234 THE THEORY OF SCREWS. [224, 



224. Properties of a Quadratic Two-system. 



The quadratic two-system is constituted of screws whose coordinates satisfy 

 three linear equations and one quadratic equation, and these screws lie 

 generally on a surface of the sixth degree ( 225). If we take the plane 

 representation of the three-system given in Chapter XV., then any conic in 

 the plane corresponds to a quadratic two-system and all the points in the 

 plane correspond to the enclosing three-system. Since any straight line in 

 the plane corresponds to a cylindroid in the enclosing system and the 

 straight line will, in general, cut a conic in the plane in two points, we have 

 the following theorem. 



A quadratic two-system has two screws in common with any cylindroid 

 belonging to the enclosing three-system. 



A pencil of four rays in the plane will correspond to four cylindroids 

 with a common screw, which we may term a pencil of cylindroids. Any 

 fifth transversal cylindroid belonging also to the same three-system will be 

 intersected by a pencil of four cylindroids in four screws, which have the 

 same anharmonic ratio whatever be the cylindroid of the three-system 

 which is regarded as the transversal. We thus infer from the well-known 

 anharmonic property of conies the following theorem relative to the screws 

 of a quadratic two-system. 



If four screws a, /3, j, 8 be taken on a quadratic two-system, and also 

 any fifth screw 77 belonging to the same system, then the pencil of cylindroids 

 (770), (77/3), (^7), (?)&) will have the same anharmonic ratio whatever be the 

 screw V). (See Appendix, note G.) 



The plane illustration will also suggest the instructive theory of Polar 

 screws which will presently be stated more generally. Let 7=0 be the conic 

 representing the quadratic two-system and let V= be the conic representing 

 the screws of zero pitch belonging to the enclosing three-system. Let P be a 

 point in the plane corresponding to an arbitrary screw 6 of the three-system. 

 Draw the polar of P with respect to U=Q and let Q be the pole of this 

 straight line Avith respect to V= 0, then Q will correspond to some screw &amp;lt;/&amp;gt; of 

 the enclosing three-system. From any given screw 6, then by the help of the 

 quadratic two-system a corresponding screw $ is determined. We may term 

 &amp;lt;/&amp;gt; the polar screw of with respect to U0. Three screws of the enclosing 

 system will coincide with their polars. These will be the vertices of the 

 triangle which is self-conjugate with respect both to U and to V. 



A possible difficulty may be here anticipated. The equation V= is itself 

 of course equivalent to a certain quadratic two-system and therefore should 

 correspond to a surface of the sixth degree. We know however ( 173) that 

 the locus of the screws of zero pitch in a three-system is an hyperboloid, so 

 that in this case the expectation that the surface would rise to the sixth 



