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Philosophy. We devote to it, accordingly, the whole of this chapter; which will form, as it were, the Geometry of the subject, embracing what can be observed or concluded with regard to actual motions, as long as the cause is not sought. In this category we shall first take up the free motion of a point, then the motion of a point attached to an inextensible cord, then the motions and displacements of rigid systems—and finally, the deformations of solid and fluid masses.
7. When a point moves from one position to another it must evidently describe a continuous line, which may be curved or straight, or even made up of portions of curved and straight lines meeting each other at any angles. If the motion be that of a material particle, however, there can be no abrupt change of velocity, nor of direction unless where the velocity is zero, since (as we shall afterwards see) such would imply the action of an infinite force. It is useful to con. sider at the oatset various theorems connected with the geometrical notion of the path described by a moving point; and these we shall now take up, deferring the consideration of Velocity to a future section, as being more closely connected with physical ideas.
8. The direction of motion of a moving point is at each instant the tangent drawn to its path, if the path be a curve; or the path itself if a straight line. This is evident from the definition of the tangent to a curve:
9. If the path be not straight the direction of motion changes from point to point, and the rate of this change, per unit of length of the curve, is called the Curvature. To exemplify this, suppose
two tangents, PT, QU, drawn to a circle, and radii OP, OQ, to the points of contact.
The angle between the tangents is the U
change of direction between P. and l, and the rate of change is to be measured by the relation between this angle and the length of the circular arc PQ. Now, if a
be the angle, s the arc, and r the radius, we see at once that (as the angle between the radii is equal to the angle between the tangents, and as the ineasure of an angle is the
0 : 1 ratio of the arc to the radius, $ 54) r=s, and therefore
is the measure of the curvature. Hence the curvature of a circle is inversely as its radius, and is measured, in terms of the proper unit of curvature, simply by the reciprocal of the radius.
10. Any small portion of a curve may be approximately taken as a circular arc, the approximation being closer and closer to the truth, as the assumed arc is smaller. The curvature at any point is the reciprocal of the radius of this circle for a small arc on each side of the point.
11. If all the points of the curve lie in one plane, it is called a plane 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 mea. sure 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 cur. vature 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 Epicyloid is eight right angles; and
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, 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 exces. sively 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 Pr
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.
(1) Suppose we have a single pulley B, about which
the flexible and inextensible cord ABP is wrapped, and P. suppose its free portions to be parallel. If (A being fixed) a point P of the cord
iP 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 B 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 Pis 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
(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 faced, 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 hyperbolā 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.
22. Let AB be any curve, PQ a portion of an involute, pP, 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 per
P pendicular to the tangent) to the curve PQ. And thus the evolute of PQ is
B a definite curve, viz. the envelop of (or line which is touched by) the normals drawn at every point of PC 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 obyious result of the
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 uniformi, 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 cal. culus 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 in t 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
= 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