937 
remain the same (see however the errata below). In §2 of comm. 4 
occasionally the question arises if none of the lines can count double 
or triple in any plane. Now the case of a double line will be dealt 
with below and for a surface #”* without the singularities mentioned, 
but with a line which counts triple in a certain plane, the total 
number of lines is always 3. 
We conclude by considering the case that the curve in a planea 
consists of a single line a and a double line 5. The normal point 
of intersection A of a and 6 has « as tangent plane. If « turns 
round a, then (according to the results of p. 983) a non-degenerated 
oval appears out of 6. The points of intersection of this oval and a 
start from A to right and left, and at these points the oval faces A 
with its convex sides (anyway at first). Obviously 6 cannot be 
intersected by a line of #* not situated in «. Hence all further lines 
intersect a. If such further lines exist, their number is at least 4, 
which brings the total up to 7. If there are still more, the oval 
degenerates in at least 4 planes through a different from a and then 
the reasoniug on p 1251 (comm. 4) shows that #? contains lines 
which do not intersect a: a contradiction. Hence for a surface F" 
with a plane section consisting of a double and a single line, the 
total number of lines is 3 or 7 (for it is easily shown that no further 
multiplicity can increase these numbers). 
ERRATA. 
In comm. 1, p. 102 |. 22 for: “straight line and isolated point”; 
read: “straight line and point-oval”. 
In comm. 2, p. 309 1. 24—34: this part can be left out. 
p. 311 1. 1214: the letter C to be replaced by D. 
In comm. 3, p. 740 1. 25 for: “Liet c be a line through A in a, 
not being tangent to the oval and not coinciding with 
a or 6”; read: “Let c be a line through A in a, not 
coinciding with a or 6 (6 is tangent to the oval)”. 
p. 742 1. 7 from bottom, for: “It follows that A must 
be cusp in every plane except a”; read: “It follows 
that A must be cusp in every plane not containing 
the line a’. 
In comm. 4, p. 1251 1. 3 from bottom, for: “Now none of these 
last 4 points can coincide with one of the first, because 
in that case a line of /* would pass through that 
point and through the point of intersection of 6, and 
b',”; read: “Now none of these last 4 lines of inter- 
61 
Proceedings Royal Acad. Amsterdam. Vol. XXIII. 
