity before collision. Let us suppose that the velocity u changes to ' by the collision ; m will accordingly have lost u u'. I will consider m as having, at the instant of collision, the velocity u', and the velocity u u'. If we suppose, in like manner, that v becomes T/ by the collision, n will have gained v' v ; I can accordingly consider it, at the instant of collision, as having the velocity ', in the direction of the actual motion, and the velocity T/ T in the opposite direction ; since, on this supposition, it will really have only the velocity i/ (i/ v) or v. As, therefore, among these four velocities there can, by sup- position, remain only u' and t/, the two others u u', and T/ T, must be destroyed in the act of collision. Now as these are directly opposite, it is necessary that the quantities of mo- tion^ which the bodies would have in virtue of these velocities, should be equal ; we have, therefore, 33 ' m (u M') = n (v' v). Now in order that u' and v' may, as we have supposed, be the velocities which the two bodies m and n have after col- lision, these velocities must be such, that the impinging body shall not have the greater action over the impinged, that is, that the two bodies shall, after col'ision, proceed in company; we have, accordingly, v' = u' ; and hence form the equation. Elastic Boides,. 177 m (u u') n (u 1 i>)j m u m u' = n u' nu, we obtain 7 _ m w + nu m+fi 5 therefore, tu/ien the bodies move in the same direction, in order to find, the velocity after collision, we take the, sum of the quantities of motion, which the bodies had before collision, and divide this sum by the sum of the masses. Thus, if m, for example, be equal to 5 ounces and n to 7, it equal to 8 feet in a second, and v to 4 feet in a second, we shall have, 5x8+7x4 40 + 28 5 + 7 ~l2~~ that is, the velocity after collision will be five feet and two thirds in a second. 288. If one of the two bodies, as n for example, were at rest before collision, we should have v = 0, and the expression of the velocity after collision would accordingly become mu m + n ' that is, we should divide the quantity of motion belonging to the impinging body by the sum of the masses of the two bodies. If, however, instead of deducing this case from the more gen- eral one, we would find it directly, we should proceed according to the same principles, and consider the impinged body as hav- ing, in consequence of the collision, a velocity u', equal to and in the direction of that which it is to have after collision, and a ve- locity u', of the same magnitude, but in the opposite direction. Thus, since it is to preserve only the first, it is necessary that in virtue of the second it should be in equilibrium with the body m, having a velocity u u' which it is to lose. Accordingly we must have Mech. 23 178 Dynamics. m {u -r u') = n u'. from which we deduce 771 U U' = - , in -j- n the same as the expression above obtained from the general for- mula. 289. When the bodies move in opposite directions, in order to find the velocity after collision, it is only necessary to suppose in the first formula, that v is negative, which gives . mu nv u' = m + n that is, when the bodies move in opposite directions, in order to find the -velocity after collision, we take the difference of the quantities of motion belonging to the bodies before collision, and divide by the sum of the masses ; and, this velocity will take place in the direction of that body which had the greater quantity of motion. We might also obtain this result directly by proceeding as in the above example. Thus the laws of the direct collision of unelastic bodies reduce themselves in all cases to this single rule ; tJie velocity after collis- ion is equal to the sum or to the difference of the quantities of motion before collision (according as the bodies move in the same or in op- posite directions), divided by the sum of the masses. Of the Force of Inertia. 290, We have supposed in what we have said, that indepen- dently of gravity, the resistance of the air, and other obstacles, one of the two bodies opposes a resistance to the other, and makes it lose a part of its velocity. But how can a body with- out gravity, and which is confined by no obstacle, oppose a re- sistance ? Does not this seem to imply that it would be capable of giving motion ? Force of Inertia. \ 79 Now every resistance does not always imply an actual motion in the residing body. If the body A, for example, be drawn at the same time by two equal and opposite forces represented by AB, AC, it would evidently have no motion. But it is not less Fig.145. evident that if a force equal to G# were to act upon it in the direction CB, this force would be destroyed by the effort AC, and the body would yield in virtue of the force AB, equal to that just applied. We do not pretend to decide whether the resistance which bodies oppose to motion, does or does not arise from a cause of this kind. However the fact may be, the resistance in question which we call the force of inertia, differs from the resistance op- posed by active forces (as that of bodies which impinge against each other in opposite directions) in this, that these last annihi- late a part of the motion ; whereas, with respect to the force of inertia, while it destroys a part of the motion in the impinging body, this motion passes wholly into the impinged body, as is clearly shown by the equation m (u u') = n (u f v), above obtained for determining the motion after collision of two bodies which move in the same direction ; for u u' is 287. the velocity lost by the impinging body, and consequently m (u u') is the quantity of motion which this body loses by collision. We have, in like manner, seen, that u' v, is the ve- locity, and n (u' v) the quantity of motion, gained by the im- pinged body. Now we have shown that these two quantities must necessarily be equal. The force of inertia, therefore, is, properly speaking, the means of the communication of motion from one body to another. Every body resists motion, and it is by resisting that it receives motion ; it receives also just so much as it destroys in the body that acts upon it. We hence see that, every obstacle being removed, however small we suppose the impinging body, and however great the mass impinged, motion will always take place upon collision. When, for example, one of the two bodies is at rest, the velocity which has for its expression 1 80 Dynamics. can never become zero, whatever be the values assigned to m, n, and u ; the only case where u can be infinitely small, is that in which n is infinitely great. Thus, if in nature we see bodies lose the motion that they have received, it is because they com- municate it to the material parts of the bodies which surround them. Now it is evident from the formula, u = that the greater the mass of the impinged body ?i, the less (other things being the same), will be the velocity M', n being consider- ed as the sum of the material particles among which m parts with its motion. It will be seen, therefore, that the velocity u' may soon be so reduced as to escape the notice of the senses, even when it is not opposed by immoveable obstacles, as friction, &c. 291. The force of inertia, being a force peculiar to matter, exists equally in every equal portion of matter, and consequent- ly in a determinate mass it takes place according to the quanti- ty of matter, or in proportion to the mass ; and as the mass is proportional to the weight, the force of inertia may be regarded as proportional to the weight. But we must take care not to infer hence, that the force of inertia arises from gravity ; it is altogether independent of it ; indeed, if while a body is falling freely, it be forced forward by the hand with a velocity greater than that of its natural descent, the hand will experience on overtaking the body, a blow, or resistance, that manifestly can- not be attributed to gravity, which acts only downward. Still less can it be ascribed to the resistance of the air ; for the resist- ance of the air, being capable of acting only on the surfaces of bodies, cannot, like the force of inertia, be proportional to the quantity of matter. The force of inertia, therefore, is a force peculiar to matter, by which every body resists a change of state, as to motion and rest. The force of inertia is proportional to the quantity of matter, and takes place in all directions according to which an effort is made to move a body* Collision ofunelastic Bodies. Application of the Principles of Collision ofunelastic Bodi es 292. The principles which we have laid down respecting the collision of unelastic bodies are applicable, whether the bodies im- pinge directly upon each other, as we have supposed, or whether they act upon each other by means of a rod which joins their centres of gravity, or whether one draws the other by a thread, provided the action is immediately and perfectly transmitted to the centre of gravity of each. If, for example, the two bodies m and n act upon each other Fig.146. by means of a thread passing over a pulley P, and we would de- termine the motion that they would receive in virtue of their gravity, we observe that gravity tends to impress the same veloc- ity upon each of the two bodies at each instant. Now as one cannot move without drawing the other, the same thing will take place with regard to the two bodies at each new action of gravi- ty, as if the two bodies drew each other in opposite directions with equal velocities ; therefore, in order to find the resulting velocity, it is necessary, calling u the velocity produced by gravity at each instant in a free body, to take the difference mu nit of the quantities of motion, and to divide it by the sum m -f- n of the masses ; we have accordingly mu nu m n or ; u m -f- n m -j- n for the actual velocity that each new action of gravity would give to the body m. We see, therefore, since m, n, and w, are constant quantities, that the body m is carried with a motion uni- formly accelerated, and that the force which actually accelerates it, is to free gravity m n m n u : u : : : 1 : : m n : m -\- n. m -f- n m -{ n Consequently, if we call g the velocity which gravity communi- cates to a free body in one second, we shall have that which it would communicate in the same time to the body m, impeded by the body n, by the proportion 182 Dynamics. n m+n :m-n : : g : -^-- g. If, therefore, we call w the velocity of m at the expiration of a 264. number t of seconds, we shall have m 267. and the space which it will have described, will be ^x which is readily found, by putting for t the given number of sec- 276. onds, and for g 32,2 feet. 293. If at the first instant the body ?i, supposed to have less mass than the other, receive an impulse or velocity T, that is, if it were struck in such a manner, that, being considered free and without gravity, it would pass over in a second a number of feet denoted by , it would divide this action with the body m which it would draw during a certain time. In order to determine how the action in question would be divided, it must be remarked, that at the first instant the action of gravity being infinitely small or nothing, the body n, urged with a velocity i>, acts upon the body m as if this last were at rest. It is necessary, therefore, in order to find the velocity remaining after the action, to divide 28. the quantity of motion n v by the sum of the masses, which gives for the velocity with which n would draw m, if gravity m -f- n did not act in the following instants. But as we have seen that it would act in such a manner as to give to the body m, in m n . , the opposite direction, the velocity g t in the time / ; it follows that, at the expiration of the time , the body n will have only the velocity Whence it will be seen, that however small n may be, and how- ever small the velocity -y, and however considerable the mass of Collision ofunelastic Bodies. 183 the body m, n will always draw m for a certain time, after which tfi will prevail, and draw n in its turn. Indeed, whatever may be the quantity of motion n T>, impress- ed upon w, so long as it is of a finite value, it is evident that it would always be necessary, in order to counteract it, that gravi- ty should act for a certain time, for it only acts by degress infi- nitely small at each instant. If we would know at the expiration of what time m will cease to ascend, we should proceed thus. Let if be the time employed by a heavy body, falling freely, in acquiring the velocity i> ; ac- cording to article 263, we shall have therefore the velocity of n will be changed to n g t' m n m -f- n m -f- n & which being put equal to zero, gives n g t' = (m n from which we deduce n t' If, for example, the velocity T, supposed to be impressed upon w, is such as a heavy body would acquire in one second, we should have t' = I". Suppose m = 100 lb , and n = I 1 *, we should have t = 1001 99 that is, the body n would draw the body m only during one ninety-ninth of a second ; still it would draw it. We see, therefore, that there is not a finite force, however small, which is not capable of overcoming the weight of a body; and that it is not possible for a body actually in motion, to be placed in equilibrium with the weight of another body, that is, with a body that has the simple tendency of gravity. The for- 184 Dynamics. mer would first draw the latter, and afterward be drawn by it ; there would indeed be an instant of rest, but it would be that in which the former had lost all the velocity impressed upon it, and this state would continue only for an instant. 294. Thus the forco of bodies in motion cannot be estimated by weights, that is, by the simple tendency of gravity in bodies destitute of local motion ; but only by other forces of the same kind, as those of heavy bodies having fallen from a certain height. Hence, in order to have an idea of the force of a body of 3 pounds, carried with a velocity of 50 feet in a second, I should seek by the method of article 277, from what height a heavy body must fall to acquire a velocity of 50 feet in a second, and I should find it to be 38,8 feet nearly. I should conclude, there- fore, that a body of 3 pounds, urged with a velocity of 50 feet in a second, must strike as if it had fallen from a height of 38,8 feet. 295. The force which bodies in motion are capable of exert- ing, is called percussion. The force of percussion cannot, therefore, in any way be compared with simple pressure, or the effort which a mass is capable of making by its weight without local motion. A blow of a hammer, though feeble, will drive a nail into a block of wood ; also a body of small mass, which by its fall had acquired but little velocity, would be attended with the same result, while fc very considerable weight would produce no effect. The reason of this difference is, that in the former case, all the degrees of velocity possessed by the body in motion, are exerted in an instant ; whereas in the latter, the weight which exerts only a pressure, receives its degrees of force successively, and imparts them in the same manner to the nail and the sur- rounding mass ; and as each of these degrees is infinitely small, it is absorbed as soon as it is received. Collision of elastic Bodies. 1 85 Of the Collision of elastic Bodies. 296. Although elastic bodies, according to the definition which we have given, must be compressible, we are not hence to infer that they must be so much the more compressible, as they are more elastic. A ball of wool, for example, is not more elastic than a ball of ivory, although it is much more compressi- ble. Be this as it may, compressibility seems to be inseparable from elasticity. A body in virtue of its compressibility, changes its figure, when a force is applied to it from without ; and in vir- tue of its elasticity, it tends to recover this figure. But among all elastic bodies, some recover their figure entirely, others only in part. These last are called imperfectly elastic bodies. As to the former, they may resume their figure more or less promptly ? and by very different degrees. But if they are such that, after being struck, they restore themselves according to the same de- grees by which they were compressed, we call them perfectly elastic bodies. In other cases they are denominated simply elastic bodies. We shall here consider only those that are perfectly elastic. We observe with respect to perfectly elastic bodies, that in collision, a resistance takes place on the part of the body which has the least velocity, and consequently a compression, and that on this account not only a restoration of the figure follows this compression, but this restoration is itself followed by a new change of figure directly contrary to the first. To this succeeds another, which reduces the body to the figure first given by the compression, and so on. In this way the parts of each body have, with respect to their centre of gravity, a vibration, or mo- tion backward and forward ; since the parts tend to return to their first figure by a motion which goes on increasing, and thus carries them beyond their former position. These changes of figure, which alternate with each other, are sensible in several elastic bodies, when struck, and particularly in those that are of the sonorous class. Mech. 24 1 8k' Dynamics. It is not, however to be supposed, that these vibrations affect 134 ^ ie ve ^ oc i t j which the bodies take after collision* They can have no influence upon the motion of the centre of gravity, since this motion takes place in each of the two bodies independently of the other. The collision of perfectly elastic bodies is to be viewed, therefore, in the following manner. When the two bodies m, n, Fig. 148. come to meet in C, the resistance which n opposes to w, causes them to be mutually compressed, until the two centres and the point of contact have all the same velocity ; thus far every thing takes place, as in the collision of hard bodies, with the exception of the change of figure, which can contribute nothing to the quantity of motion lost or gained. The change of figure is effected in such a manner, that each of the two bodies is flattened to the same degree on opposite sides ; since the parts farthest removed from contact, advancing more rapidly in the one body and less rapidly in the other, until the compression is completed, crowd very much the inter- mediate parts. The compression once finished, the parts of each body bordering upon the points of contact, support them- selves the one against the other, while the contact is transferred; and the recoil of the spring takes place toward the parts oppo- site to the point of contact, with all the force by which the bodies tend to restore their figure. It will accordingly be seen that the impinging body loses by the recoil, a velocity equal to that which it had lost by the com- pression ; and that, on the other hand, the impinged body gains by the recoil a velocity equal to that which it had gained dur- ing the compression ; and, although the two bodies do not cease to exert their elastic force when they have regained their orig- inal figure, they have no longer any action upon each other, since the force with which they go on to dilate themselves, be- ginning now to grow less, they separate from each other at this conjuncture. If, when the two bodies move in the same direction, u is the velocity of the impinging body, and v that of the impinged ; u f being supposed the common velocity which they would have Collision of elastic Bodies. 187 lafter collision, considered as unelastic, u u' would be the velocity lost by the .impinging body ; when, therefore, the recoil of the spring (taking place in a direction opposite to the motion) causes as much motion to be lost, as had already been lost by the compression, there will remain only the velocity u 2 (u u') = 2 w' u. As to the impinged body, u' v is the velocity gained by col- lision ; and we have seen that, by the recoil of the spring, it ac- quires as much more ; it will have, therefore, v 4. 2 (u' v) = 2 u' v. This case comprehends that in which one of the two bodies is at rest before collision* If the bodies move in opposite directions, the reasoning is precisely the same for the one which has the greater quantity of motion. As to the other, it would, considered as unelastic, lose its velocity by the collision, and acquire another in the op- posite direction ; u f being this velocity, we shall have v + u' for the velocity lost. Doubling this effect, on account of the bodies being elastic, and adding it to the original velocity r, we have 2 (o -f- u') v = 2 u' -f v. 297. By attending to the resulting expression in each of the above cases, it will be seen that the circumstances of the collis- ion of bodies perfectly elastic are all comprehended in this single rule ; Seek the common velocity which the two bodies would have after collision, if they were destitute of elasticity, then from double this ve- locity, take the velocity which each had before collision, and we shall have the velocity of each after collision ; it being understood that when the bodies move in opposite directions before collision, the sign is to be given to the velocity of that body which has the less quantity of motion. 298. From the principles above laid down, we might easily obtain, for the collision of elastic bodies, formulas which should contain only the masses and velocities before collision. In order 188 Dynamics. to this, it would only be necessary to substitute, in the express- ions 2 u' u and 2 u' db D, the value of u' furnished by the rules of articles 286, 288. But as these formulas would not pre- sent themselves in a manner so easy to be retained as the rules we have given, we leave this substitution to be made by those who may wish to see the result. 299. We observe that, when one of the two bodies is at rest, the velocity which it would receive by the collision is double that which it would have had, considered as non-elastic. This is an evident consequence of the general rule. 300. To give a few examples of these rules, let us suppose, in the first place, that the two bodies are equal, and that one of them is at rest ; then , which expresses the velocity after wi -f n' collision, the bodies being considered as unelastic, becomes or | u. From twice \ u or w, therefore, we subtract u to 297. obtain the velocity of the impinging body after collision, which is consequently zero. To find the velocity of the impinged body, from twice ^ u or n, we subtract 0, which it had before collision, and we have u for the velocity after collision. Hence we see that the motion of the impinging body passes wholly into the impinged. Accordingly, if several equal elastic bodies be placed in contact with each other in the same straight line, and one of them be made to impinge against the others in the direction of this line ; the only effect would be, that the one at the opposite extremity would be driven off with the same velocity. If two are made to impinge at the same time against the others, two would be detached from the other extremity, and so on. Let us suppose the two bodies to move in the same direction, one of 5 ounces, and with a velocity of 6 feet in a second, and the other of 7 ounces, with a velocity of 2 feet in a second. For the common velocity which they would have after collision, con- sidered as unelastic, we obtain 5 XG -f 7 X 2_ 44 _ -~5TT~ If, therefore, from double this quantity or 7 1, we take the veloc- ities before collision, namely, 6 and 2 respectively, we shall have Collision of elastic Bodies. 189 for the velocity of the impinging body after collision 1 |, and, for that of the impinged 5 |. If the impinged body, instead of 7 ounces, had a mass of 20 ounces ; the velocity, after collision, the bodies being considered as unelastic, would be, If from double this quantity or 5 f , we subtract the velocities be- fore collision, namely, 6 and 2 respectively, we shall have, 5 I _ 6 an( j 5 | 2, that is I and 3 for the velocities after collision, in which the sign before f indicates that the impinging body would rebound. If the two bodies are made to move in opposite directions with the same masses and the same velocities, as in the first of the above examples, the velocity after collision, the bodies being considered as unelastic, would be 5x6 7x2 30 14 12 lj& __ If from double this velocity or 2 f , we subtract the velocity 6, which the impinging body had before collision, we shall have 3 i for its velocity after collision ; it will rebound, therefore, with a velocity of 3 \ feet. As to the impinged body, it will be recollected that to twice 1 \ or 2 f , the velocity before collision is to be added, which gives 4 f for its velocity after collision. 297. 301. Since, when elastic bodies move in the same direction before collision, the velocities after collision are 297. 2 u' u and 2 u' i>, u f being the velocity which they would have, considered as une- lastic ; the difference u v of these two velocities, is the same as the difference of the velocities before collision. This difference is called the relative velocity, and is accordingly the same be- fore and after collision. 190 Dynamics** When, on the other hand, the bodies move before collision in opposite directions, their velocities after collision are, 2 u' u and 2 ' -J- *>, the difference of which is u + i>, and this was their relative velo- city, or that with which they approached each other before col- lision. Therefore the velocity with which they separate from each other after collision, is the same as that with which they approach each other before collision ; thus, with respect to elastic bodies the relative velocity is the same before and after collision. Of the Motion of Projectiles. 302. By the motion of projectiles, we understand that of bodies, which, being thrown with a certain force, are afterward left to the action of this force and that of gravity. We shall first seek the path that would be described in free space. p. , 49 From the point A, let a body be thrown in the direction AZ, and with any given velocity. If gravity were out of the ques- tion, it would move uniformly in the direction of the straight line AZ. But as gravity acts without interruption, the body will not: be in the straight line AZ, except for an instant ; instead of AZ, it will describe a curved line ABC of which AZ will be the tan- gent at the point A. since AZ is one of the instantaneous direc- Geom. * . , -11 97. tions of the moving body. 303. In order to determine the nature of this curve, let AE be the velocity communicated to the projectile, or the number of feet that it would describe in a second, if it preserved continual- ly this velocity ; and at the instant of its leaving the point A, let us suppose this velocity composed of two others one AD hori- zontal, and the other AF in a vertical direction. It is evident that the direction of gravity being vertical or perpendicular to AD, its action will not tend either to diminish or increase the velocity AD, and that consequently whatever course the body may take, it will preserve constantly the same velocity parallel to the horizon. As to the velocity in the direction AF, when the body, in virtue of its constant velocity, parallel to the horizon, shall have Motion of Projectiles. 191 advanced by a quantity equal to AP, it will not have risen to a height PJV, equal to that at which it would have arrived, uninfluenced by gravity, but to some lower point M in the same vertical PN -, because its velocity in a vertical direction, being directly opposed to that of gravity, the space which it would have described in virtue of this vertical velocity, must be dimin- ished by the space which the action of gravity would have caused the body to describe in the same time. Accordingly let v denote the velocity communicated in the direction AZ, or the number of feet that the projectile would describe uniformly each second, in virtue of this velocity, and / the time, or number of seconds or parts of a second, employed in passing from A to some point JV, we shall have AN = v t. Let g be the velocity communicated by gravity in a second, | g t 2 will be the space that a heavy body would describe in a number t of seconds. If therefore M be the point where the body will arrive at the expiration of the time J, we shall have Through the point A, draw the vertical AX, and through the point M the straight line MQ parallel to the tangent AZ. Call- ing AQ, x', and QM, which is equal to ^JV, */', we shall have x' = i g t 2 , and y' = v t. If from this last equation, we deduce the value of /, namely, and substitute it in the first, we shall obtain or D But - expresses the height from which a heavy body must o fall to acquire the velocity v ; hence, if we call this height h, we 277. 1 92 Dynamics. shall have = h, and consequently = 4 h ; therefore, 4hx' = y' 2 . We hence infer that each point M of the curve AMC has this property, that the square of the ordinate?/' or QM, parallel to the tangent AZ, is equal to the product of the abscissa AQ or x' by a constant quantity 4 h ; therefore the curve AMC is a parabola which has for a diameter the vertical line AX, and for its param- eter the quadruple of the height due to the velocity of projection, and of which the angle AQM, made by the ordinates with this Trig, diameter, is the complement of the angle of projection ZAC. 182. Tliis curve, therefore, is easily constructed, when the velocity of projection and the angle of projection are known. 304. We proceed to examine some of the properties of this curve, considered as the path traced by a projectile ; and for this purpose we refer the different points M to the horizontal line AC by drawing PM perpendicular to AC. We designate AP by a?, PM by y, and the angle of projec- Trig. 30. ti n ^^ kj a - I n tne right-angled triangle APNwe have 1 : AN :: sin NAP : PJV, :: cos NAP : AP-, whence PN = AN sin NAP = -o t sin a, and AP or x = v t cos a. Also, since MN = i g t 2 , as we have seen above, PM or y = v t sin a g t a . Deducing from the former equation the value of tf, namely, and substituting it in the latter, we shall have, = x * ina . jg* 8 ** " cos a -y 2 cos a* ' Motion of Projectiles. 1 93 or, o* v z a; sin a cos a or, putting for ^ its value 4 ft, and multiplying both members 2" o by cos a 3 , 4 Tiy cos a s = 4 hx sin a cos a a 2 , which will furnish us with the following properties. 305. As the velocity communicated to the projectile is sup- posed to be limited to a certain measure, its effect in a vertical direction must be exhausted at the end of a certain time by the action of gravity, so that at a certain point the body will cease to ascend, and thence will commence a downward motion ; but, as its horizontal velocity does not change when it has reached its highest point, as jB, it will describe the second branch BC of the same curve, and will again meet the .horizontal line AC in another point C. Now in order to determine the distance AC, called the horizontal range* of the projectile, we have only to sup- pose y = 0. We have, accordingly, 4 h x sin a cos a x 2 or x (4 h sin a cos a x) = ; which gives x = 0, and x = 4 h sin a cos a. The first value of x indicates the point A ; the second is that of AC, which may be determined by producing XA till AK is equal to 4 ft, and letting fall from the point K upon AZ the perpendicular KL, and from the point L upon AC the perpendicular LC; since we have R = 1 : sin K = sin a : : AK = 4 h : AL 4 h sin a, and #=1 : sin #LC=cos a : : AL=4 h sin a : AC 4 h siq a cos . 306. If with the same velocity of projection we would know what angle would give the greatest horizontal range, we take the differential of the value of AC, by regarding a as variable, and put this differential equal to zero ; thus c . , 4 h d a cos a 2 4 h d a sin a 2 = ; * Sometimes called also random and amplitude. Mech. 25 194 Dynamite. from which we deduce, sin a a = cos a a or sin a* cos a* , that is, trig. 8 tang a 3 = 1, and consequently, tang a = 1, in other words, the tangent of the angle of projection is in this Tri 24 case ec l ua ^ to rac * ius j accordingly this angle is equal ' to 45. Therefore, the greatest horizontal range is obtained, other things being the same, when the angle, of projection is 45. It is here supposed that Trig. 20. . - sin a = cos a = y/i ; this value substituted in the above expression for AC, gives therefore, J/ie greatest random is double the height through which a body must fall to acquire the velocity of projection. 307. If we would know to what height the body ascends, or the highest point B of the curve, we proceed thus ; in the equa- tion 4 h y cos a 2 = 4 h x sin a cos a # 2 , we put equal to zero, the differential of */, taken by regarding x only as variable, which gives 4 h dx sin a cos a 2a?e?# = 0, from which we obtain a? = 2 h sin a cos a ; 2f)5 therefore, since AC = 4 fo sin a cos a, if we suppose the perpen- dicular BD, we shall have x or AD = 2 ft sin a cos a = | Moreover, this value of x being substituted in the equation, Motion of Projectiles. 1 95 4 hy cos a 2 = 4 /i a? sin a cos a a? 2 , gives 4hy cos a 2 = 8 h a sin a 2 cos a 3 4 ft 2 sin c 2 cos a 2 , from which we obtain y or D = /i sin a 3 . This determines the vertex of the axis, since d y being zero at the point , the tangent at B is parallel to ^C, or perpendicular to BD. 308. We propose now to determine the direction AZ, to be given to a projectile in order that it may fall upon a known point Fig.160. Jlf, that is, the inclination that a mortar, for instance, must have to throw a shell upon the known point M. The perpendicular MP upon the horizontal line passing through the point A, being drawn, the distance #P, and the an- gle MAP, are to be considered as known. AP being designated by c, and the angle MAP by e, we shall have c sin e cos e : c : : sin e : MP = , cos e we have, therefore, for the point M, a? = c, and csin e ** " COS 6 Substituting these values in the equation 4hy cos a 2 = 4hx sin a cos a a; 2 , we obtain, 4 he sin e cos a 2 = 4 h c sin a cos a c 2 , cos e or 4hsm e cos a 2 = 4 ft sin a cos a cds c c cos e, or 4 /i os a (sin a cos e sin e cos a) = c cos e. Trig. 11, that is, 4 h cos a sin (a e) = c cos c ; Dynamics. or, since Trig. 27. cos a sin (a e) = (sin (a -f- a e) sin (a a -j- c) ). 4 /& i (sin (2 e) sin e) = c cos e, and 2 A 2 /i sine sin (2 a e) = - - -f c, cos e cos e which may be given by 'the following construction. Having raised upon AM the indefinite perpendicular AE ; from the middle D of ./^ = 4 /t, we erect upon w^^f the perpen- dicular Z), cutting AE in some point E, from which as a centre, and with a radius equal to EA, we describe the arc JlNN'K ; having produced PM till it meets this arc in the points JV, JV', if we draw the lines ANZ, AN'Z', these will be the directions iu which the projectile being thrown with a velocity due to the height h, it will in either case fall upon the point M. Indeed it will be readily seen that the angle EAD of the right-angled triangle ADE is equal to MAP. Therefore, since AD = 2 /t, ED = ^-r4 5 an d, since AP = c, we shall have, cos cos e consequently, ^ sin(2a e) = i:/. cos e But in the same triangle ADE, AE = ^ ; therefore, COS 6 AE sin (2 a e) = /. Let the arc KNA be produced till it meets, in Cr, the vertical GE, and from the points JV, JV 7 , draw the perpendiculars JVL, JNPX 7 . In the triangle JVEL, we have WE : JVL, or AE : El : : 1 : sin NEG ; whence, AE sin JVG = El; accordingly, Motion of Projectiles. 197 sin (2 a e) = sin NEG, and 2a e = NEG^NEA + e; consequently, -J- e. But because the angle NAM has its vertex in the circumfer- ^ ence, and AM is a tangent, NAM is equal to JV72./2 ; also the 131. angle MAP = e ; whence a = JVA/lf 4- ,/tf.tfP = NAP-, therefore the point N satisfies the question. The same may be shown with respect to the point N r . deed in the triangle N'ELf, we have N'E : JV'L', or AE : El : : 1 : sin N'ELf, : : 1 ; sin N'EG, whence AE sin N'EG = El; and, since ^2Esin (2a e) = E7, as above shown, we have sin (2 a e) = sin N'EG, and 2 a e = JV'EG = JVi^ 4- e; therefore, a = i JV^^f -f e = JV^JIf -f JIMP = 309. Thus with the same force of projection, a projectile may always be made to fall upon the same point M, according to two different directions, provided that AP does not exceed DR. The direction AN' is the most favourable for crushing buildings or other objects with shells. The direction AN is to be pre- ferred, when the purpose is simply to throw down walls and breast-works, and by rebounding to lay waste at a distance. This leads us to speak of ricochet firing ; but we shall first remark that the equation to the range or to 2 ft, we obtain 2ft This general expression for the time may be made use of in reg- ulating the fusees of bombs. We proceed now to the subject of ricochet firing. 310. By the above term is meant a motion by which a pro- jectile, after meeting with an obstacle, rebounds and commences a new motion similar to the first. The smaller the angle of ele- vation above the horizon, the greater, other things being the same is the tendency, upon rebounding, to proceed forward ; since the projectile force is exerted almost entirely in a horizontal direc- tion, and much time is required for the resistance of the air and other obstacles to destroy it. If the projectile be of an unelastic substance, and the surface upon which it falls be horizontal and unyielding, it would not bound, since upon arriving at C, accord- 'ing to any direction MC, its velocity might be decomposed into two others, of which QC, perpendicular to the surface would be simply destroyed, the rebounding in other cases being caused entirely by the elasticity, so that the other part PC remains (no account being taken of friction and the resistance of the air), and the body would move along CZ. Motion of Projectiles. 199 31 1. But if at the point C, where the body meets the surface Fig.152. there be a mound or eminence CE, the motion according to MC being decomposed into two others, one according to QC perpen- dicular to the surface CE, and the other PC in the direction of this surface, the body would proceed acccording to this latter, describing the line P, and might, after leaving the point /, des- cribe just such a curve as it would have described, if it had been projected from .E, according to C, with the same velocity ; so that it would elevate itself to a certain point, and then return td the surface in some other point /, when the motion under similar circumstances might be again renewed. 312. A ricochet motion, therefore, depends upon the position of the obstacle against which the body in question strikes. But if the obstacle be flexible or yielding like the earth, water, &c., this motion may take place even when the surface is perfectly horizontal. Indeed, by the vertical velocity QC, the body tends Fig.153 to bury itself, and does bury itself more or less, according to the nature of the obstacle ; while with the velocity PC it plows the earth, and forms a furrow, the depth of which increases till the vertical velocity QC is destroyed. Then by the remaining ve- locity in a horizontal direction, it drives before it the matter which lies in its way, and in working for itself a passage, it in- clines in the direction from which it experiences the least resist- ance, and the surface of the furrow becomes, with respect to the body, what CE was in the last case. Now as the remaining pro- Fig 152 jectile force, other things being the same, is so much the greater according as the depth of the furrow is -less, and as this depth depends upon the vertical velocity QC which will be so much the less, according as the angle .MCP, or the angle of projection RAZ, is less, it will be seen how the smallness of the angle of projec- tion is favourable to this sort of motion. 313. The figure of the body also, is of great importance. If, for example, the question related to a motion upon water, and the body were of a spherical shape, the velocity MC must be such that the vertical velocity QC may be destroyed before the vertical diameter of the body is entirely immersed, since, when the body is once covered, the resistance of the water would act equally in every direction, and there would be nothing to change* 200 Dynamics. its direction, except gravity, the tendency of which would be to prevent a ricochet. 314. As this immersion, however, takes place gradually, it will be seen that the motion of the centre must be in a curved line ; since while any part of the body remains above the sur- 'face, the direction in which the resistance acts is changing con- tinually. If, for instance, when the centre C, after having des- cribed any line PC, tends to move according to the prolongation C7, of this line, we imagine two tangents BR, Z)S, parallel to this direction, it is evident that the part BVL only would be exposed to this resistance ; and that if the body is spherical, the resultant CK of all the resistances exerted upon the different points of BVL would have a direction tending to elevate the body above C/; so that the parallelogram CIEK being formed, CE will be the course which the body would take instead of C/, no allow- ance being made for gravity. 315. Finally, if the body and the obstacle are flexible and elastic, this circumstance will further contribute to a ricochet motion. We take a very simple case, as an example ; let the Fi 155 body only be considered as flexible and elastic, and let this elas- ticity be perfect ; the body being supposed at the same time to be destitute of gravity. At the instant in which the body, pro- jected according to /2C, comes to touch the surface, its velocity is decomposed into a horizontal velocity which would remain al- ways the same, if there were no friction, and no resistance on the part of the medium in which the body moves. As to the per- pendicular or vertical velocity PC, it compresses the body, and being destroyed gradually, while the horizontal velocity contin- ues, it is evident that the centre C approaches the plane HZ by degrees, which go on decreasing, while the rate at which it ad- vances parallel to jf/Z, remains the same. Consequently, if at each instant we imagine a parallelogram having its horizontal sides to its vertical, as the horizontal velocity is to the velocity that remains in a vertical direction, the diagonal of this parallel- ogram which must mark the course of the centre each instant, will be different, and differently situated each instant, so that the centre C will approach HZ in a curve, while the compression is going on. When the compression has ceased, the centre C will Motion of Projectiles. 201 be carried for an instant in the direction of a tangent parallel to HZ; after which, the recoil taking place, the body recovers by degrees the velocity by which it tends to depart from the plane after the same manner in which the velocity was destroyed by the compression during its approach to the plane, and it will des- cribe the second part RO of the curve perfectly similar to RC. Lastly, when it shall have arrived at the point O, distant from the plane HZ by a quantity equal to the radius /C, it will move according to the tangent OT, situated like ^C; that is, the ob- lique collision of a body against an inflexible and unelastic plane, (friction being out of the question) takes place in such a manner as to make the angle* of reflection equal to the angle of incidence, these angles having for their measure the inclination to a hori- zontal plane of the tangents at the extremities C, O, of the curve described by the centre of the body during its compression and subsequent recoil. 316. If BD be the direction in which a body is thrown, re- gard being had to gravity, this body will describe the portion DC of a parabola of which BD is the tangent, until it touches the plane, then, when the compression has ceased, it will describe another portion SO of a parabola equal to the first and placed in the same manner. 317. Friction, moreover, contributes to the kind of motion under consideration, since it occasions a rotation in the body that aids it in rising above obstacles, as we have already seen. 234, 318. We conclude what we have to say on the subject of projectiles moving in an unresisting medium, with observing that, since gravity draws a body downward from the direc- tion given it by the projectile force, when we take aim at an object in shooting or in throwing any body, we should direct the sight above this object, and so much the more above it, ac- cording as it is more distant, and according also to the feebleness of the force employed. It is on this account that in fire-arms the line of sight makes an angle with the axis of the piece, so that these lines produced would meet at a point beyond the muzzle toward the mark. The projectile, ball, or bullet, propelled in the direction of the axis, commences its motion in a direction making a greater angle with the horizon than that made by the line of Meek, 26 202 Dynamics. sight ; so that the precaution is the same as if we had taken aim in the direction of the axis, but at a point above the object. 3 1 9. We remark further, that there are cases in which, al- though we have given no impulse to a body, and seem to aban- don it to gravity alone, yet this body describes a curved line common to all projectiles. A body, for example, which is suf- fered to fall from the mast-head of a vessel under sail, really des- cribes a curved line. If we attend to the point of the deck where it strikes, we shall find it just as far from the mast, other things being the same, as the point from which it started, so that the body describes a line parallel to the mast ; but with respect* to a spectator at rest, it has actually described a parabola (the resist- ance of the air not being considered), for, at the instant it was dropped, it must have had the same velocity with the vessel ; the case is therefore precisely the same, as if, the vessel being sta- tionary, we had thrown it with a velocity equal to that of the vessel, and in the same direction. It will be seen, also, at the same time, why it describes with respect to the mast a straight line parallel to this mast ; it is because they both move with the same velocity, and in the same direction; considered horizontal- ly, therefore, they must preserve the same distance from each other. 320. In the foregoing theory, we have taken it for granted; (1.) That the force of gravity is the same throughout the whole range of the projectile. (2.) That it acts in lines parallel to each other. (3.) That there is no resisting medium. The two first suppositions, although not strictly conformable to fact, are attended with no material error in practical gunnery, and those arts to which this theory is subservient. But the third is of es- sential importance to the truth of the results we have obtained. We can readily put the theory to the test of actual experiment. The initial velocity of a cannon ball, for instance, may be obtained with considerable accuracy, by either of the following methods. 321. (1.) Let the cannon together with the carriage and other weight if necessary, be suspended like a pendulum so as to move freely in the direction opposite to that in which the Motion of Projectiles. 203 ball is to be discharged.* Upon the explosion taking place, the centre of gravity will remain unchanged, that is, the quantities 134< of motion in opposite directions will be equal ; consequently, if the motion of the gun, &c., be made so slow by means of the attached weight, as to admit of its velocity being taken by actual observation, the velocity of the ball will be as much greater as its mass is less. Knowing the mass of each, we should use the following proportion ; as the mass or weight of the ball to that of the gun, carriage, &c., so is the velocity of the latter to that of the former. 322. (2.) The ball may be discharged into a large block of wood suspended so as to move freely after the manner of a pen- dulum,* and, the velocity being observed as before, we then say as the mass of the ball to that of the pendulous body, so is the velocity of the latter to that of the former. This latter method is adapted to finding the velocity at different distances from the cannon. It is thus found that the velocity of a cannon ball varies ac- cording to the quantity and quality of the powder, the size of the ball, the length of the piece, &c. At the commencement of the motion, it is ordinarily between 800 and 1600 feet in a second. 323. With a velocity equal to 800 feet in a second, the angle of projection being 45, for instance, the horizontal range, great- & c .' ' est elevation, &c., are readily determined by our formulas. We first find the height h through which a body must fall to acquire the velocity of projection 800 feet, and double this height will be the horizontal range required. Now to acquire a veloc- ity of 800 tf et in a second, a body must fall through a space equal (800) * 800 ft....log....2,90309 277. ~ 5,80618 64,4....1og....l,80889 h = 9937,75 . . K * 3,99729 2 Range = 19875,5 = 3,7 miles. * It will be seen hereafter at what point in the pendulum the im- pulse must be applied, in order that no part of it may be expended against the supports from which the pendulum is suspended. 204 Dynamics. The greatest elevation is equal to h multiplied by the sing square of the angle of projection, that is, equal to h (sin 45) 3 . h = 9937,75ft. log 3,99729 45 log sin 9,84949 9,84949 Greatest elevation = 4969 feet 3,69627 4969 wants only 31 1 feet of a mile. Moreover, according to the case supposed, we have as the expression for t the time of flight. h = 9937,75....1og....3,99729 g = 32,2 log 1,50786 2)2,48943 17",57 1,24472 2 t = 35, 14 On the supposition of a velocity of 1600 feet in a second, the angle of projection being the same, we should have for the hori- zontal range 79503 feet or 15 miles, for the greatest elevation 3,7 miles, and for the time of flight 3 minutes and 38 seconds. So great, however, is the resistance of the air, that a cannon ball, under the most favorable circumstances, is seldom known to have a range exceeding 3 miles ; the path described is not strict- ly a parabola or any known curve ; its vertex is not in the mid- dle, but more remote from the point of projection than from the other extremity ; and the path through which the body descends is less curved than that through which it ascends. This resistance increases faster than the velocity ; so that in the slower motions, there is a nearer approach to the foregoing theory, than in those which are more rapid, as is apparent to the eye in the spouting of water, and more especially of mercury from the side of a ves- sel. To treat of this resistance, and to estimate its effects, be- longs to that branch of our subject which has for its object the motion of fluids and that of bodies immersed in them. Motion dozvn inclined Planes. 205 Of the Motion of heavy Bodies down inclined Planes. A 324. A heavy body left to itself upon a plane surface KLHI, Fi 156 - inclined to a horizontal surface PIHN cannot yield entirely to its gravity. A part of the force derived from this cause, is em- 37. ployed in pressing the plane, and the other serves to bear it along the plane. It is necessary, therefore, to decompose Its gravity into two forces, one of which produces the pressure upon the plane, and the other the motion along this plane. 325. Let G be the centre of gravity of the body in, or the point in which all its action may be considered as united. Let GB be the space through which it would fall in an instant, if it were free. Let GC be drawn perpendicular to the plane ; and suppose a plane to pass through GB, GC, this plane Avill be per- pendicular to the two planes KLHI, IPNH, since it passes Georo. through the straight lines perpendicular to these planes. If therefore, we conceive DE, EF, to be the intersections of this plane with KLHI, IPNH-, DE, EF will be perpendicular to the Geom common intersection HI of these two planes. 355 - Draw GA parallel to DE, and construct the parallelogram GABCof which GB is the diagonal, and GA, GC, the sides. We may suppose that gravity, instead of urging the body accord- ing to GB, urges it at the same time according to GC with the velocity GC, and according to GA with the velocity GA. Now it is evident that GC, being perpendicular to the plane, cannot but be destroyed, if the point O where it meets the plane is at the same time a point common to the plane and the body m. As to the force GA, since it tends neither to approach toward, nor to recede from the plane, it cannot but have its full effect. GA, therefore, represents the velocity with which the body tends to move, and with which it would move in the first instant. As the force GA is in the plane of the two right lines GB, GC, it is in the plane DEF. We can therefore leave out of consid- eration the extent of the two planes KLHI, IPNH, and employ only the plane DEF represented in figure 157, so that the body may be supposed to move in the right line DE. 206 Dynamics 326. Since the force GA passes through the centre of gravity G of the body m, it must distribute itself equally to all parts of this body. Therefore, so long as friction is supposed to have no influence, the body can have no motion except that of sliding along the plane, that is, it can have no tendency to roll, whatev- er may be its figure, provided the perpendicular GB meets the plane in a point that belongs at the same time to the surface of the body. This would not be the case, however, as we have 196> seen, if the perpendicular did not meet the base of the body, or the surface by which it rests upon the plane. J"he influence of friction, moreover, tends to produce a rolling motion. 