unit of mass to move over two units of space in a unit of time, is called a double force.

A force which can cause three units of mass to move over a unit of space in a unit of time, or which can cause a unit of mass to move over three units of space in a unit of time, is called a triple force, and so on.

If we represent a unit of force by 1, a double force will be represented by 2, a triple force by 3, and so on.

In general, a force which can cause m units of mass to move over n units of space in a unit of time, will be represented by mn. Hence, forces may be compared with each other as readily as numbers, and by the same general rules.

The unit of mass, the unit of space, and the unit of time, are altogether arbitrary, but having been once assumed they must remain the same throughout the same discussion. We shall assume a mass weighing one pound at the equator as the unit of mass, one foot, as the unit of space, and one second, as the unit of time.

Let us denote any impulsive force, by f, the mass moved, by m, and the velocity which the impulse can impart to it by v. Then, since the velocity is the space passed over in one second, we shall have, from what precedes,

If we suppose m to be equal to 1, we shall have,

That is, the measure of an impulse is the velocity which it can impart to a unit of mass.

An incessant force is made of a succession of impulses. It has been agreed to take, as the measure of an incessant force, the quantity of motion that it can generate in one second, or the unit of time.

If we denote an incessant force by f, the mass moved by m, and the velocity generated in one second by v, we shall have,

f

= mv.

If we suppose m to be equal to 1, we shall have,

ƒ = v.

That is, the measure of an incessant force is the velocity which it can generate in a unit of mass in a unit of time.

If the force is of such a nature as to act equally upon every particle of a body, as gravity, for instance, the velocity. generated will be entirely independent of the mass. In these cases, the velocity that a force can generate in a unit of time, is called the acceleration due to the force. If we denote the acceleration by f, the mass acted upon by m, and the entire moving force by f', we shall have,

ƒ' == mf = mv..

Since an incessant force is made up of a succession of impulses, its measure may be assimilated to that of an impulsive force, so that both may be represented and treated in the same manner.

Forces of pressure, if not counteracted, would produce motion; and, as they differ in no other respect from the forces already considered, they also may be assimilated to impulsive forces, and treated in the same manner.

Representation of Forces.

25. It has been found convenient.in Mechanics to represent forces by straight lines; this is readily effected by taking lines proportional to the forces which they represent. Having assumed some definite straight line to represent a unit of force, a double force will be represented by a line twice as long, a triple force by a line three times as long, and so on.

A force is completely given when we have its intensity, its point of application, and the direction in which it acts. When a force is represented by a straight line, the length of the line represents the intensity, one extremity of the line represents the point

P

of application, and the direction of the

Fig. 1.

line represents the direction of the force.

Thus, in figure 1, OP represents the intensity, O the point

of application, and the direction from 0 to P is the direction of the force. This direction is gen

erally indicated by an arrow head. It is to be observed that the point of application of a force may be taken

Р

Fig. 1.

at any point of its line of direction, and it is often found convenient to transfer it from one point to another on this line.

The intensity of a force may be represented analytically by a letter, which letter is usually the one placed at the arrow head; thus, in the example just given, we should designate the force OP by the single letter P.

If forces acting in any direction are regarded as positive, those acting in a contrary direction must be regarded as negative. This convention enables us to apply the ordinary rules of analysis to the investigations of Mechanics.

Y

0

Forces situated in the same plane are generally referred to two rectangular axes, OX and OY, which are called co-ordinate axes. The direction from O towards X is that of positive abscissas; that from O towards X is that of negative abscissas. The directions from O towards Y and Y', respectively, are those of positive and negative ordiForces acting in the directions of positive abscissas and positive ordinates are positive; those acting in contrary directions, are negative.

nates.

Forces in space are referred to three rectangular co-ordinate axes, OX, OY, and OZ. Forces acting from O towards X, Y, or Z, are positive, those acting in contrary directions, are negative.

Fig. 2.

Z

Fig. 3.

CHAPTER II.

COMPOSITION, RESOLUTION, AND EQUILIBRIUM OF FORCES.

Composition of Forces whose directions coincide.

26. Composition of forces, is the operation of finding a single force whose effect is equivalent to that of two or more given forces. This single force is called the resultant of the given forces. Resolution of forces, is the operation of finding two or more forces whose united effect is equivalent to that of a given force. These forces are called components of the given force.

If two forces are applied at the same point, and act in the same direction, their resultant is equal to the sum of the two forces. If they act in contrary directions, their resultant is equal to their difference, and acts in the direction of the greater one. In general, if any number of forces are applied at the same point, some of which act in one direction, and the others in a contrary direction, their resultant is equal to the sum of those which act in one direction, diminished by that of those which act in the contrary direction; or, if we regard the rule for signs, the resultant is equal to the algebraic sum of the components; the sign of this algebraic sum makes known the direction in which the resultant acts. This principle follows immediately from the rule adopted for measuring forces.

Thus, if the forces P, P', &c., applied at any point, act in the direction of positive abscissas, whilst the forces P", P'"', &c., applied to the same point, act in the direction of negative abscissas, then will their resultant, denoted by R, be given by the equation,

R (P + P' + &c.,) (P+ P+ &c.)

2

If the first term of the second member of this equation is numerically greater than the second, R is positive, which shows that the resultant acts in the direction of positive abscissas. If the first term is numerically less than the second, R is negative, which shows that the resultant acts in the direction of negative abscissas.

If the two terms of the second member are numerically equal, R will reduce to 0. In this case, the forces will exactly counterbalance each other, and, consequently, will be in equilibrium.

Whenever a system of forces is in equilibrium, their resultant must necessarily be equal to 0. When all of the forces of the system are applied at the same point, this single condition will be sufficient to determine an equilibrium.

All of the forces of a system which act in the general direction of the same straight line, are called homologous, and their algebraic sum may be expressed by writing the expression for a single force, prefixing the symbol 2, a symbol which indicates the algebraic sum of several homologous quantities. We might, for example, write the preceding equation under the form,

This equation expresses the fact, that the resultant of a systern of forces, acting in the same direction, is equal to the algebraic sum of the forces.

Parallelogram of Forces.

R

27. Let P and Q be two forces applied to the material point O, taken as a unit of mass, and acting in the directions OP and OQ. Let OP represent the velocity generated by the force P, and OQ the velocity generated by the force Q. Draw PR parallel to OQ, and QR parallel to OP; draw also the diagonal OR.

о

Fig. 4.

From the law of inertia (Art 18), it follows that a mass acted upon by two simultaneous forces moves in the general