A Treatise on the Theory of Screws

CHAPTER VI. THE EQUILIBRIUM OF A RIGID BODY. 69. A Screw System. To specify with precision the nature of the freedom enjoyed by a rigid body, it is necessary to ascertain all the screws about which the constraints will permit the body to be twisted. When the attempt has been made for every screw in space, the results will give Us all the information conceivable with reference to the freedom of the body, and also with reference to the constraints by which the movement may be hampered. Suppose that by these trials, n screws AJt ... An have been found about each of which the body can receive a twist. It is evident, without further trial, that twisting about an infinite number of other screws must also be possible (n> 1): for suppose the body receive any n twists about tIj, ... the position attained could have been reached by a twist about some single screw A. The body can therefore twist about A. Since the amplitudes of the n twists may have any magnitude (each not exceeding an infinitely small quantity), J is merely one of an infinite number of screws, about which twisting must be possible. AU these screws, together with Au ... An, will w general form what we call a screw system of the nth order. If it be found that the body cannot be twisted about any screw which does not belong to the screw system of the «th order, then the body is said to have freedom of the nth order. It is assumed that Alt ... An are independent screws, i.e. not themselves members of a screw system of order lower than n. If this were the case, the screws about which the body could be twisted would only consist of the members of that lower screw system. Since the amplitudes of the n twists about A„...An are arbitrary, it might be thought that there are n disposable quantities in the selection of a screw 6' from a screw system of the nth order. It is, however, obvious from § 14 that the determination of the position and pitch of S depends only upon