1278 
the conie passing through A and the four points C': from this it 
ensues that the locus of B’ is composed of the straight line BA 
and a curve z°, which has a node in B, and passes through the 
points A and C9. We indicate this by the symbol z° (A,5?,4C). 
§ 2. There are consequently two curves ¢*, which have a pout 
of inflexion in B. 
Any g* has three points of inflexion /; they lie on a straight 
line j. 
As B is point of inflexion on two curves, the straight lines j 
will envelop a conic. The locus zr of the points of inflexion (curve 
of inflexion) may be generated by (*) and the system of straight 
lines j, which has the index 2, and is evidently projective with (*). 
On an arbitrary straight line the two systems determine a corre- 
spondence (1,6), on a straight line passing through A a (1,2). From 
this it is to be deduced that the locus of the points of inflexion, 
apart from the five straight lines Ab, is a curve of order seven, 
which has a quadruple point in A and nodes in the other base- 
points. We may indicate it by the symbol ¢ (A*,55*). Outside the 
basis it has no singular points *). 
On each of the two curves y*, which possess a cusp in A, two 
points / have been taken up in A; from this it ensues that the 
tangents in the quadruple point of coincide in pairs. 
$8. Let us now consider the polar curve (antitangential curve) of 
B, consequently the locus of the points of which B is the tangential 
point. It is evidently generated by (g*) and the pencil of the polar 
conics of B, consequently a figure of order five. The polar conic 
of the y*, which is composed of BA and a ¢*, contains the straight 
line BA as component part; so the real polar curve of B is a 
curve a‘. It has nodes in A and in B; in the latter point it pos- 
sesses the same tangents as the tangential curve t°. In A it is 
touched by the tangents in the two cusps; these two lines form 
with AB the three coincidences of the (1,2) in which to each pair 
of nodal tangents of the y* the tangent of the corresponding polar 
conie is associated. Symbol 2‘ (A?, b*, 4C’). 
1) With an arbitrary <? the curve 73 has in common two points lying in B, 
the point B’ and the four points C; consequently two intersections lie in the 
node A. 
2) To a general pencil (p?), consequently with 9 base-points B, belongs a curve 
of inflexion .!2 (9B), possessing apart from the basis 12 nodes, being at the same 
lime nodes of curves 9°. 
