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curve, and if it be made up of portions of straight or curved lines it is called a plane polygon. If the line do not lie in one plane, we have in one case what is called a curve of double curvature, in the other a gauche polygon. The term 'curve of double curvature' is a very bad one, and, though in very general use, is, we hope, not ineradicable. The fact is, that there are not two curvatures, but only a curvature (as above defined) of which the plane is continuously changing, or twisting, round the tangent line. The course of such a curve is, in common language, well called 'tortuous;' and the measure of the corresponding property is conveniently called Tortuosity.

12. The nature of this will be best understood by considering the curve as a polygon whose sides are indefinitely small. Any two consecutive sides, of course, lie in a plane-and in that plane the curvature is measured as above; but in a curve which is not plane the third side of the polygon will not be in the same plane with the first two, and therefore the new plane in which the curvature is to be measured is different from the old one. The plane of the curvature on each side of any point of a tortuous curve is sometimes called the Osculating Plane of the curve at that point. As two successive positions of it contain the second side of the polygon above mentioned, it is evident that the osculating plane passes from one position to the next by revolving about the tangent to the curve.

13. Thus, as we proceed along such a curve, the curvature in general varies; and, at the same time, the plane in which the curvature lies is turning about the tangent to the curve. The rate of torsion, or the tortuosity, is therefore to be measured by the rate at which the osculating plane turns about the tangent, per unit length of the curve. The simplest illustration of a tortuous curve is the thread of a screw. Compare § 41 (d).

14. The Integral Curvature, or whole change of direction, of an arc of a plane curve, is the angle through which the tangent has turned as we pass from one extremity to the other. The average curvature of any portion is its whole curvature divided by its length. Suppose a line, drawn through any fixed point, to turn so as always to be parallel to the direction of motion of a point describing the curve: the angle through which this turns during the motion of the point exhibits what we have defined as the integral curvature. In estimating this, we must of course take the enlarged modern meaning of an angle, including angles greater than two right angles, and also negative angles. Thus the integral curvature of any closed curve or broken line, whether everywhere concave to the interior or not, is four right angles, provided it does not cut itself. That of a Lemniscate, 8, is zero. That of the Epicycloid is eight right angles; and

so on.

15. The definition in last section may evidently be extended to a plane polygon, and the integral change of direction, or the angle between the first and last sides, is then the sum of its exterior angles,

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all the sides being produced each in the direction in which the moving point describes it while passing round the figure. This is true whether the polygon be closed or not. If closed, then, as long as it is not crossed, this sum is four right angles,—an extension of the result in Euclid, where all reëntrant polygons are excluded. In the star-shaped figure, it is ten right angles, wanting the sum of the five acute angles of the figure; i. e. it is eight right angles.

16. A chain, cord, or fine wire, or a fine fibre, filament, or hair, may suggest, what is not to be found among natural or artificial productions, a perfectly flexible and inextensible line. The elementary kinematics of this subject require no investigation. The mathematical condition to be expressed in any case of it is simply that the distance measured along the line from any one point to any other, remains constant, however the line be bent.

17. The use of a cord in mechanism presents us with many practical applications of this theory, which are in general extremely simple; although curious, and not always very easy, geometrical problems occur in connexion with it. We shall say nothing here about such cases as knots, knitting, weaving, etc., as being excessively difficult in their general development, and too simple in the ordinary cases to require explanation.

18. The simplest and most useful applications are to the Pulley and its combinations. In theory a pulley is simply a smooth body which changes the direction of a flexible and inextensible cord stretched across part of its surface; in practice (to escape as much as possible of the inevitable friction) it is a wheel, on part of whose circumference the cord is wrapped.

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B

(1) Suppose we have a single pulley B, about which the flexible and inextensible cord ABP is wrapped, and suppose its free portions to be parallel.

If (A being fixed) a point P of the cord
be moved to P', it is evident that each
of the portions AB and PB will be
shortened by one-half of PP'. Hence,
when P moves through any space in
the direction of the cord, the pulley B
moves in the same direction, through
half the space.

(2) If there be two cords and two pulleys, the ends AA' being fixed, and the other end of AB being attached to the pulley B'-then, if all free parts of the cord are parallel, when P is moved to P', B' moves in the same direction through half the space, and carries with it one end of the cord AB. Hence B moves through half the space B' did, that is, one fourth of PP'.

B

P

B

(3) And so on for any number of pulleys, if they be arranged in the above manner. Similar considerations enable us to determine the relative motions of all parts of other systems of pulleys and cords as long as all the free parts of the cords are parallel.

Of course, if a pulley be fixed, the motion of a point of one end of the cord to or from it involves an equal motion of the other end from or to it.

If the strings be not parallel, the relations of a single pulley or of a system of pulleys are a little complex, but present no difficulty. 19. In the mechanical tracing of curves, a flexible and inextensible cord is often supposed. Thus, in drawing an ellipse, the focal property of the curve shows us that if we fix the ends of such a cord to the foci and keep it stretched by a pencil, the pencil will trace the curve.

By a ruler moveable about one focus, and a string attached to a point in the ruler and to the other focus, and kept tight by a pencil sliding along the edge of the ruler, the hyperbola may be described by the help of its analogous focal property; and so on.

20. But the consideration of evolutes is of some importance in Natural Philosophy, especially in certain mechanical and optical questions, and we shall therefore devote a section or two to this application of Kinematics.

