102 



IVTathematics. --- ^'The qai/lraple involution of the cotangential 

 pointe' of a cubic pencil." By Professor Jan de Vries. 



(Communicated in the meeting of April 24, 1914). 



1. We consider a pencil of cnbics {j^}, with the nine base-points 

 Bf^. On the curve r/\ passing through an arbitrary point P, lie tiwee 

 points P,P',F", which have the tangential point ^) in common with 

 F ; in this way the points of the plane may be arranged in qua- 

 druples of an involution {P*) of cotangential points. We sh-dW suppose, 

 that the pencil is general, consequently contains tmelve curves with 

 a node Di,. On such a curve rf^ all the groups of the (Z^) consist 

 of two cotangential points and the point D, which must be counted 

 twice. Apparently the 12 ])oints D are the only coincidences of the 

 inxolution ; as the coiniector of the neighbouring points of /) is quite 

 indetinite, the coincidences have no detinite snpport. The points Du 

 are at the same time to be considered as singular points ; to each 

 of them an involution of pairs P,P' is associated, lying on the curve 

 éh^, which has Dh as node. 



2. The nine base-points Bk are also singular ; to each point Bu 

 a tri[»le involution of points P' , P', P" is associated, lying on a 

 curve (3^, of which we are going to determine the order. 



To each cnrve <//' we associate the line h, which touches it in B; 

 in consequence of which a projectivity arises between the pencil of 

 rays (/;) and the cubic pencil ('/^). The curve t^ produced is the 

 locus of the tangential points of B {tangential curve of B). 



The line l>, which touches a (f'^ in B, cuts it moreover in the 

 tangential point of 73; this is apparently the only point that /> has 

 in common with t^ a[)art from B. So t'' has a triple point in B ; 

 there are three lines b, which have in iJ three points in common with 

 the corresponding curve (f^ ; i.e. B is point of infection of three 

 curves 9.'. 



Let us now consider the tangential curves r\ and t\, belonging 

 to 7>\ and />*,. Both pass thi-ough the remaining seven base-points, 

 consequently have apart from the points B, three points in common ; 

 so there are three cui-ves </:", on which B^ and B^ have the same 

 tangential point. Hence it ensues that the singular curve jij belonging 

 to i>i, has triple points in each of the remaining eight points B ; 

 it does not pass through i>\ because {P^) has coincidences in D/, 



1) Tne tangential point of P is the intersection of <p'' with the straight line 

 touching it in P. 



