1253 



Fig. 4. 



To make it possible for F^ to contain, 7 {and not more) lines, it 

 is necessary that on the common line of intersection (which certainly 

 exists), corresponding points move in opposite directions. 



Let the curve in a plane a consist of 

 the lines a, h, and c, forming a triangle 

 ABC (fig. 4). Suppose a is intersected bj 

 4 lines of F*, not sitnated in «, and let 

 corresponding points on a move in the 

 same direction. We denote by e and d 

 two lines intersecting a at E and /) and 

 sitnated in the same plane. Because cor- 

 responding points on a move in the same 

 direction, BC and DE are couples of 

 points which separate each other. 



Now let the point P at which 6 and ^ 

 intersect be sitnated above « (the case 

 that P lies in the plane at infinity shall 

 be dealt with piesently). We tnrn plane n round AB in such a 

 way, that the half containing the triangle moves upwards. The lines 

 a and b are then replaced by ^n oval, which at first is divided into 

 two branches by the line at infinity. 



The [)oints of intersection of this revolving plane and the lines e 

 and d, are at first situated on different branches of the oval. How- 

 ever, in the point /-* they meet again, hence before the plane reaches 

 this positiDn, or just when passing through l\ the points mentioned 

 above must move on to the same branch again. This however can only 

 be effected, either when one of the branches retires towards the 

 line at infinity (and this is excluded, as one of the linesegmeuts EP 

 or DP must be intersected), or via a degeneration. Hence this case 

 cannot occur unless the lines b and c are also intersected by lines 

 of F*, not situated in h. 



It is not necessary to give another demonsliation for the case 

 that P is situated in the plane at infinity, or the case that C and 

 E (fig. 4) are separated l)y the point at infinity of the line, as 

 the whole argument is proof against j)rojective transformation. 



By means of a reasoning, showing strong analogy with the above, 

 the following proposition can still be proved : The caseof^{2i\\d not more) 

 lines on F"^ cannot occur, if on each of these 3 lines corresponding 

 points move in opposite directions. 



