At the point B suppose we apply two forces Q and R, each equal to P, the former in the direction of P, and the

latter in the opposite direction.

Then we may readily admit that we have made no change in the action of P.

Now P at A and R at B are equal forces acting in opposite directions; let us assume that they neutralise each other: then these two forces may be removed without disturbing the equilibrium of the body, and there will remain the force at B, that is, a force equal to P and applied at B instead of at A.

19. When we find it useful to change the point of application of a force, we shall for shortness not always state that the new point is rigidly connected with the old point; but this must be always understood.

20. We shall have occasion hereafter to assume what may be called the converse of the principle of the transmissibility of force, namely, that if a force can be transferred from its point of application to a second point without altering its effect, then the second point must be in the line of action of the force.

21. We shall frequently have to refer to an important property of a string considered as an instrument for exerting force, which we will now explain.

Let AB represent a string pulled at one end by a force P, and at the other end by a force Q, in opposite directions.

Α

P

C

B

It is clear that if the string is in equilibrium the forces must be equal.

If the force Q be applied at any intermediate point C, instead of at B, still for equilibrium we must have Q equal to P. This is sometimes expressed by saying that force transmitted directly by a string is transmitted without change.

Again, let a string ACB be stretched round a smooth peg C; then we may admit that if the string be in equilibrium the forces P and Q at its ends must be equal. This is sometimes expressed p by saying that force transmitted by a string round

с

B

a smooth peg is transmitted without change.

Or in both cases we may say briefly, that the tension of the string is the same throughout.

We suppose that the weight of the string itself may be left out of consideration.

22. Experiment shews that the weight of a certain volume of one substance is not necessarily the same as the weight of an equal volume of another substance. Thus, 7 cubic inches of iron weigh about as much as 5 cubic inches of lead. We say then that lead is denser than iron; and we adopt the following definitions:

When the weight of any portion of a body is proportional to the volume of that portion the body is said to be of uniform density. And the densities of two bodies of uniform density are proportional to the weights of equal volumes of the bodies. Thus we may take any body of uniform density as the standard and call its density unity, and then the density of any other body will be expressed by a number. Thus, suppose we take water as the standard substance; then since a cubic inch of copper weighs about as much as 9 cubic inches of water, the density of copper will be expressed by the number 9.

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EXAMPLES. I.

1. If a force which can just sustain a weight of 5 lbs. be represented by a straight line whose length is 1 foot 3 inches, what force will be represented by a straight line 2 feet long?

2. How would a force of a ton be represented if a straight line an inch long were the representation of a force of seventy pounds?

3. If a force of Plbs. be represented by a straight line a inches long, what force will be represented by a straight line b inches long?

4. If a force of P lbs. be represented by a straight line a inches long, by what straight line will a force of Qlbs. be represented?

5. A string suspended from a ceiling supports a weight of 3 lbs. at its extremity, and a weight of 6 lbs. at its middle point: find the tensions of the two parts of the string. If the tension of the upper part be represented by a straight line 3 inches long, what must be the length of the straight line which will represent the tension of the lower portion?

6. If 3 lbs. of brass are as large as 4lbs. of lead, compare the densities of brass and lead.

7. Compare the densities of two substances A and B when the weight of 3 cubic inches of A is equal to the weight of 4 cubic inches of B.

8. A cubic foot of a substance weighs 4 cwt.: what bulk of another substance five times as dense will weigh 7 cwt.?

9. Two bodies whose volumes are as 3 is to 4 are in weight as 4 is to 3: compare their densities.

10. If the weight of a cubic inches of one substance and of b cubic inches of another be in the ratio of m to n, compare the densities of the substances.

II. Parallelogram of Forces.

23. When two forces act on a particle and do not keep it in equilibrium, the particle will begin to move in some definite direction. It is clear then that a single force may be found such that if it acted in the direction opposite to that in which the motion would take place,. this force would prevent the motion, and consequently would be in equilibrium with the other forces which act on the particle. If then we were to remove the original forces, and replace them by a single force equal in magnitude to that just considered, but acting in the opposite direction, the particle would still be in equilibrium.

Hence we are naturally led to adopt the following definitions:

A force which is equivalent in effect to two or more forces is called their resultant; and these forces are called components of the resultant.

24. We have then to consider the composition of forces, that is, the method of finding the resultant of two or more forces. The present Chapter will be devoted to the case of two forces acting on a particle.

25. When two forces act on a particle in the same direction their resultant is equal to their sum and acts in the same direction.

This is obvious. For example, if a force of 5 lbs. and a force of 3 lbs. act on a particle in the same direction their resultant is a force of 8 lbs., acting in the same direction.

26. When two forces act on a particle in opposite directions their resultant is equal to their difference, and acts in the direction of the greater force.

This is obvious. For example, if a force of 5 lbs. and a force of 3 lbs. act on a particle in opposite directions their resultant is a force of 2 lbs., acting in the same direction as the force of 5 lbs.

27. We must now proceed to the case in which two forces act on a particle in directions which do not lie in the same straight line; the resultant is then determined by the following proposition:

If two forces acting on a particle be represented in magnitude and direction by straight lines drawn from the particle, and a parallelogram be constructed having these straight lines as adjacent sides, then the resultant of the two forces is represented in magnitude and direction by that diagonal of the parallelogram which passes through the particle.

This proposition is the most important in the science of Statics; it is called briefly the Parallelogram of Forces. We shall first shew how the proposition may be verified experimentally; we shall next point out various interesting results to which it leads; and finally demonstrate it.

28. To verify the Parallelogram of Forces experimentally.

Let A and B be smooth horizontal pegs fixed in a vertical wall. Let three strings be knotted together; let

O represent the knot. Let one string pass over the peg A and have a weight P attached to its end; let another string pass over the peg B and have a weight Q attached to its end; let the other string hang from O and have a weight R attached to its end. Let the system be allowed to adjust itself so as to be at rest.

By Art. 21 the pegs do not change the effects of the weights P and Q as to magnitude.

We have three forces acting on the knot at 0, and