A Treatise on the Theory of Screws

418] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 463 The theorem can otherwise be shown by drawing Figure 46. 1 4 and 3 2 are to be generators of the infinite quadric. This will show that 4 (and not 2) is the correspondent to 1, and that 2 (and not 4) is the correspondent to 3, and thus the statement of anharmonic equality, (1 QQ' 3) = (4 PP' 2), becomes perfectly definite. 1 2 and 3 4 are, of course, not generators; they are two conjugate polars of the infinite quadric. We can now see the reason of the anharmonic equality. Let PQ be a generator of the infinite quadric, as is clearly possible, for 1, 3 and 2, 4 are both generators of the opposite system. Then, since a generator of the infinite quadric must remain thereon after the displacement, it will follow that P'Q', to which PQ is displaced, must also be a generator; and thus we have four generators, 4 1, PQ, P'Q,', 2 3, on a hyperboloid of one system intersecting the two generators of another, and by the well-known property of the surface, (1 QQ' 8) = (4 PP' 2). We also see why the infinite quadric is only one of a family which remains unaltered. For, if PQ be a generator of any quadric through the tetrahedron, 1, 2, 3, 4; then, since P and Q are conveyed to P' and Q', and since the anharmonic equality holds, it follows that P'Q' will also be a generator of the quadric, i.e. a generator of the quadric will remain thereon after the displacement. It is a remarkable fact that, when the linear transformation is given, the infinite quadric is not definitely settled. We have seen how, in the first place, the linear transformation must fulfil a fundamental condition; but when that condition is obeyed, then a whole family of quadrics present themselves, any one of which is equally eligible for the infinite.