327. Since the body m must describe GA in the same time in which it would describe GB by the free action of gravity, if we conceive that at the end of the first instant, gravity acts anew ; as it communicates in equal instants equal degrees of velocity, by supposing for the second degree of velocity communicated in a vertical direction, a decomposition similar to that above made for the first instant, it is evident that the second parallelogram will be equal in all respects to the first. We accordingly in- fer, in like manner, that the force perpendicular to the plane will be destroyed, and the force parallel to the plane, and equal to GA, will be added to GA. By reasoning in the same manner for the following instants, Ave should conclude that the velocity along the inclined plane is accelerated by equal degrees ; in other words, that the motion of heavy bodies down an inclined plane is a mo- tion uniformly accelerated. Hence all that has been said upon the subject of motion uniformly accelerated, is strictly applicable to the motion that takes place down inclined planes. Consequently in this latter case, as well as in the former, the velocities are as the times, the spaces described are as the squares df the times, 264, &c. or as the squares of the acquired velocities, &c. 328. Therefore, in order to determine the motion that takes place upon a plane of a known inclination, we have only to find the ratio of the accelerating force to gravity, that is, the ratio of GA to GB. Now GA, GB, being parallel respectively to DE, DF, the angle AGB is equal to EDF, and the angle Jl being a right angle as well as the angle F, the two triangles AGB, EDF, are similar ; whence, Motion down inclined Planes. 207 DE : DF : : GB : GJl that is, the length of the inclined plane is to its height, as the velocity which gravity communicates to a free body, is to that with which it urges the body along the inclined plane. 329. Now as gravity gives to a free body, in a second of time, a velocity by which a space of 32,2 feet are described uniformly in a second, it will be easy to determine the velocity acquired 273f ' by a body in the first second of its descent down an inclined plane. If, for example, the length of the plane is double the height, the velocity acquired along the plane during the first second, will be half of 32,2 feet ; that is, at the end of the first second, if gravity ceased to act, the body would pass over 16,1 feet in a second. Having thus determined the velocity for the first second, we shall have the velocity after any proposed number of seconds, by multiplying this by the number of seconds ; also the space is found by multiplying this first velocity by half the square of 26 ~- the number of seconds. In short, it would be easy to determine all the other circumstances of the motion in question, by articles 267, &c. We hence deduce the following propositions. 330. If two heavy bodies, setting out at the same time from the point D, descend, one along the plane DE, and the other in p . 15g the direction of the perpendicular DF, and we would know in what part of the plane DE the first would be, when the second had arrived at any given point A, we have only to let fall from the point A upon DE the perpendicular AB ; and the point B will be the place sought. Indeed if we represent by g the velo- city that gravity communicates to a free body in one second, by calling t the time employed in describing DA, we shall have L on the other hand the velocity acquired in a second by the body that descends along the plane DE, is - ; accordingly by calling /' the time employed in descending from D to B, we shall have DB - g * DF X ' t' 2 - UK _ ^ X t , 208 Dynamics. whence, DA : DB :: :: DE X t 2 : DF X t' 2 . But by similar triangles, Gfjom. 213. DA : DB :: DE : DF; consequently, DE : DF : : DE X t* : DF X *' 2 ; therefore, /'2 _ j 2 or // = (f Fie.159. 331. Hence, if DG be a third plane described by a third body setting out from D at the same time with the other two, by drawing from the point A the perpendicular AC, A, B, C, are the three points at which the three would arrive in the same time. 332. If upon DA as a diameter, we describe a semicircum- ference, it will pass through the points C and B, since the angles Geom. at Q anc j are r ight angles. Consequently, the chords 1)C, DB, and the vertical diameter DA, are all described in the same time ; and as this does not depend upon the length or inclination of the chords, we may draw the general conclusion, that the time employed by a body in falling through any chord of a circle, drawn from the extremity of a vertical diameter, is the same as that employ- ed in falling through this vertical diameter. 333. We have seen that g being the velocity communicated to a free body in one second, is that given in the same time to a body that descends along DE. Let t, t', be the times employed in describing DF and DE respectively; we shall have whence, g -~^ X Motion down inclined Planes. 209 which gives, by multiplying the extremes and means and reduc- ing, 2 DP X i'* = DE x * 3 , or, DF x V* = DE x t 2 , or, taking the square root of each member, OF x t 1 = DE x f ; in other words, t : t' :: DF : DE. In like manner, if f" represent the time employed in describing DG ; we shall have t : t" :: DF : DG; whence, *' : t" : : DE : DG ; that is, the times employed in describing different planes of the same height, are to each other as the lengths of these planes. 334. The velocity of the body which descends along DF, is g t at the expiration of the time t. For a similar reason, the 2 i67. velocity of the body that descends along DE, is s * x t' at the expiration of the time t'. Accordingly, if we call w, v, the velocities acquired by the two bodies respectively upon arriv- ing at the points F, E, we shall have, whence, -ogt = ug X -=-& x *' But, as we have just seen, t : t 1 : : DF : DE, which gives Mech. 27 210 Dynamic's. DFxt' ~DE~' substituting for t this value in the above equation, we shall have, V = Urn Therefore, if several bodies descend along planes differently inclined, but of the same height, they will have the same velocity upon arriving at the same horizontal line. Of Motion along curved Surfaces. 335. If a body without gravity and without elasticity, des- cribes, in virtue of a primitive impulse, the successive sides JlB, Fjg.160. #Q - c ^ O f ar) .y. p0iyg 0nj U p 0n meeting each side it will lose a part of its velocity, which may be determined in the following manner. Let us suppose that the body moves from A toward B, and that when it is at B, its velocity is such as in a determinate time, one second for example, would cause it to describe, if it were free, the line BF in JIB produced. Having erected upon BC from the point B, the perpendicular BE, we imagine the rectan- gular parallelogram BDFE, of which BF is the diagonal, and the sides of which are in the direction of BC and BE. Instead of the velocity BF, we may suppose that the body has at the same time the two velocities BD, BE ; and as the side BC pre- vents its obeying the velocity BE, it is manifest that its velocity is reduced to BD. If from the point B, as a centre, and with a radius BF, we describe the arc FI, DI, which is the difference between BF and BD, will accordingly be the velocity lost. Now DI is the versed sine of the arc FI, or of the angle FBC, made by the two con- tiguous sides AB, BC. Therefore so long as these two sides make a finite angle, the body will lose a finite part of its veloc- ity upon meeting each of the sides. 336. But if the angle formed by the two sides is infinitely small, the velocity lost will not only not be a finite quantity, but Motion along curved Surfaces. 21 1 it will not be an infinitely small quantity of the first order, it will only be an infinitely small quantity of the second order. In es- Cal - 4 - tablishing this, the question reduces itself to showing that the versed sine of an infinitely small arc is an infinitely small quan- tity of the second order; and this may be done thus. CD being pig.161. any arc of a circle, and BD a perpendicular upon the diameter AC, we have Geom. 215. AB : BD :: BD : EC -, hence, if CD, (and for a stronger reason BD) be infinitely small, BC the versed sine of CD, will be infinitely smaller than BD, since it is contained in BD as many times as BD is contained in the infinitely greater quantity JIB. Therefore BC is infinite- ly small of the second order. 337. Accordingly, if a body without gravity move along //ieFig.162. curved surface ABC, it will have throughout the same velocity. For by considering this curve as a polygon of an infinite number of sides, since the sides make angles infinitely small with each other, the loss of velocity at the meeting of each two adjacent sides is an infinitely small quantity of the second order with respect to the original velocity. Consequently the sum of the velocities lost in passing over an infinite number of sides, that is, in passing over any arc ABC, can only form an infinitely small quantity of the first order. Therefore the velocity is not affected by this circumstance. Cal. 4. 338. We come now to the motion of heavy bodies along curved surfaces. We shall consider for the present only that which takes place in a vertical plane. 339. Accordingly, let AMB be a section of a curved surface, Fig.163. made by a vertical plane, and the path described by a body along this surface. Let us consider this curve as a polygon of an infinite number of sides, and let us suppose that the body has just described the small side LM. As its meeting with the side MN cannot occasion any loss of, velocity ; it will describe MN 335. with the velocity which it had in M, gravity being supposed no longer to act upon it. But the force of gravity being exerted according to the vertical MO, urges the body anew as it would urge one upon a plane of the same inclination. Consequently, 212 Dynamics. if we imagine the velocity MO, which gravity tends to give in an instant, decomposed into two parts, one MD perpendicular to MN, and the other DO or ME directed according to MN ; we shall see that it is by virtue of this last that the velocity of the body will be accelerated. Now by letting fall the perpendicular RN, and comparing the similar triangles MOE, MNR, we shall have, M JV : JVfi : : M : ME = ^/ O . Let us suppose that the different points of the curve JIB are referred to the vertical axis BZ. If we call, BP, x ; PM, y ; and the arc BM, s ; we shall have PQ or RN = dx ; and MN = d s. We give the sign to these quantities, because x and s go on diminishing, while the time t increases. Let g be the velocity which gravity gives to a free body in a second ; g d t will be that 267. which it would give in the instant d t. We shall therefore have the velocity represented by MO as follows, namely, MO = gdt. Calling -o the velocity which the body has when it arrives at M-, dv will denote the augmentation received during the time dt-, thus, ME = dv. Substituting the values above obtained in the equation, _ AMX.MQ MJV ' we shall have, But, by article 280, ds = v dt, or d t = - , or, s being consid- ered as decreasing while t increases, d t = - ; whence, by substitution, Motion along curved Surfaces. 213 or, vdv = gdx. The integral of this equation is, . Cal - whence, v 2 = 2 O 2gor. In order to determine the constant C, let us suppose that the point Ji from which the body begins to fall, is elevated above a horizontal line passing through B by a quantity 13Z = h. It is necessary, therefore, when v is zero, that x should be equal to h ; accordingly we have = 2C and consequently whence, by substitution, 2 go? = 2g (/i a;), = 2g X ZP. Now if a heavy body fall through the space ZP, the square of the velocity which it will have upon arriving at P, will be 2, X ZP. - Therefore, when a body descends along any curved line, it has at any point whatever, the velocity which it would have acquired by fall- ing freely through a space of the same perpendicular elevation. Thus the velocity which a body successively acquires by its gravity in descending along the concavity of a curved line, is altogether independent of the nature of this curve. 340. Hence, if the body, after having arrived at the lowest point 13 (the tangent to which I suppose to be horizontal,) meets the concavity of the same or of any other curve, touching the first in B, it will rise upon this last to a height equal to that from which it descended. 214 Dynamics. Indeed, let us suppose that the body is actually in B, or that x = ; its velocity will be such, that we shall have v 2 = 2g/i, or u 2 = 2g/i, by calling this velocity u to distinguish it from the other. Let us imagine that with this velocity it ascends along any curve BM' ; we shall find by the same reasoning as that above pursued, that its velocity in any point M', is determined by the equation, by calling v', the velocity in this case, and s* the arc BM', and observing that v' diminishes according as /', Y, and x increase 280. respectively. Consequently, putting for d t its value r , we shall have d t)' = ^___ or i/ d v' = g d x 5 and by integrating, T / 2 - 2C 2gx. But, when x = 0, the velocity t/ is u ; accordingly, u 2 = 2 C 0, and since we have Now when the body ceases to ascend, v' = which gives whence we deduce Therefore the point at which the body will have arrived in any curve BA 1 , will be at the same height as the point A. Motion of Oscillation- 2 1 5 341. As to the time employed in describing any arc AM or AB of the curve ; since d t = - , substituting for v its v equal V 2g-fc 2##, we have, __--<*'_ * ' so that it would be necesary by means of the equation of the curve to find the value of d s in x and dx, and having substituted it in the expression for d t, we should have that of t by integrat- Of the Motion of Oscillation. 342. We have seen that a heavy body having descended through any arc of a curve AB must, sotting aside the resistance of the air and friction, ascend again to the same height in a curve BA 1 which has at the point B the same horizontal tangent with BA. Accordingly, this body in returning would describe in a contrary direction the whole extent of the curve A'EA ; and thus 340, would continue to move backward and forward without end. This kind of motion is called oscillation. We have seen what is in general necessary to determine the duration of each oscillation which must evidently be double the time employed in describing the arc AB, if BA 1 is the same as AB. When the curve through which the body descends is circular, and the oscillations take place through small arcs only, they have this remarkable and important property, that their duration is not sensibly affected by the extent of the arc AB-, so that the arc Fiff 164> JIB being small, as four or five degrees only at the most, the body will always arrive at B in the same time very nearly, wheth- er it set out from the point A, or from some other point O, taken between Jl and B. Thus, retaining the denominations used above, and designat- ing by a the radius BC of the circle BAD, we shall have, by the nature of the circle, Trig. 101." y 2 = 2ax x 2 or y = 216 Dynamics. from which the value of Mm, or ds or \/dx* + dy* is readily found to be as follows, namely, Cal. 78. , adx d s = V/2 a x a: 2 But since the arc BM is small, a; is small with respect to , and x 2 may be neglected when taken in connection with Sax with- out material error, which leaves adx d s = V2 a x Substituting this value of d s in the expression for d , we shall have V4 Va Va VgVh a; or, snce adx . , ,. Now as expresses the element of an arc of a circle of which the diameter is 2 a and the abscissa x ; so in like manner, - - x expresses the element of an arc of a cir- V h x a; 1 cle whose diameter is h and abscissa a?. But the line BZ being /i, if upon BZ as a diameter we describe the semicircle BM'Z, M'm' will be this element ; so that we shall have Motion of Oscillation. 217 whence, _. *"= |J \ X + h \8a 256 a 2 But, by taking n for the ratio of the circumference of a circle to BM'Z 7i its diameter, we have - = ; accordingly, g \8a 256 a 2 Comparing this value of J' with that of t f found above, we shall have, :f;..n |J: IT ('A \5- \ ^ \8a 256 a 2 a 256 a a quantity in which - is the versed sine of the arc described dur- ing a semioscillation, radius being 1. Pendulums. 221 i Suppose t' equal to 1", and that the arcs described on each side of the vertical are 5. The versed sine of 5 is 0,0038053, radius being 1. Consequently, - = 0,0004757. With res- 9h* pect to the term ^, it is less than a unit of the sixth place. The error in each oscillation will, therefore, be i' = 1" X 0,0004757 = 0",00047i>7. Thus, if a body descend by the action of gravity along a circu- lar curve, and describe arcs infinitely small on each side of the lowest point in a second of time, the duration of each oscillation, no allowance being made for friction or the resistance of the air, would differ only 0",0004757 from that of an oscillation through an arc of 5 on each side of the lowest point, so that in a day or during 24 X 60 X 60 = 86400" vibrations, the difference would amount to 86400 X 0",0004757 or 41". Thus a pendulum of the length required to vibrate seconds, and performing its oscil- lations through arcs of 5 on each side of a vertical, would lose only 41" a day, when compared with one vibrating in arcs infi- nitely small. If the arcs described on each side of the vertical were only 1, the versed sine of which is 0,0001 523, the daily loss would be only 1",64, that is, If nearly, and for half a degree, the loss would be 0",41 or | of a second daily. Of Pendulums. 344. What we have said is particularly applicable to pendu- Fiff 16 lurns. By a pendulum, is to be understood a rod or thread sus- pended at one extremity from a fixed point, and supporting at the other extremity one or several bodies. It is called a sim- ple pendulum when it is supposed to consist merely, of a mass or weight sustained by a thread or rod without gravity, and when at the same time this mass is of a diameter very small relative to the length of the pendulum. We shall speak for the present only of the simple pendulum. 222 Dynamics. When the pendulum CB is drawn from its vertical position. the force of gravity acting according to the vertical line AM is not wholly employed in moving the body ; a part is exerted against the point C. Let therefore the whole force of gravity, represented by AM^ be decomposed into two others, represent- ed the one by #JV, directed according to G^JV, which will be destroyed, and the other by JlP which urges the body along the arc JIB. Now as the radius CA is perpendicular to the arc, it will be seen that the motion is here decomposed in the same manner as in the case above considered, where the body is supposed, without any material connection with C, to descend along the arc AB, which has for its radius the length AC of the pendulum. Accordingly every thing which we have said is ap- plicable to pendulums. The following are some of the conse- quences which are derived from the preceding investigation. 345. We have found for the duration t of an oscillation, the following expression, namely, Hence, for another pendulum whose length is a', and which is urged by a different force of gravity, or one that is capable of giving the velocity g' in a second, we shall have, by call- ing if the duration of an oscillation in this second case, hence we derive the proportion, g' Vg that is. if two pendulums of different lengths are urged by different gravities, the durations of the oscillations are as the square roots of the lengths of the pendulums , divided by tht square roots of the quan- tities which denote these gravities. 346. As gravity is the same in the same place, we shall have for pendulums of different lengths vibrating in the same place or same part of the earth, g = #, and consequently in this case the proportion becomes Pendulums. 223 t : I 1 : : \/~a~ : \TcT ; that is, in the same place the durations of the oscillations are as the square roots of the lengths of the pendulums. 347. But if the same pendulum be successively exposed to the action of two different gravities, a being equal to a', the pro- portion *:*> :: V^ : V? g 8 becomes 8 in other words, the durations of the oscillations of the same pendulum in different places are inversely as the square roots of gravity. 348. Let n be the number of oscillations or vibrations made by the pendulum a in a given time, as one hour or 3600'', we 3600" shall have t = - For the same reason, if we represent by n' the number of vibrations made by the pendulum a/ in the 3600" same time, we shall have r = - ; ; accordingly, I if 360Q " . 3600" '\ n n' ' ' T ' n ' that is, the number^ of vibrations made in the same time by two pendulums of different lengths are inversely as the durations of their respective vibrations. Consequently, since n : n' : : \f- : V -, g' g that is, the number of vibrations made in the same time by two pendu- lums of different lengths, and which are urged by different gravities, are in the inverse ratio of the square roots of the lengths of the pen- 224 Dynamics. dulums divided by the square roots of the gravities ; so that if the gravities are the same, the number of vibrations will be recipro- cally as the square roots of the lengths of the pendulums ; and if the lengths are the same, the number of vibrations will be direct- ly as the square roots of the gravities. 349. Hence if the same pendulum, carried to different parts of the earth, does not make the same number of vibrations in the same interval of time, it is to be inferred that gravity is not the same in these places, and the number of vibrations actually made in the same time by the same pendulum in two different places, will furnish the means of ascertaining the relative inten- sities of gravity at these places. It is by experiments of this kind, taken in connection with the foregoing proposition, that we are now assured of the diminution of gravity as we approach toward the equator, and on the other hand of its augmentation as we proceed from the equator toward either pole. 350. If we call t the time employed by a heavy body, falling freely, in describing the diameter ED or 2 , we shall have gt* a t* 273 2a = *_ or - = T ; Fig.164. whence g Substituting this value in the equation, we obtain, i' = i 7i t or | t' = i n /, which gives that is, the duration of the descent through any small arc AE is to the time of falling through the diameter, as the fourth of the circumference of a circle is to its diameter. But the fourth of 2$a. ' the' circumference is less than the diameter; consequently a Line of swiftest Descent. 225 body employs less time in descending along a small arc of a circle of which the inferior tangent is horizontal, than it would employ in falling through the diameter ; and since the time re- quired to pass through the diameter is the same with that re- quired to describe any chord AB, it will be seen, that a body 332 - would pass sooner from A to B, by descending along the arc AB, than by moving through the straight line AB. Therefore, al- though the straight line is indeed the shortest way from one point to another, it is not that which requires the least time for the pas- sage of a heavy body. Of the Line of swiftest Descent. 351. Not only is not a straight line that along which a heavy body would proceed in the shortest time from one point to anoth- er, out of the same vertical, but it is not the arc of a circle which answers to this description ; it is the arc of another curve which may be found in the following manner. Suppose AMR to be the curve sought, or that through which Fig.166, a heavy body would pass in the least time from a given point A, to a given point B. If we take in this curve two points M, w', infinitely near to each other, the arc Mm' must also be described in less time than any other arc passing through these same points jlf, ra', since these two points may be taken as the very points in question. Having taken the point JV* infinitely nearer to Mm' than M is to ra', suppose infinitely small straight lines MN, Nm* to be drawn ; since the time of describing M m m' must be a minimum, it follows that the difference between the time of pass- ing through Mm ml and the time through MN m!, which is the differential of the time, must be zero. Through the points JW, JV, m', draw the horizontal lines MP, m P', m' P", and through A, the vertical line AC. Call AP, x, PM, y ; AM, s, and suppose M m = m m x , or that, d s is constant. Then mr = dx, rM = dy, m/ = dx-\-ddx, r 1 m! = dy+ddy. Let u be the velocity with which the body describes Mm; it will be the velocity with which MN is described; and u -\- du will be that with which mm! and N m' will be described. Therefore Mech. 29 226 Dynamics. the time of passing through Mm will be , and the time through M 24. 11 i ds mm' will be - T -. u -f- du From the points M and m 1 as centres, and with the radii , m m') describe the arcs JVn, m t ; then comparing the trian- gles JV m n, JV" w f, with the triangles ./If m r, m m' /, we shall have nm = ./Vw X , as and Whence, and Therefore the time through MN will be m X u and the time through Nm' will be, w-f- We have, therefore, & * a 5 w + dw an equation which reduces itself to ^(*JL+ d Ji_jLlL\ = o, or d C^ = 0; d u + dn u % Line of swiftest Descent. 227 then, by integrating, we have dy ds or u O But since the velocity u is equal to that which the body would acquire in falling from the height AP, we have u* = 2gx. 277 ' Therefore, Cdy = ds V %g x i and C 2 dy 2 = 2gxds 2 = 2gx(dx* + dy 2 )-, whence we deduce To determine the constant C, it will be observed that when = C, we have dy ds, answering to the lowest point R, of the curve, where d a?=0, and xh ; therefore if we call v the velo- city which the body will have at the point where v %~gx = C, the equation Cdy = uds becomes C d s = v d s, which gives C = u. And if we call h the corresponding height ^fC, we shall have V 2 - hence, C 2 = 2g/i. Therefore, , dx/'Zx dx V ggz V h x ' is the equation of the curve. But the better to understand this curve, let us give another form to the equation. Imagine the vertical line RD drawn through the point /?, where dy = d s ; and having produced PM to O, call AD, a ; OR,x f , and OM,tf. Thenx = h tf,y = a y,da? = ^ t/ = d\f ; substituting these values, we have a?' d a? 7 x' A a;' Imagine that upon DR or fe, as a diameter, is described the semicircle DER. We shall have OE = yhx' 0/2, and the arc caui >- rr." J^iSy^ 1 we have therefore generally, OM = V -f OE + RE. To determine the constant C 7 , it must be observed that when x f = 0, we have y f = 0. Therefore, since OE and RE then be- come zero, we have C' = ; consequently OM OE + .K.E ; and the curve sought is therefore the common semicycloid, of ' which DER is the generating circle, and AD the semibase. The only thing which remains to be determined, is the quan- tity h ; for the only things given are the two points A and B, through which the body is to pass, h is determined in this man- ner. Having drawn the vertical BK, which meets in K the hori- zontal line AK passing through the point A, we describe upon AK as a semibase, the semicycloid AVT, that is, a semicy- cloid of which the generating circle has AK for the length of its semicircumference. And having drawn AB cutting this cycloid in F, we draw VK, and parallel to VK, through the point J5, we draw BD, which determines JlD for the semibase of the cycloid sought, that is, for the semicircumference of its generating circle. This construction is founded upon the circumstance, that the cy- cloids AVT, ABR, which have their bases upon AD, and the point A common, are similar, as may be easily shown.* * Since the diameters of circles are as their circumferences, or as their semicircumferences ; Geora. 287. DR : KT :: DA : KA' t but DR : KT : : DB : tf F, hence, - DB : KV : : DA : KA. Line of swiftest Descent. 229 352. We have supposed the body to have no velocity on its leaving the point A. But if it had already acquired a certain velocity in a given direction, the origin of the curve would be at some higher point. The equation Cdy = u d s, found above, gives -^j- = -~ ; whence the constant C must be such that the initial velocity being divided by it, the quotient will be equal to the sine of the angle made by the direction of this initial velocity with the vertical, a condition, which, with the other, that the body must pass through A and J2, will determine the cycloid for the case in question. 353. Besides this property of being the curve of swiftest descent in an unresisting medium, the cycloid is on several other accounts quite remarkable. It has, for example, this singular property, that whatever be the point X, from which a body be- gins to descend along the concave part of the curve, it arrives always at the lowest point R in the same time* This property is thus proved. Calling t the time, and s the arc RM corresponding to any point M) where the body is found at die end of the time , we have d t . Now designating by h' the height of X above the horizontal line OM, we have u = v 2 g (h! x') . More- 277. over, it is easy to infer from the value of d y', found above, that hence dx' x ' therefore, x /_. . 2 J Vh'x' x'z whence, reasoning as above, and calling ^ the whole time em ployed in falling from X to jR, we conclude that 230 Dynamics. Jt being the ratio of the circumference to the diameter. There- fore, that is, the time t' is independent of the height h' from which the body sets out. Of the Moment of Inertia. - Fig.167. 354. Let M. M 7 , M'', be any masses without gravity, and let them be considered as points situated in the same plane with the point F, and connected together, and with the point J 1 , in such a manner as not to admit of any change in their relative distances, or of any motion except about the point F, or about an axis pass- ing through jF, perpendicularly to the plane in which they are sit- uated. Let us suppose that these masses receive at the same time impulses according to the lines w, w', z/, directed in the above plane, and such, that if the masses were free, they would have velocities represented by these lines respectively, it is pro- posed to determine the motion that would ensue. We decompose, according to the principle of D'Alembert, the 133. velocities w , w', w", each into two others, one of which shall be effective, and the other such, that if the masses M, M', M", had res- pectively only this velocity, they would remain in equilibrium. Now it is manifest that the velocities, which the bodies are supposed to have, since they admit only of a rotation about the point F, must be perpendicular to the radii R, R', R''. Moreover, in order that these velocities may take place, that is, not mutually disturb each other, it is necessary that they should be proportion- al to these radii, or to the distances respectively from F. Accord- ingly, the communicated velocities to, a/, a>", being decomposed into 1 the effective velocities t>, i/, T", and the velocities M, w', w", with which the masses would be in equilibrium about the point F, Moment of Inertia. $3 1 we have, v : v' : : R : R', and v : v" : : R : R" (i). Letting fall from F the perpendiculars , a', a", upon the direc- 63 * tions of the velocities w, u\ w", we obtain, M u a 4- M' u' a' M" u" a" = 0. Now if we let fall also the perpendiculars c, c', c", upon the direc- tions z0, w', w", we shall have by article 62, M ' U a -f- M'V'R = M * ZO * C, or, M W a = M W * C -- M * V ' R J In like manner, M ' u' a' - M' w' c/ M' i/ R'. M" M'' a" = M" w" c" 4- M" i>" R" If from the sum of the two first of these three equations we sub- tract the last, we shall have M u a 4- M' it' a' M" u f ' a", or 0, equal to the expression below, thus, 1 = M w c 4* M' w c M" w" c" M ' V * R M' ' V ' R' M" ' v" * R". f But the above proportions f i) give v' = ^- and ?" = ^ substituting these values for v' and i>", the equation becomes = M w c + M' zi/ c' *u." ID" c" = M zt> c -f- M' w c M'' w'' c" (M R 2 -f M' R /S -f M" R") whence, M * w c 4- M ; w cf M" -w" c" M R 3 4- uf R /a -I- M' ' ' R ' a 232 Dynamics. Now the numerator of this fraction, since it expresses the sum of the moments of the forces M w ' c, &c., is equal to the moment of their resultant. If, therefore, we call this resultant g, and its distance from the point F, D ; the sum of the moments will be p X D. Moreover the denominator of the above fraction, being the sum of the products of each mass or particle into the square of its distance from F, if we represent in general any one what- ever of these masses by m, and its distance from F by r, the sum of these products may be represented by the abridged express- ionfm r 2 , (f denoting the word sum) ; we have accordingly for the velocity of any given point M, whose distance from the axis F is FM or R, the following expression, P X D v = ^ - r J mr also, X 355. Although we have supposed that all the forces, and all parts of the system are in* the same plane, it will be perceived that we should, arrive at the same result, if they were in planes parallel to each other, and perpendicular to the axis of rotation, provided that all parts of the system admit only of a rotation about a fixed axis. 356. Accordingly, as a solid body of whatever figure may be considered as an assemblage of material points, thus connected together, we may say generally, that when a body L,of whatever fig- Fig. 168. ure, and urged by forces of whatever number and magnitude, cart have no other motion, except a motion of rotation about a fixed axis AB, situated within or without the body, the velocity belonging to any given point, is found by taking the sum of the moments of all the forces (or the moment of the resultant), dividing this sum by the sum of the pro- ducts of the several parts of the body into the squares of their distan- ces respectively from the axis of rotation, and multiplying the quotient. by the distance of the point in question from this Mime axis. Fi.i69. 357" Let G be the centre of gravity of the body L, and let us suppose that while any point M, in turning, describes during an instant } the infinitely smail arc v, the centre of gravity G Moment of Inertia. 233 would describe the arc GG' perpendicular to FG ; through the point G' let the line G'K be drawn parallel and equal to GF. Instead of supposing the body to turn about F, we may imagine it carried parallel to itself with a velocity equal to GG 7 , and that at the same time its several parts turn about a moveable point G with such a velocity that by taking G'K = GF, the point K would describe the arc KF, equal to G'G ; for, on this supposition, the/ point F of the body L would still remain stationary. Now the body in this case being free, the resultant of all the motions of rotation about the moveable point G is zero. Consequently the 79. resultant of all the motions with which the body is actually urg- ed is no other than that which the body L would have, impress- ed with the velocity GG / ; that is, this force must be perpendio ular to -FG, and equal to L X GG', the mass of the body being denoted by L f Now since the parts of the body describe similar arcs, we have *$& FM : FG :: v : GG' = therefore, the resulting force of all the motions of rotation about the point F, is L x FG x v FM~ But although this resultant is the same as if, the body being free, the centre of gravity had received the velocity GG', still it will be seen that this resultant does not pass through G, but through some point O of FG produced ; since, the more remote parts having the greater force, the resultant, while it falls on the same side of F with the centre of gravity, must pass at a greater distance from F than this centre. Designating this distance FO at which the resultant passes by D 7 , we shall have for the mo- ment of the resultant L *j?f X * X >'. r M If now, at the instant when the forces M'KTC, &c., above consider- ed, begin to act upon the parts of the body, there be opposed to them, at the distance Z)', a force equal to that just determined; that is, Mech. 30 234 . Dynamics. equal to the whole effort which the abovementioned forces would exert upon the body, there would evidently be an equilibrium ; but in this case the moment = X D 1 must be equal I'M to the moment g X D ; accordingly, since/ we shall have L x FG x v v mr-. FM FM and, consequently, ^ Vt) X/mr2 X F. /mr a ~ FM x L X FGxv ~~ L 358. We hence derive the general conclusion, that, if any number whatever offerees, directed in any manner we please, in planes perpendicular to the axis of rotation, act upon a body, and are ca- pable of producing only a motion about this axis; (l.) The force thus exerted, will be equal to the mass of the body multiplied by the velocity belonging to the centre of gravity ; which velocity is deter- mined by article 357. (2). This force, will be, perpendicular to the, plane passing through the axis and the centre of gravity. (3.) Its dis- tance from the axis (always the same, whatever be the forces and their directions) will be equal to the sum of the products of the several par- ticles of the body into the squares of their distances respectively from the axis, divided by the product of the mass of the body into the dis- tance of the centre of gravity from this same axis. 359. v denoting always the velocity with which a determinate point M of the body L, tends to turn in virtue of the action of any number of forces, or of their resultant p, if we designate the distance of any particle from the axis of rotation by r, and the mass of this particle by m, since FM : v :: r : -^777-5 we shall have -^TTr for tne velocity of rotation of the particle m, and - for the force it would exert, and consequently, for the resistance 27. it would oppose to g by its inertia ; accordingly, Moment of Inertia. 235 m -or m r* v -_ w x r ' or -jw- will be the moment of this resistance ; therefore the sum of the moments of these resistances which the particles of L would op- pose to the motion of rotation, produced by p, upon these par- ticles, is n J, T V y or.^rj Tmr 2 , for the two expressions are r Jvl f JVL / the same, since v and FM do not change, whatever be the par- ticle ra, which we consider. We hence perceive, that, other things being the same, the re- sistance which the particles of a body oppose to the motion of rotation, communicated to them, is so much the greater as fm r 2 is greater. The quantity -^ - f mr 2 is called the moment of inertia of a body, and fmr 2 the exponent of the moment of inertia. 360. We shall see soon how the exponent of the moment of inertia in any body may be determined ; but when this expo- nent has been determined w r ith respect to any axis whatever, it is very easy thence to infer, what it must be with respect to any other axis parallel to the former. Let AB be any axis, and A'B' another axis parallel to it, passing through the centre of gravity G of the body. Let m be any particle of this body ; and through m suppose a plane mFF', perpendicular to the two axes AB, A'B' ; mF, m F', being drawn, and the perpendicular mP being let fall upon FF', the lines m F, m F', will be perpendicular respectively to AB, A'B 1 . 3^3 m< This being supposed, we shall have, m F*= m F 1 -f FF' -f 2 FF 1 x PP ; Geom. hence, 192. j*m- mF^fm- mF' -f f m ' FF' -f/m . 2 FF' X PP. Now, since the distance FF' is always the same, whatever be the a 2 particle m under consideration, Jm FF / is simply FF 1 Jm, or 236 Dynamics. 2 FF' x L, the mass of the body being represented by L. For the same reason/ w x ZFP x F'P is simply 2 FF'fm X PP. But/ m X F'P, being the sum of the products of the particles into their respective distances from a plane passing through A'B', 64 that is, through the centre of gravity, must be equal to zero ; we have therefore simply, fm'mF=fm'mF' + Lx FF'. Hence, knowing the exponent f m m F' of the moment of inertia with respect to an axis passing through the centre of gravity, we have the exponent with respect to any other axis parallel to this, by adding to the first the product of the mass into the square of the distance between the two axes. 354. From this result, and the expression for the velocity of rota- tion, it may be inferred that of all the axes about which a body may be made to turn in virtue of any force or impulse, those about which the velocity of rotation will be the greatest are such as pass through the centre of gravity ; since the exponent of the moment of inertia with respect to an axis passing through the centre of gravity, is less than it is with respect to any other axis. Of the Centre of Percussion and the Centre of Oscillation. 361. The foregoing propositions will be found to be of the greatest importance in many inquiries to be resumed hereafter ; we shall confine ourselves for the present to the use that may be made of them in finding the centre of percussion, and centre of F\" m. oscillation, of bodies that admit only of a rotation about a deter- minate axis or point. We understand by the centre of percus- sion, the point. O of the straight line FG, where it would be nec- essary to place a body in order that it might receive the greatest impression from the body L turning about F. Now it is evident that this point must be that through which passes the resultant of the motions of rotation of all the particles in L. The centre of percussion, therefore, is determined by the proposition of arti- cle 358. Centres of Percussion and Oscillation. 237 As to the centre of oscillation, it is the point O of a body ^ or system of bodies, whose distance from F is equal to the length which a simple pendulum must have in order to perform its os- cillations in the same time. We shall see that this centre is the same as the centre of percussion. Indeed, when the question relates to gravity, the force p, re- sulting from the action of gravity, exerted upon each material particle of a body, is equal to the whole mass multiplied by the velocity communicated by gravity in an instant to each particle ; that is, P = U X Ir, u representing this velocity. Moreover this resultant g passes through the centre of gravity ; and consequently its perpendicu- lar distance from the fixed point JF, or from the axis passing through F, is FH; hence the velocity of rotation i?, of any point JJf, when the body is left to the action of its gravity, is 354. so that, for the centre of gravity G, the velocity is . J m r 2 Now in order that a simple pendulum, whose length is FO, may make its oscillations in the same time with the body L, it is necessary, L being supposed to be drawn from a vertical po- sition by the same angular quantity, that the velocity impressed by gravity at O (fig. 172), perpendicularly to .FO, should be the same as that of the point O (fig. 171); in other words, that it should be to the velocity of G (fig. 171), as FO is to FG. Now by decomposing the velocity u or OP (fig. 172), communicated by gravity in an instant to a free body, into two others, namely OK in the direction of FO, and 00' perpendicular to FO, we shall have u : 00' : ; FO : OZ : : FG : FH; whence M : OO' ; : FG : FH, and consequently 238 Dynamics. FG We hence derive the proportion, .xra JOCLXJW x ro ::FO: re. / G ^ ?/i r 2 which gives /mr2 y m r a 'a v """ "~ **" ^ -- ** i- F-? rr n AN ~" - w x L X FHxFG L x 357. which is the same as the expression for the centre of percussion. 362. Since all the forces which act upon the body L, or up- on a system of bodies that admit only of a motion of rotation about a point or a fixed axis, cause in this body such a velocity that, for any given point Jlf, we have and since it is evident, that if this body were to turn in the op- posite direction with the same velocity, there would be an equi- librium among all these forces ; we infer, that if a body, turning with a velocity which for a determinate point M is equal to v 9 would have its motion counterbalanced by a power p, the direc- tion of which passes at a distance from F equal to 7), this power taken in connection with its distance Z), must be such that the moment p X D shall be equal to the velocity of the point M, di- vided by the distance FM^ and multiplied by the sum of the pro- ducts of the .particles into the squares of their distances respect- ively from F, or from the axis passing through F. Indeed this power must be such as will be sufficient to produce the same ve- locity in the body L, supposed at rest ; and this velocity would be v = -^ x FM, which gives FM Centres of Percussion and Oscillation. 239 363. If a body L, of any figure whatever, admitting only of Fi S- 173 - a motion about a fixed point F, or about an axis passing through F, which may in ether respects be situated as we choose, if, I say, a body L be struck by a body JV, the motion of each after collision may be determined by the principles above established. Thus, let u be the velocity of JV, before collision, accord- ing to the perpendicular TH, and u' its velocity after collision ; u u' will be the velocity, and N (u u'} the quantity of mo- tion, lost by collision, and which will pass into the body L. This 290> quantity of motion will cause in L a velocity of rotation, such that the point 7 T , for example, will turn with a velocity -') X FH 354. v = v '' - X FT (i), jf m r 2 v /' F/f being drawn perpendicular to TH. Let the infinitely small arc TT', described about the centre F, represent this velocity ; the parallelogram TA T'C being formed upon the tangent TA and the perpendicular TH, it will be seen, by substituting for TT' the velocities TA, TC, that the velocity TA cannot affect the velocity M'. which the body N must have ; but that the veloc- ity TC would impair the velocity u' if it were smaller than u' ; accordingly, since we suppose that u 1 is actually the velocity which N preserves after collision, it is necessary that TC should be equal to '. Now the similar triangles FHT, TCT. give, FT i FH : TT or v : TC, whence, and consequently u X FT V =-JIT' Substituting for u this value in the equation (i), we have vf_X_FT __ #(*-<) X FH FH fmr* from which we deduce the value of u' ; thus* 240 Dynamics. u JV X ti X FH JV X u' X FH 2 9 _ FH fmr 2 fmr u f X fm r 2 Nxu'xFH _ N X u X FH /mr 2 X F fmr 2 X FH = JV ' XU - J f m x FH ,2 f NxuX FH fmr 2 X FH FH fmr 2 + From this the value of T, or the velocity of rotation, is readily obtained. But the equation v = , or i> FJaT = u'FT, since it gives the proportion u' : v : : FH : FT, makes it evident, that u' is the velocity of rotation of the point H; from which it will be seen, that the point H turns with the veloc- ity that remains to JV after collision. 364. We hence perceive, that in order to find the motion of bodies that turn about a fixed point or axis, we must be able to determine the value ofj*m r 2 . This will always be easy, as we shall soon show, when the bodies are such as admit of being ex- pressed by equations. We may, indeed, in any case consider the body as composed of parallelepipeds, pyramids, &c., which are capable of being thus expressed ; and finding for each com- ponent part the value ofjmr 2 , take the sum of these as the to- tal value oijm r a 2 for the entire body or system of bodies. When the body is such as admits of being expressed by an equation, we proceed thus in finding the value oij'm r 2 . Centres of Percussion and Oscillation. 241 Let AB be the axis of rotation, and through AB suppose two planes PQ, AR, to pass perpendicularly to each other ; let m be p . any particle of the body in question, and having let fall the per- pendicular m F upon AB, we draw m H perpendicularly to the plane RA ; and joining FH, this line will be perpendicular to AB, and consequently to the plane PQ. The right-angled trian- glem IW gives - Hm; whence, J m X Fm orfm r 2 X The problem, therefore, reduces itself to finding the sum of the products of the particles into the squares of their distances from two planes, which pass through the axis of rotation, and are per- pendicular to each other. Now, when the algebraic expression for this sum is found with respect to one of the planes, it is easily obtained with respect to the other. Let us therefore inquire how we can find the sum of the products of the particles of a body into the squares of their distances respectively from a known plane. We will suppose the body divided into infinitely thin strata, parallel to the given plane ; and, representing the thickness of one of these strata by DD, its surface by tf, and its distance FDp. r from the plane in question by x, since the points of the surface 6 are all distant from the plane PQ by the same quantity x, we shall have x 2 a d x for the sum of the products of all the points of this surface into the squares of their distances respectively from this plane, and consequently f*x 2 <> dx for the entire sum of these products for the whole body. If we represent, in like manner, by x' the corresponding dis- tances from the plane perpendicular to PQ, and passing through the axis of rotation AB (the body being supposed to be divided into strata parallel to this second plane), and by 6' the surface of one of these strata, we shall havens/ 2 6' d x' for the sum of the products of the particles into the squares of their distances respectively for this second plane ; and accordingly Mech. 31 242 Dynamics. 2 6' dec will be the value of the sum of the products of each particle of the body into the square of its distance from the axis AE. 365. Let us now suppose, by way of illustration, that the Fig.176. b jy j n question is a rectangular parallelepiped, turning about the axis AB perpendicular to the axis of the parallelepiped, and to the side IK. By the nature of this body, the surface tf is con- stant ; thus the integral f x 2 6 d x is -, which, when x is equal to the altitude h of the parallelepiped, becomes - . In like manner, 6' being a constant quantity,^" x 2 6' dx be- comes or, MN being represented by ft', which gives x' = \ h', and, as the plane which passes through the axis divides the bod}' into two equal parts, the two halves will be , h'* 6' h' 3 6' therefore the entire sum of the products will be A 3 6 h' 3 & 35 7t If we would find the centre of percussion or of oscillation, we have only to divide this quantity by the product of the mass of the parallelepiped into the distance of its centre of gravity ; that Geom. is, by h h'f X \ h or h 2 h'f, IM being denoted by/, which gives 4054 for the distance of the centre of percussion or of oscillation 2h* 6 h' 9 6' . 2 fc h' 2 h 6 A* h'f r ~T " 6T } since <* = # and dx, becomes (z-f a) 2 X7i(<2Rz z 2 )dz, 244 Dynamics. or, by developing the whole, f7i( < 2a Cal. 85. anc [ by integrating, we have 7i (a 2 RZ Z + f flRz 3 i a 2 z 3 + |R2 4 1 2 4 i 2 S )> which, when 2 = 2 R, becomes 7i (4 a 2 R 3 + V R 4 | a 2 R 3 +8R 5 8 a R 4 y R 5 ) or w(Ja R 3 +|R 4 + f R 5 )- To find the value of fx' 2 6' d a/, it is not necessary to begin the calculation again, since from the regular figure of the sphere, it is evident that this value will be similar to the former ; we have only to suppose, therefore, that a, which expresses the distance of the plane PQ from the surface, becomes R ; that is, that this plane passes through the centre, it being supposed at the same time to be perpendicular to its first position, and we shall have 71 (f R 5 - f R 5 + | R 5 ) Or 71 X T 4 5 RS ' The two integrals being added together, make 7i (fa 2 R 3 +|R 4 -f f|R 5 ). Gcom. Since the bulk of the sphere is^XfR 3 or|7TR 3 , and the distance of its centre of gravity from the plane PQ is a + R > if we divide the above result by the product f TTR S x (a+ R) of these two quantities, we shall have the distance of the centre of oscillation and that of percussion ; thus, 71 R 3 (a + R) Centres of Percussion and Oscillation. 245 a -f- R a* -f 2aR -f R* + f a -{- R * Hence the centre of oscillation and that of percussion are below the centre of the sphere; and the centre of the sphere cannot be taken for the centre of oscillation or that of percussion, except when its radius is very small compared with the distance of the centre G from the point of suspension. If the sphere is suspended by a rod or lamina, and we would have regard to its mass, it will be recollected, that we have found - - H -- for the sum of the products of the o 1 2, particles of such a body into the squares of their distances res- pectively from the fixed point or axis. Now h is what we have represented by a ; moreover, since 6 = h'f, and & - hf = /, we shall have by substitution, 3 h'f A' 3 af . 3 12 ' this quantity and that for the sphere must be multiplied respec- tively by the specific gravities 5, S', of these bodies, if their spe- cific gravities be different ; then by adding the two products, we shall have, s j7 + s v/ + s , x (f a , R , + . -R4 + f} RS) for the sum of the products of the particles of the whole system into the squares of their distances respectively from the axis. This sum divided by the product of the masses, S a h' f+ S' f rc R 3 into the distance of the centre of gravity from the axis, gives the distance of the centre of oscillation. 