Mechanics

MECHANICS

STRICTLY speaking, the derivation of this word should have prevented the use of it as the designation of a pure science. It has been, however, employed for a long period in English speech in the identical sense that the French attach to Mécanique pure or the Germans to Reine Mechanik. These terms are all employed to denote what we should much prefer to call Abstract Dynamics,— the pure science which (as the derivation implies) treats of the action of Force upon Matter, but which is, correctly, the Science of Matter and Motion, or of Matter and Energy.

With the view of making clear from the outset the reason for the arrangement adopted in this article, we commence by stating in Newton’s own words (accompanied by a paraphrase) the Axiomata, sive Leges Motus, which form the entire basis of our subject. These laws will at once indicate the order in which the subject may most logically be treated. We defer to the end of the article the more close consideration of the idea introduced by the word “force,” as well as general remarks on “energy,” <fcc. For the present we are content to regard force as defined for us by Newton’s Laws.

Newton’s Laws of Motion.

§ 1. Lex I. Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus illud a viribus impressis cogitur statum suum mutare.

Every body continues in its state of rest, or of uniform motion in a straight line, except in so far as it is compelled by force to change that state.

Lex· II. Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.

Change of {quantity of) motion is proportional to force, and takes place in the straight line in which the force acts.

Lex III. Actioni contrariam semper et aequalem esse reactionem; sive corporum duorum actiones in se mutuo tsemper esse aequales et in partes contrarias dirigi.

To every action there is always an equal and contrary reaction; or the mutual actions of any two bodies are always equal and oppositely directed.

§ 2. In 1863 Thomson and Tait (upon whose Treatise on Natural Philosophy much of what follows is based) called attention to the fact that, as regards Lex III., Newton gives in a scholium a second sense in which the words may be interpreted. In the first sense the action and reaction are mere forces, in the second they are the rates at which forces do work. Hence, and for another reason which will appear later, the word “activity” has been introduced as the English equivalent of the word actio in Newton’s second sense. Here is the passage :—

Si aestimetur agentis actio ex ejus vi et velocitate con-junctim; et similiter resistentis reactio aestimetur con-junctim ex ejus partium singularum velocitatibus et viribus resistendi ab earum attritione, cohaesione, pondere, et acceleratione oriundis; erunt actio et reactio, in omni instrumentorum usu, sibi invicem semper aequales.

If the activity of an agent be measured by its amount and its velocity conjointly; and if, similarly, the counter-activity of the resistance be measured by the velocities of its several parts and their several amounts conjointly, ιvhether these arise from friction, molecular forces, ιveight, or acceleration; .—activity and counter-activity, in all combinations of machines, will be equal and opposite.

This may be looked upon as a Fourth Law. But, in strict logic, the First Law is superfluous, because its consequences are all implied (by negation) in the statement of the Second. (See § 8 below.) Hence there are, virtually, only three laws, so far as Newton’s system is concerned.

§ 3. These laws are to be considered as deductions from observation and experiment, and in no sense as having an α priori foundation. Their proof, so far as rigorous proof is attainable in physical matters, is commonly looked on as being furnished in the most conclusive form by observational astronomy. The Nautical Almanac, published usually about four years in advance, contains the predicted places of the sun, moon, and principal planets from day to day, in some cases from hour to hour, throughout the year. The predictions are entirely based upon the laws of motion (along with the law of gravitation), and could not possibly be accurate unless these laws are true. So thoroughly satisfactory has hitherto been the coincidence between prediction and observation that, when a deviation occurs, no one dreams of a defect in the principles of the reasoning. On the contrary, such deviations are utilized for the purpose of correcting our knowledge of the “elements” of the orbits of the moon and planets, or our estimates of the masses of these bodies; and, as in the brilliant investigations of Adams and Leverrier, they sometimes enable us to discover the existence and even assign the position of a planet never before seen.

§ 4. It is not clear in what order, or by whom, these laws were first discovered. Galileo was undoubtedly acquainted with the first two; and Huygens, Wren, Hooke, and others were acquainted with the Third Law in some of its many applications. But they were first systematized and, as we have seen, extended in a most important manner by Newton. Though they were sadly disfigured in Britain during the fifty years which elapsed after the revival of mathematics in the early part of this century, they have of late been restored to the form in which Newton gave them. This readoption of Newton’s simple but comprehensive system has of itself aided in no small degree the recent rapid advance of science.

One peculiarity of Newton’s language must be noticed here, though very briefly, as we will return to the subject towards the end of the article. A force is said to “compel” a change of state in a body; bodies are said mutually to “act” on one another, &c. Such language is, of course, in its literal acceptation, of an anthropomorphic character; but, if one thinks of the habitual use even in scientific books of such expressions as “the sun rises,” “the wind blows,” &c., it cannot be construed into an assertion that force has real objective existence.

Comments on the Lares of Motion.

§ 5. Law I. First of all this law tells us what happens to a piece of matter which is left to itself, i.e., not acted on by forces. It preserves its “state,” whether of rest or of uniform motion in a straight line. This property (which, as we shall presently show, § 7, is considerably extended by Newton himself) is commonly called the “inertia” of matter, in virtue of which it is incapable of varying in any way its state of rest or motion. It may be the sport of forces for any length of time, but so soon as they cease to act it remains in the state in which it was left until they recommence their action on it. Hence, whenever we find the state of a piece of matter changing, we conclude that it is under the action of a force or forces. Thus, for the present, we have the definition of “force” as part of this First Law :—

Force is whatever changes the state of rest or uniform motion of a body.

When a body, originally at rest, begins to move, we conclude that force is acting on it. And when a moving body is seen to change either the speed or the direction of its motion, we conclude that this is due to force.

§ 6. But there is much more than this, even in the First Law. What is “rest”? The answer must be that the term is relative. Absolute rest and absolute motion are terms to which we find it impossible to assign a meaning. Maxwell has well said (in his Matter and Motion)’.—

“All our knowledge, both of time and place, is essentially relative. When a man has acquired the habit of putting words together, without troubling himself to form the thoughts which ought to correspond to them, it is easy for him to frame an antithesis between this relative knowledge and a so-called absolute knowledge, and to point out our ignorance of the absolute position of a point as an instance of the limitation of our faculties. Any one, however, who will try to imagine the state of a mind conscious of knowing the absolute position of a point will ever after be content with our relative knowledge.”

As will be seen later, the First Law gives us also a physical definition of “time,” and physical modes of measuring it.

§ 7. Newton’s own comment on this law is as follows :—

Projectilia perseverant in motibus suis, nisi quatenus a resistentia aeris retardantur, et vi gravitatis impelluntur deorsum. Trochus, cujus partes cohaerendo perpetuo retrahunt sese a motibus rectilineis, non cessat rotari, nisi quatenus ab aere retardatur. Majora autem planetarum et cometarum corpora motus suos et progressivos et circulares, in spatiis minus resistentibus factos, conservant diutius.

It is particularly worthy of notice that we have here the undisturbed rotation of a body about an axis introduced as another of those “states” in which it will continue, in virtue of the First Law, until force acts to compel it to change that state. Also it is to be noticed that Newton adduces a hoop, whose axis is fixed in direction both in the body and in space, as an example of this new form of state maintained in virtue of inertia. Later, it will be seen that the same thing is true of a body free in space and rotating about the principal axis of greatest or of least moment of inertia through its centre of mass.

§ 8. Law II. What Newton designates by the word motus is, as he has clearly pointed out, the same as is expressed by quantitas motus, that for which we now usually employ the term “momentum.” Its numerical value depends not only on the rate of motion, but also on the amount of matter, or “mass,” of the moving body, and is directly proportional to either of these when the other is unaltered. But it is regarded by Newton as having direction as well as magnitude. It is, in fact, what in the language of quaternions is called a “vector.” The change of such a quantity may be either in numerical magnitude, or in direction, alone, or simultaneously in both. We now see what this Second Law enables us to do. For (a) Given the mass of a body, the force acting on it, and the time during which it acts, we can calculate the change of motion. This is the direct problem of dynamics of a particle.

(6) Given the mass, and the change of velocity, we can calculate the magnitude and direction of the force acting. This is the inverse problem.

(c) We can compare, and so measure, forces by the changes of motion they produce in one and the same body.

(d) We can compare the masses of different bodies by finding what changes of velocity one and the same force produces in them.

(e) We can find the one force which is equivalent, in its action, to any given set of forces. For, however many changes of motion may be produced by the separate forces, they must obviously be capable of being compounded into a single change, and we can calculate what force would produce that.

§ 9. Hitherto, we have spoken of the motion of a body,— thus implying (except, of course, in the case of Newton’s hoop or that of the earth) that all its parts are moving in exactly the same way. From this point of view every body, however large, may be treated as if it were a single particle. But when the parts of a body have different velocities, as when a rigid body is rotating, or as when a non-rigid body is suffering a change of form, the-question becomes much more complex. We cannot at this stage enter into a full explanation, but will take a couple of very simple cases to »show the nature of the new difficulties, and thence the necessity for an additional law.

Suppose a bullet to be thrown in any direction. If we know with what force the earth attracts it, the calculation of the path it will pursue depends on the Second Law, which gives all the necessary preliminary information. But let two bullets be tied together by a string: we know by trial that each moves, in general, in a manner very different from that in which it would move if free. The path of each is now, usually, a tortuous curve, while its free path would be plane. It is no longer subject to gravity alone but also to what is called the “tension” of the string. If we knew the amount of this tension on either of the bullets and its direction, we could calculate, by the help of the Second Law alone, all the circumstances of the motion of that bullet. But how are we to find this tension Ί Is it even the same for each bullet? This, if answered in the affirmative, would simplify matters considerably, but we should still require to know the amount and direction of the tension. It is clear that, without a further axiom, we cannot advance to a solution of the question.

§ 10. Law HI. Furnished with this, in addition to our previous information, we can attack the question with more hope. We see by this law that, whatever force be exerted by the string on one of the bullets, an equal and opposite force, which must therefore be in the direction of the string, is exerted on the other. Still, the magnitude of these equal forces remains to be found. But the string in no way interferes with the motion of either bullet unless it is tight, i.e., unless the distance between the bullets is equal to the length of the string. Hence, whenever the unknown force comes into play, at the same time there comes in a geometrical relation of relative position between the two bullets. This supplies the additional equation necessary for the determination of the new unknown quantity.

§ 11. As an additional illustration, suppose the string to be made of india-rubber. The Third Law tells us that the tensions it exerts on the bullets are still equal and opposite. But we no longer have the geometrical condition we had before. We have, however, what is quite sufficient, a knowledge of how the tension of the string depends on its length. Thus the tension can be calculated from the relative position of the bullets.

§ 12. Scholium to Law III. On this we will, for the present, remark only that it furnishes us with the means of studying directly the transference of energy from one body or system to another. Experiment, however, was required to complete the application of this part of Newton’s systematic treatment of the subject. What was wanted, and how it has been obtainéd, will be treated of later. The first words of the scholium, however, claim for Newton the discovery of the clause we have extracted from it. For they run thus :—Hactenus principia tradidi a mathematicis recepta, et experientia multiplici confirmata.

§ 13. What has now been said enables us to see the order in which the fundamental ideas should be taken up, so that the necessities of each should be provided for before its turn comes. An indispensable preliminary is the study of motion in the abstract, i.e., without any reference to what is moving. This is demanded in order that we may be able to apply the Second Law. The science of pure motion, without reference to matter or force, is an extension of geometry by the introduction of the idea of time and the consequent idea of velocity. Ampère suggested for it the term Cinématique, or, as we shall write it, Kinematics. We include under it all changes of form and grouping which can occur in geometrical figures or in groups of points.

We shall then be prepared to deal with the action of force on a single particle of matter, or on a body which may be treated as if it were a mere particle. Thus we have the Dynamics of a Particle. This, again, splits into two heads, Statics and Kinetics of a Particle. But all this requires the Second Law only. When we have two or more connected particles, or two particles attracting one another or impinging on one another, the Third Law is required. Next in order of simplicity come the Statics and Kinetics of a Rigid Solid. Then we have to deal with bodies whose form, <kc., are altered by forces—flexible bodies, elastic solids, fluids, <fcc. Finally, we must briefly consider the general principles, such as “conservation of energy,” “least action,” <fcc., which are deducible by proper mathematical methods from Newton’s Laws, and of which some at least, if we could more clearly realize their intrinsic nature, would probably be found to express even more simply than do Newton’s Laws the true fundamental principles of abstract dynamics.

We will not restrict ourselves to one uniform course in the application of mathematical methods. Rather, as considerations of space require to be attended to, we will vary our methods from one part of the subject to another, so as to exhibit, each at least once, all the more usual processes. And we will endeavour to make the large-type portions of the article, in which only the most elementary mathematics will be introduced, a self-contained treatise which may be read by students of very moderate mathematical knowledge.

KINEMATICS.

Position and the Means of Assigning it.

§ 14. Motion (or displacement) consists simply in “change of position.” Hence, to describe motion, we must have the means of assigning position. This is, of course, a question of Geometry {q.vf. See also Quaternions.

From these articles it appears that the position of one free point with reference to another (all these space relations are relative, as we have already said) depends on three numbers, of which one at least must involve the unit of length. In Cartesian rectangular coordinates, we denote these by x, y, z, which indicate respectively the distance of the point from each of three planes at right angles to each other, and all passing through the origin (or reference point). From another point of view they may be called “degrees of freedom.” When the value of one is assigned, say by x=a, the point is said to have lost one degree of freedom, or to have had imposed upon it one “degree of constraint.” It must now lie in a plane parallel to the first of the reference planes, and at a distance a from it. When a second degree of constraint is applied, say by y=b, another degree of freedom is lost. The point’s position is limited to lie in a second plane in a given position at right angles to the first. It must therefore lie somewhere on the straight line which consists of the series of points common to the two planes. A third degree of constraint z=c takes away its one remaining degree of freedom; and its position is now definitely assigned as the single point of intersection of three given planes.

§ 15. But constraint may be applied in other ways. Thus if we assign the condition x 2 + y 2 + z 2 = a 2 t we deprive the point of one degree of freedom by compelling it to remain at a distance a from the origin. It is now limited to the surface of a sphere, but its latitude and longitude on that sphere may be any whatever. Here again the imposition of one degree of constraint has taken away one degree of freedom.

§ 16 In general, one degree of constraint may be expressed as fx, y, *)=f∙

This, when ξ has an assigned value, is the equation of a definite surface on which the point must lie. Three such conditions determine the position of the point, and may therefore be looked upon as introducing ξ, η, ζ, another set of coordinates, which may be used in place of x, y, z. The number of such systems is, of course, unlimited; but it is often possible to choose one in which the conditions of a problem are much more simply expressed than they were when expressed in x, y, z. The whole question belongs to what is called “change of variables.” To give an elementary instance of its use,—suppose we take the ordinary simple pendulum, a pellet supported by a fine thread or wire, and oscillating in a vertical plane. If the origin be placed at the point of suspension, and the axis of z be vertical, we have two conditions :—

x 2 + y i + z 2 = α a , where a is the length of the thread; and y∕as = tanα, where a denotes the azimuth of the plane of oscillation. There is but one degree of freedom left, because two degrees of constraint have been imposed. We may choose for this either x, y, ox z ∙, but we should in each case be led to complex expressions. If, however, we consider that all the freedom left to the pendulum is to oscillate in a given plane, we may denote its sole remaining degree of freedom by θ, the angle which the string makes with the vertical; and form our dynamical equation in terms of this. When θ is found by dynamical considerations, we have ae = a sin 0 cos α , y—a sin 0 sin a , 2=acos0.

Here θ comes in as what is called a “generalized coordinate.”

If the pendulum be not limited to one plane, the azimuth, as well as the angle, of the displacement from the vertical may be any whatever. Hence there are two degrees of freedom, which are indicated by the generalized coordinates α and θ.

§ 17. In general, in any system originally with any number m of degrees of freedom, and subjected to a number n of degrees of constraint, the whole motion can be fully characterized by th - n independent quantities, called generalized coordinates, and corresponding to the degrees of freedom which remain. The elegance and simplicity of a solution often depend in a marked manner upon the choice of these; and the transformation of the general equations from Cartesian to generalized coordinates forms one of the most powerful and elegant contributions to abstract dynamics which Lagrange made in the Mécanique Analytique.

§ 18. A rigid system has onlys⅛ degrees of freedom:— three translations for any one of its points, and three independent rotations about axes passing through that point.

When one point is fixed, it loses the three translations, and has only three degrees of freedom. When a second point is fixed, it loses other two; in fact it can no longer move except by turning round the line joining the fixed points. When a third point, not in line with the other two, is fixed, there is no degree of freedom left; the system is fixed.

§ 19. It may be well to notice here that, in all cases which we shall require to consider, whatever be the relations among two different sets of variables which we employ alternatively to determine the relative positions of the parts of any system, the equations which give the relations between corresponding small increments of these variables are always linear so far as these increments are concerned. Thus, for instance, if we have as above a condition of the form

∕G≈, y, *) = ξ, we deduce from it at once m to+ m δ , + m 8s . δξ .

∖dx] ∖dy J ∖dz)

Here the differential coefficients are partial.

In such cases, any homogeneous function of the second order in δz, δy, δz, &c., will be represented by a homogeneous function, also of the second order, in δξ, δη, kx., however many be the coordinates in the separate systems. When, however, one or more of the equations of condition involves the element of time explicitly, the relations among corresponding small increments of the alternative sets of coordinates, though still linear, will not be homogeneous. Thus a homogeneous function of the second order in one set will be a function of the second order in the other set, but not homogeneous, unless the increments are produced instantaneously.

To give a single instance, suppose that the string of a simple pendulum (not necessarily oscillating in one plane) contracts uniformly. We shall now have ae ! + y 2 + 2 s =(α — et) 2 instead of the equation in the example § 15, and one of our equations among increments is xδx + yδy + zδz= -e(a-et)δt, which, though still linear, is no longer homogeneous in the increments of coordinates. Kinematics of a Point.

§ 20. The one necessary characteristic of the path described by a moving point is its continuity. There can be no break or gap in it. But, as we study kinematics, at present, solely for its physical applications, we impose a restriction on such complete generality. The path of a moving particle must be one of continuous curvature, unless either (1) the motion ceases and commences again in a different direction (in which case we have two separate and successive states of motion to consider), or (2) an infinite force is applied to the particle (a case which we need not consider). A similar remark, we may say in passing, applies to velocity also. So that, for our purpose, we may confine ourselves to the geometrical properties of the motion of a point whose rate and direction of motion change continuously, if at all, and not by fits and starts.

§ 21. If the point describe a straight line, that line gives the direction of its motion at every instant. If it describe a curve, the direction of its motion is at every instant that of the corresponding tangent to the curve.

Let A, B, C, D represent four points on the path taken in close succession, in the order in which the moving point reaches them. From A the point moves to B, so that the line joining A and B (the tangent) is the direction of motion at A. Similarly the line joining B and C gives the direction of motion at B. The points A, B, C of course lie in one plane. This is the plane in which, for two successive elements of its path, the point is moving. It is therefore that in which the change of direction of motion takes place, and is called the “osculating plane.” And, just as the straight line through A and B gives the direction of motion at A, so the circle passing through the points A, B, C determines the “curvature” of the path at A. If we apply the same reasoning to the three successive points B, C, D, we see the difference between a “plane” and a “tortuous” curve. For, if D lie in the plane ABC, the osculating plane is the same at A and at B; and if the same holds for other successive points the whole bending takes place in one plane. But if D be not in the plane ABC, BCD is the osculating plane at B, and we thus see that successive positions of the osculating plane of a tortuous curve are produced by its rotation about the tangent BC to the path; for BC is in both planes ABC and BCD. We shall not have space here to deal in detail with cases of tortuosity; but it was necessary to point out their essential nature.

§ 22. The curvature of ABC obviously depends upon the change of direction from AB to BC, and is directly proportional to it. But it is obviously greater, for the same amount of change of direction, as ABC is less. In a circle the curvature is the same at all points, and, as the radius is everywhere perpendicular to the tangent, the change of its direction is the same as that of the tangent. Hence the curvature, being the change of direction per unit length of the arc, is measured simply by the reciprocal of the radius.

Generally, if φ be the angle between the tangent at A and any fixed line in the osculating plane, and if s represent the length of the curve measured from any fixed point on it to A, we have, by the fundamental property of infinitesimals, τ δφ dφ , I.- r = -P = curvature, δs ds

(We will use, as above, the letter I. for a limit, in the sense in which that term was introduced by Newton.)

In a circle we have always (α being the radius) s=aφ, and hence the curvature

⅛ = X∙

ds a ’ so that in general the measure of curvature is the reciprocal of the radius of the circle passing through three consecutive points of the path. For other analytical expressions for curvature see vol. xiii. p. 26.

For a curve in space (whether tortuous or not) we have 1 ∕ (d 2 x y (d' 1 y ∖ i ( d' i z ∖ 2

Curvature--- √(^ 2 J ’

while the direction cosines of the radius of curvature are d?x d i y cPz P ds 2 ’ p ds 2 ’ p ds i

§ 23. The chief properties connected with the curvature of a plane curve are made very clear by the artifice of regarding it as an “involute.” This idea introduces us to the kinematics of a flexible and inextensible line. Suppose such a line, held tight, to be wrapped round a cylinder of any form, in a plane perpendicular to its axis, each point of it, when it is unwound in its own plane, will describe a curve whose form depends upon that of the transverse section of the cylinder. Let P 0 M'M (fig. I)J 3e such a section of the cylinder; MP, MP, two positions of the free part of the cord; P, P ’, the corresponding positions of a definite point of the cord; PP'P 0 the path described by that point. Then PP'P 0 is one of the involutes of MM'P 0 ; the others are the curves traced by other points of the string. But, with reference to PP'P 0 , the curve MM'P 0 is the “evolute.” The evolute of such a curve is, in fact, unique; for it is obvious that the line MP, in any of its positions, is revolving about the point of contact Μ with the evolute; so that P describes an infinitesimal arc of a circle of which Μ is the centre. Thus the evolute of a plane curve is the locus of its centre of curvature. And it is clear from the genesis of the involute that

PM = P'M' + M'M = P 0 M ’ M .

For the analytical discussion of evolutes, see vol. xiii. p. 26.

The subject of evolutes is of importance in various branches of physics, especially in optics. In mechanics its chief use is connected with the theory of the pendulum, as it shows how to cause the bob to move in a cycloid, the only path in which the time of oscillation is the same whatever be the extent of the oscillations.

§ 24. When the line, curved or straight, in which the motion takes place is given, the position of the moving point is at once assigned in terms of a single numerical quantity. In fact it has only one degree of freedom, and its position is known by the length of the arc of the curve from any fixed point to the given position. In such a case as this we are not concerned with the direction of the motion, for that is already assigned at every point of the path. We are concerned only with what we may call the “speed” of the motion. (We purposely avoid the use of the term “velocity” here, because it properly includes direction as well as speed, as will be seen later.)

§ 25. Suppose an observer to be watching the motion (as, for instance, a traveller by rail notes the telegraph posts which he passes, referring at each to his watch), and to find that at any time the moving point was at s 1 , while at time t 2 it was at s 2 .

Then it is clear that the average speed during this part of the motion is to be found by dividing the number of units of space passed over by the number of units of time employed. For it must be greater as the former is greater and less as the latter is greater. Hence the average speed is -t ‘ the speed has been uniform, during the motion observed, this average value has coincided with the actual value all through; and, if the measures of space and time are accurate, we shall get exactly the same value of this ratio whether the interval of time is small or large. Hence, if v be the speed of a uniformly moving point, the space it describes in time t is vt. But if the speed has been variable, it must at some parts of the interval have been greater, at others less, than this average. And the shorter we take the interval the less will be the difference between the greatest and least speeds during its lapse, so that the average speed will coincide more and more nearly with the actual speed. In the language of “fluxions” (which was invented for the sake of this subject) the measure of the speed at any time i 1 is when the interval t 2 - t x is shortened indefinitely. The accuracy of the preceding process depends entirely upon the limitations we have introduced for the purpose of confining ourselves to cases which can occur in ordinary physical problems. For the general reasoning on which it is based is obviously inapplicable to cases in which the speed alters by jerks—at least during the interval considered, small as it may be. But we are fortunately not required to discuss here the very delicate questions to which this may give rise. Considerable difficulty is sometimes felt by a student when he is told that at a certain part of its course a point has a speed say of 10 miles an hour, while the whole course may be only a few inches. But this arises from the novelty of the conception. It is not meant, when we speak of a speed of 10 miles per hour, that the motion necessarily lasts for an hour, or even for a second, but only that, if the then speed were to be maintained constant for an hour, the moving point's path, of τvhatev r form, would be exactly 10 miles long. In actual experience in a railway train we can judge the speed (roughly at least), and we find nothing strange in saying “Nowwe are going at twenty miles an hour,” “Now at six,” and so on. And it is clear that, after the steam is put on, the train, however short its run, must go through all rates of speed from zero to its maximum, and then through all of them to zero again, when the steam is cut off and the brake applied.

In the language of the differential calculus this becomes . _ τ δs _ ds i δZ ~dt ‘

The fluxional notation of Newton, in which the dot over a quantity expresses the rate of its increase, i.c., its differential coefficient with regard to time considered as the independent variable, is still very convenient in abstract dynamics, and is, in fact, indispensable when we come to the higher generalizations. We shall, therefore, freely employ it when it is specially useful.

§ 26. Whether uniform or variable, speed depends for its numerical value upon the units chosen for linear space and for time. Its dimensions are [LT^ 1 ], and consequently its numerical expression is increased in proportion as the unit of time is increased, and diminished in proportion as that of length is increased. Thus the speed represented by 10 in feet per second becomes

3600, O _75 5280 ’ ° - ll when expressed in miles per hour.

§ 27. The rate at which the speed (when not uniform) changes is found by a process precisely similar to that employed for the speed itself. Let the speed at time be observed to be v 1 , » t‘2 >> >> v 2 > then the average rate of increase of speed during the interval is V2~ v 1 ⅛ ~ h

The dimensions of this quantity are obviously [LT^ 2 ]. Thus its numerical value is diminished, like that of speed, in proportion as the unit of length is increased. But it is increased in the duplicate of the proportion in which the unit of time is increased. For instance a rate of increase of speed of 32 ’ 2 feet per second per second (the mere statement is enough to show the double dependence on the time unit) becomes

32'2 ^θθθ -= 79,036 nearly, when expressed in terms of miles and hours.

§ 28. When the rate of increase of speed is uniform, the above average value is its actual value throughout the interval. Hence with uniform rate of increase = a, a speed V becomes in time t v = V + αi.

Also, as it increases uniformly, its average value during time t is half way between its values at the beginning and end of that time; i.e., it is

V + Jαi.

The space described during the interval is at once found (§ 25) as the product of the interval and the average speed during its lapse;— i.e., it is s=^Vt + ⅛at 2 .

And it is easy to see from these expressions that υ 2 = V 2 + 2αs, which gives the speed acquired in terms of the space traversed.

§ 29. This is the only case in which the result can be reached without formally using the methods of the integral calculus. These expressions enable us at once to solve a great number of simple questions connected with the motion of a stone or bullet, under the action of gravity, in a vertical line. For it is found by experiment that gravity impresses, in every second, a downward speed of 32 - 2 feet per second on an unsupported body; and, by the Second Law, this is independent of the body’s previous motion.

Hence, if a stone be let fall, its speed after t seconds is 32 - 2i, and the space fallen through is 16Η 2 . Also, if it fall through s feet, it will acquire a speed whose square is v i = 64 ’ 4s.

Again, if a stone be thrown upwards with a speed of 300 feet per second, after t seconds its speed will be 300 - 32 , 2i, and the height to which it has then ascended is 300i-16 - li 2 . Thus it stops, and turns, after g^2 300 2 seconds; and the greatest height it reaches is feet.

From the statement above, putting s=∙υ, we find „ . τ v 9 - v 1 dv S=v = l>- - = — tf^ t∕ j (It

From this expression the preceding results may be at once obtained. Thus, assuming

⅛" = α , we have by integration

5 = V + at, and again s= s 0 + Vi + ⅜αi 2 .

As an instance of the indirect problem— i.e., to find the speed, and its rate of increase, when the law of the motion is given:—suppose s = αcosωi.

(This equation describes the simplest form of vibratory motion, and will be fully treated later.) We have, by taking the fluxion, s = - αωsinωi, and again s = - aω 2 cos ωt = - ω"s.

§ 30. Velocity, as we have already said, involves the ideas of speed and of direction of motion conjointly. To compound two velocities (as is required in the application of Newton’s Second Law), we have the following obvious construction. From any fixed point O (fig. 2), draw a line OA representing, in magnitude and direction, one of the two velocities. From its extremity A draw AB representing in the same way, and on the same scale, the other. Complete the triangle OAB. Then OB represents, in magnitude and direction (still on the same scale), the resultant velocity. We have called the construction obvious because one has only to think of how a point can be said to have simultaneous velocities, in order to see its truth. Thus, if OA represents the velocity of a railway train, AB that of a passenger walking in a saloon carriage, O may be looked upon as the position of the point of the carriage at which he began his walk, at the moment when he did begin it; while B represents the position of the point of the carriage which he has reached at the end of his walk, just at the moment when he did reach it. Here OA is the velocity of the carriage relative to the earth, AB that of the passenger relative to the carriage.

This proposition may be called the triangle of velocities. Another obvious mode of stating it is to complete the parallelogram of which OB is a diagonal (fig. 3); and then we have the same construction in the form :—If the two velocities to be compounded, represented by OA and OC, be taken as contiguous sides of a parallelogram, the conterminous diagonal OB represents their resultant.

§ 31. From the triangle of velocities we may pass at once to the polygon of velocities, which gives us the resultant of any number of simultaneous velocities. Thus, beginning as above at any point 0 (fig. 4), lay off OA, AB, BC (however many there may be) as successive sides of a polygon all taken in the same direction round. The separate velocities may be in one plane or not. When this is done, the final point C is easily seen to be independent of the order in which the separate velocities were taken, and is thus a perfectly definite point. 00, completing thé polygon, represents the resultant velocity. But it is taken in the opposite direction round. If C coincide with O, there is no resultant;— i.e., a point which has, simultaneously, velocities represented by the successive sides of any polygon, all taken the same way round, is at rest.

In what precedes, we have denoted the position of the moving point in its known path by the single quantity s. But if λve think of its Cartesian coordinates x, y, z, we see that in general each of these must vary during the motion. And just as we represented the whole speed by s so we may speak of x as the speed in the direction of the axis of x, &c. And now we have a hint of a most important character. For, by the ordinary laws of the differential calculus, we have three equations of the form x dx dx-

— = -ror x≈- τ -s . s ds ds

Now — is the cosine of the inclination of the tangent at s to ds the axis of x. And so with y and z. These give us ., . 0 .√ ∕ dx ∖ 2 ( dy ∖ 2 f dz ∖ 2 ∖ * 2 +^+ 22 =H>) + U) +Wγ s ∙

Hence we see that a speed in any direction may be resolved into three in any assigned directions at right angles to one another; that the speed in any one of these is determined by multiplying the whole speed by the cosine of the angle between its direction and that of its resolved part; and that the square of the whole speed is the sum of the squares of the speeds in the resolved motions. These results, however, can be obtained more directly, and in a more instructive manner, by the consideration of “velocities,” and not of mere “speeds.” But, before we take this step, let us take the second fluxions of the coordinates, and see to what they lead us. t? „ . dx .

From x = -s ds we obtain at once χ = — ÿ + s 2 ; ds ds' i or, introducing in the last term, both as a multiplier and as a divisor, the radius of curvature of the path, .. dx.. d 1 x s 2 . £ ds s + ^ds 2 p ’ with similar expressions for ÿ and z. These show that the rates of increase of speed, parallel to the three axes respectively, may be considered as made up of the resolved parts of the two directed quantities s and s 2 ∣ p. The first is in the direction of the tangent to the path, the second in the direction of the radius of curvature; and the law of resolution is, for each, multiplication by the cosine of the angle between the two directions concerned. We shall presently recognize these as the components of the acceleration.

§ 32 To resolve a velocity is of course a perfectly indefinite problem, unless the number of conditions requisite for definiteness be imposed. For, in general, it may be taken as one side of any complete polygon, whether in one plane or not; and the other sides, all taken in the opposite order round, represent its components.

The only cases which we need consider, in which the conditions are such as to ensure one definite solution, are—(1) when a velocity is to be resolved into components parallel and perpendicular to a given line; and (2) an extension of the same case to components parallel respectively to three lines at right angles to one another. In case (1) the given velocity is to be taken as the hypotenuse of a right-angled triangle of which one of the sides is parallel to the given line. In case (2) it is to be taken as the diagonal of a rectangular parallelepiped of which the edges are parallel to the three lines respectively. In either case the magnitude of each component is found by multiplying the amount of the velocity by the cosine of the angle between its direction and that of the component; and the square of the whole velocity is equal to the sum of the squares of the components. We are now prepared to take up the requisite preliminaries for the application of the Second Law. What, in fact, is “change of velocity” 1 The preceding statements at once enable us to give the answer. For let OA (fig. 5) be the velocity of a point at one instant, OB at a succeeding instant. To convert OA into OB, we must compound with it a velocity represented by AB.

AB represents the change. Hence if, during any motion whatever of a point, a line OA be constantly drawn from a fixed point O, so as to represent at every instant the magnitude and direction of the velocity of the moving point, the extremity of OA will describe a curve (plane if the original path be plane, but not otherwise, except in certain special cases) which possesses the following important but obvious properties :— (1) the tangent at A is the direction of the change of velocity in the original path; (2) the rate of motion of A is the rate of change of velocity in the original path. Hence in this auxiliary curve, called the Hodograph (<∕.v.), the velocity represents, in magnitude and direction, what is called “acceleration” in the original path. And, because the acceleration can thus be represented as a velocity, the laws of composition and resolution of velocities hold good for accelerations also.

§ 33. Hence, if we desire to know the whole acceleration in any case of motion of a point, we need only-find its components in, and perpendicular to, the tangent to the path. That in the tangent has already been found; it is V or s as in § 29. For that perpendicular to the path we may study the simple case of uniform motion in a circle.

§ 34. If a point move with uniform speed V in a circle, the hodograph is evidently a circle of radius V, and is described uniformly in the same time as the orbit (see fig. 6), Hence the speeds in the two circles are as their radii. Let R be the radius of the orbit. Then the magnitude of the acceleration in the orbit (the speed in the hodograph) is found from

A:V::V:R; that is, A = V 2 ∕R∙

The direction of this acceleration, being that of the tangent to the hodograph, is perpendicular to the corresponding radius of the hodograph, i.e., to the tangent to the orbit. Hence it is along the radius of the orbit and directed inwards to its centre.

§ 35. In other words, to compels. mass to describe an unnatural (because curved) path, it must be acted on by a force directed towards the centre of curvature of the path. We anticipate so far as to introduce here mass and force, although, strictly, we are dealing with kinematics. But the student cannot be too early warned of the dangerous error into which so many have fallen, who have supposed that a mass has a tendency to fly outwards from a centre about which it is revolving, and therefore exerts a “centrifugal force/' which requires to be balanced by a “centripetal force.” The centripetal force is required if the path is to be curved; it is required for the purpose of producing the curvature, and not because there is any tendency to fly out from the centre.

§ 36. Thus, in any motion of a point, the whole acceleration is the resultant of two parts—the first in the direction of motion and of magnitude equal to the rate of increase of speed, the second directed towards the centre of curvature and of magnitude as the curvature and the square of the speed conjointly. The sole effect of the first component is to alter the speed, of the second to alter the direction, of the motion. There is no acceleration perpendicular to the osculating plane, because two successive values of the velocity, and therefore also the corresponding change of velocity, are in that plane.

§ 37. A very convenient expression for acceleration which changes the direction of motion is furnished in terms of what is called the “angular velocity,” i.e., the rate at which direction changes. This also is properly a vector, or directed line, perpendicular to the plane in which the change of direction takes place, and of length proportional to the rate at which the angle assigning the direction changes.

§ 38. In the case of uniform motion in a circle of radius R, with speed V, the time of describing the complete circumference (2τrR) is 2τrR∕V. Hence the angular velocity is V∕R, usually denoted by ω. Thus the above expression for the acceleration in a direction perpendicular to the path of a point (§ 34) may be written in the form pω 2 , where p is the radius of curvature of the orbit and ω the angular velocity of that radius. The direction of this acceleration, as we have seen, is always towards the centre of curvature.

§ 39. The general difficulty of any question concerning acceleration is usually a purely mathematical one, involving only such physical considerations as are required for the formation of the differential equations, and for the determination of the so-called arbitrary constants or arbitrary functions involved in the integrals. We will not now discuss the various forms in which the difficulty may present itself, because in the course of the article many of the more important of these will be fully treated in connexion with motions actually observed among terrestrial or cosmical bodies.

§ 40. We have sufficiently considered (§§ 27-29) uniform acceleration in the line of motion. Let us now consider uniform acceleration in a fixed direction, whether the motion of the point be in that direction or not. This is the most general case of the motion of an unresisted projectile, on the supposition that its path is confined to a region throughout which gravity is sensibly constant alike in direction and in intensity. Two well-known properties of the parabola lead to an immediate solution of our problem.

Let fig. 7 represent a parabola, defined completely by its focus S and its directrix MN". We suppose it to be placed with its axis vertical, and vertex upwards. Take any point P, join PS, and draw PM perpendicular to the directrix.

Then

(a) If PQ bisect the angle SPM, it is the tangent to the parabola at P.

Let Q be any point in the tangent, and let QR, drawn parallel to MP, meet the curve in R. Then we have

(δ) PQ 2 = 4SP.QR.

§ 41. Now suppose a point, originally moving along PQ with uniform speed V, to have its motion accelerated in a direction parallel to MP, the acceleration being α, a constant. Then, after t seconds it would have moved along PQ through a space Nt, and parallel to MP through a space ∣αf 2 . Hence, if R be its position at that time,

PQ≈Vi, QR=⅜ α < 2 .

From these equations we find at once PQ2=¾ r .

α

This relation is of the same form as that already written for a parabola, and (as it does not involve i) it holds for every point of the path. Hence the point moves in a parabola whose axis is vertical, which touches PQ (the direction of projection) in P, and in which SP = V 2 ∕2α, But these three data determine the parabola. For we have only to draw PM vertical, make the angle QPS = QPM, and measure off the lengths PM and PS each equal to V 2 ∕2α. Μ is a point in the (horizontal) directrix, and S is the focus. Hence the path is completely determined.

It is well to notice that, as V 2 = 2αPM, Μ is the point which the projectile would just reach if it were projected vertically upwards (§ 29).

§ 42. If the speed of projection be kept constant, while the direction of PQ alters in a vertical plane, S describes a circle about P as centre. This consideration enables us easily to find the direction of projection that a given object may be struck. Let 0 (fig. 8) be the object. Join PO, and let it cut in B the circle MBS (whose centre is P). Draw ON perpendicular to the common directrix, and with radius ON describe a circle about O. This will (in general) cut MBS in two points F and F'. These are the foci of the two paths by either of which the projectile can reach 0. For by construction FO = ON, so that O lies on the path whose focus is F. Similarly for F'.

To find the most distant point along PO which can be reached, with the given speed of projection from P, we have merely to note that, as O is taken farther and farther from P, F and F' approach B, and finally coincide with it. If O be then at A, we have AT = AB, where AT is perpendicular to the directrix. Hence, if we produce AT to t so that T^ = BP, we have Ai = AP. Draw through. t a line tm parallel to TM. Then A lies on the parabola whose focus is P and directrix mt. This parabola is the envelop of all the possible paths from P. Any point within it can be reached by two different paths. These become coincident when the point lies on the curve; and no point outside it can be reached.

§ 43. Many of the most important cases of motion of a point involve acceleration whose direction is always towards a definite “centre” as it is called. In such cases the motion is obviously confined to the plane which, at any instant, contains the centre and the line of motion of the point.

Also the “moment” of the point’s velocity about the centre remains constant. Here a slight digression is necessary.

Def.— Given a directed quantity (α velocity, force, Ac.) in a line AB (fig. 9). If a perpendicular OP be drawn to AB from any point 0, the “moment” of the directed quantity about 0 is the product of its amount by the length of the perpendicular.

If the directed quantity be reversed, the sign of the moment is changed. The moment is, in fact, properly a directed quantity (or vector) perpendicular to the plane OAB. And its numerical magnitude is double the area of the triangle OAB.

[§ 44. The convention usually made as to the sign-of rotation about an axis is to regard it as positive when it is in the same sense as that in which the earth turns about its axis, as seen by a spectator above the north pole. This is in the opposite direction to that of the hands of a watch. Hence the plane angle AOP (fig. 10), representing the change of direction of a line originally coincident with OA, is positive, and is looked on as due to rotation about an axis drawn from O upwards from the plane of the figure. Thus the rotation of the sun and the orbital motions of the planets take place in the positive direction about axes drawn on the whole northwards from the plane of the ecliptic; or we may put it thus :—seizing an axis by the positive end, we must ««screw—by the negative end, we must screw—to give positive rotation. And when, later, we consider rotations about three rectangular axes, O∙r, Oy, Oz, we shall suppose them so drawn that rotation through a positive right angle about Ox changes Oy into Oz, „ Oy „ Oz „ Ox, „ Oz „ Ox „ Oy, the three letters being throughout arranged in cyclical order, xyz, yzx, &c.]

§ 45. Here we must introduce a simple geometrical proposition :—

If any point be taken in the plane of a parallelogram, and triangles be formed with the point as vertex and with contiguous sides and the conterminous diagonal as their respective bases, the sum of the areas of the first two triangles is equal to the area of the third.

Thus, in areas (fig. 11),

OAB+OAC=OAD.

If 0 lie within the angle BAG, as in fig. 12, the proposition becomes

OAC-OAB=OAD.

§ 46. Remembering that these areas represent half the moments of the bases of the respective triangles about the point 0 (that of OAB being negative in the second case above), we see that The moment of a diagonal of a parallelogram about any point in the plane of the figure is the (algebraic) sum of the moments of two conterminous sides.

Now, suppose the sides of the parallelogram to represent a velocity and its change. If the direction of the change pass through O, its moment is nil. Hence, for acceleration directed towards a fixed point the moment of the velocity about that point is constant.

This is commonly expressed by saying that the radiusvector describes equal areas in equal times about the point to zvhich the acceleration is directed. For the moment of the velocity is double the area so traced in unit of time.

Another way of expressing the same thing is to say that the angular velocity of the radius-vector is inversely as the square of its length. For the product of the square of the radius-vector and its angular velocity is double the area described by it in unit of time.

The converse of this proposition is also evidently true; i.e., when a point moves so that the moment of the velocity about a point in the plane of its motion is constant, its acceleration relative to that point (if any) is directed towards or from that point.

§ 47. Analytically: if P be the acceleration, directed towards a fixed point which we choose as origin, we have

⅛ = -Pcos0= -^Px ∣ r, ÿ= - Psin0 = - Vy ∣ r, (r and 0 being the polar coordinates of the moving point; we have already seen that the path is necessarily plane). Eliminating P, we have

O=a⅛r-⅞⅛E=∙-^(r 2 0).

Thus xÿ — yx=r 2 0 = const. = h .

This may be transformed, at once, by the methods of the differential calculus, into ps≈pv = 7ι, where p is the length of the perpendicular from the origin to the tangent to the path. Conversely, if equal areas be described by the radius-vector in equal times, we have r 2 ()≈xy-yx = h.

Whence xÿ -yx=0, or x≈Qx, y=Qy.

Hence the whole acceleration is Qr, and is directed towards or from the origin.

While we are dealing with these formulae we may investigate the general expressions for velocity and acceleration in terms of polar coordinates for a point moving in a plane.

We have ≈=rcos0, ι∕=rsin0.

From these a5 = rcos0-r0sin0,

2∕ = rsin0 + r0cos0.

Hence the speed along the radius-vector is aecos0 + ÿsin0 = r; and that perpendicular to the radius-vector (in the direction in which 0 increases) is

⅛cos0-⅛sin0 = r0.

These expressions might have been written down at once, if we note that δr and rδ0 are the resolved parts of δs along, and perpendicular to, r. But we must be careful how we carry this species of reasoning one step further. Taking the second fluxions of x and y, we have x = (r - r0 2 ) cos 0 - (2r0 + rø) sin 0, ÿ = (i i - z ’ 0 2 ) sin 0 + (2r0 + ?·0) cos 0.

Hence the acceleration along the radius-vector is xcosθ + ^sin0 = r -r0 2 , and that perpendicular to it (positive when in the direction in which 0 increases) is

ÿcos 0 - ⅛sin 0=2r0 + r'θ = ^--^-(r 2 0).

Thus, although r represents truly the speed along r, r does not represent the acceleration in that direction. It represents, in fact, only the acceleration of speed along r. But we have seen that there is acceleration along r, if its direction changes, even -when its length is constant, i.e., when the path is circular; and in that case r6' i is the quantity which we designated as pω 2 in § 38.

As a verification of these formulae, let us consider uniform motion in a straight line.

Here rcos0=α, the equation of the straight line, and

αtan0 = Nt, the condition of uniform motion. We have r=αsec0tan0.0 ,

V = a sec 2 0.0 = r 2 0 ∣ a; r = V sin 0 , i i =VcosØ.0 = c ~0 = ·

Here, although there is no acceleration, r has a definite value. But r - r0 2 = a 2 N 2 ∣ r i - α 2 V 2 ∕r 3 =0 .

From the expressions for the acceleration along and perpendicular to the radius-vector we at once obtain the result above (§ 46). For, if there be no acceleration perpendicular to the radius-vector, we have * —- (r a 0) = 0, r dt from which r 2 0 = const, = ⅛.

We have, in addition to this, the expression for the acceleration towards the origin, r-r0 2 = -P.

Eliminating 0, we have r-h 2 ∣ r i = -P.

This gives r in terms of t, and thus reduces (if we please) any case of a central orbit to a corresponding case of rectilinear motion. The difference between the accelerations in the revolving radius-vectoi’ and in the fixed line is a term depending on the inverse cube of the radius-vector. But the usual mode of proceeding is as follows.

Multiply by rdt and integrate, then r 2 + ^ = C - 2∕^Vdr; ( dr ∖ 2 h 2 Λ 2 ,, ∕∙ τ , ∖ dθ) f i + P ~ C ~ λf^ dr ∙

The left-hand member obviously represents the square of the velocity, as it is the sum of the squares of r and rØ. For we have dr . drh T = u ≈ · dθ dθr 1

This gives a relation between r and 0, which is therefore the polar equation of the path described. It is usual to employ, instead of r, its reciprocal l∕r = u. With this the equation becomes ( du ∖ 2 ∖ fPdu Ä 7τ∑ + u = c + 2 ∕ —Γ ·

∖∖ dθ J ∕ J u 2

Differentiating with regard to 0, and dividing by 2Ae 2 — , we ob- dθ tain finally d 2 u _ P

dθ 2 +U= W ’ an equation of very great importance.

When there is acceleration T perpendicular to the radius-vector, as well as - P along it, this equation takes the form

P T du d 2 u u 2 -ii 3 dθ dθ 2 + U 7 r T ' atj h 2 + 2∕ f i dθ J u i

§ 48. There are two specially important cases of central acceleration. The first is that of the gravitation law, the other that of Hooke’s law. We will take these in order, but by very different methods.

§ 49. Planetary Motion.— With the gravitation law the acceleration varies inversely as the square of the distance from the point to which it is directed. But, as we have just seen, the angular velocity of the radius-vector, i.e., of the direction of acceleration, varies according to the same law. Hence in the hodograph, the linear velocity (whose magnitude is that of the acceleration in the path) is proportional to the angular velocity of the tangent (whose direction is parallel to the acceleration).

Thus, in the hodograph, the angle between successive tangents is proportional to the arc between their points of contact; and therefore the curvature is constant;— i.e., the hodograph is a circle.

Let A (fig. 13) be the centre of this circle, O the pole of the hodograph, P any position of the tracing point. Then OP is, in magnitude and direction, the velocity in the orbit. But it may be looked on as consisting of two parts, OA and AP. Of these both are constant in magnitude; but OA is constant in direction, while AP is perpendicular to the direction of acceleration in the orbit. Hence the velocity in the orbit is the resultant of two constant parts,—one always in a fixed direction, the other always perpendicular to the radius-vector.

This gives the form of the orbit as follows :— x = α(e-ι∕∕r), y = ax∕r; so that rr = xx + yy≈ery i or r≈e(y+b)∙, where the meanings of the quantities are obvious.

But if PO cut the circle again in p, Op is proportional to the perpendicular on the tangent to the orbit from the centre of acceleration (because PO. Op is constant) and is at right angles to it (because it is in the direction of the velocity). Hence the path is such that the locus of the foot of the perpendicular from the centre of acceleration on the tangent is a circle. This property belongs exclusively to conic sections, one focus being the point from which the perpendiculars are drawn.

A third and even simpler mode of treating this most important problem is as follows. Draw OM perpendicular to PA (produced if necessary) and PN perpendicular to OA. Then OM is the resolved part of OP parallel to the tangent at P, i.e., it is the speed with which the length of the radius-vector changes. Also PN is the resolved part of OP perpendicular to the fixed line OA, i.e., it is the speed with which the moving point travels in a fixed direction. But by similar triangles OAM, PAN, we have

OM: PN:: OA: AP = a constant ratio.

Hence the increment of the radius-vector bears a constant ratio to the simultaneous increment of the distance of the moving point from a fixed line in the plane of motion. This is only a slightly altered form of statement of the focus and directrix property of conic sections.

When O is within the circle, the constant ratio is less than unity, and the conic is an ellipse; when without, the ratio is greater than unity, and we have an hyperbola. When O is on the circumference of the hodograph, the path is a parabola; for the ratio is unity.

In a subsequent section we will return to this question, and treat it from the point of view of Kepler’s Laws of Planetary Motion.

Simple as are the geometrical methods above, the direct analytical one is still simpler. For we have p ,

Γ 2 »

so that x=--⅛-cos0, ÿ=--^-sinô.

Hence, as before (§ 47), xy-yx=r 2 0 = h; and therefore, by eliminating r 2 , we have x=∙ --~cosθ . Ô , ÿ= --^-sinθ . ∂, ft ft so that X - α = - ~sinθ, ÿ - β = +-γ-cosβ.

η Λ

These give at once, by squaring and adding, (x - α) 2 + (ÿ - β) 2 = μ 2 ∣ h 2 , the equation of the circular hodograph. Also, by multiplying the first by y, and subtracting it from the second multiplied by x, we have h-βx + ay≈ rμ∕h , the equation of the orbit. This is evidently a conic section of which the origin is a focus. The directrix corresponding is the line ay - βx + h = 0, and the excentricity is h ∖ ∕a i + β" ∣ μ

From these the major axis can be calculated.

§ 50. Elliptic Motion about the Centre.— When a point moves uniformly in a circle, the motion presents very different appearances according to the spectator’s point of view. If we suppose him to be situated at a distance very great compared with the radius of the circle, he sees what is practically an orthographic projection of the orbit on a plane perpendicular to the line of sight. In general, an orthographic projection of a circle is an ellipse—whose centre is the projection of that of the circle. As equal areas are projected orthographically into equal areas, the appearance is therefore elliptic motion, in which the radius-vector from the centre describes equal areas in equal times. Hence (§ 46) the acceleration is directed towards the centre. But accelerations are projected like velocities, and like lines. Hence, as the acceleration in uniform circular motion is constant, and directed towards the centre, so in elliptic motion, with equable description of areas about the centre, the acceleration is towards the centre, and is proportional to the length of the radius-vector. But this projected orbit may again be projected orthographically, as often as we please, on different planes. It will always remain elliptical, and with the radius-vector from the centre describing equal areas in equal times. And the acceleration will always be in the same proportion as before to the radius-vector. However different in size and shape these elliptic orbits may be, they have one common property, the time of describing them is the same.

Thus we see that when the orbit is an ellipse described about its centre of figure the acceleration is central, and proportional to the radius-vector. The time of describing such an ellipse depends only upon the ratio of the acceleration to the length of the radius-vector ; or, if we choose, upon the magnitude of the acceleration at unit distance. And the converse of this proposition is also evidently true. When we look edgewise at the uniformly-described circular path with which we commenced, it is seen projected into a straight line, in which the moving point appears to oscillate. This is the case, for instance, very approximately, with the satellites of Jupiter as seen from the earth. Sun-spots, the red-spot on Jupiter, &c., all appear to move approximately in this way. But the extreme importance of this species of motion is that it is the simplest type of oscillation of a particle of matter displaced from a position of stable equilibrium. The vibrations of the ether when homogeneous plane-polarized light is passing through it, of the air when a pure musical note is sounded, the oscillations of a pendulum (through small arcs), the simplest vibrations of a pianoforte wire or a tuning-fork, the indications of a tide-gauge when the sea is calm,—all are instances of it. Hence the special necessity for studying it in detail.

§ 51. Def.— Simple harmonic motion is the resolved part, parallel to a diameter, of uniform circular motion.

Let a point P (fig. 14) move uniformly in the circle APA'. Then, drawing any diameter AOA', and PM perpendicular to it, the motion of Μ is simple harmonic.

The speed and acceleration of Μ are obviously the resolved parts, along AA', of the speed and acceleration of P. Hence if V be the speed of P we have speed of M = ™V, acceleration of M = y^ × acceleration of P

MO V 2 V 2 po ’ po~po 2 mo ∙

From these expressions we see that, if we call ω the angular velocity of OP, so that ω = V∕PO, we have speed of Μ = PM. ω , acceleration of M = MO.ω 2 .

Thus the speed of Μ increases from A to 0,—being zero at A, and V at 0; then it falls off to zero at A', and goes through the same numerical values in the opposite order, when the direction of motion is reversed at A'.

The acceleration of Μ is always directed towards 0. It has its greatest value at A and again at A', and is always proportional to the distance from 0. If T be the period of the simple harmonic motion, i.e., the period of rotation of P in the circle, we have

T=—= ⅛, ω V

where a is the radius of the circle, or, as it is also called, the “amplitude” of the simple harmonic motion. We may now write as the characteristic of this species of motion acceleration = —— × displacement; m _ /displacement or T = 2π ∕— l ι —

-∖∕ acceleration

§ 52. In our further remarks about simple harmonic motion the following terms will be found convenient. P is the position at time t of the point moving in the circle. Let E be its position at the zero of reckoning, when i = 0. Then the angle AOP may be called the “phase” of the simple harmonic motion, and AOE the “epoch.” In time units the values of the phase and epoch are found from their circular measure by dividing by ω.

If the position of the point moving with simple harmonic motion be denoted by x, we obviously have x =OM = OP cos PO A,

= OPcos(POE + EOA), = αcos(< t >i + e).

This expression is to be found, perhaps more frequently than any other, in all branches of mathematical physics. It is in terms, or series of terms, of this form that every periodic phenomenon can be described mathematically, as will be seen later. From the expressions for the longitude and radius-vector of a planet or a satellite to those of the most complex undulations whether in water, in air, or in the luminiferous medium, all are alike dependent upon it.

The results obtained geometrically above are easily reproduced from this form :—

thus x = - aω sin (ωt + e); and x = -wω 2 cos(α√ + i)= -xω 2 .

§ 53. The simplest graphical method of exhibiting the nature of any kind of rectilineal motion is to compound it with a uniform velocity in a direction perpendicular to the line in which it is executed. This is, in fact, wL·at is done in the majority of self-registering instruments, where a slip of paper is drawn by clock-work uniformly past the moving point, in a direction perpendicular to its line of motion, and a record is made by mechanical means, by a pencil, by an electric spark, or (best of all) by photographic processes. When this process is applied to a simple harmonic motion the record is of the general form of the curve in fig. 15. This curve has long been known as the “curve of sines,” or the “harmonic” curve. All its forms can be deduced from any one of them by mere extension or foreshortening in the vertical or horizontal directions in the figure. It represents the simplest forms into which a vibrating string can be thrown, as well as the instantaneous form of a section of the surface of water along which a simple series of oscillatory waves or ripples is passing. In this case the form of the section remains the same as time goes on, but the whole figure moves steadily onwards in the direction in which the waves are travelling.

This is expressed analytically by the form y≈aeos(nt -mx), where x and y are horizontal and vertical coordinates of a point at the surface of the water, y being measured from the level of the undisturbed surface. When x is constant we study, for all time, the simple harmonic rise and fall at a particular place. When t is constant we have the above-figured instantaneous glance of a section of the whole water-surface.

The rate at which the wave travels is obviously n]m ; for, if we increase t by any quantity τ, and x by the corresponding quantity nτjm, the value of y is unaltered.

§ 54. We have next to consider the result of superposing or compounding two simple harmonic motions which take place in the same line. The geometrical method amply suffices for this purpose provided the periods of the two are equal, however different may be their amplitudes and their phases. For, if we suppose PQ (fig. 16) to turn about P in the same plane and with the same angular velocity as OP about 0, the angle OPQ will remain unaltered, and therefore the triangle OPQ will remain of constant size and form while turning about 0. Thus Q describes a circle about 0 in the given period. The resolved parts of OP, PQ, along any diameter 0A, together make up the resolved part of OQ along the same line. Hence two simple harmonic motions, of the same period and in the same line, are equivalent to a single simple harmonic motion of the common period. The amplitude of the resultant simple harmonic motion is OQ, and depends only upon OP, PQ, and the angle OPQ,—the amplitudes of the two component simple harmonic motions and the supplement of the difference of their phases.

§ 55. When the difference of phase is nil, or any whole numberof circumferences, the resultant amplitude is thesum of the amplitudes of the components, which is its greatest value. When the difference of phase is an odd number of semi-circumferences, the amplitude of the resultant is the difference of those of the components.

If we produce QP to meet OA in R, we see that QOA, the phase of the resultant simple harmonic motion, is intermediate in value to the phases of the components, which are POA and QRA respectively. Its excess over the one, and its defect from the other, are the angles at Oand Q in the triangle OPQ; and their sines are to one another as the separate amplitudes QP, PO. Hence, when these amplitudes differ, the phase of the resultant coincides more nearly with that of the component whose amplitude is the greater.

Analytically the resultant motion is expressed by X = a cos (<⅛√ + ∈) + a' cos (ωt + e'),

= (α cos e + α'cose') cos «ί - («sin e+a' sine') sin ωt,

= Pcos(ω∕ + Q), provided that

P cos Q = αcose + α'cose', and PsinQ = asine+ a'sine'.

These expressions give for the amplitude of the resultant P ≈ √(α cos e + α' cos e') 2 + (a sin e + a' sin e') 2 ,

= V(a 2 + 2aa'cos(e - e') + a' 2 .

This may be put in either of the forms

√(a + a') 2 - 4aa'sin 2 ∣(e - e') or √(a - a') 2 + 4aa'cos 2 ⅛(e - e'), from which the above conclusions follow at once.

Also, for the epoch of the resultant, we have λ αsine + α'sine' tan Q = -,;.

acose + acose

When e and e' are both positive and less than ⅛ιr, this is obviously intermediate in value to tan e and tan e'.

When the periods of the components are not exactly equal, the simple artifice which follows enables us still to apply the same method of composition. We have now x = αcos (ωt + e) + a' cos (ω't + e'), = a cos (a√ + e) + a' cos (wt + e' + (a>' - ω)t) .

Hence the above values of P and Q will still satisfy the conditions if we write e'+(ω'-ω)< instead of e'. Thus we may treat the two components as being of the same period, but make the epoch of one of them steadily increase with an angular velocity equal to the difference of the angular velocities in the generating circles of the components.

The triangle OPQ will no longer preserve its form; it will pass continuously through all the various forms which we have seen would be given to it by various differences of phase in the component simple harmonic motions. The time in which it returns to a former value is evidently 2∙∏∙∕(ω' - ω), which is greater the more nearly equal are the periods of the components.

§ 56. One of the best examples of the principles we have just discussed is furnished by the tides. If there were but one tide-producing body, we should have (approximately) a simple harmonic rise and fall of the sea-level at any given place twice over in the course of about twenty-four hours, and the phase would depend simply upon the distance of the tide-producing body from the meridian (whether above or below the pole). The joint effect of the sun and moon is practically the resultant of the effects which they would separately produce. Hence, when these bodies are in conjunction or in opposition (i.e., at new or at full moon), the whole rise of the tide is the sum of the solar and lunar tides; and we have what are called “spring tides.” When the moon is in quadrature, the amplitude of the tidal rise or fall is the excess of the lunar over the solar tide, for it is low water as regards the sun when it is high water as regards the moon. In intermediate positions the effect lies between these extremes, but the joint high-tide lies nearer to the crest of the lunar than to that of the solar tide. In the first and third quarters of the moon, high tide is earlier than the high tide due to the moon alone; in the second and fourth later. This is what is called “priming” and “lagging” of the tides, and is seen at once to follow from the construction given above. Had the lunar and solar tides been of equal amplitude, spring tides would have been of double the altitude of either, and there would have been no tide at all at the time of neap.

The mode in which we have treated this special case is an illustration of the general method (above described) of combining simple harmonic motions in which the periods are slightly different.

§ 57. What we have said of the tide-waves holds of course of all waves in which the separate disturbances are so small that the joint effect is found by superposing the separate effects. Thus when, at sea, two series σf waves of equal length meefe at any place, the resultant is still a set of waves of the same length, but the altitudes and phases of the components determine those of the resultant. When crest meets crest, we have waves of the sum of the original amplitudes; when crest meets trough, the difference. In the latter case we have still water when the amplitudes of the components are equal. What is called a “jabble,”—where, for a short time, a portion of a stormy sea is almost calm, and after a little it is violently agitated,—is the result of a number of “cross seas.”

§ 58. If we now. consider the instantaneous form of a section of the surface, instead of the successive displacements of one portion of it, we can easily account for a striking phenomenon which is very frequently observed on a shelving beach. We often notice that every ninth or tenth wave or so is higher than those immediately before or after it. This is the result of superposition of two or more sets of waves in which the distance from crest to crest is different in the different sets. In the joint system we have, represented as in § 53, phenomena akin to the spring and neap tides, and the priming and lagging of the tides.

Fig. 17 shows part of the result when the amplitudes are equal, and the wave-lengths as 15 to 17. It gives also a rough approximation to the whole result when the lengths are as 7 to 8 or as 8 to 9

§ 59. To compound any number of simple harmonic motions, of equal periods, in one line, we may obviously take them two by two, and apply the preceding process over and over again till we have as final resultant another simple harmonic motion of the common period.

Or thus :—

^ = 2αcos(α√ + e) = cos α√2(α cose) - siua√2(asin e) = Pcos(a>i + Q), where P cos Q = 2(a cose), PsinQ = 2(asine).

When the separate periods are not equal, and not even nearly equal, it is only in special cases that any simplification can be effected by analytical processes. But this is not much to be regretted, because for most purposes a graphic method is sufficiently accurate, and it can always be easily carried out.

§ 60. We must now consider the composition of simple harmonic motions in directions at right angles to each other;—but for the present we confine ourselves to the case in which their periods are equal. In this case we know that the acceleration is in the same ratio to the displacement in each of the two rectangular directions. Hence by the general theorem of § 50 the motion is elliptic, with uniform description of areas about the centre.

To analyse this, suppose, at starting, that their amplitudes also are equal. Let OA, OB (fig. 18) represent the two rectangular directions. With centre O, and radius equal to·the common amplitude, describe a circle. Let AOE, BOF represent the epochs of the two components (the corresponding circular motion being supposed positive for each), then obviously EOF exceeds by a right angle the difference between the phases of the motions in OB and OA. Then if P, Q represent at time t the corresponding positions in the common circle, we have arc FQ = arc EP; and if perpendiculars be drawn, PM to OA, and QN to OB, their intersection S is the position at time t in the resultant motion. The locus of S is, by what has been proved above, an ellipse which touches the sides of the square CDC'D'.

When EOF is a right angle, i.e,, when the phases are alike, this ellipse becomes,the diagonal CO' of the square touching the circle at the extremities of AA' and BB'. When EOF is three right angles, the ellipse becomes the diagonal DD'. When it is two right angles, or four, i.e., when OB is one quarter, or three quarters, of a period in advance of OA, the ellipse becomes the circle ABA'B'. To find in any case whether it is described positively or negatively (§ 44), we have only to notice how OS turns. Now while P is near A, MS remains closely coincident with AC. If, then, Q be anywhere in the semicircle BA'B', N moves in the direetion BB' and the angle AOS diminishes. Hence the ellipse is described negatively (or in the direction of the hands of a watch) if the epoch of the motion in OB exceeds that of the motion in OA by anything up to two right angles. And similar reasoning shows that, if the excess be from two to four right angles, the ellipse is described positively.

If the amplitudes be not equal, we have only to extend or foreshorten the figure parallel to OA or to OB. The square CDC'D' becomes a rectangle, in which the orbits (all of which, with the exception of the diagonals, are now ellipses) are inscribed. Everything else is as before,

§ 61. When the periods in the two component motions are nearly, but not quite, equal, the phase of one gains gradually on the other, and the path passes continuously through the forms of all the possible ellipses, but remains possessed of the one property common to them all. It becomes a species of spiral, but in every convolution it touches, in succession, each side of the square or rectangle above discussed.

§ 62. Similar reasoning shows that the superposition of any number of simple harmonic motions in any directions and with any amplitudes and differences of phase, provided the period is the same for all, gives rise to motion in an ellipse about the centre. But this follows more easily from analysis.

Take, first, two simple harmonic motions of the same period parallel to the axes of x and y. We have aj=αcos(ωi + e), y = a'cos(ωt + e').

Eliminating t between these equations, we have at once

-⅛ - 2—,cos(e' - e) + -^j = sin 2 (e'-e), a 2 aa ’ a 2 ' ’ the equation of an ellipse.

It becomes a circle when and only when a = a', cos(e'-e) = 0, i.e., when the amplitudes are equal, and the phases differ by an odd number of right angles.

It becomes the straight line x/a - y∕a' = 0, when e' - e is zero; and x]a + y∕a ’ = 0, when e' - e is two right angles.

If SOA be called θ, we have . y α'cos(ωi + √) tan 0 = — = ∕ »

X αcos(ωi + e)

= -^-(cos(e'- e) -sin(e'- e)tan(ωi + e)) .

Hence, taking the fluxion of each side, sec 2 0. 0≈ --^-ωsin(√-e)sec 2 (ωi + e).

Thus, as before, Ô is essentially negative, i.e., the rotation in the ellipse is right-handed if e - e lie between 0 and ιr, left-handed if it lie between π and 2π.

For a simple harmonic motion, denoted by ξ = αcos(α√ + ∈), in a line whose direction cosines are I, m, n, we have the components Zξ, mξ, nξ parallel to the three axes respectively. Hence for the resultant of any number of such, all having the, same period, we have x=~2,. al cos {ωt + e) = cosωi2(αZcose) - sin»iS(aZsin«).

Thus we have three equations of the form x= Acos⅛√- A'sinωi, y = Bcoswi-B'sina√, z = C cos a>t - C' sin ωt.

If we take three quantities λ, μ, v, such that λA + j uB + vC =0, λ A , + ∕ιB ’ + vC' = 0, we have also λx + μy + vz=0.

The first two equations determine without ambiguity the ratios of μ and v to λ. Hence the third is the equation of a definite plane in which the path lies. We may now choose this plane as that of x, y. The value of z above becomes identically zero; and the elimination of t between the equations for x and y gives the ellipse as before.

§ 63. When the periods of the simple harmonic motions are not equal we have x=acos(ωt + e), y = α'cos(α>7 + e').

It is easy to trace the corresponding curve by points; but, except when there is a simple numerical ratio between ω and ω', the equation cannot be presented as an algebraic one between x and y. If 2ω' = ω, we may shift the epoch so that the equations may be written x =a cos (2ω7+a), y =a’cos ω't.

Eliminating t from the first, by the help of the second, we have x ( 2y 2 -< ∖ 2w ∕ w 2 .

— = ( -⅛ -1 )cosα ta./ 1 - ⅛sιnα.

a ∖ a 2 j a λ √ a 2

This denotes, in general, a curve of the fourth order, of a figure-of- 8 form, as in fig. 19. When a=nπ the curve is a portion of a parabola, its vertex being to the right or left as n is odd or even. This parabola corresponds, in the present case, to the straight lines in the case of § 62. When the periods differ slightly from the ratio 2:1, the path passes in succession through the forms traced, forward and backward alternately; and, each time that it opens out from the parabolic form, the tracing-point describes it in the opposite direction to that in which it described it before the path collapsed into the parabola.

§ 64. The principles already illustrated are sufficient for the examination of every case of this kind. But one or two particular cases merit special notice. The case of two uniform circular motions of equal periods, in one plane, we have already noticed (§54). Q describes its circle about P, P its circle about 0, and the result is uniform circular motion of Q about 0. The radius of this circle may be equal to the sum or difference of the radii of the separate circles, or may have any intermediate value, according to the difference of phase. If the periods be not exactly equal, the motion takes place virtually in a circle whose radius continuously oscillates between the above limits. The path is a species of spiral, which lies between two concentric circles of these radii.

§ 65. When the component circular motions are in opposite directions, we have an extremely interesting and important case. It is obvious that there must now be positions in which OP and PQ are in the same straight line. Let 0A, AB (fig. 20) be one of these. Then, in any other position, OP and PQ are equally inclined to OA. The path of Q is an ellipse, of which the major semi-axis OB is the sum of the radii, and the minor axis their difference. Hence when the radii are equal the result is simple harmonic motion in the line OBB' Thus we have the proposition, of very great importance in optics, that a simple harmonic motion may be looked upon as the resultant of two equal and opposite circular motions in one plane.

When the periods are not exactly equal, the motion may be regarded as simple harmonic motion, in a line which rotates with uniform angular velocity in a plane. This is the case of Foucault’s pendulum, and of plane polarized light passing along the axis of a crystal of quartz, or through a piece of glass or other transparent substance in the magnetic field.

§ 66. Uniform circular motions, of different periods, give epicycloids, &c. A particular case is uniform circular motion superposed on uniform rectilinear motion, in which case we have cycloids, &c. But these we merely mention.

§ 67.- By far the most important of the applications of simple harmonic analysis is summed up in what is called Fourier’s Theorem.— A complex harmonic function, with a constant term added, is the proper expression for any periodic single-valued function, and, consequently, can express any single-valued function whatever between any assigned values of the variable.

To show the importance of this in physics we need take but a single example. The one essential characteristic of a musical sound is its “periodicity.” Hence it may be analysed into a series of simple harmonic disturbances. Their respective periods are the fundamental period, its half, third, fourth part, &c. The first gives the pitch of the note; the others determine its quality. The investigation which follows is not intended to prove the theorem; it is merely introduced as readily suggesting it.

The essence of periodicity of a function f is that we must have ∕∣⅛+⅛α]=∕[aj-⅜α]

whatever be x, provided a be the period.

We may write this as ad ad

OΓ ∕ - a d ∖

(g2dz-g 2dχ )f(χ) = 0.

Now the equation

ε* f -≡-*=o has the real root

ξ = 0, and the infinite series of pairs of imaginary roots

⅜ξ= ±⅛V - 1 j where i is any integer. Hence S* ξ - S ^ iξ = ξ(l + ξ 2 ∕2 2 π 2 ) (1 + ξ 2 ∕4 2 π 2 )(l + ξ 2 ∕6 2 π 2 ) . , .

so that the differential equation for fix) gives, besides a constant term, the infinite series of terms due to solutions of equations of the second order of which the type is

(2 ⅛(⅛ρ>>=°∙ ∖ 4 4 7Γ ∖ lΛJU J J

The solution of this representative equation gives the following particular integral of the complete equation, fix) = Pj cos (2⅛ - 1 + Q,∙).

Hence the general solution is fix) = A o + S∞ P i cos (2iπxa ^ 1 + Q,∙)

= A o + 2“ Aicos2⅛∞ 1 + 2“ B i sin2⅛∞^ 1 , where the constants are to be determined by special integration, according to the process already described in the article Harmonic Analysis (q.v.).

As a single example, suppose that the value of fix) is unity from x=0 to x = a, and zero from x = a to x = 2a. This has many applications, as, for instance, to alternate heating and cooling of one surface of a solid, alternate “make and break” with a battery and a telegraph wire, &c. In this case we have fix) = JL + Â 2*> * s j n ( 2 ^ + l)⅞¾ 2 + π 2 ∙ 2i + i sm a ·

§ 68. A point describes a logarithmic spiral with constant angular velocity about the pole; find the acceleration.

Since the angular velocity of SP (fig. 21) and the inclination of this line to the tangent are each constant, the linear velocity of P is as SP. Take a length PT, equal to c.SP, to representit. Then the hodograph, the locus of p, where S∕> is parallel and equal to PT, is evidently another logarithmic spiral, similar to the former, and described with the same constant angular velocity. Hence pt, the acceleration required, is equal to eSp, and makes with Sp an angle equal to SPT. Hence, if P« be drawn parallel and equal to pt, and uv parallel to PT, the whole acceleration P« may be resolved into Pu and vu ; and Pvu is an isosceles triangle, whose base angles are each equal to the angle of the spiral. Hence Pυ and vu bear constant ratios to Pu, and therefore also to SP or PT.

The acceleration, therefore, is composed of a central acceleration proportional to the distance, and a tangential retardation proportional to the velocity. And, if the resolved part of P’s motion parallel to any line in the plane of the spiral be considered, it is obvious that in it also the acceleration will consist of two parts—one directed towards a point in the line (the projection of the pole of the spiral) and proportional to the distance from it, the other proportional to the velocity but retarding the motion. Hence a particle which, ∣ unresisted, would have a simple harmonic motion has when subject to resistance proportional to its velocity a motion represented by the resolved part of the spiral motion just described.

If a be the angle of the spiral, ω the angular velocity of SP, we have evidently PT. sinα = SP.ω.

Hence Pd=Pu∙=^=^-=-≠- PT = ~- τ - SP=n 2 . SP (suppose); 2 SP sm a sιn 2 a rr '

and mi = 2Pv . cos a = = 2Ae . PT (suppose).

Thus the central acceleration at unit distance is n 2 = ω 2 ∕sin 2 α, and the coefficient of resistance is 2k = 2ωcosα∕sinα.

The time of oscillation is evidently 2τr∕ω; but, if there had been no resistance, the properties of simple harmonic motion show that it would have been 2π ∣ n; so that it is increased by the resistance in the ratio cosecα: 1, or n: Jn 2 - k 2 ∙

The rate of diminution of SP is evidently

τ>rτι ωcosα cιτ> 7 fir)

PT . C0Sα= —-. SP=fcSP; sιnα

that is, SP diminishes in geometrical progression as time increases, the rate being k per unit of time per unit of length. By an ordinary result of arithmetic (compound interest payable every instant) the diminution of log SP in unit of time is k.

