PIERRE FRANCOIS ANDRE, a practical astronomer and geographer, was born at Laon on the 16th April 1744. His father, who was an architect, educated him for his own profession, but he accidentally became acquainted with Lalande, under whose patronage he was subsequently brought forward as an observer, surveyor, and computer. He made two voyages with M. de la Bretonnière, and assisted him in surveying some parts of the coast of France. He was afterwards employed in various computations by the Marquis de Chabert and the Duc D'Ayen. Having obtained a prize from the Academy of Sciences in 1782, for a Memoir on Comets, he became a member of the Academy the same year. About the year 1785 he undertook the publication of the Connaissance des Temps, and continued it till he was employed in geodetical operations at a distance from Paris. He was appointed member of a committee, along with Cassini, De Thury, and Legendre, to meet the English astronomers for the determination of the relative situation of the observatories which had been proposed by Cassini. It was in these operations that he first brought Borda's circle into general use. In 1791 he was appointed, in conjunction with Delambre, to execute the intentions of the Constituent Assembly with regard to the determination of a basis of linear measures. He was made director of the observatory of Paris, and he entered with great zeal on a series of observations which were to rival those of Flamsteed, Bradley, and Maskelyne; but he afterwards solicited the appointment to assist in the measurements required for the still farther extension of the arc of the meridian to the S. of Barcelona. But the secret motive for his seeking this humbler employment appears to have been a desire to remove some doubts which he entertained respecting the latitude of Barcelona, as it appeared after his death from his papers that there had been a discordance of 3° in some previous observations. Shipwreck and disease awaited him, however, and he died of fever on the 20th September 1805.

Of his publications the most important are to be found in the Memoires des Savans étrangers; also a Memoir on the Comets of 1532 and 1661, which gained the prize. In the Memoirs of the Academy, and in the Connaissance des Temps, from 1782 to 1785, there are several of his observations. Mechanics.

1. Mechanics, Applicate or Applied, is a term which, strictly speaking, includes all applications of the principles of abstract mechanics to human art. As in other branches of knowledge, so in mechanics, art is more ancient than science, many mechanical inventions having been introduced into practice ages before the general principles of their action were discovered; but also, as in other branches of knowledge, so in mechanics, science is necessary to the perfecting of art, and mechanical inventions have in all cases continued rude, wasteful, and inefficient, until their dependence on general principles has been understood. Thus have theory and practice in all ages promoted each others' advancement; and the greatest obstacle to the advancement of both has always been a popular and scholastic fallacy that they are inconsistent. Happily that fallacy is now disappearing, and its occurrence in the writings of any author may be considered as a mark either of ignorance, or of the inconsiderate use of words.

PART I.—OUTLINE OF THE THEORY OF STRUCTURES.

3. 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 the science of Hydrodynamics, and of the arts of Aeronautics and Naval Architecture, to the articles under which titles the reader is referred. 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 Transactions 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; \( \phi \) the angle of repose of the earth; then

\[ \frac{S}{E} = \left( \frac{1 + \sin \phi}{1 - \sin \phi} \right)^2. \]

To apply this to liquids, \( \phi \) must be made \( = 0 \), and then \( \frac{S}{E} = 1 \), as is well known.

4. Composition of a Structure, and Connection of its Pieces.—A structure is composed of pieces, such as the stones of a building in masonry, the beams of a timber framework, 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 treenails, pins, spikes, nails, holdfasts, screws, bolts, rivets, hoops, straps, and sockets).

5. 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.

6. 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.

7. 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 Mr Moseley, that the smallest of those alternative systems is that which will actually be ex-

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For the exposition of the principles of abstract mechanics, see the articles Dynamics and Statics. Mechanics.

Theorem.—If from the angles of the polygon of loads there be drawn lines (R₁, R₂, &c.), each of which is parallel to the resistance (as P₁P₂, &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 L₁, L₂, &c.) are applied; then will all those lines meet in one point (O), 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.

(The particular case of this theorem which relates to a system of parallel vertical loads, has long been known; but in its general form the theorem is now published in type for the first time, having appeared originally in a lithographed abstract of some lectures.)

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 O.

9. 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 equal 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.

10. 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 8 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 Mr Moseley's principle of the least resistance, the whole of the relations amongst the loads and resistances may be found.

11. 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, how great soever their number. The difference between the line of resistance and the line of pressures was first pointed out by Mr Moseley.

12. Stability of Position, and Stability of Friction.—The resistances at the several joints having been determined by the principles set forth in sections 7, 8, 9, 10, and 11, not only under the ordinary load of the structure, but... Mechanics 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.

13. 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 Mr 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:

- Retaining walls, as designed by British engineers - Retaining walls, as designed by French engineers - Rectangular piers of bridges and other buildings, and arch-stones - Rectangular foundations, firm ground - Rectangular foundations, very soft ground - Rectangular foundations, intermediate kinds of ground - Thin, hollow towers (such as furnace chimneys exposed to high winds), square - Thin, hollow towers, circular

(In the last two cases, the depth of the joint is to be understood to mean the diameter of the tower.)

Frames of timber or metal, under their ordinary or average distribution of load Frames of timber or metal, under the greatest irregularities of load

14. 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

\[ CP = CR \cdot \cos \angle PCR; \]

and the component tangential to the joint is

\[ CQ = CR \cdot \sin \angle PCR = CP \cdot \tan \angle PCR. \]

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

\[ CQ < f \cdot CP, \]

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

\[ \phi = \arctan f. \]

Now \( CQ = CP \cdot \tan \angle PCR; \)

consequently the condition of stability of friction is fulfilled if

\[ \angle PCR < \phi; \]

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.

15. 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. 3.

16. Parallel Projections of Figures.—If any figure be referred to a system of co-ordinates, rectangular or oblique; and if a second figure be constructed by means of a second system of co-ordinates, rectangular or oblique, and either agreeing with or differing from the first system in rectangularity or obliquity, but so related to the co-ordinates 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 co-ordinates shall bear, respectively, to the three corresponding co-ordinates of the corresponding point in the first figure, three ratios which are the same for every pair of corresponding points in the two figures; that pair of 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; (6.) That a parallel projection of a parallelopiped is a parallelopiped. (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 Mechanics, 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 co-ordinates parallel to any three conjugate diameters are parallel projections of each other, and of a sphere referred to rectangular co-ordinates.

17. 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.

18. Principle of the Transformation of Structures.—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 such 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 Mechanics, the before-mentioned 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 here published in its most general form for the first time in type, having previously appeared only in the lithographed abstract of some lectures. 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 projection from those of arches fitted to resist the thrust of a liquid, which is of equal intensity, horizontally and vertically.

19. 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 now referred.

PART II.—THEORY OF MACHINES.

20.—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 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.

21. 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 Mechanics, reference to the action of machines considered as moving.

In the action of a machine the following three things take place:

Firstly, Some natural source of energy communicates motion and force to a piece or pieces of the mechanism, called the receiver of power, 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 a branch of geometry called Cinematics, 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 Cinematics from Dynamics is due mainly to Monge, Ampère, and Professor Willis.

The theory of machines in the present article will be considered under the following four heads:

I. Pure Mechanism, or Applied Cinematics; 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.

22. Division of the subject.—Proceeding in the order of simplicity, the subject of Pure Mechanism, or Applied Cinematics, may be thus divided:

Division 1. Motion of a point.

Division 2. Motion of the surface of a fluid.

Division 3. Motion of a rigid solid.

Division 4. Motions of a pair of connected pieces, or of an "elementary combination" in mechanism.

Division 5. Motions of trains of pieces of mechanism.

Division 6. Motions of sets of more than two connected pieces, or of "aggregate combinations."

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.

23. Path and Direction.—The motion of a point is the change of its place relatively to some portion of space, which for the time is considered as fixed. In the theory of machines, the portion of space which is considered as fixed Mechanics, is usually that which is occupied by the frame of the machine. A moving point traces in the fixed space a line called its path, which may be straight or curved. The direction of the motion of the point at any instant is that of a tangent to the path drawn forwards from the position of the point at the instant in question, and is uniform if the path is straight—variable if it is curved.

24. Uniform Velocity.—The velocity or speed of a moving point is the length of the portion of its path which it traces in some given portion or unit of time. In calculations respecting dynamical questions, the unit of time commonly employed is the second; in considering the purposes of machines, the minute, the hour, and the day are also employed. When not otherwise specified, the second is the unit of time employed in this article. When a point continues always to trace equal lengths of its path in equal times, its velocity is said to be uniform, and is expressed numerically by dividing the length of any portion whatsoever of the path of the point by the time occupied by the point in tracing it. The unit of length employed in Britain to express velocities is usually the foot, but in some cases the mile. In the present article the foot is employed, where not otherwise specified; so that velocities will generally be stated in feet per second.

25. Varied Velocity.—When a moving point does not trace equal lengths of its path in equal times, its velocity is said to be varied. In this case it is no longer possible to compute the velocity simply by dividing the length traced between two given instants by the time occupied in tracing it, because such a computation gives different results for different pairs of instants: it gives, not the exact velocity at a particular instant, but the mean velocity between the pair of instants.

For example, let A, B denote a pair of instants of time, Δt the interval of time between them, Δs the length of path traced by the moving point in that interval, then is $\frac{\Delta s}{\Delta t}$ the mean velocity in the interval between A and B.

Now let C denote an instant of time anywhere between A and B, and let it be required to find the exact velocity of the moving point at the instant C. $\frac{\Delta s}{\Delta t}$ will be an approximation to that velocity. To find a closer approximation, take two instants A' and B', nearer to C than A and B are; let Δt' be the interval of time, and Δs' the length traced between A' and B'; then will $\frac{\Delta s'}{\Delta t'}$ be a closer approximation than $\frac{\Delta s}{\Delta t}$. Having obtained a series of such approximations,

$$\frac{\Delta s}{\Delta t}, \frac{\Delta s'}{\Delta t'}, \frac{\Delta s''}{\Delta t''}, \ldots$$

by taking pairs of instants continually nearer and nearer to C, compare their results; it will be found that they follow some law, and that each of the mean velocities denoted above consists of a constant part, and of a variable part which continually diminishes and approximates to nothing as Δt grows smaller; that is to say, as A and B approximate to C. The constant part of the mean velocity, being the limit towards which $\frac{\Delta s}{\Delta t}$ approximates as Δt diminishes, is the exact velocity at the instant C, and is thus denoted,

$$v = \frac{ds}{dt} \quad \text{(L)}$$

The process above described is well known in the differential calculus by the name of differentiation; that is to say, finding the rate of variation of one quantity s as compared with that of another quantity t.

In order to distinguish between motions in two opposite directions along the same path, one of those directions is Mechanics, called forward, and treated as positive in algebra; while the other is called backward, and treated as negative. Thus, let \( n \) denote a certain number of feet per second; then

\[ v = \frac{ds}{dt} = n, \quad \text{or} \quad +n, \]

will express that the velocity of the moving point is \( n \) feet per second forwards; and

\[ v = \frac{ds}{dt} = -n \]

will express that the velocity is of equal magnitude, but backward.

26. Direct Deviation, or Acceleration and Retardation.—When the motion of a point is said to be uniform, without further qualification, it is to be understood to be uniform both in velocity and in direction.

The word deviation in its general sense denotes the amount of any departure from uniformity of motion, whether of velocity or of direction; and the rate at which any deviation takes place is ascertained by differentiation.

The rate of direct deviation is the rate at which the velocity varies, and is denoted by

\[ \frac{dv}{dt}, \quad \frac{ds}{dt} \]

being deduced from the changes of velocity in different intervals of time in the same way that the velocity is deduced from the changes of position; that is, by differentiation. Direct deviation is acceleration if the velocity is increased, retardation if it is diminished; it is positive if forward velocity is increased, or backward velocity diminished; negative if forward velocity is diminished, or backward velocity increased.

27. Lateral Deviation or Deflection—Angular Velocity of Deviation—Revolution.—The rate of lateral deviation, being the rate at which the moving point swerves from a straight course, is found for any instant by multiplying the velocity of the moving point by the rate at which the angular direction of its path varies. Symbolically, let \( \theta \) denote the angle (expressed in length of arc to radius unity) which at any instant the path of the point makes with some fixed direction in the plane in which, for the instant, the deflection takes place; then \( \frac{d\theta}{dt} \), found as before, is the rate at which the angular direction of the path varies. Let \( p \) denote the radius of curvature of the path at the instant in question; then, by the geometry of curves, it is known that

\[ \frac{d\theta}{dt} = \frac{1}{p}, \]

consequently,

\[ \frac{d\theta}{dt} = \frac{ds}{dt} = \frac{v}{p}, \]

and the rate of lateral deviation is expressed by

\[ \frac{d\theta}{dt} = \frac{v^2}{p} = \left( \frac{d\theta}{dt} \right)^2. \]

Lateral deviation is considered as positive or negative, according to arbitrary arrangements respecting the sign of the radius of curvature of the path.

