Examples to Section I.

(1). A man rides, at a uniform pace, 15 miles in 2 hours; if his velocity be 5, and a yard be the unit of length, what is the unit of time?

In this case, we see that in accordance with the definition of the measure of velocity, the man traverses 5 yards in the unit of time.

But he goes over 15 x 1760 yards in two hours, and therefore over 5 yards in

2

3 × 1760

Hence the unit of time is

hours.

(2). A person travelling on a railway with a velocity v, wishes to fire at an object in a particular direction; supposing the course of the bullet to be rectilinear, and its velocity uniform, equal v' in the direction of the barrel of the gun, determine the direction in which he should aim.

The bullet will move under the action of two causes; the first of which, if it acted alone, would carry it along uniformly with velocity in the direction in which the carriage is moving; the other would take it uniformly with a velocity v' in the direction of the gun-barrel; its real direction then will be, by the parallelogram of velocities, in the direction of the diagonal of the parallelogram, whose sides are in the first two directions and are proportional to v and v' respectively.

To secure that this diagonal shall pass through the object required, we may make the following construction:

Supposing A to be the position of the person who fires

the gun, and P the object; take AB in the direction of the carriage's motion, proportional to v, and describe a circle about B with a radius proportional to v': let D be the point where this circle meets AP; complete the parallelogram ABDC. If the gun be fired in the direction AC, the course of the ball will evidently, from what has been said, be the line AP.

Since the circle will generally meet AP in two points, there will be two directions in which the gun may be fired in order to hit P; subject to exceptions which can be easily interpreted.

SECTION II.

FORMULE FOR RECTILINEAR MOTION.

14. We use the following symbols to represent the quantities which have been already the subject of our definitions. When we enunciate the formulæ expressive of the relations between these quantities, t generally denotes the time of an event's happening, a particular instant: it is the number of units of time, whatever that unit may be, which have elapsed at the proposed instant, since some given fixed point of time. When the contrary is not particularly stated, the interval called a second is taken as the unit of time. represents the measure of the velocity of any particle at a proposed instant: as such, its full meaning has been already given, but it may be as well here to repeat that it is the number of units of length through which it would carry the particle in a unit of time, provided it remained constant for that time. Unless the contrary be particularly stated, the length called a foot is taken as the unit of length. Thus if a particle be said to have a velocity v at a particular moment, it is meant that it would, with the velocity which it has at that moment continued constant, pass over v feet in one second.

f is generally employed to indicate the measure of an accelerating force; it is therefore (Art. 6) the velocity which the force acting upon a single particle will generate in it in a unit of time; and as the measure of velocity is a length, therefore ƒ, the measure of accelerating force, is a length: thus, using the conventional units of time and space, i. e. seconds and feet, an accelerating force, whose measure is ƒ acting for one second uniformly upon a particle which was at first at rest, will generate in it by the end of that time a velocity ƒ, or, more explicitly, a velocity which would, if continued uniform, alone carry the particle over ƒ feet in a second of time. Of

course in generating this velocity the force has moved the particle through some space; we shall see by-and-bye that this space in all cases = ƒ feet: this fact may be usefully remembered.

s is commonly taken to represent the distance which a particle may have traversed in any assigned time, or under any proposed circumstances.

Although these symbols generally stand for the quantities which have just been attributed to them, any others may of course be substituted for them at pleasure; it is only here meant that they constitute the symbols of our ordinary notation. Besides these, particular letters have been conveniently used to designate particular forces. For instance the letter g stands for the accelerating force of gravity, which is found to be always the same at the earth's surface upon every particle ; its numerical value found by experiment is 32.2 when feet and seconds are the units of length and time respectively. This means, in accordance with the explanation just given of the letter f when it stands for accelerating force, that if gravity be allowed to act upon a particle for one second, it will generate or destroy in that particle a velocity g, or a velocity which would alone carry the particle in one second over g (= 32.2) feet.

15. All these symbols are numerical, and represent the number of times which the unit, in terms of which they are estimated, enters the quantity for which they stand; they will therefore change their values when these units are for any reason changed, although the quantities denoted by them remain the same: thus any particular velocity being considered, thev which stands for it may change its numerical value for two reasons.

1st. The unit of time may be changed; for instance it may be convenient to take 1 minute as the unit in which to measure time instead of 1 second, as is ordinarily the case: now the v stands for the distance through which the velocity continued uniform would carry a particle in a unit of time; it would

manifestly carry it 60 times as far in 1 minute as in 1 second; hence in the first case the numerical value of would be 60 times that in the second, although the same velocity is represented in both cases.

2nd. The unit of distance might be altered; a yard might be employed instead of a foot: if the unit of time remained the same, as 1 second, and therefore the distance denoting the velocity the same, still its numerical value would become of what it was, because distance would be measured in terms of yards instead of feet.

If both these changes in the units were made at once, a change would generally take place in the value of v; but if the one unit were always increased in the same proportion as the other, the value of v would manifestly remain unaltered.

Again, in regard to a particular force, a change in the unit of time will make a more complicated change in its value: we have seen that it is measured by the velocity it generates in a particle in a unit of time; if then this unit be increased, manifestly the absolute velocity generated in that time is increased in the same proportion: moreover, by this increase of the unit the measure of any given velocity is increased, also in the same proportion, because by definition this measure is the distance through which the given velocity will carry the particle in the unit, and which must be greater for the same velocity, the greater the magnitude of the unit: hence, had the velocity which stands for the force remained the same after, as before, the change in the unit of time, the numerical value of the force would still be changed in direct proportion to the length of the unit; but as this velocity is itself changed in the same proportion, the numerical value of the force must be changed in proportion to the square of the unit.

To take a familiar example:

The value of g which stands for the force of gravity is the numerical value of the velocity which it will generate in a particle in one second = 32.2, because the velocity so generated will alone carry the particle over a space 32.2 feet in one second.