739 



line ends up by carrying two points of II. Besides it has a point 

 in common with either I or III converging towards C" or D" and 

 lastly it carries a point of F^ converging towards the second point 

 of intersection of D' C' and the ova! in «. Altogether four points. 



It thus appears that the second possibility is excluded and we 

 need only consider the first. 



In § 3 of the first communication we proved : If A is double 

 point in a plane u, and cusp in not more than one plane, then a is 

 tangent plane, assuming that no line of F* passes thi-ough A. Here 

 however one of the branches passing through A, is a straight line. 

 This is the only one, as we assumed that no second line of F^ 

 intersects the lifie a on which A is situated. Hence in no plane through 

 A except those passing through a, can the curve contain a line through A 

 and the demonstration of § 3 still holds. The results obtained for 

 planes through the tangents at A in « also remain valid for the 

 planes through the tangent at ^4 to the oval in a. Regarding 

 the curves in planes through the line in a however (which line 

 corresponds to the second tangent of the former case) the demon- 

 stration says nothing. These shall be dealt with later on. Also the 

 first part of § 3 where the connection of the branches is examined, 

 has to be slightly altered, but this has been done already above. 



In order to be able to use the former results here, it remaijis to 

 prove the following theorem : 



// a line of F^ passes through A, lohich line is not intersected 

 by a second one, then A cannot be cusp in more than one plane 

 (we give a fresh demonstration as the former one must be altered 

 a good deal). 



A is situated on the line a of F^ and is cusp in a plane j? which 

 of course does not contain a. Let « be an arbitrary plane through 

 a. not containing the cuspidal tangent in /? and b the line of inter- 

 section of « and |i. Line b carries except A only one point B of 

 F^. In plane « the point B cannot be isolated, as the curve in ^ 

 crosses a. Neither can the restcurve in « (that is: the curve minus 

 a), according to our assumption, consist of two lines and the only 

 remaining possibility is that the restcurve is an oval through B. 

 This oval also passes through A, because b has only the points A 

 and B in common with F' (that the oval cannot have b for tangent 

 in B follows from the same reasoning which shows that fig. 11 

 represents an impossibility.) ') 



^) Here we are not entitled to use the theorem given at the end of the first 

 communication, because this was only proved for points not situated on a line of 

 t"-^, and it is not excluded that B lies on such a line. 