The quantity above represented by \( \frac{d\theta}{dt} \) is sometimes called the angular velocity of deviation.

A point whose path deviates continually until it returns to its primitive direction is said to revolve; and the mean angular velocity of revolution is the circumference of a circle of the radius unity, divided by the time of one revolution.

28. 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.

Mr 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.

29. Resolution and Composition of Motion.—Let OPQ be part of the path of a moving point, and P the position of that point at any given instant. Let OX, OY, OZ, be any three axes, or fixed directions, traversing a fixed point O called the origin. Let three points, \( P_x, P_y, P_z \), start from O at the same instant with \( P \), moving respectively along the straight paths OX, OY, OZ, in such a manner that at each instant

\[ OP_x = x, \quad OP_y = y, \quad OP_z = z, \]

shall be respectively equal to

\[ PA, \quad PB, \quad PC, \]

the respective distances of \( P \) from the co-ordinate planes YOZ, ZOX, XOY, measured respectively in directions parallel to the three axes, OX, OY, OZ.

Then are \( P_x, P_y, P_z \), called the projections of \( P \) on the three axes respectively; their motions are called the components of the motion of \( P \) parallel to the three axes; and the operation of determining those component motions is called the resolution of the motion of \( P \) relatively to the three axes. The operation of determining the motion of \( P \), when its components are given, is called the composition of those components; and the motion of \( P \) is called the resultant of the motions of its projections. The following propositions are the main principles of the composition and resolution of motion:

(1.) The straight line joining \( P \) and O is the diagonal of the parallelopiped \( OP_xP_yP_zABC \), of which \( x, y, z \), are the edges.

(2.) If a straight line be drawn representing in direction and magnitude the velocity \( v = \frac{ds}{dt} \) of \( P \) at any instant, that straight line will be the diagonal of the parallelopiped whose edges respectively represent in direction and magnitude the component velocities of \( P_x, P_y, P_z \); viz.,

\[ \frac{dz}{dt}, \quad \frac{dy}{dt}, \quad \frac{dx}{dt}. \]

(3.) If the three respective motions of \( P_x, P_y, P_z \), during one and the same interval of time, be determined, and if a moving point, starting from O, be made to perform in three Mechanics. consecutive intervals of time, motions equal and parallel to those three motions respectively, in any order of succession, then will the last-mentioned point arrive at the position occupied by P at the end of the first-mentioned interval of time.

(The path of the last-mentioned point may be any one of the six routes from O to P along the edges of the parallelopiped.)

(4.) If a plane always parallel to one of the co-ordinate planes (such as YOZ) be made to start from O at the same instant with P, and to move with the same motion as the projection of P on one of the axes (such as P_x); and if a line in that moving plane, always parallel to another of the axes (such as OZ), be made to start from O, and move relatively to that plane with a motion parallel and equal to that of the projection of P on the remaining axis (such as P_y); and if a point in that moving line be made to start from O, and move along that line with a motion parallel and equal to that of the third projection of P (such as P_z); then will this moving point coincide at every instant with P itself.

For example, let a plane start from the position AP_xOP_y and move along with P_x to the position PB_xC; let a line in that plane start from the position OP_y, and move with a motion relatively to the plane which is equal and parallel to that of P_y, so as to arrive at the position CP_z; let a point on that line start from O, and move along the line with a motion equal and parallel to that of P_z; that point will be the point P itself.

The last-mentioned illustration of the resolution and composition of motion is capable of being exhibited by mechanism.

When the motion of P takes place entirely in one plane, two axes may be assumed in that plane; when the motion of P will have but two components along those two axes respectively, and the parallelopiped will be reduced to a parallelogram.

30. Rectangular Projection, Resolution, and Composition.—The choice of the three or two axes is a matter of convenience. When there is no special reason for making them oblique, they are made perpendicular to each other, or rectangular; and when obliquity of the axes is not specified, rectangularity is to be understood. In this case the projection of a moving point upon any one of the axes is found by letting fall a perpendicular from the point on that axis.

Let OP = r denote the distance of the point from the origin at any given instant, and x = XOX, the angle which that distance makes with any given axis; then for rectangular projection,

\[ x = \overline{OP} = r \cos x \]

Also let v be the velocity of P at any given instant, and \( x_v \) the angle which the direction of its motion, or tangent to its path, makes at the same instant with the direction of OX; then the velocity of the rectangular projection P_x at the same instant is

\[ \frac{dx}{dt} = v \cos x_v \]

Adopting an analogous notation for other two rectangular axes, we have—

\[ \begin{align*} \cos^2 x + \cos^2 y + \cos^2 z &= 1; \\ \cos^2 x_v + \cos^2 y_v + \cos^2 z_v &= 1; \\ \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2 &= v^2. \end{align*} \]

31. Resolution and Composition of Deviations.—As the Mechanics, three projections of a moving point move in straight lines, their rates of deviation, if any, are wholly direct, and are represented by

\[ \frac{dx}{dt}, \quad \frac{dy}{dt}, \quad \frac{dz}{dt}. \]

If three lines be drawn representing those three rates of deviation in direction and magnitude, the diagonal of the parallelopiped of which they are the edges is the total or resultant rate of deviation, and is the same in magnitude and direction with the diagonal of a rectangular parallelogram one of whose sides is parallel to the direction of motion of P, and represents in magnitude its rate of direct deviation \( \frac{dx}{dt} \), while the other is drawn in the direction of the radius of curvature of the path of P, and represents in magnitude its rate of lateral deviation \( \frac{v^2}{\rho} \). This is easily verified by the aid of well-known formulæ of the geometry of curved lines. If the motion of a point be uniform both in velocity and in direction, the components of that motion are uniform also; but if the motion of the point be deviated either in velocity or in direction, or in both, the components are deviated in velocity.

DIVISION II.—MOTION OF THE SURFACE OF A FLUID MASS.

32. General Principle.—A mass of fluid is used in mechanism to transmit motion and force between two or more moveable 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 v 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

\[ \sum va = 0 \]

(7.)

33. Comparative Motion of two Pistons.—If there be but two pistons whose areas are a_1 and a_2, and their velocities v_1 and v_2, their comparative motion is expressed by the equation—

\[ \frac{v_1}{a_1} = -\frac{v_2}{a_2}; \]

(8.)

that is to say, their velocities are opposite as to inwardness and outwardness, and inversely proportional to their areas.

34. 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 inclosed 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 press-plunger. (For details respecting this machine, &c., see HYDRODYNAMICS.)

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 Mechanics. When the pressure and temperature of the air can be maintained constant, this machine fulfils equation 8, 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, and are foreign to the subject of pure mechanism.

DIVISION III.—MOTION OF A RIGID SOLID.

35. Motions Classed: Shifting and Turning.—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, when every point in the body has the same motion at the same instant, and when consequently no point in the body has any motion relatively to another.

(2.) Turning or Rotation, being the only kind of relative motion amongst the points of a body which is possible without change of dimensions or figure.

(3.) Motions compounded of shifting and turning.

36. Shifting or Translation.—In a motion of simple shifting or translation all the points of the body are moving with the same velocity, and in the same direction, at the same instant. The path of a given point may be a line of any figure—straight, curved, or angular; and the path of every other point will be a line of the same figure and dimensions. Every plane and every straight line fixed in the body moves parallel to itself.

The relative motion of any two points is null; that is to say, the line joining them alters neither in length nor in direction.

The comparative motion of any two points is expressed by the ratio of equality between their velocities, and the relation of parallelism between their directions.

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 wheel-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.

37. Turning or Rotation in general.—Turning or rotation is a motion of a body such that a line fixed in or relatively to the body changes its direction. The plane of rotation is a plane parallel to those lines in the body whose change of direction is most rapid. As the rigidity of the body makes all lines in it preserve unchanged their directions relatively to each other, all lines in the body parallel to the plane of rotation change their direction by the same amount in the same interval of time. The rate of change of direction of such lines, which, if uniform, is the angle swept through by each of them in an unit of time, and, if variable, is the limit towards which that angle approximates in the operation of differentiation, is the angular velocity of rotation of the body, and is denoted by $\frac{d\theta}{dt}$, where $\theta$ is the angle made by a line fixed in the body with some fixed direction in the plane of rotation.

Each one of the parts of a rigid turning body, how small soever, turns in the same manner and in the same time with the whole body, and has obviously the same angular velocity of rotation with the whole body.

The direction of the axis of rotation is perpendicular to the plane of rotation, and is the direction of all those lines in the body whose rate of change of direction is null. This direction may either be permanent or instantaneous. It, or any line parallel to it, is sometimes called the axis simply; but the use of the word axis alone more commonly implies, in speaking of mechanism, a line in the body (such as the central line of an arbor or shaft) whose position as well as its direction is unchanged, either relatively to the frame (in which case it is a fixed axis), or relatively to some other piece of mechanism which carries the turning piece under consideration. When there is no fixed axis, the instantaneous axis is an ideal line, which at a given instant is at rest relatively to the turning body and to the frame.

Rotation is said to be right-handed relatively to an observer looking in the direction of the axis, when it is like that of the hands of a watch; left-handed when the contrary. Rotation may be alternately right and left handed, in which case it is called oscillation. Angular velocity of rotation may be deviated either by acceleration, by retardation, or by change in the direction of the axis.

38. Relative Motions of Two Points in a Turning Body.—Let A and B denote any two points in a turning body, and let it be required to express the motion of B relatively to A. Through A draw a line AO in the direction of the axis, on which let fall a perpendicular from B; let $r$ denote the length of that perpendicular. Then is the motion of B relatively to A a translation in a circular path of the radius $r$ about the axis AO; the angular velocity of deviation of B is the same with the angular velocity of rotation of the body $\frac{d\theta}{dt}$ and the linear velocity of B relatively to A is $\frac{d\theta}{dt} r$, in a direction perpendicular to the plane AOB.

The motion of A relatively to B is exactly the same with that of B relatively to A.

39. Comparative Motion of Two Points relatively to a Third.—If there be a point C at the perpendicular distance $r'$ from AO, the velocity of that point in its motion relatively to A will be $r' \frac{d\theta}{dt}$; and the comparative motion of B and C relatively to A will be expressed by the ratio $r : r'$ between their velocities, and by the angle $\frac{\Delta}{rr'}$ between their directions of motion, being the same with the angle between the planes AOB and AOC. Thus that comparative motion is independent of the angular velocity of the turning body.

40. 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.

Let the angular velocity of the rotation be denoted by then the linear velocity 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.

If the directions of motion of any two points A and B in the rotating body, at any one instant, be given, the axis may be determined by its being the line of intersection of two planes drawn through A and B, and perpendicular to the respective directions of motion of those points.

The comparative motion of A and B is expressed, like that of B and C in sec. 39, by a constant velocity-ratio, which is that of their perpendicular distances from the axis, and a constant directional relation, which is an angle equal to the angle between the two planes traversing the axis and the two points respectively. 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 axes of turning, their velocity-ratio being that of their radii.

41. 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. 5 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

\[ \frac{a}{Fm} \]

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.

42. Instantaneous Axis of a Cylinder rolling on a Cylinder.—Let a cylinder bbb, whose axis of figure is B, and angular velocity \( \gamma \), roll on a fixed cylinder aaa, whose axis of figure is A, either outside (as in fig. 6), when the rolling will be towards the same hand with the rotation, or inside (as in fig. 7), 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 traversing 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 bbb 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

\[ v_p = \gamma \cdot PT; \]

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

\[ v_B = \gamma \cdot TB; \]

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 \( \alpha \) denote the angular velocity with which the plane of axes AB rotates about the fixed axis A. Then it is evident that

\[ v_B = \alpha \cdot AB; \]

and consequently that \( \alpha = \gamma \cdot \frac{TB}{AB} \).

For internal rolling, as in fig. 7, AB is to be treated as negative, which will give a negative value to \( \alpha \), 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—

\[ \beta = \gamma - \alpha; \]

whence \( \beta = \gamma \cdot \frac{TA}{AB} \);

care being taken to attend to the sign of \( \alpha \), so that when that is negative the arithmetical values of \( \gamma \) and \( \alpha \) are to be added in order to give that of \( \beta \).

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 cymoidal 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.

