A Treatise on the Theory of Screws

418] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 461 Reverting, then, to a space of which the several points correspond to the objects of a content, we find, that for every homographic transformation which corresponds to a displacement in ordinary geometry a singly infinite family of quadrics is to remain unchanged, and the infinite quadric itself is to form one member of this family. Let us now suppose a range in the content submitted to this description of homographic transformation. Let P, Q be two objects on the range, and let X, F be the two infinite objects thereon. This range will be transformed to a new position, and the objects will now be P', Q', X', Y'. Since infinite objects must remain infinite, it follows that X' and Y' must be infinite, as well as X and Y. Also, since homographic transformation does not alter anharmonic ratio, we have (PQXY) = (P'Q'X'Y')-, whence, by Axiom V., we see that the intervene from P to Q equals the intervene from P’ to Q'; in other words, that all intervenes remain unchanged by this homographic transformation. Every homographic transformation which possesses these properties must satisfy a special condition in the coefficients. This may be found from the determinantal equation for p (p. 458), for then the following symmetric function of the four roots p2> p3, p4 must vanish: (piPi ~ Pspt) (piPs ~ p-2pi) (piPt ~ PzpsY 418. The Geometrical Meaning of this Symmetric Function. We may write the family of quadrics thus : X,X, + XZ8Z4 = 0. All these quadrics have two common generators of each kind: Æ = 0, X, = 0 (X4 = 0, X4 = 0 J and J and = 0) ^4 = 0 ^-^-2 “ 0, ^3 ~ 0. For the rays X4 = 0, X3 = 0, and X4 = 0, X4 = 0, are both contained in the plane Xlt and therefore intersect, and, accordingly, belong to the opposed system of generators. The geometrical meaning of the equation PlPz Psp4 = 0 can be also shown. The tetrahedron formed by the intersection of the two pairs of generators just referred to remains unaltered by the transformation. Any point on the edge, X4 = 0, X3- = 0, of which the co-ordinates are 0, X2, 0, X4,