680 MECHANICS Speed." Average speed. Measure ment of uniform and of variable a definite point of the cord ; FP P the path described by that point. Then PP P is one of the involutes of MMT ; the others are the curves traced by other points of the string. But, with reference to PP P , the curve MM P is the " evolute." The evolute of such a curve is, in fact, unique ; for it is obvious that the line MP, in any of its positions, is revolving about the point of contact M with the evolute ; so that P describes an infinitesimal arc of a circle of which M is the centre. Thus the evolute of a plane curve is the locus of its centre of curvature. And it is clear from the genesis of the involute that PM = P M + M M - P M M . For the analytical discussion of evolutes, see vol. xiii. p. 26. The subject of evolutes is of importance in various branches of physics, especially in optics. In mechanics its chief use is connected with the theory of the pendulum, as it shows how to cause the bob to move in a cycloid, the only path in which the time of oscillation is the same what ever be the extent of the oscillations. 24. When the line, curved or straight, in which the motion takes place is given, the position of the moving point is at once assigned in terms of a single numerical quantity. In fact it has only one degree of freedom, and its position is known by the length of the arc of the curve from any fixed point to the given position. In such a case as this we are not concerned with the direction of the motion, for that is already assigned at every point of the path. We are concerned only with what we may call the " speed " of the motion. (We purposely avoid the use of the term " velocity " here, because it properly includes direction as well as speed, as will be seen later.) 25. Suppose an observer to be watching the motion (as, for instance, a traveller by rail notes the telegraph posts which he passes, referring at each to his watch), and to find that at any time ^ the moving point was at s v while at time t 2 it was at s 2 . Then it is clear that the average speed during this part of the motion is to be found by dividing the number of units of space passed over by the number of units of time employed. For it must be greater as the former is greater and less as the latter is greater. Hence the average speed is f _ If the speed has been uniform during the motion observed, this average value has coincided with the actual value all through; and, if the measures of space and time are accurate, we shall get exactly the same value of this ratio whether the interval of time is small or large. Hence, if v be the speed of a uniformly moving point, the space it describes in time t is vt. But if the speed has been variable, it must at some parts of the interval have been greater, at others less, than this average. And the shorter we take the interval the less will be the difference between the greatest and least speeds during its lapse, so that the average speed will coincide more and more nearly with the actual speed. In the language of "fluxions" (which was invented for the sake of this subject) the measure of the speed at any time ^ is when the interval t. 2 -t l is shortened indefinitely. The accuracy of the preceding process depends entirely upon the limitations we have introduced for the purpose of con fining ourselves to cases which can occur in ordinary physical problems. For the general reasoning on which it is based is obviously inapplicable to cases in which the speed alters by jerks at least during the interval con sidered, small as it may be. But we are fortunately not required to discuss here the very delicate questions to which this may give rise. Considerable difficulty is sometimes felt by a student when he is told that at a certain part of its course a point has a speed say of 10 miles an hour, while the whole course may be only a few inches. But this arises from the novelty of the conception. It is not meant, when we speak of a speed of 10 miles per hour, that the motion necessarily lasts for an hour, or even for a second, but only that, if the then speed were to be maintained con stant for an hour, the moving point s path, of whatev r form, would be exactly 10 miles long. In actual experience in a railway train we can judge the speed (roughly at least), and we find nothing strange in saying "Now we are going at twenty miles an hour," " Now at six," and so on. And it is clear that, after the steam is put on, the train, how ever short its run, must go through all rates of speed from zero to its maximum, and then through all of them to zero again, when the steam is cut off and the brake applied. In the language of the differential calculus this becomes The fluxional notation of Newton, in which the dot over a quantity expresses the rate of its increase, i.e., its differential coefficient with regni d to time considered as the independent variable, is still very convenient in abstract dynamics, and is, in fact, indispensable when we come to the higher generalizations. We shall, therefore, freely employ it when it is specially useful. 26. Whether uniform or variable, speed depends for Dimen its numerical value upon the units chosen for linear space sions c and for time. Its dimensions are [LT* 1 ], and consequently spee its numerical expression is increased in proportion as the unit of time is increased, and diminished in proportion as that of length is increased. Thus the speed represented by 10 in feet per second becomes 3600 , _75 5280 "ll when expressed in miles per hour. 27. The rate at which the speed (when not uniform) Kate c changes is found by a process precisely similar to that chang< employed for the speed itself. Let the speed at time ^ be observed to be v l , f 11 )5 2 " " 2 then the average rate of increase of speed during the interval is t. 2 - 1, The dimensions of this quantity are obviously [LT~ 2 ]. Thus its numerical value is diminished, like that of speed, in proportion as the unit of length is increased. But it is increased in the duplicate of the proportion in which the unit of time is increased. For instance a rate of increase of speed of 32 2 feet per second per second (the mere statement is enough to show the double dependence on the time unit) becomes = 79,036 nearly, when expressed in terms of miles and hours. 28. When the rate of increase of speed is uniform, the TJnifor above average value is its actual value throughout the cnan g e interval. Hence with uniform rate of increase = a, a speed V becomes in time t tf-V + o*. Also, as it increases uniformly, its average value during time t is half way between its values at the beginning and end of that time; i.e., it is V + at . The space described during the interval is at once found ( 25) as the product of the interval and the average
speed during its lapse; i.e., it is