to their length, to be kept always the same. Then it will be impossible to take b so near to a that there shall be no point of inflexion between them. Also it is clear in this case that there is no real tangent at a; for however near we get to a, the direction of the curve sways from side to side through the same range.

If, however, the waves are so drawn that the ratio of their height to their length becomes smaller and smaller as they approach a, so that they get more and more flat without any limit, then although the proof of the rule fails as before, there is a real tangent át a, namely, the common tangent at to the two circles.

a

In both these cases our criterion for a tangent is satisfied; that is to say, there is a line at such that by taking a point x on the curve near enough to a, the angle xat can be made less than any proposed angle. Yet in one of these cases this line at is a tangent, and in the other it is not. We must therefore find a better criterion, which will distinguish between these cases.

The tangent to a circle has the following property. If we take any two points p and q between a and b, the chord pq makes with the tangent

at a an angle less than aob. For the angle between pq and at is equal to aom, where om is perpendicular to pq. Let pq be called a chord inside ab, even if p is at a or q is at b. Then we can

a

find a point b such that every chord inside ab makes with at an angle less than a proposed angle, however small. For we have only to draw the angle aob a little less than the proposed angle.

Now the second of our exceptional curves, that which really has a tangent, has also this property, that we can find a point b so near to a that every chord inside ab shall make with at an angle less than a proposed angle,

DEFINITION OF TANGENT.

47

however small. For since the waves get flatter and flatter without limit, the tangents at the successive points of inflexion make with at angles which decrease without limit. We have then only to find a point of inflexion whose tangent makes with at an angle less than the proposed angle, and take b at this point or between it and a.

But the first curve has not this property, for the inclinations of the tangents at the points of inflexion are always the same, and any one of these counts as a chord inside ab.

We shall now therefore make this definition:

When there is a line at through a point a of a curve having the property that, any angle being proposed, however small, it is always possible to find a point b so near to a on one side that every chord inside ab makes with at an angle less than the proposed angle; then this line at is called the tangent of the curve up to the point a on that side.

When there are tangents up to the point a on both sides, and these two are in one straight line, that straight line is called the tangent at a. In this case the curve is said to be elementally straight in the neighbourhood of a. It has the property that the more it is magnified, the straighter it looks.

Going back to our first and simpler definition of a tangent, as the final position of a line pq which is made to move so that p and q coalesce at a, we see that not only does it always find the tangent when there is one, but that when there is not, the final position of pq will not be determinate, but will depend upon the way in which p and q are made to coalesce at a. When therefore this method gives us a determinate line, we may be sure that that line is really a tangent.

The problem which we have now to consider is the following:-Suppose that we know the position of a point

at every instant of time during a certain period, it is required to find out how fast it is going at every instant during the same period. For example, in the simple harmonic motion described in the last chapter, we know the position of the point m by geometrical construction; namely we determine the position of p by measuring an arc ap on the circle proportional to the time, and then we draw a perpendicular pm to the diameter aa'. The The problem is to find out how fast the point m is moving when it is in any given position. The rate at which it is moving is called its velocity.

Let us now endeavour to form a clearer conception of this quantity that we have to measure; and for this purpose let us consider the simplest case, that of uniform motion in a straight line. We say of a train, or a ship, or a man walking, that they go at so many miles per hour; of sound, that it goes 1090 feet per second; of light, that it goes 200,000 miles per second. These statements seem at first to mean only that a certain space has been passed over in a certain time; that the man, for instance, has in a given hour walked so many miles. But because we know that the motion is uniform, we are hereby told not only how far the man walks in an hour, but also how far he walks in any other period of time. In walking on a French road, for example, it is convenient to walk about six kilometers per hour. Now this is one kilometer per ten minutes, and that is the same thing as one hectometer per minute. In what sense the same thing? It is not the same thing to walk a hectometer in one minute as it is to walk six kilometers in one hour. But the rate at which one is moving is the same during the minute as it is during the hour. Thus we see that to say how fast a body is going is to make a statement about its state of motion at any instant and not about its change of position in any length of time. The velocity of a moving body is an instantaneous property of it which may or may not change from instant to instant; and the peculiarity of uniform motion, in which equal spaces are traversed in equal times, is that the velocity remains constant throughout the motion; a body which moves uniformly is always going at the same rate.

ABSOLUTE MEASURE OF VELOCITY.

49

But although this rate is a property of the motion which belongs to it at a given instant, we cannot measure it instantaneously. In order to find out how fast you are walking at a particular instant, you must keep on walking. at that same rate for a definite time, and then see how far you have gone. Only, as we noticed before, it does not matter what that definite time is. Whether you find that you have walked one hectometer in a minute, or one kilometer in ten minutes, or six kilometers in an hour, the velocity so measured is the same velocity. Now for comparing velocities together, it is found convenient to refer them all to the same interval of time. Which goes faster, sound at 1090 feet per second, or a molecule of oxygen in the air at seventeen miles a minute? Clearly we must find how far the molecule of oxygen would go in a second, and compare that distance with 1090 feet. For scientific purposes the second is the period of time adopted in measuring velocities; and we may say that we know the rate at which a thing is moving when we know how far it would go in a second if it went at that same rate during the second.

A velocity, then, is measured by a certain length; namely, the distance which a body having the velocity during a second would pass over in that second. It may therefore be specified either graphically, by drawing a line to represent that length on a given scale, or by numerical approximation. When a velocity is described as so many centimeters per second it is said to be expressed in absolute measure. Thus the absolute unit of velocity is one centimeter per second. The absolute measure of six kilometers per hour is 1663. More generally we may say that the unit of velocity is one unit of length per unit of time.

This last statement is sometimes expressed in another way. Let [V] denote the unit of velocity, [L] the unit of length, and [7] the unit of time; then [V]

[L]

[T]

Here

the word per has been replaced by the sign for divided by: now it is nonsense to say that a unit of velocity is a unit of length divided by a unit of time in the ordinary

sense of the words. But we find it convenient to give a new meaning to the words "divided by," and to the symbol which shortly expresses them, so that they may be used to mean what is meant by the word per in the expression" miles per hour." This convenience is made manifest when we have to change from one unit to another. Suppose, for instance, that we want to compare the unit of velocity one centimeter per second with another unit, one kilometer per hour, We shall have

We might have got to the same result by saying that one kilometer per hour is 100,000 centimeters per 3600 seconds, that is, 277 centimeters per second. Hence if we give to the symbol of division this new meaning, and then treat it by the rules applicable to the old meaning, we arrive at right results; and we save ourselves the trouble of inventing a new symbol by using the old one in a new sense.

Another way of expressing the equation [V]=[L]: [T] is to say that velocity is a quantity of dimensions 1 in length and 1 in time.

Velocity is a directed quantity; and therefore is not fully specified until its magnitude and direction are both given. The velocity of translation of a rigid body is adequately represented by a straight line of proper length and direction drawn anywhere. Consequently it is a vector quantity, in the sense already explained.

In the uniform rectilinear motion p= a+ tẞ, the step taken in one second is B, which is therefore the velocity. When the step op from a fixed point o to the moving