928 
above, namely 1 on Z and J/// each, hence A is always double point. 
For the plane through PQ and 7’U remains only the possibility that 
A is cusp with P'U for tangent. From this follows again that A is 
double point in every plane through RS not containing TU. All 
this can at once be extended to the following: 
In every plane containing neither BD nor CE, A is double point, 
except in the planes through TU, and in these A is cusp with TU 
for tangent. 
We note that no line through A carries 2 other points of F*. 
There remain to be considered the sections in planes through BD 
or CE. If such a plane does not pass through 7'U, then it obviously 
contains a non-degenerated oval through A. For a plane through 
TU and one of the lines in @ we shall show that the entire section 
is situated on the line in «. Let us consider the plane @ through 
CE and TU. The restcurve (that is the curve minus CZ) cannot 
contain a line different from CE, for TU has only A in common 
with F* and no third line passes through A. Neither can 8 contain 
an oval not passing through A, as no line through A carries 2 other 
points of F*. If there is an oval through A, then TU is tangent at 
A. Let the branch AV of this oval (fig. 1) depart on / (above a). 
Branches leaving from A on / all depart behind the plane through 
RS and TU. This holds for every line RS inside / BAE, hence 
no branch departing from A on / has a semitangent at A situated 
outside trihedron AUCB. The branch AV cannot be isolated on 
the frontside (that is the side on which D lies) of the plane 3 
(through CE and TU), hence on that side of 8, AV is connected 
with AC. Let y be a plane through BD, such that AC and AU 
lie on different sides, then in this plane a branch departs from A 
on the subsector of I joining AV to AC and the line of intersection 
of 3 and y is tangent to this branch (as no semitangent is situated 
outside trihedron AUCB). This tangent would;have only A in common 
with #*, but in 8 it carries a second point of the oval, hence we 
arrive at a contradiction. 
We thus find that the plane through CE and 7'U contains no 
points of F* not situated on CE and the corresponding thing holds 
for the plane through BD and ZU. Hence all tangents at A are 
situated in the planes through ZU and CEor BD respectively, and 
on the other hand it is easily shown that every line through A in 
one of these planes is tangent. Hence A 2s biplanar point and in each 
of the tangent planes the entire section is situated on a single line. *) 
1) A good drawing representing such a point can be found in table II joined to 
a note of Krein, Math. Ann. 6, p. 551. 