43. Composition and Resolution of Rotations about Parallel Axes.—The plane AB may be considered as an arm rotating about the fixed axis A with the angular velocity \( \alpha \), and carrying the moving axis B, round which a body bbb rotates with the angular velocity \( \beta \) relatively to the plane or arm AB; so that the actual rotation of bbb about the instantaneous axis T, with a total or actual angular velocity \( \gamma \), is the resultant of the two component rotations about A and B respectively.

The relations between the angular velocities of the component and resultant rotations, and the distances between their axes, are expressed by the following proportions deduced from equations 12 and 13:— Let \( r_c \) denote the linear velocity of the point C. Then

\[ r_c = a \cdot \overline{CF} = \gamma \cdot \overline{CG} \]

which is one part of the solution above stated. From E draw EH \( \perp \) OB, and EK \( \perp \) OA. Then it can be shown as before that

\[ EK : EH :: OC : OD. \]

Let \( r_E \) be the linear velocity of the point E fixed in the plane of axes AOB. Then

\[ r_E = a \cdot EK. \]

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,

\[ \beta : EH = r_E = a \cdot EK \]

\[ a : \beta :: EH : EK :: OD : OC, \]

which is the remainder of the solution. Q.E.D.

The path of a point P in or attached to the rolling cone is a spherical epicycloid 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

\[ v_P = \gamma \cdot PQ. \]

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.

46. Composition of Rotations about Two Axes meeting in a Point.—The plane AOB, carrying the moving axis OB, rotates about the fixed axis OA with an angular velocity \( \alpha \), to which \( \overline{OD} \) is proportional; the cone B, or any other body, rotates about the moving axis OB with an angular velocity \( \beta \), to which \( \overline{OC} \) is proportional; then the resultant angular velocity \( \gamma \) of the body B is proportional to, and the direction of the resultant rotation coincides with, the diagonal OE of the parallelogram OCED.

In drawing lines, such as \( \overline{OC}, \overline{OD}, \) and \( \overline{OE} \), to represent in direction the axes, and in magnitude the angular velocities of rotations, care is to be taken to make each line point from the origin O in such a direction that each rotation, looked at from the quarter towards which the line denoting it points, shall be of the same kind with all the rest, viz., right-handed or left-handed. Right-handed rotation is usually selected in applying this rule.

47. Screw-like or Helical Motion.—As it has been shown in section 44 that the combination of a rotation round a given axis, with a translation of that axis in a path parallel to the plane of rotation, is equivalent simply to a rotation round a different axis, it follows that every possible movement of a rigid body parallel to one plane is either a translation or a rotation; and that the only combination of translation and rotation, in which a complex movement which is not a mere rotation is produced, occurs when the translation is 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 right or 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. 9 is right-handed. Mechanics. The translation of a body in helical motion is called its advance. Let \( v_a \) denote the velocity of advance at a given instant, which of course is common to all the particles of the body; \( \alpha \) the angular velocity of the rotation at the same instant; \( 2\pi = 6.2832 \) nearly, the circumference of a circle of the radius unity. Then

\[ T = \frac{2\pi}{\alpha} \quad (19) \]

is the time of one turn at the rate \( \alpha \); and

\[ p = v_a T = \frac{2\pi r}{\alpha} \quad (20) \]

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

\[ v_r = \alpha r \quad (21) \]

is the component of the velocity of that point in a plane perpendicular to the axis, and its total velocity is

\[ v = \sqrt{v_r^2 + v_\theta^2} \quad (22) \]

The ratio of the two components of that velocity is

\[ \frac{v_r}{v_\theta} = \frac{p}{2\pi r} = \tan \theta \quad (23) \]

where \( \theta \) denotes the angle made by the helical path of the point with a plane perpendicular to the axis.

48. To find the Motion of a Rigid Body from the Motions of Three Points in it.—When the directions and velocities of the motions of any three points in a rigid body, not being in the same straight line, are given, the motion of the whole body is determined, and may be found.

Case 1. When the velocities of the three points are equal and parallel, the motion of the whole body is an equal and parallel translation.

Case 2. When the three points are in one plane, and their velocities are perpendicular to that plane and unequal.

Draw three lines from the three points representing in length and direction their velocities. Through the extremities of those lines, draw a plane which will intersect the plane of the three points in the axis of rotation. The velocity of any one of the points being divided by its distance from that axis will give the angular velocity.

Case 3. When the motions of the three points are parallel to one plane, but not to each other.—Through any two of the three points draw planes perpendicular to their directions of motion; those planes will cut each other in the axis of rotation; and the angular velocity may be found as in Case 2.

Case 4. In every other case the motion is helical.—Let A, B, C (fig. 10) be the three points, and let the lines \( v_{aA}, v_{bB}, v_{cC} \) represent their velocities. Through any point O draw

\( OA = a \parallel v_a, OB = b \parallel v_b, OC = c \parallel v_c \).

Through \( a, b, \) and \( c \), draw a plane, on which let fall the perpendicular \( Ox \). This will represent the velocity of translation or advance. Join \( xa, xb, xc \); these lines will represent in direction and magnitude the components of the velocities of \( A, B, C \) respectively, parallel to the plane of rotation. Through any two of the three points draw planes perpendicular to these rotatory components of mechanics, their velocities; these planes will intersect in the axis of rotation. Find the angular velocity as before.

DIVISION IV.—ELEMENTARY COMBINATIONS IN MECHANISM.

49. Definitions.—An elementary combination in mechanism consists of two pieces whose kinds of motion are determined by their connection with the frame, and their comparative motion by their connection with each other; that connection 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 connection 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 connection 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 connection 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 connection 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 connection, is null. There may be several, or an indefinite number of lines of connection, or there may be but one; and a line of connection may connect either the same pair of points or a succession of different pairs.

50. General Principle.—From the definition of a line of connection it follows, that the components of the velocities of a pair of connected points along their line of connection are equal. And from this, and from the property of a rigid body, already stated in sect. 41, it follows, that the components, along a line of connection, 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 connection, 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 connection.

The general principle, stated above in different forms, serves to solve every problem in which—the mode of connection of a pair of pieces being given—it is required to find their comparative motion at a given instant, or vice versa.

51. Application to a Pair of Shifting Pieces.—In fig. 11, let \( P_1P_2 \) be the line of connection of a pair of pieces, each of which has a motion \( v_1 \) 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 connection draw a plane perpendicular to that line, cutting \( TV_1, TV_2 \), in \( V_1, V_2 \); then, Mechanics. velocity of piece 1: velocity of piece 2 :: TV₁ : TV₂. Also, TA represents the equal components of the velocities of the pieces parallel to their line of connection, and the line V₁V₂ represents their velocity relatively to each other.

52. Application to a Pair of Turning Pieces.—Let \(a_1, a_2\) be the angular velocities of a pair of turning pieces; \(θ_1, θ_2\), the angles which their line of connection makes with their respective planes of rotation; \(r_1, r_2\), the common perpendiculars let fall from the line of connection upon the respective axes of rotation of the pieces. Then the equal components, along the line of connection, of the velocities of the points where those perpendiculars meet that line, are,

\[a_1r_1 \cdot \cos θ_1 = a_2r_2 \cdot \cos θ_2;\]

consequently, the comparative motion of the pieces is given by the equation

\[\frac{a_1}{a_2} = \frac{r_1 \cdot \cos θ_1}{r_2 \cdot \cos θ_2}; \quad \ldots \quad (24)\]

53. 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 \(a_i\) be the angular velocity of the turning piece, \(r_i\) the common perpendicular of its axis of rotation and the line of connection, \(θ_i\) the angle made by the line of connection with the plane of rotation, \(θ_s\) the angle made by the line of connection with the direction of motion of the shifting piece, \(v_s\) the linear velocity of that piece. Then

\[a_i r_i \cdot \cos θ_i = v_s \cdot \cos θ_s; \quad \ldots \quad (25)\]

which equation expresses the comparative motion of the two pieces.

54. 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 Bénaudcourt, 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 Mr Willis has pointed out in his admirable treatise On the Principles of Mechanism.

The classification of Mr Willis 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 connection 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 Mr Willis into five divisions, of which the characters are as follows:

Division A: connection by rolling contact.

B: sliding contact.

C: wrapping connectors.

D: link-work.

E: reduplication.

In the present article the principle of the classification of Mr Willis is followed; but the arrangement is modified by taking the mode of connection as the basis of the primary classification, and by removing the subject of connection 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 Encyclopedia; but it is not disputed that the original arrangement may be the best for a separate treatise.

55. 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 velocities 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 in 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 connection.

56. 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 inwards, or towards its centre.

From principle III. of sect. 55 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 Mechanics. 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 Mr 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 Mr 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. 12.)

Let \( C_1, C_2 \) be the poles of a pair of rolling curves; \( T_1, T_2 \) any pair of points of contact; \( U_1, U_2 \) any other pair of points of contact. Then, for every possible pair of points of contact, the two following equations must be simultaneously fulfilled:

\[ \text{Sum of radii}, \quad C_1U_1 + C_2U_2 = C_1T_1 + C_2T_2 = \text{constant}; \]

\[ \text{arc}, \quad T_1U_1 = T_2U_2. \quad \quad (26) \]

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 radius-vectors shall be equal and contrary; or, denoting by \( r_1, r_2 \) the radius-vectors 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—

\[ \frac{dr_2}{ds} = -\frac{dr_1}{ds}; \quad \quad (27) \]

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 Mr Willis's work, already mentioned, and Mr Clerk Maxwell's paper on Rolling Curves in the Transactions of the Royal Society of Edinburgh.]

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 radius-vector of a point of contact on the wheel, \( x_1 \) the ordinate from the straight line before mentioned to the corresponding point of contact on the rack. Then

\[ \frac{dx_1}{ds} = \frac{dr_1}{ds} \quad \quad (28) \]

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 of the ellipses, and the velocity-ratio varying from \( \frac{1 + \text{eccentricity}}{1 - \text{eccentricity}} \) to \( \frac{1 - \text{eccentricity}}{1 + \text{eccentricity}} \); a 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 \( \frac{\text{eccentricity} + 1}{\text{eccentricity} - 1} \) and unity; and a parabola rotating about its focus rolls with an equal and similar parabola, shifting parallel to its directrix.

57. Conical or Bevel, and Disc Wheels.—From principles III. and VI. of sect. 55 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 mechanics axes of the wheels make with the line of contact. Hence follows the following construction (figs. 13 and 14).—Let O be the apex or point of meeting of the two axes \( OC_1, OC_2 \). The angular-velocity-ratio being given, it is required to find the line of contact. On \( OC_1, OC_2 \) take lengths \( OA_1, OA_2 \), respectively proportional to the angular velocities of the pieces on whose axes they are taken. Complete the parallelogram \( OA_1EA_2 \); 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 \( C_1OC_2 \), 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 disc for its rolling surface, as \( W_1 \) in fig. 14, in which case it is a disc wheel. The rolling surfaces of actual wheels consist of frusta or zones of the complete cones or discs, as shown by \( W_2, W_3 \) in figs. 13 and 14.

58. Sliding Contact—lateral: Skew-Bevel Wheels.—A 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. 15, 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 neither be parallel nor will they intersect each other.

The motion of two such hyperboloids, turning in contact with each other, has hitherto been classed amongst cases of rolling contact; but that classification is not strictly correct; for although the component velocities of a pair of points of contact in a direction at right angles to the line of contact are equal, still, 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. 16, let \( B_1C_1, B_2C_2 \) be the two axes; \( B_1B_2 \) their common perpendicular. Through any point \( O \) in this common perpendicular draw \( OA_1 \) parallel to \( B_1C_1 \), \( OA_2 \) parallel to \( B_2C_2 \); make those lines proportional to the angular velocities about the axes to which they are respectively parallel; complete the parallelogram \( OA_1EA_2 \), and draw the diagonal \( OE \); divide \( B_1B_2 \) in D into two parts, inversely proportional to the angular velocities about the axes which they Mechanics, 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 nor intersecting, and are called skew-bevel wheels.

In skew-bevel wheels, the properties of a line of connection 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.

59. 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 these 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.

60. Sliding Contact—direct: Teeth of Wheels, 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 which 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-circle.

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 pitch-surface is called the face; that part which lies within the pitch-surface, the flank.

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 cogwheels. A pinion is a small toothed wheel; a trundle is a pinion with cylindrical staves 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 \( t \) and \( T \) 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 \( b \)); 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 \( N \),

\[ a = N; \quad b = \frac{N}{n}; \quad c = 1. \quad \ldots \quad (29) \]

Case 2. If the greatest common divisor of \( N \) and \( a \) be \( d \), a number less than \( n \), so that \( n = md; \quad N = Md \); then

\[ a = mN = Mn = Mmd; \quad b = M; \quad c = m. \quad \ldots \quad (30) \]

Case 3. If \( N \) and \( n \) be prime to each other,

\[ a = Nn; \quad b = N; \quad c = n. \quad \ldots \quad (31) \]

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.

