a greater distance must be traversed. If nt is equal to mT, ns is equal to mS; and if nt is less than mT, ns is less than mS. Hence by Euclid's definition of proportion,

SsT: t.

Letv be the number of centimeters gone over (or described) in one second; then s : v=t: 1, or s= vt, where s is the number of centimeters described in t seconds. Here all three numbers may be incommensurable; but the algebraic formula svt supplies us with a rule for calculating s when t is known; viz., multiply t by v.

9

The curve of positions in this case is a straight line. For, if we set up the length v above the point 1, and draw through o the straight line ovq; then on drawing tq vertical through any point t, we shall have tq: v=ot: 1, or tq correctly represents the distance described in the time ot.

0

1

Uniform motion may of course take place along any path whatever. But there are two cases of special interest; when the path is a straight line and when it is a circle.

UNIFORM RECTILINEAR MOTION.

Let p be a point moving uniformly along the straight line abp, and let o be any fixed point. We shall completely describe the position of the point p at any instant, if we specify the step which must be taken to go to p from o at that instant. Now op=oa+ap. Let ab be the distance traversed in one second, then ap, being the distance traversed in t seconds, is tab. Hence we have

op = oa+t. ab,

въ

or, if we denote the step op by p, oa by a, ab by B, then p = a+ tß.

UNIFORM RECTILINEAR MOTION.

17

This is called the equation of uniform rectilinear motion. It is simply shorthand for this statement:-the steps to be taken in order to get from o to the position of p after t seconds are, first, the step a (oa) which takes us to the position at the beginning of the motion, and then t times the step B (ab).

Two uniform rectilinear motions compound into a uniform rectilinear motion.

While p moves uniformly along the line ab, let q move uniformly, relative to p, along cd; and let cd be the distance traversed in one second in the relative motion. Draw de equal and parallel to ab, then ce is the actual motion of q in one second. Draw qr parallel to ab, meeting ce pro

a

9

e

duced in r. Then, cq being traversed in the same time as ap, we must have cq: cd=ap: ab=t: 1. Now cq: cd=qr de, so that qrap. Hence r is the actual position of q at the end of the time t. It is in the straight line ce, and cr: cecq : cd=t: 1. Thus the actual motion of q is a uniform rectilinear motion.

The same thing appears by considering the equations. Let P1 be the step op, and p, the step pq; then p = p, + P2 is the step oq. Now we have

P1 = a+tẞ1, where a1 = oa, B1 = ab,

=

29

=

= ac, B2 = cd,

and therefore

p = P1+ P2 = α1 + α2 + t (ß1 + ß2), the equation to a uniform rectilinear motion.

The curve of positions of any motion whatever may be conceived to be constructed by help of a uniform rectilinear motion, in this way. Let the

a

original motion be that of a point p along the path ab.

Y

Let a point p' move along oY at the same time, so that the distance op' is at every instant equal to the distance ap measured along the path. P While this motion takes place, let the straight line oY have a uniform horizontal translation of one centimeter in every second; then by this combination of motions the point p' will describe the curve of positions oq.

t

Hence the curve of positions of any rectilinear motion is described by combining that motion with a uniform rectilinear motion of one centimeter per second in a direction at right angles to it.

UNIFORM CIRCULAR MOTION.

In uniform circular motion every point p of the moving body goes round a circle so as to describe equal arcs in equal times, and therefore proportional arcs in different times.

The radius of the circle is called the amplitude of the motion..

The time of going once round is called the period.

If the arcs measured on the circle are reckoned from a point a, and if the moving point started from e at the beginning of the time considered, the angle aoe is called the angle at epoch, or shortly the epoch. Strictly speaking, the epoch is the beginning of the time considered. The ratio of the arc ap to the whole circumference is called the phase at any instant.

Let n be the circular measure of the arc described in one second, and a the radius of the circle; so that na is the length of the arc described in one second. Then nat is the length of arc,

b

a

ep, described in t seconds, and nt is its circular measure.

UNIFORM CIRCULAR MOTION.

19

Let also e be the circular measure of aoe; then circular measure of aop

= nt + €.

We shall now obtain an expression for the step op at any instant. Draw pm, ob, perpendicular to oa. Then

ор = om + mp.

от

Now as far as

lengths are concerned,

ор

ор

and

mp

ob sin aop.

In the

in length, om- oa cos aop equation om = oa cos aop, the quantities om and oa may be regarded as steps; for as they are in the same direction, one is equal to the other multiplied by the numerical ratio cos aop. The same may be said of the equation mp = ob sin aop. Now aop=nt +e, and therefore op = oa. cos (nt + e) + ob . sin (nt + e),

or if we write p for op, ai for oa, and aj for ob, so that i, j are unit steps along oa, ob, then

p = a {icos (nt + e) + j sin (nt + e)}.

This is the equation to uniform circular motion. The angle nte is called the argument of this expression for p.

A circular motion which goes round like the hands of a clock, or clockwise, is said to be in the negative sense; one that goes round the other way, or counter-clockwise, is said to be in the positive sense.

Two uniform circular motions of the same period and the same sense compound into a uniform circular motion of that period and

sense.

Suppose the circles so placed as to have the same centre. The motions of p and q relative to o may be combined by completing the parallelogram oprq; then the

motion of r is the resultant. We may consider the parallelogram oprq to be made of four jointed rods, of which op and oq turn round o. When these motions have the

same period and the same sense, the angle poq remains always constant; therefore the shape of the parallelogram remains unchanged. Consequently or is of constant length, and makes always the same angle with op or 09. Hence r goes round uniformly in a circle of radius or.

Let op = p1, oq = P2, orp. Then, if a, b are the amplitudes, i, j unit steps at right angles to one another, P1 = ai cos (nt + €) + aj sin (nt + €),

P2 = bi cos (nt + €) + bj sin (nt + €),

P2

p=p1+p1=i{(a cose,+bcose) cosnt― (a sine,+bsin e,) sinnt} +j{(acose,+bcose) sinnt + (a sine,+bsine) cosnt},

From these two equations we must find c and e. Dividing the second by the first, we find

Squaring both sides of both equations, and adding them together, we find

These formulæ determine the amplitude and cpoch of the resultant motion. It is left to the reader to verify them by comparison with the geometrical solution.

Like the preceding theorem about uniform rectilinear motions, this theorem may be extended to any number of circular motions of the same period and sense; by first compounding the first two, then the third with their resultant, and so on. Or the extended theorem may be proved directly, either by the geometrical or by the analytical method.

HARMONIC MOTION.

While the point p moves uniformly round a circle, let a perpendicular pm be continually let fall upon a diameter