( 539 ) 



(3) is positive when N is positive, and therefore <p is imaginary. 



<I. Neither does q* = 2 give a real value for (p ; for substitution 



3 



in (1) furnishes for cos5(p the result — 1/2. 



4 



e. So we find only the real points : 



q = 0, y indefinite A' t , 



q = 1, co* 5y = 1 4\, 4',, 4',, A' 4l 4',, 



Q = — — , cos 5*/> = — 1 . . . the ten points of intersection 



a 



of the fifteen connecting lines three by three. 



4. We now consider a second case, in which the position of the 

 six basepoints is likewise a very particular one, where namely these 

 points form the vertices of a complete qnadilateral. Through these 

 six points not one genuine cubic with a cusp passes. For the three 

 pairs of opposite vertices (A lt A t ), (B lf Z? 2 ), (6\, C,) of a complete 

 quadilateral (fig. 2) form on each curve of order three, containing 



Fig. 2. 



them, three pairs of conjugate points of the same system, and these 

 do not occur on the cubic with a cusp, because through each point 

 of such a curve only one tangent touching the curve elsewhere 

 can be drawn. In this special case the locus of the cusps has broken 



