of a force to be the product of the number of units of force exerted into the number of units of space through which it acts.

Thus for illustration, if we take as the unit of efficiency or the dynamical unit, the work performed in raising 1lb. vertically through 1 foot, then the efficiency required to raise a ton through 1 yard would be 6720.

=

131. The standard of efficiency or work done assumed by Watt and adopted generally by engineers is 33000 lbs. raised through 1 foot per minute,—the agent working steadily. This is called a horse-power, and the efficiency of steam-engines and other machines is commonly expressed in terms of this unit. Thus if a machine of H horse-power can raise P lbs. through f feet in t minutes, we shall have these quantities connected by the relation of Pf=33000 H. t.

132. We have seen in the case of the simple machines, or any combination of them, that if the system be put in motion, then

P. P's displacement W. W's displacement.

=

This result shews us that however the application of a force be modified or rendered more useful by the intervention of a machine, yet no efficiency is gained thereby; and further, the same result put in the form

shews that in whatever proportion the intensity of a force be increased by means of a machine, yet the space through which the increased force will operate will be diminished in the inverse proportion as compared with the space through which the force applied must operate. Thus for the sake of illustration, suppose a weight of 500lbs. is supported on a machine

by a power equivalent to 10 lbs., then in order that the weight might be raised through one inch it would be requisite for the power to move through a space of 50 inches in the same time. This diminution of velocity in the inverse proportion of the increase of force is the foundation of the common phrase, applied to machines, that what is gained in power is lost in velocity; and we may regard it as another form of stating the principle of virtual velocities in this particular case, or the same as asserting that no efficiency is gained by the intervention of a machine.

133. Before closing this chapter we will briefly allude to the principle of the differential axle and Hunter's Screw. On the wheel and axle we have seen that (Art. 100)

P radius of axle

=

W radius of wheel' or W=P

radius of wheel
radius of axle

from which it would appear that by sufficiently diminishing the radius of the axle a given power P might be made to raise a weight W of any magnitude we please. Practically however there is a limit to the thickness of the axle; for if it be made too small, the material of which it is made will not bear the strain upon it, and it will break.

which may be increased to any extent by making the axles B and C as nearly equal as we please without unduly reducing the strength of the axle.

134. Again, in the case of the screw, it is obvious from the expression

W_circumference of circle described by P

B

(Art. 113),

B

that by diminishing the distance between two threads sufficiently, we might obtain any mechanical advantage we please; but the distance between the threads must not be less than the thickness of the threads, otherwise the companion screws could not work together; and further, if the thickness of the threads be unduly diminished, they will not be able to bear the strain upon them. This difficulty is obviated in Hunter's Screw, in which a screw A works within another screw B; thus if c be the radius of the circle described by P,-b, a the distance between two threads in the screws B, A respectively, then whilst P makes one revolution, W will descend through b, in consequence of the

screw B descending through b, but it will also rise through

a in consequence of the screw A making one turn within B; i.e. W will descend through b-a.

Whence P. P's displacement W. W's displacement gives

=

and by making b as nearly equal to a as we please, the mechanical advantage may be increased to any extent without unduly weakening the threads of the screws.

DYNAMICS.

CHAPTER I.

INTRODUCTION.

1. A MATERIAL particle has been defined to be a portion of matter indefinitely small in all its dimensions. It has therefore no determinate form or volume, but it has mass, it may be subject to the action of force, and may exert pressure on other particles. This conception of a particle is of course conventional,—a result of arbitrary definition,—so that calculations respecting such a body cannot be at once practically applied, since no bodies of which we have any experience correspond to this idea. But a particle having no parts, its motion is one and indivisible, and is therefore of a simpler kind than that of a body of finite size, different points of which might move differently. Hence we are led to consider the motion of a particle preparatory to that of bodies of finite size, and which have a real existence. The motion of such bodies can be reduced to a dependence on that of particles, by the application of suitable principles, but in the present treatise we do not propose to consider the motion of anything but particles or bodies regarded as particles; for example, a ball or a body of any kind, whenever it may occur in the following pages, will be considered, so far as its motion is concerned, as a particle coincident in position with the centre of gravity of the ball or body.