( 540 ) 



up into the four sides of the quadrilateral each of those lines counted 

 three times. For it is clear that an arbitrary point of the line A l B l C l 

 e. g., as a point of contact of this line with a conic passing through 

 A„ P s , C„ represents a cusp of the linear system of cubics. We can 

 even expect that each of the four sides must be taken into account 

 more than one time, because each of those points instead of being an 

 ordinary cusp is a point, where two continuing branches touch each 

 other. And finally the remark that the sides of the quadrilateral 

 divide the plane into four triangles e with elliptic and three quadran- 

 gles A with four hyperbolic points, so that they continue to form 

 the separation between those two domains, forces us to bring them 

 an odd number of times into account, namely three times because 

 we must arrive at a compound curve c la . 



Some more particulars with respect to the domains e and //. The 

 nodal tangents of the cubic (fig. 3) passing through the three pairs 

 of points (A x , A x ), (B lt B t ), {C\, C % ) and having in Pa node, are 

 the double rays of the involution of the pairs of lines connecting 



Fig. 3. 



P with the three pairs of points mentioned, so also the tangents in 

 P to the two conies of the tangential pencil with the sides of the 

 quadrilateral as basetangents, passing through P; now, as these two 

 conies are real or conjugate imaginary according to P lying in one 



