point of application of the force, and may draw a straight line from that point in the direction of the force, and of a length proportional to the magnitude of the force. Thus, for example, suppose a par

ticle acted on by three forces B in three different directions; and let these forces be of 3, 4, and 2 pounds respectively, Draw straight lines OA, OB, OC in the directions of these forces, and take the lengths of these straight lines proportional to the forces; that is take OB in the same proportion to OA as 4 is to 3, and take OC in the same proportion to OA as 2 is to 3: then OA, OB, and OC respectively completely represent the forces. In saying that OA represents the force we suppose that the force acts from O towards A; if the force acts from A towards O we shall say that AO represents it.

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147. Now suppose we have two or more forces acting at once on a particle, we may ask if we can find a single force which will produce the same effect as the two or more do jointly. For simplicity we will suppose two forces to be acting at once, and consider various cases.

148. Suppose two forces to act in the same direction; then they are equivalent to a single force in this direction represented by their sum. Thus if a weight of 8 pounds be hung at the eud of a string, and also a weight of 10 pounds, the effect is the same as if a single weight of 18 pounds were hung at the end. Again, suppose two forces to act in opposite directions; then they are equivalent to a single force in the direction of the greater represented by their difference. Thus if a force of 10 pounds act in one direction, and a force of 8 pounds in the opposite direction, the effect is the same as if a force of 2 pounds acted singly in the former direction.

149. When two or more forces are equivalent to a single force that single force is called the resultant of the others, and they are called components.

150. The method of finding the resultant of two forces acting on a particle, not in the same straight line, is given 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 called the Parallelogram of Forces; it is one of the most important in our subject, and we shall shew how it may be verified by experiment.

151. 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; and let a weight R be hung from O. Let the system be allowed to adjust itself so as to be at rest.

The effects of the weights P and Q are not changed as to magnitude by the passing of the strings which support them over the smooth pegs A and B. We have thus three

forces acting on the knot O, and keeping it in equilibrium; so that the effect of P along OA and of Q along OB are together just counteracted by the effect of R acting vertically downwards at O. Therefore the resultant of P along OA and of Q along OB must be equal to a force R acting

upwards at O. Now on OA take Op to contain as many inches as P contains pounds; and on OB take Og to contain as many inches as Q contains pounds; and complete the parallelogram Oqrp. Then it will be found by trial that Or contains as many inches as the weight R contains pounds, and that Or is a vertical straight line. We may change the positions of the pegs and the magnitudes of the weights employed in order to give due variety to the experiment; and the general results will afford sufficient evidence of the truth of the Parallelogram of Forces.

152. Besides the experimental verification, modes of establishing the proposition by mathematical reasoning have been given; but these are unsuitable for the present work. As we have already said in Art. 121, Newton deduces from his Laws of Motion a principle called the Parallelogram of Velocities; and from this he considers the Parallelogram of Forces to follow immediately.

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153. The case in which the directions of the two forces include a right angle deserves especial notice. Here the magnitude of the resultant force can be found by Arithmetic when the magnitudes of the components are known. Thus, if AC represents a force of 3 pounds, and AB a force of 4 pounds, and the angle BAC is a right angle, then AD, the resultant, will represent a force of 5 pounds. For the square of 5 is equal to the sum of the squares of 3 and 4: see Art. 32.

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154. If more than two forces act together on a particle we can find their resultant by repeated use of the Parallelogram of Forces. For instance, suppose there are three forces. Find the resultant of two of them by the Parallelogram of Forces; then the two may be removed and their resultant placed instead of them. Again, take this resultant and the third force, and find their resultant by the Parallelogram of Forces; we thus obtain finally a single force equivalent to the three which act together.

155. The proposition called the Parallelogram of Forces may be put in another form which expresses substantially the same fact; in this form it is called the Triangle of Forces. We may state it thus: If three forces acting on a particle keep it in equilibrium and a triangle be drawn having its sides parallel to the lines of action of the forces, the sides of the triangle will be proportional to the forces which are respectively parallel to them. Thus, for instance, in Art. 151 we have the triangle Orq; now Or is in the line of action of R, and Oq is in the line of action of Q, and rg is parallel to Op, which is in the line of action of P. And since rg is equal to Op, by Art. 16, it follows that the triangle Org has its sides proportional to the three forces R, P, and Q, which act on the knot at O and keep it in equilibrium. Any other triangle drawn so as to have its sides parallel to those of Org would be similar to Orq, and so its sides would be in the same proportion: see Art. 34.

156. As we may substitute for two or more forces a single resultant, so on the other hand we may replace a single force by two or more forces which are equivalent to it. We shall not have much occasion to use this process of resolving a force into components, as it is called, but it is of great importance and value in the higher treatises on mechanics. The most common case is that illustrated by the diagram of Art. 153; instead of any force represented by AD we may substitute the two forces represented by AC and AB, the angle BAC being a right angle.

157. It is found by experiment that a force acting on a body may be supposed applied at any point of its line of action. As a simple case suppose a heavy body hung up by a string to a support. The string may be fastened to the body on the side nearest to the support. Suppose a hole bored through the body exactly in the direction of the string, and instead of being fastened at the point of the hole nearest the support let the string be put through the hole and fastened to the point furthest from the support. The tension of the string, that is the force exerted by the string, will be found the same in the two cases: it is in fact just equal to the weight of the body. This principle

of the transmissibility of a force to any point in its line of action is frequently of great use.

158. If three forces keep a body in equilibrium and the directions of two of them meet at a point the direction of the third must pass through that point. For, consider the two forces of which the directions meet at a point; then by Art. 157 they may be supposed to act at that point: consequently they will have a resultant acting at that point, and they may be replaced by that resultant. Now it is obvious that this cannot be counteracted by the third force unless the direction of this force is exactly opposite that of the resultant. Hence the direction of the third force must pass through the point at which the directions of the other two meet. The proposition is important as affording a notion of the way in which results obtained with respect to particles are extended to the case of bodies: see Art. 144.

159. If more than two forces act on a body we may find the resultant of all the forces by the aid of the principles explained. Suppose, for example, that three forces act on a body. Take two of the forces; they may be supposed by Art. 157 to act at the point where their directions meet: find the resultant of these two forces by Art. 150, and substitute the resultant in the place of the two. Then produce the direction of this resultant to meet that of the third force, and find the resultant of these two by Art. 150. Thus we obtain a single force which is equivalent to the original three forces.

X. PARALLEL FORCES. CENTRE OF GRAVITY.

160. In the preceding Chapter we spoke of forces acting on a particle, that is on a body so small that the forces might be supposed to be applied at the same point. But it is obvious that forces may be applied at various points of a body which is too large to be considered as a particle, and we may want to know if we can find a single force equivalent to them. The question in its widest form