Now "t" and 22=t, therefore z=t. The velocity is therefore again s=nat". It appears therefore that the rule stated above applies equally whether n is an integer or a commensurable fraction.

The proof that the same rule holds good when n is negative is left as an exercise to the reader.

We may now describe shortly the process for finding s when s is given in terms of t. In the fraction

substitute the values of s, and s, in terms of t, and t; strike out any common factors from numerator and denominator; then omit the suffixes.

EXACT DEFINITION OF VELOCITY.

The same difficulties occur in regard to velocities that we have already met with in regard to tangents. When a billiard-ball is sent against a cushion and rebounds, its velocity seems to be suddenly changed into one in another direction. If this were so, we could not speak of a velocity at the instant of striking; though we might speak of a velocity up to that instant and a velocity on from it. Such an event would be indicated by a sharp point in the curve of positions, so far as sudden change in the magnitude of the velocity is concerned, or by a sharp point in the path of the moving body, in case of sudden change in direction. And still greater difficulties may be conceived, when the curve of positions is like the curves on p. 45, 46, with an infinite number of waves.

It is true that there is some reason to believe that sudden changes of velocity never actually occur in nature; that the billiard-ball, for example, compresses the cushion, and while so doing loses velocity at a very rapid rate,

CRITERION OF VELOCITY.

57

yet still not suddenly; and acquires it again as the cushion recovers its form. However, we cannot deal directly with such motions as occur in nature, but only with certain ideal motions, to which they approximate; and in these ideal motions such difficulties may occur. It is therefore necessary to find a criterion for the existence of a velocity at a given instant. In this we shall follow our previous investigation in regard to tangents.

Our first criterion was this: If ta is the tangent up to a point a, it is possible to find a point x on the curve so that the angle cat shall be less than a proposed angle, however small. Suppose the curve to be curve of positions of some rectilinear motion. Take a horizontal line ou, one centimeter

w

long; draw uv vertical, ov parallel to ta, om parallel to ха. Then the angle mov is equal to xat. Also uv is the instantaneous velocity corresponding to the point a in the curve, and um is the mean velocity in the interval corresponding to xa. If the angle xat or, which is the same thing, the angle mov, can be made less than any proposed angle, it follows that my can be made less than any proposed length. Therefore, if uv be the velocity up to a certain instant, it is possible to find an interval ending at that instant in which the mean velocity shall differ from uv less than by a proposed quantity, however small. That is, by reckoning the mean velocity in a sufficiently small interval, we can make it as close an approximation as we like to the instantaneous velocity.

To define the velocity on from an instant, we must take an interval beginning at that instant.

The more accurate criterion of the tangent is that x can be so taken that every chord inside ax shall make with at an angle less than a proposed angle. To express the corresponding criterion for a velocity, let us speak of the mean velocity in an interval of time included within a certain interval as a mean velocity inside that certain interval. Then the criterion is that if v is the velocity up to a certain instant, it is possible to find an

interval ending at that instant such that every mean velocity inside it shall differ from v less than by a proposed quantity, however small.

This criterion applies to variable velocity in rectilineal motion in the first instance; but it clearly extends to determination of the magnitude of the velocity in curvilinear motion, when that has been represented upon a straight line in the manner used for determining its curve of positions. But we may so state the criterion as to give a direct definition of velocity as a vector in all cases of motion.

Let ba be a portion of the path of a moving point, and p, q two positions either intermediate between b and a, or coinciding with either of them. Let the mean velocity from p to q (viz. the step pq divided by the time of taking it) be called a mean velocity inside ba. Let ov represent the velocity up to a in magnitude and direction, and om the mean velocity in pq. Then it

is possible to choose b so that every mean velocity inside ba shall differ

from ov less than by a proposed quantity, however small. We say that om differs from ov by the step mv, and it is meant that mv is shorter than a proposed length.

When there is a line ov having this property, there is said to be a velocity up to the point u, and ov is that velocity. The velocity on from a is defined in a similar manner. When these two are equivalent (have the same magnitude and direction) we speak of ov as the velocity at a.

The motion is then said to be elementally uniform in the neighbourhood of a.

The criterion may be illustrated by applying it to the case sat". Let to, t1, to, t be four quantities in ascending order of magnitude; we propose to shew that nat" is the velocity at the time t. We know that the mean velocity between t, and t, is

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59

The quantity between brackets consists of n terms, each of which is greater than t" and less than t1. Hence the mean velocity is greater than nat." n-1 and less than nat. The difference between these can evidently be made smaller than any proposed quantity by taking to sufficiently near to t. But the mean velocity from t, to t, differs still less from nat"-1 than nat n-1 does. Hence it is possible to choose an interval, from to to t, such that the mean velocity in every interval inside it, from t, to t,, shall differ from nat less than by a proposed quantity. Therefore nat-1 is the velocity up to the instant t. It may be shewn in the same way that it is the velocity on from that instant. Hence the motion is elementally uniform and nat"-1 is the velocity at the instant t.

COMPOSITION OF VELOCITIES.

A velocity, as a directed quantity, or vector, is represented by a step; i.e., a straight line of proper length and direction drawn anywhere. The resultant of any two directed quantities of the same kind may be defined as the resultant of the two steps which represent them. This definition is purely geometrical, and it does not of course follow that the physical combination of the two quantities will produce this geometrical resultant. In the case of velocities, however, we may now prove the following important proposition.

When two motions are compounded together, the velocity in the resultant motion is at every instant the resultant of the velocities in the component motions.

Let oA, OB be velocities in the component motions at a given instant, oC their resultant. Let also oa, ob be mean velocities of the component motions during a certain interval; then we know that their resultant oc is the mean velocity of the resultant motion during that interval, because the mean velocity is simply the step taken in the interval divided by the length of the interval, and the step taken in the resultant motion is of course the resultant of the steps taken in the component motions.

b

B

Now because oA and oB are velocities in the component motions at a certain instant, we know that an interval can be found, ending at that instant, so that the mean velocities oa and ob, for every interval inside it, differ from oA, OB respectively less than a proposed quantity; so, therefore, that Aa and Bb are always both less than the proposed quantity. Now Cc is the resultant

of Aa and Bb, and the greatest possible length of Cc is the sum of the lengths of Aa and Bb. We can secure, therefore, that Ce shall be less than a proposed quantity, by making Aa and Bb each less than half that quantity.

We can therefore find an interval ending at the given instant, every mean velocity inside which differs from o C less than by a proposed quantity, however small. Consequently oC is the velocity of the resultant motion up to the given instant.

It may be shewn in the same way that if oA and oB are velocities in the component motions on from the given instant, then oC is the velocity in the resultant motion on from the given instant; and therefore that when they are velocities at the instant, oC is the velocity at the instant in the resultant motion.

V

It is easy to shew in a similar manner that when a moving point has a velocity in any position, its parallel projection has also a velocity in that position, which is the projection of the velocity of the moving point. For let OV be the velocity of the moving point at a certain instant, OM its mean velocity in a certain interval, and let ov, om be their projections. Then the greatest possible ratio of vm to VM is that of the major axis of an ellipse,

M

which is the parallel projection of a circle in the plane OVM, to the diameter of that circle. In order therefore, to make vm less than a proposed length, we have only to make VM less than a length which is to the proposed