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of lines in « would belong to F*. Two branches, separated by a 
single one, cannot be directly connected, for if the joining sector 
were situated above «, then the sector departing above a from the 
branch in between, could not be fitted in on a twodimensional 
continuum. All this leaves only 2 possibilities for connecting the 6 
branches : 
1. Every branch is connected with the 2 surrounding ones, altern- 
ately above and below a. 
2. A representative case of this possibility is that above a. AC'is 
connected with AF, AD with AH and AA with AB, and below a. 
AD with AH, AE with AF and AB with AC (fig. 4). 
1. Suppose above a AB is connected with AC by /, AD with 
AE by III, and AF with AH by V, below a AC with AD by 
Il, AE with AF by /V and AH with AB by VI (fig. 3). To 
Fig. 3. 
begin with we assume that a branch departs from A, for instance 
on /, not having « for tangent plane. Then a plane through A 
inside / DAC can be found, in which 2 branches start from A on 
I. Besides one branch on // and V each. The branch leaving on // 
does not have « for tangent plane. Proceeding in this way we find 
that a branch leaves on /// not touching a and lastly we show 
that the same holds for /V. Then however an infinite number of 
planes can be found in which 2 branches start on 7 and JV each 
and one on // and V each. This means an infinite number of degene- 
rations, which case we exclude. 
Hence all branches starting from A have « for tangent plane. 
This means that A is point of inflexion in every plane which does 
not contain one of the lines in a, with all tangents in «e. We 
proceed to show that in no plane through one of the lines in « 
further branches depart from A. Suppose in a plane ? through BL 
a further branch leaves 4, for instance above «. Then in 8 an oval 
(with BE for tangent) leaves on / and J/J. Let 8 turn round BE, 
