A Treatise on the Theory of Screws

CHAPTER XXV . THE THEORY OF PERMANENT SCREWS*. 360. Introduction. In commencing this chapter it will be convenient to recite a well-known dynamical proposition, and then to enlarge its enunciation by successive abandonment of restrictions. Suppose a rigid body free to rotate around a fixed point. There are, as is well known, three rectangular axes about any one of which the body when once set in rotation will continue to rotate uniformly so long as the application of force is withheld. These axes are known as permanent axes. The freedom of the body in this case is of a particular nature, included in the more general type known as Freedom of the Third Order. The Freedom of the Third Order is itself merely one subdivision of the class which, including the six orders of freedom, embraces every conceivable form of constraint that can be applied to a rigid body. We propose to investigate what may be called the theory of permanent screws for a body constrained in the most general manner. The movement of the body at each moment must be a twist velocity about some one screw 0 belonging to the system of screws prescribed by the character of the constraints. In the absence of forces external to those arising from the reactions of the constraints, the movement will not, in general, persist as a twist about the same sci-ew 0. The instantaneous screw will usually shift its position so as to occupy a series of consecutive positions in the system. It must, however, be always possible to compel the body to remain twisting about 0. For this purpose a wrench of suitable intensity on an appropriate screw y may have to be applied. Without sacrifice of generality we can in general arrange that y is one of the system of screws * Trans. Roy. Irish Acad., Vol. xxix. p. 613 (1890).