514 



ciated line / and ^' lias tlierefore always two points in common 

 with /. 



To a straight line I' of points K there corresponds accordingly a 

 system »S, of oo' conies each of which cuts a line I twice and the 

 straight lines ti,, . . ., a^ once. 



Also the reverse is apparent. 



If K describes a plane n, x continues to pass through the pole 

 F of .T. 



A plane .t is the image of a system S, of oo' conies the planes 

 of which pass through a point P and tvhich cut a^, . . ., a^. Inversely 

 such a system is represented on a plane. 



To the conies of S, passing through a definite point l\ tlie points 

 of a plane curve kp lying in the polar plane .t of /-"are associated. 

 In order to find the order of kp, we try to tind the number of 

 conies of 5, through F and through a definite point Q of (/,. The 

 conies through P and Q intersecting at,a^,a^, form a surface of 

 the fourth order. For a plane through PQ contains the non-degene- 

 rate conic of this surface passing through 1\Q and the points of 

 intersection of this plane with (/.,, a^, a^, but also the straight line 

 PQ which is a double line of the surface, because together with 

 the two transversals of PQ, r/,, a^ and a^, it forms two degenerate 

 conies of the surface. As a, intersects this sui-face in four points, 

 there pass through a point Q of a, four conies of the system S, 

 of the conies cutting «,, . . . , a, and passing through P. It follows 

 from this that a plane through a', cuts the eur\e kp in foui- points 

 outside a\. Further kp has a double point on a',, which is associated 

 to the double conic of aS', lying in the plane through J* and a, and 

 passing through P and the points of intersection of this plane and 

 a,, . . , a^. The two tangents to kp at this double point are associated 

 to the straight lines joining P to the two points in which the 

 corresponding double conic cuts a^. The curve kp is accordingly of 

 the sixth order. This cui've intersects also the ten straight lines t'ik, 

 e.g. the line <',, in the point corresponding to the degenerate conic 

 consisting of /,, and the transversal of a, and <„ through P. 



The system of the conies of S, passing through a point P, is 

 accordingly represented on a plane curve of the sirth order lohich 

 has double points on a\, . . ., a,\ and which cuts the ten straight lines 

 t' t' 



As kp has six points in common with an arbitrary plane, six of 

 the planes of the conies of aS\ pass through an arbitrary point. 



The system S,' of the conies cutting a given straight line /, is 

 represented on a surface Oi- The order of this surface, i. e, the 



