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through BD (or CE) no further branches depart from A. Suppose 
a further branch leaves A in 8, then 3 contains an oval having BD 
for tangent at A. Let this oval depart from A above a. Let AF and 
AH be the branches of this oval and let AF’ depart on J, then AA 
is situated on J//, for the assumption that both lie on J at once 
gives a contradiction when we consider the section in a plane (# a) 
through RS (fig. 1). Now let 8 turn round BD, in such a way 
that the top part moves towards AC, then in every position before 
a is reached, a branch departs from A on J and we keep ovals, 
touching BD. From this it follows that in all these planes @ the 
lines through A not coinciding with BD, carry only 1 other point 
of F*. Now this is impossible as in every plane not containing BD 
or CE, A is ordinary point with tangent in a and if this tangent 
be slightly turned in that plane round A, we obtain 3 different 
points of intersection with #”. 
A point A as here described we call normal point of intersection 
of 2 lines on F*. 
We shall now consider the alternative, that a plane branch departs 
from A not having a as tangent plane at A. Let us first assume 
that a third line of F* (not situated in a) passes through A. Let this 
line depart above « on / and below « on //. Let 8 denote the 
plane through this line and BD. If every vicinity of 4 in 8 contains 
points of /J7+ JV then the curve in 8 is composed of 3 lines 
through A, which case shall be dealt with later. Remains the possi- 
bility that //74+ AE + IV is situated entirely on one side of 8. 
Then however of the 4 branches departing from A in 8, two opposite 
ones are directly connected and this is impossible, as well when one 
of the lines in 8 counts double, as when both count single. 
Let a branch departing from A and not having « for tangent 
plane, be situated on J, then through the line RS of « (fig. 1) 
planes can be found in which at least 2 branches depart from A 
on J, but in each of these planes at least 1 branch departs on JI 
and / on ZV, hence A is ordinary double point and as RS is not 
tangent, branches depart on J// and JV also, for which a is not 
tangent plane. The same now follows for J/7/. Hence if a is turned 
round a line through A in « (A BD or CE) out of its original 
position, then at first A remains double point. 
Now let « turn round RS in such a way that the back part 
moves upwards. We proceed to show that there cannot be a last 
position with double point in A and branches departing on /. Suppose 
a plane 8 formed such a last position and let AN and AM be the 
