tains on plumb-lines hanging at places near them. It is thus discovered that the weight at the end of a plumb-line is drawn a little towards a neighbouring mountain; so that the plumb-line does not hang quite in the direction in which it would hang if there were no mountain. In very accurate surveys of the earth made for the purpose of determining its exact size and shape, it is necessary to pay great attention to the deviation which the action of mountains produces in the direction of the plumb-line. 139. A very interesting example of motion is furnished by a contrivance of which the essential part is indicated by the diagram. Two heavy bodies are connected by a string which passes over a smooth peg. Here the force of gravity tends to draw each body down, while the force exerted by the string tends to draw each body up. The force exerted by the string is the same on the two bodies in agreement with the third Law of Motion, which makes the action of one body on the other equal to the reaction of the latter on the former. Experiment will shew that if the two bodies are of unequal weight and are left to themselves the heavier will descend; so that the force exerted by the string is less than the weight of the heavier body, but greater than the weight of the lighter body. When the case is examined by the aid of a little mathematics it is found that the motion is just like what would take place if a force equal to the difference of the two weights were employed to move a mass equal to the sum of the two masses. Thus if one body weighs 13 pounds, and the other weighs 12 pounds, the motion will be just like that of a body which weighs 25 pounds acted on by a force of 1 pound. Therefore the motion will be like that of a falling body but much slower, namely, at the rate of 1 foot for every 25 feet of the body falling freely. 140. The preceding example is one of those which justify our confidence in the truth of the laws of falling bodies: see Art. 89. We have here a case of motion which by the aid of sound theory we can shew to be of the same kind as that of falling bodies; while the motion is so much less rapid that it can be easily observed. A machine is made, named after its inventor, Atwood, which is furnished with appliances for performing the experiment easily, but which in principle is the contrivance of the preceding Article. The results are very satisfactory, and the student will be pleased when he has the opportunity of seeing them exhibited in a lecture-room. 141. It is usual to call the force exerted by a string, as in Art. 139, the tension of the string. There is nothing special in the nature of the force exerted in this way, but it is convenient to give it a name. 142. The solution of the problem of motion noticed in Art. 139 involves more mathematics than we assume in the reader; but it may be instructive to verify by an example the result which is asserted to hold, at least so far as to shew that it is reasonable and consistent with itself. It will be seen that both bodies move, and that by the nature of the contrivance the weights of the two bodies are set in opposition as it were; so that the motion may naturally be that which would be produced in the sum of the masses by the difference of the weights. Now in the example we say that the heavier body will descend at of the rate of a body falling freely; thus in fact of the weight of the body is taken away by the tension of the string. Again, the lighter body rises at of the rate of a body falling freely; thus in fact the weight of the body is taken away by the tension of the string and besides a force equal to of the weight exerted upwards. Thus the tension of the string must be 2 of the weight of the heavier body, and must also be 2 of the weight of the lighter body; so that our statement will not be consistent unless these two results are equal: it is easily found by trial that they are equal, each of them being 121 pounds. IX. COMPOSITION OF FORCES AT A POINT. 143. In Chapters IV. to VIII. we have discussed the motion of falling bodies, and also the Laws which relate to the connexion between force and the motion produced by it; we must now devote some Chapters to the consideration of forces not producing motion but checking the action of other forces. It is a matter of observation that forces may act on a body without putting it in motion. A man may try to lift a body and find it too heavy for him: in this case the body is acted on by the force of gravity downwards, by the resistance of the ground on which it is placed which acts upwards, and by the effort of the man which also acts upwards; and the body remains at rest. When a body remains at rest though acted on by forces, it is said to be in equilibrium; and the forces are said to counteract each other or to balance each other. 144. There are three things to consider with respect to a force acting on a body; the point of application, that is the point of the body at which the force is applied; the direction of the force; and the magnitude of the force. It is necessary for simplicity to confine ourselves for some time to the case of a very small body, which we call a particle. In this case the forces which we have to consider all act at one point, namely that at which the particle is situated. The direction of any force is the straight line along which it tends to move the particle. We have seen in Art. 123 that a similar restriction as to the size of the bodies we consider is advantageous in treating the subject of motion. 145. The magnitudes of forces are conveniently measured by the weights which they will support. Thus we speak of a force of 5 pounds; by this we mean a force which will just support a weight of 5 pounds, that is a force which will just counteract the force exerted by gravity on a body weighing 5 pounds. 146. Forces may be conveniently represented by straight lines. For we may take a point to denote the 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 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. 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 A B R 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 |