( 711 ) 
Mathematics. — “On pairs of points which are associated loith 
respect to a plane cubic.” By Prof. Jan de Vries. 
(Communicated in the meeting of February 26, 1910). 
1. By the symbolical equation 
a * = ° 
a plane cubic c* is represented. If the points X, Y, and Z are 
connected by the relation 
each of them lies on the (mixed) polar line of the other two, and 
every two of those points are harmonically separated by the polar- 
conic of the third point; they form a polar triangle of c\ 
Let us look more closely at the case that the three points lie in 
one line l ; then Z is the point of intersection of l with the polar 
line of X and Y. 
It is evident that the triplets X, Y, Z lying on l form a cubic 
involution l\ of order two having the points of intersection P,Q,R 
of c 3 with l as threefold elements. 
According to a well known property of the h we find that P, Q 
and R form at the same time a group of the l\. This is indeed 
directly to be seen ; for, the polar conic of P intersects l in P and 
in the point H, which is harmonically separated by Q and R from 
P; the polar line of Q with respect to that conic therefore passes 
through R. 
To ll belongs a neutral pair, U, Y forming with each point of 
l a triplet and therefore having l as polar line. The polar conics 
of the points lying on l form a pencil; two of those conics u 2 and 
v ’ touch l in the points V and U. 
We shall call IT and V associated points. 
Evidently each point U is associated with two points V, yiz. 
with the points which have the polar line and the polar conic of 
U in common. The associated pairs are thus arranged in an involutory 
correspondence (2,2). 
If / becomes tangent to c s , then in the point of contact L two 
threefold elements of the ll unite themselves with the two neutral 
points U, V. For, all polar conics whose poles lie on l pass through 
L and one of those curves touches / in L. So c* is curve of coin¬ 
cidence of the (2, 2) correspondence. 
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