Hence, in the resolved part of the motion, the logarithm of the amplitude is diminished, every half vibration, by ½7r∕ω.

This process of solution is only applicable to resistance of harmonic vibrations when n is greater than k. "When n is not greater than k the auxiliary curve can no longer be a logarithmic spiral, for the moving particle never describes more than a finite angle about the pole; and then the geometrical method ceases to be simpler than the analytical one.

§ 69. What we have said about composition of motions is merely a particular case of the general question of relative motion, which in its main principles is exceedingly simple. It is entirely comprehended in the following propositions,—which may be regarded as almost self-evident.

Given the motion of Ä with regard to a point 0, and that of B with regard to A, to find that of B with regard to 0.

By compounding the vectors of relative position OA, AB, we have at once the required vector OB. Thus it is obvious that we have only to add the separate components of the velocity of A with regard to O, and those of B with regard to A, to obtain those of B with regard to O. And, of course, the same rule applies to the accelerations.

If x, y, z be the coordinates of A (referred to 0) at time t ; α∕, y, √ those of B referred to parallel axes from A; ξ, ∙η, £ those of B referred to 0; we have at once

ξ=x + a∕, η=y + y'> f=z + √.

They give, by differentiation with regard to t, ζ=x+x , , &c., ξ=x+x, &c., which constitute the analytical proof of the statement above.

§ 70. Hence we have the solution of the further question: Given the motions of A and B with regard to 0, to find the relative motion of B with regard to A. In this case, of course, before compounding, the vector of A must have its sign changed.

Another very important case is that in which the motion is referred to axes which are themselves moving. So long as their directions remain unchanged, this reduces itself to the former investigation as a mere question of changed origin ; so that we need consider only the effect of the change of direction of the axes. And this is at once deducible from the results of last section. For we have only to consider, instead of the moving point, its projections on the moving axes, and find their velocities and accelerations relative to fixed axes.

Thus, if the rectangular axes of x and y be fixed, and those of ξ and η be rotating in the same plane, we have a datum of the form, θ = angle ξθx =fit), giving the position of the moving axes in terms of the time. Let P be the moving point, and PM perpendicular to Oξ (fig. 22). Then, as the polar coordinates of Μ are £, θ, we have, for its velocity, ξ along Oξ, ξ0 along MP.

But these must be combined with the velocity of P relative to Μ, which consists of ή along MP and η0 parallel to ξθ.

Thus the velocities parallel to fixed lines corresponding to the instantaneous positions of Oξ and Oη are, respectively, ζ-ηό and 17 + ξ⅛.

In the same way it is easy to see, by § 47, that the corresponding components of the acceleration are Kinematics of a Rigid Plane Figure, displaced in its own Plane.

§ 71. When a rigid plane figure is displaced anyhow in its own plane, the displacement may always be regarded as the result of a definite rotation about a definite axis perpendicular to the plane.

The proof of this follows at once from the fact that, under the assigned conditions, the figure has only three degrees of freedom; and consequently its position is determinate whenever the positions of any two of its points are given. Also, a single rotation can, in general, be found which will transfer these two points from one pair of assigned positions to another.

Let A, B, A',B' (fig. 23) be successive positions of two points of the figure. Bisect A A' by the line Oα perpendi cular to it, and let Oδ do the same for BB'. Let these perpendiculars meet in O. Then it is clear that the two triangles 0AB, OA'B' are similar and equal. Hence AB may be regarded as having passed to the position A'B' by rotation about an axis through O perpendicular to the plane of the paper. The angle of rotation is A0A' or BOB'.

The construction fails when Oα and Oδ coincide, but in this case it is evident that the required point O is the point of intersection of BA and B'A' (fig. 24). It also fails when the bisecting perpendiculars are parallel (fig. 25). But then AA' and BB' are equal and parallel, and the displacement is a pure translation, the same for every point of the plane figure, which may be regarded as an infinitely small rotation about an infinitely distant axis.

§ 72. Since any displacement in one plane corresponds in general to a rotation, any two or more rotations about parallel axes can always be compounded into a single one. Of two equal and opposite rotations the resultant is simple translation. This is evident from fig. 26. In both cases A and B are the initial positions, A' and B' the final positions of the two axes.

In the first we begin with the rotation about A, in the second with that about B.

§ 73. When these equal rotations are simultaneous instead of successive, the figure becomes a rectangle; —i.e., the translation is perpendicular to the line joining the axes. For in this case we may suppose the two rotations to be each broken up into successive equal but infinitesimal instalments. And the principles of infinitesimals show that two such instalments, either about the same or about different axes, produce the same ultimate effects whether they be applied simultaneously or successively. The general principle of which this is a particular case is" called the “principle of superposition of small motions.” It is merely an application of the fact that infinitesimals of the second order may be neglected in comparison with those of the first order.

The consideration of simultaneous rotations is very important. Suppose a plane figure to rotate in its own plane, with angular velocity ω, about the origin. Then it is obvious that τω, in a direction perpendicular to r, is the velocity of a point whose distance from the origin is r. The components are, therefore, x — -yω, 2∕=iCω.

If the rotation be about the point a, b, these become x = -(y-b)ω, i∕ = (x-a)ω.

Hence, when there is any number of simultaneous rotations about parallel axes, we have

⅛= - y∑ω + ∑(δω), 2∕ = ir∑ω-2(αa>).

If we write Ω=2α>, and α=⅛, β≈^; 2,ω 2ω

we have ⅛=-(τ∕-β)Ω, ⅛'=(aj-α)Ω.

These are the component velocities which the point x, y would have if there were only a single rotation, with angular velocity Ω, about an axis passing through the point α, β.

When 2(ω)=Ω = 0, we see that ≈ = 2(δω), #=-2(a«), so that all points of the figure have equal velocities. This is the case of pure translation. Here α and β are (in general) each infinite;— i.e., we have as resultant a vanishing angular velocity about an infinitely distant axis.

§ 74. As any displacement of a plane figure in its own plane is equivalent to a rotation, we may represent a series of displacements by a series of rotations. Also if we know the positions, in the figure itself, of the points which are successively the axes, and likewise the position which each of them occupies in space at the instant when the rotation about it takes place, we can construct the whole motion. Let them be O, A, B, C, &c., and 0, a, b, c, <fcc., respectively (fig. 27). Then the figure turns about 0 till A coincides with a. Next it turns about A (or a) till B coincides with b, and so on. Hence the motion will be represented by the rolling of the polygon OABC, fixed in the moving figure, on the polygon Oαδc fixed in the plane of the motion.

In the limit, when the axis continuously shifts its position in the figure while the rotation goes on round it, the polygons become plane curves. Thus we have the fundamental proposition that any motion of a plane figure in its own plane can be represented by the rolling of a curve attached to it, on a curve fixed in space. Both curves are situated at an infinite distance when the motion is one of pure translation.

Kinematics of a Rigid Figure.

§ 75. When a spherical cap, or skin, moves on the surface of a sphere of equal radius with which it is everywhere in contact, we may make the construction of § 71 with great circles bisecting the arcs AA' and BB'. Two great circles (unless they coincide) always intersect at the extremities of one definite diameter. The case of coincidence is met exactly as it was in § 71. Hence every motion of a spherical skin on a sphere is equivalent to a rotation about a definite axis through the centre of the sphere. Thus any number of successive or simultaneous rotations about axes passing through one point can be compounded into a single rotation about an axis passing through that point. And the construction of § 74 can be carried out with spherical polygons or curves, so that we see that any motion of a rigid figure, one point of which is fixed, can be represented by the rolling of a pyramid or cone, fixed in the figure, upon another fixed in space.

§ 76. The law of composition of simultaneous angular velocities about axes which pass through one point is precisely the same as that for simultaneous linear velocities of a moving point. The following simple geometrical process establishes the proposition for two intersecting axes; and it is easy to see that it can be extended to any number of such. Let OA and OB (fig. 28) represent, the two axes, and let the lengths of these lines (both drawn in the positive direction for the rotation about them) represent the angular velocities corre spending. Then a point P, in the angle between the positive ends of the axes, is raised above the plane of the paper by rotation about OA, but depressed below it by the rotation about OB. The amounts of the elevation and depression are pro portional to the distance from either axis, and to the angular velocity about it, conjointly. Hence they will annihilate one another if, perpendiculars PM, PN being drawn to the axes, we have

0A.PM = 0B.PN.

This is equivalent to saying that the areas of the triangles OAP, OBP, are equal,—which necessitates that P should lie on the diagonal of the parallelogram of which OA, OB are conterminous sides. Let OC be the diagonal of this parallelogram. From what has been said above it is evident that the displacement of any point in the plane is necessarily proportional to the algebraic sum of the moments of OA and OB about it, and therefore (§ 46) to the moment of OC. Hence all points in the line OC remain at rest, and the figure turns about that line with an angular velocity represented by its length. This analogy to moments shows the reason for the remarkable proposition that angular velocities, about axes which intersect, are to be compounded according to the same law as linear velocities.

§ 77. Any proposition regarding simultaneous linear velocities or accelerations has thus its counterpart in angular velocities and accelerations. Thus, as we have seen (§ 36) that under acceleration in one plane, always perpendicular to the direction of motion, a point moves with uniform velocity, so, if a figure be rotating about one axis, and have angular acceleration about a second axis always perpendicular to the first, the direction of the axis about which it rotates is changed, but not the angular velocity.

It is to be noted that in such a case the direction of the axis changes not only in space, but also in the rotating figure itself. This, however, is merely the result of § 36 in a slightly altered form.

If w z be the angular velocity of a figure about the line which, for the moment, coincides with the axis of z, the consequent displacements during time δi of a point x, y, z are (§ 73)

δx = - yω z δt , δy = xω i δt. Of course similar results hold for the angular velocities about lines for the moment coinciding with the axes of x and of y. The joint effect therefore is found by adding the various separate values obtained by permuting the letters îc, y, z in cyclical order. Thus δa3 = (zw y -yω 2 )δi, δy= (xa>z — zωχ}δt, δZ = ss lyθ}χ -” XWy)δt .

The right hand members of these equations vanish if

= ± (1).

Wχ Ct)y Wz

These correspond to the two equations of the instantaneous axis, and reproduce, in an analytical form, the result of § 76.

The angular velocity about this axis is

Ω — ∖ la>χ + α>ij + o>? .

For it is clear that the direction 'cosines of the displacement of x, y, z are proportional to Zu> y — yω z , Xo> l — Zω x , yω x — Xω,, i showing that it takes place in a line perpendicular to the plane passing through x,y,z and (1). It is therefore perpendicular to W.

Also the whole displacement is

V(δic) 2 + (δy) 2 + (δz) 2 = δZ V(zω j , - yω t y i + (xω z - Zω x ) 2 + (yω x - Xω y ) 2 ^ /—≡ ~ z ∕ „ oo (a⅛⅛ + yω y + Za> z ) 2

= δZ√ω^ + ω; + */ x 2 + y 2 + z 2 ,2, „2 . 2

V ω x + ω y -r ω z

The last factor is the distance of x, y, z from (1). Hence the second is the angular velocity about (1).

It appears at once from this result, and from the form of (1), that ω x 'φ y c⅛ Ω ’ Ω ’ Ω

are the direction cosines of the instantaneous axis.

I f the figure be rotating simultaneously about a number of axes, —say with angular velocity w 1 , about an axis whose direction eosines are l 1 , m 1 , n 1 , &c.,—we have evidently w x =S(Zω), ω z , = 2(mco), ω z =2(nω).

From these the single instantaneous axis is found immediately as above.

§ 78. Any displacement whatever of a rigid figure may be effected by means of a screw-motion, i.e., translation parallel to some definite line, accompanied by a proportionate rotation about that line. Let A and A' be successive positions of any point in the figure, and suppose the body to be brought back by a mere translation so that A' coincides again with A. Then we have seen (§ 75) that one line of the figure through A is necessarily restored to its original position. Let P be any plane section of the figure, perpendicular to this line, P' its position after displacement. These fully determine the initial and final positions of the whole figure. Shift P into the plane of P' by a translation perpendicular to either, and let P" be its position. P" can (§ 71) be brought to coincide with P' by a rotation in its own plane. Hence the proposition. There is an exceptional case when P" requires only translation to make it coincide with P'. But then the whole figure is merely translated.

§ 79. We have seen that the straight line representing an angular velocity is to be resolved by the same process as that representing a linear velocity. If we consider a figure to be rotating about axes fixed relatively to it, accelerations of angular velocity about these will be represented by changes in the lengths of the lines representing the angular velocities, and will therefore be subject to the same conditions as the angular velocities themselves. Thus, as it is obvious that a figure is rotating at any instant with the same angular velocity about an axis fixed relatively to itself, and about another axis fixed in space, which at the given instant coincides with the former, it follows that the angular accelerations about these axes are equal at that instant.

This is really the same proposition as that r is the velocity along a fixed line coinciding with the radius-vector r (§ 47). But, just as r is not the complete acceleration parallel to r, if r be rotating, so the proposition above, though true for the first fluxion of the angular velocity about a moving line, is not generally true for fluxions of higher orders.

As this subject is commonly regarded as somewhat obscure, we may give a more formal examination of it by an analytical process. Suppose ω 1 , ω 2 , ω 3 to be the angular velocities about rectangular axes 0A, OB, OC fixed relatively to a figure, and ω the angular velocity of the figure relatively to a line OS fixed in space. Let I, m, n, be the direction cosines of the latter line with regard to the former three, then Cü = Zcpj + τnω 2 d - flω 3 > and α> — "lta 2 4- wω 2 4- nw 3 4- 1<λ 2 4^ i ∣∣ m. 2 4- nω 3 >

But ll + mm + nn = 0; and if, at.a particular instant, we have Z = l, wι=0,n = 0, this gives also Z = 0, so that we have

ώ = α> 1 4- mω. 2 4- nω 3 .

Now in=cos BOS = cos θ suppose. Hence m = -sinØ. Ö .

But, at the instant in question, θ-⅛π and 0-ω 3 , so that m = — α> 3 .

In the same way we see that iι= +ω. 2 ; and thus we have ώ = ci> 1 , which is the proposition above given.

§ 80. To complete the kinematics of a rigid figure of which one point is fixed, we require to have the means of calculating its position, after the lapse of any period during which it has been rotating with given angular velocities about given axes.

If the axes about which the angular velocities are given be fixed in space, the formulae of § 77 give at once, for a unit line fixed in the figure, the expressions l= nω y — mω z , m = lω z - nu x , h = mω x - tω y .

Here I, m, n are the direction cosines of the unit line at time t ; and they satisfy, of course, the condition ll 4- mm + nfι = 0.

But, except in some special cases, these equations are intractable. This, however, is of little consequence, because in the applications to kinetics of a free rigid body the physical equations usually give the angular velocities about lines fixed in the body. Our problem, then, takes the form

§81. Given the angular velocities of a figure about each oj a system of three rectangular axes which are rigiddy attached to it, find at any time its position in space.

It is clear that, if we know the positions of the revolving axes, referred to a fixed system, with which they at one instant coincided, the corresponding position of the figure is determined. The method usually employed is as follows.

About the common origin of the two sets of axes suppose a sphere of unit radius to be described. Let X, Y, Z (fig. 29) be the traces on this sphere of the fixed axes, and A, B, C those of the revolving axes. Draw a great circle ZC so as to meet in A' the quadrant BA produced. Then it is clear that the figure can be constructed (i.e., that the data are sufficient for calculation) if we know (a) the angle XZC,—this we call ψ; (b) the arc ZC, called θ ; (c) the angle A'CA, or the arc A'A, called φ.

For X, Y, Z are given. Then

(a) shows how to draw the great circle ZC, whose pole is N on the great circle XY. Hence (δ) gives us the points C and A'. We can next draw the great circle A'N, and A and B are found on it by (c), for A'A = NB = φ. We have now only to determine these angular coordinates in terms of the angular velocities of the figure about 0A, OB, OC, which we denote by ω 1 , ω 2 , ω 3 respectively.

The velocity of C along ZC is θ. But it is produced by the rotations about A and B. Thus we have

∂ = ω 2 cos φ + ω 1 sin φ.

The velocity of C perpendicular to ZC is sinØ. ψ. This also is part of the result of the rotations about A and B, so that sin θ.φ = ω. 2 sin φ - ω 1 cos φ.

The velocity of A along AB is that of A' together with the rate of increase of A'A. Also it is entirely due to rotation about OC. Hence cos θ. ψ + Φ =ω 3 .

These three equations determine 6, ψ, φ when ω 1 , ω 2 , ω 3 are given as functions of t.

§ 82. The process above is essentially unsymmetrical. The first suggestion of a symmetrical system is due to Euler, and depends upon the general proposition of § 75. What we must seek is the single axis, and the angle of rotation about it, which (by one operation) will bring the system or figure from its initial state determined by X, Y, Z to its state at time t, determined by A, B, C.

Let I, m, n be the direction cosines of this axis, w the angle of rotation about it. Then by the elementary theorems of spherical trigonometry we find cosXA= Z 3 +(1 - ι 2 ) coszr, cos Y B = m 2 + (1 - 72i 2 ) cos w , cos ZC = n 2 + (1 - n 2 ) cos w.

Thus, as we have an independent relation among Z 2 , m 2 , n 2 , these quantities, as well as nr, can all be expressed in terms of the cosines of the three angles between the original and final directions of the three axes severally.

We have other six equations, of which only one need be written, viz. :—

cosYA = Z?n(l - cos2s∙) ÷7isin w.

§ 83. If we put w=cos∣w, a!=Zsin⅛w, τ∕=τnsin∣ot∙, a=nsin⅛w, which involve the equation of condition i0 2 + ic 2 + y 2 + 2 2 =l, the nine direction cosines of the new positions OA, OB, OC, referred to the fixed lines OX, 0Y, 0Z, become w 2 + x 2 -y 2 -z 2 , 2(wz + xy), 2(xz-wy), 2{yx - wz), w 2 -x 2 + y 2 -z 2 , 2( t yz + wx}, 2{xz + wy), 2{yz - wx) , w 2 - a; 2 — τ∕ 2 + z 2 .

These expressions, rational in terms of the four quantities w, x, y, z, are due to Rodrigues, who, however, gave them in a slightly different form.

Ifω n ω 2 , ω 3 be the angular velocities about 0A, OB, 00, respectively, we have 2w≈ — xω 1 — yω. i ~ zω 3 , 2x = Wω 2 - Zω 2 ÷T∕ω 3 , 2ÿ — Zω i + Wω 2 -Xω 3 , 2z = - yω 1 + a∙ω 2 + wω 3 .

If ω x , ω y , u l be the angular velocities about OX, 0Y, 0Z, respectively, we have

2w = xω x yω y ~zω z , 2x = ιvω x + zω y -yω z , 2y = — Za> x Wω y }-Xω z , 2z = yω x - xω y + wω z .

Each of these sets is equivalent to three independent equations only, on account of the relation ww + XX + yÿ +zz= 0.

Kinematics of a Deformable Figure. Strain.

§ 84. So far we have considered change of position of a figure of invariable form. We must now consider changes of form and volume in the figure itself. This is required for application to physical problems, such as compression of a liquid or gas, the distortion of a piece of india-rubber, &c. Any such change of volume or form is called a “strain.” The treatment of strains is entirely a kinematical question, until we come to regard them as produced in physical bodies, and consider their cause.

The system of forces which is said to produce a strain is called a “stress.” But, just as we study velocity as a preparation for the discussion of the effect of force on a free body, so we study strains as a preparation for the discussion of the effects of stress.

§ 85. In order to fix the ideas, it is convenient to suppose the figure which is to undergo strain to be cut up into an infinite number of similar, equal, and similarly situated parellelepipeds. This is effected at once by supposing it to be cut by three series of planes, those of each series being parallel to one another, and equidistant. No two of these three series must be parallel, but the distance from plane to plane need not be the same in any two of the series. If the strain be continuous there will be no finite difference of effect upon any two neighbouring parallelepipeds;—but in general their edges, which originally formed three series of parallel straight lines, will become series of curves. No two parallelepipeds of the system will in general be altered in precisely the same manner. This is called “heterogeneous strain.”

J 86. We found it convenient to study uniform speed before proceeding to consider variable speed, and so we find it convenient to take up first what is called Homogeneous Strain.— A figure is said to be homogeneously strained when all parts of it originally equal, similar, aud similarly situated remain equal, similar, and similarly situated, however much they may individually have been altered in form, volume, and position.

Now recur to our set of parallelepipeds. After a homogeneous strain these remain equal, similar, and similarly situated. Hence they must remain parallelepipeds, for they must together still continuously make up the volume of the altered figure. Thus planes remain planes, and straight lines remain straight lines. Equal parallel straight lines remain equal and parallel. Parallel planes remain parallel, ellipses remain ellipses (as is obvious from their properties relative to conjugate diameters), ellipsoids remain ellipsoids, &c.

But we can now easily see how many conditions fully determine a homogeneous strain. For if we know how each of three conterminous edges of any one of the original parallelepipeds is altered in length and direction, we can build up the whole altered system. Hence, to fully describe a homogeneous strain, we require merely to know what changes take place in the lengths and directions of three unit lines not in one plane. Three numbers are required for the altered lengths, and two (analogous, say, to altitude and azimuth, or latitude and longitude, or R.A. and N.P.D.) for each of the altered directions. Hence, in general, a homogeneous strain depends upon, and is fully characterized by, nine independent numbers.

§ 87. The simplest form of strain is that which is due to uniform hydrostatic stress acting on a homogeneous isotropic body. Here directions remain unaltered, and the lengths of all lines are altered in the same ratio. Every portion of the original figure remains similar to itself, its linear, superficial, and volume dimensions being altered as the first, second, and third powers of that ratio.

Next in order of simplicity is the case in which there are three directions, at right angles to one another, which suffer no change except as regards length. This state of things would be produced in a homogeneous isotropic body by three longitudinal extensions or compressions in lines at right angles to one another, or by hydrostatic pressure in a homogeneous non-isotropic solid. In this case, if the changes of length above spoken of are all different, an originally spherical figure becomes an ellipsoid, with three unequal axes parallel respectively to the lines whose directions remain unaltered. Every line in the body not originally parallel to one of these is altered in direction. If one of the principal changes of length be an extension, and another a shortening, there will be a cone formed of lines which are not altered in length. This is seen at once by describing from the centre of the ellipsoid a sphere equal to the original sphere. One axis of the ellipsoid being greater than the radius of the sphere, and another less, the ellipsoid and sphere must intersect; and all lines drawn from the common centre to the curve of intersection are unaltered in length (though all altered, as before remarked, in direction).

When two only of the changes of length are equal, the ellipsoid becomes one of rotation, oblate or prolate as the case may be; and if the radius of the sphere be intermediate in value to the axes of this rotation-ellipsoid, we have a right cone of rays unaltered in length.

When all three changes of length are equal we have the simplest possible case,—which has been already treated.

§ 88. The essential element in these particular cases is that three lines at right angles to one another are unaltered in direction by the strain. ' Here there is a mere change of form, and the strain is said to be “pure,” or “free from rotation.” Such a strain, in its most general form, is fully characterised by six independent numbers. For a system of three mutually perpendicular lines is fully given in direction by three numbers, and three more are required for the changes of length which they severally undergo.

But, in general, a strain is not pure. It will be shown, however, below that the principal axes of the ellipsoid into which a sphere is changed by any strain, and which is called the “strain ellipsoid,” were originally radii of the sphere at right angles to one another. Hence the strain may be looked upon as made up of two operations, viz., a pure strain, and a rotation through a definite angle about an axis in a definite position in space. The axes of the pure strain are lines fixed in the figure.

§ 89. It is useful, in farther considering the subject, to introduce along with the original strain (thus analysed), another which is called its “conjugate.” This is defined as composed of an equal pure strain with the first with an equal but opposite rotation. And the separate component operations must be taken in the opposite order in the strain and in its conjugate.

The successive application of the strain and its conjugate thus necessarily leads to the reduplication (or squaring) of the pure part of the strain, and to the annihilation of the rotation. For, call the parts, as operators, P and R. The strain and its conjugate, referred to axes fixed in space, may be either

RP and PR- 1 or P 1 RandR^ 1 P 1 , according as the pure strain or the rotation is first applied. The operations in each group are written, from right to left, in the order in which they are performed. Thus RP means the pure strain P, followed by the rotation R.

§ 90. The final results are P 2 and P’. In the first case we have the pure strain, followed by the rotation; then (by the conjugate) the rotation is undone, and the pure strain reapplied. In the second we rotate first, then apply the pure strain twice, and finally undo the rotation. Thus the student must be cautioned against the error of supposing that the results of applying PR and RP separately are generally the same. Perhaps it will be easier for the reader to consider the “reciprocal,” instead of the conjugate, of a strain. For if the strain be RP, the reciprocal is obviously P^ 1 R* 1 j if it be PR, the reciprocal is R^ 1 P^ 1 . Either pair of these, taken in either order, restores the figure to its primitive form. The one point to be noticed is that, in whatever order the direct component operations are supposed to occur in the strain, their reciprocals must be taken in the opposite order in the reciprocal strain. The reciprocal strain simply undoes the strain, and therefore differs from the conjugate by a factor, the square of the pure part of the strain.

From this we have at once, as will be seen later, the means of decomposing a given strain into its pure and its rotational factors. This is effected as soon as we can form the expression for the conjugate strain in terms of that for the strain itself.

4 91. As, in general, any strain converts a spherical portion of a figure into an ellipsoid, and as an ellipsoid has two series of parallel circular sections, it appears that in every strain there are two series of planes of no distortion. The consideration of these planes leads us to a second and very different mode of analysing a strain into simpler components. Perhaps the most elementary mode of considering this subject is by thinking of the motion of water flowing slowly down a uniform channel. We know that water, at ordinary pressures, is practically incompressible; also that the upper layers of the water in a canal flow faster than those below them. Hence the definition of a “simple shear.” Let one plane of a figure be fixed, and let the various planes parallel to it slide over it and over one another, all in the same direction, and with velocities proportional to their distances from the fixed plane. It is clear that this shear produces homogeneous strain in the figure, but it is mere change of form without change of volume. The fixed plane and all those parallel to it, are planes of no distortion. But we have seen that there must be two sets of such planes. To find the second set, let us suppose the plane of fig. 30 to be parallel to the common direction of sliding, and perpendicular to the fixed plane. This plane, so defined, is the plane of the shear. Let AB be the trace on it of the fixed plane, PQ that of one of the sliding planes, PP' the amount of its sliding. Bisect PP' in Μ by a perpendicular, meeting AB in A. Join AP, take AB = AP, and draw BQ parallel to AP, Consider the strain of the rhomboidal portion APQB of the figure. P moves to P', and Q to Q', where QQ' = PP'. Hence the rhombus remains a rhombus, for AP' = AP = AB. But the lengths of its diagonals have been interchanged. It has been subjected to an elongation of AQ, and a contraction of BP, each in the same ratio (so that their product, i.e., double the area of the rhombus, remains unaltered), while all lines perpendicular to the plane of the figure remain unaltered. From the'symmetry of the rhombus it is obvious that AQ' and BP' are the greatest and least axes of the strain ellipsoid, while AB and AP' are parallel to its circular sections. Planes originally parallel to AP and perpendicular to the paper are therefore the second set of planes of no distortion. The rotational part of the shear may be measured in terms of the angle PAM, but is given directly by PBP', and its axis is perpendicular to the plane of the figure. The most convenient measure of the shear is the ratio of PP' to AM, or, what involves the same, the angle PAP'. Another mode of measuring it is by means of the ratio BP∕AQ, = 1 + e, suppose. If e be a small quantity, as is usually the case with solids, we may write 1 ÷ e for the measure of the shear. Here e indicates the extension per unit of length along one diagonal of the rhombus, and the contraction per unit of length along the other.

§ 92. It is quite clear from what has been said that we can analyse a strain by the help of simple shears compounded with different forms of pure strain. For the shears may be taken in an infinite number of ways so as to produce the rotational part of the given strain, while also producing deformation without change of volume. The final adjustment is to be made by a pure strain, whose axes are those of the strain ellipsoid due to the shears. As a shear depends on four quantities only, two shears and a dilatation furnish the nine constants required for a homogeneous strain.

The successive application of two pure strains does not, except in special cases, give rise to a pure strain. This is, physically, a most important proposition. Thus, for instance, the instantaneous strains of each element of a perfect fluid in which there is no vortex motion are pure; and yet, if the element be followed in its motion, it will be found in general to rotate. Its motion is said to be “differentially irrotational.”

To prove this proposition by the help of a particular case is simple enough. Take, for instance, a compression in one direction, followed by an equal extension in a different direction. Only when these directions are at right angles to one another is the resultant strain pure.

§ 93. The analytical theory of strains is, at least in its elements, an immediate application of the properties of determinants, usually of the third order. We subjoin a slight sketch of it.

We have seen that it is only necessary, for the full characterizing of a strain, that we should know what becomes of three unit lines not originally coplanar. Take these parallel to the'axes (generally oblique) of x, y, z. Then if the x unit becomes a line which is the diagonal of a parallelepiped with sides a, d, g parallel to the axes, y similarly that of b, e, h, and z of c, f, i, we see at once that the coordinates of the point originally at x, y, z become a∕= ax + by + cz ) y ’ =dx+ey+fz> (A), z'= gx + hy + iz )

Here it is obvious, from the premises, that the nine quantities a, b, c; d, e, f; g, h, i are all real, and altogether independent, at least so far as kinematics is concerned. To obtain an idea of their nature from another point of view, let us suppose the axes to be rectangular. Let unit parallel to x become e 1 , in the direction given by the cosines l 1 , m 1 , n 1 . Similarly, let e 2 , Z 2 , m 2 , n 2 belong to a unit originally parallel to y, and ¾, l 3 , m 3 , ιι 3 to a unit parallel to z. Then the broken line x, y, z becomes x', y', rf, where a5 ’ =e 1 Z 1 05 + e 2 l 2 y + e 3 l 3 z ) y'≈e 1 m 1 x + e 2 m 2 y + e 3 m 3 z > . . , (A'), s' =e 1 Ji 1 a3 + e 2 n 2 y + e 3 n 3 z )

Though we have introduced three numbers e along with nine direction cosines, no greater generality is secured, for there are three necessary relations, one among each set of cosines.

To find the characteristic property of a pure strain, let us take it in its most general form. Thus let Z 1 , m 1 , n 1 now denote a line which, without change of direction, has its length altered by the strain in the ratio e 1 ∙. 1. Let Z 2 , m 2 , n 2 , e 2 and Z 3 , wι 3 , n 3 , e 3 be similar· data for the other two of the system of rectangular axes of the pure strain. Then to these axes the coordinates of x, y, z are ξ = l 1 x + m 1 y + n 1 z, η = l 2 x + m 2 y + n 2 z, ζ=l 3 x+m 3 y + n 3 z.

The strain converts ξ into ξ'=β 1 ξ, η into √ = e 2 η, and £ into C' = e 3 f. Hence the final coordinates (to the original axes) of the point originally at x, y, z are x ' ~ h⅛' + hv' + hζ' > y' = m 1 ξ' + m 2 η ’ + m 3 ζ ’ , z' = n 1 ξ' +⅝√ +w 3 C'; or, in terms of x, y, z, x'=(e 1 l 2 + e 2 Γ 2 + e 3 l 2 3 )x + (ef 1 9n 1 + β 2 l 2 m 2 + e 3 l 3 m 3 )y ∖

+ (⅛ + ⅛ ’ ⅛ + ⅞⅞⅝)z y'=(e 1 m 1 l 1 +e 2 m 2 l 2 + e 3 m 3 l 3 ')x + (e 1 ml + e 2 m 2 2 + e 3 m 2 3 )y ( s β , + (e 1 m 1 n 1 + e 2 m 2 n 2 + e 3 m 3 n 3 )z · ' ' ’ z ’ =(e 1 n 1 Z 1 + e 2 n 2 t 2 + e 3 n 3 Z 3 )aj + (e 1 n 1 m 1 + e 2 n 2 m 2 + e 3 n 3 m 3 ')y + (e 1 n 2 l +e 2 n 2 2 + e 3 n 2 2 )z . t

If we compare this with the general expression above given for a strain, we see that the coefficient of y in the value of x! is equal to that of x in the value of y'. Similarly that of z in x' is equal to that of x in z! ; and that of z in y' is equal to that of y in z ’; or finally b≈d, c=g, f=h.

Conversely, when these three conditions are satisfied, and not otherwise, the strain is pure. It is to be observed that ξ, η, ζ form a rectangular system, and thus the nine direction cosines (usually involving six arbitrary numbers) depend here on three numbers alone. Thus there are six independent numbers, corresponding to a, e, i, b, c, f, in the general expression for the. strain.

It is clear, from the elements of coordinate geometry, that the determinant [table]

represents the ratio in which the volume is increased by the strain. Let us now introduce, in succession to the strain a b c def (1), 9 h i the connected strain a d g b e h (1'), c f i which obviously produces an equal change of volume with the former.

Applying these strains in succession, we have as the final result ie” = αa∕ + <7y' + gz', v y" = bx' + ey' + hz', z" ≈ ex' + fy' + iz ’, or, substituting for x', y', their values in terms of x, y, z, ae"=(α 2 + tf 2 + y 2 )≈ + (ab + de + gh)y + (ac + df+ gι^)z, y" = (ba + ed + hg}x + (δ 2 + e 2 + Λ 2 )y + (be + ef + hι)z, z" = (ca +fd + ig)x + (cb +fe + ih)y + (c 2 +∕ 2 + i 2 )z. Thus the resultant strain is a 2 + d 2 + g 2 ab + de+gh ac+ df + gi ba + ed + hg b 2 + e 2 + h 2 bc+ ef + hl ca+fd + ig cb +fe +ih c 2 + f 2 + i 2 which, for simplicity, we will write as

α δ γ

δ e β (2).

7 β <

It will be observed that this group of nine numbers, if treated as a determinant, constitutes the product of the determinants formed of the two systems above.

This satisfies the criterion of a “pure strain,” as given above; and we thus see that in the successive application of the strains a b c a d g def and b e h g h i c f i the rotation produced by the first is annihilated by the second. i Let A, B, C, &c., be the minors of [table]

corresponding to a, b, c, &c.

Then by our original equations we have ∆x = Ax· + Dy' + Gz', ∆y = Ba∕ + Ey' + Hz', Δz=Ca∕ +Fy' + Iz'.

Thus the reciprocal of the strain a b c A∕∆ D∕∆ G∕∆

def is B∕∆ E∕∆ H∕∆

g h i C∕∆ F∕∆ Ι/Δ.

This is evident from the formulae just written. For they express that the new strain converts √, y', z' into aj, y, z.

Apply the resultant strain in succession to this reciprocal, the result is easily foreseen from separate terms like the following :—

A(a 2 + d 2 + y 2 ) + B(fflδ + de + gh) +C(αc + df+ gι) = a( Aa + Bδ + Cc) + d( Ad + Be + C∕) + g(Ag +B7⅛ + Ci)

= u∆; &c.

Hence when we apply (2) to a figure previously strained by the reciprocal of (1) the result is the strain (1'). Hence we verify that the latter is the conjugate of (1); as it possesses the properties described in § 90, all of which are thus established.

To analyse a strain in the simplest manner, we must find the axes of the strain ellipsoid (§ 88), as well as the original radii of the unit sphere which λvere distorted into them. It is more direct, however, to consider the ellipsoid which becomes a unit sphere in consequence of the strain. „The equation of that ellipsoid is (ax + by + cz) 2 + (dx + ey +fz} 2 + (gx + hy + iz) 2 =1; or, with the notation employed in (2) above, ax 2 + ∈y 2 + ιz 2 + 2Sxy + 2βyz + 2yzx = 1.

But the square of any radius-vector is x 2 + y 2 + z 2 =r 2 , suppose.

The maximum radius-vector, therefore, of the ellipsoid is found from the two equations (ax + Sy + yz~)dx + (Sx+ey + βz)dy + (yx + βy + ιz)dz = 0, xdx + ydy + zdz —0.

Hence, p being a numerical quantity to be found, ax + Sy + yz≈px, Sx + ey + βz=py, (4).

yx + βy + ιz=pz.

Multiply respectively by x, y, z, add, and take account of the two preceding undifferentiated equations. We thus have l=pr 2 , or p is the reciprocal of the square of the maximum semi-axis required.

But, if we eliminate x, y, andz simultaneously from the preceding linear equations, we have [table]

This equation is known to have three real positive roots, because the determinant is symmetrical. The roots are the squared reciprocals of the semi-axes of the ellipsoid, i.e., they are the squares of the semi-axes of the strain ellipsoid.

When the three values of p have been found from this equation, any two of the equations (4) give in an unambiguous form the corresponding values of the ratios x : y : z for each of them. Thus we know the original positions of the lines which become the axes of the strain ellipsoid. Their final positions are found from these by means of (A). And, since we thus know the original and final positions of the rectangular system, the method of Rodrigues enables us to calculate the axis and amount of the rotation.

In homogeneous strain, one direction at least is unchanged. This is an addition to, or extension of, the singular result of § 75.

For, if x, y, z be shifted to a point on its radius-vector, we must have ax + by + cz = tx Ί

dx + ey +fz≈ey I; gx + hy + iz = ez J so that [table]

a cubic equation, which must have one real root.

When the figure is rigid, the strain must be a rotation only. Hence in the formulae (A') above we have e 1 =β 2 =⅞=]. Thus the last written equation becomes [table]

or (by the properties of the direction cosines of a set of rectangular axes)

1—(Z 1 + wι 2 + n 3 )(e - e s ) — ∈ 3 = 0 i

This has, of course, the real root e = l. But we also have

1 + ∈(1 — (Z 1 + wι 2 + τ⅛)} + c 2 = 0.

This cannot have real roots if the coefficient of ∈ lie between the limits 2 or - 2. But these are its greatest and its least possible values. For, first, l 1 , m. 2 , n 3 may be each = 1 simultaneously. Here we have

(1- e) 2 = 0.

Or two of them may be each = — 1, but then (to avoid perversion) the third must be =1. Then we have

(l + e) 2 = 0.

In the first case the figure has no rotation. In the second it rotates through an angle τr about the axis of e = l.

The proposition that two pure strains succeeding one another usually give a rotational strain is proved at once by analysis. Let the pure strains be such that x'=ax + dy + cz, y ’ =dx+ ey + bz, z'=cx + by + iz∙ and x" = a! ai + d'τj + c'z ’ , y" = d'x' + e'y' + δ'z , , d'≈<!a! + b'y' + i'z'.

Then, writing only the second term of x" and the first of y" in terms of x, y, z, we have ic"= + (a'd +d'e + c'b')y+ . . . y"=(d'a + e ’ d + b ’ c)x + + . . . z ’ = ....... + + ...

It is clear that, in general, this is not a pure strain. But it is also clear that a third pure strain can be found whose application in succession to the other two will give a pure strain.

For let the last equations be written x"=a"x + b"y+c"z, y" =d"x+e"y+f"z, z! ’ = g"x + h"y + i"z; and let us apply further the pure strain x!" = ax" + Sy" + yz", y"'=Sx''+ ey" + βz", zt"=yai'+ βy" + ιz! ’ . Then we have

α∕"= + (ab'' + Se" + yh" y )y + . . . y ’ " = (Sa" + ed" + βtf ’ }χ + + . . .

√"= . . .

There are but three conditions to satisfy, that this strain may be pure. But we may accomplish this in an infinite number of ways, for we have five disposable quantities, viz., the ratios of any five of α, e, ι, β, γ, δ to the remaining one. In a precisely similar manner we may show that three pure strains can be found, such that their resultant is a mere rotation. In fact, all we have to do, since two pure strains in general produce a distortion accompanied by rotation, is to apply a pure strain to annihilate the distortion, which can of course always be done.

§ 94. In general when a figure is continuously strained, which is usually the case in physical applications, at least until cracks occur, the strain is not homogeneous. But, on account of the continuity of the strain, portions indefinitely near one another are strained indefinitely nearly alike. Hence we may treat such a case by the ordinary process for homogeneous strain, so long as we confine our attention to small regions of the figure strained. When there is discontinuity in the motion of a fluid, it is the common practice to treat the motion as continuous by the fiction of an infinitely thin vortex-sheet separating the two discontinuously moving portions. This is, in all likelihood, physically true in ordinary fluids; but, so far as the imaginary frictionless fluid of the mathematicians is concerned, it is a mere analytical artifice to enable us to carry out the investigation. See Atom and Hydromechanics, in which the mathematical theory of “vortex motion” is very fully considered.

Suppose space to be uniformly occupied by points which are displaced in a continuous mannt r. Let ξ, η, £ be the rectangular components of the displacement of a point originally situated at x, y, z. The continuity of the displacement requires merely that the differential coefficients, of all orders, of ξ, y, f, with respect to x, y, z (and any combination of them) shall be finite. That being assumed, the displacement, parallel to x, of the point whose initial coordinates were x + Sx, y + Sy, z + Sz (where δz, Sy, Sz are indefinitely small quantities of the first order) is necessarily expressed by _ 6 ⅛ - <⅛ ~ dt

ξ + ^z~Sx + - 1 Sy + -yδ^. dx dy dz

He∏ue the relative coordinate of the second point with regard to the first is changed from Sx to Sx + ⅛isx + c ⅛Sy+ ~Sz. And dx dy dz similarly for the other relative coordinates. Hence the strain in the immediate vicinity of the point x, y, z is given by 1+ ⅛ ⅛ Éf, dx dy dz ’ dη j dη dη dx dy dz ’ dζ dζ l+^.

dx dy dz'

If the differential coefficients are all small quantities, whose squares and products two and two may be neglected, i. e., if the strain is slight, we have for the ratio in which the volume is increased 1 . ⅛ , ⅜ , ⅛^. 1 dxcly ^dz

Hence the condition of no change of volume, is

⅛ + ⅛ + i K = o dx dy dz

To examine this case more closely, let us suppose that it consists of a pure strain as in § 93 (B), superposed on a rotation ω x , ω 3 ,, ω z about the axes of x, y, and z as in § 77. Let these be so small as not to interfere with one another. That compound strain would be e ι l l+ e 2 l 2+ e s l 3 edιm 1 +e 2 f 2 m 2 +e 3 Γ 3 m 3 -ωz . · · +«ÿ

e l m 1 h+e2 m 2 l 2+ e 3 m 3 r 3+ e <>z e 1 m 2 1 +e 2 m 2 2 +e 3 ml e 1 m 1 n l + . . -ω x

≡l n √l+*2 n 2 z 2+⅛ j √3- ω ⅛ + <Vχ +⅛ n 2+ f 3 n 3 ·

Comparing with the above, we find 1+ l=⅛ 3 +Mi+⅛

or, if we put ∈ for the “elongation,” so that e = l + ∈, g=e 1 Zf + e 2 Z* + e 3 Z*, with similar expressions for and . dy dz

These give df d·n dC

^s - .+ ^∙—l· -γ- = e 1 + e 2 + β 3 .

dx dy dz z z a

Again we have d·n d£ , , , .

⅛ + 0 ∕ y =2(<⅛τn 1 + c 2 Z 2 m 2 + e 3 l 3 m 3 ), — 2(∈ 1 Z 1 ^re 1 + e 2 Z 2 τjι 2 + e 3 l 3 m 3 ), with other two of the same kind.

Also we have three equations of the form [table]

These expressions show, simply, that when there is no elementary rotation the quantity ξdx + ηdy + ζdz = dφ is the complete differential of a function of three independent variables. If we combine the condition that there shall be no change of volume with those that there shall be no rotation, we can eliminate ξ, η, ζ ; and we arrive at Laplace’s equation d 2 φ d 2 φ d 2 φ dx 2 dy 2 + dz i

This shows at once how a graphical representation of stationary distributions of temperature, electric potential, &c., may be given by means of a strain.

If cZS be an element of a surface at the point x, y, z, and I, m, n the direction cosines of its normal, the rotation about the normal is obviously lwχ + f mtΛy 4- nw 3

The integral of double of this over a finite portion of surface is

Λ⅛f<⅛-⅛μ m (s-⅛) + √⅛-≤)Y

JJ ∖∖ dy dz ) ∖ dz dx) ∖ dx dy J or JJ ( ∖ dy dz J a ∖ dz dx) ∖ dx dy) }

This, as seems to have been first pointed out by Stokes, can be expressed as a simple integral in the form f (fdx + ηdy + ζdz) extended round the boundary of the surface. Hence the double integral has the same value for all finite surfaces having the same boundary; and, as a consequence, it vanishes when taken over a closed simply-connected surface. Hence we see at once that it vanishes for multiply-connected surfaces also. The proof of the equality of the single and double integral has only to be established for a mere surface element. For, when that is done, the common boundary of each pair of elements gives equal portions, with opposite signs, in the single integral.

Directly connected with the displacements of a group of points, we have the question, What is the mathematical expression of the fact that the number of points is not altered I There are many ways of answering this; but the following, which is immediately deducible from our recent investigation, seems sufficiently simple. If I, m, n be the direction cosines of the normal to an element o?S of a singly-connected closed surface, the number of points which pass through the element in the time St in consequence of the displacement ξSt, ηSt, ζSt at the point x, y, z is

(Zξ + mη + nζ)pdSSt, where p is the number of points per unit volume at x, y, z. But at every point inside the closed surface the density is altered from p to p + pSt. It will be noticed that ξ, η, ζ now stand for the x, y, z components of velocity.

Hence, if the excess of the number of points passing into the surface over those escaping be equated to the increase of the number of points included in the closed space, which is calculated from the change of density inside, we have

δZy/'(Zξ + mη + nζ~)pdS = Stfffidxdydz.

If we take for S an elementary rectangular parallelepiped, with edges Sx, Sy, Sz, this becomes at once

_ ∕⅛) + ⅛) + ⅛O ∖ ζ χ ζ ysz5t = pt x SySzSt ∖ dx dy dz ) j r a nr d ∣ ’ d (f ’ ^ d (P η ^ d ^ ∩ dt + ~d^ + ~df + ~dT~ Q '

If the arrangement is incompressible this becomes, as above, d A + d n + d i=o t dx dy dz

In any one of the last four forms the expression is called the “equation of continuity,” another of the preposterously ill-chosen terms which have been introduced with only too great success into the nomenclature of our subject.

§ 95. In the strains which we have hitherto considered all parts of a figure were regarded as capable of changing their form and volume; and the strain of any element, when not identical with that of a proximate element, was supposed to differ only infinitesimally from it. But there is another class of changes of form, for which this restriction does not hold. The most important case, and the only one we can here consider, is that of “link-work.” Here each finite piece is treated as incapable of change of form, and the change of form of the whole depends merely upon the relative motions of the parts. We will further restrict ourselves by the condition that the link-work is such that its form is determinate when the relative position of two of its parts is assigned. Thus, a jointed parallelogram is completely determined in form if the angle between two of its sides is assigned. Instead of an angle, we may assign the length of a diagonal; then the fact that the sum of the squares of the diagonals is equal to that of the squares of the sides determines the other diagonal. This gives us the kinematics of the more complex arrangement called “lazy-tongs.” The most important applications of this branch of our subject are to what is called “Mechanism.” One important practical problem in that branch was suggested by a stationary steam-engine, in which it was required to connect, by link-work of some kind, a point (of the piston-rod), which had a to-and-fro motion in a straight line, with another point (of the beam) which had a to-and-fro motion in a circular arc. Watt’s original solution of the problem depends ultimately upon the near approach to rectilinearity of the motion of any point of a rod whose extremities move in two circles in the same plane. Thus, if OP, PQ, QO' (fig. 31) be three ,bars jointed together at P and Q, having 0 and O , fixed, and the whole constrained to move in one plane, it is easy to see that the complete path of any point R of PQ is a species of figure of eight. A portion of that curve on each side of a point of inflexion (where the curvature vanishes) was found to be sufficiently straight for practical purposes.

But the rigorous solution of this problem has only been arrived at in recent times; and the beautiful device of Peaucellier, which we will briefly explain, has led to a host of remarkable investigations and discoveries in a field regarded till lately as perfectly hopeless. A simple mode of arriving at Peaucellier’s result is as follows.

Let PQ, PR (fig. 32) be equal links, and PO a link of a different length, all jointed together at P. Suppose O to be fixed, and Q and R constrained to move in a fixed straight line OQR, what is the relation between OQ and OR? We have, if PS be perpendicular to QR,

OP 2 =OS 2 + SP 2 ,

QP 2 =QS 2 + SP 2 j whence

OP 2 - QP 2 =OS 2 - QS 2 =OQ.OR.

Thus the rectangle under OQ and OR is constant ; so that, if R were to describe a straight line, Q would describe a circle having 0 on its circumference. In practical application, to keep O, Q, R in one line, the parts of the link-work are doubled symmetrically about that line, so that it takes the form of a jointed rhombus PQP'R (fig. 33) with two equal links, PO, OP' attached at the extremities of a diagonal. As a very curious result of this arrangement, if OQ have its length changed by any very small amount, the corresponding change of length of OR is directly as OR 2 or inversely as OQ 2 . Hence, as will be seen later, a constant force (towards or from 0) acting at Q will be balanced by a force (from or towards 0) acting at R and varying inversely as the square of OR.

DYNAMICS.

Definitions and General Considerations.

§ 96. We commence with a few necessary definitions. A “physical particle” is a purely abstract conception, embodying together the ideas of inertia and of a geometrical point. It is, so to speak, a mathematical fiction, embracing only those properties which are required for our temporary purpose. Any mass, however large, can be treated as a particle, provided the forces to which it is subject are exerted in lines passing through its “centre of inertia” or “centre of mass” (this term will presently be defined), so as to be incapable of setting the mass into rotation. This is, to a first approximation, true of planetary motions, but when we look more closely into that question, so as for instance to take account of the oblate forms of the planets, we have to deal with forces which produce rotatory effects, such as “precession” and “nutation.”

§97. The “quantity of matter” in a body, or the “mass,” is proportional to the “volume” and the “density” conjointly. The “density” may therefore be defined as the quantity of matter in unit volume.

If Μ be the mass, p the density, and V the volume of a homogeneous body, we have at once

M = V p , provided we so take our units that unit of mass is the mass of unit volume of a body of unit density. Hence the dimensions of p are [ML -3 ].

As will be presently explained, the most convenient unit mass is an imperial pound, or a gramme, of matter.

§ 98. The “quantity of motion,” or the “momentum,” of a moving body is proportional to its mass and velocity conjointly. As already stated this is, like velocity, a directed quantity, or “vector.” Its dimensions are, of course, [MLT -1 ].

§ 99. “Change of quantity of motion,* or “change of momentum,” is proportional to the mass moving and the change of its velocity conjointly.

Change of velocity is to be understood in the general sense of § 32. Thus, with the notation of that section, if a velocity represented by OA be changed to another represented by OB, the change of velocity is represented in magnitude and direction by AB.

§ 100. “Rate of change of momentum,” or “acceleration of momentum,” is proportional to the mass moving and the acceleration of its velocity conjointly. Thus (§ 36) the acceleration of momentum of a particle moving in a curve is Ms along the tangent, and Mv 2 ∕p in the radius of absolute curvature. The dimensions of this quantity are [MLT^ 2 ].

§ 101. The “vis viva,” or “kinetic energy,” of a moving body is proportional to the mass and the square of the speed conjointly. If we adopt the same units of mass and velocity as above, there is particular advantage in defining kinetic energy as half the product of the mass into the square of its speed. Its dimensions are [ML 2 T~ 2 ].

§ 102. “Rate of change of kinetic energy,” thus defined, is the product of the speed into the component of acceleration of momentum in the direction of motion.

For d ∕Mv 2 ∖ -, . .-,...

dt ∖ -2-J = Mvυ=r(M s ).

The dimensions are [ML a T' 3 ].

§ 103. The “space-rate of change of kinetic energy” is d ∕Mv 2 ∖ ,. dv ^ τ dv - r ..

- r - r = Mυ- r =M- r = M⅛; ds ∖ 2 ∕ ds dt and its dimensions are [MLT^ 2 ], the same as those of “force” (§ 104).

§ 104. “Force,” as we have already seen, is any cause which alters a body’s natural state of rest, or of uniform motion in a straight line.

The three elements specifying a force, or the three elements which must be known before a clear notion of the force under consideration can be formed, are—its place of application, its direction, and its magnitude. The place of application may be a surface, as when one body presses on another; or it may be throughout the whole mass of a body, as in the case of the earth’s attraction for it.

The “measure of a force” is the rate at which it produces momentum, or the momentum which it produces in unit of time, which is the same as what we have already called “rate of change of momentum.” According to this method of measurement the standard or unit force is that force which, acting on the unit of matter during the unit of time, generates the unit of velocity. The dimensions of force are therefore [MLT^ 2 ].

§ 105. Hence the British absolute unit force is the force which, acting on one pound of matter for one second, generates a velocity of one foot per second.

[According to the system followed till lately in treatises on dynamics, the unit of mass is g times the mass of the standard weight, g being the numerical value of the acceleration produced (in some particular locality) by the earth’s attraction. This definition, giving a varying unit of mass, is exceedingly inconvenient. In reality, standards of weight are masses, not forces. They are employed primarily for the purpose of measuring out a definite quantity of matter, not an amount of matter which shall be attracted by the earth with a given force.]

§ 106. To render our standard intelligible, all that has to be done is to find how many absolute units will produce, in any particular locality, the same effect as does gravity. The way to do this is to measure the effect of gravity in producing acceleration on a body unresisted in any way. The most accurate method is indirect, by means of the pendulum. The result of pendulum experiments made at Leith Fort, by Captain Kater, is that the velocity acquired by a body falling unresisted for one second is at that place 32 ’ 207 feet per second. The variation in gravity for one degree of difference of latitude about the latitude of Leith is only Ό000832 its own amount. The average value for the whole of Great Britain differs but little from 32 ’ 2; that is, the attraction of gravity on a pound of matter in the country is 32 ’ 2 times the force which, acting on a pound for a second, would generate a velocity of one foot per second. Thus, speaking very roughly, the British absolute unit of force is equal to the weight of about half an ounce. The quantity of 32'2 feet per second per second is usually called g. Its dimensions are obviously [LT^ 2 ]. And, if Μ be the mass of a body, its weight is My. In the Centimetre-Gramme-Second system of units, the absolute unit of force produces in one second, in a mass of one gramme, a velocity of one centimetre per second.

§ 107. Forces (since they involve only direction and magnitude) may be represented, as velocities are, by vectors, that is, by straight lines drawn in their directions, and of lengths proportional to their magnitudes respectively.

Also the laws of composition and resolution of any number of forces acting at the same point are, as we shall presently show (§ 117), the same as those which we have already proved to hold for velocities; so that, with the substitution of force for velocity, § 30 is still true.

§ 108. The “component” of a force in any direction is therefore found by multiplying the magnitude of the force by the cosine of the angle between the directions of the force and the component. The remaining component in this case is perpendicular bo the other.

It is very generally convenient to resolve forces into components parallel to three lines at right angles to each other, each such resolution being effected by multiplying by the cosine of the angle concerned.

The magnitude of the resultant of two or of three forces in directions at right angles to each other is the square root of the sum of their squares.

: § 109. The “centre of inertia or mass” of any system of material particles whatever (whether rigidly connected with one another, or connected in any way, or quite detached) is a point whose distance from any plane is equal to the sum of the products of each mass into its distance from the same plane, divided by the sum of the masses.

The distance from the plane yz of the centre of inertia of masses m 1 , m. 2 , Ac., whose distances from the plane are x υ x 2 , &c., is therefore ? ¾M ∣ g).

2(»i) and similarly for the other coordinates.

Hence its distance from the plane δ = λx+μy + vz -α = 0 r , ,_ I. - 1 _ ^{m(λx + μ y+ v z-a')'} 2 (mδ) is D = λx + μ y + vz - a = =7—; —- = -A— I.

2(?n) 2(to) ’ as stated above. And its velocity perpendicular to that plane is √ dδ∖ cZD 1 ( f dx dy dz ∖ ) * ∖ m dt )

Λ "S i ί + √ +, 'λ) i ~⅛H ’ from which, by multiplying by 2wι, and noting that δ is the distance of x, y, z from δ = 0, we-see that the sum of the momenta of the parts of the system in any direction is equal to the momentum in that direction of the whole mass collected at the centre of mass.

The problem of finding the centre of inertia of any given distribution of matter is a question of mere mathematics. We must confine ourselves to a few examples only. And, first, we may note that when a body is symmetrical about a plane the centre of inertia must obviously lie in that plane. Thus, as an ellipsoid and a rectangular parallelepiped have each three planes of symmetry, their centres of inertia lie at their centres of figure, where these planes meet. Again, it is obvious that, if a body can be divided into parts the centres of inertia of which lie on a straight line, the centre of inertia of the whole is in that line. Thus, as a triangular plate may be divided into strips parallel to one side, every one of which has its centre of inertia at its middle point, the centre of inertia of such a plate is the point of intersection of the bisectors of the sides. Its distance from any one side, treated as base, is therefore one-third of the height. Again, if a triangular pyramid (or tetrahedron) be divided into triangular slices by planes parallel to one face treated as base, the centres of inertia of all the slices lie in a straight line. Hence the distance of the centre of inertia from the base is one-fourth of the height. If the base be of any other form, it may be divided into triangles, and thus the whole pyramid (or cone) into tetrahedra, for each of which the same property holds. Hence the centre of inertia of a pyramid divides the line joining the vertex to the centre of inertia of the base in the ratio 3:1. All this is on the supposition that the solids treated of are of uniform density. When we deal either with more complex forms or with heterogeneous bodies, we must in general have recourse to integration.

For a continuous body we must take an element of mass, say pδzδyδz, at the point x, y, z instead of the mass m in our original formula. The sums then become integrals, and we have three expressions of the form Jffpxdxdydz fjfodxdydz ’

Here p represents the density at x, y, z; and the integration extends through the whole volume of the body.

Thus, for a homogeneous hemisphere of fàdius a we have, taking the base as the plane of yz, Γa ∕ 1r(a 2 - x 2 )xdx n √o 'sa X= ∣πα ii ^ 8

The same value would be obtained for any semiellipsoid, whatever be the diametral section, provided a be the height measured perpendicular to the base; and, in general, from the position of the centre of inertia of any body we may at once find that of the same body homogeneously strained.

Recurring to the hemisphere, suppose its density to be at every point proportional to the distance from the centre. Then we have, omitting common constant factors of numerator and denominator, _ a r∙ ι d<T t -x l ∕ xdx ∕ ' ∣ ' ∖∣ χs + τ'-dr ^ . _ Jo J* = f? .

x ' ra f Ja*-x i 5 ∕ dx ∕ r ∖ Jχ i - ∖ -r l dr Jo J^ A uniform hemispherical shell gives x≈⅛a by the well-known result due to Archimedes. From this, by taking concentric hemispherical elements, we may reproduce the preceding result for a solid hemisphere in the form

∕ 2πiC 3 cfe , x, lx r ,

∕θ 2a x =—- = -= · f a n i J 5 ∕ ς 2πx i dx. X JO

Here the first factor under each integral sign is the volume of the hemispherical element of radius x, and the second is proportional to its density.

If the density of a thin uniform spherical shell be everywhere proportional to the inverse cube of the distance from an internal point, that point is the centre of inertia. For, if a double cone of small angle be drawn, having that point as vertex, the volumes of the portions of the shell which it cuts out are as the squares of their distances from the vertex. Hence their masses are inversely as their distances from the vertex, which is thus their centre of inertia. The whole shell may be divided into pairs of elements for each of which this is true.

The reader may easily prove that, if the density of a solid sphere be inversely as the fifth power of the distance from an external point, the “electric image” of that point is the centre of inertia.

It may be proved in the last two examples that this point is not merely the centre of inertia of such distributions of matter, but that it is also a true “centre of gravity” in the sense that the whole attracts, and is attracted by, any other body whatever, as if its whole mass were concentrated in this point.

§ 110. By introducing in the definition of moment of velocity (§ 46) the mass of the moving particle as a factor, we have an important element of dynamical science, the “moment of momentum.” The laws of composition and resolution are the same as those already explained. Its dimensions are [ML 2 T^ 1 ].

§ 111. A force is said to “do work” if it moves the body to which it is applied; and the work done is measured by the resistance overcome, and the space through which it is overcome, conjointly. The dimensions of work are therefore [MLT^ 2 . I.] or [ML 2 T^ 2 ], the same as those of kinetic energy.

Thus, in lifting coals from a pit, the amount of work done is proportional to the weight of the coals lifted; that is, to the force overcome in raising them; and also to the height through which they are raised. The unit for the measurement of work, adopted in practice by British engineers, is that required to overcome the weight of a pound through the height of a foot, and is called a “foot-pound.”

In purely scientific measurements, the unit of work is not the foot-pound, but the absolute unit force (§ 105) acting through unit of length.

If the weight be raised obliquely, as, for instance, along a smooth inclined plane, the distance through which the force has to be overcome is increased in the ratio of the length to the height of the plane; but the force to be overcome is not the whole weight, but only the resolved part of the weight parallel to the plane; and this is less than the weight in the ratio of the height of the plane to its length. By multiplying these two expressions together, we find, as we might expect, that the amount of work required is unchanged by the substitution of the oblique for the vertical path.

Generally, if s be an arc of the path of a particle, S the tangential component of the applied forces, the work done on the particle between any two points of its path is ∕Sds, taken between limits corresponding to the initial and final positions. Referred to rectangular coordinates, it is easy to see, by the law of resolution of forces, § 117, that this becomes

/7 X~ + Υ^+Z~}ds , J ∖ ds ds ds J ’ where X is the component force parallel to the axis of x.

§ 112. Thus it appears that, for any force, the work done during an indefinitely small displacement of the point of application is the product of the resolved part of the force in the direction of the displacement into the displacement.

From this it follows that, if the motion of a body be always perpendicular to the direction in which a force acts on it, the force does no work. Thus the mutual normal pressure between a fixed and a moving body, the tension of the cord to which a pendulum bob is attached, the attraction of the sun on a planet if the planet describe a circle with the sun in the centre, are all cases in which no work is done by the force.

In fact the geometrical condition that the resultant of X, Y, Z shall be perpendicular to ds is „dx χ dy, 7 ^ = A ds ds ds ’ and this makes the above expression for the work vanish.

§ 113. Work done on a body by a force is always shown by a corresponding increase of kinetic energy, if no other forces act on the body which can do work or have work done against them. If work be done against any forces, the increase of kinetic energy is less than in the former case by the amount of work so done. In virtue of this, however, the body possesses an equivalent in the form of “potential energy," if its physical conditions are such that these forces will act equally, and in the same directions, when the motion of the system is reversed. Thus there may be no change of kinetic energy produced, and the work done may be wholly stored up as potential energy.

Thus a weight requites work to raise it to a height, a spring requires work to bend it, air requires work to compress it, &c.; but a raised weight, a bent spring, compressed air, <fcc., are stores of energy which can be made use of at pleasure.

As an illustration of the calculation of work, take the following question.

Suppose one end of an elastic string to be attached to a mass resting on the ground, what amount of work must be done, in raising the other end vertically, before the' mass is lifted I

If x be at any instant the length of the string, I its original length, its tension is e ⅜'∙

Hence the value of x, when the mass is just lifted, is a U-^ 1 + e J > where W is the weight of the mass.

The whole work done is the sum of all the elementary instalments of the form Ε^dχ.

These must be summed up from x = l to .τ = z 1 , so that the result required is

It is to be observed that this quantity becomes less in proportion as E is greater, i.e., the less extensible is the string.

An interesting variation of the question consists in supposing the upper end of the string to be attached to the rim of a wheel, rough enough to prevent slipping. Here the various portions of the string are wound on in a more and more stretched state as the operation proceeds.

At any stage of the operation let x be the unstretched length of the part already wound on the wheel. The tension of the free part is then e Z-(Z-≈) l —x

During the next elementary step of the process a portion dx is wound on. But its stretched length is ldx l-x' Hence the element of work is

κ 7 z ∙∙"~⅜ -

G-*) 2

This must be integrated between the limits 0 and W of e⅛∖ I—X

EZ

or from Z-⅛ = ZtoZ-z = tp , v ; and the result is

I.· τ W

√ w + ≈ 1 °8E⅛) =

which, when E is very great compared with W, gives the previous result.

Further Comments on the First Two Laιvs of Motion.

§ 114. Weave now prepared to consider, more closely than we could at starting, the bearing of the various clauses of each of Newton’s Laws. Thus, from the first law we may draw the following immediate consequences.

The times during which any particular body, not compelled by force to alter the speed of its motion, passes through equal distances are equal. And, again, every other body in the universe, not compelled by force to alter the speed of its motion, moves over equal distances in successive intervals, during which the particular chosen body moves over equal distances. The earth, in its rotation about its axis, presents us with a case of motion in which the condition of not being compelled by force to alter its speed is more nearly fulfilled than in any other which we can easily or accurately observe. Hence the numerical measurement of time practically rests on defining “equal intervals of time” as times during τυhich the earth turns through equal angles.

§ 115. It has been objected to this statement that we begin by defining uniform motion by the description of equal spaces in equal times, and then employ this definition as a mode of measuring equal times. The objection, however, is not valid; for, if we agree to measure equal intervals by the undisturbed motion of any one physical mass, we find that in the successive intervals so determined all other absolutely free physical masses describe successive equal spaces.

§ M6. Again, from the second law we see that, if we multiply the change of velocity, geometrically determined, by the mass of the body, we have the change of motion (§ 99) referred to in the law as the measure of the force which produces it. In the statement of the second law there is nothing said about the actual motion of the body before it was acted on by the force; the same force will produce precisely the same change of motion in a body whether the body be at rest or in motion with any velocity whatever. Again, nothing is said as to the body being under the action of one force only; so that we may logically put part of the second law in the following (apparently) amplified form :—

When any forces whatever act on a body, then, whether the body be originally at rest or moving with any velocity and in any direction, each force produces in the body the exact change of motion which it would have produced if it had acted singly on the body originally at rest.

§ 117. Since now forces are measured by the changes of motion they produce, and their directions assigned by the directions in which these changes are produced, and since the changes of motion of one and the same body are in the directions of and proportional to the changes of velocity, a single force, measured by the resultant change of velocity, and in its direction, will be the equivalent of any number of simultaneously acting forces. Hence The resultant of any number of forces {applied at one point) is to be found by the same geometrical process as the resultant of any number of simultaneous velocities.

From this follows at once (§ 30) the construction of the “parallelogram of forces” for finding the resultant of two forces acting at the same point, and the “polygon of forces” for the resultant of any number of forces acting at a point. And, so far as a single particle is concerned, we have at once the whole subject of Statics.

§ 118. The second law gives us the means of measuring force, and also of measuring the mass of a body.

For, if we consider the actions of various forces upon the same body for equal times, we evidently have changes of velocity produced which are proportional to the forces. The changes of velocity, then, give us in this case the means of comparing the magnitudes of different forces. Thus the speeds acquired in one second by the same mass (falling freely) at different parts of the earth’s surface give us the relative amounts of the earth’s attraction at these places.

Again, if equal forces be exerted on different bodies, the changes of velocity produced in equal times must be inversely as the masses of the various bodies. This is approximately the case, for instance, with trains of various lengths drawn by the same locomotive.

Again, if we find a case in which different bodies, each acted on bya force, acquire in thesame time the samechanges of velocity, the forces must be proportional to the masses of the bodies. This, when the resistance of the air is removed, is the case of falling bodies; and from it we conclude that the weight of a body in any given locality, or the force with which the earth attracts it, is proportional to its 'mass. This is no mere truism, but an important part of the grand Law of Gravitation. Gravity is not, like magnetism for instance, a force depending on the quality as well as on the quantity of matter in a particle.

§ 119. It appears, lastly, from this law that every theorem of kinematics connected with acceleration has its counterpart in kinetics. Thus, for instance (§ 36), we see that the force under which a particle describes any curve may be resolved into two components, one in the tangent to the curve, the other towards the centre of curvature,—their magnitudes being the rate of change of momentum in the direction of motion, and the product of the momentum into the angular velocity about the centre of curvature, respectively. In the case of uniform motion, the first of these vanishes, or the whole force is perpendicular to the direction of motion. When there is no force perpendicular to the direction of motion, there is no curvature, or the path is a straight line.

Hence, if we resolve the forces acting on a particle of mass m, whose coordinates are x, y, z, into three rectangular components X, Y, Z, we have the equations originally given by Maclaurin, viz., d 2 x _ d-y -, d 2 z „ m di 2 ~ x, m dfi~ y, m dt 2 ~ z '

In several of the examples which follow, these equations will be somewhat simplified by assuming unity as the mass of the moving particle. When this cannot be done, it is sometimes convenient to assume X, Y, Z as the component forces on unit mass, and the previous equations become m ιs ≈mλ, &c., dt 2 from which m may of course be omitted.

[Some confusion is often introduced by the division of forces into “accelerating” and “moving” forces; and it is even stated occasionally that the former are of one, and the latter of four linear dimensions. The fact is, however, that an equation such as d 2 x v dF~ X may be interpreted either as dynamical or as merely kinematical. If kinematical, the meanings of the terms are obvious; if dynamical, the unit of mass must be understood as a factor on the left-hand side, and in that case X is the ^-component, per unit of mass, of the whole force exerted on the moving body.]

If there be no acceleration, we have of course equilibrium among the forces. Hence the equations of motion of a particle are changed into those of equilibrium by putting d⅛ n t i5 I-°, &e.

§120. We have now all that is necessary for the dynamics of a single particle, with exception of the experimental laws of friction. These, very nearly as they were established by Coulomb, we will now give.

To produce sliding of one flat-faced solid on another requires a tangential force which is directly proportional to the normal pressure between the surfaces, and whose actual magnitude is found from this pressure by means of a factor called the “coefficient of statical friction.” This coefficient depends upon the nature of the solids, the roughness or smoothness of the surfaces in contact, and the amount of tallow, oil, &c., with which they have been smeared. It also depends upon the time during which they have been left in contact. It is only in extreme cases dependent on the area of the surfaces in contact.

§121. When the forces applied are insufficient to produce sliding, the whole amount of friction is not called into play; it is called out to an amount just sufficient to balance the other forces. Thus there are two quite distinct problems connected with the statics of friction :— the first, to determine the amount of friction called into play under given circumstances; the second, to find the limiting circumstances under which, with friction, equilibrium is possible. When motion is produced, there is still friction (now called “kinetic”). It follows the same laws as does statical friction, only that the coefficient, which is approximately independent of the velocity, is usually considerably less than the statical coefficient.

Statics of a Particle.

§ 122. By § 117, forces acting at the same point, or on the same material particle, are to be compounded by the same laws as velocities. Therefore the sum of their resolved parts in any direction must vanish if there is equilibrium; whence the necessary and sufficient conditions are found by resolving in three directions at right angles to one another.

They follow also directly from Newton’s statement with regard to work, if we suppose the particle to have any velocity, constant in direction and magnitude (and by § 6 this is the only general supposition we can make, since absolute rest has for us no meaning). For the work done in any time is the product of the displacement during that time into the algebraic sum of the effective components of the applied forces, and there is no change of kinetic energy. Hence this sum must vanish for every direction. Practically, as any displacement may be resolved into three, in any three directions not coplanar, the vanishing of the work for any one such set of three suffices for the criterion. But, in general, it is convenient to assume them in directions at right angles to each other.

Hence, for the equilibrium of a material particle, it is necessary, and sufficient, that the (algebraic) sums of the applied forces, resolved in any one set of three rectangular directions, should vanish.

This statement gives at once the result that, if X 1 , Y 1 , Z 1 , X∙2> ^2> ⅞> & c ·> b* 3 the components (parallel to the three axes) of the forces P 1 , P 2 , &c., acting on the particle, we must have

2(X) = 0, 2(Y) = 0, 2(Z) = 0.

When these conditions are not satisfied, there is a resultant force P, with direction cosines λ, μ, v, such that

Pλ=2(X), Pμ-2(Y), Pr=5(Z).

§ 123. By far the most extensive series of examples of the composition of forces acting on a single particle is furnished by the theory of “attraction,” where each particle of the attracting mass exerts upon the attracted particle a force in the direction of the line joining them, and of magnitude depending on their masses and their mutual distance only. See Potential.

§124. When there are but three forces acting on the particle, their directions to give equilibrium must obviously be in one plane. For, if the third were not in the plane of the other two, it would have an uncompensated component perpendicular to that plane. Hence this case is always at once reducible to the triangle or the parallelogram of forces; and the magnitudes of each of the three forces are respectively proportional to the sines of the angles between the directions of the other two.

Thus, when a pellet is supported by two strings, as in fig. 34, we may proceed as follows to determine their tensions. Let P be the pellet, of weight W, and let AP, BP be the strings attached to points A and B respectively. Let their tensions be T and T'. The remark above shows that the strings must hang in a vertical plane, since the force W acts in a vertical line. Since A, B, and the length of the strings are given, the figure is perfectly definite.

Draw Pγ vertically upwards, and make its length represent, on any assumed scale, the value of W. Draw γβ parallel to AP, and let it meet BP in β. Then βγ represents T, and P/3 represents T', in direction and also in magnitude, on the same scale in which γP represents W. This case leads to nothing but the determination of the tensions, since the form of the figure is definite.

Next, let one of the tensions be given in magnitude. To effect this, we may suppose the end of PB not to be fastened at B, but to pass over a smooth pulley and support a weight Q. Let fig. 35 represent the state of equilibrium, and let the same construction as before be made. Then we must have γP: Pβ :: W: Q; or, writing it in terms of angles, sin APB: sinAPγ:; W: Q .

A and B and the direction of γP being given, this datum suffices for the drawing of the figure; i.e., for the calculation of the angles. A little consideration will show that, however small Q be, provided the string supporting it be long enough, there is always one definite position of equilibrium. The actual calculations in such a case as this are troublesome. It was chosen mainly on that account, so as to show, in a simple case, how pure geometrical processes may occasionally save the necessity of a tedious trigonometrical investigation. But a still simpler method will be afterwards explained, viz., that, for a position of stable equilibrium, the potential energy must be a minimum. Now, to apply this to our example, we see that any downward displacement of Q produces an upward motion of P. But when AP is nearly vertical the vertical displacement of P is indefinitely smaller than that of Q, so that Q must go down. On the other hand if APB be nearly a straight line, a displacement of P produces an indefinitely smaller displacement of Q. Hence P must go down. And these results are in character independent of the relative magnitudes of P and Q, provided both be finite.

Finally, let both tensions be constant. Here we must imagine pulleys both at A and at B (fig. 36), with weights R and Q attached to the ends of the strings. But now we see that we must have the limiting condition

R + Q>W.

This is merely the geometrical condition that

Pj3 + ∕3γ>Pγ

Here the magnitudes of all three sides of Pβy are given. Hence its angles are given, and the sole position of equilibrium is at once found.

