A Treatise on the Theory of Screws

417] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 459 In general there are four, but only four double points, i.e. points which remain unaltered by the transformation. If however two of the roots of the biquadratic equation be equal, then every point on the ray connecting the two corresponding double points possesses the property of a double point. For if p! = pv, then F, : F2:: X, : Xa, and hence the point whose co-ordinates are xlt X2, 0, 0, being transformed into y2, 0, 0 remains unchanged. Let us now suppose that a certain quadric surface is to remain unaltered by the homographic transformation. At this point it seems necessary to choose the particular character of the quadric surface in the further developments to which we now proceed. The theory of any non-Euclidian geometry will of course depend on whether the surface adopted as the infinite be an ellipsoid or a double sheeted hyperboloid with no real generators or a single sheeted hyperboloid with real generators. We shall suppose the infinite, in the present theory, to be a single sheeted hyperboloid. The homographic transformation which we shall consider will transform any generator of the surface into another generator of the same system, for if it transformed the generator into one of the other system, then the two rays would intersect, which is a special case that shall not be here further considered. Let three rays Rlt Ra, Ra be generators of the first system on the hyperboloid. After the transformation these rays will be transferred to three other positions Æ/, R.!, Rs' belonging to the same system. Let Sj, be two rays of the second system. Then the intersection of Rlt Ry, Ra, Ra &c., with give two systems of homographic points. The two double points of these systems on & give two points through which two rays of the first system must pass both before and after the transformation. Two similar points can also be found on S2. These two pairs of Double points on ft and' S2 will fix a pair of generators of the first system which are unaltered by the transformation.