Def. If a flexible and inextensible string be fixed at one point of a plane curve,. and stretched along the curve, and be then unwound in the plane of the curve, its extremity will describe an Involute of the curve. The original curve, considered with reference to the other, is called the Evolute.

21. It will be observed that we speak of an involute, and of the evolute, of a curve. In fact, as will be easily seen, a curve can have but one evolute, but it has an infinite number of involutes. For all that we have to do to vary an involute, is to change the point of the curve from which the tracing-point starts, or consider the involutes described by different points of the string; and these will, in general, be different curves. But the following section shows that there is but one evolute.

P

22. Let AB be any curve, PQ a portion of an involute, pP, qQ positions of the free part of the string. It will be seen at once that these must be tangents to the arc AB at p and q. Also the string at any stage, as pP, ultimately revolves about p. Hence pP is normal (or perpendicular to the tangent) to the curve PQ. And thus the evolute of PQ is a definite curve, viz. the envelop of (or line which is touched by) the normals drawn at every point of PQ, or, which is the same thing, the locus of the centres of the circles which have at each point the same tangent and curvature as the curve PQ. And we may merely mention, as an obvious result of the

B

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mode of tracing, that the arc qp is equal to the difference of qQ and pP, or that the arc pA is equal to pP. Compare § 104.

23. The rate of motion of a point, or its rate of change of position, is called its Velocity. It is greater or less as the space passed over in a given time is greater or less: and it may be uniform, i.e. the same at every instant; or it may be variable.

Uniform velocity is measured by the space passed over in unit of time, and is, in general, expressed in feet or in metres per second; if very great, as in the case of light, it may be measured in miles per second. It is to be observed that Time is here used in the abstract sense of a uniformly-increasing quantity-what in the differential calculus is called an independent variable. Its physical definition is given in the next chapter.

24. Thus a point, which moves uniformly with velocity v, describes a space of v feet each second, and therefore vt feet in t seconds, t being any number whatever. Putting s for the space described int seconds, we have

s=vt.

Thus with unit velocity a point describes unit of space in unit of time.

25. It is well to observe here, that since, by our formula, we have generally

$ v= "

and since nothing has been said as to the magnitudes of s and t, we may take these as small as we choose. Thus we get the same result whether we derive v from the space described in a million seconds, or from that described in a millionth of a second. This idea is very useful, as it makes our results intelligible when a variable velocity has to be measured, and we find ourselves obliged to approximate to its value (as in § 28) by considering the space described in an interval so short, that during its lapse the velocity does not sensibly alter in value.

26. When the point does not move uniformly, the velocity is variable, or different at different successive instants: but we define the average velocity during any time as the space described in that time, divided by the time; and, the less the interval is, the more nearly does the average velocity coincide with the actual velocity at any instant of the interval. Or again, we define the exact velocity at any instant as the space which the point would have described in one second, if for such a period it kept its velocity unchanged.

27. That there is at every instant a definite velocity for any moving point, is evident to all, and is matter of everyday conversation. Thus, a railway train, after starting, gradually increases its speed, and every one understands what is meant by saying that at a particular instant it moves at the rate of ten or of fifty miles an hour,—although, in the course of an hour, it may not have moved a mile altogether. We may suppose that, at any instant during the motion, the steam is so adjusted as to keep the train running for some time at a uniform velocity. This is the velocity which the train had at the instant in question. Without supposing any such definite adjustment of the

driving-power to be made, we can evidently obtain an approximation. to the velocity at a particular instant, by considering (§ 25) the motion for so short a time, that during that time the actual variation of speed may be small enough to be neglected.

28. In fact, if v be the velocity at either beginning or end, or at any instant, of an interval t, and s the space actually described in

8

t

that interval; the equation = (which expresses the definition of the average velocity, § 26) is more and more nearly true, as the velocity is more nearly uniform during the interval t; so that if we take the interval small enough the equation may be made as nearly exact as we choose. Thus the set of values

Space described in one second,

Ten times the space described in the first tenth of a second,

A hundred

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and so on, give nearer and nearer approximations to the velocity at the beginning of the first second.

The whole foundation of Newton's differential calculus is, in fact, contained in the simple question, 'What is the rate at which the space described by a moving point increases?' i. e. What is the velocity of the moving point? Newton's notation for the velocity, i, e. the rate at which s increases, or the fluxion of s, is 8. This notation is very convenient, as it saves the introduction of a second letter.

29. The preceding definition of velocity is equally applicable whether the point move in a straight or a curved line; but, since, in the latter case, the direction of motion continually changes, the mere amount of the velocity is not sufficient completely to describe the motion, and we must have in every such case additional data to thoroughly specify the motion.

In such cases as this the method most commonly employed, whether we deal with velocities, or (as we shall do farther on) with accelerations and forces, consists in studying, not the velocity, acceleration, or force, directly, but its resolved parts parallel to any three assumed directions at right angles to each other. Thus, for a train moving up an incline in a N.E. direction, we may have the whole velocity and the steepness of the incline given; or we may express the same ideas thus the train is moving simultaneously northward, eastward, and upward-and the motion as to amount and direction will be completely known if we know separately the northward, eastward, and upward velocities-these being called the components of the whole velocity in the three mutually perpendicular directions N., E., and up.

30. A velocity in any direction may be resolved in, and perpendicular to, any other direction. The first component is found by multiplying the velocity by the cosine of the angle between the two directions; the second by using as factor the sine of the same angle.

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