§ 125. Now take the case of a particle resting on a surface. As we are concerned only with the portion of the surface immediately contiguous to the position of the particle, we may substitute for it its tangent plane at that point (except, of course, at singular points, where there may be an infinite number of tangent' planes; but such cases we do not consider). Hence the problem reduces itself in all cases to that of a particle resting on an inclined plane.

If the plane-be smooth, the particle cannot remain in equilibrium unless some force is present to prevent its sliding down. Let us suppose it to be supported by a force, F, acting upwards along the plane (fig. 37), Then we have three forces at work :— the weight P acting vertically downwards; the supporting pressure of the plane R, which necessarily acts perpendicularly to the surface; and the third force, just mentioned, which we see by previous considerations must be in the plane of the other two, and must therefore lie along the line of greatest slope of the plane. We might construct a triangle of forces as in the previous examples, but we will now vary the process, and resolve the forces in two directions at right angles to one another in their common (vertical) plane.

Let α be the angle of inclination of the plane to the horizon, then the algebraic sum of the components of the forces must vanish both horizontally and vertically. This gives us the two conditions

Fcosα-Rsinα = O; Fsina + Rcosa — P = 0

From these we obtain at once

F = Psinα, R = Pcosa

Now the choice of mutually perpendicular directions in which to resolve was at our option, and we see that had we chosen to resolve along and perpendicular to the plane we should have obtained the last two equations, which are equivalent to, but simpler than, the former ones, which were obtained by resolving horizontally and vertically. Theoretically speaking, it does not matter which system we choose; in practice, however, it is well to select the directions which will give the required results in the simplest form. The full value of a proper selection will not be felt till we come to the statics of a rigid solid.

If we suppose the plane to be rough, friction alone may suffice to develop the requisite force F, But the utmost value of the friction is (§ 120) μR. Hence the particle will be on the point of sliding if

μR = F = Psi∏α.

Divide the members of this equation by those of

R = Pcosα, and we find μ = tan a.

Hence, so long as the coefficient of friction is greater than the tangent of the inclination of the plane to the horizon, the friction will suffice to prevent sliding. More and more is called into play as the inclination of the plane increases, and finally when tanα = μ

the particle is just about to slide down. This simple idea, taken along with Coulomb’s results (§ 120), points to a very easy method of determining the coefficient of friction between any two substances. The limiting angle defined by

α = tan^ 1 μ

is called, on account of this property, the “angle of repose.” § 126. Let us now suppose the particle to be, in part, supported by an elastic string fixed at a point in the plane, and lying in the line of greatest slope. (This modification is introduced to show the nature of cases in which there are limits between which equilibrium is possible.) We assume “Hooke’s Law,” viz., that the tension of an elastic string, drawn out from its natural length I to length l', is expressed by e ⅛p where E is a definite constant, representing theoretically the tension which would just double the length of the string. *

Our equations are exactly the same as before, only that F consists now of two parts,—one due to friction, the other to the elasticity of the string. Thus

F = F 1 + E ^-=-^=Psinα , R=Pcosa.

υ

Now, when sliding is about to commence downwards we have

F 1 = μR,

If the particle is about to be dragged upwards,

F 1 = -μR.

Hence for the two extreme positions of equilibrium ±μPcosα + E^-γ-^=Psinα .

Hence the limiting positions of equilibrium of the particle are given by its distance from the fixed end of the string—

PZ

= + g (sinαψμcosα).

If Γ be less than the smaller of these, gravity pulls the particle down; if it be greater than the larger of them, the tension of the string pulls the particle up. In intermediate positions the full available friction is not called into play.

§ 127- Next, let a small ring P (fig. 38) be attached to one end of a string. Let the string pass round two smooth pulleys B, C, at different points, then be passed through the ring, then round two more pulleys D, E, and through the ring again, and so on,—the other end being either fastened to the ring or attached to a fixed point. It is required to find the position of equilibrium of the ring when the string is drawn tight, by operating on the lap of it behind two of the pulleys.

This is equivalent, from the physical point of view, to finding the position of equilibrium of a particle acted on by a number of equal forces each directed towards a given point. From the geometrical point of view its solution obviously answers the question, “Find the point the sum of whose distances from a number of given points is the least possible.” The points need not lie all in one plane. The solution is, from the polygon of forces, that in the equilibrium position the laps of the string, from the ring outwards, are parallel to the respective sides of a closed equilateral polygon, taken all in the same direction. That the solution is unique will be seen at once by considering a displacement of the ring, for the resultant of the forces obviously tends to diminish the displacement. When there are but three forces, their directions must be inclined at angles of 120° to one another (fig. 39). Thus we have immediately the solution of the celebrated geometrical problem, “Find the point the sum of whose distances from three given points is the least possible.”

§ 128. If, in the first problem of § 124 above, the particle were supported by three strings, instead of two, each attached to a fixed point, we should first have to assure ourselves that all three are brought into play. For, if not, the problem is reduced to the former case. The obvious condition is that, when the three strings are simultaneously tight, and the points of suspension are not in one vertical plane, the particle supported shall be situated within the triangular prism formed by vertical planes passing through each pair of points. If this condition be satisfied, the process for determining the tensions of the strings is merely to construct a parallelepiped, three of whose edges lie along the strings, while the conterminous diagonal is vertical. This leads to an obvious geometrical construction; and, when it is carried out, the lengths of the various edges are to the diagonal as the corresponding tensions to the weight of the particle. When the three points are in one vertical plane, nothing short of infinitely perfect fitting will, in general, bring all three strings simultaneously tight; and in this case the problem, mathematically considered, is indeterminate. When the strings are sufficiently extensible, all will be brought into play; and, with sufficient data, the problem is determinate.

Kinetics of a Particle with One Degree of Freedom.

§ 129. Here the motion is rectilinear, or at least takes place in some assigned curve.

The simplest case is that of a falling stone, when the effect of the resistance of the air is set aside and the acceleration due to gravity is reckoned the same at all elevations. This has already been treated with sufficient detail as a matter of pure kinematics, §§ 28, 29.

§ 130. When the particle, instead of falling freely, is constrained by a smooth inclined plane on which it slides, we see that (so long as it moves on the line of greatest slope) its weight ⅛ig has components, M<7sinα tangential to the plane and M7c0sα perpendicular to it, α being the inclination of the plane to the horizon. The latter component produces the normal pressure on the plane, and is the only contributor to it, since there is no curvature. The former produces the acceleration of the motion. Thus the acceleration is now <7sinα only; but, with this change, the results of § 29 still hold.

§ 131. If the plane be rough, with coefficient of statical friction μ, it can furnish (§ 120) a force of friction tending to prevent motion, of any amount up to

μM<7c0sα.

If this be less than Musina, motion will commence, and the force accelerating it will be

Μ y sin α - μ'M<∕C0Sa , where// isthe coefficient of kinetic friction (§121). Thusthe same results as in § 29 still hold, but with g (sιnα - μ cosa) instead of g. As we have seen that μ'<μ, accelerated motion can take place down an inclined plane in certain cases where the mass, if once at rest, would not start.

§ 132. As a slightly more complex case, let us now take again the problem of free motion in a vertical line, but allow for the diminution of gravity as the distance from the earth increases.

The weight of a particle of mass m at the earth’s surface is mg. At a distance x from the centre, it is, therefore,

R 2 where R is the radius of the earth, supposed spherical. This acts downwards, or in the direction opposite to that in which x increases. Hence, equating it, with its proper sign, to the rate of acceleration of momentum, we have for the equation of motion

R 2 mx= ·

Here the right hand member is a function of x only.

Multiply by xdt, and integrate, and we have, leaving out the extraneous factor m (the possibility of doing this showing that the motion is the same for all masses), 1 „ ~ R 2 o

⅞± 2 + C = - .

x

If V be the speed at the earth’s surface (where ae = R),

⅛V 2 + C = Ry.

Also if the particle turns, to come down again, at the height h above the surface, c =⅞⅝.

It ÷ lι

Hence

⅛ v, →(1 -i⅛)-⅛⅛∙

This shows the amount of error in the approximate formula (§ 28) for projectiles,—

⅜V 2 =^.

If the particle be supposed to have been originally at rest, at a practically infinite distance from the earth (a case which may occur with a meteorite for instance), we have x=0 when x=∞, and our formula becomes

⅜x≡=5⅛.

x

The speed with which the mass reaches the surface (where x = R) is therefore √2<∕R, i.e., that which it would acquire by falling, under constant acceleration g, through a height equal to the earth’s radius.

In this special case, the second integral is

∣x l = √¾jRi + C'.

The second integral, in its general form, is a little complex; but we may avoid it by means of a geometrical construction, founded on the results of the investigation of planetary motion soon to be given.

§ 133. Let us next take a case in which the acceleration depends upon the speed of the moving body. A sufficiently simple one is furnished by a falling raindrop, or hailstone, when the resistance of the air is taken into account. For the moderate speeds with which such bodies move, the resistance varies, at least approximately, as the square of the speed. To avoid needless complexity, we neglect here the variation of gravity due to changes of vertical height.

Suppose the particle to have been projected vertically upwards from the origin with the speed V, and let v be its speed at any time t, and x its distance from the origin at that time.

If we assume k to be the speed with which the particle must move so that the retardation due to the resistance may be equal to g, the retardation when the speed is υ will be represented by g^- .

Let the axis of x be drawn vertically upwards; then the resistance acts with gravity, and the equation of motion upwards is l--⅛< i, ÷ r, >-

or ⅛--⅛(* s+,,s >∙

Integrating, and determining the constants so that, when « = 0, t = 0 and v=V, we obtain

<∕i , 1 V x 1 V i 1 *(V - v)

4- = tan ^ 1 -j tan ^ 1 - y - = tan 1 ,., v »

k k k k-+ ∖ v 2gx 1 k' 2 + V 2

^F^ ^ log * 2 + √ 2 ’

Let T be the time at which the speed becomes zero, and ħ the corresponding value of x, then k V k a f V 2 ∖

T=— tan -1 - r -, and h = — log ( 1 + ) ·

g k ’ 2g o ∖ k- )

After this the particle begins to return; the resistance therefore acts against gravity, and the equation of motion is or dv 9 nι 2∖

v~ r = - ⅛(F - υ 2 ).

dx k 2

Integrating, and determining the constants so that, when v — 0, x=h and i = T, we obtain

⅞⅛-T)-l k y ' o k + υ 2g., ∖ ι * 2

(It must be remembered that v is now negative.)

Let U be the speed with which the particle returns to the point of projection; then, putting ae=0 in the latter equation, we obtain

TT2 2⅛γA hL=l- ε --p; k 2 or, substituting for h its value, U 2 V 2 ∕⅛ 2 * 2 - l + v 2 ∕* 2 ’ whence

U 2 V a k 2 '

It is to be observed that k is the “terminal velocity,” as it is called, i.e., the speed to which that of a falling body continually tends, whether its original speed have been greater or less than this limit.

It is to be observed also that (strictly) we should write g (1 - a) for g, where a is the specific gravity of air, to take account of the apparent loss of weight of a raindrop on account of immersion in air.

When k is very large, i.c., the absolute amount of resistance very small, as in the case of air, the general integrals in the second case above become, by expanding the logarithms, k T) k ⅛ i + ' ' ‰-,).⅞ + ≠ + ... k 1 k k i 2k i of which the terms independent of k are v= -p(i-T), υ 2 = 2g(h - x), These, if we remember that t - T is the time of fall, and h - x the space fallen through, are at once recognized as the ordinary formulae of § 28. The modification due to the resistance is shown approximately by the second terms on the right-hand side of the developments above.

The necessity for this double investigation, one part for the ascent, the other for the descent, is due to the non-conservative, or “dissipative,” character of the force of resistance.

§ 134. As an illustration of constraint by a smooth curve, let us take the case of a simple pendulum. Let O (fig. 40) be the point of suspension, P the position of the bob at any time t. Then, if PG represent the weight of the bob, and be resolved into PH, HG respectively along, and perpendicular to, the tangent at P, we see that PH produces the acceleration of the motion, while the tension of the cord balances HG and also furnishes the acceleration perpendicular to the direction of motion which is required to produce the curvature of the path. PH is (caeteris paribus') proportional to the sine of PGH, that is, of POA. Hence the acceleration is proportional to the sine of the angular displacement. When that angle is small it may be used in place of its sine. Hence, for small vibrations, the acceleration is proportional to the displacement, and the motion is “simple harmonic.” The time of vibration, being (§ 51) 2π ∙7 displacement /acceleration, is here

2ιr * ∕ ——— =2ιr * , approximately.

z V øsinø y g

The rigorous solution of the pendulum problem requires the use of elliptic functions.

§ 135. Some very curious properties of pendulum motion are easily proved by geometrical processes. The whole theory of the motion in a vertical plane of a particle attached by a weightless rod to a fixed point, whether it oscillate as a pendulum or perform continuous rotations, may be deduced from the two following propositions, which are easily established by geometrical processes in which corresponding indefinitely small motions are compared.

(1) To compare different cases of continuous rotation. Let DA (fig. 41) be taken equal to the tangent from D to the circle BPC', whose centre C is vertically under D. Let PAQ be any line through A, cutting in Q the semicircle on AC. Also make DE = DA. Then, if P move under gravity with speed due to the level of D, Q moves with speed due to the level of E, the acceleration due to gravity being reduced in the ratio AC 2 : 2BC 2 .

(2) To compare continuous rotation with oscillation. Let two circles touch one another at their lowest points O (fig. 42); compare the arcual motions of points P and p, which are always in the same horizontal line. Draw the horizontal tangent AB. Then, if P move, with speed due to g and level α, continuously in its circle, p oscillates with speed due to level AB and acceleration

αθ 2 ¾ 2 *

§ 136. Two particles are projected from the same point, in the same direction, and with the same speed, but at different instants, in a smooth circular tube of small bore whose plane is vertical, to show that the line joining them constantly touches another circle.

Let the tube be called the circle A, and the horizontal line, to the level of which the speed is due, I.. Let Μ, Μ' (fig. 43) be simultaneous positions of the particles. Suppose that MM' passes into its next position by turning about O, these two lines will intercept two indefinitely small arcs at Μ and Μ' which (by a property of the circle) are in the ratio MO: OM'. Let another circle B be described touching MM' in 0, and such that I. is the radical axis of A and B. Let MP, M'P' be drawn perpendicular to I.. Let M'M cut I. in C. Then, by the property of the radical axis, CO 2 = CM. CM'; from which we have, by geometry,

CM: CM' = 0M 2 : OM' 2 or OM 2 : 0M' 2 = PM: P'M'.

But (speed at M) 2 : (speed at M') 2 = PM: P ’ M'.

Hence the speeds of Μ and Μ' are as MO: OM', and therefore, by what we have stated above about elementary arcs at Μ and M', the proximate position of MM' is also a tangent to B, which proves the proposition.

It is easily seen from this that, if one polygon of a given number of sides can be inscribed in one circle and circumscribed about another, an infinite number can be drawn. For this we have only to suppose a number of particles moving in A with speeds due to a fall from I., and then if they form at any time -the angular points of a polygon whose sides touch B they will continue to do so throughout the motion. Fig. 44 shows two forms of a quadrilateral possessing this property.

§ 137 To find the time of fall from rest down any arc of an inverted cycloid.

Let O (fig. 45) be the point from which the particle commences its motion. Draw OA' parallel to CA, and on BA' describe a semicircle. Let P,P', P" be corresponding points of the curve, the generating circle, and the circle just drawn, and let us compare the speeds of the particle at P and the point P", Let P"T be the tangent at P".

Speed of P" element at P" Speed of P element at P

P"T P"T ∕A'B jVB^ ∕ A'B BP' BP"V AB ^2A'P"V AB '

But speed of P = √(2<∕. A'M) = ^∕jτg . A'P".

Hence speed of P"= . ∕ r‰ . A'B, a constant.

∖∕ 2AB

And, as the length of A'P"B is ⅜τr . A'B, time from A' to B in circle = time from 0 to B in cycloid

∕AB

V ⅛ '

The time of fall to the vertex from all points of the curve is therefore the same. Hence the cycloid is called a “tautochrone.”

§ 138. As an instance of cases in which the acceleration depends upon the speed and the position jointly, take the motion of a simple pendulum in a medium whose resistance varies as the velocity directly. This is the law, at least approximately, for very small speeds, whether the pendulum oscillate in a gas or in a liquid,—and even when the resistance is due to magneto-electric induction, as when the pendulum is a magnetic needle vibrating in presence of a conducting plate or a closed coil. A synthetical solution of this problem has already been given under Kinematics in § 68.

Analytically: if I be the length of the string, θ its deflexion from the vertical at time t, m the mass of the bob, we have evidently mΓθ = - mg sin θ - κ0.

The ratio κ ∣ ml may be increased (theoretically) without limit by-increasing the surface which the bob exposes, without changing its mass. But it cannot be indefinitely diminished. We will write 2k for it. If we assume the angle of oscillation to be small, we may write the equation in the form θ + 2k0 + n 2 θ=0, where n 2 =g ∣ l, and k is essentially positive, being greater as the resistance (whether on account of the viscosity of the medium or the large surface of the bob in proportion to its mass) is greater. A particular integral of this equation is evidently e≈e pf i provided p i + 2kp> + n 2 — 0, or p≈ -k± ∖ ∕k 2 -ri t ,

Hence there are two quite distinct cases of motion, distinguished by different forms of solution, depending on the relative magnitudes of k and n. These are separated from one another by the unique case in which ⅛=n.

(a) Let k > n, and let ⅛ 2 - n 2 = √ 2 < ⅛ 2 .

Then both values of p are real and negative. Thus <,-M r ' ’ + Ee”'

-S- l '(Λ e "' , + Bi- , ∙ ’ ').

If A and B have the same sign, θ diminishes, without changing sign, as t increases. But if A and B have different signs, the factor in brackets may vanish for one definite value of t, and then change sign. After that the whole reaches a maximum and then diminishes without further change of sign. Examples of these cases are furnished—(1) when the pendulum is displaced from the vertical and allowed to fall back; it then approximates asymptotically to its position of equilibrium; and (2) when it is drawn aside and flung back; in this case it may pass once through the position of equilibrium and then asymptotically return to it.

(&) Let n >k and let n 2 -k 2 = n' 2 < n 2 .

Here both values of p are imaginary, and we have e=g-^(Ag+ n '^ r î+Bs- n ' ZA/ “1 ) = Pg - * z cos(n7+Q),

This may be looked upon as a “simple harmonic motion” (§ 52), of which the amplitude diminishes in a geometric ratio with the time, the decrement depending on the resistance alone, while the period is permanently lengthened in the ratio n: n , . This ratio depends both upon the original period and the resistance, so that for the same medium and same bob it is different for strings of different lengths. This investigation gives an approximation to the gradual dying away (by internal friction or by imperfect elasticity, &c. ) of all vibratory movements. The rate of diminution of amplitude, say of torsional vibrations of a wire, is thus a valuable indication and measure of a somewhat recondite physical quantity, which, without this method, would (at present at least) be hard to measure.

(c) When n=k, i.e., in the transition case, the equation becomes θ + 2n∂ + ri 2 θ = 0 ; whose integral is known to be θ =g^ n, (A + Bi)≡g^ i '(A + Bi).

This, also, ultimately diminishes indefinitely as t increases; but, as in case (a), it may either do so continuously or after having once passed through the value zero and reached a maximum, according to the relative magnitude and the signs of A and B.

§ 139. When the path is given, the determination of the motion under given forces is, as we have seen, a mere question of integration of the equation for acceleration along the tangent. But more is required if we wish to find the normal pressure on the constraining curve. This is at once supplied by compounding the resolved part of the applied forces in the direction perpendicular to the tangent, with the additional force m ∖ r2 ∣ p acting from the centre of curvature. But, strictly speaking, all such questions require the application of Law III.

Kinetics of a Particle with Two Degrees of Freedom.

§ 140. The simplest case is that of a projectile, when gravity is supposed to be uniform and to act in parallel lines throughout the whole path, and the resistance of the air is neglected. This has been sufficiently discussed in §§ 40-42. It is merely the combination of ( 1 ) the uniformly accelerated motion of a stone let fall, with (2) a uniform velocity in a definite direction. Looked at from this point of view, it gives an interesting example of the graphic method applied in § 53 to indicate the nature of simple harmonic motion.

§ 141. We can extend these projectile results so as to take account of the alteration of direction and of intensity of gravity at different points of the path, by remembering that, as shown in § 49, the path of an unresisted projectile is an ellipse, one of whose foci is at the centre of the earth. The following, among many other analogous propositions, are easily proved.

(1) The locus of the second foci of the paths of all projectiles leaving a given point with a given speed, in a vertical plane, is a circle.

(2) The direction of projection, for the greatest range on a given line passing through the point of projection, bisects the angle between the vertical and the line.

(3) Any other point on the line which can be reached at all can be reached by two different paths, and the directions of projection for these are equally inclined to the direction which gives the maximum range.

(4) If a projectile meet the line at right angles, the point which it strikes is the vertex of the other path by which it may be reached.

(5) The envelop of all possible paths in a vertical plane is an ellipse, one of whose foci is the centre of the earth, and the other the point of projection.

To prove these propositions, let E (fig. 4G) be the earth’s centre, P the point of projection, A the point which the projectile would reach if fired vertically upwards. With centre E, and radius EA, describe a circle in the common plane of projection. This, the circle of zero velocity, corresponds to the common directrix of the parabolic paths in the ordinary theory If F be the second focus of any path, we must have EP + PF constant, because the axis major depends on the speed, not the direction, of projection. Hence (1) the locus of F is the circle AFO, centre P. Again, since, if F be the focus of the path which meets PR in Q, we must have FQ = QS, it is obvious that the greatest range Vq is to be found by the condition O7 = qs. O is therefore the second focus of this trajectory, and therefore (2) the direction of projection for the greatest range on PR bisects the angle APR. If QF = QF' = QS, F and F' are the second foci of the two paths by which Q may be reached; and, as I. F'PO= I. FPO, we see the truth of (3). If Q be a point reached by the projectile when moving in a direction perpendicular to PR, we must evidently have I. PQF' = I. PQF = I. SQR = I. EQP; i.e., EQ passes through F'. When this is the case, the ellipse whose second focus is F evidently meets PR at right angles; and that whose second focus is F' has (4) its vertex at Q. The locus of q is evidently the envelop of all the trajectories. Now Pq = PO + Oq = PA + Oq, Εq=Εs-sq= EA— Oq .

Hence Pq + Eg = PA + AE, or (5) the envelop is an ellipse, whose foci are E and P, and which passes through A.

§ 142. One of the most important problems in this branch of our subject is that of planetary motion, which forms a good typical example of the processes to be employed in the treatment of central orbits. One or two definitions, and a general property of central orbits, must be premised.

§ 143. Def. An “apse” is a point, in a central orbit, at which the path is perpendicular to the radius-vector along which the central force acts.

The length of the radius-vector is therefore, at such a point, generally a maximum or a minimum. This radius vector, drawn to an apse, is called an “apsidal line.”

A central orbit is symmetrical about every apsidal line. The simplest proof of this theorem depends upon the general principle of “reversibility,” which holds in all conservative systems. In fact if, at any instant, the velocityvector of a particle, moving under the action of a conservative system of forces, be reversed, the particle will simply retrace its previous path. For if we suppose a smooth tube, in the form of the previous path, to be employed to guide it back, the speed at every point will be of the same magnitude as before. The curvature also of the path will be the same; and thus the normal component of the applied forces will balance the so-called centrifugal force,— i.e., will suffice to produce the requisite curvature,—so that there will be no pressure on the tube, and it is not required. Hence since, at an apse, the velocity is perpendicular to the radius-vector, the two halves of the orbit on opposite sides of the apsidal line are similar and equal. Hence, however many apses there may be, there can be. at most only two apsidal distances. For the property of symmetry about each apsidal line shows that, if there be more apses than one, the fir⅝t, third, fifth, Ac., must have their apsidal distances equal, as also must the second, fourth, sixth, Ac. If there be one apse only, it may correspond either to a minimum or to a maximum value of the radius-vector; but, if there be more than one, they must be maxima and minima alternately.

§ 144. We now proceed to the gravitation case already promised. We will take, first, the direct problem as in § 49, where the force is assigned and the orbit is to be found.

A particle is projected from a given point in a given direction and with a given speed, and moves under the action of a central attraction varying inversely as the square of the distance; to determine the orbit.

We have P=μκ 2 , and therefore by the last part of § 47 d 2 w , μ n dθ 2 +u h 2 ~° , where h is double the constant area, or ⅛0 2 ( m ^ ⅛) + ( m- ⅛)°θ , the integral of which is u-~τ 2 = Acos(θ + B), or, as it is usually written, l + ecos(0-α) I (1)∙

This gives by differentiation g--A isill (β ..) (2).

Let R be the distance of the point of projection from the centre, and β the angle, and V the speed, of projection; then, when θ≈0,

1 in fldu ∖ ∞ t *-U≡d,=√

y · ·

Hence, by (1) —- 1 ≈ecosα,

μK

7ι 2 and by (2) - f -cotβ=-csinα.

μK

From these tana≈ (3 )>

μK - h i and c 2=-Δ-cosec 2 3-¾- + l (4),

But 7i 2 = V 2 R 2 sin 2 3,

. . i V s Rsin3cos0 ,γ ∖ wherefore tanct °>-V‰in 2 β^ ’

1 . e ,.⅛(⅛.r) .... (n Now (1) is the general polar equation of a conic section, focus the pole; and, as its nature depends on the value of the excentricity e, given by (4'), we see that if V 2 >2μ∕R , e > 1, and the orbit is an hyperbola; if V 2 = 2μ∕R, e= 1, and the orbit is a parabola; if V 2 < 2μ∕R, e< 1, and the orbit is an ellipse.

But the square of the speed from rest at infinity to distance R, for the law of attraction we are considering, is 2μ∕R, and the above conditions may therefore be expressed more concisely by saying that the orbit will be an hyperbola, a parabola, or an ellipse, according as the speed of projection» is greater than, equal to, or less than, the speed from infinity. Illustrations of this proposition are found in the cases of comets and of meteor swarms.

The speed of a particle moving in a circle is also often taken as the standard of comparison for estimating the velocities of bodies in their orbits. For the gravitation law of attraction the square of the speed in a circle of radius R is μ∕R ; and the above conditions may be expressed in another form by saying that the orbit will be an hyperbola, a parabola, or an ellipse, according as the speed of projection is greater than, equal to, or less than χ∕2 times the speed in a circle at the same distance.

Supposing the orbit to be an ellipse, we shall obtain its major axis and latus rectum most easily by a different process of integrating the differential equation. Multiplying it by ld~ and inte- clθ grating, we obtain

1 ( [ du ∖ i „ ) 1 „ r , τ h ∖∖ dθ) + U i =j" 2 -=C + uu.

But when μ = -^-∙ , v= V; which gives

ħeneβ λ i > I (fi ∖ ' + √> I 1 ∣ v , _ JL +μu . . ( 5 ). Δ ∖ ∖UΛ} J J Δ Δ It

Now to determine the apsidal distances, we must put

S-o; dQ and this gives us the condition 2 2μ 2μ V 2 U h? + MR M~° which is a quadratic equation whose roots are the reciprocals of the two apsidal distances. But if a be the semi-axis major, and e the excentricity, these distances are

α(l -e) and α(l + e).

Hence, as the Coefficient of the second term of (6) is the sum of the roots with their signs changed, we have

1 1 2μ .

a(l -e) α(l + e) K λ ’ 7,2 or α(l-e 2 )=- (7).

Μ

And the third term is the product of the roots, so that

_J_ = 2μ - I 2 · α 2 (l-e 2 ) Λ 2 R A 2 ’

1 9 V 2 or, by (7), -=4"- < 8 >∙

a K μ

Thus -Ly- ’ = A-A,

2 R 2α

and therefore -v 2 = ~ 0- ....... (9).

2 r 2α ’

Equations (7) and (8) give the latus rectum and major axis of the orbit, and show that the major axis is independent of the direction of projection.

Equation (9) gives a useful expression for the speed at any point, and shows that the radius of the circle of zero speed is 2α.

The time of describing any given angle is to be obtained from the formula, r ⅛ = Λ = √{μα(l -e 2 )} , by equation (7).

From this, combined with the polar equation of the ellipse about the focus, we have dt _ r 2 ∕fa ∖∖ - e 2 ) 3 ∖ 1

d0^√{μ ft (l- c *)} V ∖ μ ) '1 + ecos0) 2 ’ measuring the angle from the nearer apse.

Integrating, we find the time of describing about the focus an angle 0 measured from the nearer apse, in the ellipse or hyperbola, expressed as 2/Ä of the sectorial area ASP (figure to § 147), which might have been written down from the condition of uniform moment of momentum.

In the parabola, if d be the apsidal distance, the integral becomes [since e = l, α(l-e) = ⅛ α(l-e 2 ) = 2d] ,

. ∕2tZ 3 ∕ 0 1 . , 0 ∖

< = a ∕ —( tan —- + — tan 3 —- ) .

V Μ ∖ 2 3 2 )

From the result for the ellipse we see that the periodic time is 2ττV a 3 / μ .

In the notation commonly employed for the further development of this most important question we write

T = 2τr∕n, where n, which is called the “mean motion,” is V μ ∣ a' i .

§ 145. By laborious calculation from an immense series of observations of the planets, and of Mars in particular, Kepler was led to enunciate the following as the kinematical laws of the planetary motions about the sun.

I. The planets describe, relatively to the sun, ellipses of which the sun occupies a focus.

II. The radius vector of each planet traces out equal areas in equal times.

III. The squares of the periodic times of any two planets are as the cubes of the major axes of their orbits.

§ 146. We proceed to the inverse problem of § 8 (δ), the determination of the force from the observed motions.

From the second of the above laws we conclude that the planets are retained in their orbits by an attraction tending to the sun. If the radius-vector of a particle moving in a plane describe equal areas in equal times about a point in that plane, the resultant attraction on the particle tends to that point. For the datum is equivalent to the statement that there is no change of moment of momentum about the sun, or that the accelerations all pass through the sun viewed as a point.

From the first law it follows that the law of the intensity of the attraction is that of the inverse square of the distance.

The polar equation of an ellipse referred to its focus is

2 w=-^-(l + ecos0), where I is the latus rectum.

, r d 2 u 2e

Hence _=-- c os0, and therefore the attraction to the focus requisite for the description of the ellipse is (§ 47)

T> j ,2 √. A 2 ^ 2 2

P=Ä u ∖dθ 3 + u r~τ u ∙ Hence, if the orbit be an ellipse described about a centre of attraction at the focus, the law of intensity is that of the inverse square of the distance.

From the third law it follows that the attraction of the sun (supposed fixed) which acts on unit of mass of each of the planets is the same for each planet at the same distance.

For, in the last formula in § 144, T 2 will not vary as α 3 unless μ be constant, i.e., unless the strength of attraction of the sun be the same for all the planets.

We shall find afterwards that for more reasons than one Kepler’s laws are only approximate, but their enunciation was sufficient to enable Newton to propound the doctrine of universal gravitation, viz., that every particle of matter in the τιniverse attracts every other with an attraction whose direction is that of the line joining them, and whose magnitude is as the product of the masses directly and as the square of the distance inversely ; or, according to Maxwell’s formulation, between every pair of particles there is a stress of the nature of a tension, proportional to the product of the masses of the particles divided by the square of their distance.

If we take into account that the sun is not absolutely fixed, then, neglecting the mutual attractions of the planets, Kepler’s third law should be stated thus :—

The cubes of the major axes of the orbits are as the squares of the periodic times and the sums of the masses of the sun and the planet.

§ 147. We will now indicate, as briefly as possible, the more ordinary transformations by which the preceding formulae are adapted (for astronomical applications) to numerical calculation.

Suppose APA' (fig. 47) to be an elliptic orbit described about a centre of attraction in the focus S. Also suppose P to be the position of the particle at any time t. Draw PM perpendicular to the major axis ACA', and produce it to cut the auxiliary circle in the point Q. Let C be the common centre of the curves. Join CQ.

When the moving particle is at A, the nearest point of the orbit to S, it is said to be in “perihelion.”

The angle ASP, or the excess of the particle’s longitude over that of the perihelion, is called the ’ ∙ true anomaly.” Let us denote it by θ.

The angle ACQ is called the “excentric anomaly,” and is generally denoted by w. And, if 2π∕n be the time of a complete revolution, nt is the circular measure of an imaginary angle called the “mean anomaly it would evidently be the true anomaly if the particle’s angular velocity about S were constant.

It is easy from known properties of the ellipse to deduce the following relations between the mean and excentric, and also between the true and excentric, anomalies:—

nt=u -esinw.

’ ∙4- √(⅛) tan 4∙

By far the most important problem is to find the values of θ and r as functions of t, so that the direction and length of a planet’s radius-vector may be determined for any given time. This generally goes by the name of Kepler’s Problem.

Before indicating the systematic development of u, r, and θ in terms of t from our equations, it may be useful to remark that, if e be so small that higher terms than its square may be neglected, we may easily obtain developments correct to the first three terms. Thus u≈nt + csinii, ≈nt + esin(πi +esinn∕) nearly, = nt + esinni +⅛e 2 sin2vιt

Also — = l-ecosw, a

= 1 - ecos(ni ÷esinni),

= 1 - e cos nt + ⅜e 2 ( 1 - cos 2nt).

And r 2 ^ = √j μ α(l-e 2 )}, which may be written

α 2 (l-e 2 ) 2 dθ 2 . 1 2 .ι

——- -vr = πα 2 (l - e 2 ) i ,

(l + ecosβ) 2 dt ' or (1-e 2 )^(l + ecos0)^ 2 ^ =n.

Keeping powers of e lower than the third,

- 2ecos0 + ∣e 2 cos2θ^^∣ = n, or Mi = θ-2esinβ + fe 2 sin20j whence 0 = ni + 2esin0 - ⅜e 2 sin20, = nt + 2csin (nt + 2esin nt~) - f e 2 sin 2nt, = nt + 2e sin nt + 4e 2 cos nt smnt- ⅛e 2 sin 2nt, = nt + 2e sin nt ÷ ∣e 2 sin 2nt. Kepler’s Problem —To find r and θ as functions of t from the equations

τ∙=α(l-ecos·u) (1); θ ∕l + e. « ∕o∖

tan 2-^ √~Λ n T (% ni = u-esinu (3).

These equations evidently give r, θ, and t directly for any assigned value of u, but this is of little value in practice. The method of solution which is commonly adopted is that of Lagrange, and the general principle of it is this :—

We can develop θ from equation (2) in a series ascending by powers of a small quantity, a function of e, the coefficients of these powers involving u and the sines of multiples of u. Now by Lagrange’s theorem we may from equation (3) express u, 1 - ecos2i, sinu, sin2u, &c., in series ascending by powers of e, whose coefficients are sines or cosines of multiples of nt. Hence, by substituting these values in equation (1) and in the development of (2), we have r and θ expressed in series whose terms rapidly decrease, and whose coefficients are sines or cosines of multiples of nt. This is the complete practical solution of the problem. But we must refer the reader to special treatises for the full development of this subject. Compare § 52.

§ 148. We may take an opportunity here of giving a sketch of a particular case of the important question of “kinetic stability.” The general treatment of this subject is entirely beyond our limits. But we may investigate its conditions, in the case of a central orbit naturally circular, by a very slight modification of our equations.

Whatever be the law of central force, provided it depend on the distance alone, we can write the acceleration due to it as

μw 2 ∕(π), where u is the now reciprocal of the radius-vector, as in § 144. The kinematics of the motion is then entirely summed up in the equations + « = p>fW> an <l jr = Ä« 2 . dθ 2 h- j n dt

If l∕α be the radius of the circle, the first equation becomes simply α=⅛∕(g )∙

Now let a slight disturbance be given to the motion, such that lι is unaltered, but*that u becomes a + x. Then we have g÷ g + -⅛Λ g ÷≈)∙

Expanding to first powers of x only, and thereby assuming that X is always exceedingly small, we have g + <ι-⅞∕w)=°.

the terms independent of x vanishing by the condition for a circular orbit. By eliminating the ratio μ∕h 2 we have ^ x . +x (l - dθ 2+x ∖ l f(a))

To secure stability, x must not be capable of increasing indefinitely. This leads to the result that the multiplier of x in the above equation must be positive; i.e. t _affa) /(«)

For, if the multiplier were negative, the value of x would consist of two real exponential terms, one of which would increase indefinitely with the angle θ, and would disappear from the value of x under special conditions only.

If the multiplier were zero, x would be a linear function of θ. Hence, in the only case we need consider, we have sb =Acos(W1-⅞^+b). ∖ J ∖ a ) J

The radius-vector is therefore a maximum and minimum (i.e., apses occur) alternately as the angle θ increases by successive increments each equal to

χ∕ι _ g ∕ ’ < α ) ' fW

Suppose the force to vary as the inverse 7ith power of the distance. Here f(a) aa n ^ 2 , and we have 1 - = 1 - (n - 2) = 3 - n. Thus J ∖ G j ) n must be less than 3; i.e., a circular orbit, with the centre of force in the centre, is essentially unstable if the force vary as the inverse third, or any higher inverse power of the distance.

If n = 2, which is the gravitation case, the apsidal angle is evidently τr.

§ 149. A very curious result, due to Newton, may be indicated here, viz., that, if any central orbit be made to revolve in its own plane with angular velocity proportional at each instant to that of the radius-vector in the fixed orbit, it will still be a central orbit; and the additional force required will be inversely as the cube of the radiusvector.

Generally, in a central orbit, f- r 0 2 ≈P, r 1 0'∙h, But suppose θ to become eθ 1 , where e is a constant, and we have 0=e0 1 , which is Newton’s hypothesis. The above equations become r-r0, = P + (e 2 - l)r0f, Pc0 1 = h', or, as they may be written, r- r 0∣=P + (e 2 -l)^∕r 3 , r 2 0 1 = Λ 1 , -

From these the proposition is obvious.

Other examples of central orbits will be given when we discuss general principles, such as “least action” and “varying action.”

Special Problem. The Brachistochrone.

§ 150. A celebrated problem in the history of dynamics is that of the “curve of swiftest descent,” as it was called: —

Two points being given, which are neither in a vertical nor in a horizontal line, to find the curve joining them down which a particle sliding under gravity, and starting from rest at the higher, will reach the other in the least possible time.

The curve must evidently lie in the vertical plane passing through the points. For suppose it not to lie in that plane, project it orthogonally on the plane, and call corresponding elements of the curve and its projection σ and σ'. Then if a particle slide down the projected curve its speed at σ' will be the same as the speed in the other at σ. But σ is never less than σ', and is generally greater. Hence the time through <r' is generally less than that through σ, and never greater. That is, the whole time of falling through the projected curve is less than that through tħe curve itself. Or the required curve lies in the vertical plane through the points.

Also it is easy to see that, if the time of descent through the entire curve is a minimum, that through any portion of the curve is less than if that portion were changed into any other curve.

And it is obvious that, between any two contiguous equal values of a continuously varying quantity, a maximum or minimum must lie. [This principle, though excessively simple (witness its application to the barometer or thermometer), is of very great power, and often enables us to solve problems of maxima and minima, such as require, in analysis, not merely the processes of the differential calculus but those of the calculus of variations. The present is a good example.]

Let, then, PQ, QR and PQ', Q'R (fig. 48) be two pairs of indefinitely small sides of polygons such that the time of descending through either pair, starting from P with a given speed, may be equal. Let QQ' be horizontal, and indefinitely small compared with PQ and QR. The brachistochrone must lie between these paths, and must possess any property which they possess in common. Hence if v be the speed down PQ (supposed uniform) and v that down QB, drawing <frn, Q'n perpendicular to RQ', PQ, we must have

Qn_Q'm v v'

Now if θ be the inclination of PQ to the horizon, & that of QR, Qn = QQ'cos0, Q'wi = QQ'cos0'. Hence the above equation becomes

COS0_ COSØ' v v'

This is true for any two consecutive elements of the required curve; and therefore throughout the curve V αcos0.

But v 2 ∙ α vertical distance fallen through (§ 28). Hence the curve required is such that the cosine of the angle it makes with the horizontal line through the point of departure varies as the square root of the distance from that line,—which is easily seen to be a property of the cycloid, if we remember that the tangent to that curve is parallel to the corresponding chord of its generating circle. For in fig. 45, § 137,

AP ’ /AM

cosBP'M = cosBAP' = A⅛= ∕≡⅛ 1 oc√AM.

AB ∖ z AB v

The brachistochrone then, under gravity, is an inverted cycloid whose cusp is at the point from which the particle descends.

§ 151. Whatever be the impressed forces, reasoning similar to that in last section would show that the osculating plane of the brachistochrone always contains the resultant force, and that v' cosø= ι>cos0', where θ is now the complement of the angle between the curve and the resultant of the impressed forces.

Let that resultant = F, and let the element PQ = δs, and θ' = θ + 50, Then

√ 8 -Φ 2 =2Fδssin0, or rδr = Fδssin0.

But v<χ cosø; which gives

— = - sin0 δ0 v cos 0

Hence ^ 2 ∕⅛=-Fcos0.

OC7

But in the limit — = p, the radius of absolute curvature at Q; δ0

and Fcosø is the normal component of the impressed force. Hence we obtain the result that, in any brachistochrone, the pressure on the curve is double of that due to the force acting.

§ 152. Now for the unconstrained path from P to R we have feds a minimum (§· 202). Hence in the same way as before, <∕> being the angle corresponding to 0, vcosφ = v'cosφ' from element to element, and therefore throughout the curve, if the direction of the force be constant. Now, if the velocities in the unconstrained and brachistochrone paths be equal at any equipotential surface, they will be equal at every other. Hence, taking the angles for any equipotential surface, cos 0 cos φ = constant.

As an example, suppose a parabola with its vertex upwards to have for directrix the base of an inverted cycloid; these curves evidently satisfy the above condition, the one being the free path, the other the brachistochrone, for gravity, and the velocities being in each due to the same horizontal line. And it is seen at once that the product of the cosines of the angles which they make with any horizontal straight line which cuts both is a constant whose magnitude depends on those of the cycloid and parabola, its value being ∖ Zl ∣ ia, where I is the latus rectum of the parabola, and a the diameter of the generating circle of the cycloid.

Kinetics of a Particle Generally.

§ 153. Here we must content ourselves with a few special cases, which will be varied as much as possible.

A unit particle moves on a smooth curve, under the action of any system of forces ; find the motion.

All we know directly about the pressure R on the curve is that it is perpendicular to the tangent line at any point.

Resolve then the given forces acting upon the particle into three, —one, T, along the tangent, which in all cases in nature will be a function of x, y, z and therefore of s ; another, N, in the line of intersection of the normal and osculating planes (or radius of absolute curvature); and the third, P, perpendicular to the osculating plane.

Let the resolved parts of R in the directions of N and P be R 1 , R 2 . Now the acceleration of a point moving in any manner is com- fp s dv pounded of two accelerations, one — δ or v — along the tangent to Ctb cis υ 2 the path, and the other — towards the centre of absolute curva- p ture, the acceleration perpendicular to the osculating plane being zero; and therefore

⅞=T (i).

This equation, together with the two equations of the curve, is sufficient to determine the motion completely.

,,2

Also — = R 1 + N (2), P

R 1 and N being considered positive when acting towards the centre of absolute curvature; this equation determines R 1 .

Now R 2 is the reaction which prevents P’s withdrawing the particle from the osculating plane; and therefore

¾=-P (3)∙

(2) and (3) give the resolved parts of the pressure on the curve. Also R=√ ( R* + R; ), an d its direction makes an angle=tan - 1 ( R. 2 ∕R 1 ) with the osculating plane.

If the result of the investigation should show that at any time R could vanish, the particle must be treated as free until the equations of its free motion show that it is again in contact with the curve.

A particle moves, under given forces, on a given smooth surface; to determine the motion, and the pressure on the surface.

Let F(x, y, z) = 0 (1), be the equation of the surface, R the reaction, acting in the normal to the surface, which is the only effect of the constraint. Then, if λ, μ, V be its direction cosines, we know that ∕dF ∖ λ ∖ dχ ) (O } ∕ ∖ i d l ∖ 2 ( d I. ∖ 2 ( d EXl ' ’ ’ { ’

∖∕ ( \cfø∕ + ∖ dy J + ∖ dz ) ) with similar expressions for μ and v, the differential coefficients being partial.

If X, Y, Z be the applied forces on unit of mass, our equations of motion are, evidently, x= X + Rλ ) 0=Y + R m ( (3).

'i = Z + Rv )

Multiplying equations (3) respectively by x, ÿ, z, and adding, we obtain ix + ÿÿ+2z = îû?=Xx + Yÿ + Zz ...» (4).

R disappears from this equation, for its coefficient is

λi + μi∕ + vz and vanishes, because the line whose direction cosines are proportional to x, &c. , being the tangent to the path, is perpendicular to the normal to the surface.

If we suppose X, Y, Z to be a conservative system of forces, the integral of (4) will be of the form

⅛v 2 = ≠(ir, y, z) + C (5), and the velocity at any point will depend only on the initial circumstances of projection, and not on the form of the path pursued.

To find R, resolve along the normal, then V“lp = Xλ + Yμ + Zv + R , which gives the reaction of the surface; p being the radius of curvature of the normal section of the surface through the tangent to the path, and the mass of the particle being taken as unity.

To find the curve which the particle describes on the surface.

For this purpose we must eliminate R from equations (3). By this process we obtain X- X ._ ÿ - Y _ ⅜-Z λ μ v two equations, between which if t be eliminated, the result is the differential equation of a second surface intersecting the first in the curve described.

If there be no applied forces, or if the component of the applied force in the tangent plane coincide with the direction of motion of the particle, then the osculating plane of the path of the particle, which contains the resultant of R and the applied force, will be a normal plane, and therefore the path will be a geodesic on the surface.

Thus a particle under no forces on a smooth (or rough) surface will describe a geodesic.

§ 154. An excellent and important example is furnished by the simple pendulum, when its vibrations are not confined to one vertical plane. When the bob moves in a horizontal plane, the arrangement is called a “conical” pendulum, and it is a very simple matter, as follows, to find the motion. For the vertical component of the tension of the string must support the weight of the bob; i.e.,

Tcosα = wι<∕, where a is the inclination of the string to the vertical. Also the horizontal component of the tension must supply the force τnV 2 ∕R (§ 49) requisite for the production of the curvature of the path, i.e.,

V 2

Tsinα = m,τ~-— · ∕sι∏a

Eliminating m/T from these equations, we have cosα _ gl sin 2 a Y 2 ’

But, if τ be the time of revolution of the bob,

Vτ = 2πZsina, ·

Hence cosa

τ 3 4ιr 2 Z 2 ’

√ l cos a; 9 i.e,, the conical pendulum revolves in the period of the small vibrations of a simple pendulum whose length is the vertical component of that of the conical pendulum (§ 134).

To carry the investigation to cases in which the pendulum describes a tortuous cjιrve, we require (except for approximate results) the use of elliptic functions. We thus obtain, among others, the following results :—

The motion will be comprised between two horizontal circles. Let the depths of these circles below the centre be b + c and b - c; then the vertical motion of the bob of the pendulum will be the same as that of a point on a simple pendulum of length l i ∕c performing complete revolutions in the same periodic time as the spherical pendulum.

But for one of the most important applications the deflexion from the vertical is always very small, and it is easy to obtain a sufficiently accurate working approximation without the use of elliptic functions. If we put p and q for the semidiameters of the small elliptic orbit which will then be described by the pendulum bob, we find for the apsidal angle 7 √1 + ⅛2 + A ..

2√ + 8α 2+ ' , , ) ‘

Hence, when a pendulum is slightly disturbed in any way, the motion is to a first approximation elliptic as in § 50. But the second approximation shows that this ellipse rotates in its own plane, and in the same sense as that in which it is described, with an angular velocity proportional to its area. Hence the necessity for extreme care, in making Foucault’s experiment (presently to be described), lest the path should even slightly deviate from a vertical plane.

§ 155. Another very important and useful example is furnished by Blackburn’s pendulum, which is simply a pellet supported by three threads or fine wires knotted together at one point C (fig. 50). The two other ends of two of them are attached to fixed points A and B, and the third supports the pellet P. The motion of P is virtually executed on a smooth surface, whose principal curvatures near the lowest point are 1/CP in the plane of the three threads, and 1/PE in the plane perpendicular to them,—E being the intersection of the vertical through C with the line AB. Hence for small disturbances of this system, P has a simple harmonic motion in the plane of the paper whose period is 2τr √ z CP∕y, and another at right angles to it, with period 2τr √ ’ PE∕y. The amplitudes of these motions are arbitrary, and, with the difference of phase, depend entirely on the initial disturbance. Thus we have a very simple mechanical means of producing the combinations treated in § 63; for we have only to make

PE: PC:: ω 2 : ω' 2 , and give the bob its proper initial motion.

§ 156. When CE is very small compared, with CP, we have a realization of the case of § 61, in which the orbit is (at any instant) an ellipse, but in which the ellipse gradually changes its form and position, so as to be always inscribed in a definite rectangle. This experimental arrangement is exceedingly instructive. To avoid as far as may be the effects of resistance of the air, the vibrations should be slow, i.e., the wires should be as long as possible. The bob should be a ball of lead, containing a tube full of ink which slowly escapes from a fine orifice at its lower end, so as to make a permanent record of the path on a sheet of paper placed below the plane of motion of the bob, but parallel and very close to it. Or, the bob may be furnished with a spike at its lower end, from which induction sparks may be taken so as to pierce a sheet of paper laid on a copper plate below it.

By mere alterations of the point of suspension A, the ratio of ω, ω' may be varied at pleasure, provided that AC and BC are long enough compared with CP.

§ 157. Lissajoux produced similar curves by attaching plane mirrors to the legs of tuning-forks, and allowing a ray of light, after successive reflexions from two such mirrors, to fall on a screen. But it seems to have been first pointed out much earlier by Sang, and afterwards developed by Wheatstone, that the same result is obtained by fixing firmly one end of a steel rod, and setting the free end in vibration. There are two planes of greatest and least flexural rigidity (§ 274) in all wires, however carefully drawn. These are at right angles to one another; and the motion of the free end of the wire when slightly disturbed is therefore precisely that of the bob of the Blackburn pendulum. Another interesting mode of producing the same result is by causing a ray of sunlight to be reflected in succession from four mirrors, all attached, nearly at right angles, to parallel axes. One pair is made to rotate, the two in opposite directions, with one angular velocity. A ray reflected in succession from these is (§ 65) made to oscillate according to the simple harmonic law, in a plane which can be varied at pleasure by altering the relative position of the normals to the two mirrors. The other pair of mirrors supplies the other simple harmonic motion, also in any desired plane.

§ 158. We must next consider the effect of the earth’s rotation upon the motion of a simple pendulum. Strange to say it was left for Foucault to point out, in February 1851, that the plane of vibration of a simple pendulum suspended at either pole would appear to turn through four right angles in twenty-four hours,—the plane, in fact, remaining constant in position while objects beneath the pendulum were carried round by the diurnal rotation. At the equator, it was pretty obvious that no such effect would occur, at least if the original plane of vibration was east and west. By some process, pf which he gives no account, Foucault arrived at the result that the plane of oscillation must, in any latitude, appear to make a complete revolution in 24 a × cosec latitude. This curious result has been amply verified by experiment.

The equations of motion of the pendulum, referred to rectangular axes fixed in direction in space and drawn from the earth’s centre, the polar axis being that of z, are obviously d?~x —x-a„

--T—+ ∞ χ , with similar expressions in y and z {a, b, c being the coordinates of the point of suspension, T the tension, I the length of the string, and X, Y, Z the components of gravity).

The equations of motion referred to a new set of axes parallel to the former, but drawn through the point of suspension, are dfix-a) rr χ ~ a , (V <Pa ∖∖

* n -⅛~ ^- τ -. . . (1).

&c. = &c. )

Let us now refer the motion to axes turning with the earth, but drawn from the point of suspension. If the axis of ξ be drawn vertically, and the axes of η, £ respectively southwards and eastwards; and if u>t be the angle at time t between the planes of xz and ξη, λ being the co-latitude of the point of suspension, we have (assuming that ξ intersects z)

cos xξ = sin λ cos ωt, cosyξ = siuλsin ωt, coszξ = cosλ, &c.

By means of these expressions we can at once find the values of X - a, y-b, z - c in terms of ξ, y, £, t, as follows:

X — α = ξsinλcosωi + ηcosλcosωi - fsinα√, y- b = ξsinλsinα√ + ηcosλsinα√+ ^cosωi, z-c =ξcosλ -ηsinλ.

Let 7 be the acceleration due to the attraction of gravity alone, and V the angle (nearly equal to λ) which its direction makes with the polar axis. [We have above in effect assumed that its direction lies in the plane of zξ, as we have assumed that the axis of ξ intersects the polar axis, while we know that the centrifugal force lies in their common plane.] Let r be the distance of the point of suspension from the earth’s centre, p. the angle its direction makes; with the polar axis. Then

α = rsinμcosω∕, b = rsinμsinωi, c≈=rcosμ.

With these data we transform equations (1) from x, y, z to ξ, y, £ The equations immediately obtained are inconveniently long for our columns. But they are easily simplified as follows.

We contemplate small vibrations only; so we may treat ξ as being practically equal to - I, and omit its differential coefficients. We also omit powers and products of y, ζ, and all terms in α> 2 , except those in which it is multiplied by a large quantity. For it is known that the centrifugal force at the equator is about l∕289th of gravity, or that approximately r w 2 =y∕289.

With these considerations, and the condition that to the degree of approximation desired we have Υ=mg, we still further simplify our equations. We are led to recognize that γcos ∣z = <∕cosλ; and thus we have finally d?y o dζ. g λ ∖ ~rp, — 2ωco⅛'κ- + ~γ y — 0 ) dt i dt I I d 2 ζ n dy q . . ( dt  dt I >

These are the equations of the motion of the bob, referred to a horizontal plane fixed to the earth. The middle terms obviously depend upon the earth’s rotation.

To interpret equations (2) it is convenient to employ a second change of coordinates—to refer the motion to axes revolving uniformly in the plane of η, £ with angular velocity Ω. If p,? be the coordinates referred to the new axes, we have by analytical geometry

η = pcos∩Z - ^sin∩i, £=psin4M + ^cos∩i, the substitution of which in (2) leads to the equations √f + t~ 0 > ^ + τ~ 0 provided we take ∩=-α>cosλ (4), and omit as before terms of the order ω 2 .

(4) shows that the new axes rotate, in the opposite direction to that of the earth, with the component of the earth’s angular velocity about the vertical at the place. And, in the plane so revolving, we see by (3) that the bob of the pendulum describes an approximately elliptic orbit, of which a straight line is a particular case.

A circular path being obviously possible, let us assume as particular integrals of (2)

p = ccos(pi + α), £=csin(pi + a).

The substitution of these values gives the same result p 2 + 2up cos λ - g ∣ l =0

in each of equations (2).

Put g ∣ l=n 2 , then the values of p are, to the degree of approximation above employed, ±n-ωcosλ, so that the (apparent) angular velocity of a conical pendulum is increased or diminished by ωcosλ according as its direction of rotation is negative or positive.

§ 159. The preceding problem is a particular case of the following general one. To find the motion of a particle subjected to the action of given forces and under varying constraint. It would lead us to details incompatible with our limits to enter upon a full discussion of so wide a question, but we give one or two simple and useful cases to snow the commoner forms of procedure.

A particle under any forces, and resting on a smooth horizontal plane, is attached by an inextensible string to a point which moves in a given manner in that plane; to determine the motion of the particle.

Let x, y, X, ÿ be the coordinates, at time t, of the particle and point, a the length of the string, R the tension of the string, and m the mass of the particle.

For the motion of the particle we have d-x „ .,x - Ae « m ≈ nιX - R j dl 2 a I

3

dt-a j with the condition {x - x) 2 + (y - ÿ) 2 = α 2 .

Now x, ÿ are given functions of t. Take from both sides of the • · c!^jc (lι~ ïj · ι

equations (1) the quantities -, wι Tτ3> respectively, and we du du have the equations of relative motion d-(x -x) „ r ,x - x d 2 x V m jj ⅞ =^mX - R r ∣ ι- i - 7 J

dt 2 a dt 2 I f . . . . (∙2).

cP(y-y) xr τ,ν -ÿ d 2 y ∖ m — 7 ,o - - = mV - R-—- - m~ ) dt 2 a dt 2 1

These are precisely the equations we should have had if the point had been fixed, and in addition to the forces X, Y, and R acting on the particle, we had applied, reversed in direction, the accelerations of the point’s motion with the mass as a factor. It is evident that the same theorem will hold in three dimensions. The accelerations , —are known as functions of t, and therefore the dC 2 ’ dt 2 equations of relative motion are completely determined.

Let there be no impressed forces, and suppose first that the point moves with constant velocity in a straight line.

Here — , — are constant, and therefore no terms are intro- dt dt duced in the equations of motion.

Again, suppose the point's motion to be rectilinear, but uniformly accelerated.

The relative motion will evidently be that of a simple pendulum from side to side of the point’s line of motion. In certain cases, when the angular velocity exceeds a certain limit, we shall have the string occasionally untended; and this will give rise to an impact when it is again tended. While the string is untended the particle moves, of course, in a straight line.

Suppose the point to move, with constant angular velocity ω, in a circle whose radius is r and centre origin.

Here, supposing the point to start from the axis of x, x = reosωt, y = rsinωt.

Hence the equations of motion are, since d 2 x 2 . d 2 i) dt 2 , dt 2 d 2 ( x - x) τ , ¾-¾ x

—,- 2 — — - K F- ω i x, dt 2 a d^{y-y) v y-y , rfi . - dt 2 a (x-χ) 2 + (y-y) 2 ≈a 2 .

, vι z .,d 2 (y-y) . ..d ∖ x-i}

Whence {x - x)--‰ 2 (y - y) -^ a - -

=ω 2 { {χ-χ}9-(y-y)^ or, in polar coordinates, for the relative motion, ⅜-( a 2 ^- ∖ = - ω¾rsin(θ - ut), dt ∖ v dt J d 2 (θ -ωt) t ,r .. . n .. or — ’ - = - ω~ — Sin (θ - ωt). dt 2 a

Now θ - ωt is the inclination of the string to the radius passing through the point; call it <£>, and we have rZ 2 √> „ r .

- « 2 —sιn≠, dt 2 a the equation of motion of a simple pendulum whose length is .

The particle therefore moves, with reference to the uniformly revolving radius of the circle described by the point, just as a simple pendulum with reference to the vertical.

A particle moves in a smooth straight tube which revolves with constant angular velocity round a vertical axis to which it is perpendicular; to determine the motion.

Here, referring the particle to polar coordinates in the plane of motion of the tube, we have 0 = constant = «, P = 0 (§47), and thus for the acceleration along the tube r - rω i ≈ 0; whence r = A S ω * + B g ^^ ωt .

Suppose the motion to commence at time ∕ = 0 by the cutting of a string, length a, attaching the particle to the axis. The velocity of the particle at that instant along the tube is zero. Hence at t='θ r=a = A, + B, r=0 = A- B; so that A = B = ⅜α, and r=⅜α(g ωi +6 -ωz ).

In fig. 51 let OM be the initial position of the tube and A that of the particle, and let OL and Q be the tube and particle at t ime t. Th en 0 A = a, arc AP = au>t, OQ = r, and we have arcAP arcAP OQ = ⅛OA(ε oa +g - OA ).

From this we see that OQ and the arc AP are corresponding values of the ordinate and abscissa of a catenary whose parameter is OA. Here the vertical pressure on the tube is equal to the weight of the particle, while the horizontal pressure is

- = ~ = - τnw 2 α(g ωi - g ~ ωt ).

From this equation, combined with the value of r, we easily deduce for the horizontal pressure the value c Lnω 2 ∖ ∕ (r 2 - α 2 ), and it is therefore proportional at any instant to the tangent drawn from Q to the circle APN.

Let the tube be in the form of a circle, turning with constant angular velocity about a vertical diameter. Let AO (fig. 52) be the axis, P the position'of the particle at any time. Let POA = θ denote the particle’s position, and R the pressure on the tube in the direction of OP. tit ι cf 2 cos θ τ χ

We have a— j - 5 - ≈g- RcosØ, dt 2 d' 2 sinθ „ . n - r > · „ a — 3 -, ω-asιn θ = - Rsin θ . dt 2

Eliminating R, ct 2 θ a-~^ - aω 2 Mnθcosθ= - gs> ∖ nθ .... (1).

The position of equilibrium will therefore be given by sinβ = 0j or by 0 = y, where cosy=-^ 2 ·

Integrating (1), ∕√fl ∖ 2 ∖rfΓ∕ = C + 2ω 2 CθSyCθs0 - ω 2 C0S 2 β . . . (2).

Suppose the particle to pass through the lowest point with velocity aa l , we have =a> * ^^ 2" 2cos 7(1 - cosØ) + ω 2 sin 2 0

Î2 I

(1 - cosγ) 2 + - (cos θ - cosγ) 2 , and — can never vanish if ⅛>4cosγ, or ω 2 i >⅛, that is, if the dt ω 2 l a velocity at the lowest point be greater than that due to the level of the highest point.

If ω -≈<⅛7 the particle will oscillate; and, if ^ = 0, when0=α, α dt then (⅛7 ) = ^ c ° s ^ - cosα ) - ω2 (cos2 0 - c°s 2a ) »

= ω 2 (cos0-cosa)^-¾j-cosa - cos0^ , = ω 2 (cos θ - cos a)( 2 cos y - cos a - cos θ); and therefore, if 2cosy — cosa>l, the particle will oscillate through the lowest point.

If 1 >2cosy-cosα> -1, then, putting

2cosy-cosa = cos)3, (z70∖2

— ) = ω 2 (cos0-cosa)(cosj8-cos0), at J and the particle will oscillate on one side of the vertical diameter.

In each of these three cases the complete solution of the problem can be exhibited in terms of elliptic functions. In the last two cases, when the arcs of oscillation are very small, a sufficient solution may easily be obtained by the usual methods of approximation. This is a particularly instructive example.

§ 160. As a final example of constrained motion of a particle, let us find the form of a curve such that a particle will slide down any arc of it, from the origin, in the same time as down the chord of that arc. If OA, OB (fig. 53) be any two chords, it is plain that the difference of the times down these chords must be equal to the time of describing the arc AB. But, if OA make an angle θ with the vertical, the time of descent along it is / 2ÖA

∖ ' gcosθ ,

And the velocity at A is √2yOAcos0, so that the time of describing AB (considered as infinitesimal) is AB∕λ∕2<∕OAcos0.

If we put r for OA, our condition gives at once ds d ∕ 2r _ dθ dθ ∖ gcosθ ∣ ^∕2grcosθ where s is the length of the arc OA. This equation is easily integrated, and the resulting relation is r 2 =α 2 sin20, which belongs to the well-known lemniscate of Bernoulli. From its form we see that the vertical line from which θ is measured is a tangent at 0; so that the motion in arc commences vertically

§ 161. To complete this elementary sketch of the dynamics of a single particle we take an instance or two of “disturbed motion.” The essence of this question is usually that the disturbing forces are, at any instant, small in comparison with the forces producing the motion; so that, during any brief period, the motion is practically the same as if no disturbing cause had been at work. But, in time, the effects of the disturbance may become so great as entirely to change the dimensions and form of the orbit described. The mathematical method which has been devised to meet this question depends upon what has just been said. The character of the path is not, at any particular instant, affected by the disturbance; but its form and dimensions are. Hence, as the first depends upon the form of the equations which represent it, while the latter depend upon the actual and relative magnitudes of the constants involved, we settle, once for all, the form of the equation as if no disturbing cause had acted. But we are thus entitled to assume that the constants which it involves are quantities which vary with the time in consequence of the slight, but persistent, effects of the disturbance. And, as we know that, if at any moment the disturbance were to cease, the motion would forthwith go on for ever in the orbit then being described, we may assume that in the expressions for the components of the velocity no terms occur depending on the rate of alteration of the values of the constants. This, as will be seen below, very much simplifies the mathematical treatment of such questions.

Suppose a cycloidal pendulum, or a simple pendulum vibrating through very small arcs, to be subjected to a simple harmonic disturbance in the direction of its motion. The equation of motion will obviously be of the form θ + ri 2 θ≈ A. cos mt, where ri i = l∕g, as in § 134.

The integral of this equation is

0 = Pcos(ni + Q) cosmi.

m 2 - n 2

We see then that the result is the superposition of a new simple harmonic motion on the natural simple harmonic motion of the undisturbed bob, and that it is altogether independent of the amplitude and phase of the undisturbed motion. So long as the disturbance is very small, this new part of the motion may be neglected, unless m is very nearly equal to n. For in that case the amplitude of the disturbance may become much greater than that of the original motion. When m is equal to n, the integral changes its form, and we have θ = P cos (nt + Q) + A θsin nt. . ∆ib

This shows that, in the special case of a disturbance of the same period as the undisturbed motion, the nature of the motion is entirely changed. Thus, suppose the pendulum to be at rest at its lowest point when the disturbance is applied; then we have merely θ= A—sinni, 2n ’ a simple harmonic motion whose amplitude increases in proportion to the time elapsed since the disturbance commenced.

§ 162. As another illustration, suppose the point of suspension of a simple pendulum to have a simple harmonic motion of small amplitude in a horizontal line.

Here the equations of motion are (to horizontal and vertical axes)

∏ 1 ≈ - £

mx = -1 ——, rτ , V my=mg-Υ-f .

But if we suppose the oscillations to be small, we may write x - ξ = lθ, y=l, where I is the length of the pendulum, and θ the angle it makes with the vertical. Then we have x=lθ + ξ = lθ + Acosmt, suppose, andÿ=0.

Hence mg≈ r Γ, and Z0 + Acoswzi = -gθ, which is precisely the equation of the preceding investigation.

We see from this how to explain the somewhat puzzling phenomenon which we observe when we produce complete rotations of a stone in a sling by a comparatively trifling motion of the hand. All that is necessary is that the hand should have a slight to and fro horizontal motion, in a period nearly equal to that in which the sling and stone would vibrate as a pendulum. This result of particle kinetics is (like that in § 161) of great value in other branches of physics, especially sound, light, and radiant heat.

To illustrate the general principle, let us take the case of one degree of freedom. Then the equation of motion of an unit mass must be of the form

0 = Θ + Θ 1 , where Θ represents the normal force, and Θ 1 the abnormal or disturbing force. Leaving out Θ 1 for the moment, let the integral of 0 = Θ be

β=∕(α, ß, t), in which α and β are two arbitrary constants. We may now suppose a and β to be variable in such a way that the equation shall still be satisfied by this value of θ when the disturbing forces are included. This imposes only one condition on the two independent quantities a and β, so that to determine them completely we must impose a second. This we do, as already explained, by making the expression for the speed independent of the rates of alteration of a and β, and we gain the advantage that our solution will accord at every instant with what would be the actual future motion if the disturbance were suddenly to cease. The speed is 6≈f ’ (a)h+f'(β)β+f'(t).

We therefore assume f(af a +f(β)β = ().

Taking account of this and differentiating again, we have ¼Λ0∙*+⅛Z(<)∙4+∕"ω.

Hence we have, for the determination of a and β, the equations ∕'(α⅛+∕'(3)f = θ, d , da ι d dβ λ dJ^ · dt + dβ^^ · dt ^θ 1 '

These give the values of ⅛ and , and so completely solve the o at at problem.

§ 163. In a somewhat similar way we may treat the effects of a slight disturbance, made once for all, in the motion of a particle describing a definite path under given forces. A single example must suffice.

Thus, we have in an elliptic orbit about the focus, § 144 (9),

_L·,2 = A .

2 r 2a

At the end of the major axis farthest from the focus this becomes V2≈J± 1 ~ 6 .

a 1 ÷e

Now if at this point V be made V + δV, without change of direction, we have the condition that in the new orbit α(l + c) shall have the same value as in the old, since this will still be the apsidal distance.

Hence δ(V 2 ) = δf T- - λ ) , v ' ∖ a Υ+eJ and δ I α(l + e) ∣∙ = 0;

.∙. 2VδV=-- .-⅜- , a 1 +e 0r δc=-2^∕ jy(l-e 2 )JδV, and δα=--^-δc,

1 + e

-√(f⅛>.

which determine the increase of the major axis and the diminution of the exceutricity; and the same method is applicable to more complicated cases.

A very excellent series of examples of the elementary geometrical treatment of disturbed orbits is to be found in Airy’s Gravitation. Third Law Kinetics of Two or More Particles.

§ 164. We have, by means of the first two laws, arrived at a definition and a measure of force, and have found how to compound, and therefore how to resolve, forces, and also how to investigate the conditions of equilibrium or motion of a single particle subjected to given forces. But more is required before we can completely understand the more complex cases of motion, especially those in which we have mutual actions between or amongst two or more bodies,—such as, for instance, tensions or pressures or transference of energy in any form. This is perfectly supplied by the third law, on which Newton comments nearly as follows.

§ 165. If one body presses or draws another, it is pressed or drawn by this other with an equal force in the opposite direction. If any one presses a stone with his finger, his finger is pressed with an equal force in the opposite direction by the stone. A horse, towing a boat on a canal, is dragged backwards by a force equal to that which he impresses on the towing-rope forwards. By whatever amount, and in whatever direction, one body has its “motion” changed by impact upon another, this other body has its “motion” changed by the same amount in the opposite direction; for at each instant during the impact they exerted on each other equal and opposite pressures. When neither of the two bodies has any rotation, whether before or after impact, the changes of velocity which they experience are inversely as their masses. When one body attracts another from a distance, this other attracts it with an equal and opposite force.

§ 166. We shall for the present take for granted that the mutual action between two particles may in every case be imagined as composed of equal and opposite forces in the straight line joining them, two such equal and opposite forces constituting a “stress” between the particles. From this it follows that the sura of the quantities of motion, parallel to any fixed direction, of the particles of any system influencing one another in any possible way, remains unchanged by their mutual action; also that the sum of the moments of momentum of all the particles round any line in a fixed direction in space, and passing through any point moving uniformly in a straight line in any direction, remains constant. From the first of these propositions we infer that the centre of mass of any system of mutually influencing particles, if in motion, continues moving uniformly in a straight line, except in so far as the direction or speed of its motion is changed by stresses between the particles and some other matter not belonging to the system; also that the centre of mass of any system of particles moves just as all their matter, if concentrated in a point, would move under the influence of forces equal and parallel to the forces really acting on its different parts. From the second we infer that the axis of resultant rotation through the centre of mass of any system of particles, or through any point either at rest or moving uniformly in a straight line, remains unchanged jn direction, and the sum of moments of momentum round it remains constant, if the system experiences no force from without, or only »forces whose resultant passes through the centre of inertia of the system. This principle is sometimes called “conservation of areas,” a very misleading designation.

§ 167. Newton’s scholium, which we treat as a fourth law, points out that resistances against acceleration are to be reckoned as reactions equal and opposite to the actions by which the acceleration is produced. Thus, if we consider any one material point of a system, its reaction against acceleration must be equal and opposite to the resultant of the forces which that point experiences, whether by the actions of other parts of the system upon it, or by the influence of matter not belonging to the system. In other words, it must be in equilibrium with these forces. Hence Newton’s view amounts to this, that all the forces of the system, with the reactions against acceleration of the material points composing it, form groups of equilibrating systems for these points considered individually. Hence, by the principle of superposition of forces in equilibrium, all the forces acting on points of the system form, with the reactions against acceleration, an equilibrating set of forces on the whole system. This is the celebrated principle first explicitly stated and very usefully applied by D’Alembert in 1742, and still known by his name.

§ 168. Thus Newton lays, in an admirably distinct and compact manner, the foundations of the abstract theory of “energy,” which recent experimental discovery has raised to the position of the grandest of known physical laws. He points out, however, only its application to mechanics. The actio agentis, as he defines it, which is evidently equivalent to the product of the effective component of the force into the velocity of the point at which it acts, is simply, in modern English phraseology, the rate at which the agent works, called the “power” of the agent. The subject for measurement here is precisely the same as that for which Watt, a hundred years later, introduced the practical unit of a “horse-power,” or the rate at which an agent works when overcoming 33,000 times the weight of a pound through the distance of a foot in a minute,—that is, producing 550 foot-pounds of work per second. The unit, however, which is most generally convenient is that which Newton’s definition implies, namely, the rate of doing work in which the unit of work or energy is produced in the unit of time.

§ 169 Looking at Newton’s words in this light, we see that they may be converted into the following :—

“Work done on any system of bodies (in Newton’s Statement, the parts of any machine) has its equivalent in work done against friction, molecular forces, or gravity, if there be no acceleration; but if there be acceleration, part of the work is expended in overcoming the resistance to acceleration, and the additional kinetic energy developed is equivalent to the work so spent.”

When part of the work is done against molecular forces, as in bending a spring, or against gravity, as in raising a weight, the recoil of the spring and the fall of the weight are capable, at any future time, of reproducing the work originally expended. But in Newton’s day, and long afterwards, it was supposed that work was absolutely lost by friction.

§ 170. If a system of bodies, given either at rest or in motion, be influenced by no forces from without, the sum of the kinetic energies of all its parts is augmented iu any time by an amount equal to the whole work done in that time by the stresses which we may imagine as taking place between its points. When the lines in which these stresses act remain all unchanged in length, the sum of the kinetic energies of the whole system remains constant. If, on the other hand, one of these lines varies in length during the motion, the stress in that line will do work or will consume work, according as the distance varies with or against it.

§ 171. Experiment has shown that the mutual actions between the parts of any system of natural bodies always perform, or always consume, the same amount of work during any motion whatever, by which the system can pass from one particular configuration to another; so that each configuration corresponds to a definite amount of kinetic energy. Hence no arrangement is possible in which a gain of kinetic energy can be obtained when the system is restored to its initial configuration. In other words, “the perpetual motion” is impossible.

The “potential energy” (§ 113) of such a system, in the configuration which it has at any instant, is the amount of work that its mutual forces perform during the passage of the system from any one chosen configuration to the configuration at the time referred to. It is generally convenient so to fix the particular configuration chosen for the zero of reckoning of potential energy that the potential energy in every other configuration practically considered shall be positive.

As particular instances of this we may notice many of the results already given: for instance, the ordinary expression for the velocity acquired by a falling stone (§ 28), ⅜v 2 = gx ; for here ^mv 2 is the kinetic energy acquired, while mg.x is the work done by the weight (m<∕) during the fall. Similarly, we have in the motion of a planet, the ex-

9 ( 2 1 ∖ . . , . , v l -υ ∖ mμ . pression v i = ∕d I, which leads to m - = — (r 1 - r).

Here — is the “mean value” of the force for distances from rr 1 r to r v and therefore the right-hand side is the work done by the force, while the left-hand side is the increase of kinetic energy produced.

