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

B

from ou 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 a, 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 s = at". 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

CRITERION OF VELOCITY.

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," 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 nat1 than nat," 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

C

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 Cc 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 oC 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

M

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,

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

PROJECTION OF VELOCITY.

61

one as the diameter of the circle to the major axis of its projection. Hence an interval can be taken such that the mean velocity of the projected motion for every included interval shall differ from ov less than by a proposed quantity; or ov is the instantaneous velocity of the projected motion.

For an example of the last proposition, we may consider the simple harmonic motion, which is an orthogonal projection of uniform circular motion on a line in its plane. The velocity of p is na where a is the radius of the circle, and it is in the direction tp. The horizontal component of this is the velocity of m. The horizontal component is na sin aop

na cos ptm

=

=

na sin (nt + €)

= na cos (nt + e + 1⁄2π).

Hence if

we find

t

a

s = a cos (nt + €),

s = na cos (nt + € + 1π),

The

and the rule is the same as in circular motion. velocity is evidently = n.pm, by which representation the changes in its magnitude are rendered clear. The same result may be obtained by means of the tangent to the harmonic curve, p. 43.

The velocity in elliptic harmonic motion may be found either by composition of two simple harmonic motions, or directly by projection from the circle. We thus find that when

p =

= a cos (nt + e) + B sin (nt + e), then p = nx cos (nt + e + 1π)+ nß sin (nt +€ + 1π),

In the parabolic motion p=a+tß+ty we may now see that

p = B + 2ty.

the rule being to multiply each term by the index of t and then reduce this index by unity. Thus we can always find the velocity when the position-vector is a rational integral function of t.

FLUXIONS.

A quantity which changes continuously in value is called a fluent. It may be a numerical ratio, or scalar quantity (capable of measurement on a scale); or it may be a directed quantity or vector; or it may be something still more complex which we have yet to study. In the first case the quantity, being necessarily continuous because it changes continuously, can only be adequately specified by a length drawn to scale, or by an angle; and we may always suppose an angle to be specified by the length of an arc on a standard circle. Let one end of the length which measures the quantity be kept fixed, then as the quantity changes the other end must move. The velocity of that end is the rate of change of the quantity. Thus we may say that water is poured into a reservoir at the rate of x gallons per minute. Let the contents of the reservoir be represented on a straight line, so that every centimeter stands for a gallon; and let the change in these contents be indicated by moving one end of the line. Then this end will move at the rate of x centimeters per minute. w is the number of gallons in the reservoir, it is also the distance of the moveable end of the line from the

If