1279 
§ 4. If in the tangential point / of a point P the tangent is 
drawn, the latter intersects the y* laid through P in the second 
tangential point P" of P. Going on in this way we arrive at the 
definition of the n! tangential point P. The locus of the point 
Be is called the n tangential curve x, of B. Its order will be 
indicated by ¢,; by ay, Db, cn will be indicated how many times it passes 
through the points 4,5, C, We shall now deduce four relations 
which the numbers ty, dn, %,, Cn must satisfy. 
In the first place we observe that two of the tangents coincide 
in B with the two stationary tangents of curves ¢*, on which B 
is point of contact. Each of the remaining (/,—2) tangents belongs 
to a p° on which B coincides with the n” tangential point B. 
The curves tr, and z* have, besides the base-points, only the 
(6,—2) points Br-t in common, of which the tangential point, 
consequently 5%), coincides with B. In B they have (25, 4 +2) 
points in common, because the tangents at z* are at the same time 
tangents of 1,1. As A is node of a* and this curve passes singly 
through the four points C, we find the relation 
hear eden Abr van. velt) 
Now we pay attention to the intersections of the r,—1 belonging 
to 5 with the polar curve of another base-point C. Apart from the 
basis they have only the c¢, points L —) in common, for which 
Bo lies in C. In B we find b,;, in the remaining base-points 
(De 1 + 3¢,_1) intersections, consequently is 
A es + Oya Dena dOr on on ar (2) 
In order to obtain a third relation we pay attention to the points, 
which tv, has in common with an arbitrary g°. As apart from 
the base they can intersect only in the point Bt lying on g°, 
we find: 
Ne Aen 7 ls: ee Se (3) 
A fourth relation is produced by the property that the curve rt, 
must be rational, because each point 6” may be associated to the 
straight line that touches the y* laid through Bv. Consequently is 
(tn—1) (42 —2) = anlan—l) + bulbn— 1) + 4enlen—l) . (4) 
From the first three relations we deduce 
i= En Ön +] ot of) TELT (5) 
Cp sene send 1 CIDE GN Ge EE c (6) 
NEE tees oe ey 
From (5) and (6) it ensues moreover 6,—c, = — (b,-1—¢n_1); 
consequently is 