To put this in an analytical form, we have merely to notice that, by what has just been said, the value of 2 ∕^∕χ^ + γ⅛ +z JV

J ∖ as as ds J is independent of the paths pursued from the initial to the final positions, and therefore that

S(Xdx + Υdy + Zdz) is a complete differential. If, in accordance with what has just been said, this be called -dV, V is the potential energy, and y dV 1 dx 1 ’

Also, by the second law of motion, if m 1 be the mass of a particle of the system whose coordinates are x 1 , y 1 , z 1 we have to .⅛ = X &c. = &c.

* dt i 1 ’ and

’{*•@549+»$) I *-*M"+Y⅜+W--Λ

The integral is

⅛S(≡ 2 ) + V = H, that is, the sum of the kinetic and potential energies is constant. This is called the “conservation of energy

In abstract dynamics, with which alone this article is concerned, there is loss of energy by friction, impact, &c. This we simply leave as loss, to be accounted for by Thermodynamics.

§ 172. Hitherto, as we have been dealing with the motion of a single particle only, we have not required the assistance of even the third law. For, in those cases, already treated, in which one of the forces was not given, it was at all events due to a given constraint, and the geometrical circumstances of the constraint supplied the means of determining it. In fact we were not, in any case, concerned with reaction; or, to use the more modern form of expression, we were engaged with one half, only, of a stress. When a stone’s motion was investigated, no account was taken of the stone’s attraction for the earth ∖ when we dealt with central forces, the centre was supposed to be fixed; and, even in the cases in which variable constraint was supposed, the curve which produced it was assumed to move in a manner absolutely determined beforehand, and in no way affected by the reaction of the mass acted upon.

But, in nature, circumstances are not so simple. Though, for all practical purposes, we may calculate the motion of an ordinary projectile as if its attraction had no influence upon the motion of the earth, we cannot do so in the case of the motion of the moon about the earth. The mass of the moon is about ∙ g ⅛th of that of the earth, and its gravitation effects on the motion of the earth cannot be neglected. The moon, in fact, moves faster round the earth than would a projectile of less mass, though moving in precisely the same relative orbit (§ 146). If the earth’s motion were not accelerated by the reaction of the moon, the sole crest of the lunar tide-wave would be on the side of the earth next the moon, and there would be full-tide once only in a single rotation of the earth about its axis. We need not give further instances here; they will present themselves in almost every case we investigate.

§ 173. To give a general notion of the applications of, and necessity for, the third law, we choose a few special cases, selected so as to give, in short compass, a sufficiently general glance at the whole subject.

We take, first, the case of two stones or bullets connected by an inextensible string passing over a smooth pulley. Let their masses be m and m'. Our physical condition is that the tension of the string, whatever be its value, is the same throughout; and this is accompanied by the geometrical condition that the length of the string is constant, or that the speeds of the two masses are equal but in opposite directions. Hence the amounts of increase of momentum in a given time are as the masses. But they are also as the forces, by the second law. Thus m: m' :: T - mg: m’g - T.

This gives, at once, τ 2 mm' . m + m' ’ so that the whole downward force on m' is , ,m' - m mg-Γ = m-. g, i, m + nd ’ and the whole upward force on m is m ... m' - m I - mg = m —- g . m + m

The motion of the system is therefore of precisely the same character as that of a free mass falling in a vertical line, but the acceleration is less, in the ratio of the difference of the two masses to their sum.

§ 174. This is the essence of the arrangement called Atwood’s Machine, which used to be employed for the demonstration (in a rough way) of the first and second laws of motion, in certain simple cases. The main feature of the method is the artificial reduction of the acceleration, so that the motion of the falling body is rendered slow enough to be followed by the eye with some degree of accuracy. To prove the first law, a bar of metal was laid across one of two equal masses suspended as in the example; and the system was allowed to move under acceleration until the preponderating mass passed through a ring which arrested the bar. The subsequent motion, with no acceleration, was then observed by noting the passage of the falling mass in front of a vertical scale, while the observer also listened to the ticking of a pendulum escapement. For the verification of the second law, so far as uniform force is concerned, the apparatus was adjusted by trial so that the extra load was detached from the preponderating mass after 1, 2, 3, &c., beats of the pendulum; and the subsequent uniform speed was found to be nearly in proportion to these numbers. And, again, to prove that momentum acquired is, caeteris paribus, proportional to the force, the effects of bars of different masses were compared by the same process.

If x and l-x be the portions of the string on opposite sides of the pulley at time t, we have d?x m m— - = mg - I = mi, αi w 7m'∣I.(Z -x) = m'g - T = - m'x.

Hence by elimination of T we have m - m' g —χ , m + m and by elimination of x rτ , 2mm ’ , r T = ,g, as before.

m + 771

When one of the masses is vibrating pendulum-wise, the problem assumes a very much more difficult aspect. We will take it later as an example of the application of Lagrange’s general method.

§ 175. Let us now suppose these masses, so connected, to be thrown like a chain-shot. We see by § 166 that their centre of inertia moves as if the masses were concentrated there. Also that the moment of momentum is unaffected. Hence we have only to find the initial position and motion of the centre of inertia, and the plane and amount of the initial moment of momentum; and the complete determination of the motion follows. This case is precisely the same as that of a well-thrown quoit, the rotation of which is about its axis of symmetry. It is, so far as § 166 goes, the case of an ill-thrown quoit, which appears to wabble about in an irregular manner. But these are matters properly to be treated under Kinetics of a Rigid System.

§ 176. Suppose, next, two masses m l and τn 2 to be connected together by an elastic string, the extension of the string being proportional to the tension. Let m 1 be held in the hand, while m. 2 hangs at rest. Then let the system be allowed to fall. What is the nature of the motion? Without mathematical investigation it is easy to see that, the moment the masses are left free to fall, the tension of the stretched string will gradually draw them together. When it has thus contracted to its normal length, I, the relative speed of the two masses will have a definite value. This will continue to be the relative speed until they have passed one another and again arrived at a mutual distance I. At that instant the tension of the string comes into play again; the relative speed becomes less and less, finally vanishing when the distance between the masses is what it was at starting. Then the relative speed becomes again one of approach, increasing steadily till the distance between the masses is I. This maximum speed of approach continues till, after again passing one another, the particles once more reach the relative distance I. And so on. All this time, however, their common centre of inertia has been steadily falling with uniformly accelerated speed, as if the masses had been concentrated at it into one. Since I is the unstretched length of the string, if we call E its modulus of elasticity, its tension at any other length, λ, is

T=E λ ~ z by Hooke’s law. Hence, if initially m 1 were at the origin, and the axis of x be taken vertically downwards, we have for the initial coordinate of ττz 2

<⅛⅛-(ψ + ι)∣.

When the masses are moving, the third law informs us that the tension of the string acts equally and in opposite directions on them. Thus the equations of motion are m 1 x 1 = m 1 g + T, m 2 H 2 = m 2 g-Υ.

By eliminating T we have at once m 1 X + m 2 x 2 = (m 1 + m 2 )g.

But m 1 x 1 + m 2 x 2 = (m 1 + m 2 )ξ , if ξ be the coordinate of the centre of inertia of the two masses. Hence ξ≈g> the ordinary equation for the fall of a stone. Thus m 1 x 1 + τn 2 ¾ = A + Bi + ⅛(τn 1 + m 2 )gt 2 .

Since αq=0 , ½ 1 = 0, x 2 = ( ¾- + 1 ρ, ½ 2 = 0, when i = 0, we have ∖ ∕

A=τn 2 (⅛ + 1 y, B = 0, and thus m 1 x 1 + m 2 x 2 ^m. 2 ^ 7 ~ + 1 K + ⅛(τn 1 + τn 2 )^ 2 .

So long as x 2 - x 1 > I we have also

-1 y

Hence, multiplying the first of the equations of motion by m 2 , and the second by τn 1 , and taking the différence, we have

τn 1 τn 2 (¾ - ⅛ι) = - (τra 1 + m 2 )E^- 2 --~ X1 - 1 ) .

The integral is

α ’ 2 - x 1 = I + P cos(ni + Q), , 9 7‰+777 2i -, where n 2 =∙- i i E.

iτn 1 7n 2

Also, by the data at starting, we have _ λ -r. ι m i ,ql

Q = 0, P≈-⅛ l ∙

Hence, finally, ae,=—-—+ 1 V + ⅛(∞ι + ∞ 2 )^ 2 - w√--=^cos7ii∖

1 7n 1 + 777 2 ∖ 8 ∖ E ∕ E J = + v∕ —; ( 1 - cos nt), 2j E(77l 1 + 7ft 2 )

whence the value of x 2 can easily be found.

As soon as we have nt > ⅛τr these values cease to represent the coordinates of the two masses, because they are deduced from equations involving constraint which, in the case supposed, has ceased for a time.

At the instant nt = ⅛τr the relative speed of the masses is _ m 2 gl ∕ (77t 1 + 77i 2 )E i

E 'v m l ιn 2 l and their distance I.

This distance diminishes thenceforward with the above speed until the uppermost stone, having passed the lower one, falls below it to a distance I. We must, in order to trace the next part of the motion, reapply the differential equations above,—integrating them, and determining the Constants by the new conditions. This we leave to the reader.

§ 177. Next let us take the case of a Complex Pendulum,— the motion of two or more pellets attached, at different points, to the same thread, supported at one end. The general solution of this question presents considerable difficulties, but if we confine our attention to slight disturbances it is easily treated by very elementary processes. In fact, just as a simple pendulum, slightly disturbed in a vertical pl⅛ne, has simple harmonic motion which may be regarded as the resolved part of conical pendulum motion, so we may treat of a complex conical pendulum, and resolve its motions parallel to any vertical plane.

If there be but two masses attached to the string·, it is clear that they must, if the motion is to be a persistent conical one, be always in one vertical plane with the point of suspension. And there are obviously two dispositions of the string which are consistent with kinetic stability.

Let A (fig. 54) be the point of suspension, then the masses may rotate steadily in either of the two configurations sketched. To keep either mass moving in a horizontal circle, all that is required is that the resultant force on it shall be horizontal, directed towards the centre, and producing an acceleration equal to V 2 ∕R, as in § 34. Let the whole system turn with angular velocity ω, and let the lengths of the strings be a and δ, their directions making angles 0, <∕> with the vertical.

We will treat only the case in which these angles are so small that the arcs may be written in place of their sines. Then m requires a horizontal resultant force ma()of directed towards the axis, and Μ requires M(α0 + δ<∕>)ω 2 similarly directed. Also, as the strings are both very nearly vertical, the tension of the lower string may be taken as the weight of Μ, and that of the upper as the sum of the weights of Μ and m. Treating it, then, as a statical problem, we have for the mass m maθω i = (Μ + m)gθ - ⅛lgφ; and for Μ

Μ(a# + Z>0)ω s = + Mgφ.

These formulae correspond to the first configuration, but a change of sign of φ adapts them to the second.

These two equations involve three unknown quantities ω 2 , θ, φ. But the ratio, only, of θ and φ is involved, so that two equations are sufficient. [We have confined ourselves to small values of θ and φ, but have not assigned any limit to their smallness; so that their ratio has still an infinite range of values.]

Eliminate the ratio φ ∣ θ between the two equations; and we have, putting ψ for g∕ω 2 , Maψ=(ψ - δ)((M + m)ψ -ma), or (ψ-a)(ψ-δ) = -~ab.

M + wι

It is clear that, because the right-hand member is essentially less than ab, there are always two real values of ψ, both positive, but one greater than the greater of a, b, the other less than the lesser. These correspond to two values of φ ∣ θ, one positive, the other negative.

§ 178. The most general motion, then, of the double complex pendulum, when it vibrates in one plane, consists (for each of the masses) of the resultant of two simple harmonic motions, whose periods are

2ιr∕ω - t λπ*dφ ∣ g, ψ having one or other of the two positive values given by the equation above, and being therefore the length of the equivalent simple pendulum. Thus the double complex pendulum supplies at once the mechanical means of tracing (by ink, sand, electric sparks, &c., § 156) a graphical representation of the composition of two simple harmonic motions, of different periods, in one line.

Analytically thus. For any displacement in one plane we have, θ and φ being, as before, the deflexions and T, T' the tensions of the strings, ∕ cl ∖ 2 mal — ) sιn0= - TsinØ + T'sinø, ∖ dt ∕ ∕ y ∖2

mal — 1 cos θ=m g - T cos θ + T' cos φ,

∕ √ ∖2

Mi — ) (αsind + ⅛in<∕>)= -T'sin≠

∖ (ht J f cl ∖ 2 M( — ) (acosθ + δcos∂) = Mσ-T'cos<∕>, ∖ dt j four equations to determine θ, φ, T, and T'. They become much more manageable if we assume that θ and φ are so small that their squares may be neglected. Foι ’ then we have sin0 = 0, cos0-=l, &c., and the equations become maθ≈ -Ύθ + Τφ, O = my-T + T', ⅛i(aθ + bφ)= - T'<∕>, 0 = My-T'.

Thus T'=My, T=(Μ + m)g , and we have mat) = - (Μ + m)gθ + My<∕>, aθ + bφ = - gφ .

Introducing an arbitrary multiplier λ, we have f n ∖2 ( ι < )

( —— I > (wι + λ) aθ + κbφ > = — o , l(M + m) θ + (λ — Μ) ό !. ∖ at ] ( ) . ( )

If we choose λ so that κb λ - Μ

(wr + λ)α M + m~ e, ' ’ the equation can be put in the form

('4γ(s +i ≠).-β+≡>(9+ert .

∖ dt∕ (m + λ)a x j

Now (1) is a quadratic equation in λ, and has obviously real roots, a positive root greater than Μ, and a negative root numerically less than m·. Write (1) as the equation of an hyperbola, in the form λb λ - Μ

(m + λ,)a M + wι ’

and we see that λ + m = 0 is an asymptote. The branch on the positive side of this asymptote lies mainly below the axis of λ. But μ is positive for λ=M, and also for λ = 0.

Hence μ must pass through the value zero while λ is greater than Μ, and for another value of λ between zero and -m. But it is obvious that, for each of these values of λ, m + K is positive. Hence the equation may be written (⅛) ( 6, + c ≠)= ~n ∖ θ + cφ), where e and n have two sets of real values given as above; and thus we have the complete solution, with the four requisite arbitrary constants, in the form θ + c i φ = P 1 cos (ra 1 i + Q 1 ), θ + e 2 φ = P 2 cos (n 2 t + Q 2 ) .

This applies to every possible set of values of a, b, m, Μ; for, as we have seen, the two values of λ are essentially different, at least so long as neither of the masses becomes zero. Thus, in this particular case, we are not met by the difficulty of equal roots. But it is very interesting to contrast this case, when m is much greater than Μ, and a=b, with the case discussed in § 162 where the point of suspension of a simple pendulum has a horizontal simple harmonic motion of the period of the pendulum and in the vertical plane in which it vibrates. There the oscillations increase indefinitely; here they are in all cases essentially finite, in accordance with the assumptions made. There is, in fact, no increase of the energy of the system.

A very slight modification of the process gives us the result of small displacements not in one plane.

Kinetics of a System of Free Particles. g 179. A system of free particles is subject only to their mutual attractions; to investigate the motion of the system.

Let, at time t, x n , y n , 2» be the coordinates of the particle whose mass is m n , and let ≠'(D) be the law of attraction. Let p r q express the distance between the particles m p and m q ; then we have, for the motion of m y , [table]

with similar equations for each of the others, the summations being taken throughout the system. Before we can make any attempt at a solution of these equations, we must know their number, and the laws of attraction between the several pairs of particles. But some general theorems, independent of these data, may easily be obtained.

First, we have Conservation of Momentum. In the expression for m p '^-' p , we have a term m p m q φ'( p r q ') a ⅛ — ⅛ , dt 2 p r q and in m q c LfS we have m q m p φ' { q r p } xp ~ Xq , dt l q r p

Hence, if we add all the equations of the form (1) together, the result will be rf⅛ 1 d-χ i , n m ∙^ +m ^ + · · · · - 0; d 2 x ∖ if⅛-, z x n " 2 (∕⅛j-⅛ r ,

Similarly ^≡(wι)-0, ami fs{m) = 0, ((o (CL· where (§ 109) x, y, z is the centre of inertia of the system.

These equations show that the speed of the centre of inertia parallel to each of the coordinate axes remains invariable during the motion; that is, that the centre of inertia of the system remains at rest, or moves with constant speed in a straight line.

Next we have Conservation of Moment of Momentum. For if we multiply in succession equation (1) by y 1 , and equation (2) by x 1 , and subtract, and take the sum of all such remainders through the system of equations of the forms (1) and (2), we have 2[m(xÿ-yx)]=0 .

Integrating once, we have '½ ∖ m{xy - yi)] = 2A 3 , where the left-hand member is the moment of momentum of the system about the axis of z.

This equation shows (since xy is any plane) that generally in the motion of a free system of particles, subject only to their mutual attractions, the moment of momenttιm about every axis remains constant.

Finally, we have Conservation of Energy. Multiply (1) by ⅛- , (2) by i - y -- t (3) by -⅛-j and, treating similarly all the other dt dt dt equations, add them all together.

Let us consider the result as regards the term on the right-hand side involving the product m p m q .

Written at length it is m p m q φf p r q } j dx p ^ ( ajp - a , q )⅛ ρϊ* q ∖ (It (Lt

+ similar terms in y and z∣; and the portion in brackets is equal to

(α⅛ - a⅛)-(a? 4 - x p } + similar terms in y, z ! ; dr ∖ or - pT q - ∖ ^ p r q } ; l _ ( (dx d 2 x dy d 2 y dz cPz ∖ ) hence 2 » 37 τπτ + 17 m + n7 ( ( ∖ dt dt i dt dt i dt dt 1 J ) S et ) m p m q φX p r q )- (p r,) =0; therefore, on integration, ⅛2(pιv 2 ) + ⅛{m p m q φ( p r q ')} =H.

We see therefore that the change in the kinetic energy of the system in any time depends only on the relative distances of the particles at the beginning and end of that time.

Another general expression for the kinetic energy of a system of particles, in terms of a function of the mutual forces, and the constraining forces if there be such, is readily found as follows.

If X. y, z be the coordinates, at time t, of the particle ?zi, we have ∕ d ∖2

(di ) 2wι (χ2 + 2∕ 2 + ≡ 2 )= 22wι(x 2 + if + s 2 ) + 22m(αx + yj) + zz).

But if X, Y, Z be the components of the forces (of whatever kind) acting on m, we have (§ 119)

τnx=X, mj)=Y, mz = Z.

Thus ^nι(x a +y a +z 2 )=22m(x a +g 2 +z 2 ) + 2S(Xx+Yy+Zz) .

This expression was originally devised by Clausius for application to the kinetic theory of gases. The quantity 2m(x 2 + y 2 + z 2 ) is obviously half the sum of the three principal moments of inertia of the group of particles about the origin (§ 234).

In all cases of motion of a group, in which this sum is either constant or oscillates in an extremely short period about a constant value, the left-hand side may be regarded as (on the average at least) a vanishing quantity. Thus an equivalent of the kinetic energy is expressible as -⅛2(Xx + Yy + Zz).

This expression is called the “virial.”

In so far as it arises from the mutual action between two particles m p and m q , its value is (in the notation above)

- √ m p m q φ'{ p r q ) 3 ^Hχ p + m q m p φ ∖ p r q ) 3 ^S x ∖ ∖ „ p r <ι P r t J with corresponding terms in y and z, altogether

=⅜w p m q φ ∖ p r q } p r q .

Hence if we write, generally, r for the distance between two of the particles, and R for the stress between them as depending on their mutual action, the corresponding part of the virial is

⅜2(Rr).

This is positive when the stresses are of the nature of tension. When the mutual action is due to gravity only, .,r X 1

≠ (f> r v) ∑Γ > p' «

and the part of the virial corresponding to this is

⅜7∕ι p m q ∣ p r q , expressing half the exhaustion of the potential energy of the system.

When the particles are in very great numbers, and enclosed in a vessel from the sides of which they rebound—as is supposed in the kinetic gas theory—the pressure p, per unit of surface, on the walls of the vessel must be taken into account. If I, m, n be the direction cosines of the normal to the element dS of the wall of the vessel whose coordinates are x, y, z, the corresponding part of the virial is ⅛pffilSflr + my + nz)

extended over the whole internal surface. We here assume that p is constant. But lx+my+nz is the perpendicular from the origin on the plane of dS, so that the integral expresses three times the volume V of the vessel. Hence this part of the virial is

⅛V.

Thus, in the case of a gas not acted on by external forces, the kinetic energy is

⅛V + ⅜S(Rr).

Impact of Smooth Spheres.

§ 180. There remains to be treated, so far as particle dynamics is concerned, the self-contained subject of Impact. In connexion with it we must once more refer to the second and third of Newton’s laws. We are now dealing with forces which produce, in finite masses, finite changes of momentum in excessively short periods of time. It is clear from this statement that their effects may be treated altogether independently of finite forces, which may be acting along with them, but which produce during the very short periods in question only infinitesimal results. And, as in general we have no knowledge of the actual force exerted at any instant during the impact, nor of the time during which the action lasts, we confine ourselves to the quantity, called the “impulse,” which measures the amount of momentum lost by one of the impinging bodies and acquired by the other.

§ 181. When two balls of glass or ivory impingeon one another, the portions of the surfaces immediately in contact are disfigured and compressed until the molecular reactions thus called into play are sufficient to resist further distortion and compression. At this instant it is evident that the points in contact are moving with the same velocity. But as solids in general possess a certain degree of elasticity both of form and of volume, the balls tend to recover their spherical form, and an additional impulse is generated. This is proportional, as Newton found by experiment, to that exerted during the compression, provided neither of the bodies is permanently distorted. The coefficient of proportionality is a quantity determinable by experiment, and may be conveniently termed the “coefficient of resti tution.” It is always less than unity.

§ 182. The method of treating questions involving actions of this nature will be best explained by taking as an example the case of direct impact of one spherical ball on another. It is evident that in the case of direct impact of smooth or non-rotating spheres we may consider them as mere particles, since everything is symmetrical about the line joining their centres. If the impinging masses are of large dimensions, of the size of the earth, for instance, we cannot treat the effects of the impact independently of the other forces involved; for the duration of collision in such a case may be one of hours instead of fractions of a second.

§ 183. Suppose that a sphere of mass Μ, moving with a speed v l overtakes and impinges on another of mass M', moving in the same straight line with speed v', and that, at the instant when the mutual compression is completed, the spheres are moving with a common speed V. Let R bé the impulse during the compression, then

M(t>-V) = M'(V-√) = Rj γ - ιr Mv + MV , τ , MM' , ,, , 1 .

whence V- , and R-j^O-∙> · · (»·

From these results we see that the whole momentum after impact is the same as before, and that the common speed is that of the centre of inertia before impact. The quantity V can vanish only if

Mv+MV=0 , that is, if the momenta were originally equal and opposite.

This is the complete solution of the problem if the balls be inelastic, or have no tendency to recover their original form after compression.

§ 184. If the balls be elastic, there will be generated, by their tendency to recover their original forms, an additional impulse proportional to R.

Let e be the coefficient of restitution, and v v v 1 ' the speeds of the balls when finally separated. Then, as before,

M(V→∙ 1 ) = e R, M'(v 1 '-V) = eR; whence Mr 1 = Μ 11 ⅛2L1L _ e (υ _ √)

1 M + M' M + M a ' , 1 (Μ - eM')r + M'(l +e)υ' Μ' , 1 l v ,.

and *' rar—--'-m+m-< 1+ ' x,, -' ’ > ; with a similar expression for v l '.

These results may be more easily obtained by the simple consideration that the whole impulse is (1 +e)R; for this gives at once M(w - v 1 ) = M'(v 1 ' - √) = ( 1 + e)R.

If Μ' be infinite, and √ = 0, we have the result of direct impact on a fixed surface, viz., v - v 1 = (1 + e ∖ v or v l = - ev. The ball rebounds from the fixed surface with a speed e times that with which it impinged.

§ 185. Suppose, now, Μ = M', e= 1; that is, let the balls be of equal mass, and their coefficient of restitution unity (or, in the usual but most misleading phraseology, suppose the balls to be “perfectly elastic”); then 2R = M(v - v'); v 1 = v, and similarly v 1 ' = v, or the balls, whatever be their speeds, interchange them, and the motion is the same as if they had passed through one another without exerting any mutual action whatever.

Thus if a number of equal solitaire balls or billiard balls be arranged in contact in a horizontal groove, and another equal ball impinge on one extremity of the row, it is reduced to rest, and the ball at the other end of the row goes off with the original speed of impact. If two impinge, two go off, and so on.

§ 186. We may write the above expressions in terms of the impulse, thus

R(l + e) ∙x t , ι = v - ∩

I (2).

, . R(l+e)

Hence Mv 1 + M⅛ 1 '= Mv + MW, whatever e be, or there is no momentum lost. This is, of course, a direct consequence of the third law of motion.

Again ⅜MvJ + ⅜M⅛'J = ⅛Mυ 2 + ⅜MV s

-R(l + e)(t,-√) + ⅛R≡(l + β )*M^'

= ⅛Mv 2 + ⅛M'√ 2 - ⅛R 2 (1 - e 2 )⅛^-'

= ⅛Mt> 2 + ⅛M'√ 2 - ⅜(1 - - ®') s ·

The last term of the right hand side is therefore the kinetic energy apparently destroyed by the impact. When e = 0, its magnitude is greatest, and equal to ig⅛∙-∙') ’-*B(∙-*')-

When e=l, its magnitude is zero; that is, when the coefficient of restitution is unity no kinetic energy is lost.

The kinetic energy which appears to be destroyed in any of these cases is, as we see from § 171, only transformed— partly it may be into heat, partly into sonorous vibrations, as in the impact of a hammer on a bell. But, in spite of this, the elasticity may be “perfect.” Hence the absurdity of the designation alluded to in § 185. Also by (2)

z r τ> ∕ t * ∖ Μ + Μ

1∙ 1 -v 1 = v -v + R(l+e)--—-

lulVL

= e(v-v ’ ) by (1).

Hence the velocity of separation is e times that of approach.

§ 187. Two smooth spheres, moving in given paths and with given speeds, impinge; to determine the impulse and the subsequent motion.

Let the masses of the spheres be Μ, M', their speeds before impact v and v', and let the original directions of motion make with the line which joins the centres at the instant of impact the angles a, a, which may be calculated from the data, if the radii of the spheres be given.

Since the spheres are smooth, the entire impulse takes place in the line joining the centres at the instant of impact, and the future motion of each sphere will be in the plane passing through this line and its original direction of motion.

Let R be the impulse, e the coefficient of restitution; then, since the speeds in the line of impact are v cos α and √cosα', we have for their final values v 1 , v 1 ', after restitution, by § 184 the expressions u. = vcosα- v =M rr .(1 + e)(vcosa- √cosa'), Μ + Μ

v 1 ' = √cσsa' + —(1 +β)(∙vcosa-√cosa'), 1 Μ + Μ'

and the value of R is

. .(l + β)(vc°sα-√c°sα ’) .

Μ + Μ

Hence, the sphere Μ has finally a speed in the line joining the centres, and a speed vsinα in a known direction perpendicular to this, namely, in the plane through this and its original direction of motion. And similarly for the sphere M'. Thus the consequences of the impact are completely determined.

§ 188. When a sphere of mass Μ impinges directly, with speed V, on another Μ' at rest, the speed acquired by Μ' is MV (l + e)

M + M' ·

But, if another sphere of mass μ,, also at rest, be interposed between them, Μ' will acquire a speed μ MV(l+e) 8

(M + μ )(M' + μ ) ‘

This is greatest when μ is the geometric mean of Μ and M', and its value is then MV (l+c) s

(√M + √M') a ·

The ratio of this to the speed which Μ' would have acquired without the interposition of the third sphere is

I.·l·e

2√MM 7 * 1+ M + M'

There is thus a gain by the interposition if, and only if, 2√MM' e> M + M ’ '

This condition is always satisfied when the coefficient of restitution is unity, except in the special case of equal masses. If an infinite number of spheres be interposed between Μ and M', so adjusted as to give the greatest possible speed to M', that greatest speed is V a ∕M'∕M, provided we have e = 1.

Continuous' Succession of Indefinitely Small Impacts.

§ 189. We may now consider the case of a continuous series of indefinitely small impacts, whose effect is comparable with that of a finite force. One obvious method of considering such a problem is to estimate separately the changes in the velocity produced by the finite forces and by the impacts, in the same indefinitely small time δi, and compound these for the actual effect on the motion in that period.

Another way, of course, is to equate the rate of increase of momentum per unit of time to the force producing it.

A mass, under no forces, moves through a uniform cloud of little particles which are at rest. Those it meets adhere to it. Find the motion.

At time t let μ be the mass, and let x denote its position in its line of motion. Then, as there is no loss of momentum, we have 4∙(m*)=o. at

But if Μ be the original mass, μ 0 the mass of the particles picked up in unit of length, obviously μ = M + μo α.

Substitute and integrate, supposing x = 0, x = V, when t≈Q; and we get

(M + μo x)± = MV, from which x can be easily found.

It is interesting to observe that we have μ 0 M*W .

(M + μo z)≡ ’ so that the mass moves as if acted on by an attraction varying inversely as the cube of the distance from a point in its line of motion.

This problem obviously leads to the same result as the following :— A cannon-hall attached to one end of a chain, which is coiled up on a smooth horizontal plane, is projected along the plane. Determine its motion.

§ 190. Another excellent instance of the application of this process is furnished by the motion of a rocket, where the motive power depends on the fact that a portion of the mass is detached with considerable relative velocity. The increase of the momentum of the rocket due to this cause is equal to the relative momentum with which the products of combustion escape. If we suppose the rocket, originally of mass Μ, to lose eM in unit of time, projected from it with relative velocity V, the gain of momentum in time δi due to this cause is eMVδZ.

The total upward acceleration is therefore eMV M-eMi g '

Unless this be positive the rocket cannot rise. It will rise at once if V><∕∕e, and it cannot rise at all unless MV∕M' >g ∣ e, Μ' being the mass of the case, stick, <fcc., which are not burned away.

From the above data it is easy to calculate that the greatest speed acquired during the flight (the resistance of the air being left out of account) is

, r , Μ g[, M'∖

v ⅛-7(1 -Mp Dynamics of a System of Particles Generally.

§ 191. The la<V of energy, in abstract dynamics, may be expressed as follows :—the whole work done in any time, on any limited material system, by applied forces, is equal to the whole effect in the forms of potential and kinetic energy produced in the system, together with the work lost in friction. This principle may be regarded as comprehending the whole of abstract dynamics, because the conditions of equilibrium and of motion, in every possible case, may be derived from it.

§ 192. A material system, whose relative motions are unresisted by friction, is in equilibrium in any configuration if, and is not in equilibrium unless, the rate at which the applied forces perform work at the instant of passing through it is equal to that at which potential energy is gained, in every possible motion through that configuration. This is the celebrated principle of “virtual velocities,” which Lagrange made the basis of his Mécanique Analytique.

§ 193. To prove it, we have first to remark that the system cannot possibly move away from any particular configuration except by work being done upon it by the forces to which it is subject; it is therefore in equilibrium if the stated condition is fulfilled. To ascertain that nothing less than this condition can secure the equilibrium, let us first consider a system having only one degree of freedom to move. Whatever forces act on the whole system, we may always hold it in equilibrium by a single force applied to any one point of the system in its line of motion, opposite to the direction in which it tends to move, and of such magnitude that, in any infinitely small motion in either direction, it shall resist or shall do as much work as the other forces, whether applied or internal, altogether do or resist. Now, by the principle of superposition of forces in equilibrium, we might, without altering their effect, apply to any one point of the system such a force as we have just seen would hold the system in equilibrium, and another force equal and opposite to it. All the other forces being balanced by one of these two, they and it might again, by the principle of superposition of forces in equilibrium, be removed; and therefore the whole set of given forces would produce the same effect, whether for equilibrium or for motion, as the single force which is left acting alone. This single force, since it is in a line in which the point of its application is free to move, must move the system. Hence the given forces, to which the single force has been proved equivalent, cannot possibly be in equilibrium unless their whole work for an infinitely small motion is nothing, in which case the single equivalent force is reduced to nothing. But whatever amount of freedom to move the whole system may have, we may always, by the application of frictionless constraint, limit it to one degree of freedom only; and this may be freedom to execute any particular motion whatever, possible under the given conditions of the system. If, therefore, in any such infinitely small motion there is variation of potential energy uncompensated by work of the applied forces, constraint limiting the freedom of the system to only this motion will bring us to the case in which we have just demonstrated there cannot be equilibrium. But the application of constraints limiting motion cannot possibly disturb equilibrium, and therefore the given system under the actual conditions cannot be in equilibrium in any particular configuration if the rate of doing work is greater than that at which potential energy is stored up in any possible motion through that configuration.

§ 194. If a material system, under the influence of internal and applied forces, varying according to some definite law, is balanced by them in any position in which it may be placed, its equilibrium is said to be neutral. This is the case with any spherical body of uniform material resting on a horizontal plane. A right cylinder or cone, bounded by plane ends perpendicular to the axis, is also in neutral equilibrium on a horizontal plane. Practically, any mass of moderate dimensions is in neutral equilibrium when its centre of inertia only is fixed, since, when its longest dimension is small in comparison with the earth’s radius, the action of gravity is, as we shall see (§ 222), approximately equivalent to a single force through this point.

§ 195. But if, when displaced infinitely little in any direction from a particular position of equilibrium, and left to itself, it commences and continues vibrating, without ever experiencing more than infinitely small deviation, in any one of its parts, from the position of equilibrium, the equilibrium in this position is said to be stable. A weight suspended by a string, a uniform sphere in a hollow bowl, a loaded sphere resting on a horizontal plane with the loaded side lowest, an oblate body resting with one end of its shortest diameter on a horizontal plane, a plank, whose thickness is small compared with its length and breadth, floating on water, are all cases of stable equilibrium,—if we neglect the motions of rotation about a vertical axis in the second, third, and fourth cases, and horizontal motion in general in the fifth, for all of which the equilibrium is neutral.

§ 196. If, on the other hand, the system can be displaced in any way from a position of equilibrium, so that when left to itself it will not vibrate within infinitely small limits about the position of equilibrium, but will move .farther and farther away from it, the equilibrium in this position is said to be unstable. Thus a loaded sphere resting on a horizontal plane with its load as high as possible, an egg-shaped body standing on one end, a board floating edgewise in water, would present, if they could be realized in practice, cases of unstable equilibrium.

§ 197. When, as in many cases, the nature of the equilibrium varies with the direction of displacement, if unstable for any possible displacement it is practically unstable on the whole. Thus a circular disk standing on its edge, though in neutral equilibrium for displacements in its plane, yet being in unstable equilibrium for those perpendicular to its plane, is practically unstable. A sphere resting in equilibrium on a saddle presents a case in which there is stable, neutral, or unstable equilibrium according to the direction in which it may be displaced by rolling; but practically it is unstable.

§ 198. The theory of energy shows a very clear and simple test for discriminating these characters, or determining whether the equilibrium is neutral, stable, or unstable, in any case. If there is just as much potential energy stored up as there is work performed by the applied and internal forces in any possible displacement, the equilibrium is neutral, but not unless. If in every possible infinitely small displacement from a position of equilibrium there is more potential energy stored up than work done, the equilibrium is thoroughly stable, and not unless. If in any or in every infinitely small displacement from a position of equilibrium there is more work done than energy stored up, the equilibrium is unstable. It follows that if the system is influenced only by internal forces, or if the applied forces follow the law of doing always the same amount of work upon the system while passing from one configuration to another by all possible paths, the whole potential energy must be constant in all positions for neutral equilibrium, must be a minimum for positions of thoroughly stable equilibrium, and must be either a maximum for all displacements or a maximum for some displacements and a minimum for others when there is unstable equilibrium.

§ 199. We have seen that, according to D’Alembert’s principle, as explained above, forces acting on the different points of a material system, and their reactions against the accelerations which they actually experience in any case of motion, are in equilibrium with one another. Hence, in any actual case of motion, not only is the actual work by the forces equal to the kinetic energy produced in any infinitely small time, in virtue of the actual accelerations, but so also is the work which would be done by the forces, in any infinitely small time, if the velocities of the points constituting the system were at any instant changed to any possible infinitely small velocities, and the accelerations unchanged. This statement, when put into the concise language of mathematical analysis, constitutes Lagrange’s application of the “principle of virtual velocities” to express the conditions of D’Alembert’s equilibrium between the forces acting and the resistances of the masses to the acceleration. It comprehends, as we have seen, every possible condition of every case of motion. The “equations of motion” in any particular case are, as Lagrange has shown, deduced from it with great ease.

Commencing again with the equations of motion of a particle mdc=X + X', τn^=Y+Y', τnz=Z + Z', let us introduce quantities δ∙τ, &c., consistent with the conditions, otherwise perfectly arbitrary, and we have the general equation ⅛m(xδx + . . . ) = 2(Xδiκ + . . . ), in which, by D’Alembert’s principle, the forces X', Y', Z', due to the constraints, do not appear.

If the system be conservative the right-hand member of this is, of course, equal to the loss of potential energy, so that

-δV = 2(Xδic + . . . ), and therefore, quite generally in such a system,

Sm(⅛δx+ . . . )=-δV (1).

In the actual motion of any system we have, for each particle, Sx=xSt, &c., so that we have

2m (xx + ÿÿ + zz) = 2(Xx + Yÿ + Zz).

This is the complete statement of Newton’s scholium, § 2 above.

The right-hand member is the expression of the algebraic sum of the actiones agentium and of the reactiones resistentnum, so far as these depend upon gravity, friction, &c., and the left-hand member that of the reactiones due to the accelerations of the several particles.

If the system be conservative, this becomes

2m (xx + yy + zz) = > whose integral

⅜2∞(z 3 + ÿ 2 + z 2 ) + V = H

is of course, the general statement of the conservation of energy.

In Lagrange’s general equation above, as we have stated, the variations Sx, &c., are not usually independent. We must take account of the various constraints imposed on the system. If these retain the same character throughout the motion they may be expressed by a (generally finite) number of equations of the form

∕U ’ ι, Vυ 2 ι, «2» y·i, ⅞> · · · )∙=θ∙

Each of these gives rise to a purely kinematical relation affecting some one or more of the quantities δx, &c., of the form 2 f(1 τO δa3 + (ΙF¼ + · · · ) = °·

∖∖ dxj ∖ dy) a )

By introducing, as usual, a set of undetermined multipliers u, one for each of the conditions of constraint, we obtain on adding all these equations to the general equation above s ,, i (½ + ... )- 1 [(x + ι∙(^)+⅛(f )+ · ·)*»+ · ∙]ffl∙ If there be p particles of the system, there are ⅛p coordinates ae, y, z, connected by (say) q equations of constraint, so that there are 3p-q degrees of freedom, and therefore 3p-q independent coordinates.

Equating separately to zero the multipliers of δ∙r 1 , δy 1 , &c., in the resultant equation above, we have 2>p equations of which we write only one as a type, viz., mx=X + ¾u(Jθ (Si-

Taken along with the q equations of the form

∕=θ

these form a group of 3ρ + q equations, theoretically necessary and sufficient to determine the 3jt> quantities x 1 , y 1 , z 1 , &c., and the q quantities μ, in terms of t. Thus we have the complete analytical statement of the conditions, and the rest of the solution is a question of pure mathematics.

When we deal with a non-conservative system (which is equivalent in nature to saying “when we take an incomplete view of the question ”), some of the conditions may vary in character during the motion. This will be expressed analytically by the entrance of t explicitly into one or more of the equations of condition f. But, if we think of the mode of formation of Lagrange’s equation, we see that it was built up of separate equations, such as mx = X + X', which are true whether the equations of condition involve t explicitly or not. Each of these was multiplied by a quantity δα;, &c., the only limitation on which was that it should be consistent with the conditions of the system at the instant considered, whatever instant that might be. Hence equation (2) still holds good.

When, however, we introduce in that equation multipliers corresponding to the actual motion of the system, so that δx=xδt, &c., we find a remarkably simple expression for the energy given to, or withdrawn from, the system in consequence of the varying conditions. For the unintegrated equation (2) now becomes

≤(⅛Sτn(i≡+^ + z 2 )) = 3(X± + Yÿ + Zz)- 2 m ( ∣ Q , where the differential coefficient of f is partial. This follows at once from equations of the form

F(f)+(f)-∙

which are obtained by differentiating the equations of condition with regard to t. When the conditions do not vary, the quantities (⅛H ∖ &c., all vanish, and we see that the constraint does not alter ∖ dt)' the energy of the system.

Least and Varying Action.

§ 200. To complete our sketch of kinetics of a particle we will now briefly consider the important quantity called “action.” This, for a single particle, may be defined either as the space integral of the momentum or as double the time integral of the kinetic energy, calculated from any assumed position of the moving particle, or from an assigned epoch. For a system its value is the sum of its separate values for the various particles of the system. No one has, as yet, pointed out (in the simple form in which it is all but certain that they can be expressed) the true relations of this quantity. It was originally introduced into kinetics to suit the metaphysical necessity that something should be a minimum in the path of a luminous corpuscle (see an extract from Hamilton in the article Light, vol. xiv. p. 598). But there can be little doubt that it is destined to play an important part in the final systematizing of the fundamental laws of kinetics.

The importance of the quantity called action, so far as is at present known, depends upon the two principles of “least action” and of “varying action,” the first as old as Maupertuis, the other discovered by Hamilton about half a century ago.

The first is— If the sum of the potential and kinetic energies of a system is the same in all its configurations, then, of all the sets of paths by which the p>arts of the system can be guided by frictionless constraint to pass from one given configuration to another, that one for which the action is least is the natural one or requires no constraint.

§ 201. Unfortunately it is not easy to give examples of this important principle which can be satisfactorily treated by elementary methods,—except, indeed, the very simplest, such as those furnished by the corpuscular theory of light. Thus it is obvious that, as long as a medium is homogeneous and isotropic, the speed of a corpuscle in it is constant. The action is thus reduced to the product of the constant speed of the corpuscle by the length of its path. Hence the principle at once shows that the path must be a straight line. When the corpuscle is refracted from one such medium into another, the path is a broken line such that the product of each of its parts by the corresponding speed of the corpuscle is the least possible. This gives the law of the sines, but to agree with experiment the speed would have to be greater in the denser medium than in the rarer.

§ 202. The problem to find change of action as depending on change {nowhere finite') of the mode of passage from one given configuration to another (restricted by the condition already mentioned), is expressed mathematically by

δA = δ∕^ ⅛msds — δfi½m{xdx + ÿdy + zdz), while T = ⅜2τns 2 = ⅜Sm(⅛ 2 + y 2 ∙ + z 2 ) = H - V ,

H being the constant energy of the system, and the integral being taken between limits supplied by the two given configurations.

The first equation gives

δA = fl,m{dxδx + dyδy + dzδz + xdδx + ydδy + zdδz) = '½m{xδx + . . .) + f⅛m(dxδx + dyδy + dzδz - dxδx - di∕δy - dzδz) by partial integration. But the integrated part

2wι(⅛δai + yδy + zδz), (A)

obviously vanishes at both limits, because the initial and final configurations are given.

If we now take the corresponding variation of the expressions for the kinetic energy, we have

δT = ^2,msδs = ^2,m {xδx + yδιj -Y zδz), from which we have J⅛m{da,δx + dyδf∕ + dzδz) ≈JlΗdt.

Also we have dxδx + dyδy + dzδz = {χδx + yδy + zδz)dt ; so that finally

δA =y⅛[δT - 2m(xδx + yδy + zδz)] , which so far is a mere kinematical result. But it can be rendered physical by putting - δV for δ r Γ, in accordance with the above condition. This we will suppose done.

If now we desire to make δA vanish, so as to obtain what is called the “stationary condition,” we must make the factor in square brackets in the integral vanish; i.e., we must have 2m(xδx + i∕δy + zδz) + δV = 0

for all admissible simultaneous values of δx, δy, δz for the various particles of the system. But this is precisely the general equation which, as we found in § 199 (1), determines the undisturbed motion of the system.

§ 203. The expression δA=0 really signifies that any infinitesimal change from the natural mode of passage producesan infinitely smaller change in the corresponding amount of the action between the terminal configurations.

§ 204. It will be noticed that the essential characteristic of the modes of passage considered in this investigation is that all shall have the same terminal configurations, and that the system shall always have the same definite amount of energy. All, except the natural mode of passage, in general require constraint in order that they may be described. Hamilton's grand extension of the subject depended on comparing the actions in a number of natural modes of passage, differing from one another by slight changes in their terminal configurations, and slight changes in the whole initial energy.

In this new form of statement the unintegrated part of the expression for δA vanishes, since all the modes of passage contemplated are natural. The alteration of the whole energy, however, adds a special term to the equation, and we can at once write, from the expression (A) § 202, the equation for the change in the action under the new conditions, viz.,

δ A=£x?n(x δχ ÷ yδy - f ~δ~)J + ZδH, the part in brackets having to be taken between limits corresponding to the terminal configurations, and the variations δz, δy, δz at these being subject to the conditions of the system.

We cannot here consider this equation in its general form. We content ourselves with the simpler special deductions from it required for completing our sketch of Kinetics of a Particle.

The last given equation, written in full for a single particle of unit mass, is

δA = [xδx + yδy + zδz] - (⅛ 0 δz 0 + y 0 δy 0 + z 0 Sz 0 ) + ∕δll, where x 0 , y 0 , z 0 is the initial point, and x, y, z any other point, of the path. If the particle be altogether free, the seven variations on the right-hand side are independent of one another; and thus we have the following remarkable properties of the quantity A, regarded as a function of seven independent variables (the initial and final coordinates of the particle, and its constant energy), viz., id A ∖ _ dx i dA ∖ _ dx a ∖ dx J dt , ∖ dx^ ) dt ’ idA\_dy idA\_ dy a ∖ dy J dt ’ ∖ dy 0 ) dt ’ fdA ∖ ~dz fdA\_ dz a ∖ clz ) dt ’ ∖ dz 0 J dt ’ ∖ dH J t '

From these we gather at once that A satisfies the partial differential equations i^Y + i^Y + i^Y ==l ’ 2 “2 (H - y ) · · < 1 )> ∖ (Lx, j y dy j ∖ dz J

("γ√"y√"y=⅛- 2 (H-v.). . (2).

∖ dx 0 J ∖ dy 0 J ∖ dz a J

§ 205. The whole circumstances of the motion are thus dependent on the function A, called by Hamilton the “characteristic function.” The determination of this function is troublesome, even in very simple cases of motion; but the fact that such a mode of representation is possible is extremely remarkable.

§ 206. More generally, omitting all reference to the initial point, and the equation § 204 (2) which belongs to it, let us consider A simply as a function of x, y, z. Then Any function, A, which satisfies the partial differential equation

... ο, dA dA dA possesses the p>roperty that -- , — , — represent the rectangular components of the velocity of a particle in a motion possible under the forces whose potential is V.

For, by partial differentiation of (1) we have d ∕ cPx dV dA cPA dA d 2 A dA <7 2 A

dt ∖ dt j dP j ^ dx dx dx i ^ t " dy dxdy r dz dxdz ’ with other two equations of the same form.

But we have also three equations of the form d idA ∖ dx d 2 A dy cP A dz cP A dt ∖ dx J dt dx 1 ^ r dt dxdy " t ^ dt dxdz ’

Comparing, we see that dx_dA dy dA dz _dA dt dx ’ dt dy ’ dt dz satisfy simultaneously the two sets of equations.

§ 207. Also if α, β be constants, which, along with H, are involved in a complete integral of the above partial differential equation the corresponding path, and the time of its description, are given by where α 1 , 0 1 , e cure three additional arbitrary constants.

For these equations give, by complete differentiation with regard to t, d 2 A dx rf 2 A dy d-A dz . Ί

dxda dt dyda dt dzcla dt d 2 A dx d 2 A dy d' 2 A dz_ dxdβ dt ^ t ^ dydβ dt + dzdβ dt ∣ " · α ∙ cP A dx cP A dy d 2 A dz ∣ dxdH dt + dydH dt + dzdH dt J

But, differentiating § 206 (1) with respect to α, β, II respectively, we get

<f 2 A dA J 2 A dA d 2 A dA _ > ∣ dadx dx dady dy dadz dz I

d 2 A dA d 2 A dA cPA dA ∣ dβdx dx + dβdy dy dβdz dz d 2 A dA d' 2 A dA cPA_ dA _ dHdx dx + dHdy dy + dHdz dz j

The values of ⅛ , &c., in (a) are evidently equal respectively to dA those of —, &c., in (δ). Hence the proposition.

§ 208. “Equiactional surfaces,” i.c., those whose common equation is

A = const., are cut at right angles by the trajectories.

For the direction cosines of the normal are obviously proportional to

(⅞M , thatis, to ⅛, ⅛ .

∖ dx J ∖ dy J ∖ dz J dt dt dt

Thus the determination of equiactional surfaces is resolved into the problem of finding the orthogonal trajectories of a set of given curves in space, whenever the conditions of the motion are given.

The distance between consecutive equiactional surfaces is, at any point, inversely as the velocity in the corresponding path.

This may be seen at once as follows: the element of the action, which is the same at all points, is vδs (where δs, being an element of the path, is the normal distance between the surfaces).

§ 209. In consequence of the importance of the method we will take two examples of its application. First a direct example, then one depending on the equiactional surfaces.

To deduce from the principle of “varying action” the form and mode of description of a planet's orbit.

In this case it is obvious that represents the attraction of dr gravity (-μ∕r 2 ). Hence the right-hand member of § 206 (1) may be written 2(H + μ∕r).

Let us take the plane of xy as that of the orbit, then the equation § 206 (1) becomes

--(⅛) ’ ÷(¾7-K h+ ⅛) · · < 1 >∙

It is not difficult to obtain a satisfactory solution of this equation, but the operation is very much simplified by the use of polar coordinates. With this change, (1) becomes (⅛) , ÷O=Kh÷÷) · · · which is obviously satisfied by ∕dA∖ . .

(3 J)-∞"Λ"*-∙> (^Y- 2 (h + a)-≤ ∖ dr J ∖ rj r i

Hence A^aθ+Jdr ∖ f%(H+μ ∣ r)-<P ∣ r 2 .... (4).

The final integrals are therefore, by § 207, ( d ± ∖ =a≈θ-<, f dr —. . . (5), ∖ da J J γ l √2( H + μ∕r) - α 2 ∕r 2 and fyrfΙ =i + e= ∕*. .- r -.-, . . . (6).

∖dH∕ √ √2(H + μ∕r) - α s ∕r 2

These equations contain the complete solution of the problem, for they involve four constants, α 1 , a, H, e. (5) gives the equation of the orbit, and (6) the time in terms of the radius-vector.

To complete the investigation, let us assume μ∕α 2 = l∕Z, 2H∕ α 2 = (e 2 -l)∕Z 2 , where I and e are two new arbitrary constants introduced in place of α and H. With these (5) becomes . ∕* dr a.≈θ- ∕ 7-·-" "

J r 2 √(e 2 - 1 )∕Z 2 + 2/Zr - l∕r 2

= 0- Γ λ , — ≈gcos- 1 -(r~ 1 -Z~ 1 ) ,

√r 2 √e 2 ∕Z 2 -(l∕r-l∕Z) 2 e or r=r- 73 ∖ »

1 + ecos(0 - α 1 )

the general polar equation of conic sections referred to the focus.

Also by differentiating (5) with respect to r, we have adr -dß, r 2 √2(H+y∕r)-α 2 ∕r 2 from which, by (6), we immediately obtain i + c = ±∕⅛≡ * ir 2 dθ. aj V μ,U

This involves, again, the equation of equable description of areas. Compare § 144.

§ 210. In a planet’s elliptic orbit the time is measured by the area described about one focus, and the action by that described about the other.

For with the usual notation we have dA=vds=- ds , P by the result of § 47. But in the ellipse or hyperbola, p' being the perpendicular from the second focus, pp'= ±δ a .

Hence (ZA = ±j⅛p'ds » which expresses the result stated above.

It is easy to extend this to a parabolic orbit, for which, indeed, the theorem is even more simple.

§ 211. Unit particles are projected simultaneously and horizontally in all directions from every point of a vertical axis, all having the same total energy at starting; find the surfaces of equal action.

We may obviously confine ourselves to a plane section through the axis. Let x be the vertical coordinate of a particle, measured downwards from the level at which the common energy is wholly potential, k the coordinate of the point in the axis from which it was projected. Then we have, after the lapse of time t, x=k + jgt 2 ) y= ∖∣ ζ λgkt ) "

Eliminating t, we have the equation of the parabolic path—? Z∕ 2 x=⅛ + j 77 · 4⅛

To find the orthogonal trajectory (the meridian section of the surface of equal action), differentiate, put - for <~ > and eliminate k. We tty ccx thus have dy_y _ y Vae + y-Va;-y dx t ⅛ c x + V≈ 2 - y 2 Vx + y + Va? - y or - Va: + y(dx + dy) + Va: - y(dx - dy) = 0 , so that . (aι + y)⅛(a!-y)^ = const.

If we tum the axes through ⅜π in their own plane, the coordinates being now ξ and η, this equation becomes = .

In fig. 55, AM is the axis. A few of the paths are shown by full lines, and two of the sections of surfaces of equal action by dotted lines. These sections indicate cusps lying on the line AH, which makes an angle ⅛π with the vertical, and is touched by all the paths. The path whose vertex is G touches this line in H, and therefore passes through the cusp of which the branches are HK and HL. HK belongs to all paths whose vertices are above G, HL to those (such as ML) whose vertices are below G.

It is worthy of note that, by the first equations above, ≈±y = (V^≡V⅜y. i ) 2 .

by the substitution of which iu the equation of action we see how the time of reaching a particular surface of equal action depends upon the position of the starting point.

§ 212. A very interesting plane example, which has elegant applications in fluid motion, and in the conduction of electric currents in plates of uniform thickness, is furnished by assuming

A = logr, or A' = 0, where r and θ are the polar coordinates of the moving particle.

In the former, where the curves of equal action are circles with the origin as centre, we have dA . x dA . y dx r 2 dy y r 2 so that the paths are radii vectores described with velocity l∕r, Also we have 2(H _v).("Y + ("y i i ∖ dx ) ∖ dy ) r 1 so that the force is central, and its value is

_dV = _J_ dr r i

In the second case, where the curves of equal action are radii drawn from the pole and

A' = 0 = tan- 1 -^- , x we have dA' _ . _ y dA.' _ . _ x dx x r 2 ’ dy y r 2 ’

The kinetic energy is still l∕2r 2 , and the central force - l∕r 3 , but the paths are circles with the origin as centre. Thus the lines of equal action and the paths of individual particles are convertible.

We have also, in each of these cases, ∖ dxj ∖ dyj

This shows (as in § '94) that, whatever be originally the grouping of a set of particles moving all according to one or other of these conditions, the density at any part of the group remains unchanged during the motion. In fact, as it is easy to prove, A and A' are elementary solutions of the partial differential equation

(1); dx 2 + dy 2 and they are conjugate, in the sense that dA_dA' dA dA' dx dy ’ dy dx ‘

For this reason the paths belonging to the two systems are everywhere orthogonal to one another.

Also, as the differential equation (1) for A is linear, any linear function of particular integrals is an integral. Thus, for instance, we may take (p being any constant)

A=logr-pβ , with A' =p log r + θ.

These, representing orthogonal sets of logarithmic spirals, possess the same properties with regard to action as did the concentric circles and their radii, which, in fact, are the mere particular case when p = 0.

§ 213. It is easy to give graphic methods of tracing these curves of action by means of an old process recently much developed by Clerk Maxwell. The present example, though a very simple one, is quite sufficient to illustrate the process.

Draw, as in fig. 56, a set of circles whose radii are i' l , S 2α , S 3 ' , , &c., and a set of radii vectores making with the initial line the successive angles, — , 2— , 3— , Ac. , α being a quantity which p p P may have any convenient value. These lines will form a network, finer as a is smaller. Now suppose we wish to trace the curve

A= na.

We take the intersection of the circle whose radius is 6 ,α with the radius-vector corresponding to the angle (3-n)i.

Thus we have for the value of A at the point of intersection

A = logδ α - p a P = na, as required.

By marking the intersections corresponding to different values of s, we have a series of points in the required curve which, by adjustment of the value of α, may be made to lie as close together as is found necessary for tracing the curve of action through them libera manu.

In fig. 56 portions of three separate sets of mutually orthogonal logarithmic spirals have been traced, by using the intersections of the fundamental straight lines and circles.

We may pursue the subject much farther, by combining particular solutions like those given but taken from different origins. We can afford space for one only. Let P, Q (fig. 57) be points on the axis of x, distant a and -a from the origin, R any point in the plane of the figure. Let PR = r, <∕RI⅛=0, RQ = r 1 , <≈<RQiC=0 1 .

Then if

A=logr-logr 1 ,

A' = 0-0 1 , we must have, not only the equation d 2 A c^A = n dx 2 dy 2 satisfied by each of A and A', but also the conditions dA _ dA' dA _ dA' dx dy ’ dy dx

This follows at once from the fact that all the equations are linear.

§ 214. In fig. 57 we have S a = PR∕QR. Hence the locus of R is a circle, whose centre, B, is on QP produced.

Again A' = a<RPaJ-^≤,RQaj=^ιPRQ . The locus of v R is, in this case, any circle passing through P and Q.

These circles evidently cut one another orthogonally in R; for BR, which is a radius of the one, is a tangent to the other.

Thus particles moving in a plane, so that the speed at any point R is inversely as PR. RQ, may describe circles in wfιich PQ is a chord. In this case the curves of equal action are circles defined by the condition that the ratio PR: RQ is constant. Or they may move in the latter system of circles, in which case the former system gives the lines of equal action. For the equations in the preceding section give ∖ dx) ∖ dy) ∖ dx ) ∖ dy ) ~7 i r ∖ ’ from which the conclusion is obvious.

We may easily extend this example to other sets of orthogonal curves whose equations are

A 1 =pA + A ’,

A 1 = — A + pA', where A and A' have their recent values. Or we may extend the example by assuming at starting

A=wιlogr- m 1 logz∙ 1 , in which case it will be found that we must have A'=mθ -m l θ 1 .

These pairs may again be combined into pA + A ’ and - A- ∖ pA', and so on.

It will be noticed that in these examples the curves of equal action and the paths of the particles correspond in steady fluid motion to curves of equal pressure and lines of flow, and in electric conduction to equipotential lines and current lines. In such cases in fact, where there is no vortex-motion, the action is closely analogous to what is called the “velocity potential” in a fluid.

Generalized Coordinates.

§ 215. By the help of the result already obtained in connexion with least action, we may easily obtain in a simple, though indirect, way the remarkable transformation of the equations of motion of a system which was first given by Lagrange. We are not prepared to give here the transformation to G eneralized Coordinates in its most general form; but, even in the restricted form to which we proceed, it is almost invaluable in the treatment of the motion of conservative systems of particles in which the number of degrees of freedom is less than three times that of the particles. The one point to be noticed is that, when we restrict ourselves to a system of this kind, the expression for the kinetic energy, T, is necessarily a pure quadratic function of the rates of increase of the generalized coordinates. This is obvious from § 19. Repeating with generalized coordinates the investigation of § 202, we have

A = 2∕T<ft =∕(T + H - V)dt.

Hence δA= j ∕ r (δT + δH - δV)dt. Now let 0, φ, ψ, &c.j be the generalized coordinates, and we have 2T = P0 2 + 2Q0≠ + R<∕> 2 +

where P, Q, R, . . . are in general functions of 0, φ, ψ, . . . Of course V is a function of 0, φ, ψ . . . alone, and does not involve Ô, φ, φ, &c.

Thus we have, writing, for one only of the generalized coordinates, _/ ∖∖ dθ J ∖ dθ J ∖ dθ ∕ ]

√<7T ∖ ι τr Γ u JΓ(d(dΥ ∖ d r t dN ∖ *~ ∖ + <Z0 + rfe√ δθ J ·

But we saw that, for any natural motion, the unintegrated part of δA necessarily vanishes. Thus, as 0, φ, ψ . . . . , and, therefore, their variations, are by their very nature independent of one another, the vanishing of the un integrated part gives us one equation of motion for each degree of freedom, the type being in all of them the same, viz., dt ∖ d∂ ) ∖ dθ∕ ∖ dθ)

To exemplify the use of these equations we will take again a few of the more important cases of constraint already treated, and will then proceed to some others of interest as well as of somewhat greater complexity.

In the simple pendulum, I being again the length of the string, and θ the inclination to the vertical at time t, we have obviously

T = ⅛mP0 2 , V = C - mg I cos 0.

Hence = (⅞^ = 0, (- ∖ = mglsinθ. ∖ d0 ) ∖ dθ) ∖ dθ ) j

Thus the equation of motion is

τnZ 2 0 + mi√sin0=O

or 04-dLsin0 = O, as in § 134.

Suppose the same pendulum to be moving anyhow, 0 still denoting its inclination to the vertical, and φ denoting the azimuth of the plane in which it is displaced, we have

T = ⅜7∏Z 2 (0 2 + sin 2 0.√> 2 ), V=C - mgl cos 0

These give at once

(f S)"" p ' , ' (f )-"∙P≡*∞≡β ≠ i . (g). m w 9 .⅛ ≡- 0 , m-o. W∕ ’ ∖ dφ) ’ ∖ dφ)

Hence the two equations are

’ 0 - -sinØ cosθ.φ 2 +-~- sinØ = 0,

A(s in 2 0.≠) = O.

Still keeping to easy examples, suppose the cord of the ordinary simple pendulum to be extensible, according to Hooke’s law. Let λ be its length at time t. Then the tension is E(λ-Z)∕Z, and the work it can do in contracting is the integral of this with regard to λ from I to λ, i.e., Ε{κ-l') 2 ∣ 2l.

Hence we have V = C + E(λ - Z) 2 ∕2Z - mg ∖ cosθ ,

T = ⅜m,(λ 2 0 2 + λ 2 ).

Thus Lagrange’s equations become

-^-(λ 2 0) + <7λsin0 = O , at mx - mκβ i + E (λ - Z)∕Z - m<7 cos 0 = 0; equations which could be obtained immediately from the application of the second law, with the help of the kinematical expressions for acceleration perpendicular to, and along, the radius-vector of a plane curve (§47).

Instead of the complex pendulum treated in § 177, we will now take the case of two masses attached at different points to an elastic string, or light helical spring, and consider their vertical vibrations.

Let a, b be the unstretched lengths of the parts of the string, Μ and m the masses. Then if ξ, η be the vertical displacements at time t, we have

T = ⅞(Mξ 2 + m(ξ + ⅛) 2 ),

V-⅜(⅛ + ⅞)-M ii ξ-^tt + ,) i so that Lagrange’s equations are

A(Mξ + m(j + ⅛)) + E-∣--(M + m) 9 r=O, i (, m {ξ + ⅛)) + E -J - mg = 0 . (λu U

The equilibrium positions are found by supposing the accelerations to vanish, so that, if we suppose ξ and η to be measured from them, the terms in g will disappear. Hence the solution is of exactly the same nature as that already given for an apparently different problem (§ 178).

We may mention that equations practically the same as these are obtained when we consider the motions of a watch and its balancewheel, the watch being supported in a horizontal position by means of a wire, and oscillating in its own plane by the torsion-elasticity of the wire. The reader of § 242 below will have no difficulty in obtaining this result. It suggests a practical method of “setting” a watch to true time, without turning the hands forward or backward, and without letting it run down.

The following is a simple, but very instructive, example of the transference of energy (back and forward) between two parts of a system. Two bar-magnets of equal mass, length, and strength (fig. 58), are supported horizontally by pairs of parallel strings, so that when at rest they are in one line.

One of them is slightly displaced in the direction of its length, find the subsequent motion of both.

If we can, by any process, find two fundamental states of motion which, once established, will be permanent, any other possible motion of the system will be a superposition of these two. The amplitudes and phases in the components may have any values, so long as the whole disturbance is small. This follows from the fact that the system has two degrees of freedom only,—since we are concerned only with motions in the plane of the figure.

(A) Now one obviously possible motion is a simple harmonic vibration of the whole, without change of distance between the magnets. The period of this vibration is obviously the same as that of either magnet if the other were removed.

(B) Another obviously possible motion is that in which the magnets are, at every instant, equally and oppositely deflected. The period of this oscillation will be less or greater than that of the former according as the poles attract or repel one another.

Now the initial state of motion proposed evidently consists of the superposition of (A) and (B) in such a way that there is, at starting, no displacement of either mass, but a definite velocity of one of them only. This corresponds to simultaneous zero of displacement, with equal velocities, for each of (A) and (B). There is therefore at that instant no displacement of either mass; and one is at rest while the other is moving with double the assigned velocity. If 2π∕n, 2π ∣ n' be the periods of the two motions, it is obvious that after the time π ∣ (n-n') the magnets will have interchanged their states so that the arrangement will present exactly the same appearance as at first, if looked at from the other side.

Let a be the distance between the ends of the bars when all four strings are vertical. Then, if 0, φ be at time t the inclinations of the pairs of strings to the vertical, a becomes T) = a + l(φ- θ), where I is the common length of the strings. The expression for the potential energy due to magnetism is of the form ∕z∕D, where μ is positive if like poles be turned to one another.

Hence

T = ⅛mZ 2 (0 2 + ≠ 2 ),

* V= w (0 2 + Φ 2 )+-JL_.

Forming the equations as usual, and omitting powers of 0 and φ above the first, we have wi 2 0= -mglθ+^ 1 - -^(<∕>-0)) , ml 2 φ= - mglφ - 1 - - 0)); from which the results already given may be deduced.

Finally, let us take the case of Atwood’s machine (§ 173) when the masses are equal, and one of them is vibrating through small arcs.

Let r, θ be the polar coordinates of the vibrating mass; then, neglecting powers of 0 higher than the second, we have the generalized equations

2r-r0 2 = - ⅜<∕0 2 , A(r 2 0) = -grθ . at

Put ⅛gr for r, and 0√2 for 0, and we get r - r0 2 = - 0 2 , I4(r 2 0)= - 20. r at

Transform to rectangular coordinates in the plane of motion, x being vertically downwards; then x=y* ∣ x i , y=-2y)x.

This shows that the vertical acceleration of the vibrating particle is very small, but constantly downward. Hence the energy of the vibratory motion is steadily converted into energy of translation of the masses.

When both the equal masses vibrate through small arcs, it is found that the mass whose angular range is the greater has downward acceleration with diminishing angular range. Hence it would appear that, if the string be long enough, the entire motion would be periodic.

§ 216. Before leaving this subject we may form, from the complete value of δA given in last section, the generalized equations corresponding to those of Hamilton’s “varying action,” as given in § 204.

We have at once (dK ∖ (<ff ∖ ∕^A∖∕^T∖ ΛJA∖ ∖ dθ )~ ∖ d0) ’ ∖ dφ) ∖ dφ) ’ · * · ’ ∖ dttj l '

But, by the value of T, we have

(f)-W+⅛*+ · (g)-<W + W+ · · · &c. =&c.

These equations give t), φ, , . . as homogeneous linear functions of (g)√"), ···“⅛<√⅛).⅛) ∙∙∙ τh " s ∙ ∖ dθ J ∖ dφj ∖ dθ J ∖ dφj if we substitute these expressions in the equation which is obviously true, because T is a homogeneous function of Ô, φ, . . . of the second degree, we have a partial differential equation of the form

∕<ZA∖ 2 a ldA ∖ ∕ , dA ∖ (d A∖ 2 o , tr ,,.

r{d)) t ‰Λ⅛) fr ('⅛J + - 2(H ’ V) ’

f:om which A is to be found.

The coefficients p, q, r . . . are, in general, like P, Q, R, . . . functions of θ, φ, . . .

As an illustration, take again the example in last section, where two masses are attached to a helical spring, and vibrate in a vertical line. From the value of T there given we have ∕dA ∖ ∕<ZT∖ z ..

∕dA ∖ ∕ r dT ∖ {d- v r^r m ^·

From these we have the equation for A

∕<∕A∖ 2 n fdA ∖ ∕dA ∖ .- κr .∕dA ∖ 2 n ,, . ττ , r . " ’ U) -2, (⅛A⅛) +l t " ’ (⅛ - 2M »< h - v >· The value of V is given above. This equation is, of course, to be treated according to the process illustrated in § 209.

Statics of a Rigid Solid.

§ 217. A rigid body, as we have already seen, has at the utmost six degrees of freedom, three of translation and three of rotation. According to Newton’s scholium, the conditions of equilibrium of such a body, under the action of any system of forces, are that the algebraic sum of the rates of doing work by and against the forces shall be nil whatever uniform velocity of translation or of rotation the body may have. For, if this were not so, there would be work done against acceleration, and the body would gain or lose kinetic energy. And this gain or loss would take place even if the body were originally at rest, i.e., it would not be in equilibrium. To ensure equilibrium then, all that is necessary is that the sums of the components of the forces in any three non-coplanar directions shall vanish, along with the sums of their moments about any three non-coplanar lines. For simplicity it is usual to assume for these directions a system of rectangular axes, and for the lines another system parallel to them and passing through some definite point (say the centre of inertia) of the body.

Thus we have at once

2(X) = 0, 2(Y) = 0, 2(Z) = 0;

Σ(Zy-Yz) = 0, 2(Xz-Zx) = 0, 2(Yz-Xι∕) = 0, where X, Y, Z are the components, parallel to the axes, of a force acting at the point x, y, z of the body. If P, with direction cosines λ, ∕z, v, represent the force acting at x, y, z, these equations may be written in the form

2(Pλ) = 0, 2(P m ) = 0, 2(Pγ) = 0

2[P(vy-∕4z)] = 0, 2[P(λz-w)] = 0, 2[P(j ux-λy)] = 0.

These equations correspond to the six degrees of freedom involved.

It is easy to see that it is a mere matter of convenience through what point of the body we draw the lines about which moments are taken. For, if we shift it by quantities a, δ, c respectively, the moments become

2{Z(y-δ)-Y(z- c )}, &c.; but these are 2(Zy - Yz) - δ2(Z) + c2(Y), &c.; and, by the first three equations, these quantities are seen to reduce themselves to their first terms. Hence, in forming the equations of equilibrium, simplicity will be gained by choosing as origin a point through which the line of action of one or more of the applied forces passes.

Again, the point of application of any one of the forces may be shifted at will anywhere along the line in which the force acts. For the equations of the line in which the force at x, y, z acts are a/ - x y ’ - y z' -z ^X^ —W““Z~ ’

and these give Zy' - Yz ’ = 7.y - Yz , &c., so that the expressions for the moments are unaltered if the point of application of the force be shifted to any position along the line in which it acts.

§ 218. In the great majority of treatises on Statics the fundamental propositions of the subject, above given, are deduced from the assumption (as a thing to be proved experimentally) of the result just established, which is designated the “principle of the transmission of force.” Along with it are assumed the parallelogram of forces, and the principle of the “superposition of systems of forces in equilibrium.” Since the publication of the Principia, the continued use of such methods must be looked upon as a retrograde step in science.

§ 219. From this category we cannot quite except (so far at least as the usual modes of treating it are concerned) the valuable idea of the “couple,” due to Poinsot. But the term is in such common use, and the idea in its applications sometimes of such importance, that it cannot be omitted here.

A couple is a pair of equal forces acting on the same body in opposite directions and in parallel lines.

From the general conditions already given we see that a couple produces a definite moment of force about a particular axis, but that the axis is determinate merely as regards directiou, and not as regards position in space. The forces of a couple do not appear in the first three of the equations of equilibrium. On the other hand, the left hand members of the other three equations may all be regarded as moments of couples. All the properties of couples are contained in these statements. Thus, for instance, it is obvious that, so far as its eSects are concerned—

1. A couple may be shifted by translation to any other position in its own plane.

2, It may be shifted to any parallel plane.

3. In either of these it may be turned through any angle.

4. Its forces may be increased or diminished in any ratio, provided the distance between their lines of action (which is called the “arm” of the couple) be proportionately diminished or increased.

A couple is therefore completely determined by means of its “axis,” which is a line drawn perpendicular to its plane, and of length representing its moment. And two couples are obviously to be compounded by treating their axes as if they were forces acting at one point.

§ 220. We will now examine the consequences of the six conditions of equilibrium (§ 217) in some of the more common cases which present themselves. But, before doing so, it may make matters clearer if we restate these conditions in a somewhat different form.

The resultant of any number offerees, acting at any points of a rigid body, may be represented by a single force acting at the origin, and a couple of definite moment about a definite line passing through the origin. For equilibrium of the body this force and couple must separately vanish.

Thus if, in fig. 59, P, acting at Q, be any one of the forces, and O the origin (chosen at random), we may introduce at O a pair of equal and opposite forces ± P, parallel to P. The original force, taken along with - P at the origin, gives a couple; and in addition there is + P acting at the origin.

§ 221. When only two forces act on a body, the first condition above shows that they must be equal and opposite, and the second that they must act in the same line, if they are to maintain equilibrium. When only three forces act, the first condition shows that their directions must lie in one plane, the second that their lines of action must be parallel or must meet in one point, if they are to maintain equilibrium.

If their directions meet in one point we have again the problem of the equilibrium of a single particle under three forces; for there can be no moment about this point.

When the directions are parallel, one of the forces must obviously be equal to the sum of the other two, and must act in the opposite direction. Also its line of action must lie between those of the other two, for their moments about any point in it must be equal and opposite. Hence it is impossible that any single force should balance a couple, unless we adopt the mathematical fiction of an infinitely small force acting in a line everywhere at an infinite distance; so that its moment may be finite, and equal and opposite to that of the couple.

§ 222. When any number of parallel forces act simultaneously on a rigid body, their resultant is a single force equal to their algebraic sum, with a couple whose plane is obviously parallel to the common direction of the forces. The forces of this couple may be made, by lengthening or shortening the arm, equal to the resultant force. One of them will neutralize it, and the other remains the final resultant, which passes through a definite point called the “centre of parallel forces.” Thus any set of parallel forces necessarily has a single force as a resultant, excepting in the special case when their algebraic sum is zero.

§ 223. Excellent examples are furnished by heavy bodies of moderate dimensions, where the weights of their parts are forces practically in parallel lines. The single resultant force, in such cases, is the whole weight of the body. Its direction always passes through the centre of inertia (§ 109) because weight (in any one locality) is proportional to mass. For this reason all heavy bodies of moderate dimensions are said to have a “centre of gravity,” which coincides with the centre of inertia. But it must be noticed that the two ideas are radically different, and that, while every piece of matter has a true centre of inertia, it is, in general, only approximately that we can predicate of it that it has a centre of gravity. In fact a body has a true centre of gravity only when it attracts, and is attracted by, all other gravitating matter as if its whole mass were concentrated. in that point. See Potential. When there is a centre of gravity in a body, it is necessarily coincident with the centre of inertia. In gravitation cases, where bodies of moderate size are concerned, the resultant is, at least approximately, a single force. But, when we deal with large non-barycentric bodies like the earth, we find that the resultant of the sun’s attraction is a force (determining the orbit) and a couple (producing precession, &c.).

