A Treatise on the Theory of Screws

INTRODUCTION. 5 perpendicular to the force. Thus a force, and a couple in a plane perpen- dicular to the force, constitute an adequate representation of any system of forces applied to a rigid body. It is easily seen that all the forces acting upon a rigid body may, by transference to an arbitrary origin, be compounded into a force acting at the origin, and a couple. Wherever the origin be taken, the magnitude and direction of the force are both manifestly invariable; but this is not the case either with the moment of the couple or the direction of its axis. The origin, however, can always be so selected that the plane of the couple shall be perpendicular to the direction of the force. For at any origin the couple can be resolved into two couples, one in a plane containing the force, and the other in the plane perpendicular to the force. The first com- ponent can be compounded with the force, the effect being merely to transfer the force to a parallel position; thus the entire system is reduced to a force, and a couple in a plane perpendicular to that force. It is very important to observe that there is only one straight line which possesses the property that a force along this line, and a couple in a plane perpendicular to the line, is equivalent to the given system of forces. Sup- pose two lines possessed the property, then if the force and couple belonging to one were reversed, they must destroy the force and couple belonging to the other. But the two straight lines must be parallel, since each must be parallel to the resultant of all the forces supposed to act at a point, and the forces acting along these must be equal and opposite. The two forces would there- fore form a couple in a plane perpendicular to that of the couple which is found by compounding the two original couples. We should then have two couples in perpendicular planes destroying each other, which is manifestly impossible. We thus see that any system of forces applied to a rigid body can be made to assume an extremely simple form, in which no arbitrary element is involved. These two principles being established we are able to commence the Theory of Screws.