A Treatise on the Theory of Screws

 SSBI CHAPTER XXII. THE GEOMETRICAL THEORY*. advanced our knowledge of the 302. Preliminary. It will be remembered how Poinsot dynamics of a rigid system by a beautiful geometrical theory of the rotation of a rigid body about a fixed point. We now specially refer to the geometrical construction by which he determined the instantaneous axis about which the body commenced to rotate when the plane of the instantaneous couple was given. We may enunciate with a generality, increasing in successive steps, the problem which, in its simplest form, Poinsot had made classical. From the case of a rigid body which is constrained to rotate about a fixed point we advance to the wider conception of a body which has three degrees of freedom of the most general type. We can generalize this again into the case in which the body, instead of having a definite number of degrees of freedom has any number of such degrees. The range extends from the first or lowest degree, where the only movement of which the body is capable is that of twisting about a single fixed screw, up to the case in which the body being perfectly free, or in other words, having six degrees of freedom, is able to twist about every screw in space. It will, of course, be borne in mind that only small movements are to be understood. In a corresponding manner we may generalize the forces applied to the body. In the problem solved by Poinsot the effective forces are equivalent to a couple solely. For the reaction of the fixed point is capable of reducing any system of forces whatever to a couple. But in the more generalized problems with which the theory of screws is concerned, we do not restrict the forces to the specialized pair which form a couple. We shall assume that the forces are of the most general type and represented by a wrench upon a screw. Thus, by generalising the freedom of the rigid body, as well as the forces which act upon it, we may investigate the geometrical theory of the motion when a rigid body of the most general type, possessing a certain number of degrees of freedom of the most general type, is disturbed from a * Trans. Royal Irish Acad. Vol. xxi. (1897) p. 185.