When a mass is laid on a three-legged table, we find the pressure which each leg supports by simply taking moments about the line joining the upper ends of the other two. The leg is thus seen to support a fraction of the weight of the mass, whose numerator is the distance of the centre of gravity of the mass from this line, and its denominator the distance of the leg from the same line. Thus we have a physical proof of the geomet rical proposition that if any point, P, be taken in the plane of a triangle ABC (fig. 60), and perpendiculars be drawn from it and from the angles, we have

Aα BΓ'Cc '

If the mass of the table is to be reckoned, P must be taken as the centre of gravity of the system of table and load together. If the table be of uniform material, triangular, and supported by legs at its corners, similar reasoning shows that when it is unloaded (or loaded at its centre of gravity) each leg supports one third of the weight.

§ 224. Examples in which the resultant is a single couple are found in rigidly magnetized bodies placed in a uniform magnetic field. As the amounts of N. and S. magnetism in a body are always equal, there is no force of translation in a uniform field. The resultant couple depends for its magnitude on the orientation of the body, and the positions of equilibrium are those for which its moment vanishes.

§ 225. Let P at x, y, z, be one of a system of parallel forces, their direction cosines being λ, μ, v. Let Q be the resultant force, and R, with direction cosines λ', μ', v', the axis of the resultant couple. Then our conditions become

Q=S(P),

λ'R = S[P(ry-μz)], μ'R = 2[P(λz- vx)], v'R=2[P(μa!-λy)] .

The last three equations give the following conditions determining R, λ', μ', ν': —

λ'R + j α2(Pz) - v2(Py) = 0, -λ2(Pz)+√R + r2(Pz) = 0,

+ λ2(Py) - μ2(Px) + PR =0.

From these we have the equation of condition λλ' + μμ' ÷ vv = 0 , showing that the axis of the couple is at right angles to the common direction of the parallel forces.

We have also

R 2 =(2(Pz)) 2 + (2(Py)) 2 + (2(Pz)) 2 - (λ2(Px) + m 2(Pz∕) + v2(Pz)) 2 .

This expression is of the same form as that in § 77, and we therefore conclude that, if λ", μ", v" be the direction cosines of a line in the body such that

λ μ v

2(PÎ) = 2(Pÿ) = 2(K) ’

the magnitude of the resultant couple is directly as the sine of the angle between this line and the common direction of the parallel forces. In fact the mere form of the three equations above proves this result.

In the case of a body of moderate dimensions, acted on by gravity, P is the weight of the element at x, y, z, and therefore proportional to its mass, so that if the centre of inertia be taken as the origin we have

2(Pz∙) = 0, 2(Pτ∕) = 0, 2(Pz) = 0, and there is no couple. The whole effect is therefore the same as if the mass were condensed at the centre of inertia.

In the case of a magnet,

2(P)=0, and there is no translatory force. The couple, as we have seen, depends upon the orientation of the body as regards the direction of the line of the earth’s magnetic force.

§ 226. We have seen that any system of forces acting on a rigid body may be reduced to a force and a couple; also that when the force is in the plane of the couple the resultant can always be put in the form of a single force acting in a definite line in the body. When the force is not in the plane of the couple, we may resolve the couple into two components, the plane of one being parallel, of the other perpendicular, to the force. The first, when compounded with the force, merely shifts the line in which it acts. Thus any system of forces may be reduced to a single force, acting in a definite line called the “central axis,” and a couple in a plane perpendicular to it. One of the forces of the couple may now be compounded with the single force, and thus we obtain, as the resultant of any system of forces, a pair of forces in non-intersecting lines not perpendicular to one another. This is only one of an infinite number of ways in which fancy, or convenience, may lead us to represent the equivalent of a group of forces. Many very curious theorems have been met with in investigations on this subject. For instance, by compounding one of the forces of the resultant couple with the resultant force (not now necessarily perpendicular to its plane) we have a system of two forces acting in non-intersecting lines. Then we have the following curious proposition, which may easily be proved from the formulae already given :—

When a system of forces is reduced in any manner what ever to two, the volume of the tetrahedron of which these are opposite edges is constant.

§ 227. The most symmetrical pair of resultant forces is found thus. Take any point P (fig. 61) in the central axis, and draw through it a lihe APA', perpendicular to it, and bisected at P. Substitute for the single force at P its halves acting at A and A' respectively. Combine these respectively with the forces AQ, A ’ Q' of the couple when AA' is made its arm. The system is thus reduced to two equal forces AR, A'R', whose directions are interchangeable by a rotation of two right angles about the central axis.

Let P with direction cosines λ, μ, v be one of the forces, and let x, y, z be its point of application. Then

X = 2(Pλ), Y = S(P∕χ), Z = S(P^) are the components of the single force at the origin. Also I. = 2[P(vy - μz)]

M = 2[P(λz-va)] N=S[P(j uα-λy)] are the components of the resultant couple.

If we shift the origin to the point a, b, c the first three quantities are unaltered, but the couples become

I.'= I. + cY-Z>Z M'=M + αZ-cX N'=N + Z>X-αY.

The point a, b, c is on the central axis if the axis of the resultant couple be parallel to the single force, i.e., if

⅜=γ=y= e > su PP ose ; or I. = eX - cY + bZ,

Μ = cX + eY - aZ , N≈ -bX + aY + eZ.

Either of these sets gives the equations of the central axis.

The resultant force and couple are in one plane, and therefore the resultant is a single force in the central axis, when

I.'X + M'Y + N'Z = 0.

By the values of I.', M', N', above, we see that this is equivalent

LX + MY + NZ = 0.

When this last conditionis not satisfied, we see that the value of the left hand member which, from the way in which it occurs, must obviously be an invariant, is e(X 2 + Y 2 + Z 2 ), where e has the same value as in the three equations above.

§ 228. One of the most remarkable of the many curious theorems connected with the single resultant of a system of forces is that of Minding. We have seen that, in general, the resultant may be put in the form of a single force and a couple in a plane perpendicular to it. Tf we now suppose the system of forces to be shifted into a new position such that their points of application, their magnitudes, and the angles between their directions two and two, all remain unchanged, the resultant force will be of the same magnitude as before, but the couple will in general be different. Of the infinitely infinite number of possible positions which the forces may assume, an infinite number correspond to a zero couple. Minding has shown that the lines of action of these single resultants consist of all lines passing through each of two curves, fixed in the body, an ellipse and an hyperbola, in planes perpendicular to each other. The proof of the proposition gives an interesting example of the use of Rodrigues’s coordinates (§ 83).

The most obvious mode of attacking this question would be to resolve the applied forces into three groups, parallel respectively to three rectangular axes which revolve with them, and to choose those axes so that the sum of the resolved parts does not vanish parallel to any one of the three. Each of these systems of parallel forces has its own “centre” (§ 222),—so that the final resolution gives three forces, each of a given magnitude, acting in any mutually perpendicular directions at three definite points in the body. This, however, is not analytically so simple as the following.

We refer the body to fixed axes Ox, Oy, Oz, to be afterwards specified. As the origin and the directions of these axes are at our disposal, we may impose six conditions. Now suppose the forces to be resolved parallel to a set of rectangular axes Oa∕, Oy', Oz! which will be considered afterwards to rotate with them. Such a system of axes may, at starting, have any assigned position. This gives us three conditions more. Let then A, B, C be the components, parallel to the second set of axes, of the force applied at the point whose coordinates referred to the first system are a, b, c. Let the direction cosines of the second system in any of its future positions, referred to the first system, be l 1 , m 1 , n 1 ∙, l 2 , m 2 , n 2 ; l 3 , m 3 , n 3 respectively.

Then the force at a, b, c has the following components :—

AZ 1 + BZ 2 + CZ 3 , parallel to Ox Aτn 1 + Bw⅛+Cw⅛, ,, ,, Oy An 1 +Bji 2 +Cn 3 , ,, ,, Oz.

The expressions for the resultant force and couple at the origin will evidently depend upon the following twelve quantities, besides the direction cosines, viz :—

SA, SB, SC,

S(Aα), S(A⅛), S(Ac), S(Bα), S(Bδ), S(Bc), S(Cα), S(CZ>), S(Cc).

Assume SB = 0, SC = 0, i.e., let Ox ’ be always parallel to the direction of the resultant force. Next, let

S(Aα) = 0, S(Aδ) = 0, S(Ac)=0, i.e., let the origin be chosen as the “centre” (§ 222) of the forces parallel to the resultant force. As we have still four conditions to impose, we select the following:—

S(Bα) = 0, S(Bc)=0, S(Cα) = 0, S(Cδ) = 0.

These express that the plane of the couple due to the forces C passes through Oy, while that of the forces B passes through Oz.

Write now

5(A) = gι, S(Bδ) = ^8, 2(Cc) = ⅛ γ .

The force and couple at the origin are c¼, ⅜,m 1 , '*·> S 7 h »

e^(n 2 β — m 3 y), -

These are equivalent to a single force if (§ 227) (lι n 2 ~ n 1 l2)β ~ {l1 m 3 ~ nι 1 ι 3 ')y =0, or ¾ j 8-n 2 γ = 0 (1).

This is the required condition. When it is satisfied, the equations βf the line in which the single force S∙ acts are any two of n 1 η-m 1 ζ≈ n 2 β-m 3 y, } I] C ~ n ιζ = ⅞7 > f (2)> m 1 ξ — l 1 η = — l 2 β ) the condition that these three agree being (1).

Eliminate l 1 between the last two, and we get ξ(m 1 ζ-n 1 η) = l 3 yη-l 2 βζ (3).

Now introduce in (1), in the first of (2), and in (3), Rodrigues’s values of the cosines (§ 83), and they become respectively (yz - wx')β - {yz + wx~)y= 0, (xz - wy)η - (wz + xy)ζ= (yz + wx^)β — (yz — wx~)y , ζξ(wz+xy) - ξη(xz - wy) = (xz + wy)yη - (xy - wz)βζ.

Rearranging according to y, z, and yz, (β-y)yz-(β + y)wx≈O, (β - y y )y z + ( w, ι + χ C)y - ( χ v ~ w C') z +(0+ f) wx = θ >

[^(ξ + β) χ +^ξ- y)^]y +[<U- β^ w ~ r ι⅛ +7»=θ > the second of which may be put, by means of the first, in the form (wη + xζ)y - (xη - wζ)z + 2(β + y)WX = 0.

These three equations involve w, x, y, z in the form of the ratios only of the last three to the first. The last two are linear in JL Solving them, and substituting in the first we find, w ’ w finally, a biquadratic in A .

Hence, if particular values be assigned to ξ, η, ζ, we find four values of ® . Thus, in general, there are four positions of the single resultant force passing through any point.

But, without forming the biquadratic, we may easily obtain Minding’s theorem. Suppose we seek the locus of all points in which the plane ξη can be cut by the line of action of the single force. We have C=0, and the equations above are reduced to (β-y)yz -(3 + γ)wz =0, ∙η(wy - xz) + 2(3 + γ)wx= 0 , (ξ-γ)wy-(ξ + γ),τs =0. From the last two we find

-7nz=(3 + 7)(ξ-7>, ~7 γ h∕ = (3 + 7)(< + 7>, so that finally, by the first,

(3 2 - T 2 )(i 2 ξ 7 2 ) = y 2 η 2 , ° r S-β⅛= 1 ω∙

7 p 7

Had we put η = 0, we should have found, by a similar process,

3 2 3 2 -7 2 ~ 1 (5) '

(4) and (5) represent an hyperbola and an ellipse, or an ellipse and an hyperbola, respectively, according as β' 2 is greater or less than y 2 . In either case the vertices of the hyperbola coincide with the foci of the ellipse; so that the two curves are linked together.

It is now easy to see that, from any assigned point of space, the two curves will appear to intersect one another in four points. Two, or all, of these may in special cases coincide. Lines drawn to these points give the four positions of the single force which can pass through the assigned point.

Examples of Statical Methods and Theorems.

§ 229. Suppose a ladder to be leaning against a vertical wall. If there be no friction, what force, applied at the ' lower end, will just suffice to support it?

In the treatment of all questions of this kind the student should commence by making a rough sketch of the situation, indicating all the forces concerned, with the directions in which they act. As shown in fig. 62, the wall exerts an outward thrust S.on the upper end of the ladder, the ground an upward thrust R on the lower end. The only other force is gravity, which may be supposed to produce a downward force at the middle of the ladder, equal to its whole weight. Unless there be some other horizontal force to balance S, - the ladder will obviously slide down. Suppose then a horizontal force F to be applied at the lower end, and let the ladder be inclined at an angle α to the horizon. Then our conditions become horizontally S - F=0, vertically W-R=0, and for the couple in the plane of the figure, I being the length of the ladder,

⅜WZ cosα - SZsina = 0.

[The last equation is obtained by taking moments about the lower end of the ladder, this point being chosen (§ 217) because the directions of two of the forces pass through it.] From these equations we find at once

F = S = ⅜W cota.

It is to be observed that the requisite force F is very small while the ladder is nearly vertical, but increases without limit as it becomes more nearly horizontal.

§ 230. Next let us vary the question by supposing the coefficient of friction on the ground to be μ. The equations are precisely the same as before, and the limiting value of α for which equilibrium is possible is now to be found by putting F= m R- m W.

Thus 2μ = Cθta gives the smallest value of α for which equilibrium is possible. For any larger value of α less friction is called into play.

§ 231. If next we assume the wall also to be rough, a new friction force, G, comes in. The equations (for any given value of α) are

S-F≈0,

W-R-G = 0, ⅛ΛVZcosα-SZsinα-GZcosα = 0 .

Here there is a certain amount of indeterminateness which our formulae cannot escape (although of course it does not exist in nature) so long as we are not dealing with the limiting case in which motion is about to commence. In that case we have the additional conditions

G= z *S, F= m R.

Thus, in all, there are five equations. These are requisite and necessary because there are four forces S, G, R, F to be determined, as well as the special value of the angle α. The result of eliminating the four forces is + 1 - μ? tanα= c , .

<~∙∕d>

§ 232. We may still further vary the question by supposing a man of* weight w to ascend the ladder. Let e represent the fraction of the ladder’s length which he has ascended. The equations are

S-F=0,

W+w-R-G=0,

(⅛W + ew)Zcosα 1 -SZsinα 1 - GZcosa 1 = 0.

Introducing the condition that slipping is just about to commence, we obtain w (l÷μ 2 )e-μ 2 tan α 1 = W l- ι ⅛ 2 tan a t w

W

where a has the value given in § 231. Hence the limiting angle is increased or diminished by the load on the ladder according as

2(l + j u 2 )i-2μ s >l- j u 2 , i.e., 2e^l.

The ratio w/W does not appear in this condition. But it shows its importance when e is either greater or less than ∣.

Hence, when the ladder is just about to slip, a man makes it more stable if he stands anywhere on the lower half of it, but brings it down if he mounts higher. We conclude that, so far as sliding is concerned, it is advantageous to make the lower half of a ladder more massive than the upper half.

§ 233. Suppose a ladder, with its lower end resting against a wall, to be supported by a horizontal rail parallel to the wall (fig. 63). This case is chosen because it illustrates definite limits within which stability is ensured.

Let a be the half length of the ladder, a its inclination to the horizon, b the distance of the rail from the wall. Suppose the ladder in such a' position that if there were no friction it would slip downwards. Then the equations of equilibrium are

R + G cos α - S sin a = 0,

F + Scosa + Gsina-W = 0,

S&seca - Wacosa = 0.

In the third of these equations the lower end of the ladder has been chosen as the point about which moments are taken, because the lines of action of three of the forces pass through it. Here again there is indeterminateness, because there are two places at which friction comes in, and we do not know at which it is most freely exerted. But if the whole be on the point of slipping, we have as before the additional data F= m R, G= j uS.

These lead to the equation

(1 - μ 2 ) cos<x + 2μsiu α = - sec 2 a .

If we introduce an angle’ v, such that

1 - u 2 . 2u c ° sr = ’ SlΠJ∕ = τ - ii - 3

1 + μ 2 1 + μ this equation becomes o ∕ X 2l & ll V cos 2 αcos (« - v) = - 7 - ... ≈ - cos 2 - , a(l+μ 2 ) a 2

the right-hand member of which must necessarily be less than 1.

This determines the lowest position of the lower end consistent with equilibrium, and the mere change of sign of ∕z, and therefore of v, alters it into the equation for the highest. The signs of the friction terms are changed when the direction of slipping is supposed to be reversed.

Kinetics of a Rigid Solid.

§ 234. The motion of a rigid body is, as we have seen, completely determined when we know the motion of one of its points and the relative motion of the body about that point. The point usually chosen is the centre of inertia of the body, and the investigation of its motion comes under the kinetics of a particle, which we have already sufficiently discussed. For we are permitted to suppose the whole mass to be concentrated at that point, and to be acted on by all the separate forces, each unaltered in direction and magnitude. Hence we may now confine ourselves to the study of the motion about the centre of inertia which, for the moment, we may look on as fixed.

To illustrate, in a very simple manner, the new conceptions which are required for the study of this question, let us take a uniform circular ring of matter, of radius R, revolving with angular velocity ω about an axis through its centre, and perpendicular to its plane. Its moment of momentum is obviously

Μ . Rω . R or MR 2 . ω.

Its kinetic energy is

⅛M(Rω) 2 or ⅛MR 2 . ω 2 .

If it be acted on by a couple C, in its plane, C is the rate of increase of the moment of momentum, or MR 2 . ώ = 0.

The work done by the couple in time St is CωSt, and the increase of kinetic energy is MR 2 . ω<i>St.

By equating these we have (after dividing both sides by ω) the same equation as we obtained from the rate of increase of moment of momentum. It will be observed that these equations are of exactly the same form as those for the motion of a particle parallel to one of the coordinate axes, only that ω takes the place of a velocity (such as x) while the expression MR 2 takes the place of Μ, and the right-hand side is the moment of a force, not a force simply.

§ 235. Hence, generally, we define as follows :—

Def. The “moment of inertia” of a body about any axis is the sum of the products of the mass of each particle of the body into the square of its (least) distance from the axis.

The following theorem enables us at once to find the moment of inertia about any line, as axis, from that about a parallel axis through the centre of inertia.

Let the line be chosen as the axis of z, then the moment of inertia about it is

Sτn(x 2 + y 2 ).

But, if x, y be the coordinates of the centre of inertia, ξ, η the coordinates of m with reference to that centre, we have x=χ + ξ, y=y + ∙η, and the above expression for the moment of inertia becomes

2m(x a + y i + 2xξ + 2y-η + ξ 2 + η 2 ).

By the property of the centre of inertia, § 109,

2(wιξ)≈=0, 2(∞η) = 0.

Hence the above expression consists of two parts:—

SM7(ξ 2 + √)

the moment of inertia about a parallel axis through the centre of inertia, and

2(rø) . (x 2 + j7 2 )

the moment of inertia of the whole mass supposed concentrated at its centre of inertia.

§ 236. Hence we need study only the moments of inertia about axes passing through the centre of inertia. But we will commence with an origin assumed at hazard.

If the direction cosines of an axis through the origin be λ, μ,, v, the square of the distance of the mass m at x, y, z from it is x 2 + y 2 +z s - (λx + μy + vz) 2 .

Hence the moment of inertia is

$ = Sm(z 2 + y 2 + z 2 -(λx + μy + vz) 2 )

= Swι((y 2 + z 2 )λ 2 + (z 2 + x 2 )μ 2 + {x 2 + y 2 )v 2 - 2xyKμ - 2yzμy - 2zxvλ')'), which may be written

JI = Aλ 2 + 2G 3 λ j u + B j u 2 + 2Gjμv + 2G 2 rλ + Cv 2 .

If we measure off, on the axis, a quantity p whose square is the reciprocal of $, and call its terminal coordinates ξ, η, ζ, this equation becomes by multiplying both sides by p 2

1 = Aξ 2 +2G 3 ξ η + Bη 2 + 2G 1 √+ 2G 2 <ξ + C< 2 .

As the moment of inertia is essentially a positive quantity, this equation represents an ellipsoid. It must of course have three principal axes; and, when these are taken as the coordinates axes, the terms in ξη, ηζ, and ζξ in the above expression must disappear.

§ 237. Hence at every point of every rigid body there are three “principal axes” of inertia, at right angles to one another. One of them is the axis of absolute maximum moment, another that of absolute minimum.

Our equation now becomes, when referred to these axes, l = Aξ 2 + B√ + Cf 2 , or, dividing by p 2 ,

Jf-= Aλ 2 + Bμ 2 + Cj∕ 2 .

Thus the moment of inertia about any axis is found from those about the principal axes at that point by multiplying each by the square of the corresponding direction cosine, and adding the results.

For the quantity A was written originally as

Sτn(y a +z 2 ), i.e. , it is the moment of inertia about the axis of x. We see also that, at every point of a body, there are three rectangular axes such that the expressions ⅛(mxy), ⅛(myz'), 2(mzx) vanish when these are taken as coordinate axes.

To find how these axes are distributed in a body, let us suppose it referred to the principal axes through its centre of inertia, and let M⅛ 2 , M⅛ 2 , M⅛∣ be the moments of inertia about them. The quantities fc 1 , ⅛ 2 , ⅛ 3 are called the principal “radii of gyration.” Then, by the results above, the moment of inertia about a line λ, μ, v through the point a, β, y is I = Μ {α 2 + j 8 2 + γ 2 - (λα + μβ + rγ) 2 } + M(λ 2 ⅛ 1 2 + ∕* 2 fc* + F⅛ 2 ).

For a principal axis this is to be a maximum or minimum, with the sole condition

λ 2 + √ ’ +√≈=l.

Hence, if p be an undetermined multiplier, we have (I'J +jt>)λ - α(αλ + βμ ÷ yv) = 0 , (&Î +p)μ - 3(aλ + βμ + yv) = 0 , (I∣ +p)r - 7(aλ + βμ + yv) — 0 .

But, if we consider a surface of the second order ae 2 y 2 z 2 _ k t l +p + k* 1 +p + k a +p ’ confocal with the ellipsoid 3 ,2 ,.2 2 2

ιf + ⅛ + I 2 ^≈ 1, the direction cosines of its normal at x, y, z are X y z λ. μ .v . . ’ k ∖ + p ' k i a +p '

Hence, if this surface pass through the point a, β, y, we have (⅛J+p)λ=Pα, (* 2 + p ) m =P j 3, (⅛*+^> = Pγ, where P is determined by the equation βλ + 3^7-Piv⅞-÷v⅛- + ⅛>P. ∖∕q+p k i t +p kl+pj

Substitute this value of P in the preceding equations, and they become identical with those above given for determining the principal axes at a, β, 7. Hence Binet’s Theorem :—

The principal axes at any point of a body are normals to the three surfaces of the second order which pass through that point and are confocal with the ellipsoid (a).

§ 238. We will here tabulate the values of the moments of inertia about principal axes through the centre of inertia, in a few specially useful cases.

1. Plane uniform circular disk.

Divide it into concentrie rings, of radius r, of breadth Sr. Then the moment of inertia about the axis through the centre, and perpendicular to the plane, of the circle is ∕~ a p∕ 2πr 2 dr = ⅛πα 4 ρ , where a is the radius, and p the mass of a square unit, of the disk. But the mass is 1ra 2 p, so that k^=⅛a 2 .

This of course, applies to a circular cylinder. Obviously, in the disk

**-*∙-i*∙-∣<A

In fact the moment of inertia about an axis drawn perpendicular to any plane figure at any point is equal to the sum of the other two about rectangular axes which lie in the plane. The one is 3,m(x 2 + y 2 ), and the others are ½mx 2 and ⅛my 2 respectively.

2. Uniform rod of length I, p mass per unit length.

M = 7 p , MI 2 = 0, MI 2 = MI 2 = 2p∕ x 2 dx=S--, r 1 2 3 J/o 12

so that *J-fc 2 -⅛F.

3. Uniform rectangular plate, sides a and b, axis parallel to b. a Mk* = 2bp∕" i X 2 dx^p, JO 12

so that k ∖ = ⅛α s , and k 2 i = τ ⅛δ 2 .

Hence, by the remark above, ⅛j-⅛(α 2 + δ 2 ).

4. Uniform sphere, radius a, p mass per unit volume. Here 2(mx 2 ) = 2(nψ 2 ) =* S(wi2 2 ), and therefore the sum of any two is = j 5,m(x 2 + y 2 + z 2 ).

Thus MI 2 -MI 2 = MI 2 = linp^ a r i dr = τ 8 5 π∕>α 5 .

But M=⅜τrpα 3 , and thus k ∖ = k 2 = ⅛ 2 = f a 2 .

5. Plane uniform elliptic disk, semiaxes a,b; p mass of unit area. Moment of inertia about a is

ιtt∙-⅛∕*⅛ fa -i≤⅛, 1 J 0 3 4

so that k ∖ = ∣δ 2 .

From this follows immediately

6. Ellipsoid, semiaxes a, b, c, and of uniform density :— I-≡ = ⅜(δ 2 + c 2), ⅛ 2 -⅜(c 2 + α 2 ), k t 3 = t(a 2 + b 2 ).

From these we can, of course, reproduce the result for a sphere.

7. Rectangular parallelepiped, edges a, b, c :— k 2 l - ⅛ (δ 2 + c 2 ), k 2 ≈ ⅛ (c 2 + α 2 ), * 2 = ⅛ (α 2 + δ 2 ).

The determination of moments of inertia is, like that of centres of inertia, a purely mathematical matter, the full discussion of which would lead us away from the proper objects of this article.

§ 239. The simplest cases that can present themselves so far as rotation is concerned (for the translational effects on a rigid body are treated precisely as if it were a mere particle,—a process already sufficiently illustrated) are those in which there is one degree of freedom to rotate, i.e., when the body is rigidly attached to a fixed axis. Here the physical condition is simply that the rate of increase of moment of momentum is equal to the moment of the resultant couple about the axis of rotation.

§ 240. Let us recur to Atwood’s machine as a first example, and suppose the string not to slip on the pulley, so that the pulley must turn. In this case we must observe that the two free parts of the string are now, as it were, separate strings, so that we have no right to assume their tensions to be equal. In fact if they were equal there would be no acceleration of the rotation of the pulley, nor of course of the common velocity of the two masses. We assume that the pulley is symmetrical, and the axis through its centre of inertia.

Let a be the radius of the pulley, and ω its angular velocity, then α< l > is the linear velocity of either mass. Thus the linear acceleration of each of the masses is equal to a times the angular acceleration of the pulley. But the linear acceleration multiplied by the mass is the measure of the force producing it; while the angular acceleration multiplied by the moment of inertia is the measure of the moment of the couple producing it. Thus we have (Μ being the mass of the pulley, and k its radius of gyration)

MI 2 × angular acceleration = (T' - T)α, m' × linear acceleration = m'g - T', m × linear acceleration = T - mg.

Eliminating T and T', and taking account of the above relation between the accelerations, we find at once

Linear Acceleration = . m — w * ., n g ; m ’ + m + Mk 2 ∕a 2i, ’ from which, by the last two of our equations, the separate values of T and T' may be found.

If we compare this result with that obtained in § 173, on the supposition that the pulley was perfectly smooth, we see that the only difference is in the addition of M^ 2 ∕αto the sum of the two masses. Otherwise the nature of the motion remains unaffected.

§ 241. Let us next take the case of a body of any form attached to a horizontal axis which does not pass through its centre of inertia. In such a case gravity is the force producing motion, and we have what is called a “compound pendulum.” Draw through the centre of inertia a line parallel to the axis; let h be the distance between these lines, and θ the angle which their plane makes (at a given time) with the vertical. The moment producing angular acceleration is obviously

- fflgΛsiπ0.

Divide by the moment of inertia about the axis, which by a previous proposition (§ 235) is m(k 2 + h 2 )

(where k is the radius of gyration about the line drawn through the centre of inertia), and we have for the angular acceleration gh sin θ ~'k 2 + h 2 ’ In the case of a simple pendulum of length I, we saw that the angular acceleration is

øsinØ ~ ·

Hence the motion of the compound pendulum will be identical with that of the simple pendulum when, and only when,

,J⅛i s - a √⅛e.

h h

As h and k are necessarily positive (or rather signless) quantities, the smallest value of I is evidently when k — h. Hence the shortest time in which the mass can vibrate about any axis parallel to the original one corresponds to that of a simple pendulum of length 2k. When h is made either less or greater than k, the length of the equivalent simple pendulum increases, and for any assigned value of I greater than 2k there are two corresponding values of h, one less and the other greater than k. Their sum, however, as we see by the coefficient of the second term in the equation

A, 2 -ZΛ + ⅛ 2 = 0, is always equal to I.

If then we can find two parallel axes in a rigid body, lying in one plane with the centre of inertia, and on opposite sides of that point, such that the time of oscillation is the same for each, the distance between them is the length of the equivalent simple pendulum. Kater made use of this proposition in his determination of the length of the second’s pendulum, under the circumstances in which it was defined by Act of Parliament as a datum for restoring, in case of loss, the standard yard.

§ 242. Suppose now that a second body is attached to the first by an axis parallel to that about which the first is constrained to move; and, for simplicity, suppose the centre of inertia of the first body to be in the plane containing the two axes. Here we have a complex compound pendulum, and it is interesting to compare the motion with that of the complex simple pendulum of §§ 177, 178.

Let m', h ’ , φ correspond, for the second body, to m, h, θ for the first, and let a be the distance between the axes. For variety we will adopt Lagrange’s method. We have clearly

T = ⅜ (mk 2 0 2 + mh 2 0 2 + m'k' 2 φ 2 + m ∖ a 2 ∂ 2 + lι' 2 φ 2 + 2α7√cos (≠ - θ y ) φ0), V=C - mgh cos θ - nι'g(a cos θ + 7√cos φ).

These would enable us at once to write down the equations of motion, however large be the disturbance, but they are too complex for our present work. Let us then assume φ and θ to be very small, and we have {m(k 2 + 7i 2 ) + m , a 2 }θ + m'dh'φ = - (mh + m'a)gθ, τn'(Z∕ 2 + h' 2 )φ + m'alι'θ ≈ — m'gh!φ.

Combining, as before, by means of an undetermined multiplier, we have

(τn(⅛ 2 + 7i 2 ) + m ’ a 2 + κm'ah')f) + (m'alι' + λm'(7∕ 2 + lι' 2 ))φ = ~ g {(gnh + m'a)θ + ∖ m'h ’ φ) .

Thus the two values of λ are given by the equation m'dh' + λm∕(⅛' 2 + 7⅛' 2 ) κm'h ’ m(Jc 2 + Λ 2 ) + m'a 2 + κm'ah ’ mh + m'a, '

This may be written in the form

A + λ λ

B + λ~C ’ where B is greater than A; and A, B, C are all essentially positive, if the bodies have been only slightly displaced from the position of stable equilibrium. The equation gives

λ 2 + (B-C)λ-AC = 0, so that the values of λ are essentially real and of opposite signs. If we write μ - B for λ, this equation becomes √≈-(B + C) m + (B-A)C = 0, so that the values of λ + B are both positive, and therefore the motion of eitheι-mass is the resultant of two simple harmonic motions.

§ 243. A well-known puzzle in connexion with this subject used to be “How to distinguish between two hollow shells, one of gold the other of silver, if their diameters and masses be alike, and both be painted.” If we observe that the volumes of equal masses are inversely as the densities, the volume of the gold shell is seen to be less than that of the silver one, and therefore, on the whole, its mass is farther from the centre, and its moment of inertia greater. Hence any form of experiment in which the moment of inertia comes in will suffice to decide the question. Thus they might be alternately clamped tight to the end of a rod, and the system swung as a pendulum, when the gold sphere would vibrate more slowly than the silver one. Or they might be allowed to roll, not slide, down a rough plane. In this case the work done by gravity on each is the same when they have fallen through equal spaces. But its equivalent is in the form of kinetic energy, partly translational and partly rotational. The relative amounts of these two depend on the moments of inertia of the spheres, for the ratio of the translational velocity to the angular velocity is the same for each. Hence the gold sphere, having the greater moment of inertia, will have the smaller velocity of translation. Another form of this question was to have a shell with a spherical mass inside, which might be either free to rotate on gimbals, or else be keyed to the outer skin. The keying would of course retard the motion of the whole down a rough plane, for part of the energy due to gravity would then be shared by the internal mass in the form of energy of rotation, from which it would otherwise have been free. Another very instructive form is that of a spherical shell full of fluid. If the fluid be perfect, the moment of inertia is that of the shell alone; if it be infinitely viscous, the moment of inertia is that of shell and fluid as if they constituted one rigid solid; and we may have every intermediate amount. If we suppose the rotation of the outer shell to be suddenly stopped, the infinitely viscous contents would be reduced to rest also. But if they be not infinitely viscous they will not at once be brought to rest, but will be able to put the shell in rotation again if it be at once set free. Thus, in practice, we can tell a raw egg from a hard-boiled egg. The first is with difficulty made to rotate, and sets itself in motion again if it be stopped and at once let go. The second behaves, practically, like a rigid solid.

§ 244. The problem of the rolling of a sphere down a rough inclined plane is solved at once, as above, by applying the conservation of energy. For, if x be the coordinate of its centre parallel to the plane, θ the angle through which it has turned, and a its radius, we have the kinematical condition x=aθ

(due to the perfect roughness of the plane).

Also the potential energy lost is

Musina, where a is the inclination of the pl⅛ne to the horizon; and the kinetic energy gained is made up of the two parts,— ∣M⅛ 2 translational, and ∣MO 2 rotational.

Hence M(⅛ 2 + α 2 )^ 2 =2M<∕α0sinα, or x 2 = 2 τ ~-÷xsina. k i + a i

This shows that the motion is the same as that of a particle sliding down a smooth plane of the same inclination, under gravity diminished in the ratio α 2 : ½ 2 + α 2 . And it shows how friction may retard motion without producing any dissipation of energy.

§ 245. Suppose one point of a rigid plane sheet be made to move in any manner in the plane of the sheet, what will be the consequent rotation Ί

Let Μ be the mass, and ξ, η, given in terms of t, the coordinates of the point of the sheet whose motion is assigned. Let a, θ be the relative polar coordinates of the centre of inertia, then

M[ξ + α(cos0)] = X , M[^+α(sin0))=Y,

M⅛ 2 0 = - Yαcos0 + Xαsin0; where X and Y are the forces requisite to produce the motion. Eliminating them, we find (⅛ 2 + a-~)∂ = - a(ij cos θ - ∣'sin θ'), with which we can do no more until further data are specified.

Suppose ξ, η to move with uniform acceleration p in a direction assigned by α, then

ξ = pcos<x, ⅛ - =27sina, and (Jc 1 + a 2 )θ = - αp(sinαcos0- cos a sin θ)== opsin (θ - a).

The centre of inertia of the mass therefore moves, relatively to the constrained point, precisely as does a simple pendulum; but the direction of p is reversed.

Again suppose the constrained point to move uniformly in a circle of radius b, with angular velocity ω. We have

ξ = δcosa√, p = δsin<x√, and (⅛ 2 +α 2 )0 = + ω 2 aδ(sin ωt cos θ - cos ωt sin θ), or (0 —ωi) = — w 2 aδsin(0- ωt).

This is, again, the equation of motion of a simple pendulum, but the angle of displacement θ-ωt is no longer measured from a fixed line but from the uniformly rotating radius of the guide circle. Hence the mass oscillates, pendulum-wise, about this uniformly revolving line.

§ 246. Let us take, as an instance of impulse, the case of Robins’s “ballistic pendulum,”—a massive block of wood movable about a horizontal axis at a considerable distance above it,—employed to measure the velocity of a cannon or musket shot. The shot is usually fired into the block in a horizontal direction perpendicular to the axis. The impulsive penetration is so nearly instantaneous, and the mass of the block so large compared with that of the shot, that the ball and pendulum are moving on as one mass before the pendulum has been sensibly deflected from the position of equilibrium. This is the essential peculiarity of the ballistic method,—which is used also extensively in electromagnetic researches and in practical electric testing, when the integral quantity of the electricity which has passed in a current of short duration is to be measured. The line of motion of the bullet at impact may be in any direction whatever, but the only part which is effective is the component in a plane perpendicular to the axis. We may therefore, for simplicity, consider the motion to be in a line perpendicular to the axis, though not necessarily horizontal.

Let m be the mass of the bullet, v its velocity, and p the distance of its line of motion from the axis. Let Μ be the mass of the pendulum with the bullet lodged in it, and k its radius of gyration. Then, if ω be the angular velocity of the pendulum when the impact is complete, % mιp = M7t 2 ω, from which the solution of the question is easily determined. For the kinetic energy after impact is changed into its equivalent in potential energy when the pendulum reaches its position of greatest deflexion. Let this be given by the angle θ ; then the height to which the centre of inertia is raised is ⅞(1 - cosθ), if h be its distance from the axis. Thus

M^(l-cos0)=⅜M⅛≡ w ≡-⅜¾^, o . θ m p υ or 2sm —= —-Ç--—-,

2 Μ k ∖ ∕gh, an expression for the chord of the angle of deflexion. In practice the chord of the angle θ is measured by means of a light tape or cord attached to a point of the pendulum, and slipping with small friction through a clip fixed close to the position occupied by that point when the pendulum hangs at rest.

§ 247. As another example of impulse let us consider the case of a body moving in any way in a plane perpendicular to one of its principal axes. It is required to find what point of the body must be suddenly fixed in order that the whole may be brought to rest; also, what will be the consequent impulsive pressure at this point. It is easy to see that this is exactly the same question as to find the impulse, and its point of application, so that it may produce a given motion of a body in a plane perpendicular to one of its principal axes.

The impulse must obviously act in a plane passing through the centre of inertia. And the physical conditions are that the change of momentum of translation is equal to, and in the direction of, the impulse, while the change of moment of momentum about the centre of inertia is equal to the moment of the impulse. Let the impulse acting at the point ξ, η have components R, S parallel to rectangular coordinates in the plane of motion, and let ω be the angular velocity, u, v the linear velocities, generated by it. Then the physical conditions are

Mu==R, Mv=S, MZ,"ω = Sξ — Rη,

When u, v, ω are given, R and S are found from the first two equations, and the third is then the equation of the line in which the impulse must act. Similarly, when the impulse and its line of action are given, we have in terms of these data the quantities u, v, ω.

§248. Asa simple practical example, suppose one strikes a hard object with a stick in such a way that his hand is at rest at the instant*of the impact; with what part of the stick must he strike so that there may be no jar on his hand?

Let £ be measured along the stick from its centre of inertia in the direction which it has at the instant of impact. Then the kinematical condition is v≈aω, where a is the distance from the hand to the centre of inertia of the stick. Thus, as the impulse S is the sole cause of the stick’s being brought to rest, we have

Mv + S = 0, Mfc⅛∕α + Sξ=0; so that ξ=÷k 2 [a.

Hence if the stick be uniform and be held by one end, so that its length is 2a, and therefore 3λ 2 = α 2 , we have

ξ=⅛α; and a + ξ, the distance of the point of impact from the hand, is

α+ ⅛α=f. 2α , i e., it is at two-thirds of the length of the stick.

If, however,the hand be moving, at the instant of impact, perpendicularly to the stick with velocity V, the kinematical condition is v— V = αω, which introduces a corresponding change in the result.

§ 249. The reaction of the axis is easily calculated. If the axis about which the body is constrained to rotate be perpendicular to a plane though the centre of inertia about which the body is symmetrical, and if the applied forces act in that plane, it is clear that the reaction of the axis is a single force in that plane. Let its components be S and H. Then

2(wι¾)=2(X) + S, 2(w) = 2(Y) + H, '2.m(xij - yχ) = Σ(xY - yX).

Let a, θ, be the polar coordinates of the centre of inertia, then Ô, θ are the angular velocity and the angular acceleration for all particles of the mass, and we have

-Mαcos0.0 2 -Mαsin0.0 = 2(X) + ≡, -Mαsin0.0 2 + Mαcos0.0 = 2(Y) + H ,

M(fc 2 + α 2 )0 = 2(αY-τ∕X).

From the third we find θ, and the others give a and II.

When there is no plane of symmetry perpendicular to the axis, there must be two points of it at which reactions are exerted on the revolving body. Let the coordinates of these bearings be c, c , and the reactions there ≡, H, Z, S', H', Z' respectively.

Then we have the six equations (in which z = 0)

2(m⅛) = 2(X) + ≡ + S',

2(my) = 2(Y) + H + H', 2(τnz) = 2(Z) + Z + Z';

2wι(⅛ - yx) = 2(zY - 2/X), 2m(2∕z- z~U} = 2(∕∕Z - ≡Y) - cH - c ’ H ’, 2'>ι(≈8 - x'z) = 2(zX -xZ) + cs + c’a.'.

The fourth equation, as before, determines θ, and we have then four equations to determine S, S', H, H'. The remaining equation determines only the sum Z + Z ’. In fact by more or less perfect fitting we can throw more or less of the force parallel to the axis on one or other of the bearings. There is really no indeterminateness in nature, but we cannot get the information required to evaluate separately Z and Z'.

§ 250. When impulsive forces are applied to the body, exactly the same methods may be employed, with the exception that u' — u must be written for ⅛ ’, <fcc., and ω' - ω for θ. The quantities X, Y, Z, S, H, Z, S', H', Z' now denote impulses and not forces.

As a single example of the use of these formulae, take the case of a body rotating, under the action of no force, about an axis through its centre of inertia. Here 2(mae) = 0, &c., and a is constant. The first two of the six equations last written show that the pairs of forces S, S' and H, H' form couples. The fourth equation gives θ = ωt; and with this the remaining two become

2(wιz2∕).ω 2 = - cH - c'H'= - (c-c')H , 2(m).α> 2 = -c≡-c'S'= -(c-c')Ξ.

The multipliers of ω 2 are each zero if the axis of rotation be a principal axis, and thus, in this case, there is no stress perpendicular to the axis. When the axis is not a principal axis the left hand terms are generally finite but they vary as the body turns. It is easy to see, however, that together these terms constitute a constant couple always in a plane passing through the axis, rotating with the body and dependent directly on the square of the angular velocity. Thus, to analyse the factor ^2,{mzy), we note that z is constant, and

2∕ = rsin(α√ + α), where r, a were the polar coordinates of the mass m at time t = 0. Hence '2,{mzy) = sin ω∕2(wιzic 1 ) + cos ωt2(mzy 1 '), where ic 1 = rcosα and 2∕ 1 r =rsinα were the coordinates of m ati = 0. Similarly we have '2,(mzx) = cosω∕2(maq) - sin ωt2(mzy 1 ).

These expressions prove the preceding statements.

§ 251. Asa final instance of impulse in this branch of the subject, suppose that a rigid plate, moving anyhow in its own plane, has one of its points suddenly fixed, what will be the subsequent motion? Let the position in space at which the point is to be fixed be chosen as origin, and let the axis of x be chosen so as to pass through the centre of inertia at the moment of fixture. Then, if u, υ be the velocities of the centre of inertia, ω the angular velocity about it, a its distance from the point to be fixed, the conditions of the impact are u'=u - ≡∕M, √ = t>-H∕M, ω'=ω + Ha∕lΛk 2 ,

—three equations with five unknown quantities. But the conditions that the point in question is reduced to rest are evidently w = 0, √-ω'α=0.

These furnish the requisite additional data, and the solution is complete. If we eliminate H between the two equations which contain it, we have

⅛ 2 ω' + av' = k 2 ω + av, whence by the relation between v' and ω' we have (k 2 + α 2 )ω' = k 2 ω + av.

These equal quantities, each multiplied by Μ, represent respectively the moment of momentum about the point before and after its fixture.

§ 252. Thus, as a little consideration will show, we might have solved the problem at once, so far as the impulsive change of motion is concerned, by noticing that as the impulse is applied at the origin, the moment of momentum about that point will not be altered by it. In fact many problems, which present serious complexity when treated by the direct methods, are solved with comparative ease by such general considerations as the conservation of moment of momentum, or the conservation of energy. The first principle holds good when there is no resultant couple, or impulsive couple, round the origin; the second when no work on the whole is done by or against the forces or impulses.

§ 253. We have given instances of pure sliding, and of pure rolling, in one plane, and will now give a single instance of combined rolling and sliding. A common but instructive case of the problem we propose to consider is that of a hoop thrown forwards and at the same time made to rotate, so that after a time it stops, and finally rolls backwards to the hand. Other cases are furnished by a “following stroke” or a “screw-back” with a billiard ball.

Let the axis of x be parallel to the motion of translation of a sphere or cylinder moving on a horizontal plane. Then we have, if F be the friction, a the radius of the hoop or ball, the rate of change of momentum = F, and that of moment of momentum about the centre = Fα.

So long as sliding continues, F is constant, and equal to the product μ,M<∕ of the normal pressure and the coefficient of kinetic friction. Hence at time t, if ?/ 0 and ω 0 be the initial velocities of translation and of rotation, u = u 0 - μgt, k 2 < ∣ > = k 2 ω 0 — μgat.

These equations cease to be true when the sliding ceases, i.e., when we have pure rolling, of which the geometrical condition is w+aω=0.

This gives u 0 + aω 0 - μg(a 2 ∣ k 2 +1 )∕ 0 = 0 ,

∩ r f ⅞ 2 (⅝ + aa> 0 )

0 ~ μg{a 2 + k 2 )

At time t 1 and ever after we have „ _ „ k ^ u o + " ω o) _ n ≡o -

0 * 2 + α 2 * 2 + α 2 ’

αi< 0 + a 2 ω 0 k 2 ω 0 -au 0 ω = ω o~ k 2 + a 2 ⅛2 + α 2 *

Hence, if the body be projected in the positive direction, its ultimate motion will be in the negative direction if αw 0 - k 2 ω 0 be negative, i.e., if the initial angular velocity be positive, and greater than aujk 2 ; which is ⅜u 0 ∕a in the case of a sphere. Thus, at starting, the linear velocity of the point of contact with the plane must bear to that of translation of the ball a ratio of over 7:2 if it is to stop and return. In the case of a hoop this ratio must be at least 2 :1.

§ 254. We pass now to the case of a rigid body one point only of which is fixed. As we have already seen (§ 234) this has only to be compounded with the motion of the whole mass, supposed concentrated at the point, in order to give the most general motion of which a rigid body is capable. The geometrical processes which have been applied to this problem, though in many respects of great power and elegance, cannot be introduced here. We will therefore give the more important results in a brief analytical form, and then geometrically exhibit their application.

Recurring to the general equations

2nι(aφ - yχ) = 2(a≈Y - 2/X )= N, &c., we may transform the left hand members as follows.

Let ω x , ω v , ω i be the angular velocities of the body about the fixed axes of x, y, z respectively. Then (§ 77) we have x≈za> y - yω z , Ac.

From these we have three equations of the type X — Z0>y — ya> z + ZWy — yu>z = zùy - yûz + (i∕ω z - a⅛⅛)<⅛ - (zα> 2 - ≈ω x )α> z .

Thus we have xÿ - yx = (z 2 + y≡)c⅛ 2 - yzω u - XZ6s x + XZω y ω z - zyω x ω t + (ir 2 — y~}u3χ<>>y — xy(ω κ — w y ).

Now if the fixed axes be taken so as to coincide at a particular instant with the principal axes of the body passing through the point which is regarded as fixed, the terms involving factors of the form 2(mxy), &c., necessarily vanish (§ 237). Also we have

2nι(x 2 - y”) = 2nι(α 2 + ≈ 2 - y 2 - z 2 ), so that, if the principal moments of inertia of the body be A, B, C respectively, the dynamical equations become

Cα⅛ ÷ (B — A')ω x ω g = N , &c.

But it was proved in § 79 that when the angular velocities of a rigid body are referred to moving and to fixed axes, which coincide at a particular instant, not only are the angular velocities but also the angular accelerations equal at that instant in the two systems. Thus, if ω 1 , ω.>, ω 3 be the angular velocities of the body about its principal axes, the equations just obtained take the form (due to Euler)

Aω 1 + (C — B)ω 2 ω 3 ” 1 j f

Bi 2 + (A — C)ω 3 <⅛> 1 = Μ ,

Cig + (B — A)ω j ω 2 = N.

When ω l , ω.,, ω 3 are found, in any particular case, from these equations, the actual orientation of the body at any time can be calculated from them by the kinematical processes of § 81. The position in space of the point of the body which has been hitherto treated as fixed is to be calculated separately by the processes already explained for kinetics of a particle, and thus the motion of the body is completely determined—in the sense that the difficulties of the further steps are of a purely mathematical nature.

When the forces applied to the body have a single resultant, which either vanishes or passes through the origin, the right-hand terms disappear from Euler’s equations. Multiply the equations by ω 1 , ω 2 , ω 3 respectively, and add. We thus obtain

Aω 1 ω 1 + Bω 2 α> 2 + Cω 3 ω 3 = 0 , whence AωJ + Bw 2 + Cω*≈2T (1),

—the statement that the kinetic energy is constant. Again, multiply by Aω 1 , Bω 2 , Cω 3 respectively, and add. Then we have

A 2 ω 1 α> 1 + B 2 α> 2 d> 2 + C 2 ω 3 ci 3 = 0 , whence A 2 w 2 + B 2 w 2 + C 2 ω 3 = D' 2 (2), expressing the constancy σf amount of moment of momentum.

But if, in these equations, we norv choose to regard ω 1 , α∙. 2 , ω 3 as the coordinates, parallel to the principal axes, of the extremity of a line which represents, in magnitude and direction, the instantaneous axis, we see that that axis is a central vector of each of the ellipsoids (1) and (2). Hence the instantaneous axis describes, relatively to the body, a cone of the second order. Since T and D are the only arbitrary coefficients in the equations, all the ellipsoids (1) or (2) are similar and similarly situated. The curve of intersection of (1) and (2) projected on the plane of the axes A and B has the equation

A(C - A)αq + B(C - B)ω 2 = 2TC - D 2 .

This represents an ellipse if the terms on the left have the same sign, i.e., if 0 is either greater or less than each of A and B. Hence, if the body be originally rotating about an axis nearly coinciding either with the axis of greatest or that of least moment of inertia, it will continue to do so. These two cases are exemplified respectively by a quoit and by an elongated rifle bullet,—at least in so far as the resistance of the air does not interfere with their motion. But if the body be originally rotating about an axis nearly coinciding with the axis of intermediate moment of inertia, the curve indicated by the equation above is an hyperbola or a pair of straight lines through the origin; and the instantaneous axis travels, in general, far away from its first position in the body. We will henceforth look on A, B, C as in descending order of magnitude. It is obvious from the mode in which they are formed that A = B + C only when the body is a plate. Hence, generally, any two of A, B, C are together greater than the third. Also by multiplying (1) by A, and comparing it term by term with (2), we see that 2AT >D 2 ; similarly 2CT< D 2 .

To complete the examination of the immediate results of Euler’s equations in this case, let ns find how the length of the instantaneous axis, considered as a common vector-radius of the ellipsoids (1) and (2), depends on the time.

Let O s =»? + «’ + «’ (3);

ΩΩ = α> 1 ω 1 + a> 2 ci> 2 + ω 3 a> 3

∕C-B A-C B-A∖ Δ

= - Ç + -β- + — c - J <W 3 = - , where Δ is the determinant

1 1 1

ABC =-(C-B)(B-A)(A-C). A 2 B 2 C 2

Now (1), (2), (3) are linear equations in »*, ω 2 , ωf i , and from them we have three equations like

Δω’ = BC(C - B)Ω 2 + 2T( B 2 - C 2 ) + D 2 (C - B)

= BC(C-B){Ω 2 -α} , where q = 2T(B ÷ C) - D 2

BC

is, by the remark above, essentially positive. Hence

∆⅛ft = A 2 B 2 C 2 (C - B)(B - A)(A - C)(Ω 2 - α)(Ω 2 - ⅛)(Ω 2 - c) = A 2 B ι C 2 ∆(α - Ω 2 )(δ - Ω 2 )(c - Ω 2 ).

Thus Ω.2 - V(α - Ω 2 )(δ - Ω 2 )(c - Ω 2 ); whenceΩ 2 is at once found by elliptic functions. Ω known, we have ω 1 , ω 2 , ω 3 , and then by the method of § 81 we have the complete analytical determination of the position of the body in terms of the time.

§ 255. Such a solution, however, fails to give so clear a conception of the nature of the motion as is afforded by the very elegant geometrical representation discovered by Poinsot. We may arrive at it by considering the tangent plane to (1) at the extremity of the radius Ω. If x, y, z be the current coordinates of that plane, its equation is

Aω 1 (ω 1 — x) ÷ Bα∣ 2 (ω 2 — y y ) + Cω 3 (α> 3 — a) = 0 , so that the perpendicular from the origin upon it is equal, by (2), to 2T∕D, a constant. The direction cosines of this perpendicular are proportional to Aω 1 , Bω 2 , Cω 3 , the component moments of momentum. Hence it is the axis of resultant moment of momentum, and is therefore (§ 166) fixed in direction in space. The tangent plane to (1) at the extremity of the instantaneous axis is therefore a fixed plane, and the ellipsoid (1) rolls upon it as if it were perfectly rough. From this we can of course find the equation of the curve of contact with the plane, and thence that of the cone, fixed in space, on which the cone of instantaneous axes in the body rolls, as in § 75. The latter cone is of course given by the intersection of (1) and (2).

To complete this beautiful representation of the motion, all that is necessary is a method of measuring time, something to show the rate of rolling of the ellipsoid on the fixed plane. Many constructions have been given for this purpose, such as Poinsot’s “rolling and sliding cone,” &c., but none can compare in elegance with that invented by Sylvester. We can only sketch a particular case— sufficient, however, to completely solve the question.

Writing I, m, n for the direction cosines of the fixed line referred to the principal axes, we have

Z=Aω 1 ∕D, m = Bω 2 ∕D, n=C<⅛> 3 ∕D .... (4).

Our equations (1) and (2) may now be written Γ- ∣ A + τn 2 ∕B + n 2 ∕C = 2T∕D 2 (1 ),

Z 2 + m 2 + n 2 =l (2).

Introducing a factor p, of dimensions the same as 1∕A, we have from these a third equation

Z 2 (l∕A+p) + m 2 (l∕B+^) + n 2 (l∕C+p) = 2T∕D 2 + p . (5).

Now consider an ellipsoid zy»2 λ∕2 √2

+ _- ≡E c . . . (6, l∕A÷∕l∕B+∕l∕C÷j) k h which is similar, and similarly situated, to one of the ellipsoids confocal with (1) of § 254. Draw to it a tangent plane perpendicular to the line OL, whose direction cosines are (4). We obtain for the determination of the point of contact Q three equations of the form 7n -- y - n --—· (7∖ - S(l∕A+j>) ’ S(l∕B+ 2 >) ’ S(l∕C+p)

zρ2 λ∕2 ~2

wlιθrβ I ’ I. I

(1∕A + i 7) 2 + (1∕B +p) 2 4 (1/C +p) 2

Here x, y, z are the coordinates of Q, and the distance of the tangent plane from the origin is E∕S.

Now (6) gives at once by means of (7)

S 2 [Z 2 (1∕A +77) + m 2 (l∕B + p) + n 2 (l∕C +p) ] = E; whence, ly (5), ι3 ' i - 2 T∕ι⅛ '

This is constant, and. therefore the new tangent plane is fixed in space.

Let us now find the angular velocity about the fixed line OL of this plane’s point of contact Q with the ellipsoid (6).

The direction cosines of the instantaneous axis OP are as ω 1 , ω 2 , ω 3 . Those of OQ are as x, y, z.

And we have obviously by (4) and (7) [table]

Hence the line OQ lies in the plane containing the instantaneous axis OP and the fixed line OL. The motion of ellipsoid (6) is therefore one of combined sliding and rolling along the new tangent plane. To find the sliding, we must find the angular velocity of Q about the line OL. It is to that about OP, which is Ω, in the ratio of the sines of the angles POQ and QOL.

But sin 2 POQ = l - ⅞⅞⅞ ,

(z 2 + y 2 + z 2 )Ω∙ i and ⅛⅝‰1-< t y7 ’ y*y.

(≈ 2 + y 2 + z 2 )D 2

By means of the equations (1) and (7) above we find easily _

sin 2 POQ _ D⅛ 2 sin 2 QOL Ω 2

Hence the angular velocity of Q about OL is Dp, a constant. Now suppose the plane on which the ellipsoid (6) rolls and slides to become perfectly rough, and to be capable of rotating round OL as an axis, there will no longer be sliding of Q, but the plane will be made to rotate with the constant angular velocity Dp. Thus the time of any portion of the motion of the body will be measured out by the angle of forced rotation of this plane.

§ 256. As a simple example, let us take the case of a quoit, -in which A, the moment of inertia about the axis of figure, is greater than either of the equal quantities B and C, which may be referred to any two perpendicular lines in the plane of the quoit. The equations become

A⅛ 1 =0, so that ω 1 is constant,

Bci> 2 + (A — B)ω j ω 1 = 0 ,

Bα> 3 + (B — A)ω 1 ω 2 =0 .

A — B

Put for a moment - ω 1 =n, then we have

B

ώ 2 + nω 3 — 0, ci> 3 — ilω 2 = θ ·

These give by eliminating a> 3 a. 2 +n 2 a> 2 = 0.

Hence ω 2 =Pcos(>√ + Q),

ω 3 = -7i -1 ω 2 = Psin(ni + Q).

The resultant of these is an angular velocity P, about an axis in the plane of the axes of B and C, and making an angle nt + Q with the axis of B. Hence the instantaneous axis describes in the body a right cone whose axis is that of figure; it moves round it in the same direction as that in which the body is rotating, and with angular velocity n. The fixed cone in space is also, obviously, a right cone and the other rolls on it externally.

If instead of a quoit the body be a long stick or cylinder, we have A = B>C, and the equations become

Aω∣ + (C — A)α> 2 w3 — 0,

Aα∣o -f- (A — C)ω3CU3 = 0, c⅛=o.

The last gives ω 3 = constant, and, if

A-C n ~— ä —«3 > A 3 ’ the first two equations are w 1 -nω 2 = 0, α> 2 + nω 1 = 0.

Thus ω 1 + ftω 2 1 =0, ω 1 = Pcos(πi + Q), and a> 2 = - P sin (nt + Q).

This indicates a rotation of the axis of constant angular velocity P in the negative direction. Everything else is as before, but the cone fixed in the body rolls on the inside of that fixed in space.

§ 257. Next let us take the case of a pendulum bob, supported by a flexible but untwistable wire, and containing a gyroscope whose axis is in the direction of the length of the pendulum. Here we may use, for variety, Lagrange’s equations. For simplicity we suppose the centres of inertia of the bob and gyroscope to lie in the axis, and the bob to be symmetrical about the direction of the length of the pendulum.

Let the moment of inertia of the whole about the axis of symmetry be A when the gyroscope is supposed to be prevented from turning relatively to the bob, and let the other two principal moments about the point of suspension be B. Let that of the gyroscope about its axis be C. Then, if θ be the inclination to the vertical, φ the azimuth of the pendulum, and ψ a quantity denoting the position of the gyroscope with reference to a definite plane in the bob passing through its axis, we easily find

2T = A(l - cos θ) 2 φ 2 + B(0 2 + sin 2 0≠ 2 ) + C[≠ - (1 - cos 0)√>] 2 ,

V = M<7Z(l - cosØ) = V 0 (l - cos θ), suppose, where Μ is the whole mass, and I the distance from the point of suspension to the centre of inertia of the whole.

The general treatment of this complex problem cannot be attempted here. We may, however, easily obtain useful and characteristic results in some special simple cases, which will enable us to form a general idea of the nature of the motion.

Thus, suppose if possible 0 to be constant. This is the Conical Gyroscopic Pendulum. We easily find the equations φ — (1 - cos θ)φ = Ω = const.

(A(1 -cos0) + Bcos0)≠ 2 -CΩ≠=V o .

For any assigned values of Ω and 0, this shows what will be the corresponding value of φ. But it also shows that if we change simultaneously the signs only of Ω and φ, the value of 0 is unaltered. Thus, reversal of the direction of rotation of the gyroscope involves reversal of the direction of motion of the bob, if the time of rotation is to be unaltered. But to any assigned values of 0 and Ω two values of φ correspond. As 0 cannot, in the case considered, exceed ⅜τr, the multiplier of φ 2 is essentially positive. So is V 0 =M<7Z. Hence the values of φ are real; and one is positive, the other negative. Thus the pendulum, with any rate of rotation of the gyroscope, may be made to move in any horizontal circle; but the angular velocity will be greater when it is in the same sense as that of the rotation of the gyroscope than when it is in the opposite sense. When 0 is so small that 0 2 may be neglected, we have

B≠ 2 - CΩ≠=V 0 , or 2B^>=CΩ±√4BV 0 + C 2 Ω 2 .

To give a numerical example, let the mass of the gyroscope be ⅛ let B = MZ 2 , C=^-⅛⅞, then n ’ n c 2 2 ≠-¾>≈± ∕ 4 x + ∕<⅛)⅛.

nc 2 y I ∖ nc2 ∕

If n=5, c = 10, <j = 10Z (which are fair approximations to the dimensions of the ordinary form of the instrument),

ø =—± λ ∕ 10 + ———Ω 2 ψ 500 V (500) 2

Suppose the gyroscope to revolve 100 times per second, then Ω = 200π practically, and

≠ = ⅜π ± √10 + ⅜⅜π 2 = 2 · 513 ± 4 Ό39

= 6∙552 or-1'526.

The angular velocity, when the gyroscope is not rotating, would be that of the corresponding conical pendulum,

√<7∕Z= ±3'162; so that in this case the gyroscopic pendulum would rotate about twice as fast, or only about half as fast, as the ordinary conical pendulum, according as it rotated with or against the gyroscope.

If we had taken Ω = 10π we should have found

≠=3'29 or-3 ’ 04, nearly.

Thus the slower the gyroscope rotates the slower is the conical pendulum motion in the same direction, and the quicker that in the opposite direction.

Statics of a Chain or Perfectly Flexible Cord.

§ 258. Axiom.— When a body or system is in equilibrium under the action of any forces, additional constraints ιvill not disturb the equilibrium. Compare § 193.

This principle is of very great use in forming the fundamental equations of fluid equilibrium, and thence those of motion. And we find it of advantage, as will be presently seen, in reducing to elementary geometry the problem of the equilibrium of a chain, or perfectly flexible cord.

We may treat this problem, called that of a “catenary,” by any one of the following methods :—(1) by investigating, as a question of statics of a particle, the conditions of equilibrium of a single link; (2) by imagining a finite portion of the chain to become rigid in its equilibrium form,—assuming, by the axiom above, that it will remain in equilibrium, and then treating the question by the methods employed for a rigid body; (3) by employing the energy test of equilibrium as in§ 198.

§ 259. We exemplify each of these methods in the specially important case of the ordinary catenary.

A uniform chain hangs between two fixed points, find the tension at any point and the curve in which the chain hangs. First Method.— Let μ be the mass of unit length of the chain; T the tension at the point x, y, z ; s the length of the chain to x, y, z from some assigned point; and let the axis of y be taken vertically. Then we have for the equilibrium of the element δs, considered as a material particle,

T^√T⅛ + ⅛'τ^)δs') = 0, ds ∖ ds ds ∖ ds J )

T⅛. + M √T⅛ + ⅛'τ⅛>)-0, ds ∖ ds ds ∖ ds J j

τ⅛√⅛+⅛'τ⅛¼')-o.

ds ∖ ds ds ∖ ds j J

Omitting the terms which cancel one another, and dividing by δs, these become

A(τ⅛Vo, ⅜⅞)- w ≠(τ⅛')-o.

ds ∖ ds J ds ∖ ds J ds ∖ ds J

From the first and third it follows that dzjdx is constant, i.e., the chain hangs in a vertical plane. We may take it as that of xy, and the equations are reduced to the first two.

The first gives τγdX _ r p ds °’

showing that the horizontal component of the tension is constant throughout the whole length of the chain. Substituting for T in the second, it becomes a£/ ds ∖ dxj T o

Tire quantity on the right is evidently of [I. ^ 1 ] dimensions, because that on the left is so; and thus we may write sr=-> <> r T o =M0 ’ α∙

T o a

Hence a is the length of a portion of the chain whose weight is equal to the constant horizontal component of the tension.

The equation now becomes d i y _ 1 ds _ 1 ∕ ·£ + ∕dτ∆ 2 dx i a dx ay ∖ dx)

Integrating, we have

⅛ + √Γ√⅛γ-c i ∙.

dx V ∖ dxJ

If we now assume that the axis of y passes through the point at which the chain is horizontal, we have at that point x≈b, c ~ = 0, dx and therefore C = 1. Thus

√ j + (⅛Y + ⅛.,⅞.

z v ∖ dxj dx

Taking the reciprocal of each side, we have ∖ f ∖ dxj dx

Subtracting,

2‰Lg4, dx χ ≈ or 2-^-= ε « + g^ «; a no constant being added if we assume the axis of x so that y—a, ae = 0 together. This is the equation of the curve required.

If we take the sum, instead of the difference, of the above equations, we find X X

2∣ =g≈ +g^ »; dx

2, Λ

SO that- = g a _ g a a s being measured from the axis of y.

But the tension at x, y is, as above,

τ-τ 0 ⅛=τΛ= m .

dx w a

Hence the tension at any point of the chain is equal to the weight of a length of the chain equal to the ordinate at that point.

Thus it a chain of finite length be laid over two smooth parallel rails, its ends, when the whole is equilibrium, will be in the horizontal line corresponding to the axis of x in the above investigation, the middle part of the chain forming part of the catenary. When a given length 21 of chain rests in equilibrium on two smooth parallel rails at the same level, we have therefore s + y = I , while x=b, the half distance between the rails.

By the above expressions for y and s in terms of x l this leads to the equation b aζ a =l, which determines a when b and I are given. From this equation, as the minimum value of the left-hand side occurs when a = b, we see that the least length of chain for which equilibrium is possible under the conditions is equal to g times the distance between the r ails.

§ 260. Second Method. — The chain being in equilibrium, suppose any finite arc of it, as PQ (fig. 64), to become rigid. The forces acting on PQ are three—the tensions T 1 and T 2 at its ends, and its weight »W acting at its centre of inertia. But three forces in equilibrium are in one plane (§221). Hence the curve is in one vertical plane. Also, as the two tensions are not parallel to one another, the lines of action of all three forces meet in one point. Hence there is no couple, and the conditions are simply

T 2 cos θ i - T 1 cos θ 1 =0, T 2 sin0 2 -T 1 sin0 1 -W = O; where 0 1 , θ 2 are the inclinations to the horizon of the tan gents at the ends of the portion solidified. The first asserts that the horizontal part of the tension is the samt at P and Q, i.e., all through the chain. The second asserts that the difference of the vertical parts of the tensions at any two points is equal to the weight of the part of tin chain between them. By proper mathematical methods these data lead to the results already obtained. In fact the equations we have just obtained are the first integrals of the equations in § 259.

§ 261. Third Method.— The potential energy of the chain is / ds , ^V τχ dx, with the sole condition / —dx =constant, dx the limits of integration being fixed, and the same for each expression. Hence, by the rules of the calculus of variations, we have the same equations as before.

§ 262. Still supposing gravity to be the only applied force, there are many forms of important questions which can be solved by either of these methods. We will take some simple, but varied, examples.

Find how the mass of unit length of a chain must vary from point to point so that the catenary may be an assigned plane curve.

We have, as before,

T^) = 0; ds ∖ ds ∕ but our second equation is now ½(τ⅛}- l *, ds ∖ ds ) where μ is an unknown function of s. The third equation, being of the same form as the first, shows that the curve is in a vertical plane, and is thenceforth unnecessary.

As before, we have rτ ∖ dr r p i √S° ’ a " d ds ∖ dx J 1Q

Now, from the assigned equation of the catenary, the left-hand member can be expressed as a function of s, and thus we have the solution of the problem. For example, suppose the chain is to hang in an arc of an assigned circle. Referred to the lowest point of the circle, the equation is ic 2 = - y 2 , where a is the radius. From this we have dy x . s

-~ = = tan — , dx a- y a so that u = — sec 2 — .

ag a

The tension at the lowest point is thus equal to the weight of a piece of chain like that at the vertex, and of length equal to the radius of the circle; the mass per unit length becomes infinite at the end of a horizontal diameter.

Next find the form of the chain when the mass of any arc is pro portional to its horizontal projection. This is a rough approximation to the case of a suspension bridge where the roadway is uniform, and much more massive than the chains, to which it is attached throughout by vertical ties. The equations are as before, but the additional condition takes the form ds μ-r- =μ n , a constant. dx

This gives ⅛=⅛

o dx 2 T o and the chain forms a parabola whose vertex is downwards, and whose axis is vertical.

As a final example, we have what is called the catenary of uniform strength·, that is, the form in which a chain hangs when the tension at every point is proportional to the breaking stress at that peint. Here we suppose the strength to be proportional to the section, i.e., to the mass per unit length. This gives the condition

T=qu.

Hence ±( ⅛- VjL T = JL ⅛ , ds ∖ dx J eT 0 e dx

¾-±('*y.±(1+ ('⅛y∖, dxa ∖ dx J a ∖ ∖ dx∕ J where a = c]g.

This gives = tan (— + C; dx ∖ a ) and the complete integral may be written, by proper selection of origin, in the final form υ e ~, x s = sec—-. a

This curve has obviously two vertical asymptotes distant ±∣τrα from the axis of y. The quantity a is directly as the tenacity of the material; and thus we see that there is a limit (even in this simplest case) to the span of a chain, however strong, formed of any known kind of matter.

It is a very curious fact that, if we write the equation of this catenary in terms of the arc and the radius of curvature, it becomes identical with that of the common catenary in terms of Cartesian coordinates, horizontal and vertical. For we see at once that ds χ. dχ~ sec T ’ so that g T „ 1 + sin-£-)/( 1 - sin; while by the previous equations d 2 y X

Ί ~>^ O w COS

1 __ dx* 1 a ρ ( ds ∖ 3 ds a ' ⅛∕ "dx

Thus finally, o n - Λ —- = £*+£ ,, . Compare § 259.

§ 263. When the chain is not uniform, and when it is subject to the action of other forces than, or besides, gravity, the equations are

-^-(τ⅛V-p χ , ⅛'⅛‰- μ Y, z ds ∖ ds J ds ∖ ds ) ds ∖ ds J where X, Y, Z are the component forces on unit mass. These three equations involve the unknown quantities T, x, y, x, and s only; for μ is supposed to be given in terms of s. Two relations only among x, y, and 2 are wanted (for s is known in terms of x, y, z) ; so that the equations are necessary and sufficient to give these relations and the remaining unknown T.

The first members of these equations consist each of two terms :— — multiplied respectively by — , ⅛∙ , , ds 1 1 j j ds ’ ds ’ ds and ∑ multiplied by p g, ,⅛, p ⅛ ; p being the radius of curvature of the chain. Hence (§ 22) we conclude that, so far as the tension alone is concerned, the forces on an elementary unit of length of the chain are d r ι∕ds in the direction of the tangent, and T/p in the direction of the radius of absolute curvature. These must balance the corresponding components of the external forces on the element. Hence we see that the resultant of the applied forces lies, at every point, in the osculating plane. Thus we have dT ∕ „ dx . 7 dy , γ d. ∖ ς, d^- μ { x ⅛ + v ds + z *r^ s - T d 3 x - τ d-y „ d 2 z ∖ „

Here S and N are the tangential and normal components of the applied forces per unit length of the chain.

But when a unit particle moves in a curve, we have always dv dυ r ,, 1 r 2 - τ ,

- i 7 = V - τ - = S , and — = Λ , dt ds p where S' and N' are the normal and tangential components of the requisite force. If we write these in the form

£r = S/ JL = Σ' ds V ’ ρ V ’ and suppose that the curve in which the particle moves is the same as the catenary above, while the speed at each point has the same numerical value as the tension, we see that we must have s '=∆'=-q E=Σ'=-K.

V T ’ V T

or S'=-ST, N'=-NT.

Thus the catenary will be the free path of the particle provided the force applied at any point is equal to the reverse of the product of that acting on the chain by the numerical value of the tension of the chain at that point.

Conversely, if we take any case of free motion of a particle, a uniform chain will hang in the corresponding orbit under the action of the same forces each reversed, and divided by the numerical value of the speed at the corresponding point of the orbit. Thus we can at once pass from particle kinetics to corresponding cases of catenaries.

In the case of a projectile, the path is’a parabola, the force is constant and parallel to the axis, and the speed is as the squareroot of the distance from the directrix. Hence, that the parabola may be a catenary under gravity, it must be turned vertex downwards; and the mass of the chain per unit length at any point must be inversely as the square root of the distance from the directrix. It is easily found from this that the mass of any arc of the chain must be proportional to the length of its horizontal projection, as in the second problem solved in § 262.

In the case of a planet we have

ι> 2 =p(2∕rl∕α).

Hence a chain will hang in an ellipse if it be repelled from one focus by a force varying inversely as the square of the distance, the mass per unit length of the chain being directly as the square root of the distance from that focus and inversely as the square root of the distance from the other. If the chain be uniform, the law of the repulsive force from the first focus must be ~[ ∣∖ ∕r i r' instead of l∕r 8 , where r, r' are the distances from the two foci.

§ 2G1. When a chain or string is stretched over a curved surface, the surface must exert a reaction on it to keep it in its curved form. The preceding investigation has shown that the force normal to a chain per unit length at any point is balanced by T/p per unit of length, which must therefore be the magnitude of the reaction. We may establish this, however, in a very simple manner, as follows :—

Let AB (fig. 65) be a small portion of the cord, and AC, CB the tangents at its extremities; and let the (small) exterior angle at C be θ. Then, p being the pressure per unit length of the string, we have at once

2>.AB = 2Tsin⅜0 = T0 ultimately. But AB = p0, so that P= Vp·

If there be friction, and if the element of the rope be just about to slip, in consequence of the difference of the tensions at its ends, we have

T'-T=∕⅜p.AB = μ T0, so that

T' = T(1 +μθ).

This leads to the formula for the growth of a sum at compound interest at μ per cent, payable every instant. Hence for a finite angle a we have

T = S^ α T- I. a * x 0·

It is to be remarked here that neither the dimensions nor the form of the curve on which the cord is stretched, provided only it be plane, have any influence on this result, which involves only the coefficient of friction and the angle between the two free portions of the cord.

Kinetics oe a Chain or Perfectly Flexible Cord.