61. Sliding Contact—Forms of the Teeth of Spur-wheels and Racks.—A line of connection 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 §§ 52 and 53 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. 17, let \( C_1, C_2 \) be the centres of a pair of spur-wheels; \( B_1B_2', B_1'B_2' \) 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 \( D_1TB_1A_1, D_2TB_2A_2 \) 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 connection of those teeth is \( P_1P_2 \), perpendicular to their surfaces at \( T \). Let \( C_1P_1, C_2P_2 \) be perpendiculars let fall from the centres of the wheels on the line of contact. Then, by sect. 52, the angular-velocity-ratio is

\[ \frac{a_1}{a_2} = \frac{C_1P_1}{C_2P_2} \]

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 connection of the teeth cuts the line of centres at the pitch-point.

For, let \( P_1P_2 \) cut the line of centres at \( I \); then, by similar triangles,

\[ \frac{a_1}{a_2} = \frac{C_1P_1}{C_2P_2} = \frac{IC_1}{IC_2} \]

which is also the angular-velocity-ratio due to the rolling contact of the circles \( B_1B_2', B_1'B_2' \).

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 \( C_1IC_2 \), 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 connection is,

\[ a_1 \cdot C_1P_1 = a_2 \cdot C_2P_2 \]

III. The relative velocity perpendicular to \( P_1P_2 \) 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 \( C_1C_2 \) 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 \( a_2 \), relatively to the line of centres \( C_1C_2 \), 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. 42 and 43; 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 \( IT \); that is to say, its value is

\[ (a_1 + a_2) \cdot IT \]

so that it is greater the further 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 connection preserves always the same position, the path of contact coincides with it, and is straight; in other cases the path of contact is curved.

It is divided by the pitch-point \( I \) into two parts: the arc Mechanics 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, the flank \( D_1B_1 \) of the driving tooth drives the face \( D_2B_2 \) 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 \( EIT = IC_1P_1 = IC_2P_2 \).

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 formulae is, that in equation 3 the difference of the angular velocities should be substituted for their sum.

62. Involute Teeth.—The simplest form of tooth which fulfils the conditions of sect. 61 is obtained in the following manner (see fig. 18). Let \( C_1, C_2 \) be the centres of two wheels, \( B_1B_2', B_1'B_2' \) their pitch-circles, \( I \) the pitch-point; let the obliquity of action of the teeth be constant, so that the same straight line \( P_1P_2 \) shall represent at once the constant line of connection of teeth and the path of contact. Draw \( C_1P_1, C_2P_2 \) perpendicular to \( P_1P_2 \), and with those lines as radii describe about the centres of the wheels the circles \( D_1D_2' \), 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

\[ \frac{C_1P_1}{C_2P_2} = \frac{IC_1}{IC_2} \cdot \cos \text{obliquity} \]

(The obliquity which is found to answer best in practice is about 14°; its cosine is about \( \frac{3}{4} \), and its sine about \( \frac{1}{2} \). 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 \( P_1P_2 \). As the line of connection 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 \( P_1P_2 \); that point will trace, on a plane rotating along with the wheel 1, part of the involute of the base-circle \( D_1D_2' \); and on a plane rotating along with the wheel 2, part of the involute of the base-circle \( D_2'D_1' \); 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. 61.

Consequently, one of the forms suitable for the teeth of Mechanics. 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 \( P_1P_2 \), is less than the pitch in the ratio of cos \( \theta \) : 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 IV, of sect. 61, 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 \( \frac{3}{4} \) times the pitch; and with this length of path, and the obliquity already mentioned of \( 14^\circ \), the addendum is about \( \frac{3}{8} \) 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 connection, and consequently making, with the direction of motion of the rack, angles equal to the complement of the obliquity of action.

63. 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 discs 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. 61, and by equation 34. 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.

64. Teeth traced by Rolling Curves.—If any curve R be rolled on the inside of the pitch-circle BB of a wheel, it appears, from sect. 42, 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 tracing-point 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 pitch-circle.

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 EIE', as the case may be) at a fixed point I (or I').

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.

65. 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 incurred or undercut, 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. 20, let BB' be part of the pitch-circle of a wheel with epicycloidal teeth; CIC the line of centres; I the pitch-point; EIE' 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 = EID'; and the mean obliquity is one-half of that angle.

It appears from experience that the mean obliquity should not exceed \( 15^\circ \); therefore the maximum obliquity should be about \( 30^\circ \); therefore the arcs DI = 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.

66. Nearly Epicycloidal Teeth.—Mr Willis's method.

To facilitate the drawing of epicycloidal teeth in practice, Mr Willis has shown how to approximate to their figure by means of two circular arcs,—one concave, for the flank; the other convex, for the face; and each having for its radius the mean radius of curvature of the epicycloidal arc. Mr 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; \( \theta \) 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— Mechanics. For an internal epicycloid, \( p = \frac{4r}{R - r} \sin \theta \)

For an external epicycloid, \( p' = \frac{4r}{R + r} \sin \theta \) (37.)

Also, to find the position of the centres of curvature relatively to the pitch-circle, we have (denoting the chord of the describing circle \( T \) by \( c \)) \( T = c = 2r \sin \theta \); and therefore

For the flank, \( p - c = 2r \sin \theta \frac{R}{R - 2r} \)

For the face, \( p' - c = 2r \sin \theta \frac{R}{R + 2r} \) (38.)

For the proportions approved of by Mr Willis, \( \sin \theta = \frac{1}{2} \) nearly; \( r = p \) (the pitch) nearly; \( c = \frac{p}{2} \) nearly; and if \( N \) be the number of teeth in the wheel, \( \frac{6}{N} \) nearly; therefore approximately,

\[ p - c = \frac{p}{2} \cdot \frac{N}{N - 12} \]

\[ p' - c = \frac{p}{2} \cdot \frac{N}{N + 12} \] (39.)

Hence the following construction (fig. 21):—Let BB be part of the pitch-circle; \( a \) the point where a tooth is to cross it. Set off \( ab = ac = \frac{p}{2} \). Draw radii \( bd, ce \); draw \( fb, cg \), making angles of \( 75^\circ \) with those radius. Make \( bf = p - c, cg = p - c \). 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, Mr Willis has published tables of \( p - c \) and \( p' - c \), and invented an instrument called the "Odontograph."

67. 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.

68. 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 \( \frac{1}{11} \) part wider than the thickness of the tooth on the pitch-circle; that is to say,

Thickness of tooth \( = \frac{5}{11} \) pitch.

Width of space \( = \frac{6}{11} \) pitch.

The difference of \( \frac{1}{11} \) 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 \( \frac{1}{11} \)th of the pitch.

69. Stepped and Helical Teeth.—Dr 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 discs placed side by side, with the teeth of each a little behind those of the preceding disc. 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 wheel-work of this kind the contact of each pair of teeth commences at the foremost end of the helical front, and terminates at the afterward 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.

70. Teeth of Bevel-Wheels.—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 bevel-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 for 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 O (fig. 22) be the common apex of a pair of bevel-wheels; OB, OB', their pitch-cones; OC, OC', their axes; OI their line of contact. Perpendicular to OI draw \( A_1A_2 \), cutting the axes in \( A_1, A_2 \); make the outer rims of the patterns and of the wheels portions of the cones \( A_1B_1, A_2B_2 \), of which the narrow zones occupied by the teeth will be sufficiently near to a spherical surface described about O for practical purposes. To find the figures of the teeth, draw on a flat surface circular arcs \( ID_1, ID_2 \), with the radii \( A_1I, A_2I \); those arcs will be the developments of arcs of the pitch-circles \( B_1I, B_2I \), when the conical surfaces \( A_1B_1, A_2B_2 \) 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.

71. 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.

72. 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. 61 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 Mechanics, the ideal pitch-surface, which would give the same comparative motion by rolling contact that the cam gives by sliding contact.

73. 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 nut; and when a screw is spoken of without qualification, a convex screw is usually understood.

The relation between the advance and the rotation, which compose the motion of a screw working in contact with a fixed screw or helical guide, has already been demonstrated in sect. 47; and the same relation exists between the magnitudes of the rotation of a screw about a fixed axis and the advance of a shifting nut in which it rotates. The advance of the nut takes places 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 circumference; for a screw of \( n \) threads, one circumference.

Let \( r \) denote the radius of the pitch circle; \( n \) the number of threads; \( \theta \) the obliquity of the threads to the pitch circle, and of the normal helix to the axis.

\[ \frac{P_a}{n} = p_a \quad \text{the axial pitch,} \]

\[ \frac{P_d}{n} = p_d \quad \text{the divided pitch;} \]

\[ \frac{P_n}{n} = p_n \quad \text{the normal pitch,} \]

\[ \frac{P_d}{n} = p_d \quad \text{the divided pitch;} \]

\( P_c \) the circumferential pitch;

Then—

\[ p_a = p_a \cdot \cotan \theta = p_a \cdot \cosec \theta = \frac{2\pi r}{n}; \]

\[ p_d = p_d \cdot \sec \theta = p_d \cdot \tan \theta = \frac{2\pi r \cdot \tan \theta}{n}; \]

\[ p_n = p_n \cdot \sin \theta = p_n \cdot \cos \theta = \frac{2\pi r \cdot \sin \theta}{n}. \]

(40)

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 connection 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.

Dr 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 form 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.

74. 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. 23. \( C_1, C_2 \) are the axes of the two parallel shafts; \( D_1, D_2 \) two discs facing each other, fixed on the ends of the two shafts respectively; \( E_1, E_2 \), a bar sliding in a diametral groove in the face of \( D_1 \); \( E_1, E_2 \), 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 discs 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 discs and cross; the instantaneous axis of rotation of the cross at any instant is at \( I \), the point in the circle \( C_1C_2 \) 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.

75. 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 the 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. Mechanics.

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 connection 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. 52, if both pieces turn; and of sect. 53, 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 connection 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. 24, A, reverses the direction of the rotation communicated; an uncrossed belt, as in fig. 24, B, preserves that direction.

The length L of an endless belt connecting a pair of pulleys whose effective radii are \( r_1 \) and \( r_2 \), with parallel axes whose distance apart is C, is given by the following formula, 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,

\[ L = 2\sqrt{r_1^2 + (r_1 + r_2)^2} + \pi(r_1 + r_2)\left(\frac{r_1 + r_2}{c}\right) \]

(41 A)

For an uncrossed belt,

\[ L = 2\sqrt{(r_1 - r_2)^2 + \pi(r_1 - r_2)} + 2(r_1 - r_2)\sin\left(\frac{\pi}{2}\right) \]

(41 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.

76. Speed-Cones.—(See fig. 25.)—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 speed-cones 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 L must be constant, in equations 41 A or 41 B, according as the belt is crossed or uncrossed.

For a crossed belt, as in A and C, fig. 25, L 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 L in equation 41 B is to be made constant. The exact fulfillment 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 41 B, the following approximate equation:

\[ L \text{ nearly } = 2c + (r_1 + r_2) + \frac{(r_1 - r_2)^2}{c} \]

(42)

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. 25, 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, \( r_s \) 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

\[ v = r_s - r_1 + r_2 = \frac{(r_1 - r_2)^2}{2\pi} \]

(43)

\( 2\pi = 6.2832 \); 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.

77. 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 connection, 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 connection; 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 connection upon the respective axes of rotation.

If at any instant the direction of one of the crank-arms coincides with the line of connection, the common perpendicular of the line of connection 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 connection is in the plane of the two axes of rotation, in which Mechanics 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.

78. 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 connection 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 furthest from theirs.

79. 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 sect. 48; 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. 26); 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 connection 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; \( T_1T_2 \) 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 \( v_1 \) denote the velocity of \( T_1 \) at any given instant; \( v_2 \) that of \( T_2 \). To find the ratio of these velocities, produce \( C_1T_1 \); \( C_2T_2 \); till they intersect in \( K \); \( K \) is the instantaneous axis of the connecting rod, and the velocity-ratio is

\[ v_1 : v_2 = KT_1 : KT_2, \quad \ldots \quad (45). \]

Should \( K \) be inconveniently far off, draw any triangle with its sides respectively parallel to \( C_1T_1 \), \( C_2T_2 \), and \( T_1T_2 \); the ratio of the two sides first mentioned will be the velocity-ratio required. For example, draw \( C_2A \) parallel to \( C_1T_1 \), cutting \( T_1T_2 \) in \( A \); then

\[ v_1 : v_2 = CA : C_2T_2, \quad \ldots \quad (45). \]

80. Eccentric.—An eccentric circular disc fixed on a shaft, and used to give a reciprocating motion to a rod, is in Mechanics 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.

