A Treatise on the Theory of Screws

8 THE THEORY OF SCREWS. [4, 4. A Geometrical Investigation. We can now demonstrate that whenever a body admits of an indefinitely small movement of a continuous nature it must be capable of executing that particular kind of movement denoted by a twist about a screw. Let A1 be a standard position of the body, and let P be any marked point of the body initially at P1. As the body is displaced continuously to a neighbouring position, P will generally pursue a certain trajectory which, as the motion is small, may be identified with its tangent on which Pn is a point adjacent to P,. In travelling from PT to Pn, P passes through the several positions, . In a similar manner every other point, Qt, of the rigid body will pass through a series of positions, Q2, &c., to Qn. We thus have the points of the body initially at Plt Qlt Rlt respectively, and each moves along a straight line through the successive systems of positions -Pa, Qi, Ri, &c., on to the final position Pn, Qn, Rn. We may thus think of the consecutive positions occupied by the body A2, A3, &c., as defined by the groups of points P„ Q1; Rt and P2, Q.., R2, &c. We have now to show that if the body be twisted by a continuous screw motion direct from A, to An, it will pass through the series of positions yL, As, &c. It must be remembered that this is hardly to be regarded as an obvious consequence. From the initial position A1 to the final position .4(l, the number of routes are generally infinitely various, but when these situations are contiguous, it is always possible to pass by a twist about a screw from to An via the positions -4-2, -å-S • • ■ An_ Suppose the body be carried direct by a twist about a screw from the position to the position An. Since this motion is infinitely small, each point ol the body will be carried along a straight line, and as P, is to be conveyed to Pn, this straight line can be no other than the line jPjP,,. In its progress P1 will have reached the position P2, and when it is there the points Qlt j?j will each have advanced to certain positions along the lines Q,Qn and RjRn, respectively. But the points reached by Qt and J?, can be no other than the points Q., and R2, respectively. To prove this we shall take the case where P1; Rt are collinear. Suppose that when Pi has advanced to P2, shall not have reached Q.,, but shall be at the intermediate point Q„. (Fig. 1.) Then the line P^Qx will have moved to PiQo, and as can only be conveyed along while at the same time it must lie along P2Q2, it follows that the lines P2Q„ and R1R2 must intersect at the point Ro, and consequently all the lines in this figure lie in a plane, further, P2Q2 and P2Q„ are each equal to as the body is rigid, and so also P2R„ and P2R2 are equal to P1R1. Hence it follows that QjQo and R1R0 are parallel, and consequently all the points on the line P1Q1R1 are displaced in parallel directions. It would hence follow that the motion of every point in the body was in a parallel direction, and that consequently the entire