§ 265. The equations of motion of a chain, under the action of any finite forces, are at once formed from those of equilibrium by introducing the forces of resistance to acceleration according to Newton’s principle. Here we enter on a subject of extreme importance, but also (at least in the majority of cases) of great mathematical difficulty. One valuable result, however, can be obtained by very simple means.

A uniformly heavy and perfectly flexible cord, placed in the interior of a smooth tube in the form of any curve, and subject to no external forces, will exert no pressure on the tube if it have everywhere the same tension, and move with a certain definite speed.

For, as in § 264, the statical pressure due to the curvature of the rope is r ΓΘ ∣ l σ per unit of length (where σ is the length of the arc AB in that figure) directed inwards to the centre of curvature. Now, the element <r, whose mass is m<r (if m be the mass per unit of length), is moving in a curve whose curvature is θ'σ, with speed v (suppose). The requisite force is - l Lf- — mv~θ , and for unit of length mv 2 θ ∣ σ. Hence if T = m∣ 2 the theorem is true. If we suppose a portion of the tube to be straight, and the whole to be moving with speed v parallel to this line, and against the motion of the cord, we shall have the straight part of the cord reduced to rest, and an undulation, of any, but unvarying, form and dimensions, running along it with linear speed fΥ ∣ m.

Suppose the tension of the cord to be equal to the weight of W pounds, and suppose its length I feet and its own mass w pounds. Then T = ∖ Vy, lm = ιv, and the speed of the undulation is f ∖∖ ly'w feet per second.

266. As will be shown later, when such an undulation reaches a fixed point of the cord or chain, it is reflected, and runs back along the cord with the same definite speed. But the reflected form differs from the incident form in being turned about in its own plane through two right angles. When the string is fixed at both ends any disturbance runs along it, backwards and forwards, with this speed, and thus (in a piano or harp) administers periodic shocks to the sounding board, causing it to give out a musical note. The interval between these periodic shocks is of course the time taken by the disturbance in running from end to end of the string. Dividing the length I of the string by the speed above reckoned, we find for this interval the value ∖ lwl∕W r g = l ^fm ∣∖ Ng, the reciprocal of which is the number of impulses per second. It is thus seen to be directly as the square-root of the tension of the string, inversely as the square-root of its mass per unit of length, and also inversely as its length. These are well-known facts in Acoustics. It is to be observed that there is no necessity for limiting the proposition above to a plane curve, though we have treated the question as if it were such. The demonstration applies even to a knot of any form.

267. We will now consider more particularly the vibrations of a musical string, whose tension is great and its own mass small.

Forming the equations of motion as above hinted, we have three of the type

⅛T⅛)=-μ(X→).

as ∖ as j

In the special case of a tightly stretched inextensible string, performing very small transverse oscillations, we may greatly simplify these by assuming that no external forces act. This practically means that the weight of the string is negligible in comparison with the tension. If the axis of x be taken to coincide with the undisturbed position of the string, we have to the second order of small quantities s≈x.

With this the equation above written becomes

≤-o, as or the tension is the same throughout. The second and third equations now become rp <Z 2 ι∕ .. m rf 2 2

τ <z0 2 ~ w ' , ι <⅛ 2-au '

The y and z disturbances are therefore of the same general character, and perfectly independent of one another. We will therefore confine our attention to one of them. From the equations we see that T∕ j u must be of two linear dimensions, and we will therefore write for it α 2 . As this quantity must also be of two negative dimensions in time, a represents a speed. What speed will be seen immediately.

The equation in y is now „ ^ 2 '∕ .∙ “to*' whose integral is known to be y =j ∖ at - x) + F(αi + x), where/and Fare arbitrary functions. As we have already seen (§ 53), the first part of the value of y expresses a wave running with speed a along the axis of x in the positive direction; the second part a wave in the negative direction with the same speed. Thus we see that any small disturbance whatever, of a stretched string, gives rise to two series of waves propagated in opposite directions with equal speeds. Also, as the equation is linear, the sum of any two or more particular integrals is also an integral.

If we suppose one extremity of the string to be fixed at the origin, we have the condition z = 0, y=0, and therefore

0 =/(«<)+ F(n<)∙

As this holds for all values of t, the function F is sinq>ly the negative of/, so that y-∕(<tf-*) -f(at + x').

To investigate what becomes of a disturbance which runs along the cord to the fixed end, let us suppose that ∕(r) (which, by the remark above, may represent any part of a disturbance of the string) is a function which vanishes for all values of r which do not lie between the positive limits p and q, but which for values of r between these limits takes definite values. Then at time i = 0 we have y≈ -Λ*), for, by hypothesis, ∕(r) vanishes for all negative values of r. This denotes a disturbance of the string originally extending fιom x =p to x—q, which runs up to the origin. After the lapse of any interval greater than q/a we have y=f(at-x y ), for at+x has now become greater than q. This is a wave of exactly the same form as before, but the sign of the disturbance and the direction of its propagation are both reversed. Every portion of a wave is therefore reflected, with simple reversal of the displacement, as soon as it reaches the fixed end. For we may take the limits p and q as close together as we choose.

Now suppose the string to have another fixed point atz=Z. Then we have .

0 =, flat - ι) -flat + Z).

Thus /is (§ 67) a periodic function, of period 2Z∕α, andean therefore be expressed as a series of simple harmonic terms of the full period, half period, one-third period, &c. Hence we may write, the coefficient ⅛ being put in for convenience, y= ⅜S 1 ° 0 A m cosπwzZ^ 1 (αZ -x) + ⅜2 1 c0 B m sinπml~ 1 (at - x) — ⅜2 1 q0 A m cos πml~ 1 (at + x) - ⅜S 1 ° 0 B m sinτrwιZ^ 1 (αZ + x~)

= 2 1 eo A m sinτr7>ιZ ^ 1 at sin πm Z ^ ⅛ - S 1 c0 B m cosτrm I ^ 2 at sin τrml ^ ι x .

This expression contains the complete solution of the problem. To adapt it to any particular case, we must know at some definite time (say Z=0) the value of y in terms of x, i.e., the initial disturbance; also the corresponding value of y. We have then g∕ 0 = - 2 1 c0 B m sinτmZ~ 1 ic,

ÿ 0 =y⅛ 1 00 m A m sin τm l~ 1 x.

As y 0 , y 0 are given in terms of x, we can find, by the process of § 67, the values of A m and B, n , and thence the required value of y.

268. As another example, suppose a uniform chain to be suspended by one end, and to make small oscillations in a vertical plane.

We cannot enter here into details; so we simply assume that elementary persistent harmonic solutions are possible, or, what comes to the same thing, that there are permanent forms in which the chain can rotate about the vertical from the point of suspension.

If the axis of x be vertical, the equations of motion are d { rr ,dx ∖ , ... d ( rx ,dy ∖

- r τ ZΓ~ = - A t (i7 -a5 ), - 5 - T≠

ds ∖ ds J ds ∖ ds. J where μ is the mass of unit length of the chain. As the oscillations are supposed to be small, we may neglect the change in the vertical ordinate of any point of the chain, because it must be of the second order of small quantities if the horizontal displacement is of the first order. Hence we may put everywhere x for s, and therefore consider x to be independent of t. Thus the first equation becomes dT dx"~ μ 9 ’ whence T≈μg(l-x), wdιere I is the length of the chain. The second equation then becomes β - w ι⅛I.⅛Ll β .

dx 2 dx g y ’ or, if we measure x from the lower end of the chain upwards, d i y dy 1 .. dx i dx g

The complete integral of this equation would be much more general than we require, for it would express every possible small motion of the chain, however apparently irregular/ What we seek are the fundamental modes of simple harmonic oscillation, any number of which, as in the case of a musical string, may be superposed. Hence we may write

ι∕=ηsin(ni + α), where n is a numerical quantity as yet undetermined, but which is confined to one or other of a series of definite values; η, on the other hand, is a function of x only. With this value of y the equation becomes cZ¼ dy ri 2 n x~r~ζ + u—I y = 0 .

dx 2 dx g

By the usual method of undetermined coefficients we easily find the particular integral . ( 1 r?x ι π ’ ⅛ 2 n β ⅛ 3 . ∖

17 ° - ∖ 1 ^^ ~F + 2¾ 2 ~ 2 2 3¾ 3 +&C J · · · (1 )∙

This series is obviously convergent for all finite values of ri 1 x ∣ g.

The general integral is of the form V = B(η 1 + ιj 0 logaO + A¾; where ∙∏ 1 is a function of x, finite for all values of x, but which we need not determine. For it is clear that, to suit our present purpose, we must put B = 0; otherwise we should have y infinite at x = 0. Thus (1) is the expression we require under the limitations above imposed.

The quantity A represents the semi-amplitude of oscillation of the lower extremity of the chain. The condition that the upper end is fixed gives τ∕ o = 0 for χ=l, i.e., λ 1 π 2 Z n 4 Z 2 7i 6 Z 3 .

0 = 1 1^ Q., ., - θ θQ¾ ⅛ + & C · g 2 i g 2 2 2 3 2 ∕

The roots of this equation (which are all real and positive) give the values of n for the several fundamental modes of vibration.

We have ?? o = O for the following values of ri 2 l ∣ g : l - 454, 7'62, 18∙74, 34’79, &c.

From these we find for the periods of the various simple disturbances the following multiples of the period of a simple pendulum equal in length to the chain, viz., 0'83, 0 - 36, 0 - 23, 0 - 17, &c. When ri 2 l[g has the least of the above values, the chain is always entirely on one side of the vertical, and the time of a complete oscillation is to that of a simple pendulum of the same length as 5: 6 nearly.

269. When a free chain, at rest, has an impulsive tension applied at one end, the calculation of the consequent impulsive tension at different parts of the chain and the velocities generated is very simple.

For, calling the instantaneous speeds along the tangent and along the radius of absolute curvature v s and v p respectively, we have

ST = μv,8s, T∕ρ-μvp , where μ is the mass of unit length of chain at s. It is obvious that there can be no impulsive speed perpendicular to the osculating plane. The kinematical condition is simply that an elementary arc δs is not altered in length. But the tangential increment of speed alone would imply an increase of the length of δs in the ratio l-}~8t∙. 1 intime δZ. Also the impulsive speed vp would imply a diminution of its length in the ratio 1 - υ p δt∕p : 1 by virtually making it an arc of a circle of smaller radius, but subtending the same angle at the centre. Hence, neglecting the square of δZ as compared with its first power, we find for the kinematical condition ds p

This gives, by eliminating the impulsive velocities, Λ(I. ≤ r V- ∑, = o ds ∖ μ ds j μ p 2

If the chain be uniform, this becomes cZ 2 T T ds 2 p 2 ~ o '

The whole kinetic energy generated in the chain by the impulse is 1 fids ( ∕ d r ι ∖ 2 T 2 ∖ ⅜∕ —( -5- +-τ 5 β ∕ μ ∖ ∖ as J p~ J and the condition that this shall be a maximum is the differential equation above. This is a particular case of a general theorem due to Sir W. Thomson, viz. :—

A material system of any kind, given at rest, and subjected to an impulse in any specified direction and of any given magnitude, moves off so as to take the greatest amount of kinetic energy which the specified impulse can give it.

The direction in which an element of the chain begins to move is inclined to the tangent at an angle φ where + , v p T

tan φ = — = -= ∙

-r, (IT ds

270. It is to be observed that, in such questions as those just treated, the possibility of an impact’s being propagated instantaneously along the whole length of a chain depends upon its assumed inextensibility. When a wire (such as that employed for a distance-signal on railways) is regarded as extensible, there is a definite speed with which a disturbance of the nature of exτension is transmitted along it.

Thus, recurring to the equations of § 267, we see tnar for the motion of a stretched elastic string in the direction of its length we have

<ZT <Y ∙∙,

-≈-j t (X- ic ).

If there be no applied forces, X = 0. Also, if we use x instead of s to characterize a particular point of the suing, we must put x+ξ fora: and x for s, ξ being a function of x and t which denotes at any instant the displacement of that point.

The physical condition is expressed by Hooke’s Law in the form

E⅛ = T .

dx

Hence ^E^ = μt. dx 2

This expresses (as in § 267) the passage of simultaneous waves. They are now waves of condensation and extension, not of transverse displacement. The nature of the interpretation of the equation is of the same general character as before, the speed being vE∕∕t, Dynamics of an Elastic Solid.

§ 271. This subject, which is a very extensive and difficult one, and in its generality quite unsuitable for discussion here, has already been to some extent treated of under Elasticity. We therefore content ourselves with one or two examples, whose treatment is comparatively simple, while their applications are frequent and of considerable practical importance.

§ 272. Even so restricted a problem as that of determining the form assumed by a wire or thin rod of homogeneous isotropic elastic material, under the action of given forces and couples, presents somewhat formidable difficulties unless in its unstrained state the wire be straight and truly cylindrical or prismatic. And, even with these limitations, the problem again becomes formidable if we introduce the consideration of non-isotropic material; while, in any case, if the radius of curvature at each point is not very large in proportion to the thickness of the rod in the plane of bending, the problem is to no appreciable extent simplified by the limitation of form of the body. We will therefore give the comparatively simple case of the mere bending and twist of a homogeneous isotropic wire whose natural form is cylindrical or prismatic, the amounts of these from various sources being so small as to be superposable. Bending lengthens one set of lines of particles originally parallel to the axis of the wire and shortens others. Twist lengthens all but one such line, forming them into helices. The more detailed investigation, which we cannot give here, shows that there is one line of particles (the “elastic central line ”) which passes through the centre of inertia of each transverse section, and which may be treated (under our present limitations) as rigorously unchanged in length. The mutual molecular action of the parts of the wire on opposite sides of any transverse section may of course be reduced to a force and a couple, and the force may be conveniently treated as passing through the centre of inertia of the section. Also the twist and curvature of the wire near this section obviously depend on the couple and not on the force. For the moment of the couple is in general finite, while that of the force (about any point in the corresponding element of the wire) is infinitesimal.

§ 273. Let any two planes, at right angles to one another, be drawn through the elastic central line before distortion; and let them be cut in lines PR and PS by a transverse section through a point P of the central line. Also let PT be a tangent to that line. Suppose a similar construction to be carried out for every point P of the central line. Then it is clear that the form of the distorted wire will be completely determined if we know the form assumed by the central line, and the positions taken by the lines PR and PS drawn from each point in it. In their new positions P'T', P'R', and P'S' will still form (in consequence of the limitations we have imposed) a rectangular system; and the nature of the distortion will be clearly indicated by the change of position of this rectangular system as it passes from point to point of the distorted central line. The plane of rotation of P'T' is the osculating plane of the bending; its rate of rotation in that plane per unit length of the central line is the amount of bending; and the rate of rotation of the system P'R', Γ'S', about P'T', per unit length of the central line, is the rate of twist. Suppose P' to move with unit velocity along the distorted central line, and let p, σ, τ be the angular velocities of the system about Γ'R', P'S', P'T' respectively, then p represents the curvature (or bending) resolved in the plane S'P'T', σ that in R'P'T', while τ represents the twist.

Now, if the elastic forces constitute a conservative system, the amount of work done on an element of the body corresponding to a length δs of the central line is to be calculated entirely from its change of form. It must therefore be expressible in the form wδs where w is a function of p, σ, τ; which must be such that the couples producing the bending are dw 1 dw - r and - τ - , dp dσ while that producing the twist is dw dr

These again, are functions of p, σ, τ, and they must, on account of the principle of superposition, be linear and homogeneous. For, within the limits to which we have restricted ourselves, the doubling alike of bending and twist must involve the doubling of each of the couples. Thus w must be a homogeneous function of p, <r, τ of the second degree. Hence we may assume w — ⅛(Ap 2 4- B<r 2 + Cτ 2 + 2Dp<τ + 2Ecrτ + 2Fτp), where A, B, C, D, E, F are quantities depending on the form of the section of the wire and the nature of its material at each point. This gives

⅛ = Ap + D<r + Fτ, f ^ = Dp + B<r + Eτ, ⅛ , = Fp + E<r + Cτ. dp dσ dτ

Hence, when the couples are assigned, the amounts of bending and twist are at once calculated from them. But the expression above is much more general than we require for the limited case we are considering. For, if the only couples applied to a portion of the prism or cylinder considered be in planes perpendicular to its length, twist only will be produced. Thus, for ^- = 0, we ought to have also p=0, <r = 0. Hence E and F both vanish and we have simply w = ⅜(Ap 2 + 2Dp<r + Bσ 2 + Cτ 2 ). This may be reduced, by properly selecting the planes originally drawn through the elastic central line, to the form m> =⅜( Ap 2 +B<r 2 + Cτ 2 ).

Now we see that dw . dw _ dw ~ A,p , ~~z -I.><τ , ~j β Cτ ·

dp dσ dr

§ 274. In a prismatic or cylindrical wire of homogeneous isotropic material, the elastic central line is thus a torsion axis simply. Equal and opposite couples, applied to the ends of such a wire, in planes perpendicular to its length, produce twist in direct proportion to the moments of the couples. There are two planes perpendicular to one another, and passing through this line, such that, if equal and opposite couples in either of these planes be applied at any parts of the wire, the portion between is bent into a circular arc in that plane. These are the principal planes of flexure. The quantities A and B which, when multiplied by the amount of bending in either of these planes, give the moment of the corresponding couple are called the principal “flexure rigidities” of the wire. When they are equal (as in the case of a wire of circular, square, equilateral triangular, <fcc., section) any plane through the axis is a principal plane of flexure. C is the torsional rigidity of the wire. In general, when the wire is fixed at one end and a couple applied at the other, the wire assumes the form of a circular helix. The exceptions (or rather particular cases) are :—(a) when the plane of the couple contains the elastic central line, and there is mere flexure, without twist; (6) when the plane of the couple is perpendicular to the wire, and there is twist simply.

§ 275. As an example of the preceding theory, take first the case of a uniform plank clamped horizontally at one end, and otherwise unsupported. This is obviously the same as the case of a plank of double the length, supported by a trestle placed under its middle. We assume as before that the radius of curvature is always very large compared with the thickness of the plank.

Tn all such cases we may at once apply the principle of § 258, and suppose one portion of the plank up to a section P to be fixed in its equilibrium position. The curvature immediately contiguous to'P will then be simply proportional to the moment about P of the forces acting on the unfixed portion. Hence at the free end there will be no curvature, and the curvature at points near that end will be of the second order of infinitesimals; i.e., its rate of increase at the end vanishes.

Let x be the length of the fixed portion, I the whole length of the plank. Then, as the deflexion y from the horizontal is always very small, the curvature is expressed (§ 22) by d 2 y dx 2 ' so that we have at once E 0 = ^ Τχάχ ' = ^ ~ x ^ ’

where E is the “flexural rigidity” of the plank, and μ its mass per unit of length.

Successive integrations give e ⅛ = b + ⅛^(z-zΛ

and e 2∕= A - B(Z - z) - J τ μg (Z - χ) i ,

The terminal conditions are forx=0, 2∕ = θ, ⅛'=0∙

dx and for x= I, ⅛I. = q ÉÏ = 0 dx 2 ’ dx 2 fl ’ lιe last two are obviously satisfied.

The two former give

B = - μ f, A~Bl + ⅛ μ gl* = - ι μ gl* .

Hence Ey = - ⅛g(3Z 4 -4Z 3 (Z-ae) + (Z-α) 4 ).

Thus the droop of the free extremity (ic = Z) is

8 E 8E ’

where ΛV is the whole weight.

If the plank had been weightless, but loaded at the free end with a weight W, our equation would have been E ^=-W(Z-a·), and we should have had

E⅛ = B' + ⅛W(Z-x)≡, Ey = A' - B'(Z - x) - ⅜ W(Z - x) 3 .

The terminal conditions at x=0 are as before, so that

B'= - ⅛WZ-, A' = - ⅛WZ 3 + ⅜WZ 3 = - JWZ 3 ,

TV 73

and the droop of the free end is - , greater than before in the 3L

ratio of 8: 3.

If the plank be again looked on as heavy, but its free end be supported on a trestle which is pressed upwards till it acts with a force W, we find directly Ε <ζ^ = - x > ~⅛M( ι ~ χ y i > ⅛.r-W-⅛t ta (∣-⅛., Ey = A" - B" (Z - a;) + ⅜ W(Z - x)* - ⅛ μg (l - x )*.

The terminal conditions, at x = 0, are still as in the first case, and they give

B" = ⅜WZ 2 -⅞μ^, when the amount by which the free end is raised is =[z(⅜wz≡ - ⅛ μ gi 2 ) - ⅛wz 2 +⅛< 7 z<] ∕e=-^ , .

This is obviously the same as the amount of depression of the middle of a plank, of length 2Z supported by trestles at each end.

§ 276. Hence the droop of the middle of a plank resting on trestles at its ends is to that of the ends when the plank rests on a single trestle at the middle in the ratio of 5: 3.

If the equation expressing the curvature in the first or third cases above be twice differentiated, the common result is v <Fy

The simplicity of this expression leads us to seek for the most general form. Suppose the plank to be exposed to any system of forces in lines perpendicular to its length and breadth. Then, if any transverse section be made, the stress between the two portions of the plank will consist of forces ( + G) and couples ( + H) in the plane of length and thickness. Let the applied forces be N per unit of length. Suppose also, as before, that the radius of curvature is very great compared with the thickness. Then the equations of equilibrium of an element are

<ZG τ cZH „ .

- r - + N-θ - y - + G = 0.

dx dx

We have also the condition of bending, viz.,

E × curvature = E d ~ = H.

dx-

Eliminating H and G among these equations, we have v d*y <Z 2 H <ZG xτ

E- - = = = N

dx 4 dx 1 dx ' which of course includes all the previous particular cases. We may now determine (under the limits imposed) the form of a uniform plank of any length, supported in a nearly horizontal position at different points in its length, and loaded at any assigned points with any weights. The importance of this in practice is obvious.

§ 277. But we may easily take a further step, and investigate the oscillatory motion, so long at least as the acceleration parallel to the length of the plank and its rotation are negligible. For in such a case, if μ. be the mass per unit of length, the equation of motion is (§ 199)

e S^ n ^' , ≈'∙

We will consider only the case in which the applied force N may be neglected. This is practically the case of a uniform wire or flat rectangular spring. Suppose, further, that it is fixed at one end and free at the other, like Wheatstone’s “kaleidophone,” or like the tongue of a reed organ-pipe.

Then, writing n i for the fraction ∕√E, we have *~.+n*y = Q. dx i

A particular integral may obviously be found in the form y = ηcos(i 2 l ∣ n 2 + α) (1), where η (a function of x) and i/n (a constant number) have to be found; a is any constant. The substitution of this value of y leads to

⅛-Λ>-o, dx i ' ’ the complete integral of which is

η = AS'* + Bg^'∙ r + Ccos(ι⅛ + D).

Now, provided the value of i be properly determined, the motion represented by (1), with the above value of η, can exist by itself; and the most general motion of which the spring is capable (under the limits imposed) consists of superposition of a number of separate motions of a similar character. Hence this may be treated by itself. Our limiting conditions in the present case are aj =0, y = 0, = 0 at the fixed end; and x=Z, ^⅛ = 0, ^=0 at the free end.

Now, from the value of η above, we have

4- ⅛-=Ag u - Bg-"-Csin(κc+D), ι dx 1 ~ —Ag ’ ' z + Eg '∙ t -Ccos(⅛ + D), i i dx 1

4 r -¾ = Ag z - Bg- , * + Csin(⅛ + D). τ i dx t

Hence we have

0 = A +B + CcosD,

0 = A -B -CsinD,

0 = A g » + B g - il - C cos (il + D), 0=Ag' ’- Eg - i, + C sin (il + D).

Eliminating C and D, we have

0 = Ag i, + Bg ^ α + (A + B) cos?7 + (A - B)sin il, 0 = AS 7 — Bg ~ a - (A + B)siιu7 + (A - B)cosι7.

Eliminate the ratio A∕B, which is all that these equations furnish, and we have

(g ,7 + S"* 7 )cosr7 + 2 = 0.

From this equation the values of i must be determined. It is clear that the multiplier of cosι7 is always greater than 2, except in the special case of i==0, which we obviously need not consider, as it gives ÿ=0, and therefore belongs to the statical problem already considered. Hence as, to make cosri negative, il must be greater than ⅛ιr; and, as g 7 ” - + g ^^ 7r = 5 nearly, it is clear that the excess of the first value of il over ⅜π is somewhere about 0 ’ 3. The next value falls short of ⅞π by a quantity of the order τ ⅞- σ , the next exceeds ⅞τr by a quantity of the order 7i ⅛ 75 , &c. The required values arrange themselves in two groups, one of either group being taken alternately. The first group involves arcs a little greater than but rapidly approaching to the values of (47∕i+l)⅛π∙, the second consists of arcs a little less than but rapidly approaching to those of (47>i+ 3)⅛ιr.

§ 278. Some of the simplest, but at the same time most practically useful, of questions connected with elasticity of solids relate to the changes of form or volume experienced by circular cylindrical tubes or spherical shells exposed to hydrostatic pressure. A steam-boiler, the cylinders and tubes of an hydraulic press, a fowling-piece or cannon and (on a much smaller scale) Orsted’s piezometer, deep-sea thermometers, Ac., afford common instances. All that is necessary for attacking such questions is given in §§ 45, 46 of the article Elasticity. For it is there shown that, if a homogeneous isotropic elastic solid be subjected to a simple longitudinal stress P, uniform and in a definite direction throughout its whole substance, the result will be linear extension = P ( 4- + 4 ) in the directs 71 9⅛ j tion of P, and linear contraction = P ( --X) in all direc-

’ ∖ 621 9£ ∕

tions perpendicular to P. The quantities n and Z∙, as explained in the article referred to, are respectively the “rigidity” and the reciprocal of the “compressibility” of the solid operated on.

§ 279. The case of the piezometer, in which the vessel holding the liquid whose compression is to be measured is exposed both inside and outside to the same hydrostatic pressure, is seen to correspond to three equal stresses in directions at right angles to one another. These directions may be any whatever, and in each of them the linear extension is obviously pifi + ι λ >- 2 fi-Γ)μz.

( 2k J ∖ βn 9kJ ) Zk

P is negative, as the stress is a pressure. Hence the strain consists in a simple alteration of volume measured by P∕Z∙. Every part of the walls of the vessel, as well as its external bulk, and its interior content, is altered to the same extent, r § 280. In the case of a cylinder, when the internal and external pressures are different, it is clear from symmetry that the stresses may be resolved at any point of the walls ' into three at right angles to one another, the first (P 1 ) parallel to, the second (P 2 ) at right angles to, the axis, and the third (P 3 ) perpendicular to each of the other two. In a transverse section of the cylinder, the second of these is radial and the third is tangential to a coaxal cylinder passing through the element considered. We suppose the cylinder closed at both ends, and we make the further assumption (quite exact enough for practical applications, and most important from the point of view of simplicity of calculation) that all transverse sections of the cylinder remain, after distortion, transverse sections. This is equivalent to assuming P 1 to be constant throughout the walls of the cylinder. Hence, if there be interior pressure only, the value of this stress must be p _ pressure on end of interior of cylinder ∏∕(

1 area of transverse section of walls of cylinder a* -a* where ∏ is the interior hydrostatic pressure, and σ 0 , ∏ 1 are the internal and external radii. This stress represents a longitudinal tension of the walls of the cylinder.

Let us consider an element of the cylindrical wall, defined as follows in the unstrained state :—

Draw two transverse sections at distances x and x + δx from one end, two planes through the axis making an (infinitesimal) angle 0 with one another, and two cylinders of radii r and r+ δr about the common axis. In the strained state θ is unchanged, but ⅛ becomes x + ξ, and r becomes r + p. The distance between the transverse sections was δz ’; it becomes δz + ^δar, so that the linear exten-

sion parallel to the axis is . The distance between the cylinders was Sr ; it becomes δ7 ’ + ⅛δr, so that the radial extension is -~∙ ’ dr dr

The breadth of the base of the wedge-shaped element was rθ; it becomes (r + p^jθ, so that the linear extension perpendicular alike to the radial line and to the axis is - p - .

r

If we now write, for simplicity,

·_ 1 1

. C 3n ⅛k ’ "' 671 9⅛ ’

the three requisite equations between stresses and strains are at once obvious in the form

‰ eP 1 -∕P s -∕P.> dx

⅛=-∕Pl + ^ 2 -∕P3.

A=-∕P1-∕P 2 + <T 3 . τ

We have, however, four unknown quantities ξ, p, P 2 , and P 3 , so that another equation is required. This must be supplied by one of the statical conditions of equilibrium of the element above defined, when in its strained state. There is obviously equilibrium in the axial direction, and also parallel to the base of the element; but radially we have a case resembling that of an element of a cord as in § 264. Neglecting small quantities of a higher order than those retained, this consideration gives d d

P 3 δmδr=⅛P ss 7∙δ‰)δ7∙, or P 3 =-(P,r).

Solving the four equations, and taking account of the boundary conditions

P 2 = -∏when r = α 0 , and P 2 =0 when r=a l , we obtain the following values :—

ÿ = ∏ s -⅛~2∕), dx a ∖ -ao

= ∏√⅛(«- ≡∕)+⅛(«+/) Ί ·

r α--αjjL 7- J

⅛-Π Λl(i-2∕b¾f+∕) ^b dr oJ-n 7 I. v τ u J

or, reintroducing the values of e and/, dξ _ ∏α* 1 dx o, ∖ -a i a Zk r a*-al ∖ 3k r- 2nJ dp _ Πα_Μ_ _1 ∖ dr ~αf-α*∖3⅛^ r i 2n)'

§ 281. Thus the nature of the distortion produced in the walls of a cylindrical tube by internal pressure may be described as made up of a uniform dilatation, whose linear measure in every direction is

τ^T∕Υ “1 ·

- “θ——, combined with a shear in each transverse section, whose a 2 l -a 2 0 3*’

. 1 . ^∏a 2 a 2 1 1 measure is 1 ± - τ - i — 9 -A-5 χ- · § 91.

(a 2 - a 2 0 )r 2 2n

The shorter axis of this shear is radial, and the magnitude of the shear is obviously greater for smaller values of r. The inner layer of the walls is thus the most distorted. The amount of the distortion is directly as the pressure, and inversely as the area of the section of the walls.

When the walls are very thin the shear is practically the same throughout their thickness. 'When they are very thick, the shear near the inner surface is nearly 1±—, however fine be the bore.

2?i ct 2 ∏

That near the outer surface is nearly l+-⅛ —, which vanishes j a 2 1 2n when the bore is very fine. Thus it appears that, if a stout tube bursts by the shear produced by internal pressure, little is gained either by making it of extremely great thickness or by making it of very small bore.

The diagrams A and B in fig. 66 show, necessarily on a greatly exaggerated scale, the nature of the distortion produced at different parts of the wall of the tube. They represent transverse sections of small, originally spherical, elements made by planes at right angles to the axis. The radial diameters are horizontal. A is an element close to the external surface, B an element near the inner surface. The increase per unit volume of the interior of the tube is

Π«* ∕ 1 a 2 1 ∖ . α≈-α*∖ k a 2 0 n )' so that, if the tube be very thick in comparison with its bore, the increase is nearly ∏∕n. In flint glass this is approximately about τrΛ rτ>, when ∏ is a ton-weight per square inch.

§ 282. The reader who has followed the above investigation will find no difficulty in obtaining the corresponding results for a cylindrical tube, closed at both ends and exposed to external pressure ∏, in the form πα ι JL dx a 2 l -a' 2 3k , P ∏ a ∕ ’ l + ⅛n r a 2 - a 2 ∖ 3k τ i 2n) ’ dp _ ∏rς ∕ 1 ¾ 1 ∖

dr a 2 1 - a 2 ∖ 3k r t 2n J '

The only comments we need make are (1) that the signs of these distortions are now negative; (2) that they are (so far as change of volume is concerned) greater than for the internal pressure, as has taken the place of al as a factor in each term involving k ; (3) that the terms involving the rigidity are, except as regards sign, unchanged.

The change of volume of every part of the walk is

ι

a 2 -a 2 k' and the change of volume of the interior is

Πα; ∕ 1 1 ∖

α ϊ - a o ∖ ⅛ n ∕

The numerical value of the factor ∏(^-y + is about τ √ w for flint glass and about for steel, when ∏ is a ton-weight per square inch. There is, however, a peculiarity which (when the walls are thick enough) distinguishes this from the preceding case. For k is usually considerably greater than n, so that a fortiori 3k is greater than 2n. Hence, in the value of dp ∣ dr, the term in n is always greater than that in k so long as the pressure is internal. Thus the radial effect is compression at all parts of the walls. But, when the pressure is external, we may (if the walls be thick enough) find a value of r for which

3⅛ r 2 2n '

In glass, this occurs when r=l∙6α 0 nearly. At this distance from the axis there is no radial change of length; at greater distances there_ is radial compression, and at smaller radial extension. This is indicated in the diagrams C and D in fig. 66, which like the former are greatly exaggerated. They represent the distortion of small spherical elements of a thick tube,—the first at the inner wall, the second at the outer surface. As before, these are sections made by a plane perpendicular to the axis of the cylinder.

§ 283. In a spherical shell of internal and external radii a 0 and a 1 , the equations become a little more simple on account of the more complete symmetry.

Using the same notation, so far as it is now applicable, we have ⅛= -2∕P 1 + cP 2 , -J-=(e-∕)P 1 -∕P 3 ·

The statical equation is

2rP 1 =⅛ r 2p o ).

dr

With these we obtain, for external pressure ∏, the result e≡-∏√ιLΛl + ⅛M r a 2 - αθ∖ 3k r i in J ’ from which the other equation may be derived by differentiation.

§ 284. The propagation of plane waves in an elastic solid has been discussed in Elasticity, and the mechanics of fluids is discussed under Hydromechanics. General Considerations.

§ 285. The preceding view of the subject of Abstract Dynamics has been based entirely upon Newton’s Laws of Motion, which were adopted without discussion, as a complete and perfectly definite foundation; and the terms employed, as well as the mode of treatment in general, have somewhat closely followed Newton’s system. The only considerable apparent departure from that system is connected with the development of the idea of energy, and its application to the simplification of many of the methods and results. This also was, as we have seen, really introduced by Newton; but it has been immensely extended since his time both by mathematical and by experimental processes. It is time that we should now return to the laws of motion, and examine more closely, in the light of what we have learned, one or two of the more prominent ideas which they embody. To do so fairly we must go back to Newton’s own definitions of the terms which he employs. About many of these, which have already been quoted in §§ 97-113, there is no difference of opinion But it is otherwise when we come to the definition of “force” (§§ 5, 104).

There can be no doubt that the proper use of the term “force” in modern science is that which is implied in the statement of the first law of motion, as we rendered it in § 1 from Newton’s Latin. It is thus seen to be the English equivalent of the term vis impressa. Newton uses the word vis in other connexions, and with a certain vagueness inevitable at a time when the terminology of science was still only shaping itself ∙, but his idea of “force” was perfectly definite, and when this is in his mind the vague word vis is (when necessary) always qualified and rendered precise, either by the addition of impressa or in some equally unambiguous way. To render vis by “force,” wherever it stands without the impressa or its equivalents, is to introduce a quite gratuitous confusion for which Newton is not responsible. We have oιdy to think of the multitude of terms, such as vis insita (inertia), vis accéleratrix (acceleration), vis viva (kinetic energy), &c., &c., to see that all such complex expressions must be regarded as wholes, and that vis does not mean “force” in any one of them.

§ 286. Thus in Newton’s view force is whatever changes (but not “or tends to change ”) a body's state of rest or of uniform motion in a straight line.

He mentions, as instances, percussion, pressure, and central force. Under the last of these heads he expressly includes magnetic as well as gravitational force. Thus force may have different origins, but it is always one and the same; and it produces, in any body to which it is applied, a change of momentum in its own direction, and in amount proportional to its magnitude and to the time during which it acts.

§ 287. Thus, from Newton’s point of view, equilibrium is not a balancing of forces, but a balancing of the effects of forces. When a mass rests on a table, gravity produces in it a vertically downward velocity which is continually neutralized by the equal upward velocity produced by the reaction of the table; and these forces, whose origins and places of application are alike so widely different, are (as forces), in every respect except direction, similar and equal. And they are so because they produce, in equal times, equal and opposite quantities of motion.

§ 288. The idea of “force” was undoubtedly suggested by the “muscular sense’'; and there can be no question as to the vividness of the sensation of effort we experience when we try to lift a heavy weight or to open a massive gate. In this, as in other cases, it is the business of science to find what objective fact corresponds to the subjective data of sensation. It is very difficult to realize the fact, certain as it is, that light (in the sense of brightness) is a mere sensation or subjective impression, and has no objective existence. Yet we know that, beside those radiations which give us the sensation of light, there are others, in endless series both higher and lower in their refrangibility, to which our eyes are absolutely blind. And the only difference between these and the former is one of mere wave-length or of period of vibration. Similarly, it is very hard to realize the fact that sound (in the sense of noise) is only a sensation; and that outside us there is merely a series of alternate compressions and dilatations of the air, the great majority of which produce no sensible effect upon our ears. Thus because we know that we should seek in vain for brightness or noise in the external world, familiar as our senses have rendered us with these conceptions, we are driven to inquire whether the idea of force may not also be a mere suggestion of sense, corresponding (no doubt) to some process going on outside us, but quite as different from the sensation which suggests it as is a periodic shearing of the ether from brightness, or a periodic change of density of air from noise.

§ 289. So far, we have treated of force as acting on a body without inquiring whence or why; we have referred to the first and second laws of motion only, and have thus seen only one half of the phenomenon. As soon, however, as we turn to the third law, we find a new light cast on the question. Force is always dual. To every action there is always an equal and contrary reaction. Thus the weight which we lift or try to lift, and the massive gate which we open or try to open, both as truly exert force upon our hands as we do upon them. This looking to the other side of the account, as it were, puts matters in a very different aspect. “Do you mean to tell me,” said a medical man of the old school, “that, if I pull a ‘subject’ by the hand, it will pull me with an equal and opposite force?” When he was convinced of the truth of this statement, he gave up the objectivity of force at once.

§ 290. The third law, in modern phraseology, is merely this :—

Every action between two bodies is a stress.

When we pull one end of a string, the other end being fixed, we produce what is called tension in the string. When we push one end of a beam, of which the other end is fixed, we produce what is called pressure throughout the beam. Leaving out of account, for the moment, the effects of gravity, this merely amounts to saying that there is stress across every transverse section of the string or beam. But, in the case of the string, the part of the stress which every portion exerts on the adjoining portion is a pull ; in the case of the beam it is a push. And all this distribution of stress, though exerted across every one of the infinitely numerous cross sections of the string or beam, disappears the moment we let go the end. We can thus, by a touch, call into action at will an infinite number of stresses, and put them out of existence again as easily. This, of itself, is a very strong argument against the supposition that force, in any form, can have objective reality.

§ 291. We must now say a word or two on thé question of the objective realities in the physical world. If we inquire carefully into the grounds we have for believing that matter (whatever it may be) has objective existence, we find that by far the most convincing of them is what may be called the “conservation of matter.” This means that, do what we will, we cannot alter the mass or quantity of a portion of matter. We may change its form, dimensions, state of aggregation, &c., or (by chemical processes) we may entirely alter its appearance and properties, but its quantity remains unchanged. It is this experimental result which has led, by the aid of the balance, to the immense developments of modern chemistry. If we receive this as evidence of the objective reality of matter, we must allow objective .reality to anything else which we find to be conserved in the same sense as matter is conserved. Now there is no such thing as negative mass; mass is, in mathematical language, a signless quantity. Hence the conservation of matter does not contemplate the simultaneous production of equal quantities of positive and negative mass, thus leaving the (algebraic) sum unchanged. But this is the nature of conservation of momentum (§ 165) and of moment of momentum. The only other known’thing in the physical universe, which is conserved in the same sense as matter is conserved, is energy. Hence we naturally consider energy as the other objective reality in the physical universe, and look to it for information as to the true nature of what we call force.

§ 292. When we do so, the answer is easily obtained, and in a completely satisfactory form. We give only a very simple instance. When a stone, whose mass is Μ and weight W, has fallen through a space h towards the earth, it has acquired a speed v, which (§ 28) is given by the equation

⅜Mv 2 =WΛ.

This is a particular case of the conservation of energy, but the terms in which it is expressed are those suggested by Newton’s laws of motion, and are therefore based on the recognition of “force.” The first member of the equation represents the kinetic energy acquired; the second the potential energy lost, or the work done by gravity upon the stone during its fall. Both members therefore express real things, having objective existence.

But the “force” (so-called) which is said to have produced the motion, has the value

W=⅜Mv 2 ∕A, i.e., it is the rate per unit of length, at which potential energy is converted into kinetic energy during the fall. In other words, it is merely an expression for the space-rate at which energy is transformed.

§ 293. Another mode of presenting the case will make this still more clear. The average speed with which the stone falls is (§ 28) v∕2. Divide both sides of the equation above by this quantity, remembering that 2Λ∕v is the time of falling, which we call t. We have thus, as another perfectly legitimate deduction from our premises, W=Mr∕Z.

Here the (so-called) force appears in a new light. . It is now the time-rate at which momentum is generated in the falling stone.

§ 294. The statements in the last two sections are, in f ιct, merely particular cases of Newton’s two interpretations of action in the third law, which have already been discussed (§§ 165, 167).

Analytically, the whole affair is merely this: if s be the space described, v the speed of a particle, .. . dv dv ds dv s = v∙=^-.=- r .- r . = v- r . dt ds dt ds

Hence the equation of motion (formed by the second law) .ms=mv=f, which gives f as the time-rate of increase of momentum, may be written in the new form mv ^ds^ mv ^∙ f ' giving/as the space-rate of increase of kinetic energy.

§ 295. But a mere rate, be it a space-rate or a time-rate, is not a thing which has objective existence. No one would confound the bank rate of interest with a sum of money, nor the birth or death rate of a country with a group of individual human beings. These rates are, in fact, mere abstract numbers, by the help of which a man may compute interest per annum from the amount of capital, or the number of infants per annum from the amount of the population. The gradient of temperature, in an irregularly heated body, is a mere vector-rate, by the help of which we can calculate how much energy (in the form of heat) passes in a given time across any assigned surface in the body. To attribute objectivity to a rate is even more ridiculous than it would be to attribute it to a sensation, or to a thought, or to a word or phrase which we find useful in characterizing some material object.

§ 296. On the other hand, all these different kinds of rates have been introduced and continue to be employed, because they have been found to be useful. There is no harm done by retaining them, provided those who use them know that they are introduced for convenience of expression, and not because there are objective realities corresponding to them. Even such a term as “centrifugal force” is sometimes useful; but always under the proviso that he who employs it shall remember that it is only one side of the stress under which a particle of matter is compelled, in spite of its inertia, to move in a curved line. But the term must be taken, like “algebra,” “theodolite,” “Abracadabra,” or any other combination of letters whose derivation is uncertain or unknown, as one and indivisible, to which a certain definite meaning is attached, and as having nothing whatever to do with the meaning or derivation of the word centrifugal, whose embodiment in it is a perennial monument to the memory of an old error.

§ 297. The main characteristics of energy, especially from the experimental point of view, have already been discussed under Dynamics (< - ∕.z>.) and Energy (q.v.). But there is one point of importance connected with it which comes more naturally here than in either of the articles referred to.

When two measurable quantities, of any kind, are equivalent to one another, their numerical expressions must involve the same fundamental units, and in the same manner. This is obvious from the fact that an alteration of any unit alters in the inverse ratio the numerical measure of any quantity which is a mere multiple of it. And equivalent quantities must always be expressed by equal numbers when both are measured in terms of the same system of units. It appears, therefore, from the conservation of energy directly, as well as from the special data in §§111, 113, that potential energy must, like kinetic υnergy, be of dimensions [ML 2 T^ 2 ].

Now it is impossible to conceive of a truly dormant form of energy whose magnitude should depend in any way on the unit of time; and we are therefore forced to the conclusion that potential energy, like kinetic energy, depends (in some as yet unexplained, or rather unimagined, way) upon motion. For the immediate purposes of this article the question is not one of importance. We have been dealing with the more direct consequences of a very compact set of laws, exceedingly simple in themselves, originally based upon observation and experiment, and, most certainly, true. But reason cannot content itself with the mere consequences of a series of observed facts, however elegantly and concisely these may be stated by the help of new terms and their definitions. We are forced to inquire into what may underlie these definitions, and the laws which are observed to regulate the things signified by them. And the conclusion which appears inevitable is that, whatever matter may be, the other reality in the physical universe, energy, which is never found unassociated with matter, depends in all its widely varied forms upon motion of matter. In some cases we are sure, in others we can as yet only suspect, that it depends upon motions in a medium which, unlike ordinary matter, has not yet been subjected to the scrutiny of the chemist. But the question, in its generality, is one of the most obscure in the whole range of physics. In the articles Atom, Attraction, Ether, will be found nearly all that is yet known on this profoundly difficult subject. But to what is there said must be added the remark that a state of strain of the ether, whether associated with the propagation of light and radiant heat or with a statical distribution of electricity, represents so much “potential” energy, and must in its turn in some way depend on motion.

§ 298. The remarks of Clerk Maxwell on the nature of the evidence for Newton’s first law of motion raise a question, in some respects novel, but in all respects well worthy of careful study. He says :—

“Our conviction of the truth of this law may be greatly strengthened by considering what is involved in a denial of it. Given a body in motion. At a given instant let it be left to itself and not acted on by any force. What will happen? According to Newton’s law it will persevere in moving uniformly in a straight line; that is, its velocity will remain constant both in direction and magnitude.

“If the velocity does not remain constant let us suppose it to vary. The change of velocity must have a definite direction and magnitude. By the maxim that the same causes will always produce the same effects, this variation must be the same whatever be the time or place of the experiment. The direction of the change of motion must therefore be determined either by the direction of the motion itself, or by some direction fixed in the body. Let us, in the first place, suppose the law to be that the velocity diminishes at a certain rate, which, for the sake of the argument, we may suppose so slow that by no experiments on moving bodies could we have detected the diminution of velocity in hundreds of years. The velocity referred to in this hypothetical law can only be the velocity referred to a point absolutely at rest. For if it is a relative velocity, its direction as well as its magnitude depends on the velocity of the point of reference. If, when referred to a certain point, the body appears to be moving northward with diminishing velocity, we have only to refer it to another point moving northward with a uniform velocity greater than that of the body, and it will appear to be moving southward with increasing velocity. Hence the hypothetical law is without meaning, unless we admit the possibility of defining absolute rest and absolute velocity

“It may thus be shown that the denial of Newton’s law is in contradiction to the only system of consistent doctrine about space and time which the human mind has been able to form.”

This is a good example of a valuable application of a principle which, in its widest scope, is inconsistent with the true foundations of physical science. It is, in fact, the exceedingly dangerous “principle of sufficient reason ”— which requires for its legitimate use the utmost talent and knowledge on the part of the user.

§ 299. But in all methods and systems which involve the idea of force there is the leaven of artificiality. The true laws of motion, based entirely on experiments of the most extensive and most varied kinds, are those of the conservation and of the transformation of energy. With the help of kineιnatical ideas, it is easy to base the whole science of dynamics on these principles; and there is no necessity for the introduction of the word “force” nor of the sense-suggested ideas on which it was originally based.

§ 300. Nothing beyond a mere mention has been made above of virtual velocities, and of the so-called elementary machines. These belong to the subject of Applied Mechanics, separately treated below.

§ 301. The references which have been made to various grand theories, such as action, impulse in general, &c., have been illustrated by simple cases only. For a detailed examination of these theories the reader is referred to Thomson and Tait’s Natural Philosophy, chap. ii. To the same work he is referred for the general “theory of small oscillations,” the “dissipative function,” the “ignoration of coordinates,” the treatment of “gyrostatic systems” and of “kinetic stability.” All of these have been exhibited, though in mere particular instances, in the preceding pages.

Treatises on Mechanics.

§ 302. The following works on Mechanics are indispensable :—

1. Newton’s Principia (1st ed., 1687; latest ed., Glasgow, 1871). Here, for the first time, the fundamental principles were systematized, extended (as rve have seen) in a most vital particular, and applied, by the aid of a new mathematical method of immense power (based entirely on kinematical considerations), to many of the most important questions of cosmical and terrestrial dynamics.

Newton’s system was first taught in the university of Edinburgh; and, with brief intervals, his methods also have been habitually kept before the students there. From the time of Maclaurin to that of Forbes the value of the quasi-geometrical methods in giving a clear insight into the problems treated has rarely been overlooked. In Cambridge these methods were of later introduction, but they still deservedly figure as a necessary part of the reading of candidates for “mathematical honours.” It is to be feared, however, that in some other British universities the study of Newton’s methods is not prosecuted to anything like the same extent. But the very reverse seems to be the case in America, where, probably to a considerable extent on this account, mathematical physics is advancing in a most remarkable manner.

2. Lagrange’s Mécanique Analytique (1st cd., 1788). Though objections may fairly be taken to the fundamental method of this work, there can be no question as to the immense power and originality of its author. His “generalized coordinates,” and the equations of motion of a system in terms of these, form one of the most important contributions to the science since the days of Newton. The method of Lagrange, though he was not aware of the fact, is really based upon the consideration of energy; and when, in quite recent times, experiment had shown what are the grand laws of energy, Lagrange’s magnificent mathematical methods and results were ready for translation into the new language of science.

3. Hamilton’s papers in the Philosophical Transactions for 1833 and 1834. Here the principle of varying action, and the characteristic function, were first applied to mechanics;—though they had been given, some years before, to the Royal Irish Academy, in their optical applications. Grand as have been the extensions of these new ideas made by Hamilton himself, and by many others, among whom Jacobi and Lionville may be especially mentioned, they have been mainly in a purely mathematical direction. We wait for what cannot now be long delayed, the coming of the philosopher who is to tell us the true dynamical bearings of varying action and of the characteristic function.

4. If to these we add some of the works of Galileo, Huygens, Euler, Maclaurin, and D’Alembert, we have the great landmarks in the history of the subject, as distinguished from its development.

5. The mere enumeration of the more important developments which the subject has received, as distinguished from the absolutely new grand ideas and methods introduced, would require a long article. Brilliant examples of what may be done in this direction are furnished by Stokes’s “Report on Recent Researches in Hydrodynamics” and by Cayley’s “Reports on Theoretical Dynamics” (printed in the British Association Reports for 1846, and for 1857 and 1862). These should be consulted by every student who desires to trace the growth of the subject. They have been succeeded, in the same Reports (1880, 1881) by two excellent summaries, by Hicks, of “Recent Progress in Hydrodynamics.”

But Laplace’s Mécanique Celeste, Poisson’s Mécanique, Poinsot’s Théorie Nouvelle de la Rotation, kc., more or less parts of the immediate outcome of the period when France intellectually dwarfed the rest of the world, are still of far more than mere historic value.

For the English-reading student of modern times, the work of Thomson and Tait will be found suitable. The authors of this work claim the position of “restorers,” not of innovators; and they have (since 1863, when the first short sketch of their work was published) striven with success to re-establish in Britain Newton’s grand yet simple foundations of the subject. But these foundations, as stated above, are only temporarily the best. We have not, as yet, anything nearly so good.

Other modern works of value are the Analytic Mechanics of the late Professor Peirce (Boston, 1855) and Kirchhoffs Vorlesungen über Mathematische Physik (Leipsic, 1876). Both are rather of the nature of collections of short treatises on special questions than organized wholes, but both will well repay careful reading. This, in the case of Peirce’s work, is rendered extremely puzzling and laborious by the peculiar notations and modes of reference adopted by the author. It is particularly interesting to study the ways in which the fundamental principles are introduced in these works, and to compare them with the corresponding parts of the works of Newton and Lagrange. Lagrange, Peirce, and Kirchhoff construct each a system as free from anything but analysis as possible. In fact Lagrange prefaces his work by the characteristic statement, “On ne trouvera point de Figures dans cet ouvrage. Les méthodes que j’y expose ne demandent ni constructions, ni raisonnemens géométriques ou méchaniques, mais seulement des opérations algébriques, assujetties à une marche régulière et uniforme. Ceux qui aiment l’Analyse verront avec plaisir la Méchanique en devenir une nouvelle branche.” .... How far we have considered it expedient to differ from such an authority, a glance at the preceding pages will show.

A part of the detailed work of several of the examples above given in Dynamics of a Particle has been taken from the elementary treatise (with that title) of Tait and Steele. The English reader who wishes to pursue*elementary Statics may profitably consult the treatise of Minchin. The higher parts are discussed in the work of Somoff, Theoretische Mechanik (Leipsic, 1879). An excellent introduction to the use of Generalized Coordinates has been published by Watson and Burbury (1879). On Lagrange’s Generalized Equations the student should also read in Maxwell’s Treatise on Electricity and Magnetism, part iv. chap. v. And Maxwell’s brief treatise on Matter and Motion should be in the hands of every one commencing the subject, Analysis of the Preceding Article. Newton’s Laws of Motion, with Comments, assumed as the basis of the article, §§ 1 13.

Kinematics: Position, §§ 14-19; Kinematics of Point, §§ 20-70; of Plane Figure in its own Plane, §§ 71-74; of Rigid Figure, §§ 75-83; of Deformable Figure, §§ 84-95.

Dynamics of a Particle: General Considerations, §§ 96-113; Further Comments on the First Tu·o Laws of Motion, §§ 114- 119; Friction, §§ 120-121; Statics of a Particle, §§ 122-128; Kinetics of a Particle with One Degree of Freedom (Meteorite, Hailstone, Pendulum, Cycloidal and Resisted Pendulum), §§ 129-139; with Two Degrees of Freedom (Planetary Mo tion, Kepler's Laws and their Consequences, Kinetic Stability), §§ 140-149; The Brachistochrone, §§ 150-152; Kinetics of a Particle generally (Conical Pendulum, Blackburn s and Foucault's Pendulums, Varying Constraint, Disturbed Motion), §§ 153-163; Third Law, Kinetics of Two or More Particles (Atwood's Machine, Chain-shot, Complex Pendulum), §§ 164— 178; Kinetics of Free Particles generally, Virial, § 179; Impact (Continuous Series of Infinitely Small Impacts, Rocket), §§ 180-190; Dynamics of a System of Particles generally (Equilibrium— Neutral, Stable, and Unstable ; Lagrange's General Equation), §§ 191-199; Action, §§ 200-214; Gene ralized Coordinates, §§ 215, 216.

Statics of a Rigid Solid: Reduction of Forces to Force and Couple, Minding's Theorem, Examples of Statical Problems, §§ 217— 233.

Kinetics of a Rigid Solid: Moment of Inertia, Binefs Theorem, Compound Pcndul urn , Ballistic Pendulum, Rolling and Sliding of Sphere, Motion about Fixed Point, Poinsot's and Sylvester's Constructions, Quoit, Gyroscopic Pendulum, §§ 234-257.

Statics of a Chain: Common Catenary, Catenary of Uniform Strength, Kinetic Analogy, Chain Stretched on Surface, §§ 258-264.

Kinetics of a Chain: Wave Propagation, Musical Siring, Chain with One End Free, Impulsive Tension, Longitudinal JVave, §§ 265-270.

Dynamics of Elastic Solid: Flexure and Torsion of Wire, Bending of Plank, Oscillation of Flat Spring, Distortion oj Cylinders and Spheres by Internal and External Hydrostatic Pressure, §§ 271-284. τ 1 General Considerations about Force and Energy: Newtons Idea of Force, Origin of the Conception, Stress, Objective Physical Realities, True Nature of Force, Rates in General, Potential Energy in its Nature Kinetic, Maxwell on Inertia, True Laws of Motion, §§285-299.

References to Authoritative Works, §§ 301, 302. (I. G. Ί.) APPLIED MECHANICS.

1. The practical applications of mechanics may be divided into two classes, according as the assemblages of material objects to which they relate are intended to remain fixed or to move relatively to each other, —the former class being comprehended under the term 44 Theory of Structures,” and the latter under the term “Theory of Machines.” As the details of the theory of structures are dealt with in other articles, it will be treated of here to such extent only as may be necessary in order to state certain general principles applicable to all these subjects. The greater part of the article will relate to machines.

t

PART I. OUTLINE OF THE THEORY OF STRUCTURES.

2. Support of Structures.— Every structure, as a whole, is maintained in equilibrium by the joint action of its own weight, of the external load or pressure applied to it from without and tending to displace it, and of the resistance of the material which supports it. A structure is supported either by resting on the solid crust of the earth, as buildings do, or by floating in a fluid, as ships do in water and balloons in air. The principles of the support of a floating structure form an important part of Hydromechanics (q.v. ∖ The principles of the support, as a whole, of a structure resting on the land, are so far identical with those which regulate the equilibrium and stability of the several parts of that structure, and of which a summary will presently be given, that the only principle which seems to require special mention here is one which comprehends in one statement the power both of liquids and of loose earth to support structures, and which was first demonstrated in a paper “On the Stability of Loose Earth,” read to the Royal Society on the 19th of June 1856, and published in the Philosophical Trans actions for that year, viz. :—

Let E represent the weight of the portion of a horizontal stratum of earth which is displaced by the foundation of a structure, S the utmost weight of that structure consistently with the power of the earth to resist displacement, φ the angle of repose of the earth; then

S _ ∕ 1 + sin φ ∖ 2

E ∖l-sin≠∕'

α

To apply this to liquids, φ must be made =0, and then— =1, as is well known.

3. Composition of a Structure, and Connexion of its Pieces. — A structure is composed of pieces, — such as the stones of a building in masonry, the beams of a timber frame-work, the bars, plates, and bolts of an iron bridge. Those pieces are connected at their joints or surfaces of mutual contact, either by simple pressure and friction (as in masonry with moist mortar or without mortar), by pressure and adhesion (as in masonry with cement or with hardened mortar, and timber with glue), or by the resistance of fastenings of different kinds, whether made by means of the form of the joint (as dovetails, notches, mortises, and tenons) or by separate fastening pieces (as trenails, pins, spikes, nails, holdfasts, screws, bolts, rivets, hoops, straps, and sockets).

4. Stability, Stiffness, and Strength.— A structure may be damaged or destroyed in three ways :—first, by displacement of its pieces from their proper positions relatively to each other or to the earth; secondly, by disfigurement of one or more of those pieces, owing to their being unable to preserve their proper shapes under the pressures to which they are subjected; thirdly, by breaking of one or more of those pieces. The power of resisting displacement constitutes stability ; the power of each piece to resist disfigurement is its stiffness ; and its power to resist breaking, its strength.

5. Conditions of Stability.— The principles of the stability of a structure can be to a certain extent investigated independently of the stiffness and strength, by assuming, in the first instance, that each piece has strength sufficient to be safe against being broken, and stiffness sufficient to prevent its being disfigured to an extent inconsistent with the purposes of the structure, by the greatest forces which are to be applied to it. The condition that each piece of the structure is to be maintained in equilibrium by having its gross load, consisting of its own weight and of the external pressure applied to it, balanced by the resistances or pressures exerted between it and the contiguous pieces, furnishes the means of determining the magnitude, position, and direction of the resistances required at each joint in order to produce equilibrium; and the conditions of stability are, first, that the position, and, secondly, that the direction, of the resistance required at each joint shall, under all the variations to which the load is subject, be such as the joint is capable of exerting,—conditions which are fulfilled by suitably adjusting the figures and positions of the joints, and the ratios of the gross loads of the pieces. As for the magnitude of the resistance, it is limited by conditions, not of stability, but of strength and stiffness.

6. Principle of Least Resistance. — Where more than one system of resistances are alike capable of balancing the same system of loads applied to a given structure, it has been demonstrated by Moseley that the smallest of those alternative systems is that which will actually be exerted,—because the resistances to displacement are the effect of a strained state of the pieces, which strained state is the effect of the load, and when the load is applied the strained state and the resistances produced by it increase until the resistances acquire just those magnitudes which are sufficient to balance the load, after which they increase no further.

This principle of least resistance renders determinate many problems in the statics of structures which were formerly considered indeterminate.

7. Relations between Polygons of Loads and of Resistances. — In a structure in which each piece is supported at two joints only, the well-known laws of statics show that the directions of the gross load on each piece and of the two resistances by which it is supported must lie in one plane, must either be parallel or meet in one point, and must bear to each other, if not parallel, the proportions of the sides of a triangle respectively parallel to their directions, and, if parallel, such proportions that each of the three forces shall be proportional to the distance between the other two,—all the three distances being measured along one direction.

Considering, in the first place, the case in which the load and the two resistances by which each piece is balanced meet in one point, which may be called the centre of load, there will be as many such points of intersection, or centres of load, as there are pieces in the structure; and the directions and positions of the resistances or mutual pressures exerted between the pieces will be represented by the sides of a polygon joining those points, as in fig. 1, where Pi, 1*2, I*3> I*4 represent the centres of load in a structure of four pieces, and the sides of the polygon of resistances P 1 P 2 P 3 P 4 represent respectively the directions and position^ of the resistances exerted at the joints. Further, at any one of the centres of load let PL represent the magnitude and direction of the gross load, and Pa, Pb the two resistances by which the piece to which that load is applied is supported; then will those three lines be respectively the diagonal and sides of a parallelogram; or, what is the same thing, they will be equal to the three sides of a triangle; and they must be in the same plane, although the sides of the polygon of resistances may be in different planes.

According to a well-known principle of statics, because the loads or external pressures P 1 I. 1 , &c., balance each other, they must be proportional to the sides of a closed polygon drawn respectively parallel to their directions. In fig. 2 construct such a polygon of loads by drawing the lines I. 1 , &c., parallel and proportional to, and joined end to end in the order of, the gross loads on the pieces of the structure. Then from the proportionality and parallelism of the load and the two resistances applied to each piece of the structure to the three sides of a triangle, there results the following theorem [originally due to Rankine];—

If from the angles of the polygon of loads there be drawn lines (R 1 , R 2 , &c.), each of which is parallel to the resistance {as P 1 P 2 , &c.) exerted at the joint between the pieces to which the two loads represented by the contiguous sides of the polygon of loads (such as I. 1 , I. 2 , &c. ) are applied; then will all those lines meet in one point (0), and their lengths, measured from that point to the angles of the polygon, will represent the magnitudes of the resistances to which they are respectively parallel.

When the load on one of the pieces is parallel to the resistances which balance it, the polygon of resistances ceases to be closed, two of the sides becoming parallel to each other and to the load in question, and extending indefinitely. In the polygon of loads the direction of a load sustained by parallel resistances traverses the point 0.

8. How the Earth's Resistance is to be treated.— When the pressure exerted by a structure on the earth (to which the earth’s resistance is eqval and opposite) consists either of one pressure, which is necessarily the resultant of the weight of the structure and of all the other forces applied to it, or of two or more parallel vertical forces, whose amount can be determined at the outset of the investigation, the resistance of the earth can be treated as one or more upward loads applied to the structure. But in other cases the earth is to be treated as one of the pieces of the structure, loaded with a force equal and opposite in direction and position to the resultant of the weight of the structure and of the other pressures applied to it.

9. Partial Polygons of Resistance.— In a structure in which there are pieces supported at more than two joints, let a polygon be constructed of lines connecting the centres of load of any continuous series of pieces. This may be called a partial polygon of resistances. In considering its properties, the load at each centre of load is to be held to include the resistances of those joints which are not comprehended in the partial polygon of resistances, to which the theorem of section 7 will then apply in every respect. By constructing several partial polygons, and computing the relations between the loads and resistances which are determined by the application of that theorem to each of them, with the aid, if necessary, of Moseley’s principle of the least resistance, the whole of the relations amongst the loads and resistances may be found.

10. Line of Pressures—Centres and Line of Resistance.— The line of pressures is a line to which the directions of all the resistances in one polygon are tangents. The centre of resistance at any joint is the point where the line representing the total resistance exerted at that joint intersects the joint. The line of resistance is a line traversing all the centres of resistance of a series of joints,—its form, in the positions intermediate between the actual joints of the structure, being determined by supposing the pieces and their loads to be subdivided by the introduction of intermediate joints ad infinitum, and finding the continuous line, curved or straight, in which the intermediate centres of resistance are all situated, however great their number. The difference between the line of resistance and the line of pressures was first pointed out by Moseley.

11. Stability of Position, and Stability of Friction.— The resistances at the several joints having been determined by the principles set forth in sections 6, 7, 8, 9, and 10, not only under the ordinary load of the structure, but under all the variations to which the load is subject as to amount and distribution, the joints are now to be placed and shaped so that the pieces shall not suffer relative displacement under any of those loads. The relative displacement of the two pieces which abut against each other at a joint may take place either by turning or by sliding. Safety against displacement by turning is called stability of position ; safety against displacement by sliding, stability of friction.

12. Condition of Stability of Position. — If the materials of a structure were infinitely stiff and strong, stability of position at any joint would be insured simply by making the centre of resistance fall within the joint under all possible variations of load. In order to allow for the finite stiffness and strength of materials, the least distance of the centre of resistance inward from the nearest edge of the joint is made to bear a definite proportion to the depth of the joint measured in the same direction, which proportion is fixed, sometimes empirically, sometimes by theoretical deduction from the laws of the strength of materials. That least distance is called by Moseley the modulus of stability. The following are some of the ratios of the modulus of stability to the depth of the joint which occur in practice :— [table]

In the case of the towers, the depth of the joint is to be understood to mean the diameter of the tower.

13. Condition of Stability of Friction.— If the resistance to be exerted at a joint is always perpendicular to the surfaces which abut at and form that joint, there is no tendency of the pieces to be displaced by sliding. If the resistance be oblique, let JK (fig. 3) be the joint, C its centre of resistance, CR a line representing the resistance, CN a perpendicular to the joint at the centre of resistance. The angle NCR is the obliquity of the resistance. From R draw RP parallel and RQ perpendicular to the joint; then, by the principles of statics, the component of the resistance normal to the joint is—

CP = CR . cos^≤,PRC; and the component tangential to the joint is—

CQ = CR . sin^PCR = CP . tan^≤PCR.

If the joint be provided either with projections and recesses, such as mortises and tenons, or with fastenings, such as pins or bolts, so as to resist displacement by sliding, the question of the utmost amount of the tangential resistance CQ which it is capable of exerting depends on the strength of such projections, recesses, or fastenings, and belongs to the subject of strength, and not to that of stability. In other cases the safety of the joint against displacement by sliding depends on its power of exerting friction, and that power depends on the law, known by experiment, that the friction between two surfaces bears a constant ratio, depending on the nature of the surfaces, to the force by which they are pressed together. In order that the surfaces which abut at the joint JK may be pressed together, the resistance required by the conditions of equilibrium, CR, must be a thrust and not a pull ; and in that case the force by which the surfaces are pressed together is equal and opposite to the normal component CP of the resistance. The condition of stability of friction is that the tangential component CQ of the resistance required shall not exceed the friction due to the normal component; that is, that

CQ>∕. CP, where f denotes the coefficient of friction for the surfaces in question. The angle whose tangent is the coefficient of friction is called the angle of repose, and is expressed symbolically by—

φ = tan ^ 1 f.

Now CQ = CP . tan^,PCR; consequently the condition of stability of friction is fulfilled if

^PCR>≠5

that is to say, if the*obliquity of the resistance required at the joint does not exceed the angle of repose ; and this condition ought to be fulfilled under all possible variations of the load.

It is chiefly in masonry and earthwork that stability of friction is relied on.

14. Stability of Friction in Earth.— The grains of a mass of loose earth are to be regarded as so many separate pieces abutting against each other at joints in all possible positions, and depending for their stability on friction. To determine whether a mass of earth is stable at a given point, conceive that point to be traversed by planes in all possible positions, and determine which position gives the greatest obliquity to the total pressure exerted between the portions of the mass which abut against each other at the plane. The condition of stability is that this obliquity shall not exceed the angle of repose of the earth. The consequences of this principle are developed in a paper “On the Stability of Loose Earth,” already cited in sect. 2.

15. Parallel Projections of Figures.— If any figure be referred to a system of coordinates, rectangular or oblique, and if a second figure be constructed by means of a second system of coordinates, rectangular or oblique, and either agreeing with or differing from the first system in rectangularity or obliquity, but so related to the coordinates of the first figure that for each point in the first figure there shall be a corresponding point in the second figure, the lengths of whose coordinates shall bear respectively to the three corresponding coordinates of the corresponding point in the first figure three ratios which are the same for every pair of corresponding points in the two figures, these corresponding figures are called parallel projections of each other. The properties of parallel projections of most importance to the subject of the present article are the following :—

(1) A parallel projection of a straight line is a straight line.

(2) A parallel projection of a plane is a plane.

(3) A parallel projection of a straight line or a plane surface divided in a given ratio is a straight line or a plane surface divided in the same ratio.

(4) A parallel projection of a pair of equal and parallel straight lines, or plain surfaces, is a pair of equal and parallel straight lines, or plane surfaces; whence it follows

(5) That a parallel projection of a parallelogram is a parallelogram, and

(6) That a parallel projection of a parallelepiped is a parallelepiped.

(7) A parallel projection of a pair of solids having a given ratio is a pair of solids having the same ratio.

Though not essential for the purposes of the present article, the following consequence will serve to illustrate the principle of parallel projections :—

(8) A parallel projection of a curve, or of a surface of a given algebraical order, is a curve or a surface of the same order.

For example, all ellipsoids referred to coordinates parallel to any three conjugate diameters are parallel projections of each other and of a sphere referred to rectangular coordinates.

16. Parallel Projections of Systems of Forces.— If a balanced system of forces be represented by a system of lines, then will every parallel projection of that system of lines represent a balanced system of forces.

For the condition of equilibrium of forces not parallel is that they shall be represented in direction and magnitude by the sides and diagonals of certain parallelograms, and of parallel forces that they shall divide certain straight lines in certain ratios; and the parallel projection of a parallelogram is a parallelogram, and that of a straight line divided in a given ratio is a straight line divided in the same ratio.

The resultant of a parallel projection of any system of forces is the projection of their resultant; and the centre of gravity of a parallel projection of a solid is the projection of the centre of gravity of the first solid.

17. Principle of the Transformation of Structures.— Here we have the following theorem If a structure of a given figure have stability of position under a system of forces represented by a given system of lines, then will any structure whose figure is a parallel projection of that of the first structure have stability of position under a system of forces represented by the corresponding projection of the first system of lines.

For in the second structure the weights, external pressures, and resistances will balance each other as in the first structure; the weights of the pieces and all other parallel systems of forces will have the same ratios as in the first structure; and the several centres of resistance will divide the depths of the joints in the same proportions as in the first structure.

If the first structure have stability of friction, the second structure will have stability of friction also, so long as the effect of the projection is not to increase the obliquity of the resistance at any joint beyond the angle of repose.

The lines representing the forces in the second figure show their relative directions and magnitudes. To find their absolute directions and magnitudes, a vertical line is to be drawn in the first figure, of s ’ jch a length as to represent the weight of a particular portion of the structure. Then will the projection of that line in the projected figure indicate the vertical direction, and represent the weight of the part of the second structure corresponding to the beforementioned portion of the first structure.

The foregoing “principle of the transformation of structures” was first announced, though in a somewhat less comprehensive form, to the Royal Society on the 6th of March 1856. It is useful in practice, by enabling the engineer easily to deduce the conditions of equilibrium and stability of structures of complex and unsymmetrical figures from those of structures of simple and symmetrical figures. By its aid, for example, the whole of the properties of elliptical arches, whether square or skew, whether level or sloping in their span, are at once deduced by projection from those of symmetrical circular arches, and the properties of ellipsoidal and elliptic-conoidal domes from those of hemispherical and circular-conoidal domes; and the figures of arches fitted to resist the thrust of earth, which is less horizontally than vertically in a certain given ratio, can be deduced by a projection from those of arches fitted to resist the thrust of a liquid, which is of equal intensity, horizontally and vertically.

18. Conditions of Stiffness and Strength.— After the arrangement of the pieces of a structure and the size and figure of their joints or surfaces of contact have been determined so as to fulfil the conditions of stability,— conditions which depend mainly on the position and direction of the resultant or total load on each piece, and the relative magnitude of the loads on the different pieces,—the dimensions of each piece singly have to be adjusted so as to fulfil the conditions of stiffness and strength,— conditions which depend not only on the absolute magnitude of the load on each piece, and of the resistances by which it is balanced, but also on the mode of distribution of the load over the piece, and of the resistances over the joints.

The effect of the pressures applied to a piece, consisting of the load and the supporting resistances, is to force the piece into a state of strain or disfigurement, which increases until the elasticity, or resistance to strain, of the material causes it to exert a stress, or effort to recover its figure, equal and opposite to the system of applied pressures. The condition of stiffness is that the strain or disfigurement shall not be greater than is consistent with the purposes of the structure; and the condition of strength is that the stress shall be within the limits of that which the material can bear with safety against breaking. The ratio in which the utmost stress before breaking exceeds the safe working stress is called the factor of safety, and is determined empirically. It varies from three to twelve for various materials and structures.

The Strength of Materials forms the subject of a special article, to which the reader is referred.

PART II. THEORY OF MACHINES.

19. Parts of a Machine—Frame and Mechanism.— The parts of a machine may be distinguished into two principal divisions,—the frame, or fixed parts, and the mechanism, or moving parts. The frame is a structure which supports the pieces of the mechanism, and to a certain extent determines the nature of their motions. The form and arrangement of the pieces of the frame depend upon the arrangement and the motions of the mechanism; the dimensions of the pieces of the frame required in order to give it stability and strength are determined from the pressures applied to it by means of the mechanism. It appears therefore that in general the mechanism is to be designed first and the frame afterwards, and that the designing of the frame is regulated by the principles of the stability of structures and of the strength and stiffness of materials,—care being taken to adapt the frame to the most severe load which can be thrown upon it at any period of the action of the mechanism.

Each independent piece of the mechanism also is a structure, and its dimensions are to be adapted, according to the principles of the strength and stiffness of materials, to the most severe load to which it can be subjected during the action of the machine.

20. Definition and Division of the Theory of Machines.— From what has been said in the last section it appears that the department of the art of designing machines which has reference to the stability of the frame and to the stiffness and strength of the frame and mechanism is a branch of the art of construction. It is therefore to be separated from the theory of machines, properly speaking, which has reference to the action of machines considered as moving. In the action of a machine the following three things take place:—

First, Some natural source of energy communicates motion and force to a piece or pieces of the mechanism, called the receiver of poιυer or prime mover. Secondly, The motion and force are transmitted from the prime mover through the train of mechanism to the working piece or pieces, and during that transmission the motion and force are modified in amount and direction, so as to be rendered suitable for the purpose to which they are to be applied.

Thirdly, The working piece or pieces by their motion, or by their motion and force combined, produce some useful effect.

