( 776 ) 
Mathematics. — “On polar figures with respect to a plane cubic 
curve. By Prof. Jan de Vries. 
(Communicated in the meeting of March 26, 1910). 
1. If a plane cubic curve y s is represented symbolically by a®=0 
then a x a y a w = 0 represents the polar line p xy of the points X and 
Y, i.e. the polar line of X with respect to the polar conic tt ;j of Y 
and at the same time the polar line of Y with respect to the polar 
conic n x of X. 
The three polar lines p xy , p xz , and p yz will concur in one point 
W when the three conditions are satisfied 
axOifCiw — 0, a x a z a w = 0, a y a z a w = 0 . . . . (1 J| 
By elimination of the coordinates Wk we find out of it 
{abc) a x a y b x b z CyC z =0 ...... (2) 
So to two given points X, Y belongs a conic y\ y as locus of 
the point Z, it passes also through X and Y, for when Z and 
X coincide, we find 
(abc) a^yblcyCx = (cba) c x c y b^iya z eze — {abc) a x a y b x CyC x = 0- "JH 
As we can substitute {abc) a x a y c x c z b y b z = 0 for (2), thus also 
{abc)a x a y b z c z {b x c y —b y c x ) = 0, we can also represent y^ by 
{abc) axa y bzc z {bc§) = 0, where are the coordinates of the line X YM 
Consequently (2) can be replaced by 
(6e£) {ben) b z c z = 0.. (3) 
From this ensues that the conic y^ is the poloconica of the 
lines § and ij. 
So the poloconica of two lines is the locus of the points Z which 
with relation to the points of intersection X, Y of this conic with 
one of the given lines are in such a position that the polar lines 
p xz and p yz concur on the other one of the given lines, which is 
then at the same time polar line of X and Y. 
2. If Z and W are the points of intersection of with r\, it 
follows out of the symmetry of (3) in connection with the equations 
(1), that the four points X, Y, Z, W form a closed group, so that 
each side of the quadrangle determined by them is the polar line 
of the vertices not lying on it, therefore a polar quadrangle (Reye). 
Out of our considerations ensues that a polar quadrangle is deter¬ 
mined by two of its vertices, but also by two of its opposite sides. 
In the last case the vertices are determined by the poloconica of 
the given lines; in the former case we can use the poloconica be¬ 
longing to the polar line of the given points and their connecting 
line. 