246 Dynamics. 367. It may suffice in practice to divide the body into a great number of parts, and multiply each part by the square of its distance from the axis in order to obtain with sufficient exactness the value of fmr 2 . Of the actual Length of the Seconds Pendulum. 368. The number of vibrations performed in the same time by two different pendulums, urged by the same gravity, being inversely as the square roots of the lengths of these pendulums, we can find very nearly the length of the seconds pendulum for any given place by a very simple process. Having suspended to a very fine wire of at least three feet in length, a small dense body, as a ball of lead, gold, or platina, we ascertain the length of this wire and the radius of the ball with great exactness. We then cause this pendulum to vibrate by drawing it a little from a vertical position, and count the number of vibrations performed in a given time, as one hour, very carefully determined, and then make use of the proportion ; as 3600, the number of vibrations to be performed by the pendulum sought, is to the number actu- ally performed by the above pendulum, so is the square root of 346. the length of this latter pendulum to a fourth term or a?, which will be the square root of the length of the pendulum sought ; and by squaring this fourth term, we shall have very nearly the length of the pendulum required to vibrate seconds. This result would be exact only on the supposition that the wire or string is without weight, and that the ball consists only of a single particle or has its matter concentrated at the centre. 369. If we attempt to find geometrically the centre of oscil- lation of the ball and wire, we shall still be liable to some small error arising from irregularities in the form and distribution of the matter in question. We accordingly have recourse to another method, depending on a curious property of the com- pound pendulum by which the distance between the point of sus- pension and centre of oscillation, answering to the length of the simple pendulum vibrating in the same time, can be ascertained with the greatest precision. Length of the Seconds Pendulum. 247 We have obtained a general expression for the distance in question, as follows, namely, FO = /mr2 = /m ' mF 361. L xFG L x FF' But J* m m~F = f m mf?+ L X FP; whence, by substitution, -2 _ fm-mF' + Lx FF' L x FF -' that is, FO or FF' -J- F'O = ^^ t + FF whence, ' L x FF'' Thus, the distance of the centre of oscillation below the centre of gravity is equal to the sum of all the parts multiplied by the squares their respective distances from the axis drazvn through the centre of gravity, divided by the product of the mass into the distance of the centre of gravity from the axis of suspension. Now by multiplying both members of the above equation by FF', and dividing both by F'O, we shall obtain, F'F=/~" L X OF 1 ' Accordingly, if we consider the body as inverted, and make O the point of suspension, F will become the centre of oscillation, since we have the same expression as before for the distance of this point below the centre of gravity. We hence infer, that the point of suspension and centre of oscil- lation are convertible, that is, either being made the point of suspen- sion the other becomes the centre of oscillation. 248 Dynamics. Reciprocally, if two points are so chosen, or so adjusted to each other by moveable weights, that the pendulous body shall vibrate in the same time when suspended from one as when sus- pended from the other, these points are alternately the centres of oscillation and points of suspension, and the distance asunder is the length of the pendulum in question, and equal to that of a simple pendulum vibrating in the same time. The above proposi- tion was demonstrated by Huyghens, the original author of the theory of the pendulum, but it was not till very lately applied to any useful purpose. Captain Kater was the first, to perceive that it furnished a very simple aud accurate method of deter- mining the length of the compound pendulum. Figure 179 represents Captain Rater's pendulum. The axes jF, O, were adjusted by means of intermediate moveable weights C, D, and with so much accuracy that the number of oscilla- tions made in twenty four hours, F being uppermost, differed from those performed in the same time with O uppermost, less than half a vibration ; and the mean of twelve sets of observations with first one then the other uppermost, differed from each other less than the hundredth of a vibration. The length of the pendulum, as thus obtained, is stated to be 39,1386 inches. This is for the latitude of London, or 51 31' 08",04 JV, and on the supposition of the arcs of vibration being infinitely small, taking place in a vacuum, and at the level of the sea, the temperature being 62 by Fahrenheit's thermometer. This determination exceeds what was considered the most accurate result of the methods previous- ly in use by 0,00813 or nearly one hundredth of an inch, a very important difference in researches where the ten-thousandth of an inch is appreciable quantity. It may be observed, moreover, that if the two axes of the pendulum be cylindric surfaces, the points of suspension and oscillation are truly in these surfaces, and the length sought is rigorously the distance between these surfaces. This second property, so necessary to the completeness of the method, when actually applied to practice, was discovered by Laplace. See Ed. Rev. vol. 30, p. 407. Phil. Trans, for 1818. Force of Gravity. 249 370. It is not necessary to go through the same process in order to find the length of a pendulum required to vibrate in any other proposed time, as half a second, or half a minute. The principles we have investigated will enable us to solve all prob- lems of this kind with the greatest facility and exactness, when the length and time of vibration of one pendulum is known. Thus if it is proposed to find the length of a pendulum required to vi- brate half minutes, the proportion t 2 : t' 2 : : a : a', by substituting for J, t', \" and 30", and for a 39,1386, the length of the seconds pendulum, we have I 2 :: (30) 3 :: 39,1386 : a' = 39,1386 X 900 = 35224,74 inches, or 2935,39 feet. In like manner, the length and time of vibration of one pen- dulum being known, the time of vibration, in the same place, of any other pendulum whose length is given, may be determined. Suppose, for example, that it is required to find the time in which a pendulum of 20 feet, or 240 inches in length, would perform its vibrations ; by substituting the known quantities in the general proportion, Va : \/a f : : t : t', we have V39,1386 : V240" : : 1" : C = _?!2_- = 2",5 nearly. \39,1386 Measure of the Force of Gravity. 371. It will be easy now to determine through what space a liravy body must pass in the first second of its fall, the air and all other obstacles being removed. For the equation g gives, by squaring both members and transposing^ 2r 2 a = -jTT', Mech. 32 250 Dynamics. in which g represents the velocity acquired by a heavy body at the end of the first second of its fall, and which is double the height or space through which it would descend in a second from 266. a state of rest ; a is the length of the pendulum each of whose vibrations is performed in the time t'. Accordingly, if for t' we put one second, a must be 39,1386 inches for the latitude of Lon- don.* Moreover ^, the ratio of the circumference of a circle Geom. to its diameter, is equal to 3,1416 nearly ; hence g = (3,1416) 2 X 39,1386. Accordingly, 3,1416....2 log.... 39,1386 ....... log....l,59260 386,28 2,58690 The value of g, therefore, is 386,28 inches, or 32,1 9t feet, equal to 32,2 nearly ; and half this quantity or 16,1 is the space des- cribed by a heavy body in an unresisting medium at the surface of the earth in one second from the commencement of its motion. 273. We have thus fulfilled our promise. 'Application of the Pendulum to Time-Keepers. 372. The pendulum attached to clocks for the purpose of reg- Fig.iso.ulating their motions, is ordinarily a rod of metal or wood loaded *The length of the seconds pendulum, and consequently the val- ue of g, is referred to the latitude of London on account of the great accuracy of the observations that have been made at this place. The difference, however, in the length of the pendulum in different lati- tudes, at the level of the sea, is so small as to amount only to about J of an inch at the extreme, or when the places to which the obser- vations relate are the equator and the pole ; and the difference in the value of g at these places, is only about two inches, as may be easily shown by the above formula. f The most accurate observations on the length of the seconds pendulum at Paris in latitude 48 51' give for the value of g 32,182 ft. Application of the Pendulum to Time-Keepers. 251 at the lower extremity with a weight in the form of a lens, so placed as meet with as little resistance as possible from the air. The axis also or point of suspension is fitted to have very little friction. Connected with the pendulum, is a train of wheels and pinions, the teeth and leaves of which are so adapted to each other, that the motions correspond to the several divisions of time, and their axes carry indexes that show by the arcs they describe, the hours, minutes, and seconds. Around the axis at one extremity of this train of wheels, is wound a cord bearing a weight, that would put the whole system in rapid motion, but for the appendage to the pendulum CFD at the other extremity of the train of wheels, which, while the pendulum is at rest, effectually prevents all motion. But if the pendulum be made to vibrate, it will suffer one tooth to pass or escape at each vibration, while at the same time the impulse of the teeth upon the arms FC, FD, is so adjusted, by increasing or diminishing the weight, as just to overcome the friction and the resistance of the air, and thus to keep up the motion, while the action of the weight continues. The contrivance by which the train of wheels is connected with the pendulum, is called the escapement. 373. On account of the constancy of gravity the oscillations of the pendulum, other things being the same, must be equal or of the same duration. There are, however, several causes that tend to disturb this isochronism. (1 .) The air is subject to changes of density, on account of which the arcs of vibration will some- times be longer and sometimes shorter, while the maintaining power remains the same.* But if these changes are noted, or if the arcs of vibration are noted, the deviation from perfect regu- larity can be calculated, and allowance made accordingly.! It *As the air becomes more dense the pendulum is more resisted and would seem to be retarded, but the arc of vibration being dimin- ished, the clock goes faster, so that one of these causes tends to coun- 343. teract the other. In like manner, when the motion of the axis and wheels is obstructed by dust or want of oil, the impulse of the weight communicated to the pendulum is diminished, the arcs of vibration are reduced, and hence there is a tendency to increase its rate of going. f If the pendulum could be made to move in the arc of a cycloid, it will be perceived from what has been said of this curve, that all 353< 252 Dynamics. may be remarked, moreover, that the irregularity from this cause, is rendered for common purposes altogether inconsidera- ble, by making the pendulum very heavy, and the arcs of vi- bration very small. 374. (2.) A much more important source of error, in the rate of going of common clocks, is to be referred to changes in the actual length of the pendulum arising from heat and cold. A brass pendulum rod, for instance, has its length increased about two hundredths of an inch for a change of temperature of 30 of Fahrenheit's thermometer. This would seem to be a small quantity ; yet as it is continually exerting an influence, the accu- mulated effect in the course of 24 hours or 86400" amounts to more than a third of a minute. The expansion of iron is about f of that of brass. There are some kinds of wood that are sub- ject to very little variation of length, particularly in the direction of the fibres, on account of temperature. Still no substance is entirely free from these changes. The effect of any augmenta- tion or diminution of length in the pendulum may be computed 346. by means of the principles that have been investigated. 375. But we can obtain more convenient and sufficiently ex- act formulas for the variation in the rate of the going of a clock, when the changes in the length of the pendulum are very small, as those are which arise from heat and cold, a being the exact length of the seconds pendulum, or that by which the clock would keep correct time, and a' the actual length, as affected by heat and cold, if we put n for the number of oscillations in a day, performed by the former, n' for the corresponding number of the latter, and tf for the time of this latter, we shall have 346. also 348. arcs whether longer or shorter, would be described in the same time. But the practical difficulties attending all the methods hitherto pro- posed, are such as to occasion errors, that more than compensate for the theoretical advantages to be derived from a cycloidal motion. Application of the Pendulum to Time-Keepers. 253 whence and an Suppose x to be the augmentation or diminution of length in ques- tion, and y the corresponding daily loss or gain in seconds, we shall have an 2 an 2 an 2 a a' = a x = n' 2 (n^y) 2 n 2 ^p 2ny -\- y 2 j _ 2j/* = g (i *), and let y denote the daily gain or loss in seconds ; we shall have T \ -- " n 2 n* nearly. Whence, Thus, if a pendulum fitted to vibrate seconds at the equator, would, upon being carried to the pole, gain 5' or 300" a day, we should have _ 2 x 300 1 86400 144' that is, the force of gravity at the equator is to that at the pole, on this supposition, as 144 to 145. Let the difference h in the distances of the two stations from the centre of the earth be given, gravity being supposed to vary inversely as the square of the distance, the gain or loss of the clock might be readily found as follows. If we call R the distance of the centre of the earth from the first station, and g the force of gravity at this station, the pendu- lum being supposed to vibrate seconds, we shall have for the distance of the second station R /i, and for the force of grav- ty at this station, Ra (\ - - 7J g V* ^F ~n > (* nearly. Hence, putting-^- for x in the above formula, we obtain Thus, if the second station be above the first, as 1 mile for in- stance, the radius of the earth being 3956, or 4000 nearly, the Rotation of Bodies unconfined. 257 the formula gives ,,, - 86400 - - y 40-0^ the sign indicating that the clock loses. Rotation of Bodies unconfined. 379. It has been demonstrated that when a body L re-Fig.is. ceives an impulse in a direction HZ, not passing through its centre of gravity G, this impulse is transmitted entirely to 124t the centre of gravity, which moves in a direction parallel to HZ according to which the body has received the impulse ; and that the parts of this body, in the mean time, turn about the centre of gravity in the same manner as if it were fixed. Therefore, if the figure of this body, arid the forces impressed upon it (of which I suppose p to represent the resultant) are such that it can turn only about a single axis ; as this axis will necessarily pass through the centre of gravity, all that we have said on the subject ot the moment of inertia, is applicable to this case, understanding by r in fm r 2 , the distance of any particle from 354^. the axis which passes through the centre of gravity, and by p x D the moment of the force HZ, taken with respect to the same axis, or the sum of the moments of all the forces which act upon the body, taken with respect to this same axis. That is, the centre of gravity will move parallel to the direction of the force p, with a velocity = -y-, L being the mass of the body ; and if we draw JL/ GZ perpendicular to HZ, and call v the velocity of rotation of the point Z, we shall have or Of this we shall give a few applications. 380. Let us suppose that the body JV* impinges upon the body L, according to any direction whatever J2Q,m such a man-Fig.186. Mech. 33 258 Dynamics. ner as to cause no rotation in L, except about a single axis per- pendicular to the plane which passes through the centre of gravity G, and the perpendicular TZ belonging to the point of contact T\ it is proposed to determine the velocities after collis- ion, and the directions of these velocities, the body L being supposed at rest. Let us imagine a plane touching the point T, and let the ve- locity of JV, according to EQ be decomposed into two others one according to ET perpendicular to this plane, and the other according to El parallel to this same plane. If JV had no other velocity but 7, it would only touch L in passing, and would communicate to it no motion, the effect of friction being out of the question. It is therefore only in virtue of the velocity ET^ that the impulse is produced. Now as it is easy to determine ET in the parallelogram 7^7, of which all the angles and the diagonal EJi are supposed to be known, we shall consider this velocity E T as known, and we shall call it u. Let u' represent the velocity of .AT after collision, according to the direction ET or CZ ; consequently u u' is the velocity lost by collision, and N X (u u f ) is the force impressed upon the body L, which we have called p. Therefore the centre of gravity and all the parts of the body will move in the direction GM parallel to CZ, with a velocity * = ^F^ (o, calling this velocity v. But, as the force JV X (u u') does not pass through G, the centre of gravity of L, the body must turn about G, as if this 136 point were fixed. Let v' be the velocity of rotation of the point Z where GZ, perpendicular to CZ, meets the latter line ; we shall have, therefore, T/ - -^ * ( x uz or representing GZ by Z>, T)' = ^^ ^ ~~ M/ ) j mr* (2). * Rotation of Bodies unconfined. 259 It may be observed, moreover, that it is necessary, in order that the body .Af may really have the velocity w', that the point T of the body L should also have this same velocity V accord- ing to TZ. Let us now see with what velocity this point must advance according to TZ. It will have, in the first place, the velocity v common to all the parts of L. Moreover, if we suppose that the infinitely small arc TT perpendicular to G7 1 , represents the velocity of rotation of the point T, by constructing the parallelogram TCT'B upon the directions TT', TA and TZ, we shall have TC for the veloc- ity of T according to TZ in virtue of its rotation. Now the sim- ilar triangles TT'C, GTZ,give GT : GZ : : TT : TC = GZ But since -o' is the velocity of rotation of the point Z, we have ?' : TT :: GZ : GT, whence and consequently r'X GT 11 = ~~~ TC - GZ v v ' xGT - X -- therefore the total velocity of the point T belonging to the body L, according to EZ, is v + -sf ; and hence v + v' = u' (3). If from the three equations found above, in order to express the conditions of the motion, we deduce the values of w 7 , ?/, and v, we shall obtain + LD 2 ) u JVw/mr 2 LJVD* u (JV-f L)/ror* + LD* 260 Dynamics. If the distance GZ or D = ; that is, if the direction of the impulse passes through the centre of gravity G, the velocity of rotation ?/ = 0, the velocities u' and v are equal to each other and to , as indeed they ought to be, according to article 288. The velocity u' being determined, if it be compounded with the velocity El, which has suffered no alteration, we shall have the absolute velocity of JV, and its direction after collision. If the body L were in motion before collision, we should de- compose the velocity of JV before collision into two others, one of which should be equal and parallel to that of L ; this would contribute nothing to the impulse, and we should employ the second as we have employed the velocity according to EQ, con- sidering the body L as at rest. If we compare the value found above for t/, with that which 363. we before found for the velocity of rotation, by attending to the difference in the import of r in the two cases, we shall be able to determine the difference between the velocity of rotation which belongs to a free body, and that belonging to one which admits only of a rotation about a determinate point or axis. 381. From the value which we have found for i/, the veloc- ity of rotation, may be deduced a method for determining by experiment the value ofjffiiJrS and the position of the centre of gravity in a body of any figure whatever. We shall apply to a vessel what we have to say upon this subject. Let us suppose, that, by means of a weight JV and a rope at- tached near the stern, the vessel is drawn in a direction perpen- dicular to its length, the weight being small compared with the whole weight of the vessel. Let this weight pass, for instance, Fig 187 over the pulley P. The velocity of the vessel during the exper- iment, (which should continue only for a very short time, as a minute or half a minute) will be so small as to make it unneces- sary to take account of the resistance of the water. The action of gravity communicates to ^V, in the instant d /, the velocity g d t (g being the velocity acquired in a second o time), and produces in the vessel an infinitely small velocity of rotation, which I shall call dv f for the point A where the rope is Rotation of Bodies uncorifined. 261 attached. Putting, therefore, g d t for w, and d v' for ', in the value of v' found above, JV* being considered as very small or nothing compared with L, the mass of the vessel, which gives z/ = we shall have d * = f^- Let d v" be the velocity with which that point of the vessel turns which is distant one foot from the centre of gravity ; we shall have d-o' : dv" :: AG : I :: D : 1, and consequently dv' = Ddv". Substituting for d v' this value, we have and, by integrating, fmr Let z be the arc described by the point in question during _ the time / ; we shall have dz = v" dl, and consequently whence, by integrating, '' Therefore, if the rope acting always perpendicularly to the length of the vessel, be attached to another point /, and we call zf the arc described by the same point during the same time , we shall have 262 Dynamics. 2/wra ' calling jy the distance IG ; whence we have z : z' : : D : ZK : : JIG : IG ; and hence Alg.224. z z' : z : : AG IG or Now if after each experiment we measure, as may easily be done in several ways, the angles of rotation, that is, the num- ber of degrees contained in the arcs z, z 1 respectively, we may substitute these numbers instead of the arcs z, z', in the propor- tion ; and since the distance Jil is known, we readily obtain .4G, that is, the position of the centre of gravity. The value of JIG or D being determined, we calculate the 294. * length of the arc z which has 1 for radius, and of which the number of degrees is known ; then, since JV is known, and g is 271. equal to 32,2 feet; if we take care to observe the number of seconds which elapse up to the instant at which the number of degrees in z is counted, we shall know every thing except f m r 2 in the equation gWDt* but this equation gives fmr* = whence we obtain the value of j*m r 2 , which it would be very troublesome to obtain by a particular calculation of the differ- ent parts of the vessel. 382. When a body L of any figure whatever, having received ' an impulse in a direction HZ, not passing through the centre of gravity, takes the two motions of which we have spoken, it is ea- sy to see, that for an instant it may be regarded as having but one single motion, namely, a motion of rotation about a fixed point or axis F, which according to the figure of the body, and also the distance GZ at which the impulse passes from G, may be situated either within or without the body. For if, while the Rotation of Bodies unconfined. 263 line GZ is carried parallel to itself from GZ to G'Z', we imagine it to turn about the moveable point G, since the points of the body have velocities of rotation greater in proportion to their distances from G, it is manifest that there is upon the line ZG a point F which will be found to have described from F' toward I 1 , an arc equal to GG', and which may be regarded for an in- stant as a straight line ; the point F then will have retrograded as far by its motion of rotation as it has advanced by the \elo_c- ity common to all parts of the body ; this point will therefore have remained constantly in F, which, for this reason, may be considered for an instant, as a fixed point about which the body turns. If we would know the position of the point F, it will be remarked that the arcs FF', Z'/, which the points F' and Z' de- scribe in an instant, may be considered as straight lines perpen- dicular to GZ, or parallel to GG' ; now the similar triangles FF'G', G'Z'/, give G'Z' : G*P : : Z'l : FP, pr GZ : GF :: Z'/ : GG'; but we have found the velocity, GG' = --, and the velocity Z'/ = ~^- 5 hence GZ or D : GF : f m r 2 L therefore GF= /^l. 383. The point F is called the centre of spontaneous rotation, because it is a centre which the body takes as it were of itself. This point is precisely the centre of oscillation which the body L would have, if it turned about a fixed point or axis situated in Z ; for from GF = we have 264 Dynamics. fmr* + L X GZ ~GZ xL ' 'm r 3 -f L X GZ is in article 360 precisely what we have understood by^'m r 2 in article 361 ; therefore the point Fis here the same as the point O in article 361. We perceive, therefore, that the point about which a body may be considered as turning for an instant, is independent of the value of the force or forces which are applied to this body ; and generally it may be inferred from the value of FG, that this point is the more distant, according as the force in question, or the resultant of all the forces, acts at a less distance from the cen- tre of gravity. 361. 384. We have seen that when a body turns about a fixed point or axis, its centre of percussion is the same as its centre of oscillation ; whence these two centres are found by the same op- eration. It is not the same when the body is free. For, let us suppose a body whose mass is L, to turn about its centre of grav- ity with a velocity, which, for a point situated at the known dis- tance a, shall be v ; and that at the same time this centre moves with the velocity u. It is manifest, in the first place, that the re- sulting force of all the motions belonging to the different parts of this body, will have for its value L x u or L u, that is, the same as if the body had no motion of rotation. In the second place, the distance at which the resultant must pass from the centre of gravity, is evidently that at which a force equal to L w, would produce in the body a velocity of rotation equal to that which it actually has ; but this velocity has for its expression > calling D the distance sought ; we have, therefore, fmr 2 _ LuDa " ~fmr 2 '' and consequently Measure of Forces applied to Machines. 265 u La and hence we see that the distance of the centre of percussion of a free body depends on the ratio of the velocity of rotation to the velocity of the centre of gravity ; and particularly that it is nothing when the velocity of rotation is nothing, as in fact it ought to be. We may hence determine at what point to place an obstacle in order to stop a free body which has a progressive and rotato- ry motion at the same time ; namely, at the centre of percussion of this body, or the point where it would give the strongest blow or exert the greatest force. Method of estimating the Forces applied to Machines. 385. Any force has for its measure, as we have already said, the product of a determinate mass, into the velocity which the force in question is capable of giving to this mass. It seems proper, in this place, to add something by way of illustrating the application of this principle to machines. When two weights act against each other by means of a sim- ple fixed pulley, it is necessary in order to an equilibrium that their masses should be equal ; and this equilibrium once estab- lished, will always remain. But if instead of opposing a weight to a weight, we oppose the force of an animal, as that of a man, for example, although it be true that, in order to an equilibrium, this man has only to exert an effort equal to the weight to be sustained, that is, equal to the quantity of motion represented by the mass of this body multiplied into the velocity which gravity communicates in an instant ; it is, nevertheless, evident that if the man were capable of but one such effort, the equilibrium would continue only for an instant, because gravity renews each successive instant the action which was destroyed in the preceding. Mech. 34 266 Dynamics. It is not, therefore, by the mass only which the man supports, that we are to estimate his strength ; but we must consider, also, the number of times that he is able to exert an action equal to that which gravity communicates every instant to the body. Now if g represents the velocity which gravity is capable of giving to a free body in a second of time ; and d t represents an 267 - infinitely small portion of any time J, g d t will be the velocity which gravity gives during the instant d , t being supposed to be reckoned in seconds. Therefore, if m be the mass which it is proposed to sustain, mgdt will be its weight, or the quantity of 2 ' 2 ' motion which gravity gives it each instant d t ; it is accordingly the effort also which must be exerted each instant by the force which is to support m, either directly or by the aid of a pulley. Therefore, during any time /, this force must expend a quan- tity of motion equal to Pmgdt or mgt. Therefore, if t denotes the time at the end of which the agent is no longer able to support the mass ra, m g t may be regarded as the measure of his strength. We do not mean by this that he is no longer capable of exerting any effort ; but his force having become unequal to the effect to be produced, it is considered as nothing with respect to this effect. Let us, for example, suppose that in order to support a weight of 50 lb for an hour, it is pro- posed to employ a force, which acting by equal and infinitely small degrees, is known to produce in a mass of 20 lb , a velocity of 50 feet in a second, at the instant when this force is exhausted. It is manifest that this mass of 20 lb will have a quantity of motion equal to 20 lb X 50 or 1000. Let us see, then, if this quantity of motion be equal to what the quantity m g t becomes, by putting 50 lb for m, an hour or 3600" 273. for J, and 32,2 feet for g. It appears to fall far short of it ; such a force, therefore, would not support a weight of 50 lb during an hour. If we wished to know during what time, or what number of seconds, it would support it, we have only to suppose mgt = 1000; Measure of Forces applied to Machines. 267 and, putting 50 for w, and 32,2 for g, we shall have met 1000 1000 100 5 or ^ == that is, such a force would support a weight of 50 lb only about of a second. 386. Let us now suppose that it is required not only to sup- port the mass m during the time J, but also to move it during same time with a uniform and known velocity v. It is manifest that in communicating the velocity &, either succes- sively or at once, to the body m, there must have been expend- ed a quantity of motion equal to m v ; and to maintain this veloc- ity v during the time f, the action of gravity is to be resisted all the while just as if the body had remained at rest ; that is, there must have been expended an additional quantity of motion equal to vngt] therefore to maintain in the mass m the velocity v dur- ing the time , the agent must be capable of producing a quantity of motion equal to m v + mgt* 387. It is ascertained by actual trial, that a man can work at a machine like that represented in figure 97, for 8 hours success- ively, and cause the winch to make 30 turns a minute, the radi- us of the cylinder and that of the winch being each 14 inches, and the weight applied at the surface of the cylinder being 25 lb . This experiment determines the value of m v -f- m g t, and consequently the limit to be observed in estimating the force of a man working at a machine, and for a definite period of time. Indeed, since the radius of the winch and that of the cylinder are equal, the weight in this case passes through the same space with the power. Thus, the radius being 14 inches, at each turn the power passes through 28 X 3,1416, or 88 inches nearly ; ; since it makes 30 turns a minute, it describes 44 inches a second, or f | of a foot ; that is, the velocity 44 _ 11 12 3* The mass 268 Dynamics. m = 25 lb , g = 32,2 ft , and t = 8 h == 28800". The substitutions being made, we have roz> -h mg*= 2 J 5 -I- 23184000 = 23184092. By means of this number, we can judge whether the strength of a man be sufficient to produce a proposed effect. For instance, if it be asked whether it be possible for a man, with the machine above referred to, to raise a weight of 60 lb with a velocity of 10 feet in a second, during 6 hours, we shall perceive that it is not. For we should have in this case ro = 60 lb ; v= 10; g = 32,2; f = 21600"; which gives mv + mgt = 600 + 41731200 = 41731800; as this greatly exceeds 23184092, it follows that a single man is unequal to such an effect. It may be remarked that in these two examples, the velocity v with which the man is supposed to move the weight, is of very little" consequence in estimating the force required; for in the first example, the quantity of motion which answers to this veloc- ity, is 3 J 5 ; and in the second, 600; quantities which are very small compared with 23184092 and 41731800. Therefore, in the second example, if we are unable to produce the desired effect, it is not because-the velocity is greater than in the first case, but chiefly because the mass and the time during which it is to be moved, require of the agent too great a quantity of motion. While therefore the velocity required in the agent is small com- pared withg /, that is, with the velocity which a heavy body falling freely would acquire in the time during which the agent is sup- posed to be employed, we may take simply for the measure of the force in question, the quantity mgt] and we shall have mgt = 23184000. Thus, if the mass (the velocity with which it is to be moved be- ing moderate) multiplied by the velocity which a heavy body falling freely would acquire in the time during which the power Measure of Forces applied to Machines. 269 is to act, forms a product less than the constant number 23134000, or exceeding it but a little, the power may be considered as suf- ficient for the proposed effect, it being supposed to act as in the two preceding examples. But if the velocity with which the weight is to be moved, is considerable compared with g , it will be necessary to subtract from the constant number 23184092, the quantity of motion m v due to the velocity with which the body is to be moved ; and if the weight multiplied by g /, the ve- locity which a falling body would acquire in the time during which the machine is to be worked, forms a product smaller than the remainder above found, the power may be deemed sufficient. 388. In what we have now said, we have taken no account of friction. When the motion of the machine has become uni- form, (which is the state in which machines ought to be consider- ed) the effect of friction may be regarded as constant, and it may be compared to a new mass required to be moved together with the proposed mass. Thus, in the case above considered, the friction being supposed equivalent to the weight of a known part of the mass m, this resistance will require in the power a quantity of motion equal to mg /, and thus, c mv -\ -- mg t -{- mgt, c or / will be the measure of the moving force. If then, in the experiment above referred to (the axle being supposed to have a radius much less than that of the cylinder), we suppose the friction to have been T V of the weight, mv being neglected, as it may be in this case, we must augment the number 23184000 by its twelfth part; then the force of a man in similar circumstances may be represented by the number 25116000. We see, therefore, that to be able to estimate with sufficient ac- curacy the force of a man, we must previously ascertain the ra- tio of the force of friction to that of the weight, in the experiment employed, with the view of determining this force. Then if k 270 Dynamics. be the value derived from this experiment for f + 1) mgt, we shall have, neglecting m t>, when i? is small compared with g , This equa- tion will enable us to judge for any other supposed value of , whether the force of a man will be sufficient to move the weight m during the proposed time t. 389. In all that we have now said, we have regarded' the agent as acting immediately upon the weight, and as deriving no advantage from local circumstances and the nature of the machine. We may often rely upon a much greater effect than the particular considerations now presented would lead us to expect. For instance, in the use of the pulley a man may add to his own proper force the weight of his body, or a large part of it. There are, moreover, many other circumstances of which he may avail himself, and other machines which admit of similar expedients. Frequently the motion is not continued, but takes place by starts, as in the pulley ; and if there is a loss on this account, there is also this advantage, that the agent by intervals of rest is capable of exerting the same action for a longer time. We shall not dwell upon these details which it will be always easy to take into the account after all that has been said, espec- ially if we proceed according to experiments in which care has been taken to distinguish the several causes on which the action of the moving force depends, and to note what belongs to each. It is commonly said that a man can continue during about eight hours, to exert an effort equal to 25i b . It will be seen from what precedes, that such a statement does not sufficiently deter- mine the value of the force in question ; besides, it is necessary, as we shall soon undertake to show, to have regard to the velocity with which the man acts ; it is no less necessary to consider also the manner in which the action is applied, and many other cir- cumstances which we cannot now stop to enumerate. It is prop- er, when circumstances vary, to proceed in our calculations upon new experiments made with reference to these circumstances. Measure of Forc&s applied to Machines. 27 1 I 390. Although we have considered that case only, in which the weight transmits all its resistance to the power, it is not less easy, after what has been said respecting the ratio of the weight to the power in each machine, to determine whether by the aid of a particular machine, a given power will produce a proposed effect. In the wheel and axle, for instance, if the radius of the cylinder be , and none when there is no velocity, we have v or v : i/ or &' : : p : ^ -. -j - is therefore the loss of force sustained on account of moving with the velocity v'. There will accordingly remain as the effective force actually exerted against the weight. Now if D be the distance at which this force acts, r the resistance or weight raised, and tf the distance at which the resistance acts, and u its velocity ; then, when the machine has attained a uniform motion, we shall have 276 Dynamics. But, since D : tf : : v' we have, by substitution, To find the maximum we put the differential of the first member equal to zero, v' being regarded as variable ; we have thus Gal. 45. P dv/ - = > or V = 2 v' and -o' = | v. Substituting -D for v' in the above equation, we obtain r u = %pv> On the hypothesis, therefore, which we have assumed, the man will do most work when he moves with half his greatest velocity, and in this case the greatest effect will be p v. 394. It appears, however, by direct experiments, that the force of a man diminishes as the square of his velocity increases,! in other words, that the effective forces are as the squares of the diminutions of velocity from the point where the effective force is nothing. Calling />', therefore, the force answering to the velo- city i/, we shall have, according to this hypothesis, whence p'=p (j^rO" = p (T) 3 w ' putting v ~~ *' = w (n) ; and hence, since (7y\ 3 /w\* , J -D' or P(~) CD w) = r w, Taking the differential, as before, and putting it equal to zero, and suppressing the constant factor we have t See note on the measure of forces* Maximum Effect of Machines, 277 2vwdiD 3w 2 dw = 0; whence 2 v = 3 w, and w = f v. Substituting this value in equations (11), (i), we obtain or also that is, *fte work done is a maximum when the agent moves with one third part of the greatest velocity of which he is capable, and when the weight or load is of the greatest which hz is able to put in mo- tion during the whole time he is supposed to act. 395. Having thus considered the maximum effect of living agents, we shall proceed to the subject of machines, and shall take the case of a wheel and axle, as almost all other machines may be reduced to this. The powers by which a machine is put in motion, and by which that motion is kept up, are called first movers, or moving powers, or more familiarly, mechanical agents ; and when various moving powers are applied to the same machine, the resultant of them, or the equivalent force, is called the moving force. The first movers of machinery, are, the force of men and that of other animals, the force of steam, the force of wind, the force of moving water, the weight of water, the reaction of water, the descent of a weight, the elasticity of a spring, &c. If a machine be driven by two powers acting in two different directions, we must then find their resultant, and consider the machine as driv- en by the resulting force. The powers which oppose the production of motion in a ma- chine, and its continuance, are called resistances ; and the resul- tant of all the resisting forces is called the resistance. The work to be performed is, in general, the principal resist- ance to be overcome 5 but, in addition to this, we must consider 278 Dynamics. the resistance of friction, and the resistance arising from the in- ertia of all the parts of the machinery ; for a certain portion of the moving power is necessarily wasted in overcoming these obstacles to motion. The impelled point of a machine is that point at which the moving power is applied, or rather that point at which the mov- ing force is supposed to act, when this moving force is the result- ant of various powers differently applied. The working point of a machine is that point at which the resistance is overcome, or that point at which the resultant of all the resisting forces is sup- posed to act. The work performed, or the effect of a machine, is equal to the resistance multiplied by the velocity of the working point. The moment of impulse is equal to the moving force mul- tiplied by the velocity of the impelled point. 396. In proceeding to investigate general expressions for the ratio of the velocities of the impelled and working points of ma- chines, when their performance is a maximum, let D = the radius of the wheel to which the power is applied ; or, which is the same thing, the velocity of the impelled point of the machine ; d = the radius of the axle to which the resistance is applied, or the velocity of the working point of the machine ; p = the moving force applied at the impelled point; r = the resistance arising solely from the work to be per- formed ; m = the inertia of the moving power/?, or the quantity of mat- ter to which that power must communicate the velocity of the impelled point ; n = the inertia of the resistance, or the quantity of matter to be moved with the velocity of the working point be- fore any work can be performed ; / = the quantity of matter, which, if placed at the working point, would create the same resistance as friction ; Maximum Effect of Machines* 279 i = the quantity of matter, which if placed at the working point, would oppose the same resistance as the inertia of all the parts of the machinery. Since D and d are the radii of the wheel and axle, we shall have D : d : : r : , a weight equal to that part of the power p which is in equilibrium with the resistance. We have, therefore, p -- as an expression for the effective force of the power ; and as D is the distance at which this force is applied, we have p D r d to represent the force which is employed in giving a rotatory motion to the machine. The resistance which friction opposes to this force will be/d ; the moment of inertia of the power p will be as wo 2 ; the moment of inertia of the resistance as nd 2 , and the moment of inertia of the machinery will be as id 2 . Since the moving force is diminished by the resistance of friction, we shall have JOD rd /d for the moving force; and since the resistance arises from the moment of inertia of the resistance, the moment of inertia of the power, and that of the machinery, it will be as mo 2 -f nd 2 + *D 2d(r+/)) (mo 2 4- (n + i) This case takes place when the resistance to be overcome exerts a contrary strain on the machine, while it consists merely in the inertia of the impelled body ; as in driving a millstone, a fly, or in pushing a body along a horizontal plane. When/ = 0, . _ __ mr This case takes place when the friction is so small that it may be disregarded, which often happens in good wheel work, where the surfaces that touch one another are very small. When r = 0, and/ = 0, we have ~^(n + ) ._ D I m (n+i}* \ n + i ' _ D - D This case takes place when the circumstances of the two preced- ing cases are combined. When n = 0, we have +P 2 i m (r + / ) pi This case takes place in the grinding of corn, the sawing of wood, the boring of wooden or iron cylinders, &c., where the quantity of motion communicated to the flour, the saw dust, or the iron filings, is too trifling to be taken into the account. Meek. 36 282 Dynamics. When r = 0,/ = 0, and n 0, we have tf = D \OL , When m : n : : p : r, we have, d = D VP 2 (r + /) 2 + F* (r + i) "~ P (r +/) p (r + t) This case takes place when the inertia of the power and the re- sistance are proportional to their pressure ; as when water, min- erals, or any other heavy body, is raised by means of water acting by its weight in the buckets of an overshot wheel. When, in the last case, i = 0, and/ = 0, we have This case often takes place, and particularly in pulleys ; and making D = 1, and r = 1, we obtain and when p = 1, and D= 1, we have The preceding formulas will be found applicable to almost every case, which can occur; and the intelligent engineer will have no difficulty in accommodating them to any unforeseen circumstan- ces. The following table will, in many cases, save the trouble of calculation. It is computed from the formula D being supposed = 1, and r = 10. Maximum Effect of Machines. 283 Table containing the best Proportions between the Velocities of the Impelled and Working Points of a Machine, or between the Lev- ers by which the Power and Resistance act. Proportion- al value of the impelling power, or P- Value of the velocities of the working point or $, or of the lever by which the resistance acts, that of D being 1. Proportion- al value of the impelling power, or P> Value of the velocities of the working point, or J, or of the lever, by which the resist- ance acts, that of D being 1. I 0.048809 20 0.732051 2 0.095445 21 0.760682 3 0.140175 22 0.788854 4 0.183216 23 0816590 5 0.224745 24 0.843900 6 0.264911 25 0.870800 7 0.303841 26 0.897300 8 0.341641 27 0.923500 9 0.378405 28 0.949400 10 0.414214 29 0.974800 11 0.449138 30 1.000000 12 0.483240 40 1.236200 13 0.516575 50 1.449500 14 0.549193 60 1.645700 15 0.581139 70 1.828400 16 0.612451 80 2.000000 17 0.643168 90 2.162300 18 0.673320 100 2.316600 19 0.702938 In order to understand the method of using this table, let us suppose that we wish to raise two cubic feet of water in a second, bj means of the power of a stream which affords five cubic feet of water in a second, applied to a wheel and axle, the diameter of the wheel being seven feet. It is required, therefore, to find the diameter which we must give to the axle, in order to obtain a maximum effect. We have obviously p = 5, and r =. 2, and since p : r : : 5 : 2, we have p = - r ; but, in the above table, r = 10 ; hence p = - 10 = 25. Now it ap- pears from the table, that when p = 25, the diameter of the axle, or d, is 0.8708, D being 1 ; but as D = 7, the diameter of the axle must be 7 X 0.8708 = 6.0956. 284 Dynamics. 397. When a machine is already constructed^ the velocity of its impelled and working points are determined ; and therefore in order to obtain from it its maximum effect, we must seek for the best proportion between the power and the resistance, as these are the only circumstances over which we have any con- trol, without altering the machinery. In order to find the ratio of p to r, which would produce a maximum effect, it is requisite only to make r variable in the formula above given ; but it often happens, that when r varies, the mass n suffers a considerable change, although there are other cases when the change in n is too inconsiderable to be noticed. Let us, therefore, first take the case when r alone varies with- out inducing a change in n. In this case, the expression for the work performed, namely, rp D tf r 2 32 rfd 2 m D 2 + n d 2 + i d 2 will be a maximum when tf as will be readily found by differentiating, putting the differential equal to zero, and deducing the value of r. But according to the experiments of Coulomb, the friction is in general equal to of the resisting pressure. Hence we may omit /d, and 1 fi consider the resistance as = r + T l j r - r. Consequently, I O r = J^, and r M^j) X ^j. But if we consider the fraction as so near 1 , that the substitution of the latter will lb not greatly affect the result, we shall obtain, by making p = 1 and D = 1 ? r = r ; that is, the resistance should be nearly one half of the force which would keep the impelling power in equi- librium, a rule which is applicable to many cases where the matter moved by the working point of the machine is inconsider- able. Maximum Effect of Machines. 285 398. In those cases where n varies along with r, it will in general vary in the same proportion, and we may therefore re- present n by a? r, some multiple of r. For the sake of simplicity, the friction f may be considered as absorbing a certain portion of the impelling power, which will then be represented by p /; and we may also regard the inertia of the machine, or i, as ap- plied at the impelled instead of the working point ; that is, the moment of inertia may be considered as proportional to i D 3 . Now, if we make p /= 1, and D = 1, in the formula rpvd r 2 d 2 rfd 2 m D 2 -f- n d 2 + id* ' we shall obtain r d r 2 6 2 m -\- i -j- xrd 2 ' and making m + i s, we have r d r 2 d 2 s + xrd 2 for the work performed. This is a maximum when the differential, r being considered as variable, is equal to zero, which gives (d 3rd 2 ) (s -f- x rd 2 ) x d 2 (r$ r a d 2 ) = 0; or, by reducing, sd %s rd 2 x r 2 d* =0; that is, r2 j, * and by resolving this after the manner of an equation of the second degree, we obtain _____ s I s s 2 ^ _ " xd 2 " h \s<5 3 x 2 d* r s 2 s xd 2 When x = 1, we have _ \/s d -f s 2 s 286 Dynamics* This case takes place when the machine is employed in raising a weight, drawing water, &c. When i = O,/ = and m = p, then m + ior5=_p=l, and When D = <5, as in the common pulley, then =!.-, But, in the present case, p = 8, consequently 1:8:: 1.3928 : 11.1424, the value of r when p = 8. t 399. The subject of the maximum effect of machines may be considered in a very simple point of view, if we suppose, what is by no means improbable, that the moving power in machinery observes the same law that has been found to exist with regard to animal force, and also with regard to the force of fluids in motion. Upon this hypothesis, we shall have r = p fl -- ) , and the effect of the machine will be r -o' = p v' fl -- j , r d 4 which will be a maximum when = - , and when v' = \ v. D 9 r In these formulas, p is the load that is just sufficient to bring the machine to rest, or prevent it from moving, v is the greatest ve- locity of the power when no work is done, and t/ the velocity of the impelled point of the machine. The above equation, both members being multiplied by 9, is equivalent to the proportion, D is to #, or the velocity of the impelled point is to the velocity of the working point, when the effect is a maximum, as 9 r to 4 p. 394. Table of the Strength of Men, according to different Authors. Number of pounds rais- ed. Height to which the weight is raised. Time in which it is raised. Duration of the Work. Names of the Au- thors. 1000 180 feet 60 minutes Euler 60) J 25} g 170) g. 220 ( U 1 second 145 seconds 1 second 8 hours, half an hour Bernoulli Amontons Coulomb 1000 1000 330 225 60 minutes 60 minutes Desaguliers Smeaton 30 3^ 1 second 10 hours Emerson 30 2,43 feet 1 second Schulze 288 Dynamics. The following are the estimates that have been made of the relative strength of horses, asses, and men. . { Desaguliers. 