41 



X-' 4- y- — c- =: O, (/' =z O (2) 



at infinity in common, ao that their intersections mnst degenerate, 

 the considered curves are either straight lines or conies. 



A. All the curves are straight lines. Then they must either all 

 pass through one point, so that the H's degenerate into planes, or 

 (at most with the exception of one) as generatrices of the ^'s, form 

 angles of 45° with the ^-axis, hence be so-called "light lines". 



B. Not all the curves are straight lines. Through one conic pass 

 00^ H's, because these H's have also K:o , so a biquadratic spacial 



curve in common. Hence they form a sheaf: 



H, + XH.^ = (3) 



B C D ^ E 



Let us take —=:§, — :='>;, — =?, — =:t as coord, of a point in an 



Ji. A xi Jl 



R^, the image point of the corresponding H. The image point of an 

 H of the sheaf (3) then becomes -. 



1 4_A ' 1 + A ' 1 + A ' 1 + A ' 



The image points of the H's of this sheaf therefore form a sti-aight 

 line, the image line of this sheaf. 



Two arbitrary sheaves of the system have always an i/ in common, 

 viz, that passing through the base curves of these sheaves. Hence the 

 homologous straight lines intersect each other in the image point of 

 this H. The image lines of the considered sheaves form a system of 

 Qc* straight lines in R, such that tw^o of these straight lines ahvays 

 intersect. Now two cases may be distinguished : 



a. All the straight lines pass through one point. Then all the 

 sheaves have the H whose image point this is, in common, and 

 accordingly their base curves all lie on one hyperboloid. This case 

 is trivial. 



b. Not all the straight lines pass through one point. If a plane is 

 brought through two of them, every other will have the points in 

 which it intersects the two first, in common with this plane, and 

 therefore lie entirely in this plane. Then all the image lines and 

 image points lie in one plane. lf{^„ii^,;„T^), (§„ 7;,, g„ tJ, {^„^„^„r,) 

 are three of these image points, the coordinates of the others are : 



c. ^ V ziz — etc • 



if this is substituted in (1), we find for the general equation of the 

 ^'s of the system : 