Such are the phenomena of the action of a machine, arranged in the order of causation. But in studying or treating of the theory of machines, the order of simplicity is the best; and in this order the first branch of the subject is the modification of motion and force by the train of mechanism; the next is the effect or purpose of the machine; and the last, or most complex, is the action of the prime mover.

The modification of motion and the modification of force take place together, and are connected by certain laws; but in the study of the theory of machines, as well as in that of pure mechanics, much advantage has been gained in point of clearness and simplicity by first considering alone the principles of the modification of motion, which are founded upon what is now known as Kinematics, and afterwards considering the principles of the combined modification of motion and force, which are founded both on geometry and on the laws of dynamics. The separation of kinematics from dynamics is due mainly to Monge, Ampère, and Willis.

The theory of machines in the present article will be considered under the following four heads :—

I. Pure Mechanism, or Applied Kinematics ; being the theory of machines considered simply as modifying motion.

II. Applied Dynamics ; being the theory of machines considered as modifying both motion and force.

III. Purposes and Effects of Machines.

IV. Applied Energetics ; being the theory of prime movers and sources of power.

Chap. I. On Pure Mechanism.

21. Division of the Subject.— Proceeding in the order of simplicity, the subject of Pure Mechanism, or Applied Kinematics, may be thus divided :— [table]

A point is the boundary of a line, which is the boundary of a surface, which is the boundary of a volume. Points, lines, and surfaces have no independent existence, and consequently those divisions of this chapter which relate to their motions are only preliminary to the subsequent divisions, which relate to the motions of bodies.

Division 1. Motion of a Point.

22. Path and Direction.— See above, p. 679, § 21.

23. Uniform Velocity.— See p. 680, § 25.

24. Varied Velocity.— See p. 680, § 25.

25. Direct Deviation, or Acceleration and Retardation.— See pp. 680, 681, §§ 27-29.

26. Lateral Deviation or Deflexion —Angular Velocity of Deviation —Revolution.— See pp. 681, 682, §§ 31-38.

27. Comparative Motion.— The comparative motion of two points is the relation which exists between their motions, without having regard to their absolute amounts. It consists of two elements,— the velocity ratio, which is the ratio of any two magnitudes bearing to each other the proportions of the respective velocities of the two points at a given instant, and the directional relation, which is the relation borne to each other by the respective directions of the motions of the two points at the same given instant.

It is obvious that the motions of a pair of points may be varied in any manner, whether by direct or by lateral deviation, and yet that their comparative motion may remain constant, in consequence of the deviations taking place in the same proportions, in the same directions, and at the same instants for both points.

Willis has the merit of having been the first to simplify considerably the theory of pure mechanism, by pointing out that that branch of mechanics relates wholly to comparative motions.

The comparative motion of two points at a given instant is capable of being completely expressed by one of Sir William Hamilton’s Quaternions,—the “tensor” expressing the velocity ratio, and the “versor” the directional relation.

28. Resolution and Composition of Motion.— See p. 681, §§ 30, 31.

29. Rectangular Projection, Resolution, and Composition.— See p. 681, § 31.

30. Resolution and Composition of Deviations.— See p. 681, § 31.

Division 2. Motion of the Surface of a Fluid Mass.

31. General Principle.— A mass of fluid is used in mechanism to transmit motion and force between two or more movable portions (called pistons or plungers} of the solid envelope or vessel in which the fluid is contained; and, when such transmission is the sole action, or the only appreciable action of the fluid mass, its volume is either absolutely constant, by reason of its temperature and pressure being maintained constant, or not sensibly varied.

Let a represent the area of the section of a piston made by a plane perpendicular to its direction of motion, and υ its velocity, which is to be considered as positive when outward, and negative when inward. Then the variation of the cubic contents of the vessel in a unit of time by reason of the motion of one piston is va. The condition that the volume of the fluid mass shall remain unchanged requires that there shall be more than one piston, and that the velocities and areas of the pistons shall be connected by the equation—

2 . va=0 (1).

32. Comparative Motion of two Pistons.— If there be but two pistons, whose areas are α 1 and a i , and their velocities v 1 and v. 2 , their comparative motion is expressed by the equation—

⅛= -¾ (2);

Vi a 3 that is to say, their velocities are opposite as to inwardness and outwardness, and inversely proportional to their areas.

33. Applications—Hydraulic Press—Pneumatic Power-Transmitter.— In the hydraulic press the vessel consists of two cylinders, viz., the pump-barrel and the press-barrel, each having its piston, and of a passage connecting them having a valve opening towards the press-barrel. The action of the enclosed water in transmitting motion takes place during the inward stroke of the pump-plunger, when the above-mentioned valve is open; and at that time the press-plunger moves outwards with a velocity which is less than the inward velocity of the pump-plunger, in the same ratio that the area of the pump-plunger is less than the area of the pressplunger. (See Hydromechanics.)

In the pneumatic power-transmitter the motion of one piston is transmitted to another at a distance by means of a mass of air contained in two cylinders and an intervening tube. When the pressure and temperature of the air can be maintained constant, this machine fulfils equation 2, like the hydraulic press. The amount and effect of the variations of pressure and temperature undergone by the air depend on the principles of the mechanical action of heat, or Thermodynamics (g.v.), and are foreign to the subject of pure mechanism.

Division 3. Motion of a Rigid Solid.

31. Motions Classed.— In problems of mechanism, each solid piece of the machine is supposed to be so stiff and strong as not to undergo any sensible change of figure or dimensions by the forces applied to it,—a supposition which is realized in practice if the machine is skilfully designed.

This being the case, the various possible motions of a rigid solid body may all be classed under the following heads:—(1) Shifting or Translation ; (2) Turning or Rotation ; (3) Motions compounded of Shifting and Turning.

The most common forms for the paths of the points of a piece of mechanism, whose motion is simple shifting, are the straight line and the circle.

Shifting in a straight line is regulated either by straight fixed guides, in contact with which the moving piece slides, or by combinations of link-work, called parallel motions, which will be described in the sequel. Shifting in a straight line is usually reciprocating ; that is to say, the piece, after shifting through a certain distance, returns to its original position by reversing its motion.

Circular shifting is regulated by attaching two or more points of the shifting piece to ends of equal and parallel rotating cranks, or by combinations of wflιeel-work to be afterwards described. As an example of circular shifting may be cited the motion of the coupling rod, by which the parallel and equal cranks upon two or more axles of a locomotive engine are connected and made to rotate simultaneously. The coupling rod remains always parallel to itself, and all its points describe equal and similar circles relatively to the frame of the engine, and move in parallel directions with equal velocities at the same instant.

35. Rotation about a Fixed Axis—Lever, Wheel, and Axle.— The fixed axis of a turning body is a line fixed relatively to the body and relatively to the fixed space in which the body turns. In mechanism it is usually the central line either of a rotating shaft or axle having journals, gudgeons, or pivots turning in fixed bearings, or of a fixed spindle or dead centre round which a rotating bush turns; but it may sometimes be entirely beyond the limits of the turning body. For example, if a sliding piece moves in circular fixed guides, that piece rotates about an ideal fixed axis traversing the centre of those guides.

dθ

Let the angular velocity of the rotation be denoted by α=^ , then the linear velβcity of any point A at the distance r from the axis is ar ; and the path of that point is a circle of the radius r described about the axis.

This is the principle of the modification of motion by the lever, which consists of a rigid body turning about a fixed axis called a fulcrum, and having two points at the same or different distances from that axis, and in the same or different directions, one of which receives motion and the other transmits motion, modified in direction and velocity according to the above law.

In the wheel and axle, motion is received and transmitted by two cylindrical surfaces of different radii described about their common fixed axis of turning, their velocity-ratio being that of their radii.

36. Velocity Ratio of Components of Motion.— As the distance between any two points in a rigid body is invariable, the projections of their velocities upon the line joining them must be equal. Hence it follows that, if A in fig. 4 be a point in a rigid body CD, rotating round the fixed axis F, the component of the velocity of A in any direction AP parallel to the plane of rotation is equal to the total velocity of the point m, found by letting fall Fm perpendicular to AP; that is to say, is equal to α Frø .

Hence also the ratio of the components of the velocities of two points A and B in the directions AP and BW respectively, both in the plane of rotation, is equal to the ratio of the perpendiculars Fm and Fn.

37. Instantaneous Axis of a Cylinder rolling on a Cylinder.— Let a cylinder bbb, whose axis of figure is B and angular velocity 7, roll on a fixed cylinder aaa, whose axis of figure is A, either outside (as in fig. 5), when the rolling will be towards the same hand as the rotation, or inside (as in fig. 6), when the rolling will be towards the opposite hand; and at a given instant let T be the line of contact of the two cylindrical surfaces, which is at their common intersection with the plane AB tr⅜versing the two axes of figure.

The line T on the surface bbb has for the instant no velocity in a direction perpendicular to AB; because for the instant it touches, without sliding, the line T on the fixed surface aaa.

The line T on the surface δZ>⅛ has also for the instant no velocity in the plane AB; for it has just ceased to move towards the fixed surface aaa, and is just about to begin to move away from that surface.

The line of contact T, therefore, on the surface of the cylinder bbb, is for the instant at rest, and is the “instantaneous axis” about which the cylinder bbb turns, together with any body rigidly attached to that cylinder.

To find, then, the direction and velocity at the given instant of any point P, either in or rigidly attached to the rolling cylinder T, draw the plane PT; the direction of motion of P will be perpendicular to that plane, and towards the right or left hand according to the direction of the rotation of bbb ; and the velocity of P will be

⅛= γ .PT ....... (3),

PT denoting the perpendicular distance of P from T. The path of P is a curve of the kind called epitrochoids. If P is in the circumference of bbb, that path becomes an epicycloid.

The velocity of any point in the axis of figure B is

1⅛ = 7 . TB 4); and the path of such a point is a circle described about A with the radius AB, being for outside rolling the sum, and for inside rolling the difference, of the radii of the cylinders.

Let α denote the angular velocity with which the plane of axes AB rotates about the fixed axis A. Then it is evident that t⅛ = α . AB (5),

TR

and consequently that a = γ.-- (6).

-Δ.Ι5

For internal rolling, as in fig. 6, AB is to be treated as negative, which will give a negative value to a, indicating that in this case the rotation of AB round A is contrary to that of the cylinder bbb.

The angular velocity of the rolling cylinder, relatively to the plane of axes AB, is obviously given by the equation—

3 = 7-α;∖

. a .TA (7); whence β = y ∖γβ )

care being taken to attend to the sign of a, so that when that is negative the arithmetical values of 7 and a are to be added in order to give that of β.

The whole of the foregoing reasonings are applicable, not merely when aaa and bbb are actual cylinders, but also when they are the osculating cylinders of a pair of cylindroidal surfaces of varying curvature, A and B being the axes of curvature of the parts of those surfaces which are in contact for the instant under consideration.

38. Composition and Resolution of Rotations about Parallel Axes. —See above, p. 691, § 73.

39. Instantaneous Axis of a Cone rolling on a Cone. — Let Oαα (fig. 7) be a fixed cone, OA its axis, Ο&δ a cone rolling on it, OB the axis of the rolling cone, OT the line of contact of the two cones at the instant under consideration. By reasoning similar to that of sect. 37, it appears that OT is the instantaneous axis of rotation of the rolling cone.

Let 7 denote the total angular velocity of the rotation of the cone B about the instantaneous axis, β its angular velocity about the axis OB relatively to the plane AOB, and a the angular velocity with which the plane AOB turns round the axis OA. It is required to find the ratios of those angular velocities.

Solution.— In OT take any point E, from which draw EC parallel to OA, and ED parallel to OB, so as to construct the parallelogram OCED. Then

OD: OC: OE ) , 0 .

∙.∙. a ·. β ∙. y ∖ ''°>∙

Or because of the proportionality of the sides of triangles to the sines of the opposite angles, sin-≤TOB: sin^<TOA: sin^AOB I ..(8a)∙ :; α: β: ) that is to say, the angular velocity about each axis is proportional to the sine of the angle between the other two.

Demonstration. —From C draw CF perpendicular to OA, and CG perpendicular to OE.

rr , er c area ECO

Then CF = 2 × — ,

CiL·

, rιr , _ area ECO and CG = 2 × ——;

.∙. CG: CF :: CE = OD: OE .

Let v e denote the linear velocity of the point C. Then v c = a . CF = 7 . CG

. ∙. 7; α:: CF: CG:: OE: OD; which is one part of the solution above stated. From E draw EH perpendicular to OB, and EK to OA. Then it can be shown as before that

EK: EH :: OC: OD.

Let v t be the linear velocity of the point E fixed in the plane of axes AOB. Then r 1 = α . EK .

Now, as the line of contact OT is for the instant at rest on the rolling cone as well as on the fixed cone, the linear velocity of the point E fixed to the plane AOB relatively to the rolling cone is the same with its velocity relatively to the fixed cone. That is to say, β . EH = ι⅛ = α . EK; therefore a: β :: EH: EK:: OD: OC, which is the remainder of the solution.

The path of a point P in or attached to the rolling cone is a spherical epitrochoid traced on the surface of a sphere of the radius OP. From P draw PQ perpendicular to the instantaneous axis. Then the motion of P is perpendicular to the plane OPQ, and its velocity is tf p =7∙PQ (9).

The whole of the foregoing reasonings are applicable, not merely when A and B are actual regular cones, but also when they are the osculating regular cones of a pair of irregular conical surfaces, having a common apex at O.

40. Composition of Rotations about Two Axes meeting in a Point.— See p. 691, § 76.

41. Screw-like or Helical Motion.— Since (see p. 690, §§ 71, 72) any displacement in a plane can be represented in general by a rotation, it follows that the only combination of translation and rotation, in which a complex movement which is not a mere rotation is produced, occurs when there is a translation perpendicular to the plane and parallel to the axis of rotation.

Such a complex motion is called screw-like or helical motion; for each point in the body describes a helix or screw round the axis of rotation, fixed or instantaneous as the case may be. To cause a body to move in this manner it is usually made of a helical or screw-like figure, and moves in a guide of a corresponding figure. Helical motion and screws adapted to it are said to be rightor left - handed according to the appearance presented by the rotation to an observer looking towards the direction of the translation. Thus the screw G in fig. 8 is right-handed.

The translation of a body in helical motion is called its advance. Let v x denote the velocity of advance at a given instant, which of course is common to all the particles of the body; a the angular velocity of the rotation at the same instant; 2π = 6 - 2832 nearly, the circumference of a circle of the radius unity. Then

T=— (10)

α

is the time of one turn at the rate a ; and rp 2τΓfχ /1 1 ∖

p≈v x Υ≈ (11)

α

is the pitch or advance per turn,— a length which expresses the comparative motion of the translation and the rotation.

The pitch of a screw is the distance, measured parallel to its axis, between two successive turns of the same thread or helical projection.

Let r denote the perpendicular distance of a point in a body moving helically from the axis. Then v r =ar (12)

is the component of the velocity of that point in a plane perpendicular to the axis, and its total velocity is r = √{ι√ + v r 2} (13).

The ratio of the two components of that velocity is

¾=∕- = tan0 ....... (14); v r 2πr where θ denotes the angle made by the helical path of the point with a plane perpendicular to the axis.

42. To fiml the Motion of a Rigid Body from the Motions oj Three Points in it.— See p. 690, § 71, and p. 692, § 78.

Division 4. Elementary Combinations in Mechanism.

43. Definitions.— An elementary combination in mechanism consists υf two pieces whose kinds of motion are determined hy their connexion with the frame, and their comparative motion by their connexion with each other,—that connexion being effected either by direct contact of the pieces, or by a connecting piece, which is not connected with the frame, and whose motion depends entirely on the motions of the pieces which it connects.

The piece whose motion is the cause is called the driver ; the piece whose motion is the effect, the follower.

The connexion of each of those two pieces with the frame is in general such as to determine the path of every point in it. In the investigation, therefore, of the comparative motion of the driver and follower, in an elementary combination, it is unnecessary to consider relations of angular direction, which are already fixed by the connexion of each piece with the frame; so that the inquiry is confined to the determination of the velocity ratio, and of the directional relation, so far only as it expresses the connexion between forward and backward movements of the driver and follower. When a continuous motion of the driver produces a continuous motion of the follower, forward or backward, and a reciprocating motion a motion reciprocating at the same instant, the directional relation is said to be constant. When a continuous motion produces a reciprocating motion, or vice versa, or when a reciprocating motion produces a motion not reciprocating at the same instant, the directional relation is said to be variable.

The line of action or of connexion of the driver and follower is a line traversing a pair of points in the driver and follower respectively, which are so connected that the component of their velocity relatively to each other, resolved along the line of connexion, is null. There may be several or an indefinite number of lines of connexion, or there may be but one; and a line of connexion may connect either the same pair of points or a succession of different pairs.

44. General Principle.— From the definition of a line of connexion it follows that the components of the velocities of a pair of connected points along their line of connexion are equal. And from this, and from the property of a rigid body, already stated in sect. 36, it follows, that the components along a line of connexion of all the points traversed by that line, whether in the driver or in the follower, are equal ; and consequently, that the velocities of any pair of points traversed by a line of connexion are to each other inversely as the cosines, or directly as the secants, of the angles made by the paths of those points with the line of connexion.

The general principle stated above in different forms serves to solve every problem in which—the mode of connexion of a pair of pieces being given—it is required to find their comparative motion at a given instant, or vice versa.

45. Application to a Pair of Shifting Pieces.— In fig. 9, let P 1 P 2 be the line of connexion of a pair of pieces, each of which has a motion of translation or shifting. Through any point T in that line draw TV 1 , TV 2 , respectively parallel to the simultaneous direction of motion of the pieces; through any other point A in the line of connexion draw a plane perpendicular to that line, cutting TV 1 , TV 2 in V 1 , V 2 ; then, velocity of piece 1: velocity of piece 2:: TV 1 : TV 2 . Also TA represents the equal components of the velocities of the pieces parallel to their line of connexion, and the line V 1 V 2 represents their velocity relatively to each other.

46. Application to a Pair of Turning Pieces.— Let α 1 , α 2 be the angular velocities of a pair of turning pieces; θ 1 , θ 2 the angles which their line of connexion makes with their respective planes of rotation; r 1 , r 2 the common perpendiculars let fall from the line of connexion upon the respective axes of rotation of the pieces. Then the equal components, along the line of connexion, of the velocities of the points where those perpendiculars meet that line are—

α 1 r 1 cos θ 1 = a 2 r 2 cos θ 2 ; consequently, the comparative motion of the pieces is given by the equation = r ι cos g ι (15).

α 1 r 2 cos0 2

47. Application to a Shifting. Piece and a Turning Piece.— Let a shifting piece be connected with a turning piece, and at a given instant let α 1 be the angular velocity of the turning peice, r 1 the common perpendicular of its axis of rotation and the line of connexion, θι the angle made by the line of connexion with the plane of rotation, θ 2 the angle made by the line of connexion with the direction of motion of the shifting piece, v, 1 the linear velocity of that piece. Then

α 1 r 1 cos θ l — v a cos θ 2 (16); which equation expresses the comparative motion of the two pieces.

48. Classification of Elementary Combinations in Mechanism.— The first systematic classification of elementary combinations in mechanism was that founded by Monge, and fully developed by Lanz and Betancourt, which has been generally received, and has been adopted in most treatises on applied mechanics. But that classification is founded on the absolute instead of the comparative motions of the pieces, and is, for that reason, defective, as Willis has pointed out in his admirable treatise On the Principles of Mechanism.

Willis’s classification is founded, in the first place, on comparative motion, as expressed by velocity ratio and directional relation, and in the second place, on the mode of connexion of the driver and follower. He divides the elementary combinations in mechanism into three classes, of which the characters are as follows :—

Class A: Directional relation constant; velocity ratio constant.

Class B: Directional relation constant; velocity ratio varying.

Class C: Directional relation changing periodically; velocity ratio constant or varying.

Each of those classes is subdivided by Willis into five divisions, of which the characters are as follows:— [table]

In the present article the principle of Willis’s classification is followed; but the arrangement is modified by taking the mode of connexion as the basis of the primary classification, and by removing the subject of connexion by reduplication to the section of aggregate combinations. This modified arrangement is adopted as being better suited than the original arrangement to the limits of an article in an ençyclopaedia; but it is not disputed that the original arrangement may be the best for a separate treatise.

49. Rolling Contact—Smooth Wheels and Racks. —In order that two pieces may move in rolling contact, it is necessary that each pair of points in the two pieces which touch each other should at the instant of contact, be moving in the same direction with the same velocity. In the case of two shifting pieces this would involve equal and parallel velocities for all the points of each piece, so that there could be no rolling, and, in fact, the two pieces would move like one; hence, in the case of rolling contact, either one or both of the pieces must rotate.

The direction of motion of a point in a turning piece being perpendicular to a plane passing through its axis, the condition that each pair of points in contact with each other must move in the same direction leads to the following consequences :—

I. That, when both pieces rotate, their axes, and all their points of contact, lie in the same plane.

II. That, when one piece rotates and the other shifts, the axis of the rotating piece, and all the points of contact, lie in a plane perpendicular to the direction of motion of the shifting piece.

The condition that the velocity of each pair of points of contact must be equal leads to the following consequences :—

III. That the angular velocities of a pair of turning pieces hi rolling contact must be inversely as the perpendicular distances of any pair of points of contact from the respective axes.

IV. That the linear velocity of a shifting piece in rolling contact with a turning piece is equal to the product of the angular velocity of the turning piece by the perpendicular distance from its axis to a pair of points of contact.

The line of contact is that line in which the points of contact are all situated. Respecting this line, the above principles III. and IV. lead to the following conclusions :—

V. That for a pair of turning pieces with parallel axes, and for a turning piece and a shifting piece, the line of contact is straight, and parallel to the axes or axis; and hence that the rolling surfaces are either plane or cylindrical (the term “cylindrical” including all surfaces generated by the motion of a straight line parallel to itself).

VI. That for a pair of turning pieces with intersecting axes the line of contact is also straight, and traverses the point of intersection of the axes; and hence that the rolling surfaces are conical, with a common apex (the term “conical” including all surfaces generated by the motion of a straight line which traverses a fixed point).

Turning pieces in rolling contact are called smooth or toothless wheels. Shifting pieces in rolling contact with turning pieces may be called smooth or toothless racks.

VII. In a pair of pieces in rolling contact every straight line traversing the line of contact is a line of connexion.

50. Cylindrical Wheels and Smooth Racks.— In designing cylindrical wheels and smooth racks, and determining their comparative motion, it is sufficient to consider a section of the pair of pieces made by a plane perpendicular to the axis or axes.

The points where axes intersect the plane of section are called centres ; the point where the line of contact intersects it, the point of contact, or pitch-point ; and the wheels are described as circular, elliptical, &c., according to the forms of their sections made by that plane.

When the point of contact of two wheels lies between their centres, they are said to be in outside gearing ; when beyond their centres, in inside gearing, because the rolling surface of the larger wheel must in this case be turned inward or towards its centre.

From Principle III. of sect. 49 it appears that the angular velocity-ratio of a pair of wheels is the inverse ratio of the distances of the point of contact from the centres respectively.

For outside gearing that ratio is negative, because the wheels turn contrary ways; for inside gearing it is positive, because they turn the same way.

If the velocity ratio is to be constant, as in Willis’s Class A, the wheels must be circular; and this is the most common form for wheels.

If the velocity ratio is to be variable, as in Willis’s Class B, the figures of the wheels are a pair of rolling curves, subject to the condition that the distance between their poles (which are the centres of rotation) shall be constant.

The following is the geometrical relation which must exist between such a pair of curves. See fig. 10.

Let C 1 , C 2 be the poles of a pair of rolling curves; T∣, T 2 any pair of points of contact; U 3 , I. τ 2 any other pair of points of contact. Then, for every possible pair of points of contact, the two following equations must be simultaneously fulfilled:—

Sum of radii, C 1 U 1 + C 2 U 2 = C 1 T 1 + C 2 T 2 = constant; arc, T 2 U 2 = T 1 U 1 (17).

A condition equivalent to the above, and necessarily connected with it, is, that at each pair of points of contact the inclinations of the curves to their radii-vectores shall be equal and contrary; or, denoting by r 1 , r 2 the radii-vectores at any given pair of points of contact, and s the length of the equal arcs measured from a certain fixed pair of points of contact—

⅛3=-⅛ (18); ds ds which is the differential equation of a pair of rolling curves whose poles are at a constant distance apart.

For full details as to rolling curves, see Willis’s work, already mentioned, and Clerk Maxwell’s paper on Rolling Curves in the Transactions of the Royal Society of Edinburgh, 1849.

A rack, to work with a circular wheel, must be straight. To work with a wheel of any other figure, its section must be a rolling curve, subject to the condition that the perpendicular distance from the pole or centre of the wheel to a straight line parallel to the direction of the motion of the rack shall be constant, Let r 1 be the radiusvector of a point of contact on the wheel, ¾ the ordinate from the straight line before mentioned to the corresponding point of contact on the rack. Then 'ds ds μyj is the differential equation of the pair of rolling curves.

To illustrate this subject, it may be mentioned that an ellipse rotating about one focus rolls completely round in outside gearing with an equal and similar ellipse also rotating about one focus, the distance between the axes of rotation being equal to the major axis 14~ eccentricity of the ellipses, and the velocity ratio varying from 1 _ eccen t r i c ∩y to * ~ cc -- e - 1 foi≤ity . an hyperbola rotating about its further focus rolls in inside gearing, through a limited arc, with an equal and similar hyperbola rotating about its nearer focus, the distance between the axes of rotation being equal to the axis of the hyperbolas, and the velocity ratio varying between eccen ^ c ^ t y + ^ an q eccentricity—1 unity; and a parabola rotating about its focus rolls with an equal and similar parabola, shifting parallel to its directrix.

51. Conical or Bevel and Disk Wheels.— From Principles III. and VI. of sect. 49 it appears that the angular velocities of a pair of wheels whose axes meet in a point are to each other inversely as the sines of the angles which the axes of the wheels make with the line of contact. Hence follows the following construction (figs. 11 and 12).—Let 0 be the apex or point of meeting of the two axes OC n OC 2 . The angular velocity ratio being given, it is required to find the line of contact. On OC 1 , OC 2 take lengths OA∙ , OA 2 , respectively proportional to the angular velocities of the pieces on whose axes they are taken. Complete the parallelogram 0A 1 EA 2 j the diagonal OET will be the line of contact required.

When the velocity ratio is variable, the line of contact will shift its position in the plane C 1 OC 2 , and the wheels will be cones, with eccentric or irregular bases. In every case which occurs in practice, however, the velocity ratio is constant; the line of contact is constant in position, and the rolling surfaces of the wheels are regular circular cones (when they are called bevel wheels); or one of a pair of wheels may have a flat disk for its rolling surface, as W 2 in fig. 12, in which case it is a disk wheel. The rolling surfaces of actual wheels consist of frusta or zones of the complete cones or disks, as shown by W 1 , W 2 in figs. 11 and 12.

52. Sliding Contact {lateral): Skew- BevelWheels. —An hyperboloid of revolution is a surface resembling a sheaf or a dice box, generated by the rotation of a straight line round an axis from which it is at a constant distance, and to which it is inclined at a constant angle. If two such hyperboloids, equal or unequal, be placed in the closest possible contact, as in fig. 13, they will touch each other along one of the generating straight lines of each, which will form their line of contact, and will be inclined to the axes AG, BH in opposite directions. The axes will not be parallel, nor will they intersect each other.

The motion of two such hyperboloids, turning in contact with each other, has hitherto been classed - ,lg t ⅛∙ JE /7 _ 1 amongst cases of rolling contact; e ∙⅛G

but that classification is not strictly ( correct, for, although the component ι velocities of a pair of points of con- "J t tact in a direction at right angles to the line of contact are equal, still, y∣ ∣β

as the axes are neither parallel to each other nor to the line of contact, the velocities of a pair of points of contact have components along the line of contact which are unequal, and their difference constitutes a lateral sliding.

The directions and positions of the axes being given, and the required angular velocity ratio, the following construction serves to determine the line of contact, by whose rotation round the two axes respectively the hyperboloids are generated :—

In fig. 14, let B 1 C 1 , B 2 C 2 be the two axes; B 1 B 2 their common perpendicular. Through any point 0 in this common perpendicular draw OA] parallel to B 1 C 1 and OA 2 parallel to B 2 C 2 ; make those lines proportional to the angular velocities about the axes to which they are respectively parallel; complete the parallelogram 0A 1 EA 2 , and draw the diagonal OE; divide B 1 B 2 in D into two parts, inversely proportional to the angular velocities about the axes which they respectively adjoin; through D parallel to OE draw DT. This will be the line of contact.

A pair of thin frusta of a pair of hyperboloids are used in practice to communicate motion between a pair of axes neither parallel not intersecting, and are called skew-bevel wheels.

In skew-bevel wheels the properties of a line of connexion are not possessed by every line traversing the line of contact, but only by every line traversing the line of contact at right angles.

If the velocity ratio to be communicated were variable, the point D would alter its position, and the line DT its direction, at different periods of the motion, and the wheels would be hyperboloids of an eccentric or irregular cross-section; but forms of this kind are not used in practice.

53. Sliding Contact {circular): Grooved Wheels.—As the adhesion or friction between a pair of smooth wheels is seldom sufficient to prevent their slipping on each other, contrivances are used to increase their mutual hold. One of those consists in forming the rim of each wheel into a series of alternate ridges and grooves parallel to the plane of rotation; it is applicable to cylindrical and bevel wheels, but not to skew-bevel wheels. The comparative motion of a pair of wheels so ridged and grooved is the same with that of a pair of smooth wheels in rolling contact, whose cylindrical or conical surfaces lie midway between the tops of the ridges and •bottoms of the grooves, and those ideal smooth surfaces are called the pitch surfaces of the wheels.

The relative motion of the faces of contact of the ridges and grooves is a rotatory sliding or grinding motion, about the line of contact of the pitch-surfaces as an instantaneous axis.

Grooved wheels have hitherto been but little used.

54. Sliding Contact (direct): Teeth of Wheels, their Number and Pitch.— The ordinary method of connecting a pair of wheels, or a wheel and a rack, and the only method which insures the exact maintenance of a given numerical velocity ratio, is by means of a series of alternate ridges and hollows parallel or nearly parallel to the successive lines of contact of the ideal smooth wheels whose velocity ratio would be the same with that of the toothed wheels. The ridges are called teeth ; the hollows, spaces. The teeth of the driver push those of the follower before them, and in so doing sliding takes place between them in a direction across their lines of contact.

The pitch-surfaces of a pair of toothed wheels are the ideal smooth surfaces which would have the same comparative motion by rolling contact that the actual wheels have by the sliding contact of their teeth. The pitch-circles of a pair of circular toothed wheels are sections of their pitch-surfaces, made for spur-wheels (that is, for wheels whose axes are parallel) by a plane at right angles to the axes, and for bevel wheels by a sphere described about the common apex. For a pair of skew-bevel wheels the pitch-circles are a pair of contiguous rectangular sections of the pitch-surfaces. The pitch-point is the point of contact of the pitch-circles.

The pitch-surface of a wheel lies intermediate between the points of the teeth and the bottoms of the hollows between them. That part of the acting surface of a tooth which projects beyond the pitchsurface is called the face ; that part which lies within the pitchsurface, the√tonfc.

Teeth, when not otherwise specified, are understood to be made in’one piece with the wheel,—the material being generally cast-iron, brass, or bronze. Separate teeth, fixed into mortises in the rim of the wheel, are called cogs. A pinion is a small toothed wheel; a trundle is a pinion with cylindrical stoves for teeth.

The radius of the pitch-circle of a wheel is called the geometrical radius ; a circle touching the ends of the teeth is called the addendum circle, and its radius the real radius ; the difference between these radii, being the projection of the teeth beyond the pitch-surface, is called the addendum.

The distance, measured along the pitch-circle, from the face of one tooth to the face of the next, is called the pitch. The pitch and the number of teeth in wheels are regulated by the following principles :—

I. In wheels which rotate continuously for one revolution or more, it is obviously necessary that the pitch should be an aliquot part of the circumference.

In wheels which reciprocate without performing a complete revolution this condition is not necessary. Such wheels are called sectors.

II. In order that a pair of wheels, or a wheel and a rack, may work correctly together, it is in all cases essential that the pitch should be the same in each.

III. Hence, in any pair of circular wheels which work together, the numbers of teeth in a complete circumference are directly as the radii and inversely as the angular velocities.

IV. Hence also, in any pair of circular wheels which rotate continuously for one revolution or more, the ratio of the numbers of teeth and its reciprocal the angular velocity ratio must be expressible in whole numbers.

From this principle arise problems of a kind which will be referred to in treating of Trains of Mechanism.

V. Let n, N be the respective numbers of teeth in a pair of wheels, N being the greater. Let t, T be a pair of teeth in the smaller and larger wheel respectively, which at a particular instant work together. It is required to find, first, how many pairs of teeth must pass the line of contact of the pitch-surfaces before Zand T work together again (let this number be called a) ; and, secondly, with how many different teeth of the larger wheel the tooth t will work at different times (let this number be called f>); thirdly, with how many different teeth of the smaller wheel the tooth T will work at different times (let this be called c).

Case 1. If n is a divisor of N, a = N; 5 = —; c = l (20).

n Case 2. If the greatest common divisor of N and n be d, a number less than n, so that n = md, N = Mrf; then

α=mN=Mn=JW; 5=M; c=m . . . (21).

Case 3 If X and n be prime to each other,

α = wN; δ = N; c=n (22).

It is considered desirable by millwrights, with a view to the preservation of the uniformity of shape of the teeth of a pair of wheels, that each given tooth in one wheel should work with as many different teeth in the other wheel as possible. They therefore study that the numbers of teeth in each pair of wheels which work together shall either be prime to each other, or shall have their greatest common divisor as small as is consistent with a velocity ratio suited for the purposes of the machine.

55. Sliding Contact—Forms of the Teeth of Spur-wheels and Racks.— A line of connexion of two pieces in sliding contact is a line perpendicular to their surfaces at a point where they touch. Bearing this in mind, the principle of the comparative motion of a pair of teeth belonging to a pair of spur-wheels, or to a spur-wheel and a rack, is found by applying the principles stated generally in sects. 46 and 47 to the case of parallel axes for a pair of spur-wheels, and to the case of an axis perpendicular to the direction of shifting for a wheel and a rack.

In fig. 15, let C 1 , C 2 be the centres of a pair of spur-wheels; B 1 IB' 1 , B 2 lB' 2 portions of their pitch-circles, touching at I, the pitch-point. Let the wheel 1 be the driver, and the wheel 2 the follower.

Let D 1 TB 1 Aj, D 2 TB 2 A 2 be the positions, at a given instant, of the acting surfaces of a pair of teeth in the driver and follower respectively, touching each other at T; the line of connexion of those teeth is PjP 2 , perpendicular to their surfaces at T. Let C 1 P 1 , C 2 P 2 be perpendiculars let fall from the centres of the wheels on the line of contact.. Then, by sect. 46, the angular velocityratio is itg =⅛⅞ (23).

«1 2

The following principles regulate the forms of the teeth and their relative motions :—

I. The angular velocity ratio due to the sliding contact of the teeth will be the same with that due to the rolling contact of the pitch-circles, if the line of connexion of the teeth cuts the line of centres at the pitch-point.

For, let P 1 P 2 cut the line of centres at I; then, by similar triangles,

α 1 : a 2 :: C 2 P 2 : C 1 P 1 :: IC 2 :: IC 1 . . . (24); which is also the angular velocity ratio due to the rolling contact of the circles B 1 IB' 1 , B 2 1B' 2 ∙

This principle determines the forms of all teeth of spur-wheels. It also determines the forms of the teeth of straight racks, if one of the centres be removed, and a straight line EIE', parallel to the direction of motion of the rack, and perpendicular to C 1 IC 2 , be substituted for a pitch-circle.

II. The component of the velocity of the point of contact of the teeth T along the line of connexion is

α 1 . C 1 P 1 = α 2 . C 2 P 2 (25).

III. The relative velocity perpendicular to P 1 P 2 of the teeth at their point of contact,—that is, their velocity of sliding on each other,—is found by supposing one of the wheels, such as 1, to be fixed, the line of centres CιC 2 to rotate backwards round C 1 with the angular velocity a 1 , and the wheel 2 to rotate round C 2 as before, with the angular velocity α 2 relatively to the line of centres C 1 C 2 , so as to have the same motion as if its pitch-circle rolled on the pitch-circle of the first wheel. Thus the relative motion of the wheels is unchanged; but 1 is considered as fixed, and 2 has the total motion given by the principles of sects. 37 and 38,—that is, a rotation about the instantaneous axis I, with the angular velocity a 1 + a. 2 . Hence the velocity of sliding is that due to this rotation about I, with the radius 1’1’; that is to say, its value is

(<z 1 + <⅛). IT (26); so that it is greater the farther the point of contact is from the line of centres; and at the instant when that point passes the line of centres, and coincides with the pitch-point, the velocity of sliding is null, and the action of the teeth is, for the instant, that of rolling contact. ...

IV. The path of contact is the line traversing the various positions of the point T. If the line of connexion preserves always the same position, the path of contact coincides with it, and is straight; in other cases the path of contact is curved.

11 is divided by the pitch-point I into two parts, —the arc or line of approach described by T in approaching the line of centres, and the arc or line of recess described by T after having passed the line of centres.

During the approach, ↑λ ∖ c flank D 1 B 1 of the driving tooth drives the face D 2 B 2 of the following tooth, and the teeth are sliding towards each other. During the recess (in which the position of the teeth is exemplified in the figure by curves marked with accented letters), the face B ’ 1 A' 1 of the driving tooth drives the flank B' 2 A'. 2 of the following tooth, and the teeth are sliding from each other.

The path of contact is bounded where the approach commences by the addendum-circle of the follower, and where the recess terminates by the addendum-circle of the driver. The length of the path of contact should be such that there shall always be at least one pair of teeth in contact; and it is better still to make it so long that there shall always be at least two pairs of teeth in contact.

V. The obliquity of the action of the teeth is the angle EIT = IC 1 P 1 = IC 2 P 2 .

In practice it is found desirable that the mean value of the obliquity of action during the contact of teeth should not exceed 15* nor the maximum value 30°.

It is unnecessary to give separate figures and demonstrations for inside gearing. The only modification required in the formula is, that in equation 26 the difference of the angular velocities should be substituted for their sum.

56. Involute Teeth. — The simplest form of tooth which fulfils the conditions of sect. 55 is obtained in the following manner (see fig. 16). Let C 1 , C 2 be the centres of two wheels, B 1 IB' 1 , B. 2 1B' 2 their pitch-circles, I the pitch-point; let the obliquity of action of the teeth be constant, so that the same straight line P 1 IP 2 shall represent at once the constant line of connexion of teeth and the path of contact. Draw C 1 P 1 , C 2 P 2 perpendicular to P 1 IP 2 , and with those lines as radii describe about the centres of the wheels the circles D 1 D' 1 , D 2 D' 2 , called base-circles. It is evident that the radii of the base-circles bear to each other the same proportions as the radii of the pitch-circles, and also that

C 1 P 1 = IC 1 . cos obliquity ) , 97 ∙,

C 2 P 2 = IC 2 . cos obliquity ( ' ' ’

(The obliquity which is found to answer best in practice is about 14⅛ 0 ; its cosine is about ⅜⅜, and its sine about ∣. These values, though not absolutely exact, are near enough to the truth for practical purposes. )

Suppose the base-circles to be a pair of circular pulleys connected by means of a cord whose course from pulley to pulley is P 1 IP 2 . As the line of connexion of those pulleys is the same with that of the proposed teeth, they will rotate with the required velocity ratio. Now, suppose a tracing point T to be fixed to the cord, so as to be carried along the path of contact P 1 IP 2 , that point will trace on a plane rotating along with the wheel 1 part of the involute of the base-circle D 2 D' 1 , and on a plane rotating along with the wheel 2 part of the involute of the base-circle D. 2 D' 2 ; and the two curves so traced will always touch each other in the required point of contact T, and will therefore fulfil the condition required by Principle I. of sect. 55.

Consequently, one of the forms suitable for the teeth of wheels is the involute of a circle; and the obliquity of the action of such teeth is the angle whose cosine is the ratio of the radius of their base-circle to that of the pitch-circle of the wheel.

All involute teeth of the same pitch work smoothly together.

To find the length of the path of contact on either side of the pitch-point I, it is to be observed that the distance between the fronts of two successive teeth, as measured along P 1 IP 2 , is less than the pitch in the ratio of cos obliquity: 1; and consequently that, if distances equal to the pitch be marked off either way from I towards P 1 and P 2 respectively, as the extremities of the path of contact, and if, according to Principle I V. of sect. 55, the addendum-circles be described through the points so found, there will always be at least two pairs of teeth in action at once. In practice it is usual to make the path of contact somewhat longer, viz., about 2 i ⅛th times the pitch; and with this length of path, and the obliquity already mentioned of 14.⅛ o , the addendum is about ⅛ths of the pitch,

The teeth of a rack, to work correctly with wheels having involute teeth, should have plane surfaces perpendicular to the line of connexion, and consequently making with the direction of motion of the rack angles equal to the complement of the obliquity of action.

57. Teeth for a given Path of Contact—Mr Sang , s Method.— In the preceding section the form of the teeth is found by assuming a figure for the path of contact, viz., the straight line. Any other convenient figure may be assumed for the path of contact, and the corresponding forms of the teeth found by determining what curves a point T, moving along the assumed path of contact, will trace on two disks rotating round the centres of the wheels with angular velocities bearing that relation to the component velocity of T along TI, which is given by Principle II. of sect. 55, and by equation 25. This method of finding the forms of the teeth of wheels forms the subject of an elaborate and most interesting treatise by Mr Edward Sang.

All wheels having teeth of the same pitch, traced from the same path of contact, work correctly together, and are said to belong to the same set.

58. Teeth traced by Rolling Curves.— If any curve Pt (fig. 17) be rolled on the inside of the pitch-circle BB of a wheel, it appears, from sect. 37, that the instantaneous axis of the rolling curve at any instant will be at the point I, where it touches the pitch-circle for the moment, and that consequently the line AT, traced by a tracingpoint T, fixed to the rolling curve upon the plane of the wheel, will be everywhere perpendicular to the straight line TI; so that the traced curve AT will be suitable for the flank of a tooth, in which T is the point of contact corresponding to the position I of the pitch-point. If the same rolling curve R, with the same tracing-point T, be rolled on the outside of any other pitch-circle, it will have the face of a tooth suitable to work with the flank AT.

In like manner, if either the same or any other rolling curve R' be rolled the opposite way, on the outside of the pitch-circle BB, so that the tracing point T' shall start from A, it will trace the face AT' of a tooth suitable to work with a flank traced by rolling the same curve R' with the same tracing-point T' inside any other pitchcircle.

The figure of the path of contact is that traced on a fixed plane by the tracing-point, when the rolling curve is rotated in such a manner as always to touch a fixed straight line EIE (or E'ΓE', as the case may be) at a fixed point I (or Γ).

If the same rolling curve and tracing point be used to trace both the faces and the flanks of the teeth of a number of wheels of different sizes but of the same pitch, all those wheels will work correctly together, and will form a set. The teeth of a rack, of the same set, are traced by rolling the rolling curve on both sides of a straight line.

The teeth of wheels of any figure, as well as of circular wheels, may be traced by rolling curves on their pitch-surfaces; and all teeth of the same pitch, traced by the same rolling curve with the same tracing-point, will work together correctly if their pitch-surfaces are in rolling contact.

59. Epicycloidal Teeth.— The most convenient rolling curve is the circle. The path of contact which it traces is identical with itself; and the flanks of the teeth are internal and their faces external epicycloids for wheels, and both flanks and faces are cycloids for a rack.

For a pitch-circle of twice the radius of the rolling or describing circle (as it is called) the internal epicycloid is a straight line, being, in fact, a diameter of the pitch circle, so that the flanks of the teeth for such a pitch-circle are planes radiating from the axis. For a smaller pitch-circle the flanks would be convex and incurved or under-cut, which would be inconvenient; therefore the smallest wheel of a set should have its pitch-circle of twice the radius of the describing circle, so that the flanks may be either straight or concave.

In fig. 18, let BB' be part of the pitch-circle of a wheel with epi-eycloidal teeth; CIC' the line of centres’; I the pitch-point; EIE λ a straight tangent to the pitch-circle at that point; R the internal and R' the equal external describing circles, so placed as to touch the pitch-circle and each other at I. Let DID' be the path of contact, consisting of the arc of approach DI and the arc of recess ID'. In order that there may always be at least two pairs of teeth in action, each of those arcs should be equal to the pitch.

The obliquity of the action in passing the line of centres is nothing; the maximum obliquity is the angle EID = E'IDj and the mean obliquity is one-half of that angle.

It appears from experience that the mean obliquity should not exceed 15°; therefore the maximum obliquity should be about 30°; therefore the equal arcs DI and ID' should each be one-sixth of a circumference; therefore the circumference of the describing circle should be six'times the pitch.

It follows that the smallest pinion of a set in which pinion the flanks are straight should have twelve teeth.

60. Nearly Epicycloidal Teeth—Willis’s Method.— To facilitate the drawing of epicycloidal teeth in practice, Willis has shown how to approximate to their figure by means of two circular arcs,—one concave, for the flank, and the other convex, for the face,—and each having for its radius the mean radius of curvature of the epicycloidal arc. Willis’s formulae are founded on the following properties of epicycloids :—

Let R be the radius of the pitch-circle; r that of the describing circle; θ the angle made by the normal TI to the epicycloid at a given point T, with a tangent to the circle at I; that is, the obliquity of the action at T.

Then the radius of curvature of the epicycloid at T is—

• · · .R∣ — τ*∖

For an internal epicycloid, ρ = 4rsin0 —— I · · · · < 28 )∙

For an external epicycloid, p'=4rsin0 ι5 — s - ∖ r j ’ r R + 2r J

Also, to find the position of the centres of curvature relatively to the pitch-circle, we have, denoting the chord of the describing circle TI by c, c = 2rsin0; and therefore

For the flank, p-c = 2r sin0 τ , ^ ’ ∙∙ )

,i K 2, ' · . (29).

For the face, p -c=2rsin0 τς —— ∖ r R + 2r )

For the proportions approved of by Willis, sin0 = ∣ nearly; r≈p (the pitch) nearly; c=⅜p nearly; and, if N be the number of τ 6

teeth in the wheel, p = nearly; therefore, approximately,

N ∖

p ^ c 2'N-12[ , _p N i (30) ·

p C 2 ' N + 12 )

Hence the following construction (fig. 19). Let BB be part of the pitch-circle, and a the point where a tooth is to cross it. Set off ab = ac = ⅛p. Draw radii bd, ee; draw∕δ, eg, making angles of 75⅜ o with those radii. Make ⅞∕= p' - c, cg=p-c. From f, with the radius fa, draw the circular arc ah; from g, with the radius ga, draw the circular arc ak.

Then ah is the face and ak the flank of the tooth required.

To facilitate the application of this rule, Willis published tables of p-c and ρ'-c, and invented an instrument called the ‘ ’ odontograph.”

61. Trundles and Pin-Wheels. —If a wheel or trundle have cylindrical pins or staves for teeth, the faces of the teeth of a wheel suitable for driving it are described by first tracing external epicycloids, by rolling the pitch-circle of the pin-wheel or trundle on the pitch-circle of the driving-wheel, with the centre of a stave for a tracing-point, and then drawing curves parallel to, and within the epicycloids, at a distance from them equal to the radius of a stave. Trundles having only six staves will work with large wheels.

62. Backs of Teeth and Spaces.— Toothed wheels being in general intended to rotate either way, the backs of the teeth are made similar to the fronts. The space between two teeth, measured on the pitch-circle, is made about ⅜th part wider than the thickness of the tooth on the pitch-circle; that is to say,

Thickness of tooth = 1 s τ pitch;

Width of space =⅛ pitch.

The difference of 1 ⅛ of the pitch is called the back-lash. The clearance allowed between the points of teeth and the bottoms of the spaces between the teeth of the other wheel is about ⅛th of the pitch.

63. Stepped and Helical Teeth. — Hooke invented the making of the fronts of teeth in a series of steps with a view to increase the smoothness of action. A wheel thus formed resembles in shape a series of equal and similar toothed disks placed side by side, with the teeth of each a little behind those of the preceding disk. He also invented, with the same object, teeth whose fronts, instead of being parallel to the line of contact of the pitch-circles, cross it obliquely, so as to be of a screw-like or helical form. In wheelwork of this kind the contact of each pair of teeth commences at the foremost end of the helical front, and terminates at the aftermost end; and the helix is of such a pitch that the contact of one pair of teeth shall not terminate until that of the next pair has commenced.

Stepped and helical teeth have the desired effect of increasing the smoothness of motion, but they require more difficult and expensive workmanship than common teeth; and helical teeth are, besides, open to the objection that they exert a laterally oblique pressure, which tends to increase resistance, and unduly strain the machinery.

64. Teeth of BevelWheels. — The acting surfaces of the teeth of bevel-wheels are of the conical kind, generated by the motion of a line passing through the common apex of the pitch-cones, while its extremity is carried round the outlines of the cross section of the teeth made by a sphere described about that apex.

The operations of describing the exact figures of the teeth of bevel-wheels, whether by involutes or by rolling curves, are in every respect analogous to those for describing the figures of the teeth of spur-wheels, except that in the case of btvel-wheels all those operations are to be performed on the surface of a sphere described about the apex instead of on a plane, substituting poles for centres, and great circles fρr straight lines.

In consideration of the practical difficulty, especially in the case of large wheels, of obtaining an accurate spherical surface, and of drawing upon it when obtained, the following approximate method, proposed originally by Tredgold, is generally used :—

Let 0 (fig. 20) be the common apex of a pair of bevel-wheels; OB 1 I, OB 2 I their pitch cones; OC 1 , OC 2 their axes; 01 their line of contact. Perpendicular to 01 draw A 1 IA 2 , cutting the axes in A n A 2 ; make the outer rims of the patterns and of the wheels portions of the cones A 1 B 1 I, A 5 B 2 1, of which the narrow zones occupied by the teeth will be sufficiently near to a spherical surface described about 0 for practical purposes. To find the figures of the teeth, draw on a flat surface circular arcs ID 1 , 1D 2 , with the radii A 1 I, A 2 I; those arcs will be the developments of arcs of the pitch-circles B 1 I, B 2 I, when the conical surfaces A 1 B 1 I, A 2 B 2 I are spread out flat. Describe the figures of teeth for the developed arcs as for a pair of spur-wheels; then wrap the developed arcs on the cones, so as to make them coincide with the pitch-circles, and trace the teeth on the conical surfaces.

65. Teeth of Skew-Bevel Wheels.— The crests of the teeth of a skew-bevel wheel are parallel to the generating straight line of the hyperboloidal pitch-surface; and the transverse sections of the teeth at a given pitch-circle are similar to those of the teeth of a bevel-wheel whose pitch-surface is a cone touching the hyperboloidal surface at the given circle.

66. Cams.— A. cam is a single tooth, either rotating continuously or oscillating, and driving a sliding or turning piece either constantly or at intervals. All the principles which have been stated in sect. 55 as being applicable to teeth are applicable to cams; but in designing cams it is not usual to determine or take into consideration the form of the ideal pitch-surface, which would give the same comparative motion by rolling contact that the cam gives by sliding contact.

67. Screws.— The figure of a screw is that of a convex or concave cylinder, with one or more helical projections, called threads, winding round it. Convex and concave screws are distinguished technically by the respective names of male and female ; a short concave screw is called a mit ; and when a screw is spoken oi wι ι-out qualification a convex screw is usually understood.

The relation between the advance and the rotation, winch compose the motion of a screw working in contact with a fixed screw or helical guide, has already been demonstrated in sect. 41; and the same relation exists between the magnitudes of the rotation ot a screw about a fixed axis and the advance of a shifting nut in winch it rotates. The advance of the nut takes place in the opposite direction to that of the advance of the screw in the case in which the nut is fixed. The pitch or axial pitch of a screw has the meaning assigned to it in that section, viz.· the distance, measured parallel to the axis, between the corresponding points in two successive turns of the same thread. If, therefore, the screw has several equidistant threads, the true pitch is equal to the divided axial pitch, as measured between two adjacent threads, multiplied by the number of threads.

If a helix be described round the screw, crossing each turn of the thread at right angles, the distance between two corresponding points on two successive turns of the same thread, measured along this normal helix, may be called the normal pitch ; and when the screw has more than one thread the normal pitch from thread to thread may be called the normal divided pitch.

The distance from thread to thread, measured on a circle described about the axis of the screw, called the pitch-circle, may be called the circumferential pitch ; for a'screw of one thread it is one circum- . . c ., 1 one circumference ference; for a screw of n threads, — .

n

Let r denote the radius of the pitch circle; n the number of threads; θ the obliquity of the threads to the pitch circle, and of the normal helix to the axis;

P α ) ( pitch,

Pa _ > the axial < n ~^ a } ( divided pitch;

P n 1 ( pitch,

P, 1 > the normal <

^⅛^ “° P” ) ( divided pitch;

P c the circumferential pitch;

., , . 2τrr ∖

then p t =p a cotθ=p,, cos0 = -^- j

. . 2ττrtan0 I

p a =p n sec0=p e tan0 = > . . (31).

• n λ 2πr sin θ I Pn≈Pc Sin0 = n α COS0= I

n ∕

If a screw rotates, the number of threads which pass a fixed point in one revolution is the number of threads in the screw.

A pair of convex screws, each rotating about its axis, are used as an elementary combination to transmit motion by the sliding contact of their threads. Such screws are commonly called endless screws. At the point of contact of the screws their threads must be parallel; and their line of connexion is the common perpendicular to the acting surfaces of the threads at their point of contact. Hence the following principles :—

I. If the screws are both right-handed or both left-handed, the angle between the directions of their axes is the sum of their obliquities; if one is right-handed and the other left-handed, that angle is the difference of their obliquities.

II. The normal pitch for a screw of one thread, and the normal divided pitch for a screw of more than one thread, must be the same in each screw.

III. The angular velocities of the screws are inversely as their numbers of threads.

Hooke’s wheels with oblique or helical teeth are in fact screws of many threads, and of large diameters as compared with their lengths.

The ordinary position of a pair of endless screws is with their axes at right angles to each other. When one is of considerably greater diameter than the other, the larger is commonly called in practice a wheel, the name screw being applied to the smaller only; but they are nevertheless both screws in fact.

To make the teeth of a pair of endless screws fit correctly and work smoothly, a hardened steel screw is made of the figure of the smaller screw, with its thread or threads notched so as to.foπn a cutting tool; the larger screw, or “wheel,” is cast approximately of the required figure; the larger screw and the steel screw are fitted up in their proper relative position, and made to rotate in contact with each other by turning the steel screw, which cuts the threads of the larger screw to their true figure.

68. Coupling of Parallel Axes—Oldham's Coupling.—A coupling is a mode of connecting a pair of shafts so that they shall rotate in the same direction with the same mean angular velocity. If the axes of the shafts are in the same straight line, the coupling consists in so connecting their contiguous ends that they shall rotate as one piece; but if the axes are not in the same straight line combinations of mechanism are required. A coupling for parallel shafts which acts by sliding contact was invented by Oldham, and is represented in fig.

21. Cj, C 2 are the axes of the two parallel shafts; D 1 , D 2 two disks facing each other, fixed on the ends of the two shafts respectively; E 1 E 1 a bar sliding in a diametral groove in the face of D 1 j E 2 E 2 a bar sliding in a diametral groove in the face of D 2 : those bars are fixed together at A, so as to form a rigid cross. The angular velocities of the two disks and of the cross are all equal at every instant; the middle point of the cross, at A, revolves in the dotted circle described upon the line of centres C 1 C 2 as a diameter twice for each turn of the disks and cross; the instantaneous axis of rotation of the cross at any instant is at I, the point in the circle C 1 C 2 diametrically opposite to A.

Oldham’s coupling may be used with advantage where the axes of the shafts are intended to be as nearly in the same straight line as is possible, but where there is some doubt as to the practicability or permanency of their exact continuity.

69. Wrapping Connectors—Belts, Cords, and Chains.— Flat belts of leather or of gutta percha, round cords of catgut, hemp, or other material, and metal chains are used as wrapping connectors to transmit rotatory motion between pairs of pulleys and drums.

Belts (the most frequently used of all wrapping connectors) require nearly cylindrical pulleys. A belt tends to move towards that part of a pulley whose radius is greatest; pulleys for belts, therefore, are slightly swelled in the middle, in order that the belt may remain on a pulley, unless forcibly shifted. A belt when in motion is shifted off a pulley, or from one pulley on to another of equal size alongside of it, by pressing against that part of the belt which is moving towards the pulley.

Cords require either cylindrical drums with ledges or grooved pulleys.

Chains require pulleys or drums, grooved, notched, and toothed, so as to fit the links of the chain.

Wrapping connectors for communicating continuous motion are endless.

Wrapping connectors for communicating reciprocating motion have usually their ends made fast to the pulleys or drums which they connect, and which in this case may be sectors.

The line of connexion of two pieces connected by a wrapping connector is the centre line of the belt, cord, or chain; and the comparative motions of the pieces are determined by the principles of sect. 46 if both pieces turn, and of sect. 47 if one turns and the other shifts, in which latter case the motion must be reciprocating.

The pitch-line of a pulley or drum is a curve to which the line of connexion is always a tangent; that is to say, it is a curve parallel to the acting surface of the pulley or drum, and distant from it by half the thickness of the wrapping connector.

Pulleys and drums for communicating a constant velocity ratio are circular. The effective radius, or radius of the pitch-circle of a circular pulley or drum, is equal to the real radius added to half the thickness of the connector.

The angular velocities of a pair of connected circular pulleys or drums are inversely as the effective radii.

A crossed belt, as in fig. 22, A, reverses the direction of the rotation communicated; an uncrossed belt, as in fig. 22, B, preserves that direction.

The length I. of an endless belt connecting a pair of pulleys whose effective radii are r 1 , r, 2 , with parallel axes whose distance apart is C, is given by the following formulae, in each of which the first term, containing the radical, expresses the length of the straight parts of the belt, and the remainder of the formula the length of the curved parts.

For a crossed belt,—

I. = 2√c 2 - (r 1 + r 2 ) 2 +(r 1 + r 2 )^ π - 2 sin- j ^⅛J (32, A); and for an uncrossed belt,—

I.=2√{c 3 -(r 1 -r 2 ) s } + π(r 1 + r 2 ) ÷ 2(r 1 -r 2 ) sin-*^? (32,B); in which r 1 is the greater radius, and r 2 the less.

When the axes of a pair of pulleys are not parallel, the pulleys should be so placed that the part of the belt which is approaching each pulley shall be in the plane of the pulley.

70. Speed-Cones (see fig. 23).—A pair of speed-cones is a contrivance for varying and adjusting the velocity ratio communicated between a pair of parallel shafts by means of a belt. The speedcones are either continuous cones or conoids, as A, B, whose velocity ratio can be varied gradually while they are in motion by shifting the belt, or sets of pulleys whose radii vary by steps, as C, D, in which case the velocity ratio can be changed by shifting the belt from one pair of pulleys to another.

In order that the belt may fit accurately in every possible position on a pair of speed-cones, the quantity I. must be constant, in equations 32, A, or 32, B, according as the belt is crossed or uncrossed.

For a crossed belt, as in A and C, fig. 23, I. depends solely on c and on r 1 + r 2 . Now c is constant because the axes are parallel; therefore the sum of the radii of the pitch-circles connected in every position of the belt is to be constant. That condition is fulfilled by a pair of continuous cones generated by the revolution of two straight lines inclined opposite ways to their respective axes at equal angles.

For an uncrossed belt, the quantity I. in equation 32, B is to be made constant. The exact fulfilment of this condition requires the solution of a transcendental equation; but it may be fulfilled with accuracy sufficient for practical purposes by using, instead of 32, B, the following approximate equation :—

I. nearly—2c+π(r 1 + r 2 ) + -⅞^^- . . . (33).

The following is the most convenient practical rule for the application of this equation :—

Let the speed-cones be equal and similar conoids, as in B, fig. 23, but with their large and small ends turned opposite ways. Let r 1 be the radius of the large end of each, r 2 that of the small end, r 0 that of the middle; and let v be the sagitta, measured perpendicular to the axes, of the arc by whose revolution each of the conoids is generated, or, in other words, the bulging of the conoids in the middle of their length. Then

'-'∙- n F-⅞⅛ s (3⅛

2π∙ = 6 - 2832; but 6 may be used in most practical cases without sensible error.

The radii at the middle and end being thus determined, make the generating curve an arc either of a circle or of a parabola.

71. Linkwork in General.— The pieces which are connected by linkwork, if they rotate or oscillate, are usually called cranks, beams, and levers. The link by which they are connected is a rigid rod or bar, which may be straight or of any other figure; the straight figure, being the most favourable to strength, is always used when there is no special reason to the contrary. The link is known by various names under various circumstances, such as coupling-rod, connecting-rod, crank-rod, eccentric-rod, &c. It is attached to the pieces which it connects by two pins, about which it is free to turn. The effect of the link is to maintain the distance between the axes of those pins invariable; hence the common perpendicular of the axes of the pins is the line of connexion, and its extremities may be called the connected points. In a turning piece, the perpendicular let fall from its connected point upon its axis of rotation is the arm or crank-arm.

The axes of rotation of a pair of turning pieces connected by a link are almost always parallel, and perpendicular to the line of connexion; in which case the angular velocity ratio at any instant is the reciprocal of the ratio of the common perpendiculars let fall from the line of connexion upon the respective axes of rotation.

If at any instant the direction of one of the crank-arms coincides with the line of connexion, the common perpendicular of the line of connexion and the axis of that crank-arm vanishes, and the directional relation of the motions becomes indeterminate. The position of the connected point of the crank-arm in question at such an instant is called a dead-point. The velocity of the other connected point at such an instant is null, unless it also reaches a dead-point at the same instant, so that the line of connexion is in the plane of the two axes of rotation, in which case the velocity ratio is indeterminate. Examples of dead-points, and of the means of preventing the inconvenience which they tend to occasion, will appear in the sequel.

72. Coupling of Parallel Axes.— Two or more parallel shafts (such as those of a locomotive engine, with two or more pairs of driving wheels) are made to rotate with constantly equal angular velocities by having equal cranks, which are maintained parallel by a coupling-rod of such a length that the line of connexion is equal to the distance between the axes. The cranks pass their dead-points simultaneously. To obviate the unsteadiness of motion which this tends to cause, the shafts are provided with a second set of cranks at right angles to the first, connected by means of a similar coupling-rod, so that one set of cranks pass their dead points at the instant when the other set are farthest from theirs.

73. Comparative Motion of Connected Points.— As the link is a rigid body, it is obvious that its action in communicating motion may be determined by finding the comparative motion of the connected points, according to the principles laid down in §§ 71, 78, pp. 690, 692, and this is often the most convenient method of proceeding.

If a connected point belongs to a turning piece, the direction of its motion at a given instant is perpendicular to the plane containing the axis and crank-arm of the piece. If a connected point belongs to a shifting piece, the direction of its motion at any instant is given, and a plane can be drawn perpendicular to that direction.

The line of intersection of the planes perpendicular to the paths of the two connected points at a given instant is the instantaneous axis of the link at that instant; and the velocities of the connected points are directly as their distances from that axis.

In drawing on a plane surface, the two planes perpendicular to the paths of the connected points are represented by two lines (being their sections by a plane normal to them), and the instantaneous axis by a point (fig. 24); and, should the length of the two lines render it impracticable to produce them until they actually intersect, the velocity ratio of the connected points may be found by the principle that it is equal to the ratio of the segments which a line parallel to the line of connexion cuts off from any two lines drawn from a given point, perpendicular respectively to the paths of the connected points.

To illustrate this by one example. Let C 1 be the axis, and T 1 the connected point of the beam of a steam-engine; T 1 T 2 the connecting or crank-rod; T 2 the other connected point, and the centre of the crank-pin; C 2 the axis of the crank and its shaft. Let denote the velocity of T 1 at any given instant; v 2 that of T 2 . To find the ratio of these velocities, produce C 1 T 1 , C 2 T 2 , till they intersect in K; K is the instantaneous axis of the connecting rod, and the velocity ratio is υ 1 : v 2 :: KT j : KT 2 (35).

Should K be inconveniently far off, draw any triangle with its sides respectively parallel to C 1 T 1 , C 2 T 2 , and T 1 T 2 ; the ratio of the two sides first mentioned will be the velocity ratio required. For example, draw C 2 A parallel to C 1 T 1 , cutting T 1 T 2 in A; then v 1 : v 2 :: C 2 A: C 2 T 2 (36).

74. Eccentric.— An eccentric circular disk fixed on a shaft, and used to give a reciprocating motion to a rod, is in effect a crank-pin of sufficiently large diameter to surround the shaft, and so to avoid the weakening of the shaft which would arise from bending it so as to form an ordinary crank. The centre of the eccentric is its connected point; and its eccentricity, or the distance from that centre to the axis of the shaft, is its crank-arm.

An eccentric may be made capable of having its eccentricity altered by means of an adjusting screw, so as to vary the extent of the reciprocating motion which it communicates.

75. Reciprocating Pieces—Stroke—Dead-Points.— The distance between the extremities of the path of the connected point in a reciprocating piece (such as the piston of a steam-engine) is called the stroke or length of stroke of that piece. When it is connected with a continuously turning piece (such as the crank of a steam-engine) the ends of the stroke of the reciprocating piece correspond to the dead-points of the path of the connected point of the turning piece, where the line of connexion is continuous with or coincides with the crank-arm.

Let S be the length of stroke of the reciprocating piece, I. the length of the line of connexion, and R the crank-arm of the continuously turning piece. Then, if the two ends of the stroke be in one straight line with the axis of the crank,

S≈2R (37); and if these ends be not in one straight line with that axis, then S, I. - R, and I. + R, are the three sides of a triangle, having the angle opposite S at that axis; so that, if 0 be the supplement of the arc between the dead-points,

S 2 = 2(I. 2 + R 2 )-2(I. s -R 2 )cos0 )

λ 2L 2 + 2R 2 -S 2 [ . . . . (38).

cosβ- 2(I. 2 -R 2 ) )

76. Coupling of Intersecting Axes—Hooke's Universal Joint. Intersecting axes are coupled by a contrivance of Hooke s, known as ιe “universal joint,” which belongs to the class of linkwork (see g. 25). Let 0 be the point of intersection of the axes OC 1 , OC 2 , and θ their angle of inclination to each other. The pair of shafts C 1 , C 2 terminate in a pair of forks F 1 , F 2 , in bearings at the extremities of which turn the gudgeons at the ends of the arms of a rectangular cross, having its centre at 0. This cross is the link; the connected points are the centres of the bearings F 1 , F 2 . At each instant each of those points moves at right angles to the central plane of its shaft and fork; therefore the line of intersection of the central planes of the two forks at any instant is the instantaneous axis of the cross, and the velocity ratio of the points F 1 , F 2 (which, as the forks are equal, is also the angular velocity ratio of the shafts) is equal to the ratio of the distances of those points from that instantaneous axis. The mean value of that velocity ratio is that of equality, for each successive quarter-turn is made by both shafts in the same time; but its actual value fluctuates between the limits—

— = —when F 1 is the plane OC 1 C., )

«i cosø i } . . . (39).

and — = cos 0 when F 2 is in that plane ∖ «1 '

Its value at intermediate instants is given by the following equations :—let φ 1 , φ. 2 be the angles respectively made by the central planes of the forks and shafts with the plane OC 1 C 2 at a given instant; then cos0 = tan≠ 1 tan≠ 2 )

ct 2 dφ. 2 tan≠ 1 ÷cot≠ ι ( (40).

α 1 dφ 1 tan φ 2 + cot φ 2 )

77. Intermittent Linkwork —Click and Ratchet.— A click acting upon a ratchet-wheel or rack, which it pushes or pulls through a certain arc at each forward stroke and leaves at rest at each backward stroke, is an example of intermittent linkwork. During the forward stroke the action of the click is governed by the principles of linkwork; during the backward stroke that action ceases. A catch or pall, turning on a fixed axis, prevents the ratchet-wheel or rack from reversing its motion.

Division 5. Trains of Mechanism.

78. General Principles. — A train of mechanism consists of a series of pieces each of which is follower to that which drives it and driver to that which follows it.

The comparative motion of the first driver and last follower is obtained by combining the proportions expressing by their terms the velocity ratios and by their signs the directional relations of the several elementary combinations of which the train consists.

79. Trains of Wheelwork. — Let A 1 , A 2 , A 3 , &c., A m -ι, A, n denote a series of axes, and α 1 , α 2 , α 3 , &c., α m -ι, a m their angular velocities. Let the axis A 1 carry a wheel of N 1 teeth, driving a wheel of n, 2 teeth on the axis A 2 , which carries also a wheel of N 2 teeth, driving a wheel of n 3 teeth on the axis A 3 , and so on; the numbers of teeth in drivers being denoted by N’s, and in followers by n’s, and the axes to which the wheels are fixed being denoted by numbers. Their the resulting velocity ratio is denoted by U,n a 2 , , *1 · " · &C. · · · Nm-1

«1 «1 ct 2 ®>n-l · · · · · ·

that is to say, the velocity ratio of the last and first axes is the ratio of the product of the numbers of teeth in the drivers to the product of the numbers of teeth in the followers.

Supposing all the wheels to be in outside gearing, then, as each elementary combination reverses the direction of rotation, and as the number of elementary combinations m - 1 is one less than the number of axes m, it is evident that if m is odd the direction of rotation is preserved, and if even reversed.

It is often a question of importance to determine the number of teeth in a train of wheels best suited for giving a determinate velocity ratio to two axes. . It was shown by Young that, to do this with the least total number of teeth, the velocity ratio of each elementary combination should approximate as nearly as possible to 3 ’ 59. This would in many cases give too many axes; and, as a useful practical rule, it may be laid down that from 3 to 6 ought to be the limit of the velocity ratio of an elementary combination in wheelwork. The smallest number of teeth in a pinion for epicycloidal teeth ought to be twelve (see sect. 59),—but it is better, for smoothness of møtion, not to go below fifteen·, and for involute teeth the smallest number is about twenty-four.

Let g be the velocity ratio required, reduced to its least terms, and let B be greater than C. If is not greater than 6, and C lies c between the prescribed minimum number of teeth (which may be called t') and its double 2t, then one pair of wheels will answer the purpose, and B and C will themselves be the numbers required. Should B and C be inconveniently large, they are, if possible, to be resolved into factors, and those factors (or if they are too small, multiples of them) used for the number of teeth. Should B or C, or both, be at once inconveniently large and prime, then, instead of the exact ratio , some ratio approximating to that ratio, and

G

capable of resolution into convenient factors, is to be found by the method of continued fractions.

jj

Should θ be greater than 6, the best number of elementary combinations m - 1 will lie between log B - log C , log B - log C ⅝1 aiκ lo g 3

Then, if possible, B and C themselves are to be resolved each into m - 1 factors (counting 1 as a factor), which factors, or multiples of them, shall be not less than t nor greater than 6/; or if B and C contain inconveniently-large prime factors, an approximate velocity ratio, found by the method of continued fractions, is to be substituted for as before.

So far as the resultant velocity ratio is concerned, the order of the drivers N and of the followers n is immaterial; but to secure equable wear of the teeth, as explained in sect. 54, the wheels ought to be so arranged that, for each elementary combination, the greatest common divisor of N and n shall be either 1, or as small as possible.

80. Double Hooke's Coupling.— It has been shown in section 76 that the velocity ratio of a pair of shafts coupled by a universal joint fluctuates between the limits cosø and -—-. Hence one or both j cosø of the shafts must have a vibratory and unsteady motion, injurious to the mechanism and framework. To obviate this evil a short intermediate shaft is introduced, making equal angles with the first and last shaft, coupled with each of them by a Hooke’s joint, and having its own two forks in the same plane. Let α 1 , α 2 , a 3 be the angular velocities of the first, intermediate, and last shaft in this train of two Hooke's couplings. Then, from the principles of sect.

76 it is evident that at each instant -- =— , and consequently that σ 1 α 3 a 3 =ft 1 ; so that the fluctuations of angular velocity ratio caused by the first coupling are exactly neutralized by the second, and the first and last shafts have equal angular velocities at each instant.

81. Converging and Diverging Trains of Mechanism.— Two or more trains of mechanism may converge into one,—as when the two pistons of a pair of steam-engines, each through its own connecting-rod, act upon one crank-shaft. One train of mechanism may diverge into two or more,—as when a single shaft, driven by a prime mover, carries several pulleys, each of which drives a different machine. The principles of comparative motion in such converging and diverging trains are the same as in simple trains.

Division 6. Aggregate Combinations.

82. General Principles.— Willis has designated as “aggregate combinations” those assemblages of pieces of mechanism in which the motion of one follower is the resultant of component motions impressed on it by more than one driver. Two classes of aggregate combinations may be distinguished which, though not different in their actual nature, differ in the data which they present to the designer, and in the method of solution to be followed in questions respecting them.

Class I. comprises those cases in which a piece A is not carried directly by the frame C, but by another piece B, relatively to which the motion of A is given,—the motion of the piece B relatively to the frame C being also given. Then the motion of A relatively to the frame C is the resultant of the motion of A relatively to B and of B relatively to C; and that resultant is to be found by the principles already explained in division 3 of this chapter, sects. 34 to 41.

Class II. comprises those cases in which the motions of three points in one follower are determined by their connexions with two or with three different drivers, so that the motion of the follower, as a whole, is to be determined by the principles of §§ 71, 78, pp. 690, 692.

This classification is founded on the kinds of problems arising from the combinations. Willis adopts another classification, founded on the objects of the combinations, which objects he divides into two classes, viz., (1) to produce aggregate velocity, or a velocity which is the resultant of two or more components in the same path, and (2) to produce an aggregate path, that is, to make a given point in a rigid body move in an assigned path by communicating certain motions to other points in that body.

It is seldom that one of these effects is produced without at the same time producing the other; but the classification of Willis depends upon which of those two effects, even supposing them to occur together, is the practical object of the mechanism.

83. lieduplication of Cords—Differential Windlass— Blocks, Sheaves, and Tackle.— The axis C (fig. 26) carries a larger barrel AE and a smaller barrel DB, rotating as one piece with the angular velocity α 1 in the direction AE. The pulley or sheave FG has a weight W hung to its centre. A cord has one end made fast to and wrapped round the barrel AE; it passes from A under the sheave FG, and has the other end wrapped round and made fast to the barrel BD. Required the relation between the velocity of translation v 2 of W and the angular velocity α 1 of the differential barrel.