81. 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 connection is continuous with, or coincides with the crank-arm.

Let \( S \) be the length of stroke of the reciprocating piece, \( L \) the length of the line of connection, 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,

\[ S = 2R; \quad \ldots \quad \ldots \quad \ldots \quad (46), \]

and if these ends be not in one straight line with that axis, then \( S, L - R, \) and \( L + R, \) are the three sides of a triangle, having the angle opposite \( S \) at that axis; so that if \( \theta \) be the supplement of the arc between the dead points,

\[ S^2 = 2(L^2 + R^2) - 2(L^2 - R^2) \cos \theta, \]

\[ \cos \theta = \frac{L^2 + R^2 - S^2}{2(L^2 - R^2)} \quad \ldots \quad \ldots \quad \ldots \quad (47). \]

82. Coupling of Intersecting Axes: Hooke's Universal Joint.—Intersecting axes are coupled by a contrivance of Hooke's, known as the "universal joint," which belongs to the class of linkwork (see fig. 27). Let \( O \) be the point of intersection of the axes \( OC_1, OC_2, \) and \( \theta \) 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 \( O \). 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—

\[ \frac{a}{a_1} = \frac{1}{\cos \theta} \quad \text{when } F_1 \text{ is in the plane } OC_1C_2, \]

and \( \frac{a}{a_1} = \cos \theta \quad \text{when } F_2 \text{ is in that plane}. \)

Its value at intermediate instants is given by the following equations:—Let \( \phi_1, \phi_2 \) be the angles respectively made by the central planes of the forks and shafts with the plane \( OC_1C_2 \) at a given instant; then Mechanics.

\[ \cos \theta = \tan \phi_1 \cdot \tan \phi_2 \]

\[ \frac{a_2}{a_1} = \frac{d\phi_2}{d\phi_1} = \tan \phi_2 + \cotan \phi_2 \]

(49.)

83. 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 pull, turning on a fixed axis, prevents the ratchet-wheel or rack from reversing its motion.

DIVISION V.—TRAINS OF MECHANISM.

84. 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.

85. Trains of Wheelwork.—Let \( A_1, A_2, \ldots, A_m \) denote a series of axes; and \( a_1, a_2, \ldots, a_m \), their angular velocities. Let the axis \( A_1 \) carry a wheel of \( N_1 \) teeth, driving a wheel of \( n_1 \) teeth on the axis \( A_2 \), which carries also a wheel of \( N_2 \) teeth, driving a wheel of \( n_2 \) teeth on the axis \( A_3 \), and so on; the numbers of teeth in drivers being denoted by \( N_i \)'s, and in followers by \( n_i \)'s, and the axes to which the wheels are fixed being denoted by numbers. Then the resulting velocity-ratio is denoted by

\[ \frac{a_m}{a_1} = \frac{a_2}{a_1} \cdot \frac{a_3}{a_2} \cdot \ldots \cdot \frac{a_m}{a_{m-1}} = \frac{N_1}{n_1} \cdot \frac{N_2}{n_2} \cdot \ldots \cdot \frac{N_m}{n_m}; \quad (50) \]

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 ought to be, for epicycloidal teeth (see sect. 65), twelve; but it is still better, for smoothness of motion, not to go below fifteen, and for involute teeth the smallest number is about twenty-four.

Let \( \frac{B}{C} \) be the velocity-ratio required, reduced to its least terms, and let \( B \) be greater than \( C \). If \( \frac{B}{C} \) is not greater than 6, and \( C \) lies between the prescribed minimum number of teeth (which may be called \( D \)), and its double \( 2D \), 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 \( \frac{B}{C} \), some ratio approximating to that ratio, and capable of resolution into convenient factors, is to be found by the method of continued fractions.

Should \( \frac{B}{C} \) be greater than 6, the best number of elementary combinations \( m - 1 \) will lie between

\[ \log \frac{B}{6} - \log \frac{C}{3} \text{ and } \log \frac{B}{6} - \log \frac{C}{3} \]

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 6, nor greater than 66; 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 \( \frac{B}{C} \) as before.

So far as the result and 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. 60, 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.

86. Double Hooke's Coupling.—It has been shown in section 82 that the velocity-ratio of a pair of shafts, coupled by a universal joint, fluctuates between the limits \( \cos \theta \) and \( \frac{1}{\cos \theta} \). Hence one or both 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 \( a_1, a_2, a_3 \) 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. 82, it is evident that at each instant \( \frac{a_2}{a_1} = \frac{a_3}{a_2} \), and consequently that \( a_3 = a_1 \); 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.

87. 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 VI.—AGGREGATE COMBINATIONS.

88. General Principles.—Mr 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 Mechanics, principles already explained in Division III. of this chapter, sects. 35 to 47.

Class II. comprises those cases in which the motions of three points in one follower are determined by their connections 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 sect. 48.

This classification is founded on the kinds of problems arising from the combinations. Mr 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 Mr Willis depends upon which of those two effects, even supposing them to occur together, is the practical object of the mechanism.

89. Reduplication of Cords—Differential Windlass—Blocks, Sheaves, and Tackle.—The axis C carries a larger barrel AE, and a smaller barrel BD, rotating as one piece with the angular velocity \(a_1\) 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 \(v_1\) of W, and the angular velocity \(a_1\) of the differential barrel.

In this case \(v_1\) 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 \(a_1 \cdot AC\), upward motion being considered positive. The velocity of the point G is \(-a_1 \cdot CB\), downward motion being negative. Applying the principles of sect. 48, it appears that the instantaneous axis of the sheave FG is in the diameter FG, at the distance

\[ \frac{FG}{2} \cdot \frac{AC - BC}{AC + BC} \]

from the centre towards G; that the angular velocity of the sheave is

\[ a_2 = a_1 \cdot \frac{AC + BC}{FG}; \]

and that, consequently, the velocity of its centre is

\[ v_2 = a_2 \cdot \frac{AC - BC}{AC + BC} = a_1 (AC - BC), \]

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 one-half of 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 moveable 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 fall-block 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 in 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 end, 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 fall-block be positive, and from it negative.

Case 1. If the fastened end of the rope be attached to the fall-block 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 \(2v_2\). It then passes over a sheave in the fall-block; the velocity of the third ply is \(-2v_2\); then under a sheave in the running-block; the velocity of the fourth ply is \(4v_2\); and so on: the general law being this—Let \(n\) be an even number, then

\[ \begin{align*} \text{Velocity of the } n^{th} \text{ ply} &= nv_2; \\ \text{if the fall be the } (n+1)^{th} \text{ ply}, \quad v_1 &= nv_2; \\ \end{align*} \]

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, \(3v_2\); of the fourth \(-3v_2\); and generally, if \(n\) be an odd number,

\[ \begin{align*} \text{Velocity of the } n^{th} \text{ ply} &= nv_2; \\ \text{if the fall be the } (n+1)^{th} \text{ ply}, \quad v_1 &= nv_2; \\ \end{align*} \]

and generally,

\[ \frac{v_1}{v_2} = -n, \]

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 as to tackle and trains of tackle, see Ship-building.

90. Differential Screws.—On the same axis let there be two screws of the respective pitches \(p_1\) 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. 47) \(ap_1\); and of A relatively to B (according to sect. 73) \(-ap_2\); hence the velocity of A relatively to C is

\[ a(p_1 - p_2), \]

being the same with the velocity of advance of a screw of the pitch \(p_1 - p_2\). 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.

91. Epicyclic Trains.—The term epicyclic train is used by Mr 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. 42 to 46.

92. Link Motion.—Let S be the shaft of a steam-engine, O its axis, E, the forward eccentric, suitably placed for Mechanics, moving the slide-valve when the shaft rotates forwards, F C, o on the lines CT, etc., that the path of P between the Mechanics.

its centre, OF its crank-arm, C its rod, E its backward

eccentric, suitably placed for moving the slide-valve when the shaft rotates backwards, B its centre, OB its crank-arm, C its rod. L is a long narrow box called the link, jointed at T and T to the eccentric rods; R is the valve-rod which works the slide-valve, jointed to P, a slider, which, either by moving L or R, or both, can be adjusted to any required position in the link. When P is at T, the valve is said to be in full forward gearing, being acted upon by E alone. When P is at T, the valve is said to be in full backward gearing, being acted upon by E alone.

When P is placed in an intermediate position, the valve has an aggregate motion due to the joint action of E and E. 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

\[ T_P : T_P : FQ : BQ ; \]

then the valve will move nearly as if it were worked by one eccentric, having its centre at Q.

93. 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. 30 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 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 ACa. While the stroke of A is ACa extending to equal distances on either side of C, and equal to twice the chord of the arc Dd, the stroke of B is only equal to twice the sagitta; and thus A is guided through a comparatively long stroke by the sliding of B through a comparatively short stroke, and by rotatory motions at the joints C, D, B. (For details, see Steam Engine.)

94. Parallel Motion—Watt's Approximate.—(See fig. 31.)—Let CT, etc, be a pair of levers connected by a link T, oscillating about the axes Cc, between the positions marked 1 and 3. Let the middle positions of the levers CT, etc, be parallel to each other. It is required to find a point P in the link Tt, such that its middle position P, and its extreme positions P, P, shall be in the same straight line SS, perpendicular to CT, etc, and so to place the axes positions P, P, P, shall be as near as possible to a straight line.

The axes C, c are to be so placed that the middle M of the versed sine VT, and the middle m of the versed sine etc, of the respective arcs whose equal chords T, T, = t, t, represent the stroke, may each be in the line SS. Then T and T will be as far to one side of SS as T is to the other, and t and t will be as far to the latter side of SS as t is to the former; consequently, the two extreme positions of the links Tt, Tt, are parallel to each other, and inclined to SS at the same angle in one direction that the middle position of the link Tt is inclined to that line in the other direction; and the three intersections P, P, P, are at the same point on the link.

The position of the point P on the link is found by the following proportional equation:

\[ \frac{T_P}{T_P} : \frac{PT}{PT} : \frac{TV + te}{TV + te} : \frac{CM + cm}{CM + cm} \]

Suppose the axes C, c to be given, the line of stroke SS, and the length of stroke L = T, T, = t, t, and that it is required to find the dimensions of the levers and link. Let fall CM and cm = SS; then

\[ TV = \frac{L^2}{8CM} ; \quad te = \frac{L^2}{8cm} ; \]

\[ CT = CM + \frac{1}{2} TV ; \quad ct = cm + \frac{1}{2} te ; \]

\[ Tr = \sqrt{\left( \frac{(TV + te)^2}{4} \right)} \]

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 Tt 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, P, and P, 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. (For details respecting the various modes of practically applying it, see Steam Engine.)

CHAPTER II.—ON APPLIED DYNAMICS.

95. 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 Mechanics, bodies in general, by two "laws of motion," with respect to the proof of which, see Dynamics. They are as follows:

Law I. A body under the action of no force, or of mutually balanced forces, either remains at rest or moves in a straight line with an uniform velocity.

Law II. The deviation of the motion of a given mass, caused in a given time by a given unbalanced force, takes place in the direction of the force, and is proportional to the magnitude of the force and the time during which it acts directly, and inversely to the mass.

(As to the determination of deviations, see sects. 25, 26, 27.)

The law of the equality of action and reaction is sometimes added as a "third law of motion;" but it is properly to be classed as a law of force, whether balanced or unbalanced; being equivalent to the statement that every force is a pair of actions of equal magnitudes and opposite directions exerted mutually between a pair of bodies.

96. Comparison of Deviating Force with Gravity.—A body's own weight,—that is, its tendency to approach the earth,—acting unbalanced on the body, produces deviation at a rate per second denoted by the symbol $g$, whose numerical value is as follows:—Let $\lambda$ denote the latitude of the place, $h$ its elevation above the mean level of the sea; $g_1 = 32-169545$ feet per second, the value of $g$ for $\lambda = 45^\circ$ and $h = 0$; and $R = 208875400 \times (1 + 0.000164 \cos 2\lambda)$, the earth's radius at the locality of observation. Then

$$g = g_1 \cdot (1 - 0.00284 \cos 2\lambda) \cdot \left(1 - \frac{2h}{R}\right).$$

For latitudes exceeding $45^\circ$ it is to be borne in mind that $\cos 2\lambda$ is negative, and the terms containing it as their factor have their signs reversed.

