THEORY OF COUPLES. The two motions of which any rigid system is susceptible are those of TRANSLATION and of ROTATION. Each of these has this peculiarity, namely, that one particular case of its application yields the other kind of motion. Every motion of a system can, for any one instant, be resolved, at most, into a motion of translation of the whole system, combined with a motion of rotation about an axis; and every application of a system of forces to any rigid body, produces, generally speaking, this compound of translation and rotation. Also, if equal ana opposite forces, such as would produce simple translation, be applied at the same point, or if equal and oppo site forces, such as would produce rotation, be applied about the same axis, the result is that the equilibrium, or previous motion, of the system remains undisturbed.
But if the equal and opposite forces of translation be applied at different points, the result is rotation only, for the first instant ; and if the equal and opposite forces of rotation be applied about axes not coinciding, hut only parallel, the effect, at the first instant, is transla tion only. And though the doctrine of motion is now properly ex cluded from statics, yet the preceding theorems, together with others mentioned in ROTATION, should be well understood, and viewed in connection with the science of equilibrium, which is always illustrated, though it may not be demonstrated, by such considerations.
It was for a long time a curious but barren exception, that though any two forces acting in the same plane may, generally speaking, have their joint effect supplied by one single third force, yet if the two forces be equal in magnitude, and opposite in direction, no such single third force will do. If indeed they be applied in the same line, as o l and q a in the first figure, they equilibrate each other ; but if not in the same line, as o r and q a in the second figure, no one single force can be found which will either equilibrate them, or produce their effect. Some years ago, M. Poinsot, already mentioned for his beautiful theory of ROTATION, applied a remarkable theorem con nected with such pairs of forces to the establishment of the theory of the statics of rigid bodies, in a manner which has made his system rapidly take its place among the fundamental bases of the science. We shall in this article point out the manner in which this can be done, without much demonstration, with a view to draw the attention of those who have learned the doctrine of equilibrium in the old way : we cannot make it intelligible (without too great length) except to those who have learned the principles of analytical statics.
M. Poinsot called a pair of equal and opposite forces, not equili brating each other, by the name of a couple—too general a term perhaps : by it is to be understood a couple which cannot be made anything but a couple, or cannot be replaced by one force : an incom patible couple. The plane of the couple is the plane drawn through
the parallel forces; the arm of the couple is any line drawn perpen dicular to the forces from the direction of one to that of the other; the axis of the couple is any straight line perpendicular to its plane. And if we consider any axis, it will be apparent that the moment or leverage of the couple (LEVER) to turn the system about that axis is represented by the product of one of the forces and the arm. For if, with reference to the axis, x be the arm of one of the forces, x ± a is that of the other, a being the arm of the couple. Hence if r be one of the forces, the united leverage is r (x + a) — r x or ± P a. This product r a is called the moment of the conile.
The last-mentioned property will give a high probability of itself to the following theorems, which are the basis of the theory of couples, and can be proved, the first by aid of the composition of forces only, the second by the principle of the lever. Any couple may have the direction of its arm changed, and consequently of its forces, in any manner whatsoever, either in its own plane, or in any plane parallel to it, frovided only that the direction in which it tends to turn the sys tem remains unaltered. Secondly, any couple may be replaced by another which has the same moment, the plane and direction of turning remaining unaltered; that is, the arm may be shortened or lengthened in any manner, provided the forces be increased or dimi nished in the same proportion. If the system were in equilibrium before, it will remain in equilibrium, however its couples may be altered, in any manner described in the above theorems. Hence it follows that a couple is entirely given when there are given :-1. Its axis or any line perpendicular to its plane, which is also perpendicular to any of the planes into which it may be removed. 2. The moment of the couple ; specific forces or arms are unnecessary for its description, so long as their product is given. 3. The direction in which it tends to turn the system. The easiest way of describing a couple is then as follows ; suppose, for example, a horizontal one :—Take any vertical line for the axis of the couple ; on that axis lay down a line propor tional to its moment, and agree that vertical lines drawn upwards shall represent moments tending to turn the system from west to east ; and downwards, those tending to turn the system from east to west. But a sign must also be agreed upon ; positive moment must consist in tendency to turn in one direction, and negative in the other.