i03 



1. F^ is to answer the most general definition of a twodiraen- 

 sional continuum '), 



2. Every plane section of F'^ is an elementary curve of the 

 third order. 



This note is divided into two parts: 



In the first part we shall prove: The tamjents to plane sections 

 passing through an arbitrary point A of F\ not situated on a line 

 of F\ form one plane, which may be called tangent plane to F^ in 

 A Only one exceptional point is possible having the following 

 character: It is isolated in every plane except the planes through 

 one line, and in these it is cusp with that line as cuspidal tangent. 



In the second part we begin by proving some further theorems 

 concerning points of F^ not situated on a line of F\ At the end 

 we assume that no point of a certain plane section is situated on 

 a line of F^. Hy showing that this leads to contradictory results, 

 the existence of at least one straight line on F^ is established. 



First part. We divide the proposition as follows : 



§ 1. If ^ is isolated in a plane a, then « is tangent plane to 

 F^ in A or A is exceptional point. 



§ 2. Only one exceptional point is possible. 



§ 3. If A is double point in a plane a and cusp in not more 

 than one plane, then « is tangent plane. 



§ 4. If x4 is cusp in one and not more than one plane «, then a 

 is tangent plane. 



§ 5. If /I is cnsp in two different planes, then .4 is exceptional poiiit. 



§ 6. Through A passes at least one plane in which A is either 

 isolated point, double point or cusp. 



§ 1. //' A is isolated in a plane « then a is tangent plane or A 

 is exceptional point. 



The first thing to be done is to construct a plane in which A is 

 not isolated. The vicinity of A on F^ is the (1,1) continuous 

 representation of the vicinity of a point in a plane, hence a 

 sequence of points A,, A^, A^ . . . . of F"^ can be chosen having A 

 for sole limiting point. Let a be an arbitrary line through A in « 

 and /^^, /?2, (ij .... the planes passing through a. and A^, A^, A^ . . . . 

 respectivelj'. These planes have at least one limiting plane /i passing 

 through (I also. In case A is isolated in each of the planes i^i, /?,,/?,.. . 

 it can be shown that A is not isolated in (3. 



1) Brouwer, Math Ann. 71, p 97. 