1 horse is equal to 5 men, Smea * on . 1 " 1 ass 7 men, Bossut,&c. 2 men, Bossut. HYDROSTATICS. INTRODUCTORY REMARKS. 400. Hydrostatics is that part of Mechanics which treats of the equilibrium of fluids, and that of solids immersed in them. A fluid is a collection of material particles, so constituted as to yield to the smallest force employed to separate them. The fluids with which nature presents us, approach more or less tp this state of perfect fluidity. The adhesion which exists among the particles of several of these substances, and which gives rise to what is called viscidity, opposes itself to the separation of the particles ; but in the theory which we are about to un- fold, no account is taken of this adhesion, and we have reference only to the perfect fluids. We distinguish two kinds of fluids ; the one incompressible, or nearly so,t as water, mercury, alcohol, and liquids gener- ally. These are capable of taking an infinite variety of forms without any sensible change of bulk. The second kind of fluids comprehends atmospheric air, the gases generally, and vapours. These are in an eminent degree compressible ; they are also endued with a perfect elasticity, and are thus capable of changing at the same time their form and bulk, upon being com- pressed, and of recovering their figure again, when the compress- ing force is removed. Vapours differ from air and the gases, by losing the form of elastic fluids and returning to the state of liquids, when compressed to a certain degree, or when their t See note subjoined to this treatise on the compressibility of water. Mech. 37 290 Hydrostatics. temperature is sufficiently reduced ; whereas air and the gases are found, with few exceptions, to preserve always their elastic form in the state of the greatest compression and lowest tempera- ture to which they have hitherto been reduced.! 401. Although we are unable to assign the magnitude of the elementary parts of fluid bodies, we cannot doubt that these parts are material, and that the general laws of equilibrium and mo- tion, already established, are applicable to them as well as to sol- ids. But as this law of equilibrium is not the only one required, some other is to be sought on which the equilibrium depends. 402. As an equilibrium consists in destroying all the forces employed, and as we do not know how the parts of a fluid trans- mit their forces among themselves, it is only by having recourse to experiment, that we are able to establish our first principles. We begin, therefore, by stating what is most clearly and cer- tainly known upon this subject. Pressure of Fluids. 403. L?t ABCD be a canal or tube, composed of three branches AB, BC, CD, of equal diameters. Let us suppose that a heavy fluid is poured into the branch AB ; it will pass through the branch 5Cinto the branch CD ; and when we cease pouring, the surface of the fluid in the two branches will be in the same horizontal line, whatever be the inclination of the branch BC. This is a fact abundantly established and univer- sally admitted. We proceed to make known the consequences to be deduced from it. 404. If through any point E, taken at pleasure, we imagine a horizontal line EF to pass, it is evident that the weight of the fluid KBCF contributes nothing to the support of the columns AE, DF ; and that consequently the equilibrium would still be preserved, if the fluid contained in EBCF were suddenly de- prived of its gravity. This fluid, therefore, is to be regarded t See note on the condensation of gases into liquids. Pressure of Fluids. 291 simply as a medium of communication between the columns AE and DF ; so that EBCF transmits to the column DF all the pressure it receives from AE ; and reciprocally it transmits to AE all it receives from DF. It is also evident that we should arrive at the same conclusions, if instead of the columns AE, DF, we substitute two forces of the same value ; hence, as the result is not affected by any inclination of the branch BC, we conclude that, if a fluid, destitute of gravity, be contained in any vessel, cmc/ Fig.198. if, having made an opening in the vessel, we apply to this opening any given pressure, the force thus exerted will diffuse itself equally in all directions. 405. Now it will be readily seen, not only that the pressure transmits itself equally in all directions, but also that it acts at each point perpendicularly to the surface of the vessel contain- ing the fluid ; for if, acting on the surface, it did not act perpen- dicularly, its effect could not be entirely destroyed by the resist- ance of this surface; there would result therefore an action up- on the parts of the fluid itself, which, as it could not but trans- 404 - mit itself in all directions, would necessarily occasion a motion in the fluid ; it would be impossible, therefore, on such a suppo- sition, for a fluid to remain at rest in a vessel, which is contrary to experience. 406. We hence conclude, that if the parts of a fluid contain- FiJ91 ed in any vessel A.BCD, open toward AD, are urged by any forces whatever, and are notwithstanding preserved in a state of equilibrium, these forces must be perpendicular to the surface AD; for if there be an equilibrium, this equilibrium would still obtain, if a covering were applied of the same figure with the surface AD ; but we have just seen, that in this case the forces acting at the surface AD must be perpendicular to this surface. 407. Accordingly, let us suppose that the forces acting on the parts of the fluid are gravity itself; we shall infer that the direction of gravity is necessarily perpendicular to the surface of tranquil fluids ; and that consequently the parts of the same, heavy fluid must bt on a level, in order to be in equilibrium > what- ever be the figure of the vessel. 29:2 Hydrostatics. Fig 192. 4 ^ 8t Let us now suppose that the vessel ABCD, being closed on all sides, is filled with a fluid destitute of gravity, and that, having a very small opening at , we apply to it any force ; it is evident that the pressure that would hence be exerted upon the plane surface represented by BC, would not depend in any degree upon the quantity of fluid contained in the vessel, nor upon the figure of the vessel ; but that, since the pressure applied at E transmits itself equally in all directions, the pressure upon BC would be equal to that exerted upon any point of the opening JC, repeated as many times as there are points in BC. 409. For the same reason, the pressure applied at E, trans- mitting itself in all directions, would tend to raise the superior surface AD^ and the force thus exerted would be for each point equal to the pressure applied at any point of the opening E ; so that the surface J1D is pressed perpendicularly from within out- ward with a force equal to the pressure employed at any point of the opening E, repeated as many times as there are points in AD. Fig. 193. 41 * Let the vessel ABCDEF, the part CD being horizontal, be filled with a heavy fluid. We say that the pressure upon the bottom CZ), arising from the gravity of the fluid, does not depend upon the quantity of fluid contained in the vessel, but simply upon the extent of CZ), and its depth below the surface AF. To make this evident, let us suppose, the line BE being hor- izontal, that the fluid contained in BCDE, is suddenly deprived of its gravity, it is evident that a vertical filament IK, of heavy particles of the fluid contained in ABEF, would exert at the point K a pressure which must diffuse itself equally throughout the whole extent of the fluid BCDE that this pressure would be 40 exerted with equal force from below upward to repel the action of each of the other vertical filaments belonging to the several points of BE ; hence the filament IK effects, by itself, an equili- brium with all the other filaments of the mass ABEF '; therefore the mass BCDE being destitute of gravity, there w r ill result no other pressure on the bottom CD, than that arising from the fil- ament IK, which transmitting itself equally to all the points of CD causes upon CD a pressure equal to that exerted at the Pressure of Fluids. 293 point K, repeated as many times as there are points in CD. Ac- F . cordingly, if we suppose the heavy fluid contained in ACDF, divided into horizontal strata, the upper stratum would commu- nicate to the bottom CD no other action than that which would be communicated by the filament a 6; and the same being true of each stratum, the bottom CD will only receive the pressure that would arise from the sum of the filaments a b, 6 c, c d, &c. ; and since this pressure would transmit itself equally to all the points of CD, it is equal to the area of CD multiplied by the sum of the pressures exerted upon some one point by the sum of the filaments 06, fee, c d, &c. ; from all which we derive the following conclusions, namely ; 1. If the fluid ACDF be homogeneous, that is, composed of parts of the same nature, of the same gravity, ta\, the pressure upon the bottom CD will be. expressed by CD X a g ; or, in other words, will be measured by the weight of the prism or cylinder which has CD for its base and a g/br its altitude. 2. If the fluid is composed of strata of different densities, the pressure upon CD will be expressed by CD multiplied by the sum of the specific gravities of each stratum ; 1 say by the sum of the spe- cific gravities, and not by the sum of the weights ; for it is not on the quantity of the fluid contained in each stratum that the pressure depends but simply on the proper gravity of each fila- ment. It is important to observe that the above propositions hold true, whether the vessel grows larger toward the top, as in the present instance, or whether it is constructed from the bottom upward, as represented in figure 195. The pressure which the fluid contained in ACDF exerts upon CD, is the same as if the cylinder ECDG were filled with the fluid, the altitude being the same in both cases. This constitutes what is called the hydro- static paradox, and is often expressed in the following words, namely ; any quantity of water or other fluid, however small, may be made to balance and support a quantity however large. The principle is well illustrated by an instrument called the hydro- static bellows ; see figure 196, in which EF, CD represents two thick boards 16 or 18 inches in diameter, firmly connected to- gether by pliable leather attached to the edges, which allows a 294 Hydrostatics. motion like that in common bellows. Instead of a valve, a pipe AB, about three feet in length, is inserted at B, either in the upper or lower part of the bellows. Now if water be poured into this pipe at A, it will descend into the bellows and gradually separate the pieces EF, CD, from each other by rais- ing the latter ; and if several weights, 200 pounds, for instance, be placed upon the upper board, the small quantity of water in the pipe AB will balance all this weight. More water being poured in, instead of filling the pipe and running over the top A, it will descend into the bellows, and slowly raise the weights; the distance between the surface of the water in the pipe and that in the bellows remaining the same as before. It is manifest from what has been said, that the upward pressure exerted upon each point of the interior surface of CD is sufficient to support a column of fluid of the same height with that contained in the tube AB, and consequently that the whole upward pressure, or weight sustained, is equal to the weight of a cylinder of water, whose base is the area of CD, and whose altitude is that of the column of water in AB above the surface of the water in the bellows or lower surface of CD. The area of the base, for instance, being a foot and a half, and the altitude three feet, the whole mass would be 4 cubic feet, and the weight sustained would be 4| X 62t or 281 \ pounds, while the quantity contained in AB, depending on the size of the tube might weigh only one fourth of a pound or any less quantity. It is obvious that instead of the gravity of the fluid in the tube AB, any other force might be employed, as the impulse of the breath, or that exerted by a stopper or piston moving in the tube AB. Thus, by means of a lever HI, a dense fluid, as water Fig.197. for instance, might be forced through the pipe CO against a large piston supposed to be accurately fitted to the cylinder FD, arid connected with the rod or bar DE. A valvej being provid- ed at F to prevent the return of the fluid, this action might be repeated ; we should thus have an engine of almost unlimited T A cubic foot of water at the temperature of 50 weighs 1000 OZ - avoirdupois or 62l lb ' t See note on the construction of valves. Pressure of Fluids. 295 power contained at the same time within a small compass, and very simple in its construction. The diameter of the large pis- ton being 1 2 inches, for example, and that of the small one work- ed by the lever H and moving in AB, only one fourth of an inch, the proportion of the two surfaces, or of the power employed to the force exerted at E, would be as T ^ to 144, or as 1 to 2304. Now it would be easy, by means of the lever HI, to apply to the small piston a force equal to 20cwt. or one ton, in which case the piston working in FD would be moved with a force of 2304 tons. This instrument is called the Hydrostatic or BramaWs Press, Mr Bramah, an Englishman, being the first person who made use of the hydrostatic principle here involved, as a substi- tute for the screw in the construction of presses.! This machine evidently belongs to the class of mechanical powers, and is es- sentially different in its nature from those heretofore described. The principle of virtual velocities, however, is equally applicable to this ; since the greater the advantage gained in point of intensity, just so much is lost in respect to velocity. Suppose, for example, that the pipe AB, filled with fluid, is 2304 inches in length, the small piston, by moving through this whole extent, and thus forcing the entire contents of the pipe into the cylinder FD, would raise the large piston only one inch ; so that while the pressure upon the small piston is to that upon the large one, in case of an equilibrium, as 1 to 2304, the spaces described in the same time, or the velocities of the two pistons, are as 2304 to 1, and the quantity of motion in each is the same. 411. Let there be two fluids NHCBFL, EFLM of different Fig.198. densities, but each being homogeneous, considered by itself, and let them be made to act against each other at FL by means of the vessel in which they are contained. They can be in equi- librium only when the altitudes EF, IK, above the horizontal plane FL which separates them, are inversely as their specific gravities. Indeed, the fluid LFBCGO being itself in equilibri- 497. um, it is necessary that NHGO should be in equilibrium with EFLM; it follows, therefore, that the upward pressure exerted t The same property of fluids is sometimes employed very advan- tageously in a crane aud in raising water from mines. 296 Hydrostatics. by the column NHGQ upon FL, should be equal to the down- ward pressure exerted upon FL by the column EFLM. Now the pressure of NHGO upon FL is equal to the weight of a prism or cylinder of this fluid which has the surface FL for its base and IK for its altitude; moreover this weight is equal to the specific gravity multiplied by the bulk; accordingly, if we call the specific gravity S, we shall have for the expression of the weight S x IK X FL. For the same reason, if we call S f , the specific gravity of the fluid EFLM, we shall have S' X EF X FL, as the expression for the absolute gravity of this fluid or the. pressure which it exerts upon FL. Therefore S X IK X FL = S' X EF X FL, or S X IK=& X EF-, whence S : S' :: EF : IK, that is, the altitudes are inversely as the specific gravities. Thus if LFBCHN were mercury, and EFLM water ; since mercury is 13,6 or nearly 14 times as heavy as water, the altitude IK would be one fourteenth part of EF, whatever be the figure of the vessel. 412. From what has been said, it will be seen that the action of fluids is very different from that of solids. Properly speaking, it is only the part ECDG which exerts its action upon the surface CD ; and in figure 1 95, the surface CD is pressed by ACDF, as it would be by the weight of the fluid contained in the cylin- der ECDG. If, on the contrary, the fluid ACDF were sudden- ly to become a solid, by freezing, the bottom would support a pressure equal to the weight of the entire mass ACDF in figure 194, and only equal to the weight of ACDF in figure 195. 413. It is necessary here to distinguish between the force or pressure exerted on the bottom CD, and that which would be sustained by a person carrying the vessel. It is clear that if the bottom CD were moveable, the only thing necessary to keep it in its place, would be an effort equal to the weight of the cylin- Pressure of Fluids. 297 der ECDG, but in order to transport the vessel, an effort would f . J94 be required equal to the weight of the entire mass of water con- tained in the vessel. This will be demonstrated in a manner still more general after we have explained the method of estimating the pressure upon oblique plane surfaces and upon curred surfaces. 414. LetACDF be the vertical section of a vessel terminated Fig.199^ by surfaces either plane or curved, and inclined in any manner 20 - to the horizon. If we imagine an infinitely thin stratum a b d c, we can suppose it destitute of gravity, and pressed by the fluid above it. Now this pressure will be distributed equally to all points of the stratum, and will act perpendicularly and equally upon each of the points of the faces a c, b d. Accordingly, as this force is equal to that which a single filament IK would cause, the pressure exerted perpendicularly upon 6 d, will be expressed by b d X IK ; and it is evident that we should arrive at the same result, if instead of b d being considered as a small straight line, we regard it as a small surface. We hence derive the general conclusion, that the pressure exerted perpendicularly upon any infinitely small surface by a heavy homogeneous fluid, has for its expression this surface multiplied by its perpendicular distance from the level of the fluid. 415. Hence the whole pressure exerted upon any plane sur- face, situated as we please, is equal to the sum of the infinitely small parts of this surface, multiplied each by its distance from the level of the fluid. If we represent these small parts by m, ?i, o, &c., and their distances respectively from the level of the fluid by AA', BE', CC', &c., according to article 76, we shall have GG' X (m + n + o -f &c.,) = AA' X m -{- BB' X n -f CO X + &c., that is, the sum of these products is equal to the whole surface multiplied by the distance of its centre of gravity from the same horizontal plane. Therefore the pressure exerted by a heavy fluid against an oblique plane surface has for its measure the product of this surface into the distance of its centre of gravity from the line of level of the fluid. Mech. 38 298 Hydrostatics. 416. As the pressures exerted upon the several points of the same plane surface are perpendicular to the surface, and conse- quently parallel among themselves, the resultant or whole pres- sure must be parallel to the components. Now as we know how to find the resultant as well as that of each of the partial pres- sures, it will be easy to determine, as we have occasion, through what point the resultant passes ; it evidently cannot pass through the centre of gravity, but must pass through some point lower Fig.201. down. It is only in the case where the surface is infinitely small that the whole pressure passes through the centre of gravity of this inclined surface. 417. In order to find the resultant of all these pressures both in a vertical and in a horizontal direction, any body, whatever its figure, may be considered as composed of an infinite number of strata, parallel among themselves, and the surface of the pe- rimeter of each stratum may be represented by a series of trap- ezoids, of which the number is infinite, when the surface is a curve. So that in order to estimate the resultant of the pressure exerted by a fluid either upon the interior sides of a vessel, or upon the exterior surface of a solid immersed in it, we must de- termine the pressure exerted upon a trapezoid of an infinitely small altitude. Accordingly, let us suppose a trapezoid ABCD, of which the Fig.2C2. two parallel sides are AB, C/), and the altitude infinitely small compared with these sides ; and let there be applied at the cen- tre of gravity G of the trapezoid perpendicularly to its plane, a force p equivalent to the product of the surface of this trapezoid into the distance of its centre of gravity from the horizontal plane XZ. To determine the effect of this force, as well in a horizontal as in a vertical plane, suppose through the line CD a vertical plane CDFE, and through the line AB, considered as horizontal, a horizontal plane AFEB. Having drawn the vertical lines CE, DF, meeting this latter plane in E and F, we join BE, AF-^ and through the direction Gp of the force />, suppose a plane A7/I cutting CD at right angles, HGK and HI being the inter- sections of this plane with the two planes ABCD, FECD, res- pectively. The plane KIH will be perpendicular to each of Pressure of Fluids. 299 the two planes ABCD, FECD, since CD is their common inter- Geom section ; lastly, from the point K, where HK meets AB, let fall 355. the perpendicular KI upon the plane 7CJ>, this line must be perpendicular to HI. These steps being taken, I decompose the force p into two others, both being in the plane KIH produced, and of which one GL is horizontal or perpendicular to the plane FECD, and the other GM vertical. Calling these two forces q and r, and form- ing the parallelogram GMNL upon* the line GJV, taken arbitra- rily as the diagonal, we shall have p : q : r : : GJV : GL : GM, :: GJV : GL : LN. But as the triangle GLJVhas its sides perpendicular respectively to those of the triangle KIH, these two triangles are similar, and we have Geom, 209. GJV : GL : ZJV : : HK : HI : IK '; and consequently p : q : r :: HK : HI : IK. nn j /7/1 Multiplying the three last terms by -X X GG', which does not change the ratio, we obtain, p : q : r : HK X + CD X GG' : HI ft D I /^i T\ We observe now, 1. That HK X - -~ --- is the surface of the trapezoid ABCD; 2. That since CE, DF are parallel, as a D [ (77) also CD, F, CD is equal to FE ; whence IK X -* - is Geom, /] 7? i Ff 178 equivalent to IK X -~4 , and consequently is the sur- face of the trapezoid AFEB ; 3. As we suppose the altitude of 300 Hydrostatics. the trapezoid ABCD infinitely small compared with the sides AB and CD, EF which is equal to CD, may be taken instead or) I rm. consequently the horizontal forces are to each other as the sides a 6, b c, &;c. Moreover, at whatever point of these faces they are applied, as these faces are of an altitude infinitely small, the horizontal forces may be considered as applied each in the hor- izontal plane a b c d ef, perpendicularly to the middle of the side which s,erves as a base to the corresponding face of the prism in question. I say to the middle, since it will readily be seen that the resultant of the pressures exerted upon the surface of any one of the trapezoids which form the surface of the stratum, must pass through some point of the line joining the middle points of the two parallel sides, and that, accordingly, the horizontal force obtained by decomposing this resultant, must meet the line join- Pressure of Fluids. 303 ing the middle points of the two opposite sides of the correspond- ing face of the prism. The problem, therefore, is reduced to finding what must take place in any polygon, when each of its sides is drawn or pushed by a force applied perpendicularly to its middle point, and represented as to its value , by this side. We shall see that they will mutually destroy each other. Let us, in the first place, consider only the two forces/? and q, F 'g-205. applied perpendicularly to the middle points of the sides AB, AC, of the triangle ABC, these forces being represented as to their values by these sides respectively. It is clear that their resultant would pass through their common point of meeting /', which, in the present case, is the centre of a circle in whose cir- G eom. cumference the points A., B, C, are situated. We say, moreov- er, that this resultant will pass through the middle point of BC, to which it will consequently be perpendicular, and that it will be represented in magnitude by BC. For, if we decompose the force p into two others, one De parallel, and the other Dh per- pendicular to BC, by forming the parallelogram D e g h we shall have, by calling these two forces e and h respectively, p : e : h :: Dg : De : Dh :: Dg : De : ge. Now, by letting fall the perpendicular AO, the triangle g e D is similar to the triangle AOB, since their sides are respectively perpendicular. Accordingly, Dg : D e : ge : : AB : AO : BO, whence p : e : h : : AB : AO : BO. But, by supposition, the value of the force p is represented by AB ; therefore that of e is represented by AO, and that of h byJ?0. If we decompose in like manner the force q into two others, the one Im parallel, and the other Ik perpendicular, to BC, it may be shown as above, that m is represented by A.O, and k by CO. The two forces m and e are therefore equal, since they are represented by the same line AO. Moreover they act in opposite directions, and according to the same line DI parallel 304 Hydrostatics. to jBC, since D, /, are the middle points of AB, AC, respectively. Consequently, they will destroy each other. The resultant then must be the same as that of the two forces h and k ; and as these are parallel, being each perpendicular to CB, their resultant must be equal to their sum, and perpendicular to J5C; that is, (1.) It will be represented by BO + 00 or by C; and (2.) being perpendicular to BC, and passing, as we have just seen, through the centre F of the circle circumscribed about ABC, it passes through the middle point BC. Fig.206. This being premised, the resultant p of the two forces p, q, will be perpendicular to the middle of DC and be represented by DC; it will accordingly be equal and directly opposite to the force r; therefore all these forces will destroy each other. The same reasoning will evidently be appli- cable, whatever the number and magnitude of the sides. Hence we derive the general conclusion, that the efforts which result in a horizontal direction from the pressure of a heavy fluid exerted perpen- dicularly upon the surface of a body immersed in it, mutually des- troy each other. 422. The pressures upon any given surface being considered by themselves, the distance at which the resultant, or the centre of pressure passes, is readily found. The forces exerted upon the several points being as the distances respectively of these points from the surface of the fluid, they are as the forces that would arise from the motion of this surface about the intersection of its plane with the surface of the fluid as an axis. But we have seen that the distance of the resultant in this case is that of the 357 centre of percussion or oscillation. Hence the distance of the centre of pressure of any given surface from the surface of the fluid, is the same as that of the centre of percussion. Accord- Fig.neJngly, the centre of pressure on the side of a perpendicular pris- 365. matic vessel, is one third of the height from the bottom. Pressure of Fluids. 305 Moreover, since the magnitude of this pressure is equal to the surface multiplied by the distance of the centre of gravity, 41 ^ the pressure on the upright side of a vessel of the form of a par- allelepiped is to the pressure on the bottom, as half the area of Fig.176. the side to the area of the bottom, or as half the height of the side to the length of the bottom, reckoned in the direction of a perpendicular to this side. When the parallelopiped is a cube, the pressure on the side is half that on the bottom, and the pres- sure upon the four sides together, double that upon the bottom. Since the pressure of any fluid is proportional to the depth below the surface, the strain upon the sides of a sluice, and the banks of a canal, must increase uniformly from the top to the bottom ; and, when this force is to be resisted by earth or mason- ry, since the strength of the materials may be estimated in a certain proportion to the weight, commonly as a third or fourth part, if AB be the height of the bank or dike, and a third of its density be to that of water, as AB to BC, by joining AC, this line will represent the proper slope. If the bank be composed of stone or brick, the base BC must be at least equal to the altitude AB ; if it be of earth, BC should exceed the height AB by one half. Let AC, BC, represent the floodgates of a canal-lock which are opened by means of the extended arms AF, BF. When we Fl S- 209 shut the gate, AC is pressed at right angles by the water with a force proportional to AC, and which, from the principle of the moment of inertia, must exert a perpendicular effort at the end -- 2 C, as AC. The strain thence produced in the direction AB 355. will be opposed by an equal and opposite effort from the gate BC. These two forces constitute the power which closes the gates. If the angle ACB be very acute, the gates would close feebly; if on the other hand ACB be too obtuse, the gates would occasion a great strain upon the sides of the sluice. This strain in the direction CA is as AC, other things being the same ; it is also as the width of the canal or as AD, and for the reason above given, inversely as CD ; that is, the force in question is as CD Mech. 39 306 Hydrostatics. J\C or, the width of the canal being constant, as ^. We have, therefore, only to find when this is a minimum. It is evidently equal to the diameter of the circle circumscribing the triangle 030"' ACB; and since the least circumscribing circle is that whose centre is 1), we infer that the angle ACB of the meeting of the gates should be a right angle. 423. Water is sometimes conveyed in pipes, which, accord- ing to what is above said, must sustain a pressure in proportion to the height of the source. There is moreover a force exerted . . upon the interior of the pipe depending upon its diameter. ADE being a transverse section of the pipe, let A be a par- ticle of the fluid pressed by two contiguous particles B, C, and kept in equilibrium by the resistance or tenacity of the sub- 48. stance of the pipe. These three forces will be represented by the three sides of a triangle formed upon their directions, or by the three sides of the similar triangle OJBC, formed by lines drawn perpendicularly to the directions of the forces. Now if we suppose BC to become indefinitely small, or that the forces exerted by B, C, are directly opposed to each other, each will be represented by the radius of the section. Thus, the height of the source being the same, and the substance of the pipe the same, its thickness ought to be in proportion to the radius or diameter of the bore. We may apply the same conclusion to cylindrical vessels generally destined to hold fluids. Large casks are required to be made stronger than small ones, in the compound ratio of their diameter and height. The same precaution is to be observed, moreover, with regard to steam pipes and steam boilers of diffe- rent dimensions, for the above reasoning will hold true equally in the case of elastic fluids, as in that of liquids. Solids immersed in Fluids. 424. Since the efforts which a fluid makes in a horizontal direction mutually destroy each other, in order to preserve a Solids immersed in Fluids. 307 body in any given position in a fluid, it is only necessary to destroy the vertical part of the pressure ; and to effect this, two things are required, namely, (1.) A downward effort equal to that of the upward pressure of the fluid ; and (2.) A coincidence of these efforts in the same vertical line. Now the vertical upward pressure or buoyancy of the fluid is equal to the weight of the portion of fluid displaced 5 hence, if the por- tion of fluid displaced weighs more than the immersed body, the body will float, and it will elevate itself until the portion of fluid an- swering to the part immersed, shall weigh just as much as the entire body. Accordingly, if when a body floats, we add to it a certain weight, or take a certain weight from it, it will sink or rise until the increase or diminution of the fluid displaced shall become equal to this new weight. If the weight added or sub- tracted be small compared with that of the body itself, the quan- tity IK, by which the section AB is depressed or elevated, will Fig.207. be so much the less, according to the smallness of this new weight compared with the extent of the section AB. When, therefore, this new weight is inconsiderable, and the section AB is large, AB, and A'B' may be considered as equal, and the difference in the bulk of fluid displaced, occasioned by the sup- posed change of weight, may be estimated by the surface AB multiplied by IK, that is, by AB X IK. Therefore if w repre- sent the weight of a cubic foot of the fluid,! w X AB X IK will express the weight of the bulk in question, the surface AB, and the altitude IK being estimated in feet. Thus, if w' be the weight added or subtracted, we shall have w x AB x IK = a/, from which we deduce IK = that is, in the case of a vessel, for example, in order to find how much a certain addition to the cargo will sink the vessel, we divide t This in the case of fresh water is at a mean very nearly and in that of sea water 308 Hydrostatics. the value of this addition a/, by the surface of the section made at the water's edge (estimated in square feet), multiplied by the weight of a cubic foot of water. When a vessel is sheathed, the bulk of the hull is aug- mented by a quantity, the method of calculating which has Gal.l22.been made known. The effect, therefore, is the same, as if the cargo were diminished by the following quantity, namely, the weight of a mass of water equal to the augmentation of the hull, minus the weight of the materials which compose the sheathing. Accordingly, it may easily be determined how much less or more the vessel would sink. 425. The weight of a body remaining the same, its bulk may be enlarged at pleasure, by forming it so as to inclose a space. There is no substance therefore so heavy that it may not be made to float. 426. Since, when the weight of a body is diminished with- out changing its bulk, it must elevate itself with an effort which can be counterbalanced only by a weight equal to that which has been taken from it, it will be seen that this upward pressure of water may be advantageously employed in raising large mas- ses ; in drawing up vessels, for example, from the bottom of bays and rivers, by attaching them to other vessels, floating above, and deeply laden with stones or water, afterward to be thrown overboard. Specific Gravities. 427. In general, if S be the specific gravity of a floating body, or, which is the same thing, the weight in ounces of a cubic foot of this body,t the matter being supposed to be uni- formly distributed, b its bulk, S' the specific gravity of the fluid, t Water is the unit to which we refer all substances, except the gases, in estimating their specific gravities. It is immaterial what bulk be used for this purpose, or whether any particular bulk, provided the two in question be equal. A cubic foot of any sub- Specific Gravities. , 309 and V the bulk of the part immersed, the weight of this body will be Sb ; and that of the fluid displaced will be S' b' ; whence we have Sb = S'&', an equation which gives A/ Sb &' = ^r; from which it will be seen, that the weight Sb of the body re- maining the same, the part immersed will always be so much the less, according as the specific gravity pf the fluid is greater. Moreover, this same equation is equivalent to the proportion, b : V :: S' : S- that is, the bulk of the body is to that of the part immersed, in- versely as the specific gravity of the body to that of the fluid. 428. If the immersed body weigh more than an equal bulk of the fluid, it must sink ; and it can be retained only by a force equal to the excess of its weight above that of an equal bulk of the fluid. Now if we represent the specific gravity of the fluid, stance compared in point of weight with the same bulk of water would evidently give the same result as any other measure ; and correct results once obtained, we can substitute such other balks as we choose. The cubic foot has this particular advantage, that, if the measures be taken at the temperature of 50 of Fahrenheit, the same numbers which express the specific gravities, the decimal point being removed three places, or the whole being multiplied by 1000, give the absolute weight in avoirdupois ounces ; since a cubic foot of water at this temperature weighs 1000 ounces. It is usual, however, in determining specific gravities, to refer to the tempera- ture of 60, at which a cubic foot of water weighs 62,353 lb * or a little less than 1000 ounces. In the more accurate experiments, however, upon this subject, the absolute weight is first ascertained, and the cubic inch is taken as the measure, this bulk of distilled water at the tem- perature of 60, being estimated at 252,525 grains. In France the temperature preferred is that of the maximum density of water, or 39, 39. The atmosphere is employed as the unit in estimating the specific gravities in gases. 310 Hydrostatics. and that of the body immersed in it, by S', S, respectively, and the bulk of the body by b, we shall have, Sb S' 6 for the excess of the weight of the body above that of an equal bulk of the fluid. Hence, if we suppose that this body is retained by means of a thread, attached to the beam of a balance, and that w is the weight with which it is kept in equilibrium ; we shall have ro =S6_ S'b, from which we obtain, Now Sb is the absolute weight of the body, and w its weight in the fluid ; knowing, therefore, the absolute weight of a body, and its weight when immersed in a fluid, we easily determine the ratio of their specific gravities, by dividing by the absolute weight of the body the difference of these two weights If, for example, the absolute weight of a body be 6 ounces, and its weight when immersed in water 5 ounces, we divide the absolute weight 6 by the diffe- rence, and the quotient f shows that the specific gravity of the body is 6 or is to that of the fluid as 6 is to 1 . In other words, since the loss of weight sustained by a body on being immersed in a fluid is equal to the weight of its own bulk of that fluid, by proceeding as above, we shall have the weight of equal bulks of the substances in question, and consequently their relative weights or specific gravities. If the solid substance whose specific gravity is sought be light- er than water with which it is to be compared, it may be made to sink by attaching to it a heavier body, whose water-weight is known ; then, by subtracting this from the water weight of the compound body, we shall have that of the body in question. When the substance to be weighed is soluble in water, as com- mon salt, for instance, it may be covered with a thin coat of melted wax, which is very nearly, and may be made exactly of the same specific gravity with water. The body will thus be protected, and the loss of weight, on being immersed, will be the same as if water occupied the place of the covering. Specific Gravities. 311 Another method is to determine the specific gravity of the body with reference to some liquid, as alcohol or oil, in which no solution takes place, and whose specific gravity, compared with water is known. We have then simply to use the propor- tion, as the specific gravity of water, is to that of the fluid used so is the result above found to the result sought. 429. If the same body be immersed in another fluid, whose specific gravity is denoted by S", and w' represent the weight necessary to counterbalance it ; as in the former case we had w = Sb S' 6, we shall have, in like manner, w' = Sb 5" b. Now these two equations give = Sb w and S" b S b whence, dividing this last by the preceding, we obtain #~ " " Sb TO* Knowing, therefore, the absolute weight Sb of a body, and its weight w' in a fluid, and its weight w in any other fluid, we easily determine the ratio -^ of the specific gravities of these two fluids. By taking a solid glass ball and grinding it to such a size that it shall lose just a thousand grains, for instance, when weigh- ed in distilled water, at the assumed temperature, then by observ- ing the loss of weight / in grains, sustained on being weighed in any other fluid, we shall have 1000 : / :: 1 : 5 = 1000 Thus the specific gravity of the fluid in question is obtained by dividing / by 1000, or removing the decimal point three places to the left. If a larger number of decimal places were required, we should employ a ball weighing 10000 or 100000 grains, and divide accordingly. 430. The precious metals being heavier than those of less value, when they are debased, it must be by means of some sub- stance of less specific gravity, and the compound will conse- quently be lighter than the metal it is intended to represent. 312 Hydrostatics. Spirits, on the other hand, are lighter than the liquids with which they are adulterated, so that, in each case, the specific gravity becomes the test of purity. We have, hence, a curious and important problem, namely, knowing the specific gravity of the two ingredients that compose a compound, and the specific gravity of the compound, to find the proportion of the ingredi- ents. This proportion may be estimated by weight, as is usually the case with respect to metals, or by measure, which is the com- mon method where liquids are concerned. The weight of the in- gredients being sought, we put x for that of the denser, and y for that of the rarer or lighter, c representing that of the compound, and S, S', S", denoting their specific gravities respectively. We have x -f y = c ; also, on the supposition that the bulk of the compound is equal to that of the constituent parts, the specific gravity would be the same whether the compound were consid- ered as one entire mass, or as composed of two distinct parts, and the weight divided by t>he specific gravity in the one case would be equal to the weight divided by the specific gravity in the other, that is, To find #, we substitute in this equation the value of y deduced from the first, namely, y = c #, which gives x c x c ~S * S 77 " " S"' or x (S< S) c_ jSc _ _ (S 1 S") ~~ r ~ ~ " ' ~ """" C ' (S' S") _ (S f S") S _ (S" S')S ~S^ 7r ~ " (S' S) S" C '' (SS'}S" In like manner, we have _ (S . S") S' - ( S S') S" Specific Gravities. 313 Suppose a compound of gold and silver to have a specific gravity equal to 14, that of pure gold being 19,3, and that of pure silver 10,5 ; we have S 19,3, S' = 10,5, S" = 14 5 and consequently, S" S' = 3,5, S S' = 8,8, S S" = 5,3 ; whence, by the formula, the proportional weight of gold will be 3,5 X 19,3 67,55 _ 8,8 X 14 " 123,2 The formula for the value of y gives for the proportional weight of silver 5,3 X 10,5 J35,65 __ Q45 8,8 X 14 " 123,2 whatever* c may be in each case. Where the proportion of the ingredients by bulk or measure, instead of by weight, is sought, as in the case of liquids, the process is shorter. Calling 6, &', the bulks of the ingredient corresponding to the specific gravities , $', as the weight is equal to the bulk multiplied by the specific gravity, and the weight of the compound is equal to the sum of the weights of the ingredients, we have S" X (b + 60 or S" b + S" I' = S b + S' &', whence or I (S" S) = V (S f S") ; that is, 6 ; : V : : S' S" : S" S. Suppose a certain spirituous liquor to have a specific gravity equal to 0,93, that of highly rectified alcohol or pure spirit being Meek. 40 314 Hydrostatics. 0,83 and water 1 ; we have in this case S = 0,83, S' = 1, and S" = 0,93, which gives, S' S" = 0,07, S" S = 0,1, and by substitution, 6:6':: 0,07 : 0,10 ; that is, the proportion by measure of pure spirit to that of water, is as 7 to 10. In what is called proof spirit, it is required that the constituent parts of water and spirit, should be equal, which gives a specific gravity of 0,925, and the degree above or below proof is usually denoted by the number of gallons of water to be added to or taken from 100 gallons of the liquor in question, to bring it to the required standard or proof. But more expeditious methods have been devised for deter- mining the proportion of alcohol in spirits. It is evident that if a great variety of mixtures of known proportions were prepared, and hollow glass balls or beads were so adapted to each in res- pect to specific gravity, as just to remain suspended in any part of the fluid ; by marking the known proportion of alcohol and spirits in each case on the respective balls, similar unknown mixtures might be readily examined as to their proportion of alcohol. It would only be necessary to make trial of different balls until one was found which would tend neither to rise nor sink when immersed in the fluid. But instead of a large number of such balls, a single one with a long graduated stem and move- able weights, as represented in figure 211 might be employed to the same purpose. The ball itself should be of such a specific gravity as just to sink to the commencement of the graduation on the stem when suspended in alcohol, and the weights to be at- tached should be such as to cause the ball and some part of the stem to sink in any mixture of less specific gravity than pure water. Then the weight together with the divisions of the stem, which serve to subdivide the difference between two weights, would indicate, like the separate balls, the specific gravity of the Specific Gravities. 315 liquid and consequently its proportion of alcohol. An instru- ment so constructed, is called a hydrometer.] 432. In making use of the hydrometer, and in experiments generally upon specific gravities, there are several particulars to be taken into consideration. (1). Since all bodies expand with heat and contract with cold, it will be perceived that the specific gravity of bodies is modified by temperature. Thus bodies are specifically heavier in winter than in summer, and the same spirit would indicate dif- ferent proportions of alcohol at different seasons, regard not be- ing had to this circumstance. Accordingly a thermometer is a necessary appendage to a hydrometer, and the correction for temperature is applied by means of a moveable scale, containing the degrees of the thermometer, sliding upon another scale on which are placed the numbers of the several weights, including the stem. (2). When two substances are mixed together, there is often a mutual penetration of parts, whereby the specific gravity is increased, and there would seem, by the foregoing methods, to be a greater proportion of the heavier ingredient than actually ex- ists. Thus a pint of water and a pint of alcohol do not make a quart of liquid. The defect is sometimes a 40th part. On the other hand, the bulk in certain cases is augmented by compound- ing. A cubic inch of tin mixed with a cubic inch of lead will make a compound exceeding two cubic inches. No accurate allowance can be made for such changes ; consequently the methods above given, become in a degree defective, and the results to a certain extent uncertain. (3). The number of ingredients may exceed two, or the na- ture of one or more of the ingredients may be unknown. In all such cases, the problem is in its nature indeterminate. 433. Questions sometimes occur, especially in chemical re- searches, the reverse of those we have been considering, namely, t There is a great variety of instruments of this kind, all depend- ing on the same general principles. They are known also by a va- riety of names, as areometer, gravimeter, alcoholometer, pese-liqueur^ essay-instrument^ &c. 5 1 6 Hydrostatics. to find the specific gravity of a compound, by means of that of each of the ingredients ; and the rule which has generally been given, is to take the arithmetical mean of the specific gravities of the ingredients. It will be observed that, the greater the weight of a body, other things being the same, the greater the specific gravity ; also that the less the bulk, other things being the same, the greater the specific gravity ; that is, the specific gravity of one body is to that of another, as the weight of the first divided by its bulk, is to the weight of the second divided by its bulk, and hence the mean specific gravity of the two will be found by dividing the sum of the weights by the sum of the bulks. Thus, from which we obtain, b =^, and &' = ^- ; j >V also, 7, _l- A/ w J_ w ' W& +&' S + ' 6 ~"S + ^ 5S Hence, calling M the mean specific gravity sought, from the equation M = , j~ , above sho reasoning, we obtain, by substitution, equation M = , j~ , above shown to be true, by general M w S 1 a.' S = i , S-+ ' S Let gold, for example, of a specific gravity 19,36, be alloyed in equal weights with copper of a specific gravity 8,87, we shall have . (1 + 1) 19,36 8,87 _ 2 X 171,7232 ~~8^7 -f- 19,36 28,23 " whereas the arithmetical mean is Specific Gravities. 317 434. It w,ill be recollected that the specific gravity of a solid body is determined by dividing its absolute weight by the loss sustained on its being weighed in pure water, and the specific gravity of a fluid, whether liquid or gas, by comparing the loss of weight sustained by the same solid when weighed in pure water with that sustained on its being weighed in the fluid in question. According to this method, therefore, the absolute weight of a body is necessary in both cases. But it is difficult to obtain this unaffected by the fluid of the atmosphere in which we are im- mersed. The usual operation of weighing, except where the weight itself and the thing weighed happen to be equal in bulk, must, from what has been said, be more or less incorrect. But the specific gravity of the atmosphere being ascertained, on the supposition that it is always the same, or such as to admit of its changes of density being determined at all times, allowance may be made for its effect on the weight of bodies, more especially as it is an exceedingly light fluid, and scarcely requires to be noticed, except in very nice experiments, or where the bulks of bodies are very considerable. The best way of determining the specific gravity of the atmosphere, and of gases generally, is to weigh directly a vessel of known dimensions, when empty and again when filled with the fluid in question.! It is thus found that a vessel of three hundred cubic inches, for example, weighs 92,4 grains more when filled with air in its ordinary state, than t The vessel should be of considerable size, that is, sufficient to contain at least three or four hundred cubic inches. It might be of a globular shape, as represented in figure 212, with a narrow neck and nicely fitted stop-cock C. Its capacity would be best ascertained by filling it accurately with mercury, and then pouring the liquid into a prismatic vessel, which might be easily measured. The air might be expelled also in the same way, by connecting with the neck AB a tube of about three feet in length, and suffering the mercury to discharge itself through this tube, held upright with its lower end immersed in the same liquid. When the mercury had left the ball, upon turning the stop-cock we should effectually ex- clude the air ; but there would remain a small portion of the vapour of mercury. Ordinarily the air is exhausted by means of an air- pump, and, although the air cannot thus be wholly withdrawn, the small proportion which is left may be measured and allowance made accordingly, as will be shown hereafter. 318 Hydrostatics. it does when empty. This divided by 300, or f |'o 4 , gives for the absolute weight of a cubic inch of air 0,308 parts of a grain. By dividing this by 252,525, the weight in grains of a cubic inch of water, gives 0,00122 or T | 7 nearly for the specific gravity of common air, at the surface of the earth in its mean state of den- sity and moisture, the temperature being that to which specific gravities are generally referred by English philosophers, name- ly, 60 of Fahrenheit. Hence bodies weighed in air lose at a mean T 7 part of the weight lost on being weighed in water. Accordingly, if we weigh a body in air and increase this weight by 8"|o f tne difference between the air and water weight, we shall have the absolute weight very nearly ; that is, if w be the absolute weight, a/ the air weight, and w" the water weight, we shall have w = w f + j^ (w' w") very nearly.! If it were proposed to find how much a solid ball must weigh in the air in order that its absolute weight may be 1000 grains, from the equation 1000 = uf -f T io (w' w"), we should have or, / = 1000 ^ (w' "), w " being 328, for example, w' = 1000 0,4 = 999,6. t To be strictly correct, the formula should be ' + T*T (> *>"), but the object of this formula is to find tw, and when we have ol> tained it nearly, we may substitute this value, and thus approximate the true value of w to any degree of exactness. But generally speaking, the correction derived from the second approximation is very small, Thus, in the example that follows above, | (w 1 w") is 0,4 grains, and the second approximation would give only T ^ of 0,4 grains or 0,0005 of a grain. Where the results are intended to be ve- ry accurate, the coefficient F should be corrected for the particu- lar state of the atmosphere at the time of the experiment, the meth- od of doing- which depends upon instruments to be described here- after. Specific Gravities. 