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 1 and 0. 2 on that conic will 

 be unaltered (p. 2). So will R the intersection of the tangents to the conic 

 at Oj and 2 . The four double points will therefore be P, R, O l and 2 . 



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. O ly 0. 2 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 



with the essential condition 



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 U+XV 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. 



