A Treatise on the Theory of Screws

CHAPTER XI . FREEDOM OF THE SECOND ORDER. 116. The Screw System of the Second Order. When a rigid body is capable of being twisted about two screws 0 and </>> it is capable of being twisted about every screw on the cylindroid (0, <£) (§ 14). If it also appear that the body cannot be twisted about any screw which does not lie on the cylindroid, then as we know the body has freedom of the second order, and the cylindroid is the screw system of the second order by which the freedom is defined (§ 219). Eight numerical data are required for determination of a cylindroid (§75). We must have four for the specification of the nodal line, two more are required to define the extreme points in which the surface cuts the nodal line, one to assign the direction of one generator, and one to give the pitch of one screw, or the eccentricity of the pitch conic. Although only eight constants are required to define the cylindroid, yet teu constants must be used in defining two screws 0, tj>, from which the cylindroid could be constructed. The ten constants not only define the cylindroid, but also point out two special screws upon the surface (§77). 117. Applications of Screw Co-ordinates. We have shown (§ 40) that if a, ß be the two screws of a cylindroid, which intersect at right angles, then the co-ordiuates of any sciew 0, which makes an angle I with the screw a, are: «, cos I + ßi sin I,... «B cos I + ße sin I, reference being made as usual to any set of six co-reciprocals. In addition to the examples of the use of these co-ordinates already given (§ 40), we may apply them to the determination of that single screw 0 upon the cylindroid (a, ß), which is reciprocal to a given screw y.