A Treatise on the Theory of Screws

314] THE GEOMETRICAL THEORY. 339 But it is of the essence of the theory that the reactions of the constraints by which the motion of the body is limited to twists about screws on the cylindroid A must be wrenches on the reciprocal system B. So far, therefore, as the body thus constrained is concerned, the reactions of the constraints will neutralize the wrench on i/j. Thus the wrench on % is the only part of the impulsive wrench which need be considered. But we already know from the construction that an impulsive wrench on % will produce an instantaneous twist velocity about a determined screw a on A. Thus we have found the initial movement, and the investigation is geo- metrically complete. 313. Freedom of the Third Order. We next consider the case in which a rigid body has freedom of the third order. We require, as before, to find a geometrical construction for the in- stantaneous screw a corresponding to a given impulsive screw t). Let A be the system of screws of the third order about which the body is free to twist. Let B be the system of screws of the third order reciprocal to A. We must first construct the system of the third order P which consists of the impulsive screws that would have made the body, if perfectly free, twist about the several screws of A. As already explained (§ 307) we can, in one way, but only in one way, resolve the original wrench on t) into wrenches on B, and on P. The former is destroyed by the reactions of the constants. The latter gives rise to a twist velocity about a determinate screw on A. Thus the problem has been solved. 314. General Case. We can obviously extend a similar line of reasoning to the cases where the body had freedom of the remaining degrees. It will, however, be as simple to write the general case at once. Let A be a system of screws of the nth order, about which a body is free to twist, any other movements being prevented by constraints. If the body receive au impulsive wrench, on any screw it is required to determine the instantaneous screw, of course belonging to A, about which the body will commence to twist. Let B be the system of screws of the (6 — n)th order, reciprocal to A. The wrenches arising from the reaction of the constraints must, of course, be situated on the screws of the system B. Let P be the system of screws of the nth order, which, in case the body