For practical purposes connected with ordinary machines it is sufficiently accurate to assume

$$g = 32.2 \text{ feet per second nearly}.$$

If, then, a body of the weight $w$ be deviated by an unbalanced force $f$ expressed in units of weight, the deviation produced in a second will be

$$\frac{fg}{w}.$$

This is sometimes called the mass, or inertia of the body.

97. Deviating Forces Classed: Deflecting Force—Accelerating and Retarding Forces.—The forces to be specially considered, in treating of machines as distinguished from structures, act upon bodies already in motion, and may be resolved into components and classed with reference to their directions as compared with the directions of motions of the bodies.

A force acting on a body in a direction at right angles to its path, produces lateral deviation or deflection of the path alone, without change of velocity. Such a force may be called a deflecting force.

A force acting in the direction of the body's path produces acceleration or retardation according as it acts with or against the motion of the body, but no deflection.

98. 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 I. Balanced forces in machines of uniform velocity.

Division II. Deflecting forces in such machines.

Division III. Working of machines of varying velocity.

DIVISION I.—BALANCED FORCES IN MACHINES OF UNIFORM VELOCITY.

99. Application of Force to Mechanism.—Forces are expressed 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, how small soever. 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 ($q, r$), 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.

100. Forces applied to Mechanism Classed.—If $\theta$ 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 ($q, r$), $F$ may be resolved into two rectangular components, viz.:—

Along the direction of motion, $P = F \cdot \cos \theta$.

Across the direction of motion, $Q = F \cdot \sin \theta$.

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.

101. 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.

102. 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.

103. 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 $\Delta s$ of the corresponding portion of the path of the point of application; the sum

$$\sum P \Delta s$$

is the whole energy exerted. If the effort varies by insen- Mechanics. sible 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

\[ \int Pds \]

(63.)

Similar processes are applicable to the finding of the work performed in overcoming a varying resistance.

104. 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 between two ordinates represent the energy exerted \( \int Pds \), for the corresponding portion of the path of the point of effort.

105. 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 (q.v.) 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 to any number of points of a machine; let any one of these efforts be represented by \( P_1 \), 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 \( R_1 \), and the distance traversed by its point of application in the given interval of time by \( ds' \); then

\[ \sum Pds = \sum Rds' \]

(64.)

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.

106. 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

\[ \frac{\sum Rds}{\sum Pds} = \frac{U}{E} \]

(65.)

\( R_u \) 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.

107. Power and Effect.—The power of a machine is the energy exerted, and the effect, the useful work performed, Mechanics, in some interval of time of definite length, such as a second, a minute, 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.

108. 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 the energy and the useful work, is called by Mr Moseley the modulus of the machine. The general form of the modulus of a machine may be expressed thus

\[ E = U + \varphi(U, A) + \psi(A) \]

(66.)

where \( A \) denotes some quantity or set of quantities depending on the form, arrangement, weight, and other properties of the mechanism. Mr Moseley, however, has pointed out that in most cases this equation takes the much more simple form of

\[ E = (1 + A)U + B \]

(67.)

where \( A \) and \( B \) are constants, depending on the form, arrangement, and weight of the mechanism. The efficiency corresponding to the last equation is

\[ \frac{U}{E} = \frac{1}{1 + A + B} \]

(68.)

109. 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 connection 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.

110. 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. A couple of forces consists of two forces equal in magnitude and opposite in direction, but not acting in the same line (as \( P_1 \), \( P_2 \) in fig. 32). The perpendicular distance between the lines of action of the forces is the lever or arm of the couple (as \( AB \)). The tendency of a couple is to turn the body to which it is applied about an axis perpendicular to the plane containing the couple and its arm, and the magnitude of that tendency, called the moment of the couple, is measured by the product of the common magnitude of the forces into the length of their arm. That is to say, let \( P = P_1 = P_2 \), \( AB = r \); then the movement is

\[ M = Pr \]

(69.)

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 op- Mechanics. Positive 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 an interval of time, \( \omega \) the angular velocity of the piece; then \( adt \) is the angle through which it turns in the interval \( dt \), and \( da = rdt = radt \) 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

\[ Pds = Prdt = Pradt = Madt \quad \ldots \quad (70) \]

is the energy exerted by the couple \( M \) 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.

111. 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 ratio 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.

112. 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 or may not be an unbalanced component in this pressure, due to deviated motion. Its laws will be considered in the sequel.

113. 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 rubbing surfaces together in a direction perpendicular to themselves, into a specific constant already mentioned in part I., sect. 14, as the coefficient of friction, which depends on the nature and condition of the surfaces and 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. 14, 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 \( \phi \) the angle of repose; then

\[ f = \tan \phi; \quad R = fN = N \tan \phi = T \sin \phi; \quad \ldots \quad (71) \]

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:

| Surfaces | f | |---------------------------|---| | Wood on wood, dry | 0.25 to 0.5 | | Do., soaped | 0.2 | | Metals on oak, dry | 0.5 to 0.6 | | Do., wet | 0.24 to 0.25 | | Do., soaped | 0.2 | | Do., on elm, dry | 0.2 to 0.25 | | Hemp on oak, dry | 0.53 | | Do., wet | 0.33 | | Leather on oak, wet or dry| 0.27 to 0.35 | | Leather on metals, dry | 0.56 | | Do., wet | 0.36 | | Do., greasy | 0.23 | | Do., oiled | 0.15 | | Metals on metals, dry | 0.15 to 0.2 | | Do., wet | 0.30 |

Smooth Surfaces with Unguents—

| Occasionaly greased | 0.07 to 0.08 | | Well greased | 0.05 | | Do., best results | 0.03 to 0.006 |

It is to be understood that the above-stated law of friction is true for dry surfaces, only 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; and the following are some of its approximate values as inferred from the results of experience in railway, locomotive, and carriage axles:

| Velocity of rubbing in feet per second... | 1 | 2½ | 5 | |------------------------------------------|---|----|---| | Intensity of normal pressure per lb. per square inch of surface... | 392 | 224 | 140 |

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, that they may resist being forced out; and comparatively thin for light pressures, that their viscosity 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.

114. Work of Friction—Moment of Friction.—The work performed in an 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 an unit of time, for a portion of the rubbing surfaces at a given distance from the axis of rotation, may be found by multiplying together the friction of that portion, its distance from the axis, and the angular velocity. The product of the force of friction by Mechanics.

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 the 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

\[ \int pr \cdot du, \]

the total moment of friction,

\[ \int pr \cdot du; \]

and the work performed in an unit of time in overcoming friction, when the angular velocity is \( \alpha \),

\[ \int pr \cdot du. \]

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. 113, and also by conditions of strength and stiffness.

115. Total Pressure between Journal and Bearing.—A single piece rotating with an uniform velocity has for mutually balanced forces applied to it: the effort exerted on it by the piece which drives it; the resistance of the piece which follows it,—which may be considered for the purposes of the present question as useful resistance; its weight, and 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 \( \phi \), corresponding to the friction of the journal on the bearing.

And there are required—

- 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 green 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. 33, be a section of the bearing, C its axis; then the direction of the reaction, at the point where it intersects the circle AAA, must make the angle \( \phi \) with the radius of that circle; that is to say, it must be a line such as PT touching the inner circle BB, whose radius is \( r' \sin \phi \). 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. 8, and proved Mechanics in the article Statics.

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 pieces (called above, the "given force"), ought to be opposed to each other as directly as is practicable consistently with the purposes of the machine.

116. Frictions 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 a pivot is called a step or footstep. Pivots require great hardness, and are usually made of steel. The flat 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

\[ p = \frac{N}{2\pi r^2}, \]

and introducing this value into equation 72, the moment of friction of the flat pivot is found to be

\[ \frac{2}{3}fNr, \]

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—

\[ \frac{2}{3}fN\left(\frac{r^3 - r'^3}{r^2}\right), \]

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, jointly to the friction and to the velocity, or nearly so. Hence the pivots already mentioned wear unequally at different points, and tend to alter their figures. Mr 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. 34) 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 of 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 in Mechanics. T: PT = PY × second obliquity, and this is to be a constant quantity; hence the curve is that known as the *trajectory* 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 the 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 O towards X, P will be drawn after it from R towards C along the trajectory. 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.

117. 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

\[ \text{work lost} = \frac{N s}{n} \]

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 these be denoted by \( s_1 \) and \( s_2 \), so that \( s = s_1 + s_2 \). In sec. 61 the velocity of sliding at any instant has been given, viz.,

\[ u = c (a_1 + a_2), \]

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 \( a_1, a_2 \) 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

\[ u = cv \left( \frac{1}{r_1} + \frac{1}{r_2} \right) \]

To apply this to involute teeth, let \( c_1 \) be the length of the approach; \( c_2 \) that of the recess; \( u \), the mean velocity of sliding during the approach, \( u_2 \) that during the recess. Then

\[ u_1 = \frac{cv}{2} \left( \frac{1}{r_1} + \frac{1}{r_2} \right); \quad u_2 = \frac{cv}{2} \left( \frac{1}{r_1} + \frac{1}{r_2} \right) \]

also, let \( \theta \) be the obliquity of the action; then the times occupied by the approach and recess are respectively

\[ \frac{c_1}{v \cos \theta}, \quad \frac{c_2}{v \cos \theta}; \]

giving, finally, for the length of sliding between each pair of teeth,

\[ s = s_1 + s_2 = \frac{c_1^2 + c_2^2}{2 \cos \theta} \left( \frac{1}{r_1} + \frac{1}{r_2} \right). \]

which, being substituted in equation 76, 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, \( \frac{1}{r_1} - \frac{1}{r_2} \) is to be substituted for \( \frac{1}{r_1} + \frac{1}{r_2} \).

118. 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, whether 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; \( \theta \) the ratio of the length of that arc to the radius of the pulley; \( d\theta \) the ratio of an infinitely small element of that arc to the radius; \( F = T_1 - T_2 \) the total friction between the band and the pulley; \( dF \) the elementary portion of that friction due to the elementary arc \( d\theta \); \( f \) the coefficient of friction between the materials of the band and pulley.

Then, according to a principle proved in the articles Statics and Archimedes, it is known that the normal pressure at the elementary arc \( d\theta \) is \( Td\theta \), \( T \) being the mean tension of the band at that elementary arc; consequently the friction on that arc is \( dF = fTd\theta \). Now that friction is also the difference between the tensions of the band at the two ends of the elementary arc, or \( dF = dT = fTd\theta \); which equation being integrated throughout the entire arc of contact, gives the following formulae:

\[ \text{hyp. log } \frac{T_1}{T_2} = f\theta \]

\[ T_1 = e^{f\theta} \]

\[ F = T_1 - T_2 = T_1 \left( 1 - e^{-f\theta} \right) = T_2 \left( e^{f\theta} - 1 \right) \]

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—

\[ \frac{T_1 + T_2}{2} = \frac{e^{f\theta} + 1}{2(e^{f\theta} - 1)} \]

which is useful in determining the original tension required to enable a belt to transmit a given force between two pulleys.

The equations 78 and 79 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

\[ \theta = 6.2832a \]

\[ e^{f\theta} = \text{number whose common logarithm is } 2.7285fa \]

119. Stiffness of Ropes.—Ropes offer a resistance to being bent, and when bent, to being straightened again, which 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 inches, \( n = 48d^2 \) for white ropes, \( 35d^2 \) for tarred ropes; \( r \) the effective radius of the pulley in inches; \( T \) the tension in pounds. Then

\[ F = \frac{n}{r} (0.0012 + 0.001026n + 0.0012T) \]

For tarred ropes,

\[ F = \frac{n}{r} (0.006 + 0.001392n + 0.00168T) \]

(81.)

120. 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 so 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 the 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 has 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.

121. 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 recent achievements of Mr 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 Fahrenheit) is 772 foot-pounds. This constant, now designated as "Joule's Equivalent," is the principal experimental datum of the science of Thermodynamics, which treats of the relations between heat and mechanical work.

The heat produced by friction, when moderate in amount, is useful 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 Mechanics, some recent experiments to be, with good fatty and soupy unguents, 40° to 50° Fahrenheit; with good mineral unguents, about 30°.

122. 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:

- Oak upon oak: 0.006 (Coulomb.) - Lignum vitae on oak: 0.004 (Do) - Cast-iron on cast-iron: 0.002 (Tredgold.)

123. 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 experts 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 an 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 adjusting some moving weight in the mechanism, so that when the effort is in excess it may be 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 II.—DEFLECTING FORCES.

