Mathematics. — "On Elementary Surfaces of the third order'. 

 (Third communication). By Dr. B. P. Haalmeyek. (Commu- 

 nicated bv Prof. L. E. J. Brouwer). 



(Communicated in the meeting of September 29, 1917). 



It has been proved that F* cannot exist if that surface does not 

 contain at least one straight line. It will now be shown that if F* 

 contains a line'), this surface still cannot exist if in no plane through 

 that line tlie section consists of three lines. 



We start from a line a on F^ and assume that in no plane 

 through a the restcurve consists of two lines. It will be shown 

 that this assumption leads to contradictory results. 



Theorem 1 .' Every point of line a has a. tangent plane. 



Let A be an arbitrary point of a and ^ a plane through A not 

 containing a. A cannot be isolated in pf because there are points of 

 F^ on both sides of ^ inside any vicinity of A. Hence in ji a curve 

 passes through A. On this curve we choose a sequence of points 

 Ai, Aj . . . converging towards A from only one side. Let «i, «, . . . 

 be the planes passing through a and through A^, A, . . . respectively 

 and let a be their limiting plane (obviously « is the plane through 

 a and the tangent at A to the curve in /i). In every one of the 

 planes a^, «, ... is situated a curve of the second order, passing 

 respectively through A^, A^ . . . 



Three possibilities are to be considered : 



1. The curves of the second order contract towards f« or part of «. 



2. The curve in the limiting plane «■ consists of a and an oval 

 which intersects a at A. 



3. The curve in the limiting plane a consists of a and an oval 

 which has a. for tangent at A. 



1. The curves of the second order contract towards a or part of 

 a. This part of a anyway contains the point A. Each curve of 

 the second order divides the corresponding plane «„ in two regions '). 



M Again line will be used for straight line. 



~) An cannot be isolated in «„ because the curve in intersects the plane un • 

 Neither can the restcurve in y.n consist of a line counting double, as we assumed 

 that no second line of F^ intersects line a. 



