A Treatise on the Theory of Screws

CHAPTER XIX . HOMOGRAPHIC SCREW SYSTEMS*. 243. Introduction. Several of the most important parts of the Theory of Screws can be embraced in a more general theory. I propose in the present chapter to sketch this general theory. It will be found to have points of connexion with the modern higher geometry; in particular the theory of Homographic Screws is specially connected with the general theory of correspondence. I believe it will be of some interest to show how these abstract geometrical theories may be illustrated by dynamics. 244. On Plane Homographic Systems. It may be convenient first to recite the leading principle of the purely geometrical theory of homography. We. have already had to mention a special case in the Introduction. Let a be any point in a plane, and let ß be a corresponding point. Let us further suppose that the correspondence is of the one-to-one type, so that when one a is given then one ß is known, when one ß is given then it is the correspondent of a single a. The relation is not generally interchangeable. Only in very special circumstances will it be true that ß, regarded as in the first system, will correspond to a in the second system. The general relation between the points a and ß can be expressed by the following equations, where alt a2, a3 are the ordinary trilinear co-ordinates of a, and ß2, ß2, ß3, the co-ordinates of ß, /31 = (ll)a1 + (12)a2+(13) as> ßn — (21) + (22) a.j + (23) a3, ft = (31) a1 + (32)a2+(33) a3. In these expressions (11), (12), &c., are the constants defining the particular character of the homographic system. * Proc. Roy. Irish Acad. Ser, ii. Vol. in. p. 435 (1881).