letters which indicate the line; thus AB expresses that the force acts in direction of the arrow from A towards B; a force represented by BA would indicate a force of equal magnitude, but acting in the opposite direction, i.e. from B towards A.

The force P would be represented algebraically by expressing in algebraic symbols, the magnitude and position of the line AB which represents the force geometrically: thus its direction would be assigned by assigning the angle at which it is inclined to a known fixed line Ox in the same plane with AB: its magnitude will be assigned by assigning the numerical value of P, the number

N

of units of length; and the point of application A will be assigned by assigning the position of A with respect to the fixed lines Ox, Oy in the same plane with AB.

11. This mode of representing forces by lines is of great utility, as we shall see more particularly in the next chapter. We may illustrate it here by supposing several forces as P, B

Q, R to act simultaneously at

A

D

D'

P

R

the point A in the same direction: if they would be separately represented by AB, AC, AD, they will when acting simultaneously be together represented by a line AD', the length of which is equal to the sum of AB+ AC+ AD.

If one of the forces as R, acts in a direction opposite to that of the others P, Q, we shall have to subtract the line AD from the sum of the others

D

A

B C D'

P

-

AB, AC, and the three would be represented by a line AD equal in length to AB+ AC- AD. This is still the algebraic sum of the lines AB, AC, AD, if lines in one direction from A be considered positive, and lines in the opposite direction negative; and generally if any number of forces act simultaneously at a point and be affected with the sign + or as they act in a given direction or the opposite, they will be equivalent to a single force represented by the algebraic sum of the several forces; and if this sum be affected with a positive sign, the equivalent force will act in the direction which has been considered positive; and if it be affected with a negative sign, it will act in the opposite direction.

12. From the definition which has been given of equal forces (in Art. 8), it is obvious that two equal forces applied at a point in opposite directions will be in equilibrium. Further, it will readily be granted that two equal and opposite forces P, Q applied at the extremities of a straight rigid rod AB and acting in direction of the rod will be in

C

CB

equilibrium ;—for there is no reason that the rod should move in one direction rather than in another;-and this result will be true whatever be the length of the rod: from hence we infer that P will balance Q at whatever point of the rod Q be applied: in other words the effect of Q is the same at whatever point of the rod B, C,... it be applied, the direction remaining the same.

These considerations lead us to the following principle, called the principle of the transmission of force, which we shall hereafter find to be of great utility.

The effect of a force on a particle to which it is applied will

be the same, if we suppose the force applied at any point we please in the line of action, provided the point be rigidly connected with the original particle.

This principle-which is the fundamental one of the science of Statics-will hold whether we consider the particle as isolated, or as a constituent element of a body of finite size; and we shall find it of great use when we wish to transfer the point of application of a force from one point to another for convenience of calculation. We shall not think it necessary in every case where the supposition is required, to state that the system is supposed to be rigidly connected, but in any instance where this is not done the student will understand it to be so.

13. As an illustration of the above principle we may give the following. If a weight be supported by the hand by means of a string, the effort which the hand must exert will be the same whatever be the length of the string (the weight of the string being neglected), i. e. whether the force, which the hand exerts, be applied at A, or B, or C, or any point in the line of action of the force.

B

C

Obs. In this example the student will observe that the connection between the points A, B and the weight is not a rigid one, and in general when the force Q (fig. Art. 12), which we transfer from the point C to B, acts as in the upper figure, i. e. tends to draw C towards B, the connection between C and B need not be essentially rigid; but the two points may be otherwise connected, as for instance, by a fine inextensible thread; when however (as in the lower figure) the force tends to thrust B towards C, the connection must be a rigid one.

14. We have called the example above an illustration, and not a proof of the principle of Art. (12), for as this principle has been enunciated with reference to a particle, and since particles as such cannot be subjected to experiment, it would be vain to look for or expect a direct proof of this, or in fact of any other physical law. The student must be prepared to admit its truth as established by evidence similar to that by which other physical laws are established.

CHAPTER II.

OF FORCES ACTING IN ONE PLANE.

15. WHEN a system of forces acting on a particle at rest is not in equilibrium, the particle will begin to move in some definite direction, but a single force might be found of proper intensity which when applied independently to the particle and acting in the same direction would cause the particle to move in exactly the same manner; such a force is called the resultant of the system of forces; and the constituent forces of the system, with reference to this resultant, are called components.

In other words, the single force which is capable of producing the same effect on a particle or system of particles as would result from the combined action of several other forces, is called their resultant.

We do not enter into the question what the dynamical effect might be if the system of forces were not in equilibrium-but whatever it may be, the resultant is equivalent to the components.

When a system of forces acting on a particle or body is in equilibrium, the particle has

no tendency to motion, and the resultant is consequently nil. Hence when a system of forces (as P, Q, R,...) is in equilibrium,

one of them (as P) may be regarded as counterbalancing the combined action of all the rest, Q, R, S. It appears then that the remaining forces (Q, R, S) produce the same effect on the particle as would result from a single force equal and opposite