COUPLES, the name given in statics to pairs of equal parallel forces in opposite directions, and at different points of a body. It is shown in the art. l'arallel Forces (q.v.), that when two parallel forces act in opposite directions on a body, they may be replaced by one equal to their difference acting parallel to them in the direction of the greatest, at a point not between but beyond the points where they arc applied; and which point recedes the further from their points of application the nearer they approach equality, getting to an infinite distance when they become equal, and when their resultant accordingly is zero. In this limiting case, the forces constitute a couple; they have no tendency to translate the body; their action goes wholly to make it rotate about an axis passing through its center of gravity, and perpendicular to the plane in which the couple acts. Such being the case, a couple cannot be replaced or counter acted by any single force, for such a force would produce translation; it can only be replaced or balanced by other couples. The length of the straight line which meets the lines of action of the forces at right angles is called the " arm " of a couple, and the product of the force into its arm is called its "moment." Most of the leading propositions in the theory of C. are readily seen to be true, as soon as they are stated. For instance, as the axis round which a couple tends to make a body rotate passes through the body's center of gravity perpendicularly to the plane of the couple, it does not matter what position the couple occupies in its own plane. Also, supposing the body to be rigid, the couple may be moved into any plane parallel to its own, provided its new position be rigidly connected with the original position. It is also obvious, on the principle of the lever, that the efficiency of the couple depends on its moment simply, so that its arm may be shortened or lengthened at pleasure, pro vided the force be increased or diminished as the case may require, so as always to make the product of the force and arm the same. Suppose ropes fastened at the bow and stern of a ship pulling with equal force in opposite directions; they will make the ship turn round an axis through its center of gravity, at a rate depending on the force applied to the ropes. If the ropes be fastened to opposite points of the vessel nearer
midships, it will only turn round at the same rate, provided the force applied to the ropes be increased; and, on experiment, it would he found that the force must be increased so as that its product into the distance between the ropes shall equal the pro duct of the force in the first case into the length of the ship. Through this we can compound C. acting in the same plane, for we can turn them round till their arms coin cide, and then give them a common arm; their forces will then act in the same lines, when their resultant into the arm will lie the new couple. So two C. which are situated in planes inclined at any angle to each other may be replaced by a single couple (see fig.). Suppose the C. both to be moved in their respective planes till their arms coin cide with the line of intersection of the planes, CD. Bring them then to a common arm in this line, AB. At each end of we shall have a pair of forces, say P and Q. inclined to one another at the angle of inclination of the planes. Their resultant, by the composition of forces, will be a force R, acting in a line between the planes. We shall have then forces R acting at each end of the arm, and evidently in directions parallel and opposite. R X arm, AB, then, is moment of the resultant couple. Bay ing seen how to compound C. whose planes are inclined to one another. the theory of the composition of C. may be said to be complete. for if they are in parallel planes, we know we can bring them into the same plane and to a common arm, and so into a com mon couple. In statical theory, any number of forces acting on a body, and not in equilibrium, may be reduced to a single force, a single couple, or a single force and a single couple. We have shown that the C. may all be reduced to one, as well as those forces which do not produce couples. If the single force do not act perpendicularly to the plane of the couple, it can always be compounded with the forces of the couple, so as to reduce the whole to a single force: if it act perpendicularly, then it cannot be compounded with the couple, and the body will have at once a motion of translation and motion of rotation.