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4. ..It is easy to. find the equation of d 4 . 
The polar conic of Z with respect to a x — 0 is represented by 
a z a x = 0, the polar line by a z a x = 0. For the tangents out of Z to 
that qonic we have thus 
a z ajb z — a z aj) z b x . 
If these contain the given point T, then Z is a point of the curve 
d* belonging to Y. So it has, as equation (in current coordinates z): 
a^a z b z = ciybya z b z . 
From this it is again evident, that the polar line of D is double 
tangent, and that it touches d 4 in the tangential points D and D". 
•For, by combination with a y a z = 0 we find aya z . byb\ = 0. The 
same is obtained by combination with bl = 0; by this is confirmed 
that d* is touched in its points of intersection with the polar conic 
of D by c s and the polar line d. 
. Out of 
aya,b z — a y bya z b z _ _ byb z a z — bya y b z a z 
follows that the equation of d 4 can be transformed into 
i (aya.bl - b^al) (a s b t - aj, y ) - 0, 
so also into 
{a y b z — a z b y y a z b z = 0. 
How 
a y b z — a z b y — (a x b 2 ) (y x z 2 ) -j- (a 2 b 3 ) (ysz 3 ) + (<*3*1) (y&i)- 
If thus we represent the coordinates of the line YZ by the 
above equation passes into 
. (abg)'a z b z = 0. 
This equation expresses that the polar conic of Z is touched by 
the line YZ 1 ). 
5. At the same time is evident from this that the line (§) cuts its 
poloconica in two points^ r This is more closely confirmed by the 
observation that the poloconica of (I) is the locus of the points whose 
polar conics touch (§), from which ensues that it intersects (§) in two 
associated points. 
The curve d* can therefore be generated by determining the points 
of intersection of each of the lines s through 2> with the conjugate 
poloeonica a. The- poloconica describes there a system with index 2. 
For, when ,0 passes through any point X the polar conic of X is 
touched by s. And as two lines s satisfy that condition, X lies on 
two curves a. This generation of d 4 ~ with the aid of a system of 
*) Glebsch, Leeway sue ia geom4trie, t. II, p. 278. 
48 * 