124. Deflecting Force for Translation in a Curved Path.—If a body have a motion of translation with the velocity \( v \), so that each point in it moves in a curved path of the radius \( \rho \), then, as we have already seen (sect. 27), the rate of deflection of that body's motion in unity of time, which is common to all its points, is expressed by

\[ \frac{v^2}{\rho} = \rho a^2, \]

if \( a \) be substituted for \( \frac{d\theta}{dt} \) the expression in equation 3 for the angular velocity of deflection. Mechanics. To produce this deviation the body must (according to sect. 96, equation 60) be acted upon by a lateral pressure at right angles to, and towards the centre of curvature of its path, whose magnitude is given by the equation—

\[ F = \frac{wv^2}{g} = \frac{wpa^2}{g}, \ldots \ldots \quad (82.) \]

where \( w \) is the weight of the body, and \( g \) the deviation produced by gravity in a second (see sect. 96); and as this total deflecting force \( F \) is the resultant of deflecting forces acting upon each particle of the body and proportional respectively to the masses of the particles, its line of action must traverse the centre of gravity of the body.

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 deflection, and not in balancing another force.

125. Centrifugal Force.—The deflecting body reacts upon the guiding body with a lateral pressure equal and opposite to the deflecting force, and called the centrifugal force, because of its arising from the tendency of the deflected body to move in a straight line, and so to fly from the centre of its curved path; and also because of its tendency to pull the guiding body away from that centre.

In fact, as has been stated in section 95, every force is a pair of equal and opposite actions between a pair of bodies; and deflecting force and centrifugal force are but two different names for the same force, applied to it according as its action on the deflected body or on the guiding body is under consideration for the time. The action on the deflected body is to produce deflection at a rate proportional to the force; the action on the guiding body is to strain it and the framework which carries it, to be balanced by the stiffness which resists that strain, and to cause increased friction at rubbing surfaces. Hence it appears that the action of centrifugal force is in general prejudicial, and that it is desirable in well-designed machinery to diminish it as much as possible.

126. Rectangular Resolution of Centrifugal Force.—For convenience in mathematical investigation, centrifugal force may be resolved into rectangular components as follows:—In fig. 35 let \( O \) be the centre of curvature and \( p \) the radius of the path of the centre of gravity of the body \( w \), whose angular velocity of deflection is \( a \). Let \( OX, OY \) be a pair of rectangular axes in the plane of motion of \( w \), and \( x, y \) perpendiculars let fall from the centre of gravity of \( w \) upon these axes. It is evident that the centrifugal force exerted by \( w \) upon an axis at \( O \) may be resolved into two rectangular components, \( F_x, F_y \), parallel to \( OX, OY \) respectively, and bearing to each other and to the whole centrifugal force \( F \) the following proportions:—

\[ F : F_x : F_y \] \[ : : \rho : x : y ; \] consequently their values are—

\[ F_x = \frac{wx^2a^2}{g}; \quad F_y = \frac{wy^2a^2}{g}. \quad \ldots \quad (83.) \]

127. Centrifugal Force of a Rotating Body.—Let a body of any figure \( BB \) rotate about the axis of rotation \( O \), perpendicular to the plane \( XOY \) of the rectangular axes of co-ordinates before-mentioned. Let \( W \) be the entire weight of the body; let \( a \) be its angular velocity of rotation; this, as shown in section 38, is also the angular velocity of deflection of each particle of the body in its revolution round the axis. Conceive the body to be divided into an indefinite number of indefinitely small particles; denote one of them by \( dW \), and let its co-ordinates be \( x \) and \( y \).

The components of the centrifugal force exerted by it on the axis \( O \) are respectively

\[ \frac{x^2dW}{g} \text{ and } \frac{y^2dW}{g}; \]

and the components of the centrifugal force exerted by the whole body on that axis, being the sums or integrals of the centrifugal forces exerted by all the particles, are expressed by—

\[ F_x = \frac{a^2}{g} \int x^2dW, \quad \text{and} \quad F_y = \frac{a^2}{g} \int y^2dW. \]

Now, by the properties of the centre of gravity (for which see Statics), if \( x_1 \) and \( y_1 \) be the co-ordinates of the centre of gravity of the body \( BB \),

\[ \int x^2dW = x_1W; \quad \int y^2dW = y_1W; \]

consequently

\[ F_x = \frac{Wx_1a^2}{g}; \quad F_y = \frac{Wy_1a^2}{g}; \quad \ldots \quad (84.) \]

being precisely the same values which were found in section 126, equation 83, for the components of the centrifugal force due to a circular translation with a radius equal to the distance of the centre of gravity of the body from the axis of rotation.

Hence 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.

128. 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.

To determine its amount, or the amount of its components, let \( z \) denote the distance (positive towards, negative from, the spectator), of any given particle \( dW \) from the plane \( XOY \). Then the tendencies of the centrifugal force of \( dW \) to turn the axis of rotation \( O \) towards the right hand, about \( OX \), is the moment of the couple, \( -\frac{zy^2dW}{g} \), and about \( OY \), the moment of the couple, \( +\frac{zx^2dW}{g} \); and by integrating those expressions, the centrifugal couples of the whole body are found to be respectively—

---

1 This is a particular case of a more general principle, that the motion of the centre of gravity of a body is not affected by the mutual actions of its parts; for the proof of which see Dynamics. A permanent or principal axis of rotation is one for which each of the centrifugal couples, as well as the centrifugal force, is nothing; and is the only kind of axis about which a body not acted upon by an external force can steadily rotate. It can be proved that every body, of what shape soever, has at least three permanent axes at right angles to each other.

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.

129. Revolving Pendulums—Governors.—In fig. 36 AO represents an upright axis or spindle; B a weight called a bob, suspended by a rod OB from a horizontal axis at O, 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 O 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 O may balance each other.

In finding the position in which the bob will revolve with a given angular velocity \( \omega \), for most practical purposes 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,

\[ \frac{y}{z} = \frac{F}{W} = \frac{g}{a^2} \]

consequently,

\[ z = \frac{g}{a^2} \]

Or, if \( n = \frac{\omega}{2\pi} = \frac{a}{62832} \) be the number of turns or fractions of a turn in a second,

\[ z = \frac{g}{4\pi^2n^2} = \frac{0.8165 \text{ foot}}{n^2} = \frac{9.79771 \text{ inches}}{n^2} \]

\( z \) is called the altitude of the pendulum.

If the rod of a revolving pendulum be jointed, as in fig. 37, not to a point in the vertical axis, but to the end of a projecting arm C, 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 CE, DD (fig. 36), act on a lever EF, which acts upon the throttle-valve for the admission of steam so as to enlarge its opening when the speed becomes too small, and contract it when the speed becomes too great.

In a more recent kind of governors invented by Messrs Siemens, which may be called differential governors, 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. 38 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 what sort soever 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 III.—WORKING OF MACHINES OF VARYING VELOCITY.

130. 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 acceleration; 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. 105, 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. 95, as applied by the aid of the principles of sect. 95, to direct deviation, or acceleration and retardation.

131. 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 \( \Delta t \) let its velocity be increased at an uniform rate of acceleration from \( v_1 \) to \( v_2 \). The rate of acceleration will be

\[ \frac{dv}{dt} = \text{constant} = \frac{v_2 - v_1}{\Delta t}; \]

and, according to sect. 96, equation 60, to produce this acceleration a uniform effort will be required, expressed by

\[ P = \frac{w(v_2 - v_1)}{g \Delta t}. \quad (83) \]

(The product \( \frac{wv}{g} \) 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 \( v_2 \), it is to be observed that the distance through which the effort \( P \) acts during the acceleration is

\[ \Delta s = \frac{v + v_1}{2} \Delta t; \]

consequently, the energy of acceleration is

\[ P \Delta s = \frac{w(v_2 - v_1)(v_2 + v_1)}{2g} = \frac{w(v_2^2 - v_1^2)}{2g}, \quad (89) \]

being proportional to the increase in the square of the velocity, and independent of the time.

In order to produce a retardation from the greater velocity \( v \) to the less velocity \( v_1 \), 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 88, 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 89, putting \( Rds \) for \( Pds \).

132. 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. 123).

Let there be given the mean velocity \( V = \frac{v_1 + v_2}{2} \) 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 30 we have

\[ E = \frac{w(v_2^2 - v_1^2)}{2g}, \]

which, being divided by \( V = \frac{v_1 + v_2}{2} \), gives

\[ \frac{E}{V} = \frac{w(v_2 - v_1)}{g}; \]

and consequently

\[ v_2 - v_1 = \frac{gE}{Vw}. \quad (90) \]

The ratio of this fluctuation to the mean velocity, sometimes called the unsteadiness of the motion of the body, is

\[ \frac{v_2 - v_1}{V} = \frac{gE}{Vw}. \quad (91) \]

133. 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

\[ \frac{wv^2}{2g}. \quad (92) \]

This is called the actual energy of the motion of the body, and is one-half of the quantity which is called vis-viva in some treatises on mechanics.

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, as the case may be.

134. 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. 105 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.

135. Actual Energy of Circular Translation—Moment of Inertia.—Let a body of the weight \( w \) undergo translation in a circular path of the radius \( r \), with the angular velocity of deflection \( \alpha \), so that the common linear velocity of all its particles is \( v = ra \). Then the actual energy of that body is

\[ \frac{wv^2}{2g} = \frac{wa^2r^2}{2g}. \quad (93) \]

By comparing this with equation 82, sect. 124, it appears that the actual energy of a revolving body is equal to the potential energy \( \frac{Fp}{2} \) due to the action of the deflecting force along one-half of the radius of curvature of the path of the body.

The product \( \frac{wv^2}{g} \) by which the half-square of the angular- Mechanics, velocity is multiplied, is called the moment of inertia of the revolving body.

136. Actual Energy and Moment of Inertia of Rotation—Radius of Gyration.—Let a body of any figure BB (see fig. 35, sect. 126) rotate about the axis of rotation O, perpendicular to the plane XOY; let w be the entire weight of the body; let a be its angular velocity of rotation, being also the angular velocity of deflection of each of its particles in its revolution round the axis. Conceive the body to be divided into an indefinite number of indefinitely small particles; denote the weight of one of them by dW; and let its perpendicular distance from the axis be ρ. The actual energy of that particle, according to sect. 135, is

\[ \frac{a^2}{2g} \cdot dW \]

and the actual energy of the whole body, being the sum or integral of the actual energies of its particles, is

\[ \frac{a^2}{2g} \int \rho^2 \cdot dW \]

(94.)

The integral in this expression, by which the half-square of the angular velocity is multiplied, viz—

\[ \frac{1}{g} \int \rho^2 \cdot dW = I_1 \]

(95.)

is the moment of inertia of the whole body relatively to the given axis O, being the sum of the moments of inertia of all its particles. The actual energy of the body may be thus expressed:

\[ \frac{I_1}{2} \]

(96.)

If the moment of inertia be divided by the mass \( \frac{W}{g} \) of the body, the result is

\[ R^2 = \frac{I_1}{W} \]

(97.)

the square of a length called the radius of gyration; being the distance from the axis O at which, if the whole mass of the body were collected at one or more points, or in a ring, or hollow cylinder, the moment of inertia would be the same with that of the actual body.

If the given axis O do not already traverse the centre of gravity of the body, conceive an axis parallel to it to be drawn through that centre of gravity, and designated by the symbol G. Let \( \rho_1 = OG \) be the perpendicular distance between those axes. Let \( R_g \) be the radius of gyration, and \( I_g = R_g^2 W \), the moment of inertia of the body about the axis G. Then, from geometrical properties of the centre of gravity, the proof of which belongs to the subject of statics, it is known that

\[ R^2 = R_g^2 + \rho_1^2 \]

(98.)

from which it follows that

\[ I = I_g \times \frac{\rho_1^2 W}{g} \]

(99.)

that is to say, the moment of inertia of a body about any axis O, not traversing its centre of gravity, is equal to the moment of inertia of the whole body about an axis G, traversing the centre of gravity parallel to O, added to the moment of inertia due to a circular translation of the whole body with the radius OG.

From equation 99 it follows obviously, that the moment of inertia of a body about an axis traversing its centre of gravity in a given direction, is less than about any other axis parallel to that direction.

The respective moments of inertia of a body about its permanent axes of rotation (sect. 128), are called its principal moments of inertia.

137. Examples of Radii of Gyration.—The following are some examples, useful in practice, of the radii of gyration of homogeneous solids about permanent axes:

I. A sphere of the radius r rotating about a diameter,

\[ R_g^2 = \frac{2r^2}{5} \]

II. A spheroid of revolution rotating about its polar axis, its equatorial radius being r,

\[ R_g^2 = \frac{2r^2}{5} \]

III. An ellipsoid whose semi-axes are a, b, c, rotating about the axis 2a,

\[ R_g^2 = \frac{b^2 + c^2}{5} \]

IV. A cylindrical disc of the radius r, rotating about its axis of figure,

\[ R_g^2 = \frac{r^2}{2} \]

V. A cylindrical ring, or hollow cylinder, rotating about its axis of figure, the external and internal radii being r, r', respectively,

\[ R_g^2 = \frac{r^2 + r'^2}{2} \]

(This is applicable to many cases of rims of fly-wheels.)

