104 



On an arbitrary slraighl line lie iheieïove ji/te en pairs of cotmiy ent in I 

 points. 



\\y (he transformation [P, P'), the cnr\'e V\ vvliich contains the 

 linear triplets, is transformed into a figure of order 1350. It consists 

 of twice X itself, three times t^% twelve times the curves /i" and 

 seven times the singular curves c>^ For 2 X 45 -f- 3 X 12 -|- 9 X 

 'J 2 X 9 = 1098; the points D produce therefore a figure of order 

 252. From this it ensues that l^'^ \\^?, septuple points \\\{\ie,\2 ^mgnl-Av 

 points D. 



The pairs P, P' , which are collinear with a point E, lie on a 

 curve e^', on which E is a triple point ; the tangents in E go to 

 the points of the triplet of the {P), determined by E. The line ^^jt 

 cuts ^4." in 9 points P, which form with B]. i)airs of the (P^) ; hence 

 e''^ has nonuple points in Bk. 



The locus of the pairs P" , P'", belonging to the pairs 1\ P' of 

 ^•'^ we shall indicate by e,.. As E is collinear with 12 pairs of the 

 involution {P^) lying on jij", B^ is a dodecuple point of e,.. 



On an arbitrary (/.^ the cotangential points form three involutions 

 of pairs and the supports of the pairs of each of those involutions 

 envelop a curve of class three (curve of Cayley). Consequently E 

 is collinear with 9 pairs P, P' of (f\ and this curve contains 9 pairs 

 of fji-. As the two curves in />\. have moreover 9 X 12 points in 

 common, consequently 126 points in all, % is a curve of order 42. 



The curves e'' and lij" have in the points ^A; (/(; =]= 1) 8 X 9 X 3 

 points in common ; moreover they meet in 9 points of EB^ and in 

 the 12 pairs P, P' mentioned above. The remaining 48 common 

 points must lie in Dh\ so è^"^ has quadruple points in the 12 singular 

 points D. 



The curves f,;,"^ and ji/ have in jB/X^-=H 1) 8 X 12 X 3 inter- 

 sections; further they meet in the 9 pairs P", P", belonging to the 

 9 points P' lying on EB^, and in the 12 points P\ belonging to 

 the 12 pairs P, P' oi' ,i/, which are collinear with E. So they must 

 have 60 intersections in D/,; ^^' has consequently quintuple points 

 in the 12 singular points D. 



The curves e.^'- and i'' have in Bk 9X^2x3, in Ai2x5x2 

 intersections, together 444; the remaining 60 lie in points of inflection, 

 of which the harmonic polar lines pass through E. In such a point 

 of inflection 1, eJ'^ will have a triple point, for the corresponding 

 polar line h contains a linear triplet, so three pairs of 6^% so that 

 / appears three times as point of e.^. Consequently E bears 20 

 straight lines k : the harmonic polar lines of (f^ envelop a curve of 

 class twenty. 