319 435. According to what is above laid down, the specific 4 ^ J - gravity of a body multiplied by its bulk gives its weight. Now the density multiplied by the bulk gives what is called the mass of the body, which is proportional to its weight. Hence the specific gravity multiplied by the bulk, is proportional to the density multiplied by the bulk; therefore the specific gravities of bodies are proportional to their densities. Thus 5, S 7 , being the specific gravities, w, a/, the weights, 6, 6 X , the bulks, A, A 7 , the densities, and m, w 7 , the masses of two bodies, Sb = w, and Sty = a/; also, A b = m : A' b' =. m' : : w : a/ ; whence A 6 : A' b' : : S b : S 7 6 7 , or, A : A 7 : : 5 : S 7 . By employing the same unit in both cases, as water, for ex- ample, at the same temperature, the specific gravities would be equal to the densities. 436. If fluids of various densities, and not disposed to unite by any chemical affinity, be poured into a vessel, they will arrange themselves in horizontal strata, according to their respective densities, the heavier always occupying the lower place. This stratified arrangement of the several fluids will succeed, even though a mutual attraction should subsist, provided only that its operation be feeble and slow. Thus, a body of quicksilver may be at the bottom of a glass vessel, above it a layer of concen- trated sulphuric acid, next this a layer of pure water, and then another layer of alcohol. The sulphuric acid would scarcely act at all upon the mercury, and a considerable time would elapse before the water sensibly penetrated the acid, or the al- cohol the water. Bodies of different densities might remain suspended in these strata. Thus, while a ball of platina would lie at the bottom of the quicksilver, an iron ball would float on its surface ; but a ball of brick would be lifted up to the acid and a ball of beech would swim in the water, and another of cork might rest on the top of the alcohoK 320 Hydrostatics. Specific Gravities of the more remarkable Substances. Platinum, purified, . . hammered, . . laminated, . . drawn into wire, Gold, pure and cast, . . hammered, . . . Mercury, Lead, cast, Silver, pure and cast . . hammered, . . Bismuth, cast, .... Copper, cast, .... wire, Brass, cast, wire, Cobalt and nickel, cast, . Iron, cast, Iron, malleable, . . . Steel, soft, hammered, . . . Tin, cast, Zinc, cast, Antimony, cast, .... Molybdaenum .... Sulphate of barytes . . . Zircon of Ceylon, . . . Oriental ruby, .... Brazilian ruby, .... Bohemian garnet, Oriental topaz, .... Diamond, Crude manganese, . . . Flint glass, Glass of St. Gobin . . . Fluor spar, Parian marble, .... Peruvian emerald . . . Jasper, Carbonate of lime, . . . Rock crystal, .... Flint, Sulphate of lime, . . . Sulphate of soda, . . . Common salt, .... Native sulphur, .... Nitre, Alabaster, Phosphorus, 1,19 0,92 0,82 0,74 19.50 Plumbago, 1,86 20,33 Alum, 1,72 22,07 Asphaltum, 1,40 21,04 Jet, 1,24 19,26 Coal, from . . . 1.24 to 1,30 10,36 Sulphuric acid, .... 1,84 13,57 Nitric acid, 1,22 1 1,35 Muriatic acid, .... 10,47 Equal parts by weight of 10.51 water and alcohol, . . 0,93 9,82 Ice, 8,79 Strong alcohol, . . . 8,89 Sulphuric aether, . . 8.40 Naphtha, 0,71 8,54 Sea water, 1,03 7,81 Oil of sassafras, ... 1,09 7,21 Linseed oil, 0,94 7,79 Olive oil, 0,91 7^83 White sugar, 1,61 7,84 Gum arabic and honey . 1,45 7,30 Pitch 1,15 7,20 Isinglass, 1,11 4,95 Yellow amber, .... 1,08 4,74 Hen's egg, fresh laid, . 1,05 4,43 Human blood, .... 1,03 4.41 Camphor, 0,99 4,28 White wax, 0,97 3,53 Tallow, 0,94 4.19 Pearl, 2,75 4,01 Sheep's bone, .... 2,22 3,50 Ivory, 1,92 3,53 Ox's horn, 1,84 2,89 Lignum vitae, 1,33 2,49 Ebony, 1,18 3,18 Mahogany, 1,06 2,34 Dry oak, 0,93 2,78 Beech, 0,85 2.70 Ash, 0,84 2.71 Elm,. . . . from 0,80 to 0,60 2,65 Fir, . . . from 0,57 to 0,60 2,59 Poplar, 0,38 2,32 Cork, 0,24 2.20 Chlorine, 0,00302 2,13 Carbonic acid gas, . . 0,00164 2,03 Oxygen gas .... 0,09134 2,00 Atmospheric air . . 0,00122 1,87 Azotic gas, .... 0,00098 1,77 Hydrogen gas, . . . 0,00008 Spirit Level. 321 It will be seen by the foregoing table, that there are gases much lighter than the atmosphere ; accordingly, if a large quan- tity of one of these fluids were confined by a thin covering, as oiled silk, it would rise in the atmosphere as a cork does in water until it reached a region of the atmosphere of the same specific gravity with itself. It is upon this principle that balloons are construct- ed. They were at first filled with air rarefied by heat, and a fire was supported in a car placed under the balloon for this purpose. Hydrogen was afterward discovered, and proved to have a specific gravity only T \ of that of common atmospheric air. This gas, is the lightest of known fluids, and will consequently require a less bulk for a given buoyancy than any other. It is universal- ly employed in the construction of balloons. Spirit Level. 436. This instrument consists of a tube AB, nearly filled Fig.213. with alcohol or ether, the remaining space CD being occupied by a bubble of air. Being attached to the vertical or horizon- tal circle of a theodolite or other instrument, it serves to deter- mine the horizontal position of a line or plane, since if either end of the tube is higher than the other, the bubble of air, from its specific lightness, will indicate it by tending to the higher part. It is usual, where great exactness is required, to make the tube slightly curved, the curvature being circular ; the bubble will then move more readily, settle itself with more certainty, and describe equal spaces by equal changes of inclination. A good spirit level will exhibit a movement of more than half an inch for each minute of inclination, and the bubble will sensibly alter its position by a change of five seconds in the inclination. In such a tube, the radius of curvature will be about 150 feet. But lev- els have been rendered still more delicate. Lalande speaks of one, filled with ether, the bubble of which passed over fourteen inches by equal spaces of one tenth of an inch for every second. The radius of curvature was consequently 1719 feet or nearly one third of a mile.t t The number of seconds in a circle is 360 X 60 X 60 = 1296000, Mech. 41 322 Hydrostatics. Of the Equilibrium of Floating Bodies. 437. In order that a heavy body may be in equilibrium on the surface of a tranquil fluid, it is necessary that its weight should be less than that of an equal bulk of the fluid. There is an exception to this rule, however, in the case of very small bodies, the particles of which are of a nature not to exert any attraction upon the particles of the fluid, or whose attraction is much less than that of the particles of fluid for each other. In this case the fluid is depressed below its level, forming a little hollow about the floating body, which may be regarded as mak- ing a part of the bulk of the body ; and on account of this aug- mentation, it is evident that the body may float, although its spe- cific gravity should be greater than that of the fluid. It is on this account that a needle smeared with tallow may be made to float on water. Small globules of mercury also are supported in a similar manner. To render this effect of no avail, we have only to suppose the floating bodies so large, that the void space which may be formed about them (and which is always very small) may be neglected when taken in connection with a bulk so much greater. 438. The weight of a body being smaller than that of an equal bulk of the fluid, the body will sink till the weight of the fluid displaced becomes equal to that of the body ; and when these two weights are thus equal, the body will be in equilibri- um, if its centre of gravity and that of the fluid displaced, or the centre of buoyancy, are situated in the same vertical. With respect to homogeneous bodies, the centre of buoyancy coincides with that of the immersed part of the body ; and that the weight of the fluid displaced may be equal to that of the body, it is nec- essary that the densities should be in the inverse ratio of the Calling these tenths of an inch, we have, by dividing by 10 and by 12, 10800 feet for the absolute length of the circumference ; whence, 2;r or 6,2832 : 1 :: 10800 : 1719. In the best spirit levels, the requisite curvature is effected and rend- ered true by grinding with emery. Equilibrium of Floating Bodies. 323 bulks, or that the bulk of the immersed part should be to the entire bulk of the body, as its density is to that of the fluid ; it 427. follows, therefore, that the determination of the positions of equi- librium of a homogeneous body, placed on the surface of a fluid of a given density greater than that of the body, is reduced to a problem of pure geometry which may be very simply stated. It is required to cut the body by a plane in such a manner, that . the bulk of one of its segments shall be to that of the whole body in a given ratio, and that the centre of gravity of this segment and that of the body shall be situated in the same perpendicular to the cutting plane. 439. There are different kinds of equilibrium depending upon the form and position of the floating body. With respect to the sphere, for example, provided its density be less than that of the fluid, it will remain in equilibrium in any position whatev- er, since the centre of gravity and that of buoyancy continue to be in the same vertical. This will be the case, also, with res- pect to solids of revolution generally, on the supposition that the axis remains horizontal. Such an equilibrium is called an equi- librium of indifference. But when, from the form of the solid or its relative density, it tends, upon being inclined a little, to return to its position, the equilibrium is said to be stable. On the other hand, if its tendency after a slight inclination is to depart from its first position, the equilibrium is denominated unstable. 440. With respect to the different positions of equilibrium of the same solid, there is a remarkable property which may be demonstrated independently of any calculation. Let us suppose that the body in question is made to turn about a moveable axis which is kept constantly parallel to a fixed and horizontal straight line, and that it is made to pass in this way successively through all its positions of equilibrium in which the axis has this direc- tion ; we say that the positions of stable and unstable equilibrium will succeed each other alternately, so that if the body be mov- ed from a position of stable equilibrium, the next position will be unstable, the third stable, and so on till it has returned to its first position. Indeed, while the body is yet very near its first position, it will tend to return to it, this position being supposed to be stable ; 324 Hydrostatics. . but the tendency thus to return will gradually diminish as it revolves, till after a time the body will incline the other way, but before this tendency changes its sign (to borrow an expres- sion from algebra), there will be a position in which it will be nothing, and in which the body will neither incline to return to its first position, nor to depart from it ; this, therefore, will be its second position of equilibrium. Now we see that within this part of its revolution, the body tends to return to its first posi- tion, and consequently to depart from the second. Beyond this point, the body tends to depart from its first position, and at the same time from the second ; therefore the second position of equilibrium is not stable, since on each side of it the body tends to depart from it. Upon its passing this position, its tendency to depart from it diminishes continally, till it becomes nothing ; and beyond this the body tends to return toward its second position. The point where this tendency is nothing, is a third position of equilibrium, which is evidently stable ; for on each side of it, the body tends to return to it, either approaching to- ward or receding from its second position. If the third position is stable, it may be shown by the same kind of reasoning that the fourth is not, and that the fifth is, and so on. Thus, when the body returns to its first position, it will have necessarily passed through an even number of positions of equilibrium, alternately stable and unstable. 441. It is important to be able to distinguish a stable position of equilibrium in a floating body from one which is not so. In Fi "14 or ^ er to ^ s ' ^ et us su PP ose a body which admits of being divid- ed by a vertical plane HFI into two parts perfectly similar, both as to form and density. Let us suppose, moreover, that this body is made to depart from its position of equilibrium, in such a manner that this section HFI remains vertical, and that after having thus disturbed it, we leave it to itself without impressing upon it any velocity; in this way the section HFI will remain in the same vertical plane, during the whole motion of the body, for the two portions being perfectly similar in all respects, there is no reason why it should ever depart from the vertical plane in which it was supposed to be first situated. For the same reason, the centre of buoyancy will always be in the section HFI, as Equilibrium of Floating Bodies. 325 well as the centre of gravity. Let G, then, be the centre of gravity ; and the position being that of equilibrium, let B be the centre of buoyancy, and HI the intersection of the level of the fluid with the plane HFI, or the water-line ; in this position, the straight line GB, which connects the two centres, is vertical, and consequently perpendicular to the straight line /// ; it inclines generally when the body is made to depart from this position, and at the same time, the centre of buoyancy, and the water line, change their position upon the plane H FL I will, suppose there- fore, that this centre is the point B' ', and this line the straight line H'l', when the equilibrium has been disturbed; the forces which will tend to put the body in motion are the weight of the body which is directed according to the vertical GV drawn through the centre of gravity G, and the resultant of the vertical pressures of the fluid upon the surface of the body ; this result- ant is the buoyancy of the fluid, and is equal to the \vtight of the fluid displaced, and is exerted at the point B' its centre of j3 : gravity, in the direction contrary to that of gravity, or according to the vertical B'Z. This vertical and the inclined straight line GB being in the same plane, will cut each other in a certain point M called the metucentre. it is on the position of this point with respect to the centre of gravity G, that the stability of the equilibrium depends. The point M may be taken for the point 43 of application of the buoyancy of the fluid, which will then be exerted according to the straight line MZ ; the body will there- fore be acted upon by two parallel and contrary forces, applied at the extremities cf the straight line GB'. It is now proposed to determine in what direction the body will move, and whether these forces will tend to restore it to its position of equilibrium, pr to make it depart further from this position. 442. In the first place, if they be unequal, they will produce a motion of oscillation in the point G. For the centre of gravity ought 124. to move just as if the two forces were applied directly at this point ; therefore, the initial velocity being nothing, its motion will be in a vertical straight line, and in point of magnitude equal to the excess of the greater of the two forces over the less. If at the commencement of the motion, the weight of the body exceeds the buoyancy of the fluid, the point G will begin to descend ; its mo- tion will at first be accelerated, but cccording as the body sinks 326 Hydrostatics. in the fluid, the portion displaced will be greater and greater, and, consequently the buoyant effort will increase, till at length it will become equal to the weight of the body. The point G will still continue to move on in the same direction by virtue of its acquired velocity, but then the buoyancy of the fluid exceed- ing the weight of the body, its velocity will be retarded contin- ually, till finally the downward motion of G will cease, and then it will begin to return toward its first position, arid thus it will continue to oscillate till the motion is entirely destroyed by the resistance of the fluid. The extent of these oscillations will be smaller, ac- cording to the difference at the outset between the weight of the body and that of the fluid displaced, compared with the weight of the body. If the body be but little removed from its position, of equilibrium, this difference will be small, the extent of the oscillations will consequently be small, and will not materially affect the stability of the body. 443. During these oscillations of the point G, the body will turn about this centre, in precisely the same manner as if it were 137. fixed ; its motion of rotation will be produced, therefore, by the buoyancy of the fluid, which acts at the point M according to the direction JI/Z; and its state of equilibrium will be stable or unstable, according as the straight line GB tends to approach to, or recede from, the vertical. Now it is evident from inspection, that the buoyancy of the fluid will tend to restore the straight line GB to its vertical position, whenever the point M is above the point G ; on the other hand, if M or the metacentre is below the point G, as at JW, for instance, the buoyancy of the fluid, which will then be exerted according to M'Z', will cause the straight line GB to depart further from a vertical position, and tend to upset the floating body. Therefore, when the metacen- tre is below the centre of gravity, the equilibrium is unstable ; and, on the other hand, when the metacentre is above the centre of gravity, the equilibrium is stable ; at least with respect to all positions in which the plane HFI continues vertical. If, in a particular case, the metacentre coincides with the centre of grav- ity, there will be no tendency to turn one way or the other, and the straight line GB will remain stationary, whatever inclination be given to it. Equilibrium of Floating Bodies'. 327 444. When the form of the floating body is known, on the supposition that its position is very near that of equilibrium, it will be easy to determine the place of the metacentre, or rather to determine whether this point is above or below the centre of gravity of the body. Let us suppose, for instance, that this body is a homogeneous horizontal cylinder of an elliptical base, and of half the density of the fluid ; let HFIA be a vertical section Fig.215, made at equal distances from the two bases ; in the position of equilibrium, one of the two axes will be vertical; and as half of the bulk will be immersed in the fluid, it follows that the other axis will coincide with the water-line, and will represent the level of the fluid. The vertical axis JlF is the transverse in fig- ure 215, and the conjugate in figure 216. Now we say that in the first case, the metacentre is below the centre G of the ellipse, which is also the centre of gravity of the cylinder, and that it is above it in the second case. Draw through the point G, a straight line A'F' making a very small angle with AF '; let us now suppose th;it the axis AF is inclined till A'F 1 becomes vertical, and that at the same time we raise or depress a very little the centre of gravity G, so that the level of the fluid shall become the straight line H'P, perpendic- ular to the straight line A'F' at the point G 1 . In this position, the part H'F'l' of the ellipse HFIA will be immersed in the fluid ; and this part is divided into two unequal portions H'F'G' and I'F'G 1 by the straight line G'F'. Now it is evident that the centre of buoyancy will be found in some point J3', situated in the greater of these two portions, whence it is evident by looking at the two figures, that B'M parallel to the straight line F'A', will meet the straight line FA at the point JW, below the centre of gravity in the first figure, and above it in the second. Thus the cylinder which we are considering is in a stable or unstable position of equilibrium, according as the conjugate or transverse axis of its base is vertical. Supposing the body to turn about the horizontal straight line which joins the centres of the two bases, it will pass successively through four positions of equilibrium, which will be alternately stable and unstable, agree- -ably to the general position already advanced. 328 Hydrostatics. 445. To fix, in a few simple cases, the dimensions of the sol- id and its relative density to that of the fluid required for a par- ticular state of equilibrium, let the body in question be a ho- mogeneous parallelepiped, placed vertically in the fluid ; and let Fig.217. D* '*E be a section of this body through the axis parallel to one of its faces. The solid will evidently sink till the immersed part 42 ' 7 - JVF shall be to the whole height AF, as its density is to the density of the fluid ; and its centre of gravity and that of buoy- ancy will be G and B, the middle points respectively of the axis and of the depressed portion JVF. Suppose now that the body is inclined a little, shifting its water-line from the position HN1 to jff'JV/', the centre of buoyancy, changing from B to B', will des- cribe a small arc of a circle, which for the extent under considera- tion may be regarded as a straight line, and B' will be raised by a quantity which will be to the altitude PO of the centre of grav- ity of the triangle 7JV/', as the area of the rectangle JV/F is to that of the triangle 7JV/', that is, as JVF : \ II'. Moreover the horizontal motion of B will be to JVP or f JV7 in the same pro- portion. Whence JVF : 177' : : f JV/ : BB> = ** X 7// . 3 NF But, by similar triangles, A/7 v 77' : BM:: IP : JV7; accordingly we have, for the height of the metacentre above the centre of buoyancy, Let ./-IF, the height of the parallelepiped, be denoted by /*, its breadth or thickness HI by a, and its density or specific gravity by A. When the metacentre coincides with the centre of gravity, and the solid floats indifferently in any position, BM n r \TT? is equal to BG or to - ; that is, (water being the fluid in question,) since 1, the density of the fluid is to A, the density ef the solid, as AFoY h is to JVF or A /*, we shall have Equilibrium of Floating Bodies. 329 2 a 2 12 A h 2 12 A 2 ft 2 , er a 2 A 2 _ A -- . Qh* This being resolved after the manner of an equation of the sec ond degree, gives 2 a 2 If the parallelepiped become a cube, then a = ft, and we have S = * JS = * -^ = ' 5 ' 29 nearly ; that is, the two densities are 0,79 and 0,21 nearly. Between these limits there can be no stability ; but above 0,79 or below 0,21 the equilibrium becomes more and more permanent. Hence a cube of beech will float erect in water, while one of fir or cork will overset ; yet all these three cubes will stand firmly when placed upon the surface of mercury. We restrict ourselves, in this illustration, to cubes, because we cannot 'apply the remark to parallelepipeds generally. A stable equilibrium depends, as will be inferred from what has been said, not only upon the rel- ative densities of the solid and fluid, but also upon the propor- tion between the horizontal and vertical dimensions of the solid. In order to ascertain this proportion in the case of parallelopi- peds, and on the supposition of a density equal to half that of the fluid, we have only to put equal to zero the radical part of the above general formula, and we shall have 3 A 2 2 a 2 ; accordingly H, and h - = i nearly. Whence, approximative^., Mech. 42 330 Hydrostatics. ]2/i = 100, or h : a :: 10 : 12, that is, a parallelepiped of half the density of the fluid, and hav- ing its height te one side of a square base, as 10 to 12, would float indifferently. 446. But if the relative density of the parallelepiped were either greater or less than |, its equilibrium would become stable. Thus, if we suppose A equal to J, we shall have the distance of the metacentre above the centre of buoyancy, as follows, namely, BM = jfrh = 12 x ! f x 10 = " = 3 ' 6 inches ' and for the distance of the centre of gravity above the centre of buoyancy, BG = h ^~^ = fc = J 10 = 3,3 inches; so that the centre of gravity is about 0,3 of an inch below the metacentre. Therefore the equilibrium would be stable. . In like manner, if we substitute f instead of ^ in the above equations, or, which is the same thing, take half of each of the above results, we shall have half of 0,3, or 0,15 of an inch, for the distance of the centre of gravity below the metacentre. 447. These principles are well illustrated by the masses of ice which appear on the rivers of the colder climates at the opening of spring. Being ordinarily much broader than they are thick, they have a stable equilibrium in their natural position with their broad surface horizontal. But when by striking against each other, or by passing over a fall, they are thrown up sidewise, their equilibrium becomes unstable, and they soon return to their former position. Moreover a piece of ice of a cubical form will still preserve its balance, since its specific grav- ity does not come within the limits already pointed out, of an unstable equilibrium.! t The specific gravity of ice is 0,92, or compared with sea-water as unity 0,89. Equilibrium of Floating Bodies. 331 The ice-bergs that float from the polar seas down into warm- er regions, are gradually dissolved, not only by the sun's rays, but also by the currents of warm air and warm water to which they ate exposed. But these causes operate more powerfully on the sides than upon the top and botto;n, and their horizontal dimen- sions are thus reduced faster than their vertical, whereby they become unstable, and are overturned. Being still subject to the same kind of influence, they are liable to repeated and fre- quent changes of position before they are completely wasted. 443. To investigate generally the conditions of equilibrium F . g of a floating body, let HsilF represent a vertical section, the point G of the principal axis ANF being the centre of gravity of the whole, and the point B the centre of buoyancy. The solid being inclined a little, the water line HNI shifts to H'NF, and the centre of buoyancy B to B'. From what has been said, it will be perceived that the area of HFI is to the sum of the two triangles Nil', HNH', as NP is to BB' ; that is, 44<>i area HFI : NI X IF : : f JV7 : B&, but the triangles INI', BMB', being similar, IF : JV7 : : BB 1 : BM, therefore, by multiplying the terms in order, and suppressing the common factor in each of the ratios, we have area HFI : JV7 : : f JV7 : BM, or, calling the area HFI, <*, and the length of the water line } HNI, ? tf : Q a) 2 :: ia : BM = Accordingly, the equilibrium will be stable when the cube of the length of the water-line JV7, divided by 12 times the area of the section, exceeds the interval BG between the centre of gravity and that of buoyancy. If this quotient be just equal to BG, the equilibrium will be that of indifference ; and lastly, when this quotient is less than BG, the equilibrium will be unstable, and the body will be liable upon a slight inclination to overset. 332 Hydrostatics. Whatever be the figure of the section HFI, its area and con- sequently the centre of gravity and that of buoyancy may be found to any required degree of exactness by the method of article 114. Although the above formula has reference only to a single lamina, and to motion in the plane of this lamina, it is still ap- plicable to any solid whose parallel sections are equal and sim- ilar, for in this case the whole may be considered with respect to motion in a parallel plane, as concentrated in the middle sec- tion or lamina represented by HFI ; and with respect to motion in a vertical plane perpendicular to the lamina, by supposing a corresponding section, and putting & equal to the area of this section, and a' equal to the length of the water-line, we shall have the same formula to express the conditions of equilibrium as before. When the sections or laminae are unequal, we find the height of the metacentre of each lamina, and multiply it by the bulk of this lamina, and then divide the sum of the products, or moments of the several lamina?, by the sum of the laminas for the height w - of the common metacentre. 449. In the case of a merchant ship, it will furnish a tolera- ble approximation to take the section near the prow where the girth is commonly the largest. The transverse section of the hull of a ship is not materially different from the form of a parabola. Therefore, on this supposition, the height of the metacentre above Fi 219 ^ e centre f buoyancy, or BM, is equal to the cube of ///, the length of the water-line, divided by twelve times the area of Cal. 94.HFL But the area HFI is equal to f HI X JVT. Hence, , 12 tf " 12xi#/XAT 8 AT 2 AT' f " ' ' ' ' - - - " '- ' ' ' " ' " ' - - iiu - t Where great accuracy is required, the following formula may be used j namely, Equilibrium of Floating Bodies. 333 that is, BM is equal to half the parameter of the parabola. being the centre of gravity of the parabola JHF/, its height is readily found to be f NF. Therefore for the whole height of 107. the metacentre above the keel, we have 10JVF Such is the height of the metacentre above the keel, on the supposition that the vertical sections are all equal and parabolic, which is nearly the case with respect to long track-boats. But the figure of the keel in most vessels, fitted for sailing, approach- es to a semi-ellipse, which is likewise the general form of a hori- zontal section. Owing to these modifications, the metacentre is found to be depressed about one fourth part, and consequently its height above the centre of buoyancy will be 3 3 ^ 2 SJVF ~ In a ship, for example, whose water line is 40 feet, and the depth of its immersed portion 15 feet, we shall have for the height of the metacentre above the centre of buoyancy, But the centre of gravity of the immersed part is f 15 or 6 feet below the water line. Hence the metacentre is 4 feet above the water line. The ship will therefore float securely, so long as the general centre of gravity is kept under that limit. In in which f NL represents the sum of the cubes of the perpendiculars $Q, OJV, &c., of figure 50, these perpendiculars being taken at equal distances, and so near to each other that the included portions of the curve 0./V, OK, &c., may be considered as straight lines, the common distance being denoted by e, and the bulk of the immersed part of the vessel by b. The investigation of this formula is very simple, and is omitted here merely on account of its length. See Bezouf s Mecanique, art. 359. 334 Hydrostatics. loaded vessels, the centre of gravity has commonly been found to be higher than the centre of buoyancy, by about the eighth part of the extreme breadth. Accordingly, in the present in- stance, the centre of gravity of the whole mass would still be one foot below the surface of the water, or five feet lower than the metacentre, which would be amply sufficient for the stability of the ship. 450. Such is the position of the metacentre in the vertical plane at right angles to the longitudinal axis, and which reg- ulates the rolling of a vessel from side to side. But there is another similar point in the plane of the masts and keel, which determines the pitching, or the movement of alternate rising and sinking of the prow. The height of this metacentre is derived from the same formula, by substituting only the length, for the breadth of the vessel. Thus, let the keel measure 1 80 feet, and we have With such a strong tendency to stability, therefore, in the direc- tion of its course, a ship can scarcely ever founder in consequence of pitching at sea. The formula now given for computing the height of the me- tacentre above the centre of buoyancy, may, with some modifi- cation, be deemed sufficiently accurate in practice. It is best adapted, however, for cutters or frigates, and will require to be somewhat diminished in the case of merchant vessels. Mr At- wood performed a laborious calculation on the hull of the Cuff- nells, a ship built for the service of the East India Company, having divided it into 34 transverse sections, of five feet interval. The result was, that the metacentre stood only 4 feet 3 inches above the centre of buoyancy. But that ship, being designed chiefly for burthen, appears from the drawings to have been constructed after a very heavy model, its vertical sections ap- proaching much nearer to rectangles than parabolas. To suit it, the formula above given would have required to be reduced two Nl thirds, or to 7-^* Now the breadth of the principal section was Equilibrium of Floating Bodies. 335 43 feet and two inches, and its depth 22 feet 9 inches. Whence - ') = 5,1 feet, differing little from the conclusion of a strict- 9 1 er but very tedious process. 451. Since the height of the metacentre is inversely as the draught of a vessel, and directly as the square of its breadth, its stability depends mainly on its spreading shape. This property is an essential condition in the construction of life-boats. But the lowering even of the centre of gravity has been found to be sometimes insufficient to procure stability to new ships, which, after various ineffectual attempts, were rendered serviceable, by applying a sheathing of light wood along the outside, and thus widening the plane of floating. 452. It is not very difficult to determine the centre of buoy- ancy, by guaging the immersed part of the hull. A cubic foot of sea-water weighs 64 lb - avoirdupois, and 35 feet, therefore, make a ton. The load of the vessel corresponding to every draught of water may be hence computed. 453. The height of the metacentre above the centre of grav- Fi g .220 ity in a loaded vessel, may be determined by simple observation. Let a long, stiff, and light beam be projected transversely from the middle of the deck, and a heavy weight suspended from its remote end, inclining the ship to a certain angle, which is easily measured. Thus, if NL represent this lever, q the weight at- tached, M the metacentre, and GMQ the inclination produced, G being the centre of gravity, and GR a perpendicular drawn from it to the vertical L , p', as measures of their intensity, so that the forces 9, q', will be proportional to />, p f ; for the in- terior surface of the fluid which surrounds the tube BE, is the same as the interior surface of the tube AB, Consequently, the two masses, namely, the glass in/?/?,and the fluid around BE, differ only in their thickness ; but as the attraction of both these mas- ses is insensible at sensible distances, the difference of their thicknesses, provided their thicknesses are sensible, will produce no difference in the attractions. 4. The fluid in the tube AB is also acted upon by another force, namely, by the sides of the tube AB in which it is inclosed. If we conceive the column FB divided into an infinite number of elementary vertical columns, and if, at the upper extremity of one of these columns, we draw a horizontal plane, the portion of the tube comprehended between this plane and the level surface BC of the fluid, will not produce any vertical force upon the column ; consequently, the only effec- tive vertical force is that which is produced by the ring of the tube immediately above the horizontal plane. Now the vertical attraction of this part of the tube upon J5E, will be equal to that of the entire tube upon the column BE, which is equal in diam- eter, and similarly placed. This new force will therefore be represented by -j- q In combining these different forces, it is manifest that the fluid column BF is attracted upwards by the two forces -f <7? + 9> and downwards by the force if ; Capillary Attraction. 343. consequently, the force with which it is elevated will be 2 q q'. If we represent the bulk of the column BF by 6, its density by A, and the force of gravity by g, then A b will represent the weight of the elevated column ; but, as this weight is in equilibrium with the forces by which it is raised, we shall have the following equation ; g A b = 2 q 0'. If the force 2 9 is less than 9', then I will be negative, and the fluid will be depressed in the tube; but as long as 2 q is greater than or, which is the same thing, to the interior circumference of the tube. Calling c, therefore, the circumference of the tube, we shall have q = p c, p being a constant quantity representing the force of attraction of the tube AB for the fluid, in the case where the attractions of different bodies are expressed by the same function of the distance. In every case, however, p expresses a quantity dependent on the attraction of the matter of the tube, and independent of its figure and magnitude. In like manner we shall have q' = p' c ; p' expressing the same thing with regard to the attraction of the fluid for itself, that ;? expresses with regard to the attraction of the tube for the fluid. By substituting these values of . n R 2 , and altitude -. Hence, we have h' = h -{- -\ or, which O O is the same thing, the mean altitude h' is always equal to (he al- titude h of the lower point of the concavity ot the meniscus, in_ creased by 'one third of the radius, or one sixth of the diameter of the capillary tube. Now, since the contour c of the tube = 2 n R, and since the bulk b of water raised is equal to h' X n R 2 , we have, by substituting these values in the general formula (i.) g A h' n R 2 = 2 n R (2p /) and, dividing by n R a; d g A, we obtain, x g& gA R In applying this formula to M. Gay Lussac's experiments, we have 2 JP^iJL = R .V = 0,647205x(23,1634-f-0,215735)=15,1311, o or 0,023454 of an inch for the first experiment; and, since the heights are inversely as the radii or diameters, 0,023454 or its double 0,046908 is a constant quantity. In order to find the height of the fluid in thfe second tube by means of this constant quantity, we have Capillary Attraction. 345 and from which if we subtract one sixth of the diameter, or 0,3173, we have 15,5783 for the altitude h of the lowest point of the con- cavity of the meniscus, which differs only 0,0078 or 0,0003 of of an inch from 15,5861, the observed altitude. If we apply the same formula to M. Gay Lussac's experiments on alcohol, we shall find for the constant quantity 2 *PP' = 6,0825, as deduced from the first experiment, and h = 6,0725, which differs only 0,01147 or 0,00045 of an inch from 6,08397, the ob- served altitude. From these examples, it will be seen that the mean altitudes, or the values of h', are reciprocally proportional to the diame- ters of the tubes very nearly ; and that in accurate experiments, the correction made by the addition of the sixth part of the diameter of the tube is indispensably requisite. 459. If the section of the bore in which the fluid ascends is a Fig.224. rectangle, whose greater side is a, and smaller side tf, the base of the elevated column will be a its equal a d h', we shall have gAh' a$ = (2p /) X (2 a -f 2 d) ; and, dividing by a and by g A, we have **/ = 2 !ZZ.' X #A and In applying this formula to the elevation of water between two glass plates, the side a is very great compared with #, and therefore the quantity - being almost insensible, may be safely neglected. Hence the formula becomes = 2 ^--x i 8* " * By comparing this formula with the formula (in.) it is evident that water will rise to the same height between glass plates, as in a tube, provided the distance d between the two plates is equal to R, or half the diameter of the tube ; in other words, that, when the distance between the plates is equal to the diam- eter of the tube, the elevation in the former case is half that in the latter. This result was obtained by Newton, and has been confirmed by the experiments of succeeding philosophers. 2 p _ , pi As the constant quantity 2 -- is the same as that al- g A ready found for capillary tubes, we may take its value, name- ly, 15,1311, and substitute it in the preceding equation ; we shall then have 15,1311 ^ and since Capillary Attraction. 347 subtracting from h or 14,1544, we have h = 14,0397, which differs 0,4657 of a millimetre, or 0,0183 of an inch, from 457. 13,574, the observed altitude. 460. If the plates are inclined to each other at a small angle, the line of meeting being vertical, the water will rise between Figi2 25. them to different heights according to the general law above enunciated ; that is, the distances at L, G, being ZJV, G/, we shall have GH : LM : : ZJV : GI : : M : HK. But, by similar triangles, M O : H K : : FM : FH, whence GH : LM :: FM : FH, that is, the heights at different points of the curve ELGB are inversely as the distances from the line of meeting EF of the plates. Therefore, since the relation of the lines FM, FH, &c., to the lines LM, GH, &c., is the same as that of the abscissas to the ordinates in the common hyperbola, the surface of the fluid between the plates answers to this curve. 461. If the relative attraction of the parts of the fluid for itself, (in the case of tubes for example) and for the substance of the tube, F . ^^ be such that the surface of the fluid column in the tube becomes convex, instead of being concave, the effect is precisely the reverse of that above considered ; that is, when an equilibrium occurs, the filament occupying the axis of the tube and rising without the tube to the free surface, will have its extremity/ 1 depressed ; since, instead of an excess of upward attraction proceeding from a sus- tained annulus, situated above a horizontal plane passing through the extremity F, there will be a deficiency of upward attraction equal to the effect of this same annulus. Accordingly, the de- 348 Hydrostatics. pressions will, like the elevations, be inversely as the diameters of tubes, and the whole theory above given, with this single mod- ification, is strictly applicable. Between plates also, and between concentric tubes, a depression will take place corresponding to the elevation in the case where the upper surface is concave. It is to be observed, however, that the deficiency of attrac- tion in the case of mercury used in connection with glass, taken in its ordinary state, is to be ascribed to a want of contact be- tween the fluid and the substance of the tube or plate, arising from a film of moisture which ordinarily attaches itself to glass, and which being completely removed, mercury is found to pre- sent a concave surface like water, and consequently to rise in a tube and between plates above its natural level, t Indeed water may be made to exhibit the same apparent anomaly, by having the surface of the glass, whether tube or plate, smeared with a thin coat of tallow or wax. 462. The peculiar character of this theory consists in this, that it makes every thing depend upon the form of the surface. The nature of the solid body and tha' of the fluid determine simp- ly the direction of the first elements, where the fluid touches the solid, for it is at this point only that their mutual attraction is sensibly exerted. These directions being given, they become the same always for the same fluid and the same solid substance, whatever be the figure of the body itself which is composed of this substance. But beyond the first elements and beyond the sphere of action of the solid, the direction of the elements and the form of the surface are determined simply by the action of the fluid upon itself. We have seen that the elevation of a liquid between parallel plates, is half of that which takes place in tubes whose diameter is t Barometer tubes properly cleansed and freed from humidity, by having the mercury repeatedly boiled in them, will exemplify the truth of this remark. Moreover, with the knowledge of this fact, we can readily satisfy ourselves by simple inspection, whether the requisite attention was paid to this particular in the construc- tion of the instrument. Apparent Attraction and Repulsion of Floating Bodies. 349 equal to the distance of the plates. The cause which determines this ratio is to be found in the above theory. For, in the case of tubes, the action of the concave or convex surface upon the elevated or depressed column is half of the action of two spheres which have for radii the greatest and least radii of the osculating circles to the surface at the lowest point. The tube being flattened in any direction, the radius of the corresponding curvature augments, and finally becomes infinite, when the flat- tened sides of the tube become parallel plane surfaces. The first part of the attraction of the surface being inversely as this radius, will become zero, and there will remain only the term depending on the other osculating radius, and the attractive force is accordingly reduced one half. Such is the simple and rigor- ous result furnished by the theory of Laplace. 463. This theory serves to explain also, and with the same simplicity, all other capillary phenomena. Thus, the ascent of water between concentric tubes, and in conical tubes ; the curva- ture which water assumes when adhering to a glass plate ; the spherical form observed in the drops of liquids ; the motion of a drop which takes place between plates, having a small inclination to each other and to the horizon ; the force which causes drops floating on the surface of a liquid to unite; the adhesion of plates to the surface of a liquid, which is in many cases so great as to require a considerable weight to separate them ; these effects, so various, are all deduced from the same formula, not in a vague and conjectural way, but with numerical exactness. On the apparent Attraction and Repulsion observed in Bodies float- ing near each other on the Surface of Fluids. 464. (1). If two light bodies, capable of being wetted, be placed at the distance of one inch from each other on the surface of a basin of water, they will float at rest, and without approach- ing each other. But if they be placed at the distance of only a small part of an inch, as two or three tenths, they will together with an accelerated motion. 350 Hydrostatics. Fi 22-7 ( 2> ) ^ t ^ le two Bodies are f sucn a nature as not to suffer the fluid to adhere to them, as is the case with balls of iron used in connection with mercury, the same phenomena will be observed. ^ " ' (3.) But if one of the bodies is susceptible of an adhesion of the fluid, and the other not, as two balls of cork, for example, one of which has been carbonized by the flame of a lamp; the effect will be the reverse of that above stated ; that is, the bodies will seem to repel each other, when brought very near together, and with forces similar to those with which in the former case they tended to unite. Moreover, a single ball will approach to, or recede from, the side of the vessel, as it would approach to or recede from another ball, according as the substance of the vessel and that of the ball are similar or dissimilar as to their disposition to cause an adhesion of the fluid. 465. Tn these experiments the approach and recession of the floating bodies are not the effect of a real attraction or repul- sion between the bodies ; for, if the bodies, instead of being plac- ed upon the surface of a Jiquid, be suspended by long, slender threads, nothing of the kind is to be perceived. We must there- fore look for some other cause to which to refer these appearan- ces. Fig.229. If two plates of glass AB, CD, be suspended in water paral- lel to each other, and at such a distance, that the point H, where the two curves of elevated fluid meet, sha^ll be on a level with the common surface, the plates will remain in equilibrium But on being brought so near to each other, that the point H shall be above the common level of the surface, the mass of fluid thus raised will have the effect of a chain attached at its extremities to the plates, in drawing the plates toward each other. The approach of the balls to each other under similar circumstances, is to be referred to the same cause. When the point H is below the general level, on account of a want of adhesion in the parts of the fluid to the plates, the pressure of the plates inward toward each other not being coun- terbalanced by the pressure in the opposite direction, they must approach each other, and with a greater or less force, according Barometer. 351 to the depression of the point H, or the nearness of the plates to each other. This affords an explanation of the second case above stated. If one of the floating bodies, as A, for example, is susceptible of p 28 being wetted, while the other B is not, the fluid will rise around A and be depressed around B. Accordingly, when the balls are near to each other, the depression around B will not be symmetrical, and the body being thus placed as it were upon an inclined plane, its equilibrium will be destroyed, and it will move off from the other body in the direction of the least pressure. These phenomena, of which we have given only a familiar explanation, are all comprehended in Laplace's theory of ca- pillary attraction ; and the attractive and repulsive forces are capable, on that theory, of being subjected to a rigorous calcula- tion. Of the Barometer. 466. If we take a glass tube thirty-three or thirty-four inch- es in length, closed at one extremity and open at the other, and having filled it with mercury, place the finger over the open ex- tremity and thus immerse it in a basin of the same liquid with- out suffering the air to enter the tube, the mercury will settle down in the tube, leaving a vacuum above it, till its weight is exactly counterbalanced by the pressure of the atmosphere, ex- erted upon the surface of the mercury in the basin. This in- strument is called a barometer.] The perpendicular height at which the mercury is ordinarily maintained at the level of the sea, is very nearly thirty inches. From what has been said of the manner in which the pres- sure of-fluids is propagated, it will be perceived that it is imma- terial what be the extent of surface in the basin, or whether the atmospheric pressure be applied at the top, or, by means of a flexible bag containing the liquid, at the bottom and sides. If t See note on the construction and history of the barometer. 352 Hydrostatics. instead of entering a basin the tube turn up at the bottom, as in figure 230, so as to admit the air at C, the perpendicular elevation above a horizontal line coinciding with the surface at E or F, will be the measure of the atmospheric pressure. This eleva- tion, moreover, is independent of the form of the tube and the particular quantity of mercury contained in it. On the suppo- l ^' sition, however, that the base of the tube is an inch square, the pressure is equal to that of a parallelepiped of mercury 30 in- c' es in length, or, which amounts to the same thing, to the 410. weight of 30 cubic inches of mercury. Now 30 cubic inches of water is equal to 30 X 252, 525 grains, or 15,78 troy ounces. Whence 30 cubic inches of mercury is equal to 15,78 X 13,57 or 214,12 troy ounces; that is, to 234,7 ounces avoirdupois, or to 14,7 lb * We infer, therefore, that the pressure of the atmos* phere amounts to nearly 15 lbt upon every square inch of surface, or to about one ton upon every square foot. A common sized man exposes a surface of 10 or * 1 feet, and is consequently sub- jected to a pressure of as many tons' weight. The entire surface of the earth being estimated at 5575680000000000 feet, this number will express the weight nearly of the whole atmosphere in tons, a certain deduction being made for the space occupied by mountains and elevated regions. This pressure being exert- ed upon the surface of the ocean, fishes are exposed to it in addition to the weight of their natural element. But the pro- portion 1 : 13,57 :: 30 : 407,1, gives 407,1 inches, or 34 feet nearly, for the length of a column of water equivalent to that of 30 inches of mercury or the pres- sure of the atmosphere. Accordingly for every 34 feet depth a pressure is exerted of a ton upon every foot of surface, over and above that arising from the atmosphere. Now fishes are sometimes caught at the depth of 2600 or 2700 feet, where the pressure of the water amounts to nearly 80 atmospheres or 80 tons upon a square foot ; yet these fishes are not injured by such an immense weight, or sensibly impeded in their motions. The reason is, that they are filled with fluids, which from their im- penetrability oppose a sufficient resistance to this pressure, and thus preserve the most delicate membranes from being ruptured. Barometer. 