740 



Hence in every plane through a, not containing the cuspidal 

 tangent in /i, the restcurve is an oval through A. Passing on the 

 limiting case, it appears that in the plane through a and the cuspidal 

 tangent in /? the curve consists of a only, or <( together with an 

 oval through A. 



Furthermore it appears that an arbitrary line through A {=\= a) 

 carries at the utmost one point of F" different from A. But in no 

 plane can A be isolated (because a furnishes points of 7^' inside 

 any vicinity of A, on both sides of every plane not containing a), 

 hence in any plane which does not contain a the point A is either 

 cusp or double point. Concerning the planes through a it appeared 

 that A is double point in every one of these with the possible ex- 

 ception of the one through a and the cuspidal tangent in /?, in which 

 case the curve in that plane consists of a only. 



So far we only assumed A to be cusp in a single plane ^. Now 

 let A be cusp in two planes /i and y. We shall consider separately 

 the cases that only one or more than one line can be cuspidal 

 tangent at A. 



First case. A is situated on the line a of F'. Let b denote the 

 only line through ^4 which can be cuspidal tangent and let a be 

 the plane through a and b. The foregoing results show that there 

 are only two possibilities: 



I. The curve in « consists of ^7 and an oval through A. 

 11. The curve in a consists of a only. 



I. Let c be a line through A in a, not being tangent to the oval 

 and not coinciding with d or b. Only line h can be cuspidal tangent 

 at A hence in every plane through c i=\— a) A is double point, 

 but A is double point in a also, hence A would be double point 

 in eve7\y plane through c, and c cannot he tangent in any of these 

 planes because c carries besides A a second point of F^. This how- 

 ever cannot be, as may be shown in the same way as in § 3 of 

 the first communication. The fact that here A lies on a line of F^ 

 makes no difference as the demonstration merely depended on the 

 connection of the branches dictated by the assumption that F' is 

 a twodimensional continuum. 



II. Again let c be a line through A in « not coinciding with a 

 or b. In every plane through c (=|= «) A is ordinary double point 

 and in a the curve consists of a only. 



Let Ö be an arbitrary plane through c {—\= a). In this plane rf 

 the line c is tangent at the double point A, hence in d on both 

 sides of c at least one branch joins A with the line at infinity (on 



