136 



Through Y pass then two projective pencils of conies, which deter- 

 mine a quartic represented by 



\a/ a,' b^'\=0, (4) 



or also b} 



\ by- b:,' a:c' \ = (4*) 



The singular points are determined by the relations 



Cly 



0,j Oy Oy 



= (5) 



Now the curves üy^ bp = ay"- by' and ay' by'"- = ay" by' have apart 

 from the point 0^ (which is node on both) '12 points in common. 

 To them belong the three points, which ay' = and by- = have 

 in common apart from 0, ; they do not lie, however, on the curve 

 ay' by" = ay"- by'-. There are therefore, besides the singular pointe, 

 ?iine more singular points B^ ; the pairs of points, which form with 

 B]c groups of the involution (A^') lie on a curve ^h\ so that Bk is 

 a singular point of order four. 



The singular curve ^]^ is produced by two projective pencils with 

 common base points A and Bk\ it has therefore nodes in these two 

 points. From (4) and (5) it appears that this curve also passes through 

 the remaining singular points. The straight lines x, which contain 

 the p'airs X' ,X" lying on ^k\ envelop a conic. 



As ^h" passes through .4 twice, there are in (A^) two groups in 

 which the pair A,Bk occurs; so Bjc belongs twice to «^ This singular 

 curve has therefoi-e besides its quintuple point A, nine more nodes 

 Bk, is consequently of genus two and of class 18. 



On each of the 8 tangents of «% passing through A, two pairs 

 of the (A^) coincide ; from this it ensues that the straight lines s 

 on which two pairs have coincided, envelop a curve of class eight, 

 which we indicate by {s)^. 



4. We can now determine the order x of the locus A of the 

 pairs of points X' , X" , which form groups of the ^A'^) with the 

 points A of a straight line /. As «" contains eight points of /, A 

 passes eight times through A; analogously it has quadruple points 

 in Bk- The x points of intersection of ). with an other straight line 

 /* are vertices of triangles of involution, of which a second vertex 

 lies on /, so that the third vertex must be a common point of A 

 and /*. As these curves, besides in two vertices of the triangle 

 determined by the point //* and the x points mentioned, can only 

 intersect moreover in the singular points, we have for the deter- 



