A Treatise on the Theory of Screws

460 THE THEORY OF SCREWS. [417, In like manner we find two rays of the second system which are unaltered. The four intersections of these rays must be the four double points of the system. We can also prove in another way that in the homographic transforma- tion which preserves intervene, the four double points must, in general, lie on the fundamental quadric. For, suppose that one of the double points P was not on the quadric. Draw the tangent cone from P. The conic of contact will remain unaltered by the transformation. Therefore two points O2 and 02 on that conic will be unaltered (p. 2). So will R the intersection of the tangents to the conic at O1 and 02. The four double points will therefore be P, R, 01 and O2. But PR cuts the quadric in two other points which cannot change. Hence PR will consist entirely of double points, and therefore the discri- minant of the equation in p would have to vanish, which does not generally happen. Of course, even in this case, there are still four double points on the quadric, i.e. O2, 02 and the two points in which PR cuts the quadric. We may therefore generally assume that two pairs of opposite edges of the tetrahedron of double points are generators of the fundamental quadric, the latter must accordingly have for its equation = 0, with the essential condition Plp2 — PsPi- Every point on any quadric of this family will remain upon that quadric notwithstanding the transformation. Nor need we feel surprised, when in the attempt to arrange a homographic transformation which shall leave a single quadric unaltered, it appeared that if this was accomplished, then each member of a family of quadrics would be in the same predicament. Here again the resort to ordinary geometry makes this clear. In the displacement of a rigid system in ordinary space one ray remains unchanged, and so does every circular cylinder of which this ray is the axis. Thus we see that there is a whole family of cylinders which remain unchanged; and if U be one of these cylinders, and V another, then all the cylinders of the type J7+XT are unaltered, the plane at infinity being of course merely an extreme member of the series. More generally these cylinders may be regarded as a special case of a system of cones with a common vertex; and more generally still we may say that a family of quadrics remains unchanged.