< 75i ) 
Mathematics. — “The cubic involution o/ the first rank in the 
plane.” By Dr. W. van der Woude. (Communicated by Prof. 
P. H. Schoute.) 
, ' If V 1S a P lane is in different ways possible to arrange 
the points of V in groups of three in such a way, that an arbi¬ 
trary point forms a part of only one group. If P x is a point of V 
there must exist between the coordinates of P x and those of the 
other, points of the group, to which P x belongs, some relations by 
which those other points are entirely determined. It is however 
possible that P x can be chosen is such a way that one of these 
relations is identically satisfied; in that case P x forms part of an 
infinite number of groups. 
We now start from the following definition: 
The points of a plane V form a cubic involution of the first 
rank, when they are conjugate to each other in groups of three in 
such a way that hcith the exception of some definite jminh) each 
point fomis a part of only one group. 
A triangle of which the vertices belong to a selfsame group we 
call an involution triangle ; each pQint which is a verlex of more 
than one, therefore of an infinite number of involution triangles, we 
call a singular point of the involution; each point coinciding with 
one of, its conjugate points is called a double point. If one of the 
sides of an involution triangle rotates around a fixed point, then the 
third vertex of this triangle will describe a right line or a curve; 
we shall restrict ourselves in this investigation to the case, that one 
vertex of an involution triangle describes a right line, when the opposite 
side rotates around a fixed point. 
2. When the points of a plane V form a cubic involution of the 
first rank which satisfies the just mentioned condition and which we 
shall furtheron indicate by (i 3 ), we can conjugate projectively to 
each point of V the connecting line of its conjugate points. Each 
vertex of an involution triangle and its opposite side are pole and 
polar line with respect to a same conic, which in future we shall 
always call y 2 ; each involution triangle is a polar triangle of y a . 
It is clear that reversely not every polar triangle of y, is an involu¬ 
tion triangle of (4); for each point of V is a vertex of an infinite 
number of polar triangles of y„ but of only one involution triangle. 
If however S is a singular point of the involution, then S must 
be a vertex of an infinite number of involution triangles, thus each 
polar triangle of y a having S as vertex is at the same time an 
