(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, every point of it will describe an Involute of the curve. The original curve is called the Evolute of any one of the others.

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

mode of tracing, that the arc qp is equal to the difference of Q 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

S

v = t'

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 (S 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

S

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,

hundredth

A hundred 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 s. notation is very convenient, as it saves the introduction of a second letter.

This

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. Thus a point moving with velocity V up an Inclined Plane, making an angle a with the horizon, has a vertical velocity V sina and a horizontal velocity Vcos a.

Or it may be resolved into components in any three rectangular directions, each component being found by multiplying the whole velocity by the cosine of the angle between its direction and that of the component. The velocity resolved in any direction is the sum of the resolved parts (in that direction) of the three rectangular components of the whole velocity. And if we consider motion in one plane, this is still true, only we have but two rectangular components.

31. These propositions are virtually equivalent to the following obvious geometrical construction :

To compound any two velocities as OA, OB in the figure; where

B

A

and direction.

OA, for instance, represents in magnitude and direction the space which would be described in one second by a point moving with the first of the given velocities—and similarly OB for the second; from A draw AC parallel and equal to OB. Join OC: then OC is the resultant velocity in magnitude

OC is evidently the diagonal of the parallelogram two of whose sides are OA, OB.

Hence the resultant of any two velocities as OA, AC, in the figure, is a velocity represented by the third side, OC, of the triangle OAC.

Hence if a point have, at the same time, velocities represented by OA, AC, and CO, the sides of a triangle taken in the same order, it is at rest.

Hence the resultant of velocities represented by the sides of any closed polygon whatever, whether in one plane or not, taken all in the same order, is zero.

Hence also the resultant of velocities represented by all the sides of a polygon but one, taken in order, is represented by that one taken in the opposite direction.

When there are two velocities, or three velocities, in two or in three rectangular directions, the resultant is the square root of the sum of their squares; and the cosines of its inclination to the given directions are the ratios of the components to the resultant.

32. The velocity of a point is said to be accelerated or retarded according as it increases or diminishes, but the word acceleration is generally used in either sense, on the understanding that we may regard its quantity as either positive or negative: and (§ 34) is farther generalized so as to include change of direction as well as change of speed. Acceleration of velocity may of course be either

uniform or variable. It is said to be uniform when the point receives equal increments of velocity in equal times, and is then measured by the actual increase of velocity per unit of time. If we choose as the unit of acceleration that which adds a unit of velocity per unit of time to the velocity of a point, an acceleration measured by a will add a units of velocity in unit of time—and, therefore, at units of velocity in units of time. Hence if v be the change in the velocity during the interval t,

33. Acceleration is variable when the point's velocity does not receive equal increments in successive equal periods of time. It is then measured by the increment of velocity, which would have been generated in a unit of time had the acceleration remained throughout that unit the same as at its commencement. The average acceleration during any time is the whole velocity gained during that time, divided by the time. In Newton's notation & is used to express the acceleration in the direction of motion; and, if vs as in § 28, we have a=v=s.

=

34. But there is another form in which acceleration may manifest itself. Even if a point's velocity remain unchanged, yet if its direction of motion change, the resolved parts of its velocity in fixed directions will, in general, be accelerated.

Since acceleration is merely a change of the component velocity in a stated direction, it is evident that the laws of composition and resolution of accelerations are the same as those of velocities.

We therefore expand the definition just given, thus :-Acceleration is the rate of change of velocity whether that change take place in the direction of motion or not.

35. What is meant by change of velocity is evident from § 31. For if a velocity OA become OC, its change is AC, or OB.

Hence, just as the direction of motion of a point is the tangent to its path, so the direction of acceleration of a moving point is to be found by the following construction:

From any point O draw lines OP, OQ, etc., representing in magnitude and direction the velocity of the moving point at every instant. (Compare § 49.) The points, P, Q, etc., must form a continuous curve, for ($7) OP cannot change abruptly in direction. Now if Q be a point near to P, OP and OQ represent two successive values of the velocity. Hence PQ is the whole change of velocity during the interval. As the interval becomes smaller, the direction PQ more and more nearly becomes the tangent at P. Hence the direction of acceleration is that of the tangent to the curve thus described.

The magnitude of the acceleration is the rate of change of velocity, and is therefore measured by the velocity of P in the curve PQ.