A Treatise on the Theory of Screws

5] TWISTS AND WRENCHES. movement was simply a translation. But even in this case it would be impossible for the points Qo and _R0 to be distinct from Q.2 and R2, because, when a body is translated so that all its points move in parallel lines, it is impossible, if the body be rigid, for the distances traversed by each point not to be all equal. We have thus demonstrated that if a body is free to move from a position A2 to an adjacent position An by an infinitely small but continuous movement, it is also free to move through the series of positions A2, As, &c., by which it would be conveyed from At to An by a twist. We may also state the matter in a somewhat different manner, as follows:—It would be impossible to devise a system of constraints which would permit a body to be moved continuously from A2 to An, and would at the same time prohibit the body from twisting about the screw which directly conducts from A2 to An. Of course this would not be true except in the case where the motion is infinitely small. The connexion of this result with the present investigation is now obvious. When A is the standard position of the body, and /] ail adjacent position into which it can be moved, then the body is free to twist about the screw defined by A and B. 5- The canonical form of a small displacement. In the Theory of Screws we are only concerned with the small displace- ments of a system, and hence we can lay down the following fundamental statement. The canonical form to which the displacement of a rigid body can be reduced is a twist about a screw. If a body receive several twists in succession, then the position finally attained could have been reached in a single twist, which is called the resultant twist.