353 With regard to the facility and rapidity of their motions, as the incumbent weight acts equally in all directions, it neutralizes itself, by aiding just as much as it obstructs their efforts to move and turn themselves. The case is precisely similar with respect to land animals. The vessels of the animal, together with the bones, are filled with air or some other fluid capable of support* ing any weight, and whose elasticity being equal to that from without, proves an exact counterbalance to it. 467. In the barometer there is an equilibrium between the pressure of the mercury and that of the atmosphere. Now we have seen that when two fluids thus counterbalance each other, the altitudes must be inversely as the specific gravities. Accord- ingly, as the specific gravity of mercury is to that of air at the surface of the earth as 13,57 to 0,00122, we shall have 1 3 ^7 V 30 0,00122 : 13,57 : : 30 : ^ * g = 333688. We infer, therefore, that the height of the atmosphere, on the supposition of a uniform density throughout, is 333688 inches, or a little more than 5 miles. But the air being eminently elas- tic, the lower strata are compressed by the incumbent weight of those above, so that the density becomes less and less continu- ally as we ascend. Let the weight of the column of mercury which measures the pressure of the atmosphere, exerted upon a unit of surface, be denoted by g A h, g being the force of gravity, A the density of the mercury, and h the perpendicular height of the column above the level of the surface in the basin, and let the weight of the atmosphere upon the same surface be denoted by w, we shall have g A h = w. As we ascend into the atmosphere, the weight w and the height h diminish continually, and these diminutions depend upon the elevation attained, and the law according to which the densities of the atmospheric strata decrease. If this law were known, it might be made use of for the purpose of determining the differ- ence in the altitudes of two points above a common level, as the sea, or any assumed level. But in order to discover this law, it Mech. 45 354 Hydrostatics. is necessary to recur to certain experiments relating to the den- sity of the air under different pressures and at different tempe- ratures. Fig.230. 468. Take a recurved glass tube ABC. open at the extremity A and closed at the other extremity C ; pour into it a quantity of mercury just sufficient to fill the bended part up to the hori- zontal line DE, so that the air confined in the shorter branch CE may be neither more nor less pressed than that contained in the longer branch AD, which communicates with the atmo- sphere. The mercury being at the same height, therefore, in each branch, and the communication with the external air being cut off, if we introduce, by means of a fine tunnel, more mercury, we shall observe this liquid to stand higher in the branch BA than in the other, whereby the air in EC will be condensed, the compressing force being equal to the difference of the two columns. If the space EC, supposed, for example, to be 4 inch- es, were reduced one half or to E, by the pressure of a column of mercury extending to H, drawing the horizontal line EG, we should find the difference GH of the two columns exactly equal to the height of the barometer at the time of the observation ; so that the air contained in the space CE would be pressed by the weight of the atmosphere incumbent upon H and by the weight of another atmosphere represented by the column GH. A doub- le pressure, therefore, reduces the bulk one half. If we contin- ue to add to the weight by pouring in more mercury till the confined air is condensed to F' or to one third of the original space, we shall find the additional quantity necessary to this effect the same as before, that is, the column GH' will be equivalent to two atmospheres. Thus a triple pressure reduces the bulk of the confined air to one third the space. We might continue to increase the weight, and we should in every instance obtain results agreeable to the same general law. So, on the other hand, by diminishing the natural pressure exerted upon any portion of air, we shall still find the bulk in- versely proportional to the pressure. Let the tube ABC be supposed to have a bore not exceeding one tenth of an inch. A drop of mercury being introduced at the bend A, if the whole apparatus be placed under the receiver of an air pump, and the Barometer. 355 air be exhausted from the longer branch, till the pressure is re- duced successively one half, two thirds, &c., the portion of air confined by the drop of mercury will expand, driving the drop before it, and will occupy successively, double, triple, &c., of its original bulk. We infer, therefore, universally, that the space occupied by any given portion of air is reciprocally proportional to the pressure. In order that this law may hold true, however, in the strict- est sense, it is to be remarked that the air must be perfectly dry ; for the small quantity of aqueous vapour, which is ordinarily found mixed with the atmosphere, is not condensed by pressure according to the same law, as will be shown hereafter. The instrument represented by figure 230, is called a ma- nometer. It is used for the purpose of measuring tlie elastic force of other gases besides the atmosphere ; and they are all found to be condensed and expanded according to the above law. This important property was discovered by Mariotte, and is frequently referred to under the name of the law of Mariotte. 469. Recurring to the first experiment above described, the pressure exerted upon the portion EC of confined air, when the recurved part of the tube is just filled with mercury, is that of the atmosphere, or g A h. But this pressure is resisted and counterbalanced by the elasticity of the confined air, which by supposition is of the same density with that immediately sur- rounding the apparatus. We may take g A /t, therefore, as the measure of the elastic force of the air in question. This force remains the same so long as the air continues of the same densi- ty and the same temperature. If a manometer be removed from one place to another, care being taken not to change the state of the confined air, the product g A h which represents the elas- tic force does not undergo any change. But if the gravity g va- ries as we remove from one place to another, the height h of the mercury will also vary in the inverse proportion to g, the den- sity A of the fluid being supposed to remain the same. It will hence be perceived that the variations in the heights of the mer- cury in the manometer are capable of rendering sensible the variations of gravity, and may even be employed in determining the augmentations or diminutions of this force arising from changes of distance with respect to the centre of the earth. 356 Hydrostatics. 470. Let us now suppose that, the weight of the atmosphere remaining the same, the temperature of the confined air is rais- ed ; as this air expands, its bulk will be increased, and its densi- ty diminished. JNow we know by the careful experiments of M. Gay Lussac and others, (1.) That all the gases dilate uni- formly, at least from 32 to 212, or from the freezing to the boil- ing point of water. (2.) That the dilatation arising from the same increase of heat is precisely the same for all the gases, vapours, and mixtures of gases and vapours. (3.) That the bulk of con- fined gas, at the temperature of 32, being considered as unity, this common dilatation is 0,375, (or a little more than one third,) for 180, the difference between the boiling and freezing points of water ; which gives VH 5 = ^ or 0,00208 for the augmen- tation of bulk answering to 1 of Fahrenheit. Accordingly, we shall have for the bulk or space occupied by the portion of air in question 1 -|- 0,00208 n at the temperature denoted by ?i, the number of degrees above or below 32, the latter being consider- ed as negative. This bulk or volume may be reduced to its original limits, by bringing the temperature back to 32, or by increasing or diminishing the weight which compresses it, with- out altering the temperature. It would only be necessary, in this latter case, to add to, or take from the weight TO, a portion equal to w (0,00208) w, that is, to substitute for w the weight TO (1 4- 0,00208 71), which is the measure of the elastic force of the confined air re- duced to its original density. Hence, the bulk and density re- maining the same, the elastic force varies with the temperature, and in the same ratio. If the elastic force is proportional to the density when the temperature is the same, and varies with the temperature when the density is the same, it will be easy to deduce the value of this force in terms of the two elements, on the supposition that they both vary together. Thus, putting A for the density of the air in question, and n for the number of degrees which marks the temperature, and/) for its elastice force or pressure exerted upon the unit of surface, a being the ratio of the elastic force to the density at the temperature of 32, we shall have Barometer applied to the Measurement of Heights. 357 p aA(l + 0,00208 n.) (i.) The coefficient a is constant for the same elastic fluid, but is dif- ferent in different fluids, and requires to be determined in each, particular case. 471. In applying the results above stated to the mass of air which composes the atmosphere, we take into consideration only a single vertical column of air, supposed to rest upon the surface of the earth and to extend indefinitely upward. We may con- ceive of the surrounding mass or atmosphere as congealed or ren- dered solid. If it were previously in a state of equilibrium, this state will not be disturbed by such a supposition ; so that the column in question will still be in equilibrium as before* Now the force which acts upon the particles of air is gravity, which may, without sensible error, be regarded as exerting itself in the direction of the aerial column throughout its whole extent, or at least as far as it is necessary to take any account of it. Ac- cordingly, it is necessary, in order to an equilibrium, that the den- sity, the pressure, and the temperature should be considered as uniform throughout a horizontal stratum of infinitely small thick- ness. The column being composed of an infinite number of these strata or lamina, let h be the height or distance from the surface of the earth of one of these strata, A the density of this stratum, T its temperature, g / its gravity, p its elastic force, 6 its base, and d h its thickness. We shall have dp for the pressure exerted upon the inferior base, and 6 (p dp) for the pressure upon the superior base ; the difference rf dp must be equal to the weight 6 A g / d h of this stratum. Hence, by suppressing the common factor <*, we have the equation dp A g f d h, or, substituting for A its value _f deduced from equation (i), the fraction 0,00208 being for the sake of brevity represented bye, ~ d P=Xtt* dh - Whence 358 Hydrostatics. dp^ .. ' dh p " Nothing can be inferred from this equation until the value of n is given in terms of h. Now we know that the temperature de- creases as we ascend from the surface of the earth, but the law of this decrease has not been determined in a manner altogether satisfactory. Fortunately, this law has little influence upon our results in the calculation of heights by the barometer, on ac- count of the smallness of the coefficient e; and we may, in ques- tions of this kind, consider the temperature as constant, provided we take for n, in each particular case, the mean of the tempera- tures observed at the two extreme points of the height h to be determined. Moreover, R being the radius of the earth, and g the gravity at the surface, we have, at the distance R + h from the centre, / - (a + hy ' since this force varies in the inverse ratio of the square of the distance. The preceding equation becomes, by this substitu- tion, dp g R 2 d h Whence, by integrating on the supposition that n is constant, we have m being equal to 0,434295, log. denotes the common logarithm 'of/?. To determine the constant C, let -or be the value of p an- swering to h = ; arid we shall have Consequently, .by subtracting the preceding equation from this, we obtain h f . This equation, taken in connection with equation (i.), gives the values of p and A in terms of h. Thus we have equations con- Barometer applied to the Measurement of Heights. 359 taining the laws of the density and elastic force of the air which belong to a state of equilibrium in the atmosphere. 472. To make use of equation (n) for the purpose of measur- ing heights by means of the barometer, let us suppose the ba- rometric altitude at the surface of the earth and at the height h to be known by actual observation, and let them be denoted res- pectively by to, zo', the corresponding temperatures of the mer- curial columns being represented by T, x'. The expansion of mercury being g^ or 0,001025, that is, 0,001 nearly, for each degree of Fahrenheit's scale, if D be the density corresponding to the temperature T of the mercury at the first station, D (1 + 0,001 ( T 7 ) will be the density which answers to the temperature of the mercury at the second station. Accordingly we have zr = Dg zr, and p = vg' w' (1 + 0,001 (T T'). The correction for the upper barometric column on account of difference of temperature being made agreeably to this formula, we may consider w' as representing the length of this column thus corrected. Whence, dividing the first of the above equa- tions by the second, we have = !L x p g 1 substituting for g' its value ^ * ; and consequently, log. = log. + 2 log. (1 + , ( m .) since Let T, T', be the temperatures respectively of the air at the surface of the earth and at the height h ; T, T 7 , will generally differ from ", T X , the temperatures of the mercury in the barome- ter, since the latter is not ordinarily allowed sufficient time to acquire the temperature of the surrounding air. T, T', are to 360 Hydrostatics. be taken by means of a thermometer suspended in the air, while T, T', are supposed to be indicated by a thermometer at- tached to the barometer. We take n = -- 32. Moreover, the coefficient T | 7 or 0,00208, representing the elastic force, re- quires to be increased somewhat for the purpose of taking ac- count, as far as can be done, of the quantity of water in a state of vapour which is at all times mixed with the air in a greater or less quantity. Indeed, under the ordinary pressure of the atmosphere, the density of aqueous vapour is to that of air, as 10 to 14; consequently, the atmosphere is so much the lighter according as it is composed in a greater degree of this vapour. Now it contains so much the more vapour according as its tem- perature is more raised, whereby, when the air is dilated by heat, its weight must be diminished in a higher ratio than that of its augmentation of bulk. We increase the coefficient 0,00208 therefore to 0,00223f or T i , which has been found by actual trial to give the most correct results. We have, accordingly, e n = 0,00223 ' 32\ We now substitute in equation (n.) for e n the above value, and for log. - the value found in equation (in.), and we shall ob- tain log. - + 2 log. (1 + *"\= g W ' " + *' Whence x * - * a (I + 0,00223(I^ _ 32)) R +fc - 32 )) ^ jThis is the value adopted by Laplace, Poisson,Biot and others from the very extensive and careful researches of M. Ramond. That of Sir George Shuckburgh is 0,00238, which is generally employed by English writers upon this subject. The mean of the two is 0,00235. Barometer applied to the Measurement of Heights. 361 The best means of determining the coefficient of this mg formula, is to make use of a height (or rather a number of heights), well known by actual measurement, or by trigonomet- rical operations. We then substitute for h this known value, and for w, w/, T, T X , the lengths of the barometrical columns, and the temperature of the air at the two stations respectively, and for R the mean radius of the earth, namely 348 1 280 fathoms. We shall thus have an equation from which the value of is read- mg ily deduced once for all. Taking the mean result of a great number of observations conducted with the greatest care, by M. Ramond, we find , for the latitude of 46, t equal to 1 8336 J metres, or 10026 English fathoms. This is on the supposition of a temperature of 32, and agreeably to what has been said, it may be increased or diminished by adding or subtracting | or a 0,00223 part for each degree above or below 32 Q . We can therefore reduce it to 10000, instead of 10026, by supposing the temperature somewhat lower. Thus, since 26 is 0,0026 of 10000 0,00223 : 0,0026 : : 1 : 1,16. If, therefore, we subtract 1, 16 from 32, we shall have 30,84 or 31 nearly, for the temperature at which the constant coeffi- cient is 10000 fathoms. 473. Since this coefficient contains g, it must vary with g, that is, with the latitude. Now, according to the law of the va- t This coefficient was actually determined for the latitude of about 43. But the correction for small distances in latitude is so inconsiderable, that it may be regarded ~as nothing. Moreover, the coefficient, if corrected at all, would require to be diminished, and it is thought on the whole less liable to error by excess than by de- ficiency. | The coefficient deduced theoretically from the relative densi- ties of mercury and air, as determined by Biot and Arago, allowance being made for humidity, is 18334,1 metres, differing less than 2 metres, that is, less than 2 X 39,371 inches from the above, Mech. 46 362 Hydrostatics. riation of gravity in different latitudes, if g represent the value of this force in the latitude of 45, and g' that of any other lati- tude L, we shall have g 7 = g (I 0,002837 cos 2 L). t At 45, therefore, where cos 2 L = 0, g / =. g, or the correction is ; and for higher latitudes the correction is , or subtractive, and for lower latitudes it is -f- or additive. Whence, generally, = 10000 fath - (1 + 0,002837 cos. 2 L.) mg By means of this value of , substituted in equation (iv.) tThe value of g' in different latitudes depends upon the particu- lar figure of the terrestrial spheroid, the determination of which belongs to astronomy. We will merely observe in this place, that a comparison of articles 346, 347, conducts us directly to the equation o, a', being the lengths of the pendulum corresponding to the parts of the earth in which the intensities of gravity are g, g', respective- ly. Now it is found that the general expression for the length of the seconds pendulum, the day being divided into 100000 seconds, is metre. metre. a' = 0,739502 + 0,004208 (sin L) 2 . Hence, since (sin L) 2 i (1 cos 2 JL), and (sin 45) 2 i. Trig. 20- we shall have Trig. 27. JL = JL =- 0,739502 -f 0,002104 a' ~ ' g 1 0,739502 -f 0,004208 (sin Lp ? _ 1 1 _ 0,002837 cos 2 L 5 therefore, g' g (1 0,002837 cos 2 L). In the original memoir of M. Ramond, the coefficient stood 0,002845, and it was thus copied by Laplace and others. It was afterward corrected by M. Oltmanus, and the error acknowledged by the author in a separate edition of the memoir. Barometer applied to the Measurement of Heights. 363 we shall be able to determine the height h in any part of the earth, when z, a/, T, T', corresponding to the extreme points of h are known; or we may retain the coefficient 10000 unaltered, and apply the correction to the result, according to the follow- ing table ; Latitude. Correction. . . -(- 3^2 f the approximate height. 5 tf& + si* 100 . + _i_ 15 . - + T *t 20 . . + T |o 25 . . + -gi-g 30o .. -f- _i_ 35 . . + T '3o 40 . . + rf Tr 45 . . + 55 . yoV? 60o .. T i_ 65 . si? 70o . . . T i 750 . . __ __ 80 85 . - 90 . . 3!^ 474. As the fraction - is always very small, we shall have R very nearly the value of A, independently of the term containing this fraction ; by substituting the approximate value thus obtained for /i, in the fraction -, we shall have very nearly the correction due to the variation of gravity at different elevations in the same latitude; and by substituting the value of h thus corrected in the fraction - we can approximate the true height still more nearly. R But this second substitution is altogether superfluous in the cases which ordinarily occur. Indeed, except where h is very great, we may neglect- entirely, and the general formula then becomes 364 Hydrostatics. h = JL (i + 0,00223 r^-^) - 32^) log. ^. m # \ \ 2 x / => t' It will be pprceived that - - may require some modification, in order that the formula in this state should adapt itself to ob- served heights or known values of h ; and indeed the observa- tions of M. Ramond give, for the value of the co-efficient to be employed in this formula, = 18393 metres or 10031 fath- mg oms, exceeding the former by 6 fathoms. Accordingly the de- pression of the temperature below 32, required in order to change this to the more convenient form of 10000, will be found to be 1,45 ; retaining the co-efficient 10000, therefore, we have only to suppose the temperature 32 l,45 or 30,55. As this differs less than half a degree from 31, and as we can sel- dom be certain of the temperature of the air to a greater degree of accuracy, we may still use the same formula without any oth- er change than the omission of the term depending on -. We li. have hence a very simple, convenient, and for common cases, sufficiently exact formula, namely, h = 10000 (1 -f 0,00223 ( T "^ T ) 31) (log. w log. w/.) This being adapted to the latitude of 45, when the barome- terical observations relate to a place on a parallel considerably distant either north or south, it will be seen directly by the fore- going table when it is necessary to apply a correction for differ- ence of latitude, and what this correction is. It will be recol- lected that the lengths of the barometric columns w, w', which represent the weights of the atmosphere respectively at the two stations, are supposed to be reduced to the same temperature. The upper column w' is usually the coldest, and consequently too short. Now, according to the rate of expansion or contrac- tion already mentioned, as 1 inch is shortened 0,0001 for each degree, a column of 25 inches will be shortened 0,0025 of an inch for each degree of depression, and consequently 0,01 for ever y 4 and each portion of 2,5 inches will be shortened one tenth part of this, or 0,001 for the same amount of depress- Barometer applied to the Measurement of Heights. 365 ion or 4. In common chamber barometers, the lengths of the columns are read off to a 0,01 of an inch, and in the best to the j| 7 or a 0,001 of an inch. The best time for taking observations with the barometer for the purpose of calculating heights is during settled weather and at midday. Observations taken in the morning or evening are much more liable to be erroneous on account of ascending and descending currents of air which take place at these times. Moreover a course of continued observations is more likely to lead to accurate results than single observations. By means of accurate registers of the barometer, the difference of level of places, however remote from each other, and their elevation above the ocean, may be ascertained with a considerable degree of precission. It is found, for example, by numerous and care- ful observations in different parts of Europe, as stated by Biot, t that the mean height of the barometer at the level of the ocean is 0,7629 of a metre, the temperature being 1 2,8 of the centes- imal scale. This, reduced to English inches, becomes 30,035, and 1 2,8 of the centesimal scale corresponds to 55 of Fah- renheit's. Now the mean of 22 years observation, three obser- vations a day, at Cambridge, N. E. gives for the height of the barometer at the same temperature, 29,9974 Applying the for- mula to these observations, we shall have the following result, 30,035 .... log. 1,47763 29,997 . . . log. 1,47708 0,00055 Multiplying by 10000, or which is the same thing, removing the decimal point four places to the left, we obtain, for the approxi- mate difference of level, or the elevation of the place of observa- tion at Cambridge, above the ocean, 5,5 fathoms or 33 feet. Now the mean temperature at Cambridge, as ascertained by corresponding observations during the same period is 48,8 ; and if we take the temperature at Paris as the mean temperature of t Precis sur la Physique, vol. 1, p. 194. 3d. Edition. t Memoirs Am. Acad. vol. iii. p. 386. 366 Hydrostatics. the places at which the above result for the barometric pressure at the level of the ocean was found, we shall have 51,9 as an- swering to the temperature of the air at the lower station. Hence, 48,8 + 51,9 _ . Q x 0^0223 = The correction, therefore, for difference of temperature, is 33 X 0,045 = 1 ,5 feet nearly, which added to 33 gives 34,5 for the elevation of the place of observation at Cambridge above the level of the sea. Now the actual elevation of the cistern of the barometer, as carefully ascertained by levelling, is found to be 31 feet. In the calcula- tion of very small heights near the level of the ocean, it is very common to dispense with the formula and adopt the following rule, namely, as 0,1 is to the difference in the barometric columns^ so is 87 feet to the approximate difference of level required ; which is to be corrected, if necessary, for the difference from 31 of the mean temperature of the air at the two stations. Thus, 0,1 : 0,038 : : 87 : 33,1, a result agreeing very nearly with that derived from the formula* Thus, under a pressure of 30 inches of mercury at the tem- perature of 50, 0,1 of an inch of mercury answers to 87 feet of atmosphere. It will be seen moreover, that, as 0,1 of an inch of mercury is equivalent to 87 feet of air, 0,01 answers to 8,7, 0,001 to 0,87, and j^ to 1,14. Hence in a good mountain barometer, graduated to 500dths of an inch, there will be a sensible differ- ence in the pressure of the air arising from a change of altitude of less than two feet, or two thirds the length of the instrument. Formula (iv.) is essentially the same with that given by Laplace in the 10th book of the Micanique Celeste^ but simplified after the example of Poisson, and reduced to English measures. The following example will serve to illustrate every part of this for- mula. At the lower of two stations, the mercury in the barometer was observed to be 29,4 inches, and its temperature 50, that of Barometer applied to the Measurement of Heights. 367 the air being 45 ; and at the upper station, the height of the barometer was 25,19, its temperature 46, and that of the air 39, the latitude of the place being 30. In this case, we have T T'=50 4b=4* ; - and 31= cos 2 L = cos 2 X 30 = cos 60 = J, Whence w = 29 4 log. wF 25,2003 log. 11 0,00223 . 1st correction \ 0,002837 2d correction R = 3481280 - = 0,0002 R _ ) = 1,0004 3d correction 1,46835 1,40141 669,4 log.. . 3,82569 log.. . 1,04139 log.. . 3,34830 16,42 log.. . 2,21538 685,82 log.. . 2,83621 log.. . 3,15168 0,97 log.. . 1,98789 686,79 log.. . 2,83682 log.. . 6.54174 log.. . 4,29508 . 2 log... 0.00034 3,4 690,19 fathoms. Laplace's formula, applied to the same example, gives 683,97 fathoms, differing from the above only 1,22; whereas, by Sir George Shuckburgh's method, in which no account is taken of the variation of gravity, either for difference of latitude or difference of elevation in the same latitude, the result is 685,125. This corre>ponds with the approximate height derived from the first correction in the above example. 368 Hydrostatics. 475. We have already mentioned, that, unless very particu- lar precautions are taken, mercury is depressed in glass tubes, and that this depression is inversely proportional to the diame- ter of the tube. It is always indicated, moreover, when it takes place by the upper surface being convex. It is not necessary to have regard to this circumstance in the calculation of heights by the barometer, where the two observations are taken with the same instrument, since the difference in the length of the ba- rometric columns would be the same, whether they were cor- rected or not. t But in order that observations by different in- struments, liable to different capillary effects, may be strictly compared with each other, a correction should be applied, which may be readily done by means of the following table. Interior diameter of the Depression tube in English inches. of the Mercury. 0,6 0,005 0,5 0,007 0,4 0,015 0,35 0,025 3 0,036 0,25 0,050 0,2 0,067 0,15 0,092 0,1 0,140 t Also in a syphon barometer, or one in which the tube, instead of entering a basin, turns up at the bottom and continues of the same bore, as in figures 230, 232, since the capillary effect is the same in both branches, the observed altitude reckoned from the surface in the shorter branch, would not be affected by the correction. HYDRODYNAMICS. Of the Discharge of Fluids through Apertures in the Bottom and Sides of Vessels. 476. If a fluid be made to pass through a canal or tube of variable bore, kept constantly full, and the velocity be the same in every part of the same section, since for any given time the same quantity of fluid must pass through every section, this quantity must be equal to the area of the section multiplied by the velocity, rf, rf', being the areas of two sections, and t>, i/, the velocities at these sections, we shall have 6 -o = tf' i>', and hence 6 : 6' :: ^ : v, lhat is, the velocities in different sections are inversely as* the areas of the sections. The case here supposed is purely theoretical, and can never occur in practice, since on account of friction, the velocity is always greatest at the surface in a canal, and at the axis in a tube. 477. Let MNOP represent a vessel filled with a fluid to G#, CD an aperture, very small compared with the bottom MP, CIKD, the column of fluid directly above the aperture, and CABD the lowest lamina or stratum of this fluid, immediately contiguous to the aperture. Also let v denote the velocity ac- quired by a heavy body in falling freely through BD, the height of the stratum, and u the velocity which the same stratum would Mech. 47 370 Hydrodynamics. acquire in falling through the same space by the pressure of the column CIKD. If we suppose the lowest stratum ACDB, to fall as a heavy body through the height BD, the moving force will be its own weight. But if we suppose it to be urged by its own weight, together with the pressure of the incumbent column of fluid CIKD through the same space, the velocity in the former case will be to that in the latter, as the moving forces and the times in which they act, the mass moved being the same in both cases. But the moving forces are to each other, as the 27. heights BD, KD, and the times in which they act, the space 6 ' being the same, are inversely as the velocities. Accordingly, BD KD v : u : : --- : - . v u Whence v 2 : u 2 : : BD : KD, or v : u :: V#D : VKD. Now v is the velocity which a heavy body would actually ac- quire in falling through the space BD, and as the velocities, other things being the same, are as the square roots of the spa- 26g ces, u the velocity of the issuing fluid is that which a heavy body would acquire in falling through KD, the height of the fluid above the orifice. Therefore, the velocity with which a fluid is discharg- ed from the bottom of a vessel is equal to that acquired by a heavy body in falling through a space equal to the height of the fluid above the orifice. Also if a pipe Jl'B'C'D' be inserted horizontally, or inclined in any way to the horizon, it may be shown, in like manner, since the pressure of fluids is equal in all directions, that the fluid will be discharged with the same velocity as be- fore. It will accordingly ascend to the level of the fluid in the vessel, all obstructions being removed ; and it is found in fact, under the most favourable circumstances, nearly to reach this point. It follows, moreover, from what is above laid down, that if apertures be made at different distances s, s', s", below the surface, the velocities at these points, and consequently the quantities of fluid discharged at these points, from apertures of Discharge of Fluids through Apertures. 371 the same size, in the same time, the vessel being kept filled to the same level, will be as vT, vV> VV 7 "- Tne actual velocity at the distance s below the surface of the fluid in the vessel will be V/2^, and the quantity Q discharged in the time f, through an aperture whose area is tf, is as follows, namely, Q= will vary continually, but may be considered as constant during the indefinitely small time d t ; so that in this time there will escape through the orifice, a prism of the fluid having the area va ^ ue ^ or A in t ^ le ec l uat i on f article 282 we have 7i f dxy 2 = <2" " ^ V T^~x V* In any particular examples, it will be necessary to put for y its value deduced, in terms of a?, from the equation of the gener- ating curve. 484. Let ABCD be the vertical side of a vessel, EFGH a Fig.235. rectangular notch in it, and let IL i I be a rectangular parallelo- gram whose breadth li is infinitely small compared with EG. The velocity with which the fluid would escape at GH, is to the velocity with which it would escape from IL i /, as \/EG to \SEL> Discharge of Fluids through Apertures. 375 and the quantities of fluid discharged in a given time through 268. indefinitely small parallelograms at these depths are in the same ratio. But the parabolic curve EKH being drawn, having EG for its axis, we have EG : El :: GH : IK*, and consequently, Trig.l7 \/EG : VET :: GH : IK\ whence the quantities discharged through indefinitely small par- allelograms at the depths EG, El, are to each other as the ordi- nates GH, IK, and the sum of all the quantities discharged through all the parallelograms of which the rectangle EFGH is composed, is to the sum of all the quantities discharged through as many parallelograms at the depth EG, as the sum of all the elements IKk i of the parabola, to the sum of all the correspond- ing elements IL I i of the rectangle ; that is, as the area of the parabola EKHG to the area of the rectangle EFGH-, in other words, the quantity running through the notch EFGH is to the quantity running through an equal horizontal area at the depth EG, as EKHG to EGHF, that is, as 2 to 3. Therefore the cal. 94. mean velocity of the fluid in the notch is equal to two thirds of that at the greatest depth GH. 485. If a small aperture be made in the side of a vessel kept pi 2g6 filled to the same height, the fluid will spout out horizontally with the velocity acquired by a heavy body in falling freely through the height of the fluid above the aperture, and this velocity combin- 477. ed with the perpendicular velocity arising from the action of gravity, will cause each particle, and consequently the whole jet to describe a parabola. Now the velocity with which the fluid 303. is expelled from any aperture, as G, is such as would, if uniform- ly preserved, carry a particle through a space equal to 2 jBG 266- in the time of its natural descent through BG ; accordingly, if the direction of the aperture be horizontal, the action of gravity being at right angles to it will cause the particle to descend through the height GD in the same time that would be required in case of a natural descent through GD, if no other force were 376 Hydrodynamics. exerted upon the particle. Hence the squares of the times be- ing as the spaces, or the times simply as the square roots of the spaces, s/BG is to \,'GD as the time employed in describing BG to the time required to reach the horizontal plane DF. But in the time employed in describing UG, the particle would be car- ried uniformly and horizontally by the velocity thus acquired, through a space equal to 2 BG ; therefore, to find the amplitude or horizontal range DE of the jet, we have the proportion 215. = 2 \/BG -GD = 2 GH. As the same reasoning may be used with respect to any other point in J5D, if upon the height of the fluid BD as a diameter we describe a semicircle BKD, the horizontal distance to which the fluid will spout from any point will be twice the ordinate of the circle drawn through this point, the distance being measured on the plane of the bottom of the vessel. 486. It will hence be perceived, that if apertures be made at equal distances G, L, from the top and bottom of the vessel, the horizontal distances DE to which the fluid will spout from these apertures will be equal ; and that the point 7, bisect- ing the altitude, is that from which the fluid will spout to the greatest distance, this distance DF being equal to twice the radi- us of the semicircle or to the altitude BD of the fluid. 487. If the fluid issue obliquely instead of horizontally, the curve described will still be parabolic, and the horizontal range, &c. of the jet may be calculated as in the case of other projectiles. Fig.237. Let the aperture C be inclined, for example, upward at different angles. CB will be equal to s, the space through which a body must fall to acquire the velocity of projection, and equal to the dis- 477. tance CF, CF', of the foci of the several parabolas, traced by particles issuing with different angles of elevation. Hence BE Trlg.172 is the directrix to these parabolas, and the circle described from the centre C, and with the radius BC, will pass through the sev- eral foci F, F x , &c. Let CE, for instance, be the direction of the jet, and draw CF making the angle ECF equal to BCE-, let Motion of Gases. 377 fall the perpendicular FH, and take HG equal to HC, the dis- tance CG will be the horizontal range of the jet. But CG '= 2 CH - 2 CF X cos FCH = 2 O# X sin 2 EOF. Therefore, when the angle of elevation is 45, the focus of the parabola falls on the horizontal line at F', and the range CK is then the greatest possible, being double the altitude CB. Of the Motion of Gases. 488. To determine with what velocity the air or any other gas will rush into a void space, when urged by its own weight, we proceed according to a method analogous to that by which the moiion of liquids is determined. When the moving force and the mass or matter to be moved vary in the same proportion, the velocity will continue the same, since v . 28. Thus, if there be similar vessels of air and water, extending to the top of the atmosphere, on the supposition of a uniform density throughout, they will be discharged through equal and similar apertures with the same velocity; for in whatever pro- portion the quantity of matter moving through the aperture be varied by a change of density, the pressure which forces it out acting in circumstances perfectly similar will vary in the same proportion. Hence it follows that the air rushes into a void with the velocity which a heavy body would acquire by falling from the top of the atmosphere, this fluid being supposed to be of a uniform density throughout. The height of a uniformly dense or homogeneous atmosphere being 27807 feet, according to article 467, and g = 32,2, we shall have for the velocity in question v = \S2gh = V 2 X 32,2 X 27807 = 1338. m 489. But as the space into which the air rushes becomes more and more filled with air the velocity must be diminished contin- ually. Indeed whatever be the density of this rarer air, its elas- ticity varying with its density, will balance a proportional part Mech. 48 Hydrodynamics* of the pressure of the atmosphere, and it is the excess of this pressure only which constitutes the moving force, the matter to be moved being the same as before. Let D be the natural den- sity of the atmosphere, and A the density of that which opposes itself to the motion in question. Let p be the pressure of the atmosphere, or the force which impels it into a void, and -or the force with which this rarer air would rush into a void ; from the proportion D : A : : p : & = , D we shall have for the moving* force sought p - Again, let v be the velocity of air rushing into a void under the pressure p, and u the velocity of air under the same pressure rushing into rarefied air of the density A. Since the pressures are as the heights producing them, the fluid being supposed of a uniform density through, we shall have *:::v^: HZ^ : : 1 : I73F. \ l D \ D whence it = v X 11 , no allowance being made for the *J D inertia of the rarer air, which being displaced must oppose a certain resistance. 490. Let it be proposed to determine the time t in seconds in which the air will flow into a given exhausted vessel, until the air shall have acquired in the vessel a certain density A. Suppose h the height due to the velocity v, b the bulk or capacity of the vessel, and 6 the area of the aperture, the meas- ure in each case being in feet. Since the quantity of air neces- sary to fill the vessel will depend upon the size of the vessel, and also upon the density of the air, b A will represent this quantity, the differential of which is b d A. The velocity of influx at the first instant is v = \/2 g h ; and when the air in the vessel has acquired the density A, that is, at the end of the time <, the velocity is u = \/2gh Motion of Gases. 379 Hence the rate of influx, which may be measured by the infi- nitely small quantity of air passing the aperture during the instant d t with this velocity, will be denoted by D A X D0d t = 6dt+/2gi>h(D A). Putting these two values of the rate of influx equal to each other, we have tf dt A/2g-i>/i(D A) 6dA, and d t = , -- X > A Hence, by integrating, we obtain t = , ,- = X ^~^TA + C. i 6 V 2 g D h To find the constant C, it will be observed, that when t = 0, A = 0, and \/ D A = A/ D . We have, therefore, for the cor- rected integral t = - - /f X ( 4/D - \/D A \^ 491. When D =: A, the motion ceases, and the value of f, or the time of completely filling the vessel, becomes b b i or 7~7r =r T' or Art /I' nearl 7' Suppose, for example, the capacity of the vessel to be 8 cubic feet, or nearly a wine hogshead, and that the aperture by which air of the ordinary density, or 1, enters, is an inch square, or T i of a foot. In this case 4 A/ ~h 4 A/27807 = 668, nearly ; and hence ( = = = J "' 72 nearl y- 380 Hydrodynamics. If the aperture be only T i^ of a square inch, or the side T V, the time of completely filling the vessel will be 172" nearly, or a little less than 3'. If the experiment be made with an aperture, cut in a thin plate, we shall find-the time greater nearly in the ratio of 62 or 479t 63 to 100, as we have already remarked with respect to water flowing through small orifices. 492. We can find, in like manner, the time necessary for bringing the air in the vessel to any particular density, as of that of air in its ordinary state. For the only variable part of the integral, above found, is \/ D A, which in this case becomes >y/l | rr i, and gives y'o \/D A = J; hence, if the aperture were a square, each side being y 1 ^ of an inch, the time sought would be 172", or 86", nearly. 493. If the air in the vessel be compressed by a weight acting on the moveable cover AD, the velocity of the expelled Fig. 238 air may be determined thus. Let the additional pressure be denoted by q, and the density thence resulting by D' ; we shall then have p : p + q : : D : D', and P'P + qP :: D : D' D, which gives D' D Now, since the pressure which expels the air is the difference between the force which compresses the air in the vessel and that which compresses the internal air, the expelling force is q ; whence, the forces being as the quantities of motion, D' D p : p X -- : : m v : n w, m, , being the masses expelled, v the velocity with which air rushes into a void, and u the velocity required. But the masses or number of particles which issue through the same orifice in Motion of Gases. 381 an instant, are as the densities and velocities conjointly ; hence m v : n u : : D v 2 : D' u 2 , and consequently, X P/ "~ P - 'DV 2 ' D which gives ID' D u v / ^ D Moreover, from the proportion p : p -f- q : : D : D X , we obtain D (^ -f- q) and D (p -4- q) Whence DJ % D' D p p ' " P (P + g) : JP + g' P \ Substituting this value of - - in the above expression for w, we have the following simple and convenient formula, namely, I P u = v - t . >JjP + g 494. We have taken no notice of the effect of the air's elasti- city upon the velocity of influx into a void. Let JlBCD be a vessel containing air of any density D. This air is in a state of compression, and if the compressing force be removed it will ex- pand, and its elasticity will diminish with the density. Now its elasticity, in whatever state, is measured by the force which 382 Hydrodynamics. keeps it in that state ; and the force which keeps common air at its ordinary density is the pressure of the atmosphere, and equiv- alent to that of a column of mercury 30 inches in height. If, therefore, we suppose this air, instead of being confined by the top of the vessel, to be pressed down by a rnoveable piston car- rying a cylinder of mercury of the same base and 30 inches high, its elasticity will balance this pressure just as it does the pressure of the atmosphere ; and since from its fluidity the pres- sure received on any one part is propagated through every part and in every direction, it will press on any small portion of the vessel by its elasticity, as when loaded with this column. Hence, if this small portion of the vessel be removed leaving an opening into the void, the air will begin to flow out with the same velo- city as it would flow out when pressed by its own weight only, or with the velocity acquired by falling from the top of a ho- mogeneous atmosphere, or 1338 feet per second. But as soon as a portion of air had passed through the orifice, the density of that remaining in the vessel being reduced, its elasticity, and consequently the expelling force, is diminished. But the matter to be moved is diminished in the same proportion as the density, the capacity of the vessel remaining unchanged ; therefore, since the density and elasticity follow the same law, the mass moved will vary as the moving force, and the velocity will continue the same from the beginning to the end of the efflux. 495. The velocity with which the air issues out of a vessel under the circumstances above supposed, being constant, we can readily compare the velocity given by the theory with that Fig. 239. found by experiment. Let A be a cask of known capacity in the top of which is an aperture a of a known area. The tube TB, recurved at $, is soldered or screwed into the top of the cask. The aperture a is stopped while water is poured into the tube T till it is full, at which time a quantity of water will have passed out at B condensing the air in the cask till its spring is equal to the weight of the water in the tube. At this time a cock placed over the tube J 1 , sufficiently large to supply water as fast as it can descend into the vessel A, is to be opened to keep the tube constantly filled. For this purpose one person must constantly tend it, while another opens the aperture , which needs only to be closed with the finger, the seconds being Motion of Gases. 383 counted from the moment the finger is removed till the water flics out at a. Hence, knowing the capacity of the vessel and the area of the aperture, we obtain the velocity. If the tube TB should be continued nearly to the bottom of A, while A was filling with water, the length of the compressing column would be gradually diminished, and consequently the pressure constantly changing. To avoid any irregularity from this cause the open end of the tube is placed as near the top of the cask as is consistent with a free passage for the water. The vessel was made to contain 1 5 lb< 6 OZ< of water, from which its capacity is found to be 425,088 cubic inches. The area of the aperture f a through which the water is discharged was 0,0046 inches. (1.) The altitude of T above the cask being 30 inches, the time of expelling the air was found by several trials to be 33". (2.) The altitude of T being 6 feet, the time of expelling the air was 21,3". In the first experiment, 425,088, the capacity of the cask, being divided by 0,0046, the area of the aperture, gives 92410,4 inches for the length of the stream continued during 33". Hence g 2410 ' 4 = 233,3 feet, the velocity per second. 12X33 From the second experiment we deduce by a similar process, 361,6 for the velocity per second; and to show the correspond- ence of this with the- first, we use the proportion : : 233,3 : 361,8, differing from the experimental result one fifth of a foot. 496. To compare the velocity thus found by experiment with that assigned by theory, we use the proportion \/6 : V34 : : 361,6 : 860,5, the velocity with which the atmosphere would begin to enter a void. Taking the result before found, namely, 1338, and multi- plying it by 0,63, agreeably to what is laid down in article 479, 384 Hydrodynamics. we shall have 842,94, differing from the experimental result about / part. 497. Let it be proposed to find the quantity of air expelled Fig. 233. into an infinite void from the aperture C of the vessel ABCD during any time , and the density of the remaining air at the end of that time. The bulk expelled during the instant d t will be 6 d t \/Zgh) the velocity ^/2 gh being constant, and consequently the quantity will be <* D 7 d t \/ c lgh. The quantity at the beginning of the efflux is b D, 6 being as before the bulk of the vessel ; and when the air has acquired the density D', the quantity in the vessel is b D', and the quantity expelled is 6 D 60'; consequently, the quan- tity discharged during the instant d t must be the differential of 5 D 6 D X , that is, b d D'. Hence we have the equation 6 D' d t \/2 g h = b d and _ the integral of which is h. log. denoting the hyperbolic or Naperian logarithm of r/. When t = 0, D' is equal to D ; whence b c = - : -- h. losf. D : therefore, nearly. It is obvious that no finite time will be sufficient for the ves- sel to empty itself; for, as D' must in this case be equal to zero, ^ will be infinite, and its logarithm will also be infinite ; so that t will be infinite. Motion of the Air. 385 498. It is by a train of reasoning precisely similar, that we ascertain the quantity of condensed air which will make its escape from a vessel into the atmosphere in a given time. Let A be the density of the condensed air, and h' the height of a homogeneous atmosphere corresponding to it; also let D be the density of the atmosphere. The air having for its density A, will obviously, when mixing with the atmosphere, have the same velocity as though it were rushing into a vacuum with the den- sity A D. Now the height of a homogeneous fluid corres- ponding to this density is found by the proportion A: A D:: ft' : fc' ^LZ^.. From the equation v =. \/2 g s, it will be seen that the velo- cities acquired by falling from different heights are proportional to the square roots of these heights. If therefore, v be taken to represent the velocity of common air rushing into a vacuum, and h the height of the corresponding homogeneous atmosphere, we shall have, u being the velocity belonging to the height h' - -< whence Moreover the proportion D : A : : h : h', gives therefore JA A D |A ~o - . = v - . DA \ D Let A' be the density of the condensed air after the time t, b being, as before, the bulk of the vessel, and a a section of the Mech. 49 , we shall have 6v A' dt |^llIL^ = Let L.et \/A' D =a? from which we have d A' = 2 x d a?, and hence dA' 2xdx Cai. 120. Now the integral of = is equal to multiplied by an x 2 + yV V arc whose tangent is ^, radius being 1 . Whence i H ^_? x - X an arc whose tang, is ^- ^-^ + Ct When t = 0, A' is equal to A, and c = X an arc whose tang, is \ A D 6v \ D Let the former arc be represented by a and the latter by a' ; we shall have When the time is required in which the density of the air contained in the vessel shall be reduced to that of the external atmosphere ; as A x = D, in this case, it follows that a = 0, and Resis tance of Fluids. 