1251 



rated. Line a^ intersects this |)lane at a point not siiiialed in a. 

 This point mnst lie on one of the two lines in which the oval in 

 (i is degenerated. This line h^ intersects b at a point B^ different 

 from A and C. We shall consider the plane throngh a^ and 6,. 

 The cnrve in this plane contains d^ and />,, hence it consists of three 

 lines. None of these lines can be sitnated in <c becanse .4, B^ does 

 not coincide with one of the lines in a. Now the plane through a^ 

 and b-^ has a point of intersection with line c and it follows, that 

 c is intersected by one line, hence at least 4 lines of i'^'' not situated 

 in a. Hence each of the 3 lines in a is intersected bv at least 4 

 others and the total number is at least 15. 



We now come to the second case: the line a is intersected b}^ at 

 least 8 lines of F* not sitnated in «, that is : in at least 4 planes 

 through a, diffei'ent from <(, the oval degenerates. From these lines, 

 which intersect a, we choose 3: a^, «, and a^ not intersecting each other. 

 These three lines determine a scroll of (he second degree. This scroll 

 intersects at least one of the two lines 6 and c at two different points, 

 because « cannot be tangent plane both at B and 6'. For instance, 

 let the scroll have a point B, (diffeient from C) in common with b. 

 This lueans that through B^ passes a line which intersects r?,, a, 

 and a,. But then this line has 4 points in common with F\ hence 

 is entirely situated on that surface. Thus b is intersected by one line, 

 hence by at least 4 lines of F^, not situated in a. The reasoning used 

 for the first case now shows that c is also intersected by at least 

 4 lines, not situated in a, and this brings up the total to at 

 least 19. 



We now proceed to show, that if the number of lines exceeds 15, 

 it is at least 27. If there are more than 15, the above results show that 

 one line of a (for instance a) is intersected by at least 8, and each 

 of the other two by at least 4 lines of F^ not situated in a. 



Let 6i be a line not situated in a, but intersecting b, and let 6/ 

 be the third line in the i)lane tiirough b and b^. Line 6, intersects 

 4 lines, which do not intersect each other, and which all intersect a. 

 We consider the planes through />, and each of these 4 lines. These 

 planes have in common with n four different lines through the point 

 of intersection of b and b^, hence these planes intersect c at 4 

 different points. Through each of these 4 i)oints passes a line of F* 

 not situated in u. Tn the same way 6/ intersects 4 lines, which do not 

 intersect each other and each of which intersects a, and once more we 

 obtain 4 points of c through which pass lines of F\ Now none of 

 these last 4 points can coincide with one of the first, because in 

 that case a line of Z^" would pass through that point and through 



