Mathematics. — ''Cubic invo/utions of ihe jlrst class \ By Prof. 

 Jan dk Vries, 



(Communicated in llie meeling of February 23, 1918). 



1. By the "c/ass" of an involution in tlie plane we nnderstand 

 the nnmber of pairs of points on an arbilrai-y line. In a paper 

 printed in volnine XVI '), I have proved that the cubic involutions 

 of the first class may he reduced to six principal species provided 

 that it be supposed that there are no collinear tri|)lets. 



I will prove now that these involutions, with a few exceptions, 

 may be determined by nets of cubics. 



Let a net [c'] be given with six base-points CÏ-. AH c" that yet 

 pass through a point A', form a pencil (c"''), have therefore still two 

 points A" and A^" in common, wdiich form with A" a group of an 

 involution 7,. On an arbitrary straight line [c*] determines a cubic 

 involution /% of the second rank; the neutral pair consists of two 

 basepoints A', A", consequently is /, an involution of the first 

 class*). 



To [c'J belongs the y'jt, which has a nodal point in Ck. If A' is 

 chosen on this nodal y'/,-, one of the [)oints A^', A'" comes in 6'/,- ; 

 so Cjc is a singular point that forms groups of the /, with the pairs 

 of an /j, lying on the singular curve y^k- Each of the two points 

 of y'/c lying in Ck, belongs to a pair of the /, ; from this it ensues 

 that the pairs of this /, are lying on the tangents of a conic 

 {curve of involution of the I^). 



To [c*] belongs also the figure foi-nied by the conic y/, which 

 contains the points C\, 6\, C\, C\, C\, and a cei'tain straight line c,, 

 on which C^ lies. As [c*] determines on c„ the pairs A', A' of an 

 /,, c, is a singular straight line. 



The involution /, has therefore six singular points and six singular 

 straight lines. 



The points X" , which complete the pairs of the /, lying on 

 c, into triplets of the /,, lie evidently on y%. Let V" be the 

 projection of A" on c„, out of a tixed point of y% ; there exists a 

 relation (2,1) between Y" and A', so that Y" coincides three times 



1) "Cubic involutions in the plane". These Proceedings XVi, 974—987. 



~) If the rays XX', XX" are associated to each point A', a null-system St (2,2) arises. 