387 To illustrate this by an example, let it be required to find the time in which air of double the atmospheric density, confin- ed in a cubical vessel, each of whose sides is 12 feet, will expand into the atmosphere through an opening of one tenth of an inch in diameter, so as to be reduced to the common density. The above formula gives by substitution, 2 X 12 3 t =. - - X an arc whose tang, is 1, 0,007854 X 1338 9 V 1 9 3 V 1 00 z A 1Z * X an arc whose tang, is 1, 0,7854 X 1338 2 X 123 X 100 = = 4/ 1338 499. Even although the density of the confined air were infinitely greater than that of the atmosphere, the time in which it would be reduced would be a finite quantity. For A being infinitely greater than D, the IA D tan s- J-ir- is also infinite, and corresponds to an arc of 90. Hence in this case we have t = X 2 X 0,7854. o v The capacity of the vessel and the area of the aperture being the same as in the last example, we should have for the time in which this infinitely condensed air would be reduced to the same density with the atmosphere x 2 x ' 7854 = 561 " = 8/ 36// 388 Hydrodynamics. Of the Resistance of Fluids to Bodies moving in them. 500. THE force with which solid bodies moving in fluids, as water, air, &c., are impeded and retarded, is usually termed the resistance of fluids ; and as all our machines move either in water or in air, or both, it becomes a matter of importa ce in the theory of mechanics to inquire into the nature of this kind of force. We know by experience that force must be applied to a body in order that it may move through a fluid, such as air or water ; and that a body projected with any velocity is gradually retard- ed in its motion, and generally brought to rest. We also know that a fluid in motion will hurry a solid body along with it, and that force is necessary to maintain the body in its place. And as our knowledge of nature teaches us that the mutual actions of bodies are in every case equal and opposite, and that the ob- served change of motion is the only indication and measure of the changing force, we infer that the force which is necessary to keep a body immovable in a stream of water, flowing with a certain velocity, is the same with that which is requir- ed to move this body w r ith an equal velocity through stagnant water. A body in motion appears to be resisted by a stagnant fluid, because it is a law of mechanical nature that force must be employed in order to put any body in motion. Now, the body cannot move forward without putting the contiguous fluid in motion, and force is to be used to produce this motion. In like manner, a quiescent body is impelled by a stream of fluid, because the motion of the contiguous fluid is diminished by this solid obstacle ; the resistance, therefore, or impulse, differs in no respect from the ordinary communication of motion among solid bodies, at least in its nature ; although it may be far more difficult to reduce the various circumstances attending it to accu- rate computation, or to obtain all the requisite data on which to found the calculation. Resistance of Fluids. 339 501. The resistance which a body suffers from the fluid medium through which it is impelled depends on the velocity, form, and magnitude of the body, and on the inertia and tenacity of the fluid. For fluids resist the motion of bodies through them T (1 .) by the inertia of their particles ; (2.) by their tenacity, that is, the adhesion of those particles ; (3.) by the friction of the body against the particles of the fluid. In perfect fluids the two last causes of resistance are very inconsiderable, and therefore are not taken into the account ; but the first is always very con- siderable, and obtains equally in the most perfect as in the most imperfect fluids. And that the resistance varies with the velo- city, shape, and magnitude of the moving body is sufficiently obvious. We must carefully distinguish between resistance and retard- ation ; resistance is the quantity of motion, retardation the quan- tity of velocity, which is lost ; therefore, the retardations are as the resistances applied to the quantities of matter ; and in the same body the resistance and retardation are proportional. ' 502. To determine the force of fluids in motion, or the resist- ance of fluids against bodies moving in them. (I.) In fluids uniformly tenacious, the resistance is as the velocity with which the body moves. For, since the cohesion of the particles of the fluid is always the same for the same space, whatever be the velocity, the resistance from this cohesion will be as the space described in a given time ; that is, as the velocity. (2.) In a fluid whose particles move freely without disturb- ing each other's motions, and which flows in behind as fast as a plane body moves forward, so that the pressure on every part of the body is the same as if the body were at rest, the resist- ance will be as the density of the fluid. (3.) On the same hypothesis the resistance will be as the square of the velocity. For the resistance must vary as the number of particles which strike the plane in a given time, mul- tiplied into the force of each against the plane ; but both the number and the force is as the velocity, and consequently the resistance is as the square of the velocity. 390 Hydrodynamics. This proof supposes that after the body strikes a particle, the action of that particle entirely ceases; whereas the particles, after they are struck, must necessarily diverge, and act upon the particles behind them ; thus causing some difference between theory and experiment. This hypothesis, however, on account of its simplicity, is generally retained, and corrected afterwards by deductions from actual experiments. This ratio of the square of the velocity may be otherwise derived, thus. It is evident, that the resistance to a plane, moving perpen- dicularly through an infinite fluid, at rest, is equal to the pressure or force of the fluid on the plane at rest, the fluid moving with the same velocity, and in the contrary direction to that of the plane in the former case. But the force of the fluid in mo- tion must be equal to the weight or pressure which generates that motion ; and which, it is known, is equal to the weight or pressure of a column of the fluid, whose base is equal to the plane, and its altitude equal to the height through which a body must fall by the force of gravity, to acquire the velocity of the fluid ; and that altitude is, for the sake of brevity, called the alti- tude due to the velocity. So that if tf denote the surface of the plane, v the velocity, and s the specific gravity of the fluid ; then v 2 the altitude due to the velocity v being , the whole resistance o or moving force m, will be g being 32,2 feet. And hence, other things being the same, the resistance is as the square of the velocity. (4.) If the direction of the motion, instead of being perpen- dicular to the plane, as above supposed, be inclined to it at any angle, then the resistance to the plane in the direction of the motion, as assigned above, will be diminished in the triplicate ratio of radius to the sine of the angle of inclination, or in the ratio of 1 to t 3 , where i is the sine of the inclination. Resistance of Fluids. 391 For AB being the direction of the plane, and BD that of the Fig. 240. motion, ABD the angle whose sine is i ; the number of particles or quantity of the fluid which strikes the plane will be diminish- ed in the ratio of 1 to i ; and the force of each particle will likewise be diminished in the same ratio ; so that on both these accounts the resistance will be diminished in the ratio of 1 to i 2 ; that is, in the duplicate ratio of radius to the sine of ABD. But fur- ther, it must be considered that this whole resistance is exerted in the direction BE perpendicular to the plane ; and any force in a direction BE is to its effect in a direction AE, parallel to BD, as AE to BE, or as 1 to i. Consequently, on all these accounts, the resist- ance in the direction of the motion is diminished in the ratio of 1 to i 3 . And if this be compared with the result of the preced- ing step, we shall have for the whole resistance, or the moving force, on the plane, m = s v 2 5 (5.) If w represent the weight of the body whose plane sur- face tf is resisted by the absolute force m, then the retarding force f _ m __ as v 2 i 3 J ~~ w (6.) And if the body be a cylinder whose surface or end is d, and diameter D, or radius R, moving in the direction of its axis; then, because i= 1, and tf = n R 2 i n D 2 , where TT = 3,141593, the resisting force m will be 71 S D 2 V 2 71 S R 2 V 2 ^ ~~~ and the retarding force f n s o 2 v 2 / (7.) This is the value of the resistance when the end of the cylinder is a plane perpendicular to its axis, or to the direction of the motion. But were its face a conical surface, or an elliptic section, or any other figure every where equally inclined to the axis, the sine of the inclination being i ; then the number of parti- cles of the fluid striking the surface being still the same, but the force of each, opposed to the direction of the motion, diminished 392 Hydrodynamics. in the duplicate ratio of the radius to the sine of the inclination, the resisting force m would be 71 S D 2 V 2 i 2 71 S R 2 V 2 i 2 But if the body were terminated by an end or surface of any other form, as a spherical one, where every part of it has a dif- ferent inclination to the axis ; then a further investigation be- comes necessary. 503. To determine the resistance of a fluid to any body moving in it, having a curved end as a sphere, a cylinder with a hemispheri- cal end, fyc. Fig. 241. 1. Let BEAD be a section through the axis CA of the solid, moving in the direction of that axis. To any point of the curve draw the tangent EG, meeting the axis produced in G ; also draw the perpendicular ordinates EF, e/, indefinitely near to each other ; and draw a e parallel to CG. Putting CF = a?, EF = t/, BE = z, t = sine of the angle G, radius being 1 ; then 2 ny is the circumference whose radius is EF, or the circumference described by the point E, in revolving about the axis CA; and 2^t/ x EC, or %7iy d z, is the differential of the surface, or ir is the surface described by E c, in its revolution about CA ; hence $v 2 i* ^ j 7i s v 2 i 3 X Znydz, or - X ydz is the resistance on that ring, or the differential of the resistance to the body, whatever the figure of it may be ; the integral of which will be the resistance required. (2.) In the case of a spherical shape ; putting the radius CA or CB = R, we have / o ^ EF CP x J, = V(R.. -*), = - = - = -, and ?/eZzorEF X Ee = cEX ae = ndx-, therefore the general differential 3TS W 2 Resistance of Fluids. 393 becomes 71 SV 2 X 3 71 SV 2 j M '1- . . R d a? = - . x* dx; g R' g* 2 the integral of which, or ^-^- a? 4 , is the resistance to the sphcri- 4^R 2 cal surface generated by BE. And when a; or CF is = R or CA, it becomes 7CSV *'- for the resistance on the whole hemisphere ; 4# which is also equal to n * v . D where D = 2 R, the diameter. 103 (3.) But the perpendicular resistance to the circle of the same diameter D or BD, by section 6 of the preceding problem, is 71 s v * p2 . which being double the former, shows that the resist- *g ance to the sphere is just equal to half the direct resistance to a great circle of it, or to a cylinder of the same diameter. (4.) Since |#D 3 is" the magnitude of the globe; if s' denote its density or specific gravity, its weight w will be = wD 3 /, and therefore the retarding force m __ 7i s v 2 D 2 6 3sv 2 ^ * F W ~ l6g~ ' TTS'D 3 "~ SgS* D ' which is also equal to v 2 2gs' Hence 3s l 4s^>~ P and which is the space that would be described by the globe while its whole motion is generated or destroyed by a constant force equal to the resistance, if no other force acted on the globe to continue its motion. And if the density of the fluid were equal to that of the globe, the resisting force sufficient would be act- ing constantly on the globe without any other force, to generate or destroy the motion while describing the space f D, or of its diameter. Mech. 50 394 Hydrodynamics. (5.) Hence the greatest velocity that a globe acquires in de- scending through a fluid, by means of its relative weight in the fluid, will . be found by putting the resisting force equal to that weight. For, after the velocity has arrived at such a degree that the resisting force is equal to the weight that urges it, it will increase no longer, and the globe will afterwards continue to descend with that velocity uniformly. Now, s' and s being the specific gravi- ties respectively of the globe and fluid, s 7 s will be the rela- tive gravity of the globe in the fluid, and therefore w i^D 3 (s 7 s) is the weight by which it is urged ; also 71 S V 2 D 2 m = ~^j- is the resistance ; consequently 71 S V 2 D 2 , , _ = 1* D (S'_S), when the velocity becomes uniform ; from which equation is found I/ S' Sv v = (2 . f D . <^\ e s for the above uniform motion or greatest velocity. By comparing this with the general equations, v = \/g s, it will be seen that the greatest velocity is that acquired by the accelerating force , in describing the space f D, or that acquired by gravity in describing freely the space s 7 s If s' = 2 s, or the specific gravity of the globe be double that s' - s of the fluid, = 1 = the natural force of gravity ; and the globe will attain its greatest velocity in describing f D or f- of its diameter. It is further evident, that if the globe be very small, it will soon attain its greatest velocity, whatever its density may be. Resistance of Fluids. 395 If a leaden ball, for example, one inch in diameter descend in water, and in air of the usual density at the earth's surface, the specific gravities of these bodies being 11,3, 1, 0,00122, res- s / s pectively, since f of an inch is fa or O? 11 ? f a f ot > ~ ^e- comes 10,3 in the case of water, and 11,8-0,00122 = 926 ^ 0,00122 in the case of air, we shall have v = V 2 X 32,2 X 0,11 X 10,3 = 8,54 nearly, for the greatest velocity the ball can acquire per second in water ; and V/2 X 32,2 X 0,11 X 926 = 256 nearly, for the greatest velocity in air. But if the globe were only one hundredth of an inch in diam- eter, the greatest velocities would be only T V of the above re- sults, or 0,85 of a foot in water, and 25,6 in air ; and if the ball were still further diminished, the greatest velocity would be di- minished also, in the subduplicate ratio of the diameter of the ball. This is well illustrated in the fall of rain. The different sized drops descend with different degrees of rapidity, but all so gently as to cause no injury. Were this fluid so constituted as to allow the drops to form in larger masses, or were the air much less dense, tender vegetables would suffer by the shock, as they sometimes do in fact by the more rapid descent of hail. 504. It appears from the third step of the preceding article, that the resistance to the motion of a cylinder moving in the direction of its axis is double that of a globe of the same diame- ter ; and in experiments with bodies that move slowly, this will nearly hold true in water, but still more nearly in air ; because its particles move more freely than those of water, and less dis- turb each other's motions ; but when the motion is more rapid, considerable aberrations will occur, both from the mutual dis- turbance of the particles, and from the fluid not flowing in so fast behind as the body moves forward ; in the air also, a new cause of deviation will arise, from the condensation of the fluid before 396 Hydrodynamics. the body. Sir Isaac Newton supposes, that in a continuous non- elastic fluid, infinitely compressed, the resistances of a sphere and cylinder of equal diameters are equal ; but this appears to be an error in theory as well as in fact. When the motion is slow in water, the fluid may be conceived to be nearly of the nature which Newton supposes; yet the resistances are almost as coincident with theory as when the motion is in air ; thus M. Borda found the resistance of a sphere moving in water to be to that of its greatest circle as 1 to 2,508, and in air the resistances were as 1 to 2,45. The experiments of Dr. Hutton in air give the resistances at a mean as 1 to 2i. The reason that experiment gives the ratio of the resistances greater than that of 2 to 1 seems to be this ; in theory it is supposed that the action of every particle of the fluid ceases the instant it makes its impact on the solid ; but this is not actually the case, as we have before observed ; and since the particles, after impact on the sphere, slide along the curved surface, and hence escape with more facility than along the face of the cylinder, the error will be greater in the cylinder; that is, the greater resistance will ex- ceed the theory more than the less. It is also to be observed, that the difference between the resistances of the globe and cyl- inder in water is greater than in air ; and this is directly contrary to what might be inferred from Newton's reasoning, which sup- poses them equal in a continuous fluid, but in the ratio of 1 to 2 in a rare fluid. 505. To determine the relations of v^o r 'fv. space, and time, of a ball moving in a fluid in which it is projected with a given velocity. T.etw = the velocity of projection, s the space described in any time t, and v the velocity acquired. Now, by step 4, arti- cle 503, the accelerating force / = ; where s' is the den- sity of the ball, s that of the fluid, and D the diameter of the 282. ball. Therefore the general equation v dv = gfds becomes and hence dv 3s, , =: . ds = c d 5, v 8 s' D Resistance of Fluids. 397 q putting c for - . The correct integral of this is log. u log. r, or 8 S 7 D log. - = c s. Or putting e = 2,718281828, the number whose hyp. log. is 1, then - = e cs , and the velocity The velocity v at any time being the e~ cs part of the first velocity, the velocity lost in any time will be the 1 e~ cs part, or the e ^ part of the first velocity. (1.) If a globe, for example, be projected with any velocity in a medium of the same density with itself, and it describe a space equal to 3 D or 3 of its diameters ; then s = 3 D, and 3 s J5_ ^ ~ SS'D ~~ SD' therefore c s = f , and the velocity lost is c 08 1 __ 2,08 e cs ~ 3,08 ' or nearly f of the projectile velocity. (2.) If an iron ball of two inches diameter were projected with a velocity of 1200 feet per second ; to find the velocity lost after moving through any space, as 500 feet of air ; we should have D = T \ = i, u = 1200, s = 500, s' = 71, s = 0,0012 ; and therefore and __ 3 ss _ 3.0,0012.500 _ 3.12.500.3.6 _ 81 " 8 s' D " ~8 . 7 . ~ 8 . 22 . 10000~~ " 440' 1200 v - - = 998 feet per second ; having lost 202 feet, or nearly 1 of its first velocity. X 398 Hydrodynamics. (3.) If the earth revolved about the sun, in a medium as dense as the atmosphere near the earth's surface ; and it were required to find the quantity of motion lost in a year ; since the earth's mean density is about 4, and its distance from the sun 1 2000 of its diameters, we have 24000 X 3,1416 = 75398 diameters = s, and _3. 75398. 12. 2 ~87T0000.9 ' hence ^ = Illf parts are lost of the first motion in the space of a year, and only the T Vr P art remains. (4.) To find the time t ; we have ds ds e**ds at =. = = . 580. v u u Now, to find the integral of this, put z = e M ; then is c s = log. c, and , dz dz cds = , or a s = ; z cz consequently and ; hence , e cs d s z d z u u ry />CI * 1_ I -f- uc u c _ dz ~ uc c. But as t and s vanish together, and when 5 = 0, the quantity is equal to ; uc we' therefore ^ 6 es i ^ i i ^ i u c ~ cv cu c \v the time sought ; where c = 57-9 an( ^ v == "^ ^ e velocity. Resistance of Fluids. 399 506. It is proposed to determine the angle of inclination at which a current of air acting upon the sails of a windmill will pro- duce the greatest effect. Let CD represent the velocity and impulsive force of the F 'g- 24 2- wind against a plane whose section is AB. By reducing this force into the direction CE, we obtain the efficient force with which the fluid strikes against the plane; and by reducing this again into the direction CH, we obtain that force which imparts a rotatory motion to the sail. If CD be called a, and the angle CDE, a?, then CE = a sin a?, and CH a sin a; cos x. But as the number of particles which strike against the plane is also diminished on account of its incli- nation in the ratio of rad. to sin a?, it follows that the action of the fluid against the plane is expressed by a sin x 2 cos x. To find when this is a maximum, we have, by taking the dif- ferential, 2 a sin a? cos x 3 d x a sin a? 3 d x = 5 or 2 cos x 2 sin x 2 = ; hence sin # = V2 cos a?, and - or tang a?, radius being 1 , = y2= 1 ,4 1 4. Now 1,414 is the tangent of 54 44' ; therefore x = 54 44'. It is obvious that this determination is obtained upon the sup- position that the plane is at rest. We shall now proceed to in- quire at what angle the impulsive force produces the greatest effect, when the plane has acquired a determinate velocity. Let CD, as before, represent the direction and impulsive force Fig, 243. of the wind, DG the direction and velocity of the sail's motion. By reducing the velocity of the wind and sail into the direction AB perpendicular to the plane, the relative velocity of the wind, or that with which it strikes the plane, is ED DL. Hence the effect of the wind in producing a rotatory motion will be express- ed by EH DK. 400 Hydrodynamics. Let CD, as before, be represented by , and DG by c, then EH = a sin x cos #, and DK = c cos x 2 . Hence the impulsive force is expressed by a sin x cos x c cos x 2 . But, as before, the number of particles of fluid which strike against the plane is diminished in the ratio of rad. to sin a?, wherefore the whole force against the sail will be expressed by (a sin x cos x c cos x 2 ) sin x. Let sin a; = */, then cos x = y'l y 2 , and (a sin a? cos a? c cos a: 2 ) sin a? = (ay \/l y 2 c(l y 2 )) y> By taking the differential of this quantity we obtain y* c (1 y 2 )) dy = ; and dividing by dy, we have . or But fZT^ tan * 2 : wherefore 1 \ c 2 tang x 2 + ^3 . ^) - tang a? = o ; \ sm x^x a and, by reducing this equation of the second degree, we have 1 tang x = 3 r sm X Resistance of Fluids. 401 On the supposition that c = a, or, in other words, that the velocity of the sail is equal to that of the wind, we have tang x = Assuming as an approximation, in order to find x from this equation, that its value is 54 44', we obtain as a nearer approxi- mation x = 65 24'. Taking the sine of this angle, and substi. luting it as an approximation in the second number of the equa- tion, we find, as a still more correct approximation, x = 68 54'. Substituting this again for a?, we find, as a fourth approximation, x = 69 4', which we consider as the correct value. Let us again suppose that c = 2 a, or that the velocity of the sail is double the velocity of the wind, then the formula becomes tang* = (2+ (3 v-i \ \ dn 3 sin Substituting 75 as the value of a?, we obtain as a first ap- proximation x = 77 3'. Substituting again this value, we deduce as a second approximation x = 77 7', which may be considered as the true value of x in this case. The theory of winds has not arrived at that degree of per- fection which renders science subservient to the purposes of life. We can as yet only attribute them in a vague manner to the operation of a few general causes. From what we know of the expansibility of air by heat, one may easily conceive how the atmosphere must always be in a state of commotion, some parts of it being exposed to an intense temperature, while other are comparatively cold. The portion which encircles the torrid zone, by receiving the rays of a vertical sun, is much rarer than that which surrounds the higher latitudes. Hence, in order to maintain a due equilibrium, the aerial column at the equator must be higher than it is in colder climates. This was found to be the case by Mr. Casson, who observed that for the same ele- vation, the barometer falls only about half as much in the torrid as it does in the temperate zones. Mech. 51 402 Hydrodynamics, The air being thus expanded by means of heat, it is displaced by means of the denser air which rushes in from higher latitudes. The winds which are occasioned in this manner are called the trade winds, and are so uniform in their action, that they render the greatest service to those who navigate the tropical seas. From October to May, the north-east trade wind continues to blow, and from April to October the south-west. M. De Luc has advanced a very ingenious explanation of the oblique direction in which these winds approach the equator. This he has attributed to the rotation of the earth. The air at any place, when apparently calm, or in a state of rest, possesses obviously the same velocity of rotation with which that place revolves. Now the rotatory velocity of any point on the earth's surface is proportional to the parallel of latitude which passes through it, that is, to the cosine of latitude, as may be easily shown; hence, were the whole atmosphere in a tranquil state, the actual velocity of every section of it would be proportional to the cosine of its angular distance from the equator. Let us now suppose, that from the influence of heat the condensed air of the north begins to move towards the south. Its rotatory mo- tion being the same with that of the region which it has left, is less than that of the place into which it has come. Hence, as the direction of the earth's motion is from west to east, it will appear to blow from east to west with a relative velocity equal to the difference between the velocities of the parallel left and that come to. TJie same explanation obviously applies to the westerly direction of the wind which rushes towards the equator from the south. In confirmation of this theory, it may be ob- served, upon the authority of Mr. Dalton, that in sailing north- wards, the trade-winds veer more and more from the east towards the north, so that about their limit they become nearly N. E. ; and vice versa, in sailing southwards they Become at least almost S. W. These winds are seldom perceptible in a higher latitude than 30. Their place of concourse appears to be a little to the north of the equator, from the circumstance that the sun, the source of heat, is longer in the northern than in the southern hemisphere. Near the equator they blow about. 4 from the east or west points. Air-Pump. 403 This influx of air towards the equator appears to be counter- balanced by a current in the higher regions of the atmosphere towards the poles. The heated air at the equator continues to ascend until it reaches an altitude where its elasticity exceeds that of the air which surrounds it. It then begins to diffuse itself towards the poles, maintaining an equilibrium in the atmos- phere, and moderating the intense cold of the climates towards which it is wafted. Such would be, perhaps, the only fluctua- tions in the atmosphere were the whole globe covered with water, or the variations of heat in the earth's surface regular and con- stant, so as to be the same every where over the same parallel of latitude. As it is, however, we find the irregularities of heat, arising from the interspersion of sea and land, to be such that though all the parts of the atmosphere in some sort conspire to produce regular winds round the torrid zone, yet the effect of land is so great, that striking inequalities are produced ; witness the monsoons, the sea and land breezes, &c., which can be accounted for upon no other principle than that of rarefaction, because the rotatory velocity of different parallels in the torrid zone is nearly alike. On the Theory of the Air Pump, and Pumps for raising Water, 507. THE Air-pump is a machine fitted to exhaust the air from a proper vessel, and thus to produce what is called a va- cuum ; it is one of the most useful of philosophical experiments. By means of it the chief propositions relative to the weight and elasticity of the air are proved experimentally, in a simple and saiisfactory manner. EFGH represents a square table of wood, Ji, A two strong bar- Fig. 244. rels or tubes of brass, firn.ly retained in their position by the cross-piece T J 1 , which is pressed on them by screws O, O, fixed on the tops of the brass pillars JV, N. These barrels communicate with a cavity in the lower part D of the table. At the bottom within each barrel is fixed a valve, opening upwards ; and in each barrel a piston works, having a valve likewise opening up- 404 Hydrodynamics. wards. The pistons are moved by a cog-wheel in the piece TT, turned by the handle B, of which wheel the teeth catch in the racks of the pistons C, C. PQ is a circular brass plate, having near its centre the orifice K of a concealed pipe that communi- cates with the cavity at D; at V is a screw that closes the orifice of another pipe, and which is turned for the purpose of admitting the external air when required. LM is a glass vessel, from which the air is to be exhausted, and which has obtained the name of receiver, because it receives or holds the subjects on which the experiments are to be made. This receiver is placed on the plate PQ, and is accurately fitted to it by grinding, or by means of moistened or oiled leather. When the handle B is turned, one of the pistons is raised and the other depressed ; consequently a void space is left between the raised piston and the lower valve in the corresponding barrel; the air contained in the receiver LM communicating with the bar- rel by the orifice K immediately raises the lower valve by its spring, and expands into the void space ; and thus a part of the air in the receiver is extracted. The handle then, being turned the contrary way, raises the other piston, and performs the same operation in the barrel containing it ; while in the mean time the first mentioned piston being depressed, the air by its spring closes the lower valve, and raising the valve in the piston makes its escape. The motion of the handle being again reversed, the first barrel again exhausts, while the second discharges the air in its turn ; and thus during the whole time the pump is worked, one barrel exhausts the air from the receiver, while the other discharges it through the valve in its piston. Hence it is evi- dent, that the air can never be entirely exhausted ; for it is the elasticity of the air in the receiver that raises the valve, and forces it into the barrel t ; and eaeh operation can only take away f Various contrivances have been adopted to facilitate the mo- tion of the valve, and thus to allow the air, when in a state of great rarefaction, to pass from the receiver into the barrels. Smeaton had recourse to a valve presenting a broad surface, and supported on thin bars. Others have proposed to raise the valve mechanically by connecting it with the piston in such a manner that the piston shall exert its action at the moment it reaches the top of the barrel. A Air-Pump. 405 a certain part of the remaining air, which is in proportion to the quantity before the stroke, as the capacity of the barrel to the sum of the capacities of the barrel, receiver, and communicating pipe. 508. Now if we suppose no vapour from moisture, &c,, to rise in the receiver, the degree of exhaustion after any number of strokes of the piston may be determined by knowing the res- pective capacities of the barrel and of the receiver, including the pipe of communication &c. For, as we have seen above, that every stroke diminishes the density in a constant proportion, namely, as much as the whole capacity exceeds that of the cylin- der or barrel ; the exhaustion will go on in a geometrical pro- gression, the ratio of which is the same as that which the sum of the capacities of the receiver and barrel bears to that of the receiver; and this ratio of exhaustion will continue until the elasticity of the included air is so far diminished by its rarefac- tion as to render it too feeble to push up the valve of the piston. Let then the capacity of the barrel, receiver, and pipe of communication together be expressed by 6 -}- r, and that of the barrel alone by 6, and let 1 represent the primitive density of the air in the pump ; we shall have & + r : r : : 1 : - = the density after 1 stroke of the piston. method suggested itself to Dr. Prince much more perfect and more simple. It dispenses with the valve entirely by extending the bar- rel downward so as to admit of the piston's descending below the opening which communicates with the receiver, and thus allowing a free introduction of air into the barrel. The air in this case is ex- pelled through a valve at the top of the barrel, opening upward. There will still be a limit however to the exhaustion ; for the air cannot be forced through the valve at the top of the barrel, unless its elasticity, and consequently its density, produced by the motion of the piston, exceeds the density of the external air. To remove this difficulty Dr. Prince enclosed the valve in question in a vessel fur- nished with a small exhausting apparatus, by which the valve was relieved from the pressure of the atmosphere. This appendage is not necessary, except for purposes of very accurate exhaustion. 40$ Hydrodynamics. r r 2 o + f ' f , , : ,, , = density after 2 strokes, 6 + r : r : : : - = density after three strokes ; and the nth power of the ratio - r , b + r or - = D, the density after n strokes. (6 + r) From which we may easily find the density after any number of strokes of the piston necessary to rarefy the air a number of times, or to give it a certain density D, the primitive density being 1 . For the above equation, expressed logarithmically, is n or n X (log. r log. (6 + r)) = log. p; consequently log. D n = log. r log. (6 + r) ' in which expression D will be a fraction. If the number of times which the air is rarefied be expressed by N, an integer, the logarithmic equation will be log. N n = log. (6 + r) log. r A further reduction of the same theorem will furnish us with the proportion between the capacities of the receiver and barrel, when the air is rarefied to the density D' by a definite number of strokes n of the piston. For since (6 + r) if we take the nth root of both members of the equation we shall have Water-Pumps. 407 Thus, if D' be equal lo TTT V4 5? an d tue number of strokes n =. 11, we shall find so that r : 6 + r : : 1 : 3, and 6 : r : : 2 : 1 Pumps for Raising Water. 509. THE term pump is generally applied to a machine for raising water by means of the pressure of the atmosphere. Of pumps there are a great many different sorts ; we shall speak only of those in most common use, and shall give merely such a general description of their construction as will enable the student to understand the principles on which their operation depends. The pumps most generally used are, the sucking pump, the lifting pump, and the forcing pump ; these have some parts in common, and particularly the pistons and valves ; they will therefore be treated in a connected way. The piston is a body ABCD of circular base, which may be Fi s- 245 - moved through the interior part of the tube or body of the pump, filling it exactly as it moves along. The sucker or valve E is moveable about a joint in such a manner as either to permit or to prevent the passage of the water, according as it presses upwards or downwards. In figures 245, 246, there are likewise valves in the pistons. FGHK is another tube joined to the body of the pump, and is generally called the pipe or sucking pipe : its lower extremity is immersed in the water, of which we sup- pose RS to be the horizontal surface. 510. The sucking pump is represented in figure 245. In this pump if we suppose a power P applied to the hendle of the piston so as to raise it from / to C. the air contained in the space DVKHGFC tends by its spring to occupy the space that the pis- 408 Hydrodynamics. ton leaves void ; it therefore forces up the valve E, and enters into the body of the pump, its elasticity diminishing in proportion as it fills a greater space. Hence it will exert on the surface GH of the water a less effort than is made by the exterior air in its natural state upon the surrounding parts of the same surface #G, HS] and the excess of pressure on the part of the exterior air will cause the water to rise in the upper pipe GK to a certain height //JV, such that the weight of the column GJV, together with the spring of the superincumbent air shall just be a counterpoise to the pressure of the exterior air. At this time the valve E closes of itself; and if the piston be lowered, the air contained between the piston and the base IV of the body of the pump, having its density augmented as the piston is lowered, will at length have its density, and consequently its elasticity, greater than that of the exterior air ; this difference of elasticity will constitute a force sufficient to push the valve L of the piston upwards, and some air will escape till the exterior and interior air are reduced to the same density. The valve L then falls again ; and if we again elevate the piston, the water will be raised higher in FGHK, for the same reason as before. Thus, after a certain number of strokes of the piston, the water will reach the body of the pump ; where being once entered, it will be forced at each stroke of the piston through the spout X ; for the water above the piston will then press upon the valve and keep it shut while the piston is rising ; so that a cylinder of wa- ter whose height is equal to the stroke OT of the piston (or the vertical distance through which it passes) will be raised by each upward motion and forced through the aperture X, provided it is of an adequate magnitude. 511. The lifting pump is represented in figure 246. Its man- ner of operation is this ; the piston PCD is here placed below the horizontal surface RS of the water, and when it is made to descend, it produces a vacuum between the valve E (which is pushed down by the exterior air) and the base CD of the piston. The weight of the water, together with that of the exterior air about R and 5, presses up the valve L, and the water passes into the body of the pump ; and when the water ceases to enter, the weight of the valve L closes it. Then, if the piston be raised, it raises all the water above it, forces up the valve , and introduc- Water-Pumps. 409 es the water into the part IVYX* When the piston is raised to its highest position, the valve E is made to close by the super- incumbent water, and retains the fluid there until, by a fresh stroke of the piston, more water is forced upwards through the valve EJ that which was before in the upper part of the pump being expelled through a proper orifice or spout in the neigh- bourhood of Jf, in order to make way for a new supply. By continuing the operation, water is delivered at every stroke of the piston. 512. "The forcing pump unites in some measure the properties of the other two. The piston ABCD, which here has no valve, being elevated, rarefies the air in the space DGHVOC, and the water rises towards K-, the subsequent descent of the piston forces some of the air in this space through the valve L ; the next ascent of the piston closes the valve L, and raises the water in GK ; and so on till the water passes through the valve E and enters the space DIOC. Then the piston being pushed down, closes the valve E, and some of the condensed air is forced through the valve L. A further stroke raises more water into the space 7)O/C, and expels more air through L. At length the water reaches L, and the subsequent strokes raise it into the tube MO m w, whence it is carried off by a spout, as in /the other pumps. Or, if this pump be closed at mn, excepting a narrow pipe P5, when the water is raised by the process just described to o r, above the bottom S of the tube, the elastic force of the compressed air in the space nrom will compel the water to issue from the aperture P in a continued stream or jet, thus forming a fire engine or artificial fountain. 513. Let us now enquire into the fundamental properties of these machines. By means of the lifting pump, water may be elevated to any height we please, provided we employ a suffi- cient force. But the estimate of this force requires various con- siderations. We must have regard to the dimensions of the pis- ton, the barrel of the pump, the height to which the water is to be raised, and the velocity with which the water is to be moved, beside the effects of friction, &c. At present, however, we shall not examine these particulars in all their extent; but shall confine ourselves to one of them. Now it is certain that the Mech. 52 410 Hydrodynamics. power necessary to raise the water to any proposed height, must at least be capable of sustaining in equilibrium the pres- sure experienced by the base of the piston when it is kept at rest, and the fluid has attained the required height. This pres- sure, then, we proceed to estimate. In general the power must be, at least, capable of sustaining the weight of a column of water which has for its base that of the piston, and for its altitude the distance between the surface Fig.246. RS of the water in the reservoir and the upper surface XY of that in the pump. For when the base DC of the piston is below the surface RS of the water in the reservoir, it is manifest that the power has not to sustain the pressure of the water contained between US and DC; because this pressure is counterbalanced by that of the water surrounding the lower part of the pump, and which is transmitted by means of the inferior orifice of the pipe. The power, therefore, has only to sustain the pressure exerted upon the surface DC by the fluid comprehended between RS and XY; which pressure is equal to the weight of a column of water whose base is CD and altitude the vertical distance be- 414. tween RS and XY. When the piston is above R'S', the surface of the water in the reservoir, then it is evident the water contained between DC and R'S' does not press the piston downwards. But, as in this case it can only be sustained above R'S' by the pressure of the air upon the water surrounding the pump, and as this pres- sure is only capable of sustaining in equilibrium the contrary pressure of the air upon the surface XY, it follows that the sur- face DC is charged with a weight equivalent to the column which has DC for its base and CR' for its altitude. And this pressure, joined to that which is exerted upon DC by the superincumbent fluid between DC and XY, makes the whole pressure upon the piston, as before, equal to that of a column of water whose base is DC, and height the distance between XY and R'S'. 514. The sucking pump requires in its theory the aid of other principles. We must enquire if under the proposed circumstan- ces the water can possibly be raised to the piston, and made to pass through the valve L ; for in some cases the water will never pass a certain altitude, how many strokes soever we give to the Watir-Pumps. 411 piston. To understand this, suppose that the water has been actually raised to 7 1 , and that the situation of the piston in the Fig.245. figure is the lowest which can be given to it ; and, for greater simplicity, suppose that the pump is of the same internal diame- ter throughout. It is obvious that the air comprised in the space CDTZ is of the same density and elasticity as the exterior air (at least leaving out of consideration the weight of the valve L, and the friction attending its motion ;) for if its spring were less, the water would rise higher than ZT, and if it were greater, it would raise the valve L, and mix with the exterior air till both became of the same density. Suppose now that the play of the piston, or the distance through which it is raised at each stroke, is /JO; then when the base CD is raised to OQ, the air which previously occupied the space CDTZ will tend to expand and fill the space QOTZ ; and, if the water did not rise, would actually be so ex- panded. Its elastic force would then be less than that of the natural air, in the ratio of CDTZ to QOTZ, or of DT to OT. If, therefore, this elastic force, together with the weight of the col- 46g umn of water whose height is ZR, constitute a pressure equal to that of the atmosphere, or equal to the weight of a column of water of the same base, and at a mean 34 feet in height, there will be an equilibrium, and the water will not rise further ; if this joint pressure is greater than that of 34 feet of water, the water cannot be retained so high ; but if it is less than a col- umn of 34 feet, the water will continue to rise in the pump. 515. From these considerations we may readily investigate a general theorem. Let a be the altitude or vertical distance from the point O to the surface RS of the water in the reservoir,^=OZ), the play of the piston, and a: the distance OT\ then we have DT = x p, and S7 1 , the height of the point T, will be a x. Since the air contained in CDTZ has the same density and elasticity as the exterior air, its force may be measured by a column of water of the same base Z T and 34 feet high ; and because when this air is so expanded as to fill the space QOTZ, the elastic force will be less in the ratio of DT to O 7 1 , we shall have (rejecting the base of the column, as equally affecting every part of the process) this latter force expressed by the fourth term of this proportion, 412 Hydrodynamics. 4 * : x p :: 34 : (a; p.) But the force which the water, comprehended between ZT and RS, exerts in opposition to the exterior pressure of the air, is measured by the height a x ; consequently, the elastic force of the air in the space QOTZ, together with the weight of the water between ZT and RS, will be expressed by Now in order that the water may always rise, this joint pressure must be less than the weight of a column of water 34 feet high by some variable quantity, which we will call y ; so that the fol- lowing equation must always obtain, namely, 34 (a; ) - - &- + a x = 34 y. x The value of x deduced from this equation is ambiguous, being thus expressed ; x = \a + \y V((i + ly) 2 34 p.) Now, when the water ceases to rise, y vanishes, and the equation becomes x = | a \/ (-} a 2 34 p) ; of which the two values are real, so long as a 2 is greater than 34 p. Hence we conclude, that when one fourth of the square of the greatest height oj the piston above the surface of the water in the reservoir is gr .ater than 34 times the play of the piston, there are always two points in the sucking pump where the water may stop in its motion ; and the pump must be reputed bad when the lowest point to which the piston can be brought is found between these two points. But if 34 jo be greater than et 2 , the two values of #, when y is supposed = 0, become imaginary; so that in a pump so constructed it is impossible that y should vanish ; that is, the pressure of the exterior air always prevails, and the water is not arrested in its passage. Hence we conclude, secondly, that in order that the sucking pu p may infallibly produce its effect^ the square of half the greatest elevation of the piston above the wattr in the reservoir must always b& less than 34 times the play of the piston. Water-Pumps. 413 516. This general rule may also be easily deduced geomet- rically ; suppose the valve E to be placed at the surface RS of Fi 245 the water, the tube to be of a uniform bore, and YS to be the height of a column of water whose pressure is equal to that of the atmosphere ; that is, YS = 34 feet. Let the water be raised by working to JV ; then the weight of the column of water JV, together with the elasticity of the air above it, exactly balan- ces the pressure of the atmosphere YS. But the elasticity of the air in the space OM, (QO being the highest and CD the lowest DN situation of the piston,) is proportional to YS . ^r^,; and, conse- quently, in the case where the limit obtains, and the water rises no n/v/* further, we shall have YS = NS+ YS. ~ Transposing JVS, we have YS JVS or FJV = YS . whence OJV : DJV : : YS : FJV; or, OJV DNorDO : OJV : : YS FJV or JVS : YS-, consequently DO . YS = OJV . JVS. Hence we see, that if OS, the distance of the piston in its highest position from the water, and DO the length of the half-stroke, or the play of the piston, be given, there is a certain determinate height, as SJV, to which the water can be raised by the difference of the pressures of the exterior and interior air ; for YS is to be considered as a constant quantity, and, of course, when DO is given, OJV . JVS is given likewise. To ensure, therefore, the de- livery of water by the pump, the stroke must be such that the rectangle DO . YS shall be greater than any rectangle that can be made of the parts of OS ; that is, greater than the square of I OS, by a well known theorem. Hence we deduce a practical maxim of the same import as the preceding, which is, that no sucking pump can raise water effect- ually, unless the play of the piston in feet be greater than the square of the greatest height of the piston, divided by 136. 414 Hydrodynamics* 517. Resuming the equation and finding thence the value of ?/, we have _ x 2 a x -f- 34 p y - - - Fig.248.Now let AB represent the greatest height of the piston above the 2494 surface of the water in the reservoir, and AD the play of the pis- ton ; suppose the different portions AP of the line JlBio represent the successive values of a?, and lay down upon the perpendicu- lars PMthe values ofy which correspond to these assumed values of x ; so shall we have a curve MMC, which, while j a 2 is greater PJ 248( than 34/>,will cut AB in two points / and /', in such manner that the ordinates PM will lie on different sides of AB ; the ordinates which are on the right/iZ? shewing the positive values of 7/,and those which are on the left AB the negative values. We see, therefore, that so long as I a 2 is greater than 34 p the pressure of the ex- terior air is strongest, until the water has attained the height BI'. At this point I' it will stop (the motion acquired being left out of consideration,) because the value of y is = 0. But if the water by the motion it has acquired continues to rise till it reaches some point between /' and /, it will not stop there, but will des- cend, if the valve does not oppose its descending motion ; because the value of?/, being there negative, indicates that the pressure of the exterior air is weaker than the united pressures of the water and the interior air. If the water reaches the point /, it will stop there, for the same reason that it would at the point I 1 ; but if it rises above /, there is then no reason to fear that it will descend ; for all the ordinates PM between / and A being positive, shew that in that portion of the pump the pressure of the exterior air exceeds the combined efforts of the interior air and water. 518. When, on the contrary, the value of \ a 2 is less than Fig.249. tnat O f 34 p^ t h e curve will intersect the axis AB ; all the ordi- nates are positive, and consequently the pressure of the exterior air is always the strongest. This confirms and illustrates what has been laid down in article 515. Water-Pumps. 415 If the sucking pump were to be placed so high above the usual surface of the earth (as at the top of a high mountain), or so low beneath it (as in a deep mine), that the pressure of the atmosphere would be sensibly different from the assumed mean pressure equivalent to 34 feet of water, we must then in all the preceding investigation change the co-efficient 34 to that which would express the height in feet of the corresponding column of water. And these equivalent columns may always be ascertain- ed by means of the height of the mercurial column in the barom- eter ; the proportion being this ; as 30 inches, the mean altitude of the mercurial column, to 34 feet, the mean height of the column of water, so is any other mercurial column in inches to its cor- responding column of water in feet. 519. In the preceding calculation the pump has been supposed of a uniform bore throughout ; when this is not the case the so- lution is rendered somewhat more complex, but not difficult. To calculate the effort of the interior air when the water has not reached the body of the pump, having only attained the height HJV, for example, we must use this proportion ; as the space Fig. 245, QOFNMI : the space CDVNMI : : 34 feet : a fourth term, which being added to the weight of the column of water whose height is JVff, ought again to be equal to 34 ?/, as before. Be- sides, when the sucking pipe FG is of a smaller diameter than the body of the pump, if the conditions which we have before specifi- ed obtain, the pump cannot fail to produce the proper effect ; for the air is dilated with more facility in this latter case than when the whole is of the same diameter. We need only add on this point, that if the length of the stroke in a uniform pump, which is requisite to render the machine effectual, be greater than can conveniently be made, it maybe diminished by contracting the diameter of the sucking pipe in the subduplicate ratio of the diminu- tion of the length of the stroke. 520. As to the effort of which the power ought to be capable to sustain the water at a determinate height YH, it will be meas- ured according to what we have said respecting the lifting pump by the weight of a column of water whose base is equal to CD, and height that of XY above RS. Here, too, we leave out of consideration -the friction and the weight of the piston. 416 Hydrodynamics. 521. The velocity of the water flowing from the sucking pipe into the barrel should be equal to the velocity with which the piston moves. For if it be greater, less work will be done than the pump is competent to effect ; and if it be less, a vacuum will be produced below the piston, which will therefore be moved upwards with great difficulty. Of the Syphon. Fig. 250. 