Mathematics. — ‘On elementary surfaces of the third order”. 
(Fifth communication). By Dr. B. P. HAALMEIJER. (Communicated 
by Prof. Henprik DE Vries). 
(Communicated at the meeting of November 27, 1920). 
Some remarks from professor Ferix KrrIN caused me to go once 
more through my former communications on this subject *), which 
investigation showed me that several places require correcting and 
supplementing. 
$ 1. For surfaces of the third degree with 3 or 7 real lines *) 
there are systems of planes, which, as far as the real part of the 
section is concerned, have only one line in common with the surface. 
Only a very artificial interpretation would enable us to bring these 
surfaces under our definition of F* (comm. l.p. 102 and 103). This 
difficulty disappears when point 2 of that definition is replaced by 
the following condition (also preferable for other reasons): All plane 
sections which exist, are elementary curves, amongst which the third 
order occurs, but no higher order. 
With regard to the counting of lines, we prescribe the following: 
A line a is considered triple (resp. double) in a plane a, if every 
line 6 (Fa) of a is limiting element of a sequence of lines, situated 
in one plane and each of which has 3 (resp. 2, but for no sequence 3) 
points in common with F®, which points converge towards the point 
of intersection of a and 6. 
In all other cases a is considered single in a. 
The number of times a is counted on #* we put down as 1, plus 
1 for every plane in which a counts double, plus 2 for every plane 
in which a counts triple. 
Theorem: No plane section of F* is of the second order. 
Such a section would have one of the following forms: 
1. Isolated point. 
2. Oval. 
3. Two different single lines. 
+. One double line. 
') These Proc. Vol. XX, p. 101—118, 304—321, 736--748 and 1246—1253, 
*) Line stands for straight line. 