VI. A rectangular parallelopiped whose dimensions are 2a, 2b, 2c, rotating about the axis whose length is 2a,

\[ R_g^2 = \frac{b^2 + c^2}{3} \]

VII. A slender rod of uniform section and length 2l, rotating about an axis crossing it at right angles in the middle of its length,

\[ R_g^2 = \frac{l^2}{3} \]

(This case is also applicable to any system of rods of equal length l, radiating from a common axis, like the spokes of a fly-wheel.)

VIII. A system of bodies of respective weights w, w', w'', &c., rotating about a common axis, and having the respective radii of gyration \( \rho, \rho', \rho'', \) &c.,

\[ R_g^2 = \frac{\sum w\rho^2}{\sum w} \]

138. Fly-Wheels.—A fly-wheel is a rotating piece in a machine, generally shaped liked 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. 132, 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, \( a_m \) its maximum, \( a_i \) its minimum, and \( A = \frac{a_m + a_i}{2} \) its mean angular velocity. Let

\[ \frac{1}{S} = \frac{a_m - a_i}{A} \]

denote the steadiness 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—

---

1 This is a particular case of a more general proposition, that the whole actual energy of any system of masses is equal to the actual energy due to a motion of the whole of those masses with the velocity of their common centre of gravity, added to the sum of the actual energies due to the several motions of the several masses relatively to that common centre of gravity. \[ e = \frac{R^2 W (a_1^2 - a_2^2)}{2g} \]

from which, dividing both sides by \( A^2 \), we obtain the following equations:

\[ \begin{align*} e &= \frac{R^2 W}{gS} \\ \frac{R^2 WA^2}{2g} &= \frac{Se}{2} \end{align*} \]

(101.)

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, \( S = \) from 32 to 60; 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 connection 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 connection with each of those points. For example, in a machine-work, 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, riveting-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 \( \frac{1}{4} \)th to nearly \( \frac{1}{4} \)th 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 whole work performed at each separate operation.

139. Brakes.—A brake is an apparatus for stopping or diminishing the velocity of a machine by friction, such as the friction-strap already referred to in sect. 118. 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. 111, 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

\[ s = \frac{E}{F + R} \]

(102.)

140. 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 moveable body and a fixed body, so that the whole energy exerted by it is employed upon the moveable 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; \( E_1, E_2 \) the shares of the whole energy \( E \) exerted upon \( W_1, W_2 \) respectively. Then

\[ \frac{E_1}{W_1} : \frac{E_2}{W_2} = \frac{P - R_1}{W_1} : \frac{P - R_2}{W_2} \]

(103.)

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,

\[ \frac{E_1}{W_1} : \frac{E_2}{W_2} = \frac{P}{W_1} : \frac{P}{W_2} \]

(104.)

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. 127, 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, \( \frac{1}{160} \) of the energy exerted by the powder in exploding will be employed in propelling the ball, and \( \frac{159}{160} \) 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.

141. Centre of Percussion.—In order that a rigid solid body may have a given deviation imparted to it, it is sufficient that the resultant of the unbalanced force or forces applied to it should be identical in magnitude, direction, and position, with the resultant of the forces, which, if applied separately to the several particles of the body, would give each of them the deviation which it is required to have as forming a part of the body. The nearest point in the line of action of this resultant to the centre of gravity of the body is called the centre of percussion of the body for the given kind of deviation. It indicates the position and direction in which a single force must act in order to produce deviation in the required manner. (Details respecting the centre of percussion will be found in the article Dynamics.)

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... Mechanics of motion of all the particles of the body,—the centre of percussion is obviously the centre of gravity itself; and, as already stated (sect. 96, equation 60), if \( d \) be the deviation of velocity to be produced in the interval \( dt \), and \( W \) the weight of the body,

\[ P = \frac{W}{g} \cdot \frac{da}{dt} \]

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 \( dt \); \( I \) the moment of inertia of the body; then \( \frac{1}{2}Id(a^2) = Ida \) is the variation of the body's actual energy. Let \( M \) be the moment of the unbalanced couple required to produce the deviation; then, by equation 70, sect. 110, the energy exerted by this couple in the interval \( dt \) is \( Madt \), which, being equated to the variation of energy, gives

\[ M = I \frac{da}{dt} = \frac{R^3W}{g} \cdot \frac{da}{dt} \]

Now, 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 \( dt \). According to the principle of sect. 44, 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 \( a \cdot OG \); \( OG \) being the perpendicular distance between the two axes. Hence the required deviation may be regarded as compounded of a deviation of translation \( dt = OG \cdot da \), to produce which there would be required, according to equation 105, a force applied at G perpendicular to the plane OG—

\[ P = \frac{W}{g} \cdot OG \cdot \frac{da}{dt} \]

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 \( M \) given by equation 106. According to the principles of statics, the resultant of the force \( P \) applied at G perpendicular to the plane OG, and of the couple \( M \), is a force equal and parallel to \( P \), but applied at a distance \( GC \) from G, in the prolongation of the perpendicular \( OG \), whose value is

\[ GC = \frac{M}{P} = \frac{R^3}{OG} \]

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 O traversing C, the centre of percussion is at the point where the perpendicular \( OG \) meets O.

142. 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 Dynamics.) 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 of the damage which it occasions to the frame and mechanism.

CHAPTER III.—PURPOSES AND EFFECTS OF MACHINES.

143. 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 the reader to find, under special heads in this Encyclopaedia, the detailed descriptions of particular examples.

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.

144. 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 aid 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.

145. 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 chronometers) 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. (See Chronometer; Clock and Watch Work; Pendulum.)

Other counting machines count the oscillations of the beam of a steam-engine, the revolutions of the cylinder of a gas-meter, or of the wheel of a water-meter.

Others perform additions, subtractions, and multiplications, and of these the most elaborate kind (of which there are but two in existence,—the machine of Mr Babbage and that of Messrs Scheutz) compute tables of functions by the addition of differences.

146. Measuring Machines.—Measuring machines are pieces of mechanism, by means of which the motion of Mechanics, 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 which is attached to it.

Such micrometers have attained increased importance by the discovery of Mr 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 plato-meters or planimeters of Mr Sang, General Morin, and Mr 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.

147. (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 principles on the theory of the composition of rotations, is used to draw ellipses, epicycloids, epitrochooids, and other curves.

148. (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 which 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 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 iron, 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 whose 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 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.

149. (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.

150. Working Machines Classified.—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:

A: Machines for lifting or lowering solid weights. B: Machines for the horizontal transport of weights, either combined or not with lifting or lowering. C: Machines for projecting solids. D: Machines for lifting fluids. E: Machines for propelling or projecting fluids. F: Machines for dividing bodies. G: Machines for shaping bodies by removing portions of them. H: Machines for shaping bodies by pressure. I: Machines for uniting bodies into fabrics. J: Machines for printing. K: Machines for producing sound. L: Miscellaneous machines.

The author of this article does not pretend to assert that the above classification (taken to a considerable extent from the writings of Young and of Mr Babbage) exhausts all kinds of machines; he brings it forward merely as an attempt to introduce method to a certain extent into a subject which would otherwise be exceedingly confused.

151. (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.

152. (B.) Transporting Machines.—The mechanism of transporting machines consists of two parts: that by which the resistance is diminished, as the wheels and axles of vehicles; and that by which the resistance is overcome and the load propelled, comprising all kinds of locomotive and propelling machinery. Transporting machines are treated of in the articles relating specially to the lines of conveyance to which they are applied; such as Canals, Railroads, Roads, and Steam Navigation. Mechanics.

153. (C.) Machines for Projecting Solids.—This class comprehends all kinds of artillery.

154. (D.) Machines for Lifting Fluids.—(See Hydrodynamics and Pneumatics.)

155. (E.) Machines for Propelling or Projecting Fluids.—(See the same articles.)

156. (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.

157. (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; 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 Mr 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 Herscheles, of Mr Lassell, and of Lord Rosse.

158. (H.) Machines for Shaping Bodies by Pressure comprehend amongst others, rolling-mills for iron, steam-hammers, wire-drawing machines, pin-making and nail-making machines, coining and other stamping machinery, brickmaking machines, presses for packing and compressing, &c., &c.

159. (I.) Machines for Unitting 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.

160. (J.) Machines for printing are used to apply either colouring matters or matters for discharging colour to paper, cloth, and other materials.

161. (K.) Machines for Producing Sound.—(See Acoustics, and Music.)

162. (L.) 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.

163. 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 useful 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.

164. Sources of Energy Classified.—The sources of energy used in practice may be classed as follows:

A: Strength of men and animals. B: Weight of liquids. C: Motion of fluids. D: Heat. E: Electricity and magnetism.

165. (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 departures from those values diminishes the daily duty. Attempts have been made to represent by a formula the law of this diminution; 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, \( V_1 \) the velocity, and \( T_1 \) the time of working per day, which give the maximum daily duty; let \( P, V, T \) be any other set of values of those quantities. Then

\[ \frac{P}{P_1} + \frac{V}{V_1} + \frac{T}{T_1} = 3 \quad \ldots \quad (109) \]

One consequence of this formula is, that the best time Mechanics. 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:

| Animals | Weight | \( P_1 \) | \( V_1 \) | \( T_1 \) | \( P_1 V_1 T_1 \) | |---------|--------|----------|----------|----------|------------------| | Man | 150 lb | 30 | 2.5 | 8 | 75 | | Horse | 600 lb | 120 | 4.0 | 8 | 480 | | Ox | 600 lb | 120 | 2.5 | 8 | 200 | | Ass | 360 lb | 721 | 2.5 | 8 | 180 | | Mule | 600 lb | 100 | 3.5 | 8 | 350 |

Of the numbers in this table those for the draught horse are probably the most accurate. For the thorough-bred 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:

| P | V | \( T_1 \) | \( PV \) | \( PV T_1 \) | |---|---|----------|---------|-------------| | | | | | |

167. (B.) Weight of Liquids.—(C.) Motion of Fluids.—In water-wheels and other hydraulic engines the weight and motion of a liquid usually act together as sources of energy.

To determine the necessary loss of energy and the theoretical efficiency, let \( Q \) denote 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, when it ceases to act, which will properly discharge the liquid, and prevent its accumulating so as to impede the wheel or engine. Then the power or energy exerted per second is

\[ Q \left( H + \frac{V_1^2}{2g} \right); \]

the necessary loss,

\[ Q \cdot \frac{V_2^2}{2g}; \]

the theoretical effect or useful work per second—

\[ Q \left( H + \frac{V_1^2 - V_2^2}{2g} \right); \]

the theoretical efficiency—

\[ \frac{V_1^2 - V_2^2}{H + \frac{V_1^2}{2g}}. \]

(For details as to the actual efficiency and duty, and the construction of water-wheels and other hydraulic engines, see HYDRODYNAMICS.)

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. 40):

- Angle of each sail with the plane of rotation: at DE \( = 18^\circ \) - at BC \( = 7^\circ \)

OD \( = \frac{1}{3} \) of width OA.

Bar DE \( = \frac{OA}{3} \); bar BC \( = \frac{OA}{3} \).

AC \( = \) DE.

(Smeaton "On Windmills," in Tredgold's Hydraulic Tracts.)

166. 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 foot-pounds of work, because of the uncertainty of the amount in pounds of the resistance overcome. In this case, 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. 165, gives some examples of the daily duty of men and horses in units of horizontal transport; \( L \) denoting the load in lb, \( V \) the velocity in feet per second, and \( T \) the number of seconds per day of working: 168. (D.) Heat.—In sect. 163 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 of Fahrenheit in one lb. of liquid water) for every 772 foot-pounds of energy (being the constant already mentioned in sect. 121 as Joule's equivalent). This is called the conversion of heat into mechanical energy.

The efficiency 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 Fahrenheit, or 274°2 Centigrade, below the temperature of melting ice. (Temperatures so measured are called absolute temperatures.) Then maximum theoretical efficiency of the water or other fluid in a steam-engine or other heat-engine

\[ \frac{t_1 - t_2}{t_1} = \frac{\text{heat received}}{\text{heat converted}} \]

The necessary loss of heat by the fluid is \( \frac{t_2}{t_1} \) of the whole heat received by it; and any loss beyond this is waste.