522. I F we introduce into a vessel of water or other liquid, the shorter branch of a recurved tube DEF, called a syphon, and exhaust the air from this tube by the mouth or otherwise, the water will rise in the tube and flow out at F, until the surface of the fluid in the vessel shall have descended to the opening I) of the syphon. The reason of this phenomenon is, that when the contained air is withdrawn, the pressure of the interior air upon the surface JIE causes the fluid to rise into the syphon arid to flow through the branch EF. And although when the flowing has commenced, the air presses the fluid at the point F with a force very nearly equal to that which is exerted upon the surface of the water in the vessel, still a transverse lamina at F is urged downward by the entire column of water 7F; this column must therefore de- scend ; but in descending it tends to form a vacuum at /, which cannot fail of being filled by the action, always exerted, of the incumbent air upon the surface of the fluid in the vessel. It will hence be seen that during the discharge through the syphon the air acts only with an effort proportional to the differ- ence of level IF between F and the surface of the water in the vessel ; so that the flowing will be so much the more rapid according to the difference of the two branches of the syphon ; thus if F and D were on a level, no motion of the fluid would take place. We say generally, that the branch EF must be longer than the branch ED ; but in using this language it must be under- stood that the vertical height of E above F must be greater than that of above D. The absolute height is not concerned, DE may Steam-Engine. 417 rendered much longer than DF by being made crooked ; so long as the point D is higher than F, the fluid will pass until it arrives at D provided the height of E above D does not exceed 34 feet. Of the Steam-Engine. 523. THE whole theory of the steam engine is founded upon two principles, the developement of the elastic force of aqueous vapour by heat, and the sudden precipitation of this vapour by cooling. On account of the extensive uses of this machine in the arts, we shall here treat of it at some length. Although it is generally sufficient in mechanics to create any one force or motion, in order to be able thence to deduce all sorts of motions, yet for the sake of distinctness we shall suppose that it is proposed, in the first place, to draw water from a mine by means of a sucking pump T' T'. Here the point in question is Fig. 251. to raise the piston P '. For this purpose we attach the piston rod to a chain applied to one of the extremities A' of a bent lever moving about its centre C. It is evident, that if we attach to the opposite arm of the lever a similar chain represented by AD, we shall only have to pull this chain, in order to raise the piston P', and draw the water into the body of the pump by the external pressure of the atmosphere. This being done, the valves placed at the bottom of the pump will close ; and the apparatus being left to itself, if we suppose the weight of the piston P', together with that of the frame which supports it, to exceed the total weight of AD and P, it is evident that the piston P' will descend into the water by its own weight and cause the water to raise the valve opening through its centre ; and having reached the bottom of the body of the pump, will separate this water entirely from the water below. Then by pulling anew the chain AD, we shall raise this water with the piston, and at the same time draw more water into the body of the pump ; after which the piston will descend by its own weight in the same manner as before, and so on indefinitely. It remains then to give the requisite mo- tion to the chain AD. For this purpose we attach its lower ex- Mech. 53 418 Hydrodyn amics. tremity D to another piston P, moving like the first in the body of the pump TT likewise cylindrical ; but suppose the bottom of the body of the pump, instead of being immersed in water, to communicate with an air pump by means of which the air in it can be exhausted. It is manifest that after the exhaustion the pressure of the atmosphere upon the upper surface of the piston P will tend to make it descend ; and will, in fact, make it descend, ii the whole eifect of this pressure exceed the weight of the pis- ton P 7 , together with that of the column of water to be raised. Now the piston P, having descended to the bottom of the body of the pump, let the air be admitted below ; then the pressure on the two surfaces will be equal ; and the excess of the weight of the piston P' beginning to act, P will rise in the bore of the pump ; after which, if we make a new exhaustion under P, we shall cause P to descend and P' to rise, and we can repeat these mo- tions at. pleasure. But it will be seen that the air pump could hardly be em- ployed upon so large a scale. To supply its place, we introduce steam into the body of the pump TT. The simplest method of doing this, and which, (although it has not hitherto been most generally employed,) is nevertheless attended with some pecu- 252. liar advantages, is the following. Under the body of the pump JT, let there be placed a boiler FF, filled in part with boiling water, the steam of which being equal or a very little superior in elasticity to the pressure of the atmosphere, may be introduced at pleasure into the cylinder TT by turning a stop-cock J?, which opens a communication between the pump and the boiler. Let there be likewise at the bottom of the cylinder a small lateral passage FS, shut by a valve S opening outwards. Now the pis- ton P being forced to the bottom or nearly to the bottom of the cylinder TJ 1 , and the space below being filled with air, turn the stop-cock R which communicates with the boiler. The steam will rush into the cylinder, and by its impulse, together with its elastic force, will in part expel the air remaining in the cylinder by forcing it to open the valve S. In this operation a great quan- tity of steam is suddenly condensed by the cold surface of the cylinder TTand that of the piston P, and being reduced to water, is made to pass out through a descending tube EGS', recurved at the lower end and terminated by a valve 5', opening outwards. Steam Engine. 419 This condensation, and consequent loss of steam produced by cooling, will continue until the piston arid the portion of the cy- linder situated between it and the boiler, are brought to the tem- perature of the steam itself. Then, the steam preserving its elas- tic form under the piston P, counterbalances entirely or partly the pressure of the atmosphere upon its upper surface. The excess of weight in P', acting therefore without obstacle, causes A'P' to descend and AP to ascend, which tends to produce in the cylinder a vacuum into which, steam rising from the boiler, con- tinues to enter, until the piston P, having reached its highest point, the cylinder is completely filled with steam. Having obtained this limit, the steam opens the valve S and escapes, at first slowly and in the form of a cloud, on account of its mixing with the air and drops of water. According as the air is expelled this cur- rent becomes gradually stronger and more transparent. When the operator perceives that this point is attained, he turns back the stop-cock, and then the whole cavity of the pump remains filled with pure steam which only wants to be condensed by sudden cooling in order to leave a vacuum under the piston P. This condensation is effected by the introduction of cold water which is made to descend from an elevated reservoir Z through the tube ZRP, closed at R' by a stop-cock called the injection stop-cock. Upon turning this, the cold water thrown into the cylinder T7\ precipitates entirely or partly the steam contained there and it flows out through the tube EOS' with the water which results from this condensation ; then a vacuum being left under the piston P, the pressure of the atmosphere causes it to descend. This piston is again raised by the introduction of fresh steam; for if, as we have supposed, the water in the boiler is kept in a state of ebullition, the steam has an elastic force at least equal to that of the air, and consequently its introduction under the piston P is sufficient to counteract the pressure of the atmosphere ; so that the excess of weight in P 7 will raise P as before. But on the other hand, the steam, if heated too much, may by its elastic force cause the boiler to burst. To guard against this, we adapt to the top of the boiler a safety valve S", which opens outwards with a known effort. When the elastic force of the steam is equal or inferior to that of the external air, the valve remains closed; but when it becomes equal to that of the atmosphere and the resistance of the valve together, the steam escapes and 420 Hydrodynamics. no explosion is to be feared. Still, however, it is necessary that the boiler should be made stronger than this limit of resistance supposes. For when the steam rushes into the cold cylinder and is condensed there, this precipitation is so rapid that the new steam formed at the same time in the boiler is not always sufficiently instantaneous to supply its place. For a moment a vacuum is left in the boiler, and the pressure of the external air being no longer counterbalanced, the boiler may burst inward if its sides are not sufficiently thick. This accident sometimes happens, but it may always be prevented by means of a second safely valve, w r hich opens inward whenever the external pressure becomes too great. From this explanation it would seem that when the engine was once in operation nothing would remain in the piston or body of the pump but pure steam or a vacuum. But it must be re- marked that the injected water has also some air combined with it which escapes into the body of the pump ; since it is found there in a highly rarified state, being heated to a considerable degree by the great quantity of heat disengaged during the con- densation. Happily this air, being in small quantity, and con- tained in a small space, is easily expelled through the valve S by the first effort of the steam introduced into the cylinder. 524. The apparatus which we have described is not precisely the one first invented. It appears that the original attempt was simply to employ the force of steam as a moving power. But the more ingenious discovery of the method of condensing the steam by cooling was not made until 1696; and the English attribute it to Capt. Savary, who published an account of it in a treatise entitled The Miner's Friend. The mode of applying this princi- ple was still very imperfect. In 1 705, Newcomen, another Eng- lishman, gave it the form which we have described, in which, under the name of the atmospherical engine it was a long time not unprofitably employed. Nevertheless, with the progress we have made in mechanics and the natural sciences, it is easy to perceive that this engine had many theoretical defects. It was a great imperfection that it required an intelligent person to watch it for the purpose turn- ing the stop-cocks to introduce water and steam every time the Steam-Engine-. 421 piston reached its limit. A good machine ought always to tend itself by means of the first moving power, without any foreign aid. Another great inconvenience was the introduction of steam into a cold cylinder ; since a great loss was thereby occasioned, and was repeated at each stroke of the piston, the cylinder being continually cooled by the injected water necessary for the precip- itation. But these defects, which in the present state of the sci- ences we are so prompt to observe, could not at first have been so easily detected. They were perceived and corrected in 1 764, by Watt, the disciple and friend of Black. Being then at Glasgow, where he was employed as a mathematical instrument maker, he was directed to repair a small model of Newcomen's engine, which belonged to the University in that city. In the course of his attempts to make its operation satisfactory, he observed that it consumed more coal in proportion to its size than the large engines. Being curious to ascertain the cause of this dif- ference, and wishing to remedy so great a defect, Watt made numerous experiments for the purpose of determining what sub- stance is the most suitable for the cylinders ; and what are the most proper means of creating a perfect vacuum ; what is the temperature to which water rises in boiling, under different pressures ; and what the quantity of water necessary to produce a given volume of steam, under the ordinary pressure of the at- mosphere. He determined also the precise quantity of coal necessary to convert a known weight of water into vapour, and the quantity of cold water required to precipitate a given weight of steam. These several points being once exactly ascertained, he was led to perceive the defects of Newcomen's engine, and to assign the cause of these defects. He saw that the steam could not be condensed so as to produce any thing like a va- cuum, unless the cylinder and the water it contained, as well as that injected and that arising from the condensed steam, were cooled down at least to the temperature of about 33, and that at a higher temperature the steam had still an elasticity strong enough to oppose a very perceptible resistance to the weight of the atmosphere. On the other hand, when it is proposed to at- tain more perfect degrees of exhaustion, the requisite quantity of injected water is augmented in a very rapid proportion ; and hence results a great loss of steam, when the cylinder is again filled. These facts led Watt to conclude, that, in order to effect 422 Hydrodynamics. the most complete vacuum possible, with the least expense of steam, it was necessary that the cylinder should be kept con- stantly as hot as the steam itself, and that the injection of cold water should take place in a separate vessel which he called the condenser, and whose communication with the cylinder was sud- denly opened at the moment of the injection. Indeed after what is now known respecting the equilibrium of fluids, it is manifest that, if the air be exhausted from the condenser, the steam from the cylinder will enter it, on account of its own elasticity, the instant a communication is opened ; and an injection of wa- ter made at this instant, will precipitate not only the steam actually in the condenser but also on the same principle, all the steam contained in the cylinder, which, rushing into the vacuum, continually formed by precipitation in the conden- ser, is converted almost instantaneously into water. It only remains, then, to remove this water and disengage the air, in order to preserve always a vacuum in the condenser. Watt constructed a pump in such a manner as to be moved by the engine itself and which played continually in a tube void of air, the lower part being immersed in the water of the condenser. Finally, the condition of keeping the cylinder hot could not be fulfilled while there was a free admission of atmospheric air to the interior of its upper surface, which in the apparatus of New- comen, caused (he piston to descend ; especially, since in order to prevent the passage of the steam between the cylinder and piston, i he latter was ordinarily covered with a stratum of cold water which kept the interior of the cylinder wet. Watt con- ceived the bold and ingenious idea of dispensing entirely with the pressure of the atmosphere, and making the piston descend by the force of steam alone, by introducing it alternately above and below, and causing at the same time a vacuum in each case in the manner already described. Then he enclosed the rod of his piston in a collar of leather to prevent all access of air to the interior of the cylinder; and employing steam of an elas- ticity equal or even a little superior to the pressure of the at- mosphere, he obtained alternately above and below the piston a force equal, or a little superior, to the atmospheric pressure. He was then able, by substituting stiff rods for the chains AP, A'P 1 , to produce a force in both directions ; whereas, in Newcomen's engine, the time during which the piston was ascending in the cy- Steam-Engine. 423 linder, was entirely lost so far as the steam was concerned, since it was raised simply by the excess of weight on the other arm of the large lever. Here was a saving both of time and expense ; for the piston was accelerated each way by steam, and the quantity of fuel employed to keep it hot, during its ascent, was not wasted. Watt took care, moreover, to surround the cylin- der with a case of w r ood, or some other substance which is a non- conductor of heat; into the interior of which, he also occa- sionally introduced the steam as a means of keeping it warm. He likewise used so much economy in the construction of the different parts of his engine, that he succeeded in saving two thirds of the steam employed by Newcomen. The steam engine, thus improved, is represented in figure 253, the ex- planation of which will now be easily understood. FD is the boiler, in which the water is converted into steam by the heat of the furnace below. This boiler is sometimes made of copper, but more frequently of iron. Its bottom is con- cave and the flame curls around it. Towards the top, it has a safety valve S" fitted to resist a greater or less effort before opening, according to the degree of elastic force to be em- ployed. That the conversion of water into steam may be con- stant, it is necessary that the water in the boiler should be kept always at the same level, and consequently that it should be supplied as fast as it is vaporized. This is effected by a tube -o r, which supplies tht> boiler from a small reservoir r, filled with water, already heated, which the pump 1 1 takes from the condenser and forces into the lateral pipe t' i f . But in order to introduce this into the boiler only when it becomes necessary, the upper orifice of the tube v v is closed by a stopper, which is raised or lowered by means of a small lever a b ; and at the other arm b of the lever hangs a wire b ra, drawn downwards by a weight ra, which is so adjusted in the boiler as to keep on a level with the upper surface of the water. Then if the water falls below this level, the weight m, which it supports in part, will descend with it ; the lever turning will raise the stopper, and suffer the water to pass into the boiler ; but as soon as the level is re-established, the lever a b will again become horizontal, and return the stopper to its place. From the top of the boiler pro- ceeds the steam tube VV, which conveys the steam to the top of the cylinder TT : by the valve S, and to the bottom, by the 424 Hydrodynamics. valve S'. The communication between S and S' is effected by means of a curved pipe, whose plane of curvature is perpendic- ular to the plane of projection in figure 253. One part only of this pipe is represented in this figure ; in order that the two valves S", S'", may be exhibited, of which we are presently to speak ; but the whole may be seen in figure 254, where it is presented in. profile. The valves S", S'", are those by which the steam of the cylinder is brought into communication with the condenser, on both sides of the piston ; and they are opened and closed at proper times by the engine itself, with the aid of two small projections, 1, 2, attached to the rod 1 1 of the pump which serves to exhaust the condenser 0. This opening is effected a little before the piston has completed its vertical motion, and the communication is then established between the two surfaces, in order that the equality of pressure thence resulting may weaken the effort which would be made if it were exerted on one side only, and prevent the sudden jar which would follow from the piston's striking with its full force against the bottom of the cylinder. These are the principal conditions relating to the action of the steam, but there are others which relate to the manner of directing this action. Indeed, a bare inspection of the figure will show that the rod of the great piston and that of the pump which exhausts the condenser, being both inflexible, cannot be attached immediately to the large lever JIB ; for each point of this lever, describing an arc of a circle about its centre of rota- tion, would tend to change the point of attachment from a ver- tical direction, and this effort would break the engine. It is on this account, that, in the engine of Newcomen, where the piston only acted in its descent, its communication with the large lever was made by a chain applied to the arc of a circle. But in the engine under consideration, the inflexibility of the rods requires some other mode of communication. Mr Watt effected this by means of a particular assemblage of metallic bars, moving upon one another, and which compensate by their action, for the want of perfect verticality in the motion of the large lever. The figure represents also, several other useful appendages to the engine, such as flies to regulate its motion, and wheels to transmit it. There is also a very essential part designated by G, and called the governor. It consists of a vertical rod which is kept contin- Steam-Engine. 452 tially in rotation by the engine itself, and which carries at its summit a parallelogram formed of metallic plates, turning free- ly one upon the other, in such a manner that this parallelogram may open more or less in a horizontal direction, according as the plates diverge more or less from the axis. This diverg- ence is produced by the centrifugal force exerted by the axis in turning, under the influence of the engine, with greater or less rapidity 5 which causes the upper vertex of the parallelogram to be depressed when the engine moves more rapidly, and ele- evated when it moves more moderately. In order to give great- er force to this ascending and descending motion, the extremi- ties of the plates are loaded with spheres of solid metal, and exert their power upon a lever, whose other branch communi- cates with a plate placed transversely in the passage through which the steam passes from the boiler to the cylinder ; so that, when the engine works too slowly, the plate turns in such a manner as to give a freer passage to the steam ; and, on the other hand, if the engine moves too rapidly, the plate takes a position more nearly in a transverse direction, and diminishes the passage through which the steam has to pass. Thus the engine is made to govern and regulate itself, in such a manner as to preserve in its motions that uniformity which its purposes require. There are many other details which would furnish matter for curious speculation ; but these details, belonging to the mechanism, must be omitted here, to make room for some other particulars, no less important, relating to the principles of the machine. 525. The most essential of these is the determination of the temperature, at which it is most advantageous to employ the steam. In fact, the higher the temperature is, the greater will be its elastic force, and consequently the greater will be its effect upon the surface of the piston, the vacuum being always on the other side. From the experiments of Southern, Clement, Desormes, and Despretz, it has been found that the total quantity of heat necessary to change the same mass from water to a state of vapour, is very nearly if not quite the same for all tempera- tures. According to this principle, then, it will not be necessary to consume more fuel in order to form a given weight of steam of a higher temperature and more elastic, than is required Mech. 54 426 Hydrodynamics. for the same weight at a lower temperature and less elastic. But while the steam is raised to a higher temperature, it becomes also more dense ; so that considerably more weight is necessary to fill the same cylinder, than when its temperature is lower. This second circumstance, therefore, requires in the same ma- chine, a greater quantity of fuel, according as the temperature of the steam is increased ; so that it only remains to ascertain whether this increase of fuel is, or is not, compensated by a corresponding increase of elastic force. Now this point is easi- ly decided, if we reflect that the vapour of water and that of other liquids, so long as they exist in a state of vapour, are sub- ject to the same physical laws of compression and dilatation, as the permanent gases. Accordingly, let w represent the weight of a cubic inch of aqueous vapour, such as it would be, if it were capable of being reduced to the temperature of melting ice, and under a barometric pressure of 29,92 inches ;t and let w' represent the weight of an equal volume of the same vapour, as it would really exist at another temperature /, and with the elastic force jP, equal or inferior to the maximum density be- longing to this temperature. If we put t' =. t 32, or the number of degrees distant from freezing, according to the laws 470. of the dilatation of the permanent gases, we shall have wF a/ 29,92 (1-H'. 0,00208) All the experiments of philosophers upon aqueous vapour, and those of Despretz upon the vapour of several other liquids, show that this formula is really applicable to them ; so that we may safely employ it for the purpose under consideration. Now calling c' the quantity of heat necessary to convert 1 5,5 grains TroyJ of water, in a liquid state, at $2, into vapour, having the temperature t 1 ; c' w f will be the absolute quantity of heat neces- sary to form the weight a/ of such a vapour ; and consequently if n be the number of cubic inches contained in the cylinder to be filled, the whole quantity of heat necessary for this pur- pose, will be c' n w F 29,92(1 -|- t' . 0,00208) t Or, more accurately, 15,444 grains, or one gramme. { Equal to 0,76 of a metre. Steam-Engine. 427 Having thus brought into an equation all the different ele- ments from which this expenditure of heat results, we are able to analyse the several effects produced by them. In the first place, according to the experiments of Clement, Southern, and Despretz, before mentioned, c' is sensibly constant at all tempe- ratures at which we have yet been able to make observations, and accordingly the expenditure will be always the same. As to the elastic force F, we know that it augments according as the temperature is raised. But It will be seen by the formula that the density w', and consequently the expenditure of heal, increases in proportion to F. Consequently if we take away the factor (1 + /' . 0,00208), arising from the dilatation produced by the increase of temperature, the expenditure necessary to pro- duce the force F will be exactly proportional to this force, and we shall neither gain nor lose any thing by giving it a greater or less energy. But the influence of the factor (l + I' . 0,00208,) which increases as the temperature is raised, diminishes the expenditure in question, in proportion as the temperature be- comes higher ; for if we operate, for instance, at 212, this factor becomes 1,44096 ; and if at 320, it becomes 1,66560 ; so that the relative expenditure in this last case, compared with that in the first, with the same elastic force, is nearly in the ratio of 144 to 166, or nearly of 6 to 7. Such then is the saving of heat that may be made in the formation of steam in the extremes of temperature at which we have as yet operated. Accordingly if it were true that high pressure engines have, over those of the ordi- nary kind, an advantage as great as has been alleged, it must be sought in something else beside the saving of fuel. But in order to judge correctly of these engines it is necessary to take into consideration another element, namely, the ulterior developement of elastic power, which the steam thus formed is capable of fur- nishing, when, after having been employed with its primitive energy in the first cylinder, it is made to pass into another larger cylinder where it dilates, undergoing a reduction of temperature at the same time, in such a manner as always to fill the space into which it is received, until at length it is condensed into water, when it no longer has an elastic force, either equal or inferior to the pressure of the atmosphere. Jt is manifest that this ulterior developement of force must, in order that the whole effect may be appreciated, be added to the mechanical power at the 428 Hydrodynamics. commencement ; and it is no less evident that it offers a peculiar advantage to engines in which the steam, before being condensed, is employed at a high temperature. Nevertheless, the numerous experiments which have been made within a few years upon engines of this kind, some of which have been accompanied with careful measurements, have not seemed to confirm, so much as might have been expected, the favourable view we have given above ; and if they have taught us a real saving of fuel, when considered with reference to the force actually developed, this saving does not appear to exceed the narrow limits just assigned to the effect of the heat of dilatation. So small an advantage will be far from compensating for all the additional precautions required in such engines, the dangers incurred and the numer- ous causes of waste to which they are liable. It is necessary, in the first place, to make the boilers, cylinder, &c., very strong, that they may resist the expansive force exerted by the steam. It is likewise necessary to give greater perfection to the pistons, and to apply more frequently some lifbricating substance to preserve the contact. Repairs are in consequence often required, and the value of the engine is sensibly diminished on this account. Attempts have been made to remove this great inconvenience by an expedient formerly employed to perfect the first invention of Savary. It is this ; the piston, instead of being in immediate contact with the aqueous vapour, which melts and dissolves the grease with which it is impregnated, receives its motion through the intervention of a column of oil, or some other unctuous sub- stance not easily evaporated, on which the steam is made to act by pressure. For this purpose the cylinder in which the piston plays is enclosed in a larger cylinder with which it communi- cates, and which contains the oil. The oil rising and falling continually in the interior cylinder keeps it always lubricated. But although this ingenious arrangement may be sometimes adopted with advantage, it could not be used at high tempera- tures ; for, according to the judicious observation of Mr. Watt, the oil would be decomposed by the dissolving power of the steam. It has likewise been proposed to construct high pressure engines having a very great power with very little bulk, by em- ploying small boilers, made so strong as to resist the most Steam-Engine. 429 powerful pressure, while they admit of being heated, as it were, red hot; whereby steam would be obtained of an excessively high temperature, and which, developing itself by its expansive force in the great cylinder would possess even after its dilata- tion an elastic force sufficiently energetic. A similar arrange- ment was likewise attempted formerly, in order to furnish steam for Savary's engine ; but it does not appear to have been found profitable enough to be continued in use ; and after the calcula- tions we have made on the expenditure of heat employed in the formation of steam at every temperature, it seems improbable that a sufficient saving can be made in such engines to compen- sate for their inconveniences. Advantages of a different kind might be realized, if we could succeed in employing, without loss, some liquid different from water, having a much greater elastic force at the same temperature. One inevitable consequence of employing high temperatures is, that the loss of heat by radiation is much greater, and this makes an important item in calculating the results. To form an idea of the diminution of effect arising from these different circumstances we are to remember that according to Lavoisier and Laplace, 1 gramme or 15,444 grains Troy of charcoal, deve- lopes in burning 13038 degrees of heat by Fahrenheit's scale, or about 92 pn Wedgewood's. Now 15,444 grains of water at the temperature of 212, by being converted into steam absorb 1020,6 by Fahrenheat; then 15,444 grains of charcoal would reduce to steam 200 grains of water, on the supposition that no heat is lost, and that the water is already brought to the temperature of 212. But after a great number of experiments made upon the most perfect engines and the best constructed furnaces, Mr Clem- ent found that 1544,4 grains of charcoal does not produce more than 6 or 7 times as much steam, and 1544,4 grains of the best fossil coal never produces more than 6 times as much steam ; whence it will be seen that nearly half the heat is lost by radia- tion, and the conducting power of the boiler and the surrounding bodies. The loss is without doubt still more considerable in engines of a high pressure. 526. When we know the elastic force of the steam introduced under the surface of the piston, it is easy to estimate the whole pressure resulting from it ; but in this estimate it is necessary to 430 Hydrodynamics. take account of the tension of the steam which remains on the other side when the vacuum is not perfect. After all, this esti- mate fails of giving the primitive energy of the power employed, and much Jess the part which remains to be disposed of after all the friction is overcome and the different parts of the machine are put in motion. This useful part of the force can be measured a pos- teriori by the effect which the whole engine produces. Ordina- rily we compare the effective power of an engine with that of a certain number of horses of a medium strength, and the force of the engine is estimated accordingly. By a great number of ex- periments of this kind Watt and Bolton supposed that a horse of ordinary strength, working 8 hours a day, would raise 3200 Ib. avoirdupois one foot per hour. Smeaton made the estimate 2300, and Clement about 1300. If, therefore, we divide the number of pounds that a steam engine will raise one foot (or, which is the same thing, the product of the number of pounds into the number of feet elevation), by the number representing a horse power, the quotient will be the number of horses to which the engine is equivalent. There are engines of the power of 20, 30, &c., horses. The most powerful engine that has yet been constructed is that employed in the mines of Cornwall. It has the power of 1010 horses, and serves to drain by pumps a mine 590 feet deep. It is evident, that the power in question is all that needs to be estimated ; lor we can apply it to the raising of wa- ter, the turning of spindles, or to any other purpose that requires such a force. The transmission of the primitive motion can be effected by mechanical means and instruments of which we have already spoken, and which need not be again described. NOTES. I. On the Measure of Forces. THE name of living forces has been applied to the forces of bodies in motion, and that of dead forces to those, which, like a simple pressure, suppose no actual motion in the operating cause. There was, for a considerable time, a difference of opinion among mathematicians in regard to the measure of living forces, or the forces of bodies in motion. Some affirmed that these forces ought not to be measured by the product of the mass into the velo- city, according to the rule that has been given, but by the product of the mass into the square of the velocity. As this difference in the estimate of forces may be deemed of great importance in me- chanics, it will be proper to make a few remarks upon it. It is altogether unimportant whether we measure the force of bodies in motion by the product of the mass into the velocity simply, or by the product of the mass into the square of the velocity, pro- vided that in the two cases we assign different significations to the word/orce. When we assume as the measure of force, the product of the mass into the square of the velocity, we mean by the term force the number of obstacles which a moving body is capable of overcoming ; it is certain that the number of obstacles which mov- ing bodies of equal masses are capable of overcoming is proportional to the squares of the velocities. For example, if the bodj r A has Fig. 255. precisely the velocity necessary to close the spring j?CB, an equal body M will require only double this velocity to close four springs, each equal to ACB. For, in the first instant, for example, the body M advancing with a velocity double that of .#, will close the four springs, considered as one, twice as much as A would close its sin- gle spring; each of the four springs then will be closed only half as much as ACB, and consequently will have opposed during this instant a resistance only half as great as that of ACB ; all the four 432 Notes. springs, therefore, in being each reduced the same angular quantity as ACE is reduced, will oppose only double the resistance. In the same manner it may be demonstrated that, in the succeeding in- stants, the resistance opposed by the four springs in being closed each the same angular quantity, as that by which ACE is closed in one instant, is always to that opposed by the single spring ACB, in the same instant, in the ratio of the velocities belonging to the two bodies ; therefore a velocity double that required to close one spring is sufficient to close four springs. Thus the number of springs closed, which are 1 and 4, are as the squares of the velocities 1 and 2 neces- sary to close them. We see then that the number of obstacles which bodies in mo- tion are capable of overcoming increases as to the squares of the velocities. But by the term force ought we to understand the num- ber of obstacles ? Is it not much more natural to consider it as denot- ing the sum of the resistances opposed by these obstacles ? For it is not merely the number, but the value of each obstacle which des- troys motion. Now in this case each instantaneous resistance being evidently proportional to the quantity of motion destroyed by it, (on which point all are agreed,) the sum of the resistances will be proportional to the quantity of motion destroyed. If then by /orce, we understand the *wm, and not merely the number of the resistances which a moving body is capable of overcoming, the force is propor- tional to the quantity of motion. From this principle it has likewise been inferred that the number of resistances overcome are as the squares of the velocities. The question then is in reality nothing more nor less than a question about terms, and reduces itself to find- ing the meaning of the word force. As to this point we are per- fectly at liberty, provided we employ that which we take for the measure of force agreeably to the idea which we attach to the term force, we shall always arrive at the same results. We shall, therefore, continue to take for the measure of forces the product of the mass into the velocity ; and consequently by the force of a body we understand the sum total of the resistances necessary to exhaust its motion. II. On the Compressibility of Water. THE phenomenon of the transmission of sound through water and other liquids had long indicated that they were capable of being compressed. Canton, an English philosopher, clearly detected this Notes. 433 property by observing the volume occupied respectively by oil, water, and mercury, first placed in a vacuum, and afterwards ex- posed to the pressure of the atmosphere ; but the results which he obtained, though exact in themselves, were, however, liable to be affected by the accidental variations of form and tempera- ture to which the apparatus was subject. M. Oersted completely removed these difficulties by plunging the liquid to be compressed, together with the vessel containing it, into another liquid to which the pressure was applied, and through which it was made to pass to the interior liquid without changing the form of the vessel, since it acted equally within and without. M. Oersted found, likewise, that a pres- sure equal to the weight of the atmosphere produces in pure water a diminution of volume equal to 0,000045 of its original volume. The experiments of Canton gave 0,000044. M. Oersted found, by vary- ing the pressure from i of the weight of the atmosphere to 6 atmos- spheres, a change of volume sensibly proportional to the pressure. Later experiments, made by Mr Perkins, seem to show that this pro- portionality continues when the pressure amounts to 2000 atmos- pheres. Before the water, however, is entirely freed from air, the diminution of volume, produced by the pressure, is at first somewhat greater than the above ratio would indicate. HI. On the Condensation of Gases into Liquids. MR FARADAY enclosed in glass tubes, bent and sealed, different ehemical products which were capable of developing gases by their mutual combination. He introduced them into the tubes in such a manner that they remained separate in the different branches of each tube, and were not mixed until the tube had been sealed. All the gas developed in each tube was found to be confined to a fixed volume, to which it could be reduced only by the action of a con- siderable pressure ; this pressure causes it to liquify in the follow- ing cases. 1. Sulphurous acid produced by the action of sulphuric acid on mercury. 2. Sulphuretted hydrogen produced by the action of hydrochloric acid on the sulphuret of iron in fragments. 3. Car- bonic acid produced by the action of sulphuric acid on carbonate of ammonia. 4. Oxide of chlorine, produced by chlorate of potash and sulphuric acid. 5. Ammonia disengaged from the combination of Mech. 55 434 Notes. this substance with chloride of silver, &c. In each experiment the branch of the tube containing the mixture was warmed, while the other was cooled with moistened paper. IV. On the Construction of Valves. A VALVE is a kind of lid or cover to a tube or vessel, so contrived as to open one way by the impulse of any fluid against it, and to close, when the motion of the fluid is in the opposite direction, like the clapper of a pair of bellows. In the air-pump this purpose is effected by means of a strip of leather, bladder, or oiled silk, stretched over a small perforation in the piston, and ordinarity in a 507. plate at the bottom of the barrel. In common water-pumps the valve, or sucker^ as it is often called, is a thick piece of leather pressed down by a small wooden weight, and turning on the flexible leather as a hing^. It is represented at , figure 245, &c. In the best metallic pun ps for raising water the valve consists of a metallic cane with a stem and knob, as represent- ed at L in figure 247. The conical part is ground so as to fit accu- rately the rim of the opening in which it plays, and unlike other valves, it is rendered tighter by use, and is less likely to be ob- structed by foreign substances contained in the water. Mr Perkins invented a pump having a square bore, in which the valve consists of two triangular pieces of leather loaded with weights, and turning on a metallic hinge placed diagonally across the bore. Mr Evans adapted a similar kind of valve to the common pump of a circular bore, the form of the valve being a semi-ellipse. The chief advantage of this construction is, that there is very little obstruction to the motion of the water, and consequently less loss of power, than in the common pump, where the space, left for the pas- sage of the water, bears a less proportion to the whole bore. Aofw. 435 V. On the History and Construction of the Barometer. THE barometer takes its origin from the experiment of Torri- celli, who in consequence of the suggestion of Galileo with regard to the ascent of water in pumps, proceeded in 1643 to make experi- ments with a tube filled with mercury, conjecturing that as this fluid was about thirteen times heavier than water, it would stand at only one thirteenth of the height to which water rises in pumps, or at about thirty inches. He, therefore, filled a glass tube about three feet long with mercury, and upon immersing the open end in a vessel of the same fluid, he found that the mercury descended in the tube, and stood at about twenty-nine and a half Roman inches, and this vertical elevation was preserved, whether the tube was perpendicular or inclined to the horizon, according to the known laws of hydrostatical pressure. This celebrated experiment was repeated and diversified in several ways with tubes filled with other fluids, and the result was the same in all, allowance being made for difference of specific gravity, and thus the weight and pressure of the air were fully established. Such, however, was the force of prejudice that many refused to yield their assent till, at the suggestion of Pascal, the experiment was performed at different heights in the air with such results as left no longer any doubt upon the subject. Great care is necessary in the construction of the barometer. The tube after being cleansed as perfectly as possible, is to be grad- ually heated, and to be kept at a pretty high temperature for a con- siderable time, for the purpose of expelling all moisture that may be found adhering to it. The mercury is then to be introduced ; a small quantity only is first poured in by means of a fine funnel and thoroughly boiled in order to free it from air ; then another portion is added, and so on till the tube is filled. It is afterward to be care- fully inverted, and the open end immersed in a cistern of boiled mercury. The tube and cistern is enclosed in a metallic or wooden frame- work, containing the graduations and some necessary appendages. As the mercury rises and falls in the tube by the fluctuations of the atmosphere, its surface varies also in the cistern. But the gradual tion is intended to mark the exact length of the column, reckoned from this variable surface. If a horizontal section of the tube and cistern have a constant ratio to each other throughout the extent 436 embraced by these changes, a correction could be readily applied. Suppose, for instance, that a section of the cistern is one hundred times that of the tube, or that their diameters are as 1 to 10, and that the surface of the mercury in the cistern coincides with the point from which the graduations commence when the mercury in the tube stands at 30 inches. The correction would be one hun- dredth part of the difference from 30 inches, and additive or sub- tractive, according as this difference was below or above 30. It is usual, however, in the best barometers to bring the surface of the mercury in the cistern to the point from which the gradua- F lg .2S2. tions commence by means of a screw V, acting on a flexible piece of leather which forms the bottom of the cistern. We are able to tell when the desired coincidence is effected by means of a mark on a piece of ivory floating on the mercury and sliding over a fixed object having a corresponding mark. Sometimes the cistern is of glass, and the point of commencement of the graduations is marked upon it, or (which is much better) is indicated by the contact of Fig. 231. a sharp ivory pin P, inserted in the cap, and descending into the interior of the cistern. There are various kinds of portable barometers constructed for the purpose of measuring the heights of mountains. The latest and most convenient is represented in figure 232. It is of the syphon form, and was invented by Gay-Lussac. The barometer being filled, the extremity of the shorter branch Y is hermetically sealed. In this state the barometer is inaccessible to the external air, and con- sequently is incapable of indicating the changes of pressure in the atmosphere; but to open the communication, we draw out, by means of a blow-pipe, a small portion of the glass near the middle of the shorter branch, on the inside, and form a fine capillary tube, which is sufficient to admit the air but does not allow the mercury to escape, on account of the force with which it repels it in virtue of its capil- lary action. The difference of level between the two extremities S, A', of the column being observed, it is reversed, and a part of the mercury enters the longer branch CX, and fills it, the rest falls into the shorter branch CF, but cannot escape for the reason above mentioned. It may then be carried in this position, being always open to the air, but not to the mercury. This barometer may be enclosed in a cane and transported with great ease and safety. A small thermometer is appended, as in other cases, for the purpose of measuring the temperature of the mercury. By contracting the tube near the two ends we prevent all danger of its breaking by any sudden motion in the column of mercury. Notes. 437 By observing regularly the height of the barometer for a con- siderable period in the same place, we find that it does not remain constantly the same. For some time after the instrument was in- vented it was supposed that the mercury stood higher just before rain, and lower during fair weather. Reasons were assigned for this supposed fact It was said, that when it is about to rain the air is charged with water, and consequently that the weight of the atmo- sphere is more considerable ; and that, on the contrary, this weight must be less in fair weather, because the air is then relieved from a certain part of the moisture contained in it Unfortunately for this hypothesis it has been found, more recently, that the quantity of water which the air is capable of containing, increases with the temperature, so that in summer it contains, for the most part, more water than in winter, although there is less fair weather in winter than in summer. It appears also that the vapour of water is lighter than the same volume of air, when the same elastic force is exert- ed ; that is, if we substitute for a cubic foot of air, taken at a certain height in the atmosphere, a cubic foot of aqueous vapour, of the same temperature and elasticity, the vapour will weigh less than the air, and will consequently exert less pressure upon the barome- ter. We hence draw a conclusion the reverse of that which the first observers of the barometer undertook to maintain, namely, that the rise of the barometer indicates fair weather, and its fall, rain. This is in fact agreeable to observation in ordinary cases. But it must be confessed that the reason now given is but little better than that we have been combating. The variations of the barometer are different in different places. They are almost nothing upon the tops of high mountains, and be- tween the tropics ; even in the temperate zones they are never very great in calm weather. But the barometer almost always de- scends rapidly before a violent storm, great changes taking place in a few hours. On this account the instrument is particularly useful at sea. By comparing observations made at different and remote places, we discover a remarkable correspondence, which shows a simuUa- neousnessin the motions of the atmospheric strata that would hardly have been expected. Still this correspondence is far from being perfect, especially as to the quantity of the change. By examining a long series of observations made in the same place, we shall perceive amid all the accidental irregularities, that there is a general tendency, occuring periodically, to rise and fall at certain hours of the day. By a long series of observations, direct- ed to this point, M. Raymond discovered, that in France the barome- 438 Notes. ter attains its maximum elevation about 9 o'clock A, M., after which it descends till about 4 P. M., when it is at its minimum ; from this time it rises till near 11 P. M. when it reaches its maximum again, after which it commences a downward motion till 4 A. M. ; and thence it begins to return to the state first mentioned. This march is often deranged in European climates, where the atmosphere is so variable ; but under the tropics where the causes which act upon the atmosphere are more constant, the periodical changes are regu- lar, and to such a degree that, according to Humboldt, one may almost predict the hour of the change at any time from a single ob- servation ; and what is very remarkable, these changes, according to the same distinguished philosopher, are not affected by any atmo- spherical circumstance ; neither the wind, nor rain, nor fair weather, nor tempests, disturb the perfect regularity of these oscillations. They are found to be the same in all weathers and at all season?. For further particulars relative to the construction of the barometer and the theory of its fluctuations, the student is referred to Daniel's Meteorological Essays. THE END. ERRATA. Page 47, line 31, /or c AA n read AA" in both cases. 5G 21 ' similarly point,' read point similarly. 63 18 ' tf 2 ' read 6. C X 3 C 2 X 3 65 17 ' -' read : , in some copies. 69 32 fc sa' read as. _ 101 18 -' = i/ s6 ~ g fcc.' read y = 5 -^ s s 126 37 c pane' read plane, in some copies. 129 19 after 4 ' read : : instead of : . 141 22 after shall be" insert to. 143 15 after ' to' insert be. 171 26 for ' ~> read . o o 177 16 ' res-' read rest. 216 14 dele c W in the numerator. 223 19 for ' number' read numbers. 230 20 ' TO', w', w", ' read TO, TO', TO". aw aw 2 310 18 'f reorff 320 46 ' 0,09134' read 0,00134. 325 27 ' Gfi" read GM. - 344 _l,_<^!ve<^. 3 3 360 24 inclose ' J + fc' in brackets, thus (# -f- 364 7 /or ' 6' read 5. 366 4 ' 0,45' read 0,045. _ 366 - 23 -T- ' 1,14' read 1,74. 373 3 373 26 440 Errata. Page 274, line 1 insert a single bracket at the end of the line. 374 18 for ' 282' read 482. _ 381 - 13 < = P P _ 15 < BP> refl rf ^ and for JL- read I~T~ V ^p _j_ g 391 20 for l a ' read 6. 393 , 5 ^tfreadv*. 399 bottom line, ' EH DK* read DH DK. 400 2 for c EH read DH. " 21 c sin a?' read sin a: 2 , and make the same change in the succeeding parts of the so- lution. bottom line, extend the radical sign over . 419 22 for c ZRI" read ZR'I. 420 37 after* ' purpose' insert of. 427 20 for 6 1,44096' read 1,375. 20 ' 1,66560' read 1,600. 23 ' 144 to 166' read 1375 to 1600. //! 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