In this case v 2 is an aggregate velocity, produced by the joint action of the two drivers AE and BD, transmitted by wrapping connectors to FG, and combined by that sheave so as to act on the follower W, whose motion is the same with that of the centre of FG.

The velocity of the point F is α 1 .AC, upward motion being considered positive. The velocity of the point G is - a 1 .CB, downward motion being negative. Hence (p. 690, §§ 71, 72) the instantaneous axis of the sheave FG is iu the diameter FG, at the distauce

FG AC-BC

2 AC + BC

from the centre towards G; the angular velocity of the sheave is AC + BC

«2 «i · fg ; and, consequently, the velocity of its centre is

FG AC-BC α 1 (AC-BC) ,. oλ

*∙≈-¾∙-2-∙Λ0 + BC- 2 ! · <42) ·

or the mean between the velocities of the two vertical parts of the cord.

If the cord be fixed to the frame-work at the point B, instead of being wound on a barrel, the velocity of W is half that of AF.

A case containing several sheaves is called a block. A fall-block is attached to a fixed point; a running-block is movable to and from a fall-block, with which it is connected by two or more plies of a rope. The whole combination constitutes a tackle or purchase.

The two plies of a rope at opposite sides of a sheave in the fallblock have equal and opposite velocities. The two plies at opposite sides of a sheave in the running-block have velocities (as in the case of the sheave FG) differing equally iu opposite directions from the velocity of the running-block.

One end of the rope is fastened either to the fall-block or the running-block The other, or free eud, is called the fall. Let v 1 be the velocity of the fall, v. 2 that of the running-block; and let it be required to find their ratio; and let velocities towards the fallblock be positive, and from it negative.

Case 1. If the fastened end of the rope be attached to the fallblock its velocity is 0, and this also is the velocity of the first ply. The rope passes under a sheave in the running-block, so that the velocity of the second ply is c lv. 2 . It then passes over a sheave in the fall-block; the velocity of the third ply is - 2ι⅛; then under a sheave in the running-block; the velocity of the fourth ply is 4τ 2 ; and so on,—the general law being this :—let n be an even number, then the velocity of the n th ply=nv 2 )

,, (n + l) th ply= -nv. 2 > . . . (43).

= v 1 , if the fall be the (n + l) th ply )

Case 2. If the fastened end of the rope be attached to the running-block, the velocity of the first ply is v 2 ; of the second, - v. 2 ; of the third, 3r 2 ; of the fourth, - 3v 2 ; and, generally, if n be an odd number, velocity of the n th ply=n¾ )

,, (n +1 ) th ply = - nv, 2 > . . . (44).

=v 1 , if the fall be the (n + l) ,h ply )

Generally,—

-S---n (45), v 2 where n is the number of plies of rope by which the running-block hangs.

The sheaves in a block are usually made all of the same diameter, and turn on a fixed pin, and they have, consequently, different angular velocities. But by making the diameter of each sheave proportional to the velocity, relatively to the block, of the ply of rope which it is to carry, the angular velocities of the sheaves in one block may be rendered equal, so that the sheaves may be made all in one piece, and may have journals turning in fixed bearings. This is called White's tackle, from the inventor.

For details and technical terms see Shipbuilding.

84. Differential Screw. —On the same axis let there be two screws of the respective pitches p j and p 2 , made in one piece, and rotating with the angular velocity a. Let this piece be called B. Let the first screw turn in a fixed nut C, and the second in a sliding nut A. The velocity of advance of B relatively to C is (according to sect. 41) ap l , and of A relatively to B (according to sect. 67) - ap 2 ∙, hence the velocity of A relatively to C is <Pι~P i ) (46), being the same with the velocity of advance of a screw of the pitch Pi ~P∙ι∙ This combination, called Hunter's or the differential screw, combines the strength of a large thread with the slowness of motion due to a small one.

85. Epicyclic Trains.— The term epicyclic train is used by Willis to denote a train of wheels carried by an arm, and having certain rotations relatively to that arm, which itself rotates. The arm may either be driven by the wheels or assist in driving them. The comparative motions of the wheels and of the arm, and the aggregate paths traced by points in the wheels, are determined by the principles of the composition of rotations, and of the description of rolling curves, explained in sects. 37 to 40.

86. Link Motion.— Let S (fig. 27) be the shaft of a steam-engine, 0 its axis, E z the forward eccentric, suitably placed for moving the slide-valve when the shaft rotates forwards, F its centre, OF its crank-arm, C∕ its rod, E⅛ the backward eccentric, suitably placed for moving the slide-valve when the shaft rotates backwards, B its centre, OB its crank-arm, C⅛ its rod. I. is a long narrow box called the link, jointed at T∕ and T» to the eccentric rods; R is the valverod which works the slide-valve, jointed to P, a slider, which, by moving either I. or R, or both, can be adjusted to any required position in the link. When P is at T z the valve is said to be in full forward gearing, being acted upon by E z alone. When P is at T⅛ the valve is said to be in full backward gearing, being acted upon by Et alone. When P is placed in an intermediate position, the valve has an aggregate motion due to the joint action of E z and E⅛. The most exact mode of determining that motion is to make a skeleton drawing of the apparatus in various positions; but an approximation to the motion of the valve may be obtained by joining FB, and taking Q so that

T z P: T 6 P:: FQ: BQ; then the valve will move nearly as if it were worked by one eccentric, having its centre at Q.

87. Parallel Motions {exact).— A parallel motion is a combination of turning pieces in mechanism designed to guide the motion of a reciprocating piece either exactly or approximately in a straight line, so as to avoid the friction which arises from the use of straight guides for that purpose.

Fig. 28 represents an exact parallel motion, first proposed, it is believed, by Mr Scott Russell.

The arm CD turns on the axis C, and is jointed at D to the middle of the bar ADB, whose length is double of that of CD, and one of whose ends B is jointed to a slider, sliding in straight guides along the line CB. Draw BE perpendicular to CB, cutting CD produced in E, then E is the instantaneous axis of the bar ADB; and the direction of motion of A is at every instant perpendicular to EA, that is, along the straight line ACα. While the stroke of A is AC«, extending to equal distances on either side of C, and equal to twice the chord of the arc D<f, the stroke of B is only equal to twice the sagitta; and thus A is guided through a comparatively long stroke by the sliding ot 15 through a comparatively short stroke, and by rotatory motions at the joints C, D, B. (For details, see Steam-Engine. )

88. Parallel Motion—Watt's Approximate (see fig. 29).—Let CT, ct be a pair of levers connected by a link Ti, and oscillating about the axes Cc between the positions marked 1 and 3. Let the middle positions of the levers CT 2 , ct 2 be parallel to each other It is required to find a point P in the link Tf such that its middle position P 2 and its extreme positions P 1 , P 3 shall be in the same straight line SS, perpendicular to CT 2 , ci 2 , and so to place the axes C, c on. the lines CT 2 , ct 2 that the path of P between the positions Pj, P 2 , P 3 shall be as near as possible to a straight line.

The axes C, c are to be so placed that the middle Μ of the versed sine VT 2 , and the middle wz of the versed sine vt 2 , of the respective arcs whose equal chords T 1 T 3 , Z 1 ⅛ represent the stroke, may each be in the line SS. Then T 1 and T 3 will be as far to one.side of SS as T 2 is to the other, and t 1 and t 3 will be as far to the latter side of SS as t 2 is to the former; consequently, the two extreme positions of the links T 1 Z 1 , T 3 < 3 are parallel to each other, and inclined to SS at the same angle in one direction that the middle position of the link T a i 2 is inclined to that line in the other direction, and the three intersections P 1 , P 2 , P 3 are at the same point on the link.

The position of the point P on the link is found by the following proportional equation :—

Ti: PT: Pi )

:: TV + tv : TV: tv > (47).

:: CM + cτn: cm : CM )

Suppose the axes C, c to be given, the line of stroke SS, and the length of stroke I. = T 1 T 3 = i 1 i 3 , and that it is required to find the dimensions of the levers and link. Let fall CM and cm perpendicular to SS; then tv =- l2 -∙^=^∙ Ï

8CM ’ 8cτzι ’ I

CT = CM + ⅛TVj ct=cm + ⅛v, f ’ ‘ ‘ (48)1 Ti =√{Mm 2 + ⅛(TV + iυ) 2 } )

If C and c are at the same side of SS, the smaller of the two perpendiculars is to be treated as negative in the formulae, and the difference of the versed sines used instead of their sum; and the point P will lie in the prolongation of the link beyond Ti to the side of the longer lever. When the arcs of oscillation of the levers on either side of their middle positions do not exceed 20°, the intermediate portions of the path of P between P 1 , P 2 , and P 3 are near enough to a straight line for practical purposes; and that point may be used to guide a sliding piece, such as the piston-rod of a steam-engine, for which purpose this parallel motion was originally invented by Watt.

Chapter II. Ox Applied Dynamics.

89. Laws of Motion.— The action of a machine in transmitting force and motion simultaneously, or performing work, is governed, in common with the phenomena of moving bodies in general, by two “laws of motion,” for which see pp. 676 sq.

90. Comparison of Deviating Force with Gravity.— See nn. 698, 699, §§ 104-106. ιr

91. Deviating Forces Classed—Deflecting Force—Accelerating and Retarding Forces.— See p. 701, §§ 114-119.

92. Division of the Subject.— On this classification of the deviating forces in machines is founded the following division of the subject of dynamics as applied to machines :—

Division 1.—Balanced forces in machines of uniform velocity.

Division 2.— Deflecting forces in such machines.

Division 3.—Working of machines of varying velocity.

Division 1. Balanced Forces in Machines of Uniform Velocity.

93. Application of Force to Mechanism.— Forces are applied in units of weight; and the unit most commonly employed in Britain is the pound avoirdupois. The action of a force applied to a body is always in reality distributed over some definite space, either a volume of three dimensions or a surface of two. An example of a force distributed throughout a volume is the weight of the body itself, which acts on every particle, however small. The pressure exerted between two bodies at their surface of contact, or between the two parts of one body on either side of an ideal surface of separation, is an example of a force distributed over a surface. The mode of distribution of a force applied to a solid body requires to be considered when its stiffness and strength are treated of; but, in questions respecting the action of a force upon a rigid body considered as a whole, the resultant of the distributed force, determined according to the principles of statics, and considered as acting in a single line and applied at a single point, may, for the occasion, be substituted for the force as really distributed. Thus, the weight of each separate piece in a machine is treated as acting wholly at its centre of gravity, and each pressure applied to it as acting at a point called the centre of pressure of the surface to which the pressure is really applied.

94. Forces applied to Mechanism Classed.— If θ be the obliquity of a force F applied to a piece of a machine,—that is, the angle made by the direction of the force with the direction of motion of its point of application,—then by the principles of Statics, F may be resolved into two rectangular components, viz. :—

Along the direction of motion, P = Fcos0 )

Across the direction of motion, Q = F sin θ ∖ ’ ' ' ∙''

If the component along the direction of motion acts with the motion, it is called an effort ; if against the motion, a resistance. The component across the direction of motion is a lateral pressure; the unbalanced lateral pressure on any piece, or part of a piece, is deflecting force. A lateral pressure may increase resistance by causing friction; the friction so caused acts against the motion, and is a resistance, but the lateral pressure causing it is not a resistance. Resistances are distinguished into useful and prejudicial, according as they arise from the useful effect produced by the machine or from other causes.

95. Work. — Work consists in moving against resistance. The work is said to be performed, and the resistance overcome. Work is measured by the product of the resistance into the distance through which its point of application is moved. The unit of work commonly used in Britain is a resistance of one pound overcome through a distance of one foot, and is called a foot-pound.

Work is distinguished into useful work and prejudicial or lost work, according as it is performed in producing the useful effect of the machine, or in overcoming prejudicial resistance.

96. Energy—Potential Energy.—Energy means capacity for performing work. The energy of an effort, or potential energy, is measured by the product of the effort into the distance through which its point of application is capable of being moved. The unit of energy is the same with the unit of work.

When the point of application of an effort has been moved through a given distance, energy is said to have been exerted to an amount expressed by the product of the effort into the distance through which its point of application has been moved.

97. Variable Effort and Resistance.— If an effort has different magnitudes during different portions of the motion of its point of application through a given distance, let each different magnitude of the effort P be multiplied by the length ∆s of the corresponding portion of the path of the point of application; the sum

∑.P∆s . (50)

is the whole energy exerted. If the effort varies by insensible gradations, the energy exerted is the integral or limit towards which that sum approaches continually as the divisions of the path are made smaller and more numerous, and is expressed by

(51).

Similar processes are applicable to the finding of the work performed in overcoming a varying resistance.

98. Dynamometer or Indicator.— A dynamometer or indicator is an instrument which measures and records the energy exerted by an effort. It usually consists essentially, first, of a piece of paper moving with a velocity proportional to that of the point of application of the effort, and having a straight line marked on it parallel to its direction of motion, called the zero line; and, secondly, of a spring acted upon and bent by the effort, and carrying a pencil whose perpendicular distance from the zero line, as regulated by the bending of the spring, is proportional to the effort. The pencil traces on the piece of paper a line such that its ordinate perpendicular to the zero line at a given point represents the effort P for the corresponding point in the path of the point of effort, and the area betιυecn two ordinates represents the energy exerted, J^Vds, for the corresponding portion of the path of the point of effort.

99. Principle of the Equality of Energy and Work.— From the first law of motion it follows that in a machine whose pieces move with uniform velocities the efforts and resistances must balance each other. Now from the laws of statics (see above) it is known that, in order that a system of forces applied to a system of connected points may be in equilibrium, it is necessary that the sum formed by putting together the products of the forces by the respective distances through which their points of application are capable of moving simultaneously, each along the direction of the force applied to it, shall be zero, —products being considered positive or negative according as the direction of the forces and the possible motions of their points of application are the same or opposite.

In other words, the sum of the negative products is equal to the sum of the positive products. This principle, applied to a machine whose parts move with uniform velocities, is equivalent to saying that in any given interval of time the energy exerted is equal to the work performed.

The symbolical expression of this law is as follows :—let efforts be applied to one or any number of points of a machine; let any one of these efforts be represented by P, and the distance traversed by its point of application in a given interval of time by ds ; let resistances be overcome at one or any number of points of the same machine; let any one of these resistances be denoted by R, and the distance traversed by its point of application in the given interval of time by ds'∙, then

Σ.Ptfs=Σ.Rtfs' ...... (52).

The lengths ds, ds' are proportional to the velocities of the points to whose paths they belong, and the proportions of those velocities to each other are deducible from the construction of the machine by the principles of pure mechanism explained in Chapter I.

100. Efficiency.— The efficiency of a machine is the ratio of the useful work to the total work,—that is, to the energy exerted,—and is represented by

Σ. R u tfs' Σ. R u tfs' _ Σ. R u tfs' _ U z r - ι ^

Σ. Rtfs' - ∑7Rξtfs' + ∑.R i ,tfs' - Σ.Rtfs' - E ‘ ’

R,, being taken to represent useful and R p prejudicial resistances.

The more nearly the efficiency of a machine approaches to unity the better is the machine.

101. Power and Effect.— The power of a machine is the energy exerted, and the effect the useful work performed, in some interval of time of definite length, such as a second, an hour, or a day.

The unit of power, called conventionally a horse-power, is 550 foot-pounds per second, or 33,000 foot-pounds per minute, or 1,980,000 foot-pounds per hour.

102. Modulus of a Machine. — In the investigation of the properties of a machine, the useful resistances to be overcome and the useful work to be performed are usually given. The prejudicial resistances are generally functions of the useful resistances of the weights of the pieces of the mechanism, and of their form and arrangement; and, having been determined, they serve for the computation of the lost work, which, being added to the useful work, gives the expenditure of energy required. The result of this investigation, expressed in the form of an equation between this energy and the useful work, is called by Moseley the modulus of the machine. The general form of the modulus may be expressed thus—

E = U + ≠(U, A) + ≠(A) (54), where A denotes some quantity or set of quantities depending on the form, arrangement, weight, and other properties of the mechanism. Moseley, however, has pointed out that in most cases this equation takes the much more simple form of

E = (1+A)U + B (55), where A and B are constants, depending on the form, arrangement, and weight of the mechanism. The efficiency corresponding to the last equation is

U 1 f5ω

E 1 + A + B/U k ''

103. Trains of Mechanism.— In applying the preceding principles to a train of mechanism, it may either be treated as a whole, or it may be considered in sections consisting of single pieces, or of any convenient portion of the train,—each section being treated as a machine driven by the effort applied to it and energy exerted upon it through its line of connexion with the preceding section, performing useful work by driving the following section, and losing work by overcoming its own prejudicial resistances.

It is evident that the efficiency of the whole train is the product of the efficiencies of its sections.

104. Rotating Pieces—Couples of Forces. —It is often convenient to express the energy exerted upon and the work performed by a turning piece in a machine in terms of the moment of the couples of forces acting on it, and of the angular velocity. See p. 728, § 219. , . .

The ordinary British unit of moment is a foot-pound ; but it is to be remembered that this is a foot-pound of a different sort from the unit of energy and work.

If a force be applied to a turning piece in a line not passing through its axis, the axis will press against its bearings with an equal and parallel force, and the equal and opposite reaction of the bearings will constitute, together with the first-mentioned force, a couple whose arm is the perpendicular distance from the axis to the line of action of the first force.

A couple is said to be right or left handed with reference to the observer, according to the direction in which it tends to turn the body, and is a driving couple or a resisting couple according as its tendency is with or against that of the actual rotation.

. Let dt be au interval of time, a the angular velocity of the piece; then adt is the angle through which it turns in the interval dt, and ds=vdt*=radt is the distance through which the point of application of the force moves. Let P represent an effort, so that Pr is a driving couple, then l Rtfs—Pvtfi=Prαtfi = Mαtfi (57)

is the energy exerted by the couple Μ in the interval dt ; and a similar equation gives the work performed in overcoming a resisting couple. When several couples act on one piece, the resultant of their moments is to be multiplied by the common angular velocity of the whole piece.

105. Reduction of Forces to a given Point, and of Couples to the Axis of a given Piece.— In computations respecting machines it is often convenient to substitute for a force applied to a given point, or a couple applied to a given piece, the equivalent force or couple applied to some other point or piece; that is to say, the force or couple, which, if applied to the other point or piece, would exert equal energy or employ equal work. The principles of this reduction are that the ratio of the given to the equivalent force is the reciprocal of the rfttio of the velocities of their points of application, and the ratio of the given to the equivalent couple is the reciprocal of the ratio of the angular velocities of the pieces to which they are applied.

These velocity ratios are known by the construction of the mechanism, and are independent of the absolute speed.

106. Balanced Lateral Pressure of Guides and Bearings.— The most important part of the lateral pressure on a piece of mechanism is the reaction of its guides, if it is a sliding piece, or of the bearings of its axis, if it is a turning piece; and the balanced portion of this reaction is equal and opposite to the resultant of all the other forces applied to the piece, its own weight included. There may be or may not be an unbalanced component in this pressure, due to the deviated motion. Its laws will be considered in the sequel.

107. Friction—Unguents.— The most important kind of resistance in machines is the friction or rubbing resistance of surfaces which slide over each other. The direction of the resistance of friction is opposite to that in which the sliding takes place. Its magnitude is the product of the normal pressure or force which presses the Tubbing surfaces together in a direction perpendicular to themselves into a specific constant already mentioned in Part I., sect. 13, as the coefficient of friction, which depends on the nature and condition of the surfaces of the unguent, if any, with which they are covered. The total pressure exerted between the rubbing surfaces is the resultant of the normal pressure and of the friction, and its obliquity, or inclination to the common perpendicular of the surfaces, is the angle of repose formerly mentioned in sect. 13, whose tangent is the coefficient of friction. Thus, let N be the normal pressure, R the friction, T the total pressure, f the coefficient of friction, and φ the angle of repose; then

∕=tan≠ . . . ) (5 8 ).

R=∕N = Ntan≠ = Tsin≠ )

Experiments on friction have been made by Coulomb, Vince, Rennie, Wood, D. Rankine, and others. The most complete and elaborate experiments are those of Morin, published in his Notions Fondamentales de Mécanique, and republished in Britain in the works of Moseley and Gordon. The following is an exceedingly condensed abstract of the most important results, as regards machines, of these experiments :— [table]

It is to be understood that the above-stated law of friction is only true for dry surfaces when the pressure is not sufficient to indent or abrade the surfaces, and for greased surfaces when the pressure is not sufficient to force out the unguent from between the surfaces. If the proper limit be exceeded, the friction increases more rapidly than in the simple ratio of the normal pressure.

The limit of pressure for unguents diminishes as the speed increases. The following are some of its approximate values as inferred from experience in railway locomotive and carriage axles :— [table]

In pivots, the intensity of the pressure is usually fixed at about one ton per square inch.

Unguents should be comparatively thick for heavy pressures, tiιat they may resist being forced out, and comparatively thin for light pressures, that their viscidity may not add. to the resistance.

Unguents are of three classes, viz.:—

1. Fatty, consisting of animal or vegetable fixed oils, such as tallow, lard, lard-oil, seal-oil, whale-oil, olive-oil. Drying oils, which absorb oxygen and harden, are obviously unfit for unguents.

2. Soapy : composed of fatty oil, alkali, and water. The best grease of this class should not contain more than about 25 or 30 per cent, of water; bad kinds contain 40 or 50 per cent. The additional water diminishes the cost, but spoils the unguent.

3. Bituminous : composed of solid and liquid mineral compounds of hydrogen and carbon.

108. IFork of Friction—Moment of Friction. — The work performed in a unit of time in overcoming the friction of a pair of surfaces is the product of the friction by the velocity of sliding of the surfaces over each other, if that is the same throughout the whole extent of the rubbing surfaces. If that velocity is different for different portions of the rubbing surfaces, the velocity of each portion is to be multiplied by the friction of that portion, and the results summed or integrated.

When the relative motion of the rubbing surfaces is one of rotation, the work of friction in a unit of time, for a portion of the rubbing surfaces ata given distance from the axis of rotation, may be found by multiplying together the friction of that portion, its distance from the axis, aud the angular velocity. The product of the force of friction by the distance at which it acts from the axis of rotation is called the moment of friction. The total moment of friction of a pair of rotating rubbing surfaces is the sum or integral of the moments of friction of their several portions.

To express this symbolically, let du represent the area of a portion of a pair of rubbing surfaces at a distance r from the axis of their relative rotation; p the intensity of the normal pressure at du per unit of area; and f the coefficient of friction. Then the moment of friction of du is [table]

It is evident that the moment of friction, and the work lost by being performed in overcoming friction, are less in a rotating piece as the bearings are of smaller radius. But a limit is put to the diminution of the radii of journals and pivots by the conditions of durability and of proper lubrication stated in sect. 107, and· also by conditions of strength and stiffness.

109. Total Pressure between Journal and Bearing.— A single piece rotating with an uniform velocity has four mutually balanced forces applied to it :—(1) the effort exerted on it by the piece which drives it; (2) the resistance of the piece which follows it,— which may be considered for the purposes of the present question as useful resistance; (3) its weight; and (4) the reaction of its own cylindrical bearings. There are given the following data :—

The direction of the effort.

The direction of the useful resistance.

The weight of the piece and the direction in which it acts.

The magnitude of the useful resistance.

The radius of the bearing r.

The angle of repose φ, corresponding to the friction of the journal on the bearing.

And there are required the following:—

The direction of the reaction of the bearing.

The magnitude of that reaction.

The magnitude of the effort.

Let the useful resistance and the weight of the piece be compounded by the principles of statics into one force, and let this be called the given force.

The directions of the effort and of the given force are either parallel or meet in a point. If they are parallel, the direction of the reaction of the bearing is also parallel to them; if they meet in a point, the direction of the reaction traverses the same point.

Also, let AAA, fig. 30, be a section of the bearing, and C its axis; then the direction of the reaction, at the point where it intersects the circle AAA, must make the angle φ with the radius of that circle; that is to say, it must be a line such as PT touching the smaller circle BB, whose radius is r. sin <ρ. The side on which it touches that circle is determined by the fact that the obliquity of the reaction is such as to oppose the rotation.

Thus is determined the direction of the reaction of the bearing; and the magnitude of that reaction and of the effort are then found by the principles of the equilibrium of three forces already stated in Part I., sect. 7 (see also p. 702, § 124).

The work lost in overcoming the friction of the bearing is the same with that which would be performed in overcoming at the circumference of the small circle BB a resistance equal to the whole pressure between the journal and bearing.

In order to diminish that pressure to the smallest possible amount, the effort, and the resultant of the useful resistance, and the weight of the piece (called above the “given force ”) ought to be opposed to each other as directly as is practicable consistently with the purposes of the machine.

110. Friction of Pivots and Collars.— When a shaft is acted upon by a force tending to shift it lengthways, that force must be balanced by the reaction of a bearing against a pivot at the end of the shaft; or, if that be impossible, against one or more collars, or rings projecting from the body of the shaft. The bearing of the pivot is called a step or footstep. Pivots require great hardness, and are usually made of steel. Theyføi pivot is a cylinder of steel having a plane circular end as a rubbing surface. Let N be the total pressure sustained by a flat pivot of the radius r ; if that pressure be uniformly distributed, which is the case when the rubbing surfaces of the pivot and its step are both true planes, the intensity of the pressure is

N p=^ (60); and, introducing this value into equation 59, the moment of friction of the flat pivot is found to be

⅜∕Nr (61), or two-thirds of that of a cylindrical journal of the same radius under the same normal pressure.

The friction of a conical pivot exceeds that of a flat pivot of the same radius, and under the same pressure, in the proportion of the side of the cone to the radius of its base.

The moment of friction of a collar is given by the formula—

<y∙3 _ y ,z 3

⅜∕ n *⅛ (62).

where r is the external and r' the internal radius.

In the cup and ball pivot the end of the shaft and the step present two recesses facing each other, into which are fitted two shallow cups of steel or hard bronze. Between the concave spherical surfaces of those cups is placed a steel ball, being either a complete sphere or a lens having convex surfaces of a somewhat less radius than the concave surfaces of the cups. The moment of friction of this pivot is at first almost inappreciable from the extreme smallness of the radius of the circles of contact of the ball and cups, but, as they wear, that radius and the moment of friction increase.

It appears that the rapidity with which a rubbing surface wears away is proportional to the friction and to the velocity jointly, or nearly so. Hence the pivots already mentioned wear unequally at different points, and tend to alter their figures. Schiele has invented a pivot which preserves its original figure by wearing equally at all points in a direction parallel to its axis. The following are the principles on which this equality of wear depends :—

The rapidity of wear of a surface measured in an oblique direction is to the rapidity of wear measured normally as the secant of the obliquity is to unity. Let OX (fig. 31) be the axis of a pivot, and let RPC be a portion of a curve such that at any point P the secant of the obliquity to the normal of the curve nf a line parallel to the axis is inversely proportional to the ordinate PY, to which the velocity of P is proportional. The rotation of that curve round OX will generate the form of pivot required. Now let PT be a tangent to the curve at P, cutting OX iu T; ΓT= PY × secant obliquity, and this is to be a constant quantity; hence the curve is that known as the tractory of the straight line OX, in which PT=OR = constant. This curve is described by having a fixed straight edge parallel to OX, along which slides a slider carrying a pin whose centre is T. On that pin turns an arm, carrying at a point P a tracing-point, pencil, or pen. Should the pen have a nib of two jaws, like those of an ordinary drawing-pen, the plane of the jaws must pass through PT. Then, while T is slid along the axis from 0 towards X, P will be drawn after it from R towards C along the tractory. This curve, being an asymptote to its axis, is capable of being indefinitely prolonged towards X; but in designing pivots it should stop before the angle PTY becomes less than the angle of repose of the rubbing surfaces, otherwise the pivot will be liable to stick in its bearing.

The moment of friction of “Schiele’s anti-friction pivot,” as it is called, is equal to that of a cylindrical journal of the radius OR = PT the constant tangent, under the same pressure.

111. Friction of Teeth.— Let N be the normal pressure exerted between a pair of teeth of a pair of wheels; s the total distance through which they slide upon each other; n the number of pairs of teeth which pass the plane of axis in a unit of time; then nf~Ss (63)

is the work lost in unity of time by the friction of the teeth. The sliding s is composed of two parts, which take place during the approach and recess respectively. Let those be denoted by s 1 and s. 2 , so that s=s 1 + s 2 . I q sect. 55 the velocity of sliding at any instant has been given, viz., u = c(α 1 + <x 2 ), where u is that velocity, c the distance TI at any instant from the point of contact of the teeth to the pitch-point, and α 1 , α 2 the respective angular velocities of the wheels.

Let v be the common velocity of the two pitch-circles, r 1 , r 2 their radii; then the above equation becomes

( 1 1 ∖ w = cv 1 .

∖ r ι r ι ∕

To apply this to involute teeth, let c 1 be the length of the approach, c. i that of the recess, u x the mean velocity of sliding during the approach, u 2 that during the recess; then lh W± + 2‰-W± + ±)

2 ∖ r 1 r 2 ) 'l ∖ r x r 2 ) also, let 0 be the obliquity of the action; then the times occupied by the approach and recess are respectively c ι c A.

VCOSØ ’ vcosθ ’ giving, finally, for the length of sliding between each pair of teeth, c 2 l + c 2 2 ∕ f 1 1 ∖ , β . λ + = + · · · · (64), Û ∙vb V ξ Γj Γ% J which, substituted in equation 63, gives the work lost in a unit of time by the friction of involute teeth. This result, which is exact for involute teeth, is approximately true for teeth of any figure.

For inside gearing, if r 1 be the less radius and r 2 the greater,

— — is to be substituted for 1 .

r 1 r 2 r x r 2

112. Friction of Cords and Belts.— A flexible band, such as a cord, rope, belt, or strap, may be used either to exert an effort or a resistance upon a pulley round which it wraps. In either case the tangential force, whe'ther effort or resistance, exerted between the band and the pulley is their mutual friction, caused by and proportional to the normal pressure between them.

Let T 1 be the tension of the free part of the band at that side towards which it tends to draw the pulley, or from which the pulley tends to draw it; T 2 the tension of the free part at the other side; T the tension of the band at any intermediate point of its arc of contact with the pulley; 0 the ratio of the length of that arc to the radius of the pulley; dθ the ratio of an indefinitely small element of that arc to the radius; F = T 1 - T 2 the total friction between the band and the pulley; <ZF the elementary portion of that friction due to the elementary arc dθ; f the coefficient of friction between the materials of the band and pulley.

Then, according to a well-known principle in statics, the normal pressure at the elementary arc dθ is Tdθ, T being the mean tension of the band at that elementary arc; consequently the friction on that arc is dF=fTdθ. Now that friction is also the difference between the tensions of the band at the two ends of the elementary arc, or dT = dF = fTdθ ; which equation, being integrated throughout the entire arc of contact, gives the following formulae :— [table]

When a belt connecting a pair of pulleys has the tensions of its two sides originally equal, the pulleys being at rest, and when the pulleys are next set in motion, so that one of them drives the other by means of the belt, it is found that the advancing side of the belt is exactly as much tightened as the returning side is slackened, so that the mean tension remains unchanged. Its value is given by this formula—

T, + T. √ 0 + 1

—Σ (66),

2 2(√ b -1) k , which is useful in determining the original tension required to enable a belt to transmit a given force between two pulleys.

The equations 65 and 66 are applicable to a kind of brake called a friction-strap, used to stop or moderate the velocity of machines by being tightened round a pulley. The strap is usually of iron, and the pulley of hard wood.

Let a denote the arc of contact expressed in turns and fractions of a turn ; then

0 = 6 ’ 2832α ∣

e^ θ = number whose common logarithm is 2 - 7288 fa ⅛ ( ‘ ' ’

113. Stiffness of Ropes.— Ropes offer a resistance to being bent, and, when bent, to being straightened again, winch arises from the mutual friction of their fibres. It increases with the sectional area of the rope, and is inversely proportional to the radius of the curve into which it is bent.

The work lost in pulling a given length of rope over a pulley is found by multiplying the length of the rope in feet by its stiffness in pounds, that stiffness being the excess of the tension at the leading side of the rope above that at the following side, which is necessary to bend*it into a curve fitting the pulley, and then to straighten it again.

The following empirical formulae for the stiffness of hempen ropes have been deduced by Morin from the experiments of Coulomb :—

Let F be the stiffness in pounds avoirdupois; d the diameter of the rope in ipches, n=48cf 2 for white ropes and 35<f 2 for tarred ropes; r the effective radius of the pulley in inches; T the tension in pounds. Then

For white ropes, F = —(0Ό012 +0 001026n +0Ό012Τ) ) r (68).

For tarred ropes, F = — (0∙006 + 0 - 001392n + OO0168T) )

114. Friction-Couplings. — Friction is useful as a means of communicating motion where sudden changes either of force or velocity take place, because, being limited in amount, it may be so adjusted as to limit the forces which strain the pieces of the mechanism within the bounds of safety. Amongst contrivances for effecting this object are friction-cones. A rotating shaft carries upon a cylindrical portion of its figure a wheel or pulley turning loosely on it, and consequently capable of remaining at rest when the shaft is in motion. This pulley has fixed to one side, and concentric with it, a short frustum of a hollow cone. At a small distance from the pulley the shaft carries a short frustum of a solid cone accurately turned to fit the hollow cone. This frustum is made always to turn along with the shaft by being fitted on a square portion of it, or by means of a rib and groove, or otherwise, but is capable of a slight longitudinal motion, so as to be pressed into, or withdrawn from, the hollow cone by means of a lever. When the cones are pressed together or engaged, their friction causes the pulley to rotate along with the shaft; when they are disengaged, the pulley is free to stand still. The angle made by the sides of the cones with the axis should not be less than the angle of repose. In t ∖∖ o friction- clutch, a pulley loose on a shaft has a hoop or gland made to embrace it more or less tightly by means of a screw; this hoop h⅛s short projecting arms or ears. A fork or clutch rotates along with the shaft, and is capable of being moved longitudinally by a handle. When the clutch is moved towards the hoop, its arms catch those of the hoop, and cause the hoop to rotate and to communicate its rotation to the pulley by friction. There are many other contrivances of the same class, but the two just mentioned may serve for examples.

115. Heat of Friction — Unguents. — The work lost in friction is employed in producing heat. This fact is very obvious, and has been known from a remote period; but the exact determination of the proportion of the work lost to the heat produced, and the experimental proof that that proportion is the same under all circumstances, and with all materials, solid, liquid, and gaseous, are comparatively recent achievements of Joule. The quantity of work which produces a British unit of heat (or so much heat as elevates the temperature of one pound of pure water, at or near ordinary atmospheric temperatures, by one degree of I ahrenheit) is 772 foot-pounds. This constant, now designated as “Joule’s equivalent,” is the principal experimental datum of the science of thermodynamics.

The heat produced by friction, when moderate in amount, is usefιιl in softening and liquefying thick unguents; but when excessive it is prejudicial, by decomposing the unguents, and sometimes even by softening the metal of the bearings, and raising their temperature so high as to set fire to neighbouring combustible matters.

Excessive heating is prevented by a constant and copious supply of a good unguent. The elevation of temperature produced by the friction of a journal is sometimes used as an experimental test of the quality of unguents. When the velocity of rubbing is about 4 or 5 feet per second, the elevation of temperature has been found by some recent experiments to be, with good fatty and soapy unguents, 40° to 50’ Fahr.; with good mineral unguents, about 30®.

116. Rolling Resistance.— By the rolling of two surfaces over each other without sliding a resistance is caused which is called sometimes “rolling friction,” but more correctly rolling resistance. It is of the nature of a couple, resisting rotation. Its moment is found by multiplying the normal pressure between the rolling surfaces by an arm, whose length depends on the nature of the rolling surfaces, and the work lost in a unit of time in overcoming it is the product of its moment by the angular velocity of the rolling surfaces relatively to each other. The following are approximate values of the arm in decimals of a foot :— [table]

117. Reciprocating Forces—Stored and Restored Energy.— When a force acts on a machine alternately as an effort and as a resistance, it may be called a reciprocating force. Of this kind is the weight of any piece in the mechanism whose centre of gravity alternately rises and falls; for during the rise of the centre of gravity that weight acts as a resistance, and energy is employed in lifting it to an amount expressed by the product of the weight into the vertical height of its rise; and during the fall of the centre of gravity the weight acts as an effort, and exerts in assisting to perform the work of the machine an amount of energy exactly equal to that which had previously been employed in lifting it. Thus that amount of energy is not lost, but has its operation deferred; and it is said to be stored when the weight is lifted, and restored when it falls.

In a machine of which each piece is to move with a uniform velocity, if the effort and the resistance be constant, the weight of each piece must be balanced on its axis, so that it may produce lateral pressure only, and not act as a reciprocating force. But if the effort and the resistance be alternately in excess, the uniformity of speed may still be preserved by so adjusting some moving weight in the mechanism that when the effort is in excess it maybe lifted, and so balance and employ the excess of effort, and that when the resistance is in excess it may fall, and so balance and overcome the excess of resistance,—thus storing the periodical excess of energy, and restoring that energy to perform the periodical excess of work.

Other forces besides gravity may be used as reciprocating forces for storing and restoring energy,—for example, the elasticity of a spring or of a mass of air.

In most of the delusive machines commonly called “perpetual motions,” of which so many are patented in each year, and which are expected by their inventors to perform work without receiving energy, the fundamental fallacy consists in an expectation that some reciprocating force shall restore more energy than it has been the means of storing.

Division 2. Deflecting Forces.

118. Deflecting Force for Translation in a Curved Path.— In machinery, deflecting force is supplied by the tenacity of some piece, such as a crank, which guides the deflected body in its curved path, and is unbalanced, being employed in producing deflexion, and not in balancing another force.

119. Centrifugal Force.— See p. 682, § 35, and p. 701, § 119. '

120. Rectangular Resolution of Centrifugal Force.— See p. 701, §§ 117 and 119.

121. Centrifugal Force of a Rotating Body. — The centrifugal force exerted by a rotating body on its axis of rotation is the same in magnitude as if the mass of the body were concentrated at its centre of gravity, and acts in a plane passing through the axis of rotation and the centre of gravity of the body.

The particles of a rotating body exert centrifugal forces on each other, which strain the body, and tend to tear it asunder; but these forces balance each other, and do not affect the resultant centrifugal force exerted on the axis of rotation. If the axis of rotation traverses the centre of gravity of the body, the centrifugal force exerted on that axis is nothing.

Hence, unless there be some reason to the contrary, each piece of a machine should be balanced on its axis of rotation; otherwise the centrifugal force will cause strains, vibration, and increased friction, and a tendency of the shafts to jump out of their bearings.

122. Centrifugal Couples of a Rotating Body.— Besides the tendency (if any) of the combined centrifugal forces of the particles of a rotating body to shift the axis of rotation, they may also tend to turn it out of its original direction. The latter tendency is called a centrifugal couple, and vanishes for rotation about a principal axis (see p. 732, § 237).

It is essential to the steady motion of every rapidly rotating piece in a machine that its axis of rotation should not merely traverse its centre of gravity, but should be a permanent axis; for otherwise the centrifugal couples will increase friction, produce oscillation of the shaft, and tend to make it leave its bearings.

The principles of this and the preceding section are those which regulate the adjustment of the weight and position of the counterpoises which are placed between the spokes of the driving-wheels of locomotive engines.

123. Revolving Pendulum—Governors.— In fig. 32 AO represents an upright axis or spindle; B a weight called a bob, suspended by rod OB from a horizontal axis at 0, carried by the vertical axis. When the spindle is at rest the bob hangs close to it; when the spindle rotates, the bob, being made to revolve round it, diverges until the resultant of the centrifugal force and the weight of the bob is a force acting at 0 in the direction OB, and then it revolves steadily in a circle. This combination is called a revolving, centrifugal, or conical pendulum. Revolving pendulums are usually constructed with pairs of rods and bobs, as OB, Ob, hung at opposite sides of the spindle, that the centrifugal forces exerted at the point 0 may balance each other.

In finding the position in which the bob will revolve with a given angular velocity a, for most practical cases connected with machinery the mass of the rod may be considered as insensible compared with that of the bob. Let the bob be a sphere, and from the centre of that sphere draw BH=y perpendicular to OA. Let OH=z; let W be the weight of the bob, F its centrifugal force. Then the condition of its steady revolution is W: F:: z -.y; that is to say, ^- = -=; consequently z-=^⅞ (69).

Or, if η = θ - be the number of turns or fractions of a turn in a second, g 0 ’ 8165 foot 9'79771 inches ) z ⅛ λ .

# — 2 2 9 9 Ö Z · · ( ∕ 0 ) »

4π⅛- n 2 n 2 ) ∖ ∕> z is called the altitude of the pendulum.

If the rod of a revolving pendulum be jointed, as in fig. 33, not to a point in the vertical axis, but to the end of a projecting arm 0, the position in which the bob will revolve will be the same as if the rod were jointed to the point O, where its prolongation cuts the vertical axis.

A revolving pendulum is an essential part of most of the contrivances called governors, for regulating the speed of prime movers.

The earlier kinds of governors act on the prime mover by the variations of their altitude. Thus in Watt’s steam-engine governor the rods, through a combination of levers and linkwork C, c, D, d (fig. 32), act on a lever EF, which acts upon the throttle-valve for the admission of steam so as to enlarge or contract its opening when the speed becomes too small or too great.

In a more recent kind of governors invented by the Messrs Siemens, which may be called differentialgovernors, the regulation of the prime mover is effected by means of the difference between the velocity of a wheel driven by it and that of a wheel regulated by a revolving pendulum. Fig. 34 illustrates this class of governors. A is a vertical dead-centre or fixed shaft, about which the after-mentioned pieces turn; C is a pulley driven by the prime mover, and fixed to a bevel-wheel, which is seen below it; E is a bevel-wheel similar to the first, and having the same apex. To this wheel are hung the bobs B, of which there are usually four, although two only are shown. Those bobs form sectors of a ring, and are surrounded by a cylindrical casing F. When the bobs revolve with their proper velocity, they are adjusted so as nearly to touch this casing; should they exceed that velocity, they fly outwards and touch the casing, and are retarded by the friction. For practical purposes their velocity of rotation about the vertical axis may be considered constant. G, G are horizontal arms projecting from a socket which is capable of rotation about A, and carrying vertical bevel wheels which rest on E and support C, and transmit motion from C to E. There are usually four of the arms G, G with their wheels, though two only are shown. H is one of those arms which projects, and has a rod attached to its extremity to act on the throttle-valve of a steam-engine, the sluice of a water-wheel, or the regulator of the prime mover, of whatever sort it may be.

When C rotates with an angular velocity equal and contrary to that of E with its revolving pendulums, the arms G, G remain at rest; but should C deviate from that velocity, those arms rotate in one direction or the other, as the case may be, with an angular velocity equal to one half of the difference between the angular velocity of C and that of E, and continue in motion until the regulator is adjusted so that the prime mover shall impart to C an angular velocity exactly equal to that of the revolving pendulums.

There are various modifications of the differential governor, but they all act on the same principle.

Division 3. Working of Machines of Varying Velocity.

124. General Principles.— In order that the velocity of every piece of a machine may be uniform, it is necessary that the forces acting on each piece should be always exactly balanced. Also, in order that the forces acting on each piece of a machine may be always exactly balanced, it is necessary that the velocity of that piece should be uniform.

An excess of the effort exerted on any piece, above that which is necessary to balance the resistance, is accompanied with accelera tion; a deficiency of the effort, with retardation.

When a machine is being started from a state of rest, and brought by degrees up to its proper speed, the effort must be in excess; when it is being retarded for the purpose of stopping it, the resistance must be in excess.

An excess of effort above resistance involves an excess of energy exerted above work performed; that excess of energy is employed in producing acceleration.

An excess of resistance above effort involves an excess of work performed above energy expended; that excess of work is performed by means of the retardation of the machinery.

When a machine undergoes alternate acceleration and retardation, so that at certain instants of time, occurring at the end of intervals called periods or cycles, it returns to its original speed, then in each of those periods or cycles the alternate excesses of energy and of work neutralize each other; and at the end of each cycle the principle of the equality of energy and work stated in sect. 96, with all its consequences, is verified exactly as in the case of machines of uniform speed.

At intermediate instants, however, other principles have also to be taken into account, which are deduced from the second law of motion, sect. 89, as applied by the aid of the principles of sect. 90, to direct deviation, or acceleration and retardation.

125. Energy of Acceleration and Work of Retardation for a Shifting Body.— Let w be the weight of a body which has a motion of translation in any path, and in the course of the interval of time ∆i let its velocity be increased at a uniform rate of acceleration from v 1 to ¾. The rate of acceleration will be dυ . . Vov, constant =; dt ∆i and (p. 698, § 104) to produce this acceleration a uniform effort will be required, expressed by w-(p 2 - v 1 ) , 7n g&t κ j '

(The product — of the mass of a body by its velocity is called its momentum ; so that the effort required is found by dividing the increase of momentum by the time in which it is produced.)

To find the energy which has to be exerted to produce the acceleration from v 1 to ¾, it is to be observed that the distance through which the effort P acts during the acceleration is

. r a + * ∕.

∆.s ≈-∙ -i -∆i;

2

consequently, the energy of acceleration is tc(t⅛ - ⅜> 1 )(¾⅛ + v l > w{v↑ - v ∖ ) / 72) 2g 2g ' being proportional to the increase ii^. the square of the velocity, and independent of the time.

In order to produce a retardation from the greater velocity ι⅛ to the less velocity v l , it is necessary to apply to the body a resistance, connected with the retardation and the time by an equation identical in every respect with equation 71, except by the substitution of a resistance for an effort; and in overcoming that resistance the body performs work to an amount determined by equation 72, putting Rds for Pds.

126. Energy Stored and Restored by Deviations of Velocity.— Thus a body alternately accelerated and retarded, so as to be brought back to its original speed, performs work during its retardation exactly equal in amount to the energy exerted upon it during its acceleration; so that that energy may be considered as stored during the acceleration, and restored during the. retardation, in a manner analogous to the operation of a reciprocating force (sect. 117).

Let there be given the mean velocity V = ⅛(v 2 + v 1 ) of a body whose weight is w, and let it be required to determine the fluctuation of velocity v 2 -v 1 , and the extreme velocities v 1 , v 2 , which that body must have, in order alternately to store and restore an amount of energy E. By equation 72 wo have w(v i 2 -v↑) which, being divided by V = ∣(v s + v 1 ), gives

E w(v 2 - v 1 )

V “g : and consequently °∙-'Ι-⅛ ' < 73 >∙

The ratio of this fluctuation to the mean velocity, sometimes called the unsteadiness of*the motion of the body, is v a -v 1 ~ fi rE , 74 ,

V v⅛> k , '

127. Actual Energy of a Shifting Body.— The energy which must be exerted on a body of the weight w, to accelerate it from a state of rest up to a given velocity of translation v, and the equal amount of work which that body is capable of performing by overcoming resistance while being retarded from the same velocity of translation v to a state of rest, is

~ (75).

2?

This is called the actual energy of the motion of the body, and is half the quantity which in some treatises is called vis viva.

The energy stored or restored, as the case may be, by the deviations of velocity of a body or a system of bodies, is the amount by which the actual energy is increased or diminished.

128. Principle of the Conservation of Energy in Machines.— The following principle, expressing the general law of the action of machines with a velocity uniform or varying, includes the law of the equality of energy and work stated in sect. 99 for machines of uniform speed.

In any given interval during the working of a machine, the energy exerted added to the energy restored is equal to the energy stored added to the work performed.

129. Actual Energy of Circular Translation—Moment of Inertia. —Let a body of the weight w undergo translation in a circular path of the radius p, with the angular velocity of deflexion a, so that the common linear velocity of all its particles is v = aρ. Then the actual energy of that body is wr 2 wα 2 p 2 , 76) 2g 2g

By comparing this with the expression for the centrifugal force {wufpg'), it appears that the actual energy of a revolving body is equal to the potential energy Fp/2 due to the action of the deflecting force along one-half of the radius of curvature of the path of the body.

The product wp-]g, by which the half-square of the angular velocity is multiplied, is called the moment of inertia of the revolving body.

130. Actual Energy and Moment of Inertia of Rotation—Radius of Gyration.— See p. 732, §§ 234-237.

131. Examples of Radii of Gyration.— See p. 733, § 238.

132. Fly-wheels.— A fly-wheel is a rotating piece in a machine, generally shaped like a wheel (that is to say, consisting of a rim with spokes), and suited to store and restore energy by the periodical variations in its angular velocity. _ ...

The principles according to which variations of angular velocity store and restore energy are the same with those of sect. 126, only substituting moment of inertia for mass, and angular for linear velocity. .

Let W be the weight of a fly-wheel, R its radius of gyration, a i its maximum, α 1 its minimum, and A=4(c⅛ + a l ) its mean angular velocity. Let

1 t⅛-a 1

S A denote the unsteadiness of the motion of the fly-wheel; the denominator S of this fraction is called the steadiness. Let e denote the quantity by which the energy exerted in each cycle of the working of the machine alternately exceeds and falls short of the work performed, and which has consequently to be alternately stored by acceleration and restored by retardation of the fly-wheel. The value of this periodical excess is—

,- r,w W-<> (77), 2g from which, dividing both sides by A 2 , we obtain the following equations :— ,

^ = R 2 W 1

a2 ?S I . (78).

R 2 WA 2 Se (, 2g 2 J

The latter of these equations may be thus expressed in words :— The actual energy due to the rotation of the fly, with its mean angular velocity, is equal to one-half of the periodical excess of energy multiplied by the steadiness.

In ordinary machinery S = about 32; in machinery for fine purposes S = from 50 to 60.

The periodical excess e may arise either from variations in the effort exerted by the prime mover, or from variations in the resistance of the work, or from both these causes combined. When but one fly-wheel is used, it should be placed in as direct connexion as possible with that part of the mechanism where the greatest amount of the periodical excess originates; but when it originates at two or more points, it is best to have a fly-wheel in connexion with each of those points. For example, in a machinework, the steam-engine, which is the prime mover of the various tools, has a fly-wheel on the crank-shaft to store and restore the periodical excess of energy arising from the variations in the effort exerted by the connecting-rod upon the crank; and each of the slotting machines, punching machines, rivetting machines, and other tools has a fly-wheel of its own to store and restore energy, so as to enable the very different resistances opposed to those tools at different times to be overcome without too great unsteadiness of motion.

According to the computation of General Morin, the periodical excess e in steam-engines with single cranks is from τ ⅛th to nearly 4th of the energy exerted during one revolution of the crank. For a pair of steam-engines driving one shaft, with a pair of cranks at right angles to each other, the value of e is one-fourth of its value for a single cranked engine of the same kind, and of the same power with the two combined.

The ordinary radius of gyration of a steam-engine fly-wheel is from three to five times the length of the crank-arm. (For further particulars on this subject, see Steam-Engine.)

For tools performing useful work at intervals, and having only their own friction to overcome during the intermediate intervals, e should be assumed equal to the wlιole work performed at each separate operation.

133. Brakes. — A brake is an apparatus for stopping and diminishing the velocity of a machine by friction, such as the friction-strap already referred to in sect. 112. To find the distance s through which a brake, exerting the friction F, must rub in order to stop a machine having the total actual energy E at the moment when the brake begins to act, reduce, by the principles of sect. 105, the various efforts and other resistances of the machine which act at the same time with the friction of the brake to the rubbing surface of the brake, and let R be their resultant, — positive if resistance, negative if effort preponderates. Then s=i⅛ · (Z9).

r + K

134. Energy distributed between two Bodies—Projection and Propulsion.— Hitherto the effort by which a machine is moved has been treated as a force exerted between a movable body and a fixed body, so that the whole energy exerted by it is employed upon the movable body, and none upon the fixed body. This conception is sensibly realized in practice when one of the two bodies between which the effort acts is either so heavy as compared with the other, or has so great a resistance opposed to its motion, that it may, without sensible error, be treated as fixed. But there are cases in which the motions of both bodies are appreciable, and must be taken into account,—such as the projection of projectiles, where the velocity of the recoil or backward motion of the gun bears an appreciable proportion to the forward motion of the projectile; and such as the propulsion of vessels, where the velocity of the water thrown backward by the paddle, screw, or other propeller bears a very considerable proportion to the velocity of the water moved forwards and sideways by the ship. In cases of this kind the energy exerted by the effort is distributed between the two bodies between which the effort is exerted in shares proportional to the velocities of the two bodies during the action of the effort; and those velocities are to each other directly as the portions of the effort unbalanced by resistance on the respective bodies, and inversely as the weights of the bodies.

To express this symbolically, let W 1 , W 2 be the weights of the bodies; P the effort exerted between them; S the distance through which it acts; R 1 , R 2 the resistances opposed to the effort overcome by W 1 , W 2 respectively; E υ E 2 the shares of the whole energy E exerted upon W 1 , W 2 respectively. Then

E g: E 1 : E 2 )

.. W 2 (P-R 1 ) + W 1 (P-R 2 ) P-R 1 P-R 2 f. . (80).

W 1 W 2 ' W 1 : W 2 )

If R 1 = R 2 , which is the case when the resistance, as well as the effort, arises from the mutual actions of the two bodies, the above becomes,

E: E 1 : E 2 ) ,∏.> -

:: W 1 + W 2 : W 2 : WJ ’

that is to say, the energy is exerted on the bodies in shares inversely proportional to their weights; and they receive accelerations inversely proportional to their weights, according to the principle of dynamics, already quoted in a note to sect. 121, that the mutual actions of a system of bodies do not affect the motion of their common centre of gravity.

For example, if the weight of a gun be 160 times that of its ball, τ⅛τ °f thθ energy exerted by the powder in exploding will be employed in propelling the ball, and γ ⅞ t in producing the recoil of the gun, provided the gun up to the instant of the ball’s quitting the muzzle meets with no resistance to its recoil except the friction of the ball.

135. Centre of Percussion.— It is obviously desirable that the deviations or changes of motion of oscillating pieces in machinery should, as far as possible) be effected by forces applied at their centres of percussion.

If the deviation be a translation,— that is, an equal change of motion of all the particles of the body,—the centre of percussion is obviously the centre of gravity itself; and, according to the second law of motion, if dv be the deviation of velocity to be produced in the interval dt, and W the weight of the body, then

P = --.⅞ (82)

g dt is the unbalanced effort required.

If the deviation be a rotation about an axis traversing the centre of gravity, there is no centre of percussion; for such a deviation can only be produced by a couple of forces, and not by any single force. Let da be the deviation of angular velocity to be produced in the interval dt, and I the moment of the inertia of the body; then ⅜Ii∕(α 2 ) = Iαtfα is the variation of the body’s actual energy. Let Μ be the moment of the unbalanced couple required to produce the deviation; then, by equation 57, sect. 104, the energy exerted by this couple in the interval dt is Macfø, which, being equated to the variation of energy, gh 7 es Μ , da R AV da ? ο o ∖

= I-yr = -rτ ( 83 )∙ dt g dt

Now (fig. 35), let the required deviation be a rotation of the body BB about an axis O, not traversing the centre of gravity G, da being, as before, the deviation of angular velocity to be produced in the interval dt. A rotation with the angular velocity a about an axis O may be considered as compounded of a rotation with the same angular velocity about an axis drawn through G parallel to O and a translation with the velocity a. OG, OG being the perpendicular distance between the two axes. Hence the required deviation may be regarded as compounded of a deviation of translation dv — OG.da, to produce which there would be required, according to equation 82, a force applied at G perpendicular to the plane OG—

P = — . OG . ⅛ (84), g dt and a deviation da of rotation about an axis drawn through G parallel to O, to produce which there would be required a couple of the moment Μ given by equation 83. According to the principles of statics, the resultant of the force P, applied at G perpendicular to the plane OG, and the couple Μ is a force equal and parallel to P, but applied at a distance GO from G, in the prolongation of the perpendicular OG, whose value is

-÷⅛ Ä

Thus is determined the position of the centre of percussion C, corresponding to the axis of rotation O. It is obvious from this equation that, for an axis of rotation parallel to 0 traversing 0, the centre of percussion is at the point where the perpendicular OG meets 0.

136. Impact.— Impact or collision is a pressure of short duration exerted between two bodies. For the detailed investigation of its laws the reader is referred to Mechanics, p. 719, § 180 sg.

The effects of impact are sometimes an alteration of the distribution of actual energy between the two bodies, and always a loss of a portion of that energy, depending on the imperfection of the elasticity of the bodies, in permanently altering their figures, and producing heat. The determination of the distribution of the actual energy after collision and of the loss of energy is effected by means of the following principles :—

I. The motion of the common centre of gravity of the two bodies is unchanged by the collision.

II. The loss of energy consists of a certain proportion of that part of the actual energy of the bodies which is due to their motion relatively to their common centre of gravity.

Unless there is some special reason for using impact in machines, it ought to be avoided, on account not only of the waste of energy which it causes, but from the damage which it occasions to the frame and mechanism.

Chapter III. Purposes and Effects of Machines.

137. Observing Machines and Working Machines.— The present chapter must necessarily be limited to some very general observations on the principal classes into which machines may be divided, with reference to their purposes and effects, leaving details of particular examples for treatment under the respective special headings.

Machines may be divided, in the first instance, into two great divisions, viz. :—

I. Observing Machines, in which either the modification of motion alone, or the balancing of forces alone, is the object in view,—the performance of work being either null or incidental, and being limited to that which arises from the resistance of the machine.

II. Working machines, in which the performance of work is the main object.

138. Classification of Observing Machines.— Observing machines might very properly have been classed as instruments, being designed to aid the human senses and memory in obtaining and recording information. They may be divided, in the first instance, into four classes, according as the subject of observation by their ajd is number, measure, or weight, into—

A. Counting machines.

B. Measuring machines.

C. Copying and drawing machines.

D. Weighing machines.

And to these may be added a fifth class, in which the functions of the first four are more or less combined, viz. :—

E. Recording machines.

139. (A. ) Counting Machines. — The most important as well as the most common of counting machines are time-keepers, which count and indicate the numbers of oscillations of bodies which oscillate isochronously (viz., pendulums for clocks, balance-wheels for watches and marine Chronometern) so as to measure time. In constructing such machines, the objects to be aimed at are the exact isochronism of the pendulum or balance, and the equable action of the motive power, so that it shall overcome the friction of the mechanism without affecting the rate.

Other counting machines count the oscillations of the beam of a steam-engine, or the revolutions of the cylinder of a gas-meter or of the wheel of a water-meter.

Others perforin additions, subtractions, and multiplications, and of these the most elaborate kind (as, e.g., that of Babbage) compute tables of functions by the addition of differences.

140. (B.) Measuring Machines.— Measuring machines are pieces of mechanism, by means of which the motion of some body of the nature of an index through some geometrical magnitude, such as a distance or an angle, is connected with some other motion, either equal or greater or smaller in some given ratio, and capable of being more readily compared with some standard of measure.

To this class belong all those astronomical and surveying instruments in which the motion of a line of sight (generally the line of collimation of a telescope) through a given angle is connected with the motion of an index or vernier round a corresponding arc of a graduated circle; also those micrometers in which the advance of the end of a screw of fine pitch is measured by observing the simultaneous arc of rotation of a graduated circle attached to it.

Such micrometers have attained increased importance by the discovery of Whitworth,—that the mechanical magnifying of small distances by a train of screws affords a more accurate means of measurement than optical magnifying by the microscope,—and by the perfection to which that engineer has brought that art of accurate workmanship which is necessary in order to render mechanical magnifying possible.

Amongst measuring machines are included the platometers or planimeters of Sang, Morin, and Clerk Maxwell, which measure areas by means of mechanism. The amount of resistance in a measuring machine should be perfectly uniform, and sufficiently great to prevent accidental forces from disturbing the machine, without being so great as to render it inconveniently stiff. To combine these objects requires great accuracy of workmanship, together with strength and rigidity in the structure of the frame and mechanism.

141. (C.) Copying and Drawing Machines. — In copying machines for enlarging or reducing drawings there is usually a combination of levers and linkwork connecting a tracing-point, which is moved over the lines of the original figure, with a drawing-point, which draws the copy in such a manner that the velocity ratio of their motions is a given constant quantity, and that the directions of their motions make a constant angle.

Mechanism depending for its principle on the theory of the composition of rotations is used to draw ellipses, epicycloids, epitrochoids, and other curves.

142. (D. ) Weighing Machines. — In weighing machines the motion of the mechanism is used only for the purpose that its cessation, or its becoming an oscillation about a certain position, may indicate the equilibrium of the forces applied to the machine. Those forces may either be weights, which are to be compared with each other, or forces of other kinds, to be compared directly or indirectly with weights.

The machine for comparing weights which is capable of the most minute accuracy is also the simplest, being the balance, in which the equality of two weights is ascertained by their balancing each other at the ends of a lever of equal arms. In the steelyard, consisting either of one lever or of a train of levers, the unknown weight has an unchangeable point of application, and is compared with a known weight by shifting the latter along the lever to v hich it is applied until the machine is balanced; the ratio of the weights is then the reciprocal of the velocity ratio of their points of application. The steelyard is more convenient for weighing very heavy loads than the balance, but is not capable of such minute accuracy.

It is essential to accuracy in balances and steelyards that the friction should be less than the smallest admissible amount of error. To diminish : the friction as much as possible, the axes of motion are all knife-edges, as they are termed, of steel or hardened ircn, resting on hard surfaces of hardened iron or steel for ordinary purposes, and of some hard mineral, such as agate, for scientific purposes.

The weight of a∙ column of fluid is determined by balancing it against a column of fluid λvhose weight is known, as in the barometer, where the weight of a column of the atmosphere is balanced against that of a column of mercury.

Weights are compared with each other indirectly, and other forces compared with weights, by means of their effects in bending a spring,—a convenient method, but not susceptible of minute accuracy.

The elastic pressure exerted by a fluid may be compared with weight, either by balancing the pressure against the weight of a column of a liquid, or by maintaining a piston in equilibrium against that pressure, by means of a weight pressing it directly, or of a weight acting through a steelyard, or of the elasticity of a spring which has been compared with weights.

143. (E.) Recording Machines.— Recording machines may be divided into two classes :— self-registering instruments, which, by the aid of clockwork, record measurements either of space or of force, together with the instants of time at which these measurements were made; and dynamometers, already mentioned in Chap. II. of this article, which register measurements of force, together with the space through which it has acted, thus recording energy or work.

144. Working Machines Classed.— The object or purpose of working machines is to perform useful work; and their classification relatively to their objects and purposes is founded on the kind of useful work which they perform. In this point of view they may be classed as follows :— [table]

It is not pretended that the above classification (taken to a considerable extent from the writings of Young and of Babbage) exhausts all kinds of machines; it is brought forward merely as an attempt to introduce method to a certain extent into a subject which would otherwise be exceedingly confused.

145. (A.) Machines for Lifting and Lowering Solids.— the most common machines of this class are capstans, cranes, and windlasses.

They are usually worked by manual labour, but sometimes by hydraulic engines, or by steam-engines. The useful resistance, when a load is lifted, being the weight of that load, is in general greater than the effort exerted by the prime mover, so that the mechanism has to be adapted to giving the working-piece a less velocity than the piece to which the effort is applied. In lowering solid loads the weight of the load acts as the effort, and the energy exerted by it is expended in overcoming the friction of a brake in order that the speed of descent may not be excessive.

146. (B. ) Transporting Machines.— The mechanism of transporting machines consists of two parts :—that by which the resistance is diminished, as the wheels and a^les of vehicles; and that by which the resistance is overcome and the load propelled, comprising all kinds of locomotive and propelling machinery. In the present work transporting machines are treated of in the articles relating specially to the lines of conveyance to which they are applied.

147. (C. ) Machines for Projecting Solids. — This class comprehends all kinds of artillery.

148. (D.) Machines for Lifting Fluids. — See Hydromechanics.

149. (E. ) Machines for Propelling or Projecting Fluids.— See the same article.

150. (F.) Machines for Dividing Bodies.— This class comprehends all machines for separating solid masses into parts, whether by digging, cutting, sawing, grinding, tearing, crushing, pounding, pressing out fluids, or otherwise; and whether applied to earth, stones, metals, timber, fruit, grain, fibres, or other materials.

151. (G.) Machines for Shaping Bodies by Removing Portions of them.— This class of machines to a certain extent resembles the preceding. It includes machines for cutting, grinding, and polishing blocks of stone into required figures, and for shaping pieces of wood, metal, or other material, whether by turning, to produce spherical, cylindrical, and other curved surfaces,—by boring, punching, slotting, or gouging, to produce cylindrical, rectangular, or other orifices and grooves,—by screw-cutting, by planing, by grinding and polishing, to produce curved or plane surfaces. The most difficult and important of all these operations is to produce a surface truly plane; and the perfecting of this operation by Whitworth is the most important step recently made in Constructive Mechanics, or the art of making machines and instruments. Next in point of difficulty may be placed the art of forming the concave reflecting surfaces of great specula for telescopes, such as those of the Herschels, of Lassell, and of Lord Kosse.

152. (H. ) Machines for Shaping Bodies by Presslire comprehend, amongst others,' rolling-mills for iron, steam-hammers, wiredrawing machines, pinmaking and nailmaking machines, coining and other stamping machinery, brickmaking machines, presses for packing and compressing, &c.

153. (I.) Machines for Uniting Bodies into Fabrics comprise spinning machinery, whether applied to ropes, yarn, or thread, weaving machinery of all kinds, papermaking machinery, felting machinery, and sewing machinery.

151. (J.) Machines for Printing are used to apply either colouring matters or matters for discharging colour to paper, cloth, and other materials.

155. (K. ) Machines for producing Sound. — See Acoustics and Music.

156. (I..) Miscellaneous Machines.— There are numerous machines which perform processes, especially in the preparation of textile fabrics for the market, which it would be almost impossible to class. Examples of such machines will be found by referring to the articles relating to the various branches of manufacture.

Chapter IV. Applied Energetics, or Theory of Prime Movers.

157. Prime Movers in general—Their Efficiency.— Prime movers, or receivers of power, are those pieces or combinations of pieces of mechanism which receive motion and force directly from some natural source of energy. The point where the mechanism belonging to the prime mover ends and that belonging to the train for modifying the force and motion begins is somewhat arbitrary; in general, however, the mechanism belonging to the prime mover may be held to include all pieces which regulate or assist in regulating the transmission of energy from the source of energy. Thus, in the ordinary rotative steam-engine, the crank-shaft belongs to the prime mover, because it carries the eccentric which moves the valves and the fly-wheel which stores and restores the periodical excess of energy of the engine, and drives the governor (when there is one) which regulates the admission of steam.

The usefibl work of the prime mover is the energy exerted by it upon that piece which it directly drives; and the ratio which this bears to the energy exerted by the source of energy is the efficiency of the prime mover.

It is often convenient to divide the prime mover into sections, and resolve its efficiency into factors, each factor being the efficiency of one of those sections. Thus the efficiency of a steam-engine may be resolved into the following factors :—

Efficiency of the furnace and boiler,— being the proportion of the total heat of combustion of the fuel which takes effect in heating and evaporating the water.

Efficiency of the steam in driving the piston,— being the proportion of the energy exerted by the steam on the piston (called the indicated energy or power, as being measured by an indicator) to the mechanical equivalent of the heat received by the water.

Efficiency of the mechanism from the piston to the crank-shaft inclusive,— being the proportion of the effective energy transmitted by the crank-shaft to the indicated energy.

The product of those three factors is the efficiency of the engine as a whole.

In all prime movers the loss of energy may be distinguished into two parts,—one being the unavoidable effect of the circumstances under which the machine necessarily works in the case under consideration; the other the effect of causes which are, or may be, capable of indefinite diminution by practical improvements. Those two parts may be distinguished as necessary loss and waste.

The efficiency which a prime mover would have under given circumstances if the waste of energy were altogether prevented, and the loss reduced to necessary loss alone, is called the maximum or the theoretical efficiency under the given circumstances.

For some prime movers there is a combination of circumstances which makes the theoretical efficiency greater than any other combination does. The theoretical efficiency under those circumstances is the absolute maximum efficiency.

The duty of a prime mover is its useful work in some given unit of time, as a second, a minute, an hour, a day. In some cases, such as that of the work of animals, the duty can be ascertained, while the efficiency can only be inferred indirectly or conjecturally from the want of precise data as to the whole energy expended.

158. Sources of Energy Classed.— The sources of energy used in practice may be classed as follows :— [table]

159. (A.) Strength of Men and Animals.— The mechanical daily duty of a man or of a beast is the product of three quantities—the effort, the velocity, and the number of units of time per day during which work is continued. It is well known that for each individual man or animal there is a certain set of values of those three quantities which make their product, the daily duty, a maximum, and that any departure from those values diminishes the daily duty. Attempts have been made to represent by a formula the law of this diminution; but they have met with imperfect success. That which agrees on the whole best with the facts is the formula of Maschek, which is as follows :—let P 1 be the effort, V j the velocity, and T 1 the time of working per day, which give the maximum daily duty, and let P, V, T, be any other set of values of those quantities; then

PVT

⅞÷v 1 + ⅞- 3 < 86 >∙

One consequence of this formula is, that the best time of working per day for men, and for all animals, is one-third part of a day, or eight hours,—a conclusion in accordance with experience.

The best effort P 1 , and the best velocity V 1 , are much less certain,— the difficulty of determining their true mean values for particular species being rendered very great by the differences, not only between individuals, but between races or varieties of the same species. The following table of values is proposed by Maschek as approximately true :— [table]

Of the numbers in this table those for the draught horse are probably the most accurate. For the thoroughbred horse it is certain that the value of V 1 is much greater, and that of P 1 much less, than for the draught horse,—the effect being probably that the maximum daily duty P 1 V 1 T 1 is nearly the same; but experimental data are wanting to determine these quantities with precision.

The following table, chiefly extracted from the works of Poncelet and Morin, with the addition of some results of experiments by Lieutenant David Rankine and by the author of this article, shows the daily duty of men and horses under certain specified circumstances :—

[table]

160. Horizontal Transport.— When men and animals carry burdens, or draw or propel loads in certain vehicles, it is difficult, and sometimes impossible, to determine the duty performed in footpounds of work, because of the uncertainty of the amount in pounds of the resistance overcome. In this ease, for the purpose of comparing performances of the same kind with each other, a unit is employed called a foot-pound of horizontal transport, meaning the conveying of a load of 1 pound 1 foot horizontally. The following table, compiled from the sources referred to in sect. 159, gives some examples of the daily duty of men and horses in units of horizontal transport, I. denoting the load in lb, V the velocity in feet per second, and T the number of seconds per day of working:— [table]

161. (B.) Weight of Liquids.— (C.) Motion of Fluids.— In waterwheels and other hydraulic engines the weight and motion of 8 liquid usually act together as sources of energy.

To determine the necessary loss of energy and the theoretical efficiency, we have the following formulae:— [table]

where Q denotes the weight of liquid which acts on the wheel or other engine per second; H the vertical fall from the point where the liquid first begins to act directly or indirectly on the wheel or other engine to the point where it ceases to act; V 1 the velocity of the liquid when it begins to act; and V 2 the least velocity, ¼ hen it ceases to act, which will properly discharge the liquid, and prevent its accumulating so as to impede the wheel or engine.

(For details as to the actual efficiency and duty and the construction of hydraulic engines, see Hydkomechanics.)

In windmills, the air, being in motion, presses against and moves four or five radiating vanes or sails, whose surfaces are approximately helical, their axis of rotation being parallel, or slightly inclined in a vertical plane, to the direction of the wind. The best form and proportions for windmill sails, as determined experimentally by Smeaton, are as follows (see fig. 36):—

Angle of each sail with the plane of rotation—at I)E = 18°;

Do. do. do. at BC= 7°;

OD = J of whip OA; bar DE = ∣OA; bar BC = ⅞OA; AC^DE.

162. (D.) Heat. — In sect. 157 the three factors into which the efficiency of an engine moved by heat can be resolved have already been stated. The efficiency of the furnace and boiler in steam-engines varies from 0 '4 to 0 - 85. It may be considered that the loss of heat to the extent of 0’15 by the chimney is necessary in order to produce a sufficient draught; any loss beyond this is waste.

The theoretical efficiency of the steam, or other elastic fluid, which serves as the mechanism for converting heat into mechanical energy, is regulated by a law which will now be explained.

Heat acts on bodies in two ways—to elevate temperature and make the bodies hotter, and to produce mechanical changes. Heat employed in producing mechanical changes disappears or becomes latent, as it is termed, and can be reproduced by reversing those mechanical changes. When a cycle of mechanical changes, ending by the restoration of the body to its original condition, produces mechanical energy, heat disappears to an amount equal to that which would be generated by employing the mechanical energy in overcoming friction,—that is to say, a British unit of heat (or one degree Fahr, in one fb of liquid water) for every 772 foot-pounds of energy (being the constant already mentioned in sect.'115 as Joule’s equivalent'). This is called the conversion of heat into mechanical energy.

The efliciency of the fluid in a heat-engine is the proportion which the heat converted into mechanical energy bears to the whole heat received by the water or other fluid; and the theoretical or maximum value of that efficiency depends solely upon the respective temperatures at which the fluid receives heat and rejects the unconverted heat, according to the following law;—let t 1 represent the temperature at which the fluid receives heat, and t 2 the temperature at which it rejects the unconverted heat, as measured from the absolute zero, — that is, from a point 493° , 2 Fahr, or 274° C. below the temperature of melting ice (temperatures so measured are called absolute temperatures) ; then the maximum theoretical efliciency of the water or other fluid in a steam-engine or other heat-engine is t J~ t s (88).

The necessary loss of heat by the fluid is t. i ∕t i of the whole heat received by it; and any loss beyond this is waste.

The theoretical efficiency of the steam in ordinary steam-engines seldom exceeds ⅜th; the greatest actual efficiency is about ⅜th; the efficiency in good ordinary engines is about 0 - l or 0 08, and in bad and wasteful engines 0Ό4, or even less. (For details see Steam-Engine.)

163. (E.) Electricity and Magnetism.— Electricity developed by chemical action in a galvanic battery has been to a small extent used to produce mechanical energy by alternately magnetizing and unmagnetizing soft-iron bars.

The data for determining the actual efficiency of such engines are deficient. Their theoretical efficiency depends on the following law demonstrated by Joule :—

Let γ 1 denote the strength of the electric current which would be developed in the conducting wire of the battery if there were no iron bar to be magnetized; γ 2 the strength to which the current is reduced by the reaction of the iron bar, tending to induce a contrary current. Then the theoretical efficiency of the engine is

71-72 (89).

7ι

The proportion of the energy expended which is necessarily lost is 7,∕7 1 , and is employed in producing heat in the conducting cn cuit.

This law is exactly analogous to that of the theoretical efliciency of heat-engines given in equation 88.

There is reason to believe that electromagnetic engines are capable of a higher efficiency than heat-engines; but the greater cost of the materials consumed renders them much less economical commercially. (W.J. Μ. R.)