1248 



Let a be a line of F^. Tliroughout this note we assume that F'' 

 contains no infinite number of lines. In the third communication we 

 proved that to every point A of a corresponds a point B of that 

 line with the same tangent plane. Suppose A and B move in 0|»po- 

 site directions when the tangent plane turns round a. Tlien there 

 are two meeting points P and Q. We showed that F' cannot exist 

 if in no tangent plane through n the restcurve degenerates. Now it 

 may be possible that this degeneration just takes place in the tangent 

 plane of a meeting point, for instance I\ The restcurve would then 

 consist of two different lines through P (both different from n ')), 

 or two coinciding lines through /-*. 



We begin b}' showing tliat this last case is impossible. Let a be 

 the tangent plane at P. The section in <t consists of a and anolher 

 line b, counting doul)le. Now let tangent planes through a converge 

 towards <(, tlien the two corresponding points on a converge from 

 different sides towards P and the ovals in the (arigent planes 

 converge toward.-^ h. Fiom this fojiows that in each of liie two 

 semiliues in which /^ divides b, the sectors of F" meet from the 

 same side of <(, but from different sides for different semilines. 



If on the other hand we start fiom line b, then it appears 

 that P is on b also meeting point of corres|)Onding points, and line 

 a takes the part just now played by b. Hence in each of the semi- 

 lines of a sectors meet from the same side of a, but from different 

 sides for different semilines. These results, combined with the assump- 

 tion that F^ is a twodimensional continuum, cannot be fitted in 

 with any connection between the four branches meeting at P in «. 



The ^ possibilit}- might be put forward that b counts double in 

 every plane. Then however in all these planes the restcurve would 

 be a line, and F^ would contain an infinite number of lines. 



We shall now consider the case that the restcurve in « consists 

 of two different lines through P. This means that the curve in a 

 consists of three different lines a, b and c through P. Let the 

 tangent planes ((^, u, . . . . through ^if converge towards a. Then the 

 points Pj, P, .... on a converge from the one, and the correspond- 

 ing points P\, P\ .... from the other side, towards P. 



The ovals in the tangent planes must, in the long run, intersect 

 the line at infinity at two points, and are then divided by that line 



1) The possibility that the restcurve degenerates into a and a second line different 

 from a we did not consider in the third communication, because we then assumed 

 that no second line of F^ intersects a. Here however this case must be considered. 

 It then appears fairly easily tliat, anyway for a surface of the third order with a 

 finite number of lines, this case cannot occur. 



