105 



Mathematics. — ''.l cahic 'mvohuloii of the secoiul class.'' By 

 Prof. Jan ue Vries. 



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



1. By (lie class of a cubic involution in llie plane we shall 

 understand the number of pairs of points on an arbitrary straight 

 liue^). In a paper presented in the meeting of i^'ebruary 28^'', 1914 ■') 

 1 considered the cubic involutions of the first class, and proved that 

 they may be reduced to su principally ditfering sorts. 



The triangles A, which have tiie triplets of an involution of the 

 first class as vertices, belong at the same time to a cubic involution 

 of lines; the sides of each A form one of its groups. 



The cubic involutions of the second class possess the characteristic 

 quality of determining an involution of pairs i. e. an involutive 

 birational correspondence of points. For, let X, X', X" be a gi-onp 

 of an involution (A'') of the second class; on the line X'X" lies 

 another pair Y' , Y" ; the point Y, completing this pair into a triplet, 

 is apparently involutively associated to X. In the following sections 

 I shall consider a detinite (A'^) of the second class and inquii'e into 

 the associated involutive correspondence (A)'). 



2. We start from a pencil of conies y^ with the base-points 

 A, />!, B^, B^ and a pencil of cubics 7.' with the base-points i)\, B.^, 

 /ij, Ch {li = 1 to 6j. The curves 'r and 7 % which pass through an 

 arbitrary point A, intersect moreover in two points X' , X" , which 

 we associate to X. As the involutiojis 7'^ and 7% which are determined 

 on a straight line by the pencils {^r) tind ('ƒ'"), have two })airs A'', 

 A" and Y' , Y' in common, a cubic involution (A') of the second 

 class arises her-e. 



The ten base-points are singalar points, foi' they belong each to 

 00^ groups; on the other hand is a singular point certaiidy a base- 

 ])oint of one of the pencils. 



The pairs of points which with the singular point A determine 

 triangles of involution A, lie apparently on the curve «^ of the 

 pencil {(f"^), passing through A. As they are produced by the pencil 

 (ff"), they form a central involution, i.e. the straight Imes .i' = A' A" 

 pass through a point T of «' {opposite point of the quadruple 

 AB,B,B,). 



Analogously the j)airs A'', A", which are associated to Ci,. lie on 



1) This corresponds to the denomination introduced by Caporali for involutive 

 birational transformations. {Rend. Ace. NapoJi, 1879, p. 21 i2). 



-) '^Cubic involutions in the plane'. These Proceedings vol. XVI, p. 974. 



