226] FREEDOM OF THE FOURTH ORDER. 239 



then the two roots of the quadratic are equal but with opposite signs, and 

 hence ( 119) we have the following theorem. 



If the condition U^ = is satisfied by the co-ordinates of two screws 77 

 and which belong to the enclosing (n + l)-system, then these two screws 77, 

 and the tw.o screws which, lying on the cylindroid (rj, ), also belong to the 

 quadratic ?i-system U 9 = Q, will be parallel to the four rays of an harmonic 

 pencil. 



We are now to develop the conception of polar screws alluded to in 224, 

 and this may be most conveniently done by generalizing from a well-known 

 principle in geometry. 



Let be a point and S a quadric surface. Let any straight line through 

 cut the quadric in the two points X 1 and X 2 . Take on this straight line 

 a point P so that the section OXP^X^ is harmonic ; then for the different 

 straight lines through the locus of P is a plane. This plane is of course 

 the well-known polar of P. We have an analogous conception in the present 

 theory which appears as follows. 



Take any screw 77 in the enclosing (n + l)-system. Draw a pencil of n 

 cylindroids through 77, all the screws of each cylindroid lying in the enclosing 

 (n+ l)-system. Each of these cylindroids will have on it two screws which 

 belong to the quadratic w-systetn U e = 0. On each of these cylindroids a 

 screw can be taken which is the harmonic conjugate with respect to 77 

 with reference to the two screws of the quadratic n-system which are found 

 on the cylindroid. We thus have n screws of the f type, and these u screws 

 will define an n-system which is of course included within the enclosing 

 (n + l)-system. 



The equation of this n-system is obviously 



This equation is analogous to the polar of a point with regard to a 

 quadric surface. We have here within a given enclosing (n + l)-system a 

 certain re-system which is the polar of a screw 77 with respect to a certain 

 quadratic n-system. 



The conception of reciprocal screws enables us to take a further im 

 portant step which has no counterpart in the ordinary theory of poles and 

 polars. The linear equation for the co-ordinates of f, namely 



tf* = 0, 



is merely the analytical expression of the fact that f is reciprocal to the 



