A Treatise on the Theory of Screws

CHAPTER V . THE REPRESENTATION OF THE CYLINDROID BY A CIRCLE*. 50. A Plane Representation. The essence of the present chapter lies in the geometrical representation of a screw by a point. The series of screws which constitute the cylindroid correspond to, or are represented by, a series of points in a plane. By choosing a particular type of correspondence we can represent the screws of the cylindroid by the points of a circle Various problems on the cylindroid can then be studied by the aid of the corresponding circle. We commence with a very simple process for the discovery of the circle. It will in due course appear how this circular representation is suggested by the geometry of the cylindroid itself (§ 68). It has been shown (§ 13) that the positions of the several screws on the cylindroid may be concisely defined by the intersections of the pairs of planes, y = x tan 0, z — m sin 20. In these equations, 0 varies in correspondence with the several screws, while m is a parameter expressing the size of the cylindroid. In fact, the whole surface, except parts of the nodal line, is contained between two parallel planes, the distance between which is 2m. The pitch of the screw corresponding to 0 is expressed by p = pt> + m cos 20, where p0 is a constant. 1 See papers in Proceedings of the Royal Irish Academy, Ser. ii. Vol. iv. p. 29 (1883), and the Cunningham Memoirs of the Royal Irish Academy, No. 4 (1886). 2 I may refer to a paper by Professor Mannheim, in the Comptes rendus for 2nd February, 1885, entitled “Repräsentation plane relative aux déplacements d’une figure de forme invariable assujettie ä quatre conditions.” Professor Mannheim here shows how the above plane repre- sentation might also have been deduced from the instructive geometrical theory which he had brought before the Academy of Sciences on several occasions.