986 



T[ie curve {b^)^ touches llic lliree cj- (jind />,). 



To a conic corresponds in tiie correspondence [P, P') a cnrve of 

 order 14 ; it consists for the conic j:?/' passing throngii C'l, C., C\, 

 B^, B.^, 0Ï three curves (6V)^ of (^i)', {Bj- and a singular line. As 

 /?3" is the curve of involution of the involution {j)',p"), which is 

 determined bv tiiat line, it is a singuiai- line of the 8"^ order, conse- 

 quently a line c. 



15. F'or n = o a further investigation produces only a (P") with 

 siv singular points of the 3"' order and as many singular lines of 

 the 3'^^ order. Through each of those points Ci- passes one of those 

 lines, c^■. A combination of the curve {Ckf' with the curve y" makes 

 it clear that the first curve also passes through the remaining points C. 



To the conic y^^ passing through C\, C\, C^, C^, C\ corresponds a 

 figure of the 16^'' order, composed of the 5 curves {Ck}^, k -\= 6, and 

 a singular line, (\. So 7/ is the curve of involution belonging to Cg. 



This [P^) may be produced by a iwt of cubic curves with base- 

 points C\: All the curves determined by a point P form a pencil, 

 of which the missiug base-points form with /* a ti'iplet of the 

 iuNolution ^). 



16. For 71 = Q we tlnd as the only solution of the relations (10) 

 and (11) <j^ = 'S, öj = 2> ^^ = 4. But this is to be rejected. For a 

 conic would have to be transformed by {P, P') into a figure of the 

 18^" order. To the conic passing through 3 points ^(*^ and 2 points 

 i>'(3) would correspond the figure composed of 3 curves {By and 2 

 curves {BY, which is already of the 18^'' order. 



For n = 7 we find no solution at all. 



The results obtained are united in the following table 



1) This (P^) is a plane section of a bilinear congruence of twisted cubics 

 indicated by Veneroni {Rend. Palermo, XVI, 210) and amply discussed by Stuyvaert 

 (Bull. Acad. Belgique, 1907, p. 470;. 



