925 
3 or 5 (and we leave undecided whether the case of 5 can occur 
The occurrence of points as here described is again excluded in what 
follows. 
Comm. 4 p. 1247 line 14 we used some results of comm. 3. 
These were deduced for points through which passes only 1 line of 
F* and the application on points through which 2 lines pass, causes 
incompleteness. Besides the note at the bottom of p. 1248 is not 
quite correct. On these grounds comm. 4 requires alterations. One 
of these will appear to be the extending of the numbers 3, 7, 15, 27 
before mentioned to 3, 5'), 7, 9, 15, 27. 
We shall now start on a minute consideration (§ § 3 and 4) of 
points through which pass 2 or more lines of #%. This will permit 
us to deal separately with several singularities. In the last part (§ 5) 
we intend to point out further changes necessary in comm. 3 and 4. 
$ 3. On points through which pass 2 different lines of F*, but 
not 3 different lines situated in one plane. 
Through A pass the lines BD and CE in a (fig. 1). We distinguish 
2 cases : 
1. Neither line counts double in «. 
2. One of the lines counts double in a. 
In the first case a contains a third line, not passing through A 
and along each of the 4 branches departing from A in a, the sectors 
of F* obviously meet from opposite sides of «a. It is easily shown 
that each of these branches is connected with both surrounding ones, 
the sectors lying alternately above and below a. 
In the second case -let CZ count double. Then along AC the 
sectors meet from one side of @ and along AZ from the other (via 
the point at infinity this is the same side). Along AB and AD the 
sectors meet everywhere from opposite sides of «. Here also it is 
easily shown that each of the 4 branches meeting at A in a, is 
connected with both surrounding ones, 2 successive sectors now lying 
above and the other 2 below a. 
First case. Let AB be connected with AC and AD with AEF 
above a by / and /// respectively, and below @ AC with AD 
and AK with AB by J/ and JV. To begin with we assume that 
all plane branches departing from A on F* touch « at A. Then in 
every plane, containing neither BD nor CE, A is ordinary point 
with tangent in «. We proceed to show that in a plane 8 (Fa) 
1) Further investigation will probably show that the number 5 can be left out. 
