( 777 
Out of a^yOv = 0 and a J a y a z = 0 follows 
4r«y (ton 4- fi£l~) = 0. 
Here A and p can be determined in such a way that la V3 -\-pa z =a u 
relates to the point of intersection U of XT with WZ. 
As a u a x = 0 indicates the polar conic ar„ of V we find that X 
and Y according to the relation a^aya a = 0 lie harmonically with 
respect to n u . In an analogous way ensues from a^ay = 0 and 
O'v/izdx — 0 the relation a l0 a g a u — 0, according to which W and Z 
are also separated harmonically by n u . 
But then also the points T=(XZ,YW) and V=(XW,YZ) are 
conjugated with respect to x u , i.e. we have a u a 0 a t = 0. Now U,V,T 
are the diagonal points of the complete quadrangle XYZW, so that it 
is proved that the diagonal triangle of a polar quadrangle is always 
a polar triangle 1 ). 
3. When the conic y 2 degenerates we can take for Z each point 
on the line XY. To trace for this the condition, we put zy=Ju , i c -{-pyk; 
from (2) follows 
(abc) axdybzCy (Xb x + pb y ) f pcy) = 0, 
so 
A 2 (a5c) axdyblfrfiy + Ap { ahc ) 4* 
+ l[i (oic) OzflybxbyCzfiy + f* 2 (abc)ataybJ>yCy = 0. 
By exchanging two of the symbolic factors a, b,c, we see that 
three of these terms are identically zero; so we have 
Af* (abc) OaflybxCy = 0. 
For an arbitrary choice of X and Y this equation furnishes only 
A = 0 and p = 0, thus the points X and Y. It furnishes each point 
of XY, as soon as 
(abc) a^blc* — 0.(4) 
When X, Y, and Z are eollinear, the polar line p x of X and the 
polar lines p„ n p X2 concur in one point; for these three lines are 
the polar lines of X, Y, Z with respect to the polar conic st x . If 
now (4) is satisfied, then also p yz passes through that point, hence, 
the six polar lines p x , p y , p z , Pxy, pyz, Pzx concur in a point W. But 
when p x , py and p z are concurrent, the poloconica of §~XYZ, 
degenerates and § is tangent of the Cayleyana. 
From this ensues that for given Y the equation (4) will represent 
i) Mentioned without proof by Caporali (Transunti d. R. A. dei Lincei 1877^ 
p. 236). 
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