A Treatise on the Theory of Screws

10 THE THEORY OF SCREWS. [5- Although we have described the twist as a compound movement, yet in the present method of studying mechanics it is essential to consider the twist as one homogeneous quantity. Nor is there anything unnatural in such a supposition. Everyone will admit that the relation between two positions of a point is most simply presented by associating the purely metric element of length with the purely geometrical conception of a directed straight line. In like manner the relation between two positions of a rigid body can be most simply presented by associating a purely metric element with the purely geometrical conception of a screw, which is merely a straight line, with direction, situation, and pitch. It thus appears that a twist bears the same relation to a rigid body which the ordinary vector bears to a point. Each just expresses what is necessary to express the transference of the corresponding object from one gi ven position to another*. 6. Instantaneous Screws. Whatever be the movement of a rigid body, it is at every instant twisting about a screw. For the movement of the body when passing from one position to another position indefinitely adjacent, is indistinguishable from the twist about an appropriately chosen screw by which the same displacement could be effected. The screw about which the body is twisting at any instant is termed the instantaneous screw. 7. Definition of the word Wrench. It has been explained in the Introduction that a system of forces acting upon a rigid body may be generally expressed by a certain force and a couple whose plane is perpendicular to the force. We now employ the won! wrench, to denote a force and a couple in a plane perpendicular to the force. The quotient obtained by dividing the moment of the couple by the force is a linear magnitude. Everything, therefore, which could be specified about a wrench is determined (if the force be given in magnitude), when the position of a straight line is assigned as the direction of the force, and a linear magnitude is assigned as the quotient just referred to. Remembering the definition of a screw (§ 2), we may use the phrase, wrench on a screw, meaning thereby, a foi-ce directed along the screw and a couple in a plane perpendicular to the screw, the moment of the couple being equal to the product of the force and the pitch of the screw. Hence we may state that The canonical form to which a system of forces acting on a rigid body can be reduced is a wrench on a screw. Compare M. René de Saussure, American Journal of Mathematics, Vol. xvm. No. 4, p. 337.