A Treatise on the Theory of Screws

04 THE THEORY OF SCREWS. [72- found which are reciprocal to the screw system P. The theory of reciprocal screws will now prove that Q must really be a screw system of order 6 — n. In the first place it is manifest that Q must be a screw system of some order, for if a body be capable of twisting about even six independent screws, it must be perfectly free. Here, however, if a body were able to twist about the infinite number of screws embodied in Q, it would still not be free, because it would remain in equilibrium, though acted upon by a wrench about any screw of P. It follows that Q can only denote the collection of screws about which a body can twist which has some definite order of freedom. It is easily seen that that number must be 6-n, for the number of constants disposable in the selection of a screw belonging to a screw system is one less than the order of the system (§ 36). But we have seen that the constants disposable in the selection of X are 5 - n, and, therefore, Q must be a screw system of order 6 - n. We thus see, that to any screw system P of order n corresponds a reciprocal screw system Q of order 6 - n. Every screw of P is reciprocal to all the screws of Q, and vice versa. This theorem provides us with a definite test as to whether any given screw a is a member of the screw system P. Construct 6 — n screws of the reciprocal system. If then a be reciprocal to these 6 — n screws, a must in general belong to P. We thus have 6-n conditions to be satisfied by any screw when a member of a screw system of order n. 73. Equilibrium. If the screw system P expresses the freedom of a rigid body, then the body will remain in equilibrium though acted upon by a wrench on any screw of the reciprocal screw system Q. This is, perhaps, the most genera] theorem which can be enunciated with respect to the equilibrium of a rigid body. This theorem is thus provedSuppose a wrench to act on a screw y belonging to Q. If the body does not continue at rest, let it commence to twist about a. We would thus have a wrench about y disturbing a body which twists about a, but this is impossible, because a and y are reciprocal. In the same manner it may be shown that a body which is free to twist about all the screws of Q will not be disturbed by a wrench about any screw of P. Thus, of two reciprocal screw systems, each expresses the locus of a wrench which is unable to disturb a body free to twist about any screw of the other. 74. Reaction of Constraints. It also follows that the reactions of the constraints by which the move- ments of a body are confined to twists about the screws of a system P can only be wrenches on the reciprocal screw system Q, for the reactions of the