929 
The above results show that here /* contains no line not situated 
in a. If follows that none of the 3 lines in « counts double in any 
plane. If none of them ever counts triple either, then the total 
number of lines on F° is 8. 
Let H and K represent the points of intersection of the third line 
in a and CE and BD respectively. For each of these points two 
possibilities exist, namely they can be normal point of intersection 
of 2 lines or biplanar point of the same type as A. If both are 
normal points, then the above deseribed character of such points 
shows that none of the 3 lines in « counts triple in any plane. If 
on the other hand A is biplanar and K normal, then the line 
AH (= CE) counts triple in the plane through 7'U, in which plane 
then also lies the cuspidal tangent at the biplanar point H. In this case 
the total number of lines on F* would be5*). If lastly, Kis biplanar 
also, then each of the 3 lines in « counts triple in a particular plane 
and the total number of lines on F* becomes 9. 
Second case. CE counts double in a. Suppose above a AB is 
connected with AC by J and AC with AD by //. Below a AD 
with AE and AH with AB by J/I and JV respectively (fig. 2). To 
begin with we assume that every branch departing from A has « 
for tangent plane. Then A is point of inflexion in every plane not 
containing BD or CE. We consider a plane (4 a) through CE. If 
in this plane a further branch leaves A, then in this plane an oval 
touches CE at A. Let this oval depart from A above « and 
let the branch leaving in the direction AZ at first lie on /. Now in 
a plane (Ze) through the line AL of a (fig. 2), the branches 
departing on // and {V form a point of inflexion at A, but at least 
2 more branches depart on /, hence a contradiction has been obtained. 
Let us consider a plane 2 through BD. If in 3 another branch 
leaves A, then an oval touches BD at A. Let this oval 
depart from A above a. If both branches of this oval at first lie 
on /, then we get a contradiction as above. Hence one branch AF 
would have to start on / and the other AH on J//. In § now depart 
from A the branches AB and AD on the line and AF and AH on 
the oval having this line for tangent. Former results’) show that 
each of these 4 branches is connected with the 2 surrounding ones, 
alternately on opposite sides of 8. Now AB and AD are connected 
on that side of 8 where Z lies, hence AF and AH also, but then 
inside every vicinity of A, / and // would be connected by a sector 
1) We leave undecided whether this case can occur. 
8) Comm. 3 p. 738 and 739. 
