127 



On /* there lie 9 straight lines and 8 conies. 



The straight lines resting on b and /, determine a representation 

 of .V on a plane. 



A straight line /, through a point C fnts A* in two more points 

 outside C; tVora this follows that the o' cutting /,, lie on a hvper- 

 boloid : this is entirely defined bv /,, h and Q.-. Anahigonsly the 

 p' resting in a fi.xed point on b or on a straight line intersecting h, 

 form respectively a qnadric cone or a hyperboloid. 



4. A plane 'f. through / cuts .V along a curve ).' which has a 

 double point on b. In each of the three points of intersection of ).* 

 with /, ). is touched by a y'. Hence the curves q* touching a plane 

 Ó, have their points of contact on a curve d'. 



Let 5 be a point of b\ the o' through the tive points B and C'k 

 touching d, form a surface of the 10'*' order with sextuple points 

 in B and Ck ')■ There are accordingly -t {j' through B and Ck which 

 have è as a chord ; consequently b is quadruple on the locus J of 

 the p' touching the plane ö and belonging to the congruence (1,0). 

 Also it appears that J has quadruple points in C'/.-. Accoidingly 

 an arbitrary o' uf the (1,0) has 24 points in common with J, i. e. 

 zl is a surface of the S'** order. 



5. A" has the curve of contact ff' and a conic d' in common with 

 the plane ^. The curves d' and d' touch each other in 3 points; 

 there are therefore tliree curves p' which osculate the plane d. 



If revolves round /, d' describes a surface of the fourth order 

 with the single straight line /. 



On the curve 9' cutting / in R, the pencil of planes (d) defines 

 an involution; / is therefore cut by two tangents of p'. Consequently 

 through / there pass two planes in which /? is a point of the "com- 

 plementary" curve d'. Hence d' describes a surface of the fourth 

 order with the double straight line /. 



Let us now consider the relation between the points P and Q 

 which the curves d' and d' in a plane d have in common with /. 

 Through P there passes one p' ; the tangent at P defines the plane 

 Ó, hence two points Q. Through Q there pass two p', hence two 

 curves d', and two planes d each containing a curve d* ; six points 

 P are therefore associated to Q. If two homologous points Peind Q 

 coincide, there arises a double coincidence of the (6,2), for at that 

 point a p' is osculated by the plane d. On / there lie therefore four 

 points N^ for which the plane of osculation r passes through /. 



') This is easily seen from the intersection of this surface with 7 123, which 

 consists of 2 conies and 3 double straight lines. 



